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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

PAGE 1

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

PAGE 2

To my parents. Joseph and Lillian Durda.

PAGE 3

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 expres my gratitude to tho e who have helped me the most. First and foremost, I would like to thank my the is 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 s upport. I never once felt as though I were merely a graduate tudent. 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 thesi 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 Ill

PAGE 4

more than I can expres in words. Billy 's Bill yi ms have provided me with m o re entertain ment than I hav e a t time known what to d o with. I will miss them immensely! I will also mi ss my di sc u ss ions, afternoon chats. and laughs with the other graduate s tudents who have come to mean so much to me. especially Dave Kaufmann, Jaydeep Mukherjee, Caroline Simp so n Sumita Jayaraman, R o n Drimmel and Leonard Garcia. I would like to thank the office s taff for helping me with so many little problem D e bra 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 thi s di sse rtation a very large part of my life is at the same time drawing to a clo se and beginning anew The m os t wonderful part of my new life is that I will be haring it with Donna Without the l o ve and unwavering support of Mom Dad my sister Cathy and her hu s band Louie and my nephew and nieces Andrew Larissa and Jenna none of thi s would ever have happened iv

PAGE 5

ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT. CHAPTERS 1 INTRODUCTION TABLE OF CONTENTS 2 THE MAINBELT ASTEROID POPULATION Des cri ption of the Catalogued Population of Asteroids The MDS and PLS Surveys .... .... The PLS Extension in Zones I, II and III The Ob se rved Mainbelt Size Distribution 3 THE LLISIONAL MODEL .... .. Previou s Studies .............. Description of the Self -co nsistent Collisional Model V erifica tion of the Colli s ional Model . . . . . The Wave and the Size Distribution from 1 to 100 Meter s Dependence of the Equilibrium Slope on the Strength Scaling Law The Modified Scaling Law ..... 4 HIRAYAMA ASTEROID FAMILIES A Brief History of Asteroid Families .. The Zappala Classification . . Collisional Evolution of Families .. Number of Families .. ..... Evolution of Individual Families 5 IRAS AND THE ASTEROIDAL CONTRIBUTION TO THE ZODIACAL lll vu Vlll Xl 1 4 .4 6 .9 1 3 33 33 36 43 46 52 55 84 84 85 86 86 90 CLOUD .................. ... ..... .. ........ 98 The IRAS Dustband s . . 98 Mod e lin g the Dust Bands . . .... ... .. ....... ..... 99 V

PAGE 6

The R a ti o of Family to o n-Family Du s t . . . . . . . . . 102 6 SUMMARY 108 C o nclu sio n s . . . . . . . . . . . . . . . . . . 108 Future W or k . . . . . . . . . . . . . . . . . . 110 APPE DIX 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

PAGE 7

LIST OF TABLES 1: Numbers of asteroids in three PLS zo ne s ( MDS/PLS data ). . 1 6 2: Numbers of asteroids in three PLS zo ne (ca talogued/PLS data ) 17 3 : Adjusted completeness limit s for PLS zo nes ... . . ... . 1 8 4: Intrin sic collision probabilitie and encounter speeds for se veral mainbelt as teroid s. . . . . . . . . . . . . . . . . . . 62 vii

PAGE 8

LIST OF FIGURES 1: Proper inclination versus semimajor axi for all catalogued mainbelt asteroids ... .. . .. . ........ .... .. .. . ... 19 2 : Magnitude-frequency distribution for catalogued mainbelt asteroid 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 I. .. ... .. 25 8 : Ad o pted 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: 12: 13: Least-squares fit to the magnitude-frequency data for PLS zone I. Least-squares fit to the magnitude-frequency data for PLS zone II. Least-squares fit to the magnitude frequency data for PLS zone III . 29 30 3 1 14: The o bserved mainbelt size distribution .. . .. .. .... .... 3 2 15: Verification of model for steep initial s lope and mall bin ize . .... 63 v iii

PAGE 9

16: Verification of m o del for s hallow initial slope and mall bin size. 64 17: Verification of model for steep initial s l o pe and large bin s ize .. 65 1 8: Verification of model for s hallow initial lope and large bin ize. 66 19: Equilibrium slope as a function of time for various fragmentation power laws and for steep initial s l o pe. . . . . . . . . . . . 67 20: Equilibrium slope as a function of time for various fragmentation power laws and for shallow initial slo pe . . . . . . . . . . . 68 21: Equilibrium s lope as a function of time for various fragmentation power laws and for equilibrium initial slope . . . . . . . . . . 69 22: Wave-like deviations in ize distribution caused by truncation of particle population. . . . . . . . . . . . . . . . . . 70 23: Independence of the wave o n bin size adopted in model. ..... . .. 71 24: Comparison of the interplanetary dust flux found by Grun et al. (1985) and small particle cutoff used in our m o del. . . . . . . 72 25: Wave-like deviations imposed by a harp particle cutoff (x = 1.9). 73 26 : Size distribution resulting from gradual particle cutoff matching the o bserved interplane ta ry du st flux (x = 1.2). . . . . . . 74 27: Collisional relaxation of a perturbation to an equilibrium size distribution. . . . . . . . . . . . . . . . . . 7 5 28: Halftime for exponential decay t owar d 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 s lope parameter as a function of the slope of the size-s trength scaling law . . . . . . . . . . . . . . 78 31: Difference in th e eq uilibrium s lope parameters for families with different s tr engt h properties. . . . . . . . . . . . . . . 79 ix

PAGE 10

32: The Davis et al. (1985). Housen et al. (1991), and modified sca ling laws u se d in the collisional model. . . . . . . . . . . . . 80 33: The evolved s ize di s tribution after 4.5 billion years using the Hou en et al. (1991) caling law for (a) a ma ive initial population and (b) a mall initial population. . . . . . . . . . . . . . . . 81 34 : The evolved size di tribution after 4.5 billion years using the Davis et al. (1985) scaling law for (a) a massive initial population and (b) a mall initial population. . . . . . . . . . . . . . . . 82 35 : The evolved s ize 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 Zappala 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 hi s tory 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 s hown in (b), (c), and (d). . .... 106 43 : The ratio of areas of du s t associated with the entire mainbelt asteroid population and all families. . . . . . . . . . . . . . 107 X

PAGE 11

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 Chairman: Stanley F. Dermott Major Department: Astronomy December 1993 We present results of a numerical model of asteroid collisional evolution which verify the results of Dohnanyi (1969 J. Geophys. R es. 74 2531-2554) and allow us to place constraints on the impact strengths of asteroids. The s lope of the equilibrium size -frequency distribution is found to be dependent upon the shape of the size-strength sca ling 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 grad ual transition from strain-rate to gravity scaling. This result is not sensitive to the mass or s hape of the initial asteroid population but rather to the form of the strength sc aling law: scaling laws have definite observational consequences. The o bserved slope of the size distribution of the small asteroids is consistent with the value predicted by the slig htly negative slope of our modified scaling law. Wave like deviations from a strict power law equilibrium size di s tribution result if the s malle s t particles in the population are removed at a rate significantly greater than that needed to maintain a Dohnanyi eq uilibrium s lope. We find h owever, that th e o bserved s mall particle cutoff in the interplanetary dust complex is too g radual to s upport xi

PAGE 12

a ignificant wave. We s u gges t that a n y de via ti o n s from an eq uilibrium s ize di strib uti on in the asteroid population are the result of s t oc ha stic cra tering and fragmentation events whic h must occ ur during th e co ur se of collisional evo luti o n. B y determining the ratio of the area associa ted with the mainbelt as te ro id s t o that associa ted wi th the promin e nt Hirayama asteroid families o ur analysis indicate s that the e ntire mainbelt asteroid p o pulati o n produce s 3.4 0 6 times as much dust as the prominent families alone Thi s result i s co mpared with th e ratio of areas n eede d t o acco unt for th e zo di aca l ba ckgro und and th e IRAS du s t b a nd s as determin e d by analysis o f IRAS data. We conclude that the entire asteroid population is respon s ible for at le as t ~3 4 % of the du s t in the entire zo diacal cl o ud xii

PAGE 13

CHAPTER 1 I TRODUCTION Traditionally the debris of short period comets ha 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 ( Piotrow ki 1953) The gradual comminution of asteroidal debri 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 uggestion 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 (Orlin 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

PAGE 14

2 transp o rt lifetimes of asteroidal dust. Flynn (1989) has concluded that much of the dust c o llected 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 du t 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 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 m mutual collisions. Our intent is to place some constraints on the collisional processes affecting the a s teroids and to determine the total contribution made by mainbelt asteroid collision s to the dust of the zodiacal cloud. In Chapter 2 we describe the methods used to derive the size distribution of rnainbelt 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 re s ults confirming work by previous researchers. We then use the model to extend o ur assumptions beyond those of previous works and to shed some light on the impact s trengths of asteroids and the initial mass of the mainbelt. The collisional hi tory o f a s teroid families is examined in Chapter 4 providing further constraints on the e v o lution of the mainbelt and the dust production of families. In Chapter 5 we combine analy s i s o f IRAS data and the mainbelt and family s ize distributions to determine the

PAGE 15

3 r e lati ve contribution of dust upplied t o the zo dia cal cloud by as teroid co lli s i o n s Our c n c lu ions are ummarized and the problem s that mu s t be addressed in future work are discus e d in Chapter 6.

PAGE 16

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 ma s di tribution of the initial planetesimal warm in that region. Also, in order to determine the total quantity of du t that the a teroids contribute to the zodiacal cloud, we must u e 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 (o r ~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, a o nly 13 asteroids, or less than 0.2 % of the known population, are excluded o that the two sets of asteroids are essentially the ame. Figure 2 is a plot of the number of catalogued main belt asteroids per half-magnitude bin, where the absolute magnitude H, is defined as the V band magnitude of the a teroid at a di tance of 1 AU from the Earth, 1 AU from the Sun, at a pha e angle of 0 4

PAGE 17

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 o f s ome primordial, gaussian population of ast e roids 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 ( discus ed below) that the distribution of smaller asteroids is well described by a power-law, indicative of a population of particles in collisional equilibrium Unfortunately a s 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 o f such a distribution. In fact, the mainbelt population of asteroids is complete with respect to discovery d o wn 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 functi o n of their date of discovery It can be seen that as the years have progre s sed increased i nterest in the study of minor plan e t s 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 completenes s limit s For instance no asteroid s brighter than H = 7 have been discovered since about 1910 By 1940 the completeness limit was a magnitude fainter. Similarly w e 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 mainb e lt

