Citation |

- Permanent Link:
- http://ufdc.ufl.edu/AA00003237/00001
## Material Information- Title:
- The collisional evolution of the asteroid belt and its contribution to the zodiacal cloud
- Creator:
- Durda, Daniel David, 1965-
- Publication Date:
- 1993
- Language:
- English
- Physical Description:
- xii, 129 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Albedo ( jstor )
Asteroids ( jstor ) Diameters ( jstor ) Impact strength ( jstor ) Particle collisions ( jstor ) Population distributions ( jstor ) Population size ( jstor ) Power laws ( jstor ) Projectiles ( jstor ) Size distribution ( jstor ) Asteroids ( lcsh ) Astronomy thesis Ph. D Cosmic dust ( lcsh ) Dissertations, Academic -- Astronomy -- UF Zodiacal light ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1993.
- Bibliography:
- Includes bibliographical references (leaves 123-128).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Daniel David Durda.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 028257327 ( ALEPH )
31143069 ( OCLC ) AKC3735 ( NOTIS )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

THE COLLISIONAL EVOLUTION OF THE ASTEROID BELT AND ITS CONTRIBUTION TO THE ZODIACAL CLOUD By DANIEL DAVID DURDA DISSERTATION PRESENTED TO THE GRADUATE SCHOOL THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY To my parents, Joseph and Lillian Durda. ACKNOWLEDGMENTS There are a great many people who have played important roles in my life at UF, and although the room does not exist to thank them all in the manner I would like, I would at least like to express my gratitude to those who have helped me the most. First and foremost, I would like to thank my thesis advisor, Stan Dermott. has been far more than just an academic advisor. He has taught by splendid example how to proficiently lead a research team, looked after my professional interests, and given me the freedom to focus upon research without having to worry about financial support. I never once felt as though I were merely a graduate student. One could not ask for a better thesis advisor. My thanks also go to the other members of my committee, Humberto Campins, Phil Nicholson, and James Channell, for their helpful comments and review of this thesis. The advice and many laughs provided by Humberto were especially appreciated. I am also very grateful to Bo Gustafson and Yu-Lin Xu for the many discussions and helpful advice through the years. My fellow graduate students, my family away from home, kept me sane enough (or is it insane enough?) to make it this far. I will value my friendship with Dirk Terrell and Billy Cooke forever. I will probably miss most our countless discussions about literally more than I can express in words. Billy's "Billy-isms" have provided me with more entertainment than I have at times known what to do with. I will miss them immensely! will also miss my discussions, afternoon chats, and laughs with the other graduate students who have come to mean so much to me, especially Dave Kaufmann, Jaydeep Mukherjee, Caroline Simpson, Sumita Jayaraman, Ron Drimmel, and Leonard Garcia. I would like to thank the office staff for helping me with so many little problems. Debra Hunter, Elton, Suzie Hicks, Darlene Jeremiah, especially Jeanne Kerrick, deserve many thanks for helping me with travel, faxes, registration, and for brightening my days. Also, thanks go to Eric Johnson and Charlie Taylor for keeping the workstations alive. With this dissertation a very large part of my life is at the same time drawing to a close and beginning anew. The most wonderful part of my new life is that I will be sharing it with Donna. Without the love and unwavering support of Mom. Dad, my sister Cathy and her husband Louie and my nephew and nieces Andrew, Larissa, and Jenna, none of this would ever have happened. TABLE OF CONTENTS ACKNOWLEDGMENTS. LIST OF TABLES . LIST OF FIGURES . ABSTRACT. .. .. .. S S S S S S S S 1 iii .*a..* VI . . V 11 . S. S S S. x S S S S S S S S S S S S S S S S S S CHAPTERS INTRODUCTION S S S S S S S S S S S S S a a 1 THE MAINBELT ASTEROID POPULATION .. 4 Description of the Catalogued Population of Asteroids 4 The MDS and PLS Surveys The PLS Extension in Zones I, II, and III . The Observed Mainbelt Size Distribution . THE OLLISIONAL MODEL Previous Studies Description of the Self-consistent Collisional Model . S S 3 Verification of the Collisional Model . 'Wave' and the Size Distribution from 1 to 100 Meters S S S S S S S S 46 Dependence of the Equilibrium Slope on the Strength Scaling Law The Modified Scaling Law. . 4 HIRAYAMA ASTEROID FAMILIES. S S 52 * S S S S S S S S S S S S S S 55 . . 8 4 i A Brief History of Asteroid Families. The Zappalk Classification . Collisional Evolution of Families . .. 84 * S S S S 85 * S S S S S S S S 5 85 ^ . 86 Number of Families. Evolution of Individual Families IRAS AND THE ASTEROIDAL CONTRIBUTION TO THE ZODIACAL C L.JOlU Dl\ . a * 98 * S S 1 9 . 13 , S S 3 S S S S S S 5 3 The Ratio of Family to Non-Family Dust 6 SUMMARY . Conclusions Future Work APPENDIX A: APPENDIX B: APPENDIX C: S S S S S S S S 140 2 * S 5 5 S S S S S S S S S S S S 10 8 108 * S S S S S S S S S S S S S 1 100 * 5 5 5 5 5 5 5 5 5 5 5 S S S S S S S S S S S S 1 14VI/ APPARENT AND ABSOLUTE MAGNITUDES OF A TEROIDS ..............DB.. . SIZE, MASS, AND MAGNITUDE DISTRIBUTIONS . POTENTIAL OF A SPHERICAL SHELL 5 113 . 1 . S S 5 12 1 BIBLIOGRAPHY . .*. .. 123 BIOGRAPHICAL SKETCH S S S S S S S S S S 5 S S S S S S S S S S S S S S S *. S 12 9 LIST OF TABLES Numbers of asteroids in three PLS zones (MDS/PLS data). . Numbers of asteroids in three PLS zones (catalogued/PLS data). . Adjusted completeness limits for PLS zones. . Intrinsic collision probabilities and encounter speeds for several mainbelt 16 17 18 asteroids. . *. U l U U 6 2 U U U S U U U U U 62 LIST OF FIGURES Proper inclination versus semimajor axis for all catalogued mainbelt asteroids. Magnitude-frequency distribution for catalogued mainbelt asteroids. . 20 Absolute magnitude as a function of discovery date for all catalogued mainbelt asteroids.. . p a p a a a 2 1 Magnitude-frequency distribution for PLS zone I: PLS and catalogued asteroid data. Magnitude-frequency distribution for PLS zone II: PLS and catalogued asteroid data. a a a a a a a a a p p p p p a a a a a U p p p a 2 3 Magnitude-frequency distribution for PLS zone III: asteroid data. PLS and catalogued p a a.p a p a a a a a a p a a a 2 4 Adopted magnitude-frequency distribution for PLS zone I. 25 Adopted magnitude-frequency distribution for PLS zone II. 26 Adopted magnitude-frequency distribution for PLS zone III. 27 Magnitude-frequency distribution for the 1836 asteroids in Tables 7 and 8 of Van Houten et al. (1970).. . a .. 2 8 Least-squares fit to the magnitude-frequency data for PLS zone I. 29 Least-squares fit to the magnitude-frequency data for PLS zone II. 30 Least-squares fit to the magnitude-frequency data for PLS zone III. 31 . . 2 Verification of model for shallow initial slope and small bin size. 64 Verification of model for steep initial slope and large bin size. 65 Verification of model for shallow initial slope and large bin size. 66 Equilibrium slope as a function of time for various fragmentation power laws and for steep initial slope. . S 6 7 Equilibrium slope as a function of time for various fragmentation power laws and for shallow initial slope.. . .S. .S. ... 68 Equilibrium slope as a function of time for various fragmentation power laws and for equilibrium initial slope. . S S S S S S S S S S .6 9 Wave-like deviations in size distribution caused by truncation of particle population. Independence of the wave on bin size adopted in model. a S S S 7 1 Comparison of the interplanetary dust flux found by Grin et al. (198 and small particle cutoffs used in our model. Wave-like deviations imposed by a sharp particle cutoff (x= .. 73 Size distribution resulting from gradual particle cutoff matching the observed interplanetary dust flux (x = 1.2). Collisional relaxation of a perturbation to an equilibrium size distribution.. Halftime for exponential decay toward equilibrium fragmentation of a 100 km diameter asteroid. . slope following the Stochastic fragmentation of inner mainbelt asteroids of various sizes during a typical 500 million period.. Equilibrium slope parameter as a function of the slope of the size-strength scaling la. . . The Davis et al. (1985), Housen et al. (1991), and modified scaling laws used in the collisional model. . a S S S a a a. a 80 The evolved size distribution after 4.5 billion years using the Housen et al. (1991) scaling law for (a) a massive initial population and (b) a small initial population. . The evolved size distribution after 4 . S & U a a a a 8 1 billion years using the Davis et al. (1985) scaling law for (a) a massive initial population and (b) a small initial population. The evolved size distribution after 4.5 82 billion years using our modified scaling law for (a) a massive initial population and (b) a small initial population. . a a a a a a a a a S a a a a a a a a a S a a a 0 8 3 The 26 Hirayama asteroid families as defined by Zappala et al. (1984).. The collisional decay of families resulting from various-size parent asteroids as a function of time. .. . Formation of families in the mainbelt as a function of time. Modeled collisional history of the Gefion family. Modeled collisional history of the Maria family. . The solar system dust bands at 12, Sa a 94 S U S 95 Sa a a a a a a a a96 * a a a 97 60, and 100 im, after subtraction of the smooth zodiacal background via a Fourier filter. a a a 1. a a 105 (a) IRAS observations of the dust bands at elongation angles of 65.68 97.46 ,and 114.68 . Comparisons with model profiles based on prominent Hirayama families are shown in (b), (c), and (d). . 106 The ratio of areas of dust associated with the entire mainbelt asteroid population and all families.. a a a a a a a a a a a a a a a a a a a a a a a a liV. . 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 THE COLLISIONAL EVOLUTION OF THE ASTEROID BELT AND ITS CONTRIBUTION TO THE ZODIACAL CLOUD By DANIEL DAVID DURDA December, Chairman: 1993 Stanley F. Dermott Major Department: Astronomy We present results of a numerical mode verify the results of Dohnanyi (1969, J. Geophys. Res. to place constraints on the impact strengths of asteroids. of asteroid collisional evolution which 74, 2531-2554) and allow us The slope of the equilibrium size-frequency distribution is found to be dependent upon the shape of the size-strength scaling law. An empirical modification has been made to the size-strength scaling law which allows us to match the observed asteroid size distribution and indicates a more gradual transition from strain-rate to gravity scaling. This result is not sensitive to the mass or shape of the initial asteroid population, but rather to the form of the strength scaling law: scaling laws have definite observational consequences. The observed slope of the size distribution of the small asteroids is consistent with the value predicted by the slightly negative slope of our modified scaling law. Wave-like deviations from a strict power-law equilibrium size distribution result if the smallest particles in the population are removed at a rate significantly greater L.Lc-_ J.L -- .... _A A --: T' _1 .. 1 .. 1 -1- a significant wave. We suggest that any deviations from an equilibrium size distribution in the asteroid population are the result of stochastic cratering and fragmentation events which must occur during the course of collisional evolution. determining ratio of the area associated mainbelt asteroids that associated with the prominent Hirayama asteroid families, our analysis indicates that the entire mainbelt asteroid population produces 3.4 + 0.6 times as much dust as the prominent families alone. This result is compared with the ratio of areas needed to account for the zodiacal background and the IRAS dust bands as determined by analysis of IRAS data. We conclude that the entire asteroid population is responsible for at least ~ 34% of the dust in the entire zodiacal cloud. CHAPTER INTRODUCTION Traditionally, the debris of short period comets has been thought to be the source of the majority of the dust in the interplanetary environment (Whipple 1967 Dohnanyi 1976). However, it has been known for some time that inter-asteroid collisions are likely to occur over geologic time (Piotrowski 1953). The gradual comminution of asteroidal debris must supply at least some of the dust in the zodiacal cloud, though because of the lack of observational constraints the contribution made by mutual asteroidal collisions has been difficult to determine. Since the discovery of the IRAS solar system dust bands (Low et al. 1984), the contribution made by asteroids to the interplanetary dust complex has received renewed attention. The suggestion that the dust bands originate from the major asteroid families, widely thought to be the results of mutual asteroid collisions, was made by Dermott et (1984). They also suggested that if the families supply the dust in the bands, thus making a significant contribution to the zodiacal emission, then the entire asteroid belt must contribute a substantial quantity of the dust observed in the zodiacal background. Other evidence also points to an asteroidal source for at least some interplanetary dust. The interplanetary dust particle fluxes observed by the Galileo and Ulysses spacecraft indicate a population with low-eccentricity and low-inclination orbits (Grtin et al. 1991), 2 transport lifetimes of asteroidal dust, Flynn (1989) has concluded that much of the dust collected at Earth from the interplanetary dust cloud is of asteroidal origin. At first inspection it might be tempting to try to calculate the amount of produced in the asteroid belt by modeling, from first principles, the collisional grinding taking place in the present mainbelt. The features of the present asteroid population, however, are the product of a long history involving catastrophic collisions which have reduced the original mass of the belt. Unfortunately, initial mass of the belt is not known and our knowledge of the extent of collisional evolution in the mainbelt is limited by our understanding of the initial mass and the effective strengths of asteroids in mutual collisions. Our intent is to place some constraints on the collisional processes affecting the asteroids and to determine the total contribution made by mainbelt asteroid collisions to the dust of zodiacal cloud. Chapter we describe methods used derive the size distribution of mainbelt asteroids down to ~,5 km diameter. The size distribution of the asteroids represents a powerful constraint on the previous history of the mainbelt as well as the collisional processes which continue to shape the distribution. In Chapter 3 we describe the collisional model which we have developed and present results confirming work by previous researchers. We then use the model to extend our assumptions beyond those of previous works and to shed some light on the impact strengths asteroid ' asteroids families i initial s examined mass of the in Chapter mainbelt. The collisional history providing further constraints on the .- jh1*- k -^ C ~L .*. ^fjc- kL I ^ fk A ^ 4 *j~ r -- A-" n- ^ J. f j- ^ f -, C :fAJIJ i T a -k L rftj nNk a.. I- relative contribution of dust supplied to the zodiacal cloud by asteroid collisions. conclusions are summarized and the problems that must be addressed in future work are discussed in Chapter 6. CHAPTER THE MAINBELT ASTEROID POPULATION Description of the Catalogued Population of Asteroids The size-frequency distribution of the asteroids is very important in constraining the collisional processes which have influenced continue to affect the asteroid population as well as the total mass and mass distribution of the initial planetesimal swarm in that region. Also, in order to determine the total quantity of dust that the asteroids contribute to the zodiacal cloud, we must use the observed population of mainbelt asteroids to estimate the numbers of small asteroids which serve as the parent bodies of the immediate sources of asteroidal dust. In this chapter we will describe the data and methods from which we derive a reliable size distribution. Of the 8863 numbered and multi-opposition asteroids for which orbits had been determined as of December 1992, 8383 (or ~-95%) are found in the semimajor axis range 2.0 < a < 3.8 AU (Figure For reasons described below, we will limit our discussion to those asteroids in the range 2.0 a < 3.5 AU, defining what we will refer to as the "mainbelt. as only SOur conclusions are expected to be unaffected by this choice, 13 asteroids, or less than 0.2% of the known population, are excluded so that the two sets of asteroids are essentially the same. Figure 2 is a plot of the number of catalogued mainbelt asteroids per half-magnitude (Bowell et al. 1989). Immediately evident is a "hump" , or excess, asteroids at 8. f Although previous researchers have interpreted this excess as a remnant of some primordial, gaussian population asteroids altered subsequent collisional evolution (Hartmann and Hartmann 1968), the current interpretation is that it represents the preferential preservation of larger asteroids effectively strengthened by gravitational compression (Davis et al. 1989; Holsapple and Housen 1990). Other researchers, primarily Dohnanyi (1969, 1971), have noted from surveys of faint asteroids (discussed below) indicative the distribution a population smaller asteroids of particles is well described collisional equilibrium. a power-law, Unfortunately, evident in Figure 2, the number of faint asteroids in the catalogued population alone is not quite great enough to be sure of identifying the transition to, or slope of, such a distribution. In fact, the mainbelt population of asteroids is complete with respect to discovery down to an absolute magnitude of only about H = 11. We can see this quite clearly in Figure 3, which is a plot of the absolute brightness of the numbered mainbelt asteroids as a function of their date of discovery. It can be seen that as the years have progressed, increased interest in the study of minor planets and advances in astronomical imaging have allowed for the discovery of fainter and fainter asteroids. In turn, the brighter asteroids have all been discovered, defining fainter and fainter discovery completeness limits. For instance, no asteroids brighter than = 7 have been discovered since about 1910. 1940 the completeness limit was a magnitude fainter. Similarly, al a 1.,.,I I S n n t. n a a n. a a.. ..a 4. a n a I a a n n n -n r a4 a 1 I-i ___ U.rra .. I, the degree of completeness is greater than 99. history recorded in asteroid discovery circum, of discoveries in the wake of World War II. (Figure 3 is also interesting for the Quite apparent is the marked lack The large number of asteroids discovered during the Palomar-Leiden Survey appears as a vertical stripe near As pointed out above, between H 1960.) = 10 and H = 11 the mainbelt appears to make a transition to a linear, power-law size distribution. An absolute magnitude of H =11 corresponds to a diameter of about 30 km for an albedo of 0.1, approximately the mean albedo of the larger asteroids in the mainbelt population (see The Observed Mainbelt Size Distribution). Unfortunately, incompleteness rapidly sets in for H 11.5 and with so few data points the slope of the distribution cannot be well defined so that we cannot reliably use the data from the catalogued population alone to estimate the number of very small asteroids min the mainbelt (see Figur the Palomar-Leiden Survey (Van Houten et al. down to about H We have therefore used data from 1970) to extend the observed distribution = 15.25, corresponding to a diameter of roughly The MDS and PLS Surveys Palomar-Leiden Survey (Van Houten et al. 1970; hereafter referred to as PLS) was conducted in 1960 to extend to fainter magnitudes the results of the earlier McDonald Survey of 1950 through 1952 (Kuiper et al. 1958: hereafter referred to as MDS). MDS surveyed the entire ecliptic nearly twice around to a width of down to a limiting photographic magnitude of nearly In contrast, the practical plate limit for the PLS survey was about five magnitudes fainter. To survey and detect 7 prohibitive, so with the PLS it was decided that only a small patch of the ecliptic would be surveyed, and the results scaled to the MDS and the entire ecliptic belt. In 1984 a revision and small extension were made to the PLS (Van Houten et al. 1984), raising many quality 4 orbits to higher qualities, assigning orbits to some objects which previously had to be rejected, and adding 170 new objects which were identified on plates taken for purposes of photometric calibration. Our original intention was to use this extended data set to re-examine the size distribution of the smaller asteroids in zones of the belt chosen to be more dynamically meaningful than the three zones used in the MDS and PLS. However, we have decided not to embark on a re-analysis of the PLS data at this time as the magnitude distribution of asteroids in the inner region of the mainbelt was rather well defined in the original analysis, and we conclude that even the extended data set will not significantly improve the statistics in the outer region of the We therefore use the original PLS analysis of the absolute magnitude distribution in three zones of the mainbelt, with some caveats as described below. In both the MDS and PLS analysis the mainbelt was divided into three semimajor zones - zone I: 2.0 a < 2.6, zone a < 3.0, zone a < 3.5. Within each zone the asteroids were grouped in half-magnitude intervals of absolute photographic magnitude, g, and the numbers corrected for incompleteness in the apparent magnitude cutoff and the inclination cutoff of the survey (see Kuiper et al. 1958). The g absolute magnitudes given by Van Houten et al. are in the standard B band we transformed these absolute magnitudes to the H, G system by applying the - -- rn ~ ~' 'a -a (an1 o ii OO TI, a~ 1tni c'* r.~ nrro,-'en*aI nwirv, ka<"r\ nF y, ttar^in Ac' nor TIr the PLS, as described by Van Houten et al. The MDS values for the number of asteroids per half-magnitude bin are assumed until the corrections for incompleteness approach about 50% of the values themselves. Where the MDS values require correction for incompleteness, a maximum and minimum number of asteroids is calculated based upon two different extrapolations of the log N(mo) relation (Kuiper et al. 1958). In these cases the mean of the two values given in the MDS has been assumed. The correction factors for incompleteness in zone Il given in the MDS, however, are incorrect. corrected values are given in Table D-I of Dohnanyi (1971). For fainter values of H the number of asteroids is taken from Table of Van Houten et al., the values given there corrected by multiplying log N(H) by 1.38 to extend the counts to cover the asteroid belt over all longitudes to match the coverage of the MDS. Table 1 gives the adopted bias-corrected number of asteroids per half-magnitude bin (H magnitudes) for each of the three PLS zones and for the entire mainbelt as derived from the MDS and PLS data. While the MDS, which surveyed the asteroid belt over all longitudes, is regarded as complete down to an absolute magnitude of about g = 9.5, data need to be corrected for completeness at all magnitudes as the survey covered only a few percent of the area of the MDS. There have been a number of discussions regarding selection effects within the PLS and problems involved with linking up the MDS and PLS data (cf. Kresik 1971 and Dohnanyi 1971). We have taken a very simple approach which indicates that the MDS and PLS data link up quite well and that any selection effects within the PLS either cancel each other or are minor to begin with. Figures 4, C A^ ,^^ ^-/-. J. L ^ /^ Mk A N^ *J 1^ fi I/1 T 0 A^ a- Sk..q~l~ a-^ n,* *-. an h n. A^ nt* + n^ Sn *' i-^4 an j- a rT n n A vertical line indicates the completeness limit for the MDS. beyond which correction factors were adopted based on extrapolations of the observed trend of the number of asteroids per mean opposition magnitude bin. The solid vertical line indicates where the PLS data have been adopted to extend the MDS distribution. In each of the three zones completeness limit for the catalogued population roughly coincides the transition to the PLS data. Beyond the completeness limit the observed number catalogued asteroids per half-magnitude bin continues to increase (although at a decreasing level of completeness) until the numbers fall markedly. In each of the three zones the data for the catalogued population merges quite smoothly with the PLS data. This is particularly evident in zone II, where there is a significant decline in the number of asteroids with H 11, right in the transition region between the incompleteness corrected MDS data and the PLS data, producing an apparent discontinuity between the two data sets. The catalogued population, however, which is complete to about H = 11 in this zone, nicely follows the same trend, even showing the sharp upturn beyond the completeness limit between H = 11.25 and H = 11.75. With the catalogued population making a smooth transition between the MDS and PLS data in each of the three zones, we conclude that any selection effects which might exist within the PLS data are minor and that there is no problem with combining the MDS data (roughly equivalent to the current catalogued population) and PLS data as published. The PLS Extension in Zones I, II, and Im Having established that the PLS data may be directly used to extend our discussion 10 magnitude bin from the catalogued population for those bins brighter than the discovery completeness limit and from either the PLS data or catalogued population, whichever is greater, for the magnitude below the completeness limit. to sampling statistics there a V error associated each independent point incremental magnitude-frequency diagram. errors catalogued asteroid counts are determined directly from the raw numbers after the asteroids have been binned and counted. For the PLS data the errors must be determined from the number of asteroids per magnitude interval before the counts have been corrected for the apparent magnitude and inclination cutoffs. The corrected counts themselves are given Table 5 of Van Houten et al. These counts are then scaled to match coverage of the MDS as described above. Since the errors in the PLS counts are based on the uncorrected, unsealed counts, the PLS data points have a larger associated error than the corrected counts themselves would indicate. The resulting magnitude- frequency diagrams for each of the PLS zones are shown in Figures the numbers tabulated in 8, and 9 and Table The PLS data greatly extend the workable observed magnitude-frequency distrin- butions for the mainbelt asteroids. We immediately see that the inner two zones of the mainbelt display a well defined, linear power-law distribution for the fainter asteroids, with the prominent excess of asteroids at the brighter end of the distribution. bution in the outer third of the belt appears somewhat less well defined. Thi The distri- e results for the inner zones are very interesting, as the linear portions qualitatively match very well Dohnanyi' (1969, 1971) prediction of an equilibrium power-law distribution of frag- through the MDS and PLS data, found a mass index of 1.839, in good agreement with the theoretical expected value of q = 1.837 quoted in his work. His analysis, however, was performed on the cumulative distribution of the combined data from the three zones. We feel that it is more appropriate to consider only incremental frequency distributions since the data points are independent of one another and the limitations of the data set are more readily apparent. In this analysis we will also consider the three zones independently to take advantage of any information that the distributions may contain on the variation of the collisional evolution of the asteroids with location the mainbelt. Having assigned errors to the independent points in the incremental magnitude- frequency diagrams, a weighted least-squares solution can be fit through linear portions of the distributions in each of the three PLS zones. We must be cautious, however, to work within the completeness limits of the data. Figure 10 is a histogram of the number of asteroids per half-magnitude interval as derived from the data in Tables 7 and 8 of Van Houten et al. (1970). These are the 1836 asteroids for which orbits were able to be determined plus the 187 asteroids for which the computed orbits had to be discarded. The survey was complete to a mean photographic opposition magnitude of approximately 19, beyond which the numbers would need to be corrected for incompleteness. Recognizing the uncertainties involved in trying to estimate the degree of completeness for fainter asteroids on the photographic plates, work within the completeness limits of the raw data set. we prefer to Given the completeness limit 12 mean semimajor axis for each of the zones we calculate the adjusted completeness limits given in Table 3. Based on these more conservative completeness limits we may now calculate the least squares solutions for the individual zones. Zone I displays a distinctly linear distribution for absolute magnitudes fainter than about H = 11. weighted least-squares fit to the data (H = 11.25 fainter) yields a slope of a = 0.469 0.011, 1.782 0.018 (Figure 11). which corresponds to a mass-frequency slope of (If we assume that all the asteroids in a semimajor axis qz= zone have the same mean albedo we may directly convert the magnitude-frequency slope into the more commonly used mass frequency slope via q = the slope of the magnitude-frequency data. where a is See Appendix B.) Zone II shows a similar, though somewhat less distinct and shallower, linear trend beyond H = 11.25. A fit through these data yields a slope of a = 0.479 0.012 (q = 1.799 0.020, Figure In Zone III we obtain the solution a = 0.447 0.017 (q = 1.745 0.028, Figure for magnitudes fainter than H Dohnanyi equilibrium value of = 10.75. 1.833. These slopes are significantly lower than the The weighted mean slope for the three zones 1.781 0.007, essentially equal to the well determined slope for zone I. In addition to the slope, least-squares solution for each zones produces an estimate for the intercept of the linear distribution, number of asteroids in the population. With an esti which is a measure of the absolute mate of the mean albedo of asteroids in the population, the expressions derived in Appendix B allow us to use the parameters of the magnitude-frequency plots to quantify the size-frequency distributions for the three zones and for the mainbelt as a whole. * 1 + a, The Observed Mainbelt Size Distribution We may define the observed mainbelt size distribution that we will work with by combining data from the catalogued population of asteroids and the least-squares fits to the PLS data. The absolute sizes of the brightnesses numbered mainbelt asteroids may if we can estimate a value for th reconstructed e albedo (See from their Appendix A). Fortunately, an extensive set of albedos derived by IRAS is available for a great many asteroids. A recent study by Matson et al. (1990) demonstrates that asteroid diameters derived using IRAS-derived albedos show no significant difference between those found by occultation studies. Although an even larger number of asteroids exists for which no albedo measurements have been made, the IRAS data base is extensive enough to allow a statistical reconstruction of their albedos. without albedo estimates: There are two subsets of asteroids those for which a taxonomic classification is available, and, larger group, those which have not been typed. have used taxonomic types assigned by Tedesco et al. (1989) when available and by Tholen (1989, 1993 private communication) if a classification based upon an IRAS-derived albedo was not available. For those asteroids with a taxonomic type but no IRAS-observed albedo, we have estimated the albedo by assuming the mean value of other asteroids with the same classification. If no taxonomic information was available we assumed an albedo equal to that of the IRAS-observed asteroids at the same semimajor axis. The diameters for the catalogued asteroids, calculated using Equation 11 of Appendix A, are then collected distribution asteroids smaller completeness limit catalogued population has been derived using the magnitude data described previous section. Linear least-squares solutions, constrained to have same weighted mean slope of q = 1.781, were fit through the linear portions of the magnitude distributions in each of the three PLS zones. The individual distributions were then added to determine the intercept parameter (equivalent to the brightest asteroid in the power-law distribution) for the mainbelt as a whole. To convert the parameters of the magnitude-frequency distribution determined using the PLS data into a size-frequency distribution, we assume that all the asteroids in the population have the same mean albedo. Of the well-observed asteroids in the mainbelt, that is, asteroids with IRAS-determined albedos and measured B-V colors, we found mean albedos of 0.121, 0.105, and 0.074 in PLS zones I, II, and II, respectively. The weighted mean albedo for the entire mainbelt population is 0.097 . We chose to calculate the mean albedo based on those asteroids with diameters between 30 and 200 km, in order to avoid any possible selection effects which might affect the smallest and largest asteroids. With an estimate for the mean albedo the magnitude parameters may be converted directly into a size-frequency distribution using Equations 6 and 15 of Appendix B. In Figure we have combined the data from the catalogued asteroids and the PLS magnitude distributions to define the observed mainbelt size distribution. Down to approximately 30 km the distribution is determined directly from the catalogued asteroids and IRAS-derived albedos. The shaded band indicates the error associated with the catalogued population due to sampling statistics. For diameters estimated from PLS data. to smaller sizes. asteroids. We thus use the PLS data to extend the usable size distribution The dashed line is the best fit through the magnitude data for the small This size distribution is very well determined and will be used in the next chapter to place strong constraints on collisional models of the asteroids. 16 Table 1: Numbers of asteroids in three PLS zones (MDS/PLS data). Zone I a<2.6 N(H) Zone II N(H) Zone III ) N(H) I + II + III N(H) 3.25 1 1 0 2 3.75 0 1 0 1 4.25 0 0 0 0 4.75 0 0 0 0 5.25 0 2 1 3 5.75 2 1 0 3 6.25 5 4 2 11 6.75 5 4 5 14 7.25 5 15 11 31 7.75 13 20 24 57 8.25 15 39 31 5 114.5 10.25 10.75 11.25 11.75 12.25 12.75 13.25 13.75 14.25 14.75 15.25 15.75 16.25 143.93 143.93 503.75 1007.51 2254.90 4125.99 6093.04 10914.69 17151.66 287.86 791.61 551.73 1103.46 2614.73 3958.07 7532.34 6788.70 12401.97 215.89 95.95 287.86 503.75 503.75 575.72 1727.16 4941.60 5109.51 6069.05 7868.17 219.5 329.89 219.45 477.36 918.61 1439.29 1271.38 3334.37 8563.84 11322.48 17727.38 20749.91 Table 2: Numbers of asteroids in three PLS zones (catalogued/PLS data). 325 3.25 3.75 4.25 4.75 5.25 5.75 6.25 6.75 7.25 7.75 8.25 8.75 9.25 9.75 10.25 10.75 11.25 11.75 12.25 12.75 13.25 13.75 14.25 14.75 15.25 15.75 16.25 N(H) 1007.51 2254.90 4125.99 6093.04 10914.69 17151.66 Zone II N(H) 294 791.61 551.73 1103.46 2614.73 3958.07 7532.34 6788.70 12401.97 Zone III N(H) 503.75 503.75 575.72 1727.16 4941.60 5109.51 6069.05 7868.17 I + II + III .0 < a < 3.5 N(H) 938.7 1570.36 1642.45 3614.62 8563.84 11322.48 17727.38 20749.91 Zone I Table 3: Adjusted completeness limits for PLS zones. Semimajor Axis Zone Mean Semimajor Axis Completeness limit in H (AU) 2.0 < a < 2.6 2.43 15.3 2.6 < a < 3.0 2.75 14.6 3.0 < a < 3.5 3.17 13.8 * S *- * < * ./*. .t .* *; S**. .* t ft* ft ; ft* I * '.' ". 0, f * * a f t a * ft .- . " **9.* s ?' S a S a.t. *t - f t 1 * * S.-1 * ." .* t*. *t ,-ft * ft. * S . at'- / *** ft "*-0 * p.1 * I 1 * 5 .. ' ** *tr * '1 CO aaj~acj) Uo flU .OUl jodoad I *. . *- 0* . [ .% p - * * * * ** . * '. Ula. 9Pfl4!U~PN joquinfl jad 9pflt4iU~PN Lo a~iC) sqv uT. 9pnflhU Pw jod C.) I- Co j9qmLpfl t CO C', 00 < utg Gpnl[IU T3 aoqmntn jsdt TlT' ' 111I III IH I I I 11111 l 1-I0 I--i 0 * S0 0 s 0 U 0 e 0 0 il I ui. III l l li l l ll 9pfl4iU~PJ~ CV? -*d 1 I I I I 1 I I I I I uIqtUnN li- UT.E 9ppnfl.UmP jAd joquinN CO CVN U1g 9pnr142 II 1d JoqmnUfN LlO 1-ID OO Soe N0H HOH- HO- tza U.t 9pnf4.1UfP7 oad jsqmnjN CV --" UJ. 9plnliewu joqtUnN - CD c~i - v 0No I.' I.' IOA bOA-1 0 C2 t cO uiE Gpfl]qUUnM C1 O 0 r-. aod T-1f -N U1g 9pflh4UiN jdjGqLuiIN co co n U) ~-4 S0 (12n r** UI Gpnluiw UjPd NqmnN ul. ljSLmUQ!c aed joqmnn CHAPTER 3 THE COLLISIONAL MODEL Previous Studies Before describing the details of the collisional model developed in this thesis, it would be useful to review some previous studies. The collisional evolution of the asteroids and its effects on the size distribution of the asteroid population has been studied by a number of researchers both analitically and numerically. Dohnanyi (1969) solved analytically the integro-differential equation describing the evolution of a collection of particles, a which fragment due to mutual collisions. with size independent impact strengths, He found that the size distribution of the resulting debris can be described by a power-law distribution in mass of the form f(m)dm c m -dm, (3-1) where f(m)dm is the number of asteroids in the mass range m to m + dm and q is the slope index. Dohnanyi found that q = 1.833 for debris in collisional equilibrium, in agreement with the observed distribution of small asteroids as determined from MDS and PLS data. The equilibrium slope index q was found to be insensitive to the fragmentation power law 77 of the colliding bodies, provided that y <2. This is because the most important contribution to the mass range mn to m + dm comes from collisions in which the mrnct rnmacvnr narthn-lpe nre rf nmacc n^r fmhlr 34 Dohnanyi also found that for q near 2 but less than 2 the creation of debris by erosion, or cratering collisions, plays only a minor role. The steady-state size distribution is therefore dominated by catastrophic collisions. Hellyer (1970, 1971) solved the same collision equation numerically and confirmed the results of Dohnanyi. Hellyer showed that for four values of the fragmentation power law, referred to as z in his notation, (x = n 1 = 0.5, 0.6, 0.7, and 0.8), the population index of the small masses converged to an almost stationary value of about 1.825. convergence was most rapid for the largest values of x, but the asymptotic value of the population index is very close to the value obtained analytically by Dohnanyi. Although primarily interested in the behavior of the smallest asteroids, Hellyer also investigated the influence of random disruption of the largest asteroids on the rest of the system. His program was modified to allow for a small number of discrete fragmentation events among very massive particles. With the parameter z set to 0.7, the slope index of the smallest asteroids was seen to still attain the expected value (about 1.825), although there were discontinuities in the plot of the slope as a function of time at the times of the large fragmentation events. Davis et al. (1979) introduced a numerical model simulating collisional evolution of various initial populations of asteroids and compared the results with the observed distribution of asteroids in order to find those populations which evolved to the present belt. In their study they considered three different families of shapes for the initial distribution: generated by the accretional simulation of Greenberg et al. gaussian as suggested by (1978), and Anders (1965) and Hartmann and Hartmann (1968). They concluded that for power law initial populations the initial mass of the belt could not have been much larger than ~ 1Me, only modestly larger than the present belt. Both massive and small runaway growth distributions were found to evolve to the present distribution, however, placing no strong constraints on the initial size of the belt. eaussian initial distributions failed to relax to the observed distribution. The power law and runaway growth models, however, both produced a small asteroid distribution with a slope index similar to the value predicted by Dohnanyi. Another major conclusion of this study was that most asteroids a 100 km diameter are likely fractured throughout their volume and are essentially gravitationally bound rubble piles. Davis et al. (1985) introduced a revised model incorporating the increased impact strengths of large asteroids due to hydrostatic self-compression. The results from this numerical model were later extended to include (strain-rate) dependent impact strengths (Davis et al. 1989). The primary goal of these studies was to further constrain the extent of asteroid collisional evolution. They investigated a number of initial asteroid populations and concluded that a runaway growth initial belt with only times the present belt mass best satisfied the constraints of preserving the basaltic crust of Vesta and producing the observed number of asteroid families. However, other asteroid observations (such as the interpretation of M asteroids as exposed metallic cores of differentiated bodies and the apparent dearth asteroids representing the shattered 36 used to investigate the collisional history of asteroid families (Davis and Marzari 1993). Most recently, to include a Williams and Wetherill (1993) have extended the work of Dohnanyi wider range of assumptions and obtained an analytical solution for the steady-state size distribution of a self-similar collisional fragmentation cascade. Their results confirm the equilibrium value of = 1.833 and demonstrate that this value is even less sensitive to the physical parameters of the fragmentation process Dohnanyi had thought. In particular, Williams and Wetherill have explicitly treated the debris from cratering impacts (whereas Dohnanyi concluded that the contribution from cratering would be negligible and so dropped terms including cratering debris) and have more realistically assumed that the mass of the largest fragment resulting from a catastrophic fragmentation decreases with increasing projectile mass. They find a steady-state value of q = 1.83333 0.00001 which is extremely insensitive to the assumed physical parameters of the colliding bodies or the cratering and fragmentation. relative contributions of They note, however, that this result has still been obtained by assuming a self-similar system in which the strengths of the colliding particles are independent of size and that the results of relaxing the assumption of self-similarity will be explored in future work. Description of the Self-consistent Collisional Model An initial population of asteroids is distributed among a number of logarithmic size bins. The initial population may have any form and is defined by the user. actual number of bins depends on the model to be run, but for most cases in which those cases min which we are interested in modeling the collisional evolution of dust particles the number of bins can increase to over For most of the models the logarithmic increment was chosen to be 0.1, in order to most directly compare the size distributions with the magnitude distributions derived in Chapter (see Appendix B). For some models including dust size particles the bin size was increased to 0.2 to decrease the number of bins and shorten the run time. All particles are assumed to be spherical to have same density. characteristic size of the particles in each bin is determined from the total mass and number of particles per bin. This size is used along with the assumed material properties of the particles and the assigned collision rate to associate a mean collisional lifetime with each size The timescale for the collisional destruction of an asteroid of a given diameter depends on the probability of collision between the target asteroid and "field" asteroids, the size of the smallest field asteroid capable of shattering and dispersing the target, and the cumulative number of field asteroids larger than this smallest size. We shall now detail the procedure for calculating the collisional lifetime of an asteroid and examine each of these determinants in the process. The probability of collisions (the collision rate) between the target and the field asteroids has been calculated using the theory of Wetherill (1967). method, Utilizing the same Farinella and Davis (1992) independently calculated intrinsic collision rates which match our results to within a factor of 1.1. For a target asteroid with orbital S1 ....................................................i.. ................................................................................................................................ 