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
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Durda, Daniel David, 1965-
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Asteroids   ( lcsh )
Zodiacal light   ( lcsh )
Cosmic dust   ( lcsh )
Astronomy thesis Ph. D
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Thesis:
Thesis (Ph. D.)--University of Florida, 1993.
Bibliography:
Includes bibliographical references (leaves 123-128).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Daniel David Durda.

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Full Text












THE COLLISIONAL EVOLUTION OF THE
ASTEROID BELT AND ITS CONTRIBUTION TO
THE ZODIACAL CLOUD









By

DANIEL DAVID DURDA


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


















To my parents, Joseph and Lillian Durda.












ACKNOWLEDGMENTS




There are a great many people who have played important roles in my life at UF,

and although the room does not exist to thank them all in the manner I would like, I

would at least like to express my gratitude to those who have helped me the most.


First and foremost, I would like to thank my thesis advisor, Stan Dermott.


has been far more than just an academic advisor.


He has taught by splendid example


how to proficiently lead a research team, looked after my professional interests, and

given me the freedom to focus upon research without having to worry about financial


support.


I never once felt as though I were merely a graduate student.


One could not


ask for a better thesis advisor.

My thanks also go to the other members of my committee, Humberto Campins, Phil

Nicholson, and James Channell, for their helpful comments and review of this thesis.


The advice and many laughs provided by Humberto were especially appreciated.


I am


also very grateful to Bo Gustafson and Yu-Lin Xu for the many discussions and helpful


advice through the


years.


My fellow graduate students, my family away from home, kept me sane enough (or

is it insane enough?) to make it this far. I will value my friendship with Dirk Terrell and


Billy Cooke forever.


I will probably miss most our countless discussions about literally






more than I can express in words.


Billy's


"Billy-isms" have provided me with more


entertainment than I have at times known what to do with.


I will miss them immensely!


will also miss my discussions, afternoon chats, and laughs with the other graduate


students who have come to mean so much to me, especially Dave Kaufmann, Jaydeep

Mukherjee, Caroline Simpson, Sumita Jayaraman, Ron Drimmel, and Leonard Garcia.

I would like to thank the office staff for helping me with so many little problems.


Debra


Hunter,


Elton,


Suzie


Hicks,


Darlene


Jeremiah,


especially


Jeanne


Kerrick, deserve many thanks for helping me with travel, faxes, registration, and for


brightening my days.


Also, thanks go to Eric Johnson and Charlie


Taylor for keeping


the workstations alive.

With this dissertation a very large part of my life is at the same time drawing to


a close and beginning anew.


The most wonderful part of my new life is that I will


be sharing it with Donna.

Without the love and unwavering support of Mom. Dad, my sister Cathy and her

husband Louie and my nephew and nieces Andrew, Larissa, and Jenna, none of this


would ever have


happened.














TABLE OF CONTENTS


ACKNOWLEDGMENTS.

LIST OF TABLES . .

LIST OF FIGURES .

ABSTRACT. .. .. ..


S S S S S S S S 1 iii


.*a..* VI
. . . . . . . . V 11


. S. S S S. x S S S S S S S S S S S S S S S S S S


CHAPTERS


INTRODUCTION


S S S S S S S S S S S S S a a 1


THE MAINBELT ASTEROID POPULATION .. . . . 4

Description of the Catalogued Population of Asteroids . . . 4


The MDS and PLS Surveys


The PLS Extension in Zones I, II, and III .

The Observed Mainbelt Size Distribution .


THE


OLLISIONAL MODEL


Previous Studies


Description of the Self-consistent Collisional Model


. S S 3


Verification of the Collisional Model .


'Wave'


and the Size Distribution from 1 to 100 Meters


S S S S S S S S 46


Dependence of the Equilibrium Slope on the Strength Scaling Law


The Modified Scaling Law. . .

4 HIRAYAMA ASTEROID FAMILIES.


S S 52


* S S S S S S . S S S S S S S S 55

. . . . . . 8 4 i


A Brief History of Asteroid Families.
The Zappalk Classification . .
Collisional Evolution of Families .


. .. 84
* S S S S 85
* S S S S S S S S 5 85 ^


. 86


Number of Families.


Evolution of Individual Families


IRAS AND THE ASTEROIDAL CONTRIBUTION TO THE ZODIACAL
C L.JOlU Dl\ . . . . . . . a *


98


* S S 1 9

. 13


, S S 3


S S S S S S 5 3








The Ratio of Family to Non-Family Dust


6 SUMMARY .
Conclusions
Future Work

APPENDIX A:


APPENDIX B:

APPENDIX C:


S S . S S . S S S S 140 2


* S 5 5 S S S S S S S S S S S S 10 8
108
* S S S S S S S S S S S S S 1 100
* 5 5 5 5 5 5 5 5 5 5 5 S S S S S S S S S S S S 1 14VI/


APPARENT AND ABSOLUTE MAGNITUDES OF
A TEROIDS ..............DB.. .

SIZE, MASS, AND MAGNITUDE DISTRIBUTIONS .


POTENTIAL OF A SPHERICAL SHELL


5 113


. 1


. S S . 5 12 1


BIBLIOGRAPHY


. .*. .. 123


BIOGRAPHICAL SKETCH


S S S S S S S S S S 5 S S S S S S S S S S S S S S S *. S 12 9











LIST OF TABLES


Numbers of asteroids in three PLS zones (MDS/PLS data). . .

Numbers of asteroids in three PLS zones (catalogued/PLS data). .

Adjusted completeness limits for PLS zones. . . . .

Intrinsic collision probabilities and encounter speeds for several mainbelt


16

17


18


asteroids.


. *. U l U U 6 2 U U U S U U U U U 62













LIST OF FIGURES



Proper inclination versus semimajor axis for all catalogued mainbelt


asteroids.


Magnitude-frequency distribution for catalogued mainbelt asteroids.


. 20


Absolute magnitude as a function of discovery date for all catalogued


mainbelt asteroids..


. . p a p . a a a 2 1


Magnitude-frequency distribution for PLS zone I: PLS and catalogued


asteroid data.


Magnitude-frequency distribution for PLS zone II: PLS and catalogued


asteroid data.


a a a a a a a a a p p p p p a a a a a U p p p a 2 3


Magnitude-frequency distribution for PLS zone III:


asteroid data.


PLS and catalogued


p a a.p a p a a a a a a p a a a 2 4


Adopted magnitude-frequency distribution for PLS zone I. . . 25

Adopted magnitude-frequency distribution for PLS zone II. . 26

Adopted magnitude-frequency distribution for PLS zone III. . 27


Magnitude-frequency distribution for the


1836 asteroids in


Tables 7 and


8 of Van Houten et al.


(1970)..


. a . . .. . . 2 8


Least-squares fit to the magnitude-frequency data for PLS zone I. . 29

Least-squares fit to the magnitude-frequency data for PLS zone II. 30

Least-squares fit to the magnitude-frequency data for PLS zone III. 31


. . . . . . . 2








Verification of model for shallow initial slope and small bin size. . 64

Verification of model for steep initial slope and large bin size. . 65

Verification of model for shallow initial slope and large bin size. . 66


Equilibrium slope as a function of time for various fragmentation power


laws and for steep initial slope.


. . . S 6 7


Equilibrium slope as a function of time for various fragmentation power


laws and for shallow initial slope..


. . . .S. .S. ... 68


Equilibrium slope as a function of time for various fragmentation power


laws and for equilibrium initial slope.


. S S S S S S S S S S .6 9


Wave-like deviations in size distribution caused by truncation of particle


population.


Independence of the wave on bin size adopted in model.


a S S S 7 1


Comparison of the interplanetary dust flux found by Grin et al.


(198


and small particle cutoffs used in our model.


Wave-like deviations imposed by a sharp particle cutoff


(x=


.. 73


Size distribution resulting from gradual particle cutoff matching the


observed interplanetary dust flux (x


= 1.2).


Collisional relaxation of a perturbation to an equilibrium size


distribution..


Halftime for exponential decay toward equilibrium
fragmentation of a 100 km diameter asteroid. .


slope following the


Stochastic fragmentation of inner mainbelt asteroids of various sizes


during a typical 500 million period..