PAGE 18

6 the degree of co mpletenes s is greater than 99.7 % ( Figure 3 is also intere s ting for the hi story r eco rd ed in asteroid di scove ry ci rcumstance s Quite apparent i the marked l ack of discoveries in the wake of World War II. The large number of asteroids di scove r e d during the P a l o mar-Leiden Survey appears as a vertical stripe near 1960 .) As p oi nted o ut above, between H = 1 0 and H = 11 the main belt appears t o make a tran s iti o n t o a lin ear, power-law size di tribution An a bsolute magnitude of H = 11 co rr es pond s to a diameter of about 30 km for an albedo of 0 1 approximately the m ea n albedo of the larger asteroids in the mainbelt p o pulation (s ee The Ob se rved Mainbelt Size Distributi o n ). Unfortunately, incompleteness rapidly sets in for H ;::: 11. 5 and with so few data points the s lope of the di s tributi o n cannot be well defined so that we can n o t r e liably u se the data from the catalogued population alone to e s timate the number of very s mall asteroids in the mainbelt (see Figure 2). We have therefore u se d data from the Palomar Leiden Survey ( Van Houten et al. 1970 ) to extend the o bserved di s tribution down to about H = L:>.25, corresponding to a diameter of roughly 5 km The MDS and PLS Surveys The Pal o mar-Leiden Survey (Va n H o uten et al. 1970 ; hereafter r efe rr e d t o as PLS ) was conducted in 1960 to extend to fai nter magnitudes the results of the ear li e r McDonald Survey of 1950 throu g h 1952 ( Kuiper e t al. 1958 ; h ereaf t e r referred to as MDS). The MDS s urveyed the enti re ecli pti c n ear ly twi ce around t o a width of 40 down t o a limitin g phot ogra phic ma gni tud e of n ear ly 15 In co ntr ast, the pra ct i cal plate limit for th e PLS s urvey wa s a b o ut five ma g nitud es fainter. To s urv ey and detect as t eroids this faint over th e same lar ge area cove r e d by the MDS would ha ve been

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7 prohibitive, o with the PLS it was decided that only a mall patch of the ecliptic would be urveyed. and the results caled 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 ) rai ing 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 o n 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

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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 l og N(m 0 ) 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 l og 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, 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 s election effects within the PLS and problems involved with linking up the MDS and PLS data (cf. Kresak 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 e ff e cts within the PLS either cancel each other or are minor to begin with. Figure 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

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9 vertical line indi ate the completene limit for the MDS. beyond which correction factor were adopted ba ed on extrapolations of the observed trend of the number of a teroid per mean op po ition magnitude bin. The olid vertical line indicates where the PLS data ha e been adopted to extend the MDS di tribution. In each of the three zone the completeness limit for the ca tal ogued population roughly coincides with the tran ition to the PLS data. Beyond the completeness limit the ob erved number of catalogued asteroids per half-magnitude bin continues to increase (al though at a decrea ing level of completeness) until the number fall markedly. In each of the three zone the data for the catalogued population merge quite smoothly with the PLS data. This is particularly evident in zone II, where there is a significant decline in the number of a teroid 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 ets. The catalogued p o pulation. however, which is complete to about H = 11 in thi s zone, nicely follows the same trend. even hawing the sharp upturn beyond the completene s limit between H = 11 25 and H = 11.7 0. With the catalogued population making a mooth transition between the MDS and PLS data in each of the three zones. we conclude that any selection effects which might exi t within the PLS data are minor and that there is no problem with combining the MDS data (ro ughly eq uivalent to the current catalogued population ) and PLS data as published. The PLS Extension in Zones I II and ill Having established that the PLS data may be directly used to extend our discussion of the observed di s tributions to fainter asteroids, we define our working magnitude frequency di tribution for each zone by takin g the number of asteroids per half

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10 magnitude bin from the catalogued population for those bins brighter than the discovery c o mpleteness limit and from either the PLS data or catalogued population, whichever is greater, for the magnitude bins below the completeness limit. Due to sampling tatistics there will be a .JN 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 .JN 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 themselve s 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, unscaled counts, the PLS data points have a larger associated JR 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 di s tri 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 di s tri 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 fr ag ment s expected in a collisionally relaxed population. Dohnanyi using a least-square s fit

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11 through the MDS and PLS data. fo und 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 zone We feel that it is more appropriate to consider only incremental frequency distributions s ince 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 m the mainbelt. Having assigned errors to the independent points in the incremental magnirude frequency diagrams, a weighted leastsq uares 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 or bits had to be discarded The survey was complete to a mean photographic opposition magnirude 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 t o work within the completeness limit s of the raw data set. Given the completeness limit in mean opposition magnitude m 0 we can calculate the corresponding completeness limit in absolute magnitude for each of the three semimajor axis zones Based on the

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12 m ean se mimaj o r axis for eac h o f the zones we c alculate the adjusted completeness limit s given in Table 3 Based o n the se m o re conservative co mpletenes s limits we may now calc ul a te th e least s quares so luti o ns for the individual zones. Z o ne I displays a di s tinctly linear di s tribution for absolute magnitudes fainter than a b o ut H = 11. A weighted lea s tsq uare s fit t o the data (H = 11 .25 and fainter ) y i el ds a s l o pe of a = 0.469 0.011, which corres ponds t o a mas s -frequency s l o pe of q = 1. 782 0 0 1 8 ( Figure 11 ). (If we assume that all the asteroids in a semimajor axis zo ne have the s ame mean albedo we may directly co nvert the magnitude-frequenc y s l o pe into the more co mmonly u se d mass frequency s l o pe via q = 1 + a, where a i s the s lope o f the magnitude-frequency data. See Appendix B ) Zone II shows a s imilar. though so mewhat le ss di s tinct and s hall ower, linear trend beyond H = 11 25. A fit th ro ugh the se data yields a s l o pe o f a= 0 479 0.012 (q = 1.799 0.020, Figur e 1 2). In Zone III we obtain the so lution a = 0 447 0 0 17 (q = 1.745 0.028 Figure 13 ) fo r magnitude s fainter than H = 10. 75 These slopes are s ignificantly lower than the D o hnan y i 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 a ddition to the s lope the lea s ts quare s so lution for each zo nes produc es an es tim a t e for the intercept of the linear di s tribution which is a m eas ur e of the absolute number of as teroids in the populati o n With an estimate of the mean albedo of asteroids in th e p op ulati o n the expressions derived in App e ndi x B allow u s to u se th e param eter s of the m ag nitude -fre qu e n cy pl o t s t o quantify the size-fre qu ency distributions for the thr ee zo n es and for th e mainbelt as a whole.

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13 The Observed Mainbelt Size Distribution We may define the ob erved mainbelt ize di tribution that we will work with by combining data from the catalogued population of asteroids and the leastquare fit 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 et 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 cla s sification. If no taxonomic information was available we assumed an albedo equal t o that o f 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 or der to directly combine the data with those derived from the PLS magnitude data (see Appendix B)

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14 The size distribution of as teroid s s m a ll e r than th e completeness limit of the catalog u e d p o pul a ti o n h as be e n d e ri ve d u si n g th e PLS ma g nitude data described in the previ o u s sec ti o n. Linear lea s t -s quare s so luti o n s, constrained to have the same weig hted mean s l o pe of q = 1.781, were fit throu g h the linear portions of the magnitude di s tributi o n s in each of the three PLS zones. The individual distribution s were then a dded t o determine the intercept parameter (e quiv a l e nt t o the brightest asteroid in the power-law distribution) for the mainbelt as a whole. T o co n ve rt the parameter s o f th e ma g nitude-frequency di stri buti o n determined u si ng the PLS data int o a s i ze -frequ e n cy distribution, we assume that all the asteroids in the population have the s ame mean a lbedo. Of the well-observed asteroids in the mainbelt that is asteroids with b o th IRAS determined albedos and measured B-V co lor s, we found mean albedo s of 0.121, 0.105 and 0 074 in PLS zo ne s I II and II r es pecti ve ly The weighted mean albedo for the e ntir e m ai nb e lt p o pulati o n i s 0 097. W e chose to calc ulate the mean albedo ba se d o n th ose asteroids with diameters between 30 and 200 km in order to avoid any p ossi ble s election effects which might affect the s mallest and largest asteroids. With an es timate for the mean albedo the magnitude parameters may be converted directly int o a s ize-frequency distribution u si n g Equati o n s 6 and 15 of Appendix B In F ig ur e 14 we hav e combined the data from th e ca tal og ued as t eroi ds and the PLS ma gni tude di s tribution s t o define the o b serve d mainbelt size di stri buti o n Down to a pproximately 3 0 km the distribution i s d e termined directly from the ca tal og u e d as t eroids and IRAS d eri v e d albedos. The s h a d ed b a nd indi ca t es the IN erro r associated wit h the ca tal og u ed p o pul a ti o n due t o sam plin g s t a ti stics. For diameters less than abo ut 30 km th e m ai nb e lt p o pul ation is in co mpl ete a nd th e numb ers drop b e l ow tho se

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15 e timated from PLS data. We thu u e the PLS data to extend the usable ize di tributi o n to maller ize The d a hed line i the be t fit through the magnitude data for the s mall asteroids. This size distribution is very well determined and will be used in the next chapter to place strong con traint on collisional models of the a teroids

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16 T a bl e l : Numb e r s o f as teroid s in three PLS zo ne s ( MDS/PLS data ). H Z o ne I Z o ne II Zone III I + II + III 2 0 < o < 2.6 2.6 < o < 3 0 3.0 < o < 3.5 2 0 < a < 3.5 ( H) ( H ) N ( H ) N ( H ) 3 25 l 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 8 563 84 14.25 2254 90 3958 07 5109 51 11322.48 14.75 4125.99 7532 34 6 069 05 17727.38 15 25 6093.04 6788.70 7868.17 2 0749 91 15.75 10914 69 12401.97 16 25 17151.66

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17 T a bl e 2 : Numb e r s o f a teroid s in thr ee PLS zo ne s (c atal o gued/PLS data) H Z o ne I Z o ne II Z o ne III I + II + III 2 0
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18 Table 3 : Adju ted comp l etene s l imits fo r PLS zones. Semimajor Axis Zone 2.0 < a < 2.6 2.6 < o < 3.0 3.0 < a < 3.5 Mean Semimajor Axis (AU) 2.43 2.75 3.17 Completeness limit in H 15.3 14 .6 13 .8

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.,,----. (/) Q) Q) h 01) Q) Q -...__,, 0 +.J CD -~ ......-1 u 1-----1 h Q) 0 h 40 30 20 10 0 2 8383 A s teroid s 5 06 2 Numb e r e d 3 3 21 Mulli oppo s ilion .. :\ J:~i \ : -,~r \ '.= : .. ; := ,< -.:;_ __ . : : . . ; .. .. "': -. / ;\. . . ~ . ':"" 2 5 3 3.5 Semimajor Axis (AU) Figur e 1 : P ro per in c lin a ti o n versus semima j o r axis fo r a ll cata l og u ed main belt ast e roid s. ...... \0 4

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10 5 ,.....-t co 10 4 +..J ......-t 01) (lj H Q) H Q) ,..0 s 10 3 10 2 10 1 M a in b e lt catalo g u e d as t er o ids z 1 o 0 1 RB _L__~j__l__J _l 16 14 12 10 8 6 4 Absolute M ag nitud e, H 2 Fi g ur e 2 : M ag nitud efr e qu e n cy di s tribu tio n fo r ca t a l og u e d m ai nb e l t astero i ds 0 N 0

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20 ,------,---,-------r---.----,---.-----.---------,-,---.---.-------,------r-----,--.----------r--r------.---.--~ Q) 15 'U ....-4 0.0 ro 10 ::s Q) ...-----i 0 5 (/) ,..0 L 0 1800 Main belt .. 1850 . 1900 Discovery Date ., 195 0 2 000 Figure 3 : Ab so lut e m ag nitud e as a function o f di scove ry dat e for all ca tal og u e d mainb e lt asteroids N ......

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10 5 rl Zone I 0 Q) 10 4 0 2.0 < a < 2. 6 '"D 0 ;::J 0 +,-) o PLS ,,---j 0 10 3 catalogued Q() 0 ro 0 10 2 o ol e Q) N N ij 01 ~ 0 Q) 10 1 ij i i s 0 0 0 ;::J z 0 10 18 16 14 12 10 8 6 4 2 0 Absolute Magnitude, H Figure 4 : Magnitud efr e qu e n cy di s tributi o n for PLS zo n e I: PLS a nd ca tal og u e d as t ero id d ata.