4 such that the total number of particles in the asteroid belt is The population of field asteroids was chosen as a subset the catalogued mainbelt population. asteroids brighter than H = 10, just slightly brighter than the discovery completeness limit for the mainbelt, were chosen to define a bias-free set of field asteroids. In this way the selection for asteroids in the inner edge of the mainbelt is eliminated and the field population is more representative of the true distribution of asteroids. The orbital elements were taken to be the proper elements as computed by Milani and Knezevi6 (1990), which are more representative of the long-term orbital elements than are the osculating elements. The resulting intrinsic collision rates and mean relative encounter speeds for several representative mainbelt asteroids are given in Table 4. The mean intrinsic rate and relative encounter speed calculated bias-free set are 2.668 x 10-18 yr1 km-2 from and 5.88 km s1 672 asteroids of , respectively. The "final" collision probability for a finite-sized asteroid with diameter D is P1 = 4'I, (3-2) where o-' /Tr (since Pi includes the factor of 7) and cr = 7r(D/ ) is the collision cross-section (taken to be the simple geometric cross-section since the self-gravity of the asteroids is negligible here). a destructive collision, we mu To get the total probability that the asteroid will suffer st integrate the final probability over all projectiles of consequence using the size distribution function dN = CD-EdD. (3-3) Then D ,0,ta .1 fp IAT Pt= cr PiCD-'dD. (3-5) is simply the collision cross section times the intrinsic collision probability times the cumulative number of field asteroids larger than D,,i,,.) The collision lifetime, re = 1/Pt, (3-6) is then the time for which the probability of survival is 1/e. Let us now examine the determination of Din,. the smallest field asteroid capable of fragmenting and dispersing the target asteroid. To fragment and disperse the target asteroid, the projectile must supply enough kinetic energy to overcome both the impact strength of the target (defined as the energy needed to produce a largest fragment containing 50% of the mass of the original body) and its gravitational binding energy. The impact strength of asteroid-sized bodies is not well known. Laboratory experiments on the collisional fragmentation of basalt targets (Fujiwara et al. 1977) yield collisional specific energies of 7 x 106 , or an impact strength, x 10 . However, estimates by Fujiwara (1982) of the kinetic and gravitational energies of the fragments in the three prominent Hirayama families indicates that the asteroidal parent bodies had impact strengths of a few times 108 erg cm-3 greater than impact strengths for rocky materials. , an order of magnitude (Fujiwara assumed that the fraction of kinetic energy transferred from the impactor to the debris is fKE = 0.1.) In order to avoid implausible asteroidal compositions, we must conclude that the effective impact 40 from laboratory experiments to asteroid-sized bodies are reviewed by Fujiwara et al. (1989). Davis et al. (1985) concluded that large asteroids should be strengthened by gravitational self-compression and developed a size-dependant impact strength model which is consistent with the Fujiwara et al. (1977) results and produces a size-frequency distribution collision fragments consistent observed Hirayama families. Other researchers (Farinella et al. 1982; Holsapple and Housen 1986; Housen and Holsapple 1990) have developed alternative scaling laws for strengths, predicting impact strengths which decrease with increasing target size. We will discuss the various scaling laws in more detail later in the chapter. For the time being let us simply assume that there will be some body averaged impact strength, S, associated with an asteroid diameter gravitational binding energy of the debris must also be overcome in order to disperse the fragments of the collision. Consistent with the definition of a barely catastrophic collision, in which the largest fragment has 50% the mass of the original body, we take the binding energy to be that of a spherical shell of mass 1M (where M is the total mass of the target) resting on a core of mass 1M. Such a model should well approximate the circumstances of a core-type shattering collision. In this case, GM2 0.411f--- RJt (3-7) is the energy required to disperse one half the mass of the target asteroid to infinity Appendix C). Not all of the kinetic energy of the projectile is partitioned into comminution 41 projectile kinetic energy partitioned into kinetic energy of the members of the family order 0.1 was most consistent with the derived collision energies fragment sizes. Experimental determination of the energy partitioning for core-type collisions (Fujiwara and Tsukamoto 1980) showed that only about 0.3 to 3% of the kinetic energy of the projectile is imparted into the kinetic energy of the larger fragments and the comminutional energy for these fragments amounts to some 0.1% of the impact energy. We shall take tens of flE to be a parameter which may assume values of from a few to few percent. may then write for the minimum projectile kinetic energy needed fragment and disperse a target asteroid of mass M and diameter D f1E Emiz= E fKE SV GM2 +0.411 D/ (3-8) where V is the volume of the asteroid. From the kinetic energy of the projectile and the mean encounter speed calculated by the Wetherill model, we can find the minimum projectile mass and, hence, the minimum projectile diameter needed to fragment and disperse the target asteroid Emin = rm i n V2 = -PD mVe2 12C (3-9) Finally, then, _i (1 Dmin - E*Ini (3-10) irplQ 42 collision program this number is determined by simply counting, during each time step, the total number of particles in the bins larger than D,,~1,. In this way the projectile population is determined in a self consistent manner. Once a characteristic collisional lifetime has been associated with each size bin the number of particles removed from each bin during a timestep can be calculated. Instead of defining a fixed timestep, the size of a timestep, At, is determined within the program and updated continuously in order to maintain flexibility with the code. times At is chosen to be some small fraction of the shortest collision lifetime, At all 7( , where 7"., is usually the collision lifetime for bin 1. In most cases we have let At = 10 T,,""" . During a single timestep the number of particles removed from bin i is then found from the expression z= N( (3-11) with the stipulation that only an integer number of particles are allowed to be destroyed per bin per timestep: number z is rounded to the nearest whole number. small size bins this procedure gives the same results as calculated directly by Equation 3-11, since very large. is rounded up as often as down and the number of particles involved is For the larger size bins considered in this model, however, the procedure more realistically treats the particles as discrete bodies and allows for the stochastic destruction of asteroid sized fragments. When an asteroid a given is collisionally destroyed, fragments distributed into smaller size bins following a power-law size distribution given by T,,{i} 43 The exponent p is determined from the parameter b, the fractional size of the largest fragment in terms of the parent body, by the expression b3+4 (3-13) so that the total mass of debris equals the mass of the parent asteroid (Greenberg and Nolan 1989). The constant B is determined such that there is only one object as large as the largest remnant, Di.. The exponent p is a free parameter of the model, but is usually taken to be somewhat larger than the equilibrium value of (0.833 in mass units) in accord with laboratory experiments and the observed size-frequency distributions of the prominent Hirayama families (Cellino et al. 1991), although it is recognized that in reality a single value may not well represent the size distribution at all sizes. The total number of fragments distributed into smaller size bins from bin i is then just the number of fragments per bin as calculated from Equation 3-12 multiplied by the number of asteroids which were fragmented during the time step. Verification of the Collisional Model Verification of the collisional model consisted of a number of runs demonstrating that an equilibrium power-law size distribution with a slope index of 1.833 is obtained independent size, initial distribution, or fragmentation power-law, provided that we assume (as did Dohnanyi) a size-independent impact strength. we cannot present the results of all runs made during the validation phase in a short space, a representative series of results are presented here. bS+1 Dohnanyi. runs slope breakup power-law was set equal to the equilibrium value of q = 1.833, we assumed a constant impact strength scaling law, and the logarithmic size bin interval was set equal to 0.1. For the first run the initial size distribution was chosen to be a power-law distribution with a steep slope of q = 2.0. final distribution at 4.5 billion years is shown, as well as at earlier times at 1 billion year intervals. The evolved distribution very quickly (within a few hundred million years) attains an equilibrium slope equal to the expected Dohnanyi value of q = 1.833 for bodies in the size range of 1-100 meters. initial distribution with a slope of q = 1. rapidly attained the expected equilibrium The second run began with a much shallower r. The evolved distribution here as well very slope. The same two numerical experiments were repeated bin size increased to 0.2. results (Figures were identical to the first two experiments power-law evolved size distributions with equilibrium slopes of 1.833. To study the dependence of the equilibrium slope on the slope of the breakup power-law and the time evolution of the size distribution we altered the collisional model slightly to eliminate the effects of stochastic collisions. Perturbations on the overall slope of the size distribution produced by the stochastic fragmentation of large bodies may mask any finer-scale trends due to long term evolution of the size distribution, especially for a steep fragmentation power-law. We ran a series of models with various power-law initial size distributions and fragmentation power-laws spanning a range of slopes. results are shown graphically in Figures 19 through 21 where we have plotted the slope, a. of the size distribution as a function of time for the smallest bodies in the model. The 45 (1-100 meters) of a ~-60 bin model. In Figures 19, 20, and 21 the slopes of the initial size distributions are 1.88. 1.77, and 1.83, respectively. Note that the vertical scale in Figure 21 has been stretched relative to the previous two figures in order to bring out the relevant detail. In all three cases we see that the slope of the size distribution asymptotically approaches the value 1.833, than this within the age of the solar system. reaching values not significantly different The different values of the slope are only very slightly dependent upon the fragmentation power-law. For qb (r] in Dohnanyi' notation) higher than the equilibrium value the final slope converges for all practical value on slopes somewhat greater than 1.832 within 4.5 billion years. equilibrium the final slopes are less than 1.834. For qb less than Interestingly, for steep fragmentation power-laws, the slope is always seen to overshoot' on the way to equilibrium, either higher than 1.833 when the initial slope is lower, or lower than 1.833 when the initial slope is higher. We find perhaps not unexpectedly that the Dohnanyi equilibrium value is reached most rapidly when the fragmentation power-law is near 1.833. HeUllyer (1971) found the same behavior in his numerical solution of the fragmentation equation. In his work, however, Hellyer did not include models in which the fragmentation index was more steep than the equilibrium value, so we cannot compare our results concerning the equilibrium overshoot Recall that Dohnanyi (1969) concluded that the debris from cratering collisions played only a minor role in determining the slope of the equilibrium size distribution. Our numerical model was thus constructed to neglect cratering debris. The recent work -C i171^1',_ ., .. T7lL.i-.LZ11i /lflflfl\ C.^ tt-_ st... 2-^1-.i 2^ U--- -- - 46 of cratering debris the equilibrium slope may vary from the expected value of 1.833 by a very slight amount. Our numerical results seem to confirm this. The very slight deviations we however, will be shown to be insignificant compared to the variations in the slope due to relaxation of the Dohnanyi assumption of size-independent strengths. We conclude from this series of model runs that our numerical code properly reproduces the results of Dohnanyi (1969). With size independent impact strengths our model produces evolved power-law size distributions with slopes essentially equal to 1.833 independent of the numerical requirements the computer code assumptions concerning the colliding asteroids. 'Wave' and the Size Distribution from 1 to 100 Meters During the earliest phases of code validation our model produced an unexpected deviation from a strict power-law size distribution. Figure shows the size distri- button which resulted when particles smaller than those in the smallest size bin were inadvertently neglected in the model. Because of the increasing numbers of small par- ticles in a power-law size distribution, the vast majority of projectiles responsible for the fragmentation of a given size particle are smaller than the target and are usually near the lower limit required for fragmentation. model, By neglecting these particles in our we artificially increased the collision lifetimes of those size bins for which the smallest projectile required for fragmentation was smaller than the smallest size bin. The particles in these size bins then become relatively overabundant as projectiles and preferentially deplete targets in the next largest size bins. The particles in these bins .--- 4 4 4 t, I * a strict power-law distribution up through the largest asteroids in the population. same wave-like phenomenon was later independently discovered by Davis et al. (1993). The code was subsequently altered to extrapolate the particle population beyond the smallest size bin to eliminate the propagation of an artificial wave in the size distribution. However, in reality the removal of the smallest asteroidal debris by radiation forces may provide a mechanism for truncating the size distribution and generating such a wave- like feature in the actual asteroid size distribution. To study the sensitivity of features of the wave on the strength of the small particle cutoff we may impose a cutoff on the extrapolation beyond the smallest size bin to simulate the effects of radiation forces. We use an exponential cutoff of the form N(-i) =N( ) 10-x"/10 (3-14) where 3,..., N(1) is the smallest size bin, N o is the number of particles expected smaller than those in bin 1 based on an extrapolation from the two smallest size bins, and x is a parameter controlling the strength of the cutoff. Negative bin numbers simply refer to those size bins which would be present and responsible for the fragmentation of the smallest several bins actually present in the model. The number of "virtual" bins present depends upon the bin size adopted for a particular model, though in all cases extends to include particles .~ the diameter of those in bin 1 the size ratio required for fragmentation). (roughly This form for the cutoff is entirely empirical, but for our purposes may still be used to effectively simulate the increasingly efficient removal of smaller and smaller particles by radiation forces. When the parameter x is more realistic in its smooth tail-off in the number of particles runs with a sharp exponential cutoff are shown in Figure the two runs were identical, with the exception of the bin size . The results of two model The starting conditions for . To be sure the features of the wave were not a function of the bin size, the first model was run with a logarithmic interval of 0.1 while the second used a bin size twice as large. The parameter x had to be adjusted for the second model to ensure that the strength of the cutoff was identical to that in the first model. We can see that in both models a wave has propagated into the large end of the size distribution. The results of the two models have been plotted separately for clarity (with the final size distribution for the larger bin model offset to the left by one decade in size), but if overlaid would be seen to coincide precisely, thus illustrating that the wavelength and phase of the wave are not artifacts of the bin size adopted for the model run. The effect of a smooth (though sharp) particle cutoff may be seen by comparing the shape and onset of the wave in the smallest size particles between Figures 22 and 23. The amplitude of the wave has been found to be dependent upon the strength of the small particle cutoff. A significant wave will develop only if the particle cutoff is quite sharp, that is, if the smallest particles are removed at a rate significantly greater than that required to maintain a Dohnanyi equilibrium power-law. Since radiation forces do in fact remove the smallest asteroidal particles, providing a means of gradually truncating the asteroid size distribution, some researchers (Farinella et al. 1993, private communication) have suggested that such a wave might actually exist and may be responsible for an apparent steep slope index of asteroids in the 10-100 meter diameter size range. At least three independent observations seem to indicate a from the observed larger asteroids would yield. Although there is some uncertainty in the precise value, the observed slope of the differential crater size distribution on 951 Gaspra seems to be greater than that due to a population of projectiles in Dohnanyi collisional equilibrium, ranging from -3.5 to -4.0 (Belton et al. 1992). (The Dohnanyi equilibrium value is p = -3.5.) diameter range 0.5 to The crater counts are most reliable in the km; craters of this size are due to the impact of projectiles with diameters < 100 meters. The slope of the crater distribution on Gaspra is also consistent with the crater distribution observed in the lunar maria (Shoemaker 1983) and the size distribution of small Earth-approaching asteroids discovered by Spacewatch (Rabinowitz 1993). Davis et al. (1993) suggest that although the overall slope index of the asteroid population is close to or equal to the Dohnanyi equilibrium value, waves imposed on the distribution by the removal of the small particles may change the slope in specific size ranges to values significantly above or below the equilibrium value. To test the theory that a wave-like deviation from a strict, power-law size distribu- tion is responsible for the apparent upturn in the number of small asteroids as described above, we have modeled the evolution of a population of asteroids with the removal of the smallest asteroidal particles proceeding at two different rates: cutoff and one matching the observed particle cutoff. a very sharp particle To compare these removal rates with the removal of small particles actually observed in the inner solar system, we have plotted our model population and cutoffs with the observed interplanetary dust popula- tion (Figure 24). et al. Using meteoroid measurements obtained by in situ experiments, Grtin (1985) produced a model of the interplanetary dust flux for particles with masses this corresponds to particles with diameters of about 0.01 pm to 10 mm, respectively. Figure 24 shows the Grin et al. model and our modeled particle cutoffs for three values For the following models the logarithmic size interval was set equal to 0.1. 2x = 0 we have the simple case of strict collisional equilibrium with no particle removal by non-collisional effects, illustrated by the models presented in the previous section. When a sharp particle cutoff is modeled beginning at ~-100 /tm, the diameter at which the Poynting-Robertson lifetime of particles becomes comparable to the collisional life- time, the evolved size distribution develops a very definite wave (see Figure 25) with an upturn in the slope index present at ~100 m. The parameter a was set equal to 1.9 for this model to produce a "sharp" cutoff, i.e one obviously much sharper than the observed cutoff and one capable of producing a strong, detectable wave. If a wave is present in the real asteroid size distribution, however, the more gradual cutoff which is observed must be capable of producing significant deviations from a linear power-law. Over the range of projectile sizes of interest we can match the actual interplanetary dust population quite well with 1.2. Figure 26 illustrates that this rate of depletion of small particles is too gradual to support observable wave-like deviations. size distribution is nearly indistinguishable from a strict power-law. The evolved The observed cutoff is more gradual than those produced by simple models operating on asteroidal particles alone for at least two reasons. First, if the particle radius becomes much smaller than the wavelength of light, the interaction with photons changes and the radiation force becomes negligible once again. Second, in this size range there will be a significant contribution from cometary particles. The assumption in our model of a closed system 51 The input of cometary dust as projectiles in the smallest size bins may not be insignif- icant in balancing the collisional loss of asteroidal particles. We conclude that a strong wave is probably not present in the actual asteroid size distribution and cannot account for an increased slope index among 100 meter-scale asteroids. Although we stress that the wave requires further, more detailed investigation, we feel it most likely that any deviations from an equilibrium power-law distribution among the near-Earth asteroid population are the results of recent fragmentation or cratering events in the inner asteroid belt. Such stochastic events must occur during the course of collisional evolution and produce deviations from a Dohnanyi equilibrium due to the injection of a large quantity of debris produced by fragmentation with a power-law size distribution unrelated to the Dohnanyi value. Fluctuations in the local slope index and dust area would thus be expected to occur on timescales of the mean time between large fragmentation events and last with relaxation times of order of the collisional lifetimes associated with the size range of interest. To determine the relaxation timescale for an event large enough to cause the steep slope index observed among the smallest asteroids, we created a population of asteroids with an equilibrium distribution fit through the small asteroids as determined from PLS data. Beginning at a diameter of -l100 m we imposed an increased slope index of approximately matching the distribution of small asteroids determined from the Gaspra crater counts and Spacewatch data. With this population as our initial distribution, the collisional model was run for 500 million years. The initial population and the evolved distribution at 10 and 100 million years 2rp chnivxn in Fanltre* 77 Rv 100 n- millhinn i7rr the, nnnilattnn hal ueia nenrlv rntr-heri q = 2, 52 decays back to the equilibrium value exponentially, with a relaxation timescale of about 65 million years, although at earliest times the decay rate is somewhat more rapid. Such an event could be produced by the fragmentation of a 100-200 km diameter asteroid. Smaller scale fragmentation or cratering events would produce smaller perturbations to the size distribution and would decay more rapidly. For example, we see in Figure 29 the variation in the slope index during a typical period of 500 million years in a model of the inner third of the asteroid belt. The spikes are due to the fragmentation of asteroids of the diameters indicated. Associated with the increases in slope are increases in the local number density of small (1-100 meter-scale) asteroids. The fragmentation of the 89 km diameter asteroid indicated in Figure 29 increased the number density of 10 m asteroids in the inner third of the belt by a factor of just over Since the number density of fragments must increase as the volume of the parent asteroid, the fragmentation of a 200 km diameter asteroid would cause an increase in the number of 10 m asteroids in the inner belt of over a factor of 10. This is just the increase over an equilibrium population of small asteroids that Rabinowitz (1993) finds among the Earth approaching asteroids discovered by Spacewatch and could easily be accounted for by the formation of an asteroid family the size of the Flora clan. Dependence of the Equilibrium Slope on the Strength Scaling Law Dohnanyi (1969) result that the size distribution of asteroids in collisional equilibrium can be described by a power-law with a slope index of q = 1.833 was obtained analytically by assuming that all asteroids in the population have the same, - ~ determine the resulting effect on the size distribution. We have already demonstrated that our collisional model reproduces the Dohnanyi result for size-independent impact strengths Verification Collisional Model). However, strain-rate effects gravitational compression lead to size-dependent impact strengths, with both increasing and decreasing strengths with increasing target size, respectively (see discussion of strength scaling laws in the following section). With our collisional model we are able to explore a range of size-strength scaling laws and their effects on the resulting size distributions. In order to examine the effects of size-dependent impact strengths on the equi- librium slope of the asteroid size distribution we created a number of hypothetical size-strength scaling laws. As will be discussed in the following section, we assume (3-15) where S is the impact strength, D is the diameter of the target asteroid, and pg constant dependent upon material properties of the target. created with values of p Seven strength laws were ranging from -0.2 to 0.2 over the size range 10 km to 1 meter. The slope index output from our modified, smooth collisional model was monitored over the size range 1-100 m and the equilibrium slope at 4.5 billion years recorded. The results are plotted in Figure We find that the equilibrium slope of the size distribution is very nearly linearly dependent upon the slope of the strength scaling law. There seems to be an extremely weak second order dependence on /', however over Dohnanyi value of q is obtained. If the slope of the scaling law is negative, as is the case strain-rate dependent strengths such as the Housen Holsapple (1990) nominal case, the equilibrium slope has a higher value of q t 1.86. the other hand, is positive, an equilibrium slope less than the Dohnanyi value is obtained. These deviations from the nominal Dohnanyi value, although not great, are large enough that well constrained observations of the slope parameter over a particular size range should allow us to place constraints on the size dependence of the strength properties of asteroids in that size range. An interesting result related to the dependence of the equilibrium slope parameter on the strength scaling law is that populations of asteroid with different compositions and, therefore, different strength properties, can have significantly different equilibrium slopes. This could apply to the members of an individual family of a unique taxonomic or to sub-populations within the entire mainbelt, such as and C-types. Furthermore, we find the somewhat surprising result that the slope index is dependent only upon the form of the size-strength scaling law and not upon the size distribution impacting projectiles. is illustrated Figure where we show results of two models simulating the collisional evolution of an asteroid family. stochastic fragmentation model was modified to track the collisional history of a family of fragments resulting from the breakup of a single large asteroid (see Chapter 4). show the slope index of the family size distribution as a function of time for two families: family has the same arbitrary strength scaling law as the background population of projectiles (jz < 0 in this case), while the scaling law for family 2 has g' >0. significantly different than that of family or the background population, even though projectiles background which are solely responsible for fragmenting members of the family. Since the total dust area associated with a population of debris is sensitively dependent upon the slope of the size distribution, it could be possible to make use of IRAS observations of the solar system dust bands to constrain the strengths of particles much smaller in size than those that have been measured in the laboratory. If the small debris in the families responsible for the dust bands has reached collisional equilibrium, the observed slope of the size distribution connecting the large asteroids and the small particles required to produce the observed area could be used to constrain the average material properties of asteroidal dust. The Modified Scaling Law One of the most important factors determining the collisional lifetime of an asteroid is its impact strength (see Description of Collisional Model). The impact strengths of basalt and mortar targets ~10 cm in diameter have been measured in the laboratory, but unfortunately we have no direct measurements of the impact strengths of objects as large as asteroids. Hence, one usually assumes that the impact strengths of larger targets will scale in some manner from those measured in the laboratory (see Fujiwara et al. (1989) for a review of strength scaling laws). Recently, attempts have been made to determine the strength scaling laws from first principles either analytically (Housen and Holsapple 1990) or numerically through hydrocode studies (Ryan 1993). However, we have taken a different approach of using the numerical collisional model 56 constraints on the impact strengths of asteroidal bodies outside the size range usually explored in laboratory experiments. The observed size distribution of the mainbelt asteroids (see Figure 14) is very well determined and constitutes a powerful constraint on collisional models any viable model must be able to reproduce the observed size distribution. The results of the previous section demonstrate that details of the size-strength scaling relation can have definite observational consequences. Before examining the influence that the scaling laws have on the evolved size distributions, it would be helpful to review the scaling relations which have been used in various collisional models Figure Davis et al. (1985) law is equivalent to the size-independent strength model assumed by Dohnanyi (1969), but with a theoretical correction to allow for the gravitational self compression of large asteroids. In this model the effective impact strength is assumed to have two components: the first due to the material properties of the asteroid and the second due to depth-dependent compressive loading of the overburden. When averaged over the volume of the asteroid we have for the effective impact strength S=S0 irkGp2D2 (3-16) where is the material impact strength, p is the density, is the diameter. For asteroids with diameters much less about the compressive loading becomes insignificant compared to the material strength and yielding the size- independent strength of Dohnanyi. The Housen et al. (1991) law allows for a strain-rate dependence of the impact n+..anr^4tk ^*-afnnt4,l,,wiLiwr 1 nlfnn mior octcAfnrl c ix 7(snhrinr tiv-in t'ictrnetc melclured in thp lii,- S 0 So, 57 plausible physical explanation for a strain-rate strength dependence is also put forth. A size distribution of inherent cracks and flaws is present in naturally occurring rocks. When a body is impacted, a compressive wave propagates through the body and is reflected as a tensile wave upon reaching a free surface. The cracks begin to grow and coalesce when subjected to tension, and since the larger cracks are activated at lower stresses, they are the first to begin to grow as the stress pulse rises. However, since there are fewer larger flaws, they require a longer time to coalesce with each other. Thus, at low stress loading rates, material failure is dominated by the large cracks and failure occurs at low stress levels. Since collisions between large bodies are characterized by low stress loading rates, the fracture strength is correspondingly low. In this way a strain-rate dependent strength may manifest itself as a size-dependent impact strength, with larger bodies having lower strengths than smaller ones. Housen and Holsapple (1990) show that the impact strength is oc D' Vf0 35 where V is the impact speed. (3-17) Under their nominal rate-dependent model the constant which is dependent upon several material properties target, is equal -0.24 in the strength regime, where gravitational self compression is negligible. gravity regime, however, = 1.65, which we note is slightly dependence assumed Davis et al. (1985). magnitude of gravitational compression Housen et al. (1991) model was determined matching experimental results of the fragmentation compressed basalt targets the parent bodies of the Koronis, Eos, and Themis asteroid families (open dots). most recent studies, however, indicate that the laboratory results are to be taken as upper limits to the magnitude of the gravitational compression (Holsapple 1993, private communication). Both scaling laws have been used within the collisional model to attempt to place some constraints on the initial mass of the asteroid belt and the size-strength scaling relation itself. Unfortunately, the initial mass of the belt is not known. initial' we assume the same definition as used by Davis et al. (1985), that is, the mass at the time the mean collision speed first reached the current km s1 . Davis et al. (1989) present a review of asteroid collision studies and conclude that the asteroids represent a collisionally relaxed population whose initial mass cannot be found from models evolution alone. have therefore chosen to investigate extremes for an initial belt mass: a 'massive' initial population with ~-60 times the present belt mass, based upon work by Wetherill (1992, private communication) on the runaway accretion of planetesimals in the inner solar system, and a 'small' initial belt of roughly twice the present mass, matching the best estimate by Davis et al. 1989) of the initial mass most likely to preserve the basaltic crust of Vesta. (1985, Figures 33 and 34 show the results of several runs of the model with various combinations of scaling laws and initial populations. In both figures we have included the observed size distribution for comparison with model results, but have removed the error band for clarity. have found that models utilizing the strength scaling laws usually considered, particularly strain-rate laws, to reproduce features 59 the initial asteroid population: it is the form of the size-strength scaling law which most determines the resulting shape of the size distribution. A pure strain-rate extrapolation produces very weak 1-10 km-scale asteroids, leading to a pronounced "dip" number of asteroids in the region of the transition to an equilibrium power law. Davis et al. model does a somewhat better job of fitting the observed distribution in the transition region, further suggesting that a very pronounced weakening of small asteroids may not be realistic in this size regime. In addition, we have found that the magnitude of the gravitational strengthening given by the Davis et al. model (somewhat weaker than the Housen et al. model) produces a closer match to the shape of the "hump 00 km for the initial populations we have examined. Housen et al. If something nearer to the gravity scaling turns out to be more appropriate, however, this would simply indicate that the size distribution longward of -~150 km is mostly primordial. Since it is the shape of the size-strength scaling relation which seems to have greatest influence on the shape of the evolved size distribution, we have taken the approach of permitting the scaling law itself to be adjusted, allowing us to use the observed size distribution to help constrain asteroidal impact strengths. We have been able to match the observed size-frequency distribution, but only with an ad hoc modification to the strength scaling law. We have included in Figure 32 our empirically modified scaling law, which is inspired by the work of Greenberg et al. (1992, 1993) on the collisional history of Gaspra. The modified law matches the Housen et al. law for small (laboratory) size bodies where impact experiments (Davis and Ryan 1990) indicate * I.. ,-~ ~-. &.. ,- a ~-. n n 1 Z 1.. a nt. A a n n Z I.. a .. C.. .- ~ a a C A. .e 4. n nfl IT/ n ^ ^ 1.^ .- ^- model. For small asteroids an empirical modification has been made to allow for the interpretation of some concave facets on Gaspra as impact structures (Greenberg et al. 1993). If Gaspra and other similar-size objects such as Phobos (Asphaug and Melosh 1993) and Proteus (Croft 1992) can survive impacts which leave such proportionately large impact scars, they must be collisionally stronger than extrapolations of strain-rate scaling laws from laboratory-scale targets would predict. The modified law thus allows for this strengthening and in fact gives a collisional lifetime for a Gaspra-size body of about 1 billion years, matching the Greenberg et al. the 500 million year lifetime adopted by others. Us best estimate, which is longer than ing this modified scaling law in our collisional model we are able to match in detail the observed asteroid size distribution (Figure 35). After 4.5 billion years of collisional evolution we fit the "hump" at 100 the smooth transition to an equilibrium distribution at ~30 kmin, and the number of asteroids in the equilibrium distribution and its slope index. We note in particular that for the range of sizes covered by PLS data (5-30 km) the slightly positive slope of the modified scaling law predicts an equilibrium slope for that size range of about 1.78, less than the Dohnanyi value but precisely matching the value of +0.02 determined by a weighted least-squares fit to the catalogued mainbelt and PLS data. While we have no quantitative theory to account for our modified scaling law, there may be a mechanism which could explain the slow strengthening of km-scale bodies in a qualitative manner. Recent hydrocode simulations by Nolan et al. (1992) indicate that an impact into a small asteroid effectively shatters the material of the asteroid in an advancing shock front which precedes the excavated debris, so that crater the asteroid is thus reduced to rubble. Davis and Ryan (1990) have noted that clay and weak mortar targets, materials with fairly low compressive strengths such as the shattered material predicted by the hydrocode models, have impact strengths due to the poor conduction of tensile stress waves in the "lossy" material. If this mechanism indeed becomes important for objects much larger than laboratory targets but significantly smaller than those for which gravitational compression becomes important, a more gradual transition from strain-rate scaling to gravitational compression would be warranted. 62 '-o
