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The collisional evolution of the asteroid belt and its contribution to the zodiacal cloud

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Title:
The collisional evolution of the asteroid belt and its contribution to the zodiacal cloud
Creator:
Durda, Daniel David, 1965-
Publication Date:
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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 )
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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.

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University of Florida
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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.
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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










































































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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



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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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81,9(56,7< 2) )/25,'$



THE COLLISIONAL EVOLUTION OF THE
ASTEROID BELT AND ITS CONTRIBUTION TO
THE ZODIACAL CLOUD
By
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. 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!
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 ’Wave’ 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 — 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 Zappalá 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 /.¿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 65.68°,
97.46°. and 114.68°. 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 ± 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 “mainbelt.” 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 0°
4

5
(Bowell et al. 1989). Immediately evident is a “hump”, or excess, of asteroids at
H « 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 — 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 40°
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 — 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 — we transformed these absolute magnitudes to the H. G system by applying the
correction H — g — 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. Kresák 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 VÑ 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 \TÑ 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 distri¬
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 distri¬
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 frag¬
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 ± 0.011, which corresponds to a mass-frequency slope of
q = 1.782 ± 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 + §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 — 11.25. A fit
through these data yields a slope of a = 0.479 ± 0.012 (q = 1.799 ± 0.020, Figure 12).
In Zone III we obtain the solution a = 0.447 ± 0.017 (q — 1.745 ± 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 ± 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 \/Ñ 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
—
—
16.25
17151.66
—
—
—

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
—
—
16.25
17151.66
—
—
—

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
-+->
• 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
10°
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 °5 1 i 1 i '"i i 1 i 1 i 1 i 1 i 1 ¡
Zone III
104 — 3.0 < a < 3.5
° O o
° PLS
3 ° • catalogued
— ° O O
- • •
102 =- • o • -=
• 8
• * *
• o —
• •
101 - • -
• •
— o —
— • —
10° 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 ¡
—
Zone III
= o
— o
3.0 < a < 3.5
° PLS
o
0
o
o
1 Mill
• catalogued
•
• •
=- «o’
o
•
-
— •
e
•
•
•
O
•
• _
I T ITT
1
l
O —
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
—
E *i«
A
*
-
r m
— m
=
- •
â– 
i
=
Mil
-
r 'i
—
— (
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 rrn—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 = r¡ — 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 ± 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 “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). 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, P¡, 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 “final” collision probability for a finite-sized asteroid with diameter D is
Pf=cr'P„
(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
D„
(3—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 Dm¡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-£ = 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 — -T7Í.
(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 = jñTCrniu. 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, D¡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 — 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 ’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. 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 ’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 distri¬
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 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. 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 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(-i)a 10-r'/10, (3-14)
where i = 1,2,3,..., N(l) is the smallest size bin, N(—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
“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 (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 = —3.5 to -4.0 (Belton et al. 1992). (The
Dohnanyi equilibrium value is p = —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 £ 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; 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 popula¬
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 £ 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 ¡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 /¿m, 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 x 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 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 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 ~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 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
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 /¿' 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, ¡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 — 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 S„ 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 lab¬
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 ’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 ’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. (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 “dip” 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 “hump”
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 “hump” 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 ± 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 “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.

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 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; Zappalá 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 — 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 (Zappalá and Cellino 1992).
The Zappalá Classification
To date, probably the most reliable and complete classification of Hirayama family
members is the recent work of Zappalá 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 Zappalá
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 Zappalá families and have labeled some of the more
prominent ones. The Koronis, Eos, and Themis families remain the most reliable,
however Zappalá 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 “dangerous” 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 — 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
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 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 (Zappalá 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 Zappalá 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 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 volume density of
the asteroids in a family is then found by using Gauss’ perturbation equations to cal¬
culate the spread in orbital elements associated with the formation of the family (see,
e.g., Zappalá 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 family’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 thresh¬
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 (Zappalá
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 1—~\
Maria
Eunomia
Adeona
M Eos
Vesta v
H;...
.:*>Y Gefion *
Dora
- c,, £ y.ijr'
Flora Jr-.
-k w - ■ —
- A ilC /v V! • •,
;-^.r Nysa
i i i i 1 i
• .*•
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 IRAS’s observations, a decade after its mission ended.
Developed as a joint program of the United States, the Netherlands, and Great
Britain, IRAS’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 “noise” was the
warm cloud of solar system dust. In fact, the flux in the 12 and 25 /¿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 2-3°, and is flanked by a fainter
pair of bands above and below the ecliptic at latitudes of +10° and -10° (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 10° 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 Earth’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 observa¬
tions (Figure 42a) are in the 25 /¿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 114.68°. 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 65.68°. 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 Zappalá 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(¿¡ 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
• i—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 65.68°, 97.46°, and 114.68°.
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 “hump” 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 — 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% — 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 — 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'»r""". (A—1 >
or.
F\
log — = 0.4(n?9 — mi),
Ft.
or.
m 2 — m 1 = 2.5 log
F\
Fo
(A—2)
(A—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
(A—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
— M = 2.5 log (jq) = 5 l°£r — 5,
m.
(A—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 — 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 0° phase angle. Our standard distance unit is then rp = 1
and after substitution Eq. 3 reads:
m 2 — m i = 2.5
(A—7)
or.
mv — H — 5 log rp,
(A—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 °'4^
(A—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. — 0.4// — 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 = — 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 — linear or logarithmic. Differentiating
116

117
Equation 1 we obtain
dN = -pCD~v~ldD, (B—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), (B—I)
we see that the slope of the size distribution on a log<#V-log£> plot is — (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 = — pC\nlOD~pd\og D. (B-5)
This then represents an incremental size distribution with logarithmic increments. From
\ogdN = —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 — ^npD3, Equation 1 may be rewritten in terms of the
mass as
N = C
— | 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 — (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 £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 —
n3-P
6(3 — p)
D„
(B—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,
(B—10)
which yields a total geometrical cross-sectional area of
A tot —
npC
4(2 — p)
D--v
D max
D tn i a
(B—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 — OAH — logp.„, 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, (B—12)
where a is the slope and b is the y-intercept. Substituting for H we obtain
logt/TV = a(—5 log D + 2.5const — 2.5 logpv) + b
— —5a log D + (2.5aconst — 2.5a logp,, + b).
Comparing Eqs. 6 and 13 we see that
(B—13)
p = 5a
(B-14)
and
log (pC In lOc/log D) = 2.5aconst — 2.5a logp,: + b. (B—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 ¿log D is also fixed by the choice of magnitude bin size — 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 core¬
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 — -7ra3 = -7r(/?3 — 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 — 4tt(/23 - «3)- (<
Within the shell (which sits upon a core of mass 0.5M) the mass is given by
r
(C—3)
a
(C-4)
M(r) = â– 
(C—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 (C—7)
3 GM2
4(723 -fl3)
4 .3
r — a r
R3 ~ a 3
+ r
dr
(C—8)
3 2
a r
3 GM2
4(/23 - a3) [5(R3 - a3) 2(R3 - a3]
+
(C—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)
-Í2 = [0.2740079722]
(C—12)
or
(~i Ajf '2
-Í2 = 0.4110119——. (C—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.

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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 Cahápins, 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
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