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10 5 r-i cu Zon e II Q) 10 4 0 2 6 < a < 3 0 'TI 0 0 0 +.J 0 o P L S r-i 1 0 3 ca t a lo g u e d 0 tl.[) 0 ro 0 10 2 l e i I Q) --=j N \.;) F 0 Q) ,..0 10 1 s t 0 z 1 0 18 16 14 1 2 1 0 8 6 4 2 0 Ab s olute M ag nitud e, H F i g ur e 5 : M ag nitud efr e qu e n cy di s tributi o n for PL S zo n e 11 : PLS a nd ca t a l og u e d as t ero id d a t a.

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-~ Q) 'D .+-,) ~ Q[) (ij :2 H Q) H Q) ,.0 s z 10 5 ~ -i----i-,------. --Zone III 10 4 0 3.0 < a < 3 5 0 0 0 o PLS 10 3 0 catalogued O O 0 9 10 2 :1 0 e i 0 1 10 1 0 10 1 ~ 8 L-.....L__L__I 16 14 12 10 8 6 4 Absolute Magnitude H 2 0 Figure 6 : Ma gn itud efr e qu e n cy di str ibuti o n for PLS zo n e III : PLS and ca tal og u e d as t ero id data N

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10 5 ,......-1 Zone I Q) 10 4 2. 0 < a < 2 6 cu +,) .,......-1 10 3 QI) Cd r >--1 102 f Q) ::j N Ul t >--1 ! ! Q) 10 1 ...0 f f s H z 10 18 16 14 12 10 8 6 4 2 0 Absolute Ma g nitude H Fi g ur e 7 : Ad o pt e d ma g nitud e -fr e qu e n cy di s tributi o n fo r PLS zo n e I.

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r-i Q) '"CJ +,) r-i bl) tiJ Q) Q) ..0 s z 10 5 10 4 10 3 10 2 10 1 I ! I I t ! Zone II 2 6 < a < 3 0 H 10 1 ss 16 14 12 1 0 8 6 4 Absolute Magnitude, H 2 Figure 8: Adopted m ag nitud e-fr e quency distribution fo r PLS zo n e ll 0 N 0\

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10 5 ~ ,-----,---~ o::1 10 4 I I Zone III 3.0 < a < 3 5 +,) ~ bl) cu S--i Q) S--i Q) ,..0 s z 10 3 10 2 10 1 1 o 0 18 16 ! ! I ! ! 14 12 10 8 6 4 Absolute Ma g nitud e, H Figure 9: Adopted m ag nitud efrequen c y distributi o n fo r PLS zo n e JII 2 0 N --.l

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10 5 ~------.-,,......-l 10 4 +.J ,,......-l bD (ij H Q) H Q) ...0 a z 10 3 E 10 2 b10 1 b10 24 I ! 22 PLS Tables 7 and 8 j ! ! 20 18 16 14 Mean Opposition Magnitude, m 0 1 2 Figure 10 : Magnitud efr e qu e n cy di s tribution for th e 1 836 astero id s in T a bl es 7 and 8 of Van H o ut e n et a l. (1970). N 00

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10 5 ,,--j 10 4 --+,--) ,--j o.O Cd Q) Q) ,..0 s 10 3 10 2 10 1 ~~-r--r~ one I 2 0 < a < 2 6 q = 1 782 0 018 I ! ! t H z 10 1 s3~ _J_~~__J_ __ 16 14 12 10 8 6 4 Absolute Magnitude, H 2 Figure 11 : Least -s quare s fit to the magnitud e-fre qu e n cy data for PLS zo n e I. 0 N \0

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10 5 .----i 10 4 ;:j +J .----i tU) (U H Q) H Q) ,.0 s ;:j z 10 3 10 2 10 1 1 0 ._____,_ 18 0 16 0 q = 1.799 0 020 I I I t ! Zone II 2.6 < a < 3 0 H 14 12 10 8 6 4 Absolute Magnitude H 2 Fi g ure 1 2: Leasts quar es fit to the magnitude -fre qu e n cy d a ta fo r PLS zo n e IL 0 l,..) 0

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10 5 .....-i 10 4 +.:> .....-i o.O ro Q) 0.. Q) ,.0 s 10 3 10 2 10 1 0 q = 1.745 0 0 2 8 I !! ! Zone III 3 0 < a < 3 5 z 1 o 0 1 RB ~~~~*~~~__LJ.1Lh_ _l_L-l__L_J 16 14 12 10 8 6 4 Absolute Ma g nitude, H 2 0 Figure 1 3 : L east-s quar es fit t o th e ma g nitud e-fre qu e n cy data for PLS zo n e Ill. v.) .......

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5 10 E I I I I I I I 11 I I I I I I I 11 I I I I I I I B ,---t c:o h 104 Q) -+-,-) Q) s ro 10 3 ,---t h G) 10 2 h Q) ,.o 10 1 s z ' ' ' PLS D a t a ',, (qeq ~ 1.78) ',, ' ' ' M a i n b e lt Popu la tion ' IRA S D a t a 0 1 0 -~ 1 10 100 1 000 Di a m e t e r (km) Fi g ur e 14 : Th e o b se rv e d mainb e lt s i ze di str ibuti o n l,..) N

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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 studie The collisional evolution of the asteroids and its effects on the size distribution of the asteroid population has been s tudied 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 trengths, 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 rn -qd m, (3-1) where f ( rn )drn 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 71 of the colliding bodies provided that 71 < 2 This is because the most important contribution to the mass range rn to m + d m 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 cornrninution law. 33

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34 Dohnanyi also found that for q near 2 but les than 2 the creation of debris by ero ion, o r cra tering colli ions, play only a minor role The teady-state size distribution i therefore dominated by cata trophic collisions. Hellyer (1970, 1971) solved the same collision equation numerically and confirmed the results of Dohnanyi. Hellyer hawed that for four values of the fragmentation power law. referred to as x in his notation ( x = 'fl 1 = 0.5, 0 6, 0.7 and 0 8) the population index of the small ma ses converged to an alma t 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. Hi s program was modified to allow for a small number of discrete fragmentation events among very massive particles. With the parameter T se t to 0.7 the slope index of the malle s t 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 evo luti o n of various initial populations of asteroids and compared the re ults with the ob erved distribution of asteroids in order to find those populations which evolved to the pre se nt belt. In their study they considered three different families of hapes for th e initial di s tribution: 1. pow e r law 2. eg m e nt e d pow e r law imulating a runaway grow th distribution of bodi es a

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35 generated by the accretional imulation of Greenberg et al. (1978), and 3. gaussian as suggested by Ander (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~ l 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 o n 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 D o bnanyi. 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 s trength s of large asteroids due to hydrostatic se lf-compression The results from this numerical model were later extended to include s i ze (s train-rate) dependent impact stre ngths ( Davis et al. 1989) The primary goal of these studies was to further constrain the exte nt 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 time the present belt mass best sa tisfied the constraints of preserving the basaltic crust of Vesta and producing the observed number of asteroid families. However other asteroid observatio ns (such as the interpretation of M asteroids as exposed metallic cores of differentiated bodie s and the apparent dearth of asteroids representing the shattered mantle fragments from such bodies) s ug ges t that much more collisional evo lution occ urred than these models predict. The latest version of this model is currently being

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36 used to investigate the colli ional history of asteroid families (Davis and Marzari 1993 ). Mo t 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 steadytate size distribution of a elf-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 debri s) 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 s ize 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 mo t cases in which we are interested only in the larger asteroidal particles, the smallest sizes con idered are of order 1 meter in diameter and the model uses approximately 60 s i ze bin In

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37 those cases in which we are interested in modeling the collisional evolution of dust ize particles the number of bins can increase to ove r 120. For most of the models the l ogari thmic increment wa chosen to be 0 1. in order to most directly compare the size distributions with the magnitude distributions derived in Chapter 2 (se e Appendix B ). For some models including dust ize particle 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 u ed 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 "fie ld" asteroids, the s ize of the smallest field asteroid capable of s hattering and dispersing the target and the cumulative number of field asteroids larger than this s mallest size. We s hall 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 e l e m ents a e, and i, we calculate an intrinsic collision probability I' ;, which i s the collision rate with the background field of asteroids in units of yr1 km 2 normali zed

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38 uch that the total number of particles in the asteroid belt is I The populati o n of field asteroids wa 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 thi s way the election 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 or bital element were taken to be the proper ele ments as computed by Milani and Knezevi c (1990), which are more representative of the long-term orbital elements than are the osc ulating elements. The resulting intrinsic collision rates and mean relative encounter s peeds 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 1 yr 1 1an 2 and 5.88 km s 1 respectively. The "final" collision probability for a finite-sized asteroid with diameter D is ( 3 2) where a'= a/1r (s ince I' i includes the factor of 1r) and a= n(D/2) 2 is the collision cross-section (taken to be the si mple geometric cross-section since the self-gravity of the asteroids is negligible here ). To get the total probability that the asteroid will suffe r a destructive collision we must integrate the final probability over all projectil es of consequence using the size distribution function dN = CD-Pd D. (3-3) Then (3-4) D ,,,,,,

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39 or D ,, u ~.L (3-5) D 1,,,,, (I't is imply the collision cross ection times the intrinsic collision probability times the cumulative number of field a teroids larger than D min .) The c o llision lifetime, 1 c = 1 / I't (3-6 ) i s then the time for which the probability of survival is 1 / e Let us now examine the determination of D min, 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 trength 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 s pecific energies of 7 to 8 x 10 6 erg g 1 or an impact strength S 0 of 3 x 10 erg cm 3 However estimates by Fujiwara (1982) of the kinetic and gravitational energies o f the fragments in the three prominent Hirayama families indicates that the asteroidal parent bodies had impact strengths of a few times 10 8 erg cm3 an order of magnitude g reater than impact strengths for rocky materials ( Fujiwara assumed that the fraction of kinetic energy transferred fr o m the impactor to the debris is f J.:E = 0 1. ) In order to a v oid implausible asteroidal compositions we must conclude that the effective impact s tren g th of an a s teroid is a function of its size as well as its composition The difficulti es inherent in s caling the impact strength over several orders of magnitude in dimen s ion

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40 from laboratory experiments to a teroid-sized bodie are reviewed by Fujiwara et al. (1 989 ). Davis et al. (1985 ) concluded that large asteroids should be strengthened by gravitational self-compres ion and developed a size-dependant impact strength model which is consistent with the Fujiwara et al. (1977) results and produces a size-frequency di tribution of collision fragments consistent with that observed for the Hirayama families. Other researchers (Farinella et al 1982; Holsapple and Housen 1986; Hou en and Holsapple 1990) have developed alternative scaling laws for strengths, predicting impact strengths which decrease with increasing target size. We will discuss the variou caling 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, (3 7 ) 1 s the energy required to disperse one half the mass of the target asteroid to infinity ( s ee Appendix C) Not all of the kinetic energy of the projectile is partitioned into comminuti.on and kinetic energy of the large fragments of the colli ion. From reconstructi o n o f the three large s t Hirayama families Fujiwara (19 2 ) found that a fraction f h 'E f

