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Luminosity and mass functions of very young stellar clusters

University of Florida Institutional Repository

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LUMINOSITY AND MASS FUNCTIONS OF VER Y Y OUNG STELLAR CLUSTERS By A UGUST A. MUENCH A DISSER T A TION PRESENTED T O THE GRADU A TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQ UIREMENTS FOR THE DEGREE OF DOCT OR OF PHILOSOPHY UNIVERSITY OF FLORID A 2002

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Cop yright 2002 by August A. Muench

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Let us content ourselv es with the illusion of similarity b ut in truth I tell you, Sir if I may e xpress myself in prophetic tones, the interesting thing about life has al w ays been in the dif ferences, From The History of the Sie g e of Lisbon by Jos e Saramago This w ork is dedicated to Laura.

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A CKNO WLEDGMENTS I w ould lik e to ackno wledge a number of indi viduals and or g anizations that pro vided the direction, care, support and opportunity that ha v e allo wed me to enjo y and to research astronomy (in addition to completing this dissertation). After calmly listening to my description of v arious observ ational cosmology projects in which I w as interested, my dissertation advisor Dr Elizabeth Lada simply pointed out that she w as not currently w orking on an y such projects. She then proceeded to detail all the research that had constituted her career so f ar and the directions she w anted to tak e, listing project after project that w as open to me were I interested. She has not stopped listing the a v enues open to me and continues to of fer me the chance to w ork on and lead projects and for this and for her guidance and support I am grateful. Although I gre w up on T ampa Bay and my f ather shes commercially on the Bay I ha v e yet to nalize a good answer to the rst question posed to me by Dr Charles Lada on the tidal patterns in the Gulf of Me xico v ersus the Atlantic Ocean. Despite this delinquenc y I ha v e enjo yed trying to answer the innumerable other questions posed to me by him re g arding the data, models and interpretations contained within this w ork and in our other projects. I ha v e come to greatly appreciate the focus that Charlie and Elizabeth Lada emplo y when our attention turns to the lucid communication of our results through the w ords contained in our papers, and their e xcitement at the moment that implication raises its sometimes dangerous head. I w ould lik e to thank the members of my dissertation committee at the Uni v ersity of Florida for reading, re vie wing and pro viding their comments and questions on this w ork. I w ould also lik e to thank the members of my pre-doctoral committee at the i v

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Center for Astroph ysics, Drs. Alyssa Goodman and John Stauf fer who labored through my e xcessi v ely long progress reports and who g a v e me consistent and fruitful advice. At the Department of Astronomy I w ould lik e to of fer my thanks to Dr Stanle y Dermott, Department Chair who in f act made my career at the Uni v ersity of Florida possible and to Dr Richard Elston for his suggestions and guidance in using the Monte Carlo technique. It is also without question that both the Radio and Geoastronomy Di vision at the Center for Astroph ysics and the Department of Astronomy at the Uni v ersity of Florida ha v e been gifted by administrators and program assistants such as T om Mullen, Janice Douglas, and Ann Elton who with continuous and singular focus w ork to w ard creating a supporti v e en vironment in which to research our eld. I ha v e also been granted good friends and collaborators such as Jo ao Alv es, who w as my of ce mate at the CfA. I thank him for sharing his boundless e xcitement for his w ork, and I look forw ard to further collaboration and friendship with him. My fello w WIRE survi v or Lori Allen, has been a w onderful friend to me, is greatly missed, and I w onder on a re gular basis when will be our ne xt chance to w ork together T o Lauren Jones, who has belie v ed in me as a person and as an astronomer from the rst time she sa w me w aiting in the main of ce between classes, I send my con viction that she has much to of fer astronomy I w ould lik e to thank Joanna Le vine for her friendship and especially her support for me during this dissertation' s end times and to both her and Carlos Roman for their assistance with the reduction of the IC 348 images. My friends and colleagues who are unlisted b ut who ha v e put up with my spontaneous outb ursts about Pluto and white dw arfs amaze me with their lo yalty My parents, Gus and Betsy Muench and my brothers, Sam and Stephen, ha v e gi v en me their lo v e, interest and support throughout these years. And to my wife and my lo v e and my friend, Laura, I pray that I will nd some w ord or deed that can contain and mak e clear my gratitude to her for her support as I trudged through this dissertation. v

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I w as supported by the Smithsonian Predoctoral Fello wship program at the Harv ard-Smithsonian Center for Astroph ysics and as a substitute N ASA Graduate Student Research Fello w (grant NTG5-50233). My w ork w as also supported by a grant to Dr Elizabeth Lada from the National Science F oundation (grant AST -9733367). There is no question in my mind that the success of an y indi vidual researcher sits le v el upon three le gs: that of indi vidual commitment, that of unabridged opportunity and that of continuous scientic interaction. All of these aspects were enabled for me by being a Predoctoral Fello w at the CfA. I w ould lik e to e xtend my thanks to John Bally for permission to reproduce HST images of the proplyds in the T rapezium Cluster and to K e vin Luhman for data in adv ance of publication. Portions of this w ork are based on photographic data obtained using The UK Schmidt T elescope. The UK Schmidt T elescope w as operated by the Ro yal Observ atory Edinb ur gh, with funding from the UK Science and Engineering Research Council, until 1988 June, and thereafter by the Anglo-Australian Observ atory Original plate material is cop yright (c) the Ro yal Observ atory Edinb ur gh and the Anglo-Australian Observ atory The plates were processed into the present compressed digital form with their permission. The Digitized Sk y Surv e y w as produced at the Space T elescope Science Institute under US Go v ernment grant N A G W -2166. This publication mak es use of data products from the T w o Micron All Sk y Surv e y which is a joint project of the Uni v ersity of Massachusetts and the Infrared Processing and Analysis Center/California Institute of T echnology funded by the National Aeronautics and Space Administration and the National Science F oundation. The data products were circa the 2nd Incremental release (March 2000). This document w as typeset with the L A T E X 2 e formating system using the document class template ufthesis.cls (v2.0b) and written by Ron Smith (ufthesis@ufthesis.com) at the Uni v ersity of Florida. An y apparent success in the format of this document can almost certainly be attrib uted to Ron Smith' s ef forts for which I am grateful. vi

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T ABLE OF CONTENTS page A CKNO WLEDGMENTS . . . . . . . . . . . . . . . . i v LIST OF T ABLES . . . . . . . . . . . . . . . . . x LIST OF FIGURES . . . . . . . . . . . . . . . . . xi KEY T O ABBREVIA TIONS . . . . . . . . . . . . . . xi v KEY T O SYMBOLS . . . . . . . . . . . . . . . . xvi ABSTRA CT . . . . . . . . . . . . . . . . . . . xvii CHAPTER 1 INTR ODUCTION . . . . . . . . . . . . . . . . 1 2 MONTE CARLO MODELS OF Y OUNG STELLAR POPULA TIONS . . 10 2.1 Monte Carlo-Based Population Synthesis Model . . . . . . 10 2.2 Fundamental Cluster P arameters . . . . . . . . . . 11 2.2.1 Initial Mass Function . . . . . . . . . . . 11 2.2.2 The Cluster' s Star -F orming History . . . . . . . 13 2.2.3 Theoretical Mass-Luminosity Relations . . . . . . 14 2.3 Additional Cluster Characteristics and Model Inputs . . . . . 17 2.3.1 Reddening Properties . . . . . . . . . . . 18 2.3.2 Binary Fraction . . . . . . . . . . . . . 19 2.4 Model Outputs . . . . . . . . . . . . . . . 20 2.5 Numerical Experiments . . . . . . . . . . . . . 21 2.5.1 Dif ferent Pre-Main Sequence Ev olutionary Models . . . 21 2.5.2 Star F ormation History . . . . . . . . . . . 29 2.5.3 Initial Mass Function . . . . . . . . . . . 34 2.6 Discussion and an Example from the Literature . . . . . . 36 2.6.1 Results and Implications of Numerical Experiments . . . 36 2.6.2 An Example from the Literature: The T rapezium Cluster . . 37 2.7 Conclusions . . . . . . . . . . . . . . . . 44 3 THE F AMOUS TRAPEZIUM CLUSTER IN ORION . . . . . . 46 3.1 Near -Infrared Census . . . . . . . . . . . . . 47 3.1.1 Observ ations . . . . . . . . . . . . . . 48 3.1.2 Data Reduction and Photometry . . . . . . . . 52 vii

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3.1.3 Photometric Comparisons of Datasets . . . . . . . 55 3.1.4 Astrometry and the Electronic Catalog . . . . . . . 57 3.2 T rapezium Cluster K band Luminosity Function . . . . . . 59 3.2.1 Constructing Infrared Luminosity Function(s) . . . . . 61 3.2.2 Dening a Complete Cluster KLF . . . . . . . . 64 3.2.3 Field Star Contamination to the KLF . . . . . . . 67 3.3 T rapezium Cluster Initial Mass Function . . . . . . . . 70 3.3.1 Deri ving Distrib utions of Reddening . . . . . . . 70 3.3.2 Modeling the T rapezium Cluster KLF . . . . . . . 76 3.3.3 Deri v ed T rapezium Cluster IMF . . . . . . . . 83 3.4 Discussion . . . . . . . . . . . . . . . . 89 3.4.1 Structure of the T rapezium KLF and IMF . . . . . . 89 3.4.2 Sensiti vity of Results to Theoretical PMS Models . . . . 92 3.4.3 Comparison of IR-Based T rapezium IMFs . . . . . . 97 3.5 Conclusions . . . . . . . . . . . . . . . . 101 4 THE Y OUNG CLUSTER IC 348 IN PERSEUS . . . . . . . . 103 4.1 W ide-Field Near -Infrared Images of IC 348 . . . . . . . 105 4.1.1 FLAMINGOS Observ ations . . . . . . . . . 105 4.1.2 Infrared Census . . . . . . . . . . . . . 108 4.1.3 Cluster Structure . . . . . . . . . . . . 113 4.1.4 Cluster Reddening Properties . . . . . . . . . 119 4.2 Infrared Luminosity Functions of IC 348 . . . . . . . . 123 4.2.1 Constructing Infrared Luminosity Functions . . . . . 123 4.2.2 Field-Star Correction to the Cluster KLF(s) . . . . . 124 4.3 Initial Mass Function of IC 348 . . . . . . . . . . 128 4.3.1 Star F orming History of IC 348 . . . . . . . . 128 4.3.2 Cluster Distance and the Mass-Luminosity Relation . . . 129 4.3.3 Other Modeling P arameters: Reddening and Binaries . . . 133 4.3.4 Modeling the IC 348 Dif ferential KLF(s) . . . . . . 134 4.4 Discussion . . . . . . . . . . . . . . . . 138 4.4.1 The KLFs and IMFs of IC 348 and the T rapezium . . . 138 4.4.2 Radial V ariation of the IC 348 IMF . . . . . . . 142 4.5 Conclusions . . . . . . . . . . . . . . . . 146 5 THE Y OUNG OPEN CLUSTER NGC 2362 . . . . . . . . . 148 5.1 La Silla Observ ations of NGC 2362 . . . . . . . . . 150 5.2 2MASS Observ ations of NGC 2362 . . . . . . . . . 152 5.2.1 Spatial Structure of NGC 2362 . . . . . . . . . 152 5.2.2 Source Reddening for NGC 2362 . . . . . . . . 155 5.3 The NGC 2362 Cluster KLF . . . . . . . . . . . 157 5.3.1 Empirical Field Star KLF . . . . . . . . . . 157 5.3.2 NGC 2362 Dif ferential KLF(s) . . . . . . . . . 159 5.4 Comparison to other Y oung Cluster KLFs . . . . . . . . 160 viii

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5.5 Modeling the NGC 2362 KLF . . . . . . . . . . . 163 5.5.1 Deri ving a Mean Age Using a Fix ed IMF . . . . . . 163 5.5.2 Simultaneous Deri v ation of a Cluster' s Age and its IMF . . 167 5.5.3 NGC 2362 IMF Deri v ed Using a Fix ed SFH . . . . . 169 5.6 Discussion . . . . . . . . . . . . . . . . 172 5.6.1 Age and IMF of NGC2362 . . . . . . . . . . 172 5.6.2 Age and Spatial Structure of NGC 2362 . . . . . . 174 5.7 Conclusions . . . . . . . . . . . . . . . . 176 6 CIRCUMSTELLAR DISKS AR OUND Y OUNG BR O WN D W ARFS . . 178 6.1 T rapezium Bro wn Dw arfs with Infrared Excess . . . . . . 180 6.2 Discussion and Implications . . . . . . . . . . . 184 7 DISCUSSION ON THE STR UCTURE OF THE IMF . . . . . . 189 7.1 Y oung Clusters and the Global IMF . . . . . . . . . 189 7.2 Secondary Sub-Stellar Peak in the Cluster LFs . . . . . . 192 7.3 Ne w Clues to the Origin of Stars and Bro wn Dw arfs . . . . . 196 8 CONCLUSIONS AND FUTURE W ORK . . . . . . . . . . 198 8.1 On the Luminosity Functions of V ery Y oung Stellar Clusters . . 198 8.2 On the Initial Mass Functions of V ery Y oung Stellar Clusters . . 200 8.3 Future W ork . . . . . . . . . . . . . . . . 202 8.3.1 Continued Study of the IMF in Y oung Clusters . . . . 202 8.3.2 Structure of Y oung Open Clusters . . . . . . . . 203 8.3.3 Disks around Y oung Bro wn Dw arfs . . . . . . . 204 8.3.4 Model Impro v ements . . . . . . . . . . . 210 APPENDIX A T AB ULA TED BOLOMETRIC CORRECTIONS . . . . . . . . 211 B DIST ANCE T O THE TRAPEZIUM CLUSTER . . . . . . . . 217 C SUMMAR Y OF POPULA TION SYNTHESIS FOR TRAN CODE . . . 222 C.1 FOR TRAN Code . . . . . . . . . . . . . . 222 C.1.1 The Control Program . . . . . . . . . . . 222 C.1.2 Rejection Functions . . . . . . . . . . . . 226 C.1.3 The FOR TRAN Sub-routines . . . . . . . . . 228 C.2 Input P arameters and Output Files . . . . . . . . . . 231 REFERENCES . . . . . . . . . . . . . . . . . . 238 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . 247 ix

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LIST OF T ABLES T able page 2–1 Ev olutionary models used in numerical e xperiments . . . . . . 23 2–2 Cluster IMF deri v ed from the literature T rapezuim KLF . . . . . 40 3–1 Summary of infrared observ ations of the T rapezium cluster . . . . 49 3–2 FL W O-NTT near -infrared catalog . . . . . . . . . . . 60 3–3 Three po wer -la w T rapezium IMF parameters and errors . . . . . 79 3–4 Three po wer -la w T rapezium sub-stellar IMF . . . . . . . . 86 3–5 Ev olutionary models used to compare M-L relations . . . . . . 94 3–6 Comparison of published T rapezium IMFs based on IR photometry . . 100 4–1 Summary of FLAMINGOS observ ations of IC 348 . . . . . . 106 4–2 Comparison of IC 348 photometry to 2MASS catalog. . . . . . 109 4–3 IC 348 po wer -la w IMFs deri v ed from model KLFs . . . . . . 137 5–1 Age dependence of the IMF slope in NGC 2362 . . . . . . . 169 A–1 T able of bolometric corrections . . . . . . . . . . . . 214 A–1 T able of bolometric corrections . . . . . . . . . . . . 215 A–1 T able of bolometric corrections . . . . . . . . . . . . 216 B–1 Summary of published distances to the Orion 1d association . . . . 221 x

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LIST OF FIGURES Figure page 2–1 Example mass functions used in models . . . . . . . . . 12 2–2 Denition of the cluster' s star -forming history . . . . . . . . 13 2–3 Theoretical Hertzsprung-Russell diagram . . . . . . . . . 17 2–4 Model KLFs: v arying ph ysical inputs to e v olutionary models . . . 24 2–5 Model KLFs: comparing DM94 and DM97 . . . . . . . . 26 2–6 Model KLFs: v arying the initial deuterium ab undance . . . . . 28 2–7 Model KLFs: truncations in the mass-luminosity relation . . . . 30 2–8 Model KLFs: v arying the star forming history ( tDt ) . . . . . 31 2–9 Ev olution of mean K magnitude with cluster age . . . . . . . 32 2–10 Model KLFs: v arying the cluster' s age spread . . . . . . . 33 2–11 Model KLFs: v arying the initial mass function . . . . . . . 35 2–12 Application of models to literature data . . . . . . . . . 39 3–1 Comparison of recent T rapezium cluster IR surv e ys . . . . . . 48 3–2 Infrared color composite image of the T rapezium . . . . . . . 52 3–3 T rapezium cluster: ra w near -infrared luminosity functions . . . . 62 3–4 T rapezium cluster: construction of observ ed control eld KLF . . . 63 3–5 T rapezium cluster: deri ving MA V completeness limits . . . . . 65 3–6 T rapezium cluster: testing contrib ution of reddened eld star KLFs . . 68 3–7 Infrared colors of T rapezium sources . . . . . . . . . . 71 3–8 T rapezium cluster: e xtinction probability distrib ution function . . . 73 3–9 Ef fects of e xtinction on model cluster LFs . . . . . . . . . 74 3–10 T rapezium cluster: infrared e xcess probability distrib ution function . . 75 3–11 T rapezium cluster: best-tting model KLFs and 3 po wer -la w IMFs . . 78 xi

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3–12 T rapezium cluster: c 2 condence interv als for IMF parameters . . . 81 3–13 T rapezium cluster: best t model KLF to secondary KLF peak . . . 83 3–14 T rapezium cluster: o v erall deri v ed IMF . . . . . . . . . 84 3–15 T rapezium cluster: a closer look at the sub-stellar IMF . . . . . 87 3–16 T rapezium cluster: a secondary peak in T rapezium substellar IMF . . 88 3–17 Comparison of theoretical mass-luminosity relations . . . . . . 96 3–18 Comparison of theoretical M -T e f f -spectral type relations . . . . . 97 3–19 Comparison of trapezium IMFs from IR photometry . . . . . . 98 4–1 Infrared color composite image of IC 348 . . . . . . . . . 105 4–2 Near -infrared color -magnitude diagrams of IC 348 . . . . . . 111 4–3 Infrared color -color diagram of IC 348 . . . . . . . . . 112 4–4 Radial prole of the IC 348 cluster . . . . . . . . . . 115 4–5 Spatial distrib ution of sources in IC 348 . . . . . . . . . 117 4–6 Surf ace density prole of the IC 348 cluster . . . . . . . . 118 4–7 Extinction maps of the IC 348 FLAMINGOS re gion . . . . . . 120 4–8 Distrib utions of reddening for IC 348 . . . . . . . . . . 122 4–9 Ra w infrared luminosity functions for IC 348 . . . . . . . . 124 4–10 K-band luminosity functions by sub-re gion for IC 348 . . . . . 125 4–11 Field star correction to cluster KLFs in IC 348 . . . . . . . 126 4–12 Dif ferential KLFs for IC 348 . . . . . . . . . . . . 127 4–13 Star -forming history of IC 348 . . . . . . . . . . . . 130 4–14 Theoretical mass-luminosity relations of IC 348 . . . . . . . 131 4–15 Modeling the IC 348 KLF: cluster sub-re gions . . . . . . . 135 4–16 Modeling the IC 348 KLF: the composite cluster . . . . . . . 138 4–17 Comparison of IC 348 and T rapezium KLFs . . . . . . . . 139 4–18 Radial v ariation in the IC 348 IMF . . . . . . . . . . 144 5–1 Digitalized sk y surv e y image of NGC 2362 . . . . . . . . 150 xii

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5–2 Source distrib ution of NGC 2362 from 2MASS . . . . . . . 153 5–3 Radial proles of NGC 2362 . . . . . . . . . . . . 154 5–4 Infrared color -color diagrams for NGC 2362 . . . . . . . . 156 5–5 Field star and cluster KLFs of NGC 2362 . . . . . . . . . 158 5–6 Dif ferential KLF(s) of NGC 2362 . . . . . . . . . . . 159 5–7 Comparing the cluster KLFs of NGC 2362, the T rapezium and IC 348 . 161 5–8 Mean age of NGC 2362 deri v ed from the cluster KLF . . . . . 164 5–9 Model KLF at 5 Myr with T rapezium IMF t to NGC 2362 . . . . 165 5–10 Dependence of the NGC 2362 IMF slope on mean age . . . . . 168 5–11 Best t model KLFs to the NGC 2362 KLF . . . . . . . . 170 5–12 Mass Function of NGC 2362 . . . . . . . . . . . . 171 6–1 Selecting candidate bro wn dw arfs in the T rapezium . . . . . . 180 6–2 T rapezium bro wn dw arfs with near -infrared e xcess . . . . . . 183 6–3 Bro wn dw arf proplyds . . . . . . . . . . . . . . 185 6–4 L-band observ ations of bro wn dw arf candidates . . . . . . . 187 7–1 Comparison of T rapezium and s Ori IMF . . . . . . . . . 193 8–1 Comparison of 2MASS and FLAMINGOS imaging sensiti vity . . . 206 8–2 Imaging map of the Perseus GMC with FLAMINGOS . . . . . 208 C–1 Model input le: basic cluster and IMF parameters . . . . . . 232 C–2 Model input le: relati v e frequenc y probability distrib ution les . . . 233 C–3 Model input le: pointers and parameters for e v olutionary tracks . . 234 C–4 Model input le: output parameters . . . . . . . . . . 235 C–5 Example batch le . . . . . . . . . . . . . . . 236 C–6 Example(s) of output le headers . . . . . . . . . . . 237 xiii

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KEY T O ABBREVIA TIONS 2MASS T w o Micron All Sk y Surv e y A U Astronomical Unit CTTS Classical T -T auri Stars ESO European Southern Observ atory FLAMINGOS FLoridA Multi-object Imaging Near -IR Grism Observ ational Spectrometer FL W O Fred La wrence Whipple Observ atory FWHM Full W idth at Half Maximum GMC Giant Molecular Cloud H-R Hertzsprung-Russell (Diagram) HBL Hydrogen Burning Limit IDL Interacti v e Data Language IMF Initial Mass Function IRAF Image Reduction and Analysis F acility KLF K band Luminosity Function LF Luminosity Function LMS Luminosity Maximum Spik e M-L Mass-Luminosity (Relation) NIR Near -InfraRed NTT Ne w T echnology T elescope ONC Orion Neb ula Cluster PDF Probability Distrib ution Function PMS Pre-Main Sequence xi v

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PSF Point Spread Function SFH Star F ormation History SIR TF Space InfraRed T elescope F acility ZAMS Zero Age Main Sequence xv

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KEY T O SYMBOLS M l Absolute passband magnitude m l Apparent passband magnitude A V Magnitudes of visual e xtinction BC l Bolometric correction to passband magnitude (K) Dt Cluster' s age spread (in millions of years) f bin Binary fraction LUnits of solar luminosity m j Mass breakpoints in a po wer -la w mass function M J u p Units of a Jupiter mass MUnits of solar mass G i Inde x of a po wer -la w mass function t Cluster' s mean age (in millions of years) T e f f Ef fecti v e surf ace temperature (K) xvi

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Abstract of Dissertation Presented to the Graduate School of the Uni v ersity of Florida in P artial Fulllment of the Requirements for the De gree of Doctor of Philosoph y LUMINOSITY AND MASS FUNCTIONS OF VER Y Y OUNG STELLAR CLUSTERS By August A. Muench December 2002 Chair: Elizabeth A. Lada Major Department: Astronomy W e no w kno w that the star formation process results in freely-oating objects with masses spanning nearly four orders of magnitude. Ho we v er both the distrib ution of these objects' masses at birth and the precise ph ysics responsible for the shape of this initial mass function are poorly kno wn and can be impro v ed upon by focusing on v ery young star clusters just emer ging from their parental molecular clouds. In this dissertation I ha v e in v estig ated the usefulness of the observ ed luminosity function of a v ery young cluster as a tool for deri ving that cluster' s underlying mass function. I nd that a cluster' s luminosity function is an e xcellent probe of the initial mass function o v er the entire range of stellar and substellar mass and can be utilized to acquire the statistics necessary for testing the h ypothesis of a uni v ersal mass function. T o study the luminosity and mass functions of such clusters I de v eloped a Monte Carlo based population synthesis algorithm applicable to pre-main sequence stars. Using this algorithm I performed numerical e xperiments testing the sensiti vity of model luminosity functions to changes in fundamental cluster parameters. After sho wing that the luminosity function is intrinsically most sensiti v e to the form of

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the underlying mass function, I studied three young clusters, NGC 2362, IC 348 and the T rapezium, and performed deep near -infrared surv e ys to construct their K-band luminosity functions. Using the model luminosity function algorithm, I deri v ed each cluster' s underlying mass function and found them to be remarkably similar with all forming broad peaks at subsolar massses. Where these census are suf ciently deep I nd that the mass function turns o v er and declines in number throughout the substellar re gime b ut appears to contain structure near the deuterium-b urning limit. Re g ardless, I nd that bro wn dw arfs do not dominate stars either by number or total mass. Lastly I use a statistically signicant sample of candidate bro wn dw arfs to sho w that these objects appear as lik ely to ha v e been born with circumstellar disks as stars. Combining this nding with the continuity of the shape of the initial mass function across numerous en vironments suggests that a single ph ysical mechanism may dominate the star formation process. xviii

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CHAPTER 1 INTR ODUCTION Little is kno wn about the similarities or dif ferences between the star formation process that created the rst generation of stars in the uni v erse and the process that is forming stars and bro wn dw arfs in nearby stellar nurseries today A long standing h ypothesis, for e xample, is that the birth of primordial stars w as hea vily inuenced by the lo w metallicity of the early uni v erse, and w ould ha v e preferentially yielded stars more massi v e than those born today ( Y one yama 1972 ; P alla et al. 1983 ; Bromm et al. 2002 ). Therefore, one important diagnostic for studying an y e v olution of the star formation process is the statistical distrib ution of stellar masses at birth, or the stellar initial mass function 1 The deri v ation and comparison of the mass functions for stars in old glob ular clusters, in the g alactic eld, in intermediate-age open clusters such as the Pleiades and in e xtremely young clusters embedded in nearby molecular clouds might re v eal similarities or dif ferences that w ould test the notion of an uni v ersal mass function (see the discussion of Kroupa 2002 ) and perhaps a dominant star formation process, or that could bring about a better understanding of its stochastic nature ( Elme green & Mathieu 1983 ; Zinneck er 1984 ; Adams & F atuzzo 1996 ; Elme green 1 In general, we will refer to the stellar initial mass function as the number of stars per log arithmic unit of mass per unit v olume at birth. The choice of log arithmic mass units has both an observ ational and a theoretical basis. Be ginning with Eddington ( 1924 ), it has been sho wn both empirically and theoretically that the luminosity of a main-sequence star scales as a po wer -la w function of the star' s mass, e.g., LM 35 o v er most of the range of stellar mass. Since the standard unit of obser v ational astronomy the magnitude, is a log arithmic scaling of stellar ux, there e xists, therefore, a linear relationship between a star' s observ ed magnitude and its log arithmic mass. 1

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2 1997 ). If the formation of stars is a stochastic process and is hea vily dependent upon numerous parameters other than time, then the problem becomes one of studying the stellar initial mass function in a v ariety of ph ysical en vironments. Because the initial mass function (IMF) is an intrinsically statistical quantity all such comparisons require numerous samplings of the star formation process, in turn, requiring tools that can probe the stellar mass function o v er a lar ge v olume of space and time. Since v ery young, ne wly formed star clusters are found in en vironments ranging from the nearby Orion molecular clouds ( Lada 1990 ) to v ery massi v e clusters in the turb ulent Galactic Center ( Figer et al. 1999 ), the y may pro vide the ideal laboratory for testing whether the IMF is uni v ersal or stochastic. Further there are a number of other reasons wh y young star clusters may be particularly v aluable for mass function studies. F or e xample, a simple photometric census of the members of a young embedded cluster yields a statistically signicant population of stars and bro wn dw arfs (i.e., substellar – non-h ydrogen-b urning stars) sharing a common heritage (e.g., age, metallicity birth en vironment). Perhaps more important, such a census is relati v ely complete because v ery young clusters ha v e not lost signicant numbers of members to either dynamical or stellar e v olution. Hence, the observ ed mass function is the cluster' s initial mass function. Because the youngest star clusters are still embedded within their natal molecular cloud, a near -infrared (13 m) photometric census is often necessary to identify a complete cluster population. One direct product of such an infrared census is the young cluster' s stellar infrared luminosity function, which can be used as a tool for studying a cluster' s initial mass function. This may be a particularly ef fecti v e tool for studying the lo w-mass end of a cluster' s mass function because infrared luminosities are relati v ely easy to deri v e for young bro wn dw arfs in these clusters since such intrinsically red substellar sources are at brighter luminosities than at an y subsequent point in their e v olution. Further the de v elopment of lar ge format imaging arrays sensiti v e to near -infrared w a v elengths has made it possible to obtain

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3 statistically signicant and complete samplings of the near -infrared luminosity functions of v ery young embedded clusters. These recent increases in sensiti vity permit not only the study of the substellar mass functions of nearby clusters, b ut also the construction of infrared luminosity functions for distant young clusters e v en when little or nothing may be learned about these clusters from spectroscopic measurements. Thus, modern infrared cameras on e v en modest sized telescopes can ef ciently surv e y numerous young clusters, deri ving infrared luminosities for complete populations, and, potentially sampling the initial mass function of the current epoch o v er a relati v ely lar ge v olume of the local g alaxy The observ ed luminosity function for a cluster of stars is the product of the underlying mass function of the cluster members and the deri v ati v e of the appropriate mass-luminosity relation: d N d Ld N d log Md log M d L (1.1) Ho we v er until a cluster reaches an age of1 billion years, some fraction of the stars in the cluster will be in their “pre-main sequence” phase, meaning the y ha v e not yet be gun to fuse h ydrogen in their core. Since bro wn dw arfs ne v er achie v e nuclear b urning, these cluster members will ne v er reach the main sequence and will be contracting, cooling and becoming f ainter for their entire e xistence. Thus, the radiant luminosity of a bro wn dw arf or a pre-main sequence star is deri v ed from its gra vitational contraction ener gy and the mass-luminosity relation appropriate for these objects is a function of time, hence: d log M d Ld log M d Lt(1.2) F or the v ery young clusters we will be studying in this w ork (ages, t10 Myr ), nearly all of the members will be in a pre-main sequence phase. Further the timescale for assembling a star cluster is an appreciable fraction of the cluster' s mean age during

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4 this period. These f acts mean that the deri v ation of a young cluster' s underlying mass function from its luminosity function is sensiti v e to the history of star formation in the cluster Additionally the time-dependent mass-luminosity relation(s) used to con v ert between a cluster' s luminosity and mass functions is poorly kno wn. Since there are v ery fe w meaningful empirical constraints on the form of the pre-main sequence massluminosity relation, we must rely upon theoretical e v olutionary models of young stars when estimating this quantity Finally the predictions of these e v olutionary models v ary depending upon ho w the y were computed. Considering these complicated f actors, the most common approach to studying the luminosity functions of young star clusters has been to numerically inte grate these three fundamental quantities, i.e., the initial mass function, the star -forming history and the theoretical mass-luminosity relation, into synthetic luminosity functions and to use these model luminosity functions to interpret the observ ational data. V arious groups ha v e modeled the luminosity functions of young clusters using realistic stellar mass functions and appropriate mass-luminosity relationships (e.g., Zinneck er et al. 1993 ; Strom et al. 1993 ; Fletcher & Stahler 1994a ; Lada & Lada 1995 ; Me geath 1996 ). Zinneck er et al. ( 1993 ) were the rst to present model K band (22 m ) luminosity functions for v ery young clusters. F or their models the y adopted a “coe v al” star formation history in which all the stars were formed at a single instant of time. Moreo v er the y assumed black-body radiation to deri v e bolometric corrections and assumed a single form for the stellar mass function. Consequently their models were not v ery realistic, and the y did not attempt to t or directly compare their models to observ ed cluster luminosity functions. Lada & Lada ( 1995 hereafter LL95) impro v ed on this w ork by de v eloping e v olutionary models for the K band luminosity functions (KLF) of young clusters ranging in age from 10 6t10 7 yr using empirically determined bolometric corrections and allo wing for non-coe v al or continuous star formation in the clusters. Moreo v er

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5 the y directly compared their models to observ ed infrared luminosity functions of young clusters. Ho we v er similar to Zinneck er et al. Lada & Lada assumed a single underlying initial mass function for the stars (i.e., the Miller & Scalo 1979 eld star initial mass function), while emplo ying a single set of the published pre-main sequence e v olutionary tracks from ( D'Antona & Mazzitelli 1994 ). Additional luminosity function models were calculated by Strom et al. ( 1993 ) and K en yon & Hartmann ( 1995 ), both of whom compared their models to the de-reddened J (11 m) and K band luminosity functions of young stars. In these w orks, model luminosity functions were primarily used as probes of a cluster' s age, b ut were also emplo yed to test the similarity of the clusters' underlying initial mass function to that for the eld stars. All of these model luminosity functions were constructed for stars with masses between 01 and 20 M, since the e xisting e v olutionary tracks did not e xtend into the re gime of bro wn dw arfs ( M008 M). Thus, man y of their results are only v alid as long as there are no, or at least v ery fe w bro wn dw arfs in these clusters. It is some what dif cult to e v aluate the success of these early modeling w orks in de v eloping the luminosity function technique as a tool for deri ving the initial mass functions of embedded clusters. First, these models were fundamentally limited by the lack of consistent e v olutionary models that included young bro wn dw arfs. Second, the lack of independent estimates for the star -forming histories of the clusters studied meant that these authors approached the problem needing to constrain both the age and mass function; the y frequently constructed their models using a single mass function equi v alent to that for eld stars. Further their models were rarely applied dir ectly to the observ ations, instead requiring that the actual data be initially corrected for v arious observ ational ef fects such as reddening. Thus, these ef forts were ne v er intended to pro vide comprehensi v e models of real data such as one might e xpect from a true population-synthesis model. In addition, when the models were t to the data, error estimates or other quantication of the usefulness of the luminosity function method

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6 were not calculated, making it dif cult to dra w conclusions about the accurac y of this method. In part due to the limitations of these early models and partially resulting from the approach tak en by these original authors, the luminosity function method has not yet been used as a tool for deri ving and for comparing the initial mass functions of a series of young clusters. F ortunately technical impro v ements in some of these areas ha v e recently been made. F or e xample, e v olutionary sequences ha v e been calculated for bro wn dw arfs with masses as small as that of the planet Jupiter ( M J u p ). In addition, impro v ed age estimates for se v eral clusters such as the T rapezium ( Hillenbrand 1997 ) and IC 348 ( Herbig 1998 ) ha v e been made by e xamining brighter members using either optical spectra or the optical color -magnitude diagram. In light of these technical adv ances and the constraints placed upon the ages of some nearby young clusters, we undertook a systematic study to determine the usefulness of a young cluster' s near -infrared luminosity function as a tool for studying and deri ving that cluster' s initial mass function. Based upon the success of prior approaches to studying the luminosity function of a young cluster we formulated our study using three principles: 1) Our study w ould concentrate on the products of simple near -infrared surv e ys of young clusters. 2) W e w ould emplo y a set of model luminosity functions to interpret the products of these near -infrared surv e ys. 3) W e w ould study multiple young clusters to test, de v elop and e xpand our method(s). From these principles, we de v eloped a series of specic goals:Creation of a population-synthesis algorithm for young star clusters that includes all of the fundamental and observ ational characteristics rele v ant to the products of a near -infrared surv e y .Design of a series of numerical e xperiments to systematically test the sensiti vity of model luminosity functions to changes in the three fundamental quantities go v erning the form of the cluster luminosity function (e.g., the star -forming history initial mass function, and theoretical mass-luminosity relation).

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7Construction of the near -infrared luminosity functions of a series of young clusters from deep multi-w a v elength near -infrared surv e ys of these clusters.Deri v ation of the initial mass functions for these clusters through the application of our population-synthesis models to the cluster luminosity functions.Comparison of our results to those found via other methods for studying the mass function(s) of young clusters.Examination of the h ypothesis of a “uni v ersal initial mass function” for young clusters by comparing the luminosity and mass functions deri v ed for the clusters in this study W e accomplished these goals by focusing our ef forts in three distinct w ays. First, we de v eloped a e xible, Monte Carlo-based population-synthesis algorithm for simulating the observ ations of young clusters and for creating model luminosity functions that could be applied to cluster data. The second focus of our research has been a series of deep near -infrared surv e ys of three young clusters, the construction of the infrared luminosity functions for these clusters, and the deri v ation of these cluster' s mass functions. The third focus of this w ork is a discussion of e vidence that a single process dominates the formation of stars across the mass spectrum do wn to v ery small masses (a fe w times the mass of the planet Jupiter). In summary we nd that a cluster' s near -infrared luminosity function is an e xcellent probe of the initial mass function of a v ery young cluster and that the combination of deep near -infrared surv e ys with model luminosity functions can be used to accurately deri v e the initial mass function do wn to and belo w the deuterium-b urning limit in young nearby star clusters. Further the e vidence that the IMF(s) we deri v e from modeling the cluster luminosity function are rob ust relati v e to other methods suggests that KLF modeling can be applied to a much lar ger sample of young clusters o v er a considerable v olume of the local g alaxy pro viding the statistics necessary for establishing the de gree of uniformity of the initial mass function through (local) space and time.

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8 W e briey summarize the structure of this w ork. In Chapter 2 we de v elop our Monte Carlo based population-synthesis algorithm and use this algorithm to test the theoretical sensiti vity of a cluster' s luminosity function to changes in such parameters as age and initial mass function. W e then apply these models to the luminosity function for a young cluster constructed from literature data. In Chapters 3 and 4 we describe detailed studies of the luminosity and mass functions for the young T rapezium and IC 348 clusters using deep near -infrared surv e ys. Blaauw ( 1964 ) rst compared these tw o clusters as part of his discussion of OB associations and subsidiary young clusters: “T w o v ery interesting clusters with a dif ferent character do, ho we v er occur: the T rapezium Cluster in I Orion, and the cluster near o Persei in II Per [IC 348]. Their dimensions are much smaller than those of ordinary clusters, and both are of recent origin. ” In our study of these nearby clusters, we de v elop empirical recipes for including reddening into our population-synthesis models and for statistically correcting the observ ed cluster luminosity function to account for the contamination of our observ ations by non-member eld stars. W e then apply our method to the distant open cluster NGC 2362, in Chapter 5 and e xamine the usefulness of our method when little is kno wn about a cluster' s age or age spread. In Chapter 6 we present observ ational e vidence for the e xistence of circumstellar disks around bro wn dw arfs and discuss ho w the continuity of disks around young stars and bro wn dw arfs points to w ards a common origin for both. W e compare the initial mass functions we ha v e deri v ed for these three clusters, and e xamine the h ypothesis for an uni v ersal mass function for young clusters in Chapter 7 Here we combine the e vidence of a common origin for stars and bro wn dw arfs and the continuity of the mass function across a number of clusters and en vironments to discuss what processes might dominate the formation of stars and bro wn dw arfs. After summarizing our ndings in Chapter 8 we briey detail additional future w ork that will focus on the ne w questions raised by this study W e reserv e a number of the parts of our study to the appendices. Here we eng age in a

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9 brief discussion of the distance to the T rapezium Cluster and list minor details of our modeling algorithm, including our tab ulation of empirical bolometric corrections and descriptions of the computer code used in our population-synthesis algorithm.

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CHAPTER 2 MONTE CARLO MODELS OF Y OUNG STELLAR POPULA TIONS 2.1 Monte Carlo-Based P opulation Synthesis Model F or use in the interpretation of infrared luminosity functions of young stellar clusters, we created a Monte Carlo-based population synthesis algorithm for pre-main sequence stars. The underlying principle of our population synthesis model is the treatment of the fundamental cluster properties as probability distrib ution functions that are sampled and inte grated using a Monte Carlo rejection method algorithm. Thus, the algorithm w as designed to create a synthetic star cluster with members whose ages and masses are dra wn from a specied star -forming history (SFH) and underlying initial mass function (IMF). Each synthetic star' s mass and age w as con v erted to observ able quantities using mass-luminosity (M-L) relations interpolated from a set of theoretical e v olutionary models. Additional properties such as reddening due to interstellar e xtinction or by e xcess ux from circumstellar disks were also assigned to each synthetic star by using probability distrib ution functions, while other parameters such as distance and binary fraction were x ed to specic v alues for the entire cluster Further our use of a Monte Carlo formulation also allo ws us to run multiple numerical simulations of a model cluster thus gi ving us a statistical lens to use when comparing our models to real clusters, which typically contain between 100 and 1000 members. In Sections 2.2 and 2.3 we describe each of the cluster parameters and ho w it w as implemented into our models before detailing a series of numerical e xperiments in Section 2.5 aimed at testing the sensiti vity of a model cluster' s luminosity function (LF) to changes in the underlying cluster parameters. In Section 2.6 we discuss the results of these e xperiments and illustrate the ef fecti v eness of KLF modeling for constraining a cluster' s IMF by applying our technique to data tak en from the literature 10

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11 for the f amous T rapezium Cluster in Orion. In Section C.1 we briey detail each of the FOR TRAN subroutines that were written to implement this algorithm. 2.2 Fundamental Cluster P arameters 2.2.1 Initial Mass Function In our standard model, stars can ha v e masses between 80 and 0.02 M, limits set by the range of e v olutionary models a v ailable for v ery high-mass O stars and v ery lo w mass bro wn dw arfs and giant planets. W e parameterized the underlying cluster initial mass function with a number of dif ferent analytical forms. Throughout this w ork, we refer to the initial mass function as the frequenc y of stars per unit lo g mass per unit v olume. Since we may suppose that a cluster represents a single star formation e v ent, then there is no purpose in inte grating this function o v er space v olume. A simple po wer -la w function is the most common parameterization of the IMF and that originally used by Salpeter ( 1955 ), e.g., xlogM M c 1M G(2.1) where c1 is a normalization constant, and G is the po wer -la w inde x. In this form, Salpeter found that the initial mass function for stars in the eld had G 135 o v er the mass range from 1 to 10 M. Our standard parameterization of the underlying cluster IMF consisted of po wer -la w se gments, G i connected at break masses, m j F or e xample, for masses between our upper mass limit and the rst mass break m 1 the IMF is described as a po wer -la w with inde x, G 1 and from m 1 to m 2 the IMF has a po wer -la w inde x, G 2 etc. Cluster IMFs could ha v e as man y as v e (5) independent po wer -la w se gments. W e also used the log-normal distrib ution as a functional form of the IMF e.g., xlogM M c 1e xp c 2 logM M c 32 (2.2)

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12 Figure 2–1: Example mass functions used in models. The log-normal form follo ws the parameterization of Miller & Scalo ( 1979 ) and is e xtended to the lo west masses. Standard tw o (2) po wer -la w IMFs are sho wn where the high-mass IMF slope, G 1 equals -1.35 (equi v alent to Salpeter ( 1955 )) and then breaks at a mass, m 1 equal to 05 M. Belo w the break mass, the IMF is go verned by a lo w mass slope, G 2 for which we sho w v e dif ferent v alues: -1.35, -0.40, 0.00, +0.40, and +1.0. where c1 is a normalization constant, c2 equals 1 2 logs2, c3 equals l o gM M or the mean log mass of the distrib ution and s is the v ariance of this mean. Figure 2–1 illustrates these mass function parameterizations. The mean and v ariance of the log-normal IMF sho wn correspond to the eld star mass function gi v en by Miller & Scalo ( 1979 hereafter MS79), ha ving constants of c2109 and c3 102 or a mean mass of 0.0955 M 1 The e xample tw o po wer -la w IMFs sho wn in Figure 2–1 ha v e G 1 135, m 105 Mand G 2 v arying from -1.35 to +1.0. 1 This set of log-normal parameters corresponds to the MS79 deri v ation that used a maximum age of the g alactic disk equal to 12 Gyr

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13 2.2.2 The Cluster' s Star -F orming History F or most of the models presented in this w ork, we assumed a constant star for mation rate during the formation of a young cluster W e adopt this characterization partially because it is the simplest such model, and partially because the precision of observ ations which suggest that a cluster' s SFH is episodic or accelerating ( P alla & Stahler 2000 ) is certain to be strongly modied by intrinsic errors that w ould lead to e xaggerated star -forming histories ( K en yon & Hartmann 1990 ; Hartmann 2001 ). Further we assumed that there is no correlation between mass of a cluster member and when it w as formed in the cluster Figure 2–2: Denition of the cluster' s star -forming history The cluster' s mean age, t in this simple model is equi v alent to the a v erage of the ages of the oldest and youngest stars, assuming a constant star formation rate. Therefore, we parameterized the SFH using a “mean age”, t and an “age spread, ” Dt F or e xample, a coe v al cluster will ha v e no age spread and Dtt00. A cluster with the lar gest possible age spread w ould ha v e Dtt20 with star formation starting 2t years ago and continuing to the present. Figure 2–2 illustrates these denitions. W e note that these denitions of the cluster' s star -forming history are dif ferent than those used in the models of LL95 and K en yon & Hartmann ( 1995 ). F or

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14 these w orks, the age of the cluster referred to the total timespan since star formation be g an, which is also the age of the oldest cluster members. Thus, for constant star forming histories, their “age” w ould correspond to the “age spread” of our SFH and it w ould also be equal to twice our deri v ed “mean age. ” Our standard model SFH, therefore, approximates an y real SFH to rst order by using the most common age of the members and a rough age spread. The requirement of a constant star formation rate, ho we v er is not a pre-requisite of our models, and an y to y or empirical distrib ution of age can be used to dra w ages for a synthetic cluster 2.2.3 Theor etical Mass-Luminosity Relations The mass-luminosity relation appropriate for con v erting the synthetic stars' masses into observ able luminosity is dependent on the e v olutionary status of the star F or all the clusters considered here, the youngest (1510 5 years) and most massi v e cluster members ( M5 M) will ha v e already contracted on to the Zero Age Main Sequence (ZAMS) ( P alla & Stahler 1990 ). F or these O and B type members, we con v erted their mass to bolometric luminosity and ef fecti v e temperature using a theoretical ZAMS deri v ed from Schaller et al. ( 1992 ). No post-main sequence e v olution is included for the high and intermediate mass objects, since for the clusters considered here ( t10 Myr), only the O stars w ould ha v e had suf cient time to complete their core h ydrogen b urning and be gin to e v olv e into giant or super giant-type stars. The majority of the cluster members will be in the pre-main sequence phase of their e v olution. Since these stars are still contracting, the appropriate mass-luminosity relation is age dependent, and we must rely upon theoretical e v olutionary models to con v ert from the synthetic star' s masses and ages into luminosities. These e v olutionary models ha v e been calculated by a number of authors ( Hen ye y et al. 1955 ; Hayashi 1961 ; Iben 1965 ; Burro ws et al. 1993 ; P alla & Stahler 1993 ; D'Antona & Mazzitelli 1994 ; Baraf fe et al. 1998 ), who ha v e e xplored a v ariety of dif ferent ph ysical inputs and initial conditions to the models. T ypically these models track the pre-main sequence

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15 e v olution (luminosity and ef fecti v e temperature) of a star of a particular mass across what is referred to as the theoretical Hertzsprung-Russell (H-R) diagram. Unfortunately pre-main sequence (PMS) theoretical models are not typically calculated for the entire mass range from bro wn dw arfs (0.001 M) to high-mass B stars (10 M). Because of this, we often had to combine tw o dif ferent sets of PMS tracks to pro vide a complete mass range. W e took the opportunity to use dif ferent sets of PMS tracks for high and lo w mass stars to remo v e an apparent mass-age correlation found by man y authors who ha v e used PMS e v olutionary tracks to deri v e real ages and masses for stars using the H-R diagram ( Hillenbrand 1995 ; Me yer 1996 ; Hillenbrand 1997 ). These authors point out that when masses and ages are deri v ed for a cluster of real stars using PMS tracks, a correlation e xisted such that the more massi v e stars were systematically older than the lo wer mass stars. Further these authors suggested that the cause of this correlation is due to the w ay canonical PMS tracks ha v e been constructed. Canonical PMS tracks e v olv e the model stars from innite spheroids, while recent studies suggest that stars e v olv e during a proto-stellar phase along a specic mass-radius relationship referred to as the proto-stellar birthline ( Stahler 1983 ; P alla & Stahler 1990 ). Using a proto-stellar birthline as the initial condition for PMS tracks will most prominently adjust the predicted luminosities and ef fecti v e temperatures (as a function of time) for the youngest and highest mass stars, where the stars' proto-stellar (birthline) lifetimes are comparable with these stars' pre-main sequence contraction lifetimes. Rather than using canonical PMS tracks for model stars with masses greater than solar we used “accretion-scenario” PMS model calculations by P alla & Stahler ( 1993 ) and Bernasconi ( 1996 ). Accretion scenario PMS tracks better represent the location of the young intermediate mass stars on the H-R diagrams ( P alla & Stahler 1993 ; Bernasconi & Maeder 1996 ). Y et the accretion-scenario PMS tracks cannot be straightforw ardly used with subsolar mass canonical PMS tracks. W e adopted the

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16 accretion-scenario tracks listed abo v e for models abo v e 2 Mand canonical tracks belo w 1 Mtak en from D'Antona & Mazzitelli ( 1994 ) and D'Antona & Mazzitelli ( 1997 ). Between these tw o mass limits, we compared the canonical and accretionscenario calculations. W e calculated an a v erage of each accretion-scenario and canonical mass track, weighting the a v erage to result in a smooth con v ersion from the canonical (subsolar mass) to accretion-scenario (intermediate mass) re gimes. W e e xamined the theoretical H-R diagram resulting from our combination of ZAMS, accretion-scenario, a v eraged, and canonical PMS tracks. These ne w sets of tracks and resulting isochrones were found to be smooth between all re gimes and the y were used as input to the modeling algorithm. In Figure 2–3 we sho w an e xample of the distrib ution of mass tracks and isochrones in the H-R diagram. W e dene our standard set of PMS tracks to be a mer ger of the D'Antona & Mazzitelli ( 1997 ) subsolar mass and Bernasconi ( 1996 ) “accretion-scenario” intermediate mass tracks. Our modeling algorithm uses a cubic spline routine to interpolate between the mass tracks and isochrones on the H-R diagram to deri v ed luminosities and ef fecti v e temperatures for the masses and ages dra wn from the IMF and SFH. Using these luminosities and ef fecti v e temperatures we con v erted to an absolute magnitude using the formula: M lM bol 25logLL BC lT e f f(2.3) W e assumed M bol 475 and our empirical bolometric corrections were tab ulated as functions of ef fecti v e temperature and were tak en from the literature. W e list the sources of the bolometric corrections in Section A Appropriate bolometric correction tables were constructed for I K bands, allo wing for the calculation of red and near infrared colors, magnitudes and monochromatic luminosity functions. Lastly for dening the source of the mass-luminosity relation, we did not account for stars in their proto-stellar phase since the contrib ution of these e xtremely young objects to the total population of an embedded star cluster goes as the ratio of the

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17 Figure 2–3: Theoretical Hertzsprung-Russell diagram. Pre-main sequence e v olutionary tracks from 0.02 to 5 Mand isochrones from 0.5 to 10 Myr are sho wn. The mer ged tracks are from DM94 and P alla & Stahler ( 1993 ). Also sho wn is the birthline for a proto-stellar accretion rate of 105 M yr duration of the proto-stellar phase (01Myr ) to the age spread of the stars in the re gion (12 Myr ), and hence will be quite small in most cases ( Fletcher & Stahler 1994a b ). 2.3 Additional Cluster Characteristics and Model Inputs In addition to the three fundamental quantities (IMF SFH, M-L relation) that go v ern the structure of a young cluster' s luminosity function, there are a number of observ ational characteristics that must be included into our population synthesis model. Some of these parameters, the distance to a young cluster for e xample, are not easily constrained by the analysis we present here, and are subsequently assumed from literature sources, becoming a x ed parameter in our models. F or v ery young clusters, distance is often determined by association with a molecular cloud comple x whose systemic v elocity is kno wn and has been con v erted to a distance estimate. In

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18 other cases, some high-mass cluster members are optically visible and are assumed to be on the ZAMS, and these stars are used to deri v e a distance modulus. The model algorithm al w ays con v erts from the absolute passband magnitude of the stars, M l into an apparent passband magnitude m l based on the x ed distance. The cluster' s reddening properties and appropriate binary fraction are treated as free parameters, and we describe their inclusion into the modeling algorithm belo w 2.3.1 Reddening Pr operties The mean reddening estimates, e.g., EBV, used in traditional open cluster studies are inappropriate for v ery young clusters because of the lar ge, v ariable e xtinction arising from the parental molecular cloud. Although the magnitude of this e xtinction is decreased by w orking at near -infrared rather than optical w a v elengths, the reddenings are suf ciently lar ge and spatially v ariable, that a single mean e xtinction for the entire cluster w ould be inappropriate. Additionally hot dust in circumstellar disks around young stars reprocesses the stellar radiation and re-emits it at infrared w a v elengths, further reddening a young star' s intrinsic infrared colors and increasing the infrared ux observ ed. T o include these parameters into the modeling algorithm, probability distrib ution functions (PDFs) are constructed for both of these reddening properties. These PDFs can ha v e either functional (e.g., g aussian) or empirical forms. In both cases, we constrain the cluster' s reddening properties from the infrared colors obtained when a young cluster is surv e yed. Indeed, tw o goals of the current luminosity function modeling are to 1) deri v e recipes for e xtracting the distrib utions of reddening from the observ ed infrared colors themselv es, and 2) to use these distrib utions in our modeling algorithm to recreate not only the cluster' s luminosity function, b ut also to duplicate the distrib ution of sources in the cluster' s infrared color -color and color -magnitude diagrams. W e describe our deri v ation of empirical reddening PDFs in detail in Section 3.3.1

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19 2.3.2 Binary Fraction One observ ational constraint imposed on our studies of young clusters is the angular resolution limit of our surv e ys. Thus, the observ ed luminosity function can be altered by the presence of unresolv ed multiple stars, by cluster members missed because of chance projections or by confusion due to background stars. Because the clusters we study are reasonably nearby chance projections do not produce a signicant number of f alse binaries or missed cluster members. Further we are not observing clusters close to the g alactic plane, in the g alactic center or in other g alaxies so will not consider the latter ef fect of confusion in our models. The ef fects of unresolv ed binaries and higher order systems on the cluster luminosity function is a well kno wn problem and it depends partially on the underlying IMF of the primaries and of the secondaries ( Kroupa et al. 1991 ). While the typical angular resolution of our surv e ys allo ws us to identify some visual binary systems, we can typically probe only to separations of200 to 300 A U where the binary fraction is observ ed to be no more than 10-15% ( Duquenno y & Mayor 1991 ). Hence the majority of the binaries are unresolv ed and may inuence the form of the luminosity function and mass function we deri v e. Unresolv ed binaries ha v e tw o ef fects on the form of a cluster' s intrinsic luminosity function. First, binaries with mass ratios of1 will be up to 0.75 magnitudes brighter than the indi vidual members, and will shift the o v erall form of the luminosity function. Second, binaries with lo w mass ratio (lo w mass secondaries to higher mass primaries) will result in cluster members that are completely lost since the y will not contrib ute an appreciable fraction of the total ux of the unresolv ed system. W e include the e xistence of unresolv ed binaries in our Monte Carlo algorithm using a one-parameter binary fraction dened by Reipurth & Zinneck er ( 1993 ) as: fN binar ies N binar iesN singl e s t ar s (2.4)

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20 Thus we ignore higher order un-resolv ed systems (triples, quadruples). W e further mak e the simplifying assumption that the primaries and secondaries are dra wn from the initial mass function, and that the distrib ution of mass ratios is uniform. T o include binaries into our algorithm we follo w the formulation of Kroupa (pri v ate communication). Simply after N stars are sampled from the mass function, a subset are randomly paired into binary systems, (e.g., “systems, ” N sys with the remaining stars becoming “singles, ” N sing ). The number of each type is approximated in the code by the equations: N singINTN s t ar s1 f 1 f(2.5) N sysINT N s t ar sN sing 2 (2.6) Both members of a binary system are assigned the same age and e xtinction dra wn from the star -forming history and the appropriate reddening distrib ution. If the population synthesis includes ux from a circumstellar disk, each member of a binary is assigned a separate ux e xcess. The luminosities of the members of the binary system are con v erted to indi vidual magnitudes, reddened and nally mer ged (in ux units) to simulate their un-resolv ed nature. 2.4 Model Outputs Our Monte Carlo based pre-main sequence population synthesis code w as scripted to produce a number of dif ferent possible simulations. T aking adv antage of the code' s Monte Carlo nature, random samples (of N stars and M iterations) of an y or all of the input distrib utions (IMF SFH, reddening distrib utions) can be deri v ed. Further synthetic H-R diagrams, infrared color -magnitude and color -color diagrams can be produced for permutations of all of these input parameters. Finally model (binned) infrared (IJHK) luminosity functions can be created using parameters that adjust the bin sizes and bin centers. T w o model luminosity functions are standard output for each set of input parameters. The rst is the luminosity

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21 function constructed from the un-con v olv ed magnitude of e v ery indi vidual star without the ef fects of reddening or unresolv ed binaries. The second is an observ able model luminosity function which includes these ef fects. This w ould be the luminosity function used in modeling young cluster luminosity functions in later chapters, while returning both the intrinsic and observ able LFs allo ws us to mak e simple direct tests of the impact of v arious observ ational quantities. In all cases, the output les are simple ASCII les with headers containing the parameters used in that model run. 2.5 Numerical Experiments Using our Monte Carlo population synthesis code, we performed a series of numerical e xperiments aimed at e v aluating the sensiti vity of a young cluster' s luminosity function to each of the three fundamental underlying inputs: the theoretical M-L relation, the cluster' s star -forming history and the cluster' s IMF W e create a suite of model luminosity functions systematically v arying each of the three fundamental underlying relations while holding the other tw o functions constant. F or each synthetic model run, we produced model luminosity functions by binning the resulting synthetic magnitudes in half (0.5) magnitude bins as is standard for actual observ ed cluster luminosity functions. Our standard model cluster for these e xperiments contained 1000 stars and for each set of x ed parameters we produced typically 50-100 independent luminosity functions. W e computed the mean luminosity function from these realizations, and record the one sigma standard de viation of the computed mean of each model luminosity function bin. 2.5.1 Differ ent Pr e-Main Sequence Ev olutionary Models The e v olution of pre-main sequence stars across the H-R diagram and onto the main sequence is not observ ationally well constrained. Details of PMS e v olution rely hea vily upon theoretical PMS tracks. These theoretical PMS tracks v ary in their predictions depending on the numerical methods and theoretical assumptions used in their creation. Since these PMS tracks are used to con v ert from a stellar age and mass

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22 to a monochromatic magnitude, the resulting luminosity functions will depend to some de gree on the PMS e v olutionary models which are chosen. T o e v aluate ho w PMS tracks with dif ferent input ph ysics, chemical ab undances or ef fecti v e mass ranges af fect the shape and form of a model luminosity function, we constructed and compared model luminosity functions calculated assuming dif ferent PMS tracks. F or these e xperiments, we x ed the initial mass function to ha v e a log-normal distrib ution as described in Equation 2.2 W e produced a suite of model clusters with a range of mean ages from 0.2 to 15 Myr and age spreads from coe v al to twice the mean age of the model cluster F or the purposes of e v aluating the ef fects of using dif ferent input PMS tracks, we only directly compared KLF models ha ving identical star -forming histories. D'Antona & Mazzitelli (1994): Differing input ph ysics. D'Antona & Mazzitelli ( 1994 hereafter DM94) calculated four dif ferent sets of e v olutionary PMS tracks v arying tw o input ph ysical parameters, the input opacity tables and the treatments of internal con v ection. T able 2–1 summarizes the four combinations of input ph ysics and other parameters of the DM94 PMS tracks. Only one of these data sets contained stars with masses less than the h ydrogen b urning limit. Consequently we used a common range of stellar masses from 2.5 to 0.1 Mto compute dif ferent model KLFs using the four sets of DM94 PMS tracks. Figure 2–4 compares synthetic KLFs computed from the DM94 PMS tracks for coe v al models with mean ages of 1 and 7 million years, respecti v ely In Figure 2–4 dif ferent symbols correspond to dif ferent input opacity tables in the PMS tracks used. F or the 1 million year coe v al models, the tw o KLFs corresponding to PMS tracks with K urucz opacities are essentially indistinguishable, indicating that the KLFs are insensiti v e to the con v ection model used. The tw o model KLFs corresponding to PMS tracks with Ale xander opacities e xhibit a relati v ely narro w b ut signicant feature or peak between M K3-4 which is not apparent in the KLFs with K urucz opacities. The position of this spik e is dif ferent for the tw o con v ection

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23 T able 2–1. Ev olutionary models used in numerical e xperiments Source Model Opacity Con v ection [ D H ]aMass Range Name T able Model ( M M) DM94 A CM Ale xander et al. (1989) FSTb2.0 0.0182.5 DM94 AMT Ale xander et al. (1989) ML Tc2.0 0.1002.5 DM94 KCM K urucz (1991) FSTb2.0 0.1002.5 DM94 KMT K urucz (1991) ML Td2.0 0.1002.5 DM97ed1.5 Ale xander & Fer guson (1994) FSTf1.0 0.0171.5 DM97ed2.5 Ale xander & Fer guson (1994) FSTf2.0 0.0173.0 DM97ed4.5 Ale xander & Fer guson (1994) FSTf4.0 0.0173.0 aDeuterium Ab undance relati v e to Hydrogen; In units of105bFull Spectrum T urb ulence Model; Canuto & Mazzitelli ( 1991 1992 )cMixing Length Theory; 1H p12dMixing Length Theory; 1H p15eDM97 models were initially released in 1997. These models were updated in 1998. The models used were those of the updated calculations.fFull Spectrum T urb ulence Model; Canuto et al. ( 1996 ) References. — D'Antona & Mazzitelli ( 1994 DM94); D'Antona & Mazzitelli ( 1997 DM97) models used with the Ale xander opacities. This feature is due to deuterium-b urning which causes a slo wing of the stellar luminosity e v olution ( Zinneck er et al. 1993 ) and therefore results in a pile up of stars in the luminosity function. The deuterium-b urning spik e is absent in the 7 Myr coe v al model in Figure 2–4 and in all coe v al models with mean ages greater than 2-3 Myr for stars abo v e the h ydrogen b urning limit. The onset of deuterium-b urning is a function of stellar mass. Lo w mass stars contract more slo wly than higher mass stars and be gin b urning deuterium after high-mass stars. Ho we v er by 3 Myr e v en stars at the h ydrogen b urning limit w ould ha v e b urned all of their initial deuterium ab undance. A second feature of interest in the KLFs is the spik e/dip at M K2 in the 7 Myr coe v al model. It is present in all four 7 Myr KLFs and in all KLFs with mean ages greater than 3-4 Myr This feature is the result of stars reaching a luminosity maximum on radiati v e tracks before be ginning h ydrogen b urning and mo ving to

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24 Figure 2–4: Model KLFs: v arying ph ysical inputs to e v olutionary models. Each model KLF corresponds to a dif ferent combination of input ph ysics as described in DM94 (see also T able 2–1 ). These model KLFs were constructed using a log-normal IMF (see equation 2.2 ) with a lo wer mass limit of 01 Mand ha ving coe v al star formation with mean ages of 1 (top) and 7 (bottom) Myr Dif ferent symbols correspond to dif ferent input opacity tables used by the PMS tracks. Each bin' s v alue corresponds to the mean v alue of that bin for 100 independent realizations of the model KLF Each realization of the model KLF contained 1000 stars. Error bars correspond to the 1 s standard de viation of the mean v alue of that bin for the 100 iterations. lo wer luminosities on the main sequence ( Iben 1965 ). W e refer to this feature as the luminosity maximum spik e (LMS). This luminosity maximum spik e has been studied by Belik o v & Piskuno v ( 1997 ) in intermediate age (50-100 Myr) clusters and these authors ha v e used it to study the age of the Pleiades open cluster ( Belik o v et al. 1998 ). Model KLFs appear de generate in the absence of the deuterium-b urning spik e. The e xistence of a deuterium spik e remo v es the de generac y and dif ferentiates between the tw o dif ferent PMS opacity models. Moreo v er the position of the deuterium-b urning

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25 spik e can dif ferentiate between the tw o con v ection treatments b ut only for Ale xander opacities. Ho we v er only the youngest clusters e xhibit a deuterium-b urning spik e. Model KLFs computed with dif ferent PMS tracks and with mean ages greater then 2-3 Myr are essentially indistinguishable from each other and consequently insensiti v e to the input ph ysics of the PMS models. Note that the deuterium-b urning spik e is most prominent when deuterium-b urning occurs in those stars with masses at the peak of the chosen IMF which for models discussed here occurs at the h ydrogen b urning limit (mean ages 1-2 Myr). Introducing an age spread to the cluster star -forming history diminishes the dif ferences between the KLFs for all four DM94 and at an y cluster mean age. While we fully describe the ef fects of age and age spread on the model KLFs in Section 2.5.2 these result implies that e xcept in the youngest clusters, the KLF will be observ ationally insensiti v e to v ariations in the input ph ysics of the PMS models. T o further study ho w dif ferent PMS tracks af fect the model KLFs, we compared the model KLFs using the PMS tracks of DM94 with the models computed using the more recent and impro v ed calculations of D'Antona & Mazzitelli ( 1997 hereafter DM97). T able 2–1 lists the rele v ant characteristics of the DM97 tracks. W e constructed model KLFs computed with the standard mass range (0.02 to 80 M) using the DM94 A CM and DM97 d2.5 PMS tracks. These tw o PMS tracks ha v e similar deuterium ab undances b ut DM97 ha v e adv ancements to the opacity table and treatment of con v ection as well as a ne w equation of state. Figure 2–5 compares model KLFs using these tw o PMS tracks with a coe v al cluster SFH and mean ages of 0.8 and 5 million years. In general the o v erall shapes of the model KLFs from the tw o dif ferent PMS tracks are quite similar b ut some minor dif ferences can be quantied. First, the DM97 model KLFs are some what narro wer and ha v e peaks shifted to slightly brighter magnitudes than those KLFs corresponding to the DM94 A CM tracks. This w as a consistent result for all cluster ages and star -forming histories.