Q- _ 0 - C)J 4-- en- "0-en- * S ^ -o driderri
I- 50 C.) V3 C "0 2 C.) "C tn N 00 N t *O~~~e en 0U ci O '^i~ ON 0 en O hN ^mc r C.). 3 ' 0 0- 0 C C *^ U 'CC ON In nOi In o r I o^ In ON n '- N N *^ 666666666 ** o 6) o i >- 1 ci *S 'S 5C '.0 "00 e ^ n ON "0- " c NS N- e^n -^ '. 0o oom . enJ en ...* .00 -7 C g i2 "0~ Ca Ca c c U/ 1 1-. ~ ^ INGC3 ^'' r -2 00 re 00 o u ?j . 0- JIQ]UGLGajoqI ~o1 joqu.ITW jaoqmnn 1Quui9ajouDI ~o1 0O- OrH jaqumnM JI 4uGuoajouJ ~oI 0Or joqtmnN tP1u9ma9juI ~oT Gdois tuniaqijinba odois ummn q.Itnba odoi mntaqIiinb 0 C12 flqUmfl IPU9Um9JoUI ~o1 0 Cqn .19qumfl 10 0 IPmu9mUomUI ~o1 OA ui) jaqmnfN PIVGUASUJOuI ~o1 g- tO 0 oqmnj i Tmu moaoI PGoJ 0 Cq2 j9qujflw tO 0 IPI4UGLU.JoUI ~o1 jaqUinNf 0 0O JP1~9UJJDU Co OT C1 [(se8 urj GdoIS Jo8otuI J ( mmjqvpnba 79 1 I | I 1..* I I I I II I --i. .- .. .n .'* c-^ ^ _ .^w ,.n.^aF ~ ~ ~ .l ." ""J. :- *.. f 4 a --i^ *0"^^C 8 S* y 4g .e. m. 2as E? 1 am -- E-. >1 *^ J ^. ..0...J -r w w e 9 9 ** .ge ** ** * p p A esad _~ **p1 ...t /! S^t~i S _5 a I *- ** --*- .." .:ii ^ ^ I B ... . sm- r maghthe *e *me-- f^ - _^^rtaW ^^^gatat fl *11 Cd 'a"'t -. I-P M"t aga-JJHJl I^ I- I 0 . ***fH >< ^\'\0. ...1:cO c "vtr-<& ^ ^ G | ^ ^ ?! r 1 E-1 0 A : a q-4 -. 4) :'-4! '-4-'" b.." wnuqT!Tnb .ill" 0 ~ C) V r mUO SJ9) Mi4u9j4s ;suddmI -^ C )-3 ^3 .2- * % n C o o 0 0 0 -- <-- umg jaiusiQ sad jsqmuN -l" 0f C- '0 4. 4 o 0o 0 0 0 0 - O- - u~g sa~awerg sad saqurnN - - >0. So0 0 0 0 tug: JawuIPlG sad sqtunN "0 0 0 0 0 -4 -, ,- -" r- uTr jaiaurtiQ sad saqunNj '00 fl N o 0 0 0 0 0 -1 T- r4- T- T- utg aw~aw~tg i0 4 I, 0 -_ 0 0 0 0 0 --< l l T^" T-t -. urn aa~am~tQ sad saqutnN saqwnmn CHAPTER HIRAYAMA ASTEROID FAMILIES A Brief History of Asteroid Families The Hirayama asteroid families represent natural experiments in asteroid collisional processes. The size-frequency distributions of the individual families may be used to determine the mode of fragmentation of individual large asteroids and debris associated with the families may also be exploited to calibrate the amount of dust to associate with the fragmentation of asteroids in the mainbelt background population. The clustering of asteroid proper elements, clearly visible in Figure noticed by Hirayama (1918), parent asteroid. was first which he attributed to the collisional fragmentation of a Hirayama identified by eye the three most prominent families, Koronis, Eos, and Themis (which he named after the first discovered asteroid in each group), in this first study and added other, though perhaps less certain families, in a series of later papers (1919, 1923, 1928). After Hirayama's first studies, classifications of asteroids into families have been given by many other researchers (Brouwer 1951; Arnold 1969; Lindblad and Southworth 1971; Williams 1979, 1992; Zappala et al. 1990; Bendjoya et al. 1991), and a number of other families have become apparent. Some researchers claim to be able to identify more than a hundred groupings, while others feel that only the few largest families discovered asteroids, later investigators are able to identify smaller, populated families which were previously unseen), the different perturbation theories which are used to calculate the proper elements, and the different methods used to distinguish the family groupings from "background" asteroids mainbelt, which have ranged from eyeball searches to more objective cluster analysis techniques. This lack of unanimous agreement on the number of asteroid families or on which asteroids should be included in families, prompted some (Gradie et al. 1979; Carusi and Valsecchi 1982) to urge that a further understanding of the discrepancies between the different classification schemes was necessary before the physical reality of any of the families could be given plausible merit. Only in the last few years have different methods lead to a convergence in the families identified by different researchers (Zappala and Cellino 1992). The Zappala Classification To date, probably the most reliable and complete classification of Hirayama family members is the recent work of Zappal& et al. (1990). They used a set of 4100 numbered asteroids whose proper elements were calculated using a second-order (in planetary masses), fourth-degree eccentricities inclinations) secular perturbation theory (Milani and Kne2evid 1990) and checked for long-term stability by numerical integration. A hierarchical clustering technique was applied to the mainbelt asteroids to create a dendrogram of the proper elements and combined with a distance parameter related to the velocity needed for orbital change after removal from the parent 86 A significance parameter was then assigned to each family to measure its departure from a random clustering. revised proper elements become available for more numbered asteroids the clustering algorithm is easily rerun to update the classification of members in established families and to search for new, small families. et al. In their latest classification ZappalA (1993, private communication) find 26 families, of which about 20 are to considered significant and robust. In Figure 36 we have plotted the proper inclination versus semimajor axis for all 26 Zappalh families and have labeled some of the more prominent ones. Koronis, and Themis families remain the most reliable, however Zappala also considers many of the smaller, compact families such as Dora, Gefion, and Adeona quite reliable. The less secure families are usually the most sparsely populated or those which might possibly belong to one larger group and remain to be confirmed as more certain proper elements become available. The Flora family, instance, although quite populous, is considered a "dangerous" family, having proper elements which are still quite uncertain due to its proximity to the v6 secular resonance. The high density of asteroids in this region, which is likely a selection effect favoring the discover of small, faint asteroids in the inner belt, also makes the identification of individual families difficult the entire region merges into one large "clan", making it difficult to determine which of the asteroids there are genetically related. Collisional Evolution of Families Number of Families 87 initial population coupled with relatively weak asteroids would imply that nearly all the families identifiable today must be relatively young. A smaller initial belt and asteroids with large impact strengths would allow even modest-size families to survive for billions of years. To attempt to distinguish between these two possibilities and to examine the collisional history of families we modified our stochastic collisional model to allow us to follow the evolution of a family of fragments resulting from the breakup of a single large asteroid, simulating the formation of an asteroid family. At a specified time an asteroid of a specified size is fragmented and the debris distributed into the model' size bins in a power-law distribution as described in Chapter As the model proceeds, a copy of the fragmentation and debris redistribution routine is spawned off in parallel to follow the evolution of the family fragments. The projectile population responsible for the fragmentation of the family asteroids is found in a self- consistent manner from the evolving background population. Collisions between family members are neglected for the following reason. We have calculated that the intrinsic collision probability between family members may be as much as four times greater than that between family and background asteroids. For example, the intrinsic collision probability between 158 Koronis and mainbelt background asteroids is 3.687 x 10-18 km-2 13.695 , while the probability of x 10-18 yr-1 km-2 collisions with other Koronis family members is . Due to their similar inclinations and eccentricities, however, the mean encounter speed between family members is lower than with asteroids of the background population, requiring larger projectiles for fragmentation. The mean II T r .^1 rr 1 . Koronis family members and asteroids of the background projectile population. very large total number of projectiles in the background population completely swamps the small number of asteroids within the family itself, so that the collisional evolution of a family is still dominated by collisions with the background asteroid population. To determine how many of the families produced by the model should be observ- able at the present time we have defined a simple family visibility criterion which mimics the clustering algorithm actually used to find families against the background asteroids of the mainbelt (Zappala et al. 1990). We have found the volume density of non-family asteroids in orbital element space for the middle region of the belt (corresponding to zone 4 of Zappala et al. 1990). In the region 2.501 2.825, and 0.0 0.3 we found 1799 non-family asteroids which yields a mean vol- ume density typical of the mainbelt of 1799/(0.324AU x 0.3 x 0.3) = 1799/0.02916 = 61694.102 asteroids per unit volume of proper element space. the asteroids in a family is then found by using Gauss' pertu The volume density of rbation equations to cal- culate the spread in orbital elements associated with the formation of the family (see, e.g., Zappal& et al. 1984). The typical AV associated with the ejection speed of the fragments will be of the order of the escape speed of the parent asteroid, which scales as the diameter, D . The typical volume of a family must then scale as so that families formed from the destruction of large asteroids are spread over a larger volume. We computed the volume associated with the formation of a family from a parent 110 km in diameter (the size of the smallest parent asteroids we consider) to be 2.26 element units. The AV for a parent of this size is approximately 135m . Within x 10-5 |

Full Text |

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f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f:DYHf DQG WKH 6L]H 'LVWULEXWLRQ IURP WR 0HWHUV 'HSHQGHQFH RI WKH (TXLOLEULXP 6ORSH RQ WKH 6WUHQJWK 6FDOLQJ /DZ 7KH 0RGLILHG 6FDOLQJ /DZ +,5$<$0$ $67(52,' )$0,/,(6 $ %ULHI +LVWRU\ RI $VWHURLG )DPLOLHV 7KH =DSSDOÂ£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f 1XPEHUV RI DVWHURLGV LQ WKUHH 3/6 ]RQHV FDWDORJXHG3/6 GDWDf $GMXVWHG FRPSOHWHQHVV OLPLWV IRU 3/6 ]RQHV ,QWULQVLF FROOLVLRQ SUREDELOLWLHV DQG HQFRXQWHU VSHHGV IRU VHYHUDO PDLQEHOW DVWHURLGV 9OO PAGE 9 /,67 2) ),*85(6 3URSHU LQFOLQDWLRQ YHUVXV VHPLPDMRU D[LV IRU DOO FDWDORJXHG PDLQEHOW DVWHURLGV 0DJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU FDWDORJXHG PDLQEHOW DVWHURLGV $EVROXWH PDJQLWXGH DV D IXQFWLRQ RI GLVFRYHU\ GDWH IRU DOO FDWDORJXHG PDLQEHOW DVWHURLGV 0DJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU 3/6 ]RQH 3/6 DQG FDWDORJXHG DVWHURLG GDWD 0DJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU 3/6 ]RQH ,, 3/6 DQG FDWDORJXHG DVWHURLG GDWD 0DJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU 3/6 ]RQH ,,, 3/6 DQG FDWDORJXHG DVWHURLG GDWD $GRSWHG PDJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU 3/6 ]RQH $GRSWHG PDJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU 3/6 ]RQH ,, $GRSWHG PDJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU 3/6 ]RQH ,,, 0DJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU WKH DVWHURLGV LQ 7DEOHV DQG RI 9DQ +RXWHQ HW DO f /HDVWVTXDUHV ILW WR WKH PDJQLWXGHIUHTXHQF\ GDWD IRU 3/6 ]RQH /HDVWVTXDUHV ILW WR WKH PDJQLWXGHIUHTXHQF\ GDWD IRU 3/6 ]RQH ,, /HDVWVTXDUHV ILW WR WKH PDJQLWXGHIUHTXHQF\ GDWD IRU 3/6 ]RQH ,,, 7KH REVHUYHG PDLQEHOW VL]H GLVWULEXWLRQ 9HULILFDWLRQ RI PRGHO IRU VWHHS LQLWLDO VORSH DQG VPDOO ELQ VL]H 9OOO PAGE 10 9HULILFDWLRQ RI PRGHO IRU VKDOORZ LQLWLDO VORSH DQG VPDOO ELQ VL]H 9HULILFDWLRQ RI PRGHO IRU VWHHS LQLWLDO VORSH DQG ODUJH ELQ VL]H 9HULILFDWLRQ RI PRGHO IRU VKDOORZ LQLWLDO VORSH DQG ODUJH ELQ VL]H (TXLOLEULXP VORSH DV D IXQFWLRQ RI WLPH IRU YDULRXV IUDJPHQWDWLRQ SRZHU ODZV DQG IRU VWHHS LQLWLDO VORSH (TXLOLEULXP VORSH DV D IXQFWLRQ RI WLPH IRU YDULRXV IUDJPHQWDWLRQ SRZHU ODZV DQG IRU VKDOORZ LQLWLDO VORSH (TXLOLEULXP VORSH DV D IXQFWLRQ RI WLPH IRU YDULRXV IUDJPHQWDWLRQ SRZHU ODZV DQG IRU HTXLOLEULXP LQLWLDO VORSH :DYHOLNH GHYLDWLRQV LQ VL]H GLVWULEXWLRQ FDXVHG E\ WUXQFDWLRQ RI SDUWLFOH SRSXODWLRQ ,QGHSHQGHQFH RI WKH ZDYH RQ ELQ VL]H DGRSWHG LQ PRGHO &RPSDULVRQ RI WKH LQWHUSODQHWDU\ GXVW IOX[ IRXQG E\ *ULLQ HW DO f DQG VPDOO SDUWLFOH FXWRIIV XVHG LQ RXU PRGHO :DYHOLNH GHYLDWLRQV LPSRVHG E\ D VKDUS SDUWLFOH FXWRII [ f 6L]H GLVWULEXWLRQ UHVXOWLQJ IURP JUDGXDO SDUWLFOH FXWRII PDWFKLQJ WKH REVHUYHG LQWHUSODQHWDU\ GXVW IOX[ [ fÂ§ f &ROOLVLRQDO UHOD[DWLRQ RI D SHUWXUEDWLRQ WR DQ HTXLOLEULXP VL]H GLVWULEXWLRQ +DOIWLPH IRU H[SRQHQWLDO GHFD\ WRZDUG HTXLOLEULXP VORSH IROORZLQJ WKH IUDJPHQWDWLRQ RI D NP GLDPHWHU DVWHURLG 6WRFKDVWLF IUDJPHQWDWLRQ RI LQQHU PDLQEHOW DVWHURLGV RI YDULRXV VL]HV GXULQJ D W\SLFDO PLOOLRQ SHULRG (TXLOLEULXP VORSH SDUDPHWHU DV D IXQFWLRQ RI WKH VORSH RI WKH VL]HVWUHQJWK VFDOLQJ ODZ 'LIIHUHQFH LQ WKH HTXLOLEULXP VORSH SDUDPHWHUV IRU IDPLOLHV ZLWK GLIIHUHQW VWUHQJWK SURSHUWLHV PAGE 11 7KH 'DYLV HW DO f +RXVHQ HW DO f DQG PRGLILHG VFDOLQJ ODZV XVHG LQ WKH FROOLVLRQDO PRGHO 7KH HYROYHG VL]H GLVWULEXWLRQ DIWHU ELOOLRQ \HDUV XVLQJ WKH +RXVHQ HW DO f VFDOLQJ ODZ IRU Df D PDVVLYH LQLWLDO SRSXODWLRQ DQG Ef D VPDOO LQLWLDO SRSXODWLRQ 7KH HYROYHG VL]H GLVWULEXWLRQ DIWHU ELOOLRQ \HDUV XVLQJ WKH 'DYLV HW DO f VFDOLQJ ODZ IRU Df D PDVVLYH LQLWLDO SRSXODWLRQ DQG Ef D VPDOO LQLWLDO SRSXODWLRQ 7KH HYROYHG VL]H GLVWULEXWLRQ DIWHU ELOOLRQ \HDUV XVLQJ RXU PRGLILHG VFDOLQJ ODZ IRU Df D PDVVLYH LQLWLDO SRSXODWLRQ DQG Ef D VPDOO LQLWLDO SRSXODWLRQ 7KH +LUD\DPD DVWHURLG IDPLOLHV DV GHILQHG E\ =DSSDOÂ£ HW DO f 7KH FROOLVLRQDO GHFD\ RI IDPLOLHV UHVXOWLQJ IURP YDULRXVVL]H SDUHQW DVWHURLGV DV D IXQFWLRQ RI WLPH )RUPDWLRQ RI IDPLOLHV LQ WKH PDLQEHOW DV D IXQFWLRQ RI WLPH 0RGHOHG FROOLVLRQDO KLVWRU\ RI WKH *HILRQ IDPLO\ 0RGHOHG FROOLVLRQDO KLVWRU\ RI WKH 0DULD IDPLO\ 7KH VRODU V\VWHP GXVW EDQGV DW DQG ]P DIWHU VXEWUDFWLRQ RI WKH VPRRWK ]RGLDFDO EDFNJURXQG YLD D )RXULHU ILOWHU Df ,5$6 REVHUYDWLRQV RI WKH GXVW EDQGV DW HORQJDWLRQ DQJOHV RI r r DQG r &RPSDULVRQV ZLWK PRGHO SURILOHV EDVHG RQ SURPLQHQW +LUD\DPD IDPLOLHV DUH VKRZQ LQ Ef Ff DQG Gf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f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s WLPHV DV PXFK GXVW DV WKH SURPLQHQW IDPLOLHV DORQH 7KLV UHVXOW LV FRPSDUHG ZLWK WKH UDWLR RI DUHDV QHHGHG WR DFFRXQW IRU WKH ]RGLDFDO EDFNJURXQG DQG WKH ,5$6 GXVW EDQGV DV GHWHUPLQHG E\ DQDO\VLV RI ,5$6 GDWD :H FRQFOXGH WKDW WKH HQWLUH DVWHURLG SRSXODWLRQ LV UHVSRQVLEOH IRU DW OHDVW ab RI WKH GXVW LQ WKH HQWLUH ]RGLDFDO FORXG f f ;8 PAGE 14 &+$37(5 ,1752'8&7,21 7UDGLWLRQDOO\ WKH GHEULV RI VKRUW SHULRG FRPHWV KDV EHHQ WKRXJKW WR EH WKH VRXUFH RI WKH PDMRULW\ RI WKH GXVW LQ WKH LQWHUSODQHWDU\ HQYLURQPHQW :KLSSOH 'RKQDQ\L f +RZHYHU LW KDV EHHQ NQRZQ IRU VRPH WLPH WKDW LQWHUDVWHURLG FROOLVLRQV DUH OLNHO\ WR RFFXU RYHU JHRORJLF WLPH 3LRWURZVNL f 7KH JUDGXDO FRPPLQXWLRQ RI DVWHURLGDO GHEULV PXVW VXSSO\ DW OHDVW VRPH RI WKH GXVW LQ WKH ]RGLDFDO FORXG WKRXJK EHFDXVH RI WKH ODFN RI REVHUYDWLRQDO FRQVWUDLQWV WKH FRQWULEXWLRQ PDGH E\ PXWXDO DVWHURLGDO FROOLVLRQV KDV EHHQ GLIILFXOW WR GHWHUPLQH 6LQFH WKH GLVFRYHU\ RI WKH ,5$6 VRODU V\VWHP GXVW EDQGV /RZ HW DO f WKH FRQWULEXWLRQ PDGH E\ DVWHURLGV WR WKH LQWHUSODQHWDU\ GXVW FRPSOH[ KDV UHFHLYHG UHQHZHG DWWHQWLRQ 7KH VXJJHVWLRQ WKDW WKH GXVW EDQGV RULJLQDWH IURP WKH PDMRU DVWHURLG IDPLOLHV ZLGHO\ WKRXJKW WR EH WKH UHVXOWV RI PXWXDO DVWHURLG FROOLVLRQV ZDV PDGH E\ 'HUPRWW HW DO f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f FRQVLVWHQW ZLWK DQ DVWHURLGDO RULJLQ RI WKH SDUWLFOHV )URP FRPSXWHU VLPXODWLRQV RI WKH HQWU\ KHDWLQJ RI ODUJH PLFURPHWHRULWHV DQG FRPSDULVRQ RI WKH FROOLVLRQDO GHVWUXFWLRQ DQG PAGE 15 Â‘! WUDQVSRUW OLIHWLPHV RI DVWHURLGDO GXVW )O\QQ f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a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abf DUH IRXQG LQ WKH VHPLPDMRU D[LV UDQJH D $8 )LJXUH f )RU UHDVRQV GHVFULEHG EHORZ ZH ZLOO OLPLW RXU GLVFXVVLRQ WR WKRVH DVWHURLGV LQ WKH UDQJH D $8 GHILQLQJ ZKDW ZH ZLOO UHIHU WR DV WKH fPDLQEHOWf 2XU FRQFOXVLRQV DUH H[SHFWHG WR EH XQDIIHFWHG E\ WKLV FKRLFH DV RQO\ DVWHURLGV RU OHVV WKDQ b RI WKH NQRZQ SRSXODWLRQ DUH H[FOXGHG VR WKDW WKH WZR VHWV RI DVWHURLGV DUH HVVHQWLDOO\ WKH VDPH )LJXUH LV D SORW RI WKH QXPEHU RI FDWDORJXHG PDLQEHOW DVWHURLGV SHU KDOIPDJQLWXGH ELQ ZKHUH WKH DEVROXWH PDJQLWXGH + LV GHILQHG DV WKH 9EDQG PDJQLWXGH RI WKH DVWHURLG DW D GLVWDQFH RI $8 IURP WKH (DUWK $8 IURP WKH 6XQ DW D SKDVH DQJOH RI PAGE 18 %RZHOO HW DO f ,PPHGLDWHO\ HYLGHQW LV D fKXPSf RU H[FHVV RI DVWHURLGV DW + m $OWKRXJK SUHYLRXV UHVHDUFKHUV KDYH LQWHUSUHWHG WKLV H[FHVV DV D UHPQDQW RI VRPH SULPRUGLDO JDXVVLDQ SRSXODWLRQ RI DVWHURLGV DOWHUHG E\ VXEVHTXHQW FROOLVLRQDO HYROXWLRQ +DUWPDQQ DQG +DUWPDQQ f WKH FXUUHQW LQWHUSUHWDWLRQ LV WKDW LW UHSUHVHQWV WKH SUHIHUHQWLDO SUHVHUYDWLRQ RI ODUJHU DVWHURLGV HIIHFWLYHO\ VWUHQJWKHQHG E\ JUDYLWDWLRQDO FRPSUHVVLRQ 'DYLV HW DO +ROVDSSOH DQG +RXVHQ f 2WKHU UHVHDUFKHUV SULPDULO\ 'RKQDQ\L f KDYH QRWHG IURP VXUYH\V RI IDLQW DVWHURLGV GLVFXVVHG EHORZf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b )LJXUH LV DOVR LQWHUHVWLQJ IRU WKH KLVWRU\ UHFRUGHG LQ DVWHURLG GLVFRYHU\ FLUFXPVWDQFHV 4XLWH DSSDUHQW LV WKH PDUNHG ODFN RI GLVFRYHULHV LQ WKH ZDNH RI :RUOG :DU ,, 7KH ODUJH QXPEHU RI DVWHURLGV GLVFRYHUHG GXULQJ WKH 3DORPDU/HLGHQ 6XUYH\ DSSHDUV DV D YHUWLFDO VWULSH QHDU f $V SRLQWHG RXW DERYH EHWZHHQ + DQG + WKH PDLQEHOW DSSHDUV WR PDNH D WUDQVLWLRQ WR D OLQHDU SRZHUODZ VL]H GLVWULEXWLRQ $Q DEVROXWH PDJQLWXGH RI + FRUUHVSRQGV WR D GLDPHWHU RI DERXW NP IRU DQ DOEHGR RI DSSUR[LPDWHO\ WKH PHDQ DOEHGR RI WKH ODUJHU DVWHURLGV LQ WKH PDLQEHOW SRSXODWLRQ VHH 7KH 2EVHUYHG 0DLQEHOW 6L]H 'LVWULEXWLRQf 8QIRUWXQDWHO\ LQFRPSOHWHQHVV UDSLGO\ VHWV LQ IRU + DQG ZLWK VR IHZ GDWD SRLQWV WKH VORSH RI WKH GLVWULEXWLRQ FDQQRW EH ZHOO GHILQHG VR WKDW ZH FDQQRW UHOLDEO\ XVH WKH GDWD IURP WKH FDWDORJXHG SRSXODWLRQ DORQH WR HVWLPDWH WKH QXPEHU RI YHU\ VPDOO DVWHURLGV LQ WKH PDLQEHOW VHH )LJXUH f :H KDYH WKHUHIRUH XVHG GDWD IURP WKH 3DORPDU/HLGHQ 6XUYH\ 9DQ +RXWHQ HW DO f WR H[WHQG WKH REVHUYHG GLVWULEXWLRQ GRZQ WR DERXW + FRUUHVSRQGLQJ WR D GLDPHWHU RI URXJKO\ NP 7KH 0'6 DQG 3/6 6XUYH\V 7KH 3DORPDU/HLGHQ 6XUYH\ 9DQ )ORXWHQ HW DO KHUHDIWHU UHIHUUHG WR DV 3/6f ZDV FRQGXFWHG LQ WR H[WHQG WR IDLQWHU PDJQLWXGHV WKH UHVXOWV RI WKH HDUOLHU 0F'RQDOG 6XUYH\ RI WKURXJK .XLSHU HW DO KHUHDIWHU UHIHUUHG WR DV 0'6f 7KH 0'6 VXUYH\HG WKH HQWLUH HFOLSWLF QHDUO\ WZLFH DURXQG WR D ZLGWK RI r GRZQ WR D OLPLWLQJ SKRWRJUDSKLF PDJQLWXGH RI QHDUO\ ,Q FRQWUDVW WKH SUDFWLFDO SODWH OLPLW IRU WKH 3/6 VXUYH\ ZDV DERXW ILYH PDJQLWXGHV IDLQWHU 7R VXUYH\ DQG GHWHFW DVWHURLGV WKLV IDLQW RYHU WKH VDPH ODUJH DUHD FRYHUHG E\ WKH 0'6 ZRXOG KDYH EHHQ PAGE 20 SURKLELWLYH VR ZLWK WKH 3/6 LW ZDV GHFLGHG WKDW RQO\ D VPDOO SDWFK RI WKH HFOLSWLF ZRXOG EH VXUYH\HG DQG WKH UHVXOWV VFDOHG WR WKH 0'6 DQG WKH HQWLUH HFOLSWLF EHOW ,Q D UHYLVLRQ DQG VPDOO H[WHQVLRQ ZHUH PDGH WR WKH 3/6 9DQ +RXWHQ HW DO f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f 7KH J DEVROXWH PDJQLWXGHV JLYHQ E\ 9DQ +RXWHQ HW DO DUH LQ WKH VWDQGDUG % EDQG ZH WUDQVIRUPHG WKHVH DEVROXWH PDJQLWXGHV WR WKH + V\VWHP E\ DSSO\LQJ WKH FRUUHFWLRQ + FM fÂ§ %RZHOO HW DO f 7KH ELDVFRUUHFWHG QXPEHU RI DVWHURLGV SHU KDOIPDJQLWXGH ELQ LQ HDFK RI WKH ]RQHV LV D FRPELQDWLRQ RI WKH UHVXOWV RI WKH 0'6 DQG PAGE 21 WKH 3/6 DV GHVFULEHG E\ 9DQ +RXWHQ HW DO 7KH 0'6 YDOXHV IRU WKH QXPEHU RI DVWHURLGV SHU KDOIPDJQLWXGH ELQ DUH DVVXPHG XQWLO WKH FRUUHFWLRQV IRU LQFRPSOHWHQHVV DSSURDFK DERXW b RI WKH YDOXHV WKHPVHOYHV :KHUH WKH 0'6 YDOXHV UHTXLUH FRUUHFWLRQ IRU LQFRPSOHWHQHVV D PD[LPXP DQG PLQLPXP QXPEHU RI DVWHURLGV LV FDOFXODWHG EDVHG XSRQ WZR GLIIHUHQW H[WUDSRODWLRQV RI WKH ORJ 1Pf UHODWLRQ .XLSHU HW DO f ,Q WKHVH FDVHV WKH PHDQ RI WKH WZR YDOXHV JLYHQ LQ WKH 0'6 KDV EHHQ DVVXPHG 7KH FRUUHFWLRQ IDFWRUV IRU LQFRPSOHWHQHVV LQ ]RQH ,,, JLYHQ LQ WKH 0'6 KRZHYHU DUH LQFRUUHFW 7KH FRUUHFWHG YDOXHV DUH JLYHQ LQ 7DEOH ', RI 'RKQDQ\L f )RU IDLQWHU YDOXHV RI + WKH QXPEHU RI DVWHURLGV LV WDNHQ IURP 7DEOH RI 9DQ +RXWHQ HW DO WKH YDOXHV JLYHQ WKHUH FRUUHFWHG E\ PXOWLSO\LQJ ORJ 1+f E\ WR H[WHQG WKH FRXQWV WR FRYHU WKH DVWHURLG EHOW RYHU DOO ORQJLWXGHV WR PDWFK WKH FRYHUDJH RI WKH 0'6 7DEOH JLYHV WKH DGRSWHG ELDVFRUUHFWHG QXPEHU RI DVWHURLGV SHU KDOIPDJQLWXGH ELQ + PDJQLWXGHVf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Â£N DQG 'RKQDQ\L f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f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f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Â“ HUURU DVVRFLDWHG ZLWK HDFK LQGHSHQGHQW SRLQW LQ DQ LQFUHPHQWDO PDJQLWXGHIUHTXHQF\ GLDJUDP 7KH HUURUV IRU WKH FDWDORJXHG DVWHURLG FRXQWV DUH GHWHUPLQHG GLUHFWO\ IURP WKH UDZ QXPEHUV DIWHU WKH DVWHURLGV KDYH EHHQ ELQQHG DQG FRXQWHG )RU WKH 3/6 GDWD WKH ?7Â“ HUURUV PXVW EH GHWHUPLQHG IURP WKH QXPEHU RI DVWHURLGV SHU PDJQLWXGH LQWHUYDO EHIRUH WKH FRXQWV KDYH EHHQ FRUUHFWHG IRU WKH DSSDUHQW PDJQLWXGH DQG LQFOLQDWLRQ FXWRIIV 7KH FRUUHFWHG FRXQWV WKHPVHOYHV DUH JLYHQ LQ 7DEOH RI 9DQ +RXWHQ HW DO 7KHVH FRXQWV DUH WKHQ VFDOHG WR PDWFK WKH FRYHUDJH RI WKH 0'6 DV GHVFULEHG DERYH 6LQFH WKH HUURUV LQ WKH 3/6 FRXQWV DUH EDVHG RQ WKH XQFRUUHFWHG XQVHDOHG FRXQWV WKH 3/6 GDWD SRLQWV KDYH D ODUJHU DVVRFLDWHG ?9 HUURU WKDQ WKH FRUUHFWHG FRXQWV WKHPVHOYHV ZRXOG LQGLFDWH 7KH UHVXOWLQJ PDJQLWXGH IUHTXHQF\ GLDJUDPV IRU HDFK RI WKH 3/6 ]RQHV DUH VKRZQ LQ )LJXUHV DQG DQG WKH QXPEHUV WDEXODWHG LQ 7DEOH 7KH 3/6 GDWD JUHDWO\ H[WHQG WKH ZRUNDEOH REVHUYHG PDJQLWXGHIUHTXHQF\ GLVWULn EXWLRQV IRU WKH PDLQEHOW DVWHURLGV :H LPPHGLDWHO\ VHH WKDW WKH LQQHU WZR ]RQHV RI WKH PDLQEHOW GLVSOD\ D ZHOO GHILQHG OLQHDU SRZHUODZ GLVWULEXWLRQ IRU WKH IDLQWHU DVWHURLGV ZLWK WKH SURPLQHQW H[FHVV RI DVWHURLGV DW WKH EULJKWHU HQG RI WKH GLVWULEXWLRQ 7KH GLVWULn EXWLRQ LQ WKH RXWHU WKLUG RI WKH EHOW DSSHDUV VRPHZKDW OHVV ZHOO GHILQHG 7KH UHVXOWV IRU WKH LQQHU ]RQHV DUH YHU\ LQWHUHVWLQJ DV WKH OLQHDU SRUWLRQV TXDOLWDWLYHO\ PDWFK YHU\ ZHOO 'RKQDQ\LnV f SUHGLFWLRQ RI DQ HTXLOLEULXP SRZHUODZ GLVWULEXWLRQ RI IUDJn PHQWV H[SHFWHG LQ D FROOLVLRQDOO\ UHOD[HG SRSXODWLRQ 'RKQDQ\L XVLQJ D OHDVWVTXDUHV ILW PAGE 24 WKURXJK WKH 0'6 DQG 3/6 GDWD IRXQG D PDVV LQGH[ RI T fÂ§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f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f \LHOGV D VORSH RI D s ZKLFK FRUUHVSRQGV WR D PDVVIUHTXHQF\ VORSH RI T s )LJXUH f ,I ZH DVVXPH WKDW DOO WKH DVWHURLGV LQ D VHPLPDMRU D[LV ]RQH KDYH WKH VDPH PHDQ DOEHGR ZH PD\ GLUHFWO\ FRQYHUW WKH PDJQLWXGHIUHTXHQF\ VORSH LQWR WKH PRUH FRPPRQO\ XVHG PDVV IUHTXHQF\ VORSH YLD T _D ZKHUH D LV WKH VORSH RI WKH PDJQLWXGHIUHTXHQF\ GDWD 6HH $SSHQGL[ %f =RQH ,, VKRZV D VLPLODU WKRXJK VRPHZKDW OHVV GLVWLQFW DQG VKDOORZHU OLQHDU WUHQG EH\RQG + fÂ§ $ ILW WKURXJK WKHVH GDWD \LHOGV D VORSH RI D s T s )LJXUH f ,Q =RQH ,,, ZH REWDLQ WKH VROXWLRQ D s T fÂ§ s )LJXUH f IRU PDJQLWXGHV IDLQWHU WKDQ + 7KHVH VORSHV DUH VLJQLILFDQWO\ ORZHU WKDQ WKH 'RKQDQ\L HTXLOLEULXP YDOXH RI T 7KH ZHLJKWHG PHDQ VORSH IRU WKH WKUHH ]RQHV LV T s HVVHQWLDOO\ HTXDO WR WKH ZHOO GHWHUPLQHG VORSH IRU ]RQH ,Q DGGLWLRQ WR WKH VORSH WKH OHDVWVTXDUHV VROXWLRQ IRU HDFK ]RQHV SURGXFHV DQ HVWLPDWH IRU WKH LQWHUFHSW RI WKH OLQHDU GLVWULEXWLRQ ZKLFK LV D PHDVXUH RI WKH DEVROXWH QXPEHU RI DVWHURLGV LQ WKH SRSXODWLRQ :LWK DQ HVWLPDWH RI WKH PHDQ DOEHGR RI DVWHURLGV LQ WKH SRSXODWLRQ WKH H[SUHVVLRQV GHULYHG LQ $SSHQGL[ % DOORZ XV WR XVH WKH SDUDPHWHUV RI WKH PDJQLWXGHIUHTXHQF\ SORWV WR TXDQWLI\ WKH VL]HIUHTXHQF\ GLVWULEXWLRQV IRU WKH WKUHH ]RQHV DQG IRU WKH PDLQEHOW DV D ZKROH PAGE 26 7KH 2EVHUYHG 0DLQEHOW 6L]H 'LVWULEXWLRQ :H PD\ GHILQH WKH REVHUYHG PDLQEHOW VL]H GLVWULEXWLRQ WKDW ZH ZLOO ZRUN ZLWK E\ FRPELQLQJ GDWD IURP WKH FDWDORJXHG SRSXODWLRQ RI DVWHURLGV DQG WKH OHDVWVTXDUHV ILWV WR WKH 3/6 GDWD 7KH VL]HV RI WKH QXPEHUHG PDLQEHOW DVWHURLGV PD\ EH UHFRQVWUXFWHG IURP WKHLU DEVROXWH EULJKWQHVVHV LI ZH FDQ HVWLPDWH D YDOXH IRU WKH DOEHGR 6HH $SSHQGL[ $f )RUWXQDWHO\ DQ H[WHQVLYH VHW RI DOEHGRV GHULYHG E\ ,5$6 LV DYDLODEOH IRU D JUHDW PDQ\ DVWHURLGV $ UHFHQW VWXG\ E\ 0DWVRQ HW DO f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f ZKHQ DYDLODEOH DQG E\ 7KROHQ SULYDWH FRPPXQLFDWLRQf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f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f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caÂ“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f + =RQH =RQH ,, =RQH ,,, ,, ,,, D D D D 1+f 1+f 1+f 1+f f PAGE 30 7DEOH 1XPEHUV RI DVWHURLGV LQ WKUHH 3/6 ]RQHV FDWDORJXHG3/6 GDWDf + =RQH D 1+f =RQH ,, D 1+f =RQH ,,, D 1+f ,, ,,, D 1+f PAGE 31 7DEOH $GMXVWHG FRPSOHWHQHVV OLPLWV IRU 3/6 ]RQHV 6HPLPDMRU $[LV =RQH 0HDQ 6HPLPDMRU $[LV $8f &RPSOHWHQHVV OLPLW LQ + D D D PAGE 32 P D &' *2 &' 4 e R 6K &' 2 2K 7 WfÂ§UaU L LfÂ§U ,rf f9 ÂœL Yn f f ; f f f < r W Y f f f L U rf !f f } f f P B f fÂ§ ff n f 9 f f f n :2U:Ufn Â f Â‘ LU A rb rb f : f YW nf} I f f Â nL f A 9 L f f ? f rr UMO? fr fr f V n!f L 9n& ? f Wn ff LYLL f f f r rÂ‘ F r f ? !AY nrÂ‘ f ff f ff B f f B B f f f f f f f P M f r f f fb f rfff f fffÂ‘ r 9 f f f Y f r} L A f f 7 r f fÂ§ L Lr f f A A fÂ§fÂ§ L L b L L L $VWHURLGV 1XPEHUHG 0XOWLRSSRVLWLRQ / 6HPLPDMRU $[LV $8f )LJXUH 3URSHU LQFOLQDWLRQ YHUVXV VHPLPDMRU D[LV IRU DOO FDWDORJXHG PDLQEHOW DVWHURLGV PAGE 33 1XPEHU SHU 0DJQLWXGH %LQ b 0DLQEHOW FDWDORJXHG DVWHURLGV $EVROXWH 0DJQLWXGH + )LJXUH 0DJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU FDWDORJXHG PDLQEHOW DVWHURLGV PAGE 34 'LVFRYHU\ 'DWH )LJXUH $EVROXWH PDJQLWXGH DV D -XQFWLRQ RI GLVFRYHU\ GDWH IRU DOO FDWDORJXHG PDLQEHOW DVWHURLGV PAGE 35 1XPEHU SHU 0DJQLWXGH %LQ $EVROXWH 0DJQLWXGH + )LJXUH 0DJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU 3/6 ]RQH 3/6 DQG FDWDORJXHG DVWHURLG GDWD PAGE 36 1XPEHU SHU 0DJQLWXGH %LQ (a I L U 7 S L U ( fÂ§ =RQH ,, fÂ§ R R r D fÂ§ R R Q R 3/6 f FDWDORJXHG fÂ§ fÂ§ R f f f fÂ§ f f f f r f f fÂ§ f f D fÂ§ f f fÂ§ f Z 2 f n L / L / f R f f f AL $EVROXWH 0DJQLWXGH + )LJXUH 0DJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU 3/6 ]RQH ,, 3/6 DQG FDWDORJXHG DVWHURLG GDWD PAGE 37 1XPEHU SHU 0DJQLWXGH %LQ $EVROXWH 0DJQLWXGH + L L W U U U _ fÂ§ =RQH ,,, R D fÂ§ R R R fÂ§ fÂ§ R R SOV f FDWDORJXHG fÂ§ r R R fÂ§ f f f fÂ§ B fÂ§ f R f D P fÂ§ B f f Z f 2 f f f fÂ§ f f fÂ§ 2 fÂ§ BB@ f fÂ§ R )LJXUH 0DJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU 3/6 ]RQH ,,, 3/6 DQG FDWDORJXHG DVWHURLG GDWD PAGE 38 1XPEHU SHU 0DJQLWXGH %LQ Â =RQH D $EVROXWH 0DJQLWXGH + )LJXUH $GRSWHG PDJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU 3/6 ]RQH , PAGE 39 1XPEHU SHU 0DJQLWXGH %LQ U L L L L m L f Â U =RQH ,, D L L L $EVROXWH 0DJQLWXGH + )LJXUH $GRSWHG PDJQLWXGHIUHTXHQF\ GLVWULEXWLRQ IRU 3/6 ]RQH ,, PAGE 40 1XPEHU SHU 0DJQLWXGH %LQ R An n fÂ§ rm =RQH ,,, D fÂ§ fÂ§ r fÂ§ LQ P P P P fÂ§ P L fÂ§ fÂ§ nL 7 L O O O Â‘MLL M L / $EVROXWH 0DJQLWXGH + )LJXUH $GRSWHG PDJQLWXGHIUHTXHQF\ GLVWULEXWLRQ ORU 3/6 ]RQH ,,, PAGE 41 3/6 7DEOHV DQG 0HDQ 2SSRVLWLRQ 0DJQLWXGH P UR RF IRU WKH DVWHURLGV LQ 7DEOHV DQG RI 9DQ +RXWHQ HW DO f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f VROYHG DQDO\WLFDOO\ WKH LQWHJURGLIIHUHQWLDO HTXDWLRQ GHVFULELQJ WKH HYROXWLRQ RI D FROOHFWLRQ RI SDUWLFOHV DOO ZLWK VL]H LQGHSHQGHQW LPSDFW VWUHQJWKV ZKLFK IUDJPHQW GXH WR PXWXDO FROOLVLRQV +H IRXQG WKDW WKH VL]H GLVWULEXWLRQ RI WKH UHVXOWLQJ GHEULV FDQ EH GHVFULEHG E\ D SRZHUODZ GLVWULEXWLRQ LQ PDVV RI WKH IRUP IPfGP RF PaTGPf ZKHUH IPfGP LV WKH QXPEHU RI DVWHURLGV LQ WKH PDVV UDQJH P WR P GP DQG T LV WKH VORSH LQGH[ 'RKQDQ\L IRXQG WKDW T IRU GHEULV LQ FROOLVLRQDO HTXLOLEULXP LQ DJUHHPHQW ZLWK WKH REVHUYHG GLVWULEXWLRQ RI VPDOO DVWHURLGV DV GHWHUPLQHG IURP 0'6 DQG 3/6 GDWD 7KH HTXLOLEULXP VORSH LQGH[ T ZDV IRXQG WR EH LQVHQVLWLYH WR WKH IUDJPHQWDWLRQ SRZHU ODZ Uc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f VROYHG WKH VDPH FROOLVLRQ HTXDWLRQ QXPHULFDOO\ DQG FRQILUPHG WKH UHVXOWV RI 'RKQDQ\L +HOO\HU VKRZHG WKDW IRU IRXU YDOXHV RI WKH IUDJPHQWDWLRQ SRZHU ODZ UHIHUUHG WR DV [ LQ KLV QRWDWLRQ [ Uc fÂ§ DQG f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f DOWKRXJK WKHUH ZHUH GLVFRQWLQXLWLHV LQ WKH SORW RI WKH VORSH DV D IXQFWLRQ RI WLPH DW WKH WLPHV RI WKH ODUJH IUDJPHQWDWLRQ HYHQWV 'DYLV HW DO f LQWURGXFHG D QXPHULFDO PRGHO VLPXODWLQJ WKH FROOLVLRQDO HYROXWLRQ RI YDULRXV LQLWLDO SRSXODWLRQV RI DVWHURLGV DQG FRPSDUHG WKH UHVXOWV ZLWK WKH REVHUYHG GLVWULEXWLRQ RI DVWHURLGV LQ RUGHU WR ILQG WKRVH SRSXODWLRQV ZKLFK HYROYHG WR WKH SUHVHQW EHOW ,Q WKHLU VWXG\ WKH\ FRQVLGHUHG WKUHH GLIIHUHQW IDPLOLHV RI VKDSHV IRU WKH LQLWLDO GLVWULEXWLRQ SRZHU ODZ VHJPHQWHG SRZHU ODZ VLPXODWLQJ D UXQDZD\ JURZWK GLVWULEXWLRQ RI ERGLHV DV PAGE 48 JHQHUDWHG E\ WKH DFFUHWLRQDO VLPXODWLRQ RI *UHHQEHUJ HW DO f DQG JDXVVLDQ DV VXJJHVWHG E\ $QGHUV f DQG +DUWPDQQ DQG +DUWPDQQ f 7KH\ FRQFOXGHG WKDW IRU SRZHU ODZ LQLWLDO SRSXODWLRQV WKH LQLWLDO PDVV RI WKH EHOW FRXOG QRW KDYH EHHQ PXFK ODUJHU WKDQ a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f LQWURGXFHG D UHYLVHG PRGHO LQFRUSRUDWLQJ WKH LQFUHDVHG LPSDFW VWUHQJWKV RI ODUJH DVWHURLGV GXH WR K\GURVWDWLF VHOIFRPSUHVVLRQ 7KH UHVXOWV IURP WKLV QXPHULFDO PRGHO ZHUH ODWHU H[WHQGHG WR LQFOXGH VL]H VWUDLQUDWHf GHSHQGHQW LPSDFW VWUHQJWKV 'DYLV HW DO f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f VXJJHVW WKDW PXFK PRUH FROOLVLRQDO HYROXWLRQ RFFXUUHG WKDQ WKHVH PRGHOV SUHGLFW 7KH ODWHVW YHUVLRQ RI WKLV PRGHO LV FXUUHQWO\ EHLQJ PAGE 49 XVHG WR LQYHVWLJDWH WKH FROOLVLRQDO KLVWRU\ RI DVWHURLG IDPLOLHV 'DYLV DQG 0DU]DUL f 0RVW UHFHQWO\ :LOOLDPV DQG :HWKHULOO f KDYH H[WHQGHG WKH ZRUN RI 'RKQDQ\L WR LQFOXGH D ZLGHU UDQJH RI DVVXPSWLRQV DQG REWDLQHG DQ DQDO\WLFDO VROXWLRQ IRU WKH VWHDG\VWDWH VL]H GLVWULEXWLRQ RI D VHOIVLPLODU FROOLVLRQDO IUDJPHQWDWLRQ FDVFDGH 7KHLU UHVXOWV FRQILUP WKH HTXLOLEULXP YDOXH RI T DQG GHPRQVWUDWH WKDW WKLV YDOXH LV HYHQ OHVV VHQVLWLYH WR WKH SK\VLFDO SDUDPHWHUV RI WKH IUDJPHQWDWLRQ SURFHVV WKDQ 'RKQDQ\L KDG WKRXJKW ,Q SDUWLFXODU :LOOLDPV DQG :HWKHULOO KDYH H[SOLFLWO\ WUHDWHG WKH GHEULV IURP FUDWHULQJ LPSDFWV ZKHUHDV 'RKQDQ\L FRQFOXGHG WKDW WKH FRQWULEXWLRQ IURP FUDWHULQJ ZRXOG EH QHJOLJLEOH DQG VR GURSSHG WHUPV LQFOXGLQJ FUDWHULQJ GHEULVf DQG KDYH PRUH UHDOLVWLFDOO\ DVVXPHG WKDW WKH PDVV RI WKH ODUJHVW IUDJPHQW UHVXOWLQJ IURP D FDWDVWURSKLF IUDJPHQWDWLRQ GHFUHDVHV ZLWK LQFUHDVLQJ SURMHFWLOH PDVV 7KH\ ILQG D VWHDG\VWDWH YDOXH RI T s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f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fILHOGf DVWHURLGV WKH VL]H RI WKH VPDOOHVW ILHOG DVWHURLG FDSDEOH RI VKDWWHULQJ DQG GLVSHUVLQJ WKH WDUJHW DQG WKH FXPXODWLYH QXPEHU RI ILHOG DVWHURLGV ODUJHU WKDQ WKLV VPDOOHVW VL]H :H VKDOO QRZ GHWDLO WKH SURFHGXUH IRU FDOFXODWLQJ WKH FROOLVLRQDO OLIHWLPH RI DQ DVWHURLG DQG H[DPLQH HDFK RI WKHVH GHWHUPLQDQWV LQ WKH SURFHVV 7KH SUREDELOLW\ RI FROOLVLRQV WKH FROOLVLRQ UDWHf EHWZHHQ WKH WDUJHW DQG WKH ILHOG DVWHURLGV KDV EHHQ FDOFXODWHG XVLQJ WKH WKHRU\ RI :HWKHULOO f 8WLOL]LQJ WKH VDPH PHWKRG )DULQHOOD DQG 'DYLV f LQGHSHQGHQWO\ FDOFXODWHG LQWULQVLF FROOLVLRQ UDWHV ZKLFK PDWFK RXU UHVXOWV WR ZLWKLQ D IDFWRU RI )RU D WDUJHW DVWHURLG ZLWK RUELWDO HOHPHQWV D H DQG W ZH FDOFXODWH DQ LQWULQVLF FROOLVLRQ SUREDELOLW\ 3c ZKLFK LV WKH FROOLVLRQ UDWH ZLWK WKH EDFNJURXQG ILHOG RI DVWHURLGV LQ XQLWV RI \Un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f ZKLFK DUH PRUH UHSUHVHQWDWLYH RI WKH ORQJWHUP RUELWDO HOHPHQWV WKDQ DUH WKH RVFXODWLQJ HOHPHQWV 7KH UHVXOWLQJ LQWULQVLF FROOLVLRQ UDWHV DQG PHDQ UHODWLYH HQFRXQWHU VSHHGV IRU VHYHUDO UHSUHVHQWDWLYH PDLQEHOW DVWHURLGV DUH JLYHQ LQ 7DEOH 7KH PHDQ LQWULQVLF UDWH DQG UHODWLYH HQFRXQWHU VSHHG FDOFXODWHG IURP WKH DVWHURLGV RI WKH ELDVIUHH VHW DUH [ a \U NP DQG NP Vn UHVSHFWLYHO\ 7KH fILQDOf FROOLVLRQ SUREDELOLW\ IRU D ILQLWHVL]HG DVWHURLG ZLWK GLDPHWHU LV 3I FUn3O f ZKHUH Dn DU VLQFH 3 LQFOXGHV WKH IDFWRU RI WWf DQG D LU'f LV WKH FROOLVLRQ FURVVVHFWLRQ WDNHQ WR EH WKH VLPSOH JHRPHWULF FURVVVHFWLRQ VLQFH WKH VHOIJUDYLW\ RI WKH DVWHURLGV LV QHJOLJLEOH KHUHf 7R JHW WKH WRWDO SUREDELOLW\ WKDW WKH DVWHURLG ZLOO VXIIHU D GHVWUXFWLYH FROOLVLRQ ZH PXVW LQWHJUDWH WKH ILQDO SUREDELOLW\ RYHU DOO SURMHFWLOHV RI FRQVHTXHQFH XVLQJ WKH VL]H GLVWULEXWLRQ IXQFWLRQ G1 &'aSG' 7KHQ f PAGE 52 3W LV VLPSO\ WKH FROOLVLRQ FURVV VHFWLRQ WLPHV WKH LQWULQVLF FROOLVLRQ SUREDELOLW\ WLPHV WKH FXPXODWLYH QXPEHU RI ILHOG DVWHURLGV ODUJHU WKDQ 'PcQf 7KH FROOLVLRQ OLIHWLPH f LV WKHQ WKH WLPH IRU ZKLFK WKH SUREDELOLW\ RI VXUYLYDO LV H /HW XV QRZ H[DPLQH WKH GHWHUPLQDWLRQ RI 'PLQ WKH VPDOOHVW ILHOG DVWHURLG FDSDEOH RI IUDJPHQWLQJ DQG GLVSHUVLQJ WKH WDUJHW DVWHURLG 7R IUDJPHQW DQG GLVSHUVH WKH WDUJHW DVWHURLG WKH SURMHFWLOH PXVW VXSSO\ HQRXJK NLQHWLF HQHUJ\ WR RYHUFRPH ERWK WKH LPSDFW VWUHQJWK RI WKH WDUJHW GHILQHG DV WKH HQHUJ\ QHHGHG WR SURGXFH D ODUJHVW IUDJPHQW FRQWDLQLQJ b RI WKH PDVV RI WKH RULJLQDO ERG\f DQG LWV JUDYLWDWLRQDO ELQGLQJ HQHUJ\ 7KH LPSDFW VWUHQJWK RI DVWHURLGVL]HG ERGLHV LV QRW ZHOO NQRZQ /DERUDWRU\ H[SHULPHQWV RQ WKH FROOLVLRQDO IUDJPHQWDWLRQ RI EDVDOW WDUJHWV )XMLZDUD HW DO f \LHOG FROOLVLRQDO VSHFLILF HQHUJLHV RI WR [ HUJ Jn RU DQ LPSDFW VWUHQJWK 6 RI [ n HUJ FPn +RZHYHU HVWLPDWHV E\ )XMLZDUD f RI WKH NLQHWLF DQG JUDYLWDWLRQDO HQHUJLHV RI WKH IUDJPHQWV LQ WKH WKUHH SURPLQHQW +LUD\DPD IDPLOLHV LQGLFDWHV WKDW WKH DVWHURLGDO SDUHQW ERGLHV KDG LPSDFW VWUHQJWKV RI D IHZ WLPHV HUJ FP DQ RUGHU RI PDJQLWXGH JUHDWHU WKDQ LPSDFW VWUHQJWKV IRU URFN\ PDWHULDOV )XMLZDUD DVVXPHG WKDW WKH IUDFWLRQ RI NLQHWLF HQHUJ\ WUDQVIHUUHG IURP WKH LPSDFWRU WR WKH GHEULV LV NH f ,Q RUGHU WR DYRLG LPSODXVLEOH DVWHURLGDO FRPSRVLWLRQV ZH PXVW FRQFOXGH WKDW WKH HIIHFWLYH LPSDFW VWUHQJWK RI DQ DVWHURLG LV D IXQFWLRQ RI LWV VL]H DV ZHOO DV LWV FRPSRVLWLRQ 7KH GLIILFXOWLHV LQKHUHQW LQ VFDOLQJ WKH LPSDFW VWUHQJWK RYHU VHYHUDO RUGHUV RI PDJQLWXGH LQ GLPHQVLRQ PAGE 53 IURP ODERUDWRU\ H[SHULPHQWV WR DVWHURLGVL]HG ERGLHV DUH UHYLHZHG E\ )XMLZDUD HW DO f 'DYLV HW DO f FRQFOXGHG WKDW ODUJH DVWHURLGV VKRXOG EH VWUHQJWKHQHG E\ JUDYLWDWLRQDO VHOIFRPSUHVVLRQ DQG GHYHORSHG D VL]HGHSHQGDQW LPSDFW VWUHQJWK PRGHO ZKLFK LV FRQVLVWHQW ZLWK WKH )XMLZDUD HW DO f UHVXOWV DQG SURGXFHV D VL]HIUHTXHQF\ GLVWULEXWLRQ RI FROOLVLRQ IUDJPHQWV FRQVLVWHQW ZLWK WKDW REVHUYHG IRU WKH +LUD\DPD IDPLOLHV 2WKHU UHVHDUFKHUV )DULQHOOD HW DO +ROVDSSOH DQG +RXVHQ +RXVHQ DQG +ROVDSSOH f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b WKH PDVV RI WKH RULJLQDO ERG\ ZH WDNH WKH ELQGLQJ HQHUJ\ WR EH WKDW RI D VSKHULFDO VKHOO RI PDVV ?0 ZKHUH 0 LV WKH WRWDO PDVV RI WKH WDUJHWf UHVWLQJ RQ D FRUH RI PDVV ?0 6XFK D PRGHO VKRXOG ZHOO DSSUR[LPDWH WKH FLUFXPVWDQFHV RI D FRUHW\SH VKDWWHULQJ FROOLVLRQ ,Q WKLV FDVH A *0 + fÂ§fÂ§fÂ§ 5 f LV WKH HQHUJ\ UHTXLUHG WR GLVSHUVH RQH KDOI WKH PDVV RI WKH WDUJHW DVWHURLG WR LQILQLW\ VHH $SSHQGL[ &f 1RW DOO RI WKH NLQHWLF HQHUJ\ RI WKH SURMHFWLOH LV SDUWLWLRQHG LQWR FRPPLQXWLRQ DQG NLQHWLF HQHUJ\ RI WKH ODUJH IUDJPHQWV RI WKH FROOLVLRQ )URP UHFRQVWUXFWLRQ RI WKH WKUHH ODUJHVW +LUD\DPD IDPLOLHV )XMLZDUD f IRXQG WKDW D IUDFWLRQ RI PAGE 54 SURMHFWLOH NLQHWLF HQHUJ\ SDUWLWLRQHG LQWR NLQHWLF HQHUJ\ RI WKH PHPEHUV RI WKH IDPLO\ RI RUGHU ZDV PRVW FRQVLVWHQW ZLWK WKH GHULYHG FROOLVLRQ HQHUJLHV DQG IUDJPHQW VL]HV ([SHULPHQWDO GHWHUPLQDWLRQ RI WKH HQHUJ\ SDUWLWLRQLQJ IRU FRUHW\SH FROOLVLRQV )XMLZDUD DQG 7VXNDPRWR f VKRZHG WKDW RQO\ DERXW WR b RI WKH NLQHWLF HQHUJ\ RI WKH SURMHFWLOH LV LPSDUWHG LQWR WKH NLQHWLF HQHUJ\ RI WKH ODUJHU IUDJPHQWV DQG WKH FRPPLQXWLRQDO HQHUJ\ IRU WKHVH IUDJPHQWV DPRXQWV WR VRPH b RI WKH LPSDFW HQHUJ\ :H VKDOO WDNH NH WR EH D SDUDPHWHU ZKLFK PD\ DVVXPH YDOXHV RI IURP D IHZ WR IHZ WHQV RI SHUFHQW :H PD\ WKHQ ZULWH IRU WKH PLQLPXP WRWDO SURMHFWLOH NLQHWLF HQHUJ\ QHHGHG WR IUDJPHQW DQG GLVSHUVH D WDUJHW DVWHURLG RI PDVV 0 DQG GLDPHWHU (PLQ 6Y rAUf f Bf ZKHUH 9 LV WKH YROXPH RI WKH DVWHURLG )URP WKH NLQHWLF HQHUJ\ RI WKH SURMHFWLOH DQG WKH PHDQ HQFRXQWHU VSHHG FDOFXODWHG E\ WKH :HWKHULOO PRGHO ZH FDQ ILQG WKH PLQLPXP SURMHFWLOH PDVV DQG KHQFH WKH PLQLPXP SURMHFWLOH GLDPHWHU QHHGHG WR IUDJPHQW DQG GLVSHUVH WKH WDUJHW DVWHURLG )LQDOO\ WKHQ f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n LOO WLO ZKHUH UF LV XVXDOO\ WKH FROOLVLRQ OLIHWLPH IRU ELQ L ,Q PRVW FDVHV ZH KDYH OHW $W 'XULQJ D VLQJOH WLPHVWHS WKH QXPEHU RI SDUWLFOHV UHPRYHG IURP ELQ L LV WKHQ IRXQG IURP WKH H[SUHVVLRQ ] 1^LffÂ§fÂ§ f 7FOf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aYG' f PAGE 56 7KH H[SRQHQW S LV GHWHUPLQHG IURP WKH SDUDPHWHU E WKH IUDFWLRQDO VL]H RI WKH ODUJHVW IUDJPHQW LQ WHUPV RI WKH SDUHQW ERG\ E\ WKH H[SUHVVLRQ f VR WKDW WKH WRWDO PDVV RI GHEULV HTXDOV WKH PDVV RI WKH SDUHQW DVWHURLG *UHHQEHUJ DQG 1RODQ f 7KH FRQVWDQW % LV GHWHUPLQHG VXFK WKDW WKHUH LV RQO\ RQH REMHFW DV ODUJH DV WKH ODUJHVW UHPQDQW 'cU 7KH H[SRQHQW S LV D IUHH SDUDPHWHU RI WKH PRGHO EXW LV XVXDOO\ WDNHQ WR EH VRPHZKDW ODUJHU WKDQ WKH HTXLOLEULXP YDOXH RI LQ PDVV XQLWVf LQ DFFRUG ZLWK ODERUDWRU\ H[SHULPHQWV DQG WKH REVHUYHG VL]HIUHTXHQF\ GLVWULEXWLRQV RI WKH SURPLQHQW +LUD\DPD IDPLOLHV &HOOLQR HW DO f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f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f DWWDLQV DQ HTXLOLEULXP VORSH HTXDO WR WKH H[SHFWHG 'RKQDQ\L YDOXH RI T IRU ERGLHV LQ WKH VL]H UDQJH RI PHWHUV 7KH VHFRQG UXQ EHJDQ ZLWK D PXFK VKDOORZHU LQLWLDO GLVWULEXWLRQ ZLWK D VORSH RI T 7KH HYROYHG GLVWULEXWLRQ KHUH DV ZHOO YHU\ UDSLGO\ DWWDLQHG WKH H[SHFWHG HTXLOLEULXP VORSH 7KH VDPH WZR QXPHULFDO H[SHULPHQWV ZHUH UHSHDWHG ZLWK WKH ELQ VL]H LQFUHDVHG WR 7KH UHVXOWV )LJXUHV DQG f ZHUH LGHQWLFDO WR WKH ILUVW WZR H[SHULPHQWV fÂ§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f RI D a ELQ PRGHO ,Q )LJXUHV DQG WKH VORSHV RI WKH LQLWLDO VL]H GLVWULEXWLRQV DUH DQG UHVSHFWLYHO\ 1RWH WKDW WKH YHUWLFDO VFDOH LQ )LJXUH KDV EHHQ VWUHWFKHG UHODWLYH WR WKH SUHYLRXV WZR ILJXUHV LQ RUGHU WR EULQJ RXW WKH UHOHYDQW GHWDLO ,Q DOO WKUHH FDVHV ZH VHH WKDW WKH VORSH RI WKH VL]H GLVWULEXWLRQ DV\PSWRWLFDOO\ DSSURDFKHV WKH YDOXH UHDFKLQJ YDOXHV QRW VLJQLILFDQWO\ GLIIHUHQW WKDQ WKLV ZLWKLQ WKH DJH RI WKH VRODU V\VWHP 7KH GLIIHUHQW YDOXHV RI WKH VORSH DUH RQO\ YHU\ VOLJKWO\ GHSHQGHQW XSRQ WKH IUDJPHQWDWLRQ SRZHUODZ )RU Tc LQ 'RKQDQ\LfV QRWDWLRQf KLJKHU WKDQ WKH HTXLOLEULXP YDOXH WKH ILQDO VORSH FRQYHUJHV IRU DOO SUDFWLFDO YDOXH RQ VORSHV VRPHZKDW JUHDWHU WKDQ ZLWKLQ ELOOLRQ \HDUV )RU Tc OHVV WKDQ HTXLOLEULXP WKH ILQDO VORSHV DUH OHVV WKDQ ,QWHUHVWLQJO\ IRU VWHHS IUDJPHQWDWLRQ SRZHUODZV WKH VORSH LV DOZD\V VHHQ WR fRYHUVKRRWf RQ WKH ZD\ WR HTXLOLEULXP HLWKHU KLJKHU WKDQ ZKHQ WKH LQLWLDO VORSH LV ORZHU RU ORZHU WKDQ ZKHQ WKH LQLWLDO VORSH LV KLJKHU :H ILQG SHUKDSV QRW XQH[SHFWHGO\ WKDW WKH 'RKQDQ\L HTXLOLEULXP YDOXH LV UHDFKHG PRVW UDSLGO\ ZKHQ WKH IUDJPHQWDWLRQ SRZHUODZ LV QHDU +HOO\HU f IRXQG WKH VDPH EHKDYLRU LQ KLV QXPHULFDO VROXWLRQ RI WKH IUDJPHQWDWLRQ HTXDWLRQ ,Q KLV ZRUN KRZHYHU +HOO\HU GLG QRW LQFOXGH PRGHOV LQ ZKLFK WKH IUDJPHQWDWLRQ LQGH[ ZDV PRUH VWHHS WKDQ WKH HTXLOLEULXP YDOXH VR ZH FDQQRW FRPSDUH RXU UHVXOWV FRQFHUQLQJ WKH HTXLOLEULXP RYHUVKRRW 5HFDOO WKDW 'RKQDQ\L f FRQFOXGHG WKDW WKH GHEULV IURP FUDWHULQJ FROOLVLRQV SOD\HG RQO\ D PLQRU UROH LQ GHWHUPLQLQJ WKH VORSH RI WKH HTXLOLEULXP VL]H GLVWULEXWLRQ 2XU QXPHULFDO PRGHO ZDV WKXV FRQVWUXFWHG WR QHJOHFW FUDWHULQJ GHEULV 7KH UHFHQW ZRUN RI :LOOLDPV DQG :HWKHULOO f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f :LWK VL]H LQGHSHQGHQW LPSDFW VWUHQJWKV RXU PRGHO SURGXFHV HYROYHG SRZHUODZ VL]H GLVWULEXWLRQV ZLWK VORSHV HVVHQWLDOO\ HTXDO WR LQGHSHQGHQW RI WKH QXPHULFDO UHTXLUHPHQWV RI WKH FRPSXWHU FRGH DQG WKH DVVXPSWLRQV FRQFHUQLQJ WKH FROOLGLQJ DVWHURLGV 7KH n:DYHf DQG WKH 6L]H 'LVWULEXWLRQ IURP WR 0HWHUV 'XULQJ WKH HDUOLHVW SKDVHV RI FRGH YDOLGDWLRQ RXU PRGHO SURGXFHG DQ XQH[SHFWHG GHYLDWLRQ IURP D VWULFW SRZHUODZ VL]H GLVWULEXWLRQ )LJXUH VKRZV WKH VL]H GLVWULn EXWLRQ ZKLFK UHVXOWHG ZKHQ SDUWLFOHV VPDOOHU WKDQ WKRVH LQ WKH VPDOOHVW VL]H ELQ ZHUH LQDGYHUWHQWO\ QHJOHFWHG LQ WKH PRGHO %HFDXVH RI WKH LQFUHDVLQJ QXPEHUV RI VPDOO SDUn WLFOHV LQ D SRZHUODZ VL]H GLVWULEXWLRQ WKH YDVW PDMRULW\ RI SURMHFWLOHV UHVSRQVLEOH IRU WKH IUDJPHQWDWLRQ RI D JLYHQ VL]H SDUWLFOH DUH VPDOOHU WKDQ WKH WDUJHW DQG DUH XVXDOO\ QHDU WKH ORZHU OLPLW UHTXLUHG IRU IUDJPHQWDWLRQ %\ QHJOHFWLQJ WKHVH SDUWLFOHV LQ RXU PRGHO ZH DUWLILFLDOO\ LQFUHDVHG WKH FROOLVLRQ OLIHWLPHV RI WKRVH VL]H ELQV IRU ZKLFK WKH VPDOOHVW SURMHFWLOH UHTXLUHG IRU IUDJPHQWDWLRQ ZDV VPDOOHU WKDQ WKH VPDOOHVW VL]H ELQ 7KH SDUWLFOHV LQ WKHVH VL]H ELQV WKHQ EHFRPH UHODWLYHO\ RYHUDEXQGDQW DV SURMHFWLOHV DQG SUHIHUHQWLDOO\ GHSOHWH WDUJHWV LQ WKH QH[W ODUJHVW VL]H ELQV 7KH SDUWLFOHV LQ WKHVH ELQV DUH QRW SUHVHQW LQ VXIILFLHQW TXDQWLWLHV WR IUDJPHQW ODUJH QXPEHUV RI SDUWLFOHV LQ WKH QH[W ODUJHVW VL]H ELQV DQG VR RQ 7KLV SDWWHUQ LV UHSHDWHG LQ D ZDYHOLNH GHYLDWLRQ IURP PAGE 60 D VWULFW SRZHUODZ GLVWULEXWLRQ XS WKURXJK WKH ODUJHVW DVWHURLGV LQ WKH SRSXODWLRQ 7KH VDPH ZDYHOLNH SKHQRPHQRQ ZDV ODWHU LQGHSHQGHQWO\ GLVFRYHUHG E\ 'DYLV HW DO f 7KH FRGH ZDV VXEVHTXHQWO\ DOWHUHG WR H[WUDSRODWH WKH SDUWLFOH SRSXODWLRQ EH\RQG WKH VPDOOHVW VL]H ELQ WR HOLPLQDWH WKH SURSDJDWLRQ RI DQ DUWLILFLDO ZDYH LQ WKH VL]H GLVWULEXWLRQ +RZHYHU LQ UHDOLW\ WKH UHPRYDO RI WKH VPDOOHVW DVWHURLGDO GHEULV E\ UDGLDWLRQ IRUFHV PD\ SURYLGH D PHFKDQLVP IRU WUXQFDWLQJ WKH VL]H GLVWULEXWLRQ DQG JHQHUDWLQJ VXFK D ZDYHn OLNH IHDWXUH LQ WKH DFWXDO DVWHURLG VL]H GLVWULEXWLRQ 7R VWXG\ WKH VHQVLWLYLW\ RI IHDWXUHV RI WKH ZDYH RQ WKH VWUHQJWK RI WKH VPDOO SDUWLFOH FXWRII ZH PD\ LPSRVH D FXWRII RQ WKH H[WUDSRODWLRQ EH\RQG WKH VPDOOHVW VL]H ELQ WR VLPXODWH WKH HIIHFWV RI UDGLDWLRQ IRUFHV :H XVH DQ H[SRQHQWLDO FXWRII RI WKH IRUP 1^Lf 1LfD OU[n f ZKHUH L 1Of LV WKH VPDOOHVW VL]H ELQ 1fÂ§Lf LV WKH QXPEHU RI SDUWLFOHV H[SHFWHG VPDOOHU WKDQ WKRVH LQ ELQ EDVHG RQ DQ H[WUDSRODWLRQ IURP WKH WZR VPDOOHVW VL]H ELQV DQG [ LV D SDUDPHWHU FRQWUROOLQJ WKH VWUHQJWK RI WKH FXWRII 1HJDWLYH ELQ QXPEHUV VLPSO\ UHIHU WR WKRVH VL]H ELQV ZKLFK ZRXOG EH SUHVHQW DQG UHVSRQVLEOH IRU WKH IUDJPHQWDWLRQ RI WKH VPDOOHVW VHYHUDO ELQV DFWXDOO\ SUHVHQW LQ WKH PRGHO 7KH QXPEHU RI fYLUWXDOf ELQV SUHVHQW GHSHQGV XSRQ WKH ELQ VL]H DGRSWHG IRU D SDUWLFXODU PRGHO WKRXJK LQ DOO FDVHV H[WHQGV WR LQFOXGH SDUWLFOHV a WKH GLDPHWHU RI WKRVH LQ ELQ URXJKO\ WKH VL]H UDWLR UHTXLUHG IRU IUDJPHQWDWLRQf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b WKH ODUJH HQG RI WKH VL]H GLVWULEXWLRQ 7KH UHVXOWV RI WKH WZR PRGHOV KDYH EHHQ SORWWHG VHSDUDWHO\ IRU FODULW\ ZLWK WKH ILQDO VL]H GLVWULEXWLRQ IRU WKH ODUJHU ELQ PRGHO RIIVHW WR WKH OHIW E\ RQH GHFDGH LQ VL]Hf EXW LI RYHUODLG ZRXOG EH VHHQ WR FRLQFLGH SUHFLVHO\ WKXV LOOXVWUDWLQJ WKDW WKH ZDYHOHQJWK DQG SKDVH RI WKH ZDYH DUH QRW DUWLIDFWV RI WKH ELQ VL]H DGRSWHG IRU WKH PRGHO UXQ 7KH HIIHFW RI D VPRRWK WKRXJK VKDUSf SDUWLFOH FXWRII PD\ EH VHHQ E\ FRPSDULQJ WKH VKDSH DQG RQVHW RI WKH ZDYH LQ WKH VPDOOHVW VL]H SDUWLFOHV EHWZHHQ )LJXUHV DQG 7KH DPSOLWXGH RI WKH ZDYH KDV EHHQ IRXQG WR EH GHSHQGHQW XSRQ WKH VWUHQJWK RI WKH VPDOO SDUWLFOH FXWRII $ VLJQLILFDQW ZDYH ZLOO GHYHORS RQO\ LI WKH SDUWLFOH FXWRII LV TXLWH VKDUS WKDW LV LI WKH VPDOOHVW SDUWLFOHV DUH UHPRYHG DW D UDWH VLJQLILFDQWO\ JUHDWHU WKDQ WKDW UHTXLUHG WR PDLQWDLQ D 'RKQDQ\L HTXLOLEULXP SRZHUODZ 6LQFH UDGLDWLRQ IRUFHV GR LQ IDFW UHPRYH WKH VPDOOHVW DVWHURLGDO SDUWLFOHV SURYLGLQJ D PHDQV RI JUDGXDOO\ WUXQFDWLQJ WKH DVWHURLG VL]H GLVWULEXWLRQ VRPH UHVHDUFKHUV )DULQHOOD HW DO SULYDWH FRPPXQLFDWLRQf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f 7KH 'RKQDQ\L HTXLOLEULXP YDOXH LV S f 7KH FUDWHU FRXQWV DUH PRVW UHOLDEOH LQ WKH GLDPHWHU UDQJH WR NP FUDWHUV RI WKLV VL]H DUH GXH WR WKH LPSDFW RI SURMHFWLOHV ZLWK GLDPHWHUV e PHWHUV 7KH VORSH RI WKH FUDWHU GLVWULEXWLRQ RQ *DVSUD LV DOVR FRQVLVWHQW ZLWK WKH FUDWHU GLVWULEXWLRQ REVHUYHG LQ WKH OXQDU PDULD 6KRHPDNHU f DQG WKH VL]H GLVWULEXWLRQ RI VPDOO (DUWKDSSURDFKLQJ DVWHURLGV GLVFRYHUHG E\ 6SDFHZDWFK 5DELQRZLW] f 'DYLV HW DO f VXJJHVW WKDW DOWKRXJK WKH RYHUDOO VORSH LQGH[ RI WKH DVWHURLG SRSXODWLRQ LV FORVH WR RU HTXDO WR WKH 'RKQDQ\L HTXLOLEULXP YDOXH ZDYHV LPSRVHG RQ WKH GLVWULEXWLRQ E\ WKH UHPRYDO RI WKH VPDOO SDUWLFOHV PD\ FKDQJH WKH VORSH LQ VSHFLILF VL]H UDQJHV WR YDOXHV VLJQLILFDQWO\ DERYH RU EHORZ WKH HTXLOLEULXP YDOXH 7R WHVW WKH WKHRU\ WKDW D ZDYHOLNH GHYLDWLRQ IURP D VWULFW SRZHUODZ VL]H GLVWULEXn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n WLRQ )LJXUH f 8VLQJ PHWHRURLG PHDVXUHPHQWV REWDLQHG E\ LQ VLWX H[SHULPHQWV *UWLQ HW DO f SURGXFHG D PRGHO RI WKH LQWHUSODQHWDU\ GXVW IOX[ IRU SDUWLFOHV ZLWK PDVVHV J P fJ :LWK D SDUWLFOH PDVV GHQVLW\ RI J FPn *ULLQ HW DO f PAGE 63 WKLV FRUUHVSRQGV WR SDUWLFOHV ZLWK GLDPHWHUV RI DERXW P WR PP UHVSHFWLYHO\ )LJXUH VKRZV WKH *ULLQ HW DO PRGHO DQG RXU PRGHOHG SDUWLFOH FXWRIIV IRU WKUHH YDOXHV RI [ )RU WKH IROORZLQJ PRGHOV WKH ORJDULWKPLF VL]H LQWHUYDO ZDV VHW HTXDO WR )RU [ ZH KDYH WKH VLPSOH FDVH RI VWULFW FROOLVLRQDO HTXLOLEULXP ZLWK QR SDUWLFOH UHPRYDO E\ QRQFROOLVLRQDO HIIHFWV LOOXVWUDWHG E\ WKH PRGHOV SUHVHQWHG LQ WKH SUHYLRXV VHFWLRQ :KHQ D VKDUS SDUWLFOH FXWRII LV PRGHOHG EHJLQQLQJ DW fÂ§ LP WKH GLDPHWHU DW ZKLFK WKH 3R\QWLQJ5REHUWVRQ OLIHWLPH RI SDUWLFOHV EHFRPHV FRPSDUDEOH WR WKH FROOLVLRQDO OLIHn WLPH WKH HYROYHG VL]H GLVWULEXWLRQ GHYHORSV D YHU\ GHILQLWH ZDYH VHH )LJXUH f ZLWK DQ XSWXUQ LQ WKH VORSH LQGH[ SUHVHQW DW a P 7KH SDUDPHWHU [ ZDV VHW HTXDO WR IRU WKLV PRGHO WR SURGXFH D fVKDUSf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n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a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f DVWHURLGV 7KH IUDJPHQWDWLRQ RI WKH NP GLDPHWHU DVWHURLG LQGLFDWHG LQ )LJXUH LQFUHDVHG WKH QXPEHU GHQVLW\ RI P DVWHURLGV LQ WKH LQQHU WKLUG RI WKH EHOW E\ D IDFWRU RI MXVW RYHU 6LQFH WKH QXPEHU GHQVLW\ RI IUDJPHQWV PXVW LQFUHDVH DV WKH YROXPH RI WKH SDUHQW DVWHURLG WKH IUDJPHQWDWLRQ RI D NP GLDPHWHU DVWHURLG ZRXOG FDXVH DQ LQFUHDVH LQ WKH QXPEHU RI P DVWHURLGV LQ WKH LQQHU EHOW RI RYHU D IDFWRU RI 7KLV LV MXVW WKH LQFUHDVH RYHU DQ HTXLOLEULXP SRSXODWLRQ RI VPDOO DVWHURLGV WKDW 5DELQRZLW] f ILQGV DPRQJ WKH (DUWK DSSURDFKLQJ DVWHURLGV GLVFRYHUHG E\ 6SDFHZDWFK DQG FRXOG HDVLO\ EH DFFRXQWHG IRU E\ WKH IRUPDWLRQ RI DQ DVWHURLG IDPLO\ WKH VL]H RI WKH )ORUD FODQ 'HSHQGHQFH RI WKH (TXLOLEULXP 6ORSH RQ WKH 6WUHQJWK 6FDOLQJ /DZ 7KH 'RKQDQ\L f UHVXOW WKDW WKH VL]H GLVWULEXWLRQ RI DVWHURLGV LQ FROOLVLRQDO HTXLOLEULXP FDQ EH GHVFULEHG E\ D SRZHUODZ ZLWK D VORSH LQGH[ RI T ZDV REWDLQHG DQDO\WLFDOO\ E\ DVVXPLQJ WKDW DOO DVWHURLGV LQ WKH SRSXODWLRQ KDYH WKH VDPH RU VL]HLQGHSHQGHQW LPSDFW VWUHQJWK 2WKHU UHVHDUFKHUV :LOOLDPV DQG :HWKHULOO f KDYH H[SUHVVHG WKH LQWHQW WR FRQVLGHU GHYLDWLRQV IURP VHOIVLPLODULW\ DQDO\WLFDOO\ WR PAGE 66 GHWHUPLQH WKH UHVXOWLQJ HIIHFW RQ WKH VL]H GLVWULEXWLRQ :H KDYH DOUHDG\ GHPRQVWUDWHG WKDW RXU FROOLVLRQDO PRGHO UHSURGXFHV WKH 'RKQDQ\L UHVXOW IRU VL]HLQGHSHQGHQW LPSDFW VWUHQJWKV VHH 9HULILFDWLRQ RI &ROOLVLRQDO 0RGHOf +RZHYHU VWUDLQUDWH HIIHFWV DQG JUDYLWDWLRQDO FRPSUHVVLRQ OHDG WR VL]HGHSHQGHQW LPSDFW VWUHQJWKV ZLWK ERWK LQFUHDVLQJ DQG GHFUHDVLQJ VWUHQJWKV ZLWK LQFUHDVLQJ WDUJHW VL]H UHVSHFWLYHO\ VHH GLVFXVVLRQ RI VWUHQJWK VFDOLQJ ODZV LQ WKH IROORZLQJ VHFWLRQf :LWK RXU FROOLVLRQDO PRGHO ZH DUH DEOH WR H[SORUH D UDQJH RI VL]HVWUHQJWK VFDOLQJ ODZV DQG WKHLU HIIHFWV RQ WKH UHVXOWLQJ VL]H GLVWULEXWLRQV ,Q RUGHU WR H[DPLQH WKH HIIHFWV RI VL]HGHSHQGHQW LPSDFW VWUHQJWKV RQ WKH HTXLn OLEULXP VORSH RI WKH DVWHURLG VL]H GLVWULEXWLRQ ZH FUHDWHG D QXPEHU RI K\SRWKHWLFDO VL]HVWUHQJWK VFDOLQJ ODZV $V ZLOO EH GLVFXVVHG LQ WKH IROORZLQJ VHFWLRQ ZH DVVXPH WKDW D f ZKHUH 6 LV WKH LPSDFW VWUHQJWK LV WKH GLDPHWHU RI WKH WDUJHW DVWHURLG DQG ILn LV D FRQVWDQW GHSHQGHQW XSRQ PDWHULDO SURSHUWLHV RI WKH WDUJHW 6HYHQ VWUHQJWK ODZV ZHUH FUHDWHG ZLWK YDOXHV RI n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n FRUUHVSRQGLQJ WR D VL]HLQGHSHQGHQW VWUHQJWK PAGE 67 WKH 'RKQDQ\L YDOXH RI T LV REWDLQHG ,I WKH VORSH RI WKH VFDOLQJ ODZ LV QHJDWLYH DV LV WKH FDVH ZLWK VWUDLQUDWH GHSHQGHQW VWUHQJWKV VXFK DV WKH +RXVHQ DQG +ROVDSSOH f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f :H VKRZ WKH VORSH LQGH[ RI WKH IDPLO\ VL]H GLVWULEXWLRQ DV D IXQFWLRQ RI WLPH IRU WZR IDPLOLHV IDPLO\ KDV WKH VDPH DUELWUDU\ VWUHQJWK VFDOLQJ ODZ DV WKH EDFNJURXQG SRSXODWLRQ RI SURMHFWLOHV cLn LQ WKLV FDVHf ZKLOH WKH VFDOLQJ ODZ IRU IDPLO\ KDV Pn 7KH VORSH LQGH[ IRU IDPLO\ LV DSSURSULDWH IRU WKH SDUWLFXODU YDOXH RI n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f 7KH LPSDFW VWUHQJWKV RI EDVDOW DQG PRUWDU WDUJHWV a FP LQ GLDPHWHU KDYH EHHQ PHDVXUHG LQ WKH ODERUDWRU\ EXW XQIRUWXQDWHO\ ZH KDYH QR GLUHFW PHDVXUHPHQWV RI WKH LPSDFW VWUHQJWKV RI REMHFWV DV ODUJH DV DVWHURLGV +HQFH RQH XVXDOO\ DVVXPHV WKDW WKH LPSDFW VWUHQJWKV RI ODUJHU WDUJHWV ZLOO VFDOH LQ VRPH PDQQHU IURP WKRVH PHDVXUHG LQ WKH ODERUDWRU\ VHH )XMLZDUD HW DO f IRU D UHYLHZ RI VWUHQJWK VFDOLQJ ODZVf 5HFHQWO\ DWWHPSWV KDYH EHHQ PDGH WR GHWHUPLQH WKH VWUHQJWK VFDOLQJ ODZV IURP ILUVW SULQFLSOHV HLWKHU DQDO\WLFDOO\ +RXVHQ DQG +ROVDSSOH f RU QXPHULFDOO\ WKURXJK K\GURFRGH VWXGLHV 5\DQ f +RZHYHU ZH KDYH WDNHQ D GLIIHUHQW DSSURDFK RI XVLQJ WKH QXPHULFDO FROOLVLRQDO PRGHO WR DVN ZKDW WKH VWUHQJWK VFDOLQJ UHODWLRQ PXVW EH LQ RUGHU WR UHSURGXFH WKH REVHUYHG VL]H GLVWULEXWLRQ RI WKH DVWHURLGV 7KH UHVXOWV DOORZ XV WR SODFH VRPH REVHUYDWLRQDO PAGE 69 FRQVWUDLQWV RQ WKH LPSDFW VWUHQJWKV RI DVWHURLGDO ERGLHV RXWVLGH WKH VL]H UDQJH XVXDOO\ [SORUHG LQ ODERUDWRU\ H[SHULPHQWV 7KH REVHUYHG VL]H GLVWULEXWLRQ RI WKH PDLQEHOW DVWHURLGV VHH )LJXUH f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f 7KH 'DYLV HW DO f ODZ LV HTXLYDOHQW WR WKH VL]HLQGHSHQGHQW VWUHQJWK PRGHO DVVXPHG E\ 'RKQDQ\L f EXW ZLWK D WKHRUHWLFDO FRUUHFWLRQ WR DOORZ IRU WKH JUDYLWDWLRQDO VHOI FRPSUHVVLRQ RI ODUJH DVWHURLGV ,Q WKLV PRGHO WKH HIIHFWLYH LPSDFW VWUHQJWK LV DVVXPHG WR KDYH WZR FRPSRQHQWV WKH ILUVW GXH WR WKH PDWHULDO SURSHUWLHV RI WKH DVWHURLG DQG WKH VHFRQG GXH WR GHSWKGHSHQGHQW FRPSUHVVLYH ORDGLQJ RI WKH RYHUEXUGHQ :KHQ DYHUDJHG RYHU WKH YROXPH RI WKH DVWHURLG ZH KDYH IRU WKH HIIHFWLYH LPSDFW VWUHQJWK Ua?nf V 6R LUN*S ) f ZKHUH 6 LV WKH PDWHULDO LPSDFW VWUHQJWK S LV WKH GHQVLW\ DQG LV WKH GLDPHWHU )RU DVWHURLGV ZLWK GLDPHWHUV PXFK OHVV WKDQ DERXW NP WKH FRPSUHVVLYH ORDGLQJ EHFRPHV LQVLJQLILFDQW FRPSDUHG WR WKH PDWHULDO VWUHQJWK DQG a 6 \LHOGLQJ WKH VL]H LQGHSHQGHQW VWUHQJWK RI 'RKQDQ\L 7KH +RXVHQ HW DO f ODZ DOORZV IRU D VWUDLQUDWH GHSHQGHQFH RI WKH LPSDFW VWUHQJWK HIIHFWLYHO\ PDNLQJ ODUJHU DVWHURLGV ZHDNHU WKDQ WDUJHWV PHDVXUHG LQ WKH ODEn RUDWRU\ 7KH WKHRU\ LV GHVFULEHG LQ GHWDLO LQ +RXVHQ DQG +ROVDSSOH f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f VKRZ WKDW WKH LPSDFW VWUHQJWK LV RF efO9n f ZKHUH 9H LV WKH LPSDFW VSHHG 8QGHU WKHLU QRPLQDO UDWHGHSHQGHQW PRGHO WKH FRQVWDQW n ZKLFK LV GHSHQGHQW XSRQ VHYHUDO PDWHULDO SURSHUWLHV RI WKH WDUJHW LV HTXDO WR LQ WKH VWUHQJWK UHJLPH ZKHUH JUDYLWDWLRQDO VHOI FRPSUHVVLRQ LV QHJOLJLEOH ,Q WKH JUDYLW\ UHJLPH KRZHYHU WKH\ ILQG WKDW MM ZKLFK ZH QRWH LV VOLJKWO\ OHVV WKDQ WKH GHSHQGHQFH DVVXPHG E\ 'DYLV HW DO f 7KH PDJQLWXGH RI WKH JUDYLWDWLRQDO FRPSUHVVLRQ LQ WKH +RXVHQ HW DO f PRGHO ZDV GHWHUPLQHG E\ PDWFKLQJ H[SHULPHQWDO UHVXOWV RI WKH IUDJPHQWDWLRQ RI FRPSUHVVHG EDVDOW WDUJHWV LQGLFDWHG E\ WKH VROLG GRWV LQ )LJXUH f VLPXODWLQJ WKH RYHUEXUGHQ RI ODUJH DVWHURLGV DQG HVWLPDWHV RI LPSDFW VWUHQJWKV )XMLZDUD f GHWHUPLQHG IURP UHFRQVWUXFWLRQV RI PAGE 71 WKH SDUHQW ERGLHV RI WKH .RURQLV (RV DQG 7KHPLV DVWHURLG IDPLOLHV RSHQ GRWVf 7KH PRVW UHFHQW VWXGLHV KRZHYHU LQGLFDWH WKDW WKH ODERUDWRU\ UHVXOWV DUH WR EH WDNHQ DV XSSHU OLPLWV WR WKH PDJQLWXGH RI WKH JUDYLWDWLRQDO FRPSUHVVLRQ +ROVDSSOH SULYDWH FRPPXQLFDWLRQf %RWK VFDOLQJ ODZV KDYH EHHQ XVHG ZLWKLQ WKH FROOLVLRQDO PRGHO WR DWWHPSW WR SODFH VRPH FRQVWUDLQWV RQ WKH LQLWLDO PDVV RI WKH DVWHURLG EHOW DQG WKH VL]HVWUHQJWK VFDOLQJ UHODWLRQ LWVHOI 8QIRUWXQDWHO\ WKH LQLWLDO PDVV RI WKH EHOW LV QRW NQRZQ %\ nLQLWLDOn ZH DVVXPH WKH VDPH GHILQLWLRQ DV XVHG E\ 'DYLV HW DO f WKDW LV WKH PDVV DW WKH WLPH WKH PHDQ FROOLVLRQ VSHHG ILUVW UHDFKHG WKH FXUUHQW a NP Vn 'DYLV HW DO f SUHVHQW D UHYLHZ RI DVWHURLG FROOLVLRQ VWXGLHV DQG FRQFOXGH WKDW WKH DVWHURLGV UHSUHVHQW D FROOLVLRQDOO\ UHOD[HG SRSXODWLRQ ZKRVH LQLWLDO PDVV FDQQRW EH IRXQG IURP PRGHOV RI WKH VL]H HYROXWLRQ DORQH :H H[WUHPHV IRU DQ LQLWLDO EHOW PDVV D nPDVVLYHf LQLWLDO SRSXODWLRQ ZLWK a WLPHV WKH SUHVHQW EHOW PDVV EDVHG XSRQ ZRUN E\ :HWKHULOO SULYDWH FRPPXQLFDWLRQf RQ WKH UXQDZD\ DFFUHWLRQ RI SODQHWHVLPDOV LQ WKH LQQHU VRODU V\VWHP DQG D nVPDOOn LQLWLDO EHOW RI URXJKO\ WZLFH WKH SUHVHQW PDVV PDWFKLQJ WKH EHVW HVWLPDWH E\ 'DYLV HW DO f RI WKH LQLWLDO PDVV PRVW OLNHO\ WR SUHVHUYH WKH EDVDOWLF FUXVW RI 9HVWD )LJXUHV DQG VKRZ WKH UHVXOWV RI VHYHUDO UXQV RI WKH PRGHO ZLWK YDULRXV FRPELQDWLRQV RI VFDOLQJ ODZV DQG LQLWLDO SRSXODWLRQV ,Q ERWK ILJXUHV ZH KDYH LQFOXGHG WKH REVHUYHG VL]H GLVWULEXWLRQ IRU FRPSDULVRQ ZLWK PRGHO UHVXOWV EXW KDYH UHPRYHG WKH 9Â“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fGLSf LQ WKH QXPEHU RI DVWHURLGV LQ WKH UHJLRQ RI WKH WUDQVLWLRQ WR DQ HTXLOLEULXP SRZHU ODZ 7KH 'DYLV HW DO PRGHO GRHV D VRPHZKDW EHWWHU MRE RI ILWWLQJ WKH REVHUYHG GLVWULEXWLRQ LQ WKH WUDQVLWLRQ UHJLRQ IXUWKHU VXJJHVWLQJ WKDW D YHU\ SURQRXQFHG ZHDNHQLQJ RI VPDOO DVWHURLGV PD\ QRW EH UHDOLVWLF LQ WKLV VL]H UHJLPH ,Q DGGLWLRQ ZH KDYH IRXQG WKDW WKH PDJQLWXGH RI WKH JUDYLWDWLRQDO VWUHQJWKHQLQJ JLYHQ E\ WKH 'DYLV HW DO PRGHO VRPHZKDW ZHDNHU WKDQ WKH +RXVHQ HW DO PRGHOf SURGXFHV D FORVHU PDWFK WR WKH VKDSH RI WKH fKXPSf DW NP IRU WKH LQLWLDO SRSXODWLRQV ZH KDYH H[DPLQHG ,I VRPHWKLQJ QHDUHU WR WKH +RXVHQ HW DO JUDYLW\ VFDOLQJ WXUQV RXW WR EH PRUH DSSURSULDWH KRZHYHU WKLV ZRXOG VLPSO\ LQGLFDWH WKDW WKH VL]H GLVWULEXWLRQ ORQJZDUG RI a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f RQ WKH FROOLVLRQDO KLVWRU\ RI *DVSUD 7KH PRGLILHG ODZ PDWFKHV WKH +RXVHQ HW DO ODZ IRU VPDOO ODERUDWRU\f VL]H ERGLHV ZKHUH LPSDFW H[SHULPHQWV 'DYLV DQG 5\DQ f LQGLFDWH WKDW VWUDLQUDWH VFDOLQJ EHVW GHVFULEHV WKH IUDJPHQWDWLRQ RI PRUWDU WDUJHWV *UDYLWDWLRQDO VWUHQJWKHQLQJ VHWV LQ IRU ODUJH DVWHURLGV PDWFKLQJ WKH PDJQLWXGH RI WKH 'DYLV HW DO PAGE 73 PRGHO )RU VPDOO DVWHURLGV DQ HPSLULFDO PRGLILFDWLRQ KDV EHHQ PDGH WR DOORZ IRU WKH LQWHUSUHWDWLRQ RI VRPH FRQFDYH IDFHWV RQ *DVSUD DV LPSDFW VWUXFWXUHV *UHHQEHUJ HW DO f ,I *DVSUD DQG RWKHU VLPLODUVL]H REMHFWV VXFK DV 3KRERV $VSKDXJ DQG 0HORVK f DQG 3URWHXV &URIW f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f $IWHU ELOOLRQ \HDUV RI FROOLVLRQDO HYROXWLRQ ZH ILW WKH fKXPSf DW NP WKH VPRRWK WUDQVLWLRQ WR DQ HTXLOLEULXP GLVWULEXWLRQ DW a NP DQG WKH QXPEHU RI DVWHURLGV LQ WKH HTXLOLEULXP GLVWULEXWLRQ DQG LWV VORSH LQGH[ :H QRWH LQ SDUWLFXODU WKDW IRU WKH UDQJH RI VL]HV FRYHUHG E\ 3/6 GDWD NPf WKH VOLJKWO\ SRVLWLYH VORSH RI WKH PRGLILHG VFDOLQJ ODZ SUHGLFWV DQ HTXLOLEULXP VORSH IRU WKDW VL]H UDQJH RI DERXW OHVV WKDQ WKH 'RKQDQ\L YDOXH EXW SUHFLVHO\ PDWFKLQJ WKH YDOXH RI T s GHWHUPLQHG E\ D ZHLJKWHG OHDVWVTXDUHV ILW WR WKH FDWDORJXHG PDLQEHOW DQG 3/6 GDWD :KLOH ZH KDYH QR TXDQWLWDWLYH WKHRU\ WR DFFRXQW IRU RXU PRGLILHG VFDOLQJ ODZ WKHUH PD\ EH D PHFKDQLVP ZKLFK FRXOG H[SODLQ WKH VORZ VWUHQJWKHQLQJ RI NPVFDOH ERGLHV LQ D TXDOLWDWLYH PDQQHU 5HFHQW K\GURFRGH VLPXODWLRQV E\ 1RODQ HW DO f LQGLFDWH WKDW DQ LPSDFW LQWR D VPDOO DVWHURLG HIIHFWLYHO\ VKDWWHUV WKH PDWHULDO RI WKH DVWHURLG LQ DQ DGYDQFLQJ VKRFN IURQW ZKLFK SUHFHGHV WKH H[FDYDWHG GHEULV VR WKDW FUDWHU H[FDYDWLRQ WDNHV SODFH LQ HIIHFWLYHO\ XQFRQVROLGDWHG PDWHULDO 7KH UHPDLQLQJ ERG\ RI PAGE 74 WKH DVWHURLG LV WKXV UHGXFHG WR UXEEOH 'DYLV DQG 5\DQ f KDYH QRWHG WKDW FOD\ DQG ZHDN PRUWDU WDUJHWV PDWHULDOV ZLWK IDLUO\ ORZ FRPSUHVVLYH VWUHQJWKV VXFK DV WKH VKDWWHUHG PDWHULDO SUHGLFWHG E\ WKH K\GURFRGH PRGHOV PD\ KDYH YHU\ KLJK LPSDFW VWUHQJWKV GXH WR WKH SRRU FRQGXFWLRQ RI WHQVLOH VWUHVV ZDYHV LQ WKH fORVV\f PDWHULDO ,I WKLV PHFKDQLVP LQGHHG EHFRPHV LPSRUWDQW IRU REMHFWV PXFK ODUJHU WKDQ ODERUDWRU\ WDUJHWV EXW VLJQLILFDQWO\ VPDOOHU WKDQ WKRVH IRU ZKLFK JUDYLWDWLRQDO FRPSUHVVLRQ EHFRPHV LPSRUWDQW D PRUH JUDGXDO WUDQVLWLRQ IURP VWUDLQUDWH VFDOLQJ WR JUDYLWDWLRQDO FRPSUHVVLRQ ZRXOG EH ZDUUDQWHG PAGE 75 7DELF ,QWULQVLF FROOLVLRQ SUREDELOLWLHV DQG HQFRXQWHU VSHHGV IRU VHYHUDO PDLQEHOW DVWHURLGV $VWHURLG 3URSHU 6HPLPDMRU D[LV $8f 3URSHU (FFHQWULFLW\ 3URSHU ,QFOLQDWLRQ GHJUHHVf ,QWULQVLF &ROOLVLRQ 3UREDELOLW\ n \U NPf (QFRXQWHU 6SHHG NP Vnf &HUHV 3DOODV 9HVWD )ORUD 7KHPLV %UXQKLOG .RURQLV (RV 7LVLSKRQH PAGE 76 , OLOLO ,QLWLDO 6ORSH 'LDPHWHU NPf 21 )LJXUH 9HULILFDWLRQ RI PRGHO IRU VWHHS LQLWLDO VORSH DQG VPDOO KLQ VL]H PAGE 77 &' U4 D &' 46 R A %LQVL]H )LQDO UL L LQ P P ,QLWLDO 6ORSH 3UHGLFWHG T 'LDPHWHU NPf , , ,,, )LJXUH 9HULILFDWLRQ RI PRGHO ORU VKDOORZ LQLWLDO VORSH DQG VPDOO ELQ VL]H PAGE 78 r 'LDPHWHU NPf 21 /Q )LJXUH 9HULILFDWLRQ RI PRGHO IRU VWHHS LQLWLDO VORSH DQG ODUJH ELQ VL]H PAGE 79 Q 7 U777777 UU &' FW D &' &' 6O X D WLR R R %LQVL]H )LQDO T ,QLWLDO 6ORSH 3UHGLFWHG T L L L LOO O ///, ,, 'LDPHWHU NPf , ,, )LJXUH 9HULILFDWLRQ RI PRGHO IRU VKDOORZ LQLWLDO VORSH DQG ODUJH ELQ VL]H PAGE 80 7LPH %\UVf )LJXUH (TXLOLEULXP VORSH DV D IXQFWLRQ RI WLPH IRU YDULRXV IUDJPHQWDWLRQ SRZHU ODZV DQLO IRU VWHHS LQLWLDO VORSH PAGE 81 (TXLOLEULXP 6ORSH 7LPH %\UVf )LJXUH (TXLOLEULXP VORSH XV D XQFWLRQ R WLPH WRU YDULRXV IUDJPHQWDWLRQ SRZHU ODZV DQG ORU VKDOORZ LQLWLDO VORSH PAGE 82 (TXLOLEULXP 6ORSH O , O /B , , B -B 7LPH %\UVf 21 nY& )LJXUH (TXLOLEULXP VORSH DV D IXQFWLRQ RI WLPH IRU YDULRXV IUDJPHQWDWLRQ SRZHU ODZV DQG IRU HTXLOLEULXP LQLWLDO VORSH PAGE 83 Â+ t FG LG 6K EIO R 'LDPHWHU NPf )LJXUH :DYHOLNH GHYLDWLRQV LQ VL]H GLVWULEXWLRQ FDXVHG E\ WUXQFDWLRQ RI SDUWLFOH SRSXODWLRQ PAGE 84 } 'LDPHWHU NPf )LJXUH ,QGHSHQGHQFH RI WKH ZDYH RQ ELQ VL]H DGRSWHG LQ PRGHO PAGE 85 ORJ ,QFUHPHQWDO 1XPEHU W &2 'LDPHWHU NPf )LJXUH &RPSDULVRQ RI WKH LQWHUSODQHWDU\ GXVW OOX[ IRXQG E\ *ULLQ HW DO f DQG VPDOO SDUWLFOH FXWRIIV XVHG LQ RXU PRGHO PAGE 86 } 6+ &' U4 c] D &' &' 2 &-IO R 'LDPHWHU NPf )LJXUH :DYHOLNH GHYLDWLRQV LPSRVHG E\ D VKDUS SDUWLFOH FXWRII [ f PAGE 87 )LJXUH 'LDPHWHU NPf PAGE 88 &' FWL e &' &' R WX2 2 77777M L UUUUU 7 T Df fÂ§ 0\U fÂ§ 0\U T A L L L L L O 'LDPHWHU NPf )LJXUH &ROOLVLRQDO UHOD[DWLRQ RI D SHUWXUEDWLRQ WR DQ HTXLOLEULXP VL]H GLVWULEXWLRQ PAGE 89 )LJXUH 7LPH 0\Uf +DOIWLPH IRU H[SRQHQWLDO GHFD\ WRZDUG HTXLOLEULXP VORSH IROORZLQJ WKH IUDJPHQWDWLRQ RI D NP GLDPHWHU DVWHURLG PAGE 90 6ORSH 3DUDPHWHU 7LPH 0\Uf )LJXUH 6WRFKDVWLF IUDJPHQWDWLRQ RI LQQHU PDLQEHOW DVWHURLGV RI YDULRXV VL]HV GXULQJ D W\SLFDO PLOOLRQ SHULRG PAGE 91 (TXLOLEULXP )LJXUH (TXLOLEULXP VORSH SDUDPHWHU DV D IXQFWLRQ RI WKH VORSH RI WKH VL]HVWUHQJWK VFDOLQJ ODZ PAGE 92 FU FU Z LfÂ§LfÂ§LfÂ§U LfÂ§LfÂ§L U } f f f r D } f ff %DFNJURXQG , / > f f f D f D f f f f fÂ‘ )DPLO\ )DQQO\ O / / 7LPH %\UVf )LJXUH 'LIIHUHQFH LQ WKH HTXLOLEULXP VORSH SDUDPHWHUV IRU IDPLOLHV ZLWK GLIIHUHQW VWUHQJWK SURSHUWLHV PAGE 93 ,PSDFW 6WUHQJWK HUJ &2 777777,, PX IP L QWLLL R )XMLZDUD 0RGLILHG 6FDOLQJ /DZ +RXVHQ HW DL f // ,, 'LDPHWHU NPf )LJXUH 7KH 'DYLV HW DO f +RXVHQ HW DO f DQG PRGLILHG VFDOLQJ ODZV XVHG LQ WKH FROOLVLRQDO PRGHO PAGE 94 1XPEHU SHU 'LDPHWHU %LQ RR )LJXUH 7KH HYROYHG VL]H GLVWULEXWLRQ DOWHU ELOOLRQ \HDUV XVLQJ WKH +RXVHQ HW DO f VFDOLQJ ODZ IRU Df D PDVVLYH LQLWLDO SRSXODWLRQ DQG Ef D VPDOO LQLWLDO SRSXODWLRQ PAGE 95 1XPEHU SHU 'LDPHWHU %LQ O 'LDPHWHU NPf )LJXUH 7KH HYROYHG VL]H GLVWULEXWLRQ DOWHU ELOOLRQ \HDUV XVLQJ WKH 'DYLV HW DO f VFDOLQJ ODZ IRU Df D PDVVLYH LQLWLDO SRSXODWLRQ DQG Kf D VPDOO LQLWLDO SRSXODWLRQ PAGE 96 1XPEHU SHU 'LDPHWHU %LQ r &I& /N! )LJXUH 7KH HYROYHG VL]H GLVWULEXWLRQ DIWHU ELOOLRQ \HDUV XVLQJ RXU PRGLILHG VFDOLQJ ODZ IRU Df D PDVVLYH LQLWLDO SRSXODWLRQ DQG Ef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f ZKLFK KH DWWULEXWHG WR WKH FROOLVLRQDO IUDJPHQWDWLRQ RI D SDUHQW DVWHURLG +LUD\DPD LGHQWLILHG E\ H\H WKH WKUHH PRVW SURPLQHQW IDPLOLHV .RURQLV (RV DQG 7KHPLV ZKLFK KH QDPHG DIWHU WKH ILUVW GLVFRYHUHG DVWHURLG LQ HDFK JURXSf LQ WKLV ILUVW VWXG\ DQG DGGHG RWKHU WKRXJK SHUKDSV OHVV FHUWDLQ IDPLOLHV LQ D VHULHV RI ODWHU SDSHUV f $IWHU +LUD\DPDfV ILUVW VWXGLHV FODVVLILFDWLRQV RI DVWHURLGV LQWR IDPLOLHV KDYH EHHQ JLYHQ E\ PDQ\ RWKHU UHVHDUFKHUV %URXZHU $UQROG /LQGEODG DQG 6RXWKZRUWK :LOOLDPV =DSSDOÂ£ HW DO %HQGMR\D HW DO f DQG D QXPEHU RI RWKHU IDPLOLHV KDYH EHFRPH DSSDUHQW 6RPH UHVHDUFKHUV FODLP WR EH DEOH WR LGHQWLI\ PRUH WKDQ D KXQGUHG JURXSLQJV ZKLOH RWKHUV IHHO WKDW RQO\ WKH IHZ ODUJHVW IDPLOLHV WR EH FRQVLGHUHG UHDO 7KH GLVDJUHHPHQWV DULVH IURP WKH GLIIHUHQW VWDUWLQJ VHWV RI DVWHURLGV FRQVLGHUHG HDUO\ FODVVLILFDWLRQV LQFOXGHG IHZHU DVWHURLGV ZLWK PRUH PAGE 98 GLVFRYHUHG DVWHURLGV ODWHU LQYHVWLJDWRUV DUH DEOH WR LGHQWLI\ VPDOOHU OHVV SRSXODWHG IDPLOLHV ZKLFK ZHUH SUHYLRXVO\ XQVHHQf WKH GLIIHUHQW SHUWXUEDWLRQ WKHRULHV ZKLFK DUH XVHG WR FDOFXODWH WKH SURSHU HOHPHQWV DQG WKH GLIIHUHQW PHWKRGV XVHG WR GLVWLQJXLVK WKH IDPLO\ JURXSLQJV IURP WKH EDFNJURXQG DVWHURLGV RI WKH PDLQEHOW ZKLFK KDYH UDQJHG IURP H\HEDOO VHDUFKHV WR PRUH REMHFWLYH FOXVWHU DQDO\VLV WHFKQLTXHV 7KLV ODFN RI XQDQLPRXV DJUHHPHQW RQ WKH QXPEHU RI DVWHURLG IDPLOLHV RU RQ ZKLFK DVWHURLGV VKRXOG EH LQFOXGHG LQ IDPLOLHV SURPSWHG VRPH *UDGLH HW DO &DUXVL DQG 9DOVHFFKL f WR XUJH WKDW D IXUWKHU XQGHUVWDQGLQJ RI WKH GLVFUHSDQFLHV EHWZHHQ WKH GLIIHUHQW FODVVLILFDWLRQ VFKHPHV ZDV QHFHVVDU\ EHIRUH WKH SK\VLFDO UHDOLW\ RI DQ\ RI WKH IDPLOLHV FRXOG EH JLYHQ SODXVLEOH PHULW 2QO\ LQ WKH ODVW IHZ \HDUV KDYH GLIIHUHQW PHWKRGV OHDG WR D FRQYHUJHQFH LQ WKH IDPLOLHV LGHQWLILHG E\ GLIIHUHQW UHVHDUFKHUV =DSSDOÂ£ DQG &HOOLQR f 7KH =DSSDOÂ£ &ODVVLILFDWLRQ 7R GDWH SUREDEO\ WKH PRVW UHOLDEOH DQG FRPSOHWH FODVVLILFDWLRQ RI +LUD\DPD IDPLO\ PHPEHUV LV WKH UHFHQW ZRUN RI =DSSDOÂ£ HW DO f 7KH\ XVHG D VHW RI QXPEHUHG DVWHURLGV ZKRVH SURSHU HOHPHQWV ZHUH FDOFXODWHG XVLQJ D VHFRQGRUGHU LQ WKH SODQHWDU\ PDVVHVf IRXUWKGHJUHH LQ WKH HFFHQWULFLWLHV DQG LQFOLQDWLRQVf VHFXODU SHUWXUEDWLRQ WKHRU\ 0LODQL DQG .QH]HYLF f DQG FKHFNHG IRU ORQJWHUP VWDELOLW\ E\ QXPHULFDO LQWHJUDWLRQ $ KLHUDUFKLFDO FOXVWHULQJ WHFKQLTXH ZDV DSSOLHG WR WKH PDLQEHOW DVWHURLGV WR FUHDWH D GHQGURJUDP RI WKH SURSHU HOHPHQWV DQG FRPELQHG ZLWK D GLVWDQFH SDUDPHWHU UHODWHG WR WKH YHORFLW\ QHHGHG IRU RUELWDO FKDQJH DIWHU UHPRYDO IURP WKH SDUHQW ERG\ )DPLOLHV ZHUH WKHQ LGHQWLILHG E\ FRPSDULQJ WKH PDLQEHOW GHQGURJUDP ZLWK RQH JHQHUDWHG IURP D TXDVLUDQGRP GLVWULEXWLRQ RI RUELWV VLPXODWLQJ WKH DFWXDO GLVWULEXWLRQ PAGE 99 $ VLJQLILFDQFH SDUDPHWHU ZDV WKHQ DVVLJQHG WR HDFK IDPLO\ WR PHDVXUH LWV GHSDUWXUH IURP D UDQGRP FOXVWHULQJ $V UHYLVHG SURSHU HOHPHQWV EHFRPH DYDLODEOH IRU PRUH QXPEHUHG DVWHURLGV WKH FOXVWHULQJ DOJRULWKP LV HDVLO\ UHUXQ WR XSGDWH WKH FODVVLILFDWLRQ RI PHPEHUV LQ HVWDEOLVKHG IDPLOLHV DQG WR VHDUFK IRU QHZ VPDOO IDPLOLHV ,Q WKHLU ODWHVW FODVVLILFDWLRQ =DSSDOÂ£ HW DO SULYDWH FRPPXQLFDWLRQf ILQG IDPLOLHV RI ZKLFK DERXW DUH WR EH FRQVLGHUHG VLJQLILFDQW DQG UREXVW ,Q )LJXUH ZH KDYH SORWWHG WKH SURSHU LQFOLQDWLRQ YHUVXV VHPLPDMRU D[LV IRU DOO =DSSDOÂ£ IDPLOLHV DQG KDYH ODEHOHG VRPH RI WKH PRUH SURPLQHQW RQHV 7KH .RURQLV (RV DQG 7KHPLV IDPLOLHV UHPDLQ WKH PRVW UHOLDEOH KRZHYHU =DSSDOÂ£ DOVR FRQVLGHUV PDQ\ RI WKH VPDOOHU FRPSDFW IDPLOLHV VXFK DV 'RUD *HILRQ DQG $GHRQD TXLWH UHOLDEOH 7KH OHVV VHFXUH IDPLOLHV DUH XVXDOO\ WKH PRVW VSDUVHO\ SRSXODWHG RU WKRVH ZKLFK PLJKW SRVVLEO\ EHORQJ WR RQH ODUJHU JURXS DQG UHPDLQ WR EH FRQILUPHG DV PRUH FHUWDLQ SURSHU HOHPHQWV EHFRPH DYDLODEOH 7KH )ORUD IDPLO\ IRU LQVWDQFH DOWKRXJK TXLWH SRSXORXV LV FRQVLGHUHG D fGDQJHURXVf IDPLO\ KDYLQJ SURSHU HOHPHQWV ZKLFK DUH VWLOO TXLWH XQFHUWDLQ GXH WR LWV SUR[LPLW\ WR WKH VHFXODU UHVRQDQFH 7KH KLJK GHQVLW\ RI DVWHURLGV LQ WKLV UHJLRQ ZKLFK LV OLNHO\ D VHOHFWLRQ HIIHFW IDYRULQJ WKH GLVFRYHU RI VPDOO IDLQW DVWHURLGV LQ WKH LQQHU EHOW DOVR PDNHV WKH LGHQWLILFDWLRQ RI LQGLYLGXDO IDPLOLHV GLIILFXOW fÂ§ WKH HQWLUH UHJLRQ PHUJHV LQWR RQH ODUJH fFODQf PDNLQJ LW GLIILFXOW WR GHWHUPLQH ZKLFK RI WKH DVWHURLGV WKHUH DUH JHQHWLFDOO\ UHODWHG &ROOLVLRQDO (YROXWLRQ RI )DPLOLHV 1XPEHU RI )DPLOLHV 2QH FRQVWUDLQW RQ WKH FROOLVLRQDO KLVWRU\ RI WKH PDLQEHOW LV WKH QXPEHU RI IDPLOLHV ZKLFK KDYH EHHQ SURGXFHG DQG UHPDLQ YLVLEOH DW WKH SUHVHQW WLPH $ YHU\ PDVVLYH PAGE 100 LQLWLDO SRSXODWLRQ FRXSOHG ZLWK UHODWLYHO\ ZHDN DVWHURLGV ZRXOG LPSO\ WKDW QHDUO\ DOO WKH IDPLOLHV LGHQWLILDEOH WRGD\ PXVW EH UHODWLYHO\ \RXQJ $ VPDOOHU LQLWLDO EHOW DQG DVWHURLGV ZLWK ODUJH LPSDFW VWUHQJWKV ZRXOG DOORZ HYHQ PRGHVWVL]H IDPLOLHV WR VXUYLYH IRU ELOOLRQV RI \HDUV 7R DWWHPSW WR GLVWLQJXLVK EHWZHHQ WKHVH WZR SRVVLELOLWLHV DQG WR H[DPLQH WKH FROOLVLRQDO KLVWRU\ RI IDPLOLHV ZH PRGLILHG RXU VWRFKDVWLF FROOLVLRQDO PRGHO WR DOORZ XV WR IROORZ WKH HYROXWLRQ RI D IDPLO\ RI IUDJPHQWV UHVXOWLQJ IURP WKH EUHDNXS RI D VLQJOH ODUJH DVWHURLG VLPXODWLQJ WKH IRUPDWLRQ RI DQ DVWHURLG IDPLO\ $W D VSHFLILHG WLPH DQ DVWHURLG RI D VSHFLILHG VL]H LV IUDJPHQWHG DQG WKH GHEULV GLVWULEXWHG LQWR WKH PRGHOn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n NPn ZKLOH WKH SUREDELOLW\ RI FROOLVLRQV ZLWK RWKHU .RURQLV IDPLO\ PHPEHUV LV [ \Un NPn 'XH WR WKHLU VLPLODU LQFOLQDWLRQV DQG HFFHQWULFLWLHV KRZHYHU WKH PHDQ HQFRXQWHU VSHHG EHWZHHQ IDPLO\ PHPEHUV LV ORZHU WKDQ ZLWK DVWHURLGV RI WKH EDFNJURXQG SRSXODWLRQ UHTXLULQJ ODUJHU SURMHFWLOHV IRU IUDJPHQWDWLRQ 7KH PHDQ HQFRXQWHU VSHHG EHWZHHQ PHPEHUV RI WKH .RURQLV IDPLO\ IRU LQVWDQFH LV DSSUR[LPDWHO\ NP Vn VLJQLILFDQWO\ ORZHU WKDQ WKH URXJKO\ NP Vn HQFRXQWHU VSHHG EHWZHHQ O PAGE 101 .RURQLV IDPLO\ PHPEHUV DQG DVWHURLGV RI WKH EDFNJURXQG SURMHFWLOH SRSXODWLRQ 7KH YHU\ ODUJH WRWDO QXPEHU RI SURMHFWLOHV LQ WKH EDFNJURXQG SRSXODWLRQ FRPSOHWHO\ VZDPSV WKH VPDOO QXPEHU RI DVWHURLGV ZLWKLQ WKH IDPLO\ LWVHOI VR WKDW WKH FROOLVLRQDO HYROXWLRQ RI D IDPLO\ LV VWLOO GRPLQDWHG E\ FROOLVLRQV ZLWK WKH EDFNJURXQG DVWHURLG SRSXODWLRQ 7R GHWHUPLQH KRZ PDQ\ RI WKH IDPLOLHV SURGXFHG E\ WKH PRGHO VKRXOG EH REVHUYn DEOH DW WKH SUHVHQW WLPH ZH KDYH GHILQHG D VLPSOH IDPLO\ YLVLELOLW\ FULWHULRQ ZKLFK PLPLFV WKH FOXVWHULQJ DOJRULWKP DFWXDOO\ XVHG WR ILQG IDPLOLHV DJDLQVW WKH EDFNJURXQG DVWHURLGV RI WKH PDLQEHOW =DSSDOÂ£ HW DO f :H KDYH IRXQG WKH YROXPH GHQVLW\ RI QRQIDPLO\ DVWHURLGV LQ RUELWDO HOHPHQW VSDFH IRU WKH PLGGOH UHJLRQ RI WKH EHOW FRUUHVSRQGLQJ WR ]RQH RI =DSSDOÂ£ HW DO f ,Q WKH UHJLRQ D H DQG VLQ] ZH IRXQG QRQIDPLO\ DVWHURLGV ZKLFK \LHOGV D PHDQ YROn XPH GHQVLW\ W\SLFDO RI WKH PDLQEHOW RI $8 [ [ f DVWHURLGV SHU XQLW YROXPH RI SURSHU HOHPHQW VSDFH 7KH YROXPH GHQVLW\ RI WKH DVWHURLGV LQ D IDPLO\ LV WKHQ IRXQG E\ XVLQJ *DXVVf SHUWXUEDWLRQ HTXDWLRQV WR FDOn FXODWH WKH VSUHDG LQ RUELWDO HOHPHQWV DVVRFLDWHG ZLWK WKH IRUPDWLRQ RI WKH IDPLO\ VHH HJ =DSSDOÂ£ HW DO f 7KH W\SLFDO $9 DVVRFLDWHG ZLWK WKH HMHFWLRQ VSHHG RI WKH IUDJPHQWV ZLOO EH RI WKH RUGHU RI WKH HVFDSH VSHHG RI WKH SDUHQW DVWHURLG ZKLFK VFDOHV DV WKH GLDPHWHU 7KH W\SLFDO YROXPH RI D IDPLO\ PXVW WKHQ VFDOH DV =! VR WKDW IDPLOLHV IRUPHG IURP WKH GHVWUXFWLRQ RI ODUJH DVWHURLGV DUH VSUHDG RYHU D ODUJHU YROXPH :H FRPSXWHG WKH YROXPH DVVRFLDWHG ZLWK WKH IRUPDWLRQ RI D IDPLO\ IURP D SDUHQW NP LQ GLDPHWHU WKH VL]H RI WKH VPDOOHVW SDUHQW DVWHURLGV ZH FRQVLGHUf WR EH [ fÂ§ HOHPHQW XQLWV 7KH $. IRU D SDUHQW RI WKLV VL]H LV DSSUR[LPDWHO\ P Vn :LWKLQ WKH PRGHO WKH IDPLO\ YROXPH DVVRFLDWHG ZLWK D SDUHQW DVWHURLG RI DQ\ VSHFLILHG VL]H PAGE 102 LV WKHQ VFDOHG IURP WKLV YDOXH 7KH QXPEHU RI WHOHVFRSLFDOO\ YLVLEOH DVWHURLGV VSUHDG WKURXJKRXW WKLV YROXPH LV WKHQ XVHG WR FRPSXWH WKH IDPLO\nV YROXPH GHQVLW\ 7KH W\SLFDO FRPSOHWHQHVV OLPLW IRU IDPLOLHV LQ WKH PLGGOH PDLQEHOW LV a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n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Â£ HW DO f 2WKHU IDPLOLHV VXFK DV 9HVWD %LQ]HO DQG ;X f DQG SRVVLEO\ 7KHPLV :LOOLDPV f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f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f L 0DULD f (XQRPLD f f Â‘ ffr!r $GHRQD f ,D f f f W f (RV 9HVWD Y f W AY *HILRQ 'RUD )ORUD !f U9frn U ff f f \\ Â‘ 9L 1\VD ? .RURQLV 7KHPLV A QUn Nr frn f f Y 6HPLPDMRU $[LV $8f 7KH +LUDYDPD DVWHURLG IDPLOLHV PAGE 107 'HQVLW\ 5HODWLYH WR %DFNJURXQG 7LPH %\UVf )LJXUH 7KH FROOLVLRQDO GHFD\ RI IDPLOLHV UHVXOWLQJ IURP YDULRXVVL]H SDUHQW DVWHURLGV DV D IXQFWLRQ RI WLPH PAGE 108 )LJXUH )RUPDWLRQ RI IDPLOLHV LQ WKH PDLQEHOW DV D IXQFWLRQ RI WLPH PAGE 109 9& 21 )LJXUH 0RGHOHG FROOLVLRQDO KLVWRU\ RI WKH *HILRQ IDPLO\ PAGE 110 'LDPHWHU NPf )LJXUH 0RGHOHG FROOLVLRQDO KLVWRU\ RI WKH 0DULD IDPLO\ PAGE 111 &+$37(5 ,5$6 $1' 7+( $67(52,' $/ &2175,%87,21 72 7+( =2',$&$/ &/28' 7KH ,5$6 'XVWEDQGV 7KH ,QIUDUHG $VWURQRPLFDO 6DWHOOLWH ,5$6f ZDV FDUULHG IURP 9DQGHQEXUH $LU )RUFH %DVH WR LWV QHDUSRODU 6XQV\QFKURQRXV RUELW E\ D 'HOWD URFNHW RQ -DQXDU\ WK )RU PRQWKV WKH RQHWRQ VDWHOOLWH UHWXUQHG D ZHDOWK RI GDWD VXUYH\LQJ QHDUO\ b RI WKH VN\ DW DQG P EHIRUH LWV VXSSO\ RI OLTXLG KHOLXP FRRODQW UDQ GU\ VHH 0DWVRQ HW DO IRU D GHWDLOHG GHVFULSWLRQ RI WKH PLVVLRQf ,W LV D WHVWDPHQW WR WKH TXDQWLW\ DQG TXDOLW\ RI WKH GDWD UHWXUQHG E\ WKH WHOHVFRSH WKDW UHVHDUFKHUV DUH VWLOO PDNLQJ QHZ GLVFRYHULHV IURP ,5$6fV REVHUYDWLRQV D GHFDGH DIWHU LWV PLVVLRQ HQGHG 'HYHORSHG DV D MRLQW SURJUDP RI WKH 8QLWHG 6WDWHV WKH 1HWKHUODQGV DQG *UHDW %ULWDLQ ,5$6fV SULPDU\ PLVVLRQ ZDV WR VWXG\ VWDUIRUPLQJ UHJLRQV WKH SUHVHQFH RI FROG GXVW\ PDWHULDO LQ WKH JDOD[\ DQG WKH LQIUDUHG HPLVVLRQ IURP H[WUDJDODFWLF REMHFWV +RZHYHU RQH RI WKH PDLQ IDFWRUV FRQWULEXWLQJ WR WKH REVHUYDWLRQDO fQRLVHf ZDV WKH ZDUP FORXG RI VRODU V\VWHP GXVW ,Q IDFW WKH IOX[ LQ WKH DQG ÂP ZDYHEDQGV LV QHDUO\ FRPSOHWHO\ GRPLQDWHG E\ HPLVVLRQ IURP WKH ]RGLDFDO FORXG ,5$6 PDGH WKH VXUSULVLQJ GLVFRYHU\ RI WKUHH UHODWLYHO\ QDUURZ EDQGV RI LQIUDUHG HPLVVLRQ VXSHULPSRVHG RQ WKH EURDG ]RGLDFDO HPLVVLRQ /RZ HW DO 1HXJHEDXHU HW DO f 7KH PRVW SURPLQHQW EDQG OLHV QHDU WKH HFOLSWLF DW ODWLWXGHV RI r DQG LV IODQNHG E\ D IDLQWHU SDLU RI EDQGV DERYH DQG EHORZ WKH HFOLSWLF DW ODWLWXGHV RI r DQG r VHH )LJXUH f 7KH EDQGV FDQ DOVR EH VHHQ LQ WKH DQG ÂP GDWD DOWKRXJK DW D ORZHU PAGE 112 LQWHQVLW\ &RORUWHPSHUDWXUH FDOFXODWLRQV /RZ HW DO f \LHOG YDOXHV EHWZHHQ DQG FRQVLVWHQW ZLWK WKH WHPSHUDWXUH RI D UDSLGO\ URWDWLQJ JUD\ ERG\ ORFDWHG EHWZHHQ DQG $8 7KLV GLVWDQFH PDWFKHV HVWLPDWHV RI WKH ORFDWLRQ RI WKH EDQG HPLVVLRQ DW $8 REWDLQHG E\ SDUDOOD[ PHDVXUHPHQWV *DXWLHU HW DO +DXVHU HW DO 'HUPRWW HW DO f 7KH HVWLPDWHG ORFDWLRQ RI WKH EDQG SDLUV ZLWKLQ WKH DVWHURLG EHOW VXJJHVWHG WR /RZ HW DO f WKDW WKH EDQG HPLVVLRQ DURVH IURP WKH GXVW\ GHEULV SURGXFHG E\ FROOLVLRQV EHWZHHQ DVWHURLGV 'HUPRWW HW DO f GHPRQVWUDWHG WKDW WKH EDQGV DUH OLNHO\ DVVRFLDWHG ZLWK DVWHURLG IDPLOLHV QRWLQJ WKDW WKH ODWLWXGHV RI WKH GXVW EDQGV PDWFK WKH LQFOLQDWLRQV RI WKH WKUHH PRVW SURPLQHQW +LUD\DPD DVWHURLG IDPLOLHV 7KH\ OLQNHG WKH FHQWUDO GXVW EDQG ZLWK WKH 7KHPLV DQG .RURQLV IDPLOLHV DQG WKH r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f 0RGHOLQJ WKH 'XVW %DQGV 7R DQDO\]H WKH ,5$6 REVHUYDWLRQV RI WKH ]RGLDFDO HPLVVLRQ DQG WR GHWHUPLQH WKH GLVWULEXWLRQ RI GXVW ZLWKLQ WKH ]RGLDFDO FORXG 'HUPRWW DQG 1LFKROVRQ f GHYHORSHG D WKUHHGLPHQVLRQDO QXPHULFDO PRGHO 6,08/ ZKLFK SHUPLWV WKH FDOFXODWLRQ RI WKH GLVWULEXWLRQ RI VN\ EULJKWQHVV DV VHHQ E\ WKH ,5$6 WHOHVFRSH DVVRFLDWHG ZLWK DQ\ PAGE 113 SDUWLFXODU GLVWULEXWLRQ RI GXVW SDUWLFOH RUELWV 0RGLILFDWLRQV WR LPSURYH WKH PRGHO DQG LQFUHDVH LWV YHUVDWLOLW\ KDYH VLQFH EHHQ PDGH VHH IRU H[DPSOH ;X HW DO f 7KH 6,08/ PRGHO FRQVLVWV RI WKUHH PDMRU FRPSRQHQWV OfD UHSURGXFWLRQ RI WKH H[DFW YLHZLQJ JHRPHWU\ RI ,5$6 LQFOXGLQJ WKH HIIHFWV RI WKH HFFHQWULFLW\ RI WKH (DUWKfV RUELW f WKH GLVWULEXWLRQ RI RUELWDO HOHPHQWV RI WKH GXVW SDUWLFOHV LQ VSDFH DQG f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f $V DQ H[DPSOH ,5$6 REVHUYDWLRQV RI WKH GXVWEDQGV DW WKUHH GLIIHUHQW HORQJDWLRQV DQJOHV DUH FRPSDUHG ZLWK WKH IOX[HV SUHGLFWHG XVLQJ WKH 6,08/ PRGHO 7KH REVHUYDn WLRQV )LJXUH Df DUH LQ WKH ÂP ZDYHEDQG DQG LOOXVWUDWH WKH UDQJH RI DPSOLWXGHV DQG VKDSHV SURGXFHG E\ WKH YDULDEOH YLHZLQJ JHRPHWU\ GXULQJ WKH ,5$6 PLVVLRQ 7KH PRGHO SURILOHV LOOXVWUDWHG LQ )LJXUHV E F DQG G ZHUH SURGXFHG XVLQJ GXVW IURP VL[ SURPLQHQW IDPLOLHV 7KHPLV .RURQLV (RV 1\VD 'RUD DQG *HILRQ 7KH FURVVVHFWLRQDO DUHDV RI GXVW DVVRFLDWHG ZLWK WKH IDPLOLHV ZHUH WUHDWHG DV IUHH SDUDPHWHUV DQG DGMXVWHG WR ILW WKH REVHUYDWLRQV DW HORQJDWLRQ DQJOH r ([DFWO\ WKH VDPH SDUWLFOH GLVWULEXWLRQ PAGE 114 ZDV XVHG IRU WKH RWKHU WZR HORQJDWLRQV ZLWK WKH H[FHSWLRQ WKDW WKH WRWDO DUHD KDG WR EH DGMXVWHG GRZQZDUG VOLJKWO\ IRU HORQJDWLRQ r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f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a [ NP 7KLV LV IRXQG WR EH b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a NP 7KLV H[FHVV HIIHFWLYH YROXPH UHSUHVHQWV D UHPQDQW RI WKH LQLWLDO DVWHURLG SRSXODWLRQ ZKLFK KDV QRW \HW UHDFKHG FROOLVLRQDO HTXLOLEULXP DQG GRHV QRW FRQWULEXWH WR WKH PDLQEHOW GXVW DUHD 7R DFWXDOO\ FDOFXODWH WKH UDWLR RI IDPLO\ WR QRQIDPLO\ DUHDV ZH ILW DQ HTXLOLEULXP GLVWULEXWLRQ WKURXJK WKH FRPELQHG PDJQLWXGH GLVWULEXWLRQ RI DOO WKH =DSSDOÂ£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f LQ HDFK SRSXODWLRQ WKH\ QHHG QRW DFWXDOO\ EH FDOFXODWHG WR REWDLQ WKH UDWLR ZH VHHN VLQFH ERWK DUH UHODWHG WR WKH HIIHFWLYH YROXPHV ZKLFK LQ WXUQ DUH GHWHUPLQHG E\ WKH LQWHUFHSWV RI WKH OHDVWVTXDUHV VROXWLRQV :H WDNH DV D PHDVXUH RI WKH LQWHUFHSW WKH GLDPHWHU RI WKH ODUJHVW DVWHURLG 'PD[ ZKLFK ZRXOG EH SUHVHQW LQ WKH PAGE 117 HTXLOLEULXP GLVWULEXWLRQ ILW WR WKH GDWD )RU WKH HQWLUH PDLQEHOW SRSXODWLRQ ZH ILQG WKDW 'PD [ 2,p NP ZKLOH IRU WKH FRPELQHG IDPLO\ GLVWULEXWLRQ 'PD[ J NP )RU D VSHFLILHG FXWRII VL]H 'PWQ WKH WRWDO JHRPHWULFDO FURVVVHFWLRQDO DUHD DVVRFLDWHG ZLWK WKH GHEULV LV WKHQ FDOFXODWHG GLUHFWO\ IURP (TXDWLRQ RI $SSHQGL[ % )RU -\PLQ P WKH FURVVVHFWLRQDO DUHDV IRU WKH HQWLUH PDLQEHOW SRSXODWLRQ DQG DOO IDPLOLHV FRPELQHG DUH [ DQG [ 8 NPf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b RI WKH WRWDO ]RGLDFDO HPLVVLRQ ,I WKH JUDGXDO FRPPLQXWLRQ RI QRQIDPLO\ DVWHURLGV LQ WKH PDLQEHOW SURGXFHV DERXW WLPHV DV PXFK GXVW DV WKDW DVVRFLDWHG ZLWK IDPLOLHV WKHQ WKH HQWLUH PDLQEHOW DVWHURLG SRSXODWLRQ PXVW EH UHVSRQVLEOH IRU DW OHDVW D WKLUG RI WKH GXVW SDUWLFOHV LQ WKH ]RGLDFDO FORXG PAGE 118 (FOLSWLF /DWLWXGH 7KH )RXULHU ILOWHU PAGE 119 7RWDO )OX[ -\6Uf (FOLSWLF /DWLWXGH )LJXUH Df ,5$6 REVHUYDWLRQV RI WKH GXVW EDQGV DW HORQJDWLRQ DQJOHV RI r r DQG r &RPSDULVRQV ZLWK PRGHO SURILOHV EDVHG RQ SURPLQHQW +LUD\DPD IDPLOLHV DUH VKRZQ LQ Ef Ff DQG Gf PAGE 120 1XPEHU SHU 'LDPHWHU %LQ )LJXUH 7KH UDWLR RI DUHDV RI GXVW DVVRFLDWHG ZLWK WKH HQWLUH PDLQEHOW DVWHURLG SRSXODWLRQ DQG DOO IDPLOLHV PAGE 121 &+$37(5 6800$5< &RQFOXVLRQV :H PD\ VXPPDUL]H WKH PDLQ FRQFOXVLRQV RI WKLV ZRUN f 'DWD IURP WKH 3DORPDU/HLGHQ 6XUYH\ RI IDLQW DVWHURLGV KDV EHHQ XVHG WR VXSSOHPHQW GDWD IURP WKH FDWDORJXHG SRSXODWLRQ RI DVWHURLGV WR H[WHQG WKH VL]HIUHTXHQF\ GLVWULEXWLRQ RI WKH PDLQEHOW WR GLDPHWHUV RI a NP 7KH REVHUYHG VL]H GLVWULEXWLRQ GLVSOD\V D PDUNHG fKXPSf DW VL]HV QHDU NP DQG PDNHV D JUDGXDO WUDQVLWLRQ WR D GLVWLQFWO\ OLQHDU GLVWULEXWLRQ IRU GLDPHWHUV OHVV WKDQ DERXW NP 7KH REVHUYHG VORSH RI WKH OLQHDU SRUWLRQ LV VOLJKWO\ WKRXJK VWDWLVWLFDOO\ VLJQLILFDQWO\ OHVV WKDQ WKH HTXLOLEULXP VORSH SUHGLFWHG E\ 'RKQDQ\L f 7KH REVHUYHG GLVWULEXWLRQ LV TXLWH ZHOO GHWHUPLQHG DQG FRQVWLWXWHV D VWURQJ FRQVWUDLQW RQ FROOLVLRQDO PRGHOV RI WKH DVWHURLG SRSXODWLRQ f :H KDYH GHYHORSHG D QXPHULFDO PRGHO WR VWXG\ WKH FROOLVLRQDO HYROXWLRQ RI WKH DVWHURLGV ZKLFK FRQILUPV WKH HDUOLHU UHVXOWV RI 'RKQDQ\L f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f :KHQ XVHG ZLWKLQ RXU FROOLVLRQDO PRGHO WKH VL]HLQGHSHQGHQW DQG VWUDLQUDWH VFDOLQJ ODZV RI 'DYLV HW DO f DQG +RXVHQ HW DO f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f :KHQ WKH VHOIVLPLODULW\ RI WKH RULJLQDO 'RKQDQ\L f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f $QDO\VLV RI ,5$6 GDWD KDV VKRZQ WKDW DOWKRXJK WKH VRODU V\VWHP GXVW EDQGV DUH RQO\ DERXW b WKH VWUHQJWK RI WKH EURDG ]RGLDFDO HPLVVLRQ D VLJQLILFDQW SRUWLRQ RI WKH GXVW UHVSRQVLEOH IRU WKH EDQGV FRQWULEXWHV WR WKH EURDG EDFNJURXQG VR WKDW WKH SURPLQHQW IDPLOLHV DFWXDOO\ VXSSO\ DERXW b RI WKH GXVW LQ WKH ]RGLDFDO FORXG 2XU FRPSDULVRQ RI WKH HIIHFWLYH YROXPHV RI WKH IDPLOLHV DQG WKH SRUWLRQ RI WKH PDLQEHOW SRSXODWLRQ LQ FROOLVLRQDO HTXLOLEULXP VKRZV WKDW WKH QRQIDPLO\ PDLQEHOW DVWHURLGV SURGXFH DSSUR[LPDWHO\ WLPHV DV PXFK GXVW DV WKH SURPLQHQW IDPLOLHV $OO PDLQEHOW DVWHURLGV PXVW WKHQ VXSSO\ DW OHDVW b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f ZLOO \LHOG TXDQWLWDWLYH LQIRUPDWLRQ RQ WKH UDWH RI WUDQVSRUW RI DVWHURLGDO SDUWLFOHV WR WKH LQQHU VRODU V\VWHP DQG WKH FRPPLQXWLRQ RI WKH DVWHURLGV $SSUR[LPDWHO\ b RI WKH DVWHURLGDO SDUWLFOHV SDVVLQJ WKH (DUWK DUH PAGE 124 ,OO WHPSRUDULO\ IRU a \HDUVf WUDSSHG LQ UHVRQDQW ORFN ZLWK WKH SODQHW ,I WKH PDVV LQSXW UHTXLUHG WR VXSSO\ WKH REVHUYHG ULQJ FDQ EH GHWHUPLQHG WKH SURGXFWLRQ UDWH RI GXVW\ DVWHURLGDO GHEULV RYHU DW OHDVW WKH ODVW f LQ WKH PDLQEHOW ZLOO EH TXDQWLWLHG DQG ZLOO SURYLGH DQ H[WUHPHO\ VWURQJ FRQVWUDLQW RQ FROOLVLRQDO PRGHOV RI WKH PDLQEHOW $OWKRXJK ZH KDYH FRQFOXGHG WKDW WKH ZDYH LQGXFHG E\ WKH UHPRYDO RI WKH VPDOOHVW SDUWLFOHV LQ WKH SRSXODWLRQ LV SUREDEO\ QRW DQ LPSRUWDQW IHDWXUH RI WKH DFWXDO DVWHURLG VL]H GLVWULEXWLRQ ZH FDXWLRQ WKDW PRUH ZRUN QHHGV WR EH GRQH RQ WKH SUREOHP $ b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f 2QH UHVROXWLRQ RI WKLV GLVFUHSDQF\ PD\ OLH LQ D PAGE 125 FORXG RI DVWHURLGDO SDUWLFOHV ZKRVH HIIHFWLYH DUHD LQFUHDVHV ZLWK GHFUHDVLQJ KHOLRFHQWULF GLVWDQFH DV PLJKW EH H[SHFWHG IRU SDUWLFOHV XQGHUJRLQJ FRQWLQXDO FROOLVLRQDO HYROXWLRQ FRQFXUUHQW ZLWK RUELWDO GHFD\ GXH WR UDGLDWLRQ HIIHFWV :RUN RQ WKLV SUREOHP KDV DOUHDG\ EHDXQ *XVWDIVRQ HW DO f 7KH UHVXOWV ZLOO DOVR \LHOG D GHVFULSWLRQ RI WKH YDULDWLRQ RI WKH SDUWLFOH VL]H GLVWULEXWLRQ ZLWK KHOLRFHQWULF GLVWDQFH ,I D FORXG RI DVWHURLGDO SDUWLFOHV LV VKRZQ WR FRQWULEXWH PRUH WR WKH EDFNJURXQG IOX[ DW KLJK HFOLSWLF ODWLWXGHV WKH WRWDO FRQWULEXWLRQ PDGH WR WKH ]RGLDFDO FORXG E\ IDPLOLHV ZRXOG LQFUHDVH WR JUHDWHU WKDQ WKH SUHVHQW HVWLPDWH RI ab 7KH WRWDO VXSSO\ RI GXVW PDGH E\ WKH PDLQEHOW DVWHURLG SRSXODWLRQ ZRXOG WKHQ EH JUHDWHU WKDQ b fÂ§ LI WKH IDPLO\ FRQWULEXWLRQ VLPSO\ GRXEOHG WR b WKH WRWDO DVWHURLGDO FRPSRQHQW ZRXOG LQFUHDVH WR QHDUO\ b UHYHUVLQJ WKH SUHVHQWO\ HVWLPDWHG UDWLR RI DVWHURLG WR FRPHW GXVW PAGE 126 $33(1',; $ $33$5(17 $1' $%62/87( 0$*1,78'(6 2) $67(52,'6 7KH PDJQLWXGHV RI VRODU V\VWHP REMHFWV DUH GHVFULEHG XVLQJ WKH VDPH V\VWHP DV LQ VWHOODU DVWURQRP\ fÂ§ QDPHO\ WKHUH LV D IDFWRU RI LQ IOX[ DVVRFLDWHG ZLWK D PDJQLWXGH GLIIHUHQFH RI XQLWV ,Q RWKHU ZRUGV PDJQLWXGH O2fn :LWK WKLV GHILQLWLRQ PDJ O2f PDJ O2f DQG LQ JHQHUDO WKH UDWLR RI IOX[HV IURP WZR REMHFWV ZLWK PDJQLWXGHV P? DQG PR LV )? )R YP"Q $fÂ§f 1RWH WKDW LQ WKH ODVW HTXDWLRQ WKH LV H[DFW DQG QRW URXQGHG RII $Q DEVROXWH PDJQLWXGH PD\ EH GHILQHG DV WKH DSSDUHQW PDJQLWXGH REVHUYHG ZKHQ WKH REMHFW LV DW VRPH VWDQGDUG GLVWDQFH )URP WKH LQYHUVHVTXDUH ODZ RI OLJKW SURSDJDWLRQ ZH KDYH IRU WKH IOX[HV RI WZR LGHQWLFDO REMHFWV REVHUYHG DW GLIIHUHQW GLVWDQFHV $fÂ§f )RU VWHOODU VRXUFHV WKH VWDQGDUG GLVWDQFH LV SDUVHFV \LHOGLQJ DIWHU VXEVWLWXWLQJ (T ZLWK UL IRU )L)R LQ (Tf WKH IDPLOLDU P fÂ§ 0 ORJ ORJU fÂ§ $fÂ§f PAGE 127 ZKHUH 0 LV WKH DSSDUHQW PDJQLWXGH DW SDUVHFV 6LPLODUO\ ZH FDQ GHILQH DQ DEVROXWH PDJQLWXGH IRU VRODU V\VWHP REMHFWV ,I ZH OHW U EH WKH GLVWDQFH RI WKH REMHFW IURP WKH 6XQ DQG S EH WKH GLVWDQFH IURP WKH (DUWK LQ $VWURQRPLFDO 8QLWVf (T \LHOGV $f VLQFH WKH REMHFW DSSHDUV GLPPHU GXH WR ERWK LWV LQFUHDVHG GLVWDQFH IURP WKH 6XQ OHVV LQWHUFHSWHG OLJKWf DQG IURP WKH (DUWK GHFUHDVHG IOX[f 7KLV LV VLPLODU WR WKH UHDVRQ WKDW WKH VWUHQJWK RI D UDGDU VLJQDO GHWHFWHG IURP DQ REMHFW YDULHV LQYHUVHO\ ZLWK WKH IRXUWK SRZHU RI LWV GLVWDQFH fÂ§ WKHUH LV D OU GHFUHDVH LQ IOX[ LQ ERWK WKH WUDQVPLWWHG EHDP DQG WKH UHIOHFWHG VLJQDOf 7KH DEVROXWH PDJQLWXGH RI D VRODU V\VWHP REMHFW LV GHILQHG WR EH WKH DSSDUHQW PDJQLWXGH LW ZRXOG KDYH LI REVHUYHG ZKHQ $8 IURP WKH (DUWK $8 IURP WKH 6XQ DQG DW r SKDVH DQJOH 2XU VWDQGDUG GLVWDQFH XQLW LV WKHQ US DQG DIWHU VXEVWLWXWLRQ (T UHDGV PR $f + fÂ§ ORJUS $fÂ§f ZKHUH + LV WKH 9EDQG DEVROXWH PDJQLWXGH )URP (T ZH VHH WKDW WKH REVHUYHG IOX[ RI DQ DVWHURLG LV SURSRUWLRQDO WR r$+ %XW WKH IOX[ IURP WKH DVWHURLG GHSHQGV RQ LWV FURVV VHFWLRQDO DUHD D ODUJH DVWHURLG DSSHDUV EULJKWHU WKDQ D VPDOO DVWHURLGf DQG LWV JHRPHWULF DOEHGR DQ DVWHURLG ZLWK D EULJKW VXUIDFH LV PRUH UHIOHFWLYH WKDQ DQ DVWHURLG ZLWK D GDUN VXUIDFHf :H WKHQ KDYH WKDW )OX[ H[ S' D rn+ $fÂ§f PAGE 128 ZKHUH SY LV WKH JHRPHWULF DOEHGR DQG LV WKH GLDPHWHU RI WKH DVWHURLG 7KHQ ORJS ORJ! FRQVW fÂ§ 2$+ $f ORJ FRQVW fÂ§ fÂ§ ORJS $fÂ§f ZKLFK LV WKH H[SUHVVLRQ JLYHQ E\ =HOOQHU f 1RWH WKDW f % 9f 9Uf +f 7KH FRQVWDQW KDV WKH YDOXH DQG LV GHULYHG NQRZLQJ WKH DSSDUHQW PDJQLWXGH RI WKH 6XQ PAGE 129 $33(1',; % 6,=( 0$66 $1' 0$*1,78'( ',675,%87,216 7KH GLVWULEXWLRQ RI VL]HV DQG PDVVHV RI DVWHURLGV PD\ EH SUHVHQWHG LQ D QXPEHU RI ZD\V FXPXODWLYH SORWV RI WKH QXPEHU ODUJHU RU PRUH PDVVLYH WKDQ Y LQFUHPHQWDO SORWV QXPEHU SHU VL]H RU PDVV ELQf ZLWK OLQHDU LQFUHPHQWV DQG LQFUHPHQWDO SORWV ZLWK ORJDULWKPLF LQFUHPHQWV 7KLV QRWH LV PHDQW WR GHWDLO WKH UHODWLRQVKLSV EHWZHHQ WKH YDULRXV SORWV DQG WR GHULYH H[SUHVVLRQV IRU WKH WRWDO PDVV DQG FURVVVHFWLRQDO DUHD LQ WKH IUDJPHQWV LQ WKH GLVWULEXWLRQ ,W LV ZHOO NQRZQ WKDW PDQ\ IUDJPHQWDWLRQ HYHQWV LQ QDWXUH SURGXFH D SRZHU ODZ VL]H RU PDVVf GLVWULEXWLRQ RI IUDJPHQWV $ SRZHU ODZ GLVWULEXWLRQ KDV WKH IRUP 1 &'a3 %fÂ§ f ZKHUH 1 LV WKH FXPXODWLYH QXPEHU ODUJHU WKDQ GLDPHWHU DQG & LV D FRQVWDQW %\ WDNLQJ WKH FRPPRQ ORJDULWKP RI ERWK VLGHV ORJ 1 S ORJ I ORJ & %f ZH VHH WKDW WKH SRZHU ODZ H[SRQHQW S LV WKH QHJDWLYH VORSH LQ D ORJ$0RJ' SORW DQG WKH FRQVWDQW & GHILQHV WKH \LQWHUFHSW 7R VHH PRUH FOHDUO\ DQ\ FRQFHQWUDWLRQV RU GHSOHWLRQV RI SDUWLFOHV LQ FHUWDLQ VL]H UDQJHV DQ LQFUHPHQWDO SORW LV PRUH XVHIXO :H PXVW EH FDUHIXO KRZHYHU WR FOHDUO\ GHILQH WKH NLQG RI LQFUHPHQW ZKLFK KDV EHHQ FKRVHQ fÂ§ OLQHDU RU ORJDULWKPLF 'LIIHUHQWLDWLQJ O LK PAGE 130 (TXDWLRQ ZH REWDLQ %fÂ§f ZKHUH G1 LV WKH QXPEHU LQ WKH OLQHDU LQFUHPHQW RI ZLGWK G' 7KH QHJDWLYH VLJQ VLPSO\ IRUPDOO\ LQGLFDWHV WKDW WKH QXPEHU SHU ELQ GHFUHDVHV ZLWK LQFUHDVLQJ GLDPHWHU ZH DUH LQWHUHVWHG LQ WKH PDJQLWXGH RI WKH FKDQJH VR WKDW WKH QHJDWLYH VLJQ PD\ EH LJQRUHG 6HH DOVR (TXDWLRQ EHORZf 7DNLQJ WKH ORJDULWKP RI ERWK VLGHV ?RJG1 fÂ§ S f ORJ ORJ ^S&G'f %f ZH VHH WKDW WKH VORSH RI WKH VL]H GLVWULEXWLRQ RQ D ?RJG1?RJ' SORW LV fÂ§ ^S f ,I WKH FXPXODWLYH SORW KDG D VORSH RI WKH LQFUHPHQWDO SORW ZLWK OLQHDU LQFUHPHQWV ZRXOG KDYH D VORSH RI 1RZ VLQFH LORJU ZH PD\ UHZULWH (TXDWLRQ DV G1 fÂ§ fÂ§ S&?QO2'aSG ORJ %fÂ§f W 7KLV WKHQ UHSUHVHQWV DQ LQFUHPHQWDO VL]H GLVWULEXWLRQ ZLWK ORJDULWKPLF LQFUHPHQWV )URP ORJ G1 fÂ§S ORJ ORJ S&OQO2G ORJ 'f %fÂ§f ZH VHH WKDW WKH VORSH LV WKH VDPH DV WKDW IRU D FXPXODWLYH SORW QDPHO\ S ,Q PDQ\ FDVHV WKH IUDJPHQW GLVWULEXWLRQ LV GHVFULEHG LQ WHUPV RI PDVV UDWKHU WKDQ VL]H 6LQFH 0 AQS5 _QS' (TXDWLRQ PD\ EH UHZULWWHQ LQ WHUPV RI WKH PDVV DV &0a &0L %f 7KH VORSH RI D FXPXODWLYH PDVV GLVWULEXWLRQ SORW LV WKHQ VLPSO\ RQH WKLUG WKDW RI WKH FXPXODWLYH VL]H GLVWULEXWLRQ SORW $V EHIRUH LI WKH VORSH RI D FXPXODWLYH PDVV PAGE 131 GLVWULEXWLRQ SORW LV T WKH FRUUHVSRQGLQJ LQFUHPHQWDO SORW ZLWK OLQHDU LQFUHPHQWV ZLOO KDYH D VORSH RI fÂ§IT f DQG WKH LQFUHPHQWDO SORW ZLWK ORJDULWKPLF LQFUHPHQWV ZLOO KDYH D VORSH RI T 'RKQDQ\L f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nG' %f VLQFH 0 MS' (TXDWLRQ LV LQWHJUDWHG RYHU WKH VL]H UDQJH IURP WKH VPDOOHVW WR ODUJHVW SDUWLFOHV SUHVHQW $IWHU FDUU\LQJ RXW WKH LQWHJUDWLRQ WKH ILQDO H[SUHVVLRQ IRU WKH WRWDO PDVV LV 0 WRW QSS& Sf 'aS KL D [ UQ WX %fÂ§f 7KH WRWDO FURVVVHFWLRQDO DUHD LV IRXQG LQ D VLPLODU PDQQHU E\ LQWHJUDWLQJ WKH FURVV VHFWLRQ RI D VLQJOH SDUWLFOH A' RYHU WKH VL]H GLVWULEXWLRQ $ WRW 7 'S&'aSOG' %fÂ§f ZKLFK \LHOGV D WRWDO JHRPHWULFDO FURVVVHFWLRQDO DUHD RI $ WRW r3& '9 Sf I 8; WQ _I %OOf PAGE 132 5HFDOO WKDW LQ WKHVH H[SUHVVLRQV S DQG & DUH WKH QHJDWLYH VORSH DQG FRQVWDQW IRU WKH FXPXODWLYH VL]H GLVWULEXWLRQf :KHQ GLVFXVVLQJ WKH DVWHURLGV ZH RIWHQ DOVR XVH WKH IUHTXHQF\ GLVWULEXWLRQ RI PDJQLWXGHV LQ OLHX RI WKH VL]H GLVWULEXWLRQ 7KH PDJQLWXGHV DUH ELQQHG WKH 3/6 XVHV KDOIPDJQLWXGH ELQV IRU LQVWDQFHf DQG WKH QXPEHU RI DVWHURLGV SHU ELQ LV SUHVHQWHG LQ D ?RJG10DJ SORW 5HPHPEHULQJ WKDW ORJ FRQVW fÂ§ 2$+ fÂ§ ORJSf ZH VHH WKDW VXFK D SORW LV HTXLYDOHQW WR DQ LQFUHPHQWDO VL]HIUHTXHQF\ GLVWULEXWLRQ SORW ZLWK ORJDULWKPLF LQFUHPHQWV VLQFH DQ LQFUHPHQW RI [ LQ DEVROXWH PDJQLWXGH + FRUUHVSRQGV WR DQ LQFUHPHQW RI [ LQ ORJ' :H FDQ WKHQ GHULYH H[SUHVVLRQV ZKLFK ZLOO DOORZ GLUHFW FDOFXODWLRQ RI WKH WRWDO PDVV RU WRWDO FURVVVHFWLRQDO DUHD IURP WKH PDJQLWXGHIUHTXHQF\ SORW &RQVLGHU D ?RJG1+ SORW RI WKH IRUP ORJ G1 D+ E %fÂ§f ZKHUH D LV WKH VORSH DQG E LV WKH \LQWHUFHSW 6XEVWLWXWLQJ IRU + ZH REWDLQ ORJ G1 DfÂ§ ORJ FRQVW fÂ§ ORJ!ff E fÂ§ fÂ§D ORJ DFRQVW fÂ§ D ORJSf Ef &RPSDULQJ (TV DQG ZH VHH WKDW %fÂ§f %fÂ§f DQG ORJS&OQ O2FORJ'f DFRQVW fÂ§ DORJSf E %fÂ§f )RU D SRSXODWLRQ LQ FROOLVLRQDO HTXLOLEULXP ZLWK D VORSH SDUDPHWHU S WKH VORSH RI WKH PDJQLWXGH GLVWULEXWLRQ LV WKHQ D f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fÂ§ G ORJ G+f PAGE 134 $33(1',; & 327(17,$/ 2) $ 63+(5,&$/ 6+(// :H ZLVK WR ILQG WKH JUDYLWDWLRQDO ELQGLQJ HQHUJ\ RI D VSKHULFDO VKHOO RI PDVV 0 FRYHULQJ D VSKHUH RI PDVV 0 7KLV LV WKH HQHUJ\ QHHGHG WR GLVSHUVH WKH IUDJPHQWV RI D EDUHO\ FDWDVWURSKLF FROOLVLRQ ZKLFK E\ GHILQLWLRQ KDV b RI WKH PDVV RI WKH WDUJHW VKDWWHUHG DQG GLVSHUVHGf DQG LV SUREDEO\ D IDLUO\ JRRG DSSUR[LPDWLRQ WR D FRUHn W\SH VKDWWHULQJ FROOLVLRQ $ WDUJHW DVWHURLG ZLWK WRWDO PDVV 0 DQG UDGLXV 5 KDV b RI LWV PDVV FRQWDLQHG LQ D VSKHULFDO VKHOO ZLWK UDGLXV U D f n 5 DSSUR[LPDWHO\ "f WR U 5 7KH YROXPH RI WKH VKHOO LV JLYHQ E\ &Of ,I ZH DVVXPH WKDW WKH PDVV LV XQLIRUPO\ GLVWULEXWHG ZLWKLQ WKH VKHOO ZH FDQ ZULWH 3 +M0f W fÂ§ Df &fÂ§f :LWKLQ WKH VKHOO ZKLFK VLWV XSRQ D FRUH RI PDVV 0f WKH PDVV LV JLYHQ E\ U 0Uf OR $[[ G[ ?fÂ§0 &f D &f &f PAGE 135 DQG ,0 Uf 0U Â Df &f 7KH ELQGLQJ HQHUJ\ FDQ QRZ EH FDOFXODWHG 5 *0UfG0Uf 5 GU $I 0 fÂ§ D I D 0U Â D GU D D &f 0 fÂ§ Df UA fÂ§ DrU D U GU D &fÂ§f *0 U DrU Df > Df Df U D &fÂ§f *0 fÂ§ Df D fÂ§ Df Df D= fÂ§ Df AD &f *0 " f s I f -U i fA 7KH ODVW WZR WHUPV LQ WKH EUDFNHWV FDQFHO OHDYLQJ &f D A > @ &fÂ§f &fÂ§f 7KLV FRPSDUHV WR WKH ELQGLQJ HQHUJ\ RI D XQLIRUP VSKHUH WKH HQHUJ\ QHHGHG WR GLVSHUVH DQ HQWLUH VSKHUH RI PDVV 0 DQG UDGLXV 5f LQ ZKLFK FDVH WKH FRQVWDQW LV f 7KXV DV H[SHFWHG LW WDNHV VRPHZKDW OHVV HQHUJ\ WR GLVSHUVH D VKHOO RI RQHKDOI WKH WRWDO PDVV RII RI D WDUJHW DVWHURLG PAGE 136 %,%/,2*5$3+< $UQROG 5 $VWHURLG IDPLOLHV DQG fMHW VWUHDPV f $VUURQ $VSKDXJ ( DQG + 0HORVK 7KH 6WLFNQH\ LPSDFW RQ 3KRERV $ G\QDPLFDO PRGHO ,FDUXV %HOWRQ 0 6 DQG WKH *DOLOHR ,PDJLQJ 7HDP *DOLOHR HQFRXQWHU ZLWK *DVSUD )LUVW SLFWXUHV RI DQ DVWHURLG 6FLHQFH %HQGMR\D 3K ( 6OH]DN DQG &O )URHVFKO 7KH ZDYHOHW WUDQVIRUP $ QHZ WRRO IRU DVWHURLG IDPLO\ GHWHUPLQDWLRQ $VWURQ $VWURSK\V %LQ]HO 5 3 DQG 6 ;X &KLSV RII RI DVWHURLG 9HVWD (YLGHQFH IRU WKH SDUHQW ERG\ RI EDVDOWLF DFKRQGULWH PHWHRULWHV 6FLHQFH %RZHOO ( % +DSNH 'RPLQJXH /XPPL 3HOWRQLHPL DQG $ : +DUULV $SSOLFDWLRQ RI SKRWRPHWULF PRGHOV WR DVWHURLGV ,Q $VWHURLGV ,, 5 3 %LQ]HO 7 *HKUHOV DQG 0 6 0DWWKHZV (GVf SS 8QLY RI $UL]RQD 3UHVV 7XFVRQ %URXZHU 6HFXODU YDULDWLRQV RI WKH RUELWDO HOHPHQWV RI PLQRU SODQHWV $VWURQ &DUXVL $ DQG % 9DOVHFFKL 2Q DVWHURLG FODVVLILFDWLRQV LQ IDPLOLHV $VWURQ $VWURSK\V &HOOLQR $ 9 =DSSDOÂ£ DQG 3 )DULQHOOD 7KH VL]H GLVWULEXWLRQ RI PDLQEHOW DVWHURLGV IURP ,5$6 GDWD 0RQ 1RW 5 $VWURQ 6RF &URIW 6 3URWHXV *HRORJ\ VKDSH DQG FDWDVWURSKLF GHVWUXFWLRQ ,FDUXV 'DYLV 5 & 5 &KDSPDQ 5 *UHHQEHUJ DQG $ : +DUULV &ROOLVLRQDO HYROXWLRQ RI DVWHURLGV 3RSXODWLRQV URWDWLRQV DQG YHORFLWLHV ,Q $VWHURLGV 7 *HKUHOV (Gf SS 8QLY RI $UL]RQD 3UHVV 7XFVRQ 'DYLV 5 & 5 &KDSPDQ 6 :HLGHQVFKLOOLQJ DQG 5 *UHHQEXUJ &ROOLVLRQDO KLVWRU\ RI DVWHURLGV (YLGHQFH IURP 9HVWD DQG WKH +LUD\DPD IDPLOLHV ,FDUXV PAGE 137 'DYLV 5 3 )DULQHOOD 3 3DROLFFKL DQG $ & %DJDWLQ 'HYLDWLRQV IURP WKH VWUDLJKW OLQH %XPSV DQG JULQGVf LQ WKH FROOLVLRQDOO\ HYROYHG VL]H GLVWULEXWLRQ RI DVWHURLGV /XQDU 3ODQHW 6FL &RQI ;;,9 'DYLV 5 DQG ) 0DU]DUL &ROOLVLRQDO HYROXWLRQ RI DVWHURLG IDPLOLHV ,Q SUHSDUDWLRQ 'DYLV 5 DQG ( 9 5\DQ 2Q FROOLVLRQDO GLVUXSWLRQ ([SHULPHQWDO UHVXOWV DQG VFDOLQJ ODZV ,FDUXV 'DYLV 5f 6 :HLGHQVFKLOOLQJ 3 )DULQHOOD 3 3DROLFFKL DQG 5 3 %LQ]HO $VWHURLG FROOLVLRQDO KLVWRU\ (IIHFWV RQ VL]HV DQG VSLQV ,Q $VWHURLGV ,, 5 3 %LQ]HO 7 *HKUHOV DQG 0 6 0DWWKHZV (GVf SS 8QLY RI $UL]RQD 3UHVV 7XFVRQ 'HUPRWW 6 ) ' 'XUGD % $ 6 *XVWDIVRQ 6 -D\DUDPDQ < / ;X 5 6 *RPHV DQG 3 1LFKROVRQ D 7KH RULJLQ DQG HYROXWLRQ RI WKH ]RGLDFDO GXVW FORXG ,Q $VWHURLGV &RPHWV 0HWHRUV ( %RZHOO DQG $ +DUULV (GVf SS /XQDU DQG 3ODQHWDU\ ,QVWLWXWH +RXVWRQ 'HUPRWW 6 ) 5 6 *RPHV ' 'XUGD % $ 6 *XVWDIVRQ 6 -D\DUDPDQ < / ;X DQG 3 1LFKROVRQ E '\QDPLFV RI WKH ]RGLDFDO FORXG ,Q &KDRV 5HVRQDQFH DQG &ROOHFWLYH '\QDPLFDO 3KHQRPHQD LQ WKH 6RODU 6\VWHP 6 )HUUD]0HOOR (Gf : rr SS .OXZHU $FDGHPLF 3XEOLVKHUV 'RUGUHFKW 'HUPRWW 6 ) 6 -D\DUDPDQ < / ;X DQG & /LRX ,5$6 REVHUYDWLRQV VKRZ WKDW WKH HDUWK LV HPEHGGHG LQ D VRODU ULQJ RI DVWHURLGDO SDUWLFOHV LQ UHVRQDQW ORFN ZLWK WKH SODQHW 6XEPLWWHG WR 1DWXUH 'HUPRWW 6 ) DQG 3 1LFKROVRQ ,5$6 GXVW EDQGV DQG WKH RULJLQ RI WKH ]RGLDFDO FORXG +LJKOLJKWV RI $VWURQRP\ SS 'HUPRWW 6 )f 3 1LFKROVRQ $ %XPV DQG 5 +RXFN 2ULJLQ RI WKH VRODU V\VWHP GXVW EDQGV GLVFRYHUHG E\ ,5$6 1DWXUH SS 'HUPRWW 6 ) 3 1LFKROVRQ 5 6 *RPHV DQG 5 0DOKRWUD 0RGHOOLQJ WKH ,5$6 VRODU V\VWHP GXVWEDQGV $GY 6SDFH 5HV SS 'RKQDQ\L 6 &ROOLVLRQDO PRGHO RI DVWHURLGV DQG WKHLU GHEULV *HRSK\V 5HV 'RKQDQ\L 6 )UDJPHQWDWLRQ DQG GLVWULEXWLRQ RI DVWHURLGV ,Q 3K\VLFDO 6WXGLHV RI 0LQRU 3ODQHWV 7 *HKUHOV (Gf SS 1$6$ 63 'RKQDQ\L 6 6RXUFHV RI LQWHUSODQHWDU\ GXVW $VWHURLGV ,Q ,$8 &ROORTLXP 1R PAGE 138 ,QWHUSODQHWDU\ 'XVW DQG =RGLDFDO /LJKW + (OVDVVHU DQG + )HFKWLJ (GVf SS 6SULQJHU9HUODJ %HUOLQ )DULQHOOD 3 DQG 5 'DYLV &ROOLVLRQ UDWHV DQG LPSDFW YHORFLWLHV LQ WKH PDLQ DVWHURLG EHOW ,FDUXV )DULQHOOD 3 3 3DROLFFKL DQG 9 =DSSDOÂ£ 7KH DVWHURLGV DV RXWFRPHV RI FDWDVWURSKLF FROOLVLRQV ,FDUXV )O\QQ $WPRVSKHULF HQWU\ KHDWLQJ $ FULWHULRQ WR GLVWLQJXLVK EHWZHHQ DVWHURLGDO DQG FRPHWDU\ VRXUFHV RI LQWHUSODQHWDU\ GXVW ,FDUXV )XMLZDUD $ &RPSOHWH IUDJPHQWDWLRQ RI WKH SDUHQW ERGLHV RI 7KHPLV (RV DQG .RURQLV SDUHQW ERGLHV ,FDUXV A )XMLZDUD $ 3 &HUURQL 'DYLV ( 5\DQ 0 'L 0DUWLQR +ROVDSSOH DQG +RXVHQ ([SHULPHQWV DQG VFDOLQJ ODZV IRU FDWDVWURSKLF FROOLVLRQV ,Q $VWHURLGV ,, 5 3 %LQ]HO 7 *HKUHOV DQG 0 6 0DWWKHZV (GVf SS 8QLY RI $UL]RQD 3UHVV 7XFVRQ )XMLZDUD $ .DPLPRWR DQG $ 7VXNDPRWR 'HVWUXFWLRQ RI EDVDOWLF ERGLHV E\ KLJKYHORFLW\ LPSDFW ,FDUXV )XMLZDUD $ DQG $ 7VXNDPRWR ([SHULPHQWDO VWXG\ RQ WKH YHORFLW\ RI IUDJPHQWV LQ FROOLVLRQDO EUHDNXS ,FDUXV *DXWLHU 7 1 +DXVHU 0 DQG ) /RZ 3DUDOOD[ PHDVXUHPHQWV RI WKH ]RGLDFDO GXVW EDQGV ZLWK WKH ,5$6 VXUYH\ %XOO $PHU $VWURQ 6RF *UDGLH & & 5 &KDSPDQ DQG :LOOLDPV )DPLOLHV RI PLQRU SODQHWV ,Q $VWHURLGV 7 *HKUHOV (Gf SS 8QLY RI $UL]RQD 3UHVV 7XFVRQ *UHHQEHUJ 5 DQG 0 & 1RODQ 'HOLYHU\ RI DVWHURLGV DQG PHWHRULWHV WR WKH LQQHU VRODU V\VWHP ,Q $VWHURLGV ,, 5 3 %LQ]HO 7 *HKUHOV DQG 0 6 0DWWKHZV (GVf SS 8QLY RI $UL]RQD 3UHVV 7XFVRQ *UHHQEHUJ 5 0 & 1RODQ : ) %RWWNH 5 $ .ROYRRUG 9HYHUND DQG WKH *DOLOHR ,PDJLQJ 7HDP &ROOLVLRQDO DQG G\QDPLFDO HYROXWLRQ RI *DVSUD %XOO $PHU $VWURQ 6RF *UHHQEHUJ 5f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f SS /XQDU DQG 3ODQHWDU\ ,QVWLWXWH +RXVWRQ +DUWPDQQ : DQG $ & +DUWPDQQ $VWHURLG FROOLVLRQV DQG HYDOXDWLRQ RI DVWHURLGDO PDVV GLVWULEXWLRQ DQG PHWHRULWH IOX[ ,FDUXV +DXVHU 0 7 1 *DXWLHU *RRG DQG ) /RZ ,5$6 REVHUYDWLRQV RI WKH LQWHUSODQHWDU\ GXVW HPLVVLRQ ,Q 3URSHUWLHV DQG ,QWHUDFWLRQV RI ,QWHUSODQHWDU\n 'XVW 5 *LHVH DQG 3 /DP\ (GVf SS A 5HLGHO 'RUGUHFKW +HOO\HU % 7KH IUDJPHQWDWLRQ RI WKH DVWHURLGV 0RQ 1RW 5 $VWURQ 6RF +HOO\HU % 7KH IUDJPHQWDWLRQ RI WKH DVWHURLGV fÂ§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Â£N / 2UELWDO VHOHFWLRQ HIIHFWV LQ WKH 3DORPDU/HLGHQ DVWHURLG VXUYH\ ,Q 3K\VLFDO 6WXGLHV RI 0LQRU 3ODQHWV 7 *HKUHOV (Gf SS 1$6$ 63 .XLSHU 3 < )XJLWD 7 *HKUHOV *URHQHYHOG .HQW 9DQ %LHVEURHFN DQG & 9DQ +RXWHQ 6XUYH\ RI DVWHURLGV $VWURSK\V 6XSSO /LQGEODG % $ DQG 5 % 6RXWKZRUWK $ VWXG\ RI DVWHURLG IDPLOLHV DQG VWUHDPV E\ FRPSXWHU WHFKQLTXHV ,Q 3K\VLFDO 6WXGLHV RI 0LQRU 3ODQHWV 7 *HKUHOV (Gf SS 1$6$ 63 /RZ ) $ %HLWHPD 7 1 *DXWLHU ) & *LOOHWW & $ %HLFKPDQ 1HXJHEDXHU ( PAGE 141 7HGHVFR ( ) :LOOLDPV / 0DWVRQ 9HHGHU & *UDGLH DQG / $ /HERLnVN\ 7KUHHSDUDPHWHU DVWHURLG WD[RQRP\ FODVVLILFDWLRQV ,Q $VWHURLGV ,, 5 3 %LQ]HO 7 *HKUHOV DQG 0 6 0DWWKHZV (GVf SS 8QLY RI $UL]RQD 3UHVV 7XFVRQ 7KROHQ $VWHURLG WD[RQRPLF FODVVLILFDWLRQV ,Q $VWHURLGV ,, 5 3 %LQ]HO 7 *HKUHOV DQG 0 6 0DWWKHZV (GVf SS 8QLY RI $UL]RQD 3UHVV 7XFVRQ 9DQ +RXWHQ & 3 +HUJHW DQG % 0DUVGHQ 7KH 3DORPDU/HLGHQ VXUYH\ RI IDLQW PLQRU SODQHWV &RQFOXVLRQ ,FDUXV 9DQ +RXWHQ & 9DQ +RXWHQ*URHQHYHOG 3 +HUJHW DQG 7 *HKUHOV 7KH 3DORPDU/HLGHQ VXUYH\ RI IDLQW PLQRU SODQHWV $VWU $VWURSK\V 6XSSO A :HWKHULOO : &ROOLVLRQV LQ WKH DVWHURLG EHOW *HRSK\V 5HV :KLSSOH ) / 2Q PDLQWDLQLQJ WKH PHWHRQWLF FRPSOH[ 6PLWKVRQ $VWURSK\V 2EV 6SHF 5HSW 1R SS fÂ§ :LOOLDPV 5 DQG : :HWKHULOO 6L]H GLVWULEXWLRQ RI FROOLVLRQDOO\ HYROYHG DVWHURLGDO SRSXODWLRQV $QDO\WLFDO VROXWLRQ IRU VHOIVLPLODU FROOLVLRQ FDVFDGHV 6XEPLWWHG WR ,FDUXV :LOOLDPV 3URSHU HOHPHQWV DQG IDPLOLHV PHPEHUVKLSV RI WKH DVWHURLGV ,Q $VWHURLGV 7 *HKUHOV (Gf SS 8QLY RI $UL]RQD 3UHVV 7XFVRQ :LOOLDPV $VWHURLG IDPLOLHV fÂ§ DQ LQLWLDO VHDUFK ,FDUXV ;X < / 6 ) 'HUPRWW ' 'XUGD % $ 6 *XVWDIVRQ 6 -D\DUDPDQ DQG & /LRX 7KH ]RGLDFDO FORXG &KLQHVH $FDGHP\ RI 6FLHQFHV %HLMLQJ LQ SUHVV =DSSDOÂ£ 9 DQG $ &HOOLQR $VWHURLG IDPLOLHV 5HFHQW UHVXOWV DQG SUHVHQW VFHQDULR &HO 0HFK DQG '[Q $VW fU =DSSDOÂ£ 9 $ &HOOLQR 3 )DULQHOOD DQG = .QH]HYLF $VWHURLG IDPLOLHV ,GHQWLILFDWLRQ E\ KLHUDUFKLFDO FOXVWHULQJ DQG UHOLDELOLW\ DVVHVVPHQW $VWURQ =DSSDOÂ£ 9 3 )DULQHOOD = .QH]HYL DQG 3 3DROLFFKL &ROOLVLRQDO RULJLQ RI WKH DVWHURLG IDPLOLHV 0DVV DQG YHORFLW\ GLVWULEXWLRQV ,FDUXV =HOOQHU % $VWHURLG WD[RQRP\ DQG WKH GLVWULEXWLRQ RI WKH FRPSRVLWLRQDO W\SHV ,Q $VWHURLGV 7 *HKUHOV (Gf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y DANIEL DAVID DURDA 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 1993 To my parents, Joseph and Lillian Durda. ACKNOWLEDGMENTS There are a great many people who have played important roles in my life at UF, and although the room does not exist to thank them all in the manner I would like. I would at least like to express my gratitude to those who have helped me the most. First and foremost. I would like to thank my thesis advisor, Stan Dermott. Stan has been far more than just an academic advisor. He has taught by splendid example how to proficiently lead a research team, looked after my professional interests, and given me the freedom to focus upon research without having to worry about financial support. I never once felt as though I were merely a graduate student. One could not ask for a better thesis advisor. My thanks also go to the other members of my committee, Humberto Campins, Phil Nicholson, and James Channell, for their helpful comments and review of this thesis. The advice and many laughs provided by Humberto were especially appreciated. I am also very grateful to Bo Gustafson and Yu-Lin Xu for the many discussions and helpful advice through the years. My fellow graduate students, my family away from home, kept me sane enough (or is it insane enough?) to make it this far. I will value my friendship with Dirk Terrell and Billy Cooke forever. I will probably miss most our countless discussions about literally everything. I have enjoyed exploring the underwater caves of north Florida with Dirk more than I can express in words. Billyâ€™s "Billy-isms" have provided me with more entertainment than I have at times known what to do with. I will miss them immensely! 1 will also miss my discussions, afternoon chats, and laughs with the other graduate students who have come to mean so much to me, especially Dave Kaufmann, Jaydeep Mukherjee. Caroline Simpson. Sumita Jayaraman. Ron Drimmel, and Leonard Garcia. I would like to thank the office staff for helping me with so many little problems. Debra Hunter, Ann Elton, Suzie Hicks, Darlene Jeremiah, and especially Jeanne Kerrick. deserve many thanks for helping me with travel, faxes, registration, and for brightening my days. Also, thanks go to Eric Johnson and Charlie Taylor for keeping the workstations alive. With this dissertation a very large part of my life is at the same time drawing to a close and beginning anew. The most wonderful part of my new life is that I will be sharing it with Donna. Without the love and unwavering support of Mom. Dad, my sister Cathy and her husband Louie and my nephew and nieces Andrew, Larissa, and Jenna, none of this would ever have happened. IV TABLE OF CONTENTS ACKNOWLEDGMENTS iii LIST OF TABLES vii LIST OF FIGURES viii ABSTRACT xi CHAPTERS 1 INTRODUCTION 1 2 THE MAINBELT ASTEROID POPULATION 4 Description of the Catalogued Population of Asteroids 4 The MDS and PLS Surveys 6 The PLS Extension in Zones I, II, and III 9 The Observed Mainbelt Size Distribution 13 3 THE JLLISIONAL MODEL 33 Previous Studies 33 Description of the Self-consistent Collisional Model 36 Verification of the Collisional Model 43 The â€™Waveâ€™ and the Size Distribution from 1 to 100 Meters 46 Dependence of the Equilibrium Slope on the Strength Scaling Law 52 The Modified Scaling Law 55 4 HIRAYAMA ASTEROID FAMILIES 84 A Brief History of Asteroid Families 84 The ZappaD Classification 85 Collisional Evolution of Families 86 Number of Families 86 Evolution of Individual Families 90 5 IRAS AND THE ASTEROIDAL CONTRIBUTION TO THE ZODIACAL CLOUD 98 The IRAS Dustbands 98 Modeling the Dust Bands 99 The Ratio of Family to Non-Family Dust 102 6 SUMMARY 108 Conclusions 108 Future Work 110 APPENDIX A: APPARENT AND ABSOLUTE MAGNITUDES OF ASTEROIDS 113 APPENDIX B: SIZE, MASS. AND MAGNITUDE DISTRIBUTIONS .... 116 APPENDIX C: POTENTIAL OF A SPHERICAL SHELL 121 BIBLIOGRAPHY 123 BIOGRAPHICAL SKETCH 129 vi LIST OF TABLES 1: Numbers of asteroids in three PLS zones (MDS/PLS data) 16 2: Numbers of asteroids in three PLS zones (catalogued/PLS data) 17 3: Adjusted completeness limits for PLS zones 18 4: Intrinsic collision probabilities and encounter speeds for several mainbelt asteroids 62 vii LIST OF FIGURES 1: Proper inclination versus semimajor axis for all catalogued mainbelt asteroids 19 2: Magnitude-frequency distribution for catalogued mainbelt asteroids. . . 20 3: Absolute magnitude as a function of discovery date for all catalogued mainbelt asteroids 21 4: Magnitude-frequency distribution for PLS zone I: PLS and catalogued asteroid data 22 5: Magnitude-frequency distribution for PLS zone II: PLS and catalogued asteroid data 23 6: Magnitude-frequency distribution for PLS zone III: PLS and catalogued asteroid data 24 7: Adopted magnitude-frequency distribution for PLS zone 1 25 8: Adopted magnitude-frequency distribution for PLS zone II 26 9: Adopted magnitude-frequency distribution for PLS zone III 27 10: Magnitude-frequency distribution for the 1836 asteroids in Tables 7 and 8 of Van Houten et al. (1970) 28 11: Least-squares fit to the magnitude-frequency data for PLS zone 1 29 12: Least-squares fit to the magnitude-frequency data for PLS zone II. ... 30 13: Least-squares fit to the magnitude-frequency data for PLS zone III. ... 31 14: The observed mainbelt size distribution 32 15: Verification of model for steep initial slope and small bin size 63 viii 16: Verification of model for shallow initial slope and small bin size 64 17: Verification of model for steep initial slope and large bin size 65 18: Verification of model for shallow initial slope and large bin size 66 19: Equilibrium slope as a function of time for various fragmentation power laws and for steep initial slope 67 20: Equilibrium slope as a function of time for various fragmentation power laws and for shallow initial slope 68 21: Equilibrium slope as a function of time for various fragmentation power laws and for equilibrium initial slope 69 22: Wave-like deviations in size distribution caused by truncation of particle population 70 23: Independence of the wave on bin size adopted in model 71 24: Comparison of the interplanetary dust flux found by Griin et al. (1985) and small particle cutoffs used in our model 72 25: Wave-like deviations imposed by a sharp particle cutoff (x = 1.9). ... 73 26: Size distribution resulting from gradual particle cutoff matching the observed interplanetary dust flux (x â€” 1.2) 74 27: Collisional relaxation of a perturbation to an equilibrium size distribution 75 28: Halftime for exponential decay toward equilibrium slope following the fragmentation of a 100 km diameter asteroid 76 29: Stochastic fragmentation of inner mainbelt asteroids of various sizes during a typical 500 million period 77 30: Equilibrium slope parameter as a function of the slope of the size-strength scaling law 78 31: Difference in the equilibrium slope parameters for families with different strength properties 79 IX 32: The Davis et al. (1985). Housen et al. (1991), and modified scaling laws used in the collisional model 80 33: The evolved size distribution after 4.5 billion years using the Housen et al. (1991) scaling law for (a) a massive initial population and (b) a small initial population 81 34: The evolved size distribution after 4.5 billion years using the Davis et al. (1985) scaling law for (a) a massive initial population and (b) a small initial population 82 35: The evolved size distribution after 4.5 billion years using our modified scaling law for (a) a massive initial population and (b) a small initial population 83 36: The 26 Hirayama asteroid families as defined by ZappalÃ¡ et al. (1984). . 93 37: The collisional decay of families resulting from various-size parent asteroids as a function of time 94 38: Formation of families in the mainbelt as a function of time 95 39: Modeled collisional history of the Gefion family 96 40: Modeled collisional history of the Maria family 97 41: The solar system dust bands at 12. 25. 60, and 100 /.Â¿m, after subtraction of the smooth zodiacal background via a Fourier filter 105 42: (a) IRAS observations of the dust bands at elongation angles of 65.68Â°, 97.46Â°. and 114.68Â°. Comparisons with model profiles based on prominent Hirayama families are shown in (b), (c), and (d) 106 43: The ratio of areas of dust associated with the entire mainbelt asteroid population and all families 107 X 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 THE COLLISIONAL EVOLUTION OF THE ASTEROID BELT AND ITS CONTRIBUTION TO THE ZODIACAL CLOUD By DANIEL DAVID DURDA December, 1993 Chairman: Stanley F. Dermott Major Department: Astronomy We present results of a numerical model of asteroid collisional evolution which verify the results of Dohnanyi (1969, J. Geophys. Res. 74, 2531-2554) and allow us to place constraints on the impact strengths of asteroids. The slope of the equilibrium size-frequency distribution is found to be dependent upon the shape of the size-strength scaling law. An empirical modification has been made to the size-strength scaling law which allows us to match the observed asteroid size distribution and indicates a more gradual transition from strain-rate to gravity scaling. This result is not sensitive to the mass or shape of the initial asteroid population, but rather to the form of the strength scaling law: scaling laws have definite observational consequences. The observed slope of the size distribution of the small asteroids is consistent with the value predicted by the slightly negative slope of our modified scaling law. Wave-like deviations from a strict power-law equilibrium size distribution result if the smallest particles in the population are removed at a rate significantly greater than that needed to maintain a Dohnanyi equilibrium slope. We find, however, that the observed small particle cutoff in the interplanetary dust complex is too gradual to support xi a significant wave. We suggest that any deviations from an equilibrium size distribution in the asteroid population are the result of stochastic cratering and fragmentation events which must occur during the course of collisional evolution. By determining the ratio of the area associated with the mainbelt asteroids to that associated with the prominent Hirayama asteroid families, our analysis indicates that the entire mainbelt asteroid population produces 3.4 Â± 0.6 times as much dust as the prominent families alone. This result is compared with the ratio of areas needed to account for the zodiacal background and the IRAS dust bands as determined by analysis of IRAS data. We conclude that the entire asteroid population is responsible for at least ~34% of the dust in the entire zodiacal cloud. xii CHAPTER 1 INTRODUCTION Traditionally, the debris of short period comets has been thought to be the source of the majority of the dust in the interplanetary environment (Whipple 1967: Dohnanyi 1976). However, it has been known for some time that inter-asteroid collisions are likely to occur over geologic time (Piotrowski 1953). The gradual comminution of asteroidal debris must supply at least some of the dust in the zodiacal cloud, though because of the lack of observational constraints the contribution made by mutual asteroidal collisions has been difficult to determine. Since the discovery of the IRAS solar system dust bands (Low et al. 1984). the contribution made by asteroids to the interplanetary dust complex has received renewed attention. The suggestion that the dust bands originate from the major asteroid families, widely thought to be the results of mutual asteroid collisions, was made by Dermott et al. (1984). They also suggested that if the families supply the dust in the bands, thus making a significant contribution to the zodiacal emission, then the entire asteroid belt must contribute a substantial quantity of the dust observed in the zodiacal background. Other evidence also points to an asteroidal source for at least some interplanetary dust. The interplanetary dust particle fluxes observed by the Galileo and Ulysses spacecraft indicate a population with low-eccentricity and low-inclination orbits (Grim et al. 1991), consistent with an asteroidal origin of the particles. From computer simulations of the entry heating of large micrometeorites and comparison of the collisional destruction and 1 9 transport lifetimes of asteroidal dust. Flynn (1989) has concluded that much of the dust collected at Earth from the interplanetary dust cloud is of asteroidal origin. At tirst inspection it might be tempting to try to calculate the amount of dust produced in the asteroid belt by modeling, from tirst principles, the collisional grinding taking place in the present mainbelt. The features of the present asteroid population, however, are the product of a long history involving catastrophic collisions which have reduced the original mass of the belt. Unfortunately, the initial mass of the belt is not known and our knowledge of the extent of collisional evolution in the mainbelt is limited by our understanding of the initial mass and the effective strengths of asteroids in mutual collisions. Our intent is to place some constraints on the collisional processes affecting the asteroids and to determine the total contribution made by mainbelt asteroid collisions to the dust of the zodiacal cloud. In Chapter 2 we describe the methods used to derive the size distribution of mainbelt asteroids down to ~5 km diameter. The size distribution of the asteroids represents a powerful constraint on the previous history of the mainbelt as well as the collisional processes which continue to shape the distribution. In Chapter 3 we describe the collisional model which we have developed and present results confirming work by previous researchers. We then use the model to extend our assumptions beyond those of previous works and to shed some light on the impact strengths of asteroids and the initial mass of the mainbelt. The collisional history of asteroid families is examined in Chapter 4, providing further constraints on the evolution of the mainbelt and the dust production of families. In Chapter 5 we combine analysis of IRAS data and the mainbelt and family size distributions to determine the 3 relative contribution of dust supplied to the zodiacal cloud by asteroid collisions. Our conclusions are summarized and the problems that must be addressed in future work are discussed in Chapter 6. CHAPTER 2 THE MAINBELT ASTEROID POPULATION Description of the Catalogued Population of Asteroids The size-frequency distribution of the asteroids is very important in constraining the collisional processes which have influenced and continue to affect the asteroid population as well as the total mass and mass distribution of the initial planetesimal swarm in that region. Also, in order to determine the total quantity of dust that the asteroids contribute to the zodiacal cloud, we must use the observed population of mainbelt asteroids to estimate the numbers of small asteroids which serve as the parent bodies of the immediate sources of asteroidal dust. In this chapter we will describe the data and methods from which we derive a reliable size distribution. Of the 8863 numbered and multi-opposition asteroids for which orbits had been determined as of December 1992. 8383 (or ~95%) are found in the semimajor axis range 2.0 < a < 3.8 AU (Figure 1). For reasons described below, we will limit our discussion to those asteroids in the range 2.0 < a < 3.5 AU, defining what we will refer to as the â€œmainbelt.â€ Our conclusions are expected to be unaffected by this choice, as only 13 asteroids, or less than 0.2% of the known population, are excluded so that the two sets of asteroids are essentially the same. Figure 2 is a plot of the number of catalogued mainbelt asteroids per half-magnitude bin, where the absolute magnitude, H, is defined as the V-band magnitude of the asteroid at a distance of 1 AU from the Earth, 1 AU from the Sun. at a phase angle of 0Â° 4 5 (Bowell et al. 1989). Immediately evident is a â€œhumpâ€, or excess, of asteroids at H Â« 8. Although previous researchers have interpreted this excess as a remnant of some primordial, gaussian population of asteroids altered by subsequent collisional evolution (Hartmann and Hartmann 1968), the current interpretation is that it represents the preferential preservation of larger asteroids effectively strengthened by gravitational compression (Davis et al. 1989; Holsapple and Housen 1990). Other researchers, primarily Dohnanyi (1969, 1971), have noted from surveys of faint asteroids (discussed below) that the distribution of smaller asteroids is well described by a power-law. indicative of a population of particles in collisional equilibrium. Unfortunately, as evident in Figure 2, the number of faint asteroids in the catalogued population alone is not quite great enough to be sure of identifying the transition to. or slope of, such a distribution. In fact, the mainbelt population of asteroids is complete with respect to discovery down to an absolute magnitude of only about H = 11. We can see this quite clearly in Figure 3, which is a plot of the absolute brightness of the numbered mainbelt asteroids as a function of their date of discovery. It can be seen that as the years have progressed, increased interest in the study of minor planets and advances in astronomical imaging have allowed for the discovery of fainter and fainter asteroids. In turn, the brighter asteroids have all been discovered, defining fainter and fainter discovery completeness limits. For instance, no asteroids brighter than H = 7 have been discovered since about 1910. By 1940 the completeness limit was a magnitude fainter. Similarly, we may see that the current limit of completeness is approximately H = 11. Even if a dozen mainbelt asteroids brighter than this remain to be discovered in the mainbelt. 6 the degree of completeness is greater than 99.7%. (Figure 3 is also interesting for the history recorded in asteroid discovery circumstances. Quite apparent is the marked lack of discoveries in the wake of World War II. The large number of asteroids discovered during the Palomar-Leiden Survey appears as a vertical stripe near 1960.) As pointed out above, between H â€” 10 and H = 11 the mainbelt appears to make a transition to a linear, power-law size distribution. An absolute magnitude of H = 11 corresponds to a diameter of about 30 km for an albedo of 0.1, approximately the mean albedo of the larger asteroids in the mainbelt population (see The Observed Mainbelt Size Distribution). Unfortunately, incompleteness rapidly sets in for H 2 11.5 and with so few data points the slope of the distribution cannot be well defined so that we cannot reliably use the data from the catalogued population alone to estimate the number of very small asteroids in the mainbelt (see Figure 2). We have therefore used data from the Palomar-Leiden Survey (Van Flouten et al. 1970) to extend the observed distribution down to about H = 15.25, corresponding to a diameter of roughly 5 km. The MDS and PLS Surveys The Palomar-Leiden Survey (Van Flouten et al. 1970; hereafter referred to as PLS) was conducted in 1960 to extend to fainter magnitudes the results of the earlier McDonald Survey of 1950 through 1952 (Kuiper et al. 1958; hereafter referred to as MDS). The MDS surveyed the entire ecliptic nearly twice around to a width of 40Â° down to a limiting photographic magnitude of nearly 15. In contrast, the practical plate limit for the PLS survey was about five magnitudes fainter. To survey and detect asteroids this faint over the same large area covered by the MDS would have been 7 prohibitive, so with the PLS it was decided that only a small patch of the ecliptic would be surveyed, and the results scaled to the MDS and the entire ecliptic belt. In 1984 a revision and small extension were made to the PLS (Van Houten et al. 1984), raising many quality 4 orbits to higher qualities, assigning orbits to some objects which previously had to be rejected, and adding 170 new objects which were identified on plates taken for purposes of photometric calibration. Our original intention was to use this extended data set to re-examine the size distribution of the smaller asteroids in zones of the belt chosen to be more dynamically meaningful than the three zones used in the MDS and PLS. However, we have decided not to embark on a re-analysis of the PLS data at this time as the magnitude distribution of asteroids in the inner region of the mainbelt was rather well defined in the original analysis, and we conclude that even the extended data set will not significantly improve the statistics in the outer region of the belt. We therefore use the original PLS analysis of the absolute magnitude distribution in three zones of the mainbelt. with some caveats as described below. In both the MDS and PLS analysis the mainbelt was divided into three semimajor zones â€” zone I: 2.0 < a < 2.6, zone II: 2.6 < a < 3.0, and zone III: 3.0 < a < 3.5. Within each zone the asteroids were grouped in half-magnitude intervals of absolute photographic magnitude, g, and the numbers corrected for incompleteness in the apparent magnitude cutoff and the inclination cutoff of the survey (see Kuiper et al. 1958). The g absolute magnitudes given by Van Houten et al. are in the standard B band â€” we transformed these absolute magnitudes to the H. G system by applying the correction H â€” g â€” 1 (Bowell et al. 1989). The bias-corrected number of asteroids per half-magnitude bin in each of the zones is a combination of the results of the MDS and 8 the PLS. as described by Van Houten et al. The MDS values for the number of asteroids per half-magnitude bin are assumed until the corrections for incompleteness approach about 50% of the values themselves. Where the MDS values require correction for incompleteness, a maximum and minimum number of asteroids is calculated based upon two different extrapolations of the log N(m0) relation (Kuiper et al. 1958). In these cases the mean of the two values given in the MDS has been assumed. The correction factors for incompleteness in zone III given in the MDS, however, are incorrect. The corrected values are given in Table D-I of Dohnanyi (1971). For fainter values of H the number of asteroids is taken from Table 5 of Van Houten et al., the values given there corrected by multiplying logN(H) by 1.38 to extend the counts to cover the asteroid belt over all longitudes to match the coverage of the MDS. Table 1 gives the adopted bias-corrected number of asteroids per half-magnitude bin (H magnitudes) for each of the three PLS zones and for the entire mainbelt as derived from the MDS and PLS data. While the MDS, which surveyed the asteroid belt over all longitudes, is regarded as complete down to an absolute magnitude of about g = 9.5, the PLS data need to be corrected for completeness at all magnitudes as the survey covered only a few percent of the area of the MDS. There have been a number of discussions regarding selection effects within the PLS and problems involved with linking up the MDS and PLS data (cf. KresÃ¡k 1971 and Dohnanyi 1971). We have taken a very simple approach which indicates that the MDS and PLS data link up quite well and that any selection effects within the PLS either cancel each other or are minor to begin with. Figures 4, 5, and 6 show the combined MDS/PLS magnitude-frequency data for zones I, II, and III. respectively, superimposed upon the data for the catalogued asteroids. The dashed 9 vertical line indicates the completeness limit for the MDS. beyond which correction factors were adopted based on extrapolations of the observed trend of the number of asteroids per mean opposition magnitude bin. The solid vertical line indicates where the PLS data have been adopted to extend the MDS distribution. In each of the three zones the completeness limit for the catalogued population roughly coincides with the transition to the PLS data. Beyond the completeness limit the observed number of catalogued asteroids per half-magnitude bin continues to increase (although at a decreasing level of completeness) until the numbers fall markedly. In each of the three zones the data for the catalogued population merges quite smoothly with the PLS data. This is particularly evident in zone II. where there is a significant decline in the number of asteroids with H ^ 11. right in the transition region between the incompleteness corrected MDS data and the PLS data, producing an apparent discontinuity between the two data sets. The catalogued population, however, which is complete to about H = 11 in this zone, nicely follow's the same trend, even showing the sharp upturn beyond the completeness limit between H = 11.25 and H = 11.75. With the catalogued population making a smooth transition between the MDS and PLS data in each of the three zones, we conclude that any selection effects which might exist within the PLS data are minor and that there is no problem with combining the MDS data (roughly equivalent to the current catalogued population) and PLS data as published. The PLS Extension in Zones I. II, and III Having established that the PLS data may be directly used to extend our discussion of the observed distributions to fainter asteroids, we define our working magnitude- frequency distribution for each zone by taking the number of asteroids per half- 10 magnitude bin from the catalogued population for those bins brighter than the discovery completeness limit and from either the PLS data or catalogued population, whichever is greater, for the magnitude bins below the completeness limit. Due to sampling statistics there will be a VÃ‘ error associated with each independent point in an incremental magnitude-frequency diagram. The errors for the catalogued asteroid counts are determined directly from the raw numbers after the asteroids have been binned and counted. For the PLS data the \TÃ‘ errors must be determined from the number of asteroids per magnitude interval before the counts have been corrected for the apparent magnitude and inclination cutoffs. The corrected counts themselves are given in Table 5 of Van Houten et al. These counts are then scaled to match the coverage of the MDS as described above. Since the errors in the PLS counts are based on the uncorrected, unsealed counts, the PLS data points have a larger associated \//V error than the corrected counts themselves would indicate. The resulting magnitude- frequency diagrams for each of the PLS zones are shown in Figures 7, 8. and 9 and the numbers tabulated in Table 2. The PLS data greatly extend the workable observed magnitude-frequency distriÂ¬ butions for the mainbelt asteroids. We immediately see that the inner two zones of the mainbelt display a well defined, linear power-law distribution for the fainter asteroids, with the prominent excess of asteroids at the brighter end of the distribution. The distriÂ¬ bution in the outer third of the belt appears somewhat less well defined. The results for the inner zones are very interesting, as the linear portions qualitatively match very well Dohnanyi's (1969, 1971) prediction of an equilibrium power-law distribution of fragÂ¬ ments expected in a collisionally relaxed population. Dohnanyi, using a least-squares fit through the MDS and PLS data, found a mass index of q = 1.839, in good agreement with the theoretical expected value of q = 1.837 quoted in his work. His analysis, however, was performed on the cumulative distribution of the combined data from the three zones. We feel that it is more appropriate to consider only incremental frequency distributions since the data points are independent of one another and the limitations of the data set are more readily apparent. In this analysis we will also consider the three zones independently to take advantage of any information that the distributions may contain on the variation of the collisional evolution of the asteroids with location in the mainbelt. Having assigned errors to the independent points in the incremental magnitude- frequency diagrams, a weighted least-squares solution can be fit through the linear portions of the distributions in each of the three PLS zones. We must be cautious, however, to work within the completeness limits of the PLS data. Figure 10 is a histogram of the number of asteroids per half-magnitude interval as derived from the data in Tables 7 and 8 of Van Houten et al. (1970). These are the 1836 asteroids for which orbits were able to be determined plus the 187 asteroids for which the computed orbits had to be discarded. The survey was complete to a mean photographic opposition magnitude of approximately 19, beyond which the numbers would need to be corrected for incompleteness. Recognizing the uncertainties involved in trying to estimate the degree of completeness for fainter asteroids on the photographic plates, we prefer to work within the completeness limits of the raw data set. Given the completeness limit in mean opposition magnitude, m0, we can calculate the corresponding completeness limit in absolute magnitude for each of the three semimajor axis zones. Based on the 12 mean semimajor axis for each of the zones we calculate the adjusted completeness limits given in Table 3. Based on these more conservative completeness limits we may now calculate the least squares solutions for the individual zones. Zone I displays a distinctly linear distribution for absolute magnitudes fainter than about H = 11. A weighted least-squares fit to the data (H = 11.25 and fainter) yields a slope of a = 0.469 Â± 0.011, which corresponds to a mass-frequency slope of q = 1.782 Â± 0.018 (Figure 11). (If we assume that all the asteroids in a semimajor axis zone have the same mean albedo we may directly convert the magnitude-frequency slope into the more commonly used mass frequency slope via (2 = 1 + Â§a, where a is the slope of the magnitude-frequency data. See Appendix B.) Zone II shows a similar, though somewhat less distinct and shallower, linear trend beyond H â€” 11.25. A fit through these data yields a slope of a = 0.479 Â± 0.012 (q = 1.799 Â± 0.020, Figure 12). In Zone III we obtain the solution a = 0.447 Â± 0.017 (q â€” 1.745 Â± 0.028, Figure 13) for magnitudes fainter than H = 10.75. These slopes are significantly lower than the Dohnanyi equilibrium value of q = 1.833. The weighted mean slope for the three zones is q = 1.781 Â± 0.007, essentially equal to the well determined slope for zone I. In addition to the slope, the least-squares solution for each zones produces an estimate for the intercept of the linear distribution, which is a measure of the absolute number of asteroids in the population. With an estimate of the mean albedo of asteroids in the population, the expressions derived in Appendix B allow us to use the parameters of the magnitude-frequency plots to quantify the size-frequency distributions for the three zones and for the mainbelt as a whole. 