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41 projectile kinetic energy partitioned into kinetic energy of the members of the family o f 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 f hE 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 1 ( G M 2 ) Emin = fr.: E SV + 0 .4 11 D / 2 (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 Finally then 1 2 1 3 2 Emin = 2 m'fflin, = 12 1rp D min D min = ( 1 2Emin ) 7r p,2 (3 9 ) ( 3 10 ) To finally determine the collision lifetime characteristic of each si z e bin we need o nly specify the cumulative number of field asteroids larger than D m in Within th e

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42 collision program this number is determined by simply counting, during each time tep, the total number of particles in the bins larger than D mi nIn 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, tlt, is determined within the program and updated continuously in order to maintain flexibility with the code. At all times tlt is chosen to be some small fraction of the shortest collision lifetime, T r I lj tl l where T c ., ;,, is usually the collision lifetime for bin i = 1. In most cases we have let tlt = / 0 T c ,, ,,,, During a single timestep the number of particles removed from bin i is then found from the expression tlt z = N(i) ( .) T c '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 v e ry 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 s ize bins foll o wing a power law si z e distribution given by (3 1 2)

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43 The exponent p is determined from the parameter b, the fractiona l size of the large s t fragment in terms of the parent body. by the expression b 3 + 4 P = b 3 + 1 (3 13) so that the total mass of debris equals the mass of the parent asteroid (Greenberg and Nolan 1989 ). The constant Bis determined such that there is only one object as large as the largest remnant D1r. 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

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44 D o hnan yi. In both run the l o p e o f the breakup p owe r-law wa s et equal t o the equilibri um val u e of q = l. 33. we ass um ed a co n sta nt imp ac t trength eating law and the l oga rithmic s i ze bin inter va l was se t eq ual t o 0.1. F o r the fir s t run the initial i ze di s tributi o n was c hosen to be a p owe r law di s tributi o n with a s teep s lope o f q = 2 0. The final di s tributi o n at 4.5 billi o n years i s s h ow n. as well as at earlier time s at 1 billi o n year interval s. The evo lved di stri buti o n very quickly (wi thin a few hundred million years ) at tain s an eq uilibrium lope eq u a l to the expected D o hnanyi value of q = l.833 fo r b o die s in the s ize ran ge of 1 -100 meters Th e seco nd run began with a much s hall o wer initial distributi o n with a s l o pe of q = 1. 7 The evolved distribution here as well very rapidly attained the expected e quilibrium s l o pe The ame two numerical ex perim e nt s were repeated with the bin size increased to 0.2. The result s ( Figure s 17 and 1 8) were identical t o the first tw o experiments p owe r-law evo lved size di s tributi o n with eq uilibrium s l o pe s of 1.833. T o s tud y the dependence of the equilibrium s l o pe o n the s lope of the breakup p owe r-law and the time evolution of the s ize di s tribution we altered the co llisi o nal m o d e l li g htly to eliminate the effects of stochastic collisions. P e rturbati o n s on the ove rall l o p e of the s i ze di s tributi o n produced by the s t oc h astic fragmentation of lar ge b o die s may mask any finer-scale trend s due t o l o ng term evo luti o n of th e s i ze di stri buti o n e p ec i ally for a s t ee p fragmentation p owerlaw We ran a series of m o dels with vario u s p owerl aw initial size di s tribution s and fra g mentation p owe r law pannin g a ran ge of s l o p e The results are s h ow n gra phi ca lly in Figures 19 th ro u g h 2 1 where we hav e pl otte d th e l o p e q of the ize di stri buti o n a a function of tim e for the mallest b odies in the model. The l ope i determined a t eac h time t ep by a l ea t quares fit to the 20 rnall e t ize bin

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45 (1 100 meters) of a ~60 bin model. In Figures 19. 20, and 21 the slopes of the initial i z e distributions are 1.88. 1.77, and 1.83. respectively Note that the vertical caJe in Figure 21 has been tretched relative to the previous two figures in order to bring o ut the relevant detail. In all three case we ee that the slope of the size distribution a ympt o tically approaches the value 1.833, reaching values not ignificantly different than thi within the age of the olar system. The different values of the lope are only very lightly dependent upon the fragmentation power-law. For q b (17 in Dohnanyi's notation ) higher than the equilibrium value the final lope converges for all practical value o n slopes somewhat greater than 1.832 within 4 5 billion years. For q b 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 s lope 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 teep than the equilibrium value so we cannot compare our results concerning the equilibrium overshoot. Recall that Dobnanyi ( 1969) concluded that the debris from cratering colli ions 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 o f Williams and Wetherill (1993) confirm that the details of cratering mechanics are unimportant in determining the equilibrium slope, although without the balancing input

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46 of cratering debris the equilibrium lope may vary from the expected value of 1.833 by a very light 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 eries 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 re ulted 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 tho e size bins for which the smallest projectile required for fragmentation was smaller than the smallest size bin. The particles in these ize bins then become relatively overabundant as projectiles and preferentially deplete targets in the next largest size bins. The particles in these bin 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

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47 a trict power-law di tribution up through the large t a teroids in the population. The ame wave-like phenomenon was later independently discovered by Davis et al. (1993). The code was subsequently altered to extrapolate the particle population beyond the malle t ize bin to eliminate the propagati on of an artificial wave in the size distribution. However. in reality the removal of the smal lest asteroidal debri by radiation force may provide a mechanism for truncating the ize di tribution and generating such a wave like feature in the actual a teroid size distribution. To s tudy the sensitivity of feature of the wave on the strength of the s mall particle cutoff we may impose a cutoff on the extrapolation beyond the smalle t ize bin t o si mulate the effects of radiation force We use an exponential cutoff of the form ( i) = ( i) 10.r' / 10 0 (3-14) where i = 1, 2 3 .. N(l ) is the smallest size bin, N ( i ) 0 is the number of particles expected smal ler than those in bin 1 based o n an extrapolation from the two smallest ize bins. and x is a parameter controlling the s trength 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 vi rtual bins present depends upon the bin ize adopted for a particular model though in all cases extends to include particles ~ 1 ~ the diameter of those in bin 1 (ro ughly the ize ratio required for fragmentation ) This form for the cutoff is entirely empirical. but for o ur purposes may still be used t o effectively s imulate the increasingly efficient removal of s maller and s maller particles by radiation forces. When the parameter x is ufficiently large the imposed cutoff is essentially the sa me as the inadvertent truncation of the size distribution which lead to the results illustrated in Figure 22, although it i

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48 more realistic in it mooth 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 run were identical, with the exception of the bin size To be sure the feature 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 ize 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 ma y 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, ome 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 ize range than an equilibrium extrapolation

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49 from the b erved larger a teroid would yield. Although there is ome uncertainty in the precise value, the observed lope of the differential crater ize distribution on 951 Ga pra seems to be greater than that due to a population of projectiles in Dohnanyi colli ional equilibrium, ranging from p = 3.5 to 4.0 (Belton et al. 1992). (The Dohnanyi equilibrium value i p = -3 .~ .) The crater counts are mo t reliable in the diameter range 0.5 to 1 km; craters of this size are due to the impact of projectile with diameters .::: 100 meters. The slope of the crater di tribution on Gaspra is al o 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). Davi et al. (1993) suggest that although the overall slope index of the a teroid population is close to or equal to the Dohnanyi equilibrium value, waves impo ed on the distribution by the removal of the small particles may change the lope in specific size ranges to values significantly above or below the equilibrium value. To te t the theory that a wave-like deviation from a strict, power-law size distribu ti o n i re ponsible 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 rate 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. Grtin et al. (1985) produced a model of the interplanetary dust flux for particles with masses 10 1 g .::: m .::: 10 2 g. With a particle mass den ity of 2.7 g cm 3 (Grtin et al. 1985)

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50 this corresponds to particles with diameter of about 0.01 m to 10 mm, respectively. Figure 24 hows the Grtin et al. model and our modeled particle cutoffs for three values f x For the following models the logarithmic size interval was set equal to 0.1. For x = 0 we have the irnple case of strict collisional equilibrium with no particle removal by non-collisional effects illustrated by the models presented in the previous section. When a harp particle cutoff is modeled beginning at ~ 100 1 1, 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 mall 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 y tern with no input into size bin other than collisional debri from larger bins break down.

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51 The input of cometary dust as projectiles in the smallest size bins may not be in ignificant 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-Eaith 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 s mall 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

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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 s ize distribution of asteroids in colli s i o nal e quilibrium can be described by a power-law with a slope index of q = 1. 833 wa s o btained analytically by assuming that all asteroids in the population have the ame or s ize independent impact strength Other researchers (Williams and Wetherill 199 3) have expr ess ed the intent to consider deviations from self similarity a nalyti c ally t o

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53 determine the resulting effect on the s ize distribution. We have already demonstrated that o ur collisional m odel reproduce the D o hnanyi result for size-independent impact stre ngth ( ee Verification of Collisional Model ). However strain-rate effects and gravitational compression lead to s ize-dependent impact strengths, with both increasing and decreasing strengths with increasing target size, respectively (see discussion of s trength scaling laws in the following sectio n ) With our collisional model we are able t o explore a range of size-st rength scali ng laws and their effects on the re s ulting size distributions. In or der to examine the effects of sizedependent impact strengths on the equi librium s lope of the asteroid size distribution we created a number of hypothetical size-streng th scali ng law s. As will be discussed in the following section, we assume that (315 ) where S is the impact s trength D is the diameter of the target asteroid, and 1/ 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 ove r the size range 10 km to 1 meter. The s lope index output from our modified s mooth collisional model was monitored over the size range 1 100 m and the equilibrium s lope at 4.5 billion years recorded The results are plotted in Figure 30. We find that the equilibrium s lope of the s i ze distribution is very nearly linearly dependent upon the slope of the strength scaling law. There see ms to be an extremely weak seco nd order dependence on however over the range of plausible a linear fit with a s lope of approximately -0.13 is seen to fit th e data s ufficiently well. When = 0, corresponding to a size-independent s tren gth

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54 the Dohnanyi value of q is obtained. If the slope of the scaling law is negative. as i the ca e with train-rate dependent strengths such as the Housen and Hol apple (1990 ) nominal case, the equilibrium slope has a higher value of q 1.86. If, o n the other hand. it' i 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 ize range should allow us to place constraints on the ize dependence of the strength properties of asteroids in that size range. An interesting re ult 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 Sand 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 tochastic fragmentation model was modified to track the colli ional history of a family of fragments resulting from the breakup of a single large asteroid ( see Chapter 4 ). We how the slope index of the family size distribution as a function of time for two familie : family 1 has the same arbitrary strength scaling law as the background population of projectiles ( 1t' < 0 in this case), while the scaling law for family 2 has 1/ > 0 The slope index for family 2 is appropriate for the particular value of 1-L' chosen and i

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55 significantly different than that of family 1 or the background population, even though it is the projectiles in the background which are olely responsible for fragmenting members of the family. Since the total du t area associated with a population of debris is ensitively dependent upon the slope of the ize distribution. it could be possible to make use of IRAS observation of the solar sy tern 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 propertie of asteroidal dust. The Modified Scaling Law One of the most important factors determining the collisional lifetime of an asteroid is its impact strength (s ee 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 e t 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 s ize distribution of the asteroids The re s ult s allow us to place so me observational