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26 Figure 2–5: Model KLFs: comparing DM94 and DM97. Sho wn are model KLFs computed using the A CM model from DM94 and the d2.5 model from DM98 (see T able 2–1 ). These tw o PMS e v olutionary models dif fer in basic input ph ysics such as opacity table, equation of state and treatment of con v ection, ho we v er the y co v er similar mass ranges and ha v e identical deuterium ab undances. Upper panel: t08 Myr Lo wer panel: t50 Myr Both panels correspond to model KLFs for clusters with a coe v al star -forming history Error bars are the same as those in Figure 2–4 Second, the lar gest dif ferences between the model KLFs occur at the f aint end. This is where DM97 describe the lar gest dif ferences in their PMS tracks with respect to the DM94 PMS tracks. DM97 PMS tracks ha v e a v ery dif ferent resulting mass-ef fecti v e temperature relation for lo w mass stars and bro wn dw arfs than DM94. Since the K band bolometric correction is f airly insensiti v e to ef fecti v e temperature for stars cooler than 3500 K (see Section A ), these changes do not radically af fect the model KLF Further DM97 PMS tracks ha v e lar ger luminosities for the lo w mass stars and young bro wn dw arfs compared to DM94. Lik e wise the DM97 model KLFs are shifted to

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27 brighter magnitudes with respect to DM94 for the f aint end of the KLF Ho we v er these dif ferences in the KLFs are relati v ely small and it w ould be dif cult, observ ationally to distinguish between them. D'Antona & Mazzitelli (1997): v ariations in deuterium ab undance. The DM97 PMS tracks were specically created to study the ef fects of v arying the initial deuterium ab undance for PMS e v olutionary calculations. It is unclear ho w much deuterium pre-main sequence stars might contain as the y e v olv e from the birthline to w ard the main sequence. And there is little observ ational e vidence to constrain this parameter so it should be considered as an ambiguity in modeling the KLFs. W e studied the ef fects of the deuterium ab undance on the KLFs by e xperimenting with the three PMS tracks presented by DM97. The opacities used by DM97 in their PMS tracks are adv ancements to those in the DM94 PMS tracks which produced a deuterium-b urning spik e in the KLFs of Figure 2–4 DM97 input ph ysics and deuterium ab undances are summarized in T able 2–1 W e produced model KLFs using the three DM97 PMS tracks, d1.5, d2.5, and d4.5, so labeled by their respecti v e deuterium ab undance ratios, e.g., the d1.5 set of tracks has a deuterium ab undance of 10105 Respecti v ely these three sets of PMS tracks ha v e deuterium ab undances of one half, one and tw o times the interstellar deuterium ab undance, which isDH020105 Figure 2–6 compares model KLFs deri v ed from these PMS tracks for mean ages of 2 and 7 Myr and both coe v al and Dtt20 age spreads. Comparing the coe v al models it is clear that increasing the [D/H] ab undance shifts the deuterium-b urning spik e to brighter magnitudes and increases its size. The deuterium-b urning peak disappeared from the d1.5 KLF by 3 Myr the d2.5 KLFs by 10 Myr and from the d4.5 KLFs not until be yond 10 Myr F or model KLFs sho wn in Figure 2–6 with the maximum age spread, v ariations in the KLFs due to changes in the initial deuterium ab undance are too small to be observ able. The main result here is that v ariations in the

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28 Figure 2–6: Model KLFs: v arying the initial deuterium ab undance. Each panel compares model KLFs computed with dif ferent deuterium ab undances at the onset of pre-main sequence contraction. Labels (DH) correspond to the ratio of the deuterium to h ydrogen ab undance in units of105 and represent one half, equal to and twice the measured interstellar mediumDH. The model KLFs use log-normal IMF sampled o v er the entire mass range a v ailable for the DM97 PMS models and each panel corresponds to a specic SFH. Error bars are the same as those in Figure 2–4 .DHratio only produce signicant (i.e., observ able) dif ferences in the model KLFs of coe v al (no age spread) clusters. F or these clusters v ariations in deuterium ab undance af fects the location and size of the deuterium-b urning feature and this occurs only in younger ( t3 Myr) clusters or for the highest deuterium ab undances. Once stars ha v e under gone deuterium-b urning, their KLFs are identical. Ag ain, the presence of an

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29 age spread dilutes the deuterium-b urning feature rendering the form of the cluster KLF independent of theDHratio. Effecti v e mass ranges f or PMS models. W e in v estig ated the ef fects of using dif ferent IMF mass ranges by comparing model KLFs with the standard mass range (0.02 to 80 M) to model KLFs with a truncated mass range (0.1 to 2.5 M), i.e., one e xcluding bro wn dw arfs, intermediate or high-mass stars. This e xperiment is useful for comparing our model LFs to prior LF modeling by other authors who typically did not include stars belo w the h ydrogen b urning limit or did not include high-mass stars. Figure 2–7 compares model KLFs with truncated and standard mass ranges for tw o dif ferent star -forming histories. F or a coe v al SFH (upper panel, mean age 3.0 Myr), a truncation in the mass range produces a truncation in the model KLFs at the highest and lo west magnitude bins. Ho we v er with an age spread (lo wer panel, same mean age, Dtt20), the truncated model KLF is decient in stars o v er a wider range of magnitudes, and the tw o KLFs are similar only o v er a narro w range of magnitudes. The form of the cluster KLF is clearly v ery sensiti v e to the adopted mass range of the underlying IMF 2.5.2 Star F ormation History As sho wn in the e xperiments of Section 2.5.1 mean age and age spread ha v e an important ef fect on the KLF T o more fully e xplore this, we created model KLFs with a range of mean ages and age spreads, using a single underlying mass function, and a x ed set of PMS tracks. F or these e xperiments, we used the same log-normal IMF as in Section 2.5.1 (see Equation 2.2 ). As in the pre vious section, we considered tw o mass ranges for the IMF one range with stars do wn to the 010 Mand one including bro wn dw arfs with masses do wn to 002 M. W e adopted our standard PMS e v olutionary models described abo v e, i.e., our combination of DM97 d2.5 PMS models, Bernasconi ( 1996 ) intermediate-mass PMS tracks, and Schaller et al. ( 1992 )

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30 Figure 2–7: Model KLFs: truncations in the mass-luminosity relation. Model KLFs testing the inclusion into model KLFs of high and intermediate mass stars as well as stars at the h ydrogen b urning limit and bro wn dw arfs. The mass to luminosity relation w as e xtracted from the A CM PMS model of DM94, intermediate mass PMS tracks from P alla & Stahler ( 1993 ) and a ZAMS from Schaller et al. ( 1992 ). T w o dif ferent mean ages and SFH histories are sho wn for illustration. Upper panel: coe v al star -forming history and a mean age of 3.0 Myr Lo wer panel: continuous star formation o v er the age of the cluster with a mean age of 5 Myr Error bars are the same as those in Figure 2–4 ZAMS models. W e compared the ef fects of changing the mean age and age spread by studying ho w model KLFs e v olv e with time. Figure 2–8 compares model KLFs with dif ferent mean ages and cluster age spreads. Each panel simultaneously displays a one, three and ten million year mean age cluster KLF for a specic Dtt F or a gi v en age spread, the models clearly shift to f ainter magnitudes with increasing mean age. F or small age spreads, the deuteriumb urning feature also e v olv es to f ainter magnitudes with time appearing at M K35

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31 Figure 2–8: Model KLFs: v arying the star forming history ( tDt ). Each panel displays a dif ferent Dtt for three mean ages of 1, 3 and 10 million years. Note that from panel to panel, features in the model KLF caused by inections in the M-L relation are smoothed by the increased age spread. The appar ent do wnw ard break in the last bin of the model KLFs is primarily due to incompleteness in that bin due to the lo wer mass limit of the M-L relation at 002 M. Please see Section 2.5.1 for further e xplanation of the ef fects of an articial truncation in the mass-luminosity relation. Error bars are the same as those in Figure 2–4 at 1 Myr and M K55 at 3 Myr and M K8 at 10 Myr T o quantify the KLF e v olution with time, we calculated the mean K magnitude of the model KLF at each mean age from 0.5 to 10 Myr and for a range of age spreads. In Figure 2–9 we plot the KLF mean magnitude v ersus the cluster mean age and plot this quantity for the tw o e xtrema of Dtt T w o sets of curv es are plotted, the upper corresponding to an

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32 underlying cluster IMF with bro wn dw arfs (standard IMF mass range) and the lo wer to an underlying IMF that truncates at 0.1 M. Figure 2–9: Ev olution of mean K magnitude with cluster age. The KLF mean refers to the arithmetic mean of the K magnitudes for all synthetic cluster members. T w o sets of v alues are plotted for KLFs ha ving tw o dif ferent underlying IMFs. ”W ith Bro wn Dw arfs” contains stars belo w 01 Mand ”W ithout Bro wn Dw arfs” has no objects less than 01 M. F or each set of curv es, the KLF mean w as plotted for the tw o e xtrema of the cluster' s age spread, Dtt0 and 2 Error bars are not sho wn b ut are within the size of the plotting symbols for a cluster of 1000 stars. The mean K magnitude of the model KLFs e v olv es o v er 2 magnitudes in the rst 10 Myr of the cluster lifetime, re g ardless of the age spread or the mass range o v er which the IMF w as considered. Age spread has little ef fect e xcept to slightly shift the KLFs to brighter magnitudes. The e v olution of the mean K magnitude proceeds most quickly in the rst 3 million years where the models e v olv e by 1 full magnitude. The model KLFs without bro wn dw arfs naturally ha v e signicantly brighter mean v alues b ut for these KLFs the mean K magnitude e v olv es similarly to the standard models. This indicates that the KLFs are more sensiti v e to changes in the underlying IMF than to changes in the cluster star -forming history W e also studied the width of the model KLFs and found that KLFs widen systematically with time as w as sho wn by LL95

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33 Figure 2–10: Model KLFs: v arying the cluster' s age spread. Synthetic cluster KLFs with mean ages of t1and 6 Myr are sho wn in top and bottom panels, respecti v ely In each panel the same four age spreads sho wn in Figure 2–8 are o v er -plotted. Upper panel: t1 Myr. Lo wer panel: t6 Myr. Increased age spread erases features in the model KLFs caused by inections in the mass-luminosity relation. Error bars are the same as those in Figure 2–4 V ariations in the mean cluster age produce more signicant changes in the the model KLFs than do changes in the cluster age spread. W e sho w in Figure 2–10 model KLFs for tw o mean ages and for both of these mean ages we sho w the four dif ferent age spreads from Figure 2–8 F or a gi v en mean age, it w ould be dif cult to observ ationally distinguish clusters with dif fering age spreads. In detail, models with dif fering age spreads do e xhibit dif ferences in the prominence of the deuterium-b urning spik e and the maximum luminosity dip/spik e. At what point can one distinguish a coe v al model KLF from a model KLF with an age spread? T o answer this question, we compared model KLFs with increasing age spread to a coe v al model of the same mean

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34 age. Using the c 2 test, we distinguished the age spread at which the models KLFs no longer appear to be coe v al. The general trend from our test is that for an increasing mean age, we require a steadily increasing age spread to distinguish the models from a coe v al KLF F or mean ages up to 5 Myr we could not distinguish model KLFs with age spreads from their coe v al counterparts until the age spread e xceeded the cluster' s mean age ( Dtt1). This changes some what between 5 and 10 Myr since the deuterium-b urning feature is present among the bro wn dw arfs b ut is not v ery prominent in the model KLFs. Thus, only a v ery small age spread is required to erase it from the model KLFs and thus the models no longer appear ”coe v al” with only a small amount of age spread. Once the deuterium-b urning feature is lost from the M-L relation, the models require v ery lar ge and probably unrealistic age spreads for them to signicantly dif fer from a coe v al model of the same mean age. 2.5.3 Initial Mass Function W e v aried the underlying initial mass function of a young cluster to test the inuence of the input IMF on the model KLFs. In pre vious sections we used a single IMF equi v alent to the log-normal ( MS79 ) mass function and only changed the mass limits to this IMF T o test the sensiti vity of the KLF to v ariations in the underlying IMF we adopted a tw o se gment po wer -la w IMF as dened in Section 2.2.1 and in Equation 2.1 In these e xperiments, we v aried G 1 v alues from -2.5 to -0.25, m 1 from 006 to 15 Mand G 2 v alues from -1.35 to +2.0. Figure 2–11 displays some of the model KLFs and the corresponding underlying IMFs. The cluster star -forming history used for these models has a mean age of 5 Myr and a Dtt = 1.0, or an age spread of 5 Myr W e sho w model KLFs normalized to the bright end of the KLF where the underlying IMF po wer -la w indices ha v e identical G 1 slopes equal to -1.35. This e xample uses a m 105 Mand v e G 2 v alues equal to -1.35, -0.40, 0.0, +0.40 and +1.35. The most

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35 Figure 2–11: Model KLFs: v arying the initial mass function. This plot illustrates the sensiti vity of the model KLFs to changes in the form of the underlying po wer -la w IMF (see Equation 2.1 ). The dif ferent model KLFs are normalized to their bright LF slopes where their underlying IMFs are identical. The left panel sho ws the model KLFs corresponding to the underlying IMFs sho wn in the right hand panel. Symbols are identical for underlying IMFs and the resulting model KLF steeply rising KLF corresponds to a single Salpeter po wer -la w IMF o v er the entire mass range. Model KLFs display v ariations due to changes in all three parameters of the tw o po wer -la w IMF In Figure 2–11 the ef fects of changing G 2 are lar ge and the dif ferences between KLFs with a slightly rising and a slightly f alling IMF belo w the break mass are signicant. V arying the m 1 produces shifts in the peak of the model KLFs. Another result of these tests is that o v er the range of K magnitudes go v erned by a single underlying IMF po wer -la w the model KLF tends to be characterized by a po wer -la w lik e slope. This is true both for the bright and f aint slopes of the model KLFs a w ay from the turno v er caused by the m parameter in the model IMF

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36 Other than a steep do wnw ard drop seen in the last bin of the model KLFs, the model KLFs closely mimic the underlying IMF decreasing or increasing in number where the IMF is rising or f alling. The drop in the last bin of the model KLFs is a byproduct of the limits of the PMS tracks and can be understood by re vie wing the comparisons of truncated and e xtended M-L relations in Section 2.5.1 and Figure 2–7 Simply this turno v er is the result of truncating the mass range for the underlying IMF at 002 M. In summary these calculations clearly sho w that the shape of the model KLF is v ery sensiti v e to v ariations in the underlying cluster IMF Indeed, modest v ariations in the cluster IMF produce signicantly greater responses in the model KLFs than do v ariations in the SFH and PMS model input ph ysics. 2.6 Discussion and an Example fr om the Literatur e 2.6.1 Results and Implications of Numerical Experiments From these numerical e xperiments which e v aluate the sensiti vity of the K-band luminosity function to v ariations in three of its fundamental ph ysical parameters: its underlying IMF its star -forming history and its mass-to-luminosity relation, we nd that the KLF of a young cluster is more sensiti v e to v ariations in its underlying IMF than to either v ariations in the star -forming history or the PMS mass-to-luminosity relation. W e also nd that v ariations in the cluster mean age can produce a signicant response in the KLF of a young cluster In particular we nd that the KLF systematically e v olv es with time. Both the mean magnitude and the width of the KLF increase with increasing mean age, conrming the results of earlier modeling (LL95). At the same time, v ariations in the cluster age spread are found to ha v e a small ef fect on the form of the KLF and w ould lik ely be dif cult to distinguish observ ationally Except for the youngest and purely coe v al clusters, we nd that the synthetic KLFs appear relati v ely insensiti v e to the adopted PMS e v olutionary models (at least for the range of PMS models considered here). In the youngest coe v al clusters, the

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37 location and size of the deuterium-b urning spik e in the KLF w as found to depend sensiti v ely on the PMS tracks adopted for the underlying stars. Ho we v er we nd that e v en a small amount of age spread broadens the spik e and w ould mak e it observ ationally dif cult to detect. W e conclude from these e xperiments that the KLF of a young stellar population can be used to place interesting constraints on the form of the cluster' s underlying IMF pro vided an independent estimate of the cluster mean age is a v ailable. The most direct method of determining the mean age of a young cluster is to obtain optical or infrared spectra and place the objects on the H-R diagram. Through comparison to theoretical PMS tracks, the ages of the stars are determined and a mean age for the cluster deri v ed. From spectroscopic observ ations, one can also simultaneously deri v e the indi vidual masses of the stars and with complete spectra for all cluster members, an independent and more direct determination of the IMF results. Ho we v er because of spectroscopic sensiti vity limits, the determination of masses is usually only possible for the bright stellar population. Since the monochromatic K magnitude of the cluster members can be acquired for stars much f ainter than the limit of spectroscopic methods, the analysis of the near -infrared (NIR) luminosity function is a particularly po werful tool for in v estig ating the IMF of f aint stars in distant clusters or stars at and belo w the h ydrogen b urning limit in nearby clusters. Determining the fraction of cluster members at and belo w the h ydrogen b urning limit is a holy grail of present stellar research. The application of the luminosity function method to a nearby populous cluster w ould pro vide a rst glimpse into the bro wn dw arf population formed at the time of a typical open cluster' s birth. 2.6.2 An Example fr om the Literatur e: The T rapezium Cluster The T rapezium cluster is a e xcellent system for e v aluating the KLF modeling techniques de v eloped in this paper It is the most densely populated and best studied

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38 nearby (D400-450pc; see Section B ) cluster and the central part of a much lar ger cluster kno wn as the Orion Neb ula Cluster (ONC). The ONC has recently been studied by Hillenbrand ( 1997 ), who used optical spectroscop y to obtain a mean age for the cluster of 0810 6 years and to construct an IMF for stars with masses primarily in e xcess of the h ydrogen b urning limit (HBL). In addition, infrared imaging surv e ys ha v e been made of both the T rapezium cluster ( Zinneck er et al. 1993 ; McCaughrean et al. 1995 ) and the ONC ( Ali & Depo y 1995 ) enabling the construction of the cluster KLF from these literature data. F or comparison with our models, we consider only the KLF for the T rapezium cluster the 5' by 5' central core of the ONC. W e constructed a KLF of the T rapezium by combining the cluster KLFs published by Zinneck er et al. ( 1993 ) and McCaughrean et al. ( 1995 ). Our adopted KLF for the T rapezium is sho wn in the top panel of Figure 2–12 The Zinneck er et al. KLF includes the bright stars b ut is not complete at and belo w the HBL. The McCaughrean et al. KLF e xtends to v ery f aint magnitudes, well belo w the HBL for a one million year old cluster 400pc distant, b ut because of source saturation, is incomplete for and does not include bright stars. Neither of these referenced cluster KLFs were corrected for contamination by fore ground or background eld stars. In addition, neither w as corrected for the ef fects of neb ular contamination which w ould confuse the completeness of the surv e ys. Ho we v er we compared this combined T rapezium KLF to the literature KLF from the Ali & Depo y ( 1995 ) surv e y of the entire ONC and found good agreement in the location of the turno v er bright and f aint ends of the tw o KLFs, although the Ali & Depo y surv e y w as not as sensiti v e as that represented by the McCaughrean et al. KLF W e reiterate that the e xtent to which this literature based KLF represents the true T rapezium KLF is uncertain because we cannot account for eld star or neb ular contamination. Here our goal is to nd the simplest functional form of an underlying IMF whose resulting model KLF best ts the observ ed KLF W e constrained the star -forming

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39 Figure 2–12: Application of models to literature data. T op panel: Literature T rapezium KLF compared to the best t model KLF (t from M K 05 to 65). Also sho wn: a model KLF created using a single po wer -la w Salpeter IMF The cluster KLF error bars are 1 s counting statistics. The model KLF error bars are described in Figure 2–4 Lo wer panel: KLF deri v ed T rapezium IMF compared to the Orion Neb ula Cluster IMF deri v ed by Hillenbrand ( 1997 ) using an optical spectroscopic study (histogram). Also sho wn: the Salpeter IMF and the mass completeness limit of the optical analysis. F or comparison, model IMF (g) is scaled to the same number of stars as the Hillenbrand IMF abo v e the latter completeness limit. Error bars for the Hillenbrand IMF reect 1 s counting statistics. history of the T rapezium cluster by using the mean age from Hillenbrand ( 1997 ) i.e., 0.8 million years. W e allo wed an age spread of 1.2 million years ( Dtt = 1.5) about this mean age, corresponding to constant star formation from 0.2 to 1.4 million years ago. W e inspected the observ ed KLF and determined that a single po wer -la w IMF could not satisfy the observ ations since the KLF has a peak and turno v er well abo v e the completeness limits of the tw o surv e ys. Therefore we be g an with a simple

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40 T able 2–2. Cluster IMF deri v ed from the literature T rapezuim KLF N GaName c 2 Prob G 1 m 1 G 2 m 2 G 3 2 a 0.38 -0.50 0.10 +1.00 3 b 0.71 -0.75 0.25 0.00 0.10 +0.75 3 c 0.86 -1.00 0.40 0.00 0.08 +1.00 3 d 0.88 -1.00 0.60 -0.25 0.10 +1.00 3 e 0.93 -0.75 0.25 -0.25 0.10 +0.75 3 f 0.99 -1.00 0.70 -0.25 0.08 +1.00 3 g 0.99 -1.35 0.80 -0.25 0.08 +1.35 4bh 0.96 -1.70 1.00 -0.20 0.10 +0.75 4bi 0.99 -1.70 1.00 -0.20 0.08 +1.00 aNumber of po wer -la ws, G in the deri v ed IMF .bAbo v e 10 M, this IMF has a G 0 equal to -1.30. G 1m 1and G 2 were x ed. 2 po wer -la w IMFs. W e ne xt used a three po wer -la w IMF with a at (zero slope) IMF in the middle. F or symmetry the tw o outer po wer -la w slopes were set to ha v e equal b ut opposite sign slopes. W e v aried these outer slopes to ha v e absolute v alues between 0.25 and 2.00 and adjusted the mass range o v er which the middle slope of the IMF w as at. Finally as a third set of e xperiments, we allo wed the slope of the middle po wer -la w to v ary still holding the outer tw o slopes to ha v e equal b ut opposite sign slopes. W e produced a suite of model KLFs for these dif ferent IMFs and compared them to the combined T rapezium KLF using a chi-square tting procedure. Simply we normalized model KLFs to the observ ed KLF such that the model and observ ed KLFs contain the same number of stars between absolute K magnitudes, M K0 and 65. W e then calculated the c 2 statistic and probability o v er this K magnitude range. T o deri v e a best t, we compared a suite of model KLFs v arying a single IMF parameter e.g, the middle slope G 2 or one of the m j v alues and then determining the c 2 minima for that v ariable. Model KLFs were created for a range of possible IMF parameters and compared to the T rapezium KLF in this w ay

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41 Best t model IMFs for each of the tested functional forms of the IMF are listed in T able 2–2 T w o po wer -la w ts in general were not good. Symmetric at topped IMFs t better and nally a slightly rising IMF across the middle pro vided a best t with c 21. Some v ariation in each of the parameters still allo wed for a t of c 21 and e xamples are listed in T able 2–2 The IMFs (f) and (g) produced best ts to the data and for purposes of discussion, we adopt IMF (g) as representati v e of the T rapezium IMF and repeat its parameters here: d N d log M M G; G 135 : 008 M M 025 : 080 M M 008 M 135 : M 080 M(2.7) The model KLF corresponding to IMF (g) is sho wn in the top panel of Figure 2–12 compared to the combined T rapezium KLF and compared to a model KLF calculated with the single po wer -la w slope Salpeter eld star IMF o v er the entire standard mass range. From our modeling of the observ ed KLF for the T rapezium cluster we nd that the predicted IMF has a rising slope for intermediate mass stars, attens around a solar mass, reaches a peak near the HBL and turns o v er belo w the h ydrogen b urning limit. There are se v eral comparisons between the observ ed and modeled T rapezium KLF and between the ONC IMF deri v ed by Hillenbrand and our deri v ed IMF (g) which should be made. First, there e xists a signicant ”tail” to the observ ed T rapezium luminosity function which is not accounted for in the model KLFs. No attempt w as made to account for these v ery f aint stars as cluster members because if the y were, the y w ould require ages much older than the distrib ution suggested by the H-R diagram or lo wer masses than pro vided by our standard PMS tracks we are using. W e instead suggest that these are either e xtremely embedded cluster members or hea vily e xtincted background eld stars (A V2030). W e base these suppositions on the f act that the T rapezium is at the core of a blister H II re gion on the front of a dense molecular

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42 cloud, and because secondary peaks in young cluster luminosity functions are often e vidence of a background population seen in projection to w ards the cluster Either of these possibilities w ould in turn imply that our deri v ed IMF is in f act an upper limit to actual IMF belo w the h ydrogen b urning limit. Experiments studying the ef fects of e xtinction on the model KLF by Me geath ( 1996 ) and Comer on et al. ( 1996 ) found that while e xtinction tended to shift a luminosity function to f ainter magnitudes, the slope(s) of the KLF were preserv ed. Thus, the steeply f alling slope at the lo w mass end of the deri v ed T rapezium IMF is reecti v e of the actual underlying IMF Ho we v er the true IMF may turno v er at a lar ger mass than that implied by our present models. In the lo wer panel of Figure 2–12 the mass function deri v ed from the T rapezium KLF is compared to that deri v ed from spectroscopic observ ations by Hillenbrand ( 1997 ). The tw o mass functions are generally v ery similar In particular these tw o mass functions agree v ery well at the high-mass end ( M 20 M). F or masses in the range 20 M M 05 Mthe IMF deri v ed from modeling the luminosity function contains more stars than that deri v ed by Hillenbrand It is not, ho we v er clear ho w signicant this dif ference is gi v en the possible systematic uncertainties in v olv ed in both methods of determining the IMF Further these tw o IMFs sample dif ferent v olumes of the Orion Neb ula re gion. F or masses belo w M 01 M, the IMF deri v ed from the KLF modeling also contains considerably more stars than the spectroscopic IMF Ho we v er this dif ference is also not lik ely to be signicant either since the spectroscopic IMF of Hillenbrand ( 1997 ) is not complete belo w 0.1 M. Lastly we can in v estig ate whether the eld star IMF (FSIMF) could also produce a KLF which reasonably matched the literature T rapezium KLF T o test this, we used the recent eld star IMF parameterization from Scalo ( 1998 ). Scalo ( 1998 ) suggested a

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43 multiple po wer -la w IMF with the form: d N d log M M G; G 130 : M 1000 M 170 : 1000 M M 100 M 020 : 100 M M 010 M(2.8) Comparing the IMF in Equation 2.7 to the eld star IMF in Equation 2.8 one nds that these tw o IMFs are quite similar although for stars in the range of 100MM 10, the Scalo IMF is steeper than the IMF in Equation 2.7 In addition, the Scalo FSIMF does not e xtend belo w the h ydrogen b urning limit. T o f acilitate comparison to the T rapezium data, we added a fourth po wer -la w to the Scalo IMF to account for the f aintest stars. W e v aried m 2 the mass at which the fourth po wer -la w be gins, between 006 and 01 M. In addition, we v aried the slope of the fourth po wer la w G 4 between -1.0 and +2.0. The best ts with this IMF are also listed in T able 2–2 Using this modied eld star IMF did yield a c 21 with an IMF that breaks near the h ydrogen b urning limit and f alls with a similar steep slope as in the prior IMF ts. T o the e xtent that our adopted KLF represents the true KLF of the cluster our modeling suggests that the IMF for bro wn dw arfs in the T rapezium cluster f alls relati v ely steeply with decreasing mass. Ho we v er because contamination due to reddened background stars and incompleteness due to neb ular confusion has not properly been tak en into account in the construction of this literature T rapezium KLF the form of the deri v ed IMF belo w the h ydrogen b urning limit should be re g arded with appropriate caution. As sho wn in Lada & Lada ( 1995 ) and Lada et al. ( 1996 ), one can use control-eld observ ations (which are not a v ailable for this dataset) to g auge the completeness and membership at the f aint end of the LF Also, our present modeling has not included the ef fects of e xtinction and infrared e xcess. Hillenbrand et al. ( 1998 ), using the (I-K) diagnostic, found an a v erage K band e xcess of 0.35 among identied optically visible cluster members. This a v erage e xcess is smaller then the bins we ha v e used to construct the T rapezium KLF and therefore should ha v e only a minor ef fect.

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44 Ov erall, we conclude from our modeling that the IMF of the T rapezium cluster is well represented by a three po wer -la w mass function with a high-mass slope between -1.00 and -1.7, a break in slope between 1 and 06 Mfollo wed by a relati v ely at or slightly rising slope to the h ydrogen b urning limit. From our luminosity function modeling, we then found, for the rst time that the T rapezium IMF f alls with a steep slope 1 into the bro wn dw arf re gime. 2.7 Conclusions After de v eloping a Monte-Carlo based model luminosity function algorithm, we performed a series of e xperiments aimed at studying ho w the pre-main-sequence mass-to-luminosity relation, star -forming history and initial mass function each af fect the form of the luminosity function for populations of young pre-main sequence stars. Using models of the near -infrared luminosity function and v arying these primary inputs, we ha v e deri v ed the follo wing simple conclusions about model near -infrared luminosity functions: 1. W e nd that the KLF of a young cluster is considerably more sensiti v e to v ariations in its underlying IMF than to either v ariations in the star -forming history or the PMS mass-to-luminosity relation. 2. PMS luminosity functions e v olv e in a systematic manner with increasing mean age and age spread. The y e v olv e to f ainter magnitudes and widen systematically with age. 3. The KLFs of young stellar populations are found to be generally insensiti v e to v ariations in the adopted PMS mass-to-luminosity relations. In the youngest, coe v al clusters, the presence of deuterium-b urning can produce signicant features in the KLF which are sensiti v e to the adopted mass to luminosity relation. Ho we v er e v en a small departure from a purely coe v al star -forming history will render these features dif cult to detect observ ationally W e then undertook a preliminary e xamination of the T rapezium Cluster using data tak en from the literature. W e apply our models to the K band luminosity function of the T rapezium and are able to deri v e an underlying T rapezium IMF which spans a

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45 range of stellar mass from 5 Mto 0.02 M, well into the bro wn dw arf re gime. The IMF we deri v e is the simplest multiple po wer -la w function which can reproduce the observ ed luminosity function of the cluster gi v en the mean age and star -forming history deri v ed from pre vious optical spectroscopic studies ( Hillenbrand 1997 ). The deri v ed IMF for the T rapezium cluster consists of three po wer la w se gments, has a peak near the h ydrogen b urning limit and steadily decreases belo w the h ydrogen b urning limit and throughout the bro wn dw arf re gime. W e deri v e a bro wn dw arf mass spectrum of the form dN/dlogmm1 (0.08MM 0.02). Ho we v er the form of the IMF belo w the h ydrogen b urning limit must be re g arded with caution since the f aint end of the observ ed cluster KLF has not been adjusted for the possible ef fects of background star and neb ular contamination. Abo v e the h ydrogen b urning limit, the T rapezium IMF we deri v e from its KLF also appears consistent with that recently adv ocated for eld stars by Scalo ( 1998 ).

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CHAPTER 3 THE F AMOUS TRAPEZIUM CLUSTER IN ORION In Section 2.6.2 we e xplored the monochromatic K band luminosity function for the well-studied T rapezium Cluster in Orion, which we constructed from literature sources. While we found good agreement between the mass function deri v ed from modeling the cluster' s luminosity function and that IMF found for this cluster using a spectroscopic analysis of the optically visible members, luminosity function modeling enabled the deri v ation of the cluster' s substellar IMF which w as not possible from the optical/spectroscopic analysis. W e concluded from the application of these rst-order models to the T rapezium Cluster KLF that model luminosity functions are indeed useful for studying the mass functions of young clusters. Ho we v er the models we applied to the T rapezium cluster did not include other observ ational characteristics of a young cluster that may af fect the con v ersion between the luminosity and mass functions. Ha ving only the monochromatic T rapezium KLF tak en from the literature with no color or completeness information pre v ented our studying these observ ational ef fects in detail. Further we concluded that we could not t our models to the entire luminosity range of the literature KLF because of structure that we attrib uted to hea vily reddened cluster members or background eld stars. T o impro v e upon this modeling and to standardize the formula for applying the model luminosity functions to the products of a deep near -infrared surv e y of a young embedded cluster we ha v e constructed o v er a three year period of observ ations a multiepoch, multi-w a v elength near -infrared census of the T rapezium Cluster that we describe in Section 3.1 Using this detailed near -infrared census of the T rapezium, we ha v e e xpanded our analysis of this cluster' s K band luminosity function and its underlying Initial Mass Function. In Section 3.2 we construct the cluster' s KLF e xploring both 46

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47 the contrib ution of background eld stars, and the completeness of our surv e y as it probes the cluster' s parental molecular cloud. W e rederi v e the cluster' s underlying IMF in Section 3.3 rening our techniques to include the ef fects of source reddening and to t the model KLFs to the data. In our re vised analysis we are able to probe the cluster' s KLF to f ainter magnitudes and deri v e the cluster' s mass function do wn to the deuterium-b urning limit. W ith these ne w results, we discuss in Section 3.4 the relationship between the form of a cluster' s KLF and its deri v ed IMF and we compare our T rapezium IMF deri v ed in this chapter and in Chapter 2 to the T rapezium IMF deri v ed by other authors using dif ferent methods. W e illustrate the relati v e rob ustness of the pre-main sequence mass-luminosity relation as predicted by dif ferent theoretical e v olutionary models of young stars. 3.1 Near -Infrar ed Census T o deri v e a complete multi-w a v elength census of the sources in the T rapezium Cluster we performed infrared observ ations during 1997 December 1998 No v ember and 2000 March using tw o telescopes: the 1.2m telescope at the Fred La wrence Whipple Observ atory (FL W O) at Mt. Hopkins, Arizona (USA) and the European Southern Observ atory' s (ESO) 3.5m Ne w T echnology T elescope (NTT) in La Silla, Chile. These observ ations yielded the multi-epoch, multi-w a v elength FL W O-NTT infrared catalog that contains1000 sources. Subsets of this catalog ha v e been published pre viously in the Lada et al. ( 2000 ) and Muench et al. ( 2001 ) studies of the frequenc y of circumstellar disks around stars and bro wn dw arfs in the T rapezium Cluster W e detail belo w the observ ations, data reduction, and photometry in v olv ed with the construction of the catalog. W e also include summaries of the photometric qualities of the datasets and an e xplanation of the electronic v ersion of the nal FL W O-NTT infrared catalog.

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48 3.1.1 Obser v ations W e summarize in table 3–1 the characteristics of the three observing runs used to obtain the infrared photometry that comprise the FL W O-NTT Near -Infrared Catalog of the T rapezium Cluster W e compare the area(s) co v ered by the FL W O-NTT catalog to those of other recent IR surv e ys in gure 3–1 Figure 3–1: Comparison of recent T rapezium cluster IR surv e ys. The tw o shaded re gions represent the 6 56 5 FL W O surv e y and the 5 5NTT surv e y presented in this w ork. Also sho wn are the HST -NICMOS surv e y ( Luhman et al. 2000 solid black border), the K eck surv e y ( Hillenbrand & Carpenter 2000 solid white border), and the UKIR T surv e y ( Lucas & Roche 2000 brok en black border). The locations of luminous cluster members, spectral types B3 and earlier are sho wn as white stars. Whipple Obser v atory – 1997 and 1998: 1.2m JHK-bands. Initial infrared observ ations of the T rapezium Cluster re gion were made on 14, 15, 16 December 1997 with the FL W O 1.2m telescope at Mt. Hopkins, Arizona using the STELIRcam dual channel infrared camera. The STELIRcam instrument allo ws simultaneous infrared

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49 T able 3–1. Summary of infrared observ ations of the T rapezium cluster Observ atoryaDate P assbandbPlate ScalecYYYY / MM / DD Beamsize FL W O 1997 / 12 / 14 H 0.596 / 3.58 FL W O 1997 / 12 / 14 K 0.596 / 3.58 FL W O 1997 / 12 / 15 H 0.596 / 3.58 FL W O 1997 / 12 / 15 K 0.596 / 3.58 FL W O 1997 / 12 / 16 J 0.596 / 3.58 FL W O 1998 / 11 / 04 J 0.596 / 3.58 FL W O 1998 / 11 / 04 H 0.596 / 3.58 FL W O 1998 / 11 / 04 L 0.596 / 3.58 NTT 2000 / 03 / 14 K s 0.288 / 1.73 NTT 2000 / 03 / 14 H 0.288 / 1.73 NTT 2000 / 03 / 14 J 0.288 / 1.73 NTT 2000 / 03 / 14 K s 0.288 / 1.73 aFL W O: Fred La wrence Whipple Observ atory; NTT : Ne w T echnology T elescope.bFilter central w a v elength l m : FL W OJ) 125, H) 165; K) 220; L) 350; NTT J) 125; H) 165; K s ) 216.cPlate scale: arcsec/pix el; Beamsize: diameter of photometry beam (arcsec)

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50 observ ations using tw o 256256 pix el InSb arrays and emplo ying a dichroic mirror to di vide w a v elengths long-w ard and short-w ard of 1.9 m. A cold lens assembly allo ws three changeable elds of vie w and for all our FL W O observ ations the camera w as congured to ha v e 2 52 5 eld of vie w with a plate scale of 0 6 /pix el. W e surv e yed the T rapezium Cluster re gion in a 33 mosaic pattern, centering on the bright O7 star HD 37022 ( Q 1c Orionis) and o v erlapping34 between mosaic positions. Our observ ational technique w as to observ e 3 on-cluster mosaic positions follo wed by 1-2 non-neb ulous of f-elds which were used both for the creation of accurate sk y/at elds and for eld star estimation. These of f-elds were centered at at R.A. = 05 h 26; DEC. =0600(J2000) and were determined to be free of molecular material by inspection of the P alomar Sk y Surv e y Plates and the 100 micron dust opacity maps of W ood et al. ( 1994 ). On 14 December 1997, H barr165 mand K barr22 mimages were obtained for all 9 mosaic positions, and 7 of the 9 mosaic positions were repeated at H and K band on 15 December J barr125 mimages of all 9 mosaic positions were obtained on 16 December 1997. Each mosaic position w as observ ed with nine dithers of 1 minute each (4 co-additions of 15 seconds) and with 12 spacing, yielding an ef fecti v e inte gration time of 9 minutes per eld. The T rapezium Cluster re gion w as observ ed at optimal airmass ( 125secz 150 ). The resulting JHK mosaics mutually co v ered an on-cluster area of approximately 6 56 5. Conditions were photometric throughout all three nights with seeing estimates ranging from 1217 arc-seconds (FWHM). T o impro v e the photometry of bright sources and increase the dynamic range of our data, we used STELIRcam at the FL W O 1.2m telescope to obtain additional short e xposure J and H band images on 4 and 5 No v ember 1998. The T rapezium Cluster re gion w as ag ain observ ed in a 33 mosaic b ut with the telescope in nodding mode taking a single 12 second (12 co-additions of 1 second each) image at each mosaic position follo wed by an identical of f-eld e xposure at a nod position 450to the west.

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51 After nishing all 9 mosaic positions, the center of the mosaic w as shifted by a small random amount (510 ) and the pattern w as repeated. Nine repetitions of the mosaic yielding a total ef fecti v e inte gration time of 108 seconds per band and these images were observ ed at transit, with a range of airmasses of 1.24 1.28. The resulting JH mosaic images co v ered an area of 7 7or slightly lar ger than the FL W O 1997 observ ations. Conditions were ag ain photometric with seeing estimated at 1618 In this dataset only the brightest 8 stars (all OB spectral types) were saturated. Eur opean Souther n Obser v atory – 2000: 3.5m JHK s -bands. Our NTT images of the T rapezium Cluster were obtained under conditions of superb seeing (05 FWHM) on 14 March 2000 using the SOFI infrared spectrograph and imaging camera. The NTT telescope uses an acti v e optics platform to achie v e ambient seeing and high image quality and the SOFI camera emplo ys a lar ge format 10241024 pix el Ha w aii HgCdT e array T o obtain a single wide eld image of the T rapezium Cluster we congured SOFI to ha v e a 4 954 95 eld of vie w with a plate scale of 0 29 /pix el. Each e xposure consisted of 9 separate dithers each randomly f alling within 20 of the observ ation center Each indi vidual dither w as the co-a v erage of eight 1.2 second e xposures, yielding an total ef fecti v e inte gration time of 86.4 seconds for each combined image. W e display a JHKs color composite image of the NTT re gion in Figure 3–2 W e observ ed the T rapezium Cluster with identical sequential pairs of on and of f-cluster dithered images. During one hour on 14 March 2000, we obtained four image pairs of the T rapezium Cluster and of f-cluster positions. These were, in temporal order at K s2162 m, H165 m, J125 mand ag ain at K s and the on-cluster images had FWHM estimates of 0.53 0.55 0.61 and 0.78 Seeing estimates of stars in the paired non-neb ulous of f-cluster image(s) yielded similar if not mar ginally higher resolution point spread functions (PSF). Observ ations were tak en near transit with a v ery small range of airmass ( 1138secz 1185).

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52 Figure 3–2: Infrared color composite image of the T rapezium. T ak en with SOFI at the ESO NTT telescope, La Silla, Chile, March 2000. North is up and east is left and the eld of vie w is 5 5. 3.1.2 Data Reduction and Photometry Data reduction of the FL W O and NTT images w as performed using routines in the Image Reduction and Analysis F acility (IRAF) and Interacti v e Data Language (IDL). Our standard data reduction algorithm w as described in Lada et al. ( 2000 ) for the FL W O images, and it w as subsequently used for the NTT images. Simply indi vidual dithered frames were reduced using sk y and at eld images deri v ed from the non-neb ulous of f-cluster dithered images which were interspersed with the oncluster images. Each set of reduced dithered frames were then combined using a standard “shift-andadd” technique. While all the FL W O data w as linearized after

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53 dark-subtraction using a system supplied linearity correction, linearization coef cients were not obtained for the NTT data. “Sk y” at-elds constructed from the NTT images were compared to system at-elds which are re gularly tak en and monitored by the NTT staf f. While the NTT system at-elds were found to v ary by only 2-3% o v er long periods of time, when we compared our sk y at-elds to the system atelds, signicant small scale v ariations (5-10%) were re v ealed across the array W e concluded this w as due to our relati v ely short NTT inte gration times which results in poor sampling of the intrinsically non-at SOFI array Therefore, we substituted the system supplied at-elds into our reduction procedure. The high resolution of our NTT images results in moderate under -sampling of the point spread functions; we tested to see if sub-pix el linear reconstruction (drizzling) of our images w ould impro v e our data quality Since our images ha v e only a fe w dithers (9), the drizzle algorithm did not impro v e our result o v er standard inte ger “shift-and-add. ” Each reduced image w as characterized with a FWHM estimate of the stellar PSF and an estimate of the pix el to pix el noise. The stellar FWHM w as estimated by selecting 10-20 stars per image using IMEXAM and a v eraging their “enclosed” FWHM measurements. Roughly thirty 100 pix el box es were placed randomly across each image from which to measure the pix el-to-pix el noise. While a single pix el to pix el noise estimate for an neb ulous image is not lik ely accurate, we used it in the IRAF D A OFIND algorithm to search for objects 5 s abo v e the noise threshold. The found sources were then mark ed on the images, and each source w as inspected by e ye to remo v e ob vious f alse detections and include objects missed by D A OFIND. This manual check and selection process w as bolstered by using the numerous repeat observ ations in our data set to ensure a source' s v alidity W e use our of f-eld non-neb ulous images to estimate the formal detection sensiti vity and nd 10 s limits of: 18.5 at J, 17.7 at H and 16.8 at K for our deep 1997 FL W O observ ations; 15.3 at J and 15.1 at H for our

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54 shallo w 1998 FL W O observ ations, and 19.75 at J, 18.75 at H and 18.10 at K s for our 2000 NTT observ ations. The 1997 and 1998 FL W O observ ations all had FWHM estimates between 2.2 and 3.0 pix els and are, therefore, mar ginally sampled. W e emplo yed the IRAF D A OPHO T ( Stetson 1987 ) point spread function (PSF) tting routine to deri v e photometry for these sources. Our procedure w as to perform multi-aperture (2-10 pix el radii) photometry on all the sources on each image, to select 20 stars on each image from which to deri v e a PSF and in an iterati v e f ashion, to create the PSF subtract nearest neighbors and to re-create the PSF until a good PSF w as deri v ed. Final PSF photometry w as e xtracted using the ALLST AR routine and the subtracted images were visually inspected for f aint stars missed near bright stars. W e used a PSF t radius of 3 pix els or a beam of 36 for our PSF photometry and set the sk y annulus to a 10 pix el radius. Our PSF procedure emplo yed the sk y-tting routines ( P ark er 1991 ) implemented in the D A OPHO T package which we found in articial star tests to decrease our photometric errors in neb ular re gions by a f actor of tw o. The 2000 NTT images had FWHM estimates ranging between 1.8 and 2.1 pix els, and these images are therefore mar ginally under -sampled and not easily suitable for PSF photometry Further the SOFI eld of vie w suf fers from coma-lik e geometric distortions on the northern 1015% of the array F or these tw o reasons, we decided to perform only aperture photometry on the NTT images. Multi-aperture photometry w as performed on sources detected in the NTT image using annuli with radii from 2 to 10 pix els. The sk y w as measured from the mode of the distrib ution of pix el v alues in an annuli from 10 to 20 pix els. From inspection of the curv es of gro wth of both isolated and neb ulous sources, we chose a 3 pix el radius (18 beam) for most of our NTT sources. Additionally the choice of small apertures allo wed us to minimize the ef fects of neb ular contamination and cro wding on the stellar PSF F or f aint sources in v ery confused or highly neb ulous re gions, we repeated the photometry with a 2 pix el

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55 aperture and a sk y annulus from 7 to 12 pix els. The change in sk y annulus does not signicantly af fect our photometry because the fraction of the stellar PSF be yond 7 pix els contains less than 5% of the ux, and the errors resulting from including this ux in the sk y estimate are smaller than the errors introduced from using too distant a sk y annulus on the neb ulous background. Aperture corrections were deri v ed for our data by choosing15 relati v ely bright stars as free of neb ular contamination as possible. W e performed multi-aperture photometry on them and using the IRAF MKAPFILE routine to visually inspect the stellar curv es of gro wth and calculate corrections. Since small apertures were used to minimize the ef fects of the bright neb ular background, the resulting corrections which constituted a some what substantial fraction of the stellar ux. Aperture corrections were carefully check ed by comparing the corrections deri v ed for on (neb ulous) and of f-cluster positions, which are interspersed in time with the on-cluster frames, and found to agree or to correlate with changes in seeing. Because the 1997 and 1998 FL W O observ ations were performed on the same photometric system and under similar conditions, their aperture corrections were similar and f airly constant between mosaic positions. The a v erage aperture correction from the 3 pix el tting radius to the 10 pix el sk y radius w as -0.35 magnitudes. F or the NTT images photometered using apertures, a typical 3 pix el aperture correction w as -0.14 magnitudes and for those stars photometered using a 2 pix el aperture, a correction of -0.34 w as used. 3.1.3 Photometric Comparisons of Datasets W e report in the electronically published catalog all the photometry from the FL W O and NTT observ ations. Further we e xplored an y photometric dif ferences between the FL W O and NTT observ ations because both systems will be mer ged to construct the cluster' s luminosity function, since the y do not ha v e the same dynamic range. These dif ferences include the lter systems, the methods and ef fecti v e beamsizes of the photometry and the epochs of the observ ations. W e tested if an y color

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56 terms were present due to dif fering photometric (lter) systems, we compared the magnitudes and colors of 504 sources common to both the NTT and FL W O photometry .W e compared the (J H) and (H K) colors of the NTT photometry to the FL W O photometry and t these comparisons with linear relations. The (J-H) colors were well t by a linear relation (slope1); ho we v er we found an of fset, DJH 010 magnitudes between the tw o systems. A similar comparison to the photometry of sources in the T w o Micron All Sk y Surv e y (2MASS) catalog 1 indicated this of fset w as at J band and w as restricted to the FL W O sources. Comparison of 2MASS photometry to the NTT photometry re v ealed no systematic of fsets. A comparison of the FL W O and NTT (H K) colors w as also well t by a linear relation (slope0.97) though this slope suggests that for the reddest sources, the NTT (H K s ) color is bluer than the FL W O (H K) color Further it w as e vident from these comparisons that while the global lter systems are quite similar the dif ference in the NTT and FL W O photometry of indi vidual sources w as lar ger than e xpected from formal photometric errors 2 From our f ak e star e xperiments and from the photometry of sources in o v erlap re gions on mosaick ed frames, we determined our measur ed photometric error is 5% for the majority of our sources increasing up to 15% for the sources at our completeness limit. Ho we v er when comparing sources common to both the FL W O and NTT data (well abo v e our completeness limit), we deri v ed 1 s standard de viations of022 for magnitudes and018 for colors. V ery similar dispersions were deri v ed when comparing our FL W O photometry to the Hillenbrand et al. ( 1998 ) or 2MASS catalogs or when comparing our 1 A current un-restricted search of the 2MASS First and Second Incremental Point Source and Extended Source Catalogs currently returns only 171 sources. 2 the quadratic sum of uncertainties from aperture corrections, zeropoint and airmass corrections, at elding error and sk y noise

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57 NTT data to the Hillenbrand & Carpenter ( 2000 ) H and K band dataset. W e attrib ute a portion of this additional photometric noise between the dif ferent datasets to the intrinsic infrared v ariability of these pre-main sequence sources which has been found for stars in this cloud to ha v e an a v erage of 0.2 magnitudes at infrared w a v elengths ( Carpenter et al. 2001 ). W e note that the dif ference in the beamsize used for the FL W O and NTT photometry and by the v arious other published data sets will also contrib ute a de gree of added photometric noise due to the presence of the strong neb ular background, thus making the NTT photometry preferable to the FL W O data for its higher angular resolution. 3.1.4 Astr ometry and the Electr onic Catalog Astrometry with reasonably high precision w as performed by matching the XY pix el locations of a lar ge number (50) of the observ ed sources to the equatorial positions of these sources listed on the 2MASS w orld coordinate system and deri ving full plate solutions using the IRAF CCMAP routine. Mosaic positions of the 1997 and 1998 observ ations were shifted to f all onto a common XY pix el grid dened by the K band FL W O 1997 mosaic images. T o create the common K band XY grid, sources in the o v erlap re gions between mosaic positions were matched and global of fsets calculated. The tw o camera arrays of the FL W O STELIRcam instrument are not centered precisely on the sk y and the J and H band coordinates were transformed using the IRAF GEOMAP routine into the K band XY coordinate grid. The NTT positions were aligned to the NTT J band image. F or the FL W O plate solution, 161 2MASS sources were matched to the FL W O XY coordinates yielding a plate scale of 0.596 ”/pix el and an astrometric solution with rms errors of010 An independent solution of 82 NTT sources matched to the 2MASS database yielded a plate scale of 0.288 ”/pix el and an astrometric solution ha ving rms errors007 W e construct the electronic v ersion of the FL W O-NTT catalog based upon all the sources detected by our FL W O and NTT observ ations, and we compliment our

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58 electronic catalog by including sources identied in other catalogs and f alling within our surv e y area, b ut that were saturated, undetected or unresolv ed by our observ ations. Since our nal catalog co v ers a substantially dif ferent area than comparable deep infrared surv e ys and includes numerous ne w sources, we chose to assign ne w source designations for our nal catalog. These are based upon the IA U standard format that includes a catalog acron ym, a source sequence, and source specier F or the catalog acron ym, we chose MLLA, based upon the initials of the last names of the authors. This acron ym is currently unused in the Dictionary of Celestial Nomenclature. W e chose to sequence the catalog using a running number incremented from 00001 to 01010. W e use a specier only where necessary to distinguish unresolv ed sources, typically emplo ying the designations (A), (B), etc. NTT astrometry is preferentially used in the nal catalog. F or undetected or unresolv ed sources, we made e v ery ef fort to include astrometry from the source' s identifying catalog if the original catalog could be globally aligned to the FL W O-NTT catalog. W e list cross-references based on the most comprehensi v e or deep surv e ys; these include the Hillenbrand ( 1997 ), Hillenbrand & Carpenter ( 2000 ), Luhman et al. ( 2000 ) and McCaughrean & Stauf fer ( 1994 ) designations. F or sources lacking cross-references in these catalogs, we list their 2MASS designations (circa the 2nd Incremental 2MASS Point Source Catalog) where possible. The LR2000 designations are based on their deri v ed equatorial coordinates and due to signicant astrometric errors do not correspond to the positions we deri v e in the FL W O-NTT catalog. F or e xample, we nd of f-sets of042 in RA and 044 in DEC between our positions and those of LR2000. After remo ving these of fsets, we still nd median residuals of 044 between our coordinates and those of LR2000 with errors as lar ge as 1 ; this is in contrast to the rms residuals of 01 between our catalog and the 2MASS and HC2000 positions. Hence we do not list the LR2000 position-dependent designations e xcept where necessary to identify sources undetected by nal catalog.

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59 The entire FL W O-NTT T rapezium Cluster catalog has been published electronically in the recent w ork, Muench et al. ( 2002 ). T o illustrate what w as publically released in that catalog, we ha v e supplied a sample table here, consisting of only a subset of the sources a v ailable in the electronic v ersion. 3.2 T rapezium Cluster K band Luminosity Function W e restrict our subsequent analysis of the cluster' s luminosity and mass function to the area surv e yed by our deeper NTT observ ations. Our observ ations detected 749 sources within this re gion. The completeness of this sample at the f aintest magnitudes is dif cult to quantify because of the spatially v ariable neb ular background. The formal 10 s detection limits of our catalog in the NTT re gion are 19.75 at J, 18.75 at H and 18.10 at K s based upon the pix el to pix el noise in non-neb ulous of f-cluster observ ations that were tak en adjacent in time to the on-cluster images. T o better estimate our actual completeness limits, we performed articial star e xperiments by constructing a stellar PSF for each of our images and using the IRAF ADDST AR routine to place synthetic stars in both the of f-cluster and the neb ulous on-cluster images. A small number of synthetic stars (30-70) with a range of input magnitudes were randomly added across each image and were then reco v ered using the D A OFIND routine. This w as repeated a lar ge number of times (40-200) to achie v e suf cient signal to noise for these tests. In of f-cluster images, the deri v ed 90% completeness limits agreed well with the estimated 10 s detection limits. In the on-cluster images, the completeness limits were reduced to 90% completeness limits of J1815, H178, and K s175 with slightly brighter limits in the dense central core (0.5radius from q 1 C Orionis). W e also carefully compared our source list to those published by other recent surv e ys for the NTT re gion. T o our resolution limit, we detected all the sources found by the Hillenbrand & Carpenter ( 2000 hereafter HC2000) K eck surv e y e xcept for one, all b ut tw o sources from the Luhman et al. ( 2000 ) Hubble Space T elescope NICMOS surv e y b ut we could not identify nine sources listed in Lucas & Roche

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60T able 3–2. FL W O-NTT near -infrared catalog Seq. Spec. R.A. Dec. FL W O (Mag) FL W O (Err) NTT (Mag) NTT (Err) Phot H97 HC2000 Other (J2000) (J2000) J H K L J H K L J H K s J H K s Flag ID ID ID 00001 5 35 22.45 -5 26 10.9 14.59 13.68 99.00 12.40 0.02 0.06 -1.00 0.26 0 00002 5 35 26.57 -5 26 09.6 16.24 15.41 99.00 99.00 0.04 0.01 -1.00 -1.00 0 00003 5 35 11.65 -5 26 09.0 13.51 11.63 10.56 9.21 0.04 0.01 0.02 0.02 0 00004 5 35 15.97 -5 26 07.4 14.12 12.25 99.00 10.30 0.02 0.01 -1.00 0.04 0 00005 5 35 09.21 -5 26 05.7 17.01 16.18 15.65 0.05 0.04 0.06 0 00006 5 35 20.13 -5 26 04.2 14.57 13.01 12.27 0.01 0.01 0.01 0 0535201-052604(4) 00007 5 35 04.48 -5 26 04.1 12.14 11.25 10.89 10.15 0.03 0.04 0.03 0.03 0 0535044-052604(4) 00008 5 35 05.18 -5 26 03.4 13.54 12.88 12.52 12.04 0.01 0.01 0.03 0.12 0 262 00009 5 35 11.48 -5 26 02.6 9.91 9.06 8.84 8.47 0.05 0.01 0.02 0.02 0 365 00010 5 35 22.57 -5 26 02.1 17.18 16.56 16.37 0.09 0.08 0.08 0 00011 5 35 06.92 -5 26 00.5 13.48 13.14 12.54 11.43 0.02 0.05 0.02 0.06 13.39 12.56 12.19 0.01 0.01 0.01 0 299 00012 5 35 24.34 -5 26 00.3 13.09 12.42 12.12 11.38 0.08 0.03 0.02 0.10 13.05 12.37 12.02 0.01 0.01 0.01 0 3101 00013 5 35 10.48 -5 26 00.3 13.01 12.25 11.82 11.11 0.03 0.03 0.02 0.05 12.92 12.07 11.66 0.01 0.01 0.01 0 3104 00014 5 35 10.76 -5 26 00.0 15.54 13.61 12.60 11.72 0.04 0.08 0.02 0.09 15.42 13.60 12.57 0.01 0.01 0.01 0 00015 5 35 15.42 -5 25 59.5 13.72 12.84 12.38 11.56 0.01 0.02 0.02 0.06 13.68 12.78 12.26 0.01 0.01 0.02 0 3103 Note. — FL W O: Fred La wrence Whipple Observ atory Mt. Hopkins, Arizona; NTT : Ne w T echnology T elescope, European Southern Observ atory La Silla, Chile. References. — Hillenbrand ( 1997 H97); Hillenbrand & Carpenter ( 2000 HC2000)

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61 ( 2000 hereafter LR2000) UKIR T surv e y Further it is our nding of 58 ne w sources within our NTT re gion and un-reported by prior catalogs that adds support to the deep and v ery sensiti v e nature of our census. 3.2.1 Constructing Infrar ed Luminosity Function(s) The FL W O and NTT observ ations o v erlap considerably in dynamic range with 504 stars ha ving multi-epoch photometry F or our analysis, we preferentially adopt infrared luminosities from the NTT photometry because it has higher angular resolution and it is an essentially simultaneous set of near -infrared data. F or 123 stars that are saturated in one or more bands on the NTT images, the FL W O photometry w as used. This transition from NTT to FL W O photometry is at approximately J = 11.5, H = 11.0, and K = 11.0. F or the brightest 5 OB stars, saturated on all our images, we used JHK photometry from the Hillenbrand et al. ( 1998 ) catalog. Photometric dif ferences between the FL W O and NTT datasets are small (see section 3.1.3 ) and will not af fect our construction of the T rapezium Cluster infrared luminosity function(s). In Figure 3–3 we present the ra w infrared T rapezium Luminosity Functions (LFs). W e use relati v ely wide bins (0.5 magnitudes) that are much lar ger than the photometric errors. In Figure 3–3 (a), we compare the J and H band LFs for stars in this re gion. In the Figure 3–3 (b), we compare the K band LF of the NTT re gion to that K band LF constructed in Section 2.6.2 As w as observ ed in pre vious studies of the T rapezium Cluster the cluster' s infrared luminosity function (J, H, or K) rises steeply to w ard f ainter magnitudes, before attening and forming a broad peak. The LF steadily declines in number belo w this peak b ut then rapidly tails of f a full magnitude abo v e our completeness limits. F or our current deri v ation of the T rapezium IMF we use the T rapezium K band Luminosity Function, rather than the J or H LFs. W e do so in order to minimize the ef fects of e xtinction on the luminosities of cluster members (see Section 3.3.1 ), to maximize our sensiti vity to intrinsically red, lo w luminosity bro wn dw arf members of this

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62 Figure 3–3: T rapezium cluster: ra w near -infrared luminosity functions. A) T rapezium Cluster J and H band Luminosity Functions. The T rapezium HLF is the open histogram and the T rapezium JLF is the shaded histogram. Completeness (90%) limits are mark ed by a solid v ertical line at 18.15 (J) and a brok en v ertical line at 17.8 (H). B) T rapezium Cluster K band Luminosity Function. The T rapezium KLF constructed from the FL W O-NTT catalog is compared to the literature KLF constructed in Section 2.6.2 The K=17.5 90% NTT completeness limit is demark ed by a v ertical brok en line. cluster and to mak e detailed comparisons to our study of the literature T rapezium KLF in Section 2.6.2 F or e xample, the ne w FL W O-NTT T rapezium KLF contains signicantly more stars at f aint (K14) magnitudes than the literature KLF constructed in Section 2.6.2 Interestingly a secondary peak near K = 15 seen in that KLF (see Figure 2–12 ) (originally McCaughrean et al. 1995 ) is much more signicant and peaks at K=15.5 in the ne w FL W O-NTT KLF Similar peaks are not apparent in the J or H band LFs constructed here, though Lucas & Roche ( 2000 ) report a strong secondary peak in their T rapezium HLF Such secondary peaks in young cluster luminosity functions ha v e often been e vidence of a background eld star population contrib uting to the source counts (e.g., Luhman et al. 1998 ; Luhman & Riek e 1999 ). T o account for the possible eld star contamination, we systematically obtained images of control elds a w ay from the cluster and of f of the Orion Molecular Cloud. The FL W O of f-cluster eld(s) were centered at approximately R.A. = 05 h 26 m ; DEC.

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63 Figure 3–4: T rapezium cluster: construction of observ ed control eld KLF A) The tw o histograms are the of f-eld KLFs obtained as part of the FL W O and NTT observ ations. The NTT of f-elds are approximately 2 magnitudes deeper than the FL W O of f-elds, b ut the FL W O of f-elds co v ered twice the area of the NTT of f-elds. Both are scaled to the size of the T rapezium NTT re gion. The inset diagram sho ws the distrib ution of H-K colors for these tw o of f-elds. Their similar narro w widths indicate the y are free of interstellar e xtinction; B) The weighted a v erage of the FL W O and NTT eld stars KLFs is compared to the T rapezium Cluster KLF constructed in Figure 3–3 (b). =0600(J2000) and were roughly twice the area of the NTT of f-elds. The NTT of f-cluster re gion w as centered at R.A. = 05 h 37 m 43s 7; DEC.=015542 7 (J2000). Figure 3–4 (a) displays the tw o of f-eld KLFs (scaled to the same area) from these observ ations and in the inset, their (H K) distrib utions. The relati v ely narro w (H K) distrib utions indicate that the tw o of f-elds sample similar populations and that the y are un-reddened. W e constructed an observ ed eld star KLF by a v eraging these luminosity functions, weighting (by area) to w ard the FL W O of f-elds for K brighter than 16th magnitude, the completeness limit of that dataset, and to w ard the more sensiti v e NTT of f-elds for f ainter than K = 16. In Figure 3–4 (b), we compare the resulting eld star KLF to the T rapezium KLF of the NTT re gion. It is plainly apparent from the ra w control eld observ ations that while eld stars may contrib ute to the T rapezium Cluster

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64 IR luminosity function o v er a range of magnitudes, their numbers peak at magnitudes f ainter than the secondary peak of the on-cluster KLF and do not appear suf cient in number to e xplain it. 3.2.2 Dening a Complete Cluster KLF W e determine the completeness of our FL W O-NTT T rapezium Cluster KLF by constructing and by analyzing the cluster' s infrared (H K) v ersus K color -magnitude diagram. F or the purposes of our analysis, we adopt the follo wing parameters for the T rapezium: a cluster mean age of 0.8 Myr ( Hillenbrand 1997 ) and a cluster distance of 400 pc As seen in Figure 3–5 (a), the luminosities of the T rapezium sources form a continuously populated sequence from the bright OB members (K5) through sources detected belo w our completeness limits. T o interpret this diagram, we compare the locations of the FL W O-NTT sources in color -magnitude space to the cluster' s mean age isochrone as deri v ed from theoretical pre-main sequence (PMS) calculations. Because the DM97 models include masses and ages representati v e of the T rapezium Cluster we will use these tracks to dene a complete cluster sample from Figure 3–5 (a). Dif ferences among pre-main sequence tracks should not ha v e signicant ef fect upon our analysis of the color -magnitude diagram (see Section 3.4.3 ). It is clear from this diagram that the cluster sources are reddened a w ay from the theoretical 0.8 Myr isochrone, which forms a satisf actory left hand boundary to the sources in this color -magnitude space. This isochrone, ho we v er does not span the full luminosity range of the observ ations and a number (40) sources lie belo w the f aint end of the DM97 isochrone. As a result, our subsequent analysis that mak es use of the DM97 models will be restricted to considering only those sources whose luminosities, after correction for e xtinction, w ould correspond to masses greater than the mass limit of the DM97 tracks, i.e., 0.017 Mor roughly 17 times the mass of Jupiter ( M J u p ). Despite the lo wer mass limit imposed by these PMS

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65 Figure 3–5: T rapezium cluster: deri ving MA V completeness limits. A) T rapezium Cluster (H K) / K color -magnitude diagram for the NTT re gion. Stars selected to f all into our mass & e xtinction limited sample are indicated by lled circles. The distrib ution of sources in this color -magnitude space is compared to the location of the pre-main sequence 0.2 and 0.8 Myr isochrones from DM97. Reddening v ectors (A V17) sho wn for 2.50, 0.08 and 0.02 Mstars at the cluster' s mean age (0.8 Myr). The zero-age main sequence ( K en yon & Hartmann 1995 ; Bessell 1995 ) is sho wn for 03-M6.5 stars at a distance of 400pc. B) Ef fects of mass/e xtinction limits on the cluster KLF Comparison of the MA V limited KLF deri v ed from (A) to the ra w T rapezium KLF (see Figure 3–3 b). Sensiti vity (K = 18.1) and completeness (K = 17.5) limits are sho wn as v ertical brok en lines. tracks, our infrared census spans nearly three orders of magnitude in mass, illustrating the utility of studying the mass function of such rich young clusters. Extinction acts to redden and to dim sources of a gi v en mass to a brightness belo w our detection limits. T o determine our ability to detect e xtincted stars as a function of mass, we dra w a reddening v ector from the luminosity (and color) of a particular mass star on the mean age isochrone until it intersects the 10 s sensiti vity limit of our census. W e can detect the 1 Myr old Sun seen through A V60 limits magnitudes of e xtinction or a PMS star at the h ydrogen b urning limit seen through35 magnitudes. F or v ery young bro wn dw arfs at our lo wer mass limit (17 M J u p ), we probe the cloud to A V17 magnitudes. W e use this latter reddening v ector as a boundary to which

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66 we are complete in mass, and we dra w a mass and e xtinction ( MA V ) limited subset of sources bounded by the mean age isochrone and the A V17 reddening v ector and mark these as lled circles in Figure 3–5 (a). Our MA V limits probe the v ast majority of the cluster population, including 81% of the sources the color -magnitude diagram. In Figure 3–5 (b) we present the MA V limited KLF containing 583 sources. Thirty-tw o sources, detected only at K band (representing only 4% of our catalog), were also e xcluded from our further analysis. The median K magnitude of these sources is K = 15, and we e xpect that these are lik ely hea vily reddened objects. W e compare the MA V limited KLF to the un-ltered T rapezium KLF Clearly hea vily reddened sources contrib uted to the cluster KLF at all magnitudes and their remo v al results in a narro wer cluster KLF Ho we v er the structure (e.g., peak, slope, inections, etc.) of the KLF remains lar gely unchanged. The secondary peak of the cluster KLF between K1417 seems to be real since it is present in both the ra w and the MA V limited KLFs, though we ha v e not yet corrected for background eld stars. There are at least three possible sources of incompleteness in our mass/e xtinction limited sample. The rst arises because sources that are formally within our mass and e xtinction limits may be additionally reddened by infrared e xcess from circumstellar disks and, hence, be left out of our analysis. Ho we v er this bias will af fect sources of all masses equally because infrared e xcess appears to be a property of the young T rapezium sources o v er the entire luminosity range ( Muench et al. 2001 ). Second, the T rapezium Cluster is not fully coe v al and our use of the cluster' s mean age to dra w the MA V sample means that cluster members at our lo wer mass limit (17 M J u p ) b ut older than the cluster' s mean age ( t08 Myr ) will be f ainter than the lo wer boundary and left out of our sample. Further sources younger than the mean age b ut belo w 17 M J u p will be included into the sample. This “age bias” will af fect the lo west mass sources, i.e.,20 M J u p Third, because of the strong neb ular background, our

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67 true completeness limit (see Section 3.1 ) is brighter than our formal 10 s sensiti vity for approximately 60% of the area surv e yed. The resulting sample incompleteness only af fects our sensiti vity to sources less than 30 M J u p and with A V10. W e do not correct the T rapezium KLF to account for these ef fects or biases. 3.2.3 Field Star Contamination to the KLF The lack of specic membership criteria for the embedded sources in the T rapezium Cluster requires an estimate of the number of interloping non-cluster eld stars in our sample. Some published studies, for e xample LR2000 and Luhman et al. ( 2000 ), assume that the parental molecular cloud acts as a shield to background eld stars; whereas HC2000 suggests that the background contrib ution is non-ne gligible. HC2000 estimates the eld star contrib ution using an empirical model of the infrared eld star population and con v olving this model with a local e xtinction map deri v ed from a molecular line map of the re gion. This approach may suf fer from its dependence upon a eld star model that is not calibrated to these f aint magnitudes and that does not include v ery lo w mass eld stars. As we sho w there are also considerable uncertainties in the con v ersion of a molecular line map to an e xtinction map. F or our current study we use our observ ed K band eld star luminosity function (see Figure 3–4 ) to test these prior methodologies and to correct for the eld star contamination. W e point out that no such estimate can account for contamination due to young, lo w mass members of the fore ground Orion OB1 association. W e compare in Figure 3–6 the ef fects of six dif ferent e xtinction models upon our observ ed eld star KLF In panels A and B, we tested simple Gaussian distrib utions of e xtinction centered respecti v ely at A V10 and 25 magnitudes with s5 magnitudes. In both cases, the reddened eld star KLF contains signicant counts abo v e our completeness limit and “background e xtinction shields” such as these do not pre v ent the inltration of eld stars into our counts. In the second pair of reddened of f-elds (panels C and D), we follo wed the HC2000 prescription for

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68 Figure 3–6: T rapezium cluster: testing contrib ution of reddened eld star KLFs. P anels A & B: The observ ed of f-eld KLF (Figure 3–4 b) reddened by “background shields” of e xtinction in the form of g aussian distrib utions centered at A V10 (panel A) and 25 (panel B); P anels C & D: The observ ed of feld KLF reddened by an e xtinction map con v erted from a C 18 O map. The tw o panels represent the v ariation in the reddened of f-eld as a function of the uncertainty in the C 18 O to A V con v ersion; P anels E & F: The same e xperiment as performed in C & D, b ut these ha v e been ltered to reect the actual contrib ution due to the MA V limits.

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69 estimating background eld stars by con v olving our observ ed eld star KLF with the C 18 O map from Goldsmith et al. ( 1997 ) con v erted from column density to dust e xtinction. W e note that there is substantial uncertainty in the con v ersion from C 18 O column density to dust e xtinction. There is at least a f actor of 2 v ariation in this con v ersion v alue in the literature, where Frerking et al. ( 1982 ) deri v ed a range from 0724in units of 10 14 cm 2 mag1and Goldsmith et al. ( 1997 ) estimated a range of v alues from 173. Either the result of measurement uncertainty or the product of dif ferent en vironmental conditions, this v ariation produces a f actor of 2 uncertainty in the e xtinction estimates from the C 18 O map. In short, we nd that a C 18 OA V ratio of 3.0 (panel C) results in twice as man y interloping background eld stars as w ould a v alue of 1.7 (panel D; equi v alent to that used by HC2000 ). In panels E and F of Figure 3–6 we deri v e the same reddened of f-eld KLFs as in the prior pair b ut the y ha v e been ltered to estimate the actual contrib ution of eld-stars to our MA V limited sample. These lters, which were based upon on the K brightness of the lo wer mass limit of our PMS models and on the deri v ed e xtinction limit of the MA V sample, were applied during the con v olution of the eld star KLF with the cloud e xtinction model such that only reddened eld stars that w ould ha v e A V20 and unreddened K magnitudes16 w ould be counted into ltered reddened of f-eld KLF The e xtinction limit w as e xpanded from 17 to 20 magnitudes to account for the dispersion of the H-K distrib ution of un-reddened eld-stars (02). A f actor of 2 uncertainty remains. Alv es et al. ( 1999 ) deri v e a more consistent estimate of the C 18 OA V ratio from near -infrared e xtinction mapping of dark clouds, suggesting a median ratio of 21. Adopting C 18 OA V21, we estimate there are2010 eld stars in our MA V limited KLF From these e xperiments, we nd, ho we v er that both the ra w and reddened of f-eld KLFs al w ays peak at fainter ma gnitudes than the secondary peak of the T rapezium KLF and that the subtraction of these eld-star corrections from the T rapezium KLF do not remo v e this secondary peak.