Equilibrium slope parameter as a function of the slope of the
size-strength scaling la. . . . . . . .







The Davis et al.


(1985),


Housen et al.


(1991), and modified scaling laws


used in the collisional model.


. a S S S a a a. a 80


The evolved size distribution after 4.5 billion years using the Housen et
al. (1991) scaling law for (a) a massive initial population and (b) a small


initial population. . . .

The evolved size distribution after 4


. . S & U a a a a 8 1


billion years using the Davis et al.


(1985) scaling law for (a) a massive initial population and (b) a small


initial population.

The evolved size distribution after 4.5


82


billion years using our modified


scaling law for (a) a massive initial population and (b) a small initial


population.


. a a a a a a a a a S a a a a a a a a a S a a a 0 8 3


The 26 Hirayama asteroid families as defined by Zappala et al.


(1984)..


The collisional decay of families resulting from various-size parent


asteroids as a function of time. . .. . .

Formation of families in the mainbelt as a function of time.


Modeled collisional history of the Gefion family.

Modeled collisional history of the Maria family. .


The solar system dust bands at 12,


Sa a 94


S U S 95


Sa a a a a a a a a96

* a a a 97


60, and 100 im, after subtraction


of the smooth zodiacal background via a Fourier filter.


a a a 1. a a 105


(a) IRAS observations of the dust bands at elongation angles of 65.68


97.46


,and 114.68


. Comparisons with model profiles based on


prominent Hirayama families are shown in (b), (c), and (d).


. . 106


The ratio of areas of dust associated with the entire mainbelt asteroid


population and all families..


a a a a a a a a a a a a a a a a a a a a a a a a liV. .












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

THE COLLISIONAL EVOLUTION OF THE ASTEROID BELT
AND ITS CONTRIBUTION TO THE ZODIACAL CLOUD

By

DANIEL DAVID DURDA


December,


Chairman:


1993


Stanley F. Dermott


Major Department:


Astronomy


We present results of a numerical mode


verify the results of Dohnanyi (1969, J. Geophys. Res.
to place constraints on the impact strengths of asteroids.


of asteroid collisional evolution which


74, 2531-2554) and allow us
The slope of the equilibrium


size-frequency distribution is found to be dependent upon the shape of the


size-strength


scaling law. An empirical modification has been made to the size-strength scaling law
which allows us to match the observed asteroid size distribution and indicates a more


gradual transition from strain-rate to gravity scaling.


This result is not sensitive to the


mass or shape of the initial asteroid population, but rather to the form of the strength


scaling law: scaling laws have definite observational consequences.


The observed slope


of the size distribution of the small asteroids is consistent with the value predicted by
the slightly negative slope of our modified scaling law.
Wave-like deviations from a strict power-law equilibrium size distribution result
if the smallest particles in the population are removed at a rate significantly greater


L.Lc-_ J.L -- .... _A A --: T' _1 .. 1 .. 1 -1-







a significant wave.


We suggest that any deviations from an equilibrium size distribution


in the asteroid population are the result of stochastic cratering and fragmentation


events


which must occur during the course of collisional evolution.


determining


ratio of the area associated


mainbelt asteroids


that associated with the prominent Hirayama asteroid families,


our analysis indicates


that the entire mainbelt asteroid population produces


3.4 + 0.6 times


as much dust as


the prominent families alone.


This result is compared with the ratio of areas needed to


account for the zodiacal background and the IRAS dust bands as determined by analysis


of IRAS data.


We conclude that the entire asteroid population is responsible for at least


~ 34% of the dust in the entire zodiacal cloud.












CHAPTER


INTRODUCTION


Traditionally, the debris of short period comets has been thought to be the source


of the majority of the dust in the interplanetary environment (Whipple 1967


Dohnanyi


1976). However, it has been known for some time that inter-asteroid collisions are likely


to occur over geologic time (Piotrowski


1953).


The gradual comminution of asteroidal


debris must supply at least some of the dust in the zodiacal cloud, though because of the

lack of observational constraints the contribution made by mutual asteroidal collisions

has been difficult to determine.