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56 constraints o n the impact strengths of asteroidal bodies out ide the size range u ually exp l o red in lab oratory experiments The ob erved ize di tribution of the mainbelt asteroid ( ee 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 di tribution. The results of the previous section demonstrate that details of the size-strength eating relation can have definite observational con equences. 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 co mpression of large asteroids. In this model the effective impact stre ngth is assumed to have two components: the first due to the material properties of the asteroid and the seco nd due to depth-dependent compressive loading of the overburden. When averaged over the volume of the asteroid we have for the effective impact strength 1rkGp 2 D 2 S=S o +----, 1 5 (3-16 ) where S 0 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 S ::::: S 0 yielding the 1ze independent strength of Dohnanyi The Housen et al. (1991) law allows for a S !rain-rate dependence of the impact trength. effectively making larger asteroids weaker than targets mea s ured in the lab ora tory. The theory i described in detail in Housen and Holsapple (1990) where a

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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 (3-17) where V e is the impact speed. Under their nominal rate-dependent model the constant f-l' 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 / .t 1 = 1.65 which we note is slightly less than the D 2 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

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58 the par e nt b o di es of the K oro ni s, E os, and Themis a teroid families (o pen dots ). The m t recent s tudies, h oweve r, indicate that the laboratory result s are to be taken a upper limit s t o the magnitude o f the gravitational compres ion (Holsapple 1993 private co mmunication ). Both scal ing law have been u ed within the co llisional model t o attempt to place o me constraints on the initial ma ss of the asteroid belt and the s izes trength sca ling relation itself. Unfortunately, the initial ma ss of the belt i s not known. By initial' we assume the s ame definiti o n as used by Davi s et al (1985), that i s, the mas s at the time the mean collision s peed first reached the current ~5 km s 1 Davis et al. ( 1989 ) pre s ent a review of asteroid collision s tudies and conclude that the as teroid s represent a collisionally relaxed p o pulati o n wh ose initial mass cannot be fo und from m o del s of the size evolution alone. We have therefore chosen t o inve s tigate tw o ex tremes for an initial belt mass: a ma ss ive initial p o pulation with ~6 0 times the pre ent belt mass based upon work by Wetherill (1992, private communication) on the runaway accretion of planetesimals in the inner so lar system, and a 's mall' initial belt of roughly twice the pre se nt ma ss, matching the best estimate by Davis e t al. (1985, 1989 ) o f the initial mass m os t likely to preserve the basaltic crust of Ve s ta Figures 33 a nd 34 s how the re s ults of seve ral run s of the model with various combination of eati n g law s a nd initial populations. In both figure s we have included the observed s i ze di tribution for comparison with model result s, but have removed the -/N error band for c larity We have found that model s utilizin g the s trength sca lin g law s u uall y co n s id ere d particularly th e pur e s train rat e law fail to r e produc e feature of the o b e rv e d di tributi o n This co nclu sio n i not particularly ensitive to th e d etai l of

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59 the initial asteroid population: it is the form of the sizetrength caling law which mo t 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 a teroids 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 asteroid 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) o n the collisional history of Gaspra. The modified law matches the Housen et al. law for s mall (laboratory) size bodies where impact experiments (Davis and Ryan 1990) indicate that strain rate scaling best describes the fragmentation of mortar targets. Gravitational s trengthening sets in for large asteroids matching the magnitude of the Davis et al.

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60 model. For mall asteroids an empirical modification has been made to allow for the interpretation of some concave facets on Gaspra a impact structures (Greenberg et al. 1993 ). If Gaspra and other imilar-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 train-rate sca ling laws from laboratory-scale targets would predict. The modified law thus allow 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 s lope 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 s low 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 precede s the excavated debris so that crater excava ti o n takes place in effectively unconsolidated material The remaining body of

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61 the a teroid i s thus reduced t o rubble Davis and Ryan (1990) have noted that clay a nd weak m o rtar targets. materials with fairly low co mpressive s trengths such as the hattered material predicted by the hydrocode models, may have very high impact trengths 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 ignificantly s maller than those for which gravitational compression become s important, a more gradual tran s ition from s train-rate sca ling to gravitational compression would be warranted.

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Table 4 : Intrin s ic co lH s ion probabilitie s and encounter s peed s for severa l mainb e lt as t eroi d s Asteroid Proper Semirnajor Proper Eccentricity Proper Inclination Int1insic Collision Encounter Speed axis (AU) (degrees) Probability (10 18 (km s 1 ) yr 1 km 2 ) 1 Ceres 2.767 0.115 9.660 3.146 5.4 2 Palla s 2.769 0.252 34.771 l.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 Themi s 3 134 0 152 1 .083 2.843 6.3 123 Brunhild 2.695 0 110 7.296 3.340 5 1 0\ N 158 Koroni s 2.869 0.045 2.149 3.766 4.5 221 Eos 3.012 0.077 9.939 2.818 5 2 466 Ti s iph o n e 3 367 0.075 20.359 1 1 34 6 9

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15 h Q) ,D s Z 10 cu ..+-) Q) s Q) 5 h u ...--, 01) 0 T o Binsize = 0 1 Final q = 1.833 Predicted q = 1 833 Initi a l Sl ope 2 0 0 ~ 10 Diameter (km) Figure 15 : V e rifi ca ti o n o f m o d e l for s t eep initj a l s l ope a nd s m aJ I bin s i ze 10 3 0\ v.)

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15 h Q) ,D s ;:j Z 10 cu +-> Q) s Q) 5 h u 01) 0 0 ---= 10 Binsize = 0.1 Final q = 1 833 Predicted q = 1 833 10 1 Initial Slope = 1.7 Diameter (km) Fi g ur e 1 6 : V e rifi ca ti o n o f m o d e l for s h a ll ow initi a l s l o p e and s mall bin size. 10 3 0\ +:>,.

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15 h G) ,..0 ~ T 0 Initi a l S lop e = 2 0 s ;:j Z 10 ro -+--,) G) s G) 51 h Bin s ize = 0. 2 u f--l t Final q = 1 833 Q[) Predicted q = 1 833 0 0 ~ 10 Diam e ter (km) Fi g ur e 17 : V e rificati o n o f m o d e l for s t ee p initial s l o p e a nd lar ge bin s i ze 10 3 0\ u-,

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15 r----i------nnmr-i------nnm1111TTTTTr-7------rTTTTTT.----r--.---.T"TTTT"~~H Q) ...0 s ;:j Z 10 ......-4 ro -+-,) Q) s 5 u 1----i till 0 ......-4 0 -,_ 10 To Binsize = 0 2 Final q = 1.833 Predicted q = 1.833 I nitial Slope Diameter (km) 1.7 Figure 18 : V er ifi ca ti o n of m o d e l fo r s hall ow initial s l o p e a nd l arge bin s i ze. 10 3 0-, 0-,

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1 88 r-i-~---i---i--r---r~-----r----,-------,-------i--.-.---,----.----.------r~~~~ Q) ~1.86 0 ,----l 1/) s 1.84 r---i >--4 ,.0 ,,---i ,----l ,,---i er 1.82 Initial S lop e 1 .88 qb = 1.65 = 1 83 q b = 1.94 1 80 L...--L------'---____J_____J___L___L.__j____J__l____j____L__L__L _L_L___L__l___L__j__j__J_...J_j 0 1 2 3 4 Time (B y r s ) Figur e 1 9 : Eq uilibrium slope as a fun c ti o n of time fo r various fr ag m e ntati on p o wer l a ws a nd for s t ee p initi al s l o p e. 0\ -..l

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1. 8 8 r-7""~-i---r-11-r---,------,-----,--, --,---r----.-----.-,-----,------.------.----~~ G) ~1.86 Initial Slope = 1.77 0 ,-----i Cf) s :::; 1.84 L ,,,-----__ ai--. = 1 94 .---i H ,..0 1/ qb = 1 83 O'\ r---i 00 ,-----i r---i ~1 82 w I / I qb = 1.65 1.80 L--...JJ..--1-__L__LL___j___L____L_____L__L____L_____l____L____l____L_j___l_.l..__L_L__L_l__j_J 0 1 2 3 4 Time (Byrs) Fi g ur e 20: Equi librium s l o p e as a fun c ti o n of tim e for various fr agme nt a ti o n power l aws a nd for s h a ll ow init.ial s l o pe

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1 836 ~~~-,---~~~~~~~~~~~~-,----.-,-, Q) 1.834 L qb = 1.65 p_, 0 i qb = 1.83 if) S 1 832 .,......., 1.830 r/ I 0\ \0 ....--1 g 1.828 l / ah = 1.94 Initial Sl op e = 1.8 3 1 8 2 6 ~------'---'-------'--...L_L___L_L______L____L__L__L___.l____L_J__L_ j__[__J__J__J__j_] 0 1 2 3 4 Time (Byrs) Figure 2 1 : Equ ilib r iu m s l o p e as a fun c ti o n of tim e fo r various fragmentation power l aws and fo r eq uilibriu m initial s l o pe

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25 H G) s 20 ~ ~ To Truncat e d S i ze Di s tribu t ion ;::j z 15 cu +..) [ 4 5 Byr G) s 10 cJ -...) G) 0 H u 1-----t 5 oO 0 0 10 6 10 4 10 2 10 Diam e ter (km) F i g ur e 22 : W a v elik e d e viation s in s i ze di s tributi o n ca u se d by trun ca tion o f p artic l e p o pul a ti o n

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25 """' """' """' 1111111 h (1) ,..o 20 s z 15 CD ..+-) (1) 10 L J -.J ....... h u 1----i 5 oD 0 0 111111111 111111111 1 11111111 111111111 1 1111 1111 111111111 111111111 1 11 11 1 111 1151111111_ ... ,111111 1 0 B 1 0 4 1 0 2 1 0 Diameter (km) Figur e 23: Ind e pend e n ce o f the wave o n bin s i ze adopt e d in m o d e l.

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,...----... 0 ('I') I s _____, r X = 0 h Q) 5 ,..0 s ;=j z 10 (ij +,) I I T / \ 7 -...l Q) N s Q) 15 h u o.O Q 201 111 1 11111 1 1111111 1 111 1 1 1 111 I l l l ll l l l f l 1 1 11 1111 1 I 1 1111 111 1 1 1111 1 11 10 12 10 10 10 8 10 6 Diameter (km) Figure 2 4 : Compari s on of th e interplan e tary dust flux found by Grtin e t a l. (1 98 5 ) and s m a ll parti c l e c ut o ff s u se d in o ur m o d e l.

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25 h G) s 20 ~ \ Shar p Pa r t i c l e Cu toff z 15 cu ....,.) [ 4 5 By r G) s 1 o c7 -.J G) \.,.) h u 5 01) 0 0 R 1 0 4 1 0 2 Di a m e t e r (km) Fi g ur e 25: W ave lik e d e v i a ti o n s i m p os e d b y a s harp parti cl e c u toff (x = 1.9 ).

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25 Q) ,..o 20 Is Observ e d Particle Cutoff ;j z 15 (U [ 4 5 B y r Q) s 10 CJ -.J Q) +:>u I--! 5 01) 0 ,----l 0 ~ 10 4 10 2 Diam e ter (k!Il) Fi g ur e 26 : Si ze di s tributi o n r es ultin g fr o m gradu a l parti c l e c ut o ff m a t c hin g th e o b se rv e d int e rplan etary du s t flu x ( x = 1. 2)

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H 14 G) T 0 (q = 2) s ::::J 12 z ,......-t (1j -+,-) 10 100 Myr G) s G) I "" I -..} H u, u 8 01) r q = 1.83 3 0 6 ,......-t I I I I I II II I I I I I II II I I I I I I I I I I 10 3 10 2 10 1 10 Diam e ter (km) Fi g ur e 27: Collisional r e lax a ti o n of a p e rturb a ti o n t o a n eq uilibrium size distribution.

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0 ~-----.--------,-------,--,-----,--,---,----,------,--.---------.-----,---.------,---,-------.-------,----,-------,-----,--,---,-----.--------, 1 ,---, -t / T ,...----,. 2 q = qoe (Y) (Y) T = 65 Myr CXJ -3 I ..__,.. -4 CD L J -..) L.____J 0\ -5 -6 -7'-'--'--"----.______._-'--"----.______._-'--"----.______._-'-_,__.______._-'-_,__.______._~_,__~ 0 1 2 3 4 5 Time (100 Myr) Figure 28 : Halftime for exponential decay toward equilibrium s l ope following the fragmentation of a 100 km diam e t er asteroid.

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1 9 0 ~-,----.--,--,--,----.--,--,--,----.-----,r--,--,----.-----,r--,--,----.-----,-----,-----,----.-----,---, CT 1.88 89 (T = 23 Myr) H Q) ..+-:l Q) s CD H CD Q) 0 ,.......-l Cf) 70 1.86 56 56 56 1.84 1.82 1. 8 0 '-----'---'----'---L__J___L__L__L__L_L___L___L_j___l___j___JL___L_l__L_JL..J.__J__L_J_j 0 1 2 3 4 5 Time (100 Myr) Figure 29: Stoch as tic fragmentati o n of inner mainb e lt asteroids of various s i zes during a t yp i ca l 500 million period -..J -..J

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1. 8 8 r-r-11~11-----r~11~~r-,----,----,---r-----,-------.----.------.---,--.-,----r-, 1.86 s .--I 1.84 ,.0 .--I ,........, .--I 1.82 Dohnanyi q ( constant strength) I 1 8 0 '-'-~--:-'-____..1.___j______J____.l______l___l_l_..L____j___j___J_L__j___L_J_L_L_.l_____l_L_j::::J 0 2 0 .1 0 0 .1 0 .2 Slope of Scaling Law Figure 30: Equilibrium slo p e param e t er as a function of the s lop e o f the size-strength sc aling law -.J 00

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1.9 1.85 o-4 s ~ 1.8 ,.0 ...-t ...-t o-4 1.75 ; I I I I = I f,~,,_~,c=-_ 1,.:,c .=:/) : ~ //_ l 1 t .. t l '.""l I Family 1 / I Family 2 ---I . I : 1 1 . . t .. 1 .. ': i ~ ;l '. M '. i : ( ~ ~lt ~ ~ ) j!~ L\/ . . . :: I 1. Ll : . I J S : f I J ; it J ., J . I r 'f 1 I I \ I 1.7 0 1 2 3 4 Time (Byrs) Figure 31: Difference in the equilibrium slope parameters for families with different strength properties -...J

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1010 ,,----.._ C") I s g u 10 01) H Q) 10 8 +..:> 01) Q) H +..:> Cf) 10 7 +..:> 06 u 1 co s F I 11 1 1 1 1 11 I II I IIIII I 1 1111 111 I 11 1 1111 1 1 I 1 111111 I 11 1 1 1111 I 1 1 1 1 "" I ii 11 111 I I 1 1 11111 I I I 111.lD o Fujiwara (198 2 ) Housen e t al. ( 1 9 91 ) S ex: 0 o 24 Davis e t al. (1985) I Modifi e d Sca lin g L a w Housen e t al. ( 1991) 1-----1 10 5 1 111 1 11 111 1 11 1 1 1111 11 1 11 1 111 11 111 1111 1 1 1111111 11111 1111 111 1 11111 111 111111 1 1 1 1 11 111 1111 111 11 1111 1 1111 11 1 1 11 11 1 10 9 10 7 10 5 10 3 10 1 10 1 10 3 Diameter (km) Fi g ur e 3 2 : Th e D av i s e t a l. ( 1 98 5 ) H o u se n e t a l. ( 1 99 1 ) and m o difi e d s ca lin g l a w s u se d in th e c o lli s i o n a l m o d e l. 00 0

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10 6 .::: ...... II1 s.... 10 4 Q) ...., Q) E 10 3 (lj ....... Cl s.... Q) 10 2 0.. s.... Q) .D 10 1 E ;:l z 10 1 (a) 10 100 1000 Diameter (km) 106 .::: ...... II1 s.... 10 Q) ...., Q) E 10 3 (lj 8 s.... Q) 10 2 0.. s.... Q) .D 10' e ;:l z 10 1 T 0 (small} 10 100 Diameter (km) Figure 33: Th e evolved size di s tribution after 4.5 billion years u si ng the H o u se n e t al (199 1 ) sc aling l aw for (a) a massiv e initial p o pulation and ( b ) a s mall initial p o pu l ation. 1000 00

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106 5 a:l 4 s.... 10 Q) ...., Q) E 0 3 cu 1 iS 10 2 0. s.... Q) ..o o' E 1 ;::l z 00 1 I T. (massive) 10 100 Diameter (km) 1000 10 6 r:: ill F s.... 10 4 Q) ...., Q) 103 iS s.... Q) 102 0. s.... Q) ..o 10 1 E ;::l z 100 1 (b) Mainbell Populalion 10 100 Diameter (km) Figure 34: The evo lv ed size distribution after 4.5 billion years u sing the Davis et al. (1985) scali n g law for (a) a massive irutial population and (b) a sma ll irutial population. 1000 00 N

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10 6 h 10 4 (a) Q.) ..., Q.) E lOJ 10 2 0. h Q.) ..0 lOl E ;:l z 10 1 T, (massive) Malnbelt Populalion 10 100 Diameter (km) 1000 106 iE F h 10 4 Q.) ..., Q.) lOJ .... Cl h Q.) 10 0. h Q.) ..o 10 1 E ;:l z 100 1 (b) 10 100 Diameter (km) Figure 35 : The evo l ved size distribution after 4 5 billion years u sing o ur modified sca lin g l aw for (a) a massive initial population and (b) a sma ll inWal population 1000 00 l>)

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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, Koroni 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 re earchers claim to be able to identify more than a hundred groupings while others feel that only the few large t familie are to be considered "real". The disagreements ari e from the different tarting et of asteroids considered (early classifications included fewer asteroids with more 84

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85 discovered asteroids, later investigators are able to i e ntify 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 (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 Zappala 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 g enerated from a quasi-random distribution of orbits simulating the actual distribution.

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86 A ignificance 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 Zappala 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 Zappala families and have labeled some of the more prominent ones. The Koronis Eos 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 b e 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 1 i 5 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 g enetically related. Collisional Evolution of Families Number of Families One constraint on the collisional history of the mainbelt is the numb e r of famili e which hav e been produced and remain vi s ible at the present time A very mas iv e

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7 initial population coupled with relatively weak a teroids would imply that nearly all the families identifiable today mu t be relatively young. A smaller initial belt and asteroids with large impact strengths would allow even modestize families to survive for billions of years. To attempt to distingui h between the e two pos ibilities and to examine the collisional history of familie we modified our stochastic collisional model to allow u to follow the evolution of a family of fragments resulting from the breakup of a ingle large asteroid. simulating the formation of an a teroid family At a specified time an asteroid of a pecified ize is fragmented and the debris di tributed into the model' ize bins in a power-law distribution as described in Chapter 3. A the model proceeds. a copy of the fragmentation and debris redistribution routine pawned 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 consi tent 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 101 yr 1 km2 while the probability of collisions with other Koronis family members is 13 69 5 x 101 8 yr 1 krn 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

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88 Kor ni family members and asteroids of the background projectile population. The very large total number of projectiles in the background population completely swamps the s mall 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 50 1 a 2 825, 0 0 e 0. 3 and 0 0 s in i 0 3 we found 1799 non-family asteroids which yields a mean vol ume density typical of the mainbelt of 1799 /( 0 32 4 A U x 0 3 x 0 3 ) = 1799 / 0. 02 916 = 616 94. 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. Zappala et al 1984). The typical 6.V associated with the ejection speed of the fragments will be of the order of the escape speed of the parent asteroid, which scale s as the diameter D. The typical volume of a family must then scale as D 3 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 2 6 x 10 5 element units. The 6. V for a parent of this ize is approximately 135 m s 1 Within the model the family volume associated with a parent a teroid of any pecified ize

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89 is then caled from thi value. The number of tele copically visible asteroid pread throughout this volume i then u ed to compute the family volume density. The typical completeness limit for families in the middle mainbelt is ~30 km We imply count the number of family member in s ize bins larger than this when computing the volume density. As our colli ional m ode l proceeds for a certain parent body ize. we then monitor the volume den si ty of the o b servab le asteroid 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 h ow the volume density of families derived from variousize parent asteroids decays with time as the member asteroids are s ubsequently ground away due to further collisions. The dashed horizontal line indicates the den ity of the mainbelt background, the thresh o ld density for detection. For parent bodie s not significan tly 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 so lar system. Our detection criterion then allows us to estimate how many visible families o ur 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 thi s l og plot reflects the size and s pacing of the s ize bins used in our model. We can see that the rate of formation of families decreases noticeably with time especially for the s maller parent bodies Families to the left of the dashed boundary are ground t o undetectibility by pre se nt techniques Our nominal collision model predicts about 30 families, not greatly more than the roughly two dozen families presently recognized

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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 Distribution 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 o rder to compare our m o deled families with tho se observed in the mainbelt we mu s t define the size distributions of the observed families. We proceed as in Chapter 2 when rec o nstructing the s izes of other mainbelt asteroids. For those family members with IRAS albedos the diameters were calculated based on their V-band absolute magnitudes fr o m Equation 11 of Appendix A. For family asteroids without a measured albedo we assumed the mean albedo of o ther IRAS-observed asteroids within the family Ten of the most s parse 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 o ther mainbelt asteroids at the same semimajor axis The final size distributi o n s are then presented as cumulative size-frequency distribution s. The diameters down to which families are considered complete with respect to discovery have been calculated a s uming the current mean opposition magnitude completeness limit for the mainbelt of approximately 16. The Gefi o n Family For the pre se nt time we have chosen to limit o ur m o delin g of individual familie t o a few of th e smal ler families Features of the ize di stri bution s of some of the

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91 large t families imply elf-gravitational reaccumulation on the largest remnant ( Zappala et al 1984 ). Other families. such as Vesta ( Binzel and Xu 1993) and pos ibly Themis ( William 1992 ) may represent very large cratering events The families resulting from uch events are sufficiently difficult to model with our imple power-law fragmentation routine that we feel they are best left to future, more ophisticated model The ize di tribution of some of the more mode tize families however, ugge t imple 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 asteroid 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 Maria family in contrast to the Gefi o n family i very well explained if that family is quite old. The best fit is obtained with the de truction of a 17 5 180 km parent asteroid 3 billion years ago. Figure 40 shows the results of the collisional model averaged over five model runs to eliminate stochastic variation s from run to run The family has been modeled assuming the same mean

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92 intrinsic c o llisi o n probability and encounter speed used within the mainbelt model. In reality, the high inclination of the Maria 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 tructure 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

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,,,-.._ (/) Q) Q) h tu) Q) 'D ___,, 0 ,......., +...:> Cd .,......., u 1--4 h Q) 00 h 20 ,----,-~ ------.-----r-----,-----.----------,--r---,----.-----,--,--,----,--, 15 Maria ~t :'.!. ... :! .. ,: .... \'~ :: : .. Eunom1 a ( " : '! 10 .:Adeona . Eo s . t' . .. . . .. :" ii, .,. G e fiori Vest a ,J ,ij, : Dora 5 ~ ~ ... :.f; : __ .. ,... ,. ~\. Flo ra ., : v . . ii, . ...... ,,,.~ :. . .. ";'(:.' r :~; :,,. . Koronis Th em is , f:~~~/,;: ..:,Id:,_;~ _. Ny s a ~..., ~/ , ' ,. . . ~... :~ .. . .. ,. ; ~ :,: . 0 '--'------'-____._ __ .____.____._____._ __ .____.____._ __._~ _.____._~ 2.0 2.5 3.0 3 .5 Semimajor Axis (AU) Figur e 36: Th e 26 Hirayama a s t e roid famrn es a s d e fin e d b y Z a pp a l a e t al ( 1 98 4 ). I.O w

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'""d 5~~ TT~TTT7~,,-~~TfTTT7---,rr,,-..-~H b.O 4 u CD O 3 +,) Q) > r-1 +,) CD ..Q) >-, +,) r-1 (/) 2 1 Collisional Decay of Fam ilies 110 km 221, 278 km Q) QI I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I Q 0 1 2 3 Time (Byr s ) Figure 37: Th e collisional d ecay of fantilies resulting from vario u s-size parent astero id s as a function of time I.O +>

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1000 .---.-----,------,----,--i-,-----,---.-.---,--,-,r--r------.------,-----.------,---------,,--------,-------.-~~ 700 s 500 1 \ I \ ......._,,,, I \ H Q) 300 t I \ s .. "ro ...... 2 00 Q ....... ... ~ .. -----.. .. .... . ------~---100 ~---'-_J_---'-_L_jl__L_l__L___j___l__j___L____L__l__j__j_____J::::::,,__i__L_L__j_J 0 1 2 3 4 T im e (E y r s ) F i g ur e 38 : Fo rm a ti o n of f am ili es in th e m ai nb e lt as a f un c ti o n o f ti m e. \0 U\

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Q) a z Q) >-~ +> ro a u 10 4 10 3 1 By ~ ~ 10 Myr Gefion Family 150 km Parent 10 2 10 1 10 __l 1 3 "" ~ 10 30 100 Diameter (km) Figure 39: M ode l ed co lli s i o nal hi sto ry of the Gefion famjly. j ID 0\ 3 00

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10 4 H 10 3 ~\ Maria Family Q) ,.0 175 km Parent s z Q) 10 2 3 Byr >- ....-i +.J E .. -~ ~"" j Cu \0 -.J ,---i s 10 1 u 10 1 l_____l_____~~___u_J 3 10 30 100 300 Diameter (km) Figur e 40 : Mod e l e d co lli s ionaJ hi s t ory of lh e Maria family.

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CHAPTER 5 IRAS AND THE ASTEROIDAL 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. F o r 11 months the one-ton satellite returned a wealth of data surveying nearly 96% of the sky at 12 25, 60, and 100 1.1, 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 mad e the urpri s ing discovery of three relatively narrow bands of infrared emission s uperimpo sed o n the broad zodiacal emission ( Low et al. 1984 ; Neugebauer et al 1984). The mo t 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 latitud es of + 10 and -10 (see Figure 41 ). The bands can also be seen in the 60 and 100 m data although at a l ower 98

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99 intensity. Color-temperature calculations ( Low et al. 1984 ) yield value 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 matche estimate of the location of the band emi sion at 2 3-2.5 AU obtained by parallax measurements ( Gautier et al. 1984; Hau er et al 1985 ; Dermott et al. 1990 ). The estimated location of the band pairs within the a teroid belt suggested to Low et al. (1984) that the band emission arose from the du ty 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 f amilie They linked the central du t band with the Themis and Koronis familie s 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 ignificant 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 di s tribution of sky brightness, as seen by the IRAS telescope, associated with any

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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: (1) 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 gravitati o nal 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 f. l 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: Tbemis Koronis Eos, Nysa, Dora, and Gefion. The cross sectional areas of dust associated with the families were treated as free parameters and adju ted to fit the observations at elongation angle 114 68 Exactly the same particle distribution

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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 o b ervations. 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 tronger 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 sever al 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 i s 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

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102 family flux is added to the remaining background. When this total s ignal is p assed through the filter a seco nd time the re s ulting s mooth background and band profile match very clo ely the o riginal obse rved background and band profile s The total area a soc iated with the families needed to model the du s t band s is found to be ~3 x 10 9 krn 2 This is found to be ~10 % of the area needed to m ode l t he n o n-family contribution to the s m oo th zodiacal background. The Ratio of Family to Non-Family Dust Having es tabli s hed that the IRAS du st bands are a socia ted with the prom inent Hirayama asteroid families and ha v ing determined the extent of their contribution to the total flux in the zo diacal cloud if the particle production rate of family as teroid s is no different than that of other mainbelt asteroids we may use the families t o 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 i s hi ghly 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 s h ow n th a t despite the uncertaintie s associated with individual fragmentations, the equilibrium size frequency distributi o n can still be well-described by a s imple power-law di strib uti on In Chapter 3 we showed that regardless of hape of the initial populati o n of debri the size distribution of a family of fra g ments will evolve to an eq uilibrium di tribution with a half time of order of the fragmentation lif e time of the lar ges t debris bein g con idered If th e i ze-freq uency di s tribution of debris in the families can be de sc ribed b y a impl e

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103 equilibrium power-law, then we may relate the urface area of the du s t in the band to the total volume of the family fragments by the expre ions 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 a the large t a teroids may not contribute to the population of colli ion fragments. In fact, we can ee 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 doe s n o t contribute to the mainbelt dust area. To actually calculate the ratio of family to non -f amily areas we fit an equilibrium distribution through the combined magnitude distribution of all the Zappala families and through the linear portion of the mainbelt asteroid population. We have plotted these distribution in Figure 43 with the equilibrium fits obtained by constrained least-squares o lutions Although we work directly with the magnitude data for convenience the abscis a 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 1.1,m) 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 tum are determined by the intercepts of the least -s quares solutions. We take as a measure of the intercept the diameter of the largest asteroid, Dma x which would be pre se nt in the

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104 equilibrium distribution fit to the data. For the entire mainbelt population we find that Dmn z = 308~1~ km, while for the combined family distribution D,,, w.i = 1 89~f 0 km For a pecified cutoff s ize D mi 11, the total geometrical cross-sectional area associated with the debris is then calculated directly from Equation 11 of Appendix B F o r Drnin = 10 11 m the cross-sectional areas for the entire mainbelt population and all families combined are 1. 45 x 10 11 and 4.25 x 10 10 km 2 respectively. The ratio of the cumulative area of dust in the entire mainbelt p o pulation to that associated with all the families is then approximately 3.--! : 1. Due to standard errors in the values of the intercepts from the leastsq uares solution and uncertainties i n 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 t o the zo diacal cloud as the prominent families. In the previous ection 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 mu s t be re s ponsible for at least a third of the dust particles in the zodiacal cloud.

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10 fl00m if) 8 co 0 6 ,--l ......._., >< 4 ....---4 .I \I _.. ~! 0 25um UI r---1 ro 2 'tJ -~ (/) Q) 0 cc: 20 10 0 10 2 0 Ecliptic Latitude Figure 41 : The solar s y s t e m dust band s at 1 2, 25 60 and 100 aft e r s ubtra c ti o n o f th e s m oo th zo dia c al b ac k gro und via a F o uri e r filt e r.

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4 2 0 I 3 f-' I\ / or;, u: o -j i {b) 97 46 Observa tio n Mode l ... 1.0 2 ,--... 1--, 1 I (/) ,,.._ /fV Iv\ / \-\ I 0 0 ::,.... 0 .....,, (0 0 ...-t 1.0 .__, 20 -10 0 10 20 -2 0 10 0 10 2 0 >< 2 0 ;:J .. (c) 114.68 Observatiol 4 o t (d) 65 68 Observ atio n Model Mod el (U IA 1.0 3 0 0 0 0\ E2 0 0 0~~ V 1.0 0 0 1.0 -1. 0 -2 0 -10 0 10 20 -2 0 10 0 10 20 Ecliptic Latitude Figure 42: (a) IRAS observations of the du s t band s at elongation angles of 65.68 97.46 and 114 .68 Compari so ns with model profiles based on prominent Hirayama familie s are s h ow n in ( b ) (c), and ( d )

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. ,--t h Q) +,) Q) s ro ,-1 Q 10 5 10 4 10 3 h 0 2 Q) 1 h Q) ,.{) 0 l s 1 z 0 0 0 0 0 0 100 I I 1 I I I 0 Ratio of Cumulative Areas of Du st Entire Mainbelt / Families = 3 4 0 6 0 0 En tire Mainbelt I All Families ! I I I I I I I -~-J.1 .,, J: 4 l, I I I I I I I I I 1 4 1 I I ,1 10 100 1 000 Diameter (km) Figur e 43 : 1l1 e ratio o f areas of du st assoc iat e d with th e e ntir e mainbelt asteroid p o pul atio n a nd a ll familie s. 0 -..)

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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 sl o pe 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 fr o m that of the background asteroids. Observations of the du t areas associated with particular 108

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109 f arnilies and models of the collisional history of tho e 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 caling 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 1 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 uggested 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.

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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. o that the prominent families actually upply about 10 % of the dust in the zodiacal cloud. Our comparison of the effective volumes of the familie 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 tronger 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 olar system and the comminution o f the asteroids. Approximately 20 % of the a teroidal particles passing the Earth are

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111 temporarily ( for ~ 10 4 years) trapped in resonant l ock 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 10 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 m o re 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 s ize 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 cl o ud and dust bands although there are other observations for which our model must al s o account but which at present are problematical. In particular although a single size particl e model of main belt asteroidal dust can explain the observed inclination and nodes o f the z odiacal cloud the model predicts a total flux at high ecliptic latitudes which i s far too low (Dermott et al. 1992b). One resolution of this discrepancy may lie in a

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112 clo ud of asteroidal particle who e effective area increases with decreasing heli oce ntric di tan ce, as might be expected for particle s under go ing continual collisional evolution conc urr e nt with o rbital decay due to radiation effects. Work on this problem ha already begun ( Gustafson et al. 1992). The re s ult s will also yield a description of the variation of the particle size distribution with heliocentric distance If a cloud of a teroid a l particles is s hown to contribute more to the background flux at high ecliptic latitude s the total contribution made to the zo diacal cloud by families would increase to g re ater than the pre se nt estimate of ~ 10 %. The total s upply of dust made by the main belt asteroid population would then be grea ter than 30 % if the family contribution s impl y doubled to 20%, the total asteroidal component would increase to nearly 70 %, reversing the presently estimated ratio o f asteroid to cornet du s t.