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70 These ndings suggest that the secondary KLF peak is a real feature in the T rapezium Cluster' s infrared luminosity function. 3.3 T rapezium Cluster Initial Mass Function W e analyze the T rapezium Cluster' s K band luminosity function constructed in section 3.2 using our model luminosity function algorithm described in Section 2 Our goal is to deri v e the underlying mass function or set of mass functions whose model luminosity functions best t the T rapezium Cluster KLF W e ha v e impro v ed our modeling algorithm by including statistical distrib utions of the reddening properties of the cluster W e ha v e also impro v ed our analysis by applying the background eld star correction from Section 3.2.3 and by emplo ying impro v ed tting techniques for deri ving IMF parameters and condence interv als. Before deri ving the cluster IMF we use the e xtensi v e color information a v ailable from the FL W O-NTT catalog to e xplore the reddening (e xtinction and infrared e xcess) properties of the T rapezium sources. In Section 3.3.1 we use this information to create recipes for deri ving the probability distrib utions functions of e xtinction and e xcess which can be folded back into our modeling algorithm during our deri v ation of the T rapezium IMF W e present the ne w model luminosity functions and tting techniques in Section 3.3.2 and summarize the deri v ed IMF in Section 3.3.3 3.3.1 Deri ving Distrib utions of Reddening Extinction pr obability distrib ution function. W e use the e xtensi v e color information pro vided by our FL W O-NTT catalog to construct a probability distrib ution function of the intra-cluster e xtinctions (hereafter referred to as the Extinction Probability Distrib ution Function or EPDF) based upon the color e xcesses of indi vidual T rapezium sources. Because the stellar photospheric (H K) color has a v ery narro w distrib ution of intrinsic photospheric v alues it should be the ideal color from which to deri v e line of sight e xtinction estimates, as sho wn, for e xample, in the Alv es et al. ( 1998 ) study of the structure of molecular clouds. In Figure 3–7 (a) we sho w the

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71 Figure 3–7: Infrared colors of T rapezium sources. A) Histogram of the observ ed (H K) color for the FL W O-NTT T rapezium sources. The subset of these sources which lack J band measurements are indicated by the shaded histogram; B) T rapezium Cluster (H K) vs (J H) color -color diagram for the NTT re gion. Symbols indicate if the source' s colors were tak en from the FL W O catalog (lled circles, JHK) or the NTT catalog (open circles, JHK s ). histogram of observ ed (H K) color for all our T rapezium Cluster sources. This histogram peaks at (H K) = 0.5 and is quite broad especially when compared to the narro w unreddened photospheric (eld-star) (H K) distrib utions seen in Figure 3–4 (a). This broad distrib ution may be in part the result of e xtinction; ho we v er as recently sho wn in Lada et al. ( 2000 ) and Muench et al. ( 2001 ), approximately 50% of the these T rapezium Cluster sources, independent of luminosity display infrared e xcess indicati v e of emission from circumstellar disks. This is illustrated in Figure 3–7 (b) where it is clear that there are both hea vily reddened sources (A V35) and sources with lar ge infrared e xcesses (f alling to the right of the reddening band for main sequence objects). If the (H K) color e xcess were assumed to be produced by e xtinction alone without accounting for disk emission, the resulting e xtinction estimates w ould be too lar ge. Me yer et al. ( 1997 ) sho wed that the intrinsic infrared colors of stars with disks are conned to a locus (the classical T -T auri star locus or cTTS locus) in the (H K)/(J

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72 H) color -color diagram. W e deri v e indi vidual A V estimates for sources in the (H K)/(J H) color -color diagram by dereddening these stars back to this cTTS locus along a reddening v ector dened by the Cohen et al. ( 1981 ) reddening la w Sources without J comprise20% of the catalog and as sho wn in Figure 3–7 their (H K) colors appear to sample a more hea vily embedded population, implying e xtinctions as high as A V60. A V estimates are deri v ed for these sources by assigning a typical star -disk (H K) color = 0.5 magnitudes, and de-reddening that source. Sources near to b ut belo w the cTTS locus are assigned an A V0. The indi vidual e xtinctions are binned into an e xtinction probability distrib ution function (EPDF) as sho wn in Figure 3–8 Also sho wn are the ef fects of changing the typical star -disk (H K) color assumed for those stars without J band. Little change is seen. Compared to the cloud e xtinction distrib ution function, which w as inte grated o v er area from the C 18 O map, the cluster EPDF is v ery non-g aussian and peaks at relati v ely lo w e xtinctions, A V25, ha ving a median A V475 and a mean A V92. Further the cluster EPDF is not well separated from the reddening distrib ution pro vided by the molecular cloud. Rather the cluster population signicantly e xtends to e xtinctions as high as A V1025, near and be yond the peak of the cloud e xtinction function. Ancillary e vidence of this signicant population of hea vily reddened stars is seen in the color -color diagram (Figure 3–7 b) which clearly illustrates the e xtension of the cluster to re gions of the molecular cloud with A V10. Lastly it is clear that the deep nature of our surv e y has allo wed us to sample both the majority of the embedded cluster and the cloud o v er the full range of density In our re vised model luminosity function algorithm, we randomly sample the cluster' s EPDF to assign an A V to each articial star in the model LF The ef fect of the EPDF on the model luminosity function is w a v elength and reddening la w ( Cohen et al. 1981 in this case) dependent. In Figure 3–9 we construct model I, J, H, and K luminosity functions, reddening each by the T rapezium Cluster EPDF The ef fect of the

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73 Figure 3–8: T rapezium cluster: e xtinction probability distrib ution function. Plotted are three v ariations in the EPDF under dif ferent assumptions of the typical (HK) (star -disk) color for the 20% of the stars lacking J band measurements. See Section 3.3.1 for deri v ation. It is compared to the e xtinction probability distrib ution function inte grated from the C 18 OA V map. Note that the y are not well separated distrib utions. A brok en v ertical line indicates the A V17, MA V limit. EPDF on the intrinsic I and J band LFs is profound, rendering the reddened I band LF almost unrecognizable. Y et at longer w a v elengths, specically at K band, the ef fects of e xtinction are minimized. W e note that the o v erall form of the reddened model K band luminosity function has not been changed by the T rapezium EPDF in a signicant w ay e.g., the peak of the model KLF is not signicantly blurred and the f aint slope of the KLF has not been changed from f alling to at. This suggests that our modeling of the literature T rapezium KLF in Section 2.6.2 which did not account for reddening due to e xtinction, is generally correct. Ho we v er we lik ely deri v ed too lo w of a turno v er

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74 Figure 3–9: Ef fects of e xtinction on model cluster LFs. Model luminosity functions of the T rapezium (using the T rapezium IMF of Equation 2.7 and deri v ed in Section 2.6.2 ) is con v olv ed with the T rapezium Cluster EPDF at four dif ferent w a v elengths. Reddening ef fects are most signicant at I and J bands and are minimized at K band. mass for the T rapezium IMF because reddening shifted the intrinsic LF to f ainter magnitudes. Infrar ed excess pr obability distrib ution function. Because we wish to use the T rapezium K band LF to minimize the ef fects of e xtinction, we must also account for the ef fects of circumstellar disk emission at K band. The frequenc y distrib ution of the resulting e xcess infrared ux is not a well kno wn quantity and when pre viously deri v ed, it has depended signicantly upon additional information deri v ed from the spectral classication of cluster members ( Hillenbrand et al. 1998 ; Hillenbrand & Carpenter 2000 ). One of the goals of this present w ork is to construct

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75 Figure 3–10: T rapezium cluster: infrared e xcess probability distrib ution function. The deri v ed H-K s color e xcess distrib ution function is assumed to reect a magnitude e xcess at K band alone. a recipe for deri ving the K band e xcess distrib ution directly from the infrared colors of the cluster members. T o deri v e a rst-order infrared e xcess probability distrib ution function (IXPDF) for the T rapezium Cluster sources, we simply assume that an y e xcess (H K) color (abo v e the photosphere, after remo ving the ef fects of e xtinction) reects an e xcess at K band alone realizing this may underestimate the infrared e xcess of indi vidual sources. W e only use the sources ha ving JHK measurements and lying abo v e the cTTS locus in the color -color diagram. W e remo v e the ef fects of e xtinction from each source' s observ ed (H K) color using the same method described abo v e, i.e., dereddening back to the cTTS locus. Ho we v er the photospheric (H K) color for each star cannot be discreetly remo v ed from this data alone. The photospheric infrared colors of pre-main

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76 sequence stars appear to be mostly dw arf-lik e ( Luhman 1999 ), and therefore, we used the observ ed eld star (H K) distrib ution sho wn in Figure 3–4 (a) as a probability distrib ution of photospheric v alues. W e deri v e the IXPDF by binning the de-reddened (H K) colors into a probability function and then subtracting the distrib ution of photospheric colors using a Monte Carlo inte gration. The T rapezium Cluster IXPDF is sho wn in Figure 3–10 The IXPDF peaks near 0.2 magnitudes with a mean = 0.37, a median = 0.31, and a maximum e xcess of20 magnitudes. Probabilities of ne g ati v e e xcesses were ignored. The IXPDF is similar to the (H K) e xcess distrib ution sho wn in HC2000 and deri v ed in Hillenbrand et al. ( 1998 ) yet e xtends to some what lar ger e xcess v alues. Each articial star in our models is randomly assigned a K band e xcess (in magnitudes) dra wn from the IXPDF 3.3.2 Modeling the T rapezium Cluster KLF T o model the T rapezium Cluster KLF we apply the appropriate eld star correction deri v ed in section 3.2.3 to the MA V limited KLF constructed in Section 3.2.2 W e x the T rapezium Cluster' s star -forming history and distance to be identical to that used in Section 2.6.2 Specically these are a distance of 400pc (m-M=8.0; see appendix B ) and a star -forming history characterized by constant star formation from 1.4 to 0.2 Myr ago, yielding a cluster mean age of 0.8 Myr ( Hillenbrand 1997 ) and an age spread of 1.2 Myr W e adopt our standard set of mer ged theoretical pre-main sequence tracks from Section 2.2.3 3 Our mer ged standard set of tracks span a mass range from 60 to 0.017 M, allo wing us to construct a continuous IMF within this range. W e incorporated the cluster' s reddening distrib utions deri v ed in Section 3.3.1 3 Our standard set of theoretical tracks are a mer ger of e v olutionary calculations including a theoretical Zero Age Main Sequence (ZAMS) from Schaller et al. ( 1992 ), a set of intermediate mass (1-5 M) “accretion-scenario” PMS tracks from Bernasconi ( 1996 ), and the lo w mass standard deuterium ab undance PMS models from D'Antona & Mazzitelli ( 1997 ) for masses from 1 to 0.017 M

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77 into our modeling algorithm and chose our standard functional form of the cluster IMF; specically an IMF constructed of po wer -la w se gments, G i connected at break masses, m j W e nd that an underlying 3 po wer -la w IMF produced model KLFs that t the observ ations o v er most of the luminosity range, corresponding to masses from 5 to 0.03 M. In Section 5 we utilize our c 2 minimization routine to identify those 3 po wer -la w IMFs that best t the observ ed KLF within this mass range, and we estimate condence interv als for these IMF parameters in Section 5 W e nd that that the f aint T rapezium bro wn dw arf KLF corresponding to masses less than 0.03 M, contains structure and a secondary peak that are not well t by the 3 po wer -la w IMF models. In Section 5 we model this secondary KLF peak using a corresponding break and secondary feature in the cluster bro wn dw arf IMF between 0.03 and 0.01 M. Results of c 2 tting: best t thr ee po wer -law IMFs. Our c 2 minimization procedure calculates the reduced c 2 statistic and probability for a particular model KLF t to the T rapezium KLF o v er a range of magnitude bins. P arameters for the underlying three po wer -la w IMF are tak en from the best t model KLFs, and we t both reddened and unreddened model KLFs. The 3 po wer -la w IMF deri v ed from these ts is summarized in T able 3–3 W e found that the results of our model ts were dependent upon the dynamic range of K magnitude bins o v er which the models were minimized. Specically we nd that our results are v ery sensiti v e to the formation of a secondary peak in the T rapezium KLF at K = 15.5, which remains despite the subtraction of the eld star KLF W e deri v e good model KLF tsc 2 prob1when tting between the K75 bin and the K145 bin (see Figure 3–11 a), the same luminosity range we modeled in Section 2.6.2 W ithin this t range, we nd an optimal T rapezium IMF nearly identical to that found in Equation 2.7 e v en after accounting for reddening. The deri v ed IMF rises steeply from the most massi v e stars with G 1 13 before breaking to a shallo wer IMF slope of G 2 02 at 06 M(log m 1 02). The deri v ed IMF

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78 Figure 3–11: T rapezium cluster: best-tting model KLFs and 3 po wer -la w IMFs. T op panels: the MA V limited, background subtracted T rapezium KLF (histogram) and best t reddened model KLFs (unconnected lled circles). Bottom panels: the resulting underlying IMFs and corresponding chi-sq probabilities. P anel (A) sho ws models t between K=7.5 and 14.5, the same range t in Section 2.6.2 and Figure 2.7 (a). P anel (B) sho ws three po wer -la w IMF ts to the secondary peak in the cluster KLF at K=15.5, which correspond to lo w c 2 probability due to the presence of structure and the secondary KLF peak. peaks near the h ydrogen b urning limit (010008 Mor log m 2 1011) and then breaks and f alls steeply throughout the bro wn dw arf re gime with G 3 10. W e also deri v e good ts to K=15 (just before the secondary peak in the cluster KLF), with the resulting IMF peaking at slightly higher masses (013010 M) and f alling with a slightly shallo wer slope, G 3 07 to 08. The unreddened luminosities of this t range correspond to a mass range from 5.0 to 0.03 M. Ho we v er we cannot produce model KLFs based upon a three po wer -la w IMF that adequately t the secondary peak in the T rapezium KLF F or e xample, our best t to the secondary peak in Figure 3–11 (b) is inconsistent with the o v erall form of the f aint KLF being unable to replicate both the f alling KLF at K = 14.5 nor the

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79 T able 3–3. Three po wer -la w T rapezium IMF parameters and errors P arameteraRange m KbBest Fitc() Best Fitd() G 110 20 14.5 -1.16 0.16 -1.24 0.20 log m 101 11 14.5 -0.17 0.10 -0.19 0.13 G 204 04 14.5 -0.24 0.07 -0.16 0.15 log m 201 14 14.5 -1.05 0.05 -1.00 0.13 G 304 20 14.5 1.10 0.25 1.08 0.38 G 115.0 -1.13 0.16 -1.21 0.18 log m 115.0 -0.19 0.11 -0.22 0.11 G 215.0 -0.24 0.15 -0.15 0.17 log m 215.0 -1.00 0.10 -0.92 0.13 G 315.0 0.82 0.15 0.73 0.20 log m 215.5 -0.89-0.77G 315.5 0.300.30 log m 216.5 -0.72G 316.5 0.23 aThe parameters G i are the po wer -la w indices of the IMF which here is dened as the number of stars per unit logM M The parameters m j are the break masses in the po wer -la w IMF and are in units of logM M .bF aintest KLF bin t by Model KLF .cModel ts without Source Reddening.dModel ts accounting for Source Reddening. Note. — All tab ulated ts deri v ed using our standard set of PMS tracks (primarily from DM97).

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80 secondary peak at K = 15.5. Such structure in the f aint T rapezium KLF implies similar non-po wer la w structure in the underlying IMF while our current models based upon a three po wer -la w IMF essentially assign a single po wer -la w IMF slope for the entire bro wn dw arf re gime. W e will e xplore this structure in the f aint bro wn dw arf KLF and IMF in Section 5 b ut rst we e xamine the condence interv als for the deri v ed 3 po wer -la w IMFs. Results of c 2 tting: range of permitted thr ee po wer -law IMFs. Our c 2 tting routine also allo ws us to in v estig ate the range of permitted cluster IMFs from modeling the cluster KLF W e illustrate the range of IMFs and the ef fects of source reddening on our ts in Figure 3–12 and summarize the corresponding constraints on the IMF parameters in T able 3–3 In each panel, we plot the contours of c 2 probability for tw o of the 5 dependent IMF parameters while restricting the other three parameters to a best t model. In each panel we also display contours for ts with (solid) and without (dashed) source reddening, and we e xamine the dependence of these parameters for models t to the K=14.5 and K=15.0 bins. In all our tting e xperiments (here and in Section 2.6.2 ), the high-mass slope of the cluster IMF G 1 w as well constrained with slopes measured between -1.0 and -1.3. Based on this result, we x G 1 to equal -1.3. P anels (a) (c) in Figure 3–12 display the ranges of the other 4 IMF parameters when tting to a K limit145. P anel (a) plots the dependence of the tw o break masses, m 1 and m 2 The ts for these parameters are well beha v ed with 90% contours ha v e a typical width of 0.1-0.2 de x in units of log mass. Source reddening has tw o clear ef fects upon our t results. When source reddening is included, the high-mass break, m 1 decreases and the lo w mass break, m 2 increases. The second ef fect is that the size of the 90% condence contour increases when source reddening is included into the model ts. P anel (b) displays the dependence of the lo w mass break, m 2 on the middle po wer -la w slope, G 2 G 2 is f airly well constrained to be slightly rising to lo wer masses, and the permitted range of m 2

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81 Figure 3–12: T rapezium cluster: c 2 condence interv als for IMF parameters. Contours of c 2 probability for the 5 parameters of the underlying three po wer -la w IMF T w o parameters are compared in each panel while xing the other three to a best t v alue. Solid contours are best t ranges from models that include source reddening. Dashed contours are from best t models without source reddening. Contour le v els are sho wn at interv als 95, 90, 70, 50 and 30% condence. P anels (A) to (C) are sho wn for ts to K=14.5 and panel (D) is sho wn for ts to K=15. is ag ain roughly 0.1 0.2 de x, centered near 0.1 M(log m 2 1). Accounting for source reddening ag ain shifts the lo w-mass break to slightly higher masses, increases the size of the 90% contour and in this case, attens the central po wer -la w P anel (c) displays the dependence of G 3 upon the second break mass, m 2 Though m 2 is f airly well constrained to ha v e v alues between 013 and 008 M, the lo w mass po wer -la w slope, G 3 has a lar ge range of possible slopes from 0.50 to 1.50 within the 90% c 2 contour for models with source reddening. P anel (d) plots the same parameters

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82 as panel (c) b ut for ts to the K limit15. These ts gi v e some what atter G 3 slopes and some what higher mass m 2 breaks, b ut are actually slightly better constrained. As discussed in the pre vious section, our model KLFs emplo ying a 3 po wer -la w IMF do not pro vide good ts to the secondary peak in the KLF As the t range shifts to f ainter magnitudes, G 3 attens, b ut the total c 2 condence depreciates due to the secondary peak. W e e xplore the IMF parameters necessary to t this secondary peak in the ne xt section. Fitting the secondary peak in the T rapezium cluster KLF In contrast to our e xpectations when we interpreted the literature T rapezium KLF in Section 2.6.2 the departure from a po wer -la w decline and the formation of a secondary peak at the f aint end of the T rapezium KLF remains after correcting for reddened background eld stars. When we attempt to t the f aint KLF using an underlying three po wer -la w IMF we nd that our model KLFs, while producing e xcellent ts o v er the majority of the T rapezium KLF could not simultaneously reproduce the formation of the secondary peak. Since there is no kno wn corresponding feature in the mass-luminosity relation (see Section 3.4.2 ), we h ypothesize that the KLF' s break from a single continuous declining slope at K145 (M30 M J u p ) and the formation of a secondary KLF peak directly imply a similar break and feature in the cluster IMF Further the rapid tailing of f of the cluster KLF belo w this secondary peak also directly implies a similar rapid decline or truncation in the underlying IMF as w as also discussed in LR2000 W e modeled the secondary KLF peak by adding a fourth, truncated, po wer -la w se gment, G 4 to the three po wer -la w IMFs deri v ed in section 5 The truncation of the fourth po wer -la w se gment enabled us to model the rapid tailing of f of the cluster KLF belo w the secondary peak, b ut w as also dictated by the articial lo w mass cut of f present in the adopted mer ged PMS tracks, which for the substellar re gime come from DM97. As such, the truncation mass of the model IMF w as arbitrarily set to 0.017 M. W e found that this 4 po wer -la w truncated IMF produced good c 2 model KLF

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83 Figure 3–13: T rapezium cluster: best t model KLF to secondary KLF peak. The T rapezium MA V limited KLF t o v er the entire luminosity range using a four po wer -la w IMF with a lo wer mass truncation (17 M J u p ). The bro wn dw arf IMF breaks from a steady decline ( G 3073) between 0.03 and 0.02 Mand rises to the mass truncation. Abo v e the m 3 mass break the IMF is that described in T able 3–3 for ts to K = 15. ts to the secondary KLF peak. The best t model KLF sho wn in Figure 3–13 has an underlying bro wn dw arf IMF that breaks from a steady decline at m 30025 Mand then rises steeply with G 4 5 before truncating at the lo wer mass limit. Examination of the condence interv als for the m 3 and G 4 parameters sho wed that higher mass breaks ( m 30035) required atter G 4 slopes, b ut the y had w orse c 2 and peak ed before the observ ed peak at K=15.5. This suggests that were the slope of the massluminosity relation continuous (and constant) to w ard lo wer masses, the e xact location of the secondary IMF peak w ould shift to some what lo wer masses than we can deri v e using the truncated PMS tracks. 3.3.3 Deri v ed T rapezium Cluster IMF

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84 Figure 3–14: T rapezium cluster: o v erall deri v ed IMF Hatched areas are deri v ed from the range of 90% condence contours for KLF ts. Solid line is the best t T rapezium IMF listed in Equation 3.1 The T rapezium IMF is also compared to the log-normal Miller & Scalo ( 1979 ) eld star IMF The o v erall cluster IMF Figure 3–14 sho ws our o v erall best t T rapezium IMF and graphically displays the range of cluster IMFs permitted by our modeling of the T rapezium KLF using our standard set of mer ged PMS tracks. W e adopt the follo wing four po wer -la w function with a truncation at the lo west masses for the underlying IMF of the T rapezium: d N d log M M G; G 121 : M 0600 M 015 : 0600 M M 0120 M 073 : 0120 M M 0025 M 500 : 0025 M M 0017 M0 ; M 0017 M(3.1)

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85 W e nd that despite the use of deeper more complete observ ations, the application of detailed mass and e xtinction limits to the cluster sample, the inclusion of source reddening into the model luminosity function algorithm and the correction of our infrared census for reddened eld stars, our deri v ed T rapezium IMF is not a signicant re vision o v er that found by studying the literature KLF W e nd that the inclusion of source reddening into our modeling algorithm, while pro viding a more accurate representation of the cluster properties, results in cluster IMFs that ha v e v ery similar po wer -la w slopes and break masses as ts without source reddening, especially when t to the same luminosity range. Source reddening does indeed blur the precision of the IMFs we can deri v e. F or e xample, our T rapezium IMF in Equation 2.7 deri v ed without accounting for source reddening is some what broader and peaks to slightly lo wer mass than the IMF deri v ed here with source reddening. Though the o v erall deri v ed IMF has not signicantly changed from our ndings in Section 2.6.2 our more complete infrared census and impro v ed understanding of the eld-star population do allo w us to e xplore the T rapezium IMF at lo wer masses than Section 2.6.2 W e nd that the secondary peak of the observ ed T rapezium KLF is not the result of background eld stars, and we deri v e a corresponding secondary peak in the lo w mass bro wn dw arf IMF between 10 and 30 M J u p Ho we v er because of the restriction imposed by the lo w mass limit of the PMS tracks, both the precise location and amplitude of the secondary peak and the precise form of the IMF belo w 17 M J u p are some what uncertain. A closer look at the substellar IMF T o better dene the secondary peak in the T rapezium IMF we consider only the substellar re gime of the T rapezium KLF (K13), where we can emplo y a dif ferent set of PMS tracks that co v er the corresponding bro wn dw arf re gime b ut also e xtend to masses less than the limit of our standard mer ged PMS tracks. The Burro ws et al. ( 1997 hereafter B97) PMS models are a v ailable from 0.12 to 0.001 M(1 M J u p ), and for the rele v ant age range of the T rapezium. While the mass to K luminosity relation is relati v ely rob ust between

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86 T able 3–4. Three po wer -la w T rapezium sub-stellar IMF IMF P arameter Fit to K=16.0 Fit to K=16.5 ( j )j1 s median( j )j1 s median( j ) Fits to MA V KLF w/o Of f-eld Correction G 1 +1.39 0.22 +1,30 +1.36 0.18 +1.30 m 1 0.0214 0.0022 0.020 0.0220 0.0025 0.020 G 2 -4.87 1.52 -5.00 -5.50 2.01 -5.00 m 2 0.0125 0.0012 0.012 0.0137 0.0005 0.014 G 3 +3.70 2.60 +4.00 +5.70 1.59 +6.00 Fits to MA V Of f-eld Corrected KLFa G 1 +1.53 0.13 +1,60 +1.51 0.14 +1.60 m 1 0.0217 0.0024 0.020 0.0228 0.0025 0.025 G 2 -5.43 1.95 -6.00 -5.04 2.00 -4.00 m 2 0.0130 0.0008 0.013 0.0137 0.0008 0.014 G 3 +3.87 2.47 +4.00 +5.61 2.25 +6.00 aMA V KLF has limits of A V9 and M001 MNote. — Fits to Sub-Stellar T rapezium KLF using B97 tracks. dif ferent sets of PMS tracks, the B97 PMS models do display a some what atter massluminosity relation for substellar objects than do DM97. W e re-deri v ed the cluster' s MA V limited, background corrected T rapezium KLF follo wing our prescription in Section 3.2.2 b ut using the B97 isochrone. Since the neb ular background decreases our surv e y' s completeness to hea vily reddened (A V10), v ery lo w mass (M003 M) bro wn dw arfs, we dra w this MA V sample to our completeness limit rather than our 10 s sensiti vity limit to ensure the precision of the substellar IMF The resulting MA V limited KLF from the e xtended B97 isochrone samples the cluster population to a predicted mass limit of 001 Mand an A V9. Ne w model KLF ts (see Figure 3–15 a) that emplo y the B97 tracks and use a three po wer -la w underlying substellar IMF yield a po wer -la w bro wn dw arf IMF f alling with a similar b ut some what steeper slope than our standard tracks (see summarized

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87 Figure 3–15: T rapezium cluster: a closer look at the sub-stellar IMF A) The T rapezium Sub-Stellar MA V KLF is compared to best t model KLFs using the B97 tracks. T w o ts are sho wn: with eld star correction (observ ed solid histogram; model lled circles); and without eld star correction (observ ed dashed histogram; model open circles). Error bars for the observ ed KLFs are from counting statistics. Those for the model KLFs are the 1 s bin v ariation from 50 model iterations. The v ertical dotted line demarks the K=17.5 completeness limit. B) Deri v ed Sub-Stellar T rapezium IMF using the B97 tracks. Our mass completeness limit (0.01 M) is displayed as a v ertical dotted line. The IMF t range allo wed by the KLF modeling is shaded with the best t 3 po wer -la w IMFs from ts with and without background correction are sho wn (see also T able 3–4 ). IMF parameters in T able 3–4 ) 4 Similar to DM97, the B97 tracks require the presence of a signicant secondary peak that departs from the po wer -la w function at m 3002 Mand rises v ery steeply with G 4 5 as sho wn in Figure 3–15 (b). Further the e xtended mass range of the B97 tracks allo ws us to resolv e the location 4 Fitting the substellar sample dra wn to an A V17 with model LFs using B97 tracks yielded a po wer -la w decline closer to that deri v ed from our standard mer ged models

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88 of the secondary peak: our ts require a peak near the deuterium-b urning limit, i.e., 1314 M J u p follo wed by a rapidly declining IMF with a slope of G 5 5 do wn to 10 M J u p The sharp decline in the substellar IMF belo w this peak is not the result of the application of the of f-eld correction. Fits to T rapezium substellar KLF without correcting for background eld stars yield nearly identical cluster IMFs (see Figure 3–15 (a) and T able 3–4 ). Further the sharp decline in our deri v ed substellar IMF belo w the deuterium-b urning limit and independent of the background correction conrms a similar straightforw ard interpretation of the observ ed rapid turn do wn in the cluster KLF and dearth of sources in the (H K)/K color magnitude diagram nearly a full magnitude abo v e our completeness limits (see Figure 3–5 ). Figure 3–16: T rapezium cluster: a secondary peak in T rapezium substellar IMF Sho wn is a Monte Carlo simulation of the T rapezium Sub-Stellar IMF from fty (50) samples of 150 bro wn dw arfs dra wn from the best t T rapezium Sub-Stellar IMF deri v ed from B97 tracks. The plotted histogram is the a v erage of the iterations, and the error bars represent the deri v ed 1 s standard de viation of each IMF bin from the Monte Carlo simulation.

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89 Lastly this feature in the IMF appears to be a statistically signicant departure from the po wer -la w decline of the bro wn dw arf IMF as w as implied by our c 2 tting in Section 5 W e ran a Monte Carlo simulation of the deri v ed T rapezium substellar IMF for a population of 150 bro wn dw arfs. In Figure 3–16 we sho w the resulting histogram form of the a v erage simulated cluster IMF Using equally sized bins in log mass units, we calculated the statistical v ariation in an IMF bin as a function of 50 dra wn samples. From these plotted 1 s error bars, it is clear that the deri v ed secondary peak is a signicant statistical result. In addition, these results imply that a statistically signicant identication of such a feature at the tail of the IMF requires the e xamination of a rich substellar population such as that pro vided by the T rapezium Cluster 3.4 Discussion 3.4.1 Structur e of the T rapezium KLF and IMF The stellar r egime. From our careful construction and impro v ed modeling of the T rapezium Cluster KLF we are able to deri v e the underlying T rapezium Cluster IMF spanning the entire mass range from OB stars to substellar objects near the deuterium-b urning limit. W e nd that the stellar T rapezium IMF rst rises steeply with a Salpeter -lik e po wer -la w slope from high-mass stars to near 0.6 Mwhere the IMF attens and forms a broad peak e xtending to the h ydrogen b urning limit. There the IMF turns o v er and declines into the bro wn dw arf re gime. From our modeling e xperiments in Chapter 2 we kne w that where an underlying IMF has a po wer -la w form, the young cluster' s model KLF also has a po wer -la w form. Further we found that peaks in the model KLFs can arise both due to peaks in the underlying IMF and from features in the M-L relation. From our current modeling of the T rapezium KLF we nd that these conclusions about the relationship between the structure of the KLF and IMF are unchanged by the presence of source reddening. The po wer -la w slope of the bright end (K115) of the cluster KLF reects the po wer -la w slope of the

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90 deri v ed IMF The formation of the primary KLF peak is also similar to the structure of the underlying IMF we deri v e. The broad main peak of the T rapezium KLF is formed by a combination of a peak in the underlying stellar IMF and a feature in the theoretical mass-luminosity relation due to deuterium-b urning. Moreo v er we nd that our KLF modeling has allo wed us to disentangle these tw o ef fects. The main KLF peak at K11115 corresponds to PMS stars between 0.4 and 0.2 M, which according to the DM97 PMS models are under going deuterium-b urning at the mean age of this cluster while the deri v ed IMF has a broad peak at some what lo wer masses (0.2-0.08 M) than the KLF peak w ould to rst order imply Lastly our detailed KLF modeling has determined that the turn-o v er and decline in the cluster KLF does reect a similar turn-o v er and decline of the T rapezium IMF across the h ydrogen b urning limit and is not solely a product of the deuterium-b urning spik e (e.g., Zinneck er et al. 1993 ). The substellar r egime. As in our w ork in Section 2.6.2 our KLF modeling technique has permitted us to deri v e the T rapezium substellar IMF while the impro v ed depth of our IR census has allo wed us to e xtend this deri v ation from 003 Mdo wn to near the deuterium-b urning limit. Our KLF modeling that no w includes source reddening conrms that the steady decline of the cluster KLF between K = 12 and K = 15 reects a steady po wer -la w decline in the substellar IMF Independent of our modeling results, ho we v er no more than 2242 % of the sources are substellar objects 5 The secondary peak in the bro wn dw arf re gime of the cluster KLF at K=15.5 and the subsequent rapid decline of the cluster KLF ho we v er do not correspond to an y kno wn features in the theoretical mass-luminosity relations we ha v e e xamined (see Section 3.4.2 and Figure 3–17 ). Moreo v er detailed KLF modeling using tw o dif ferent 5 Error based upon the uncertainty in the h ydrogen b urning limit due distance and cluster mean age.

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91 sets of PMS tracks require both the presence of a break from a single po wer -la w decline of the T rapezium bro wn dw arf IMF around 002003 Mand the formation of a signicant, secondary IMF peak. Using the B97 tracks, this IMF peak is located near the deuterium-b urning limit, 13-14 M J u p and is follo wed by a rapid decline to lo wer masses. Although both sets of PMS tracks suggest the presence of a secondary peak, the precise details (e.g., location and amplitude) may be track dependent. F or e xample, in the T rapezium substellar IMF found using the B97 tracks, 36% of the bro wn dw arfs in the cluster are found in the secondary IMF peak while 64% ha v e their mass distrib ution go v erned by the po wer -la w re gime. F or the IMF found using our standard mer ged tracks, these number are 15 and 85%, respecti v ely ho we v er the truncation of the tracks at the lo west masses will slightly sk e w these latter percentages. One pro viso to the deri v ation of a signicant IMF peak at the deuterium-b urning limit is the contamination of our IR census by non-cluster members. Though, we ha v e accounted for the background eld star contrib ution to the cluster KLF we ha v e also sho wn that there is reasonable uncertainty in the cloud e xtinction properties. Additionally the lar ge beamsize of the C 18 O map may mask lo w e xtinction holes in the molecular cloud. Since there are75 sources in the secondary peak of the MA V limited KLF before background subtraction, our current background eld star estimate w ould ha v e to be of f more than a f actor of tw o to remo v e an y feature from the IMF at these lo w masses; it w ould ha v e to be of f by a f actor of 4, ho we v er to account for all of the bro wn dw arf members. Alternately our IR census may be contaminated by the presence of lo w mass members from the interv ening b ut only slightly older Orion OB1c association. While these sources cannot create such a secondary peak, the y could contrib ute the o v er all KLF and IMF Since these sources are some what older the y w ould appear at f ainter magnitudes, sk e wing the KLF and IMF to lo wer masses and producing an o v er -estimate of the number of bro wn dw arfs. Though spectroscopic follo w up of a fe w of these f aint sources w ould separate out

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92 background stars (and pro vide a good test of the C 18 OA V con v ersion), members of the fore ground OB1 association w ould be dif cult to spectroscopically separate from actual T rapezium cluster members because e v en at these older ages the y will not ha v e e v olv ed v ery much in temperature or surf ace gra vity Ho we v er as we ha v e sho wn, the deri v ed turn do wn in the “sub-bro wn dw arf ” IMF belo w the deuterium-b urning limit appears independent of background correction. W e conclude, therefore, that if the mass-luminosity relation for lo w mass bro wn dw arfs is reasonably rob ust and does not contain a pre viously unidentied feature, and our estimate of the contamination of our infrared census by non-cluster members is accurate, then the e xisting structure of the f aint cluster KLF can only be created by a break from a single declining po wer -la w bro wn dw arf IMF the formation of a corresponding peak in the underlying T rapezium IMF near the deuterium-b urning limit, and a rapid decline of the IMF into the planetary mass re gime. 3.4.2 Sensiti vity of Results to Theor etical PMS Models The accurac y of an IMF deri v ed for a young stellar cluster is intrinsically dependent upon the rob ustness of the con v ersion from observ ables to a mass function (or indi vidual masses) pro vided by the theoretical e v olutionary models. In Chapter 2 we came to the some what surprising conclusion that model KLFs were f airly insensiti v e to dif ferences in the e v olutionary PMS models from which the mass-luminosity relations were dra wn. This w as despite that f act that the detailed ph ysics (e.g., opacities, model atmospheres, internal con v ection theory and initial conditions) in v olv ed with calculating the theoretical PMS e v olutionary models are poorly constrained and that changes in the assumed ph ysics of these models ha v e been sho wn to produce signicant differences in the locations of e v olutionary tracks and isochrones on the theoretical HR diagram ( D'Antona & Mazzitelli 1994 1997 ; Baraf fe et al. 1998 ; D'Antona 1998 ; Siess et al. 2000 ; Baraf fe et al. 2002 ). Our ndings in Chapter 2 w ould also appear to disagree with recent summaries of the IMF in young clusters which concluded,

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93 based upon the track v ariations in the HR diagram, that the accurac y of current PMS models are the primary uncertainty to the form of the deri v ed IMF ( Me yer et al. 2000 ). Therefore, we e xplore in more detail the dependence of the theoretical mass-luminosity relation rele v ant for luminosity function modeling upon the dif ferent PMS tracks. In Figure 3–17 we compare the theoretical mass-infrared luminosity (K magnitude) relations con v erted from six sets of theoretical PMS models for a progressi v e series of young cluster mean ages. In T able 3–5 we summarize the dif ferent input ph ysics and parameters used by v arious PMS models. The sets of theoretical PMS models were tak en from literature sources and con v erted to observ ables using a single set of bolometric corrections (see Section A ). Remarkably the theoretical mass-K magnitude relations are f airly de generate between the dif ferent PMS models, and those dif ferences that do e xist are the lar gest at v ery young ages ( t1 Myr), agreeing with the recent analysis of ( Baraf fe et al. 2002 ). Consequently this implies that for the T rapezium Cluster there will be some uncertainty in our deri v ed mass function due to the PMS tracks. On the other hand, while the models of B97 and Siess et al. ( 2000 ) display the most signicant v ariations in their predicted M-L relations, we ha v e sho wn in Section 3.3.3 that the substellar T rapezium IMF deri v ed from KLF modeling using the B97 tracks is not signicantly dif ferent than that found using the DM97 tracks. As we concluded in Section 2.5.1 most dif ferences in the mass-luminosity relations due to dif ferences in input ph ysics are much smaller than we could e v er observ e and will not impact our modeling results. This result may be understood by considering the f act that the luminosity of a PMS star is determined by v ery basic ph ysics, simply the con v ersion of gra vitational potential ener gy to radiant luminosity during the K e vinHelmholtz contraction. And this primarily depends on the general ph ysical conditions in the stellar interior (e.g., whether the interior is radiati v e or partially to fully con v ecti v e). The luminosity e v olution at the youngest ages (1 Myr) will depend, ho we v er

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94T able 3–5. Ev olutionary models used to compare M-L relations ModelatminbMmaxMminInitialc[D H]dOpacityeEOSfCon v ectiongAtmospherehName (Myr) ( M) ( M) Conditions T able Model Model DM94 0.1 2.50 0.020 Canonical 2.0 OP AL92+Ale x89 MHD FST -1 Gray B97 0.1 0.15 0.001 Canonical 2.0 ? SC ? Non-Gray DM97 0.1 3.00 0.017 Canonical 2.0 OP AL92+Ale x94 MHD+OP AL FST -2 Gray BCAH98 1.0 1.40 0.020 Canonical 2.0 OP AL96+Ale x94 SCVH ML T -1.0 Non-Gray(Ne xtGen) PS99 0.1 6.00 0.100 Birthline — OP AL96+Ale x94 PTEH ML T -1.5 Gray SDF00 0.1 7.00 0.100 Canonical 2.0 OP AL96+Ale x94 PTEH(r) ML T -1.6 Analytic Fit aAll models used standard solar metallicity (Z=0.02).bMinimum Age (Myr) listed by the models.cInitial Ph ysical Conditions from which the tracks are e v olv ed. Canonical: the model stars are e v olv ed from “innite” spheroids; Birthline: the PS99 models be gin with spherically accreting (single accretion rate, 105Myr1) protostars e v olving along the birthline before be ginning their PMS contraction phase.dDeuterium ab undance relati v e to h ydrogen, in units of 105. The initial D/H for the PS99 models w as 20105, ho we v er this is signicantly modied by the b urning of deuterium during the model' s proto-stellar phase.eOpacity T able (interior not atmosphere): Ale x89 ( Ale xander et al. 1989 ); OP AL92 ( Rogers & Iglesias 1992 ); Ale x94 ( Ale xander & Fer guson 1994 ); OP AL96 ( Rogers et al. 1996 )fCon v ection Model: FST (Full Spectrum T urb ulence. FST -1: Canuto & Mazzitelli ( 1991 1992 ), FST -2: Canuto et al. ( 1996 )); ML T -1.X (Mixing Length Theory 1.X = a = 1/Hp)gEquation of State: MHD ( Mihalas et al. 1988 ); SC ( Saumon & Chabrier 1991 1992 ); SCVH ( Saumon et al. 1995 ); PTEH ( Pols et al. 1995 r: Re vised by SDF), OP AL ( Rogers et al. 1996 )hT reatment of Atmosphere: Analytic Fit (SDF00) is a 1D t of Ttto atmosphere models; Ne xtGen ( Hauschildt et al. 1999 )

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95 on the initial conditions of the contracting PMS star as it e xits its proto-stellar stage, though these dif ferences are quickly erased ( Baraf fe et al. 2002 ). This is signicantly dif ferent than the situation for mass functions deri v ed by placing the stars on the theoretical HR diagram using spectroscopic and photometric observ ations. Because most young stars ha v e late type K-M spectral types, the y are on nearly v ertical Hayashi contraction tracks in the HR diagram. As a result a star' s mass deri v ed from the HR diagram is primarily a function of its assigned ef fecti v e temper ature, i.e., its observ ed spectral type. W e illustrate this dependence in Figure 3–18 (a) where we plot the predicted ef fecti v e temperatures as a function of mass for stars in a 1 Myr old cluster In contrast to the quite similar mass-luminosity relations, a star' s mass deri v ed based upon its spectral type is uncertain due to dif ferences in the PMS models by f actors of 3 (or more). The con v ersion from spectral type to mass is made w orse by the uncertain con v ersion of spectral type to ef fecti v e temperatures for late type sources, resulting from their sub-giant gra vities ( Luhman 1999 ). Such uncertainties will undoubtedly result in spectroscopically deri v ed IMFs that v ary substantially as a function of PMS tracks used (compare, for e xample, the Luhman et al. 2000 deri v ation of the T rapezium IMF from DM97 and BCAH98 tracks). Further these track dif ferences, while decreasing with time, are not resolv ed by 5 Myr as sho wn in Figure 3–18 (b). In summary the uncertainties in the PMS models primarily manifest themselv es in v ariations in the predicted ef fecti v e temperatures of the young stars rather than the predicted luminosities. W e do not surmise that luminosity function modeling, which emplo ys massluminosity relations, is free from systematic dependencies. As concluded in Chapter 2 a cluster' s mean age must be kno wn in order to deri v e a cluster' s initial mass function from its luminosity function; this can only be deri v ed from placing the stars on the theoretical HR diagram. In general, ho we v er age is a function of luminosity for lo w mass stars on the HR diagram and will be more or less similar when deri v ed

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96 Figure 3–17: Comparison of theoretical mass-luminosity relations. Theoretical pre-main sequence mass-luminosity (infrared K band) relations were e xtracted from six dif ferent sets of e v olutionary models (see T able 3–5 ). In all cases the intrinsic model quantities (luminosity ef fecti v e temperature) were con v erted to K magnitudes using a single T e f f -bolometric correction relationship. Sho wn for 6 sets of cluster mean ages: 0.1 (a); 0.5 (b); 1 (c); 2 (d); 3 (e), and 5 (f) Myr Baraf fe et al. ( 1998 ) models do not include models for ages less than 1 Myr

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97 Figure 3–18: Comparison of theoretical M -T e f f -spectral type relations. (Figure adopted and re vised from D'Antona 1998 ). The spectral type ef fecti v e temperature mass relationships were tak en directly from the 6 sets of PMS models at 1 (A) and 5 (B) Myr Also sho wn is the gra vity (dw arf vs sub-giant) dependence of the spectral type to ef fecti v e temperature calibration for late type PMS stars. Because v ery young subsolar mass stars and bro wn dw arfs are primarily on v ertical Hayashi contraction tracks in the HR diagram, there is theoretically a close correspondence between ef fecti v e temperature and mass. The ef fecti v e temperature to spectral type scale (right hand y-axis) is a cool dw arf scale ( summed from K en yon & Hartmann 1995 ; Bessell 1995 ; W ilking et al. 1999 ). The inset spectral sequence is the hotter sub-giant temperature-spectral scale tuned by Luhman & Riek e ( 1999 ). from these PMS models. The e xception ag ain occurs at the youngest ages, where the denition of a star' s age may dif fer if the models include the proto-stellar lifetimes. Ev en in the case of the P alla & Stahler ( 1999 hereafter PS99) models, which be gin as protostars accreting along an initial mass-radius relationship or birthline in the HR diagram ( Stahler 1983 ), the mass-luminosity relations are not substantially di v er gent from canonical theoretical models e xcept at the v ery youngest ages. 3.4.3 Comparison of IR-Based T rapezium IMFs In addition to our initial modeling in of the T rapezium in Section 2.6.2 and our present study a number of other authors ha v e recently deri v ed T rapezium IMFs based upon deep infrared observ ations. While all of these deri v ations mak e use of the same set of theoretical pre-main sequence models for con v erting observ ations to mass

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98 (functions), the y use some what dif ferent cluster parameters and emplo y a v ariety of dif ferent methodologies. Systematic uncertainties might arise due to v arying of cluster parameters such as distance, due to dif ferent assumptions about the cluster population such as eld star contamination or from simple observ ational ef fects such as surv e y area or the w a v elength re gime analyzed. Further it is not understood ho w closely dif ferent methods can arri v e at the same IMF Figure 3–19: Comparison of T rapezium IMFs from IR photometry All deri v ations used the DM97 PMS tracks for masses less than 1 M. T able 3–6 summarizes dif ferences among the deri v ation methods. The HC2000 T rapezium IMF corresponds to their A V10 limited sample. The “MLL2000” IMF is t (g) from T able 2–2 In Figure 3–19 we compare the IMFs deri v ed by us in Section 2.6.2 and in Section 3.3 to those deri v ed by LR2000 HC2000 and Luhman et al. ( 2000 ). Globally these IMFs are remarkably similar The y all ha v e Salpeter -lik e high-mass slopes, all reach a broad peak at subsolar masses and all decline in frequenc y with decreasing mass belo w the h ydrogen b urning limit with bro wn dw arf IMF slopes between +1

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99 and +0.5. After inspecting the dif ferent methods and cluster parameters used by these authors, which we summarize in T able 3–6 this result should be in part e xpected. When dif ferent methods use the same PMS tracks (in this case DM97; M1 M) and essentially the same star -forming histories, the resulting IMFs should basically agree. The cluster parameters used by these w ork ers are not e xactly homogeneous. Further there are slight v ariations between these IMFs that might be due in part to observ ational ef fects. F or e xample, the truncation or turn do wn in the high-mass end of the LR2000 and HC2000 IMFs is due to bright source saturation in these surv e ys, not to a real IMF feature. At the lo w mass end, the IMF deri v ations appear to di v er ge belo w 30 M J u p (-1.5 in log solar mass units) with a “spik e” in the LR2000 IMF b ut no feature in the Luhman et al study Because LR2000 surv e ys the lar gest area while the Luhman et al. ( 2000 ) study surv e ys the smallest area, one might suspect that this dif ference is due to an increase in eld star contamination or perhaps, counting statistics for the smaller study The latter is the most lik ely e xplanation since both our study and that of HC2000 surv e y similar lar ge areas and apply eld star corrections while nding substellar IMFs that contain either a secondary peak or a plateau at the lo west masses. Lastly the methodologies emplo yed range from a purely statistical approach (e.g., our LF modeling) to the deri v ation of indi vidual masses of the stars via a h ybrid combination of spectroscop y and infrared colors (e.g., Luhman et al.). It is unclear ho w to mak e detailed comparisons of these methods, ho we v er in general, the LR2000 HC2000 and our LF modeling primarily depend upon the theoretical mass-luminosity relation e xtracted from the PMS tracks. The stellar portion of the Luhman et al IMF depends upon the theoretical HR diagram, while the substellar depends upon the predicted infrared colors and magnitudes. One apparent dif ference between the resulting IMFs that might be related to the dif ferent methods is the e xact location of the IMF' s “peak, ” or what is sometimes termed the “characteristic” mass. This

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100T able 3–6. Comparison of published T rapezium IMFs based on IR photometry W ork IMF Distance tDtaExtinction?bIR Excess?bField StarbAVLimit? IMF Peak AreacCommentsdName Method (pc), (m-M) (Myr) Correction? (M M) ( pc) chapter 2 Model KLF 400, 8.00 0.8, 1.2 Not Not Not None 0.08 0.34 Literature Fit Included Included Included K only HC2000 (H-K)/K 480, 8.40 0.4, 1.0 Deri v ed Empirical Reddened AV25100.15 0.35 K eck Galaxy Model no limit H, K LR2000 MJM440, 8.22 1.0, 0.0 Deri v ed Assumed Assumed (J-H)15 0.40 0.50 UKIR T None None I, J, H Luhmen IR Spectra 450, 8.27 0.4,1 Deri v ed Assumed Assumed AH14 0.25 0.073 NICMOS et al + Colors None None F110, F160 chapter 3 Model KLF 400, 8.00 0.8, 1.2 EPDF IXPDF Reddened AV17 0.10 0.34 FL W O + NTT Fit Deri v ed Deri v ed Obs. KLF (0.57) J, H, K aCluster mean age and age spread used by authors. F or Luhman et al. ( 2000 ), an empirical star forming history w as used by the authors and those tab ulated here are approximate characterizations.bListed if and ho w these quantities: e xtinction, e xcess and the contrib ution of background eld stars, were included into that w ork' s IMF deri v ation.cSize (area) of surv e ys in parsec assuming D = 400pc. The tw o v alues for this w ork are for the NTT/FL W O o v erlap re gion and the lar ger FL W O re gion.dComments include location of observ ation(s) and broadband lters used. Note. — All IMF deri v ations used the DM97 pre-main sequence models for masses less than 1 M. References. — HC2000: Hillenbrand & Carpenter ( 2000 ); LR2000: Lucas & Roche ( 2000 ); Luhmen et al: Luhman et al. ( 2000 )

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101 “peak” mass v aries between IMF deri v ations by 0.7 de x in log solar mass units. It is not immediately apparent that internal uncertainties in the IMF deri v ations could cause this scatter F or e xample, the “peak” of the T rapezium IMF from Section 2.6.2 is re vised only 0.1 de x by the inclusions of source reddening. F or methods that depend upon mass-luminosity relations, the resulting IMF will be dependent upon the assumed cluster distance and age; modest changes in these parameters should result in slightly dif ferent M-L relations and slightly dif ferent IMFs. Ho we v er there is no strict correlation between “peak” mass and the cluster age or distance used. Hence, we conclude that specic IMF details such as the e xact location of “peak” mass cannot be securely identied by these methods; although, we can conclude that the T rapezium IMF peaks at subsolar masses some where between 0.3 Mand the h ydrogen b urning limit. 3.5 Conclusions Using a ne w and v ery complete near -infrared census of the T rapezium Cluster we ha v e performed a detailed analysis of the T rapezium Cluster' s K band luminosity function and its underlying mass function. F ollo wing our earlier w ork in Chapter 2 we e xpanded our luminosity function modeling to include the ef fects of source reddening, and we studied in detail the eld star contrib ution to the cluster KLF W e applied our ne w models to the T rapezium KLF to e xplore its structure and to deri v e the cluster' s initial mass function. From this analysis we dra w the follo wing conclusion(s): 1. The T rapezium Cluster IMF rises in number with decreasing mass and forms a broad peak at subsolar masses between 0.3 Mand the h ydrogen b urning limit before declining into the bro wn dw arf re gime. Independent of modeling details, no more than22% of the young sources f all belo w the h ydrogen b urning limit, placing a strict limit on the bro wn dw arf population in this cluster 2. The T rapezium Cluster substellar IMF breaks from a single declining po wer -la w slope between 0.02 and 0.03 Mand forms a signicant secondary peak near the

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102 deuterium-b urning limit (13 M J u p ). W e deri v e these results through detailed analysis of the lik ely eld star contamination and from our modeling of the cluster' s f aint KLF using tw o dif ferent sets of theoretical mass-luminosity relations, although the precise details of this peak do depend upon the PMS models. Re g ardless, this peak may contain between 15 and 36% of all the substellar objects in this cluster Belo w this peak the substellar IMF declines rapidly to w ard lo wer masses suggesting that the yield of freely oating, planetary mass objects during the formation of the T rapezium Cluster w as e xtremely lo w 3. W e nd that source reddening (due to infrared e xcess and e xtinction) has only modest ef fects upon our modeling of the T rapezium cluster' s luminosity function. Source reddening tends to broaden the IMFs deri v ed and blur the precision with which we can deri v e IMF parameters. Ho we v er the T rapezium IMF we deri v e here after accounting for source reddening and eld stars is not a substantial re vision o v er that T rapezium IMF we deri v ed in Chapter 2 4. Pre-main sequence luminosity e v olution and the resulting age dependent massluminosity relations are relati v ely rob ust results of most modern PMS e v olutionary models, e xcept at the v ery youngest ages where the models are af fected by initial conditions. Con v ersely the predicted ef fecti v e temperatures, hence predicted spectral types, are considerably less rob ust. This suggests that modeling a cluster' s K band luminosity function is lik ely to produce a f aithful representation of the true IMF of the cluster Further we nd that the dif ferent published methodologies used for deri ving the T rapezium IMF from near -infrared photometry produce nearly identical results, although the precise location of a “peak” or characteristic mass for the T rapezium cannot be securely identied.

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CHAPTER 4 THE Y OUNG CLUSTER IC 348 IN PERSEUS Infrared detector technology has often pro vided the impetus for ne w sensiti v e, and re v ealing studies of young clusters and star -forming re gions. At the v ery early stages of this technology for e xample, the young partially embedded IC 348 cluster which is the focus of this chapter w as in v estig ated in the infrared using a single channel photometer by Strom et al. ( 1974 ) who used this instrument to disco v er hea vily embedded sources at the interf ace of the cluster with the Perseus Molecular Cloud. The de v elopment of array format IR cameras yielded v ery lar ge surv e ys of giant molecular clouds (GMCs) (e.g., the Orion B GMC; Lada et al. 1991 ) and young clusters such as IC 348. These intensi v e surv e ys often consisted of lar ge mosaics containing tens to hundreds of indi vidual frames; the Lada & Lada ( 1995 hereafter LL95) surv e y of IC 348 required, for e xample, twenty four separate tiles to co v er the cluster' s central parsec. Recent adv ancements to infrared detector technology ha v e permitted the de v elopment of wide-eld infrared cameras that pro vide a moti v ation to re visit lar ge area IR surv e ys of GMCs and young clusters. The v ery sensiti v e and contiguous co v erage pro vided by these wide-eld cameras allo ws, for e xample, the simultaneous co v erage of lar ge cluster areas and the ef cient census taking of candidate young bro wn dw arfs as small 10 Jupiter masses ( M J u p ) found deeply embedded (A V510) in the parental cloud. Combining such a wideeld IR census with tools for studying the characteristics of the stars and bro wn dw arfs in young clusters such as the construction and modeling of their luminosity functions ( Muench et al. 2000 2002 ), permits the testing of v arious h ypotheses about the cluster' s initial structure and mass function. F or e xample, one often cited adv antage to studying the mass function of v ery young clusters is that fe w members will ha v e been 103

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104 lost due to the cluster' s dynamical e xpansion. Recent N-body e xperiments, ho we v er ha v e sho wn that considerable dynamical e v olution may occur in clusters within one million years after the e xpelling of the molecular g as ( Kroupa et al. 2001 ). Other simulations suggest that some mass se gre g ation could be primordial ( Bonnell et al. 2001 ). Thus, the combination of wide-eld infrared imaging with luminosity function modeling may pro vide important clues about the spatial distrib ution of stellar masses while the cluster is still embedded in the parental cloud. IC 348 has been the tar get of a number of wide-eld studies since the Lada & Lada IR surv e y W ide-eld optical ( Luhman 1999 ), H a ( Herbig 1998 ) and X-ray imaging ( Preibisch et al. 1996 ) ha v e all reinforced the LL95 nding that IC 348 is spatially e xtended on the sk y and partially embedded at the edge of the Perseus GMC. In our current wide-eld near -IR imaging, we are able to surv e y in a single image an area encompassing nearly all the boundaries of these past surv e ys while probing v ery lo w mass sources (001 M) o v er a much lar ger cluster v olume (lar ger e xtinction) than prior studies. W e use the results of our infrared census to e xamine the cluster' s structure, reddening and relationship to the Perseus Molecular Cloud in Section 4.1 F ollo wing from our analysis of the T rapezium cluster' s luminosity and mass function in Chapter 3 we construct the K band luminosity function for IC 348 in Section 4.2 In Section 4.3 we emplo y our model luminosity function algorithm to deri v e and to study the IMF of IC 348, encompassing with our wide-eld surv e ys much lar ger area than past IMF studies which ha v e concentrated on the central core of the cluster ( Luhman et al. 1998 ; Najita et al. 2000 ). From this analysis, we compare our results for IC 348 and the T rapezium and discuss in Section 4.4 the impact of spatial and statistical IMF v ariations on meaningful comparisons of the IMFs for dif ferent clusters.

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105 Figure 4–1: Infrared color composite image of IC 348. F or orientation, north is up while east is left and the eld of vie w is 205205. The bright blue star to the north by northwest is o Persei. The cluster' s interf ace with the Perseus Molecular Cloud is clearly outlined by a series of neb ular features and enshrouded or hea vily reddened sources along the southern edge of the image. The series of blue-green-red sources to the south by southwest of the cluster center is the asteroid 545 Messalina. 4.1 W ide-Field Near -Infrar ed Images of IC 348 4.1.1 FLAMINGOS Obser v ations W e obtained wide-eld near -infrared images of the IC 348 cluster using the FLoridA Multi-object Imaging Near -IR Grism Observ ational Spectrometer (FLAMINGOS, Elston 1998 ) on the 2.1m telescope at the Kitt Peak National Observ atory

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106 T able 4–1. Summary of FLAMINGOS observ ations of IC 348 T ar get Filter Julian Date Exp. Dithers T otal Exp. Airmass Seeing 5 s Flat-Field (sec) (sec) ( ) (mag) type IC 348 K 2452257.82961 20 49 980 1.139 1.60 16.81 dome IC 348 H 2452257.85944 20 48 960 1.262 1.81 17.69 sk y IC 348 J 2452257.88446 20 46 920 1.414 1.87 17.70 sk y Of f 1 K 2452260.81462 30 30 900 1.112 2.10 16.97 dome IC 348 K s 2452313.60532 60 16 960 1.007 1.48 17.72 dome IC 348 H 2452313.62027 60 14 840 1.020 1.49 18.04 sk y IC 348 J 2452313.68697 60 24 1440 1.176 1.67 18.82 sk y Of f 2 K s 2452316.58424 60 16 960 1.001 1.59 17.74 dome Arizona (USA) during December 2001 and February 2002. The FLAMINGOS instrument emplo ys a 2K HgCdT e “HA W AII-2” imaging array which when congured on the 2.1m Kitt Peak telescope yields a 205 205eld of vie w with a deri v ed plate scale at K band of0608 pix el. On both 14 December 2001 and 08 February 2002 dithered sets of IC 348 images were obtained with FLAMINGOS in the J, H and K (or K s ) passbands. These image sets were also obtained within a narro w windo w of time (1.5 2.5 hrs) and a restricted range of airmass (sec z14). W e list the details of these observ ations in table 4–1 Briey we emplo yed a lar ge number (15-50) of short (20-60 sec) non-repeating dithers to yield total inte gration times of approximately 14-24 minutes, depending upon the passband. Conditions on both nights appeared photometric, with no apparent cloud co v er stable background sk y counts, and seeing estimates between 16 and 19 on 14 December 2001 and 1517 on 08 February 2002. T w o wide-eld of f-cluster K (or K s ) images were similarly obtained on 17 December 2001 and 11 February 2002. These tw o non-o v erlapping re gions lie approximately 1 de gree east of IC 348, along a line of constant g alactic latitude with the young cluster Their equatorial eld centers were: 1) 03 h 48 m 21s 9; DEC.=313806 7 (J2000); 2) 03 h 48 m 19s 4; DEC.=310834 7 (J2000). Additional details of these of f-eld observ ations are also listed in table 4–1

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107 The sets of dithered images were reduced using the April 2002 v ersion of the FLAMINGOS data reduction pipeline (Elston et al, in preparation), using rele v ant dark frames obtained on each observing night and either local sk y or dome at-elds, depending upon the lter Local sk y at-elds were used for J and H bands, while dome ats were used for K or K s passbands. Briey the FLAMINGOS data reduction pipeline is based upon a tw o-pass, object masking routine that permits the creation of star -free median sk y frames from the tar get images while follo wing standard techniques for the reduction of near -infrared data. T o tak e adv antage of the relati v ely lar ge number of dithers in our datasets, the pipeline emplo ys the drizzle IRAF routine ( Fruchter & Hook 2002 ) to allo w for linear sub-pix el image reconstruction during the nal combination of the dithered frames. In summary 6 reduced cluster images (2 at each passband) were obtained, in addition to tw o K band of f-eld images. In Figure 4–1 we display a (f alse) color composite near -infrared image of IC 348 using the J, H, and K s FLAMINGOS images from our February 2002 observ ations. A number of interesting cluster features are outlined by lo w-le v el neb ulosity These include the cluster core, which displays deep red neb ulosity suggesti v e of some what higher e xtinction, and the interf ace between the IC 348 cluster and the molecular cloud all along the southern edge of the image. This interf ace re gion also includes numerous signposts of v ery recent star formation including the HH-211 infrared jet ( McCaughrean et al. 1994 ), the enshrouded IR source deemed the “Flying Ghost” neb ula ( Strom et al. 1974 ; Boulard et al. 1995 ), a dark lane suggesti v e of a ared edge-on disk-lik e structure, and a number of bright infrared sources detected only in the K band. A fe w image artif acts can also be seen in this gure. These include geometric distortions of the stars in the northeast corner of the image, red and green glint features due to internal reections along the south edge of the cluster and coma lik e ghosts south-southwest of the cluster center resulting from a long time constant in one

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108 amplier of the HA W AII-2 array On the other hand, the alternating blue-green-red source south-southwest of the cluster center is not an image artif act b ut is instead the asteroid 545 Messalina. 4.1.2 Infrar ed Census Photometry and calibration. W e characterized each reduced image by deri ving estimates of the FWHM of the stellar point spread function and the pix el-to-pix el noise in the background sk y using the IMEXAMINE IRAF routine, although the pix el-topix el noise is correlated in drizzled images. The resulting seeing estimates and 5 s detection limits are listed in table 4–1 W e found that the slightly better seeing and longer e xposure times of the February data yielded detection limits approximately 1 magnitude f ainter than the December observ ations. Sources were initially identied on each reduced image using the stand alone S-Extractor package ( Bertin & Arnouts 1996 ). In an iterati v e f ashion, accurate centroids were calculated for the detected sources using the CENTER IRAF routine, and mark ed on the reduced images, after which the images were manually searched to identify f alse detections or to add sources missed near bright stars. The source lists for each of the 6 on-cluster images were then cross-correlated to identify and check those sources not appearing on all the images. Multi-aperture photometry w as performed on the sources using the APPHO T IRAF package and the instrumental magnitudes were corrected out to the be ginning of the sk y annulus using photometric curv es of gro wth calculated from2030 bright stars using the MKAPFILE IRAF routine. From the corrected multi-aperture photometry we chose to use the smallest beamsize that simultaneously g a v e the most consistent photometry when compared to lar ger apertures. Because of a spatially v arying PSF due to geometric distortions present in the nal images, we resorted to using a rather lar ge aperture (radius = 5 pix els; beamsize = 6 ), which yielded aperture corrections typically of order 006 magnitudes. Absolute calibration of the instrumental photometry w as performed using zeropoint and airmass corrections

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109 T able 4–2. Comparison of IC 348 photometry to 2MASS catalog. Date T ar get P assband MagnitudeaNumber of 1 s MedianbRange Matches Noise Of fset 14 Dec 2001 IC 348 J 11001550 343 0.067 0.050 14 Dec 2001 IC 348 H 11001450 345 0.053 0.049 14 Dec 2001 IC 348 K 11001375 280 0.051 0.075 17 Dec 2001 Of f K 11001375 103 0.033 0.073 08 Feb 2002 IC 348 J 12001550 334 0.068 0.085 08 Feb 2002 IC 348 H 12001450 267 0.066 0.086 08 Feb 2002 IC 348 K s 11751375 210 0.078 0.079 11 Feb 2002 Of f K s 11751375 98 0.078 0.077 aF or the magnitude range compared for the FLAMINGOS and 2MASS photometry of IC 348, the bright limit depended upon the source saturation in the FLAMINGOS images while the f aint limit depended upon the increase in the RMS noise of the 2MASS data with magnitude.bMedian Of fset = 2MASS FLAMINGOS absol u t e deri v ed from Persson et al. ( 1998 ) standard stars which were observ ed on the same night as the tar gets. F or secondary calibration, we calculated and remo v ed median of fsets of order004008 magnitudes between our absolute photometry and the 2MASS photometric system. Accuracy W e quantied our photometric accurac y by matching our data to the 2MASS catalog, yielding o v er 670 matches in the cluster re gion and more than 500 sources matched in each of the of f-eld images. W e also directly compared our December and February on-cluster data. W e list the results of these photometric comparisons in table 4–2 This table includes the magnitude range of the comparisons based on the FLAMINGOS saturation and 2MASS photometric limits, the 1 s dispersion between the datasets and the secondary calibration of fsets applied in Section 4.1.2 While the 17 December of f-eld data display a dispersion of3% at K band relati v e to the 2MASS catalog, we nd JHK dispersions in the December 14 on-cluster data are lar ger than e xpected from purely photometric errors, with the K band scatter

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110 4% lar ger in the on-cluster frame than that found in the of f-cluster frame tak en three days later under similar sk y conditions. W e attrib ute some of this additional scatter to the intrinsic v ariability of young PMS stars in IC 348 ( Herbst et al. 2000 ), which in the infrared typically peaks at J band ( Carpenter et al. 2001 ) (i.e., for our IC 348 data s K005 v ersus s J007 mag). Between 03 11 February 2002, oil from the primary mirror cell contaminated the FLAMINGOS de w ar windo w This had a ne gligible ef fect on our J and H band photometry since these data were at-elded with local, contemporaneous sk y ats. Ho we v er the K data were at-elded using non-contemporaneous dome at-elds and display a 46% increase in the photometric scatter relati v e to the FLAMINGOS data obtained in December and to the 2MASS catalog. This scatter does not af fect the analysis or conclusions of this paper Results. After considering the dif ferent saturation, detection and noise limits of the December and February IC 348 FLAMINGOS data, we produced a single w orking source list with photometry dra wn from both FLAMINGOS datasets. Simply we chose to use the December K band data do wn to m K155. Belo w this v alue we transition to the intrinsically deeper February K s data. In the J and H bands we a v eraged the tw o datasets between the saturation and the detection limits of these respecti v e catalogs, recording the 1 s standard de viation between the tw o observ ations. F or the brightest 39 stars, which were saturated in one or more bands on all of our images, we substituted the 2MASS photometry for our FLAMINGOS photometry Finally for the f aintest objects (m J185; m H175; m K170) we used smaller aperture photometry (radius = 3.5 pix els, aperture correction = 012 mag). In Figure 4–2 we present the infrared color -magnitude diagrams (J-H vs H and H-K vs K) for the FLAMINGOS IC 348 re gion without ltering for photometric error W e compare the distrib ution of sources in these diagrams to the location of pre-main sequence isochrones tak en from the D'Antona & Mazzitelli ( 1997 ) and

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111 Figure 4–2: Near -infrared color -magnitude diagrams of IC 348. No radial or photometric criteria ha v e been applied to the sources in these diagrams. The locations of the sources in these diagrams are compared to the location of the DM97 and Baraf fe et al. ( 1998 ) pre-main sequence isochrones for 2 and 5 Myr at 320 pc. Reddening v ectors ( Cohen et al. 1981 ) with length A V7 are dra wn for 1.4, 0.08 and 0.02 MPMS objects at the cluster' s mean age. A) J-H/H color -magnitude diagram containing 1534 sources (82% of total); B) H-K/K color -magnitude diagram containing 1739 sources (93% of total). Baraf fe et al. ( 1998 ) e v olutionary models. F or these comparisons we assume a cluster mean age of 2 Myr and a distance of 320 pc (see sections 4.3.1 and 4.3.2 for further discussion of these parameters). Using these cluster parameters, we nd that we are sensiti v e to a 2 Myr old 80 M J u p bro wn dw arf seen through30 magnitudes of e xtinction or a 10 M J u p source near the deuterium-b urning limit seen at an A V7 (using Burro ws et al. 1997 ). Three general characteristics of the sources projected to w ards our IC 348 FLAMINGOS re gion are clearly seen: 1) The color magnitude diagrams indicate a cluster re gion ha ving only modest reddenings relati v e to other star -forming re gions with the v ast majority of the sources ha ving A V7; 2) There is a density of sources between 10m Km H15 that is closely outlined by the PMS isochrones and represents a range of magnitude space that is dominated by lik ely cluster members; 3) Belo w m Km H15 the color -magnitude distrib ution

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112 appears to become dominated by eld-stars (and lik ely g alaxies) as indicated by a rapid increase and steady broadening in the density of sources on these plots. All of these features were seen in indi vidual datasets (December vs February) and are not modied if we change an y of the parameters for mer ging the data. Figure 4–3: Infrared color -color diagram of IC 348. Diagram is ltered on magnitude (m K15) b ut not by photometric error This observ ed color distrib ution is compared to the intrinsic colors of eld dw arfs (O-M9, Bessell & Brett ( 1988 ); Kirkpatrick et al. ( 1999 )), classical T -T auri stars with optically thick disks ( Me yer et al. 1997 ), and giants.12% of the sources display e xcess infrared emission in this diagram. A reddening v ector of length A V7 illustrates the modest reddenings seen by the majority of the cluster In Figure 4–3 we display the infrared H-K vs J-H color -color diagram for sources in the IC 348 re gion. W e include sources most lik ely to be cluster members by using characteristics (2) and (3) of the color -magnitude diagrams listed abo v e to apply a some what arbitrary m K15 magnitude limit. There are 580 sources with m K15 of

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113 which 563 ha v e JHK photometry; we display all of these in the color -color diagram without ltering for error Ag ain, it is clear that the cluster re gion is only mar ginally reddened with the nearly all of the sources ha ving A V7. The o v erwhelming majority of the sources f all within a reddening band bounded by the reddening v ectors for the giant branch and the tip of the M dw arf sequence (M9). While 1 source has IR colors signicantly to the left of the reddening band, 66 sources ha v e infrared colors which f all to the right of the reddening v ector for M9 dw arfs. These sources f all into a re gion of infrared e xcess in the color -color diagram, which because IC 348 is a v ery young cluster are considered to be lik ely cluster members with optically thick circumstellar disks. Finally 42 of these sources ha ving infrared e xcesses greater than their 1 s photometric noise, and we note that these ltered sources are uniformly distrib uted as a function of H band magnitude. W e no w use these basic results to e xamine the structure of the IC 348 cluster 4.1.3 Cluster Structur e F or our subsequent analysis of the luminosity and mass function of IC 348 we wish to select that portion of the FLAMINGOS wide-eld image that pro vides the best sampling of the o v erall cluster Studying a re gion some what similar in size to the FLAMINGOS area, LL95 used a surf ace density analysis to sho w that IC 348 could be brok en into 9 apparent sub-clusterings spread across their surv e y re gion. Although subsequent wide-eld H a ( Herbig 1998 ) and optical ( Luhman 1999 ) surv e ys co v ered areas similar to LL95 most studies of the IMF of IC 348 ha v e concentrated on the central LL95 sub-cluster IC348a ( Herbig 1998 ; Luhman et al. 1998 ; Najita et al. 2000 ) and ha v e not included the other LL95 sub-clusters. While these optical and H a studies may systematically under -estimate the cluster size due to e xtinction or miss members that do not display H a emission, the y ha v e conrmed that the cluster is in f act spread o v er an area lar ger than the IC348a re gion. Using the deeper wide-eld near -infrared imaging pro vided by our FLAMINGOS observ ations, we re-in v estig ated

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114 the structure of the IC 348 cluster by calculating the cluster' s radial prole and by e xamining the spatial distrib ution of sources. Cluster radius. In Figure 4–4 we construct the radial prole of sources in the FLAMINGOS IC 348 re gion, centering on the IC 348a sub-cluster W e use only those sources m K15 and calculate the surf ace density in stars per square de gree using both annuli of equal width and annuli ha ving constant areas. W e compare the resulting radial proles to the eld star surf ace density calculated from our of f-eld data. Clearly the cluster e xceeds the unreddened background surf ace density o v er most of the FLAMINGOS re gion. Further the cluster e xtends o v er an area considerably lar ger than the IC348a sub-cluster whose radius w as gi v en as 047 pc (505) in LL95 and o v er an area lar ger than the radius of 4calculated by Herbig The cluster appears to dip to the unreddened background surf ace density at a radius of1011; ho we v er since this is also the radius at which the prole be gins to clip the edge of the surv e y re gion, we cannot condently rule out a lar ger cluster radius using the current FLAMINGOS images. At the distance of IC 348, this translates to a cluster radius of1 pc, similar to the ef fecti v e radius of 119 pc deri v ed by Carpenter ( 2000 ). T o pro vide a means for comparing our study to that of other authors, we brok e our FLAMINGOS cluster re gion into tw o sub-di visions based upon the radial prole and further assigned these tw o sub-re gions primarily functional names, e.g., the cluster “core” sub-re gion with a radius = 5and the cluster “halo” re gion between the cluster “core” and a radius of 1033, corresponding to the lar gest unclipped radius permitted by the current FLAMINGOS surv e y The core re gion is approximately the IC 348a sub-cluster b ut is slightly lar ger than that area studied by Luhman et al. ( 1998 ) and Najita et al. ( 2000 ), while the halo re gion co v ers an area approximately 3.3 times that of the core and encompasses the stars contained in the LL95 sub-clusters b-i. Although our construction of the cluster' s radial prole has allo wed us to di vide the spatial e xtent of IC 348 within our surv e y re gion, we found that it follo wed

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115 Figure 4–4: Radial prole of the IC 348 cluster Surf ace density (in stars per square de gree) is measured in circular annuli, centering on the location of the LL95 sub-group IC348a. Proles are calculated using annuli of equal area (histogram with bins of decreasing width) and annuli with equal radial steps (hea vy solid line with error bars). First annuli width w as R 0175. Proles are compared to the unreddened background surf ace density (lightly shaded band; width = 2the 1 s de viation of background, see te xt.) and the background surf ace density reddened by A V4. Also sho wn: the di vision of cluster into sub-re gions (v ertical dashed lines, see te xt); a 1/r prole t to the entire cluster re gion (dot-dashed line; c 249104 )); and a King prole t to the cluster core (dotted line; r cor e025pc ; c 21). Note: bins with R1033ha v e been geometrically corrected to account for the surv e y boundaries and the con v ersion from angular to ph ysical scale (upper x-axis) is calculated for a distance of 320 pc. neither a simple r1 nor a King prole. W e t both analytical proles to the cluster' s constant area annuli prole, v arying the radial e xtent of the ts, and for the King prole, allo wing both the core and tidal radii to v ary freely Neither analytic prole pro vided reasonable c 2 ts to the entire cluster prole, although the core is well t by

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116 a King prole with r cor e025pc As displayed in Figure 4–4 the tail of the cluster prole, i.e., the halo sub-re gion, is much atter than either of the analytic proles. One interpretation, for e xample, is that the LL95 sub-clusters that constitute the cluster' s halo are actual separate entities, rather than simply being statistical uctuations on a 1/r cluster prole. Our radial prole and ts lik ely suf fer from tw o problems. First, we use circular annuli although as we will sho w in Figure 4–6 the cluster is elliptically elong ated in the N-S direction. Second, we rely upon an empirical estimate of the eld-star surf ace density from a re gion near to IC 348 b ut that, in principle, could uctuate between this location and the background rele v ant to IC 348. Indeed, the surf ace densities of the tw o of f-cluster locations uctuate more than e xpected purely from counting statistics despite their similar g alactic latitudes. This spatial uctuation, ho we v er is e xceedingly small relati v e to the e xcess surf ace density of the cluster halo and will not lik ely af fect the cluster boundary we deri v e. On the other hand, we cannot rule out a some what lar ger cluster radius because the obscuration of background eld stars by the parental molecular cloud will lo wer the e xpected eld star surf ace density as we illustrate in Figure 4–4 Reddening the background by the a v erage e xtinction in the IC 348 re gion(A V4; see Section 4.1.4 ) only e xpands the cluster radius by 12, ho we v er and the cluster probably does not e xceed a radius of 15(1.4pc), which is a boundary traced v ery clearly by wide-eld X-ray detections ( Preibisch et al. 1996 ). Spatial distrib ution of sour ces. W ith the purpose of characterizing our tw o cluster sub-re gions, we e xamined the spatial distrib ution of the IC 348 FLAMINGOS sources, plotting them using separate symbols for dif ferent luminosity ranges in Figure 4–5 a. By se gre g ating the tw o sub-re gions, we sho w for e xample that there are as man y members with B & A spectral types within the cluster core as there are in the cluster halo, although o Persei is not a lik ely cluster member A similar conclusion is reached about relati v ely bright sources in IC 348 with as man y sources with m K10 in the

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117 Figure 4–5: Spatial distrib ution of sources in IC 348. Locations of sources in dif fer ent magnitude ranges are sho wn in an equatorial tangent projection of the wide-eld FLAMINGOS IC 348 re gion. The tw o plots correspond to dif ferent samples: A) all sources; B) sources m K15. Lar ge lled stars correspond to sources with spectral types A5 and earlier including the B0III giant o Persei. In (A), lar ge lled circles correspond to sources m K10, smaller lled circles, m K15, and small open circles, m K15. In (B), sources m K15 are sho wn as open circles and sources displaying infrared e xcess in Figure 4–3 are lled diamonds. The R5and R1033boundaries of the cluster sub-re gions are sho wn as concentric circles. Note the decrease in surf ace density near the cluster -cloud interf ace in (A) and the anti-correlation of this decrease with the surf ace density of infrared e xcess sources in (B). cluster core as in the cluster halo. On the other hand, the f aintest sources (m K15) do sho w some spatial v ariations, with a sharp decrease in these sources along the cluster' s southern edge, clearly outlining the interf ace with the molecular cloud. In Figure 4–5 b we plot the spatial distrib ution of the 580 sources with m K15 and that were used to construct the cluster' s radial prole. The cluster core is clearly dened, while unlik e the f aint sources there is no change in the surf ace density of bright halo stars to w ard the molecular cloud. W e further e xamine the locations of the 66 sources displaying IR e xcess. As w as found by LL95 the majority of these sources lie outside of the cluster core. Moreo v er the surf ace density of these IR e xcess sources appears to increase to w ard the southern interf ace with the Perseus Molecular Cloud and

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118 Figure 4–6: Surf ace density prole of the IC 348 cluster The surf ace density of IC 348 sources is sho wn ltered by a Nyquist sampled box (width200 see upper left) that has an area similar to that used in the bins of the radial prole. Contours are in multiples of the surf ace density deri v ed by reddening the of f-eld by A V40, yielding a surf ace density of 2305 stars/sq. de g, the same surf ace density sho wn as the shaded band in Figure 4–4 Contours steps at 1, 2, 5, and 10 times the background are labeled. opposite to the beha vior of the f aintest objects. This w ould suggest that man y of the IR e xcess sources are correlated to and lik ely embedded within the molecular cloud, and may be associated with the most recent star formation in the IC 348 re gion. Lastly in gure 4–6 we construct a surf ace density plot of the m K15 stars used in the radial prole (Figure 4–4 ) and in Figure 4–5 b W e use a Nyquist sampled box lter that had an area roughly equi v alent to the area of the annuli used in the radial prole. W e use this relati v e lar ge box to match the noise in the radial prole as a function of constant area and at this resolution, only a smooth N-S elong ated cluster is seen with no signicant sub-clusterings. W e conrmed, ho we v er that if we

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119 used the same spatial resolution as the Lada & Lada study (90 ), we reco v er most of the sub-clusters identied by LL95 and which no w lie in the cluster' s halo. W e note, ho we v er that neither map displays an y apparent correlation of the surf ace density to the location of the cluster -cloud interf ace re gion. 4.1.4 Cluster Reddening Pr operties W e also used the sources observ ed within the IC 348 re gion as line of sight probes of the parental molecular cloud, allo wing us to characterize the reddening that w ould be seen by cluster members or background eld-stars. Building upon the method of Alv es et al. ( 1998 ) and our recipe(s) described in Chapter 3 we calculated indi vidual e xtinction estimates for each source by de-reddening the sources' infrared colors to a locus of assumed intrinsic colors in the (H-K) vs (J-H) color -color diagram. W e then e xamined reddening maps created from these indi vidual e xtinctions and binned the A V estimates into an e xtinction probability distrib ution functions (EPDFs) which we will use when estimating the number of interloping eld-stars and when calculating luminosity function models to interpret the observ ations. Extinction estimates and maps. W e calculated indi vidual A V for sources in tw o dif ferent luminosity ranges, di viding them into sets that are lik ely dominated by either cluster members or background eld-stars. These samples were selected from the (H-K) vs K color -magnitude diagram and were separated into “bright” and “f aint” sources by the reddening v ector of a source at m K15 (H-K = 0.35). The “f aint” sample w as also limited to objects brighter than a reddening v ector for a source at m K17 (H-K = 0.4). The tw o magnitude samples were de-reddened back to dif ferent loci of assumed intrinsic colors. The “bright” sources were assumed to be young PMS stars, and those with JHK colors (95%) were de-reddened back to the classical T -T auri Star (cTTS) locus in the JHK color -color diagram ( Me yer et al. 1997 slope = 0.58; J-H intercept = 0.52), while the remaining 5% were assigned an intrinsic H-K color =

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120 Figure 4–7: Extinction maps of the IC 348 FLAMINGOS re gion. Indi vidual source A V estimates deri v ed in Section 4.1.4 are con v erted to an e xtinction map using a Nyquist sampled box lter (width200 ; see upper left; same as pre vious gure). Contours are in steps of A V1 from A V1 to 20; we label the A V2510 contours. Left hand panel: A V map deri v ed from the bright, lik ely cluster stars; Right hand panel: A V map of the e xtinction seen by the f ainter lik ely background stars. See te xt for sample selection and e xplanation of reddening estimates. 0.5 and A V estimates deri v ed. The “f aint” sources were assumed to be dominated by eld M dw arfs and those with JHK colors (85%) were de-reddened back to a linear approximation of the M dw arf branch in the color -color diagram (from K6 to M9, slope = 0.16; J-H intercept = 0.61) 1 while the remaining 15% lacking J band were assigned an intrinsic H-K = 0.16. In Figure 4–7 we present the resulting e xtinction maps deri v ed from these tw o samples. Both maps clearly dene the location of the cluster -cloud interf ace along the re gion' s southern border while the y also outline a NE-SW band of reddening that f alls across the cluster' s core, similar to the cluster' s N-S elong ation. One straightforw ard conclusion from the spatial v ariations in either reddening map is that the IC 348 re gion 1 These cTTS and M dw arf approximations result in nearly all eld giants being de-reddened to the colors of the K1 K2III spectral class.

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121 cannot be characterized by a single mean A V v alue. Although the tw o A V maps are ph ysically v ery similar the reddening seen by the background stars is in general lar ger than that fore ground to the cluster members, suggesting, for e xample, that there is considerably more material behind the cluster core then in front of it. Extinction pr obability distrib ution functions (EPDFs). In Figure 4–8 we plot the normalized histograms of the (H-K) color and of A V for sources in the tw o magnitude ranges dened in the pre vious section and separate them further by cluster sub-re gion. As e xpected from the A V maps, the f aint sources' color and A V distrib utions are broader and redder than the bright lik ely cluster stars, although the EPDFs of the lik ely cluster members in both re gions appear some what similar Applying a tw o-sided K olmogoro v-Smirno v test to the A V v alues deri v ed from bright and f aint sources in the cluster core, we found that it is unlik ely that the y are dra wn from the same A V distrib ution, ha ving a KS probability of only 0.00024. This is in contrast to the halo sub-re gion, where the bright and f aint stars ha v e a 0.23 probability of being dra wn from the same A V distrib ution. Similarly the bright stars in the core and halo ha v e a 0.043 probability of being dra wn from the same distrib ution while the f aint stars in both re gions cannot, ruled out at the 20000108 probability T ak en together the EPDFs and these statistical tests support tw o basic conclusions about the reddening seen to w ards IC 348: 1) there is a measurable dif ference in the reddening seen by the background stars between the core and halo sub-re gions, o wing to the substantial material appearing behind the cluster core (also see Figure 4–8 b); 2) the bright, lik ely cluster members of the core and halo appear to ha v e f airly similar reddenings, despite the projection of the entire re gion onto v arious pieces of the Perseus GMC. The normalized A V histograms (hereafter kno wn as EPDFs) are generally sk e wed to A V5 and are quite non-g aussian, ag ain indicating that a singleA Vv alue is inappropriate to describe the cluster reddening. On a v erage we nd reddenings to the

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122 Figure 4–8: Distrib utions of reddening for IC 348. P anels (A) and (B): the fractional distrib ution of H-K color di vided by sub-re gion. P anels (C) and (D): the probability distrib utions of A V deri v ed from de-reddening sources in the (H-K)/(J-H) color -color diagram. In all panels, the distrib utions are di vided into the results from magnitude limited “bright” and ”f aint” samples. See te xt for sample selection and e xplanation of de-reddening. background stars ofA Vcor e721 s63a vg. de viation = 4.6, median = 5.3) andA Vhal o461 s43a vg. de viation = 3.0, median = 3.4), while to the lik ely cluster members these a v erages areA Vcor e491 s42a vg. de viation = 2.9, median = 3.8) andA Vhal o421 s40a vg. de viation = 2.6, median = 3.3). These latter a v erages are roughly the same as the A K05 assumed by LL95 for the entire IC 348 re gion and their sk e w to lo wer A V is e vident in the better agreement of their medians to the medianA V 28 deri v ed by Herbig using spectral types for cluster members. Since our observ ations probe a v olume of the cluster re gion much lar ger than these prior studies, we will use these EPDFs when correcting for

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123 background eld-stars in Section 4.2.2 and in our modeling of the luminosity function in Section 4.3.4 4.2 Infrar ed Luminosity Functions of IC 348 4.2.1 Constructing Infrar ed Luminosity Functions In Figure 4–9 we construct the ra w infrared luminosity functions (LFs) for sources in the FLAMINGOS IC 348 re gion. W e restrict the sources to the area bounded by the 1033radius and use (0.5 mag) bins wider than our photometric uncertainties. W e compare the J, H and K band IC 348 LFs and nd that all of them display a double peak ed structure: the rst peak lying at J=H=K=13135 and the second between J = 17.5 and K = 16.5. F ollo wing our analysis of the cluster' s color -magnitude diagrams in Section 4.1.2 we interpret the brighter of these peaks to be sources in the IC 348 cluster and the f ainter peak to be dominated by background eld-stars and g alaxies. In Figure 4–10 we display the K band LFs of the IC 348 “core” and the “halo” sub-re gions, scaling them to stars per square de gree and comparing them to the unreddened eld-star KLF constructed from our of f-cluster datasets (see also Section 4.2.2 ). Conrming our preliminary study of the cluster' s color -magnitude diagrams and radial prole, both sub-re gions display a considerable e xcess of sources relati v e to the eld-star KLF for m K15. Both sub-re gions KLFs reach bright peaks although the y occur in some what dif ferent locations with the core KLF for e xample, peaking 1.5 magnitudes brighter than the halo. Belo w these bright peaks, both KLFs atten or turno v er before rising ag ain, parallel to the eld-star KLF While the f aint KLF peak of the tw o sub-re gions ha v e nearly identical size and structure, the y appear to be smaller and shifted to f ainter magnitudes then the eld-star KLF Such dif ferences could certainly be caused by the reddening and obscuration of background eld-stars due to the molecular cloud. In the ne xt section we use our detailed study of the cluster reddening properties from Section 4.1.4 to estimate the size of the eld-star contrib ution to the observ ed KLFs.

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124 Figure 4–9: Ra w infrared luminosity functions for IC 348. W e compare the J, H and K band LFs for sources in the entire IC 348 Cluster FLAMINGOS re gion. Note that all three ha v e similar tw o peak ed structure. 4.2.2 Field-Star Corr ection to the Cluster KLF(s) T o statistically estimate the eld-star contrib ution to the ra w IC 348 KLF(s), we rst scaled the observ ed eld-star KLF to the area of the cluster and then con v olv ed it with a reddening probability distrib ution function that characterizes the ef fects of the molecular cloud in that re gion. W e used the e xtinction probability distrib ution functions deri v ed in section 4.1.4 from the f aint stars de-reddened to the M dw arf locus and treated the cluster' s core and halo re gions separately In Figures 4–11 ab we compare the ra w KLFs to the reddened eld-star KLFs appropriate to that sub-re gion. The reddened eld-star KLFs v ery closely match the ra w KLFs for m K15 in both sub-re gions, although the y e xceed the cluster KLFs at the f aintest magnitudes

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125 Figure 4–10: K-band luminosity functions by sub-re gion for IC 348. The KLFs are scaled to stars per square de gree and are compared to the observ ed eld star KLF deri v ed by combining the tw o of f-cluster elds. Both the cluster core and halo sub-re gions appear to dominate the un-reddened of f-eld counts for m K15. Error bars are 1 s counting statistics. because the y were not ltered to match our detection limits. W e subtracted these reddened eld-star KLFs from the ra w cluster KLFs, and display the resulting differential KLFs in Figures 4–11 cd, constructing error bars that are the 1 s counting statistics of the sum of the ra w and eld star KLFs. The structure of the dif ferential KLFs is signicant for the bins containing m K1675, belo w which the eld-star correction clearly o v er -estimates the observ ations. This magnitude limit corresponds to a 10 M J u p bro wn dw arf at the 2 Myr mean age we assume for IC 348. The tw o subre gions ha v e a nearly identical number of members, with the core containing 15316 sources with m K15 (17224; m K17) and the halo containing 15023 sources

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126 with m K15 (17640; m K17). While the sub-re gion KLFs are well populated for m K15, at f ainter magnitudes the substantial eld-star corrections yield v ery lar ge uncertainties in the KLF structure. Figure 4–11: Field star correction to cluster KLFs in IC 348 P anels (A) and (B): comparison of the ra w sub-re gion KLFs to the observ ed eld star KLF reddened by the f aint star A V distrib utions deri v ed in Section 4.1.4 and displayed in Figure 4–8 The reddening eld star KLF(s) were scaled to the area of the cluster sub-re gion and subtracted to yield the dif ferential KLFs displayed in panels (C) and (D). The dif ferential KLFs are signicant for m K165, belo w which we o v er -estimate the eld star correction. Error bars are the square root of the sum of the observ ed counts and the predicted eld star counts for each bin. In Figure 4–12 we directly compare the sub-re gion dif ferential KLFs and sum them to construct the o v erall cluster KLF Ag ain, the dif ferences in the structure of the tw o sub-re gion KLFs are ob vious to the e ye, with the peak of the halo KLF sk e wing

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127 signicantly to f ainter magnitudes, before strongly turning o v er While the peaks of the sub-re gion KLFs are clearly distinct, the KLFs are similar to within their 1 s error bars in most bins b ut separated by 2 to 3 s in the m K11 and m K13 bins. A tw o sample chi-square test of the tw o histograms (range: m K815) yields a probability of 0.04 that the y are dra wn from the same parent distrib ution, indicating that these sub-re gion KLFs are dif ferent at the 2 s le v el. Figure 4–12: Dif ferential KLFs for IC 348. A) Comparison of the dif ferential IC 348 KLFs for the tw o cluster sub-re gions. B) Sum of sub-re gion dif ferential KLFs into the composite IC 348 KLF P anel B compares the sum of the sub-re gion dif ferential KLFs from tw o dif ferent background corrections. The “Red” correction is the eld star KLF reddened by the EPDF for the background stars; the “Blue” correction is the eld star KLF reddened by the EPDF for the bright cluster stars. When the tw o sub-re gion KLFs are summed together the complete cluster KLF displays a v ery broad peak between m K115 and 13 before decreasing v ery sharply to m K15. In the complete cluster KLF we nd that despite the lar ge uncertainties at f aint magnitudes the tw o sub-re gion KLFs sum to yield a statistically signicant number of cluster members with m K15. Further we nd that if we increase the size of the eld-star correction by using a bluer e xtinction distrib ution function (one

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128 rele v ant for the cluster stars), the main cluster KLF characteristics are not substantially altered, although the size of the v ery f aint population is almost halv ed. 4.3 Initial Mass Function of IC 348 T o analyze the IC 348 dif ferential K band luminosity function(s) constructed in Section 4.2 we used our model luminosity function algorithm presented in Chapter 2 and e xpanded in chapter 3 Our goal is to place constraints on the initial mass function of IC 348 by deri ving that mass function or set of mass functions whose model luminosity functions best t the cluster KLF F or the purpose of comparing of our w ork to other studies, we indi vidually analyzed the KLFs of the tw o cluster sub-re gions as well as the composite cluster KLF Since a young cluster' s luminosity function is the product of an age dependent mass-luminosity relation and the cluster' s IMF we detail the star -forming history of IC 348 in Section 4.3.1 and the appropriate theoretical mass-luminosity relations in section 4.3.2 before xing these quantities and deri ving the cluster IMF in Section 4.3.4 4.3.1 Star F orming History of IC 348 T o deri v e a mean age and age spread appropriate for our luminosity function modeling, we e xamine published studies of the apparent star -forming history of IC 348. In Figure 4–13 we plot a histogram of the ages deri v ed by Herbig ( 1998 ) using the dereddened VV -I color -magnitude diagram for candidate pre-main sequence members in an 112 square arc-minute re gion of IC 348. W e mer ge the ages deri v ed by Herbig for sources with and without detectable H a emission and deri v e an ensemble cluster mean age of2 Myr W e approximate the age spread of IC 348 to be3 Myr corresponding to constant star formation from 0.5 to 3.5 Myr ago, and note that this age spread is 2.5 times longer than that we used for the younger T rapezium Cluster W e sho w in Figure 4–13 that our assumed star -forming history closely approximates the b ulk of the Herbig SFH, b ut clips the “older” tail of this distrib ution. W e do not include this “older” tail in our SFH of IC 348 for the follo wing reasons: 1)

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129 Observ ational errors in the deri v ation of de-reddened color -magnitude or theoretical HR diagrams can lead to e xaggerated star -forming histories, specically resulting in articially inated cluster' s age spreads ( Hartmann 2001 ; K en yon & Hartmann 1990 ); 2) A number of the “oldest” IC 348 objects analyzed in the Najita et al. ( 2000 ) Hubble Space T elescope NICMOS study were also found to be the reddest cluster objects, suggesting that the y are background interlopers; 3) Our assumed age spread also v ery closely approximates that SFH deri v ed by P alla & Stahler ( 2000 ), who found that the star formation in IC 348 be g an approximately 3 Myr ago. Further P alla & Stahler found ne gligible star formation at ages greater than 3 Myr and also found no dependence of the SFH on location within IC 348. Thus, we will use the same star formation history when modeling both the ensemble cluster and the cluster' s “core” and “halo” sub-re gion KLFs. 4.3.2 Cluster Distance and the Mass-Luminosity Relation Herbig ( 1998 ) included an e xtensi v e discussion on the distance to IC 348 based upon literature sources e xisting at that time. Ar guing that closer distances (260 pc) were systematic under -estimates, he chose a distance for IC 348 (of 320 pc) based in part upon the f act that within the current uncertainties, one could not dif ferentiate between the distance to IC 348 (31622 pc Strom et al. 1974 ) and to the Perseus OB2 association Bor gman & Blaauw (32230 pc 1964 ). T o understand ho w such distance uncertainties might af fect our results, we e xamined more recent Hipparcosbased proper motion and parallax studies of the cluster and association and then studied the impact of this systematic on the rele v ant mass-luminosity relation. The recent use of Hipparcos data, ho we v er does not appear to ha v e resolv ed the distance uncertainty between the Perseus OB2 association and the IC 348 cluster de Zeeuw et al. ( 1999 ) deri v ed a distance of 31727 pc for 17 members of the Perseus OB2 association spread o v er a projected 3737 pc area, while also statistically estimating the number of interlopers. On the other hand, when Scholz et al.

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130 Figure 4–13: Star -forming history of IC 348. The star formation history of IC 348 from Herbig ( 1998 ) is compared to that used in our model luminosity function algorithm. The Herbig SFH is the mer ger of the ages deri v ed for stars with and without detectable H a emission using the de-reddened optical color -magnitude diagram. The SFH assumed for our models has a mean age, t20 Myr with constant star formation from 0.5 to 3.5 Myr ago. ( 1999 ) performed a recent proper motion study of the IC 348 re gion, the y deri v ed a distance of261 pc using a dif ferent set of 9 Hipparcos sources. This latter distance estimate to IC 348 should probably be treated with some caution. Since more than half of the 9 sources with parallax es were at projected distances of 2.5 8 pc from the cluster center the y f all well outside an y cluster outer radius we ha v e discussed here and may not be actual members, especially since no statistical estimate of the number of non-members w as performed for this sample. Further Ripepi et al. ( 2002 ) v ery recently reported the disco v ery of an F star within the IC 348 boundaries that displays

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131 Figure 4–14: Theoretical mass-luminosity relations of IC 348. Mass-K magnitude relations for dif ferent ages, distances and set of theoretical e v olutionary models are compared to illustrate the sensiti vity of our method to assumptions about cluster distance and age. Mass-K magnitude relations from D'Antona & Mazzitelli ( 1997 ) are sho wn at t2,3 and 10 Myr for a distance of 320 pc and at t3 Myr for a distance of 260 pc. Mass-K magnitude relations from the Baraf fe et al. ( 1998 ) tracks are also compared at 2 and 10 Myr F or all of these comparisons, the PMS tracks were con v erted to observ ables using a single set of bolometric corrections. rapid d Scuti-lik e v ariability that is interpreted as the pulsation of a PMS star while in its instability strip ( Marconi & P alla 1998 ). The deri v ed pulsation period strongly f a v ors a lar ger distance to IC 348 of320 pc. Thus, it remains unclear if the distance to IC 348 can be separated from the distance to the OB association. F or our modeling we adopt the distance of 320 pc (m-M = 7.5) to IC 348 for consistenc y with the w ork of LL95 and Herbig ( 1998 ) b ut we e xamined ho w such a distance uncertainty could af fect our results.

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132 Such distance uncertainties may translate, for e xample, into systematics in the deri v ation of cluster properties such as mean age and possibly the IMF F or e xample, Herbig sho wed that assigning IC 348 a closer distance yielded a systematically older cluster mean age since the cluster appears intrinsically f ainter when compared to pre-main sequence e v olutionary models. In Figure 4–14 we e xamine the net ef fect of this age-distance uncertainty on the theoretical mass-luminosity relations rele v ant for our luminosity function modeling. Shifting the distance from 320 to 260 pc produces a shift in the mean age from 2 to34 Myr (see also Haisch et al. 2001a ). Ho we v er the change in the distance modulus (042 magnitudes) is roughly equi v alent to the a v erage luminosity e v olution of stars (1.0 0.1 M) between 2 and 3 Myr ( d K 035). By comparing the mass-K magnitude relation at 2 Myr and 320pc to that at 3 Myr and 260 pc, we nd that abo v e the h ydrogen b urning limit, our deri v ed IMF will ha v e little systematic dependence upon the cited age-distance uncertainty for IC 348 and should be a f aithful representation of the true cluster IMF On the other hand, the slope of the substellar mass-K magnitude relation is systematically af fected, in part due to a lar ger mass range under going deuterium-b urning, and our deri v ed substellar IMF will be less reliable until this age-distance uncertainty is resolv ed. Lastly in Figure 4–14 we compare the theoretical mass-luminosity relations tak en from tw o sets of e v olutionary calculations. As we found in chapter 3 the current theoretical mass-K magnitude relations are v ery consistent between current sets of PMS tracks, meaning that the deri v ed IMF will be mostly independent of which set of modern PMS models we use, although future updates to the input ph ysics may change these conclusions. T o pro vide consistenc y between our studies of v arious young clusters, we will deri v e the cluster IMF using our standard set of PMS tracks, which are based on the D'Antona & Mazzitelli ( 1997 ) tracks and fully described in Section 2.2.3

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133 4.3.3 Other Modeling P arameters: Reddening and Binaries Three additional characteristics of young stars that we can include into our model luminosity function algorithm are the reddening of the cluster members by the parental molecular cloud, e xcess infrared ux due to optically thick disks around the cluster members, and the frequenc y of un-resolv ed companions. First, to account for the reddening of the cluster by the Perseus Molecular Cloud, we used the e xtinction distrib ution functions deri v ed in Section 4.1.4 for the bright cluster stars de-reddened to the cTTS locus. Second, in Section 4.1.2 we conrmed the LL95 nding of60 infrared e xcess sources distrib uted across the IC 348 cluster re gion. Correcting for eld stars and di viding the e xcess sources by sub-re gion, we nd infrared e xcess fractions of 1411%22153for the core sub-re gion, 2417%36150for the cluster halo and 1910%58303for the composite cluster If we restrict these estimates to those 42 sources which ha v e IR e xcess greater than their 1 s photometric errors, these percentages drop to 9%14% and 11%, respecti v ely Because these e xcess fractions are v ery small, we chose to not account for IR e xcess in our KLF models of IC 348. It is possible, ho we v er that a more substantial fraction of sources could ha v e small K band e xcesses ( d K ir x01) that are not apparent in the color -color diagram especially since Haisch et al. ( 2001a ) found that 65% of the IC 348 members ha v e inner circumstellar disks as traced at 3 m W e ar gue such small e xcesses will not substantially af fect the cluster KLF and hence the deri v ed cluster IMF since this ux e xcess is much smaller than the bins used to create the cluster luminosity function. The last issue is if we should include the ef fects of un-resolv ed binaries, which in our models are instituted using a binary fraction, f bin and assuming random pairing from a single parental IMF T o parallel our analysis of the T rapezium Cluster in Chapter 3 ho we v er we chose to not include un-resolv ed binaries into our modeling routine. Thus the IC 348 IMF we deri v e is the “primary” or “single” star IMF

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134 Duch ˆ ene et al. ( 1999 ) found that the binary fraction in IC 348 is v ery similar to that found in other young clusters and to the eld. Further the consistenc y of the binary frequenc y in stellar clusters o v er a lar ge range of cluster age ( P atience et al. 2002 ), including young clusters lik e IC 348, suggests that the single star IMFs of star clusters can be readily deri v ed and compared without correction for binaries. A related issue in our studies of dif ferent cluster IMFs is whether or not a signicant fraction of r esolved wide binaries ha v e been included into the cluster KLFs. F or e xample, the ph ysical resolution of our T rapezium study w as240 au, while in IC 348 it is480 au. Since the binary fraction at these lar ge separations is quite small ( f01), the fe w resolv ed binary systems that will be included into these cluster KLFs should not signicantly modify the single star IMFs we deri v e. 4.3.4 Modeling the IC 348 Differ ential KLF(s) Using these cluster parameters, we produced a suite of model luminosity functions by v arying the underlying initial mass functions and then t these models to the observ ed cluster KLF(s). Our standard IMF parameterization consists of po wer -la w se gments, G i connected a break masses, m j and for our analysis of IC 348 we used 2 and 3 se gment IMFs. W e independently calculated and t model luminosity functions for the cluster' s “core” and “halo” sub-re gion KLFs and for the composite cluster KLF v arying the e xtinction distrib ution function appropriately for each re gion. Our tting technique calculates the c 2 statistic and probability between the model KLFs and observ ed KLF o v er a range of magnitude bins and from these statistical measures, the parameters (mean, standard de viation) of the underlying IMF are deri v ed. W e summarize these ts and the resulting IMFs for the tw o sub-re gions in section 4.3.4 and for the o v erall cluster in Section 4.3.4 The IC 348 cor e and halo sub-r egion KLFs. In the left hand panels of Figure 4–15 we display the sub-re gion dif ferential KLFs compared to the best t model KLFs deri v ed from our c 2 tting technique. The underlying po wer -la w IMFs are displayed

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135 Figure 4–15: Modeling the IC 348 KLF: cluster sub-re gions. Left hand panels display model KLFs best t to the dif ferential IC 348 KLFs of each sub-re gion (upper: core; lo wer: halo). The best t model KLFs are normalized to the observ ations o v er a range from m K814, corresponding to a mass range from 25 to 004 M. Right hand panels display the underlying tw o and three po wer -la w IMFs corresponding to the specic model KLFs (by symbol). Deri v ed IMF parameters are listed in table 4–3 in the right hand panels of this gure and we list our deri v ed IMF parameters in table 4–3 W e found that model KLFs constructed using 2 po wer -la w IMFs pro vided satisf actory ts to both sub-re gion KLFs, while we were able to obtain slightly better ts to the “halo” KLF using a 3 se gment IMF Further by v arying the range of KLF bins t by our c 2 routine, we are able to deri v e good KLF ts o v er the luminosity range from m K814, corresponding to a mass range from 25 to 004 M 40 M J u p ). Fits do wn to the m K15 bin are also moderately constrained in the core

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136 sub-re gion, corresponding to a lo w mass limit of25 M J u p Owing to the lar ge statistical uncertainties at the f aintest magnitudes due to the eld star contamination, we could not deri v e reasonable ts to the dif ferential KLF(s) for m K15. F or both sub-re gions, the m 1 mass break is v ery strongly constrained by the location of the bright KLF peaks in the 2 po wer -la w IMF ts. The G 1 IMF slope rises steeply with decreasing mass in the “'core” sub-re gion, b ut is much shallo wer in the “halo” re gion. In our 3 po wer -la w ts to the “halo” sub-re gion, we found a modest c 2 minima that indicated there is an inection in this high-mass slope near 080 M, yielding a G 1 similar to that found in the 2 se gment “core” IMF As pointed out before, the tw o sub-re gions ha v e distinctly dif ferent IMF peaks, with the core m 1 mass break occurring at 056018 Mand the peak of the halo IMF strongly constrained to lie at 010002 M(see table 4–3 Belo w their peaks, both sub-re gion IMFs steadily f all with decreasing mass do wn to our m KM t limit. The G 2 slope is more tightly constrained for the cluster core IMF than in the halo, where the rapid change in KLF slope between m K12 and 15 yields considerable uncertainty in this IMF parameter Despite these limitations of our tting procedure, we nd that the resulting substellar IMF slopes are v ery dif ferent in the tw o sub-re gions, although as we discuss in Section 4.4.2 the y yield similar fractions of bro wn dw arfs when inte grated o v er the entire mass range. The composite IC 348 KLF W e e xamine the best t model KLFs to the complete cluster KLF in the left hand panel of Figure 4–16 Unlik e the simpler structure of the tw o sub-re gion KLFs, the composite cluster KLF displays a v ery broad main peak follo wed by a sharp turno v er at f ainter magnitudes. This structure required the use of model KLFs based upon a 3 po wer -la w underlying IMF and we tab ulate the deri v ed IMF parameters as function of the t range in table 4–3 W ith these 3 se gment underlying IMFs we were able to t the composite cluster KLF do wn to m K15, constraining the IMF from 25 Mdo wn to25 M J u p

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137T able 4–3. IC 348 po wer -la w IMFs deri v ed from model KLFs Cluster NGmKMM bc2IMF P arameterscRe gionaLimit Limit Prob G11 s m11 s G21 s m21 s G31 s core 2 140 0040 093143 038 056 018043 025 core 2 150 0025 083137 035 047 013059 017 core 2 165001 037147 038 061 019032 019 halo 2 140 0040 076072 014 010 002198 046 halo 2 150 0025 041075 016 010 001225 026 halo 2 165001001 halo 3 140 0040 081123 039 083 028055 014 0093 001189 045 halo 3 150 0025 040125 039 082 029053 014 0092 001220 027 halo 3 165001001 cluster 3 140 0040 070149 030 079 025019 018 0089 002175 053 cluster 3 150 0025 063153 028 083 023023 014 0086 001228 022 cluster 3 165001001 T rap 3 0025 099121 018 060 016015 017 0120 004073 020 aNumber of po wer -la ws, Gi, used in the underlying IMF of the model KLFs.bCon v ersion of the mKl imi tto MMusing the 2e6 Myr DM97 isochrone. F or mKl imi t165, we con v erted to mass using the Burro ws et al. ( 1997 ) and Baraf fe et al. ( 2002 ) e v olutionary models.cUnits of IMF parametes: Giare slopes for an IMF dened as the number of stars per unit logMM ; mjare mass breaks gi v en in units of linear solar mass ( MM). Note. — c2probability calculated for best t model KLF o v er the range from mK8mKl imi t. A v erage IMF parameters calculated within the 0.35 condence contour

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138 Figure 4–16: Modeling the IC 348 KLF: the composite cluster The left hand panel compares the best t model KLFs to the composite dif ferential IC 348 KLF The right hand panel compares the IMFs of these model KLFs to the IMF deri v ed for the T rapezium cluster from Chapter 3 although the T rapezium is only the R03 pc core of the lar ger Orion Neb ula Cluster ( Hillenbrand 1997 ). Model KLFs and IMFs for IC 348 ha v e corresponding symbols. Ov er this mass range, our KLF ts yield an IC 348 IMF ha ving a broad peak do wn to the h ydrogen b urning limit before rolling o v er and decreasing sharply into the substellar re gime. The high-mass G 1 slope is moderately constrained with an inde x of 1503Salpeter 135before attening at m 1080 M. The G 2 slope and the m 2 mass break are both strongly constrained by the broad KLF peak and the sharp KLF turno v er at m K13, with a v ery slo wly rising G 2 02015 across much of the IMF before peaking at the h ydrogen b urning limit. Similar to the situation for the “halo” KLF/IMF the G 3 slope is v ery steeply f alling b ut poorly constrained, with an inde x of 2004. 4.4 Discussion 4.4.1 The KLFs and IMFs of IC 348 and the T rapezium The stellar r egime. W e e xamine the structure of the deri v ed IC 348 KLF and IMF by comparing these cluster characteristics to those we deri v ed for the T rapezium

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139 Figure 4–17: Comparison of IC 348 and T rapezium KLFs. P anel (A) compares the K band LFs of IC 348 and the T rapezium, shifted to absolute magnitudes. No reddening corrections ha v e been included b ut the T rapezium KLF has been scaled to contain the same number of stars as IC 348. P anel (B) compares the IC 348 dif ferential KLF to tw o model KLFs representing the T rapezium e v olv ed to the age of IC 348. The models use the star -forming history and reddening for IC 348 b ut substitute the T rapezium IMF deri v ed in Chapter 3 and use tw o dif ferent sets of PMS tracks ( D'Antona & Mazzitelli 1997 ; Burro ws et al. 1997 DM97 and Bur97). This illustrates the predicted location and size of the secondary KLF peak of the T rapezium were this cluster the age of IC 348. cluster in Chapter 3 By comparing the T rapezium IMF (also reproduced in table 4–3 ) to the deri v ed IC 348 IMF we nd that these tw o v ery young clusters ha v e nearly identical IMFs throughout the stellar re gime. As illustrated in Figure 4–16 both clusters ha v e IMFs that rise in number with decreasing mass into the subsolar re gime, with Salpeter lik e po wer -la w slopes ( G 1T r a p 121 and G 1IC 348 153). Their IMFs both atten around 07 M, ha ving slopes of G 2 02 and forming v ery broad shallo w peaks at subsolar masses. The “peak” or mode of their IMFs v aries between 015 and 008 Mwith the IC 348 IMF sk e wing to slightly lo wer masses than the T rapezium. The strong similarities between these clusters' stellar IMFs e xist despite the signicant apparent dif ferences between their cluster KLFs. In Figure 4–17 a, we

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140 compare the T rapezium and IC 348 KLFs, shifting them by their respecti v e distance moduli to absolute magnitudes and scaling the T rapezium population to match that of IC 348. The IC 348 KLF is clearly broader and shifted to f ainter magnitudes relati v e to the T rapezium KLF with their primary KLF peaks dif fering by almost 2 magnitudes. These dif ferences, ho we v er are precisely those predicted by the e v olution of the luminosity function with age ( Lada & Lada 1995 ; Muench et al. 2000 ). Indeed if we e v olv e a model KLF of the younger T rapezium to the age of IC 348 and compare it to the IC 348 KLF in Figure 4–17 b, we nd it agrees with the observ ed IC 348 KLF e xtremely well do wn to m K125, near the unreddened h ydrogen b urning limit for IC 348 ( t2 Myr m KH BL127) 2 F ainter than this magnitude the e v olv ed T rapezium KLF moderately under -estimates the IC 348 KLF between m K13 and 135, indicati v e of the slight sk e wing of the IC 348 IMF mode to lo wer masses. Belo w m K13, both the observ ed IC 348 KLF and the e v olv ed T rapezium KLF steeply f all in number marking the transition to the substellar re gime that we e xplore in the ne xt section. The substellar r egime. Re vie wing Figure(s) 4–16 b and 4–17 one immediate similarity between the substellar KLFs and IMFs of IC 348 and the T rapezium is their mutual steep decline to w ards f ainter magnitudes and lo wer masses. Although the IC 348 IMF mode sk e ws to lo wer masses than the T rapezium, it turns o v er and decreases in a much steeper manner than the T rapezium, i.e., G BDsIC 34820 while G BDsT r a p07. This is also illustrated by the w ay that the e v olv ed T rapezium KLF 2 This quantity is f airly independent of current PMS tracks, see Figure 4–14 From v arious models we nd for IC 348 that m KH BL ( D'Antona & Mazzitelli 1994 )1274, m KH BL ( DM97 )1267, m KH BL ( Burro ws et al. 1997 )1255, and m KH BL ( Baraf fe et al. 1998 )1283. F or a mean age of 3 Myr at a distance of 260 pc,m KH BL 1268.

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141 seems to consistently o v er -estimate the number f aint objects in IC 348, although the y do agree within the lar ge statistical uncertainties. A second similarity between the substellar KLFs of IC 348 and the T rapezium, is the formation of a modest b ut statistically signicant secondary KLF peak, which in IC 348 contains 4229 sources between m K15 and 17. Further the secondary KLF peak for IC 348 occurs precisely in the magnitude range predicted by the e v olv ed T rapezium KLF suggesting the y are related features corresponding to sources in the mass range from 1020 M J u p As w as the case for our T rapezium modeling, such KLF structure rejected our tting of these magnitude bins using models based upon 3 se gment po wer -la w IMFs. This is because we are trying to t this non-po wer la w KLF structure (a dip or g ap follo wed by a secondary peak) with model KLFs that are essentially a po wer -la w throughout the bro wn dw arf re gime because the theoretical mass-luminosity relations that we ha v e used are smooth and do not contain an y signicant e v olutionary features in this mass range. Considering the statistical uncertainties due to the background correction we did not attempt to e xplicitly t additional IMF se gments to the IC 348 peak although the size of the predicted peak using the T rapezium IMF closely approximates this feature. The similarity of the predicted size of the secondary peak in IC 348 relati v e to the T rapezium and the similar mass range represented by this feature suggests that the y are intrinsically related and may be e vidence conrming that such structure e xists in the cluster KLF at these ages. W e discuss whether this similarity supports our h ypothesis of a secondary IMF peak near the deuterium-b urning limit or suggests a dif ferent origin of this secondary KLF peak in Chapter 7 One meaningful constraint re g ardless of detailed IMF structure is the fr action of cluster members that are substellar F or e xample, in the T rapezium we found that 2242 % of the clusters members fell between 80 and 17 M J u p ha ving the interesting implication that only1 in 4 of the cluster members were bro wn dw arfs! Estimating

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142 this fraction for IC 348 is more dif cult because the background correction mak es directly counting all the substellar sources dif cult. Rather than directly counting sources, we rst inte grated the deri v ed IMF to calculate the fraction of substellar sources do wn to the mass limit of our t. W e nd that bro wn dw arfs between 80 and 25 M J u p (the lo wer mass limit corresponding to our model ts to m K15 and the be ginning of the secondary KLF peak) constitute only 14% of the members with 2% uncertainty due to the v ariation in our ts, and 9% uncertainty due to the counting statistics as dominated by the background correction. F or comparison, inte grating the T rapezium IMF o v er the same mass range yields a bro wn dw arf fraction of 20%. When we count the number of sources contained in the secondary peak of the IC 348 KLF and not included into the KLF ts, we are able to e xtend the range of substellar masses do wn to 10 M J u p ho we v er we also nd that the precision of our estimate is signicantly w orsened by the lar ge eld star contamination. Since this secondary KLF peak contains between 26 and 42 sources depending upon the size of the correction for eld stars, the total substellar fraction increases to 2025% for IC 348, although the corresponding error bar also increases to14%. Thus despite the uncertainties due to the eld star correction, we nd that these fractions indicate that IC 348 has an ywhere from a moderate dearth to an similar bro wn dw arf fraction as that found the T rapezium and further that bro wn dw arfs are not nearly as populous as stars in either cluster 4.4.2 Radial V ariation of the IC 348 IMF Our di vision of the cluster into tw o sub-re gions based on the cluster' s radial prole (see Section 4.1.3 ) allo ws us to mak e important comparisons to past studies of the IC 348 IMF which ha v e primarily focused on the cluster' s core. F or e xample, when we compare the IMF deri v ed by Najita et al. ( 2000 ) for around 100 members of the IC 348a sub-cluster to that we deri v e for the cluster core in Figure 4–18 we nd the y agree remarkably well o v er the mass range from 0.25 to 0.025 M, although the

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143 NICMOS IMF is atter at higher masses lik ely due to source saturation or possibly statistical uctuations since the y are not precisely the same samples. Such a strong similarity is not found, ho we v er when we compare the IMF of the cluster' s core to that we deri v ed for the cluster halo. P aralleling the apparent ph ysical dif ferences seen in the KLFs of the IC 348 sub-re gions (see gures 4–10 and 4–12 ), we nd that the IMF of the halo sk e ws to lo wer masses relati v e to the IMF of the cluster' s core. This v ariation of the IMF is a real representation of the dif ferences in the tw o sub-re gions and is not the product of a v ariation in some other ph ysical quantity Reddening due to the molecular cloud and specic to each sub-re gion is included into these ts, although no meaningful dif ferences e xist between the rele v ant EPDFs of the tw o re gions (see Section 8 ). Although we did not include the ef fects of infrared e xcess, an y such ef fect w ould actually incr ease the IMF dif ferences, since the lar ger e xcess fraction of the halo and w ould require the halo IMF peak to shift to lo wer masses. Further the lar ger e xcess fraction in the halo might imply that the halo is younger than the core. Ho we v er accounting for such an age dif ference w ould ag ain shift the IMF peaks in opposite directions from one another and amplify the IMF v ariations we deri v e. Lastly although it w ould seem to contradict the distrib ution of infrared e xcess and H a sources and w as not corroborated by the study of P alla & Stahler ( 2000 ), Herbig ( 1998 ) reported a slight age gradient in IC 348, nding an increasing mean age at lar ger radii. Such a gradient does act in the correct direction to account for some of the KLF dif ferences, ho we v er e v en if this age gradient is real, it is considerably too small ( t145 Myr at R4to 2.8 Myr at R10) to bring the IMFs of the sub-re gions into agreement. T o align the sub-re gion IMF peaks gi v en the observ ed KLFs w ould require tw o distinct populations where the halo is 5-10 Myr older than the core. This is an dif cult h ypothesis to accept considering the populations are of equal size, and while such a model might align the peak of the IMFs by shifting the halo to higher masses, it w ould further imply that the IMF of the halo is

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144 Figure 4–18: Radial v ariation in the IC 348 IMF Comparison of the Core and Halo Sub-Re gion IMFs for IC 348. Sho wn are Monte Carlo simulations of the deri v ed IMFs for the IC 348 sub-re gions and calculated for a sample of 150 stars. Error bars, calculated as the 1 s v ariation in each IMF bin from 100 iterations, are sho wn e v ery 0.3 de x in log mass for clarity These sub-re gion IMFs are also compared to the IMF of the IC 348a sub-cluster deri v ed by Najita et al. ( 2000 ) using NICMOS narro w band imaging, with error bars tak en directly from this w ork. The IC 348a sub-cluster is approximately the same area as the IC 348 core sub-re gion. radically truncated belo w 03 M. As w as originally found in the model ts of LL95 we conclude that such a tw o age population model cannot e xplain the IC 348 KLF Since the dif ferences between the core and halo sub-re gions can only be related to their underlying IMFs and these IMF dif ferences also appear statistically signicant as illustrated in gure 4–18 this means that IC 348 displays radial IMF v ariations on scales of the order of 1pc. Further these radial IMF dif ferences e xist primarily at subsolar masses in IC 348, unlik e typical scenarios for mass se gre g ation in v ery young

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145 clusters in which only the higher mass stars ( M1 M) are thought to be preferentially af fected. W ith these cluster(s) being only a fe w crossing times old, there is only enough time for the massi v e stars to sink to the cluster core ( Kroupa et al. 2001 ); otherwise, the massi v e stars are preferentially born in the cluster center ( Bonnell et al. 2001 ). Radial v ariations of a cluster' s subsolar mass IMF ha v e also been reported for the Orion Neb ula Cluster(ONC), of which the T rapezium is the core. While this cluster displays additional e vidence for the se gre g ation of high-mass stars to the cluster core ( Hillenbrand & Hartmann 1998 ), the lo w mass ONC IMF v aries between the central core and outer cluster halo. This has been sho wn by Hillenbrand ( 1997 ) and Hillenbrand & Carpenter ( 2000 ) who found that while the IMF of the central r035 pc T rapezium core peaks around 02 M, the addition of the cluster' s halo (r t id al25 pc) produces a composite ONC IMF that continues to rise do wn to 01 M, the completenesss limit of the Hillenbrand spectroscopic surv e y of the entire ONC cluster While the dif ference in the sk e w of the subsolar IMF as a function of radius appears smaller in the T rapezium than what we ha v e found for IC 348, it proceeds in the same direction, i.e., both the cluster' s halo and composite cluster IMFs sk e w to lo wer masses than the IMF that w ould be deri v ed if only the cluster core' s were e xamined. Lastly it is important to e xamine ho w the substellar sources are radially distrib uted. Since dynamical ef fects that could produce the radial IMF v ariation may operate on the lo west mass sources by sk e wing them to lar ger radii, bro wn dw arfs may correlate with the distrib ution of lo w mass sources and be systematically located further from the cluster center While the slope of the substellar IMF radically v aries between the IC 348 core and halo, the percentage of sources that are substellar does not. W e nd that sources in the mass range from 8025 M J u p constitute roughly 14% of the sources in the cluster core and a similar 16% in the cluster halo. Unless bro wn

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146 dw arfs preferentially populate the e xtreme outer re gions of IC 348 be yond the r1 pc radius of our surv e y this nding suggests that the mechanism that is breaking the uni v ersality of the IMF on small size scales, whether it is primordial IMF v ariations or dynamical mass se gre g ation, does not appear to be signicantly acting upon the substellar population. Interestingly the size of the bro wn dw arf population of the outer r035 pc re gion of the ONC is currently unkno wn and a wider -eld IR surv e y could re v eal a similar distrib ution of the substellar sources. In combination with our results for IC 348 such observ ations might be used to ascertain the origin of this radial v ariation of the subsolar IMF in clusters that are still embedded in their parental molecular clouds. 4.5 Conclusions Using wide-eld near -infrared images pro vided by the FLAMINGOS camera on the Kitt Peak 2.1m telescope, we performed a detailed census of the young 2 Myr IC 348 cluster located on the northeastern end of the Perseus Molecular Cloud. Using the multi-color infrared photometry pro vided by our observ ations, we e xplored this cluster' s structure, reddening, and relationship to the parental molecular cloud, and then used these results to construct and to analyze the IC 348 KLF and to correct it for eld star contamination. Using our model luminosity function algorithm described in Muench et al. ( 2000 ) and Muench et al. ( 2002 ), we deri v ed the cluster' s initial mass function, pro viding detailed ts and error estimates. From our analysis of the cluster' s structure and luminosity function and by comparison to our earlier study of the T rapezium cluster we dra w the follo wing conclusions about the KLF and IMF of IC 348: 1. W e deri v e an IMF for the composite IC 348 cluster spanning the mass range from 25 to 0025 M. Further we nd that the IC 348 IMF we deri v e is nearly identical to the the T rapezium IMF we deri v ed in chapter 3 : The tw o clusters IMFs rise with decreasing mass, ha ving a Salpeter -lik e slopes before attening belo w 07 M. W ithin

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147 the current uncertainties in PMS e v olutionary models, we nd that the mode of these cluster IMFs appear to f all at a mass of01 M. 2. Further we nd that the relati v e size of the substellar population is v ery similar in both clusters within the uncertainties of our method, re v ealing that bro wn dw arfs constitute only 1525% of the members of either cluster 3. IC 348 forms a modest b ut statistically signicant secondary KLF peak, corresponding to sources in the same mass range that we found responsible for a similar secondary KLF peak in the T rapezium. The similar KLF features in the T rapezium and IC 348 may signify either the presence of a secondary peak in the substellar IMF between 10 and 20 M J u p as we deri v ed for the T rapezium in Chapter 3 or be the result of a pre viously undocumented feature in the bro wn dw arf mass-luminosity relation, as we discuss further in Chapter 7 4. Radial v ariations are found in the KLF and IMF of IC 348 on the parsec scale, with a sk e wing of the KLF to f ainter sources and the IMF to lo wer mass stars in the cluster' s halo, a portion of the cluster whose IMF w as pre viously undetermined. This radial v ariation in the subsolar IMF is similar to what w as found pre viously for the subsolar mass stars in the Orion Neb ula Cluster It is unclear what process is breaking the uni v ersality of the cluster' s IMF on small spatial scales, b ut it appears dif ferent from dynamical mass se gre g ation which primarily acts upon higher mass stars at these young ages. Further while the slope of the substellar IC 348 IMF v aries substantially as a function of radius, the percentage of sources that are bro wn dw arfs does not. Finally we dra w the general conclusion that the e xistence of radial v ariations of the IMF on parsec scales e v en at v ery young ages (1 Myr) may mean that wide-eld imaging surv e ys are a pre-requisite to making meaningful IMF comparisons between dif ferent embedded clusters, a f act that has clearly been true in older open clusters for some time.

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CHAPTER 5 THE Y OUNG OPEN CLUSTER NGC 2362 In addition to probing the lo west mass populations of nearby clusters, luminosity function studies ha v e the utility that the y can be applied to deep surv e ys of more distant clusters. Using optical luminosity functions, Phelps & Janes ( 1993 ), for e xample, performed a study of the intermediate solar mass members of a lar ge sample of some what older distant clusters. Such studies of distant clusters are vital for determining the de gree of uniformity of the initial mass function through space and time, as the y are dra wn from a much lar ger v olume of the local g alaxy and almost certainly sample more v aried cluster en vironments that those found within the local kiloparsec. Ho we v er because of their distance, only the brightest, most massi v e cluster members can be studied spectroscopically Coupled with the f act that at these young ages fe w if an y such massi v e stars ha v e be gun to e v olv e, this means that the ages of these clusters are often poorly kno wn. NGC 2362 is a classic v ery young open cluster lying at a distance of 1500 pc ( Balona & Lane y 1996 ). Despite containing v arious e v olutionary signposts suggesting an age younger than 10 Myr the cluster is not embedded in a molecular cloud. Indeed, as we will sho w there we could not nd e vidence of remnant molecular material an ywhere near the cluster Morphologically NGC 2362 is dominated by the O9Ib super -giant star t Canis Majoris, often referred to as lying at the cluster' s center The cluster has an e xtensi v e B star population which denes the standard observ ational Zero-Age Main Sequence ( Johnson & Mor g an 1953 ) and which has been used to deri v e a cluster distance of1500 pc ( Johnson 1950 ; Balona & Lane y 1996 ; Moitinho et al. 2001 ). Modern optical studies ha v e re v ealed lo wer mass stars in the cluster ( W ilner & Lada 1991 ; Moitinho et al. 2001 ), yet kno wledge of the 148

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149 cluster' s age has until recently depended upon the post-main sequence e v olution of a single star: t Canis Majoris. Fits of post-main sequence e v olutionary models to the location of t Canis Majoris in the HR diagram ha v e yielded a cluster age of approximately 5 Myr ( Balona & Lane y 1996 ; Moitinho et al. 2001 ), yet such ts are v ery uncertain, mostly because t Canis Majoris is actually a spectroscopic quadruple ( v an Leeuwen & v an Genderen 1997 ; Stickland et al. 1998 ). While the age of NGC 2362 has been impro v ed by a v ery recent, deep optical study of the premain sequence star locus for the cluster ( Moitinho et al. 2001 ), we will consider the uncertainty in the age of NGC 2362 when studying this cluster' s luminosity function. This should allo w us to quantify the usefulness of our techniques when applied to re gions where e v en less is kno wn about the age of the cluster T o construct the infrared luminosity function for NGC 2362 we collaborated with J. Alv es who surv e yed a 10 10re gion around t Canis Majoris ( Alv es et al. 2001 ). This infrared census (hereafter referred to as the La Silla surv e y) reached a depth of m K17 and is briey detailed in Section 5.1 T o estimate the contamination of the cluster KLF by eld stars, of f-cluster images, totally30% of the on-cluster surv e y area were also obtained to equi v alent depth at K band. These of f-elds were located 1from the cluster along a line of constant g alactic latitude with the cluster In addition, we undertook an archi v al study of NGC 2362 using an e xisting all-sk y infrared surv e y to e xamine the cluster o v er an area much lar ger than that of the La Silla observ ations. In Section 5.2 we use this wide-eld surv e y to dene the cluster boundaries, to e xamine the reddening properties to w ard the cluster and to impro v e our estimate of the eld star contamination. In Section 5.3 we combine this study with the products of the La Silla surv e y and construct the dif ferential cluster KLF for NGC 2362. In Section 5.4 we compare the NGC 2362 cluster KLF to those KLFs we ha v e deri v ed for the T rapezium and IC 348 clusters. Finally we model the NGC 2362

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150 Figure 5–1: Digitalized sk y surv e y image of NGC 2362. F or orientation, north is up while east is left and the eld of vie w is 600600on this Schmidt projection of the red “J” band image from the UK Schmidt telescope. Epoch: 1973-1974. KLF in Section 5.5 e xploring the dependence of our results on the cluster' age and deri ving the cluster' s underlying mass function. 5.1 La Silla Obser v ations of NGC 2362 Between 27-29 December 1996, a near -infrared (JHK) surv e y of the central 10 10re gion of NGC 2362 w as completed by J. Alv es using the IRA C-2b infrared camera on the 2.2 meter telescope at the European Southern Observ atory (ESO) in La Silla, Chile. The surv e y consisted of 225 indi vidual images in a 55 mosaic

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151 grid centered on t Canis Majoris and the IRA C-2b camera w as congured to ha v e a 2 22 2 eld of vie w and a 051 pix el plate scale. The image reduction w as performed by J. Alv es using the IRAF package(s), and follo wing standard techniques in the infrared. Briey nightly dark frames and median local sk y frames were subtracted from the indi vidual images before the y were at-elded using local sk y ats. F or each mosaic position, indi vidual dithers were stack ed and combined using the shift-and-add technique. Final images had an ef fecti v e inte gration time of 13.5 minutes, FWHM estimates of12 and nearly 2200 sources were detected at K band in the cluster re gion. Similarly 72 indi vidual images in 8 dif ferent, non-o v erlapping control elds were observ ed at K band to pro vide statistical estimates of the eld star population. The elds were located one de gree from t Canis Majoris along a line of constant g alactic latitude with four each at higher and lo wer g alactic longitude relati v e to NGC 2362. Figure 5–2 in Section 5.2 illustrates the spatial location of these of f-elds. Aperture photometry w as performed using a beamsize of 4 and w as corrected out to the sk y annulus using aperture corrections deri v ed for each reduced mosaic position. Final calibration (zeropoint and airmass corrections) w as performed by comparison to Elias et al. ( 1982 ) infrared standards observ ed throughout each observing night and chosen to sample the range of airmass of the observ ations. The data were calibrated to this natural system of the IRA C-2b camera and color corrections to a standard system were not subsequently applied. Source e xtraction and photometry were performed on each indi vidual frame before being mosaick ed together using the centers of stars in o v erlap re gions. Astrometry with reasonably high precision w as performed by matching the XY pix el locations of a lar ge number (200) of the observ ed sources on the mosaick ed grid to the equatorial positions of these sources listed on the 2MASS w orld coordinate system and deri ving full plate solutions using the IRAF CCMAP This process w as completed in tw o steps. Rough coordinates were deri v ed by manually

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152 matching 3 sources both on the grid and in the 2MASS catalog, yielding a linear solution applied to all the sources. The rough La Silla database w as then matched in full to the 2MASS catalog, yielding a lar ger number of matches and higher order corrections in the CCMAP routine. The nal coordinates ha v e residual errors of02 5.2 2MASS Obser v ations of NGC 2362 5.2.1 Spatial Structur e of NGC 2362 T o surv e y NGC 2362 o v er an area lar ger than that co v ered in the La Silla obser v ations, we made use of archi v ed data in the public domain. The T w o Micron All Sk y Surv e y (2MASS) is an infrared surv e y of the entire sk y at J, H and K s w a v elengths (122 m ). This surv e y w as completed in December 2000 with the nal data release planned for September 2002. The resulting 2MASS source catalog is being released incrementally; for the purposes of this w ork we will use the 2MASS products from the Second Incremental Data Release (circa March 2000). From the 2MASS catalog, we do wnloaded positions and (JHK s ) magnitudes for sources that ha v e K s magnitudes brighter than 14 and lie within 1of t Canis Majoris. This f aint limit w as conserv ati v ely chosen to be brighter than the nominal (tar get) completeness limit (K s = 14.3) of the 2MASS surv e y although the 2MASS surv e y is more complete than their tar get limit in most areas of the sk y A digitalized sk y surv e y of this area of NGC 2362 is sho wn in gure 5–1 and the spatial distrib ution of these 2MASS sources around NGC 2362 is sho wn in Figure 5–2 Sources f alling near the locations of e xtremely bright stars or along these stars' dif fraction spik es ha v e been withheld from the pubic domain until the nal catalog release; the resulting g aps in the 2MASS spatial co v erage are e vident across the surv e y eld. Indeed, within the boundaries of the La Silla infrared observ ations of NGC 2362, the 2MASS catalog is signicantly incomplete with respect to the La Silla observ ations because of the e xtreme brightness of t Canis Majoris.

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153 Figure 5–2: Source distrib ution of NGC 2362 from 2MASS. Surf ace plot of 2MASS sources surrounding NGC 2362. Lar ge empty areas are re gions incomplete in the current 2MASS catalog due to bright star artif acts. The re gion sur v e yed by the La Silla observ ations is indicated by the central square box. The circle, centered on the cluster represents the area outside of which the 2MASS surv e y w as used to calculate the eld star surf ace density The tw o La Silla of f-eld locations are indicated by small box es at the periphery of the 2MASS surv e y Using this dataset, we calculated radial proles for the source distrib ution in NGC 2362 in an attempt to dene the spatial e xtent of this cluster W e used t Canis Majoris as the cluster' s center and deri v ed radial proles of the 2MASS surv e y re gion using the same tw o methods used in Chapter 4 : 1) by counting the stars in successi v e annuli of equal radial separation and 2) counting them in successi v e annuli ha ving equal areas. While the former is a more common method, the latter gi v es a better estimate of the noise in the eld star surf ace density for a x ed area. In Figure 5–3 we plot both of these radial proles from the 2MASS surv e y further constructing three sets of proles for dif ferent magnitude interv als. At lar ge radii from the cluster

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154 Figure 5–3: Radial proles of NGC 2362. Three radial proles (referenced to the location of t Canis Majoris) are displayed corresponding to three dif ferent magnitude ranges (see plot). In all cases the radius of the rst annulus equaled 6. The radial prole(s) of 2MASS sources are sho wn on the full axis; those of the inner 01La Silla dataset are are sho wn in the inset using an R ini t15and for all sources with m K14. Articial dips in the 2MASS radial prole(s) at q r045 and 10 result from the g aps in the 2MASS co v erage (see Figure 5–2 ). Three horizontal dotted lines are sho wn for each prole and correspond to the median eld star surf ace density (for that magnitude range) and the1 s v ariation in the of f-eld surf ace density in area equi v alent to the NGC 2362 La Silla surv e y V er tical dashed lines demark the limit of the La Silla surv e y and an outer boundary where the radial prole intercepts the median eld star surf ace density the surf ace density is well beha v ed, with a mean(s) of 486 stars per square de gree (m K11), 1564 stars per square de gree (m K125) and 5188 stars per square de gree (m K14). Because the 2MASS catalog is v ery incomplete in the cluster core, we used the La Silla IR catalog to nd that the central q r002(r=0.5 pc) reaches a surf ace

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155 density of2510 4 stars (m K14) per square de gree (see inset in Figure 5–3 ) and does appear to hit the background at a radius of15 pc. Despite the incompleteness of the current 2MASS catalog in the cluster core, there appears to be a clear central concentration of stars within the ESO surv e y boundaries. This is true for either radial prole style and an y magnitude range. Gaps in the 2MASS co v erage e xtend be yond the cluster core, ho we v er with prominent dips occurring at q r045 and 10. Despite these features, there is a suggestion from all of the 2MASS cluster prole(s) that NGC 2362 e xtends be yond the ESO surv e y re gion. This is inspite of the f act that the central core does appear to go to the background at smaller radii. In all the radial proles the bins between q r010 and 035appear ele v ated relati v e to the median surf ace density and while not all the bins e xceed the 1 s noise estimate in the background, most of the structur e of these bins is constant between dif ferent magnitude ranges. Since one goal of this archi v al surv e y is to impro v e the eld star correction by surv e ying a lar ger cluster -free area, we wish to e xclude an y re gion that could contain cluster members. While the noise in these proles do not allo w us to determine a precise outer radius for the cluster we do break the NGC 2362 2MASS into three some what arbitrarily named re gions based on this prole: 1) the eld star population, corresponding to the area well outside an y cluster boundary ( q r035); 2) a cluster “halo” re gion, corresponding to the 2MASS sources between the eld star population and that area surv e yed by our La Silla observ ations (01q035); 3) The cluster “core” ( q01), which w as surv e yed to much greater depths (and completeness) by the La Silla observ ations. 5.2.2 Sour ce Reddening f or NGC 2362 F ollo wing the procedures used in chapters 3 and 4 we e xamine the infrared color -color diagram for NGC 2362 to determine the reddening of cluster sources due to interstellar e xtinction, the parental molecular cloud or by infrared e xcess from disks. In Figure 5–4 we construct the (H K s ) vs (J H) color -color diagrams for each of

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156 Figure 5–4: Infrared color -color diagrams for NGC 2362. These color -color diagrams were constructed from the 2MASS surv e y of NGC 2362 for each of the three cluster sub-re gions: A) the cluster core; B) the cluster “halo”; C) the eld star population. In the last case, only 2000 randomly chosen sources (of the 13000 stars in this re gion) are sho wn. The reddening la w used is that from Cohen et al. ( 1981 ). the three sub-re gions of NGC 2362. Remarkably neither the ef fects of interstellar e xtinction nor infrared e xcess are e vident in an y of the diagrams. Not only is there little e xtinction along the line of sight to w ards NGC 2362, there is v ery little if an y seen to w ards the eld stars within the cluster' s 1neighborhood. W e attempted to conrm this lack of e xcess by e xamining the digital sk y surv e y plates and the dust opacity maps of W ood et al. ( 1994 ) and could nd no bright or dark neb ula, and an apparent “hole” of lo w opacity appears on the IRAS maps. Further the lack of infrared e xcess sources in the cluster core implies that nearly all of the stars ha v e lost their inner disks. This observ ation w as rst found by Alv es et al. ( 2001 ) who studied the color -color diagram of the deep La Silla observ ations and found that this dearth of infrared e xcess sources continues do wn to that surv e y' s completeness limit (m K165). It has been subsequently conrmed by Haisch et al. ( 2001b ) who found no sources do wn to m K14 with 3 m e xcess. These ndings ha v e tw o immediate implications for our analysis. First, our modeling procedure will not need to include the ef fects of source reddening on the cluster members. Second, estimating the contrib ution of eld stars to the cluster KLF will not be complicated by patch y e xtinction as w as the case in our study of

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157 IC 348. On the other hand, the lack of a lar ge background molecular cloud (as w as the case for the T rapezium) means the eld star contamination will be signicant. Thus our statistical subtraction a eld star KLF from the on-cluster KLF will result in a meaningful KLF only for that magnitude range where cluster dominates the eld star density No w we combine this wide-eld surv e y with the deeper La Silla observ ations to construct the cluster and eld star K band luminosity functions for NGC 2362. 5.3 The NGC 2362 Cluster KLF 5.3.1 Empirical Field Star KLF In Figure 5–5 (a) we construct the eld star KLF rele v ant for the NGC 2362 cluster re gion. In histogram form we sho w the KLF (scaled to stars per square de gree) of 2MASS sources between the 035cluster boundary and the 1surv e y boundary W e compare this “eld star” KLF to the KLF deri v ed from our La Silla of f-cluster observ ations, which co v ers a mere 1/350th the area of the corresponding 2MASS surf ace area. Brighter than m K12, the La Silla of f-elds do not co v er suf cient area to properly sample the bright population; this is a moti v ating f actor for our use of the 2MASS of f-cluster surv e y Between m K12 and 14, where the La Silla and 2MASS eld star KLFs o v erlap and are both well sampled, the y agree e xtremely well and ha v e v ery similar slopes, suggesting the tw o of f-elds can be straightforw ardly combined. Belo w m K14, ho we v er the slope of the deeper La Silla eld star KLF appears change and atten relati v e to the La Silla/2MASS KLF for m K14. W e illustrate this change in slope by tting each KLF with a po wer -la w function and comparing their resulting slopes. From m K714, the 2MASS KLF can be characterized as a po wer la w with a slope of 0348800056, while between m K12 and 165, the La Silla of f-cluster KLF has a slope of 0278700146. F or the purposes of our subsequent analysis, we will assume this change in slope is a real feature of the eld star KLF W e construct the eld star KLF rele v ant for

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158 Figure 5–5: Field star and cluster KLFs of NGC 2362. A) Construction of an empirical eld star KLF for NGC 2362. The deep La Silla of f-eld KLF is compared to the 2MASS eld star KLF Error bars are from 1 s counting statistics, though the f actor of 350 between the 2MASS surv e y and La Silla of f-cluster areas results in smaller statistical errors bars for the 2MASS of f-elds relati v e to those f ainter than m K14. Also sho wn are po wer -la w indices of the 2MASS and La Silla of f-eld KLFs, illustrating the attening belo w m K14. B) Comparison of NGC 2362 cluster and eld star KLFs (in stars per square de gree). The cluster dominates the e xpected eld star counts do wn to the m K165 completeness limit. Error bars are from counting statistics. NGC 2362 by combining the 2MASS eld star KLF brighter than m K14 with the La Silla of f-cluster KLF for m K14 and f ainter The po wer -la w inde x of this combined eld star KLF is 0319800059. Uncertainties in the indi vidual bins of the combined KLF are assigned based on the area of the surv e y from which the y were dra wn. This means that the error bars for the f ainter bins tak en from the La Silla of f-elds are lar ger than at the brighter magnitude bins deri v ed from 2MASS. There are a fe w concerns with our construction of the eld star KLF for NGC 2362; specically that the change in slope of the eld star KLF could represent some systematic error in its construction. Such an error could come about if, for e xample, the La Silla observ ations coincidentally struck a lo w surf ace density re gion or a small patch of molecular material. Y et there is no e vidence of reddening in this

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159 re gion and the La Silla observ ations came from more than one location relati v e to the cluster Further the mean surf ace density of stars (m K14) in the La Silla and 2MASS re gions agree to within 5% as do their KLFs between m K12 and 14. 5.3.2 NGC 2362 Differ ential KLF(s) In Figure 5–5 (b), we compare our empirical eld star KLF to the NGC 2362 cluster KLF deri v ed from the La Silla observ ations. Both are scaled to stars per square de gree for this comparison. The cluster counts dominate the eld star counts do wn to and belo w our completeness limit, ho we v er the eld star KLF appears to be some what more sensiti v e. T o construct the cluster KLF for NGC 2362, we scaled the empirical eld star KLF to the size of the cluster re gion and subtracted it directly from the oncluster KLF The resulting dif ferential NGC 2362 KLF is sho wn in Figure 5–6 (a). One sigma error bars on indi vidual bins carry the quadratic sum of the counting uncertainty in the on-eld counts and the eld star uncertainties described abo v e and dependent upon the source of eld star KLF Figure 5–6: Dif ferential KLF(s) of NGC 2362. A) The dif ference of the on-cluster and the scaled eld star KLF 1 s error bars carry the quadratic sum of the cluster counting statistics and the uncertainty in the eld star KLF which depends upon the size of its original surv e y (See te xt and Figure 5–5 b ). B) The dif ference of the 2MASS cluster“halo” KLF and the scaled eld star KLF Error bars represent the total counting uncertainty for this area.