13 The Observed Mainbelt Size Distribution We may define the observed mainbelt size distribution that we will work with by combining data from the catalogued population of asteroids and the least-squares fits to the PLS data. The sizes of the numbered mainbelt asteroids may be reconstructed from their absolute brightnesses if we can estimate a value for the albedo (See Appendix A). Fortunately, an extensive set of albedos derived by IRAS is available for a great many asteroids. A recent study by Matson et al. (1990) demonstrates that asteroid diameters derived using IRAS-derived albedos show no significant difference between those found by occultation studies. Although an even larger number of asteroids exists for which no albedo measurements have been made, the IRAS data base is extensive enough to allow a statistical reconstruction of their albedos. There are two subsets of asteroids without albedo estimates: those for which a taxonomic classification is available, and. the larger group, those which have not been typed. We have used the taxonomic types assigned by Tedesco et al. (1989) when available and by Tholen (1989, 1993 private communication) if a classification based upon an IRAS-derived albedo was not available. For those asteroids with a taxonomic type but no IRAS-observed albedo, we have estimated the albedo by assuming the mean value of other asteroids with the same classification. If no taxonomic information was available we assumed an albedo equal to that of the IRAS-observed asteroids at the same semimajor axis. The diameters for the catalogued asteroids, calculated using Equation 11 of Appendix A, are then collected and binned with a logarithmic increment of 0.1 in order to directly combine the data with those derived from the PLS magnitude data (see Appendix B). 14 The size distribution of asteroids smaller than the completeness limit of the catalogued population has been derived using the PLS magnitude data described in the previous section. Linear least-squares solutions, constrained to have the same weighted mean slope of q = 1.781, were fit through the linear portions of the magnitude distributions in each of the three PLS zones. The individual distributions were then added to determine the intercept parameter (equivalent to the brightest asteroid in the power-law distribution) for the mainbelt as a whole. To convert the parameters of the magnitude-frequency distribution determined using the PLS data into a size-frequency distribution, we assume that all the asteroids in the population have the same mean albedo. Of the well-observed asteroids in the mainbelt, that is, asteroids with both IRAS-determined albedos and measured B-V colors, we found mean albedos of 0.121. 0.105, and 0.074 in PLS zones I, II, and II, respectively. The weighted mean albedo for the entire mainbelt population is 0.097. We chose to calculate the mean albedo based on those asteroids with diameters between 30 and 200 km. in order to avoid any possible selection effects which might affect the smallest and largest asteroids. With an estimate for the mean albedo the magnitude parameters may be converted directly into a size-frequency distribution using Equations 6 and 15 of Appendix B. In Figure 14 we have combined the data from the catalogued asteroids and the PLS magnitude distributions to define the observed mainbelt size distribution. Down to approximately 30 km the distribution is determined directly from the catalogued asteroids and IRAS-derived albedos. The shaded band indicates the \/Ã‘ error associated with the catalogued population due to sampling statistics. For diameters less than about 30 km the mainbelt population is incomplete and the numbers drop below those 15 estimated from PLS data. We thus use the PLS data to extend the usable size distribution to smaller sizes. The dashed line is the best tit through the magnitude data for the small asteroids. This size distribution is very well determined and will be used in the next chapter to place strong constraints on collisional models of the asteroids. 16 Table 1: Numbers of asteroids in three PLS zones (MDS/PLS data). H Zone I 2.0 < a < 2.6 N(H) Zone II 2.6 < a < 3.0 N(H) Zone III 3.0 < a < 3.5 N(H) I + II + III 2.0 < a < 3.5 N(H) 3.25 1 1 0 2 3.75 0 1 0 1 4.25 0 0 0 0 4.75 0 0 0 0 5.25 0 2 1 3 5.75 2 1 0 3 6.25 5 4 2 11 6.75 5 4 5 14 7.25 5 15 11 31 7.75 13 20 24 57 8.25 15 39 31 85 8.75 24 51 39.5 114.5 9.25 24 62 67 153 9.75 19 68.5 132 219.5 10.25 28 86 215.89 329.89 10.75 28 95.5 95.95 219.45 11.25 71.5 118 287.86 477.36 11.75 127 287.86 503.75 918.61 12.25 143.93 791.61 503.75 1439.29 12.75 143.93 551.73 575.72 1271.38 13.25 503.75 1103.46 1727.16 3334.37 13.75 1007.51 2614.73 4941.60 8563.84 14.25 2254.90 3958.07 5109.51 11322.48 14.75 4125.99 7532.34 6069.05 17727.38 15.25 6093.04 6788.70 7868.17 20749.91 15.75 10914.69 12401.97 â€” â€” 16.25 17151.66 â€” â€” â€” 17 Table 2: Numbers of asteroids in three PLS zones (catalogued/PLS data). H Zone I 2.0 < a < 2.6 N(H) Zone II 2.6 < a < 3.0 N(H) Zone III 3.0 < a < 3.5 N(H) I + II + III 2.0 < a < 3.5 N(H) 3.25 1 1 0 2 3.75 0 0 0 0 4.25 0 1 0 1 4.75 0 0 0 0 5.25 0 2 1 3 5.75 3 3 1 7 6.25 3 3 2 8 6.75 6 4 7 17 7.25 9 15 11 35 7.75 11 28 25 64 8.25 14 38 39 91 8.75 23 52 38 113 9.25 29 62 73 164 9.75 17 72 95 184 10.25 33 73 118 224 10.75 50 91 195 336 11.25 63 133 301 497 11.75 141 294 503.75 938.75 12.25 275 791.61 503.75 1570.36 12.75 515 551.73 575.72 1642.45 13.25 784 1103.46 1727.16 3614.62 13.75 1007.51 2614.73 4941.60 8563.84 14.25 2254.90 3958.07 5109.51 11322.48 14.75 4125.99 7532.34 6069.05 17727.38 15.25 6093.04 6788.70 7868.17 20749.91 15.75 10914.69 12401.97 â€” â€” 16.25 17151.66 â€” â€” â€” Table 3: 18 Adjusted completeness limits for PLS zones. Semimajor Axis Zone Mean Semimaior Axis (AU) Completeness limit in H 2.0 < a < 2.6 2.43 15.3 2.6 < a < 3.0 2.75 14.6 3.0 < a < 3.5 3.17 13.8 Figure 1: Proper inclination versus semimajor axis for all catalogued mainbelt asteroids Number per Magnitude Bin Absolute Magnitude, H Figure 2: Magnitude-frequency distribution for catalogued mainbelt asteroids. 20 CD ic XJ 10 2 -+-> â€¢ r-H C ttf) 10 CD O C/] X) < 0 1800 1850 1900 1950 2000 Discovery Date Figure 3: Absolute magnitude as a function of discovery date for all catalogued mainbelt asteroids. Number per Magnitude Bin Absolute Magnitude, H ro to Figure 4: Magnitude-frequency distribution for PLS zone I: PLS and catalogued asteroid data. Number per Magnitude Bin 105 io4 io3 io2 io1 10Â° 18 16 14 12 10 8 6 4 2 0 Absolute Magnitude, H ro Lk) Figure 5: Magnitude-frequency distribution for PLS zone II: PLS and catalogued asteroid data. Number per Magnitude Bin 1 Â°5 1 i 1 i '"i i 1 i 1 i 1 i 1 i 1 Â¡ Zone III 104 â€” 3.0 < a < 3.5 Â° O o Â° PLS 3 Â° â€¢ catalogued â€” Â° O O - â€¢ â€¢ 102 =- â€¢ o â€¢ -= â€¢ 8 â€¢ * * â€¢ o â€” â€¢ â€¢ 101 - â€¢ - â€¢ â€¢ â€” o â€” â€” â€¢ â€” 10Â° i -J i I i J l I i L i l^j I i I i 18 16 14 12 10 8 6 4 2 0 Absolute Magnitude, H Figure 6: Magnitude-frequency distribution for PLS zone Ill: PLS and catalogued asteroid data. e 1 1: 1 1 1 1 1 1 "i r i 1 i r Â¡ â€” Zone III = o â€” o 3.0 < a < 3.5 Â° PLS o 0 o o 1 Mill â€¢ catalogued â€¢ â€¢ â€¢ =- Â«oâ€™ o â€¢ - â€” â€¢ e â€¢ â€¢ â€¢ O â€¢ â€¢ _ I T ITT 1 l O â€” i *L,. 1 i 1 i 10 10 10 10 10 10 16 Zone 2.0 < a i * 14 12 10 8 6 4 Absolute Magnitude, H I C 2.6 2 0 Figure 7: Adopted magnitude-frequency distribution for PLS zone I. 10 10 10 10 10 10 = ' I 1 n I r 1 1 I 1 I Zone II 2.6 < a < 3.0 1 I I 3 C i J I L J I L 8 16 14 12 10 8 6 4 2 0 Absolute Magnitude, H N> ON Figure 8: Adopted magnitude-frequency distribution for PLS zone II. 10 10 10 10 10 10 r1-1 1 i"1 i 1 i 1 i 1 1 1 1 = Zone III - ET~ 2 3.0 < a < 3.5 â€” E *iÂ« A * - r m â€” m = - â€¢ â– i = Mil - r 'i â€” â€” ( l 1 l 1 1 1 1 1 1 1 1 ',11 .i 1 i 1 8 16 14 12 10 8 6 4 2 0 Absolute Magnitude, H to Figure 9: Adopted magnitude-frequency distribution for PLS zone III. 10 10 10 10 10 10 Mean Opposition Magnitude, m0 -frequency distribution for the 1836 asteroids in Tables 7 and 8 of Van Houten et al. (1970). 10 10 10 10 10 10 Absolute Magnitude, H to VO 11: Least-squares lit to the magnitude-frequency data for PLS zone I. Number per Magnitude Bin Figure 12: Least-squares fit to the magnitude-frequency data for PLS zone II. 10 10 10 10 10 io' 16 14 12 10 8 6 4 2 0 Absolute Magnitude, H 13: Least-squares fit to the magnitude-frequency data for PLS zone 111. 10 10 10 10 10 10 i rrnâ€”ttttt PLS Data \ (qeq ~ 1.78) \ J i i i i i 10 Diameter (km) U) ro Figure 14: The observed mainbelt size distribution. CHAPTER 3 THE COLLISIONAL MODEL Previous Studies Before describing the details of the collisional model developed in this thesis, it would be useful to review some previous studies. The collisional evolution of the asteroids and its effects on the size distribution of the asteroid population has been studied by a number of researchers both analitically and numerically. Dohnanyi (1969) solved analytically the integro-differential equation describing the evolution of a collection of particles, all with size independent impact strengths, which fragment due to mutual collisions. He found that the size distribution of the resulting debris can be described by a power-law distribution in mass of the form f(m)dm oc m~qdm(3-1) where f(m)dm is the number of asteroids in the mass range m. to m + dm and q is the slope index. Dohnanyi found that q = 1.833 for debris in collisional equilibrium, in agreement with the observed distribution of small asteroids as determined from MDS and PLS data. The equilibrium slope index q was found to be insensitive to the fragmentation power law of the colliding bodies, provided that r? < 2. This is because the most important contribution to the mass range m to ??? + dm comes from collisions in which the most massive particles are of mass near m. The number of such particles produced depends on the number of collisions and not on the slope of the comminution law. 33 34 Dohnanyi also found that for q near 2 hut less than 2 the creation of debris by erosion, or cratering collisions, plays only a minor role. The steady-state size distribution is therefore dominated by catastrophic collisions. Hellyer (1970. 1971) solved the same collision equation numerically and confirmed the results of Dohnanyi. Hellyer showed that for four values of the fragmentation power law, referred to as x in his notation, (x = rÂ¡ â€” 1 = 0.5, 0.6, 0.7, and 0.8), the population index of the small masses converged to an almost stationary value of about 1.825. The convergence was most rapid for the largest values of x, but the asymptotic value of the population index is very close to the value obtained analytically by Dohnanyi. Although primarily interested in the behavior of the smallest asteroids, Hellyer also investigated the influence of random disruption of the largest asteroids on the rest of the system. His program was modified to allow for a small number of discrete fragmentation events among very massive particles. With the parameter x set to 0.7, the slope index of the smallest asteroids was seen to still attain the expected value (about 1.825), although there were discontinuities in the plot of the slope as a function of time at the times of the large fragmentation events. Davis et al. (1979) introduced a numerical model simulating the collisional evolution of various initial populations of asteroids and compared the results with the observed distribution of asteroids in order to find those populations which evolved to the present belt. In their study they considered three different families of shapes for the initial distribution: 1. power law, 2. segmented power law, simulating a runaway growth distribution of bodies as 35 generated by the accretional simulation of Greenberg et al. (1978), and 3. gaussian as suggested by Anders (1965) and Hartmann and Hartmann (1968). They concluded that for power law initial populations the initial mass of the belt could not have been much larger than ~ 1 M_, only modestly larger than the present belt. Both massive and small runaway growth distributions were found to evolve to the present distribution, however, placing no strong constraints on the initial size of the belt. The gaussian initial distributions failed to relax to the observed distribution. The power law and runaway growth models, however, both produced a small asteroid distribution with a slope index similar to the value predicted by Dohnanyi. Another major conclusion of this study was that most asteroids > 100 km diameter are likely fractured throughout their volume and are essentially gravitationally bound rubble piles. Davis et al. (1985) introduced a revised model incorporating the increased impact strengths of large asteroids due to hydrostatic self-compression. The results from this numerical model were later extended to include size (strain-rate) dependent impact strengths (Davis et al. 1989). The primary goal of these studies was to further constrain the extent of asteroid collisional evolution. They investigated a number of initial asteroid populations and concluded that a runaway growth initial belt with only 3 to 5 times the present belt mass best satisfied the constraints of preserving the basaltic crust of Vesta and producing the observed number of asteroid families. However, other asteroid observations (such as the interpretation of M asteroids as exposed metallic cores of differentiated bodies and the apparent dearth of asteroids representing the shattered mantle fragments from such bodies) suggest that much more collisional evolution occurred than these models predict. The latest version of this model is currently being 36 used to investigate the collisional history of asteroid families (Davis and Marzari 1993). Most recently, Williams and Wetherill (1993) have extended the work of Dohnanyi to include a wider range of assumptions and obtained an analytical solution for the steady-state size distribution of a self-similar collisional fragmentation cascade. Their results confirm the equilibrium value of q = 1.833 and demonstrate that this value is even less sensitive to the physical parameters of the fragmentation process than Dohnanyi had thought. In particular, Williams and Wetherill have explicitly treated the debris from cratering impacts (whereas Dohnanyi concluded that the contribution from cratering would be negligible and so dropped terms including cratering debris) and have more realistically assumed that the mass of the largest fragment resulting from a catastrophic fragmentation decreases with increasing projectile mass. They find a steady-state value of q = 1.83333 Â± 0.00001 which is extremely insensitive to the assumed physical parameters of the colliding bodies or the relative contributions of cratering and fragmentation. They note, however, that this result has still been obtained by assuming a self-similar system in which the strengths of the colliding particles are independent of size and that the results of relaxing the assumption of self-similarity will be explored in future work. Description of the Self-consistent Collisional Model An initial population of asteroids is distributed among a number of logarithmic size bins. The initial population may have any form and is defined by the user. The actual number of bins depends on the model to be run, but for most cases in which we are interested only in the larger asteroidal particles, the smallest sizes considered are of order 1 meter in diameter and the model uses approximately 60 size bins. In 37 those cases in which we are interested in modeling the collisional evolution of dust size particles the number of bins can increase to over 120. For most of the models the logarithmic increment was chosen to be 0.1. in order to most directly compare the size distributions with the magnitude distributions derived in Chapter 2 (see Appendix B). For some models including dust size particles the bin size was increased to 0.2 to decrease the number of bins and shorten the run time. All particles are assumed to be spherical and to have the same density. The characteristic size of the particles in each bin is determined from the total mass and number of particles per bin. This size is used along with the assumed material properties of the particles and the assigned collision rate to associate a mean collisional lifetime with each size bin. The timescale for the collisional destruction of an asteroid of a given diameter depends on the probability of collision between the target asteroid and â€œfieldâ€ asteroids, the size of the smallest field asteroid capable of shattering and dispersing the target, and the cumulative number of field asteroids larger than this smallest size. We shall now detail the procedure for calculating the collisional lifetime of an asteroid and examine each of these determinants in the process. The probability of collisions (the collision rate) between the target and the field asteroids has been calculated using the theory of Wetherill (1967). Utilizing the same method, Farinella and Davis (1992) independently calculated intrinsic collision rates which match our results to within a factor of 1.1. For a target asteroid with orbital elements a, e, and i, we calculate an intrinsic collision probability, PÂ¡, which is the collision rate with the background field of asteroids in units of yr'1 km"2 normalized 38 such that the total number of particles in the asteroid belt is 1. The population of field asteroids was chosen as a subset of the catalogued mainbelt population. All asteroids brighter than H = 10. just slightly brighter than the discovery completeness limit for the mainbelt. were chosen to define a bias-free set of field asteroids. In this way the selection for asteroids in the inner edge of the mainbelt is eliminated and the field population is more representative of the true distribution of asteroids. The orbital elements were taken to be the proper elements as computed by Milani and Knezevic (1990), which are more representative of the long-term orbital elements than are the osculating elements. The resulting intrinsic collision rates and mean relative encounter speeds for several representative mainbelt asteroids are given in Table 4. The mean intrinsic rate and relative encounter speed calculated from the 672 asteroids of the bias-free set are 2.668 x 10~16 yr"1 km 2 and 5.88 km s'1, respectively. The â€œfinalâ€ collision probability for a finite-sized asteroid with diameter D is Pf=cr'Pâ€ž (3-2) where a' = a/zr (since Pi includes the factor of 7r) and a = ir(D/2)~ is the collision cross-section (taken to be the simple geometric cross-section since the self-gravity of the asteroids is negligible here). To get the total probability that the asteroid will suffer a destructive collision, we must integrate the final probability over all projectiles of consequence using the size distribution function dN = CD~vdD. (3-3) Then Dâ€ž (3â€”4-) 39 or D.nax Pt= I a'P,CD~r(ID. (3-5) D (Pt is simply the collision cross section times the intrinsic collision probability times the cumulative number of field asteroids larger than DmÂ¡n.) The collision lifetime, tc = 1 /Pt, (3-6) is then the time for which the probability of survival is 1/e. Let us now examine the determination of Dmin, the smallest field asteroid capable of fragmenting and dispersing the target asteroid. To fragment and disperse the target asteroid, the projectile must supply enough kinetic energy to overcome both the impact strength of the target (defined as the energy needed to produce a largest fragment containing 50% of the mass of the original body) and its gravitational binding energy. The impact strength of asteroid-sized bodies is not well known. Laboratory experiments on the collisional fragmentation of basalt targets (Fujiwara et al. 1977) yield collisional specific energies of 7 to 8 x 106 erg g'1, or an impact strength, S0, of 3 x 10' erg cm'3. However, estimates by Fujiwara (1982) of the kinetic and gravitational energies of the fragments in the three prominent Hirayama families indicates that the asteroidal parent bodies had impact strengths of a few times 108 erg cm"3, an order of magnitude greater than impact strengths for rocky materials. (Fujiwara assumed that the fraction of kinetic energy transferred from the impactor to the debris is //v-Â£ = 0.1.) In order to avoid implausible asteroidal compositions, we must conclude that the effective impact strength of an asteroid is a function of its size as well as its composition. The difficulties inherent in scaling the impact strength over several orders of magnitude in dimension 40 from laboratory experiments to asteroid-sized bodies are reviewed by Fujiwara et al. (1989). Davis et al. (1985) concluded that large asteroids should be strengthened by gravitational self-compression and developed a size-dependant impact strength model which is consistent with the Fujiwara et al. (1977) results and produces a size-frequency distribution of collision fragments consistent with that observed for the Hirayama families. Other researchers (Farinella et al. 1982; Holsapple and Housen 1986; Housen and Holsapple 1990) have developed alternative scaling laws for strengths, predicting impact strengths which decrease with increasing target size. We will discuss the various scaling laws in more detail later in the chapter. For the time being let us simply assume that there will be some body averaged impact strength, S, associated with an asteroid of diameter D. The gravitational binding energy of the debris must also be overcome in order to disperse the fragments of the collision. Consistent with the definition of a barely catastrophic collision, in which the largest fragment has 50% the mass of the original body, we take the binding energy to be that of a spherical shell of mass \M (where M is the total mass of the target) resting on a core of mass \M. Such a model should well approximate the circumstances of a core-type shattering collision. In this case. -n = 0.411^^ (3-7) is the energy required to disperse one half the mass of the target asteroid to infinity (see Appendix C). Not all of the kinetic energy of the projectile is partitioned into comminution and kinetic energy of the large fragments of the collision. From reconstruction of the three largest Hirayama families, Fujiwara (1982) found that a fraction Jke of 41 projectile kinetic energy partitioned into kinetic energy of the members of the family of order 0.1 was most consistent with the derived collision energies and fragment sizes. Experimental determination of the energy partitioning for core-type collisions (Fujiwara and Tsukamoto 1980) showed that only about 0.3 to 3% of the kinetic energy of the projectile is imparted into the kinetic energy of the larger fragments and the comminutional energy for these fragments amounts to some 0.1% of the impact energy. We shall take $ke to be a parameter which may assume values of from a few to few tens of percent. We may then write for the minimum total projectile kinetic energy needed to fragment and disperse a target asteroid of mass M and diameter D (3-8) where V is the volume of the asteroid. From the kinetic energy of the projectile and the mean encounter speed calculated by the Wetherill model, we can find the minimum projectile mass and. hence, the minimum projectile diameter needed to fragment and disperse the target asteroid Emin â€” -T7Ã. (3-9) Finally, then. (3-10) To finally determine the collision lifetime characteristic of each size bin, we need only specify the cumulative number of field asteroids larger than Dmin. Within the 42 collision program this number is determined by simply counting, during each time step, the total number of particles in the bins larger than D,In this way the projectile population is determined in a self consistent manner. Once a characteristic collisional lifetime has been associated with each size bin the number of particles removed from each bin during a timestep can be calculated. Instead of defining a fixed timestep, the size of a timestep. At, is determined within the program and updated continuously in order to maintain flexibility with the code. At all times At is chosen to be some small fraction of the shortest collision lifetime, rCintii, where rCrniu is usually the collision lifetime for bin i = 1. In most cases we have let At = jÃ±TCrniu. During a single timestep the number of particles removed from bin i is then found from the expression z = N(i)^~ (3-11) with the stipulation that only an integer number of particles are allowed to be destroyed per bin per timestep: the number z is rounded to the nearest whole number. For small size bins this procedure gives the same results as calculated directly by Equation 3-11, since z is rounded up as often as down and the number of particles involved is very large. For the larger size bins considered in this model, however, the procedure more realistically treats the particles as discrete bodies and allows for the stochastic destruction of asteroid sized fragments. When an asteroid of a given size is collisionally destroyed, its fragments are distributed into smaller size bins following a power-law size distribution given by dN = BD~V(1D. (3-12) 43 The exponent p is determined from the parameter b. the fractional size of the largest fragment in terms of the parent body, by the expression P = 63 + 4 63 + r (3-13) so that the total mass of debris equals the mass of the parent asteroid (Greenberg and Nolan 1989). The constant B is determined such that there is only one object as large as the largest remnant, DÂ¡r. The exponent p is a free parameter of the model, but is usually taken to be somewhat larger than the equilibrium value of 2.5 (0.833 in mass units) in accord with laboratory experiments and the observed size-frequency distributions of the prominent Hirayama families (Cellino et al. 1991), although it is recognized that in reality a single value may not well represent the size distribution at all sizes. The total number of fragments distributed into smaller size bins from bin i is then just the number of fragments per bin as calculated from Equation 3-12 multiplied by z, the number of asteroids which were fragmented during the time step. Verification of the Collisional Model Verification of the collisional model consisted of a number of runs demonstrating that an equilibrium power-law size distribution with a slope index of 1.833 is obtained independent of the bin size, initial size distribution, or fragmentation power-law, provided that we assume (as did Dohnanyi) a size-independent impact strength. As we cannot present the results of all runs made during the validation phase in a short space, a representative series of results are presented here. Figures 15 and 16 show the evolved size distributions for two separate runs of the numerical model, illustrating that the model reproduces the results predicted by 44 Dohnanyi. In both runs the slope of the breakup power-law was set equal to the equilibrium value of q = 1.833, we assumed a constant impact strength scaling law, and the logarithmic size bin interval was set equal to 0.1. For the first run the initial size distribution was chosen to be a power-law distribution with a steep slope of q = 2.0. The final distribution at 4.5 billion years is shown, as well as at earlier times at 1 billion year intervals. The evolved distribution very quickly (within a few hundred million years) attains an equilibrium slope equal to the expected Dohnanyi value of q = 1.833 for bodies in the size range of 1-100 meters. The second run began with a much shallower initial distribution with a slope of q = 1.7. The evolved distribution here as well very rapidly attained the expected equilibrium slope. The same two numerical experiments were repeated with the bin size increased to 0.2. The results (Figures 17 and 18) were identical to the first two experiments â€” power-law evolved size distributions with equilibrium slopes of 1.833. To study the dependence of the equilibrium slope on the slope of the breakup power-law and the time evolution of the size distribution we altered the collisional model slightly to eliminate the effects of stochastic collisions. Perturbations on the overall slope of the size distribution produced by the stochastic fragmentation of large bodies may mask any liner-scale trends due to long term evolution of the size distribution, especially for a steep fragmentation power-law. We ran a series of models with various power-law initial size distributions and fragmentation power-laws spanning a range of slopes. The results are shown graphically in Figures 19 through 21 where we have plotted the slope. q, of the size distribution as a function of time for the smallest bodies in the model. The slope is determined at each timestep by a least-squares fit to the 20 smallest size bins 45 (1-100 meters) of a ~60 bin model. In Figures 19. 20. and 21 the slopes of the initial size distributions are 1.88. 1.77, and 1.83. respectively. Note that the vertical scale in Figure 21 has been stretched relative to the previous two figures in order to bring out the relevant detail. In all three cases we see that the slope of the size distribution asymptotically approaches the value 1.833. reaching values not significantly different than this within the age of the solar system. The different values of the slope are only very slightly dependent upon the fragmentation power-law. For q(77 in Dohnanyi's notation) higher than the equilibrium value the final slope converges for all practical value on slopes somewhat greater than 1.832 within 4.5 billion years. For less than equilibrium the final slopes are less than 1.834. Interestingly, for steep fragmentation power-laws, the slope is always seen to â€™overshootâ€™ on the way to equilibrium, either higher than 1.833 when the initial slope is lower, or lower than 1.833 when the initial slope is higher. We find perhaps not unexpectedly that the Dohnanyi equilibrium value is reached most rapidly when the fragmentation power-law is near 1.833. Hellyer (1971) found the same behavior in his numerical solution of the fragmentation equation. In his work, however. Hellyer did not include models in which the fragmentation index was more steep than the equilibrium value, so we cannot compare our results concerning the equilibrium overshoot. Recall that Dohnanyi (1969) concluded that the debris from cratering collisions played only a minor role in determining the slope of the equilibrium size distribution. Our numerical model was thus constructed to neglect cratering debris. The recent work of Williams and Wetherill (1993) confirm that the details of cratering mechanics are unimportant in determining the equilibrium slope, although without the balancing input 46 of cratering debris the equilibrium slope may vary from the expected value of 1.833 by a very slight amount. Our numerical results seem to confirm this. The very slight deviations we see, however, will be shown to be insignificant compared to the variations in the slope due to relaxation of the Dohnanyi assumption of size-independent strengths. We conclude from this series of model runs that our numerical code properly reproduces the results of Dohnanyi (1969). With size independent impact strengths our model produces evolved power-law size distributions with slopes essentially equal to 1.833 independent of the numerical requirements of the computer code and the assumptions concerning the colliding asteroids. The â€™Wave' and the Size Distribution from 1 to 100 Meters During the earliest phases of code validation our model produced an unexpected deviation from a strict power-law size distribution. Figure 22 shows the size distriÂ¬ bution which resulted when particles smaller than those in the smallest size bin were inadvertently neglected in the model. Because of the increasing numbers of small parÂ¬ ticles in a power-law size distribution, the vast majority of projectiles responsible for the fragmentation of a given size particle are smaller than the target and are usually near the lower limit required for fragmentation. By neglecting these particles in our model, we artificially increased the collision lifetimes of those size bins for which the smallest projectile required for fragmentation was smaller than the smallest size bin. The particles in these size bins then become relatively overabundant as projectiles and preferentially deplete targets in the next largest size bins. The particles in these bins are not present in sufficient quantities to fragment large numbers of particles in the next largest size bins, and so on. This pattern is repeated in a wave-like deviation from 47 a strict power-law distribution up through the largest asteroids in the population. The same wave-like phenomenon was later independently discovered by Davis et al. (1993). The code was subsequently altered to extrapolate the particle population beyond the smallest size bin to eliminate the propagation of an artificial wave in the size distribution. However, in reality the removal of the smallest asteroidal debris by radiation forces may provide a mechanism for truncating the size distribution and generating such a waveÂ¬ like feature in the actual asteroid size distribution. To study the sensitivity of features of the wave on the strength of the small particle cutoff we may impose a cutoff on the extrapolation beyond the smallest size bin to simulate the effects of radiation forces. We use an exponential cutoff of the form N(-i) = N(-i)a 10-r'/10, (3-14) where i = 1,2,3,..., N(l) is the smallest size bin, N(â€”i)0 is the number of particles expected smaller than those in bin 1 based on an extrapolation from the two smallest size bins, and x is a parameter controlling the strength of the cutoff. Negative bin numbers simply refer to those size bins which would be present and responsible for the fragmentation of the smallest several bins actually present in the model. The number of â€œvirtualâ€ bins present depends upon the bin size adopted for a particular model, though in all cases extends to include particles ~ the diameter of those in bin 1 (roughly the size ratio required for fragmentation). This form for the cutoff is entirely empirical, but for our purposes may still be used to effectively simulate the increasingly efficient removal of smaller and smaller particles by radiation forces. When the parameter x is sufficiently large, the imposed cutoff is essentially the same as the inadvertent truncation of the size distribution which lead to the results illustrated in Figure 22, although it is 48 more realistic in its smooth tail-off in the number of particles. The results of two model runs with a sharp exponential cutoff are shown in Figure 23. The starting conditions for the two runs were identical, with the exception of the bin size. To be sure the features of the wave were not a function of the bin size, the first model was run with a logarithmic interval of 0.1 while the second used a bin size twice as large. The parameter x had to be adjusted for the second model to ensure that the strength of the cutoff was identical to that in the first model. We can see that in both models a wave has propagated into the large end of the size distribution. The results of the two models have been plotted separately for clarity (with the final size distribution for the larger bin model offset to the left by one decade in size), but if overlaid would be seen to coincide precisely, thus illustrating that the wavelength and phase of the wave are not artifacts of the bin size adopted for the model run. The effect of a smooth (though sharp) particle cutoff may be seen by comparing the shape and onset of the wave in the smallest size particles between Figures 22 and 23. The amplitude of the wave has been found to be dependent upon the strength of the small particle cutoff. A significant wave will develop only if the particle cutoff is quite sharp, that is, if the smallest particles are removed at a rate significantly greater than that required to maintain a Dohnanyi equilibrium power-law. Since radiation forces do in fact remove the smallest asteroidal particles, providing a means of gradually truncating the asteroid size distribution, some researchers (Farinella et al. 1993, private communication) have suggested that such a wave might actually exist and may be responsible for an apparent steep slope index of asteroids in the 10-100 meter diameter size range. At least three independent observations seem to indicate a greater number of small asteroids in this size range than an equilibrium extrapolation 49 from the observed larger asteroids would yield. Although there is some uncertainty in the precise value, the observed slope of the differential crater size distribution on 951 Gaspra seems to be greater than that due to a population of projectiles in Dohnanyi collisional equilibrium, ranging from p = â€”3.5 to -4.0 (Belton et al. 1992). (The Dohnanyi equilibrium value is p = â€”3.5.) The crater counts are most reliable in the diameter range 0.5 to 1 km; craters of this size are due to the impact of projectiles with diameters Â£ 100 meters. The slope of the crater distribution on Gaspra is also consistent with the crater distribution observed in the lunar maria (Shoemaker 1983) and the size distribution of small Earth-approaching asteroids discovered by Spacewatch (Rabinowitz 1993). Davis et al. (1993) suggest that although the overall slope index of the asteroid population is close to or equal to the Dohnanyi equilibrium value, waves imposed on the distribution by the removal of the small particles may change the slope in specific size ranges to values significantly above or below the equilibrium value. To test the theory that a wave-like deviation from a strict, power-law size distribuÂ¬ tion is responsible for the apparent upturn in the number of small asteroids as described above, we have modeled the evolution of a population of asteroids with the removal of the smallest asteroidal particles proceeding at two different rates; a very sharp particle cutoff and one matching the observed particle cutoff. To compare these removal rates with the removal of small particles actually observed in the inner solar system, we have plotted our model population and cutoffs with the observed interplanetary dust populaÂ¬ tion (Figure 24). Using meteoroid measurements obtained by in situ experiments. Grim et al. (1985) produced a model of the interplanetary dust flux for particles with masses 10_18g < m Â£ 102g. With a particle mass density of 2.7 g cm'3 (Griin et al. 1985) 50 this corresponds to particles with diameters of about 0.01 Â¡im to 10 mm, respectively. Figure 24 shows the Grtin et al. model and our modeled particle cutoffs for three values of x. For the following models the logarithmic size interval was set equal to 0.1. For x = 0 we have the simple case of strict collisional equilibrium with no particle removal by non-collisional effects, illustrated by the models presented in the previous section. When a sharp particle cutoff is modeled beginning at ~ 100 /Â¿m, the diameter at which the Poynting-Robertson lifetime of particles becomes comparable to the collisional lifeÂ¬ time, the evolved size distribution develops a very definite wave (see Figure 25) with an upturn in the slope index present at ~100 m. The parameter x was set equal to 1.9 for this model to produce a â€œsharpâ€ cutoff, i.e one obviously much sharper than the observed cutoff and one capable of producing a strong, detectable wave. If a wave is present in the real asteroid size distribution, however, the more gradual cutoff which is observed must be capable of producing significant deviations from a linear power-law. Over the range of projectile sizes of interest we can match the actual interplanetary dust population quite well with x = 1.2. Figure 26 illustrates that this rate of depletion of small particles is too gradual to support observable wave-like deviations. The evolved size distribution is nearly indistinguishable from a strict power-law. The observed cutoff is more gradual than those produced by simple models operating on asteroidal particles alone for at least two reasons. First, if the particle radius becomes much smaller than the wavelength of light, the interaction with photons changes and the radiation force becomes negligible once again. Second, in this size range there will be a significant contribution from cometary particles. The assumption in our model of a closed system with no input into size bins other than collisional debris from larger bins breaks down. 51 The input of cometary dust as projectiles in the smallest size bins may not be insignifÂ¬ icant in balancing the collisional loss of asteroidal particles. We conclude that a strong wave is probably not present in the actual asteroid size distribution and cannot account for an increased slope index among 100 meter-scale asteroids. Although we stress that the wave requires further, more detailed investigation, we feel it most likely that any deviations from an equilibrium power-law distribution among the near-Earth asteroid population are the results of recent fragmentation or cratering events in the inner asteroid belt. Such stochastic events must occur during the course of collisional evolution and produce deviations from a Dohnanyi equilibrium due to the injection of a large quantity of debris produced by fragmentation with a power-law size distribution unrelated to the Dohnanyi value. Fluctuations in the local slope index and dust area would thus be expected to occur on timescales of the mean time between large fragmentation events and last with relaxation times of order of the collisional lifetimes associated with the size range of interest. To determine the relaxation timescale for an event large enough to cause the steep slope index observed among the smallest asteroids, we created a population of asteroids with an equilibrium distribution fit through the small asteroids as determined from PLS data. Beginning at a diameter of ~100 m we imposed an increased slope index of q = 2, approximately matching the distribution of small asteroids determined from the Gaspra crater counts and Spacewatch data. With this population as our initial distribution, the collisional model was run for 500 million years. The initial population and the evolved distribution at 10 and 100 million years are shown in Figure 27. By 100 million years the population has very nearly reached equilibrium once again. Figure 28 shows that the slope index in the range 1-100 m 52 decays back to the equilibrium value exponentially, with a relaxation timescale of about 65 million years, although at earliest times the decay rate is somewhat more rapid. Such an event could be produced by the fragmentation of a 100-200 km diameter asteroid. Smaller scale fragmentation or cratering events would produce smaller perturbations to the size distribution and would decay more rapidly. For example, we see in Figure 29 the variation in the slope index during a typical period of 500 million years in a model of the inner third of the asteroid belt. The spikes are due to the fragmentation of asteroids of the diameters indicated. Associated with the increases in slope are increases in the local number density of small (1-100 meter-scale) asteroids. The fragmentation of the 89 km diameter asteroid indicated in Figure 29 increased the number density of 10 m asteroids in the inner third of the belt by a factor of just over 2. Since the number density of fragments must increase as the volume of the parent asteroid, the fragmentation of a 200 km diameter asteroid would cause an increase in the number of 10 m asteroids in the inner belt of over a factor of 10. This is just the increase over an equilibrium population of small asteroids that Rabinowitz (1993) finds among the Earth approaching asteroids discovered by Spacewatch and could easily be accounted for by the formation of an asteroid family the size of the Flora clan. Dependence of the Equilibrium Slope on the Strength Scaling Law The Dohnanyi (1969) result that the size distribution of asteroids in collisional equilibrium can be described by a power-law with a slope index of q = 1.833 was obtained analytically by assuming that all asteroids in the population have the same, or size-independent, impact strength. Other researchers (Williams and Wetherill 1993) have expressed the intent to consider deviations from self-similarity analytically to 53 determine the resulting effect on the size distribution. We have already demonstrated that our collisional model reproduces the Dohnanyi result for size-independent impact strengths (see Verification of Collisional Model). However, strain-rate effects and gravitational compression lead to size-dependent impact strengths, with both increasing and decreasing strengths with increasing target size, respectively (see discussion of strength scaling laws in the following section). With our collisional model we are able to explore a range of size-strength scaling laws and their effects on the resulting size distributions. In order to examine the effects of size-dependent impact strengths on the equiÂ¬ librium slope of the asteroid size distribution we created a number of hypothetical size-strength scaling laws. As will be discussed in the following section, we assume that SocD'C (3-15) where S is the impact strength, D is the diameter of the target asteroid, and // is a constant dependent upon material properties of the target. Seven strength laws were created with values of /Â¿' ranging from -0.2 to 0.2 over the size range 10 km to 1 meter. The slope index output from our modified, smooth collisional model was monitored over the size range 1-100 m and the equilibrium slope at 4.5 billion years recorded. The results are plotted in Figure 30. We find that the equilibrium slope of the size distribution is very nearly linearly dependent upon the slope of the strength scaling law. There seems to be an extremely weak second order dependence on /T, however over the range of plausible // a linear fit with a slope of approximately -0.13 is seen to fit the data sufficiently well. When //' = 0, corresponding to a size-independent strength. 