Since the discovery of the IRAS solar system dust bands (Low et al.


1984), the


contribution made by asteroids to the interplanetary dust complex has received renewed


attention.


The suggestion that the dust bands originate from the major asteroid families,


widely thought to be the results of mutual asteroid collisions, was made by Dermott et


(1984).


They also suggested that if the families supply the dust in the bands, thus


making a significant contribution to the zodiacal emission, then the entire asteroid belt

must contribute a substantial quantity of the dust observed in the zodiacal background.

Other evidence also points to an asteroidal source for at least some interplanetary dust.

The interplanetary dust particle fluxes observed by the Galileo and Ulysses spacecraft


indicate a population with low-eccentricity and low-inclination orbits (Grtin et al.


1991),






2

transport lifetimes of asteroidal dust, Flynn (1989) has concluded that much of the dust

collected at Earth from the interplanetary dust cloud is of asteroidal origin.


At first inspection it might be tempting to


try to calculate


the amount of


produced in the asteroid belt by modeling, from first principles, the collisional grinding


taking place in the present mainbelt.


The features of the present asteroid population,


however, are the product of a long history involving catastrophic collisions which have


reduced the original mass of the belt.


Unfortunately,


initial mass of the


belt is


not known and our knowledge of the extent of collisional evolution in the mainbelt is

limited by our understanding of the initial mass and the effective strengths of asteroids


in mutual


collisions.


Our intent is to place some constraints on the collisional processes affecting the

asteroids and to determine the total contribution made by mainbelt asteroid collisions


to the dust of


zodiacal cloud.


Chapter


we describe


methods


used


derive the size distribution of mainbelt asteroids down to ~,5 km diameter.


The size


distribution of the asteroids represents a powerful constraint on the previous history of

the mainbelt as well as the collisional processes which continue to shape the distribution.

In Chapter 3 we describe the collisional model which we have developed and present


results confirming work by previous researchers.


We then


use the model to extend


our assumptions beyond those of previous works and to shed some light on the impact


strengths


asteroid


' asteroids

families i


initial


s examined


mass of the


in Chapter


mainbelt.


The collisional history


providing further constraints


on the


.- jh1*- k -^ C ~L .*. ^fjc- kL I ^ fk A ^ 4 *j~ r -- A-" n- ^ J. f j- ^ f -, C :fAJIJ i T a -k L rftj nNk a..


I-








relative contribution of dust supplied to the zodiacal cloud by asteroid collisions.


conclusions are summarized and the problems that must be addressed in future work

are discussed in Chapter 6.












CHAPTER


THE MAINBELT


ASTEROID POPULATION


Description of the Catalogued Population of Asteroids


The size-frequency distribution of the asteroids is very important in constraining


the collisional


processes


which


have


influenced


continue


to affect the


asteroid


population as well as the total mass and mass distribution of the initial planetesimal


swarm in that region.


Also, in order to determine the total quantity of dust that the


asteroids contribute


to the


zodiacal


cloud,


we must use


the observed


population of


mainbelt asteroids to estimate the numbers of small asteroids which serve as the parent


bodies of the immediate sources of asteroidal dust.


In this chapter we will describe the


data and methods from which we derive a reliable size distribution.

Of the 8863 numbered and multi-opposition asteroids for which orbits had been


determined as of December


1992, 8383 (or


~-95%) are found in the semimajor axis


range 2.0 <


a < 3.8 AU (Figure


For reasons described below,


we will limit our


discussion to those asteroids in the range 2.0


a < 3.5


AU, defining what we will


refer to as the "mainbelt.


as only


SOur conclusions are expected to be unaffected by this choice,


13 asteroids, or less than 0.2% of the known population, are excluded so that


the two sets of asteroids are essentially the same.

Figure 2 is a plot of the number of catalogued mainbelt asteroids per half-magnitude








(Bowell et al.


1989).


Immediately evident is a


"hump"


, or excess,


asteroids at


8. f


Although previous researchers have interpreted this excess as a remnant of


some


primordial,


gaussian


population


asteroids


altered


subsequent collisional


evolution (Hartmann and Hartmann 1968), the current interpretation is that it represents

the preferential preservation of larger asteroids effectively strengthened by gravitational


compression


(Davis et al.