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160 The NGC 2362 dif ferential KLF contains 53760 stars abo v e the completeness limit of m K165. The cluster KLF rises in a nearly po wer -la w manner before attening belo w m K145 and peaking at the completeness limit. W e also computed the dif ferential KLF for sources in the 2MASS cluster halo. As seen in Figure 5–6 (b), this yields a net positi v e dif ferential halo KLF supporting the radial prole suggestion that the cluster e xtends be yond the La Silla surv e y W e estimate that there are 11461 sources m K14 in the NGC 2362 cluster halo re gion, which is roughly the same size population as the cluster core to this depth. 5.4 Comparison to other Y oung Cluster KLFs W e can compare the NGC 2362 dif ferential KLF constructed here to the K band luminosity functions of the other young clusters that we ha v e studied in detail in this w ork: IC 348 and the T rapezium. As we ha v e sho wn, dif ferences between cluster KLFs can be attrib uted to dif ferences in either the clusters' mean ages or dif ferent underlying IMFs with a small inuence from the cluster reddening properties. In Figure 5–7 a we rst compare the NGC 2362 KLF to the that of the T rapezium Cluster W e shifted the T rapezium MA V KLF from Section 3.2.2 to the distance of NGC 2362 and brightened it by 0.5 magnitudes to account for the median e xtinction of the T rapezium sources (A V475; A K043). W e did not modify the T rapezium KLF to account for the ef fects of sources with infrared e xcess in that cluster Clearly the tw o KLFs dif fer both in the peak and breadth with the T rapezium KLF peaking at much brighter magnitudes and being some what broader than the NGC 2362 KLF The dif ferences in the peaks of the KLFs could be easily e xplained by dif ferences in the ages of these clusters. Based simply upon the e v olutionary signposts in NGC 2362, e.g., the lack of a parental molecular cloud or stars with infrared e xcess, the1 Myr T rapezium Cluster is almost certainly younger than NGC 2362, and therefore should peak at brighter magnitudes. The dif ference in the KLF widths, though, is the opposite of that e xpected from our numerical e xperiments and is due to

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161 Figure 5–7: Comparing the cluster KLFs of NGC 2362, the T rapezium and IC 348. A) Comparison of the NGC 2362 dif ferential KLF to the T rapezium Cluster KLF The T rapezium Cluster MA V KLF from Section 3.2.2 w as shifted to the distance of NGC 2362 (1500pc) and w as brightened by A V475 (A K043) to account for the median e xtinction seen in the T rapezium. B) Comparison of NGC 2362 dif ferential KLF to the IC 348 Cluster KLF The IC 348 cluster' s background corrected dif ferential KLF from Section 4.2 w as shifted to the distance of NGC 2362 and w as brightened by A V42 (A K038) to account for the mean e xtinction seen in IC 348. the truncated sensiti vity of the NGC 2362 La Silla surv e y Since we found in Section 2.5.2 that cluster KLFs broaden with age, the NGC 2362 KLF should be broader than the T rapezium KLF if it were older and the tw o clusters are dra wn from the same underlying IMF Since our observ ations of NGC 2362 KLF are not nearly as sensiti v e in absolute K magnitude as the T rapezium and because infrared e xcess has lik ely broadened the T rapezium KLF some what we can discard this dif ference in clusters widths. Only if the tw o clusters ha v e quite similar ages could these dif ference in their KLF widths be used to infer something about the cluster' s IMF In Figure 5–7 (b), we compare the NGC 2362 KLF with the KLF of the IC 348 cluster as deri v ed in Section 4.2 Ag ain we shifted the IC 348 cluster to the distance of NGC 2362 and brightened it to correct for the mean e xtinction seen in IC 348 (A V42; A K038) The KLFs of the IC 348 and NGC 2362 clusters closely agree,

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162 with the IC 348 KLF slightly shifted to brighter magnitudes b ut with the NGC 2362 KLF ha ving a fe w more v ery bright stars than IC 348. As discussed in Section 4.3.1 IC 348 has an approximate mean age of 2 Myr or slightly older than the T rapezium. Thus the comparison of these tw o cluster KLFs means that NGC 2362 is slightly older than IC 348 pro vided the y ha v e similar underlying IMFs. Put a dif ferent w ay NGC 2362 cannot be signicantly older ( t10 Myr) than IC 348 and ha v e a similar IMF to IC 348. If NGC 2362 does ha v e a much older age, this KLF comparison w ould indicate that the IMF of NGC 2362 w ould be signicantly sk e wed to higher masses than IC 348. On the other hand e v en if these tw o clusters ha v e similar ages, then does appear to be slight e xcess of bright stars in NGC 2362 relati v e to IC 348. The observ ed e xcess occurs between m K95 and 12 (at the distance of NGC 2362) and this luminosity range corresponds to main sequence stars in the mass range of29 Mfor either cluster This e xcess, ho we v er does not con vincingly imply that there is a statistically preference for high-mass stars in NGC 2362. T w o issues are important to remember First, let us assume an uni v ersal IMF between these three clusters similar to the IMF we deri v e for the T rapezium and IC 348. Because the high-mass stars are out on the tail of this probability distrib ution, the lar ger the cluster population the greater the statistical lik elihood of creating higher mass stars. Since NGC 2362 contains more stars than IC 348 and this e xcess constitutes only a handful of v ery massi v e members, we cannot condently conrm a v ariation in the underlying IMFs from this luminosity function comparison alone. Second, we cannot be certain that the bright end of the luminosity function (or the high-mass portion of the IMF) has not been inuenced by mass se gre g ation, although we ha v e co v ered a v ery lar ge area of NGC 2362 ( r24pc).

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163 5.5 Modeling the NGC 2362 KLF T o apply our model luminosity functions to the NGC 2362 KLF we constructed in Section 5.3 we adopt our standard PMS e v olutionary models based on a zero age main sequence, and sets of pre-main sequence e v olutionary models. W e used a x ed cluster distance of 1500pc and did not include an y distrib ution functions of source reddening. W e performed a multi-part analysis of the NGC 2362 KLF using our model KLFs, v arying both the assumed age and IMF for this cluster In section 5.5.1 we used our modeling code to deri v e the cluster' s mean age from its KLF by adopting an underlying IMF In Section 5.5.2 we e xamined the dependence of the deri v ed IMF slope on the assumed age of the cluster and nd that both parameters can be simultaneously constrained for this cluster Finally in section 5.5.3 we adopt a mean age for the cluster and deri v e the NGC 2362 IMF spanning a mass range from 10 to 02 M. 5.5.1 Deri ving a Mean Age Using a Fixed IMF Since we concluded from our numerical e xperiments in Section 2.5 that the tw o most inuential parameters on the form of a cluster' s LF are the underlying initial mass function and the cluster' s mean age, one of these parameters must be x ed to deri v e strong constraints upon the other parameter As we summarized earlier in this chapter a mean age may not be kno wn for a distant young cluster simply because no signposts of its age can be quantied. T o e xamine the usefulness of model luminosity functions for studying such young distant clusters, we rst e xamined NGC 2362 as if kno wledge of its mean age were not kno wn. By e xpanding upon the procedure used in Lada & Lada ( 1995 ), we will use model luminosity functions and a x ed underlying IMF to deri v e a mean age for NGC 2362. T o study the mean age of NGC 2362, we created a suite of model luminosity functions emplo ying the T rapezium IMF described in Equation 3.1 and v arying the cluster' s mean age by increments of 0.5 Myr W e also ran this e xperiment for tw o

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164 dif ferent age spreads (coe v al and 2 Myr) and for tw o dif ferent binary fractions (no binaries and f bin040). In Figure 5–8 we sho w the results of c 2 ts of the suite of model KLFs to the NGC 2362 KLF Note, the ts were applied o v er the entire luminosity range of the observ ed NGC 2362 KLF Figure 5–8: Mean age of NGC 2362 deri v ed from the cluster KLF Each circle represents the t of a single model iteration to the NGC 2362 dif ferential KLF The solid black line is the t the mean model KLF to the NGC 2362 KLF for that set of parameters. A) Dt01 Myr (essentially coe v al), f000; B) Dt01 Myr f040; C) Dt20 Myr f000; D) Dt20 Myr f040. All of these ts ha v e clear c 2 minima and all suggest a mean age for NGC 2362 between 4 and 6.5 Myr though none of these ts approach a c 2 probability of 1. Similar to our numerical modeling e xperiments, we nd that the age spread in the

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165 cluster has v ery little af fect on the ts or on the mean age we deri v e. On the other hand, the deri v ed mean age is slightly dependent upon the assumed binary fraction. The inclusion of unresolv ed binaries broadens the c 2 minima deri v ed from tting model KLFs, and yields a slightly older mean age. F or e xample, the mean age deri v ed from models without binaries is between 3.5 and 5.5 Myr while the ts that include binaries yield ages between 4 and 6.5 Myr Lastly we use the statistical sampling that our method naturally pro vides to e xamine if simple uctuations in the cluster KLF could mask the true age. Fitting the NGC 2362 KLF to 50 indi vidual iterations of the model KLF rather than the mean KLF model suggests that the accurac y of this method is 1 2 Myr for a cluster of 500 stars. Figure 5–9: Model KLF at 5 Myr with T rapezium IMF t to NGC 2362. The t w as performed o v er the luminosity range from m K95 to m K165. Error bars on the model KLF correspond to the 1 s statistical uctuation in that KLF bin for 50 iterations of a cluster of 500 stars.

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166 W e are able to deri v e a mean age of52 Myr for NGC 2362 using our model luminosity function techniques and assuming the IMF of the T rapezium. This age agrees v ery well with a cluster age of512 Myr deri v ed from the e v olution of t Canis Majoris or from the cluster' s pre-main sequence locus in the color -magnitude diagram ( Balona & Lane y 1996 ; Moitinho et al. 2001 ). In gure 5–9 we compare the NGC 2362 KLF to a model KLF constructed using the T rapezium IMF at an age of 5 Myr This model KLF passes through most of the bins of the NGC 2362 KLF abo v e its its completeness limit and at this mean age, the m K165 completeness limit translates to a mass limit of02 M. Belo w our completeness limit, the model KLF sharply declines, reecting the f alling substellar IMF of the T rapezium. W ere NGC 2362 to ha v e a similar substellar IMF as that of the T rapezium, then the model KLF mak es a clear prediction of what deeper observ ations w ould re v eal. T w o interesting predictions arise from this simple KLF modeling. The rst is that the non-po wer la w structure of the bright portion of the NGC 2362 KLF has corresponding structure in the model of the T rapezium KLF at t5 Myr This structure in the model KLFs is due to the LMS spik e discussed in Section 2.5.1 The small LF peak/dip is the result of a double v aluing of the theoretical M-L relation as stars on radiati v e tracks in the HR diagram reach a maximum luminosity before settling onto the ZAMS. The LMS spik e is predicted to be a strong feature in young clusters with mean ages greater than 2-3 Myr and that it will e v olv e to f ainter magnitudes as the cluster ages. This is because as the cluster ages the subset of stars on radiati v e tracks in the HR diagram shifts to lo wer and lo wer masses, while, assuming a normal IMF the size of this subset of stars also gro ws. If no such feature e xisted in the theoretical mass luminosity relation, then it w ould be v ery dif cult to t the bright KLF with a single underlying po wer -la w as we ha v e done here. The second nding is that there appears to be a slight e xcess of the brightest stars in NGC 2362 relati v e to the model e v olv ed T rapezium KLF W e discuss this e xcess in sections 5.5.2 and 5.6.1

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167 5.5.2 Simultaneous Deri v ation of a Cluster' s Age and its IMF Since a young cluster' s LF is systematically dependent upon the cluster' s age and underlying mass function, it is important to determine ho w an uncertainty in one of these quantities maps into the deri v ation of the other In this section, we allo w both the mean age and the IMF of the cluster to v ary and use our modeling techniques to simultaneously constrain both of these quantities for NGC 2362. As seen in Figure 5–6 the NGC 2362 dif ferential KLF has roughly a po wer -la w increase do wn to m K145, b ut attens to w ards the completeness limit. W e learned from our numerical e xperiments that where a cluster KLF has a po wer -la w form, the underlying IMF also has a po wer -la w form. W e use this information to infer that o v er some mass range, the underlying NGC 2362 IMF is a po wer -la w T o test the dependence of the deri v ed inde x of this po wer -la w function on the inferred mean age of NGC 2362, we created a suite of model luminosity functions, systematically v arying the slope of the po wer -la w IMF and the mean age. W e v aried the IMF slope from at to steeply rising, i.e., G0 2, in steps of 0.05. W e also v aried the cluster' s mean age from 1 to 10 Myr in steps of 0.25 Myr W e x ed the age spread at 2 Myr b ut calculated model KLFs with binary fractions of 0.0 and 0.4. Finally we t these models to the observ ed KLF o v er the luminosity range do wn to where the cluster KLF attens, i.e.,from m K95 to 145. From these ts we plot the dependence of the po wer -la w IMF on the cluster' s mean age in Figure 5–10 W e also sho w this dependence for tw o dif ferent binary fractions. T w o results are immediately clear: 1) as e xpected, there is a systematic relationship between these tw o parameters (age and IMF); 2) both parameters are simultaneously constrained to a narro w range of best t v alues. These results are unchanged by the v ariation in the binary fraction, though it does slightly af fect the nal result.

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168 Figure 5–10: Dependence of the NGC 2362 IMF Slope on mean age. W e plot contours of c 2 probability for ts to the NGC 2362 KLF v arying the cluster mean age and po wer -la w IMF slope. Contours range from 0.1 to 1.0 in steps of 0.1. The horizontal lines correspond to the Salpeter eld star IMF slope (solid line) and to the high-mass po wer -la w IMF slope deri v ed for the T rapezium (dashed line; from Section 3.3.3 ) Corresponds to a mass range from 10 Mto between 1 and 0.2 Mdepending upon the cluster' s age (see table 5–1 ). A) Fits without binaries, f00. B) Fits with binaries, f040. Examining ts with c 2 probabilities greater than 0.6, we nd that our technique constrains the mean age of NGC 2362 to lie between 3.5 and 5.5 Myr with a best t ( c 2 prob095) v alue of 446051median45Myr for ts without binaries. Including binaries actually sk e ws the mean age range v ery slightly to young er ages ( t wbinar ies441064 Myr). The deri v ed IMF slope is clearly constrained to lie between -0.8 and -1.3 and has a best t v alue of G 108008median 105. W e nd, though, that including unresolv ed binaries results in a slightly atter IMF slope, G wbinar ies 099009. Because of the PMS nature of the stars in NGC 2362, the luminosity range t by these models (m K95145) translates into dif ferent mass ranges for the po wer -la w IMF as a function of age. W e summarize this dependence in T able 5–1 W e nd that the bright end of this luminosity t range al w ays corresponds to early B type stars

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169 T able 5–1. Age dependence of the IMF slope in NGC 2362 Cluster Mean Age IMF Mass Rangea M IMF Slopebt (Myr) ZAMS DM97 BCAH98 PS99 G c 2 Prob 1.09 0.20 0.23 0.23 -0.70 0.10 3.0 . 0.56 0.50 0.49 -0.90 0.45 5.0 . 0.78 0.66 0.60 -1.00 0.83 7.0 . 0.88 0.75 0.80 -1.80 10.0 . 0.95 0.86 0.90 -1.95 aCorresponding to the K luminosity range t. K = 9.5 corresponds to a B2.5V main sequence star at the cluster' s distance, which con v erts to a mass of 9 M(using K en yon & Hartmann 1995 ; W inkler 1997 ) K = 14.5 corresponds to pre-main sequence stars whose mass depends upon the cluster' s age and the set of PMS tracks.bF or ts without binaries. There is no c 2 po wer at ages7 Myr References. — D'Antona & Mazzitelli ( 1997 DM97); Baraf fe et al. ( 1998 BCAH98); P alla & Stahler ( 1999 PS99) with masses9 while the f aint end of the luminosity range al w ays corresponds to PMS stars. F or the range of best t v alues of the mean age of the cluster this lo w mass limit is between 05 and 08 Mand depends v ery slightly on the set of tracks used for con v ersion. W ithin this corresponding mass range (101 M) we nd that po wer -la w slope for NGC 2362 is constrained to be slightly atter than that found for the T rapezium or the eld star IMF according to Salpeter under an y v ariations in the assumed mean age or binary fraction. 5.5.3 NGC 2362 IMF Deri v ed Using a Fixed SFH Since there is independent e vidence of the age of NGC 2362 a v ailable from optical color -magnitude diagrams we can directly deri v e the cluster IMF W e assume a mean age of 5 Myr and assign an age spread of 2 Myr based on the spread in age (luminosity) seen in the color -magnitude diagram ( Moitinho et al. 2001 ) and because this age spread is typical of the other young clusters in our sample. W e used multiple

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170 po wer -la w se gments for the underlying IMF and v aried the t range o v er which the y were applied. W e only performed these ts using model KLFs that did not include unresolv ed binaries. Figure 5–11: Best t model KLFs to the NGC 2362 KLF The t w as performed o v er the luminosity range from m K95 to 165. Error bars on the model KLF correspond to the 1 s statistical uctuation in that KLF bin for 50 iterations of a cluster of 500 stars. Single po wer -la w IMFs did not pro vide good ts o v er the entire luminosity range from K = 9.5 to 16.5. Fits do wn to m K145 ( M1008 M) were well t by a single po wer -la w G 116021 for this x ed age and similar to our ndings from the pre vious section. The break and attening of the KLF at f ainter magnitudes, ho we v er rejected a single po wer -la w o v er the entire mass/luminosity range. T w o po wer -la w IMFs pro vided better ts to the form of the cluster KLF The best tting model KLF using this 2 po wer -la w IMF is sho wn in Figure 5–11 compared to the NGC 2362 dif ferential KLF The model t is e xcellent, passing through nearly

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171 Figure 5–12: Mass Function of NGC 2362. The deri v ed IMF for NGC 2362 is compared to the T rapezium IMF do wn to the 02 Mlimit. The hatched re gion is a graphical representation of the range of IMFs permitted from by the KLF models. e v ery bin of the dif ferential KLF although there is still a modest e xcess of bright (m K10) sources relati v e to model KLF From these ts with our model KLFs we nd a best t IMF ha ving the form: d N d log M G 1 : -1.100.28 m 1 : 0.690.18 MG 2 : -0.110.11 (5.1) In Figure 5–12 we compare the deri v ed NGC 2362 IMF to that found for the T rapezium (from Equation 3.1 ) and IC 348 (from T able 4–3 ). W e also graphically display the range of IMFs permitted by the NGC 2362 dif ferential KLF Clearly the deri v ed NGC 2362 IMF is v ery similar to that IMF we deri v ed for the other tw o younger clusters. All three ha v e po wer -la w inclines to w ard lo wer masses, break at

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172 around 0608 Mand atten at subsolar masses. The NGC 2362 high mass IMF slope, G 1 is slightly atter than in the T rapezium, which is also atter than that deri v ed for IC 348, although all of these dif ferences are smaller than the dispersion in our ts for the high mass slope for each of the clusters. 5.6 Discussion 5.6.1 Age and IMF of NGC2362 From our construction and modeling of the NGC 2362 KLF we nd that the stellar IMF of this cluster is quite similar to that IMF we found for the IC 348 and T rapezium clusters. While all three rise with po wer -la w slopes into the subsolar re gime, the KLF and IMF of NGC 2362 both suggest a slight e xcess of bright/massi v e stars relati v e to these other clusters. The IMF slope in the high to intermediate mass range v aries from NGC 2362 G 11 to the T rapezium G 12 to IC 348 G 153, relati v e to Salpeter G 135, although all of these slopes are the same within their respecti v e errors. The NGC 2362 IMF we deri v ed attens belo w1 M, b ut continues to rise do wn to our completeness limit of02 M, similar to our ndings for IC 348 and the T rapezium. Our observ ations cannot resolv e the mode of the NGC 2362 IMF so we cannot compare this measure to that found in the other clusters. Further we are not able to probe the bro wn dw arf re gime of NGC 2362, though we nd that if the substellar IMF of NGC 2362 were similar to that in IC 348 and the T rapezium, i.e. f alling, then it w ould be v ery dif cult to deri v e from constructing a deeper luminosity function. This is because the statistical noise from the eld star surf ace density w ould lik e sw amp the counts in the substellar re gime. A dif ferent mechanism for identifying cluster members, lik ely in the form of optical-infrared color -magnitude diagrams, will be necessary for determining the size of the bro wn dw arf population in NGC 2362. Re g ardless, we note that the lo wer mass limit of our study of NGC 2362 is nearly the

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173 same as the sensiti vity limit of the Hillenbrand ( 1997 ) spectroscopic study of the Orion Neb ula Cluster despite the f act that NGC 2362 is four times further a w ay In addition to deri ving the IMF of NGC 2362, we undertook a series of e xper iments to quantify the usefulness of model luminosity functions for studying distant young clusters where little is kno wn about the cluster' s mean age. Expanding on the w ork of Lada & Lada ( 1995 ), we found that by assuming an underlying IMF for NGC 2362, we could deri v e a mean age for NGC 2362 of 5 Myr in close agreement with other age estimates that ha v e used the optical color -magnitude diagram. Further the intrinsic sampling error in this age estimate is 1-2 Myr set by the statistical size of the cluster From these e xperiments, we conrmed our earlier ndings in Section 2.5.2 that the assumed age spread for the cluster has little ef fect on the deri v ed age. Ho we v er the inclusion of unresolv ed binaries does af fect the deri v ed mean age, shifting the cluster to some what older ages for a x ed IMF This can be understood because unresolv ed binaries, dra wn from an IMF that increases with decreasing mass, will act to brighten the intrinsic KLF thus, requiring an older age to match the observ ations. Lastly we calculated a series of numerical models of the NGC 2362 KLF in which we made no a priori assumptions about NGC 2362 e xcept for its distance and the functional form of the underlying IMF basing this latter assumption on the po wer la w appearance of the KLF Simultaneous ts of the cluster' s mean age and po wer -la w IMF slope were performed, yielding narro w constraints on both properties for this cluster The success of these ts can be understood in part by re vie wing our numerical e xperiments illustrating the e v olution of the cluster KLF with mean age. Recalling Figure(s) 2–8 and 2–9 we sho wed that cluster KLFs e v olv e as much during their rst 3 Myr of e xistence as the y do in the ne xt 10 Myr This means that as the cluster ages, the uncertainty in the IMF due to the uncertainty in the age decreases. As we ha v e sho wn, the systematic errors for a 4-5 Myr old cluster are quite small, 0.5 Myr in age and 0.1 de x in the slope of the IMF Considering the blurring ef fects of reddening

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174 on our model KLF tting technique, such an analysis of distant, younger embedded clusters will lik ely yield less accurate constraints on both the cluster' s age and IMF Ho we v er the results of such an analysis w ould become meaningful when looking for radical v ariations in the IMF e v en if the IMF/Mean Age errors we deri v e for a 4 5 Myr old open cluster were doubled when studying embedded clusters. 5.6.2 Age and Spatial Structur e of NGC 2362 In our study of IC 348 we learned that despite the youth of the cluster' s tar geted by our study the spatial structure of a young cluster can inuence the form of the IMF we deri v e because radial v ariations in the cluster IMF e xist e v en at these young ages. Independent of whether these IMF v ariations are indicati v e of dynamical or primordial mass se gre g ation or are simply the product of small samplings of the underlying IMF their e xistence implies that the comparison of cluster IMFs as we did in Figure 5–12 requires some idea of the relati v e areas of the cluster surv e yed. The radii of our surv e ys of these three clusters are r03 pc for the T rapezium, r05 pc for IC 348 and r22 pc for NGC 2362. F or IC 348, this radius is congruent with our radial prole analysis which does not suggest that the cluster is considerably lar ger than this size. The T rapezium, ho we v er is only the core of the lar ger Orion Neb ular Cluster whose outer radius is r t id al25 pc ( Hillenbrand & Hartmann 1998 ) and which does appear to ha v e radial IMF v ariations on these scales. Our surv e y of NGC 2362 co v ers an equi v alent area to the entire ONC, ho we v er because of its adv anced age the cluster may ha v e e xpanded be yond these boundaries. If NGC 2362 were simply the core of an unbound association, then with an initial v elocity dispersion of 1 2 km/sec imprinted on the stars by the molecular cloud members could ha v e scattered to a radii of 5 10 pc. Indeed, we can conclude that there is some e vidence in the radial prole of the cluster (Figure 5–3 ) and the dif ferential 2MASS KLF of the cluster “halo” re gion (Figure 5–6 b) that NGC 2362 e xtends be yond the boundary of the La Silla surv e y and to radial distances of 5 10 pc.

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175 It is not necessary that the lar ger size of NGC 2362 be the result of the e xpansion of an unbound cluster Indeed, the presence of O stars and the apparently rapid clearing of the parental molecular cloud do not pre v ent the cluster from emer ging as a bound entity as w as sho wn in the recent dynamical modeling of Kroupa et al. ( 2001 ). Using N-body models that included the potential ener gy from the molecular cloud and tuned to ha v e initial conditions (e.g., central stellar density number of members) similar to the ONC, Kroupa et al. found that despite the rapid remo v al of the molecular g as by O stars and with v ery mar ginal star formation ef ciencies (i.e.,03) a bound cluster can form containing approximately 30% of the original members (i.e. 70% of the sources are either ejected or tidally stripped in the rst 100 Myr of the cluster' s e v olution). These models also sho w that such a cluster under goes v ery rapid changes in its structure (i.e., core radius and outer boundary) within the r st fe w Myr after the cloud is remo v ed. Since NGC 2362 is near the dynamical relaxation time for the ONC (6.5 Myr Hillenbrand & Hartmann 1998 ) and both contain O stars, it is possible that NGC 2362 represents a dynamically e v olv ed ONC. In such a model, we already ha v e tw o pieces of information from our surv e y First, the surf ace density of stars within a radius of 20 pc will ha v e decreased from the 164 stars pc2 found by Hillenbrand & Hartmann after a v eraging o v er the entire ONC to 11 stars pc2 found in our La Silla surv e y Second, there are as man y stars, one solar mass and greater in the cluster halo as in the cluster core, although much of the dynamical mass se gre g ation to the halo should consist of subsolar mass stars ( Kroupa et al. 2001 ). A v ery wide-eld optical study of NGC 2362, mer ged with a complete 2MASS catalog, w ould impro v e upon this rough e vidence for a dynamically e xpanding open cluster Such observ ations w ould allo w for indi vidual cluster members to be separated from eld stars by their location in the optical and optical-infrared color -magnitude diagrams (e.g. Adams et al. 2001 ; Moitinho et al. 2001 ). This w ould alle viate the need for lar ge background corrections as well as allo w for the construction of more

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176 accurate radial proles which could be t with analytical proles (e.g., King models) to determine cluster parameters such as core and tidal radii. Further the adv ent of wide eld infr ar ed imaging such as our use of the FLAMINGOS instrument when studying IC 348, allo ws us to probe the entire cluster halo to subsolar depths in both the optical and the infrared. Thus, NGC 2362 may be an ideal template for studying the dynamical e v olution of young clusters as the y emer ge from their parental molecular cloud. 5.7 Conclusions Combining model luminosity functions with a near -infrared census of the young, distant open cluster NGC 2362, we e xplored the usefulness of the luminosity function method when little is kno wn about the age of an open cluster W e found that we are able to place constraints on both the mean age and IMF of this cluster by xing only basic quantities such as cluster distance. When we assumed a mean age obtained from the cluster' s optical color -magnitude diagram, we were able to directly deri v e the cluster' s IMF From these modeling results and by comparison of the NGC 2362 KLF and IMF to those functions deri v ed for the T rapezium and IC 348 in chapters 3 and 4 we dra w the follo wing basic conclusions: 1. From our modeling of the dif ferential NGC 2362 KLF using a x ed mean age, we deri v e a cluster IMF that spans a mass range from 10 to 0.2 M. Ov er this range, the cluster' s IMF is v ery similar to the stellar portion of the IMFs that we deri v ed for the other clusters in this study all of which appear to form broad peaks at subsolar masses. Although our current study of NGC 2362 does not probe the substellar re gime, we nd that we are able to probe the IMF do wn to a mass limit v ery similar to the sensiti vity of spectroscopically based IMF studies in nearby clusters, although NGC 2362 is 1.5 kpc a w ay 2. W e nd that under the assumption of an uni v ersal IMF we can deri v e the age of NGC 2362 from its luminosity function and that this KLF deri v ed age is v ery similar

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177 to that found from placing the cluster on the color -magnitude diagram. W e nd that this method has an intrinsic accurac y of 1-2 Myr set by the statistical size of the cluster The assumed age spread in a cluster has little ef fect on the deri v ation of the mean age, ho we v er ignoring unresolv ed binaries will tend to produce too young of an age. 3. Simultaneous ts of the cluster' s mean age and po wer -la w IMF slope yield reasonable constraints on both parameters. This result is due in part to the f act that LF e v olution (to f ainter magnitudes) slo ws with increasing age, reducing the intrinsic uncertainty in the deri v ed IMF Such a modeling e x ercise suggests that luminosity function is a po werful tool for studying clusters e v en when little is kno wn about the cluster' s age.

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CHAPTER 6 CIRCUMSTELLAR DISKS AR OUND Y OUNG BR O WN D W ARFS Among the most fundamental issues raised by the e xistence of bro wn dw arfs is the question of their origin and genetic relationship to planets and stars. Are bro wn dw arfs giant planets or small, f ailed stars, or something else altogether dif ferent? The critical test needed to resolv e this question is to determine whether bro wn dw arfs primarily form within circumstellar disks as companions to stars, similar to planets, or from their o wn indi vidual cloud cores or fragments, lik e stars. T o date, the most important observ ations bearing on this question ha v e been: 1) the observ ed lack of close bro wn dw arf companions found in radial v elocity surv e ys of nearby eld stars (the so-called bro wn dw arf desert, e.g., Marc y & Butler 1998 ) and 2) the e xistence of free oating bro wn dw arfs in young clusters (e.g., Bouvier et al. 1998 ). Both f acts w ould appear to implicate a stellar (non-planet lik e) origin for these objects, i.e., formation from independent, contracting fragments of the parental molecular cloud. Ho we v er our understanding of the origin of sub-stellar objects is f ar from complete. F or e xample, an alternati v e formation scenario has been recently proposed by Reipurth & Clark e ( 2001 ) who suggest that most freely oating bro wn dw arfs did not form from their o wn proto-stellar fragments, b ut instead were initially formed as companions to other protostars and then were dynamically ejected via 3 body encounters before the y could gro w into stellar mass objects. The most direct w ay to address the question of the origin and nature of bro wn dw arfs is to in v estig ate the properties of e xtremely young sub-stellar objects in re gions of acti v e star and planet formation. F or e xample, nding young bro wn dw arfs surrounded by their o wn circumstellar accretion disks w ould lik ely implicate a stellar -lik e formation mechanism (from indi vidual cloud fragments) and place strong 178

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179 constraints for the theoretical models of their origin (e.g., Reipurth & Clark e 2001 ). Moreo v er such a nding w ould raise the interesting question of whether planetary companions can form around such objects. Recently Lada et al. ( 2000 ) used near -infrared (13 m ) color -color diagrams to sho w that a lar ge fraction (8085 %) of the stars in the young T rapezium Cluster display thermal infrared e xcess indicati v e of circumstellar disks. Further the y found that the fraction of stars with disks remained high with decreasing mass to near the h ydrogen b urning limit. Belo w this limit, their observ ations became incomplete. Does the incidence of circumstellar disks also continuously e xtend across the h ydrogen b urning limit to sub-stellar mass objects? As we ha v e sho wn in chapters 2 and 3 deeper infrared observ ations ha v e re v ealed a substantial population of f aint sources which appear to be free oating sub-stellar objects in this cluster ( McCaughrean et al. 1995 ; Muench et al. 2000 ; Lucas & Roche 2000 ; Hillenbrand & Carpenter 2000 ; Luhman et al. 2000 ; Muench et al. 2002 ). As we e xtensi v ely discussed in Section 3.2.3 ho we v er the identications of these sources as sub-stellar objects are not secure because our observ ations of nearby un-reddened control elds re v ealed signicant numbers of eld stars in the corresponding brightness range (see also Figure 3–4 ), suggesting that eld star contamination could be a se v ere problem, especially for the f aintest candidates. Our attempts in Section 3.2.3 to account for the ef fects of the screen of e xtinction pro vided by the molecular cloud behind the T rapezium do suggest the v ast majority of the bro wn dw arf candidates are not reddened eld stars, despite the considerable uncertainties intrinsic to that statistical analysis (also see our discussion in Section 3.4.1 on the ef fects of membership uncertainties on the deri v ation of the cluster' s IMF). Thus, independent conrmation of membership is clearly important and could be pro vided by indications of e xtreme youth, such as the presence of infrared e xcess and dusty disks surrounding these objects.

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180 In this chapter we present an observ ational analysis of the colors of the bro wn dw arf population for the T rapezium cluster using the deep NTT infrared data described in Section 3.1 In Section 6.1 we nd that a relati v ely lar ge fraction of the candidate bro wn dw arfs e xhibit infrared e xcess indicati v e of circumstellar disks. As we discuss in Section 6.2 this conrms both their membership in the cluster and their status as sub-stellar objects and perhaps suggests an origin for them that is more stellar -lik e than planetary-lik e. 6.1 T rapezium Br o wn Dwarfs with Infrar ed Excess Figure 6–1: Selecting candidate bro wn dw arfs in the T rapezium. W e plot the infrared color -magnitude diagrams for only those T rapezium sources ha ving NTT observ ations and JH and K s magnitudes. The sources are compared to the location of the 1 Myr (at 400pc) isochrone from the BCAH98 e v olutionary models and atmospheres. Candidate bro wn dw arfs (lled circles) were selected by their H band luminosities (and colors) and are mark ed in both color -magnitude diagrams. Reddening v ectors for 1, 0.08 and 0.02 Mobjects are dra wn at visual e xtinctions of 20, 20 and 10 magnitudes, respecti v ely Stars with spectral typesM6 are identied as lled stars. a) J-H/H color -magnitude diagram. b) H-K s /K s color -magnitude diagram.

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181 In Figure 6–1 we construct the infrared color -magnitude diagrams for those NTT T rapezium sources which were simultaneously detected at JHK s w a v elengths. Thus, the sample sho wn here dif fers from that sho wn in Figure 3.2.2 (a) by not including sources that were not detected at J band. In these diagrams, we compare the locations of these sources to the location of the theoretical isochrone from the Baraf fe et al. ( 1998 , BCAH98) non-gre y e v olutionary models at the assumed mean age (1 Myr) and distance (400 pc, see appendix B ) of the T rapezium cluster The BCAH98 theoretical isochrone closely follo ws the near -IR colors of the T rapezium sources, forming an e xcellent left-hand boundary to the source distrib ution in this color magnitude space. The T rapezium sources are reddened a w ay from this boundary with implied e xtinctions of A V135 mag. W e identied candidate bro wn dw arfs in the T rapezium Cluster by comparing the infrared luminosities of detected sources to those predicted by the theoretical e v olutionary models. W e selected all the NTT sources in the J-H/H diagram (Figure 6–1 a) f ainter than the predicted luminosity of the h ydrogen b urning limit (hereafter HBL;008 M) 1 b ut brighter than the luminosity of an 0.02 Mobject. This lo wer limit w as chosen because the current theoretical e v olutionary models do not e xtend much belo w this mass, and because we wish to e xclude cooler lo wer mass objects whose intrinsic colors are not well constrained. Between these tw o mass/luminosity limits we identied 112 candidate bro wn dw arfs in the J-H/H diagram. W e also indicate in Figure 6–1 the locations of 19 T rapezium Cluster members with spectral types equal to or later than M6 tak en from Hillenbrand ( 1997 ), Lucas et al. ( 2001 ) or Luhman (pri v ate communication). The spectral type M6 is an important boundary 1 The predicted colors and magnitudes of the h ydrogen b urning limit for this distance/age combination are essentially identical to those for a younger assumed age (0.4 Myr) b ut at a lar ger distance (470pc).

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182 because recent spectroscopic studies ha v e suggested that it represents the h ydrogen b urning limit in v ery young ( t10Myr) clusters ( Luhman 1999 ). In Figure 6–1 (a), these late type sources are on a v erage 1 magnitude brighter than our adopted h ydrogen b urning limit. The f aintness of our IR selected bro wn dw arfs relati v e to these late type sources conrms that we are lik ely selecting sources belo w the HBL. W e rene our selection of bro wn dw arf candidates by plotting the J-H/H candidates in the H-K s /K s color -magnitude diagram in Figure 6–1 (b). In this diagram a number of candidates are brighter and redder than the h ydrogen b urning limit. W e retain these as lik ely bro wn dw arf candidates because the y ha v e photometric errors which are much too small to ha v e scattered them to this location, because the y are f ainter than most of the M6+ dw arfs, and because e xcess 2 m ux from circumstellar disks could act to brighten and redden such sources out of the bro wn dw arf re gime in the H-K s /K s color -magnitude diagram. A fe w v ery f aint candidates scatter belo w the 0.02 Mlimit in the H-K s /K s diagram, and we e xclude these sources from our nal sample. In Figure 6–2 we plot the H-K s /J-H color -color diagram for the 109 candidate bro wn dw arfs. W e also plot for comparison the loci of colors for giants and for mainsequence dw arfs from Bessell & Brett ( 1988 ). W e e xtended the loci of M dw arf colors in Figure 6–2 from M6 to M9 using the empirical bro wn dw arf colors compiled in Kirkpatrick et al. ( 2000 ). The predicted ef fecti v e temperatures of 1 Myr bro wn dw arfs from the BCAH98 e v olutionary models are quite w arm, e.g. T e f f2500K for masses greater than our 0.02 Mlimit. Therefore, we e xpect that the intrinsic infrared colors of such young bro wn dw arfs are no redder than those of M9 dw arfs (J-H = 0.72; HK s = 0.46 Kirkpatrick et al. 2000 ) which agree well with the H-K colors predicted by

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183 Figure 6–2: T rapezium bro wn dw arfs with near -infrared e xcess. H-K s /J-H color color diagram for the 109 sources with NTT JHK s magnitudes and which f all into the bro wn dw arf re gime of Figure 6–1 The candidate bro wn dw arfs are compared to the intrinsic colors of giants and A0-M6 dw arfs from Bessell & Brett ( 1988 ), the late M (M6 M9) color sequence from Kirkpatrick et al. ( 2000 ) and the Classical T -T auri locus from Me yer et al. ( 1997 ). Appropriate reddening v ectors ( Cohen et al. 1981 ) are dra wn for giants, for M6 stars and for M9 stars. Colors of T rapezium sources with v ery late (M6+) kno wn spectral types are sho wn as stars. Circled sources ha v e color errors of less than 10% and the size of 15% uncertainties in color are illustrated at the upper right. current model atmospheres of lo w surf ace gra vity 2600 K sources ( Allard et al. 2001 2 ) W e nd 65%15% (71/109) of our candidates f all to the right of the reddening band for M dw arfs and into the infrared e xcess re gion of the color -color diagram. W e 2 see also nrn !#"$$%

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184 further determine that 54% of the candidates ha v e an infrared e xcess that is greater than their 1 s photometric uncertainties in color In addition to normal photometric uncertainties the measured colors of these sources could be corrupted by the presence of the strong neb ular background, and we performed an e xtensi v e set of articial star photometry e xperiments to test this possibility W e found that neb ular contamination can introduce some additional scatter to a star' s measured J-H color and this can e xplain in part the J-H colors of25% of the e xcess sources which are bluer than e xpected for late type sources (J-H06). Ho we v er blue w ard J-H scatter can produce a f alse e xcess fraction (1020%) only for the the f aintest articial stars, i.e., H = K =16 mag. Further we nd that such neb ular contamination ne v er produces as lar ge a dispersion of the H-K s colors as found in our observ ations of the candidate bro wn dw arfs, and we conclude that the observ ed infrared e xcesses are an intrinsic property of these objects. 6.2 Discussion and Implications From analysis of their near -infrared colors, we nd that50% of the candidate bro wn dw arfs in the T rapezium cluster display signicant near -infrared e xcess. This is similar to the beha vior of the stellar population of this cluster and suggests the e xtreme youth of these lo w luminosity sources. This, in turn, pro vides independent conrmation of their membership in the cluster and their nature as bona de substellar objects. As is the case for the more massi v e stellar members, the most lik ely e xplanation for the observ ed near -infrared e xcesses around the bro wn dw arfs in this cluster is the presence of circumstellar disks. Strong, independent support for the disk interpretation deri v es from the f act that we nd 21 of the candidate bro wn dw arfs to be spatially coincident with optically identied HST “proplyds” ( O' dell & W ong 1996 ; Bally et al. 2000 ) which are kno wn to be photo-e v aporating circumstellar disks. W e illustrate HST images from Bally et al. of three of the IR luminosity selected bro wn dw arfs that display the proplyd characteristics in Figure 6–3 W e note that the proplyd

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185 bro wn dw arfs display a JHK e xcess fraction of 71%, while the bro wn dw arf candidates unassociated with kno wn proplyds ha v e a slightly lo wer e xcess fraction of 63%. The proplyd bro wn dw arfs also display bluer J-H colors than the remaining bro wn dw arf candidates and account for half the e xcess sources with J-H color06. Despite their relati v ely blue J-H colors, the proplyd nature of these sources af rms the h ypothesis that the observ ed JHK infrared e xcess is intrinsic and a signature of the presence of a circumstellar disk. Figure 6–3: Bro wn dw arf proplyds. HST images (F656N lter) of three infrared selected bro wn dw arf candidates which are also optical proplyds. Data from Bally et al. ( 2000 ) with permission. The h ypothesis that the observ ed near -IR e xcess is caused by circumstellar disks is further supported by observ ations of bro wn dw arf candidates in other clusters. Late-type bro wn dw arf candidates in the r Ophiuchi cluster were identied by their w ater v apor absorption features and display e vidence for v eiling in their infrared spectra as well as e vidence for infrared e xcesses in their H-K/J-H color -color diagrams ( W ilking et al. 1999 ; Cushing et al. 2000 ). ISO (67 m ) observ ations re v eal 4 bro wn dw arf candidates with mid-infrared e xcesses in Chamaeleon ( Comer on et al. 2000 ). Luhman ( 1999 ) identied 7 bro wn dw arf candidates in the IC 348 cluster which, after de-reddening, f all to the right of the main-sequence reddening band b ut belo w the cTTS locus similar to the locus of bro wn dw arfs identied here. Luhman also identied strong H a emission (WH a 10 A ) in a number of these sources and suggested that these are not simply passi v e circumstellar disks, b ut that these bro wn

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186 dw arfs are under going accretion. Finally po werful e vidence for accretion disks around v ery young bro wn dw arfs w as found by Muzerolle et al. ( 2000 ) who identied an asymmetric H a emission line prole for the M6 PMS object V410 Anon 13 in T aurus and successfully used magnetospheric accretion models to sho w that this bro wn dw arf candidate w as indeed accreting b ut at a rate much lo wer than has been found in higher mass stars. Compared to these other studies, our sample of T rapezium Cluster bro wn dw arfs is the rst population that is suf ciently lar ge to statistically estimate the frequenc y of sub-stellar objects born with circumstellar disks. Indeed, there are no w more bro wn dw arfs identied in the T rapezium cluster than are presently kno wn in all other star forming re gions combined. Ho we v er our estimate of the disk frequenc y from the JHK diagram could underestimate the true disk frequenc y for a number of reasons. First, JHK observ ations trace the innermost re gions of disks and the particular disk geometry (inclination, presence of inner disk holes, etc.) can act to reduce the ef cienc y of detecting disks from JHK photometry especially for late type sources ( Lada & Adams 1992 ; Hillenbrand et al. 1998 ). F or e xample, the50% e xcess fraction we nd for bro wn dw arfs is nearly identical to the e xcess fraction found in the JHK diagram of Lada et al. ( 2000 ) for objects in the cluster which are abo v e the h ydrogen b urning limit. Ho we v er by emplo ying 3 m photometry these authors found a much higher ,85%, disk frequenc y e v en for the f aintest members the y detected. Similarly of the 19 M6-M9 sources with kno wn spectral types and displayed in Figure 6–1 7 display v ery small near -infrared e xcesses37%. But of the 14 such sources with spectral types and detections at L band, all b ut one source displays much stronger e vidence for 3 m e xcess as sho wn in Figure 6–4 although the intrinsic K-L colors for M6-M9 stars are some what uncertain. Second, as a result of selecting candidate bro wn dw arfs at all reddenings we may ha v e included reddened background eld stars in the sample which w ould act

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187 Figure 6–4: L-band observ ations of bro wn dw arf candidates. The fraction of bro wn dw arfs displaying infrared e xcess is lik ely a lo wer limit to the true disk fraction. Longer w a v elength observ ations (L band, 3 m ) such as those used in Lada et al. ( 2000 ) are necessary to better understand the spectral ener gy distrib utions of these sources and to deri v e a more secure estimate of the disk fraction. F or e xample, of the 19 late spectral type (M6+) sources, 14 were detected at L band (3 m). Though 37% of these sources display IR e xcess at K band 2 m, 13 of 14 ha v e K s L colors indicati v e of cool disks. to decrease the deri v ed disk fraction. When we select candidates at lo w reddenings (A V5 relati v e to the isochrone in the Figure 6–1 a) to e xclude background eld stars, we nd 77% (57/74) of this sample display infrared e xcess. Further this sample is an e xtinction limited sample which is complete at all masses in our selected range and therefore is lik ely representati v e of the population as a whole. W e conclude from our current study and from the ndings of Lada et al. ( 2000 ) that circumstellar disks are present around a high fraction of T rapezium Cluster members acr oss the entir e mass spectrum This implies that bro wn dw arfs and higher mass

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188 stars form via a similar mechanism, e.g., from indi vidual contracting fragments of the parental molecular cloud which, via conserv ation of angular momentum, form a central star accompanied by a circumstellar disk ( Shu et al. 1987 ). Lo w & L ynden-Bell ( 1976 ) sho wed that within the conditions of molecular clouds, the minimum Jeans mass for a cloud fragment could be as small as 0.007 M, well belo w the mass necessary to create the T rapezium bro wn dw arfs. The free-oating nature of these bro wn dw arfs rules out their formation as companions in a circumstellar disk. Thus our results seem to implicate a formation mechanism for bro wn dw arfs in which such objects are formed with circumstellar disks from indi vidual protosub stellar cores. Consequently e v en sub-stellar objects may be capable of forming with systems of planetary companions. Conrmation of our h ypothesis that a substantial fraction of the bro wn dw arfs in the T rapezium are surrounded by circumstellar disks requires additional data. Deep 3 m ground-based observ ations such as those used by Lada et al. ( 2000 ) are necessary to permit a more accurate measurement of the e xcess fraction for the entire bro wn dw arf population. Longer w a v elength infrared observ ations, such as those that will be possible with Space InfraRed T elescope F acility (SIR TF), w ould enable the construction of more complete SEDs for these sources which could then be compared directly to theoretical disk predictions. Estimates of the masses of the disks w ould ha v e interesting implications for the possibility of forming planetary companions around bro wn dw arfs. Finally high resolution spectra of these objects w ould enable searches for accretion indicators, such as H a emission, v eiling, etc., which could yield accretion rates and information about the gro wth and early e v olution of these interesting objects.

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CHAPTER 7 DISCUSSION ON THE STR UCTURE OF THE IMF 7.1 Y oung Clusters and the Global IMF From our current w ork and by comparison of our w ork to that of other authors, the general structure of the IMF in these three young stellar clusters is readily apparent: a continuous IMF that rises with a relati v ely steep slope to w ard subsolar masses, attens and forms a broad peak between 03 Mand the h ydrogen b urning limit before turning o v er and declining into the bro wn dw arf re gime. This IMF structure is roughly half-g aussian, though not e xactly log-normal (see Figure 3–14 b). Further it is quite consistent with current deri v ations of the IMF in other star clusters. Mass functions for open clusters such as the Pleiades ( Bouvier et al. 1998 ) and M 35 ( Barrado y Na v ascu es et al. 2001 ) rise with similar po wer -la w slopes and form broad peaks at subsolar masses before apparently rolling o v er and declining into the bro wn dw arf re gime. Additionally the color -magnitude diagrams of v ery luminous clusters such as NGC 3603 ( Brandl et al. 1999 ) and NGC 6231 ( Baume et al. 1999 ) all display e vidence of IMFs that peak at subsolar masses, though a more complete discussion a w aits detailed deri v ations of their subsolar and substellar IMFs. Extending this comparison to glob ular clusters using the study of P aresce & De Marchi ( 2000 ) we nd that e v en these older more populous clusters all ha v e IMFs that form broad peaks at subsolar masses similar to the open and embedded star clusters. On the other hand, the mode of these glob ular cluster IMFs is consistently around 03 Mor some what higher than that we deri v e in IC 348 or the T rapezium e v en after accounting for the range of IMFs permitted by our models. Perhaps more interesting, the form of the IMF in all of these star clusters is v ery similar to the initial mass function deri v ed for stars in the g alactic eld ( Salpeter 1955 ; Miller & Scalo 1979 ; Scalo 189

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190 1986 ; Kroupa et al. 1993 ; Scalo 1998 ; Kroupa 2001 ; Chabrier 2001 ). F or e xample, both the Scalo ( 1986 ) and Kroupa et al. ( 1993 ) eld star IMFs rise in number with decreasing mass in a manner nearly identical to the young clusters, ha ving intermediate (110 M) mass slopes of G 1 13 to17. Further these eld star IMFs atten between 1 and 05 M(we deri v e07 M) ha ving subsolar po wer -la w slopes of G 2 03 to02, similar ag ain to that IMF structure we nd for the young clusters. The e xistence of a peak (or mode or “characteristic mass”) in the eld star IMF w as rst suggested by Miller & Scalo ( 1979 ) b ut it is not yet clear at what mass this IMF mode lies. The most recent study of the V band luminosity function for eld stars does not nd e vidence for a mode or turno v er at masses greater 01 M( Chabrier 2001 ), which w ould mean that the mode of the IMF in the eld is signicantly dif ferent from and shifted to lo wer masses relati v e to that IMF found for glob ular clusters. In this re g ard, the eld star IMF is more similar to our ndings for young embedded clusters than to the older glob ulars; ho we v er we also learned that the determination of the IMF mode in these young clusters is not straightforw ard. First, we found in section 3.4.3 that dif ferent deri v ations of the T rapezium IMF yielded IMF modes that v aried more than the e xpected uncertainties from an y one method and had a dispersion that spanned the range from the glob ular cluster IMF mode to the current constraints on the IMF in the eld. Second, and perhaps more important, we found and discussed in Section 4.4.2 that there is e vidence for radial v ariations in the subsolar IMFs of both the IC 348 and the Orion Neb ula Cluster of which the T rapezium is the core. These radial v ariations, re g ardless of origin, yield dif ferent IMF modes for a cluster depending upon what area of the cluster is sampled. Only when we e xamine these young cluster' s composite IMFs do the modes appear to sk e w to w ard the h ydrogen b urning limit, resulting in IMFs that appear more similar to the constraints placed on the eld than to the IMF mode found in the glob ulars.

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191 Re g ardless of these potentially meaningful details, if we broadly compare these IMFs we nd that all of these stellar systems display remarkable consistenc y in the form of their IMFs; a similar conclusion has recently been reached by Kroupa ( 2002 ). It is not clear ho we v er the e xtent to which this documented continuity between the cluster and eld star IMFs e xtends across the h ydrogen b urning limit and throughout the substellar re gime. The aforementioned Pleiades cluster IMF rolls o v er and declines into the substellar mass re gime with a slope slightly atter G bd s 05 ( Bouvier et al. 1998 ; Hambly et al. 1999 ), than that we ha v e deri v ed for these tw o embedded clusters G bd s 0720, although the substellar IMFs in these embedded clusters are in f act better populated than in the Pleiades. Although IC 348 and the T rapezium IMFs each decline into the substellar re gime with a dif ferent IMF slope, the y in f act ha v e nearly identical percentages of bro wn dw arfs as members, i.e.,25%. On the other hand, the IMF for eld bro wn dw arfs is not yet rob ustly kno wn as reected in the tw o some what contradictory results of Reid et al. ( 1999 ) and Chabrier ( 2002 ). While Reid et al. suggests a rising substellar IMF (implying bro wn dw arfs dominate stars by number), Chabrier ( 2002 ) determines that a at substellar IMF is an upper limit and that at most the space density of bro wn dw arfs equals that of stars (i.e.,01102 pc3 ), although this is still twice the frequenc y of bro wn dw arfs seen in young clusters. T w o ndings in star -forming re gions also mak e it dif cult to conrm the uni v er sality or stochastic nature of the substellar IMF First, current surv e ys nd an apparent dearth of bro wn dw arfs in the isolated star -forming re gion T aurus ( Luhman 2000 ). If this nding is conrmed and is not the product of small number statistics, then the lack of bro wn dw arfs in a star -forming re gion that primarily creates only handfuls of solitary or binary stars rather than rich clusters w ould be solid proof for a v ariation in the IMF as a function of star -forming en vironment.

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192 Second, our nding of peaks in the substellar K band luminosity functions of IC 348 and the T rapezium may be v ery important for testing a h ypothesis of an uni v ersal IMF dra wn from a dominate star formation process. This is because, as we discuss further in the ne xt section, these peaks can be e xplained by tw o alternate h ypotheses: 1) that a secondary peak forms in the substellar initial mass function near the deuterium-b urning limit, suggesting that an alternati v e mechanism may form the lo west mass objects, or 2) that a pre viously unkno wn feature in the e v olution of lo w mass bro wn dw arfs e xits and is not included into current theoretical models on which all deri v ations of the substellar IMF rely 7.2 Secondary Sub-Stellar P eak in the Cluster LFs As we deri v ed in Chapter 3 the secondary peak in the substellar re gime of the T rapezium KLF can be attrib uted to a corresponding IMF peak near the deuteriumb urning limit using current e v olutionary models. Our nding in Chapter 4 of an IC 348 KLF feature ha ving a similar size and corresponding to the same mass range, 1020 M J u p lends some support to this conclusion. In addition, the nearby 350 pc5 Myr s Orionis open cluster ( W alter et al. 1997 ) has been the tar get of recent imaging surv e ys that are suf ciently sensiti v e to detect such IMF structure. From an optical and near -infrared surv e y of 847 arcmin 2 of this cluster B ejar et al. ( 2001 ) deri v ed a v ery at substellar IMF ha ving a slope of only G bd s 02, b ut that appeared to be rapidly rising near the deuterium b urning limit. T aking the J band data from this w ork, we construct the s Ori JLF in Figure 7–1 a. Belo w a bright LF peak that corresponds to 01 Mat the cluster' s mean age and that is lik ely incomplete due to source saturation in the B ejar et al. surv e y the s Ori JLF f alls in number with increasing magnitude before forming a v ery broad plateau. T o mak e detailed comparisons between the T rapezium and s Orionis substellar IMFs, we re-deri v ed the s Orionis IMF using the J band data published in B ejar et al. W e follo wed their prescription of using a coe v al mass-luminosity relation for a 5

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193 Figure 7–1: Comparison of T rapezium and s Ori IMF A) The J band LF of the Sigma Orionis substellar members (photometry from B ejar et al. ( 2001 )) B) T rapezium IMF (using B97 tracks) compared to s Ori IMF The s Ori IMF w as deri v ed by directly computing mass estimates from a coe v al mass-luminosity relation at 5 Myr In this panel this M-L relation w as from the Chabrier et al. ( 2000 ); Baraf fe et al. ( 2002 ) tracks. C) T rapezium IMF (using B97 tracks) compared to s Ori IMF (using DM97 tracks). Myr cluster at 352 pc to deri v e indi vidual masses for the sources and then binned the resulting lo garithm of their masses into a e v enly spaced histogram. W e used the mass-J magnitude relations from the v ery recent DUSTY PMS tracks of Chabrier et al. ( 2000 ) (with updates from Baraf fe et al. 2002 ) because the B97 mass-luminosity relations display a v ery strong inection at 5 Myr that is unseen in other PMS tracks (see Figure 3–17 e). As sho wn in Figure 7–1 bc, the T rapezium and s Orionis substellar IMFs

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194 decline in number with decreasing mass in a nearly identical f ashion do wn to around 002 to 003 M. What at rst appeared to be lar ge dif ferences in the slope of the substellar IMF of the T rapezium and s Orionis as deri v ed by B ejar et al. turned out to be only phenomenological dif ferences due to the random binsizes used by B ejar et al. in the construction of the s Orionis IMF and their approximation of this clearly nonpo wer la w IMF by a single po wer -la w inde x. Belo w this 30 M J u p the substellar IMFs of both clusters break and form either a secondary peak or a broad plateau at the deuterium-b urning limit. As we ha v e already concluded, the morphological details of the secondary peak in the cluster IMF at the deuterium-b urning limit are some what model dependent. F or e xample, a stronger secondary peak forms in the s Orionis IMF at the deuterium-b urning limit if we use the DM97 tracks (Figure 7–1 c). Further the assumption of coe v ality for a non-coe v al cluster w ould articially broaden an y potentially sharp secondary peak at the deuterium-b urning limit. T ak en as a sum, we ha v e sho wn that three young clusters at three dif ferent ages (1,2 and 5 Myr) all ha v e luminosity functions that form peaks or broad plateaus that could correspond to similar IMF features in the mass range from 10 20 M J u p Ho we v er since the observ ed luminosity function is the product of the IMF and the slope of the mass-luminosity r elation it may also be the case that a subtle feature or inection e xists in the empirical mass-luminosity relation that has not been resolv ed by e v en the most recent theoretical models of bro wn dw arfs that we ha v e used (e.g., Chabrier et al. 2000 ; Baraf fe et al. 2002 ). Such an undocumented M-L feature could also produce a peak in the luminosity function independent of the structure of the underlying IMF as w as pre viously found, for e xample, in the Kroupa et al. ( 1990 1993 ) studies of the Hand H 2 opacity features in the optical MM V relation for eld stars. There is some observ ational e vidence that seems to point to w ard an undocumented feature in the e v olution of bro wn dw arfs. After in v estig ating the color -magnitude

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195 diagrams of a number of open clusters with a wide range of ages (1 100 Myr) and including s Orionis, Jameson ( 2002 ) pointed out the presence of a g ap in the substellar re gime of the clusters' loci on the observ ational HR diagram. Interestingly this g ap occurred at nearly constant color as a function of age, suggesting it is related to the source' s temperature and not related to the mass function. This is because after 10 Myr v ery lo w mass stars and bro wn dw arfs be gin to e v olv e to signicantly cooler temperatures with increasing age, and such a color feature, if it were related to the mass function, should also e v olv e to redder or cooler colors with age. From this data Jameson h ypothesized the e xistence of a pre viously unkno wn feature in the massluminosity and mass-color relations for substellar objects, a feature that could be responsible for the secondary JLF and KLF features that e xist in s Ori, IC 348 and the T rapezium. Re g ardless, it is clearly uncertain which ef fect (IMF M-L relation) will in f act dominate the nature of the secondary LF peaks and features we ha v e observ ed. Another conclusion from the Kroupa et al studies that is rele v ant to our deciphering the secondary LF peaks is that until an empirical mass-luminosity feature is accurately constrained, the size of the resulting LF peak due only to the inection in the mass-luminosity relation (hence independent of the IMF), will be v ery uncertain ( Kroupa & T out 1997 ). Therefore it remains up to future observ ations to disentangle these tw o ef fects for the v ery lo west masses in these young clusters and to determine whether a substellar IMF peak may e xist. Ho we v er since whiche v er mechanism that is producing this LF peak appears to operate only at the v ery lo west masses in these v ery young clusters, it seems unlik ely that it will modify the composite cluster IMFs we deri v e in the stellar re gime and do wn to and across the h ydrogen b urning limit. Further a ne w M-L feature at the deuterium-b urning limit, for e xample, probably will not adjust (inate or decrease) the percentage of sources that are bro wn dw arfs in these

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196 clusters. Indeed, in the case of the T rapezium, such a feature cannot change the direct counting of sources that appear substellar 7.3 New Clues to the Origin of Stars and Br o wn Dwarfs The continuity of the structure of the IMF across so man y en vironments and the lack of meaningful de viations from this global IMF structure (to which our models w ould be sensiti v e) suggests that a single star formation process may be responsible for producing the majority of the mass spectrum. The alternate interpretation of a relati v ely uni v ersal IMF form is that the number of v ariables or contrib uting processes to the IMF is suf ciently lar ge that the y almost al w ays conspire to produce a single “uni v ersal” IMF no matter ho w the y indi vidually v ary ( Adams & F atuzzo 1996 ). T w o important ndings from our w ork may shed light on the origin of the IMF in the conte xt of these h ypotheses. First, we found in Chapter 6 that stars and bro wn dw arfs form with similarly high initial frequencies of circumstellar disks. Combined with the nding that these bro wn dw arfs disks appear to ha v e v ery similar properties to those disks found around stars ( Natta & T esti 2001 ), this e vidence suggests that stars and bro wn dw arfs form via a similar ph ysical mechanism, i.e. as contracting fragments of the molecular cloud. While it may be the case that v arious ph ysical processes might inuence the ne details (e.g., the IMF' s “peak” or mode, for e xample) of the IMF' s nal form ( e.g., Adams & F atuzzo 1996 ), the original fragmentation distrib ution function of a turb ulent molecular cloud probably dominates the nal form of the stellar and substellar IMF ( Klessen 2001 ) do wn to v ery lo w masses. The second nding from this w ork that concerns the origin of the IMF is the suggestion that this continuous, globally consistent mass function breaks and forms a secondary peak near the deuterium-b urning limit in a number of the youngest clusters. W ere a secondary peak in the IMF of the lo west mass bro wn dw arfs conrmed, then it may pro vide e vidence for a secondary competing formation mechanism for these

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197 lo w mass objects. Indeed, the transition in the substellar IMF at 30 M J u p from a steady po wer -la w decline to the secondary peak at the deuterium b urning limit may represent the transition from the formation of bro wn dw arfs as indi vidual fragments of the molecular cloud to their formation, for e xample, as truncated stellar embryos that were dynamically ejected from hierarchical proto-stellar systems before the y had a chance to accrete into higher mass objects ( Reipurth & Clark e 2001 ). Alternately the determination that this secondary LF peak is due to an e v olutionary feature for bro wn dw arfs and not the IMF should not inhibit the search for such a transition in the formation mechanisms of bro wn dw arfs. The Reipurth & Clark e h ypothesis for e xample, mak es the prediction that proto-bro wn dw arfs that are ejected from the initially bound proto-stellar system will ha v e truncated disks with small radii ( r1020au ) and, thus will not ha v e similar disk lifetimes or disk structures relati v e to those qualities for the disks around stars. Thus, the frequenc y and characteristics of circumstellar disks around these v ery lo w mass bro wn dw arfs and planetary mass objects may pro vide an essential test of their formation from indi vidual presub -stellar cores or via some entirely dif ferent mechanism.

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CHAPTER 8 CONCLUSIONS AND FUTURE W ORK 8.1 On the Luminosity Functions of V ery Y oung Stellar Clusters W e ha v e conducted an e xtensi v e in v estig ation into the usefulness of the stellar luminosity function as a tool for in v estig ating v ery young ( t10 Myr) star clusters and for deri ving the underlying mass functions of these clusters. T o accomplish this general goal, we constructed a Monte Carlo based pre-main sequence populations synthesis algorithm for modeling young stellar populations with realistic mass functions and the direct ef fects of v arious observ ational quantities. After performing a series of numerical e xperiments and through application of this modeling code to the observ ations of three young clusters, we nd that the observ ed infrared luminosity function for a young cluster is an e xcellent tool for deri ving and comparing the form of the underlying IMFs of young clusters. A cluster' s observ ed luminosity function is also useful for making qualitati v e comparisons between clusters of dif ferent ages and as such can act as a probe of v ery distant clusters that cannot be e xamined via other methods. Summarizing the results of our construction of the population synthesis algorithm for pre-main sequence star clusters, the numerical e xperiments using these models, and the application of these models to the observ ed LFs from our observ ations, we dra w the follo wing general conclusions about model luminosity functions: 1. Numerical e xperiments using our model luminosity function algorithm re v eal that the intrinsic cluster luminosity function is most sensiti v e to the form of the underlying mass function and to a lesser de gree on the cluster' s assumed mean age. 2. The cluster' s model luminosity function e v olv es to f ainter magnitudes as the cluster' s age increases. This e v olution progresses as much during the cluster' s rst 3 Myr as it 198

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199 does during the time from 3 to 10 Myr meaning that uncertainties in the deri v ed IMF due to uncertainties in the estimate of the mean age decrease for older clusters. This w as seen in the case of our NGC 2362 study where reasonable constraints on both parameters (age and IMF) can be obtained pro vided only basic assumptions such as the cluster' s distance. 3. Model luminosity functions do not appear to depend v ery much upon the assumed set of theoretical mass-luminosity relations used to deri v e the mass-luminosity relation. The LF dif ferences resulting from changing sets of M-L relations are much smaller than can lik ely be observ ed and should not impact our ndings. W e nd that this is because dif ferent sets of theoretical e v olutionary models for canonical pre-main sequence e v olution predict remarkably similar mass-infrared luminosity relations despite signicant v ariations in the fundamental input ph ysics. This is in contrast to the ef fecti v e temperatures predicted by these e v olutionary models, which are v ery sensiti v e to input ph ysics. 4. The technique of modeling the cluster luminosity function appears v ery adept at dis-entangling kno wn mass-luminosity relation features from the form of the underlying IMF Thus, if future updates to the theoretical mass-luminosity relations change our pre vious conclusion about the rob ustness of the M-L relation by including ne w pre viously unkno wn M-L features, our modeling procedure has the ability to easily include these re visions and resolv e the cluster IMF Further we deri v ed a deep near -infrared census of sources in three young star clusters using sensiti v e near -infrared imaging of these clusters. From these observ ations we construct the clusters' dif ferential K band luminosity function by correcting the observ ed luminosity functions for the statistical contrib ution of interloping eld stars. From the construction and comparison of the luminosity functions of these three clusters we dra w the follo wing general conclusions about the near -infrared luminosity function of a young stellar cluster:

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200 1. The luminosity function of a v ery young star cluster can be constructed for sources do wn to and belo w the deuterium-b urning mass limit 001 M using the products of deep near -infrared surv e ys. 2. Although young clusters display a number of features in their observ ed luminosity functions that are ph ysically related to kno wn features in the mass-luminosity relation, we nd from our luminosity function modeling of these observ ations that the o v erall structure of the cluster' s near -infrared luminosity function directly reects the form of the underlying initial mass function. 3. Three young clusters that ha v e been the tar gets of infrared surv e ys sensiti v e do wn to the deuterium-b urning limit all display structure in their substellar luminosity function in the form of a secondary peak or plateau. These LF features occur in magnitude ranges much brighter than the sensiti vity limits of the photometric surv e ys and cannot be e xplained by uncertainties in the cluster membership. By comparing clusters of dif ferent ages, this feature appears to e v olv ed to f ainter magnitudes with increasing age and thus, appears to be an intrinsic feature of these clusters' infrared luminosity functions. 8.2 On the Initial Mass Functions of V ery Y oung Stellar Clusters Using our population synthesis models of the young stars and their luminosity functions and deep near -infrared surv e ys of three young (1-5 Myr) cluster we are able to deri v e the underlying initial mass functions for these clusters. In tw o of these clusters, these IMFs constitute nearly the complete range of stellar and substellar mass, from B stars do wn to the deuterium-b urning limit. In a third cluster we nd that we can deri v e the IMF to a mass sensiti vity similar to spectroscopic observ ations of nearby clusters although our tar get open cluster is at a distance of 1500pc, four to v e times the distance to these nearby clusters. Thus, luminosity function modeling has the ability to study the IMF o v er a v olume nearly tw o orders of magnitude greater than the use of spectroscopic measurements, illustrating our method' s usefulness as a probe of

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201 the uni v ersality of the IMF From these detailed studies, we dra w the follo wing basic conclusions about our modeling technique and the IMF of young clusters: 1. W e nd that the IMF deri v ed from modeling a cluster' s infrared luminosity function is in good agreement with that IMF determined either by using spectra to place the stars on the theoretical HR diagram or by other methods that rely upon IR photometry of cluster members. 2. The IMFs we deri v e for the three young clusters in our surv e y are remarkably consistent with one another and in good agreement with the IMF found in other open clusters, in glob ular clusters and in the eld. From our current study and through these comparisons to other re gions we nd no reason to reject the h ypothesis for a globally consistent structure of the IMF 3. W e nd that the three v ery young clusters that ha v e structure and secondary peaks in their substellar LFs may reect the e xistence of a break from the continuity of the IMF and the formation of a secondary IMF peak near the deuterium-b urning limit. Alternately this feature reect an unkno wn feature in the mass-luminosity relation of lo w mass bro wn dw arfs. 4. The combination of the sensiti v e products of the deep infrared surv e ys performed for this w ork and the model luminosity function algorithm ha v e allo wed us to constrain the IMF of cluster to much greater mass sensiti vity than other methods based on the use of spectroscopic measurements. Thus, we are able to use a cluster' s luminosity function to sample a much greater v olume of the local Galaxy Finally when we combine the rob ust IMFs we deri v e for the young clusters and their agreement with the IMF found in other stellar systems with e vidence that star and bro wn dw arfs, independent of mass, are born with circumstellar disks, we conclude that not only is there e vidence for a globally consistent formulation of the initial mass function, b ut that a single star formation process is lik ely responsible for nearly the entire range of mass. Whether these conclusions e xtend to the IMF and

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202 formation mechanism of v ery lo w mass bro wn dw arfs near the planetary mass re gime, i.e., M10 M J u p will be the tar get of the ne xt generation of IMF studies. 8.3 Futur e W ork There are a number of important questions raised by this w ork. F or e xample, it is clear that the slope of the substellar IMF is not nearly as rob ustly determined as the stellar portion of the global IMF Could it display signicant v ariations from re gion to re gion as suggested by the Luhman T aurus result? Further what is the nature of the secondary LF peak near the deuterium-b urning limit and is there other e vidence for a transition in the formation mechanism of bro wn dw arfs at these lo west masses? W e outline a fe w future projects we are undertaking that might answer similar such questions, as well as future w ork rele v ant to the luminosity function models presented here. 8.3.1 Continued Study of the IMF in Y oung Clusters Clearly the luminosity function is a tool that can be ef ciently applied to a much lar ger sample of clusters than the three clusters studied here. Simply increasing the number of clusters studied and the v olume of the local Galaxy surv e yed w ould rapidly produce a suf cient sample of homogeneously deri v ed IMFs that can be compared in a statistically meaningful f ashion. F or e xample, we can follo w the e xample set by Kroupa ( 2001 ) and ask if, the v ariations in the G 1 high-mass slope between NGC 2362, the T rapezium and IC 348 are real and reect something dif ferent about the clusters or if the y are simply statistical uctuations. While such uctuations might be e xpected out on the high-mass tail of the IMF the y are unlik ely to be found near the peak or mode of the IMF since clearly this mass (or mass range) is that portion of the IMF that is most statistically signicant for an y sized population. W ith a suf ciently lar ge sample of clusters, surv e yed o v er suf ciently lar ge areas to a v oid radial IMF v ariations, one could easily calculate the rob ustness of the IMF mode in clusters and compare it in detail to models of the fragmentation of molecular clouds. Further a statistical

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203 comparisons of the IMF mode in clusters, the eld and glob ular clusters might pro vide further e vidence for stochastic v ariations in the IMF Lastly a lar ger sample of LF deri v ed cluster IMFs w ould greatly impro v e the uncertainty in the substellar IMF 8.3.2 Structur e of Y oung Open Clusters The e xistence of the radial IMF v ariations in IC 348 and the Orion Neb ula Cluster has important consequences for making meaningful comparisons between the IMFs of dif ferent clusters in our current IMF study The origin of these radial IMF v ariations, while be yond the scope of the current w ork, has important consequences, none the less, on the general understanding of the IMF F or e xample, if the radial IMF v ariations are primordial, then the y represent a breaking of the uni v ersality of the IMF on spatial scales less than a parsec in area, implying that dif fering processes are acting on the fragmentation of the g as to gi v e dif ferent locale IMFs. If the origin of the radial IMF v ariations is dynamical, then a study of this phenomena in a set of young clusters with a range of ages w ould re v eal what corrections must be applied to con v ert the observ ed mass function into the initial mass function for a dynamically e v olv ed cluster On the other hand, determination of the mass function of a cluster' s halo re v eals one additional piece of information: if most stars are born in clusters then the halo IMF of a cluster constitutes that portion of the IMF being donated to the eld in the current epoch. W e will be gin such a study of the structural e v olution of young clusters by e xploring NGC 2362 o v er a much lar ger area than the La Silla surv e y First we must determine if the cluster' s halo as suggested by the 2MASS radial prole is real and what is the subsolar IMF of this halo. This will be accomplished by emplo ying wide-eld optical and infrared cameras and the 2MASS catalog to construct opticalinfrared color -magnitude diagrams for the cluster halo with the goal of selecting members and estimate masses and mass functions. Since numerical models suggest that signicant structural e v olution happens within the rst 10 Myr after the dispersal of the molecular cloud ( Kroupa et al. 2001 ) we will e xtend this study to tw o other clusters

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204 of similar youth: NGC 3293 (7 Myr) ( Herbst & Miller 1982 ) and NGC 1502. Combining these three clusters with a wide-eld IR surv e y of the Orion Neb ula Cluster to determine the mode of this cluster' s IMF and the substellar IMF in the cluster halo, we will be able to study ho w clusters dynamically e v olv e and are dispersed into the g alactic eld. 8.3.3 Disks ar ound Y oung Br o wn Dwarfs One possible a v enue for answering questions about the formation mechanism of bro wn dw arfs is to in v estig ate the substellar population of an entire Giant Molecular Cloud (GMC). Such an in v estig ation w ould identify bro wn dw arfs forming in a v ariety of en vironments in a GMC and in suf cient numbers to be gin to pro vide statistical answers to these questions. The Perseus Molecular Cloud is a good candidate cloud for a number of reasons. First, it is kno wn to contain both embedded clusters, e.g., NGC 1333 and IC 348, which we ha v e studied here in Chapter 4 and isolated star forming cores (e.g., LDN 1448). Because Perseus is quite nearby (d300 pc ), v ery young bro wn dw arfs across the mass spectrum from the h ydrogen to the deuterium-b urning limit can be studied both by ground based near -ir imaging and spectroscop y and by mid-infrared imaging using SIR TF Most vital then is the f act that the Perseus GMC is the tar get of a complete SIR TF Le g ac y Science mapping from 3 to 70 m The SIR TF Le g ac y projects are designed to complete lar ge space-based surv e ys that could not be completed by an y single general observ er and the y will pro vide their data products to the general community for archi v al research immediately after acquisition and processing. F or these reasons, a current study of the bro wn dw arf population of the Perseus GMC has the potential to answer vital questions about the origin and initial mass function of bro wn dw arfs. W ith the goal of obtaining a complete census of young bro wn dw arfs in a molecular cloud, I am emplo ying the wide-eld imaging and spectroscopic capabilities of the FLAMINGOS camera to conduct a ground based photometric and spectroscopic surv e y

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205 for v ery lo w mass stars and bro wn dw arfs forming throughout the Perseus Molecular Cloud. I am performing this near -infrared surv e y in adv ance of the anticipated midinfrared mapping of this cloud by the SIR TF Le g ac y Science project entitled, “From Molecular Cores to Planet-F orming Disks” ( Ev ans et al. 2001 ). The fundamental goal of the proposed research is to identify the substellar population forming within the Perseus Molecular Cloud and to tab ulate the composite near and mid-infrared properties of these sources. The scientic objecti v es of mer ging a ground based wide-eld near -infrared surv e y with the products of SIR TF Le g ac y Science are:Identifying where bro wn dw arfs predominantly form within a Giant Molecular Cloud. By combining near and mid-infrared imaging with spectral classication, lo w mass and substellar sources can be identied throughout the molecular cloud. Is the nascent Perseus bro wn dw arf population concentrated in young embedded clusters such as NGC 1333 and IC 348, or in a more distrib uted population throughout the cloud? Ho w distrib uted bro wn dw arfs distrib uted in and around clusters? Do dynamical or primordial mass se gre g ation produce a bro wn dw arf population that is primarily found in the halos of these young clusters?Determining if the Initial Mass Function of lo w mass stars and bro wn dw arfs (from 0.1 to 0.01 M) v aries between the young clusters and the isolated star forming sites within a GMC. Masses for indi vidual young sources will be estimated by comparing luminosities and spectral classications to theoretical e v olutionary models, and the lo w mass initial mass functions of the Perseus embedded clusters and the distrib uted population can be compared. Is the formation frequenc y of bro wn dw arfs a function of initial stellar density or location within a GMC?Establishing the initial properties of circumstellar disks around bro wn dw arfs. Spectral ener gy distrib utions from 1 to 24 m will be constructed for young substellar sources across the mass range from the h ydrogen to the deuterium b urning limits, allo wing for detailed modeling of the bro wn-dw arf/disk properties. Are the basic ph ysical characteristics (frequenc y accretion rate, size and mass) of disks around bro wn dw arfs the same as for lo w mass stars? Or do dif ferences e xist as a function of mass that might indicate a transition in the formation mechanism from stars to bro wn dw arfs? The basic methodology of this project is to deri v e a complete near -ir census of stellar and substellar objects forming within the Perseus Molecular Cloud will be

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206 combined with the SIR TF Le g ac y observ ations. In addition to this near -infrared census, I will conduct an e xtensi v e near -infrared spectroscopic study of young embedded substellar sources in this cloud identied both from the near -infrared census and it' s mer ger with the SIR TF Le g ac y observ ations. Figure 8–1: Comparison of the 2MASS and FLAMINGOS imaging sensiti vity near infrared surv e ys of the Perseus Molecular Cloud. The T rapezium Cluster infrared color -magnitude diagram (from Chapter 3 w as shifted to the distance of the Perseus Molecular Cloud to act as a template embedded cluster 2MASS observ ations will allo w the cloud to be probed to A V5 for objects do wn to 0.05 M. The deeper FLAMINGOS imaging will probe sources o v er the entire substellar range to A V10. Based on the nominal 10 s sensiti vities e xpected from the SIR TF Le g ac y Science mapping of the Perseus Cloud, spectral ener gy distrib utions will be constructed for objects across the substellar mass range, from the h ydrogen to the deuterium-b urning limit. During this project' s rst year I will compile a sensiti v e v olume-limited (e xtinction-limited) near -infrared catalog of candidate pre-main sequence stars and

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207 young substellar objects (10001 M) in the cloud. This has already been be gun with the wide-eld study of IC 348 in Chapter 4 and will be ef ciently completed by combining the e xisting T w o Micron All-Sk y Surv e y (2MASS) of the Perseus Cloud with additional deep imaging that emplo ys the wide-eld capabilities of FLAMINGOS. As sho wn in Figure 8–1 the sensiti vities of the e xtant 2MASS database are suf cient to catalog young 1 Myr sources do wn to 0.05 Min Perseus, though only to relati v ely lo w reddenings for the substellar population. Thus, 2MASS can be immediately used to surv e y the lo w e xtinction re gions of the cloud. T o detect substellar sources from 0.05 0.01 Mand to probe the more embedded population, I will use FLAMINGOS with a 21eld of vie w on the 2.1m KPNO telescope to map 36 pointings of the Perseus Cloud in the JHK broadband lters. As sho wn in Figure 8–2 the FLAMINGOS surv e y is restricted to the embedded clusters and isolated dense cores as traced by the 13 CO g as ( Bachiller & Cernicharo 1986 ) and will be obtained in collaboration with the allocated NO A O Surv e y project, “T o w ard a Complete Near -Infrared Spectroscopic and Imaging Surv e y of Giant Molecular Clouds” (E. Lada, P .I.). The combined 2MASS and FLAMINGOS catalogs of Perseus will pro vide a v olume-limited census of young stars do wn to the h ydrogen b urning limit seen through 30 magnitudes of e xtinction, and bro wn dw arfs as small as 10-20 M J u p seen through reddenings of A V10 magnitudes. During the project' s second year I will be gin using the multi-object spectroscopic capabilities of the FLAMINGOS camera to obtain spectra of the sources in the 2MASS+FLAMINGOS catalog. FLAMINGOS pro vides simultaneous spectroscopic measurements of 50 to 100 sources per pointing, and I will use FLAMINGOS to obtain spectra from 1.25 to 2.5 m with spectral resolutions of R1000. F or e xample, during one hour on the KPNO 4m, FLAMINGOS can measure the spectra of an unreddened 1 Myr 0.02 MPerseus bro wn dw arf to a S/N = 50. Thus, I will be able to deri v e spectral types for sources across h ydrogen b urning limit and do wn to

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208 Figure 8–2: Imaging map of the Perseus GMC with FLAMINGOS. Indi vidual pointings are o v erlapped by 3. Stars mark the locations of the young embedded IC 348 and NGC 1333 clusters. The SIR TF Le g ac y Science observ ations will map the entire 60 de g40 de g cloud re gion including both the dense molecular g as and e xtensi v e of f-cloud areas. The FLAMINGOS mapping concentrates upon the embedded stellar clusters and dense molecular cores and the 2MASS database will be used to catalog potential pre-main sequence stars and young bro wn dw arfs in of f-cloud and in lo w column density re gions of the molecular g as. Deep near -infrared imaging is necessary to catalog the youngest bro wn dw arfs deeply embedded within the molecular cores and young clusters in the Perseus GMC. The combination of the 2MASS and FLAMINGOS datasets will pro vide an e xtinction limited sample of sources sensiti v e to a 20 M J u p bro wn dw arf seen through 10 magnitudes of e xtinction. The SIR TF observ ations will be able to detect (at the 20 s le v el) the 5 m photospher e of this same reddened lo w mass bro wn dw arf.