54 the Dohnanyi value of q is obtained. If the slope of the scaling law is negative, as is the case with strain-rate dependent strengths such as the Housen and Holsapple (1990) nominal case, the equilibrium slope has a higher value of q ~ 1.86. If, on the other hand, Â¡j! is positive, an equilibrium slope less than the Dohnanyi value is obtained. These deviations from the nominal Dohnanyi value, although not great, are large enough that well constrained observations of the slope parameter over a particular size range should allow us to place constraints on the size dependence of the strength properties of asteroids in that size range. An interesting result related to the dependence of the equilibrium slope parameter on the strength scaling law is that populations of asteroid with different compositions and, therefore, different strength properties, can have significantly different equilibrium slopes. This could apply to the members of an individual family of a unique taxonomic type or to sub-populations within the entire mainbelt, such as the S- and C-types. Furthermore, we find the somewhat surprising result that the slope index is dependent only upon the form of the size-strength scaling law and not upon the size distribution of the impacting projectiles. This is illustrated in Figure 31, where we show the results of two models simulating the collisional evolution of an asteroid family. The stochastic fragmentation model was modified to track the collisional history of a family of fragments resulting from the breakup of a single large asteroid (see Chapter 4). We show the slope index of the family size distribution as a function of time for two families: family 1 has the same arbitrary strength scaling law as the background population of projectiles (// < 0 in this case), while the scaling law for family 2 has > 0. The slope index for family 2 is appropriate for the particular value of //' chosen and is 55 significantly different than that of family 1 or the background population, even though it is the projectiles in the background which are solely responsible for fragmenting members of the family. Since the total dust area associated with a population of debris is sensitively dependent upon the slope of the size distribution, it could be possible to make use of IRAS observations of the solar system dust bands to constrain the strengths of particles much smaller in size than those that have been measured in the laboratory. If the small debris in the families responsible for the dust bands has reached collisional equilibrium, the observed slope of the size distribution connecting the large asteroids and the small particles required to produce the observed area could be used to constrain the average material properties of asteroidal dust. The Modified Scaling Law One of the most important factors determining the collisional lifetime of an asteroid is its impact strength (see Description of Collisional Model). The impact strengths of basalt and mortar targets ~10 cm in diameter have been measured in the laboratory, but unfortunately we have no direct measurements of the impact strengths of objects as large as asteroids. Hence, one usually assumes that the impact strengths of larger targets will scale in some manner from those measured in the laboratory (see Fujiwara et al. (1989) for a review of strength scaling laws). Recently, attempts have been made to determine the strength scaling laws from first principles either analytically (Housen and Holsapple 1990) or numerically through hydrocode studies (Ryan 1993). However, we have taken a different approach of using the numerical collisional model to ask what the strength scaling relation must be in order to reproduce the observed size distribution of the asteroids. The results allow us to place some observational 56 constraints on the impact strengths of asteroidal bodies outside the size range usually explored in laboratory experiments. The observed size distribution of the mainbelt asteroids (see Figure 14) is very well determined and constitutes a powerful constraint on collisional models â€” any viable model must be able to reproduce the observed size distribution. The results of the previous section demonstrate that details of the size-strength scaling relation can have definite observational consequences. Before examining the influence that the scaling laws have on the evolved size distributions, it would be helpful to review the scaling relations which have been used in various collisional models (see Figure 32). The Davis et al. (1985) law is equivalent to the size-independent strength model assumed by Dohnanyi (1969), but with a theoretical correction to allow for the gravitational self compression of large asteroids. In this model the effective impact strength is assumed to have two components: the first due to the material properties of the asteroid and the second due to depth-dependent compressive loading of the overburden. When averaged over the volume of the asteroid we have for the effective impact strength S = S0 + 7rkGp2D2 i5~~ (3-16) where Sâ€ž is the material impact strength, p is the density, and D is the diameter. For asteroids with diameters much less than about 10 km the compressive loading becomes insignificant compared to the material strength and 5 ~ Sa, yielding the size- independent strength of Dohnanyi. The Housen et al. (1991) law allows for a strain-rate dependence of the impact strength, effectively making larger asteroids weaker than targets measured in the labÂ¬ oratory. The theory is described in detail in Housen and Holsapple (1990) where a 57 plausible physical explanation for a strain-rate strength dependence is also put forth. A size distribution of inherent cracks and Haws is present in naturally occurring rocks. When a body is impacted, a compressive wave propagates through the body and is reflected as a tensile wave upon reaching a free surface. The cracks begin to grow and coalesce when subjected to tension, and since the larger cracks are activated at lower stresses, they are the first to begin to grow as the stress pulse rises. However, since there are fewer larger flaws, they require a longer time to coalesce with each other. Thus, at low stress loading rates, material failure is dominated by the large cracks and failure occurs at low stress levels. Since collisions between large bodies are characterized by low stress loading rates, the fracture strength is correspondingly low. In this way a strain-rate dependent strength may manifest itself as a size-dependent impact strength, with larger bodies having lower strengths than smaller ones. Housen and Holsapple (1990) show that the impact strength is 5 oc D^'V;0-35, (3-17) where Ve is the impact speed. Under their nominal rate-dependent model the constant /i\ which is dependent upon several material properties of the target, is equal to -0.24 in the strength regime, where gravitational self compression is negligible. In the gravity regime, however, they find that /V = 1.65, which we note is slightly less than the D2 dependence assumed by Davis et al. (1985). The magnitude of the gravitational compression in the Housen et al. (1991) model was determined by matching experimental results of the fragmentation of compressed basalt targets (indicated by the solid dots in Figure 32), simulating the overburden of large asteroids, and estimates of impact strengths (Fujiwara 1982) determined from reconstructions of 58 the parent bodies of the Koronis, Eos, and Themis asteroid families (open dots). The most recent studies, however, indicate that the laboratory results are to be taken as upper limits to the magnitude of the gravitational compression (Holsapple 1993, private communication). Both scaling laws have been used within the collisional model to attempt to place some constraints on the initial mass of the asteroid belt and the size-strength scaling relation itself. Unfortunately, the initial mass of the belt is not known. By â€™initial' we assume the same definition as used by Davis et al. (1985), that is, the mass at the time the mean collision speed first reached the current ~5 km s'1. Davis et al. (1989) present a review of asteroid collision studies and conclude that the asteroids represent a collisionally relaxed population whose initial mass cannot be found from models of the size evolution alone. We have therefore chosen to investigate two extremes for an initial belt mass: a â€™massiveâ€™ initial population with ~60 times the present belt mass, based upon work by Wetherill (1992, private communication) on the runaway accretion of planetesimals in the inner solar system, and a â€™smallâ€™ initial belt of roughly twice the present mass, matching the best estimate by Davis et al. (1985, 1989) of the initial mass most likely to preserve the basaltic crust of Vesta. Figures 33 and 34 show the results of several runs of the model with various combinations of scaling laws and initial populations. In both figures we have included the observed size distribution for comparison with model results, but have removed the x/Tv error band for clarity. We have found that models utilizing the strength scaling laws usually considered, particularly the pure strain-rate laws, fail to reproduce features of the observed distribution. This conclusion is not particularly sensitive to the details of 59 the initial asteroid population: it is the form of the size-strength scaling law which most determines the resulting shape of the size distribution. A pure strain-rate extrapolation produces very weak 1-10 km-scale asteroids, leading to a pronounced â€œdipâ€ in the number of asteroids in the region of the transition to an equilibrium power law. The Davis et al. model does a somewhat better job of fitting the observed distribution in the transition region, further suggesting that a very pronounced weakening of small asteroids may not be realistic in this size regime. In addition, we have found that the magnitude of the gravitational strengthening given by the Davis et al. model (somewhat weaker than the Housen et al. model) produces a closer match to the shape of the â€œhumpâ€ at 100 km for the initial populations we have examined. If something nearer to the Housen et al. gravity scaling turns out to be more appropriate, however, this would simply indicate that the size distribution longward of ~ 150 km is mostly primordial. Since it is the shape of the size-strength scaling relation which seems to have the greatest influence on the shape of the evolved size distribution, we have taken the approach of permitting the scaling law itself to be adjusted, allowing us to use the observed size distribution to help constrain asteroidal impact strengths. We have been able to match the observed size-frequency distribution, but only with an ad hoc modification to the strength scaling law. We have included in Figure 32 our empirically modified scaling law, which is inspired by the work of Greenberg et al. (1992, 1993) on the collisional history of Gaspra. The modified law matches the Housen et al. law for small (laboratory) size bodies where impact experiments (Davis and Ryan 1990) indicate that strain-rate scaling best describes the fragmentation of mortar targets. Gravitational strengthening sets in for large asteroids matching the magnitude of the Davis et al. 60 model. For small asteroids an empirical modification has been made to allow for the interpretation of some concave facets on Gaspra as impact structures (Greenberg et al. 1993). If Gaspra and other similar-size objects such as Phobos (Asphaug and Melosh 1993) and Proteus (Croft 1992) can survive impacts which leave such proportionately large impact scars, they must be collisionally stronger than extrapolations of strain-rate scaling laws from laboratory-scale targets would predict. The modified law thus allows for this strengthening and in fact gives a collisional lifetime for a Gaspra-size body of about 1 billion years, matching the Greenberg et al. best estimate, which is longer than the 500 million year lifetime adopted by others. Using this modified scaling law in our collisional model we are able to match in detail the observed asteroid size distribution (Figure 35). After 4.5 billion years of collisional evolution we fit the â€œhumpâ€ at 100 km, the smooth transition to an equilibrium distribution at ~30 km, and the number of asteroids in the equilibrium distribution and its slope index. We note in particular that for the range of sizes covered by PLS data (5-30 km) the slightly positive slope of the modified scaling law predicts an equilibrium slope for that size range of about 1.78, less than the Dohnanyi value but precisely matching the value of q = 1.78 Â± 0.02 determined by a weighted least-squares fit to the catalogued mainbelt and PLS data. While we have no quantitative theory to account for our modified scaling law, there may be a mechanism which could explain the slow strengthening of km-scale bodies in a qualitative manner. Recent hydrocode simulations by Nolan et al. (1992) indicate that an impact into a small asteroid effectively shatters the material of the asteroid in an advancing shock front which precedes the excavated debris, so that crater excavation takes place in effectively unconsolidated material. The remaining body of 61 the asteroid is thus reduced to rubble. Davis and Ryan (1990) have noted that clay and weak mortar targets, materials with fairly low compressive strengths such as the shattered material predicted by the hydrocode models, may have very high impact strengths due to the poor conduction of tensile stress waves in the â€œlossyâ€ material. If this mechanism indeed becomes important for objects much larger than laboratory targets but significantly smaller than those for which gravitational compression becomes important, a more gradual transition from strain-rate scaling to gravitational compression would be warranted. Table 4: Intrinsic collision probabilities and encounter speeds for several mainbelt asteroids. Asteroid Proper Semimajor axis (AU) Proper Eccentricity Proper Inclination (degrees) Intrinsic Collision Probability (10 1H yr 1 km"2) Encounter Speed (km s'1) 1 Ceres 2.767 0.115 9.660 3.146 5.4 2 Pallas 2.769 0.252 34.771 1.905 11.8 4 Vesta 2.361 0.099 6.356 2.733 5.5 8 Flora 2.201 0.145 5.371 2.113 5.7 24 Themis 3.134 0.152 1.083 2.843 6.3 123 Brunhild 2.695 0.110 7.296 3.340 5.1 158 Koronis 2.869 0.045 2.149 3.766 4.5 221 Eos 3.012 0.077 9.939 2.818 5.2 466 Tisiphone 3.367 0.075 20.359 1.134 6.9 OS Figure 15: Verification of model for steep initial slope and small bin size. Figure 16: Verification of model for shallow initial slope and small bin size. ON Lf\ Figure 17: Verification of model for steep initial slope and large bin size. Figure 18: Verification of model for shallow initial slope and large bin size. Equilibrium Slop Time (Byrs) Figure 19: Equilibrium slope as a function of time for various fragmentation power laws and for steep initial slope. quilibrium Slop Figure 20: Equilibrium slope as a function of time for various fragmentation power laws and for shallow initial slope. Equilibrium Slope Time (Byrs) Figure 21: Equilibrium slope as a function of time for various fragmentation power laws and for equilibrium initial slope. Figure 22: Wave-like deviations in size distribution caused by truncation of particle population. Diameter (km) Figure 23: Independence of the wave on bin size adopted in model. log Incremental Number Figure 24: Comparison of the interplanetary dust flux found by Griin et al. (1985) and small particle cutoffs used Figure 25: Wave-like deviations imposed by a sharp particle cutoff (x = 1.9). 25 20 15 10 5 0 = 12). Diameter (km) Figure 27: Collisional relaxation of a perturbation to an equilibrium size distribution. Figure 28: Halftime for exponential decay toward equilibrium slope following the fragmentation of a 100 km diameter asteroid. Slope Parameter, Time (100 Myr) Figure 29: Stochastic fragmentation of inner mainbelt asteroids of various sizes during a typical 500 million period. Equilibrium Figure 30: Equilibrium slope parameter as a function of the slope of the size-strength scaling law. Figure 31: Difference in the equilibrium slope parameters for families with different strength properties. Figure 32: The Davis et al. (1985), Housen et al. (1991), and modified scaling laws used in the collisional model. Number per Diameter Bin Figure 33: The evolved size distribution after 4.5 billion years using the Housen et al. (1991) scaling law for (a) a massive initial population and (b) a small initial population. Number per Diameter Bin Diameter (km) x to Figure 34: The evolved size distribution after 4.5 billion years using the Davis et al. (1985) scaling law for (a) a massive initial population and (b) a small initial population. Number per Diameter Bin Figure 35: The evolved size distribution after 4.5 billion years using our modified scaling law for (a) a massive initial population and (b) a small initial population. CHAPTER 4 HIRAYAMA ASTEROID FAMILIES A Brief History of Asteroid Families The Hirayama asteroid families represent natural experiments in asteroid collisional processes. The size-frequency distributions of the individual families may be used to determine the mode of fragmentation of individual large asteroids and debris associated with the families may also be exploited to calibrate the amount of dust to associate with the fragmentation of asteroids in the mainbelt background population. The clustering of asteroid proper elements, clearly visible in Figure 1, was first noticed by Hirayama (1918), which he attributed to the collisional fragmentation of a parent asteroid. Hirayama identified by eye the three most prominent families, Koronis, Eos. and Themis (which he named after the first discovered asteroid in each group), in this first study and added other, though perhaps less certain families, in a series of later papers (1919, 1923, 1928). After Hirayamaâ€™s first studies, classifications of asteroids into families have been given by many other researchers (Brouwer 1951; Arnold 1969; Lindblad and Southworth 1971; Williams 1979, 1992; ZappalÃ¡ et al. 1990; Bendjoya et al. 1991), and a number of other families have become apparent. Some researchers claim to be able to identify more than a hundred groupings, while others feel that only the few largest families are to be considered "real". The disagreements arise from the different starting sets of asteroids considered (early classifications included fewer asteroids â€” with more 84 85 discovered asteroids, later investigators are able to identify smaller, less populated families which were previously unseen), the different perturbation theories which are used to calculate the proper elements, and the different methods used to distinguish the family groupings from the "background" asteroids of the mainbelt, which have ranged from eyeball searches to more objective cluster analysis techniques. This lack of unanimous agreement on the number of asteroid families or on which asteroids should be included in families, prompted some (Gradie et al. 1979: Carusi and Valsecchi 1982) to urge that a further understanding of the discrepancies between the different classification schemes was necessary before the physical reality of any of the families could be given plausible merit. Only in the last few years have different methods lead to a convergence in the families identified by different researchers (ZappalÃ¡ and Cellino 1992). The ZappalÃ¡ Classification To date, probably the most reliable and complete classification of Hirayama family members is the recent work of ZappalÃ¡ et al. (1990). They used a set of 4100 numbered asteroids whose proper elements were calculated using a second-order (in the planetary masses), fourth-degree (in the eccentricities and inclinations) secular perturbation theory (Milani and Knezevic 1990) and checked for long-term stability by numerical integration. A hierarchical clustering technique was applied to the mainbelt asteroids to create a dendrogram of the proper elements and combined with a distance parameter related to the velocity needed for orbital change after removal from the parent body. Families were then identified by comparing the mainbelt dendrogram with one generated from a quasi-random distribution of orbits simulating the actual distribution. 86 A significance parameter was then assigned to each family to measure its departure from a random clustering. As revised proper elements become available for more numbered asteroids the clustering algorithm is easily rerun to update the classification of members in established families and to search for new, small families. In their latest classification ZappalÃ¡ et al. (1993, private communication) find 26 families, of which about 20 are to be considered significant and robust. In Figure 36 we have plotted the proper inclination versus semimajor axis for all 26 ZappalÃ¡ families and have labeled some of the more prominent ones. The Koronis, Eos, and Themis families remain the most reliable, however ZappalÃ¡ also considers many of the smaller, compact families such as Dora. Gefion, and Adeona quite reliable. The less secure families are usually the most sparsely populated or those which might possibly belong to one larger group and remain to be confirmed as more certain proper elements become available. The Flora family, for instance, although quite populous, is considered a â€œdangerousâ€ family, having proper elements which are still quite uncertain due to its proximity to the 0q secular resonance. The high density of asteroids in this region, which is likely a selection effect favoring the discover of small, faint asteroids in the inner belt, also makes the identification of individual families difficult â€” the entire region merges into one large â€œclanâ€, making it difficult to determine which of the asteroids there are genetically related. Collisional Evolution of Families Number of Families One constraint on the collisional history of the mainbelt is the number of families which have been produced and remain visible at the present time. A very massive 87 initial population coupled with relatively weak asteroids would imply that nearly all the families identifiable today must be relatively young. A smaller initial belt and asteroids with large impact strengths would allow even modest-size families to survive for billions of years. To attempt to distinguish between these two possibilities and to examine the collisional history of families we modified our stochastic collisional model to allow us to follow the evolution of a family of fragments resulting from the breakup of a single large asteroid, simulating the formation of an asteroid family. At a specified time an asteroid of a specified size is fragmented and the debris distributed into the model's size bins in a power-law distribution as described in Chapter 3. As the model proceeds, a copy of the fragmentation and debris redistribution routine is spawned off in parallel to follow the evolution of the family fragments. The projectile population responsible for the fragmentation of the family asteroids is found in a self- consistent manner from the evolving background population. Collisions between family members are neglected for the following reason. We have calculated that the intrinsic collision probability between family members may be as much as four times greater than that between family and background asteroids. For example, the intrinsic collision probability between 158 Koronis and mainbelt background asteroids is 3.687 x 10-18 yr'1 km'2, while the probability of collisions with other Koronis family members is 13.695 x 10_1S yr'1 km'2. Due to their similar inclinations and eccentricities, however, the mean encounter speed between family members is lower than with asteroids of the background population, requiring larger projectiles for fragmentation. The mean encounter speed between members of the Koronis family, for instance, is approximately 1.3 km s'1, significantly lower than the roughly 4.5 km s'1 encounter speed between 88 Koronis family members and asteroids of the background projectile population. The very large total number of projectiles in the background population completely swamps the small number of asteroids within the family itself, so that the collisional evolution of a family is still dominated by collisions with the background asteroid population. To determine how many of the families produced by the model should be observÂ¬ able at the present time we have defined a simple family visibility criterion which mimics the clustering algorithm actually used to find families against the background asteroids of the mainbelt (ZappalÃ¡ et al. 1990). We have found the volume density of non-family asteroids in orbital element space for the middle region of the belt (corresponding to zone 4 of ZappalÃ¡ et al. 1990). In the region 2.501 < a < 2.825, 0.0 < e < 0.3, and 0.0 < sinz < 0.3 we found 1799 non-family asteroids which yields a mean volÂ¬ ume density typical of the mainbelt of 1799/(0.324AU x 0.3 x 0.3) = 1799/0.02916 = 61694.102 asteroids per unit volume of proper element space. The volume density of the asteroids in a family is then found by using Gaussâ€™ perturbation equations to calÂ¬ culate the spread in orbital elements associated with the formation of the family (see, e.g., ZappalÃ¡ et al. 1984). The typical AV" associated with the ejection speed of the fragments will be of the order of the escape speed of the parent asteroid, which scales as the diameter, D. The typical volume of a family must then scale as ZT3, so that families formed from the destruction of large asteroids are spread over a larger volume. We computed the volume associated with the formation of a family from a parent 110 km in diameter (the size of the smallest parent asteroids we consider) to be 2.26 x 10_J element units. The AK for a parent of this size is approximately 135 m s'1. Within the model the family volume associated with a parent asteroid of any specified size 89 is then scaled from this value. The number of telescopically visible asteroids spread throughout this volume is then used to compute the familyâ€™s volume density. The typical completeness limit for families in the middle mainbelt is ~30 km. We simply count the number of family members in size bins larger than this when computing the volume density. As our collisional model proceeds for a certain parent body size, we then monitor the volume density of the observable asteroids in that family and compare this density to that of the background. When the family density drops to that of the background, we assume that the family is no longer observable. Figure 37 shows how the volume density of families derived from various-size parent asteroids decays with time as the member asteroids are subsequently ground away due to further collisions. The dashed horizontal line indicates the density of the mainbelt background, the threshÂ¬ old density for detection. For parent bodies not significantly larger than 100 km, the resulting families drop below the detection threshold after 1-1.5 billion years. Families formed from parents larger than 250 km may remain detectable for 3.5 billion years, nearly the lifetime of the solar system. Our detection criterion then allows us to estimate how many visible families our model predicts. We have noted in Figure 38 the times at which asteroids larger than 100 km have been collisionally destroyed in our model. The regular spacing on this log plot reflects the size and spacing of the size bins used in our model. We can see that the rate of formation of families decreases noticeably with time, especially for the smaller parent bodies. Families to the left of the dashed boundary are ground to undetectibility by present techniques. Our nominal collision model predicts about 30 families, not greatly more than the roughly two dozen families presently recognized. 90 especially considering the complexity of the Flora region and the likely number of small families that remain to be detected in the outer belt. Evolution of Individual Families Size Distributions Our model, of course, allows us to study the collisional history of individual families as they evolve due to collisions with the evolving background population. In order to compare our modeled families with those observed in the mainbelt we must define the size distributions of the observed families. We proceed as in Chapter 2 when reconstructing the sizes of other mainbelt asteroids. For those family members with IRAS albedos the diameters were calculated based on their V-band absolute magnitudes from Equation 11 of Appendix A. For family asteroids without a measured albedo we assumed the mean albedo of other, IRAS-observed asteroids within the family. Ten of the most sparse families contained no members for which IRAS-derived albedos were available. In these cases we assumed that the family members had the same albedo as other mainbelt asteroids at the same semimajor axis. The final size distributions are then presented as cumulative size-frequency distributions. The diameters down to which families are considered complete with respect to discovery have been calculated assuming the current mean opposition magnitude completeness limit for the mainbelt of approximately 16. The Gefion Family For the present time, we have chosen to limit our modeling of individual families to a few of the smaller families. Features of the size distributions of some of the 91 largest families imply self-gravitational reaccumulation on the largest remnant (ZappalÃ¡ et al. 1984). Other families, such as Vesta (Binzel and Xu 1993) and possibly Themis (Williams 1992). may represent very large cratering events. The families resulting from such events are sufficiently difficult to model with our simple power-law fragmentation routine that we feel they are best left to future, more sophisticated models. The size distributions of some of the more modest-size families, however, suggest simple, complete fragmentation of the parent asteroid which our model approximates quite well. The Gefion family is best modeled by the destruction of a 150-160 km parent body approximately 500 million years ago (Figure 39). The dashed vertical line indicates the diameter down to which the family is considered complete. There is quite a range of uncertainty in the age due to the stochastic nature of the fragmentation of individual asteroids, especially in small families where the total number of large asteroids is small. Some models with a slightly larger parent body may match the observed distribution after as long as 1 billion years. Smaller parents may produce a much younger family. Most models, however, consistently give a best match at a few to several hundred million years with a parent 150-160 km across. The results for the Dora family are very similar. The Maria Family The rather distinctive size distribution of the Mana family, in contrast to the Gefion family, is very well explained if that family is quite old. The best fit is obtained with the destruction of a 175-180 km parent asteroid 3 billion years ago. Figure 40 shows the results of the collisional model averaged over five model runs to eliminate stochastic variations from run to run. The family has been modeled assuming the same mean 92 intrinsic collision probability and encounter speed used within the mainbelt model. In reality, the high inclination of the Mana family results in a slighter higher mean encounter speed with mainbelt projectiles, making it possible for smaller projectiles to disrupt Maria targets. Although this could result in a slightly higher rate of collisional evolution than we have modeled, the slight increase could not decrease the age of the family to less than about 2 billion years. The family also displays a much less compact structure than, for example, the Eos family. This could be due to a significant loss of family members from collisional fragmentation, also suggesting a great age. Proper Inclination (degrees) 20 1 1 1 1 1 1 1 1 1 1 1â€”~\ Maria Eunomia Adeona M Eos Vesta v H;... .:*>Y Gefion * Dora - c,, Â£ y.ijr' Flora Jr-. -k w - â– â€” - A ilC /v V! â€¢ â€¢, ;-^.r Nysa i i i i 1 i â€¢ .