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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 = 100 1 / 5 = (10 2 ) l / 5 = 10 2 / 5 10.4 = 2 512 ... With this definition 2 mag= (10 .4 ) 2 3 mag= (10.4) 3, and m general the ratio of fluxes from two objects with magnitudes m 1 and m 2 is : F1 ( 0A) m 2 mt (A 1 ) F 10 2 or F1 ( A 2 ) l og = 0 .4( m2 rn. 1 ) F 2 o r, F1 (A 3) m2 m 1 = 2 .5 l og p 2 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 :: = G:) 2 (A-4 ) For s tellar sources the standard distance is 10 parsecs yielding (after substituting Eq 4 with r1 = 10 for Fi/ F2 in Eq 3) the familiar 2 m M = 2 .5 l og (; 0 ) = 5 l o g r -5, (A 5) 11 3

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114 where M i 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 A tronomical Units ), Eq. 4 yields: ( A-6 ) ince 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 /r 2 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 r p = 1 and after substitution Eq. 3 reads: ( rp ) 2 m.2 rn. 1 = 2 .5 l og 1 ( A 7 ) o r, m. 0 H = 5 l ogrp, ( A 8 ) where H is the V-band absolute magnitude From Eq. 1 we s ee that the observed flux of an asteroid is prop o rtional to 10 -0A H But the flux from the asteroid depends on its cross sectional area (a large asteroid appear brighter than a s mall asteroid) and its geometric albedo ( an asteroid with a bright urface more reflective than an asteroid with a dark urface) We then have that Flux 10 -0A H ( A -9)

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115 where p 1 i the geo metric albedo and D is the diameter of the asteroid. Then, logp 1 + 2 log D = co n t. 0.4H ( A-10 ) o r, 2 l og D = co n t. OAH log p 1 ( A-11 ) which i the expre i o n given by Zellner (1979). ( o te that B(l 0) (B V) = V( l.O ) = H.) The constant ha the value 6.241 and i derived knowing the apparent magnitude of the Sun.

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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 x, incremental plots ( number per s ize or mass bin ) with linear increments and incremental pl o ts 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 siz e (o r mass) distribution of fragments A power law distribution has the form ( B 1) where N is the cumulative number larger than diameter D and C i s a constant. By taking the common logarithm of both sides, l ogN = p l ogD +logC ( B 2) we see that the power law exponent p is th e n ega tive slope in a l ogNlo g D pl o t an d the cons tant C, define s the y-intercept. To see mor e clearly any concentrations o r d e pl e tions of particle s in certain 1 ze ranges an incremental plot is more u se ful. We mu s t be careful, how eve r t o c learl y define the kind of increment which ha s b ee n c ho se n lin ear or lo g arithmic Diff erentiati n g 11 6

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117 Equation 1 we o btain d = pcn-p-ldD, ( B -3) where dN is the number in the linear increment of width dD. ( The negative s ign si mply fo rmally indicates that the number per bin decrea ses with increasing diameter ; we are interested in the magnitude of the change o that the negative s ign may be ign o red See also Equation 5 b e l ow ) Taking the l ogari thm of b o th si de s, l og dN = (p + 1 ) log D + log (pCd D), ( B -4) we ee that the s l o pe of the size distribution o n a l ogdNl og D pl o t is -(p + 1). If th e c umulative plot had a s l o p e of -2.5, the incremental pl o t with linear increments would have a s l o pe of -3 .5. Now s ince d l og x = ln~o d:r, we may rewrite Equation 3 as dN = pC lnl OD-vd l og D ( B-5 ) This then represents an incr e m e ntal size di stri buti on with l ogari thmic increments. From l og dN = -p l og D + l og (pC ln 1 Od log D) ( B 6 ) we ee that the s lope i s the same as that for a cumulative plot namely -p. In many cases the fragment di s tributi o n is de sc ribed in term s of ma ss rather than 1ze. Since M = r pR 3 = r pD 3 Equation 1 may be rewritten in t erms of the mass as ( B -7) Th e s l o pe of a cumulative mas s distribution pl o t is then simply o ne third th a t of the c umulativ e size distribution pl o t. As before, i f the s l o pe of a cumulative mass

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118 distribution plot is -q, the corresponding incremental plot with linear increments will have a s lope 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 colli ional 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 ize. For a continuous distribution of sizes it is the particle ma ss integrated over the size-frequency distribution: ( B-8 ) since M = i p D 3 Equation 8 is integrated over the size range from the smalle t to largest particles present. After carrying out the integration, the final expression for the total mass is 1f p p C 3p I D,u u ,; Mt o t = -D 6 ( 3 p) D ,..., (B 9 ) The total cross sectional area is found in a similar manner by integrating the era ection of a single particle, f D 2 over the size distribution: A j 1r n 2 c n p1 d D t o t 4 P ( B 10 ) which yields a total geometrical cross-sectional area of 7rpC 2 I D ,uur A --D -p llJ t = 4(2 p) D ...... ( B 11 )

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119 ( Recall that in the e expressions p and C are the negative lope 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 u e half-magnitude bins for instance) and the number of a teroids per bin i presented in a logdN-Mag plot. Remembering that 2 l og D = ca n s t 0.4 H l o g pu, we see that uch a plot is equivalent to an incremental size-frequency di tribution plot with logarithmic increments since an increment of x in ab olute 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 logdN H plot of the form l ogdN = a H + b, where a is the slope and b is the y-intercept. Substituting for H we obtain l ogd = a( 5 l og D + 2. 5co n t 2 .5 l ogp(/) + b = -5a l og D + (2 5aco n t 2.5a l ogp(/ + b). Comparing Eqs. 6 and 13 we see that p = 5a and l og ( p C 1n 10d l og D ) = 2.5aco n t 2 .5 a lo g p l!+ b. (B 12 ) (B 13 ) (B 14 ) (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 .). ) Once we have assumed a mean albedo and con s tant for the distribution of asteroid s under consideration these expressions allow

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120 u s t o u se the parameter s of the magnitude-frequency pl o t to find the quantitie s p and C fo r the s i ze di s tributi o n which m ay then be u se d in Equations 9 and 11 to find the t o tal ma ss and area assoc iated with the di s tributi o n ( The constants a and b in Equation 15 dep e nd o n the s ize of the ma g nitude bin which ha s been chosen Therefore the value of d log D is also fixed by the c hoice of ma g nitude bin s ize d log D = 0 2dH.)

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APPENDIX C POTENTIAL OF A SPHERICAL SHELL We wish to find the gravitational binding energy of a spherical shell of mass O SM covering a sphere of mass O.SM 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 % o f its mass contained in a spherical shell with radius r = a = ( 0. 5) 1 13 R ( approximately 0.79R ) to r = R. The volume of the shell is given by 4 3 4 3 4 ( 3 3 ) V = -1r R -1r a = 1r R a 3 3 3 (C 1 ) If we assume that the mass is uniformly distributed within the shell we can write ( C-2 ) Within the shell (which sits upon a core of mass 0.5M) the mass is given by r M (r)= 4 1r x 2 dx+ -M 1 J 1 2 2 (C 3 ) a [ 3Mx 2 l,. 1 M (r) = 2( fl 3 a3) a+ 2M (C-4 ) (C 5 ) 1 2 1

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122 and 3Mr 2 d 1 \1(r) = ( 3 3 ). 2 R er The binding energy can now be calculated ( C 6 ) R R -n = J G M (r ) d M(r) dr =J G [! M r3 a3 + M ] Mr dr ( C 7 ) r 2 R 3 a 3 2 2 R 3 a 3 a a = 3GM 2 j~ [r a 3 r + r ] dr 4(R 3 a 3 ) R 3 a 3 ( C 8 ) a J [ "" 3 J 2 ] R 3 GM r<> a r r = 4( R 3 a 3 ) 5( R 3 a 3 ) 2( R 3 a 3 ) + 2 a (C 9 ) ( C 10 ) [ 5 1 3 ( 2 l 2 ) 2 ] 3 GM2 R 5 ( 0 5)3 R 5 2R R ( 0. 5) 3 R R 2 (0.5)3R2 2 R 3 fl 3 R 3 + 2 2 ( C 11 ) The last two terms in the brackets cancel, leaving GM 2 -n = 3 2 R 3 [ 0. 2 7 4 0079R 2 ] ( C 12 ) o r ( C 1 3) 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 ) Thu a s e xpected it takes somewhat le ss e ner g y t o disperse a shell of one-half the total ma o ff of a target a s teroid.

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128 Tedesco. E. F., J. G. William D L. Matson. G J. Veeder. J. C. Gradie, and L. A. Leb ofs ky 1989. Three-parameter asteroid taxonomy classificati o ns. In Ast e roid s II ( R. P. Binzel. T. Gehrels, and M. S Matthews Eds. ) pp. 1151 1161. Univ. o f Ariz o na Pre s s Tucson. Th o len. D. J 1989 A teroid taxonomic cla sifications. In Ast e roids II ( R. P Binzel T. Gehrels and M. S. Matthew s, Ed s.), pp 1139-1150 Univ o f Arizona Pre ss, Tucson. Van Houten C. J P. Herget, and B. G. Marsden 1984 The Palomar-Leiden survey o f faint minor planets: Conclusion. Icarus 59, 1-19 Van Houten C. J., I. Van Houten-Groeneveld P. Herget, and T. Gehrels 1970 The Palomar-Leiden urvey of faint minor planets. Astr. Astrophys Suppl. 2 339-448. Wetherill G. W. 1967. Collisions in the asteroid belt. J G eo ph y s. R e s 72 2429 2444 Whipple F. L. 1967. On maintaining the meteoritic complex. Smithson. Astroph y s Obs. Spec. R e pt No. 239 pp 1-46. Williams D. R. and G. W. Wetherill 1993. Size distribution of collisionally ev o lved asteroidal populations: Analytical solution for self-similar collision ca cade s. Submitted to Icarus. Williams. J G. 1 9 79 Proper elements and families memberships of the asteroid In Asteroids (T. Gehrels Ed.) pp. 1040-1063 Univ. of Arizona Press Tucson. Williams, J G. 1992. Asteroid families an initial search. Icarus 96 251-280 Xu Y. L. S. F. Dermott D. D. Durda, B. A. S. Gustafson S Jayaraman, and J. C Liou 1993 The zodiacal cloud. Chinese Academy of Sciences Beijing. in pr ess Zappala V and A. Cellino 1992. Asteroid families : Recent results and present s cen ario. C e l. Mech. and D y n. Ast 54 207-227. Zappala V A. Cellino, P. Farinella, and Z. Knezevic 1990 Asteroid familie I. Identification by hierarchical clustering and reliability assessment. Astron. J. 100 2030 2046. Zappala V. P Farinella, Z. Knezevic, and P. Paolicchi 1984. Collisional o rigin o f th e a teroid families : Mass and velocity di tributions. Icarus 59 261-285. Zellner B 1979. A teroid taxonomy and the di tribution of the compo iti o nal t y pe In Ast e roids ( T. Gehrels Ed.), pp. 783 806. Univ of Arizona Pre s Tu cso n

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BIOGRAPHICAL SKETCH Daniel David Durda was born 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

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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 P~ ,:;;i,-n.; lv '.J 7 } -0/v ,-vv.~ 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 ~a ~~ == ===== -Humberto Cins Cochair Associate Professor of Astronorp.y 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 Jam 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. Philip D. Nicholson A s sociate Professor of Astronomy Cornell University

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Thi di ertation was s ubmitted t o the Graduate Faculty of the Department of A tr o n omy in the C o llege of Liberal Arts a nd S cie n ces and to th e Graduate School and wa acce pted a partial fulfillment of the requirement s fo r the degree of D oc t or o f Phil o o ph y. D ecember 199 3 Dean Graduate Sch oo l


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