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209 approximately002 M. Spectra obtained at J and K w a v elengths will be also be searched for e vidence of continuum v eiling due to disk accretion, and for the P a b128 mand Br g217 mnear -infrared h ydrogen emission lines which ha v e been sho wn to be good tracers of the accretion luminosity/rate for embedded T -T auri stars ( Muzerolle et al. 1998 ). Prior to the SIR TF Le g ac y mapping, I will concentrate the spectra surv e y on re gions of kno wn star formation such as NGC 1333 and IC 348, or sources selected from the color -color diagram by ha ving infrared colors that lie to the right of the reddening band for M6 stars. This should pro vide a rst cut at isolating both late type stars and bro wn dw arfs (i.e., those M6 or later), and identifying young sources which display near -infrared e xcess. The rst pipeline products from the SIR TF Le g ac y Science project will become publically a v ailable be ginning roughly 7 months after SIR TF launch, currently scheduled for December 2002. Since this is after the rst year of my current project, the near -infrared imaging campaign will be completed and the 2MASS+FLAMINGOS catalog fully tab ulated, while the spectroscopic surv e y will be underw ay The initial SIR TF Le g ac y Science data products will include source photometry at 3.3, 4.5, 5.8, 8, and 24 m and are projected to ha v e nominal 10 s sensiti vities to sources at and some what below the limits of the near -infrared catalog. F or e xample, the current predicted sensiti vities of the Le g ac y Science observ ations will be suf cient to detect (at the 20 s le v el) the photospher e at 5 m of a 1 Myr 20 M J u p bro wn dw arf reddened by 10 magnitudes of e xtinction. Thus, when the near -infrared catalog and the SIR TF data products are mer ged, near to mid-infrared spectral ener gy distrib utions can be instantly calculated for e v ery source within the e xtinction-limited near -ir catalog, allo wing for ef cient separation of young pre-main sequence stars and bro wn dw arfs from background eld dw arfs and giants. After this mer ger of the SIR TF data with the

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210 2MASS-FLAMINGOS near -infrared catalog, the FLAMINGOS spectroscopic observ ations will be focused on the re vised sample of candidate bro wn dw arfs in Perseus. This follo w-up will continue through the third year of this project. 8.3.4 Model Impr o v ements A nal, follo wup to this dissertation is the addition of a fe w minor impro v ements to the population synthesis algorithm. The primary goal of these impro v ements is to mak e the output more realistic and more v aried, gi ving the code additional uses. These primary impro v ements include the addition of a photometric noise model for more realistic simulations of color -color and color -magnitude diagrams and impro v ed output parameters for simulation of unresolv ed binaries so that the colors and magnitudes of indi vidual components can be retrie v ed. Some secondary impro v ements will also be implemented to mak e v arious model parameters more “random. ” This includes dra wing random cluster size from a poissonian distrib ution or dra wing random binary fractions. Lastly the code itself can be impro v ed by instituting a better interpolation scheme for mo ving between the mass tracks and isochrones of the pre-main sequence e v olutionary models. T ypically these models are not well spaced, either in mass or age, and the cubic spline routine we are using could be impro v ed upon with a 2-D interpolator for e xample.

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APPENDIX A T AB ULA TED BOLOMETRIC CORRECTIONS Bolometric corrections, which were used to con v ert model stars' luminosities and ef fecti v e temperatures into monochromatic magnitudes, were interpolated from an empirical table of basic stellar properties deri v ed from literature sources. The follo wing formulae were used to con v ert from these theoretical quantities to the monochromatic pass band magnitude. M lM bol BC l (A.1) M bol M bol 25logLL (A.2) M lM bol 25logLL BC l (A.3) W e ha v e adopted the near -infrared colors of dw arf stars. While the temperature scales for young pre-main sequence stars f all some where between the temperature scales for main sequence dw arfs and giants, their near -infrared colors are more dw arf-lik e ( Luhman 1999 ). W e constructed our initial table of stellar properties be ginning with those compiled in K en yon & Hartmann ( 1995 hereafter KH95) W e adjusted this tab ulation to reect the lar ge temperature range of our model stars, to update it with additional observ ations and to in v estig ate some of the dependencies of our models on this tab ulation. F or spectral types O3 to B0.5, corresponding to T e f f from 30000 to 50000 K we used V band bolometric corrections and ef fecti v e temperatures from V acca et al. ( 1996 ). O star colors were assumed de generate at all near -infrared bands and assigned the colors of Johnson ( 1966 ). B star red and near -infrared colors were tak en from a recent tab ulation by W inkler ( 1997 ). There is presently a signicant study in the 211

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212 literature of the color -T e f f -spectral type relation for cool stars with T e f f less than 3500 K ( Le ggett et al. 1996 ). Because we related the stellar ef fecti v e temperature directly to the stellar color and bolometric correction and do not assign spectral types to our model stars, we do not need to dene an y particular spectral sequence. F or the color -T e f f -BC V relation of M dw arfs, we used the relations compiled by Bessell ( 1991 1995 ); Bessell et al. ( 1998 ). W e used the bolometric magnitudes deri v ed by T inne y et al. ( 1993 ) for v ery late type M dw arfs to e xtend these bolometric corrections to belo w2000 K the approximate temperature of a 20 M J u p bro wn dw arf at 10 Myrs. W e check ed our tab ulation ag ainst other recent compilations of stellar properties and cool stars observ ations in the literature. W e compared our T e f f -BC V relation to those polynomial ts recently deri v ed by Flo wer ( 1996 ) and Hillenbrand ( 1997 ). W e found that our tab ulation w as in systematic disagreement with these ts. The cause w as traced to the original Aller et al. ( 1982 ) T e f f -BC V tab ulation used in the KH95 compilation. W e refer to the discussion in Bessell et al. ( 1998 there Appendix D) as to the source of this discrepanc y and follo w their prescription to add +0.12 to the Aller et al. ( 1982 ) BC V scale. Combined with our choice of M bol equal to 4.75, this yielded a solar BC V equal to -0.07 and an absolute M V magnitude of 4.81. W e then smoothed the T e f f -BC V relation to match those of Hillenbrand ( 1997 ) and Flo wer ( 1996 ). Our tab ulation of bolometric corrections as a function of the log arithm of the ef fecti v e temperature is gi v en in T able A–1 A more important concern is the accurac y of our tab ulation belo w 3500 K W e tested our tab ulation for cool stars by compiling the observ ed colors, ef fecti v e temper ature determinations and bolometric luminosities for a lar ge number of M dw arfs from the literature including published data by Berriman et al. ( 1992 ); T inne y et al. ( 1993 ); Jones et al. ( 1994 ); Le ggett et al. ( 1996 ). W e used repeat observ ations and deri v ations of stellar properties (e.g. T e f f ) of the same M dw arfs b ut by dif ferent authors as

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213 reecting dif ferent spectral type to ef fecti v e temperature scales and v ariability among late-type stars, as well as fundamental uncertainties in the computation of these v alues. W e also compared our bolometric correction tables to those BCs predicted by recent model atmosphere and e v olutionary calculations at 1 Myrs and 10 Gyrs ( Baraf fe et al. 1998 ), although the predicted broadband ux es from model atmospheres ha v e been found to be lar gely inaccurate ( Le ggett 1992 ; Graham et al. 1992 ; Kirkpatrick et al. 1993 ; T inne y et al. 1993 ; Le ggett et al. 1996 ). From the comparison of our compilation, the models and the observ ed M dw arf colors, bolometric corrections, and ef fecti v e temperatures, we found three main conclusions. First, our comparison indicated that the compiled bolometric corrections were consistent with the observ ed M dw arf v alues. Second, although the bolometric corrections inferred from the model atmospheres agreed v ery well for the coolest objects, the v ery young models, i.e. those with ages in the range of 1 to 10 Myrs, predicted bolometric corrections that were in substantial disagreement with the observ ed M dw arfs for temperatures greater than 3500 K Most important, our comparison sho wed that for the near -infrared bands, specically the K-band, the bolometric corrections are f airly insensiti v e to the ef fecti v e temperature scale for lo w mass stars and bro wn dw arfs.

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214 T able A–1. T able of bolometric corrections Log T e f f BC I BC J BC H BC K 4.70952 -4.92 -5.29 -5.40 -5.50 4.68726 -4.76 -5.13 -5.24 -5.34 4.66389 -4.60 -4.97 -5.08 -5.18 4.63909 -4.42 -4.79 -4.90 -5.00 4.61289 -4.24 -4.61 -4.72 -4.82 4.58490 -4.04 -4.40 -4.52 -4.62 4.55509 -3.83 -4.18 -4.30 -4.40 4.52297 -3.58 -3.94 -4.05 -4.13 4.50596 -3.44 -3.78 -3.89 -3.97 4.40483 -2.87 -3.19 -3.29 -3.37 4.34242 -2.48 -2.75 -2.84 -2.91 4.27184 -2.03 -2.26 -2.34 -2.40 4.23045 -1.87 -2.09 -2.14 -2.21 4.18752 -1.51 -1.71 -1.77 -1.83 4.14613 -1.24 -1.40 -1.46 -1.50 4.11394 -1.03 -1.17 -1.22 -1.26 4.07555 -0.77 -0.90 -0.94 -0.98 4.02119 -0.45 -0.52 -0.54 -0.56 3.97864 -0.17 -0.17 -0.17 -0.17 3.96520 -0.08 -0.04 -0.04 -0.03 3.95279 0.02 0.09 0.10 0.11 3.94052 0.04 0.14 0.17 0.18 3.92737 0.08 0.22 0.25 0.27 3.92169 0.12 0.28 0.34 0.36

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215 T able A–1—Continued Log T e f f BC I BC J BC H BC K 3.91381 0.21 0.34 0.41 0.44 3.89487 0.28 0.40 0.48 0.51 3.87967 0.35 0.48 0.57 0.60 3.86864 0.40 0.53 0.64 0.67 3.85733 0.46 0.58 0.71 0.74 3.84819 0.47 0.61 0.76 0.79 3.83822 0.48 0.65 0.81 0.84 3.82866 0.49 0.70 0.88 0.92 3.81889 0.51 0.76 0.97 1.01 3.80889 0.53 0.82 1.05 1.09 3.80346 0.56 0.85 1.15 1.19 3.79796 0.59 0.95 1.24 1.29 3.79239 0.63 0.97 1.27 1.32 3.78640 0.65 0.99 1.29 1.34 3.78032 0.66 1.00 1.31 1.36 3.77415 0.66 1.02 1.33 1.38 3.76790 0.66 1.02 1.34 1.39 3.76567 0.67 1.04 1.37 1.42 3.76343 0.67 1.07 1.39 1.45 3.76118 0.68 1.08 1.44 1.50 3.75587 0.68 1.09 1.49 1.55 3.75051 0.69 1.17 1.56 1.62 3.74194 0.69 1.16 1.57 1.64 3.73320 0.68 1.15 1.58 1.65

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216 T able A–1—Continued Log T e f f BC I BC J BC H BC K 3.72016 0.64 1.22 1.67 1.75 3.70586 0.66 1.26 1.73 1.82 3.69020 0.66 1.28 1.78 1.87 3.67486 0.63 1.34 1.88 1.97 3.66181 0.61 1.41 1.99 2.09 3.63849 0.64 1.41 2.02 2.13 3.62377 0.62 1.39 2.02 2.14 3.61278 0.62 1.37 2.03 2.18 3.59106 0.59 1.60 2.27 2.44 3.57749 0.55 1.62 2.28 2.46 3.55630 0.55 1.64 2.30 2.50 3.52634 0.50 1.81 2.45 2.68 3.49554 0.40 1.93 2.55 2.82 3.46982 0.20 2.03 2.65 2.98 3.45637 0.05 2.02 2.66 3.01 3.44248 -0.20 2.00 2.66 3.04 3.43136 -0.50 1.98 2.68 3.10 3.23000 -1.75 1.70 2.72 3.33

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APPENDIX B DIST ANCE T O THE TRAPEZIUM CLUSTER Mink o wski opens his re-analysis of T rumpler ( 1931 ) deri v ation of the distance to Orion with a sentence still applicable today “ All published v alues of the distance of the Orion neb ula are open to some criticism. ” This uncertainty can be best seen in the de Zeeuw et al. ( 1999 ) Hipparcos study of the re gion. In what should be the best distance estimate to Orion, de Zeeuw et al. had dif culties nding the OB associations due to their mostly radial motion a w ay from the Sun. The f act that there are numerous and often distinct associations ( Blaauw 1964 ) in a v ery lar ge re gion on the sk y naturally mak es precise distance distance estimates dif cult since the re gion could be as deep along the line of sight as it is across the sk y This w as in f act one positi v e result from the de Zeeuw et al. Hipparcos study who found that at least one of the associations w as 50-100 pc in front of the others. W e performed a simple historical re vie w of the v alues for the distance to the Orion Molecular Cloud or the Orion Neb ula Cluster of which, the T rapezium Cluster studied in Chapter 2 and Chapter 3 is the core. W e attempted where possible to detail the method used to deri v e the distance and if error estimates were documented, ho we v er a number of the references were dif cult to locate using the resources at hand. W e also included a number of commonly cited papers that are not estimates of the distance to this re gion, e.g., Jones & W alk er ( 1988 ). F or our modeling, we chose to use a distance of 400 pc to the T rapezium and the Orion Neb ula Cluster By inspection of the descriptions and the table belo w the reader will note that this is on the near side of the distance to this re gion, and for e xample, disagrees to a dif ference of 80 pc with the often used Genzel et al. ( 1981 ) distance. The distance we chose to use for the T rapezium places it at distance in agreement with 217

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218 the Orion 1c population as measured by Anthon y-T w arog ( 1982 ), W olf f ( 1990 ), and Bro wn et al. ( 1994 ). W e point out that there is not only a lar ge range in these distance estimates b ut each is also accompanied by a lar ge error bar This includes an error bar of 80 pc for the Genzel et al. ( 1981 ) measurement to the BNKL object(s), meaning our assumed distance is well within the associated error bars of an y of other preferred measurement e v en if 400pc is a systematically closer distance than that assumed by other authors (see table 3–6 ). One primary problem that persists in more accurately separating the distance to the Orion Neb ula Cluster from the Orion 1c association is that the y projected along the same line of sight and the 1c association is primarily concentrated along and parallel to the Orion A Molecular cloud which contains the ONC and the T rapezium. Indeed, the y are aligned with such agreement that if we re vie w the Hillenbrand ( 1997 ), Reb ull ( 2001 ), and Carpenter et al. ( 2001 ) wide-eld studies of this re gion we nd that the y sho w no morphological signatures that can separate the tw o entities other then their age (the Orion 1c association is of order 2-5 Myrs see Bro wn et al. ( 1994 )). Thus, the distance spread between the 1c association and the BNKL is interesting since the y w ould ha v e to be separated by 80 pc, yet ph ysically aligned. W e lea v e such a problem to impro v ed radial v elocities and proper motions of stars seen to w ard and within the Orion A molecular cloud. T rumpler ( 1931 ): In this w ork, T rumpler used a number of diagnostics to estimate the distance of the Orion Neb ula. Using the method that will be repeated by numerous authors in their later studies of the distance to Orion, T rumpler deri v ed a distance modulus of 8.5 magnitudes by comparison of the assumed main sequence magnitudes for the B stars in this cluster Mink o wski ( 1946 ): Mink o wski re-calculates the total absorption seen to w ards the three brightest stars in the T rapezium and deri v es a distance modulus of 7.38 magnitudes. Shar pless ( 1952 ): Performs a surv e y of B type stars throughout the entire Orion re gion and deri v es distances using assumed absolute magnitudes for B type stars.

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219 A distance modulus of 8.5 w as deri v ed to all the B stars in Orion with a distance modulus of 8.6 deri v ed for stars near the Orion Neb ula. An error estimate of03 w as gi v en. This result w as re vised by Sharpless ( 1962 ). P ar enago ( 1954 ): A detailed cataloging of stars in the Orion Neb ula. A distance modulus of 8.0 w as deri v ed by this author according to Strand ( 1958 ). (I need to read this paper and determine ho w it w as performed). J ohnson & Hiltner ( 1956 ): In this re-calibration of the standard main sequence deri v ed by Johnson & Mor g an ( 1953 ) using young stellar clusters, Johnson & Hiltner recognize that some luminosity e v olution a w ay from a zero-age main sequence will occur between clusters of dif ferent ages. Using their re-calibrated zero-age main sequence, the y calculated a distance modulus of 8.0 by de-reddening the data from Sharpless ( 1954 ) and stars with spectral types B8-A0 to their re vised main sequence. Ho we v er their ascertain that A0 stars are on the main sequence is certainly not correct for the Orion Neb ula Cluster leading to an underestimate of the distance. Strand ( 1958 ): A distance of 520pc w as deri v ed for the ONC using radial v elocities of the 6 OB stars and 135 stars with proper motions. The use of radial v elocities of the OB stars in the ONC, which ha v e an a v erage of 1.5 members per star ( Preibisch et al. 1999 ), lik ely produces signicant distance errors. Shar pless ( 1962 ): Although the analysis w as performed on the Orion re gion by breaking it into tw o sub-re gions, the Belt and the Sw ord, a single distance modulus of 8.2 magnitudes w as calculated. No error bars are gi v en. Blaauw ( 1964 ): A distance of 460pc (distance modulus of 8,3) is quoted in this discussion of the Orion OB association. No distance is deri v ed in this paper instead referring to a w ork by Bor gman & Blaauw ( 1964 ). J ohnson ( 1965 ): A distance of 380 pc to the ONC re gion using 21 stars both with good radial v elocities and proper motions. W alk er ( 1969 ): Fit the zero-age main sequence of Johnson ( 1963 ) to the B stars. P enston ( 1973 ): Fit of the Johnson ( 1963 ) ZAMS to the B stars. Re vised by Penston et al. ( 1975 ). W arr en & Hesser ( 1978 ): Stromgren photometry of B stars in the Orion Neb ula. Mermilliod ( 1981 ): The well-cited comparison of open cluster color -magnitude diagrams and main sequences. No methodology or errors w as listed for the Orion distance in particular b ut in general distances were from the tting of the zero age main sequence.

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220 Genzel et al. ( 1981 ): By observing H 2 O masers embedded within the BNKL objects, these authors were able to calculate a proper motion and a distance to this re gion. Anthony-T war og ( 1982 ): Re vised W arren & Hesser ( 1978 ) estimate using a dif ferent H b calibration. Also used dif ferent combinations of sub-groups. In general, these distance estimates are 40-80 pc closer than the original W arren & Hesser ndings. W alk er ( 1983 ): F ound no radial v elocity dependence on spectral type (in opposition to that found by Johnson ( 1965 )) Slight re vision to mean v elocity dispersion relati v e to Johnson ( 1965 ) (higher). J ones & W alk er ( 1988 ): This w as not a distance estimate to the Orion Neb ula Cluster These authors sho wed that 470 pc is a distance more consistent with the rejection of fore ground objects via the proper motions distrib ution than a distance of 250 pc. W olff ( 1990 ): Used H g and the Balmer discontinuity to determine T e f f surf ace gra vity and the absolute bolometric magnitude of B stars in Orion. The y deri v ed distance estimates to all four OB associations in Orion. Br o wn et al. ( 1994 ): Used VLUBW photometry and ZAMS tting to deri v ed distance to stars in each sub-re gion. Also compared the A V vs 100 m emission from IRAS to estimate a distance to the Orion A molecular cloud.

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221 T able B–1. Summary of published distances to the Orion 1d association W ork Date Re gionaDistance Error MethodbNumber Name Pub Desig. Modulus Used of Stars Sharpless 1952 ONC 8.50 0.30 B Stars ? P arenago 1954 ONC 8.00 Unkno wn ? Johnson & Hiltner 1956 ONC 8.00 ZAMS ? Strand 1958 ONC 8.60 PM / R V 6/135 Blaauw 1964 1d 8.00 0.10 ZAMS ? Johnson 1965 1d 7.90 PM / R V 21 W alk er 1969 1d 8.37 0.05 ZAMS ? Penston 1973 1d 7.80 0.15 ZAMS ? Penston et al 1975 ONC 8.10 0.13 ZAMS(1) ? . 1975 ONC 7.71 0.21 ZAMS(2) ? . 1975 ONC 7.98 0.12 ZAMS(3) ? W arren & Hesser 1978 1d 8.40 0.53 B stars ? . 1978 1d+1c* 8.28 0.50 B stars ? Mermilliod 1981 ? 8.20 0.15 ZAMS ? Genzel et al 1981 BNKL 8.41 0.40 H20 Masers Anthon y-T w orog 1982 1c 7.86 0.09 B stars 41 . 1982 1d+1c* 8.02 0.12 B stars 23 . 1982 1d+1c* 8.19 0.10 B stars 15 W olf f 1990 1c 7.70 0.50 B Stars ? . 1990 1d 8.20 0.03 B Stars 2 Bro wn 1994 1c 8.00 0.49 VBLUW + log(g)/T e f f 34 . 1994 1d 7.90 0.25 VBLUW + log(g)/T e f f 3 . 1994 cloud 8.10 0.48 A V vs 100 m IRASde Zeeuw et al 1999 1c 8.52 0.25 Hipparcos ? aRe gion corresponded to a v ariety of “samples. ” These included the general Orion re gion, the Orion A Molecular Cloud, the Orion OB1 associations, the Orion OB1 sub-associations, e.g., 1c or 1d, and the Orion Neb ula Cluster itself.bWhere possible the origin of the distance estimate is gi v en. the listing of `B stars' or to `ZAMS' refers to comparison of observ ed quantities for these stars to theoretical or empirical estimates to deri v e distances. PM: proper motions; R V : radial v elocities. Since PM and R V are gi v en together the number of stars used in each estimate is gi v en in the Stars column.

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APPENDIX C SUMMAR Y OF POPULA TION SYNTHESIS FOR TRAN CODE C.1 FOR TRAN Code W e detail the FOR TRAN programs written for implementing the population synthesis algorithm for young stars and bro wn dw arfs. W e pro vide the basic algorithm of the lumfunc.f control program and a short summary of an y independent subroutines used, including an y dependencies and the code' s origin if it w as scripted from an e xternal source. The structure of input, batch and output les are listed in Section C.2 The control program and subroutines were compiled into binaries on a desktop computer with an Intel 400 Mhz Pentium II central processor running RedHat Linux (v ersion 6.0, k ernel-2.2.16-3). Binaries were compiled using g77 (GNU project F ortran Compiler v ersion(s) e gcs-2.91.66 19990314/Linux (e gcs-1.1.2 release) (from FSF-g77 v ersion 0.5.24-19981002)) with theO3 optimization. On this same machine the program had a runtime of60 sec (with writing to disk disabled) for 100 iterations of a cluster of 1000 stars. C.1.1 The Contr ol Pr ogram The control program lumfunc.f w as designed to produce from 1 to N simulations a synthetic cluster of stars using a single set of input parameters passed to it from an e xternal ASCII le. It w as also designed to yield a v ariety of stellar parameters as output. Further to allo w for multiple sets of inputs without ha ving multiple input les, batch v ersions of the control program were written to allo w some parameters to be issued from the command line rather than from an input le. At the core of the control program algorithm is the random sampling of a series of probability distrib ution functions to obtain the fundamental (mass, age) or observ ational (e xtinction, e xcess) properties of the synthetic stars in the cluster W e sampled these 222

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223 distrib utions using the Monte Carlo rejection method algorithm scripted into a series of logical rejection functions. The rejection method algorithm operates by simply selecting tw o random numbers from uniform distrib utions: an abscissa or the v ariable in question (let us say mass for e xample) and an ordinate or a “probability” that ranges between 0 and 1. Since the rejection method is simply the inte gration of the probability distrib ution function (it w ould be the initial mass function in this e xample), one simply must ask if the random ordinate (probability) lies abo v e or belo w the probability distrib ution function(IMF) at the abscissa (mass). If it lies belo w the abscissa is accepted as a v alid mass; the program iterates until the full number of stars ha v e been assigned v alues (masses). The rejection functions written for our populations synthesis model simply return logical v ariables containing TR UE or F ALSE if the random v alue is accepted or not according to the probability distrib ution function (PDFs). Rejection functions can sample either analytical PDFs, e.g, g aussian, or the y can use normalized histograms that list the relati v e frequenc y as a function of the v alue (referred to as RELFREQs). The type of rejection function for each parameter (age, mass, e xtinction, e xcess) is hard-wired into the code, ho we v er and to change the rejection function type, the user is required to edit and recompile the code. Each analytical rejection function is controlled by a set of parameters that are read in from the input le and passed to the rejection function. When the RELFREQ rejection function is used, the code loads and samples a relati v e frequenc y histogram contained in an e xternal ASCII le and which can be crafted into an y random shape required. W e briey document the steps used in the control program algorithm belo w and all of the rejection functions are summarized in Section C.1.2 1. P arameter Input. All rele v ant cluster parameters are read from an e xternal input le that is passed to the code at the command line. In batch mode, v ariable parameters are read from screen, otherwise the program reads x ed parameters

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224 from the ASCII le. The batch v ersion of the control program is also adjusted to ignore those parameters which are read from screen. 2. P arameter Documentation. The input parameters are echoed to screen (screen echoing can be disabled) and then con v erted to string v ariables which will be written to the output le headers. This is one of the tw o locations in the program that must be manually changed by the user before compiling the binary The user changes a documentation v ariable to indicate which rejection functions will be used. The user must also sets logical v ariables if an y of the rejection functions are relati v e frequenc y distrib utions. 3. Pre-iteration setup. The program determines the initial seed for the random number generator opens and reads e xternal les documenting which e v olutionary mass tracks to use, loads the theoretical zero age main sequence and bolometric correction tables, reads the les containing relati v e frequenc y distrib ution histograms (if needed) and con v erts the binary fraction and total number of cluster members into the number of binary systems and solitary stars in the nal cluster The output le is opened and the ASCII header written to it. 4. Be gin Iterations. The program creates a single synthetic cluster per iteration, sequentially writing the deri v ed cluster quantities to the output le. 5. Monte Carlo Sampling. F our parameters, the ages, masses, e xtinctions and K band e xcesses of the synthetic stars in the cluster are randomly sampled from probability distrib utions using Monte Carlo based rejection functions. This is the second location in the control program that must be adjusted by the user for each compiled binary since the rejection functions are hard-wired into the code. The user can select from a set of analytical rejection function which are adjusted by parameters in the input le. Alternately the user can use a “Relati v e Frequenc y” rejection function (see RELFREQ belo w) which samples a changeable frequenc y histogram contained in an e xternal ASCII le. Ev ery star is assigned a mass

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225 (and an IR e xcess if instituted), ho we v er binaries are assigned the same age (and e xtinction) if necessary 6. Creation of Isochrones. The program creates a theoretical isochrone at the age of e very synthetic star by interpolating along the mass tracks of the e v olutionary models. The program uses a cubic spline routine, interpolates as a function of log age, and treats the luminosity and ef fecti v e temperature separately One potential future impro v ement to the models is to implement a nearest neighbor non-linear tw o dimensional spline to simultaneously deri v e these quantities. 7. Interpolation along the indi vidual isochrones. F or the mass of each star the program interpolates along its calculated isochrone and between the kno wn mass v alues gi v en by the tracks. Ag ain, this interpolation uses a cubic spline routine, interpolates as a function of log mass, and treaties the luminosity and ef fecti v e temperature separately There are more signicant potential adv antages to implementing a nearest neighbor non-linear 2D interpolation scheme because of the lar ge separations of the mass tracks pro vided by the e v olutionary models. Note that an input parameter is emplo yed to set the maximum mass interpolated from the pre-main sequence e v olutionary models. F or higher masses, the luminosity and ef fecti v e temperature are interpolated from the zero-age main sequence. 8. Con v ersion to Magnitudes. The luminosity of e v ery synthetic star is rst con v erted to an absolute bolometric magnitude and then to a passband magnitude using bolometric corrections interpolated as functions of ef fecti v e temperature from look-up tables Extinction is then added to the absolute magnitude as a function of the reddening la w (a parameter in the input le), and an IR e xcess, in magnitudes, is subtracted at K band. The program then creates tw o arrays for each passband lter using four of them to store the IJHK magnitudes of e v ery star The program then tak es the passband magnitudes of indi vidual members

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226 of binary systems, con v erts them to passband ux, adds them, co v erts back to passband magnitudes and places these “un-resolv ed” magnitudes into the second four arrays. All the arrays are then shifted by the distance modulus. 9. Binning Luminosity Functions. Re g ardless of the type of output requested, the program bins both the single star arrays and the unresolv ed binary arrays into luminosity functions, according to the binning parameters listed in the input le. 10. Writing to Output File. F or each iteration, the program writes to le whate v er output data is requested. The output type is specied in the input le and can include tables of an y of the star' s parameters calculated throughout the model run. The program then be gins the ne xt iteration as necessary The program closes the output le after completing the last iteration. C.1.2 Rejection Functions W e describe each of the rele v ant rejection functions written for this code. F or the analysis undertak en in this w ork, the IMF w as e xclusi v ely sampled by the LOGNORMAL function for the e xperiments in Chapter 2 and by the PO WERIMF for tting data in Chapters 2 3 4 and 5 Since the star formation rate w as al w ays assumed constant, the SFH w as sampled by the UNIFORM function. The e xtinction distrib ution functions (EPDF) and infrared e xcess distrib utions functions (IXPDF) used in tting the KLFs of the T rapezium and IC 348 were empirical relati v e frequenc y distrib ution functions deri v ed directly from the data and sampled with RELFREQ. W e point out that nearly an y probability distrib ution could be loaded and emplo yed when using the RELFREQ rejection function, although the sampled function will only be as smooth as the size of the bins in the frequenc y histogram. UNIFORM. Rejection function checks if a random abscissa lies between a minimum and maximum v alue. COEV AL. Rejection function that al w ays returns a true v alue. Initially used to deri v e coe v al populations, a similar result can be obtained by using the UNIFORM

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227 function and making the minimum age nearly identical (b ut NO T the e xact same) as the maximum age. GA USS. Rejection function that tests a g aussian distrib ution. The function is sampled between tw o limits and can ha v e up to six terms corresponding to a g aussian distrib ution plus a quadratic function, e.g., pd fx p 0 p 1x p 2x 2 p 3e x p xp 4 2p 52(C.1) where, p 0 is a constant term, p 1 is a linear term, p 2 is a quadratic term, p 3 is the normalization of the g aussian, p 4 is the mean of the g aussian, and p 5 is the width of the g aussian. LOGNORMAL. Rejection function that tests a LOGNORMAL distrib ution as gi v en in Equation 2.2 PO WERIMF Rejection function that constructs a probability distrib ution function consisting of four po wer -la ws connected at three break masses. All the se gments are normalized together and the rejection function determines which po wer la w se gment go v erns the random abscissa in question before testing the ordinate. RELFREQ Rejection function that uses a binned histogram containing the relati v e frequenc y of the abscissa. This normalized relati v e frequenc y histogram is searched to nd which bin the abscissa w ould f all before testing the ordinate v ersus the relati v e probability of that bin. The only requirement is that the bins ha v e equal widths and that the bin centers (in units of the abscissa) increase in v alue. NONEN. Rejection function that al w ays returns a true v alue. Used when disabling either the EPDF or IXPDF O THERS. Ad hoc combinations of some of these rejection functions were written for testing and are in f act a v ailable in the code. These include: MSPO WER, a combination of a Miller -Scalo log-normal IMF breaking at some mass and changing to

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228 a po wer -la w to the lo west masses; and SALIMF a sampling of the Salpeter eld star IMF o v er some mass range. C.1.3 The FOR TRAN Sub-r outines Ele v en additional sub-routines were emplo yed in the population synthesis model. These include both ne wly written sub-routines to handle the loading of v arious e xternal ASCII les and sub-routines borro wed from Numerical Recipes in F ortran; 2nd ed. Press et al. 1986. W e list and summarize each of these sub-routines belo w ext pms.f Small subroutine to open and read a simple ASCII le containing a single e v olutionary mass track. Each mass track must be a four column array where the rst column is a running number the second is the log of the age (log t ), the third is the log of the luminosity in solar units (log LL), and the fourth is the log of the ef fecti v e surf ace temperature (log T e f f ). Each mass track le must be sorted by increasing time and the user should note the maximum and minimum ages for each mass track. This e xtraction routine is used by lumfunc.f to read in each of the mass tracks. Called by: lumfunc.f; Requir es: none ext bck.f Small subroutine to open and read a simple ASCII le containing the tw o column bolometric correction table, where the rst column is the log of T e f f and the second is the bolometric correction. Bolometric correction le must be sorted by increasing ef fecti v e temperature. Called four times by lumfunc.f, once each for the four passbands used in the models. Called by: lumfunc.f; Requir es: none ext hist.f Subroutine to open and read a simple ASCII le containing a probability distrib ution function in histogram form that will act as a “Relati v e Frequenc y” rejection function for a Monte Carlo inte gration. The input histogram must ha v e equally sized bins whose centers must increase in v alue. Subroutine normalizes the probability distrib ution, if necessary and returns the bin centers, frequencies, and

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229 limits of the resulting probability distrib ution function. Used by lumfunc.f when e v er the ”RELFREQ” Rejection Function is used to sample the IMF SFH, Extinction or Infrared Excess distrib ution functions. Called by: lumfunc.f; Requir es: none spltime.f A simple wrapper subroutine to run the cubic spline interpolation routines when lumfunc.f is constructing isochrones from the pre-main sequence tracks at the age of each synthetic star Splines each mass track (luminosity and ef fecti v e temperature separately) vs log time (log t ) to yield an isochrone. Hardwires the rst deri v ati v e estimates used in the spline routine. Called by: lumfunc.f; Requir es: spline.f inter p.f splmass.f A simple wrapper subroutine to run the cubic spline interpolation routines on the indi vidual isochrones constructed by spltime.f for each of the model stars. Splines each isochrone (luminosity and ef fecti v e temperature separately) vs log mass (log M ) for each indi vidual model star Hardwires the rst deri v ati v e estimates used in the spline routine. Called by: lumfunc.f; Requir es: spline.f inter p.f spline.f Subroutine to calculate the second deri v ati v e at each point along anx iny inseries for use in the cubic spline interpolation of this series. Requires boundary conditions in the form of rst deri v ati v e estimates. These are hardwired in the wrapper routines (spltime, splmass) used by lumfunc.f. From Numerical Recipes in F ortran; 2nd ed. Press et al. 1986, pg 109. Called by: spltime.f splmass.f; Requir es: none inter p.f Subroutine to tak e an array ofx iny inv alues and an array of the second deri v ati v e at each point and perform the cubic spline interpolation of y ou t for an array of x ou t From From Numerical Recipes in F ortran; 2nd ed. Press et al. 1986, pg 110. Called by: spltime.f splmass.f; Requir es: none

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230 splbck.f Subroutine to linearly interpolate on a table of bolometric corrections. Interpolation is performed as a function of the log of the surf ace temperature, log T e f f It interpolates all the model stars of an iteration at one time and is called by lumfunc.f four times (once for each of the four passbands) per iteration. Called by: lumfunc.f; Requir es: locate.f mainseq.f Subroutine to linearly interpolate along the zero age main sequence. The interpolation is performed as a function of log M Is called if lumfunc.f determines that the mass of a specic model star f alls outside (greater or less than) the mass range appropriate for the input pre-main sequence tracks. Note, this mass range is set by the user This subroutine is run twice in lumfunc.f for each model star once to interpolate luminosity as a function of log M and ag ain to interpolate surf ace temperature as a function of log M Called by: lumfunc.f; Requir es: locate.f locate.f Subroutine to search an ordered list and determine between which tw o elements and input v alue lies. Returns the subscript of the position in the list that is less than the input v alue. From Numerical Recipes in F ortran; 2nd ed. Press et al. 1986, pg 111. Called by: mainseq.f splbck.f; Requir es: none bin.f This simple subroutine con v erts a list of v alues into a binned distrib ution. It has three parameters: the binsize and the minimum (dmin) and maximum(dmax) of the sampling range. Bins are created between dmin and dmax and ha v e centers whose v alue aredmin n1 2 binsizeand n123 V alues are considered to f all into a specic bin if the y are:bin centerbinsize 2 v alue bin centerbinsize 2(C.2) The subroutine returns a list of bin centers and bin counts. Called by: lumfunc.f; Requir es: none

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231 C.2 Input P arameters and Output Files Model parameters were passed to the compiled binary using a simple ASCII input le. This input le contains 1-7 parameters per line and is fully commented. W e detail this input le in gure(s) C–1 C–2 C–3 and C–4 breaking it into four parts and discussing an y rele v ant details. When the program w as setup to run in batch mode, simple scripts were written to echo those parameters that were being v aried, while the remaining x ed inputs were read from a normal input le. W e gi v e an e xample of one line of a batch input in gure C–5 Lastly the output les from the population synthesis code consisted of simple ASCII te xt with informational headers. The results of indi vidual iterations were listed sequentially in the output les and tw o e xample output le headers are gi v en with e xplanation in gure C–6 A current decienc y of the code is that the indi vidual luminosity function simulations cannot be processed and combined by the code into a single luminosity function (and accompan ying statistical information, i.e., standard de viations) that can be directly compared to data. Such a procedure w ould permit smaller output les whose size currently depends upon the type of output chosen and the number of cluster iterations.

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232 !******************************************************************************* INPUT FILE FOR LUMFUNC MODELING PROGRAM. VERSION: 06/29/2002. A.Muench (@UF) !******************************************************************************* !------------------------------------------------------------------------------!******************************************************************************* BASIC POPULATION PARAMETERS !******************************************************************************* !------------------------------------------------------------------------------! N_STARS M_ITERS %% SAMPLING PARAMETERS. Max Nstars = 18000 !------------------------------------------------------------------------------00150 0100 !------------------------------------------------------------------------------! MAX_AGE MIN_AGE %% IF COEVAL SFH, ENTER AGE AT MAX_AGE !------------------------------------------------------------------------------03.50e+06 00.50e+06 !------------------------------------------------------------------------------! DM [m-M(mag)] BIN_FRAC [nbin/(nsing+nbin)] %% BIN_FRAC < 1/N_STARS = NO BIN !------------------------------------------------------------------------------08.00 0.00001 !------------------------------------------------------------------------------! AV_MIN AV_MAX NORM 1SIG OFFSET LINEAR QUAD %% AV_PARAMS !------------------------------------------------------------------------------00.000 03.000 01.00 00.30 00.3000 00.0000 00.0000 00.0000 !------------------------------------------------------------------------------! Ai/Av Aj/Av Ah/Av Ak/Av %% REDDENING LAW [ = Cohen et al 81] !------------------------------------------------------------------------------00.600 00.265 00.155 00.090 !------------------------------------------------------------------------------! IRX_MIN IRX_MAX NORM 1SIG OFFSET LINEAR QUAD %% IRX_PARAMS !------------------------------------------------------------------------------00.250 50.000 01.00 09.75 02.6845 01.0417 -00.0615 00.0000 !------------------------------------------------------------------------------!******************************************************************************* IMF PARAMETERS. %% P1-P7 = SEVEN PARAMS DEPENDING ON IMF MODEL !******************************************************************************* %% POWERIMF: P1,P3,P5,P7 = SLOPES, P2,P4,P6 = MASS BREAK [Both linear mass] %% LOGNORMAL: P1-P2 = [MEAN, SIGMA] P1 P2 P3 P4 P5 P6 P7 !-------------------------------------------------------------------------------2.25 00.820 -1.53 00.092 1.20 00.010 -1.00 !------------------------------------------------------------------------------! RANGE OF MASS TO SAMPLE IMF %% MUST FIX THE PMS MASS LIMITS BELOW IMF_MAX IMF_MIN %% [log mass] !------------------------------------------------------------------------------00.4000 -01.6000 !------------------------------------------------------------------------------Figure C–1: Model input le: basic cluster and IMF parameters. This subset of model parameters includes the number of stars in the cluster and the number of iterations to perform in addition to x ed cluster parameters such as distance, binary fraction, and reddening la w The parameters for functional forms of the cluster' s star formation history IMF e xtinction and e xcess distrib utions are listed and these parameters are interpreted dif ferently depending upon the type of rejection function chosen in the compiled binary or the y are ignored if the code is set up to use relati v e frequenc y distrib utions which are listed as inputs in gure C–2 The important parameters, IMF MIN/MAX, specify the range of mass sampled in the functional form of the IMF

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233 !******************************************************************************* RELFREQ CALLS: MUST SET LOGICAL IN CODE. ALL FILE NAME LENGTHS LIMITED TO A80 MUST SET PROPER NUMBER OF COMMENTS (SEE BOTTOM) !******************************************************************************* !------------------------------------------------------------------------------! IMF RELFREQ FILE. %% IMF PARAMS ABOVE IGNORED !------------------------------------------------------------------------------/home/aamn/Models/Distributions/imf_hist.dat !------------------------------------------------------------------------------! STAR FORMING HISTORY RELFREQ FILE. %% SFH PARAMS ABOVE IGNORED !------------------------------------------------------------------------------/home/aamn/Models/Distributions/sfh_hist.dat !------------------------------------------------------------------------------! EXTINCTION DISTRIBUTION FUNCTION RELFREQ FILE. %% AV_PARMS ABOVE IGNORED !------------------------------------------------------------------------------/home/aamn/Models/Distributions/EDF/Trap/trap.edf_lim.hist !------------------------------------------------------------------------------! IXDF RELFREQ FILE. ASSUMED FOR K ONLY. %% IRX_PARAMS ABOVE IGNORED !------------------------------------------------------------------------------/home/aamn/Models/Distributions/trap.ixdf.hist !------------------------------------------------------------------------------! NUMBER OF COMMENTS IN RELFREQ FILES. %% INORDER: IMF,SFH,AV,IRX !------------------------------------------------------------------------------00 00 32 20 !------------------------------------------------------------------------------Figure C–2: Model input le: relati v e frequenc y probability distrib ution les. These parameters are les containing binned histograms that describe the relati v e frequenc y of the gi v en parameter (mass, age, A V or IR e xcess). This histograms must ha v e been created with equal sized bins with the bin center in the rst column. The bin centers must be listed with positi v e increasing v alue b ut their v alues (gi v en in the second column) need not be normalized. When RELFREQ distrib utions are used for a specic cluster parameter for e xample, e xtinction, the other e xtinction parameters gi v en in the input le are ignored.

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234 !******************************************************************************* EVOLUTIONARY TRACK / BOLOMETRIC CORRECTIONS FILE POINTERS / PARAMETERS !******************************************************************************* !------------------------------------------------------------------------------! FILE CONTAINING CHARACTER NAMES OF MASS TRACKS. %% [MAXLEN OF FILENAME = A80] !------------------------------------------------------------------------------/home/aamn/Models/PMS/BMDM/d2.5/char_mass !------------------------------------------------------------------------------! FILE CONTAINING REAL MASSES OF EACH MASS TRACK. %% [MAXLEN OF FILENAME = A80] !------------------------------------------------------------------------------/home/aamn/Models/PMS/BMDM/d2.5/real_mass !------------------------------------------------------------------------------! PREFIX TO MASS TRACK FILE. %% [MAXLEN = A50] !------------------------------------------------------------------------------/home/aamn/Models/PMS/BMDM/d2.5/Comp_ !------------------------------------------------------------------------------! SUFFIX TO MASS TRACK FILE. %% MASS TRACK FILE=PREFIX+CHAR_MASS[i]+SUFFIX !------------------------------------------------------------------------------.25 !------------------------------------------------------------------------------! IJHK BOLOMETRIC CORRECTION FILES. %% (Teff,BC) LINEAR INTERP. [MAXLEN = A80] !------------------------------------------------------------------------------/home/aamn/Models/BCS/bci_comp_4.dat /home/aamn/Models/BCS/bcj_comp_4.dat /home/aamn/Models/BCS/bch_comp_4.dat /home/aamn/Models/BCS/bck_comp_4.dat !------------------------------------------------------------------------------! MAIN SEQUENCE FILE. %% LOGMASS vs (L, Teff). LINEAR INTERP. [MAXLEN = A80] !------------------------------------------------------------------------------/home/aamn/Models/PMS/MainSeq/mainseq_4 !------------------------------------------------------------------------------! RANGE OF MASS TO USE PMS MODELS. %% ABOVE PMS_MAX = MAIN_SEQUENCE PMS_MAX PMS_MIN %% REQ: PMS_MIN LE IMF_MIN. [log mass] !------------------------------------------------------------------------------00.5000 -01.7000 !------------------------------------------------------------------------------Figure C–3: Model input le: pointers and parameters for e v olutionary tracks. The e v olutionary models, bolometric corrections and theoretical main sequence are all stored in separate ASCII les. F or e xample in the case of the e v olutionary models, indi vidual ASCII les e xist for indi vidual mass tracks. This set of parameters and pointers are used when opening and reading in these e v olutionary les while the code is running. Further since there are an unkno wn number of e v olutionary tracks, each corresponding to a specic mass object, the code is passed lists of the indi vidual masses that it uses in both opening the les and in the interpolation of the tracks. An important parameter is the PMS MIN/MAX parameters which set the range of the tracks to use relati v e to the range of masses sampled from the IMF The user is required to ensure that the IMF is not sampled less than the minimum mass of the e v olutionary models, else interpolation will f ail. F or masses greater than PMS MAX, the code interpolates along the theoretical zero-age main sequence.

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235 !******************************************************************************* OUTPUT PARAMETERS !******************************************************************************* !------------------------------------------------------------------------------! OUTPUT TYPE. %% SEE README or CODE HEADER !------------------------------------------------------------------------------JHKLF !------------------------------------------------------------------------------! BIN_SIZE BIN_MIN BIN_MAX %% OUTPUT LUMFUNC BINNING PARAMETERS !------------------------------------------------------------------------------00.500 03.250 21.250 !------------------------------------------------------------------------------! OUTPUT FILE. %% [MAXLEN = A80] !------------------------------------------------------------------------------/home/aamn/Models/Data/example.dat !------------------------------------------------------------------------------! SCREEN INFORMATION TURNS OFF/ON SCREEN WRITING INFORMATION. !------------------------------------------------------------------------------ON !------------------------------------------------------------------------------! EXPERIMENT RUN TITLE OR OTHER INFO. %% [MAXLEN = A50] !------------------------------------------------------------------------------Example of Input File !------------------------------------------------------------------------------!******************************************************************************* USER NAME OR OTHER USER INFO %% MAXLEN + A50 !******************************************************************************* A. Muench (@UF) !------------------------------------------------------------------------------! Figure C–4: Model input le: output parameters. The ASCII output les can contain a v ariety of ph ysical or observ able cluster properties depending upon the setting of the OUTPUT TYPE parameter All iterations are written sequentially to the OUTPUT FILE, while other comments, titles or username are also added to the output le. If screen writing is enabled, all input parameters are echoed to screen.

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236 Example of one line of batch file input to luminosity function modeling: echo -e -2.30 10.00 -2.30 01.00 -2.30 00.10 -2.30\n 2.00e+06 1.00e+06\n 10.88 0.40\n /home/aamn/Models/Data/Fits/NGC2362/ngc2362.T02.DT2.all_lf\n /home/aamn/Models/IFiles/lf_batch_age_imf.param" | /home/aamn/Models/lumfunc_batch_age_imf_v0 which translates into: echo -e The seven IMF parameters \n Maximum and Minimum Ages \n Distance Modulus and Binary Fraction \n Output Filename \n Input Filename containing other static parameters | binary file Figure C–5: Example batch le. The control program lumfunc.f w as modied to accept certain parameters echoed to the command line from scripts while operating in batch mode. In this e xample, the parameters of a four se gment po wer -la w IMF the star forming history the distance, the binary frequenc y and the output le are all being v aried in batch mode. The other parameters are x ed and listed in a normal input le. Echoed v ariables are listed b ut ignored in the input le in batch mode.

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237 Example 1: No Extinction or Excess used, although parameters listed. JHK LFs are the output. # Output from lumfunc.f FORTRAN program # # Last written: 20:43:24 # On date of: 6/29/2002 # By user: A. Muench (@UF) # Run title: Example of Output File # # FUNCTIONS: imf[POWERIMF] sfh[UNIFORM] av[NONEN] irex[NONEN] # GENPARAMS: N: 150 M: 100 DM: 8.0000 # BINPARAMS: Nsing: 150 Nbin: 0 Nsystem: 150 # SFHPARAMS: Min Age: 0.50E+06 Max Age: 0.35E+07 # IMFPARAMS: -2.250000 0.820000 -1.530000 0.092000 1.200000 0.010000 # AVPARAMS : 0.000 3.000 1.000 0.300 0.300 0.00000 0.00000 0.00000 # IXPARAMS : 0.250 50.000 1.000 9.750 2.684 1.04170 -0.06150 0.00000 # DATAFILE : JHKLF /home/aamn/Models/Data/example.dat Example 2: Extinction and Excess histograms were used. Binary fraction was non-negligable. Ouput included masses, ages, luminosities and effective temperatures. # Output from lumfunc.f FORTRAN program # # Last written: 2:22:59 # On date of: 5/16/2002 # By user: A. Muench (@UF) # Run title: Testing IMF # # FUNCTIONS: imf[POWERIMF] sfh[UNIFORM] av[RELFREQ] irex[RELFREQ] # GENPARAMS: N: 1000 M: 50 DM: 10.9000 # BINPARAMS: Nsing: 818 Nbin: 91 Nsystem: 909 # SFHPARAMS: Min Age: 0.25E+07 Max Age: 0.35E+07 # IMFPARAMS: -2.350000 0.100000 -1.000000 0.020000 -1.000000 0.010000 # AVPARAMS : hist file: /home/aamn/Models/Distributions/EDF/Trap/trap.edf.hist # IXPARAMS : hist file: /home/aamn/Models/Distributions/trap.ixdf.hist # DATAFILE : MSAGE_LT /home/aamn/Models/Data/check_imf.dat Figure C–6: Example(s) of output le headers. The program writes simple ASCII le containing the sequentially listed output from each sampling iteration of a model run. Simple headers containing rele v ant information are appended to the be ginning of each output le. T w o e xample headers are gi v en, containing v ery dif ferent sets of input parameters.

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BIOGRAPHICAL SKETCH August w as born in the city of T ampa in the state of Florida during August of 1973 to Elizabeth Ealer and August Albert Muench, Jr He spent most of his current life in the tin y tomato-infested to wn of Ruskin on the eastern shores of T ampa Bay where he attended East Bay High School by some natural twist of Florida f ate. T w o astronomical high points during this period are w orth noting. The rst w as his memory of staring into the steamy scintillating Florida sk y trying in v ain to see Halle y' s Comet in 1986. The second w as respectfully telling his f ather that he had purchased, as a w onderful gift for his sons, the wrong kind of telescope. In September 1991, August be g an as an under graduate at the Geor gia Institute of T echnology in Atlanta, Geor gia. Despite the good intention of attending this school to become a chemical engineer and to actually be gin to earn a decent salary by the age of 23, August instead became disenchanted by the probabilistic nature of the equations used to determine reaction rates for v arious chemical processes. Seeking a higher standard of precision, he shifted his studies to those of Ph ysics although it seems lik ely at the 2 s le v el that during this paradigm shift his subconscious w as shifting to w ard the opposite end of the precision scale. That being to w ard astronomy After four years in Atlanta, August w as admitted under generous prete xts to the master' s program of the Department of Astronomy at the Uni v ersity of Florida. Roughly tw o years later after another paradigm shift from observ ational cosmology to infrared star and planet formation and being admitted to the doctoral program as a graduate student w orking with Dr Elizabeth Lada, he departed Gainesville to be g an as a Smithsonian Predoctoral Fello w at the Harv ard-Smithsonian Center for Astroph ysics (CfA) in Cambridge, Massachusetts where he w ork ed with Dr Charles Lada. Thus, the

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248 CfA w as where August spent three years enjo ying the pedestrian (only in the best w ay) city of Boston, li ving in v arious attics in Somerville, and enjo ying the hearty winters. After nishing this fello wship b ut not sadly his dissertation, he left Boston to return to Gainesville (i.e., Florida) while his ancee' Laura Nasrallah also left Harv ard and Cambridge to be gin as an assistant professor in Religious Studies at Occidental Colle ge in Los Angeles (i.e., California). A bit less then a year later Laura and August were married in Baltimore, Maryland and after a trip to T uscan y the y became MuenchNasrallah via the progressi v e courts of Gainesville. The last year of his dissertation w as spent with alternating periods w orking from their home in LA and long stretches spent in Gainesville. After the defense of this dissertation, August will be gin a postdoctoral research position at the Space InfraRed T elescope F acility (SIR TF) Science Center in P asadena, California.


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Title: Luminosity and mass functions of very young stellar clusters
Physical Description: Mixed Material
Creator: Muench, August A. ( Author, Primary )
Publication Date: 2002
Copyright Date: 2002

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Holding Location: University of Florida
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Permanent Link: http://ufdc.ufl.edu/UFE0000578/00001

Material Information

Title: Luminosity and mass functions of very young stellar clusters
Physical Description: Mixed Material
Creator: Muench, August A. ( Author, Primary )
Publication Date: 2002
Copyright Date: 2002

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
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LUMINOSITY AND MASS FUNCTIONS
OF VERY YOUNG STELLAR CLUSTERS















By

AUGUST A. MUENCH


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2002

































Copyright 2002

by

August A. Muench






















Let us content ourselves with the illusion of similarity, but in
truth I tell you, Sir, if I may express myself in prophetic tones,
the interesting thing about life has always been in the differences,

From The History of the Siege of Lisbon by Jose Saramago





This work is dedicated to Laura.















ACKNOWLEDGMENTS

I would like to acknowledge a number of individuals and organizations that

provided the direction, care, support and opportunity that have allowed me to enjoy and

to research astronomy (in addition to completing this dissertation).

After calmly listening to my description of various observational cosmology

projects in which I was interested, my dissertation advisor, Dr. Elizabeth Lada

simply pointed out that she was not currently working on any such projects. She

then proceeded to detail all the research that had constituted her career so far and the

directions she wanted to take, listing project after project that was open to me were I

interested. She has not stopped listing the avenues open to me and continues to offer

me the chance to work on and lead projects and for this and for her guidance and

support I am grateful.

Although I grew up on Tampa Bay and my father fishes commercially on the Bay,

I have yet to finalize a good answer to the first question posed to me by Dr. Charles

Lada on the tidal patterns in the Gulf of Mexico versus the Atlantic Ocean. Despite

this delinquency, I have enjoyed trying to answer the innumerable other questions

posed to me by him regarding the data, models and interpretations contained within this

work and in our other projects. I have come to greatly appreciate the focus that Charlie

and Elizabeth Lada employ when our attention turns to the lucid communication of our

results through the words contained in our papers, and their excitement at the moment

that implication raises its sometimes dangerous head.

I would like to thank the members of my dissertation committee at the University

of Florida for reading, reviewing and providing their comments and questions on this

work. I would also like to thank the members of my pre-doctoral committee at the









Center for Astrophysics, Drs. Alyssa Goodman and John Stauffer, who labored through

my excessively long progress reports and who gave me consistent and fruitful advice.

At the Department of Astronomy, I would like to offer my thanks to Dr. Stanley

Dermott, Department Chair, who in fact made my career at the University of Florida

possible and to Dr. Richard Elston for his suggestions and guidance in using the Monte

Carlo technique. It is also without question that both the Radio and Geoastronomy

Division at the Center for Astrophysics and the Department of Astronomy at the

University of Florida have been gifted by administrators and program assistants such as

Tom Mullen, Janice Douglas, and Ann Elton who with continuous and singular focus

work toward creating a supportive environment in which to research our field.

I have also been granted good friends and collaborators such as Joao Alves, who

was my office mate at the CfA. I thank him for sharing his boundless excitement

for his work, and I look forward to further collaboration and friendship with him.

My fellow WIRE survivor, Lori Allen, has been a wonderful friend to me, is greatly

missed, and I wonder on a regular basis when will be our next chance to work together.

To Lauren Jones, who has believed in me as a person and as an astronomer from the

first time she saw me waiting in the main office between classes, I send my conviction

that she has much to offer astronomy. I would like to thank Joanna Levine for her

friendship and especially her support for me during this dissertation's end times and

to both her and Carlos Roman for their assistance with the reduction of the IC 348

images. My friends and colleagues who are unlisted but who have put up with my

spontaneous outbursts about Pluto and white dwarfs amaze me with their loyalty.

My parents, Gus and Betsy Muench and my brothers, Sam and Stephen, have

given me their love, interest and support throughout these years. And to my wife

and my love and my friend, Laura, I pray that I will find some word or deed that can

contain and make clear my gratitude to her for her support as I trudged through this

dissertation.









I was supported by the Smithsonian Predoctoral Fellowship program at the

Harvard-Smithsonian Center for Astrophysics and as a substitute NASA Graduate

Student Research Fellow (grant NTG5-50233). My work was also supported by a grant

to Dr. Elizabeth Lada from the National Science Foundation (grant AST-9733367).

There is no question in my mind that the success of any individual researcher sits level

upon three legs: that of individual commitment, that of unabridged opportunity and that

of continuous scientific interaction. All of these aspects were enabled for me by being

a Predoctoral Fellow at the CfA.

I would like to extend my thanks to John Bally for permission to reproduce HST

images of the proplyds in the Trapezium Cluster, and to Kevin Luhman for data in

advance of publication. Portions of this work are based on photographic data obtained

using The UK Schmidt Telescope. The UK Schmidt Telescope was operated by the

Royal Observatory Edinburgh, with funding from the UK Science and Engineering

Research Council, until 1988 June, and thereafter by the Anglo-Australian Observatory.

Original plate material is copyright (c) the Royal Observatory Edinburgh and the

Anglo-Australian Observatory. The plates were processed into the present compressed

digital form with their permission. The Digitized Sky Survey was produced at the

Space Telescope Science Institute under US Government grant NAG W-2166. This

publication makes use of data products from the Two Micron All Sky Survey, which

is a joint project of the University of Massachusetts and the Infrared Processing and

Analysis Center/California Institute of Technology, funded by the National Aeronautics

and Space Administration and the National Science Foundation. The data products

were circa the 2nd Incremental release (March 2000). This document was typeset with

the ILTEX 2 formating system using the document class template ufthesis.cls (v2.0b)

and written by Ron Smith (ufthesis@ufthesis.com) at the University of Florida. Any

apparent success in the format of this document can almost certainly be attributed to

Ron Smith's efforts for which I am grateful.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ...................... . iv

LIST OF TA BLES . . . . . . . . x

LIST OF FIGURES . . . . . . . . xi

KEY TO ABBREVIATIONS ............................ ..xiv

KEY TO SYM BOLS ................................ ..xvi

A B STR A CT . . .. . . . . . . .. xvii

CHAPTER

1 INTRODUCTION ........................ . .. 1

2 MONTE CARLO MODELS OF YOUNG STELLAR POPULATIONS .... 10

2.1 Monte Carlo-Based Population Synthesis Model .. . 10
2.2 Fundamental Cluster Parameters ..................... .. 11
2.2.1 Initial Mass Function ............. . 11
2.2.2 The Cluster's Star-Forming History ... ........ 13
2.2.3 Theoretical Mass-Luminosity Relations .... . ..14
2.3 Additional Cluster Characteristics and Model Inputs . 17
2.3.1 Reddening Properties ...................... .. 18
2.3.2 Binary Fraction . . . . . . .. 19
2.4 M odel Outputs . . . . . . . .. 20
2.5 Numerical Experiments ..... . . . . 21
2.5.1 Different Pre-Main Sequence Evolutionary Models . 21
2.5.2 Star Formation History . . . . . 29
2.5.3 Initial Mass Function . . . . . 34
2.6 Discussion and an Example from the Literature . ..... 36
2.6.1 Results and Implications of Numerical Experiments . 36
2.6.2 An Example from the Literature: The Trapezium Cluster . 37
2.7 Conclusions . . . . . . . . 44

3 THE FAMOUS TRAPEZIUM CLUSTER IN ORION . . 46

3.1 Near-Infrared Census . . . . . . 47
3.1.1 O observations . . . . . . . 48
3.1.2 Data Reduction and Photometry . . . . 52










3.1.3 Photometric Comparisons of Datasets . . . 55
3.1.4 Astrometry and the Electronic Catalog . 57
3.2 Trapezium Cluster K band Luminosity Function . . . 59
3.2.1 Constructing Infrared Luminosity Function(s) . . 61
3.2.2 Defining a Complete Cluster KLF . . . . 64
3.2.3 Field Star Contamination to the KLF . . . 67
3.3 Trapezium Cluster Initial Mass Function . . . . 70
3.3.1 Deriving Distributions of Reddening . . . 70
3.3.2 Modeling the Trapezium Cluster KLF . . . 76
3.3.3 Derived Trapezium Cluster IMF . . . . 83
3.4 D discussion . . . . . . . 89
3.4.1 Structure of the Trapezium KLF and IMF .. . . 89
3.4.2 Sensitivity of Results to Theoretical PMS Models . . 92
3.4.3 Comparison of IR-Based Trapezium IMFs .. . . 97
3.5 C conclusions . . . . . . . . 101

4 THE YOUNG CLUSTER IC 348 IN PERSEUS... . . . 103


4.1 Wide-Field Near-Infrared Images of IC 348 . .
4.1.1 FLAMINGOS Observations . .
4.1.2 Infrared Census . .............
4.1.3 Cluster Structure . ....
4.1.4 Cluster Reddening Properties . . .
4.2 Infrared Luminosity Functions of IC 348 . ...
4.2.1 Constructing Infrared Luminosity Functions .
4.2.2 Field-Star Correction to the Cluster KLF(s) .
4.3 Initial Mass Function of IC 348 . ........
4.3.1 Star Forming History of IC 348 . ..


4.3.2 Cluster Distance and the Mass-Luminosity Relation
4.3.3 Other Modeling Parameters: Reddening and Binaries
4.3.4 Modeling the IC 348 Differential KLF(s) . .
4.4 D discussion . . . . . . .
4.4.1 The KLFs and IMFs of IC 348 and the Trapezium .
4.4.2 Radial Variation of the IC 348 IMF . .....
4.5 C conclusions . . . . . . .

5 THE YOUNG OPEN CLUSTER NGC 2362 .. ..........


5.1 La Silla Observations of NGC 2362 .
5.2 2MASS Observations of NGC 2362 .
5.2.1 Spatial Structure of NGC 2362 .
5.2.2 Source Reddening for NGC 2362 .
5.3 The NGC 2362 Cluster KLF . .
5.3.1 Empirical Field Star KLF . .
5.3.2 NGC 2362 Differential KLF(s) .
5.4 Comparison to other Young Cluster KLFs .


. . 105
. . 105
. . 108
. . 113
. . 119
. . 123
. . 123
. . 124
. . 128
. . 128
. . 129
. . 133
. . 134
. ... 138
. . 138
. . 142
. . 146


. . .. . 150
. ... . 152
... . 152
. . 155
. .. . 157
. .. . 157
... . 159
... . 160










5.5 Modeling the NGC 2362 KLF . . . . .
5.5.1 Deriving a Mean Age Using a Fixed IMF . .
5.5.2 Simultaneous Derivation of a Cluster's Age and its IMF
5.5.3 NGC 2362 IMF Derived Using a Fixed SFH . .
5.6 D discussion . . . . . .
5.6.1 Age and IMF of NGC2362 . . . .
5.6.2 Age and Spatial Structure of NGC 2362 . .
5.7 C conclusions . . . . . . .