*â€¢ Koronis . l i 1 ( ' I 1 i 15 10 - o 2.0 2.5 3.0 Semimajor Axis (AU) 3.5 Figure 36: The 26 Hirayama asteroid families as defined by Zappala et al. (1984). Density Relative to Background Time (Byrs) Figure 37: The collisions! decay of families resulting from various-size parent asteroids as a function of time. Diameter (km) Time (Byrs) Figure 38: Formation of families in the mainbelt as a function of time. 10 10 10 10 10 3 10 30 100 300 Diameter (km) Figure 39: Modeled collisional history of the Gefion family. so Os 10 10 10 10 101 Figure 40: Modeled collisional history of the Maria family. CHAPTER 5 IRAS AND THE ASTEROID AL CONTRIBUTION TO THE ZODIACAL CLOUD The IRAS Dustbands The Infrared Astronomical Satellite (IRAS) was carried from Vandenburg Air Force Base to its near-polar. Sun-synchronous orbit by a Delta rocket on January 25th, 1983. For 11 months the one-ton satellite returned a wealth of data, surveying nearly 96% of the sky at 12, 25, 60, and 100 //m before its supply of liquid helium coolant ran dry (see Matson et al. 1989 for a detailed description of the mission). It is a testament to the quantity and quality of the data returned by the telescope that researchers are still making new discoveries from IRASâ€™s observations, a decade after its mission ended. Developed as a joint program of the United States, the Netherlands, and Great Britain, IRASâ€™s primary mission was to study star-forming regions, the presence of cold, dusty material in the galaxy, and the infrared emission from extragalactic objects. However, one of the main factors contributing to the observational â€œnoiseâ€ was the warm cloud of solar system dust. In fact, the flux in the 12 and 25 /Â¿m wavebands is nearly completely dominated by emission from the zodiacal cloud. IRAS made the surprising discovery of three relatively narrow bands of infrared emission superimposed on the broad zodiacal emission (Low et al. 1984; Neugebauer et al. 1984). The most prominent band lies near the ecliptic, at latitudes of 2-3Â°, and is flanked by a fainter pair of bands above and below the ecliptic at latitudes of +10Â° and -10Â° (see Figure 41). The bands can also be seen in the 60 and 100 //m data, although at a lower 98 99 intensity. Color-temperature calculations (Low et al. 1984) yield values between 165 and 200 K. consistent with the temperature of a rapidly rotating gray body located between 2.2 and 3.2 AU. This distance matches estimates of the location of the band emission at 2.3-2.5 AU obtained by parallax measurements (Gautier et al. 1984; Hauser et al. 1985; Dermott et al. 1990). The estimated location of the band pairs within the asteroid belt suggested to Low et al. (1984) that the band emission arose from the dusty debris produced by collisions between asteroids. Dermott et al. (1984) demonstrated that the bands are likely associated with asteroid families, noting that the latitudes of the dust bands match the inclinations of the three most prominent Hirayama asteroid families. They linked the central dust band with the Themis and Koronis families and the 10Â° band pair with the Eos family. Firmly establishing a connection between the solar system dust bands and specific asteroid families would provide conclusive evidence that asteroids are a significant source of dust in the zodiacal cloud and would imply that the gradual comminution of background asteroids in the mainbelt population makes a significant contribution to the broader zodiacal emission. A number of papers have since been published detailing the progress which has been made in relating the geometry of the dust bands and the orbital elements of asteroid families (Dermott et al. 1985; 1990; 1992a; 1992b). Modeling the Dust Bands To analyze the IRAS observations of the zodiacal emission and to determine the distribution of dust within the zodiacal cloud. Dermott and Nicholson (1989) developed a three-dimensional numerical model, SIMUL. which permits the calculation of the distribution of sky brightness, as seen by the IRAS telescope, associated with any 100 particular distribution of dust particle orbits. Modifications to improve the model and increase its versatility have since been made (see, for example, Xu et al. 1993). The SIMUL model consists of three major components: (l)a reproduction of the exact viewing geometry of IRAS, including the effects of the eccentricity of the Earthâ€™s orbit, (2) the distribution of orbital elements of the dust particles in space, and (3) the contribution to the total brightness from a single orbit. The distribution of dust particle orbits is determined by starting with a postulated source of dust particles, either asteroidal or cometary or other, and then describing the orbital evolution of the particles under the influence of Poynting-Robertson light drag, radiation pressure, solar wind, and gravitational perturbations. Once the structure of the cloud has been specified in terms of the distribution of orbits and the thermal properties of the particles, SIMUL calculates the flux observed in any direction and at any observing time. The result is a model profile of the brightness distribution as a function of ecliptic latitude, observed in a given waveband as the telescope sweeps through the model cloud at a given elongation angle (defined as the angle between the Sun. spacecraft, and spacecraft line-of-sight). As an example, IRAS observations of the dustbands at three different elongations angles are compared with the fluxes predicted using the SIMUL model. The observaÂ¬ tions (Figure 42a) are in the 25 /Â¿m waveband and illustrate the range of amplitudes and shapes produced by the variable viewing geometry during the IRAS mission. The model profiles illustrated in Figures 42b, c, and d were produced using dust from six prominent families: Themis. Koronis, Eos, Nysa. Dora, and Gefion. The cross-sectional areas of dust associated with the families were treated as free parameters and adjusted to fit the observations at elongation angle 114.68Â°. Exactly the same particle distribution 101 was used for the other two elongations, with the exception that the total area had to be adjusted downward, slightly, for elongation 65.68Â°. Still, the very good fits to the observations, reproducing the complex shapes and amplitudes of the bands, are nearly conclusive evidence that the dust bands are associated with specific asteroid families. To obtain the band profiles illustrated in Figures 41 and 42 and to determine the total area associated with bands, the much stronger and broader zodiacal background must be separated from the weak band emission. This involves using a Fourier filter to find the spatial frequency distribution of the flux signal and separating the high- frequency region, associated with the band emission, from the low-frequency region, associated with the broad background. The resulting band profiles are only a small portion of the total contribution to the observed flux made by the families, however. Determining the total contribution made by the families to the observed flux involves several iterative steps and is complicated by the fact that some of the flux in the smooth zodiacal background is contributed by the bands themselves (Xu 1993, private communication). Very briefly, the process involves adding to the observed smooth zodiacal background the modeled total flux associated with the prominent families. This combined observed and modeled total flux is greater than the observed total flux, of course, because a portion of the original observed smooth background contains a contribution from the families. This new profile is passed through the same Fourier filter used for the observations to generate a new smooth background and band profile. The difference between the new band profile and the modeled total family contribution is the portion of the family flux which contributes to the smooth zodiacal background. This contribution is subtracted from the new smooth background, and the modeled 102 family flux is added to the remaining background. When this total signal is passed through the filter a second time, the resulting smooth background and band profile match very closely the original observed background and band profiles. The total area associated with the families needed to model the dust bands is found to be ~3 x 109 km2. This is found to be ~10% of the area needed to model the non-family contribution to the smooth zodiacal background. The Ratio of Family to Non-Family Dust Having established that the IRAS dust bands are associated with the prominent Hirayama asteroid families and having determined the extent of their contribution to the total flux in the zodiacal cloud, if the particle production rate of family asteroids is no different than that of other mainbelt asteroids we may use the families to calibrate the extent of the non-family asteroidal component of the interplanetary dust complex. Unfortunately, the amount of dust generated in a single asteroid collision is highly uncertain. Although we assume for simplicity that the fragmentation debris can be described by a simple power-law size distribution, in reality a single value for the slope may not well represent the distribution at all sizes and the mode of fragmentation can be expected to be highly variable from event to event. Fortunately, we have shown that despite the uncertainties associated with individual fragmentations, the equilibrium size- frequency distribution can still be well-described by a simple power-law distribution. In Chapter 3 we showed that regardless of shape of the initial population of debris, the size distribution of a family of fragments will evolve to an equilibrium distribution with a half-time of order of the fragmentation lifetime of the largest debris being considered. If the size-frequency distribution of debris in the families can be described by a simple 103 equilibrium power-law. then we may relate the surface area of the dust in the bands to the total volume of the family fragments by the expressions derived in Appendix B. Similarly, we may calculate the total area associated with the entire mainbelt asteroid population. Our calculation of the total mainbelt dust area cannot be directly related to the equivalent volume of the mainbelt population, however, as the largest asteroids may not contribute to the population of collision fragments. In fact, we can see directly from the observed size distribution derived in Chapter 2 that relative to an equilibrium power-law distribution, there is an excess of asteroids for diameters larger than ~30-40 km. This excess effective volume represents a remnant of the initial asteroid population which has not yet reached collisional equilibrium and does not contribute to the mainbelt dust area. To actually calculate the ratio of family to non-family areas we lit an equilibrium distribution through the combined magnitude distribution of all the ZappalÃ¡ families and through the linear portion of the mainbelt asteroid population. We have plotted these distributions in Figure 43 with the equilibrium fits obtained by constrained least-squares solutions. Although we work directly with the magnitude data, for convenience the abscissa has been labeled in kilometers by converting the magnitude data to diameters by assuming for each distribution the mean albedo of family and mainbelt asteroids. Although we have calculated the actual cumulative areas (down to a minimum cutoff size of 10 /am) in each population, they need not actually be calculated to obtain the ratio we seek, since both are related to the effective volumes, which in turn are determined by the intercepts of the least-squares solutions. We take as a measure of the intercept the diameter of the largest asteroid, Dmax, which would be present in the 104 equilibrium distribution fit to the data. For the entire mainbelt population we find that = 308^2 km. while for the combined family distribution Dmax = 189+2(Â¿Â¡ km. For a specified cutoff size. Dmrn, the total geometrical cross-sectional area associated with the debris is then calculated directly from Equation 11 of Appendix B. For Dnun = 10 nm the cross-sectional areas for the entire mainbelt population and all families combined are 1.45 x 1011 and 4.25 x 1010 km2, respectively. The ratio of the cumulative area of dust in the entire mainbelt population to that associated with all the families is then approximately 3.4 : 1. Due to standard errors in the values of the intercepts from the least-squares solutions and uncertainties in the mean albedos assigned to each group, there is an uncertainty in this ratio of about 0.6. Since the entire mainbelt population as we have defined it includes the contribution from families, the non-family mainbelt asteroids must contribute about 2.4 times as much dust to the zodiacal cloud as the prominent families. In the previous section we found that analysis of IRAS data indicates that the prominent families associated with the dust bands are responsible for about 10% of the total zodiacal emission. If the gradual comminution of non-family asteroids in the mainbelt produces about 2.4 times as much dust as that associated with families, then the entire mainbelt asteroid population must be responsible for at least a third of the dust particles in the zodiacal cloud. S-H CO o T 1 X cO P "0 â€¢ iâ€”H cn CD Ecliptic Latitude o Ln Figure 41: The solar system dust bands at 12, 25, 60, and 100 gm, after subtraction of the smooth zodiacal background via a Fourier filter. Total Flux (106 Jy/Sr) o ON Figure 42: (a) IRAS observations of the dust bands at elongation angles of 65.68Â°, 97.46Â°, and 114.68Â°. Comparisons with model profiles based on prominent Hirayama families are shown in (b), (c), and (d). Number per Diameter Bin Diameter (km) Figure 43: The ratio of areas of dust associated with the entire mainbelt asteroid population and all families. CHAPTER 6 SUMMARY Conclusions We may summarize the main conclusions of this work: (1) Data from the Palomar-Leiden Survey of faint asteroids has been used to supplement data from the catalogued population of asteroids to extend the size-frequency distribution of the mainbelt to diameters of ~5 km. The observed size distribution displays a marked â€œhumpâ€ at sizes near 100 km and makes a gradual transition to a distinctly linear distribution for diameters less than about 30 km. The observed slope of the linear portion is slightly, though statistically significantly less than the equilibrium slope predicted by Dohnanyi (1969). The observed distribution is quite well determined and constitutes a strong constraint on collisional models of the asteroid population. (2) We have developed a numerical model to study the collisional evolution of the asteroids which confirms the earlier results of Dohnanyi (1969) for size-independent impact strengths. If the strengths of asteroids are allowed to vary with size, however, we find that the slope of the equilibrium size distribution is dependent upon the slope of the size-strength scaling relation. We further find that the equilibrium slope does not depend on the size distribution of the projectile population. These results imply that it is possible for an asteroid family with material properties different from that of the average background population to have an equilibrium size distribution distinct from that of the background asteroids. Observations of the dust areas associated with particular 108 109 families and models of the collisional history of those families might be combined to place constraints on the impact strengths of particles of sizes much smaller than have been measured in the laboratory. (3) When used within our collisional model, the size-independent and strain-rate scaling laws of Davis et al. (1985) and Housen et al. (1991) yield evolved size distributions which fail to match the observed mainbelt distribution. We find the results to be not greatly sensitive to the mass or shape of the initial asteroid population, but rather to the shape of the scaling law. The form of the size-strength scaling relation has definite observational consequences and cannot be neglected when considering the results of collisional models. We have therefore taken the empirical approach of using the observed asteroid size distribution to determine the shape of the size-scaling law. The results indicate a much more gradual transition to the gravity-scaling regime than predicted by current scaling laws. (4) When the self-similarity of the original Dohnanyi (1969) fragmentation problem is broken by allowing the smallest particles in the population to be removed by radiation forces, wave-like deviations from a strict power-law size distribution result. The amplitude of the wave is found to be strongly dependent on the strength of the small particle cutoff. When we model the empirically derived interplanetary dust flux we find the small particle cutoff too gradual to support observable deviations â€” the wave is unlikely responsible for the increase in slope suggested to exist for asteroids smaller than approximately 100 meters. We suggest, instead, that stochastic fragmentation events, which must occur in the course of collisional evolution, are more likely responsible for any observed deviations from an equilibrium distribution. 110 (5) Analysis of IRAS data has shown that although the solar system dust bands are only about 2-3% the strength of the broad zodiacal emission, a significant portion of the dust responsible for the bands contributes to the broad background, so that the prominent families actually supply about 10% of the dust in the zodiacal cloud. Our comparison of the effective volumes of the families and the portion of the mainbelt population in collisional equilibrium shows that the non-family mainbelt asteroids produce approximately 2.4 times as much dust as the prominent families. All mainbelt asteroids must then supply at least 34% of the dust in the zodiacal cloud. Future Work Some of the results of the collisional model immediately suggest the need for follow-up study. Our model neglects the contribution of debris created by cratering impacts. To what degree will the equilibrium slope dependence upon the slope of the strength scaling law be affected by the inclusion of cratering debris? Will the deviation from the Dohnanyi equilibrium become more or less severe with increasingly stronger size-dependence? Might the relation become more strongly non-linear? With regard to our modified scaling law, could the shape of the evolved size distribution be significantly affected by small cratering fragments? One would think not. since volumetrically, catastrophic collisions dominate the mass input into any size range, but the details remain to be tested within our model. The recent discovery of a ring of asteroidal particles trapped in corotational resonance with Earth (Dermott et al. 1993) will yield quantitative information on the rate of transport of asteroidal particles to the inner solar system and the comminution of the asteroids. Approximately 20% of the asteroidal particles passing the Earth are temporarily (for ~104 years) trapped in resonant lock with the planet. If the mass input required to supply the observed ring can be determined, the production rate of dusty asteroidal debris over at least the last 1()4 in the mainbelt will be quantified and will provide an extremely strong constraint on collisional models of the mainbelt. Although we have concluded that the wave induced by the removal of the smallest particles in the population is probably not an important feature of the actual asteroid size distribution, we caution that more work needs to be done on the problem. A more realistic treatment of the removal of the small particles, by actually computing the removal rate by Poynting-Robertson drag and light pressure as a function of particle size, is necessary. How important is the role played by cometary particles in negating the assumption that the only mass input into the smallest size bins is due to the comminution of larger asteroidal particles? Taking the problem beyond a simple particle-in-a-box model might indicate whether the strength of any induced wave is dependent upon location in the belt. We might speculate that wave-like deviations would be strongest in the outer mainbelt, where small particles removed by radiation forces are not as rapidly replaced by a constant influx of particles being transported from beyond, as in the inner mainbelt. Only further, more refined models can answer these questions. We have had some success in accounting for important features of the zodiacal cloud and dust bands, although there are other observations for which our model must also account but which at present are problematical. In particular, although a single-size particle model of mainbelt asteroidal dust can explain the observed inclination and nodes of the zodiacal cloud, the model predicts a total flux at high ecliptic latitudes which is far too low (Dermott et al. 1992b). One resolution of this discrepancy may lie in a cloud of asteroidal particles whose effective area increases with decreasing heliocentric distance, as might be expected for particles undergoing continual collisional evolution concurrent with orbital decay due to radiation effects. Work on this problem has already begun (Gustafson et al. 1992). The results will also yield a description of the variation of the particle size distribution with heliocentric distance. If a cloud of asteroidal particles is shown to contribute more to the background flux at high ecliptic latitudes, the total contribution made to the zodiacal cloud by families would increase to greater than the present estimate of ~10%. The total supply of dust made by the mainbelt asteroid population would then be greater than 30% â€” if the family contribution simply doubled to 20%. the total asteroidal component would increase to nearly 70%, reversing the presently estimated ratio of asteroid to comet dust. APPENDIX A APPARENT AND ABSOLUTE MAGNITUDES OF ASTEROIDS The magnitudes of solar system objects are described using the same system as in stellar astronomy â€” namely, there is a factor of 100 in flux associated with a magnitude difference of 5 units. In other words, 1 magnitude = 1001/5 = (lO2)1^ = 102//5 = 100-4 = 2.512... With this definition 2 mag = (l004)~, 3 mag = (lO04)3, and in general, the ratio of fluxes from two objects with magnitudes m.\ and m.2 is: = (io'Â»r""". (Aâ€”1 > or. F\ log â€” = 0.4(n?9 â€” mi), Ft. or. m 2 â€” m 1 = 2.5 log F\ Fo (Aâ€”2) (Aâ€”3) Note that in the last equation the 2.5 is exact and not 2.512 rounded off. An absolute magnitude may be defined as the apparent magnitude observed when the object is at some standard distance. From the inverse-square law of light propagation we have for the fluxes of two identical objects observed at different distances, 2 Fi Ft (Aâ€”4) For stellar sources the standard distance is 10 parsecs, yielding (after substituting Eq. 4 with ri = 10 for F\/Ft in Eq.3) the familiar â€” M = 2.5 log (jq) = 5 lÂ°Â£r â€” 5, m. (Aâ€”5) 113 114 where M is the apparent magnitude at 10 parsecs. Similarly, we can define an absolute magnitude for solar system objects. If we let r be the distance of the object from the Sun and p be the distance from the Earth (in Astronomical Units), Eq. 4 yields: F\ Fo r2p2 r\pi (A-6) since the object appears dimmer due to both its increased distance from the Sun (less intercepted light) and from the Earth (decreased flux). (This is similar to the reason that the strength of a radar signal detected from an object varies inversely with the fourth power of its distance â€” there is a 1/r2 decrease in flux in both the transmitted beam and the reflected signal.) The absolute magnitude of a solar system object is defined to be the apparent magnitude it would have if observed when 1 AU from the Earth, 1 AU from the Sun, and at 0Â° phase angle. Our standard distance unit is then rp = 1 and after substitution Eq. 3 reads: m 2 â€” m i = 2.5 (Aâ€”7) or. mv â€” H â€” 5 log rp, (Aâ€”8) where H is the V-band absolute magnitude. From Eq. 1 we see that the observed flux of an asteroid is proportional to 10_o But the flux from the asteroid depends on its cross sectional area (a large asteroid appears brighter than a small asteroid) and its geometric albedo (an asteroid with a bright surface is more reflective than an asteroid with a dark surface). We then have that Flux oc pvD2 oc 10 Â°'4^ (Aâ€”9) 115 where pv is the geometric albedo and D is the diameter of the asteroid. Then, log/><, + 21og/9 = const. - 0.4// (A-10) or. 2 log D = const. â€” 0.4// â€” logp,.,. (A-l 1) which is the expression given by Zellner (1979). (Note that 5(1.0) - (B - V) = Vr(1.0) = H.) The constant has the value 6.241 and is derived knowing the apparent magnitude of the Sun. APPENDIX B SIZE, MASS, AND MAGNITUDE DISTRIBUTIONS The distribution of sizes and masses of asteroids may be presented in a number of ways: cumulative plots of the number larger or more massive than .v, incremental plots (number per size or mass bin) with linear increments, and incremental plots with logarithmic increments. This note is meant to detail the relationships between the various plots and to derive expressions for the total mass and cross-sectional area in the fragments in the distribution. It is well known that many fragmentation events in nature produce a power law size (or mass) distribution of fragments. A power law distribution has the form N = CD~P, (B-l) where N is the cumulative number larger than diameter D and C is a constant. By taking the common logarithm of both sides, log N = â€” p log D + log C, (B-2) we see that the power law exponent, p, is the negative slope in a logAMogD plot and the constant, C, defines the y-intercept. To see more clearly any concentrations or depletions of particles in certain size ranges an incremental plot is more useful. We must be careful, however, to clearly define the kind of increment which has been chosen â€” linear or logarithmic. Differentiating 116 117 Equation 1 we obtain dN = -pCD~v~ldD, (Bâ€”3) where dN is the number in the linear increment of width dD. (The negative sign simply formally indicates that the number per bin decreases with increasing diameter; we are interested in the magnitude of the change, so that the negative sign may be ignored. See also Equation 5 below.) Taking the logarithm of both sides, \ogdN = -(p + 1) log D + log (pCdD), (Bâ€”I) we see that the slope of the size distribution on a log<#V-logÂ£> plot is â€” (p + 1). If the cumulative plot had a slope of -2.5. the incremental plot with linear increments would have a slope of -3.5. Now since r/logr = \^pdx, we may rewrite Equation 3 as dN = â€” pC\nlOD~pd\og D. (B-5) This then represents an incremental size distribution with logarithmic increments. From \ogdN = â€”p log D + log (pClnlOd log D) (B-6) we see that the slope is the same as that for a cumulative plot, namely -p. In many cases the fragment distribution is described in terms of mass rather than size. Since M = ^KpR3 â€” ^npD3, Equation 1 may be rewritten in terms of the mass as N = C â€” | 3 CM~3 =C'M~q. npj (B-7) The slope of a cumulative mass distribution plot is then simply one third that of the cumulative size distribution plot. As before, if the slope of a cumulative mass 118 distribution plot is -q. the corresponding incremental plot with linear increments will have a slope of â€” (q + 1) and the incremental plot with logarithmic increments will have a slope of -q. Dohnanyi (1969) has shown that the theoretical value for the slope of a cumulative mass distribution of fragments in collisional equilibrium is q = 0.833. The corresponding value for the cumulative size distribution is then p = 2.5. Let us now derive expressions for the total mass and total geometrical cross- sectional area contained in fragments which are distributed according to a particular size distribution. The total mass of a collection of particles of various sizes is just the sum of the masses of individual particles of each size multiplied by the number of particles of that size. For a continuous distribution of sizes it is the particle mass integrated over the size-frequency distribution: Mtot = I Â£prfpCD-f-'dD, (B-8) since M = ppD'K Equation 8 is integrated over the size range from the smallest to largest particles present. After carrying out the integration, the final expression for the total mass is Mtot â€” n3-P 6(3 â€” p) Dâ€ž (Bâ€”9) The total cross-sectional area is found in a similar manner by integrating the cross- section of a single particle, ^D2, over the size distribution: Atat = j JtfpCD-v-hlD, (Bâ€”10) which yields a total geometrical cross-sectional area of A tot â€” npC 4(2 â€” p) D--v D max D tn i a (Bâ€”11) 119 (Recall that in these expressions p and C are the negative slope and constant for the cumulative size distribution.) When discussing the asteroids, we often also use the frequency distribution of magnitudes in lieu of the size distribution. The magnitudes are binned (the PLS uses half-magnitude bins, for instance) and the number of asteroids per bin is presented in a \ogdN-Mag plot. Remembering that 2 log D = const â€” OAH â€” logp.â€ž, we see that such a plot is equivalent to an incremental size-frequency distribution plot with logarithmic increments, since an increment of x in absolute magnitude H corresponds to an increment of 0.2x in logD. We can then derive expressions which will allow direct calculation of the total mass or total cross-sectional area from the magnitude-frequency plot. Consider a \ogdN-H plot of the form \ogdN = aH + b, (Bâ€”12) where a is the slope and b is the y-intercept. Substituting for H we obtain logt/TV = a(â€”5 log D + 2.5const â€” 2.5 logpv) + b â€” â€”5a log D + (2.5aconst â€” 2.5a logp,, + b). Comparing Eqs. 6 and 13 we see that (Bâ€”13) p = 5a (B-14) and log (pC In lOc/log D) = 2.5aconst â€” 2.5a logp,: + b. (Bâ€”15) (For a population in collisional equilibrium with a slope parameter p = 2.5, the slope of the magnitude distribution is then a = 0.5.) Once we have assumed a mean albedo and constant for the distribution of asteroids under consideration, these expressions allow 120 us to use the parameters of the magnitude-frequency plot to find the quantities p and C for the size distribution, which may then be used in Equations 9 and 11 to find the total mass and area associated with the distribution. (The constants a and b in Equation 15 depend on the size of the magnitude bin which has been chosen. Therefore the value of Â¿log D is also fixed by the choice of magnitude bin size â€” dlog D = 0.2dH.) APPENDIX C POTENTIAL OF A SPHERICAL SHELL We wish to find the gravitational binding energy of a spherical shell of mass 0.5M covering a sphere of mass 0.5M. This is the energy needed to disperse the fragments of a barely catastrophic collision (which, by definition, has 50% of the mass of the target shattered and dispersed) and is probably a fairly good approximation to a coreÂ¬ type shattering collision. A target asteroid with total mass M and radius R has 50% of 1 /3 its mass contained in a spherical shell with radius r = a = (0.5) ' R (approximately 0.79R) to r = R. The volume of the shell is given by V = -7r/23 â€” -7ra3 = -7r(/?3 â€” a3). 3 3 3 v ' (C-l) If we assume that the mass is uniformly distributed within the shell, we can write 3 l\M) P â€” 4tt(/23 - Â«3)- (< Within the shell (which sits upon a core of mass 0.5M) the mass is given by r (Câ€”3) a (C-4) M(r) = â– (Câ€”5) 121 122 and 3A/r- f/M(r) 2(R3-a3)' (C-6) The binding energy can now be calculated -0 - GM(r)dM(r] dr = G 1 r3 - a3 1 2MW^? + 2M 3 Mr 2 723 - a3 dr (Câ€”7) 3 GM2 4(723 -fl3) 4 .3 r â€” a r R3 ~ a 3 + r dr (Câ€”8) 3 2 a r 3 GM2 4(/23 - a3) [5(R3 - a3) 2(R3 - a3] + (Câ€”9) 3 GM'2 4(723 - a3) 725 - a5 a3 (722 - a'2) R2 - a2 5(R3 - a3) 2(R3 - a3) (C-10) 3 GM2 2 R3 725 - (0.5)^/25 4723(/22 - (0.5)"/?2) ^2 _ (0.5)f 7?2 |723 /23 The last two terms in the brackets cancel, leaving (C-ll) -Ã2 = [0.2740079722] (Câ€”12) or (~i Ajf '2 -Ã2 = 0.4110119â€”â€”. (Câ€”13) This compares to the binding energy of a uniform sphere (the energy needed to disperse an entire sphere of mass M and radius R), in which case the constant is 5/3 (0.6). Thus, as expected, it takes somewhat less energy to disperse a shell of one-half the total mass off of a target asteroid. BIBLIOGRAPHY Arnold, J. R. 1969. Asteroid families and â€œjet streams.â€ Astron. J. 74, 1235-1242. Asphaug E. and H. J. Melosh 1993. The Stickney impact on Phobos: A dynamical model. Icarus 101, 144-164. Belton. M. J. S. and the Galileo Imaging Team 1992. Galileo encounter with 951 Gaspra: First pictures of an asteroid. Science 257, 1647-1652. Bendjoya, Phâ€ž E. Slezak, and Cl. FroeschlÃ© 1991. The wavelet transform: A new tool for asteroid family determination. Astron. Astrophys. 251, 312-330. Binzel, R. P. and S. Xu 1993. Chips off of asteroid 4 Vesta: Evidence for the parent body of basaltic achondrite meteorites. Science 260, 186-191. Bowell, E.. B. Hapke, D. Domingue, K. Lummi, J. Peltoniemi, and A. W. Harris 1989. Application of photometric models to asteroids. In Asteroids II (R. P. Binzel. T. Gehrels, and M. S. Matthews, Eds.), pp. 524-556. Univ. of Arizona Press. Tucson. Brouwer, D. 1951. Secular variations of the orbital elements of minor planets. Astron. J. 56. 9-32. Carusi, A. and G. B. Valsecchi 1982. On asteroid classifications in families. Astron. Astrophys. 115, 327-335. Cellino, A., V. ZappalÃ¡, and P. Farinella 1991. The size distribution of main-belt asteroids from IRAS data. Mon. Not. R. Astron. Soc. 253, 561-574. Croft, S. K. 1992. Proteus: Geology, shape, and catastrophic destruction. Icarus 99. 402-419. Davis. D. R., C. R. Chapman. R. Greenberg, and A. W. Harris 1979. Collisional evolution of asteroids: Populations, rotations, and velocities. In Asteroids (T. Gehrels, Ed.), pp. 528-557. Univ. of Arizona Press, Tucson. Davis, D. R., C. R. Chapman, S. J. Weidenschilling, and R. Greenburg 1985. Collisional history of asteroids: Evidence from Vesta and the Hirayama families. Icarus 62. 30-53. 123 124 Davis. D. R., P. Farinella, P. Paolicchi. and A. C. Bagatin 1993. Deviations from the straight line: Bumps (and grinds) in the collisionally evolved size distribution of asteroids. Lunar Planet. Sci. Conf. XXIV. Davis. D. R. and F. Marzari 1993. Collisional evolution of asteroid families. In preparation. Davis, D. R. and E. V. Ryan 1990. On collisional disruption: Experimental results and scaling laws. Icarus 83. 156-182. Davis. D. R., S. J. Weidenschilling, P. Farinella. P. Paolicchi. and R. P. Binzel 1989. Asteroid collisional history: Effects on sizes and spins. In Asteroids II (R. P. Binzel. T. Gehrels, and M. S. Matthews, Eds.), pp. 805-826. Univ. of Arizona Press. Tucson. Dermott, S. F., D. D. Durda. B. A. S. Gustafson. S. Jayaraman. Y. L. Xu. R. S. Gomes, and P. D. Nicholson 1992a. The origin and evolution of the zodiacal dust cloud. In Asteroids. Comets, Meteors 1991 (E. Bowell and A. Harris, Eds.), pp. 153-156. Lunar and Planetary Institute, Houston. Dermott, S. F., R. S. Gomes, D. D. Durda, B. A. S. Gustafson, S. Jayaraman, Y. L. Xu. and P. D. Nicholson 1992b. Dynamics of the zodiacal cloud. In Chaos, Resonance, and Collective Dynamical Phenomena in the Solar System (S. Ferraz-Mello, Ed.), pp. 333-347. Kluwer Academic Publishers, Dordrecht. Dermott, S. F., S. Jayaraman. Y. L. Xu. and J. C. Liou 1993. IRAS observations show that the earth is embedded in a solar ring of asteroidal particles in resonant lock with the planet. Submitted to Nature. Dermott, S. F. and P. D. Nicholson 1991. IRAS dust bands and the origin of the zodiacal cloud. Highlights of Astronomy 8. pp. 259-266. Dermott, S. F., P. D. Nicholson, J. A. Bums, and J. R. Houck 1984. Origin of the solar system dust bands discovered by IRAS. Nature 312, pp. 505-509. Dermott, S. F., P. D. Nicholson, R. S. Gomes, and R. Malhotra 1990. Modelling the IRAS solar system dustbands. Adv. Space Res. 10, pp. 165-172. Dohnanyi, J. S. 1969. Collisional model of asteroids and their debris. J. Geophys. Res. 74, 2531-2554. Dohnanyi, J. S. 1971. Fragmentation and distribution of asteroids. In Physical Studies of Minor Planets (T. Gehrels, Ed.), pp. 263-295. NASA SP-267. Dohnanyi, J. S. 1976. Sources of interplanetary dust: Asteroids. In IAU Colloqium No. 125 31 Interplanetary Dust and Zodiacal Light (H. Elsasser and H. Fechtig, Eds.), pp. 187-205. Springer-Verlag, Berlin. Farinella. P. and D. R. Davis 1992. Collision rates and impact velocities in the main asteroid belt. Icarus 97, 111-123. Farinella, P., P. Paolicchi, and V. ZappalÃ¡ 1982. The asteroids as outcomes of catastrophic collisions. Icarus 52, 409-433. Flynn, G. J. 1989. Atmospheric entry heating: A criterion to distinguish between asteroidal and cometary sources of interplanetary dust. Icarus 77, 287-310. Fujiwara. A. 1982. Complete fragmentation of the parent bodies of Themis, Eos, and Koronis parent bodies. Icarus 52, 434^443. Fujiwara, A., P. Cerroni. D. Davis, E. Ryan, M. Di Martino, K. Holsapple, and K. Housen 1989. Experiments and scaling laws for catastrophic collisions. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 240-265. Univ. of Arizona Press, Tucson. Fujiwara, A., G. Kamimoto. and A. Tsukamoto 1977. Destruction of basaltic bodies by high-velocity impact. Icarus 31, 277-288. Fujiwara, A. and A. Tsukamoto 1980. Experimental study on the velocity of fragments in collisional breakup. Icarus 44, 142-153. Gautier. T. N., Hauser. M. G., and F. J. Low 1984. Parallax measurements of the zodiacal dust bands with the IRAS survey. Bull. Amer. Astron. Soc. 16, 442. Gradie, J. C., C. R. Chapman, and J. G. Williams 1979. Families of minor planets. In Asteroids (T. Gehrels, Ed.), pp. 359-390. Univ. of Arizona Press, Tucson. Greenberg, R. and M. C. Nolan 1989. Delivery of asteroids and meteorites to the inner solar system. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 778-804. Univ. of Arizona Press, Tucson. Greenberg, R., M. C. Nolan, W. F. Bottke, R. A. Kolvoord, J. Veverka, and the Galileo Imaging Team 1992. Collisional and dynamical evolution of Gaspra. Bull. Amer. Astron. Soc. 24, 933. Greenberg, R., M. C. Nolan, W. F. Bottke, and R. A. Kolvoord 1993. Collisional history of Gaspra. Submitted to Icarus. Greenberg, R., J. F. Wacker, W. K. Hartmann, and C. R. Chapman 1978. Planetesimals to planets: Numerical simulation of collisional evolution. Icarus 35, 1-26. 126 Griin, E.. M. Baguhl. H. Fechtig, J. Kissel, D. Linkert, G. Linkert, N. Siddique, M. S. Hanner. B.-A. Lindblad, J. A. M. McDonnell, G. E. Morfill. G Schwehm. and H. A. Zook 1991. Interplanetary dust observed by Galileo and Ulysses. In The 23 rd Annual Meeting of the Division for Planetary Sciences, pp. 23. Griin, E.. H. A. Zook, H. Fechtig, and R. H. Giese 1985. Collisional balance of the meteoritic complex. Icarus 62, 244-272. Gustafson, B. A. S., E Griin, S. F. Dermott, and D. D. Durda 1992. Collisional and dynamical evolution of dust from the asteroid belt. In Asteroids, Comets, Meteors 1991 (E. Bowell and A. Harris, Eds.), pp. 223-226. Lunar and Planetary Institute. Houston. Hartmann, W. K. and A. C. Hartmann 1968. Asteroid collisions and evaluation of asteroidal mass distribution and meteorite flux. Icarus 8, 361-381. Hauser, M. G., T. N. Gautier, J. Good, and F. J. Low 1985. IRAS observations of the interplanetary dust emission. In Properties and Interactions of Interplanetary Dust (R. Giese and P. Lamy, Eds.), pp. 43-48. D. Reidel, Dordrecht. Hellyer. B. 1970. The fragmentation of the asteroids. Mon. Not. R. Astron. Soc. 148. 383-390. Hellyer, B. 1971. The fragmentation of the asteroids â€” II. Numerical calculations. Mon. Not. R. Astron. Soc. 154, 279-291. Hirayama, K. 1918. Groups of asteroids probably of common origin. Astron. J. 31. 185-188. Hirayama, K. 1919. Further notes on the families of asteroids. Proc. Phys.-Math. Soc. Japan 3, 52-59. Hirayama, K. 1923. Families of asteroids. Japan J. Astron. Geophys. 1, 55-93. Hirayama, K. 1928. Families of asteroids. Second paper. Japan J. Astron. Geophys. 6. 137-162. Holsapple, K. A. and K. R. Housen 1986. Scaling laws for the catastrophic collisions of asteroids. Mem. S. A. It. 57, 65-85. Housen, K. R. and K. A. Holsapple 1990. On the fragmentation of asteroids and planetary satellites. Icarus 84, 226-253. Housen. K. R., Schmidt, R. M., and K. A. Holsapple 1991. Laboratory simulations of large scale fragmentation events. Icarus 94, 180-190. 127 KresÃ¡k, L. 1971. Orbital selection effects in the Palomar-Leiden asteroid survey. In Physical Studies of Minor Planets (T. Gehrels. Ed.), pp. 197-210. NASA SP-267. Kuiper. G. P.. Y. Fugita, T. Gehrels. I. Groeneveld. J. Kent. G. Van Biesbroeck. and C. J. Van Houten 1958. Survey of asteroids. Astrophys. J. Suppl. 3. 289-334. Lindblad. B. A. and R. B. Southworth 1971. A study of asteroid families and streams by computer techniques. In Physical Studies of Minor Planets (T. Gehrels. Ed.), pp. 337-352. NASA SP-267. Low. F. J.. D. A. Beitema. T. N. Gautier. F. C. Gillett. C. A. Beichman, G. Neugebauer. E. Young. H. H. Auman, N. Boggess. J. P. Emerson. H. J. Habing, M. G. Hauser. J. R. Houck. M. Rowan-Robinson. B. T. Soifer. R. G. Walker, and P. R. Wesselius. Infrared cirrus: New components of the extended infrared emission. Astrophys. J. Lett. 278. L19-L22. Matson. D. L.. G. L. Veeder. and E. F. Tedesco 1990. Accuracy of IRAS asteroidal size distributions. Bull. Am. Astron. Soc. 22. 1117. Matson, D. L., G. L. Veeder. E. F. Tedesco. and L. A. Lebofsky 1989. The IRAS asteroid and comet survey. In Asteroids II (R. P. Binzel. T. Gehrels, and M. S. Matthews. Eds.), pp. 778-804. Univ. of Arizona Press. Tucson. Milani. A and Z. Knezevic 1990. Secular perturbation theory and computation of asteroid proper elements. Cel. Mech. and Dyn. Ast. 49, 347â€”411. Neugebauer, G., C. A. Beichman, B. T. Soifer, H. H. Auman. T. J. Chester, T. N. Gautier. F. C. Gillett, M. G. Hauser, J. R. Houck. C. J. Lonsdaley, F. J. Low, and E. T. Young. Early results from the Infrared Astronomical Satellite. Science 224, 14-21. Nolan. M. C., E. Asphaug, and R. Greenberg 1992. Numerical simulation of impacts on small asteroids. Bull. Am. Astron. Soc. 24. 959-960. Piotrowski, S. I. 1953. The collisions of asteroids. Acta Astron. Ser. A 6, 115-138. Rabinowitz, D. L. 1993. The size distribution of the earth-approaching asteroids. Astrophys. J. 407, 412â€”427. Ryan, E. V. 1993. Asteroid collisions: Target size effects and resultant velocity distributions. Lunar Planet. Sci. Conf. XXTV. pp. 1227-1228. Shoemaker, E. M. 1983. Asteroid and comet bombardment of the earth. Ann. Rev. Earth Planet. Sci. 11, 461â€”494. 128 Tedesco, E. F., J. G. Williams, D. L. Matson, G. J. Veeder, J. C. Gradie, and L. A. Leboi'sky 1989. Three-parameter asteroid taxonomy classifications. In Asteroids II (R. P. Binzel. T. Gehrels, and M. S. Matthews, Eds.), pp. 1151-1161. Univ. of Arizona Press, Tucson. Tholen. D. J. 1989. Asteroid taxonomic classifications. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 1139-1150. Univ. of Arizona Press, Tucson. Van Houten, C. J.. P. Herget. and B. G. Marsden 1984. The Palomar-Leiden survey of faint minor planets: Conclusion. Icarus 59, 1-19. Van Houten, C. J., I. Van Houten-Groeneveld, P. Herget, and T. Gehrels 1970. The Palomar-Leiden survey of faint minor planets. Astr. Astrophys. Suppl. 2. 339^148. Wetherill, G. W. 1967. Collisions in the asteroid belt. J. Geophys. Res. 72, 2429-2444. Whipple, F. L. 1967. On maintaining the meteontic complex. Smithson. Astrophys. Ohs. Spec. Rept. No. 239, pp. 1â€”46. Williams, D. R. and G. W. Wetherill 1993. Size distribution of collisionally evolved asteroidal populations: Analytical solution for self-similar collision cascades. Submitted to Icarus. Williams, J. G. 1979. Proper elements and families memberships of the asteroids. In Asteroids (T. Gehrels, Ed.), pp. 1040-1063. Univ. of Arizona Press, Tucson. Williams, J. G. 1992. Asteroid families â€” an initial search. Icarus 96, 251-280. Xu, Y. L., S. F. Dermott, D. D. Durda, B. A. S. Gustafson. S. Jayaraman. and J. C. Liou 1993. The zodiacal cloud. Chinese Academy of Sciences. Beijing, in press . ZappalÃ¡ V. and A. Cellino 1992. Asteroid families: Recent results and present scenario. Cel. Mech. and Dyn. Ast. 54, 207-227. ZappalÃ¡ V., A. Cellino, P. Farinella, and Z. Knezevic 1990. Asteroid families. I. Identification by hierarchical clustering and reliability assessment. Astron. J. 100. 2030-2046. ZappalÃ¡ V., P. Farinella, Z. KnezeviÃ©, and P. Paolicchi 1984. Collisional origin of the asteroid families: Mass and velocity distributions. Icarus 59, 261-285. Zellner, B. 1979. Asteroid taxonomy and the distribution of the compositional types. In Asteroids (T. Gehrels, Ed.), pp. 783-806. Univ. of Arizona Press. Tucson. BIOGRAPHICAL SKETCH Daniel David Durda was bom in Detroit, Michigan, on October 26th, 1965. In 1978 he moved to the small, northern Michigan town of Alger, where most of his awe of the natural world was cultivated. He graduated from Standish-Sterling Central High School in 1983. He attended the University of Michigan, earning a B.S. in astronomy, and graduated with distinction in 1987. In the fall of that year he began his graduate studies at the University of Florida. He received his M.S. in astronomy in 1989 and joined the solar system dynamics group to begin research for his Ph.D. thesis. He will receive his doctorate in astronomy in December of 1993. 129 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Stanley F. Dermott, Chair Professor of Astronomy I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Humberto CahÃ¡pins, Cochair Associate Professor of Astronomy I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. JamesJ^. Channell Professor of Geology I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. (P. ^ Philip D. Nicholson Associate Professor of Astronomy Cornell University This dissertation was submitted to the Graduate Faculty of the Department of Astronomy in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1993 Dean. Graduate School UNIVERSITY OF FLORIDA 3 1262 08553 9368 xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID ENBV2FZAL_U77VOR INGEST_TIME 2017-07-12T21:07:24Z PACKAGE AA00003237_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES |