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LUMINOSITY AND MASS FUNCTIONS
OF VERY YOUNG STELLAR CLUSTERS
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
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.
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
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 (email@example.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
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
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 . ..............
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
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
A TABULATED BOLOMETRIC CORRECTIONS ..
B DISTANCE TO THE TRAPEZIUM CLUSTER .. ..
. . . 2 17
C SUMMARY OF POPULATION SYNTHESIS FORTRAN CODE
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
. . 247
LIST OF TABLES
. . 23
. . 40
. . 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 . ..........
LIST OF FIGURES
3-5 Trapezium cluster: deriving M- Av completeness limits .
-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 .
. . 30
. . 62
. . 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 . ...
. . 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
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-
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)
New Technology Telescope
Orion Nebula Cluster
Probability Distribution Function
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
August A. Muench
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.
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
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
dN dN d log M
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:
-dL( = d z() (1.2)
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
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
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.
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
We also used the log-normal distribution as a functional form of the IMF, e.g.,
(log(--)) = cl*exp(-c2*(log( )- c3)2)),
MS79 Log-normal IMF ..........
Various 2 Power Law IMFs
1 0 -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.
2.2.2 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
Post tbegnSF endSF -now
Age Spread: ATs egSFbegnSF ndSF
Cluster Age: Tcluster -beginSF
Cluster Mean Age:
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
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
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,
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
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)
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
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
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
S ACM & AMT
-A- KCM & KMT
4 6 8
ute K Maanitude
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 2 4 6 8 10
Absolute K Magnitude
S= 5.0 Myrs, Coeval
-- DM94 ACM
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
2.0 Myrs, Coeval
2 4 6 8
Absolute K Magnitude
/ / .u Iviy YuU v
c 0.0 ------
S0 2 4 6 8
Absolute K Magnitude
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
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
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
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
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
0 2 4 6 8
Absolute K Magnitude
0 2 4 6 8
Absolute K Magnitude
0 2 4 6 8
Absolute K Magnitude
0 2 4 6 8
Absolute K Magnitude
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
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
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
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
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
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
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
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
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
Absolute K Mc
1 -- KLF fit IMF (g)
H97 ONC IMF
--" Salpeter IMF
H97 Completeness Limit
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
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
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)
-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.
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
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).
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.
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.
83.88 83.86 83.84 83.82
R.A. ( J2000 )
83.80 83.78 83.76
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
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
(a)FLWO: Fred Lawrence Whipple Observatory; NTT: New
(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
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)
(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.
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
= -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
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
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
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
10 15 20
5 10 15
10 15 20
10 15 20
5 10 15
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.
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,
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
. . 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
0 10 20 30 40 50
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
''''''''''''''''''' ''''''''''''''''''' '''''''''''
(H-K)sta-disk = 0.2
(H-K)stor-disk = 0.5
(H-K).tor-disk = 1.0
5 10 15 20
5 10 15
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
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
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
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
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 )
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
-0.6 -0.4 -0.2 0.0 0.2 0.4
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
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