6 CIRCUMSTELLAR DISKS AROUND YOUNG BROWN DWARFS .


6.1 Trapezium Brown Dwarfs with Infrared Excess . .
6.2 Discussion and Implications . ..............


. 180
. 184


7 DISCUSSION ON THE STRUCTURE OF THE IMF . . 189

7.1 Young Clusters and the Global IMF . . . . 189
7.2 Secondary Sub-Stellar Peak in the Cluster LFs . . 192
7.3 New Clues to the Origin of Stars and Brown Dwarfs . . 196

8 CONCLUSIONS AND FUTURE WORK . . . . . 198


8.1 On the Luminosity Functions of Very Young Stellar Clusters
8.2 On the Initial Mass Functions of Very Young Stellar Clusters


. 198
. 200


8.3 Future
8.3.1
8.3.2
8.3.3
8.3.4


W ork . . . .
Continued Study of the IMF in Young Clusters
Structure of Young Open Clusters . ...
Disks around Young Brown Dwarfs . ..
Model Improvements . ..........


. . 202
. . 202
. . 203
. . 204
. . 210


APPENDIX

A TABULATED BOLOMETRIC CORRECTIONS ..

B DISTANCE TO THE TRAPEZIUM CLUSTER .. ..


. . . 2 17


C SUMMARY OF POPULATION SYNTHESIS FORTRAN CODE


222


C.1 FORTRAN Code . ......
C.1.1 The Control Program .
C.1.2 Rejection Functions . .
C.1.3 The FORTRAN Sub-routines
C.2 Input Parameters and Output Files .


REFERENCES . .........

BIOGRAPHICAL SKETCH . ....


. . . . 2 2 2
. . .. . 222
. . .. . 22 6
. . . 228
... . . 2 3 1


238


. . 247


. 163
. 163
. 167
. 169
. 172
. 172
. 174
. 176

















LIST OF TABLES


Table

2-1

2-;

3-1

3-;
3-:

3-2

3-4

3-(

4-1

4-;

4-2

5-1

A-

A-

A-


page

. . 23

. . 40

. 49

. . 60

. . 79

. . 86

. . 94

tometry . 100

. . 106

. . 109

. . 137

. . 169

. . 214

. . 2 15

. . 216


B-1 Summary of published distances to the Orion Id association


Evolutionary models used in numerical experiments .

Cluster IMF derived from the literature Trapezuim KLF

Summary of infrared observations of the Trapezium cluster

FLWO-NTT near-infrared catalog . ..

Three power-law Trapezium IMF parameters and errors

Three power-law Trapezium sub-stellar IMF .

Evolutionary models used to compare M-L relations .

Comparison of published Trapezium IMFs based on IR pho

Summary of FLAMINGOS observations of IC 348 .

Comparison of IC 348 photometry to 2MASS catalog. .

IC 348 power-law IMFs derived from model KLFs ..

Age dependence of the IMF slope in NGC 2362 .

Table of bolometric corrections . ..........

Table of bolometric corrections . ..........

Table of bolometric corrections . ..........


l1


















LIST OF FIGURES


Figur

2-

2-

2-

2-

2-

2-

2-

2-

2-

2-

2-

2-

3-

3-

3-

3-


3-5 Trapezium cluster: deriving M- Av completeness limits .


e


-6 Trapezium cluster: testing contribution of reddened field star KLFs .

-7 Infrared colors of Trapezium sources . .

-8 Trapezium cluster: extinction probability distribution function .

-9 Effects of extinction on model cluster LFs ........ . .....

-10 Trapezium cluster: infrared excess probability distribution function

-11 Trapezium cluster: best-fitting model KLFs and 3 power-law IMFs .


1 Example mass functions used in models . ........

2 Definition of the cluster's star-forming history . .....

3 Theoretical Hertzsprung-Russell diagram . ........

4 Model KLFs: varying physical inputs to evolutionary models .

5 Model KLFs: comparing DM94 and DM97 . ......

6 Model KLFs: varying the initial deuterium abundance . .

7 Model KLFs: truncations in the mass-luminosity relation .

8 Model KLFs: varying the star forming history (, A) .

9 Evolution of mean K magnitude with cluster age . ...

10 Model KLFs: varying the cluster's age spread .

11 Model KLFs: varying the initial mass function . .....

12 Application of models to literature data . .........

1 Comparison of recent Trapezium cluster IR surveys .

2 Infrared color composite image of the Trapezium . ...

3 Trapezium cluster: raw near-infrared luminosity functions .

4 Trapezium cluster: construction of observed control field KLF .


. 68

. 71

. 73

. 74

. 75

. 78


page

. 12

. 13

. 17

. 24

. 26

. 28

. . 30

. 31

. 32

. 33

. 35

. 39

. 48

. 52

. . 62

. 63

. . 65


. .










-12 Trapezium cluster: X2 confidence intervals for IMF parameters .

-13 Trapezium cluster: best fit model KLF to secondary KLF peak .

-14 Trapezium cluster: overall derived IMF .......... .....

-15 Trapezium cluster: a closer look at the sub-stellar IMF . ..

-16 Trapezium cluster: a secondary peak in Trapezium substellar IMF .


3-17 Comparison of theoretical mass-luminosity relations


-18 Comparison of theoretical M-Teff-spectral type relations .

-19 Comparison of trapezium IMFs from IR photometry .

-1 Infrared color composite image of IC 348 . ....

-2 Near-infrared color-magnitude diagrams of IC 348 .

-3 Infrared color-color diagram of IC 348 . .

-4 Radial profile of the IC 348 cluster . ........

-5 Spatial distribution of sources in IC 348 . .....

-6 Surface density profile of the IC 348 cluster . ...

-7 Extinction maps of the IC 348 FLAMINGOS region .

-8 Distributions of reddening for IC 348 . .

-9 Raw infrared luminosity functions for IC 348 . ..

-10 K-band luminosity functions by sub-region for IC 348 .

-11 Field star correction to cluster KLFs in IC 348 . ..

-12 Differential KLFs for IC 348 . ...........

-13 Star-forming history of IC 348 . ..........

-14 Theoretical mass-luminosity relations of IC 348 . .

-15 Modeling the IC 348 KLF: cluster sub-regions . ..

-16 Modeling the IC 348 KLF: the composite cluster . .

-17 Comparison of IC 348 and Trapezium KLFs . ...

-18 Radial variation in the IC 348 IMF . ..

-1 Digitalized sky survey image of NGC 2362 . ...


. .


. 81

. 83

. 84

. 87

. 88


. . 96

. . 97

. . 98

. . 105

. . 1 1 1

. . 112

. . 115

. . 117

. . 118

. . 120

. . 122

. . 124

. . 125

. . 126

. . 127

. . 130

. . 13 1

. . 135

. . 138

. . 139

. . 144

. . 150









-2 Source distribution of NGC 2362 from 2MASS .... . . 153

-3 Radial profiles of NGC 2362 .............. . . 154

-4 Infrared color-color diagrams for NGC 2362.... . . 156

-5 Field star and cluster KLFs of NGC 2362 .... . . 158

-6 Differential KLF(s) of NGC 2362 ................. .. ..159

-7 Comparing the cluster KLFs of NGC 2362, the Trapezium and IC 348 161

-8 Mean age of NGC 2362 derived from the cluster KLF . . ... 164

-9 Model KLF at 5 Myr with Trapezium IMF fit to NGC 2362 . . 165

-10 Dependence of the NGC 2362 IMF slope on mean age . . 168

-11 Best fit model KLFs to the NGC 2362 KLF . . 170

-12 M ass Function of NGC 2362 . . . . . . 171

-1 Selecting candidate brown dwarfs in the Trapezium . ..... 180

-2 Trapezium brown dwarfs with near-infrared excess . . 183

-3 Brown dwarf proplyds .................. ......... .. 185

-4 L-band observations of brown dwarf candidates . . ..... 187

-1 Comparison of Trapezium and ( Ori IMF . . . . 193

-1 Comparison of 2MASS and FLAMINGOS imaging sensitivity . 206

-2 Imaging map of the Perseus GMC with FLAMINGOS . . 208

-1 Model input file: basic cluster and IMF parameters . ..... 232

-2 Model input file: relative frequency probability distribution files . 233

-3 Model input file: pointers and parameters for evolutionary tracks . 234

-4 Model input file: output parameters . . . . . .235

-5 Example batch file . . . . . . . .236

-6 Example(s) of output file headers . . . . . .237















KEY TO ABBREVIATIONS


2MASS


Two Micron All Sky Survey


AU Astronomical Unit

CTTS Classical T-Tauri Stars

ESO European Southern Observatory

FLAMINGOS FLoridA Multi-object Imaging Near-IR Grism Observational Spec-
trometer

FLWO Fred Lawrence Whipple Observatory

FWHM Full Width at Half Maximum

GMC Giant Molecular Cloud

H-R Hertzsprung-Russell (Diagram)

HBL Hydrogen Burning Limit

IDL Interactive Data Language

IMF Initial Mass Function

IRAF Image Reduction and Analysis Facility

KLF K band Luminosity Function

LF Luminosity Function

LMS Luminosity Maximum Spike

M-L Mass-Luminosity (Relation)


NIR


NTT

ONC

PDF

PMS


Near-InfraRed


New Technology Telescope

Orion Nebula Cluster

Probability Distribution Function

Pre-Main Sequence









PSF Point Spread Function

SFH Star Formation History

SIRTF Space InfraRed Telescope Facility

ZAMS Zero Age Main Sequence















KEY TO SYMBOLS


Mk Absolute passband magnitude

mk Apparent passband magnitude

Av Magnitudes of visual extinction

BCk Bolometric correction to passband magnitude (K)

AT Cluster's age spread (in millions of years)

fbin Binary fraction

Lo Units of solar luminosity

mj Mass breakpoints in a power-law mass function

Mjup Units of a Jupiter mass

Me Units of solar mass

Fi Index of a power-law mass function

z Cluster's mean age (in millions of years)

Teff Effective surface temperature (K)















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

LUMINOSITY AND MASS FUNCTIONS
OF VERY YOUNG STELLAR CLUSTERS

By

August A. Muench

December 2002

Chair: Elizabeth A. Lada
Major Department: Astronomy

We now know that the star formation process results in freely-floating objects

with masses spanning nearly four orders of magnitude. However, both the distribution

of these objects' masses at birth and the precise physics responsible for the shape of

this initial mass function are poorly known and can be improved upon by focusing on

very young star clusters just emerging from their parental molecular clouds. In this

dissertation I have investigated the usefulness of the observed luminosity function of a

very young cluster as a tool for deriving that cluster's underlying mass function. I find

that a cluster's luminosity function is an excellent probe of the initial mass function

over the entire range of stellar and substellar mass and can be utilized to acquire the

statistics necessary for testing the hypothesis of a universal mass function.

To study the luminosity and mass functions of such clusters I developed a Monte

Carlo based population synthesis algorithm applicable to pre-main sequence stars.

Using this algorithm I performed numerical experiments testing the sensitivity of

model luminosity functions to changes in fundamental cluster parameters. After

showing that the luminosity function is intrinsically most sensitive to the form of









the underlying mass function, I studied three young clusters, NGC 2362, IC 348 and

the Trapezium, and performed deep near-infrared surveys to construct their K-band

luminosity functions. Using the model luminosity function algorithm, I derived each

cluster's underlying mass function and found them to be remarkably similar, with

all forming broad peaks at subsolar massses. Where these census are sufficiently

deep I find that the mass function turns over and declines in number throughout the

substellar regime but appears to contain structure near the deuterium-burning limit.

Regardless, I find that brown dwarfs do not dominate stars either by number or total

mass. Lastly, I use a statistically significant sample of candidate brown dwarfs to

show that these objects appear as likely to have been born with circumstellar disks

as stars. Combining this finding with the continuity of the shape of the initial mass

function across numerous environments suggests that a single physical mechanism may

dominate the star formation process.


xviii















CHAPTER 1
INTRODUCTION

Little is known about the similarities or differences between the star formation

process that created the first generation of stars in the universe and the process that

is forming stars and brown dwarfs in nearby stellar nurseries today. A long standing

hypothesis, for example, is that the birth of primordial stars was heavily influenced by

the low metallicity of the early universe, and would have preferentially yielded stars

more massive than those born today (Yoneyama, 1972; Palla et al., 1983; Bromm et al.,

2002). Therefore, one important diagnostic for studying any evolution of the star

formation process is the statistical distribution of stellar masses at birth, or the stellar

initial mass function1 The derivation and comparison of the mass functions for stars

in old globular clusters, in the galactic field, in intermediate-age open clusters such as

the Pleiades and in extremely young clusters embedded in nearby molecular clouds

might reveal similarities or differences that would test the notion of an universal mass

function (see the discussion of Kroupa, 2002) and perhaps a dominant star formation

process, or that could bring about a better understanding of its stochastic nature

(Elmegreen & Mathieu, 1983; Zinnecker, 1984; Adams & Fatuzzo, 1996; Elmegreen,



1 In general, we will refer to the stellar initial mass function as the number of
stars per logarithmic unit of mass per unit volume at birth. The choice of logarith-
mic mass units has both an observational and a theoretical basis. Beginning with
Eddington (1924), it has been shown both empirically and theoretically that the lu-
minosity of a main-sequence star scales as a power-law function of the star's mass,
e.g., L = M35 over most of the range of stellar mass. Since the standard unit of obser-
vational astronomy, the magnitude, is a logarithmic scaling of stellar flux, there exists,
therefore, a linear relationship between a star's observed magnitude and its logarithmic
mass.









1997). If the formation of stars is a stochastic process and is heavily dependent upon

numerous parameters other than time, then the problem becomes one of studying the

stellar initial mass function in a variety of physical environments. Because the initial

mass function (IMF) is an intrinsically statistical quantity, all such comparisons require

numerous samplings of the star formation process, in turn, requiring tools that can

probe the stellar mass function over a large volume of space and time.

Since very young, newly formed star clusters are found in environments ranging

from the nearby Orion molecular clouds (Lada, 1990) to very massive clusters in the

turbulent Galactic Center (Figer et al., 1999), they may provide the ideal laboratory

for testing whether the IMF is universal or stochastic. Further, there are a number

of other reasons why young star clusters may be particularly valuable for mass

function studies. For example, a simple photometric census of the members of a young

embedded cluster yields a statistically significant population of stars and brown dwarfs

(i.e., substellar non-hydrogen-burning stars) sharing a common heritage (e.g., age,

metallicity, birth environment). Perhaps more important, such a census is relatively

complete because very young clusters have not lost significant numbers of members to

either dynamical or stellar evolution. Hence, the observed mass function is the cluster's

initial mass function. Because the youngest star clusters are still embedded within

their natal molecular cloud, a near-infrared (1 3/,m) photometric census is often

necessary to identify a complete cluster population. One direct product of such an

infrared census is the young cluster's stellar infrared luminosity function, which can be

used as a tool for studying a cluster's initial mass function. This may be a particularly

effective tool for studying the low-mass end of a cluster's mass function because

infrared luminosities are relatively easy to derive for young brown dwarfs in these

clusters since such intrinsically red substellar sources are at brighter luminosities than

at any subsequent point in their evolution. Further, the development of large format

imaging arrays sensitive to near-infrared wavelengths has made it possible to obtain









statistically significant and complete samplings of the near-infrared luminosity functions

of very young embedded clusters. These recent increases in sensitivity permit not only

the study of the substellar mass functions of nearby clusters, but also the construction

of infrared luminosity functions for distant young clusters even when little or nothing

may be learned about these clusters from spectroscopic measurements. Thus, modem

infrared cameras on even modest sized telescopes can efficiently survey numerous

young clusters, deriving infrared luminosities for complete populations, and, potentially

sampling the initial mass function of the current epoch over a relatively large volume

of the local galaxy.

The observed luminosity function for a cluster of stars is the product of the

underlying mass function of the cluster members and the derivative of the appropriate

mass-luminosity relation:

dN dN d log M
S-x (1.1)
dL d log M dL

However, until a cluster reaches an age of 1 billion years, some fraction of the

stars in the cluster will be in their "pre-main sequence" phase, meaning they have

not yet begun to fuse hydrogen in their core. Since brown dwarfs never achieve

nuclear burning, these cluster members will never reach the main sequence and will

be contracting, cooling and becoming fainter for their entire existence. Thus, the

radiant luminosity of a brown dwarf or a pre-main sequence star is derived from its

gravitational contraction energy, and the mass-luminosity relation appropriate for these

objects is a function of time, hence:

dlogM dlogM
-dL( = d z() (1.2)
dL dL

For the very young clusters we will be studying in this work (ages, T < 10 Myr), nearly

all of the members will be in a pre-main sequence phase. Further, the timescale for

assembling a star cluster is an appreciable fraction of the cluster's mean age during









this period. These facts mean that the derivation of a young cluster's underlying mass

function from its luminosity function is sensitive to the history of star formation in the

cluster. Additionally, the time-dependent mass-luminosity relations) used to convert

between a cluster's luminosity and mass functions is poorly known. Since there are

very few meaningful empirical constraints on the form of the pre-main sequence mass-

luminosity relation, we must rely upon theoretical evolutionary models of young stars

when estimating this quantity. Finally, the predictions of these evolutionary models

vary depending upon how they were computed. Considering these complicated factors,

the most common approach to studying the luminosity functions of young star clusters

has been to numerically integrate these three fundamental quantities, i.e., the initial

mass function, the star-forming history and the theoretical mass-luminosity relation,

into synthetic luminosity functions and to use these model luminosity functions to

interpret the observational data.

Various groups have modeled the luminosity functions of young clusters using

realistic stellar mass functions and appropriate mass-luminosity relationships (e.g.,

Zinnecker et al., 1993; Strom et al., 1993; Fletcher & Stahler, 1994a; Lada & Lada,

1995; Megeath, 1996). Zinnecker et al. (1993) were the first to present model K band

(2.2 pm) luminosity functions for very young clusters. For their models they adopted

a coevall" star formation history in which all the stars were formed at a single instant

of time. Moreover, they assumed black-body radiation to derive bolometric corrections

and assumed a single form for the stellar mass function. Consequently, their models

were not very realistic, and they did not attempt to fit or directly compare their models

to observed cluster luminosity functions.

Lada & Lada (1995, hereafter, LL95) improved on this work by developing evo-

lutionary models for the K band luminosity functions (KLF) of young clusters ranging

in age from 106 < T < 107 yr, using empirically determined bolometric corrections

and allowing for non-coeval or continuous star formation in the clusters. Moreover,









they directly compared their models to observed infrared luminosity functions of young

clusters. However, similar to Zinnecker et al., Lada & Lada assumed a single underly-

ing initial mass function for the stars (i.e., the Miller & Scalo, 1979, field star initial

mass function), while employing a single set of the published pre-main sequence evo-

lutionary tracks from (D'Antona & Mazzitelli, 1994). Additional luminosity function

models were calculated by Strom et al. (1993) and Kenyon & Hartmann (1995), both

of whom compared their models to the de-reddened J (1.1pm) and K band luminosity

functions of young stars. In these works, model luminosity functions were primarily

used as probes of a cluster's age, but were also employed to test the similarity of the

clusters' underlying initial mass function to that for the field stars. All of these model

luminosity functions were constructed for stars with masses between 0.1 and 20 M.,

since the existing evolutionary tracks did not extend into the regime of brown dwarfs

(M < 0.08 M). Thus, many of their results are only valid as long as there are no, or

at least very few, brown dwarfs in these clusters.

It is somewhat difficult to evaluate the success of these early modeling works in

developing the luminosity function technique as a tool for deriving the initial mass

functions of embedded clusters. First, these models were fundamentally limited by

the lack of consistent evolutionary models that included young brown dwarfs. Second,

the lack of independent estimates for the star-forming histories of the clusters studied

meant that these authors approached the problem needing to constrain both the age and

mass function; they frequently constructed their models using a single mass function

equivalent to that for field stars. Further, their models were rarely applied directly

to the observations, instead requiring that the actual data be initially corrected for

various observational effects such as reddening. Thus, these efforts were never intended

to provide comprehensive models of real data such as one might expect from a true

population-synthesis model. In addition, when the models were fit to the data, error

estimates or other quantification of the usefulness of the luminosity function method









were not calculated, making it difficult to draw conclusions about the accuracy of this

method. In part due to the limitations of these early models and partially resulting

from the approach taken by these original authors, the luminosity function method has

not yet been used as a tool for deriving and for comparing the initial mass functions of

a series of young clusters.

Fortunately, technical improvements in some of these areas have recently been

made. For example, evolutionary sequences have been calculated for brown dwarfs

with masses as small as that of the planet Jupiter (Mjup). In addition, improved age

estimates for several clusters such as the Trapezium (Hillenbrand, 1997) and IC 348

(Herbig, 1998) have been made by examining brighter members using either optical

spectra or the optical color-magnitude diagram. In light of these technical advances

and the constraints placed upon the ages of some nearby young clusters, we undertook

a systematic study to determine the usefulness of a young cluster's near-infrared

luminosity function as a tool for studying and deriving that cluster's initial mass

function.

Based upon the success of prior approaches to studying the luminosity function of

a young cluster, we formulated our study using three principles: 1) Our study would

concentrate on the products of simple near-infrared surveys of young clusters. 2) We

would employ a set of model luminosity functions to interpret the products of these

near-infrared surveys. 3) We would study multiple young clusters to test, develop and

expand our methodss. From these principles, we developed a series of specific goals:


Creation of a population-synthesis algorithm for young star clusters that includes
all of the fundamental and observational characteristics relevant to the products of
a near-infrared survey.

Design of a series of numerical experiments to systematically test the sensitivity
of model luminosity functions to changes in the three fundamental quantities
governing the form of the cluster luminosity function (e.g., the star-forming
history, initial mass function, and theoretical mass-luminosity relation).









Construction of the near-infrared luminosity functions of a series of young
clusters from deep multi-wavelength near-infrared surveys of these clusters.

Derivation of the initial mass functions for these clusters through the application
of our population-synthesis models to the cluster luminosity functions.

Comparison of our results to those found via other methods for studying the mass
functions) of young clusters.

Examination of the hypothesis of a "universal initial mass function" for young
clusters by comparing the luminosity and mass functions derived for the clusters
in this study.

We accomplished these goals by focusing our efforts in three distinct ways. First, we

developed a flexible, Monte Carlo-based population-synthesis algorithm for simulating

the observations of young clusters and for creating model luminosity functions that

could be applied to cluster data. The second focus of our research has been a series

of deep near-infrared surveys of three young clusters, the construction of the infrared

luminosity functions for these clusters, and the derivation of these cluster's mass

functions. The third focus of this work is a discussion of evidence that a single process

dominates the formation of stars across the mass spectrum down to very small masses

(a few times the mass of the planet Jupiter). In summary, we find that a cluster's

near-infrared luminosity function is an excellent probe of the initial mass function

of a very young cluster, and that the combination of deep near-infrared surveys with

model luminosity functions can be used to accurately derive the initial mass function

down to and below the deuterium-burning limit in young nearby star clusters. Further,

the evidence that the IMF(s) we derive from modeling the cluster luminosity function

are robust relative to other methods suggests that KLF modeling can be applied to a

much larger sample of young clusters over a considerable volume of the local galaxy,

providing the statistics necessary for establishing the degree of uniformity of the initial

mass function through (local) space and time.









We briefly summarize the structure of this work. In Chapter 2 we develop our

Monte Carlo based population-synthesis algorithm and use this algorithm to test the

theoretical sensitivity of a cluster's luminosity function to changes in such parameters

as age and initial mass function. We then apply these models to the luminosity

function for a young cluster constructed from literature data. In Chapters 3 and 4 we

describe detailed studies of the luminosity and mass functions for the young Trapezium

and IC 348 clusters using deep near-infrared surveys. Blaauw (1964) first compared

these two clusters as part of his discussion of OB associations and subsidiary young

clusters: "Two very interesting clusters with a different character do, however, occur:

the Trapezium Cluster in I Orion, and the cluster near o Persei in II Per [IC 348].

Their dimensions are much smaller than those of ordinary clusters, and both are of

recent origin." In our study of these nearby clusters, we develop empirical recipes for

including reddening into our population-synthesis models and for statistically correcting

the observed cluster luminosity function to account for the contamination of our

observations by non-member field stars. We then apply our method to the distant open

cluster, NGC 2362, in Chapter 5 and examine the usefulness of our method when little

is known about a cluster's age or age spread. In Chapter 6 we present observational

evidence for the existence of circumstellar disks around brown dwarfs and discuss

how the continuity of disks around young stars and brown dwarfs points towards a

common origin for both. We compare the initial mass functions we have derived for

these three clusters, and examine the hypothesis for an universal mass function for

young clusters in Chapter 7. Here we combine the evidence of a common origin for

stars and brown dwarfs and the continuity of the mass function across a number of

clusters and environments to discuss what processes might dominate the formation of

stars and brown dwarfs. After summarizing our findings in Chapter 8 we briefly detail

additional future work that will focus on the new questions raised by this study. We

reserve a number of the parts of our study to the appendices. Here we engage in a







9

brief discussion of the distance to the Trapezium Cluster, and list minor details of our

modeling algorithm, including our tabulation of empirical bolometric corrections and

descriptions of the computer code used in our population-synthesis algorithm.















CHAPTER 2
MONTE CARLO MODELS OF YOUNG STELLAR POPULATIONS

2.1 Monte Carlo-Based Population Synthesis Model

For use in the interpretation of infrared luminosity functions of young stellar

clusters, we created a Monte Carlo-based population synthesis algorithm for pre-main

sequence stars. The underlying principle of our population synthesis model is the

treatment of the fundamental cluster properties as probability distribution functions that

are sampled and integrated using a Monte Carlo rejection method algorithm. Thus,

the algorithm was designed to create a synthetic star cluster with members whose ages

and masses are drawn from a specified star-forming history (SFH) and underlying

initial mass function (IMF). Each synthetic star's mass and age was converted to

observable quantities using mass-luminosity (M-L) relations interpolated from a set

of theoretical evolutionary models. Additional properties such as reddening due to

interstellar extinction or by excess flux from circumstellar disks were also assigned to

each synthetic star by using probability distribution functions, while other parameters

such as distance and binary fraction were fixed to specific values for the entire cluster.

Further, our use of a Monte Carlo formulation also allows us to run multiple numerical

simulations of a model cluster, thus giving us a statistical lens to use when comparing

our models to real clusters, which typically contain between 100 and 1000 members.

In Sections 2.2 and 2.3, we describe each of the cluster parameters and how it

was implemented into our models before detailing a series of numerical experiments

in Section 2.5 aimed at testing the sensitivity of a model cluster's luminosity function

(LF) to changes in the underlying cluster parameters. In Section 2.6 we discuss the

results of these experiments and illustrate the effectiveness of KLF modeling for

constraining a cluster's IMF by applying our technique to data taken from the literature









for the famous Trapezium Cluster in Orion. In Section C. 1 we briefly detail each of the

FORTRAN subroutines that were written to implement this algorithm.

2.2 Fundamental Cluster Parameters

2.2.1 Initial Mass Function

In our standard model, stars can have masses between 80 and 0.02 M, limits set

by the range of evolutionary models available for very high-mass O stars and very low

mass brown dwarfs and giant planets. We parameterized the underlying cluster initial

mass function with a number of different analytical forms. Throughout this work, we

refer to the initial mass function as the frequency of stars per unit log mass per unit

volume. Since we may suppose that a cluster represents a single star formation event,

then there is no purpose in integrating this function over space volume.

A simple power-law function is the most common parameterization of the IMF

and that originally used by Salpeter (1955), e.g.,


4(log( )) = cl*M, (2.1)

where cl is a normalization constant, and F is the power-law index. In this form,

Salpeter found that the initial mass function for stars in the field had F = -1.35 over

the mass range from 1 to 10 MQ. Our standard parameterization of the underlying

cluster IMF consisted of power-law segments, Fi, connected at break masses, mj. For

example, for masses between our upper mass limit and the first mass break ml, the

IMF is described as a power-law with index, F1, and from ml to m2, the IMF has a

power-law index, F2, etc. Cluster IMFs could have as many as five (5) independent

power-law segments.

We also used the log-normal distribution as a functional form of the IMF, e.g.,


(log(--)) = cl*exp(-c2*(log( )- c3)2)),


(2.2)











10.0


M/Me
1.0


0.10


MS79 Log-normal IMF ..........
Various 2 Power Law IMFs

1 0 -1
LOG M/Me


Figure 2-1:


Example mass functions used in models. The log-normal form follows the
parameterization of Miller & Scalo (1979) and is extended to the lowest
masses. Standard two (2) power-law IMFs are shown where the high-mass
IMF slope, F1, equals -1.35 (equivalent to Salpeter (1955)) and then breaks
at a mass, ml, equal to 0.5M0. Below the break mass, the IMF is gov-
erned by a low mass slope, F2, for which we show five different values:
-1.35, -0.40, 0.00, +0.40, and +1.0.


where cl is a normalization constant, c2 equals 1/(2log(c)2), c3 equals log(-j-) or

the mean log mass of the distribution and c is the variance of this mean.

Figure 2-1 illustrates these mass function parameterizations. The mean and

variance of the log-normal IMF shown correspond to the field star mass function

given by Miller & Scalo (1979, hereafter, MS79), having constants of c2 = 1.09 and

c3 = -1.02 or a mean mass of 0.0955 M1 The example two power-law IMFs shown

in Figure 2-1 have 71 = -1.35, mi = 0.5 M and F2 varying from -1.35 to +1.0.




1 This set of log-normal parameters corresponds to the MS79 derivation that used a
maximum age of the galactic disk equal to 12 Gyr.


2





0
c
3 1




0



_3
O -1










2.2.2 The Cluster's Star-Forming History

For most of the models presented in this work, we assumed a constant star for-

mation rate during the formation of a young cluster. We adopt this characterization

partially because it is the simplest such model, and partially because the preci-

sion of observations which suggest that a cluster's SFH is episodic or accelerating

(Palla & Stahler, 2000) is certain to be strongly modified by intrinsic errors that would

lead to exaggerated star-forming histories (Kenyon & Hartmann, 1990; Hartmann,

2001). Further, we assumed that there is no correlation between mass of a cluster

member and when it was formed in the cluster.

C ster Dating Parameters



<----Age Spread---->

Star Formation

Post tbegnSF endSF -now


Age Spread: ATs egSFbegnSF ndSF
Cluster Age: Tcluster -beginSF
Cluster Mean Age: cluster =beginSF (TSF)/2



Figure 2-2: Definition of the cluster's star-forming history. The cluster's mean age, z,
in this simple model is equivalent to the average of the ages of the oldest
and youngest stars, assuming a constant star formation rate.


Therefore, we parameterized the SFH using a "mean age", T, and an "age spread,"

AT. For example, a coeval cluster will have no age spread and AT/T = 0.0. A cluster

with the largest possible age spread would have AT/T = 2.0 with star formation

starting 2 x t years ago and continuing to the present. Figure 2-2 illustrates these

definitions. We note that these definitions of the cluster's star-forming history are

different than those used in the models of LL95 and Kenyon & Hartmann (1995). For









these works, the age of the cluster referred to the total timespan since star formation

began, which is also the age of the oldest cluster members. Thus, for constant star-

forming histories, their "age" would correspond to the "age spread" of our SFH and

it would also be equal to twice our derived "mean age." Our standard model SFH,

therefore, approximates any real SFH to first order by using the most common age of

the members and a rough age spread. The requirement of a constant star formation

rate, however, is not a pre-requisite of our models, and any toy or empirical distribution

of age can be used to draw ages for a synthetic cluster.

2.2.3 Theoretical Mass-Luminosity Relations

The mass-luminosity relation appropriate for converting the synthetic stars' masses

into observable luminosity is dependent on the evolutionary status of the star. For all

the clusters considered here, the youngest (1 5 x 105 years) and most massive cluster

members (M > 5M) will have already contracted on to the Zero Age Main Sequence

(ZAMS) (Palla & Stahler, 1990). For these O and B type members, we converted their

mass to bolometric luminosity and effective temperature using a theoretical ZAMS

derived from Schaller et al. (1992). No post-main sequence evolution is included

for the high and intermediate mass objects, since for the clusters considered here

(T < 10Myr), only the O stars would have had sufficient time to complete their core

hydrogen burning and begin to evolve into giant or supergiant-type stars.

The majority of the cluster members will be in the pre-main sequence phase of

their evolution. Since these stars are still contracting, the appropriate mass-luminosity

relation is age dependent, and we must rely upon theoretical evolutionary models to

convert from the synthetic star's masses and ages into luminosities. These evolutionary

models have been calculated by a number of authors (Henyey et al., 1955; Hayashi,

1961; Iben, 1965; Burrows et al., 1993; Palla & Stahler, 1993; D'Antona & Mazzitelli,

1994; Baraffe et al., 1998), who have explored a variety of different physical inputs and

initial conditions to the models. Typically these models track the pre-main sequence









evolution (luminosity and effective temperature) of a star of a particular mass across

what is referred to as the theoretical Hertzsprung-Russell (H-R) diagram.

Unfortunately, pre-main sequence (PMS) theoretical models are not typically

calculated for the entire mass range from brown dwarfs (0.001M) to high-mass B

stars (10M). Because of this, we often had to combine two different sets of PMS

tracks to provide a complete mass range. We took the opportunity to use different sets

of PMS tracks for high and low mass stars to remove an apparent mass-age correlation

found by many authors who have used PMS evolutionary tracks to derive real ages and

masses for stars using the H-R diagram (Hillenbrand, 1995; Meyer, 1996; Hillenbrand,

1997). These authors point out that when masses and ages are derived for a cluster

of real stars using PMS tracks, a correlation existed such that the more massive stars

were systematically older than the lower mass stars. Further, these authors suggested

that the cause of this correlation is due to the way canonical PMS tracks have been

constructed. Canonical PMS tracks evolve the model stars from infinite spheroids,

while recent studies suggest that stars evolve during a proto-stellar phase along a

specific mass-radius relationship referred to as the proto-stellar birthline (Stahler,

1983; Palla & Stahler, 1990). Using a proto-stellar birthline as the initial condition

for PMS tracks will most prominently adjust the predicted luminosities and effective

temperatures (as a function of time) for the youngest and highest mass stars, where

the stars' proto-stellar (birthline) lifetimes are comparable with these stars' pre-main

sequence contraction lifetimes.

Rather than using canonical PMS tracks for model stars with masses greater

than solar, we used "accretion-scenario" PMS model calculations by Palla & Stahler

(1993) and Bemasconi (1996). Accretion scenario PMS tracks better represent the

location of the young intermediate mass stars on the H-R diagrams (Palla & Stahler,

1993; Bernasconi & Maeder, 1996). Yet the accretion-scenario PMS tracks cannot

be straightforwardly used with subsolar mass canonical PMS tracks. We adopted the









accretion-scenario tracks listed above for models above 2 Me and canonical tracks

below 1M taken from D'Antona & Mazzitelli (1994) and D'Antona & Mazzitelli

(1997). Between these two mass limits, we compared the canonical and accretion-

scenario calculations. We calculated an average of each accretion-scenario and

canonical mass track, weighting the average to result in a smooth conversion from

the canonical (subsolar mass) to accretion-scenario (intermediate mass) regimes. We

examined the theoretical H-R diagram resulting from our combination of ZAMS,

accretion-scenario, averaged, and canonical PMS tracks. These new sets of tracks

and resulting isochrones were found to be smooth between all regimes and they were

used as input to the modeling algorithm. In Figure 2-3 we show an example of the

distribution of mass tracks and isochrones in the H-R diagram. We define our standard

set of PMS tracks to be a merger of the D'Antona & Mazzitelli (1997) subsolar mass

and Bemasconi (1996) "accretion-scenario" intermediate mass tracks.

Our modeling algorithm uses a cubic spline routine to interpolate between the

mass tracks and isochrones on the H-R diagram to derived luminosities and effective

temperatures for the masses and ages drawn from the IMF and SFH. Using these

luminosities and effective temperatures we converted to an absolute magnitude using

the formula:

Mk = Mbol, 2.5 x log(L/L) BC((Teff) (2.3)

We assumed Mbol, = 4.75 and our empirical bolometric corrections were tabulated

as functions of effective temperature and were taken from the literature. We list the

sources of the bolometric corrections in Section A. Appropriate bolometric correction

tables were constructed for I K bands, allowing for the calculation of red and near-

infrared colors, magnitudes and monochromatic luminosity functions.

Lastly, for defining the source of the mass-luminosity relation, we did not account

for stars in their proto-stellar phase since the contribution of these extremely young

objects to the total population of an embedded star cluster goes as the ratio of the

































Figure 2-3: Theoretical Hertzsprung-Russell diagram. Pre-main sequence evolutionary
tracks from 0.02 to 5 Meand isochrones from 0.5 to 10 Myr are shown.
The merged tracks are from DM94 and Palla & Stahler (1993). Also
shown is the birthline for a proto-stellar accretion rate of 10-5 M
yr

duration of the proto-stellar phase (' 0.1Myr) to the age spread of the stars in the

region (~ 1 2Myr), and hence will be quite small in most cases (Fletcher & Stahler,

1994a,b).

2.3 Additional Cluster Characteristics and Model Inputs

In addition to the three fundamental quantities (IMF, SFH, M-L relation) that

govern the structure of a young cluster's luminosity function, there are a number

of observational characteristics that must be included into our population synthesis

model. Some of these parameters, the distance to a young cluster, for example, are

not easily constrained by the analysis we present here, and are subsequently assumed

from literature sources, becoming a fixed parameter in our models. For very young

clusters, distance is often determined by association with a molecular cloud complex

whose systemic velocity is known and has been converted to a distance estimate. In









other cases, some high-mass cluster members are optically visible and are assumed to

be on the ZAMS, and these stars are used to derive a distance modulus. The model

algorithm always converts from the absolute passband magnitude of the stars, Mk

into an apparent passband magnitude mk based on the fixed distance. The cluster's

reddening properties and appropriate binary fraction are treated as free parameters, and

we describe their inclusion into the modeling algorithm below.

2.3.1 Reddening Properties

The mean reddening estimates, e.g., E(B-V), used in traditional open cluster

studies are inappropriate for very young clusters because of the large, variable ex-

tinction arising from the parental molecular cloud. Although the magnitude of this

extinction is decreased by working at near-infrared rather than optical wavelengths, the

reddenings are sufficiently large and spatially variable, that a single mean extinction

for the entire cluster would be inappropriate. Additionally, hot dust in circumstellar

disks around young stars reprocesses the stellar radiation and re-emits it at infrared

wavelengths, further reddening a young star's intrinsic infrared colors and increasing

the infrared flux observed.

To include these parameters into the modeling algorithm, probability distribution

functions (PDFs) are constructed for both of these reddening properties. These PDFs

can have either functional (e.g., gaussian) or empirical forms. In both cases, we

constrain the cluster's reddening properties from the infrared colors obtained when

a young cluster is surveyed. Indeed, two goals of the current luminosity function

modeling are to 1) derive recipes for extracting the distributions of reddening from the

observed infrared colors themselves, and 2) to use these distributions in our modeling

algorithm to recreate not only the cluster's luminosity function, but also to duplicate

the distribution of sources in the cluster's infrared color-color and color-magnitude

diagrams. We describe our derivation of empirical reddening PDFs in detail in Section

3.3.1.









2.3.2 Binary Fraction

One observational constraint imposed on our studies of young clusters is the

angular resolution limit of our surveys. Thus, the observed luminosity function can

be altered by the presence of unresolved multiple stars, by cluster members missed

because of chance projections or by confusion due to background stars. Because

the clusters we study are reasonably nearby, chance projections do not produce a

significant number of false binaries or missed cluster members. Further, we are not

observing clusters close to the galactic plane, in the galactic center or in other galaxies

so will not consider the latter effect of confusion in our models.

The effects of unresolved binaries and higher order systems on the cluster

luminosity function is a well known problem and it depends partially on the underlying

IMF of the primaries and of the secondaries (Kroupa et al., 1991). While the typical

angular resolution of our surveys allows us to identify some visual binary systems, we

can typically probe only to separations of ~ 200 to 300 AU where the binary fraction

is observed to be no more than 10-15% (Duquennoy & Mayor, 1991). Hence the

majority of the binaries are unresolved and may influence the form of the luminosity

function and mass function we derive.

Unresolved binaries have two effects on the form of a cluster's intrinsic luminosity

function. First, binaries with mass ratios of ~ 1 will be up to 0.75 magnitudes brighter

than the individual members, and will shift the overall form of the luminosity function.

Second, binaries with low mass ratio (low mass secondaries to higher mass primaries)

will result in cluster members that are completely lost since they will not contribute an

appreciable fraction of the total flux of the unresolved system.

We include the existence of unresolved binaries in our Monte Carlo algorithm

using a one-parameter binary fraction defined by Reipurth & Zinnecker (1993) as:

f = Nbinaries (
Nbinaries + Nsinglestars









Thus we ignore higher order un-resolved systems (triples, quadruples). We further

make the simplifying assumption that the primaries and secondaries are drawn from the

initial mass function, and that the distribution of mass ratios is uniform.

To include binaries into our algorithm we follow the formulation of Kroupa

(private communication). Simply, after N stars are sampled from the mass function,

a subset are randomly paired into binary systems, (e.g., "systems," Nsys with the

remaining stars becoming "singles," Nsing). The number of each type is approximated

in the code by the equations:


Nsing = INT (Ntars ) (2.5)
1. +f
Nsys = INT( (Nstars Nsing)/2.) ) (2.6)


Both members of a binary system are assigned the same age and extinction

drawn from the star-forming history and the appropriate reddening distribution. If the

population synthesis includes flux from a circumstellar disk, each member of a binary

is assigned a separate flux excess. The luminosities of the members of the binary

system are converted to individual magnitudes, reddened and finally merged (in flux

units) to simulate their un-resolved nature.

2.4 Model Outputs

Our Monte Carlo based pre-main sequence population synthesis code was scripted

to produce a number of different possible simulations. Taking advantage of the code's

Monte Carlo nature, random samples (of N stars and M iterations) of any or all of

the input distributions (IMF, SFH, reddening distributions) can be derived. Further,

synthetic H-R diagrams, infrared color-magnitude and color-color diagrams can be

produced for permutations of all of these input parameters.

Finally, model (binned) infrared (IJHK) luminosity functions can be created using

parameters that adjust the bin sizes and bin centers. Two model luminosity functions

are standard output for each set of input parameters. The first is the luminosity









function constructed from the un-convolved magnitude of every individual star without

the effects of reddening or unresolved binaries. The second is an observable model

luminosity function which includes these effects. This would be the luminosity

function used in modeling young cluster luminosity functions in later chapters, while

returning both the intrinsic and observable LFs allows us to make simple direct tests of

the impact of various observational quantities. In all cases, the output files are simple

ASCII files with headers containing the parameters used in that model run.

2.5 Numerical Experiments

Using our Monte Carlo population synthesis code, we performed a series of

numerical experiments aimed at evaluating the sensitivity of a young cluster's lumi-

nosity function to each of the three fundamental underlying inputs: the theoretical

M-L relation, the cluster's star-forming history and the cluster's IMF. We create a suite

of model luminosity functions systematically varying each of the three fundamental

underlying relations while holding the other two functions constant. For each synthetic

model run, we produced model luminosity functions by inning the resulting synthetic

magnitudes in half (0.5) magnitude bins as is standard for actual observed cluster

luminosity functions. Our standard model cluster for these experiments contained 1000

stars and for each set of fixed parameters we produced typically 50-100 independent lu-

minosity functions. We computed the mean luminosity function from these realizations,

and record the one sigma standard deviation of the computed mean of each model

luminosity function bin.

2.5.1 Different Pre-Main Sequence Evolutionary Models

The evolution of pre-main sequence stars across the H-R diagram and onto the

main sequence is not observationally well constrained. Details of PMS evolution

rely heavily upon theoretical PMS tracks. These theoretical PMS tracks vary in their

predictions depending on the numerical methods and theoretical assumptions used in

their creation. Since these PMS tracks are used to convert from a stellar age and mass









to a monochromatic magnitude, the resulting luminosity functions will depend to some

degree on the PMS evolutionary models which are chosen. To evaluate how PMS

tracks with different input physics, chemical abundances or effective mass ranges affect

the shape and form of a model luminosity function, we constructed and compared

model luminosity functions calculated assuming different PMS tracks.

For these experiments, we fixed the initial mass function to have a log-normal

distribution as described in Equation 2.2. We produced a suite of model clusters with

a range of mean ages from 0.2 to 15 Myr and age spreads from coeval to twice the

mean age of the model cluster. For the purposes of evaluating the effects of using

different input PMS tracks, we only directly compared KLF models having identical

star-forming histories.

D'Antona & Mazzitelli (1994): Differing input physics. D'Antona & Mazzitelli

(1994, hereafter DM94) calculated four different sets of evolutionary PMS tracks vary-

ing two input physical parameters, the input opacity tables and the treatments of

internal convection. Table 2-1 summarizes the four combinations of input physics and

other parameters of the DM94 PMS tracks. Only one of these data sets contained stars

with masses less than the hydrogen burning limit. Consequently, we used a common

range of stellar masses from 2.5 to 0.1 M to compute different model KLFs using the

four sets of DM94 PMS tracks. Figure 2-4 compares synthetic KLFs computed from

the DM94 PMS tracks for coeval models with mean ages of 1 and 7 million years,

respectively. In Figure 2-4, different symbols correspond to different input opacity

tables in the PMS tracks used. For the 1 million year coeval models, the two KLFs

corresponding to PMS tracks with Kurucz opacities are essentially indistinguishable,

indicating that the KLFs are insensitive to the convection model used. The two model

KLFs corresponding to PMS tracks with Alexander opacities exhibit a relatively narrow

but significant feature or peak between MK 3-4 which is not apparent in the KLFs

with Kurucz opacities. The position of this spike is different for the two convection









Table 2-1. Evolutionary models used in numerical experiments
Source Model Opacity Convection [_](a) Mass Range
Name Table Model (-)
DM94 ACM Alexander et al. (1989) FST(b) 2.0 0.018 -- 2.5
DM94 ACM Alexander et al. (1989) FSTN 2.0 0.018 2.5
DM94 AMT Alexander et al. (1989) MLT(C) 2.0 0.100 -> 2.5
DM94 KCM Kurucz (1991) FST(b) 2.0 0.100 -> 2.5
DM94 KMT Kurucz (1991) MLT(d) 2.0 0.100 -- 2.5
DM97(e) dl.5 Alexander & Ferguson (1994) FST(f) 1.0 0.017 -- 1.5
DM97(e) d2.5 Alexander & Ferguson (1994) FST() 2.0 0.017 -> 3.0
DM97(e) d4.5 Alexander & Ferguson (1994) FST(f) 4.0 0.017 -- 3.0

(a)Deuterium Abundance relative to Hydrogen; In units of x 10-5
(b)Full Spectrum Turbulence Model; Canuto & Mazzitelli (1991, 1992)
(c)Mixing Length Theory; 1/Hp = 1.2
(d)Mixing Length Theory; 1/Hp = 1.5
(e)DM97 models were initially released in 1997. These models were updated in
1998. The models used were those of the updated calculations.
(fFull Spectrum Turbulence Model; Canuto et al. (1996)

References. D'Antona & Mazzitelli (1994, DM94); D'Antona & Mazzitelli
(1997, DM97)


models used with the Alexander opacities. This feature is due to deuterium-burning

which causes a slowing of the stellar luminosity evolution (Zinnecker et al., 1993) and

therefore results in a pile up of stars in the luminosity function. The deuterium-burning

spike is absent in the 7 Myr coeval model in Figure 2-4, and in all coeval models

with mean ages greater than 2-3 Myr for stars above the hydrogen burning limit. The

onset of deuterium-burning is a function of stellar mass. Low mass stars contract

more slowly than higher mass stars and begin burning deuterium after high-mass stars.

However by 3 Myr, even stars at the hydrogen burning limit would have burned all of

their initial deuterium abundance.

A second feature of interest in the KLFs is the spike/dip at MK = 2 in the 7

Myr coeval model. It is present in all four 7 Myr KLFs and in all KLFs with mean

ages greater than 3-4 Myr. This feature is the result of stars reaching a luminosity

maximum on radiative tracks before beginning hydrogen burning and moving to










Comparing DM94 PMS Models
A = 1.0 Myrs, Coeva









2 4 6 8
Absolute K Maqnitude


Myrs, Coev


S ACM & AMT
-A- KCM & KMT

4 6 8
ute K Maanitude


Figure 2-4:


Model KLFs: varying physical inputs to evolutionary models. Each model
KLF corresponds to a different combination of input physics as described
in DM94 (see also Table 2-1). These model KLFs were constructed using
a log-normal IMF (see equation 2.2) with a lower mass limit of 0. 1M
and having coeval star formation with mean ages of 1 (top) and 7 (bottom)
Myr. Different symbols correspond to different input opacity tables used
by the PMS tracks. Each bin's value corresponds to the mean value of that
bin for 100 independent realizations of the model KLF. Each realization
of the model KLF contained 1000 stars. Error bars correspond to the loc
standard deviation of the mean value of that bin for the 100 iterations.


lower luminosities on the main sequence (Iben, 1965). We refer to this feature as the

luminosity maximum spike (LMS). This luminosity maximum spike has been studied

by Belikov & Piskunov (1997) in intermediate age (50-100 Myr) clusters and these

authors have used it to study the age of the Pleiades open cluster (Belikov et al., 1998).

Model KLFs appear degenerate in the absence of the deuterium-burning spike.

The existence of a deuterium spike removes the degeneracy and differentiates between

the two different PMS opacity models. Moreover, the position of the deuterium-burning









spike can differentiate between the two convection treatments but only for Alexander

opacities. However, only the youngest clusters exhibit a deuterium-burning spike.

Model KLFs computed with different PMS tracks and with mean ages greater then 2-3

Myr are essentially indistinguishable from each other and consequently insensitive to

the input physics of the PMS models. Note that the deuterium-burning spike is most

prominent when deuterium-burning occurs in those stars with masses at the peak of

the chosen IMF, which for models discussed here occurs at the hydrogen burning limit

(mean ages 1-2 Myr).

Introducing an age spread to the cluster star-forming history diminishes the

differences between the KLFs for all four DM94 and at any cluster mean age. While

we fully describe the effects of age and age spread on the model KLFs in Section

2.5.2, these result implies that except in the youngest clusters, the KLF will be

observationally insensitive to variations in the input physics of the PMS models.

To further study how different PMS tracks affect the model KLFs, we compared

the model KLFs using the PMS tracks of DM94 with the models computed using the

more recent and improved calculations of D'Antona & Mazzitelli (1997, hereafter

DM97). Table 2-1 lists the relevant characteristics of the DM97 tracks. We constructed

model KLFs computed with the standard mass range (0.02 to 80 Me) using the DM94

ACM and DM97 d2.5 PMS tracks. These two PMS tracks have similar deuterium

abundances but DM97 have advancements to the opacity table and treatment of

convection as well as a new equation of state. Figure 2-5 compares model KLFs

using these two PMS tracks with a coeval cluster SFH and mean ages of 0.8 and

5 million years. In general the overall shapes of the model KLFs from the two

different PMS tracks are quite similar but some minor differences can be quantified.

First, the DM97 model KLFs are somewhat narrower and have peaks shifted to

slightly brighter magnitudes than those KLFs corresponding to the DM94 ACM

tracks. This was a consistent result for all cluster ages and star-forming histories.










Comparing DM94 d DM98 PMS Models
7 = 0.8 Myrs, Coeval
.0-

.5

.0 -

.5


0 2 4 6 8 10
Absolute K Magnitude


S= 5.0 Myrs, Coeval




.0

-- DM94 ACM


Figure 2-5:


Model KLFs: comparing DM94 and DM97. Shown are model KLFs com-
puted using the ACM model from DM94 and the d2.5 model from DM98
(see Table 2-1). These two PMS evolutionary models differ in basic input
physics such as opacity table, equation of state and treatment of convec-
tion, however, they cover similar mass ranges and have identical deuterium
abundances. Upper panel: T = 0.8 Myr Lower panel: T = 5.0 Myr Both
panels correspond to model KLFs for clusters with a coeval star-forming
history. Error bars are the same as those in Figure 2-4.


Second, the largest differences between the model KLFs occur at the faint end. This

is where DM97 describe the largest differences in their PMS tracks with respect to the

DM94 PMS tracks. DM97 PMS tracks have a very different resulting mass-effective

temperature relation for low mass stars and brown dwarfs than DM94. Since the K

band bolometric correction is fairly insensitive to effective temperature for stars cooler

than 3500K (see Section A), these changes do not radically affect the model KLF.

Further, DM97 PMS tracks have larger luminosities for the low mass stars and young

brown dwarfs compared to DM94. Likewise the DM97 model KLFs are shifted to









brighter magnitudes with respect to DM94 for the faint end of the KLF. However, these

differences in the KLFs are relatively small and it would be difficult, observationally, to

distinguish between them.

D'Antona & Mazzitelli (1997): variations in deuterium abundance. The

DM97 PMS tracks were specifically created to study the effects of varying the initial

deuterium abundance for PMS evolutionary calculations. It is unclear how much

deuterium pre-main sequence stars might contain as they evolve from the birthline

toward the main sequence. And there is little observational evidence to constrain

this parameter, so it should be considered as an ambiguity in modeling the KLFs.

We studied the effects of the deuterium abundance on the KLFs by experimenting

with the three PMS tracks presented by DM97. The opacities used by DM97 in their

PMS tracks are advancements to those in the DM94 PMS tracks which produced

a deuterium-burning spike in the KLFs of Figure 2-4. DM97 input physics and

deuterium abundances are summarized in Table 2-1. We produced model KLFs

using the three DM97 PMS tracks, dl.5, d2.5, and d4.5, so labeled by their respective

deuterium abundance ratios, e.g., the dl.5 set of tracks has a deuterium abundance of

1.0 x 10-5. Respectively, these three sets of PMS tracks have deuterium abundances of

one half, one and two times the interstellar deuterium abundance, which is [D/H]o =

2.0 x 10-5.

Figure 2-6 compares model KLFs derived from these PMS tracks for mean ages

of 2 and 7 Myr and both coeval and AT/T = 2.0 age spreads. Comparing the coeval

models it is clear that increasing the [D/H] abundance shifts the deuterium-burning

spike to brighter magnitudes and increases its size. The deuterium-burning peak

disappeared from the dl.5 KLF by 3 Myr, the d2.5 KLFs by 10 Myr and from the

d4.5 KLFs not until beyond 10 Myr. For model KLFs shown in Figure 2-6 with the

maximum age spread, variations in the KLFs due to changes in the initial deuterium

abundance are too small to be observable. The main result here is that variations in the










-- T7
E 2.5

uo 2.0
4-
0
1.5

E 1.0
z0.5

S0.0
O :
0
-j 0


2.0 Myrs, Coeval


2 4 6 8
Absolute K Magnitude


-T =
o 2.5

co 2.0
4I-
0
1.5

E 1.0
z0.5

0, 0.0
0




-jT
E 2.5

u 2.0
4-
0
1.5
1)
I 10
E 1.0

0.5

O- 0.0
oj


/ / .u Iviy YuU v
S2.5 ,

0 \

1.5

E 1.0

0.5

c 0.0 ------
0
S0 2 4 6 8
Absolute K Magnitude


Figure 2-6:


0 2 4 6 8
Absolute K Magnitude

7.0 Myrs, T/T =








-*- [D/H] = 1.0
------ [D/H] = 2.0
-0- [D/H] = 4.0

0 2 4 6 8
Absolute K Magnitude


2.0


Model KLFs: varying the initial deuterium abundance. Each panel com-
pares model KLFs computed with different deuterium abundances at the
onset of pre-main sequence contraction. Labels ([D/H]) correspond to
the ratio of the deuterium to hydrogen abundance in units of x 10-5 and
represent one half, equal to and twice the measured interstellar medium
[D/H]. The model KLFs use log-normal IMF sampled over the entire mass
range available for the DM97 PMS models and each panel corresponds to
a specific SFH. Error bars are the same as those in Figure 2-4.


[D/H] ratio only produce significant (i.e., observable) differences in the model KLFs

of coeval (no age spread) clusters. For these clusters variations in deuterium abundance

affects the location and size of the deuterium-burning feature and this occurs only in

younger (t < 3 Myr) clusters or for the highest deuterium abundances. Once stars

have undergone deuterium-burning, their KLFs are identical. Again, the presence of an


2.0 Myrs, AT/T


2.0









age spread dilutes the deuterium-burning feature rendering the form of the cluster KLF

independent of the [D/H] ratio.

Effective mass ranges for PMS models. We investigated the effects of using

different IMF mass ranges by comparing model KLFs with the standard mass range

(0.02 to 80 Me) to model KLFs with a truncated mass range (0.1 to 2.5 Me), i.e., one

excluding brown dwarfs, intermediate or high-mass stars. This experiment is useful

for comparing our model LFs to prior LF modeling by other authors who typically did

not include stars below the hydrogen burning limit or did not include high-mass stars.

Figure 2-7 compares model KLFs with truncated and standard mass ranges for two

different star-forming histories. For a coeval SFH (upper panel, mean age 3.0 Myr), a

truncation in the mass range produces a truncation in the model KLFs at the highest

and lowest magnitude bins. However, with an age spread (lower panel, same mean

age, AzT/ = 2.0), the truncated model KLF is deficient in stars over a wider range of

magnitudes, and the two KLFs are similar only over a narrow range of magnitudes.

The form of the cluster KLF is clearly very sensitive to the adopted mass range of the

underlying IMF.

2.5.2 Star Formation History

As shown in the experiments of Section 2.5.1, mean age and age spread have

an important effect on the KLF. To more fully explore this, we created model KLFs

with a range of mean ages and age spreads, using a single underlying mass function,

and a fixed set of PMS tracks. For these experiments, we used the same log-normal

IMF as in Section 2.5.1 (see Equation 2.2). As in the previous section, we considered

two mass ranges for the IMF, one range with stars down to the 0.10 OM and one

including brown dwarfs with masses down to 0.02M. We adopted our standard

PMS evolutionary models described above, i.e., our combination of DM97 d2.5 PMS

models, Bernasconi (1996) intermediate-mass PMS tracks, and Schaller et al. (1992)









Truncations in the M-L Relation
7 = 3.0 Vyrs, Coeva


4 6


Figure 2-7:


Model KLFs: truncations in the mass-luminosity relation. Model KLFs
testing the inclusion into model KLFs of high and intermediate mass stars
as well as stars at the hydrogen burning limit and brown dwarfs. The mass
to luminosity relation was extracted from the ACM PMS model of DM94,
intermediate mass PMS tracks from Palla & Stahler (1993) and a ZAMS
from Schaller et al. (1992). Two different mean ages and SFH histories
are shown for illustration. Upper panel: coeval star-forming history and a
mean age of 3.0 Myr. Lower panel: continuous star formation over the age
of the cluster with a mean age of 5 Myr. Error bars are the same as those
in Figure 2-4.


ZAMS models. We compared the effects of changing the mean age and age spread by

studying how model KLFs evolve with time.

Figure 2-8 compares model KLFs with different mean ages and cluster age

spreads. Each panel simultaneously displays a one, three and ten million year mean

age cluster KLF for a specific AT/T. For a given age spread, the models clearly shift

to fainter magnitudes with increasing mean age. For small age spreads, the deuterium-

burning feature also evolves to fainter magnitudes with time appearing at MK = 3.5










AT/- = 0.0 Coeval


o 2.0

0 1.5
ci)
E 1.0
z
" 0.5
-j


0 2 4 6 8
Absolute K Magnitude


2.5

o 2.0

0 1.5
Q)
E 1.0
z
S0.5
0
-j


0 2 4 6 8
Absolute K Magnitude


0 2 4 6 8
Absolute K Magnitude


0 2 4 6 8
Absolute K Magnitude


Figure 2-8:


Model KLFs: varying the star forming history (z,AT). Each panel displays
a different AT/T for three mean ages of 1, 3 and 10 million years. Note
that from panel to panel, features in the model KLF caused by inflections
in the M-L relation are smoothed by the increased age spread. The appar-
ent downward break in the last bin of the model KLFs is primarily due to
incompleteness in that bin due to the lower mass limit of the M-L relation
at 0.02M. Please see Section 2.5.1 for further explanation of the effects
of an artificial truncation in the mass-luminosity relation. Error bars are
the same as those in Figure 2-4.


at 1 Myr and MK = 5.5 at 3 Myr, and MK = 8 at 10 Myr. To quantify the KLF

evolution with time, we calculated the mean K magnitude of the model KLF at each

mean age from 0.5 to 10 Myr and for a range of age spreads. In Figure 2-9, we plot

the KLF mean magnitude versus the cluster mean age and plot this quantity for the

two extrema of AT/T. Two sets of curves are plotted, the upper corresponding to an


AT/7 = 0.5










underlying cluster IMF with brown dwarfs (standard IMF mass range) and the lower to

an underlying IMF that truncates at 0.1 Me.


t No Brown Dwar


Figure 2-9:


Evolution of mean K magnitude with cluster age. The KLF mean refers to
the arithmetic mean of the K magnitudes for all synthetic cluster members.
Two sets of values are plotted for KLFs having two different underlying
IMFs. "With Brown Dwarfs" contains stars below 0.1M and "Without
Brown Dwarfs" has no objects less than 0.1M0. For each set of curves,
the KLF mean was plotted for the two extrema of the cluster's age spread,
AT/T = 0 and 2 Error bars are not shown but are within the size of the
plotting symbols for a cluster of 1000 stars.


The mean K magnitude of the model KLFs evolves over 2 magnitudes in the first

10 Myr of the cluster lifetime, regardless of the age spread or the mass range over

which the IMF was considered. Age spread has little effect except to slightly shift the

KLFs to brighter magnitudes. The evolution of the mean K magnitude proceeds most

quickly in the first 3 million years where the models evolve by 1 full magnitude. The

model KLFs without brown dwarfs naturally have significantly brighter mean values

but for these KLFs the mean K magnitude evolves similarly to the standard models.

This indicates that the KLFs are more sensitive to changes in the underlying IMF than

to changes in the cluster star-forming history. We also studied the width of the model

KLFs and found that KLFs widen systematically with time as was shown by LL95.


* AT/T = 0.0, Coeval
E AT/T = 2.0









Increasing the Age Spread
T 1.0 Mvrs


Figure 2-10:


Absolute K Magnitude

Model KLFs: varying the cluster's age spread. Synthetic cluster KLFs
with mean ages of = land6 Myr are shown in top and bottom panels,
respectively. In each panel the same four age spreads shown in Figure
2-8 are over-plotted. Upper panel: T = 1 Myr. Lower panel: T = 6 Myr.
Increased age spread erases features in the model KLFs caused by inflec-
tions in the mass-luminosity relation. Error bars are the same as those in
Figure 2-4.


Variations in the mean cluster age produce more significant changes in the the

model KLFs than do changes in the cluster age spread. We show in Figure 2-10,

model KLFs for two mean ages and for both of these mean ages we show the four

different age spreads from Figure 2-8. For a given mean age, it would be difficult to

observationally distinguish clusters with differing age spreads. In detail, models with

differing age spreads do exhibit differences in the prominence of the deuterium-burning

spike and the maximum luminosity dip/spike. At what point can one distinguish a

coeval model KLF from a model KLF with an age spread? To answer this question, we

compared model KLFs with increasing age spread to a coeval model of the same mean









age. Using the x2 test, we distinguished the age spread at which the models KLFs no

longer appear to be coeval. The general trend from our test is that for an increasing

mean age, we require a steadily increasing age spread to distinguish the models from

a coeval KLF. For mean ages up to 5 Myr, we could not distinguish model KLFs with

age spreads from their coeval counterparts until the age spread exceeded the cluster's

mean age (AzT/ ~ 1). This changes somewhat between 5 and 10 Myr, since the

deuterium-burning feature is present among the brown dwarfs but is not very prominent

in the model KLFs. Thus, only a very small age spread is required to erase it from the

model KLFs and thus the models no longer appear coevall" with only a small amount

of age spread. Once the deuterium-burning feature is lost from the M-L relation, the

models require very large and probably unrealistic age spreads for them to significantly

differ from a coeval model of the same mean age.

2.5.3 Initial Mass Function

We varied the underlying initial mass function of a young cluster to test the

influence of the input IMF on the model KLFs. In previous sections we used a single

IMF equivalent to the log-normal (MS79) mass function and only changed the mass

limits to this IMF. To test the sensitivity of the KLF to variations in the underlying

IMF, we adopted a two segment power-law IMF as defined in Section 2.2.1 and in

Equation 2.1

In these experiments, we varied Fi values from -2.5 to -0.25, ml from 0.06 to 1.5 M

and F2 values from -1.35 to +2.0. Figure 2-11 displays some of the model KLFs and

the corresponding underlying IMFs. The cluster star-forming history used for these

models has a mean age of 5 Myr and a AT/T = 1.0, or an age spread of 5 Myr. We

show model KLFs normalized to the bright end of the KLF where the underlying

IMF power-law indices have identical F1 slopes equal to -1.35. This example uses a

ml = 0.5M and five F2 values equal to -1.35, -0.40, 0.0, +0.40 and +1.35. The most










Model KLFs


2.5



2.0

C
0

0 1.5
+


E
: 1.0

0
_j


0 2 4 6 8 10 1.0 0.5 0.0 -0.5 -1.0 -1.5
Absolute K Magnitude LOG M/Me

Figure 2-11: Model KLFs: varying the initial mass function. This plot illustrates the
sensitivity of the model KLFs to changes in the form of the underly-
ing power-law IMF (see Equation 2.1). The different model KLFs are
normalized to their bright LF slopes where their underlying IMFs are
identical. The left panel shows the model KLFs corresponding to the
underlying IMFs shown in the right hand panel. Symbols are identical for
underlying IMFs and the resulting model KLF.


steeply rising KLF corresponds to a single Salpeter power-law IMF over the entire

mass range.

Model KLFs display variations due to changes in all three parameters of the

two power-law IMF. In Figure 2-11, the effects of changing F2 are large and the

differences between KLFs with a slightly rising and a slightly falling IMF below the

break mass are significant. Varying the ml produces shifts in the peak of the model

KLFs. Another result of these tests is that over the range of K magnitudes governed

by a single underlying IMF power-law, the model KLF tends to be characterized by

a power-law like slope. This is true both for the bright and faint slopes of the model

KLFs away from the turnover caused by the m parameter in the model IMF.









Other than a steep downward drop seen in the last bin of the model KLFs, the

model KLFs closely mimic the underlying IMF, decreasing or increasing in number

where the IMF is rising or falling. The drop in the last bin of the model KLFs is a

byproduct of the limits of the PMS tracks and can be understood by reviewing the

comparisons of truncated and extended M-L relations in Section 2.5.1 and Figure

2-7. Simply this turnover is the result of truncating the mass range for the underlying

IMF at 0.02MQ. In summary, these calculations clearly show that the shape of the

model KLF is very sensitive to variations in the underlying cluster IMF. Indeed, modest

variations in the cluster IMF produce significantly greater responses in the model KLFs

than do variations in the SFH and PMS model input physics.

2.6 Discussion and an Example from the Literature

2.6.1 Results and Implications of Numerical Experiments

From these numerical experiments which evaluate the sensitivity of the K-band

luminosity function to variations in three of its fundamental physical parameters: its

underlying IMF, its star-forming history, and its mass-to-luminosity relation, we find

that the KLF of a young cluster is more sensitive to variations in its underlying IMF

than to either variations in the star-forming history or the PMS mass-to-luminosity

relation.

We also find that variations in the cluster mean age can produce a significant

response in the KLF of a young cluster. In particular, we find that the KLF systemat-

ically evolves with time. Both the mean magnitude and the width of the KLF increase

with increasing mean age, confirming the results of earlier modeling (LL95). At the

same time, variations in the cluster age spread are found to have a small effect on the

form of the KLF and would likely be difficult to distinguish observationally.

Except for the youngest and purely coeval clusters, we find that the synthetic

KLFs appear relatively insensitive to the adopted PMS evolutionary models (at least

for the range of PMS models considered here). In the youngest coeval clusters, the









location and size of the deuterium-burning spike in the KLF was found to depend

sensitively on the PMS tracks adopted for the underlying stars. However, we find that

even a small amount of age spread broadens the spike and would make it observation-

ally difficult to detect.

We conclude from these experiments that the KLF of a young stellar population

can be used to place interesting constraints on the form of the cluster's underlying

IMF, provided an independent estimate of the cluster mean age is available. The most

direct method of determining the mean age of a young cluster is to obtain optical or

infrared spectra and place the objects on the H-R diagram. Through comparison to

theoretical PMS tracks, the ages of the stars are determined and a mean age for the

cluster derived.

From spectroscopic observations, one can also simultaneously derive the individual

masses of the stars and with complete spectra for all cluster members, an independent

and more direct determination of the IMF results. However because of spectroscopic

sensitivity limits, the determination of masses is usually only possible for the bright

stellar population. Since the monochromatic K magnitude of the cluster members

can be acquired for stars much fainter than the limit of spectroscopic methods, the

analysis of the near-infrared (NIR) luminosity function is a particularly powerful tool

for investigating the IMF of faint stars in distant clusters or stars at and below the

hydrogen burning limit in nearby clusters. Determining the fraction of cluster members

at and below the hydrogen burning limit is a holy grail of present stellar research.

The application of the luminosity function method to a nearby populous cluster would

provide a first glimpse into the brown dwarf population formed at the time of a typical

open cluster's birth.

2.6.2 An Example from the Literature: The Trapezium Cluster

The Trapezium cluster is a excellent system for evaluating the KLF modeling

techniques developed in this paper. It is the most densely populated and best studied









nearby (D ~ 400-450pc; see Section B) cluster, and the central part of a much larger

cluster known as the Orion Nebula Cluster (ONC). The ONC has recently been studied

by Hillenbrand (1997), who used optical spectroscopy to obtain a mean age for the

cluster of 0.8 x 106 years and to construct an IMF for stars with masses primarily in

excess of the hydrogen burning limit (HBL). In addition, infrared imaging surveys have

been made of both the Trapezium cluster (Zinnecker et al., 1993; McCaughrean et al.,

1995) and the ONC (Ali & Depoy, 1995) enabling the construction of the cluster KLF

from these literature data. For comparison with our models, we consider only the KLF

for the Trapezium cluster, the 5' by 5' central core of the ONC.

We constructed a KLF of the Trapezium by combining the cluster KLFs pub-

lished by Zinnecker et al. (1993) and McCaughrean et al. (1995). Our adopted KLF

for the Trapezium is shown in the top panel of Figure 2-12. The Zinnecker et al.

KLF includes the bright stars but is not complete at and below the HBL. The

McCaughrean et al. KLF extends to very faint magnitudes, well below the HBL

for a one million year old cluster, 400pc distant, but because of source saturation, is

incomplete for and does not include bright stars. Neither of these referenced cluster

KLFs were corrected for contamination by foreground or background field stars. In

addition, neither was corrected for the effects of nebular contamination which would

confuse the completeness of the surveys. However, we compared this combined

Trapezium KLF to the literature KLF from the Ali & Depoy (1995) survey of the

entire ONC and found good agreement in the location of the turnover, bright and faint

ends of the two KLFs, although the Ali & Depoy survey was not as sensitive as that

represented by the McCaughrean et al. KLF. We reiterate that the extent to which

this literature based KLF represents the true Trapezium KLF is uncertain because we

cannot account for field star or nebular contamination.

Here our goal is to find the simplest functional form of an underlying IMF whose

resulting model KLF best fits the observed KLF. We constrained the star-forming










Trapezium KLF


Absolute K Mc


Trapezium IMF


/-M
/ !






1 -- KLF fit IMF (g)
H97 ONC IMF
--" Salpeter IMF
H97 Completeness Limit


Figure 2-12:


Application of models to literature data. Top panel: Literature Trapezium
KLF compared to the best fit model KLF (fit from MK = -0.5 to 6.5).
Also shown: a model KLF created using a single power-law Salpeter
IMF. The cluster KLF error bars are la counting statistics. The model
KLF error bars are described in Figure 2-4. Lower panel: KLF derived
Trapezium IMF compared to the Orion Nebula Cluster IMF derived by
Hillenbrand (1997) using an optical spectroscopic study histogramm). Also
shown: the Salpeter IMF and the mass completeness limit of the optical
analysis. For comparison, model IMF (g) is scaled to the same number
of stars as the Hillenbrand IMF above the latter completeness limit. Error
bars for the Hillenbrand IMF reflect la counting statistics.


history of the Trapezium cluster by using the mean age from Hillenbrand (1997)

i.e., 0.8 million years. We allowed an age spread of 1.2 million years (AT/T = 1.5)

about this mean age, corresponding to constant star formation from 0.2 to 1.4 million

years ago. We inspected the observed KLF and determined that a single power-law

IMF could not satisfy the observations since the KLF has a peak and turnover well

above the completeness limits of the two surveys. Therefore we began with a simple









Table 2-2. Cluster IMF derived from the
Nr(a)Name X2 Prob. F1 ml

2 a 0.38 ... ...
3 b 0.71 -0.75 0.25
3 c 0.86 -1.00 0.40
3 d 0.88 -1.00 0.60
3 e 0.93 -0.75 0.25
3 f 0.99 -1.00 0.70
3 g 0.99 -1.35 0.80
4(b) h 0.96 -1.70 1.00
4(b) i 0.99 -1.70 1.00


literature Trapezuim KLF
F2 m2 F3

-0.50 0.10 +1.00
0.00 0.10 +0.75
0.00 0.08 +1.00
-0.25 0.10 +1.00
-0.25 0.10 +0.75
-0.25 0.08 +1.00
-0.25 0.08 +1.35
-0.20 0.10 +0.75
-0.20 0.08 +1.00


(a)Number of power-laws, F, in the derived IMF.

(b)Above 10M, this IMF has a Fo equal to -1.30.
F1, mi, andF2 were fixed.

2 power-law IMFs. We next used a three power-law IMF with a flat (zero slope) IMF

in the middle. For symmetry, the two outer power-law slopes were set to have equal

but opposite sign slopes. We varied these outer slopes to have absolute values between

0.25 and 2.00 and adjusted the mass range over which the middle slope of the IMF

was flat. Finally as a third set of experiments, we allowed the slope of the middle

power-law to vary, still holding the outer two slopes to have equal but opposite sign

slopes.

We produced a suite of model KLFs for these different IMFs and compared them

to the combined Trapezium KLF using a chi-square fitting procedure. Simply, we

normalized model KLFs to the observed KLF such that the model and observed KLFs

contain the same number of stars between absolute K magnitudes, MK = 0 and 6.5. We

then calculated the X2 statistic and probability over this K magnitude range. To derive

a best fit, we compared a suite of model KLFs varying a single IMF parameter, e.g,

the middle slope F2 or one of the mj values and then determining the X2 minima for

that variable. Model KLFs were created for a range of possible IMF parameters and

compared to the Trapezium KLF in this way.









Best fit model IMFs for each of the tested functional forms of the IMF are listed

in Table 2-2. Two power-law fits in general were not good. Symmetric flat topped

IMFs fit better and finally a slightly rising IMF across the middle provided a best

fit with x2 1. Some variation in each of the parameters still allowed for a fit of

X2 ~ 1 and examples are listed in Table 2-2. The IMFs (f) and (g) produced best fits

to the data and for purposes of discussion, we adopt IMF (g) as representative of the

Trapezium IMF and repeat its parameters here:

+1.35 : 0.08 MO > M,

dl--gNM = M ; F = -0.25 0.80 M > M, > 0.08 Me (2.7)

-1.35 : M, > 0.80M

The model KLF corresponding to IMF (g) is shown in the top panel of Figure 2-12

compared to the combined Trapezium KLF and compared to a model KLF calculated

with the single power-law slope Salpeter field star IMF over the entire standard mass

range.

From our modeling of the observed KLF for the Trapezium cluster we find that

the predicted IMF has a rising slope for intermediate mass stars, flattens around a

solar mass, reaches a peak near the HBL and turns over below the hydrogen burning

limit. There are several comparisons between the observed and modeled Trapezium

KLF and between the ONC IMF derived by Hillenbrand and our derived IMF (g)

which should be made. First, there exists a significant "tail" to the observed Trapezium

luminosity function which is not accounted for in the model KLFs. No attempt was

made to account for these very faint stars as cluster members because if they were,

they would require ages much older than the distribution suggested by the H-R diagram

or lower masses than provided by our standard PMS tracks we are using. We instead

suggest that these are either extremely embedded cluster members or heavily extincted

background field stars (Av > 20 30). We base these suppositions on the fact that

the Trapezium is at the core of a blister H II region on the front of a dense molecular









cloud, and because secondary peaks in young cluster luminosity functions are often

evidence of a background population seen in projection towards the cluster. Either of

these possibilities would in turn imply that our derived IMF is in fact an upper limit

to actual IMF below the hydrogen burning limit. Experiments studying the effects of

extinction on the model KLF by Megeath (1996) and Comeron et al. (1996) found that

while extinction tended to shift a luminosity function to fainter magnitudes, the slope(s)

of the KLF were preserved. Thus, the steeply falling slope at the low mass end of the

derived Trapezium IMF is reflective of the actual underlying IMF. However, the true

IMF may turnover at a larger mass than that implied by our present models.

In the lower panel of Figure 2-12, the mass function derived from the Trapezium

KLF is compared to that derived from spectroscopic observations by Hillenbrand

(1997). The two mass functions are generally very similar. In particular, these two

mass functions agree very well at the high-mass end (M, > 2.0 Mu). For masses in

the range 2.0 MQ > M, > 0.5 MQ the IMF derived from modeling the luminosity

function contains more stars than that derived by Hillenbrand. It is not, however,

clear how significant this difference is given the possible systematic uncertainties

involved in both methods of determining the IMF. Further, these two IMFs sample

different volumes of the Orion Nebula region. For masses below M, < 0.1 Me, the

IMF derived from the KLF modeling also contains considerably more stars than the

spectroscopic IMF. However, this difference is also not likely to be significant either

since the spectroscopic IMF of Hillenbrand (1997) is not complete below 0.1 MQ.

Lastly, we can investigate whether the field star IMF (FSIMF) could also produce

a KLF which reasonably matched the literature Trapezium KLF. To test this, we used

the recent field star IMF parameterization from Scalo (1998). Scalo (1998) suggested a









multiple power-law IMF with the form:

-1.30 : M. > 10.00 M

d log = M F = -1.70 : 10.00 M > M. > 1.00 M (2.8)
dlogM
-0.20 : 1.00M > M, > 0.10AM

Comparing the IMF in Equation 2.7 to the field star IMF in Equation 2.8, one finds

that these two IMFs are quite similar, although for stars in the range of 10.0 >

M/M > 1.0, the Scalo IMF is steeper than the IMF in Equation 2.7. In addition,

the Scalo FSIMF does not extend below the hydrogen burning limit. To facilitate

comparison to the Trapezium data, we added a fourth power-law to the Scalo IMF to

account for the faintest stars. We varied m2, the mass at which the fourth power-law

begins, between 0.06 and 0.1 M. In addition, we varied the slope of the fourth power-

law, F4 between -1.0 and +2.0. The best fits with this IMF are also listed in Table 2-2.

Using this modified field star IMF did yield a X2 ~ 1 with an IMF that breaks near the

hydrogen burning limit and falls with a similar steep slope as in the prior IMF fits.

To the extent that our adopted KLF represents the true KLF of the cluster, our

modeling suggests that the IMF for brown dwarfs in the Trapezium cluster falls

relatively steeply with decreasing mass. However, because contamination due to

reddened background stars and incompleteness due to nebular confusion has not

properly been taken into account in the construction of this literature Trapezium KLF,

the form of the derived IMF below the hydrogen burning limit should be regarded with

appropriate caution. As shown in Lada & Lada (1995) and Lada et al. (1996), one can

use control-field observations (which are not available for this dataset) to gauge the

completeness and membership at the faint end of the LF. Also, our present modeling

has not included the effects of extinction and infrared excess. Hillenbrand et al. (1998),

using the (I-K) diagnostic, found an average K band excess of 0.35 among identified

optically visible cluster members. This average excess is smaller then the bins we have

used to construct the Trapezium KLF, and therefore should have only a minor effect.









Overall, we conclude from our modeling that the IMF of the Trapezium cluster is

well represented by a three power-law mass function with a high-mass slope between

-1.00 and -1.7, a break in slope between 1 and 0.6M followed by a relatively flat

or slightly rising slope to the hydrogen burning limit. From our luminosity function

modeling, we then found, for the first time that the Trapezium IMF falls with a steep

slope ~ +1 into the brown dwarf regime.

2.7 Conclusions

After developing a Monte-Carlo based model luminosity function algorithm,

we performed a series of experiments aimed at studying how the pre-main-sequence

mass-to-luminosity relation, star-forming history and initial mass function each affect

the form of the luminosity function for populations of young pre-main sequence stars.

Using models of the near-infrared luminosity function and varying these primary

inputs, we have derived the following simple conclusions about model near-infrared

luminosity functions:

1. We find that the KLF of a young cluster is considerably more sensitive to variations

in its underlying IMF than to either variations in the star-forming history or the PMS

mass-to-luminosity relation. 2. PMS luminosity functions evolve in a systematic

manner with increasing mean age and age spread. They evolve to fainter magnitudes

and widen systematically with age. 3. The KLFs of young stellar populations are

found to be generally insensitive to variations in the adopted PMS mass-to-luminosity

relations. In the youngest, coeval clusters, the presence of deuterium-burning can

produce significant features in the KLF which are sensitive to the adopted mass to

luminosity relation. However even a small departure from a purely coeval star-forming

history will render these features difficult to detect observationally.

We then undertook a preliminary examination of the Trapezium Cluster using data

taken from the literature. We apply our models to the K band luminosity function of

the Trapezium and are able to derive an underlying Trapezium IMF which spans a









range of stellar mass from 5 M to 0.02 M., well into the brown dwarf regime. The

IMF we derive is the simplest multiple power-law function which can reproduce the

observed luminosity function of the cluster given the mean age and star-forming history

derived from previous optical spectroscopic studies (Hillenbrand, 1997). The derived

IMF for the Trapezium cluster consists of three power law segments, has a peak near

the hydrogen burning limit and steadily decreases below the hydrogen burning limit

and throughout the brown dwarf regime. We derive a brown dwarf mass spectrum of

the form dN/dlogm ~ m+1 (0.08 > M/M > 0.02). However, the form of the IMF

below the hydrogen burning limit must be regarded with caution since the faint end of

the observed cluster KLF has not been adjusted for the possible effects of background

star and nebular contamination. Above the hydrogen burning limit, the Trapezium IMF

we derive from its KLF also appears consistent with that recently advocated for field

stars by Scalo (1998).















CHAPTER 3
THE FAMOUS TRAPEZIUM CLUSTER IN ORION

In Section 2.6.2 we explored the monochromatic K band luminosity function for

the well-studied Trapezium Cluster in Orion, which we constructed from literature

sources. While we found good agreement between the mass function derived from

modeling the cluster's luminosity function and that IMF found for this cluster using a

spectroscopic analysis of the optically visible members, luminosity function modeling

enabled the derivation of the cluster's substellar IMF, which was not possible from the

optical/spectroscopic analysis. We concluded from the application of these first-order

models to the Trapezium Cluster KLF that model luminosity functions are indeed

useful for studying the mass functions of young clusters.

However, the models we applied to the Trapezium cluster did not include other

observational characteristics of a young cluster that may affect the conversion between

the luminosity and mass functions. Having only the monochromatic Trapezium KLF

taken from the literature with no color or completeness information prevented our

studying these observational effects in detail. Further, we concluded that we could

not fit our models to the entire luminosity range of the literature KLF because of

structure that we attributed to heavily reddened cluster members or background field

stars. To improve upon this modeling and to standardize the formula for applying the

model luminosity functions to the products of a deep near-infrared survey of a young

embedded cluster, we have constructed over a three year period of observations a multi-

epoch, multi-wavelength near-infrared census of the Trapezium Cluster that we describe

in Section 3.1. Using this detailed near-infrared census of the Trapezium, we have

expanded our analysis of this cluster's K band luminosity function and its underlying

Initial Mass Function. In Section 3.2, we construct the cluster's KLF, exploring both









the contribution of background field stars, and the completeness of our survey as it

probes the cluster's parental molecular cloud. We rederive the cluster's underlying

IMF in Section 3.3, refining our techniques to include the effects of source reddening

and to fit the model KLFs to the data. In our revised analysis we are able to probe

the cluster's KLF to fainter magnitudes and derive the cluster's mass function down

to the deuterium-burning limit. With these new results, we discuss in Section 3.4 the

relationship between the form of a cluster's KLF and its derived IMF, and we compare

our Trapezium IMF derived in this chapter and in Chapter 2 to the Trapezium IMF

derived by other authors using different methods. We illustrate the relative robustness

of the pre-main sequence mass-luminosity relation as predicted by different theoretical

evolutionary models of young stars.

3.1 Near-Infrared Census

To derive a complete multi-wavelength census of the sources in the Trapezium

Cluster, we performed infrared observations during 1997 December, 1998 November

and 2000 March using two telescopes: the 1.2m telescope at the Fred Lawrence

Whipple Observatory (FLWO) at Mt. Hopkins, Arizona (USA) and the European

Southern Observatory's (ESO) 3.5m New Technology Telescope (NTT) in La Silla,

Chile. These observations yielded the multi-epoch, multi-wavelength FLWO-NTT

infrared catalog that contains ~ 1000 sources. Subsets of this catalog have been

published previously in the Lada et al. (2000) and Muench et al. (2001) studies of

the frequency of circumstellar disks around stars and brown dwarfs in the Trapezium

Cluster. We detail below the observations, data reduction, and photometry involved

with the construction of the catalog. We also include summaries of the photometric

qualities of the datasets and an explanation of the electronic version of the final

FLWO-NTT infrared catalog.











3.1.1 Observations

We summarize in table 3-1 the characteristics of the three observing runs used to

obtain the infrared photometry that comprise the FLWO-NTT Near-Infrared Catalog of

the Trapezium Cluster. We compare the areas) covered by the FLWO-NTT catalog to

those of other recent IR surveys in figure 3-1.

-5.32


-5.34


-5.36


-5.38


0 -5.40


-5.42
E M...


83.88 83.86 83.84 83.82
R.A. ( J2000 )


83.80 83.78 83.76


Figure 3-1:


Comparison of recent Trapezium cluster IR surveys. The two shaded
regions represent the 6'5 x 6!5 FLWO survey and the 5' x 5' NTT
survey presented in this work. Also shown are the HST-NICMOS
survey (Luhman et al., 2000, solid black border), the Keck survey
(Hillenbrand & Carpenter, 2000, solid white border), and the UKIRT
survey (Lucas & Roche, 2000, broken black border). The locations of lu-
minous cluster members, spectral types B3 and earlier, are shown as white
stars.


Whipple Observatory 1997 and 1998: 1.2m JHK-bands. Initial infrared

observations of the Trapezium Cluster region were made on 14, 15, 16 December 1997

with the FLWO 1.2m telescope at Mt. Hopkins, Arizona using the STELIRcam dual

channel infrared camera. The STELIRcam instrument allows simultaneous infrared


FLWO Region NTT Region -B3 Stars
5.46 I I I I I I I .





















Table 3-1. Summary of infrared observations of the Trapezium cluster
Observatory(a) Date Passband (b)Plate Scale ()
YYYY / MM / DD Beamsize

FLWO 1997 / 12 / 14 H 0.596 / 3.58
FLWO 1997 / 12 / 14 K 0.596 / 3.58
FLWO 1997 / 12 / 15 H 0.596 / 3.58
FLWO 1997 / 12 / 15 K 0.596 / 3.58
FLWO 1997 / 12 / 16 J 0.596/ 3.58


FLWO
FLWO
FLWO
NTT
NTT
NTT
NTT


1998
1998
1998
2000
2000
2000
2000


0.596
0.596
0.596
0.288
0.288
0.288
0.288


3.58
3.58
3.58
1.73
1.73
1.73
1.73


(a)FLWO: Fred Lawrence Whipple Observatory; NTT: New
Technology Telescope.
(b)Filter central wavelength )(/um)): FLWO- J) 1.25, H)
1.65; K) 2.20; L) 3.50; NTT- J) 1.25; H) 1.65; Ks) 2.16.

(c)Plate scale: arcsec/pixel; Beamsize: diameter of photom-
etry beam (arcsec)









observations using two 256 x 256 pixel InSb arrays and employing a dichroic mirror

to divide wavelengths long-ward and short-ward of 1.9 um. A cold lens assembly

allows three changeable fields of view and for all our FLWO observations the camera

was configured to have 2'5 x 2'5 field of view with a plate scale of 0."6 /pixel. We

surveyed the Trapezium Cluster region in a 3 x 3 mosaic pattern, centering on the

bright 07 star HD 37022 (0 Ic Orionis) and overlapping ~ 34" between mosaic

positions. Our observational technique was to observe 3 on-cluster mosaic positions

followed by 1-2 non-nebulous off-fields which were used both for the creation of

accurate sky/flat fields and for field star estimation. These off-fields were centered at at

R.A. = 05h26/; DEC. = -0600' (J2000) and were determined to be free of molecular

material by inspection of the Palomar Sky Survey Plates and the 100 micron dust

opacity maps of Wood et al. (1994). On 14 December 1997, Hbarr(1.65 um) and

Kbarr(2.2 um) images were obtained for all 9 mosaic positions, and 7 of the 9 mosaic

positions were repeated at H and K band on 15 December. Jbarr(1.25 um) images

of all 9 mosaic positions were obtained on 16 December 1997. Each mosaic position

was observed with nine dithers of 1 minute each (4 co-additions of 15 seconds) and

with 12" spacing, yielding an effective integration time of 9 minutes per field. The

Trapezium Cluster region was observed at optimal airmass ( 1.25 < sec(z) < 1.50

). The resulting JHK mosaics mutually covered an on-cluster area of approximately

6'5 x 6'5. Conditions were photometric throughout all three nights with seeing

estimates ranging from 1.2 1.7 arc-seconds (FWHM).

To improve the photometry of bright sources and increase the dynamic range of

our data, we used STELIRcam at the FLWO 1.2m telescope to obtain additional short

exposure J and H band images on 4 and 5 November 1998. The Trapezium Cluster

region was again observed in a 3 x 3 mosaic but with the telescope in nodding mode

taking a single 12 second (12 co-additions of 1 second each) image at each mosaic

position followed by an identical off-field exposure at a nod position 450' to the west.









After finishing all 9 mosaic positions, the center of the mosaic was shifted by a small

random amount (5 10") and the pattern was repeated. Nine repetitions of the mosaic

yielding a total effective integration time of 108 seconds per band and these images

were observed at transit, with a range of airmasses of 1.24 1.28. The resulting JH

mosaic images covered an area of 7' x 7' or slightly larger than the FLWO 1997

observations. Conditions were again photometric with seeing estimated at 1.6 1.8".

In this dataset only the brightest 8 stars (all OB spectral types) were saturated.

European Southern Observatory 2000: 3.5m JHKs-bands. Our NTT

images of the Trapezium Cluster were obtained under conditions of superb seeing

(~ 0.5" FWHM) on 14 March 2000 using the SOFI infrared spectrograph and imaging

camera. The NTT telescope uses an active optics platform to achieve ambient seeing

and high image quality, and the SOFI camera employs a large format 1024 x 1024

pixel Hawaii HgCdTe array. To obtain a single wide field image of the Trapezium

Cluster, we configured SOFI to have a 4'95 x 4'95 field of view with a plate scale

of 0"'29 /pixel. Each exposure consisted of 9 separate dithers each randomly falling

within 20" of the observation center. Each individual dither was the co-average of eight

1.2 second exposures, yielding an total effective integration time of 86.4 seconds for

each combined image. We display a JHKs color composite image of the NTT region in

Figure 3-2.

We observed the Trapezium Cluster with identical sequential pairs of on and

off-cluster dithered images. During one hour on 14 March 2000, we obtained four

image pairs of the Trapezium Cluster and off-cluster positions. These were, in temporal

order, at Ks (2.162 um), H (1.65 um), J (1.25 um) and again at Ks, and the on-cluster

images had FWHM estimates of 0.53", 0.55", 0.61"and 0.78". Seeing estimates of

stars in the paired non-nebulous off-cluster images) yielded similar if not marginally

higher resolution point spread functions (PSF). Observations were taken near transit

with a very small range of airmass ( 1.138 < sec(z) < 1.185).







































Figure 3-2: Infrared color composite image of the Trapezium. Taken with SOFI at the
ESO NTT telescope, La Silla, Chile, March 2000. North is up and east is
left and the field of view is 5' x 5'.

3.1.2 Data Reduction and Photometry

Data reduction of the FLWO and NTT images was performed using routines in

the Image Reduction and Analysis Facility (IRAF) and Interactive Data Language

(IDL). Our standard data reduction algorithm was described in Lada et al. (2000)

for the FLWO images, and it was subsequently used for the NTT images. Simply,

individual dithered frames were reduced using sky and flat field images derived from

the non-nebulous off-cluster dithered images which were interspersed with the on-

cluster images. Each set of reduced dithered frames were then combined using a

standard "shift-and- add" technique. While all the FLWO data was linearized after









dark-subtraction using a system supplied linearity correction, linearization coefficients

were not obtained for the NTT data. "Sky" flat-fields constructed from the NTT

images were compared to system flat-fields which are regularly taken and monitored

by the NTT staff. While the NTT system flat-fields were found to vary by only 2-3%

over long periods of time, when we compared our sky flat-fields to the system flat-

fields, significant small scale variations (5-10%) were revealed across the array. We

concluded this was due to our relatively short NTT integration times which results

in poor sampling of the intrinsically non-flat SOFI array. Therefore, we substituted

the system supplied flat-fields into our reduction procedure. The high resolution of

our NTT images results in moderate under-sampling of the point spread functions; we

tested to see if sub-pixel linear reconstruction (drizzling) of our images would improve

our data quality. Since our images have only a few dithers (9), the drizzle algorithm

did not improve our result over standard integer "shift-and-add."

Each reduced image was characterized with a FWHM estimate of the stellar

PSF and an estimate of the pixel to pixel noise. The stellar FWHM was estimated by

selecting 10-20 stars per image using IMEXAM and averaging their "enclosed" FWHM

measurements. Roughly thirty 100 pixel boxes were placed randomly across each

image from which to measure the pixel-to-pixel noise. While a single pixel to pixel

noise estimate for an nebulous image is not likely accurate, we used it in the IRAF

DAOFIND algorithm to search for objects 5 0 above the noise threshold. The found

sources were then marked on the images, and each source was inspected by eye to

remove obvious false detections and include objects missed by DAOFIND. This manual

check and selection process was bolstered by using the numerous repeat observations

in our data set to ensure a source's validity. We use our off-field non-nebulous images

to estimate the formal detection sensitivity and find 10 ( limits of: 18.5 at J, 17.7 at H

and 16.8 at K for our deep 1997 FLWO observations; 15.3 at J and 15.1 at H for our









shallow 1998 FLWO observations, and 19.75 at J, 18.75 at H and 18.10 at Ks for our

2000 NTT observations.

The 1997 and 1998 FLWO observations all had FWHM estimates between

2.2 and 3.0 pixels and are, therefore, marginally sampled. We employed the IRAF

DAOPHOT (Stetson, 1987) point spread function (PSF) fitting routine to derive

photometry for these sources. Our procedure was to perform multi-aperture (2-10 pixel

radii) photometry on all the sources on each image, to select 20 stars on each image

from which to derive a PSF, and in an iterative fashion, to create the PSF, subtract

nearest neighbors and to re-create the PSF until a good PSF was derived. Final PSF

photometry was extracted using the ALLSTAR routine and the subtracted images were

visually inspected for faint stars missed near bright stars. We used a PSF fit radius

of 3 pixels or a beam of 3.6" for our PSF photometry, and set the sky annulus to a

10 pixel radius. Our PSF procedure employed the sky-fitting routines (Parker, 1991)

implemented in the DAOPHOT package which we found in artificial star tests to

decrease our photometric errors in nebular regions by a factor of two.

The 2000 NTT images had FWHM estimates ranging between 1.8 and 2.1 pixels,

and these images are therefore marginally under-sampled and not easily suitable for

PSF photometry. Further, the SOFI field of view suffers from coma-like geometric

distortions on the northern 10- 15% of the array. For these two reasons, we decided

to perform only aperture photometry on the NTT images. Multi-aperture photometry

was performed on sources detected in the NTT image using annuli with radii from 2

to 10 pixels. The sky was measured from the mode of the distribution of pixel values

in an annuli from 10 to 20 pixels. From inspection of the curves of growth of both

isolated and nebulous sources, we chose a 3 pixel radius (1.8" beam) for most of our

NTT sources. Additionally, the choice of small apertures allowed us to minimize the

effects of nebular contamination and crowding on the stellar PSF. For faint sources in

very confused or highly nebulous regions, we repeated the photometry with a 2 pixel









aperture and a sky annulus from 7 to 12 pixels. The change in sky annulus does not

significantly affect our photometry because the fraction of the stellar PSF beyond 7

pixels contains less than 5% of the flux, and the errors resulting from including this

flux in the sky estimate are smaller than the errors introduced from using too distant a

sky annulus on the nebulous background.

Aperture corrections were derived for our data by choosing t 15 relatively

bright stars as free of nebular contamination as possible. We performed multi-aperture

photometry on them and using the IRAF MKAPFILE routine to visually inspect the

stellar curves of growth and calculate corrections. Since small apertures were used to

minimize the effects of the bright nebular background, the resulting corrections which

constituted a somewhat substantial fraction of the stellar flux. Aperture corrections

were carefully checked by comparing the corrections derived for on (nebulous) and

off-cluster positions, which are interspersed in time with the on-cluster frames, and

found to agree or to correlate with changes in seeing. Because the 1997 and 1998

FLWO observations were performed on the same photometric system and under similar

conditions, their aperture corrections were similar and fairly constant between mosaic

positions. The average aperture correction from the 3 pixel fitting radius to the 10

pixel sky radius was -0.35 magnitudes. For the NTT images photometered using

apertures, a typical 3 pixel aperture correction was -0.14 magnitudes and for those stars

photometered using a 2 pixel aperture, a correction of -0.34 was used.

3.1.3 Photometric Comparisons of Datasets

We report in the electronically published catalog all the photometry from the

FLWO and NTT observations. Further, we explored any photometric differences

between the FLWO and NTT observations because both systems will be merged to

construct the cluster's luminosity function, since they do not have the same dynamic

range. These differences include the filter systems, the methods and effective beam-

sizes of the photometry and the epochs of the observations. We tested if any color









terms were present due to differing photometric (filter) systems, we compared the mag-

nitudes and colors of 504 sources common to both the NTT and FLWO photometry.We

compared the (J H) and (H K) colors of the NTT photometry to the FLWO photom-

etry and fit these comparisons with linear relations. The (J-H) colors were well fit by a

linear relation (slope 1); however, we found an offset, A(J-H) 0.10 magnitudes

between the two systems. A similar comparison to the photometry of sources in the

Two Micron All Sky Survey (2MASS) catalog 1 indicated this offset was at J band

and was restricted to the FLWO sources. Comparison of 2MASS photometry to the

NTT photometry revealed no systematic offsets. A comparison of the FLWO and NTT

(H K) colors was also well fit by a linear relation (slope ~ 0.97) though this slope

suggests that for the reddest sources, the NTT (H Ks) color is bluer than the FLWO

(H K) color.

Further, it was evident from these comparisons that while the global filter systems

are quite similar, the difference in the NTT and FLWO photometry of individual

sources was larger than expected from formal photometric errors 2 From our fake

star experiments and from the photometry of sources in overlap regions on mosaicked

frames, we determined our measured photometric error is 5% for the majority of our

sources increasing up to 15% for the sources at our completeness limit. However,

when comparing sources common to both the FLWO and NTT data (well above our

completeness limit), we derived loc standard deviations of 0.22 for magnitudes and

~ 0.18 for colors. Very similar dispersions were derived when comparing our FLWO

photometry to the Hillenbrand et al. (1998) or 2MASS catalogs or when comparing our



1 A current un-restricted search of the 2MASS First and Second Incremental Point
Source and Extended Source Catalogs currently returns only 171 sources.
2 the quadratic sum of uncertainties from aperture corrections, zeropoint and airmass
corrections, flat fielding error and sky noise









NTT data to the Hillenbrand & Carpenter (2000) H and K band dataset. We attribute

a portion of this additional photometric noise between the different datasets to the

intrinsic infrared variability of these pre-main sequence sources which has been found

for stars in this cloud to have an average of 0.2 magnitudes at infrared wavelengths

(Carpenter et al., 2001). We note that the difference in the beamsize used for the

FLWO and NTT photometry and by the various other published data sets will also

contribute a degree of added photometric noise due to the presence of the strong

nebular background, thus making the NTT photometry preferable to the FLWO data for

its higher angular resolution.

3.1.4 Astrometry and the Electronic Catalog

Astrometry with reasonably high precision was performed by matching the XY

pixel locations of a large number (> 50) of the observed sources to the equatorial

positions of these sources listed on the 2MASS world coordinate system and deriving

full plate solutions using the IRAF CCMAP routine. Mosaic positions of the 1997

and 1998 observations were shifted to fall onto a common XY pixel grid defined by

the K band FLWO 1997 mosaic images. To create the common K band XY grid,

sources in the overlap regions between mosaic positions were matched and global

offsets calculated. The two camera arrays of the FLWO STELIRcam instrument are

not centered precisely on the sky and the J and H band coordinates were transformed

using the IRAF GEOMAP routine into the K band XY coordinate grid. The NTT

positions were aligned to the NTT J band image. For the FLWO plate solution, 161

2MASS sources were matched to the FLWO XY coordinates yielding a plate scale of

0.596 "/pixel and an astrometric solution with rms errors of ~ 0.10". An independent

solution of 82 NTT sources matched to the 2MASS database yielded a plate scale of

0.288 "/pixel and an astrometric solution having rms errors 0.07".

We construct the electronic version of the FLWO-NTT catalog based upon all

the sources detected by our FLWO and NTT observations, and we compliment our









electronic catalog by including sources identified in other catalogs and falling within

our survey area, but that were saturated, undetected or unresolved by our observations.

Since our final catalog covers a substantially different area than comparable deep

infrared surveys and includes numerous new sources, we chose to assign new source

designations for our final catalog. These are based upon the IAU standard format that

includes a catalog acronym, a source sequence, and source specifier. For the catalog

acronym, we chose MLLA, based upon the initials of the last names of the authors.

This acronym is currently unused in the Dictionary of Celestial Nomenclature. We

chose to sequence the catalog using a running number incremented from 00001 to

01010. We use a specifier only where necessary to distinguish unresolved sources,

typically employing the designations (A), (B), etc. NTT astrometry is preferentially

used in the final catalog. For undetected or unresolved sources, we made every effort

to include astrometry from the source's identifying catalog if the original catalog

could be globally aligned to the FLWO-NTT catalog. We list cross-references based

on the most comprehensive or deep surveys; these include the Hillenbrand (1997),

Hillenbrand & Carpenter (2000), Luhman et al. (2000) and McCaughrean & Stauffer

(1994) designations. For sources lacking cross-references in these catalogs, we list their

2MASS designations (circa the 2nd Incremental 2MASS Point Source Catalog) where

possible. The LR2000 designations are based on their derived equatorial coordinates

and due to significant astrometric errors do not correspond to the positions we derive

in the FLWO-NTT catalog. For example, we find off-sets of -0.42" in RA and 0.44"

in DEC between our positions and those of LR2000. After removing these offsets,

we still find median residuals of 0.44" between our coordinates and those of LR2000

with errors as large as 1"; this is in contrast to the rms residuals of 0.1" between our

catalog and the 2MASS and HC2000 positions. Hence we do not list the LR2000

position-dependent designations except where necessary to identify sources undetected

by final catalog.









The entire FLWO-NTT Trapezium Cluster catalog has been published electron-

ically in the recent work, Muench et al. (2002). To illustrate what was publically

released in that catalog, we have supplied a sample table here, consisting of only a

subset of the sources available in the electronic version.

3.2 Trapezium Cluster K band Luminosity Function

We restrict our subsequent analysis of the cluster's luminosity and mass function

to the area surveyed by our deeper NTT observations. Our observations detected 749

sources within this region. The completeness of this sample at the faintest magnitudes

is difficult to quantify because of the spatially variable nebular background. The

formal 10 ( detection limits of our catalog in the NTT region are 19.75 at J, 18.75

at H and 18.10 at Ks based upon the pixel to pixel noise in non-nebulous off-cluster

observations that were taken adjacent in time to the on-cluster images. To better

estimate our actual completeness limits, we performed artificial star experiments by

constructing a stellar PSF for each of our images and using the IRAF ADDSTAR

routine to place synthetic stars in both the off-cluster and the nebulous on-cluster

images. A small number of synthetic stars (30-70) with a range of input magnitudes

were randomly added across each image and were then recovered using the DAOFIND

routine. This was repeated a large number of times (40-200) to achieve sufficient signal

to noise for these tests. In off-cluster images, the derived 90% completeness limits

agreed well with the estimated 10 a detection limits. In the on-cluster images, the

completeness limits were reduced to 90% completeness limits of J ~ 18.15, H ~ 17.8,

and Ks ~ 17.5 with slightly brighter limits in the dense central core (0.5' radius from

01C Orionis). We also carefully compared our source list to those published by other

recent surveys for the NTT region. To our resolution limit, we detected all the sources

found by the Hillenbrand & Carpenter (2000, hereafter, HC2000) Keck survey except

for one, all but two sources from the Luhman et al. (2000) Hubble Space Telescope

NICMOS survey, but we could not identify nine sources listed in Lucas & Roche

























Table 3-2. FLWO-NTT near-infrared catalog
Seq Spec RA Dec FLWO (Mag) FLWO (Err) NTT (Mag) NTT (Err) Phot H97 HC2000 Other
(J2000) (J2000) J H K L J H K L J H Ks J H Ks Flag ID ID ID


00001 5 35 2245
00002 5 35 2657
00003 5 35 1165
00004 5 35 1597
00005 5 35 0921
00006 5 35 20 13
00007 5 35 0448
00008 5 35 05 18
00009 5 35 1148
00010 5 35 2257
00011 5 35 0692
00012 5 35 2434
00013 5 35 1048
00014 5 35 1076
00015 5 35 1542


-5 26 109 1459 1368 9900 1240 002 006 -1 00 026 0
-5 26 096 1624 1541 9900 9900 004 001 -1 00 -1 00 0
-5 26 090 1351 11 63 1056 921 004 001 002 002 0
-5 26 074 1412 1225 9900 1030 002 001 -1 00 004 0
-5 26 057 1701 1618 1565 005 004 006 0
-5 26 042 1457 1301 1227 001 001 001 0
-5 26 041 1214 1125 1089 1015 003 004 003 003 0
-5 26 034 1354 1288 1252 1204 001 001 003 012 0 262
-5 26 026 991 906 884 847 005 001 002 002 0 365
-5 26 021 17 18 1656 1637 009 008 008 0
-5 26 005 1348 13 14 1254 11 43 002 005 002 006 1339 1256 12 19 001 001 001 0 299
-5 26 00 3 1309 12 42 12 12 11 38 008 003 002 010 1305 12 37 12 02 001 001 001 0 3101
-5 26 003 1301 1225 11 82 11 11 003 003 002 005 1292 1207 11 66 001 001 001 0 3104
-5 26 000 1554 1361 1260 11 72 004 008 002 009 1542 1360 1257 001 001 001 0
-5 25 595 1372 1284 1238 11 56 001 002 002 006 1368 12 78 1226 001 001 002 0 3103


Note FLWO Fred Lawrence Whipple Observatory, Mt Hopkins, Arizona, NTT New Technology Telescope, European Southern Observatory, La Silla, Chile


References Hillenbrand (1997, H97), Hillenbrand & Carpenter (2000, HC2000)


0535201-052604(4)
0535044-052604(4)









(2000, hereafter, LR2000) UKIRT survey. Further, it is our finding of 58 new sources

within our NTT region and un-reported by prior catalogs that adds support to the deep

and very sensitive nature of our census.

3.2.1 Constructing Infrared Luminosity Function(s)

The FLWO and NTT observations overlap considerably in dynamic range with

504 stars having multi-epoch photometry. For our analysis, we preferentially adopt

infrared luminosities from the NTT photometry because it has higher angular resolution

and it is an essentially simultaneous set of near-infrared data. For 123 stars that are

saturated in one or more bands on the NTT images, the FLWO photometry was used.

This transition from NTT to FLWO photometry is at approximately J = 11.5, H =

11.0, and K = 11.0. For the brightest 5 OB stars, saturated on all our images, we used

JHK photometry from the Hillenbrand et al. (1998) catalog. Photometric differences

between the FLWO and NTT datasets are small (see section 3.1.3) and will not affect

our construction of the Trapezium Cluster infrared luminosity functionss.

In Figure 3-3, we present the raw infrared Trapezium Luminosity Functions

(LFs). We use relatively wide bins (0.5 magnitudes) that are much larger than the

photometric errors. In Figure 3-3(a), we compare the J and H band LFs for stars in

this region. In the Figure 3-3(b), we compare the K band LF of the NTT region to that

K band LF constructed in Section 2.6.2. As was observed in previous studies of the

Trapezium Cluster, the cluster's infrared luminosity function (J, H, or K) rises steeply

toward fainter magnitudes, before flattening and forming a broad peak. The LF steadily

declines in number below this peak but then rapidly tails off a full magnitude above

our completeness limits.

For our current derivation of the Trapezium IMF, we use the Trapezium K band

Luminosity Function, rather than the J or H LFs. We do so in order to minimize the

effects of extinction on the luminosities of cluster members (see Section 3.3.1), to max-

imize our sensitivity to intrinsically red, low luminosity brown dwarf members of this
























4 6 8 10 12 14 16


Figure 3-3: Trapezium cluster: raw near-infrared luminosity functions. A) Trapezium
Cluster J and H band Luminosity Functions. The Trapezium HLF is the
open histogram and the Trapezium JLF is the shaded histogram. Com-
pleteness (90%) limits are marked by a solid vertical line at 18.15 (J) and a
broken vertical line at 17.8 (H). B) Trapezium Cluster K band Luminosity
Function. The Trapezium KLF constructed from the FLWO-NTT catalog is
compared to the literature KLF constructed in Section 2.6.2. The K=17.5
90% NTT completeness limit is demarked by a vertical broken line.


cluster, and to make detailed comparisons to our study of the literature Trapezium KLF

in Section 2.6.2 For example, the new FLWO-NTT Trapezium KLF contains signifi-

cantly more stars at faint (K > 14) magnitudes than the literature KLF constructed in

Section 2.6.2. Interestingly, a secondary peak near K = 15 seen in that KLF (see Fig-

ure 2-12) (originally McCaughrean et al., 1995) is much more significant and peaks at

K=15.5 in the new FLWO-NTT KLF. Similar peaks are not apparent in the J or H band

LFs constructed here, though Lucas & Roche (2000) report a strong secondary peak in

their Trapezium HLF. Such secondary peaks in young cluster luminosity functions have

often been evidence of a background field star population contributing to the source

counts (e.g., Luhman et al., 1998; Luhman & Rieke, 1999).

To account for the possible field star contamination, we systematically obtained

images of control fields away from the cluster and off of the Orion Molecular Cloud.

The FLWO off-cluster fields) were centered at approximately R.A. = 05h26m; DEC.



























Figure 3-4:


6 8 10 12 14 16 18 4 6 8 10 12 14 16 18
K Magnitude K Magnitude
Trapezium cluster: construction of observed control field KLF. A) The
two histograms are the off-field KLFs obtained as part of the FLWO and
NTT observations. The NTT off-fields are approximately 2 magnitudes
deeper than the FLWO off-fields, but the FLWO off-fields covered twice
the area of the NTT off-fields. Both are scaled to the size of the Trapez-
ium NTT region. The inset diagram shows the distribution of H-K colors
for these two off-fields. Their similar narrow widths indicate they are free
of interstellar extinction; B) The weighted average of the FLWO and NTT
field stars KLFs is compared to the Trapezium Cluster KLF constructed in
Figure 3-3(b).


= -0600' (J2000) and were roughly twice the area of the NTT off-fields. The NTT

off-cluster region was centered at R.A. = 05h37m43s7; DEC.= -01055'42"'7 (J2000).

Figure 3-4(a) displays the two off-field KLFs (scaled to the same area) from these

observations and in the inset, their (H K) distributions. The relatively narrow (H K)

distributions indicate that the two off-fields sample similar populations and that they are

un-reddened. We constructed an observed field star KLF by averaging these luminosity

functions, weighting (by area) toward the FLWO off-fields for K brighter than 16th

magnitude, the completeness limit of that dataset, and toward the more sensitive NTT

off-fields for fainter than K = 16. In Figure 3-4(b), we compare the resulting field star

KLF to the Trapezium KLF of the NTT region. It is plainly apparent from the raw

control field observations that while field stars may contribute to the Trapezium Cluster









IR luminosity function over a range of magnitudes, their numbers peak at magnitudes

fainter than the secondary peak of the on-cluster KLF and do not appear sufficient in

number to explain it.

3.2.2 Defining a Complete Cluster KLF

We determine the completeness of our FLWO-NTT Trapezium Cluster KLF by

constructing and by analyzing the cluster's infrared (H K) versus K color-magnitude

diagram. For the purposes of our analysis, we adopt the following parameters for the

Trapezium: a cluster mean age of 0.8 Myr (Hillenbrand, 1997) and a cluster distance

of 400pc. As seen in Figure 3-5(a), the luminosities of the Trapezium sources form a

continuously populated sequence from the bright OB members (K ~ 5) through sources

detected below our completeness limits.

To interpret this diagram, we compare the locations of the FLWO-NTT sources in

color-magnitude space to the cluster's mean age isochrone as derived from theoretical

pre-main sequence (PMS) calculations. Because the DM97 models include masses

and ages representative of the Trapezium Cluster we will use these tracks to define a

complete cluster sample from Figure 3-5(a). Differences among pre-main sequence

tracks should not have significant effect upon our analysis of the color-magnitude

diagram (see Section 3.4.3). It is clear from this diagram that the cluster sources are

reddened away from the theoretical 0.8 Myr isochrone, which forms a satisfactory left

hand boundary to the sources in this color-magnitude space. This isochrone, however,

does not span the full luminosity range of the observations and a number (~ 40)

sources lie below the faint end of the DM97 isochrone. As a result, our subsequent

analysis that makes use of the DM97 models will be restricted to considering only

those sources whose luminosities, after correction for extinction, would correspond to

masses greater than the mass limit of the DM97 tracks, i.e., 0.017 Me or roughly 17

times the mass of Jupiter (MJup). Despite the lower mass limit imposed by these PMS










4

6

8

10

S12
2
14

16

18




Figure 3


C


3-5:


1 2 3 4 4 6 8 10 12 14 16 18
( H K ) Color K Magnitude

Trapezium cluster: deriving M Av completeness limits. A) Trapezium
Cluster (H K) / K color-magnitude diagram for the NTT region. Stars
selected to fall into our mass & extinction limited sample are indicated
by filled circles. The distribution of sources in this color-magnitude space
is compared to the location of the pre-main sequence 0.2 and 0.8 Myr
isochrones from DM97. Reddening vectors (Av = 17) shown for 2.50,
0.08 and 0.02 Me stars at the cluster's mean age (0.8 Myr). The zero-age
main sequence (Kenyon & Hartmann, 1995; Bessell, 1995) is shown for
03-M6.5 stars at a distance of 400pc. B) Effects of mass/extinction limits
on the cluster KLF. Comparison of the M Av limited KLF derived from
(A) to the raw Trapezium KLF (see Figure 3-3b). Sensitivity (K = 18.1)
and completeness (K = 17.5) limits are shown as vertical broken lines.


tracks, our infrared census spans nearly three orders of magnitude in mass, illustrating

the utility of studying the mass function of such rich young clusters.

Extinction acts to redden and to dim sources of a given mass to a brightness below

our detection limits. To determine our ability to detect extincted stars as a function

of mass, we draw a reddening vector from the luminosity (and color) of a particular

mass star on the mean age isochrone until it intersects the 10(y sensitivity limit of our

census. We can detect the 1 Myr old Sun seen through Av ~ 60 limits magnitudes of

extinction or a PMS star at the hydrogen burning limit seen through 35 magnitudes.

For very young brown dwarfs at our lower mass limit (17 Mjup), we probe the cloud

to Av = 17 magnitudes. We use this latter reddening vector as a boundary to which


(A)
i *





S 17 Sp





DM9 02 Myrs
DM97 @ 0 Myrs
S AMS, D=400p









we are complete in mass, and we draw a mass and extinction (M- Av) limited subset

of sources bounded by the mean age isochrone and the Av = 17 reddening vector

and mark these as filled circles in Figure 3-5(a). Our M- Av limits probe the vast

majority of the cluster population, including 81% of the sources the color-magnitude

diagram.

In Figure 3-5(b) we present the M- Av limited KLF, containing 583 sources.

Thirty-two sources, detected only at K band (representing only 4% of our catalog),

were also excluded from our further analysis. The median K magnitude of these

sources is K = 15, and we expect that these are likely heavily reddened objects. We

compare the M Av limited KLF to the un-filtered Trapezium KLF. Clearly, heavily

reddened sources contributed to the cluster KLF at all magnitudes and their removal

results in a narrower cluster KLF. However, the structure (e.g., peak, slope, inflections,

etc.) of the KLF remains largely unchanged. The secondary peak of the cluster KLF

between K i 14 17 seems to be real since it is present in both the raw and the

M- Av limited KLFs, though we have not yet corrected for background field stars.

There are at least three possible sources of incompleteness in our mass/extinction

limited sample. The first arises because sources that are formally within our mass and

extinction limits may be additionally reddened by infrared excess from circumstellar

disks and, hence, be left out of our analysis. However, this bias will affect sources

of all masses equally because infrared excess appears to be a property of the young

Trapezium sources over the entire luminosity range (Muench et al., 2001). Second, the

Trapezium Cluster is not fully coeval and our use of the cluster's mean age to draw

the M- Av sample means that cluster members at our lower mass limit (17Mjup)

but older than the cluster's mean age (T > 0.8Myr) will be fainter than the lower

boundary and left out of our sample. Further, sources younger than the mean age but

below 17Mjup will be included into the sample. This "age bias" will affect the lowest

mass sources, i.e., < 20 Mjp. Third, because of the strong nebular background, our









true completeness limit (see Section 3.1) is brighter than our formal 10C sensitivity

for approximately 60% of the area surveyed. The resulting sample incompleteness

only affects our sensitivity to sources less than 30Mjup and with Av > 10. We do not

correct the Trapezium KLF to account for these effects or biases.

3.2.3 Field Star Contamination to the KLF

The lack of specific membership criteria for the embedded sources in the Trapez-

ium Cluster requires an estimate of the number of interloping non-cluster field stars in

our sample. Some published studies, for example LR2000 and Luhman et al. (2000),

assume that the parental molecular cloud acts as a shield to background field stars;

whereas HC2000 suggests that the background contribution is non-negligible. HC2000

estimates the field star contribution using an empirical model of the infrared field

star population and convolving this model with a local extinction map derived from a

molecular line map of the region. This approach may suffer from its dependence upon

a field star model that is not calibrated to these faint magnitudes and that does not

include very low mass field stars. As we show, there are also considerable uncertainties

in the conversion of a molecular line map to an extinction map. For our current study,

we use our observed K band field star luminosity function (see Figure 3-4) to test

these prior methodologies and to correct for the field star contamination. We point out

that no such estimate can account for contamination due to young, low mass members

of the foreground Orion OB1 association.

We compare in Figure 3-6 the effects of six different extinction models upon our

observed field star KLF. In panels A and B, we tested simple Gaussian distributions

of extinction centered respectively at Av = 10 and 25 magnitudes with C = 5

magnitudes. In both cases, the reddened field star KLF contains significant counts

above our completeness limit and "background extinction shields" such as these

do not prevent the infiltration of field stars into our counts. In the second pair of

reddened off-fields (panels C and D), we followed the HC2000 prescription for



























10 15 20
K Magnitude


10 15 20
K Magnitude


5 10 15
K Magnitude


10 15 20
K Magnitude


10 15 20
K Magnitude


5 10 15
K Magnitude


Trapezium cluster: testing contribution of reddened field star KLFs. Panels
A & B: The observed off-field KLF (Figure 3-4b) reddened by "back-
ground shields" of extinction in the form of gaussian distributions centered
at Av = 10 (panel A) and 25 (panel B); Panels C & D: The observed off-
field KLF reddened by an extinction map converted from a C180 map.
The two panels represent the variation in the reddened off-field as a func-
tion of the uncertainty in the C180 to Av conversion; Panels E & F: The
same experiment as performed in C & D, but these have been filtered to
reflect the actual contribution due to the M- Av limits.


Figure 3-6:









estimating background field stars by convolving our observed field star KLF with

the C180 map from Goldsmith et al. (1997) converted from column density to dust

extinction. We note that there is substantial uncertainty in the conversion from C180

column density to dust extinction. There is at least a factor of 2 variation in this

conversion value in the literature, where Frerking et al. (1982) derived a range from

0.7 2.4 (in units of 1014 cm2 mag-1) and Goldsmith et al. (1997) estimated a range

of values from 1.7 3. Either the result of measurement uncertainty or the product of

different environmental conditions, this variation produces a factor of 2 uncertainty in

the extinction estimates from the C180 map. In short, we find that a C180 -+ Av ratio

of 3.0 (panel C) results in twice as many interloping background field stars as would a

value of 1.7 (panel D; equivalent to that used by HC2000).

In panels E and F of Figure 3-6 we derive the same reddened off-field KLFs

as in the prior pair, but they have been filtered to estimate the actual contribution of

field-stars to our M- Av limited sample. These filters, which were based upon on the

K brightness of the lower mass limit of our PMS models and on the derived extinction

limit of the M- Av sample, were applied during the convolution of the field star KLF

with the cloud extinction model such that only reddened field stars that would have

Ay < 20 and unreddened K magnitudes < 16 would be counted into filtered reddened

off-field KLF. The extinction limit was expanded from 17 to 20 magnitudes to account

for the dispersion of the H-K distribution of un-reddened field-stars (~ 0.2). A factor

of 2 uncertainty remains. Alves et al. (1999) derive a more consistent estimate of the

C180 -+ Av ratio from near-infrared extinction mapping of dark clouds, suggesting a

median ratio of 2.1. Adopting C180 -+ Av = 2.1, we estimate there are ~ 20 10

field stars in our M- Av limited KLF. From these experiments, we find, however,

that both the raw and reddened off-field KLFs always peak at fainter magnitudes

than the secondary peak of the Trapezium KLF, and that the subtraction of these

field-star corrections from the Trapezium KLF do not remove this secondary peak.









These findings suggest that the secondary KLF peak is a real feature in the Trapezium

Cluster's infrared luminosity function.

3.3 Trapezium Cluster Initial Mass Function

We analyze the Trapezium Cluster's K band luminosity function constructed in

section 3.2 using our model luminosity function algorithm described in Section 2.

Our goal is to derive the underlying mass function or set of mass functions whose

model luminosity functions best fit the Trapezium Cluster KLF. We have improved our

modeling algorithm by including statistical distributions of the reddening properties

of the cluster. We have also improved our analysis by applying the background field

star correction from Section 3.2.3 and by employing improved fitting techniques for

deriving IMF parameters and confidence intervals. Before deriving the cluster IMF, we

use the extensive color information available from the FLWO-NTT catalog to explore

the reddening (extinction and infrared excess) properties of the Trapezium sources.

In Section 3.3.1, we use this information to create recipes for deriving the probability

distributions functions of extinction and excess which can be folded back into our

modeling algorithm during our derivation of the Trapezium IMF. We present the new

model luminosity functions and fitting techniques in Section 3.3.2 and summarize the

derived IMF in Section 3.3.3.

3.3.1 Deriving Distributions of Reddening

Extinction probability distribution function. We use the extensive color in-

formation provided by our FLWO-NTT catalog to construct a probability distribution

function of the intra-cluster extinctions (hereafter referred to as the Extinction Prob-

ability Distribution Function or EPDF) based upon the color excesses of individual

Trapezium sources. Because the stellar photospheric (H K) color has a very narrow

distribution of intrinsic photospheric values it should be the ideal color from which

to derive line of sight extinction estimates, as shown, for example, in the Alves et al.

(1998) study of the structure of molecular clouds. In Figure 3-7(a) we show the




























Figure 3-7: Infrared colors of Trapezium sources. A) Histogram of the observed (H
K) color for the FLWO-NTT Trapezium sources. The subset of these
sources which lack J band measurements are indicated by the shaded his-
togram; B) Trapezium Cluster (H K) vs (J H) color-color diagram for
the NTT region. Symbols indicate if the source's colors were taken from
the FLWO catalog (filled circles, JHK) or the NTT catalog (open circles,
JHKs).


histogram of observed (H K) color for all our Trapezium Cluster sources. This

histogram peaks at (H K) = 0.5 and is quite broad especially when compared to the

narrow unreddened photospheric (field-star) (H K) distributions seen in Figure 3-4(a).

This broad distribution may be in part the result of extinction; however, as recently

shown in Lada et al. (2000) and Muench et al. (2001), approximately 50% of the these

Trapezium Cluster sources, independent of luminosity, display infrared excess indica-

tive of emission from circumstellar disks. This is illustrated in Figure 3-7(b) where it

is clear that there are both heavily reddened sources (Av ~ 35) and sources with large

infrared excesses (falling to the right of the reddening band for main sequence objects).

If the (H K) color excess were assumed to be produced by extinction alone without

accounting for disk emission, the resulting extinction estimates would be too large.

Meyer et al. (1997) showed that the intrinsic infrared colors of stars with disks are

confined to a locus (the classical T-Tauri star locus or cTTS locus) in the (H K)/(J


SA) All Sources
[ Sources w/o J


SB)











I J


LWO Photometry


. . i . . . . . . . . . . ..' ' ' '


I .. .. I ./* .' "









- H) color-color diagram. We derive individual Av estimates for sources in the (H

- K)/(J H) color-color diagram by dereddening these stars back to this cTTS locus

along a reddening vector defined by the Cohen et al. (1981) reddening law. Sources

without J comprise ~ 20% of the catalog and as shown in Figure 3-7, their (H K)

colors appear to sample a more heavily embedded population, implying extinctions as

high as Av 60. Av estimates are derived for these sources by assigning a typical

star-disk (H K) color = 0.5 magnitudes, and de-reddening that source. Sources near

to but below the cTTS locus are assigned an Av = 0. The individual extinctions

are binned into an extinction probability distribution function (EPDF) as shown in

Figure 3-8. Also shown are the effects of changing the typical star-disk (H K) color

assumed for those stars without J band. Little change is seen. Compared to the cloud

extinction distribution function, which was integrated over area from the C180 map, the

cluster EPDF is very non-gaussian and peaks at relatively low extinctions, Av = 2.5,

having a median Av = 4.75 and a mean Av = 9.2. Further, the cluster EPDF is not

well separated from the reddening distribution provided by the molecular cloud. Rather

the cluster population significantly extends to extinctions as high as Av = 10 25,

near and beyond the peak of the cloud extinction function. Ancillary evidence of this

significant population of heavily reddened stars is seen in the color-color diagram

(Figure 3-7b) which clearly illustrates the extension of the cluster to regions of the

molecular cloud with Av > 10. Lastly, it is clear that the deep nature of our survey

has allowed us to sample both the majority of the embedded cluster, and the cloud over

the full range of density.

In our revised model luminosity function algorithm, we randomly sample the

cluster's EPDF to assign an Av to each artificial star in the model LF. The effect of the

EPDF on the model luminosity function is wavelength and reddening law (Cohen et al.,

1981, in this case) dependent. In Figure 3-9 we construct model I, J, H, and K

luminosity functions, reddening each by the Trapezium Cluster EPDF. The effect of the










140


120


100


80


60


-I
- A)















- .........


--I


Figure 3-8:


0 10 20 30 40 50
Av

Trapezium cluster: extinction probability distribution function. Plotted are
three variations in the EPDF under different assumptions of the typical (H-
K)(star-disk) color for the 20% of the stars lacking J band measurements.
See Section 3.3.1 for derivation. It is compared to the extinction probabil-
ity distribution function integrated from the C180 -+ Av map. Note that
they are not well separated distributions. A broken vertical line indicates
the Av = 17, M-Av limit.


EPDF on the intrinsic I and J band LFs is profound, rendering the reddened I band LF

almost unrecognizable. Yet at longer wavelengths, specifically at K band, the effects

of extinction are minimized. We note that the overall form of the reddened model K

band luminosity function has not been changed by the Trapezium EPDF in a significant

way, e.g., the peak of the model KLF is not significantly blurred and the faint slope of

the KLF has not been changed from falling to flat. This suggests that our modeling of

the literature Trapezium KLF in Section 2.6.2, which did not account for reddening due

to extinction, is generally correct. However, we likely derived too low of a turnover


I I..


''''''''''''''''''' ''''''''''''''''''' '''''''''''


I'


(H-K)sta-disk = 0.2
(H-K)stor-disk = 0.5
(H-K).tor-disk = 1.0
C180->Av EDF
C180-Av=2.1


1













2.0 ILF


t 1.5
0
1.0
E
z
S0.5

0.0


2.0 JLF

0
^ 1.5
0
1.0
E

0.5-


0.0


5 10 15 20
I Magnitude


Figure 3-9:


5 10 15
J Magnitude


2.0
co
S1.5
0
S1.0
E
z
0.5
0
0.0


5 10 15 20 5 10 15 20
H Magnitude K Magnitude


Effects of extinction on model cluster LFs. Model luminosity functions
of the Trapezium (using the Trapezium IMF of Equation 2.7 and derived
in Section 2.6.2) is convolved with the Trapezium Cluster EPDF at four
different wavelengths. Reddening effects are most significant at I and J
bands and are minimized at K band.


mass for the Trapezium IMF because reddening shifted the intrinsic LF to fainter

magnitudes.


Infrared excess probability distribution function. Because we wish to


use the Trapezium K band LF to minimize the effects of extinction, we must also

account for the effects of circumstellar disk emission at K band. The frequency


distribution of the resulting excess infrared flux is not a well known quantity, and


when previously derived, it has depended significantly upon additional information

derived from the spectral classification of cluster members (Hillenbrand et al., 1998;

Hillenbrand & Carpenter, 2000). One of the goals of this present work is to construct









100 I I
B)

80 I

80

^ 60-
0

Q)
E 40
z


20




0.0 0.5 1.0 1.5
( H K ) Excess = K Excess

Figure 3-10: Trapezium cluster: infrared excess probability distribution function. The
derived H-Ks color excess distribution function is assumed to reflect a
magnitude excess at K band alone.


a recipe for deriving the K band excess distribution directly from the infrared colors of

the cluster members.

To derive a first-order infrared excess probability distribution function (IXPDF)

for the Trapezium Cluster sources, we simply assume that any excess (H K) color

(above the photosphere, after removing the effects of extinction) reflects an excess at K

band alone, realizing this may underestimate the infrared excess of individual sources.

We only use the sources having JHK measurements and lying above the cTTS locus

in the color-color diagram. We remove the effects of extinction from each source's

observed (H K) color using the same method described above, i.e., dereddening back

to the cTTS locus. However, the photospheric (H K) color for each star cannot be

discreetly removed from this data alone. The photospheric infrared colors of pre-main









sequence stars appear to be mostly dwarf-like (Luhman, 1999), and therefore, we used

the observed field star (H K) distribution shown in Figure 3-4(a) as a probability

distribution of photospheric values. We derive the IXPDF by inning the de-reddened

(H K) colors into a probability function and then subtracting the distribution of

photospheric colors using a Monte Carlo integration.

The Trapezium Cluster IXPDF is shown in Figure 3-10. The IXPDF peaks near

0.2 magnitudes with a mean = 0.37, a median = 0.31, and a maximum excess of ~ 2.0

magnitudes. Probabilities of negative excesses were ignored. The IXPDF is similar

to the (H K) excess distribution shown in HC2000 and derived in Hillenbrand et al.

(1998) yet extends to somewhat larger excess values. Each artificial star in our models

is randomly assigned a K band excess (in magnitudes) drawn from the IXPDF.

3.3.2 Modeling the Trapezium Cluster KLF

To model the Trapezium Cluster KLF, we apply the appropriate field star correc-

tion derived in section 3.2.3 to the M- Av limited KLF constructed in Section 3.2.2.

We fix the Trapezium Cluster's star-forming history and distance to be identical to

that used in Section 2.6.2. Specifically, these are a distance of 400pc (m-M=8.0; see

appendix B) and a star-forming history characterized by constant star formation from

1.4 to 0.2 Myr ago, yielding a cluster mean age of 0.8 Myr (Hillenbrand, 1997) and

an age spread of 1.2 Myr. We adopt our standard set of merged theoretical pre-main

sequence tracks from Section 2.2.3 3 Our merged standard set of tracks span a mass

range from 60 to 0.017 M, allowing us to construct a continuous IMF within this

range. We incorporated the cluster's reddening distributions derived in Section 3.3.1



3 Our standard set of theoretical tracks are a merger of evolutionary calcula-
tions including a theoretical Zero Age Main Sequence (ZAMS) from Schaller et al.
(1992), a set of intermediate mass (1-5 MQ) "accretion-scenario" PMS tracks from
Bemasconi (1996), and the low mass standard deuterium abundance PMS models from
D'Antona & Mazzitelli (1997) for masses from 1 to 0.017 MQ









into our modeling algorithm and chose our standard functional form of the cluster

IMF; specifically, an IMF constructed of power-law segments, Fi connected at break

masses, mj. We find that an underlying 3 power-law IMF produced model KLFs that

fit the observations over most of the luminosity range, corresponding to masses from

5 to 0.03 M. In Section 5, we utilize our X2 minimization routine to identify those 3

power-law IMFs that best fit the observed KLF within this mass range, and we estimate

confidence intervals for these IMF parameters in Section 5. We find that that the faint

Trapezium brown dwarf KLF, corresponding to masses less than 0.03 M0, contains

structure and a secondary peak that are not well fit by the 3 power-law IMF models.

In Section 5 we model this secondary KLF peak using a corresponding break and

secondary feature in the cluster brown dwarf IMF between 0.03 and 0.01 M.

Results of X2 fitting: best fit three power-law IMFs. Our X2 minimization

procedure calculates the reduced X2 statistic and probability for a particular model

KLF fit to the Trapezium KLF over a range of magnitude bins. Parameters for the

underlying three power-law IMF are taken from the best fit model KLFs, and we fit

both reddened and unreddened model KLFs. The 3 power-law IMF derived from these

fits is summarized in Table 3-3. We found that the results of our model fits were

dependent upon the dynamic range of K magnitude bins over which the models were

minimized. Specifically, we find that our results are very sensitive to the formation

of a secondary peak in the Trapezium KLF at K = 15.5, which remains despite the

subtraction of the field star KLF.

We derive good model KLF fits (x2 prob ~ 1) when fitting between the K = 7.5

bin and the K = 14.5 bin (see Figure 3-11a), the same luminosity range we modeled

in Section 2.6.2. Within this fit range, we find an optimal Trapezium IMF nearly

identical to that found in Equation 2.7, even after accounting for reddening. The

derived IMF rises steeply from the most massive stars with F1 = -1.3 before breaking

to a shallower IMF slope of F2 = -0.2 at 0.6 M (log mi ~ -0.2). The derived IMF


















4 16


Figure 3-11:


Log Mass (M/Me) Log Mass (M/Me)

Trapezium cluster: best-fitting model KLFs and 3 power-law IMFs. Top
panels: the M Av limited, background subtracted Trapezium KLF (his-
togram) and best fit reddened model KLFs (unconnected filled circles).
Bottom panels: the resulting underlying IMFs and corresponding chi-sq
probabilities. Panel (A) shows models fit between K=7.5 and 14.5, the
same range fit in Section 2.6.2 and Figure 2.7(a). Panel (B) shows three
power-law IMF fits to the secondary peak in the cluster KLF at K=15.5,
which correspond to low X2 probability due to the presence of structure
and the secondary KLF peak.


peaks near the hydrogen burning limit (0.10 0.08 M or log m2 = -1.0- 1.1) and

then breaks and falls steeply throughout the brown dwarf regime with F3 +1.0. We

also derive good fits to K=15 (just before the secondary peak in the cluster KLF), with

the resulting IMF peaking at slightly higher masses (0.13 0.10M) and falling with

a slightly shallower slope, F3 ~ +0.7 to 0.8. The unreddened luminosities of this fit

range correspond to a mass range from 5.0 to 0.03 Me.

However, we cannot produce model KLFs based upon a three power-law IMF

that adequately fit the secondary peak in the Trapezium KLF. For example, our best

fit to the secondary peak in Figure 3-1 (b) is inconsistent with the overall form of

the faint KLF, being unable to replicate both the falling KLF at K = 14.5 nor the


- prob = 03 24
Sprob = 024

















Table 3-3. Three power-law Trapezium IMF parameters and errors
Parameter (a) Range mK (b)Best Fit (c) () Best Fit (d) ()

F1 -1.0 -+ -2.0 14.5 -1.16 0.16 -1.24 0.20
logim +0.1 -+ -1.1 14.5 -0.17 0.10 -0.19 0.13
F2 -0.4 +0.4 14.5 -0.24 0.07 -0.16 0.15
log m2 +0.1 -+ -1.4 14.5 -1.05 0.05 -1.00 0.13
F3 -0.4 +2.0 14.5 1.10 0.25 1.08 0.38
F1 -- 15.0 -1.13 0.16 -1.21 0.18
logml -.- 15.0 -0.19 0.11 -0.22 0.11
F2 -.- 15.0 -0.24 0.15 -0.15 0.17
log m2 -- 15.0 -1.00 0.10 -0.92 0.13
F3 -.- 15.0 0.82 0.15 0.73 0.20
log m2 -- 15.5 -0.89 .- -0.77 ..
F3 -.- 15.5 0.30 -.- 0.30 .
log m2 -- 16.5 ... -. -0.72 --
F3 .-- 16.5 ... ... 0.23 .


(a)The parameters Fi are the power-law indices of the IMF which here
is defined as the number of stars per unit log(--). The parameters mj
are the break masses in the power-law IMF and are in units of log( -).

(b)Faintest KLF bin fit by Model KLF.

(c)Model fits without Source Reddening.
(d)Model fits accounting for Source Reddening.


Note. All tabulated fits derived using our standard set of PMS
tracks (primarily from DM97).









secondary peak at K = 15.5. Such structure in the faint Trapezium KLF implies similar

non-power law structure in the underlying IMF, while our current models based upon

a three power-law IMF essentially assign a single power-law IMF slope for the entire

brown dwarf regime. We will explore this structure in the faint brown dwarf KLF

and IMF in Section 5, but first we examine the confidence intervals for the derived 3

power-law IMFs.

Results of X2 fitting: range of permitted three power-law IMFs. Our X2

fitting routine also allows us to investigate the range of permitted cluster IMFs from

modeling the cluster KLF. We illustrate the range of IMFs and the effects of source

reddening on our fits in Figure 3-12 and summarize the corresponding constraints

on the IMF parameters in Table 3-3. In each panel, we plot the contours of X2

probability for two of the 5 dependent IMF parameters while restricting the other three

parameters to a best fit model. In each panel we also display contours for fits with

(solid) and without (dashed) source reddening, and we examine the dependence of

these parameters for models fit to the K=14.5 and K=15.0 bins.

In all our fitting experiments (here and in Section 2.6.2), the high-mass slope

of the cluster IMF, F1, was well constrained with slopes measured between -1.0 and

-1.3. Based on this result, we fix F1 to equal -1.3. Panels (a) (c) in Figure 3-12

display the ranges of the other 4 IMF parameters when fitting to a K limit = 14.5.

Panel (a) plots the dependence of the two break masses, ml andm2. The fits for these

parameters are well behaved with 90% contours have a typical width of 0.1-0.2 dex in

units of log mass. Source reddening has two clear effects upon our fit results. When

source reddening is included, the high-mass break, ml, decreases and the low mass

break, m2, increases. The second effect is that the size of the 90% confidence contour

increases when source reddening is included into the model fits. Panel (b) displays the

dependence of the low mass break, m2, on the middle power-law slope, F2. F2 is fairly

well constrained to be slightly rising to lower masses, and the permitted range of m2
























-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2
m, ( Log M/Me )


Figure 3-12:


7 f ... II ...... II 1111 .. II III .. II III.. II III. II ....


-1.30 -1.20 -1.10 -1.00 -0.90 -0.80 -1.30 -1.20 -1.10 -1.00 -0.90 -0.80
m2 ( Log M/Me ) m2 ( Log M/Me )

Trapezium cluster: X2 confidence intervals for IMF parameters. Contours
of x2 probability for the 5 parameters of the underlying three power-law
IMF. Two parameters are compared in each panel while fixing the other
three to a best fit value. Solid contours are best fit ranges from models
that include source reddening. Dashed contours are from best fit mod-
els without source reddening. Contour levels are shown at intervals 95,
90, 70, 50 and 30% confidence. Panels (A) to (C) are shown for fits to
K=14.5 and panel (D) is shown for fits to K=15.


is again roughly 0.1 0.2 dex, centered near 0.1 M (logm2 ~ -1). Accounting for

source reddening again shifts the low-mass break to slightly higher masses, increases

the size of the 90% contour, and in this case, flattens the central power-law.

Panel (c) displays the dependence of F3 upon the second break mass, m2. Though

m2 is fairly well constrained to have values between 0.13 and 0.08 M, the low mass

power-law slope, F3, has a large range of possible slopes from 0.50 to 1.50 within the

90% X2 contour for models with source reddening. Panel (d) plots the same parameters


0.70
(B)
0.80

0.90








1.40
-0.6 -0.4 -0.2 0.0 0.2 0.4
F2









as panel (c) but for fits to the K limit = 15. These fits give somewhat flatter F3 slopes

and somewhat higher mass m2 breaks, but are actually slightly better constrained. As

discussed in the previous section, our model KLFs employing a 3 power-law IMF do

not provide good fits to the secondary peak in the KLF. As the fit range shifts to fainter

magnitudes, F3 flattens, but the total X2 confidence depreciates due to the secondary

peak. We explore the IMF parameters necessary to fit this secondary peak in the next

section.

Fitting the secondary peak in the Trapezium cluster KLF. In contrast to our

expectations when we interpreted the literature Trapezium KLF in Section 2.6.2 the

departure from a power-law decline and the formation of a secondary peak at the faint

end of the Trapezium KLF remains after correcting for reddened background field

stars. When we attempt to fit the faint KLF using an underlying three power-law IMF,

we find that our model KLFs, while producing excellent fits over the majority of the

Trapezium KLF, could not simultaneously reproduce the formation of the secondary

peak. Since there is no known corresponding feature in the mass-luminosity relation

(see Section 3.4.2), we hypothesize that the KLF's break from a single continuous

declining slope at K > 14.5 (M < 30Mjup) and the formation of a secondary KLF

peak directly imply a similar break and feature in the cluster IMF. Further, the rapid

tailing off of the cluster KLF below this secondary peak also directly implies a similar

rapid decline or truncation in the underlying IMF, as was also discussed in LR2000.

We modeled the secondary KLF peak by adding a fourth, truncated, power-law

segment, F4, to the three power-law IMFs derived in section 5. The truncation of

the fourth power-law segment enabled us to model the rapid tailing off of the cluster

KLF below the secondary peak, but was also dictated by the artificial low mass cut off

present in the adopted merged PMS tracks, which for the substellar regime come from

DM97. As such, the truncation mass of the model IMF was arbitrarily set to 0.017

M. We found that this 4 power-law truncated IMF produced good X2 model KLF