1989;


Holsapple and Housen


1990).


Other researchers,


primarily Dohnanyi (1969, 1971), have noted from surveys of faint asteroids (discussed


below)


indicative


the distribution


a population


smaller asteroids


of particles


is well described


collisional equilibrium.


a power-law,


Unfortunately,


evident in Figure 2, the number of faint asteroids in the catalogued population alone

is not quite great enough to be sure of identifying the transition to, or slope of, such

a distribution.

In fact, the mainbelt population of asteroids is complete with respect to discovery


down to an absolute magnitude of only about H = 11.


We can see this quite clearly in


Figure 3, which is a plot of the absolute brightness of the numbered mainbelt asteroids

as a function of their date of discovery. It can be seen that as the years have progressed,

increased interest in the study of minor planets and advances in astronomical imaging


have allowed for the discovery of fainter and fainter asteroids.


In turn, the brighter


asteroids have all been discovered, defining fainter and fainter discovery completeness


limits.


For instance,


no asteroids brighter than


= 7 have been discovered since


about 1910.


1940 the completeness limit was a magnitude fainter.


Similarly,


al a


1.,.,I


I S
n n t. n a a n. a a.. ..a 4. a n a I a a n n n -n r a4 a 1 I-i ___ U.rra .. I,






the degree of completeness is greater than 99.


history recorded in asteroid discovery circum,

of discoveries in the wake of World War II.


(Figure 3 is also interesting for the


Quite apparent is the marked lack


The large number of asteroids discovered


during the Palomar-Leiden Survey appears as a vertical stripe near


As pointed out above, between H


1960.)


= 10 and H = 11 the mainbelt appears to make


a transition to a linear, power-law size distribution.


An absolute magnitude of H


=11


corresponds to a diameter of about 30 km for an albedo of 0.1, approximately the mean


albedo of the larger asteroids in the mainbelt population (see


The Observed Mainbelt


Size Distribution).


Unfortunately, incompleteness rapidly sets in for H


11.5 and with


so few data points the slope of the distribution cannot be well defined so that we cannot

reliably use the data from the catalogued population alone to estimate the number of


very small asteroids min the mainbelt (see Figur

the Palomar-Leiden Survey (Van Houten et al.


down to about H


We have therefore used data from


1970) to extend the observed distribution


= 15.25, corresponding to a diameter of roughly


The MDS and PLS Surveys


Palomar-Leiden Survey


(Van Houten et al.


1970;


hereafter referred


to as


PLS) was conducted in 1960 to extend to fainter magnitudes the results of the earlier


McDonald Survey of 1950 through


1952 (Kuiper et al.


1958: hereafter referred to as


MDS).


MDS surveyed the entire ecliptic nearly twice around to a width of


down


to a


limiting photographic


magnitude of nearly


In contrast,


the practical


plate limit for the PLS survey was about five magnitudes fainter.


To survey and detect






7

prohibitive, so with the PLS it was decided that only a small patch of the ecliptic would

be surveyed, and the results scaled to the MDS and the entire ecliptic belt.

In 1984 a revision and small extension were made to the PLS (Van Houten et al.

1984), raising many quality 4 orbits to higher qualities, assigning orbits to some objects

which previously had to be rejected, and adding 170 new objects which were identified


on plates taken for purposes of photometric calibration.


Our original intention was to


use this extended data set to re-examine the size distribution of the smaller asteroids in

zones of the belt chosen to be more dynamically meaningful than the three zones used

in the MDS and PLS. However, we have decided not to embark on a re-analysis of the

PLS data at this time as the magnitude distribution of asteroids in the inner region of the

mainbelt was rather well defined in the original analysis, and we conclude that even the

extended data set will not significantly improve the statistics in the outer region of the


We therefore use the original PLS analysis of the absolute magnitude distribution


in three zones of the mainbelt,


with some caveats as described below.


In both the MDS and PLS analysis the mainbelt was divided into three semimajor


zones


- zone


I: 2.0


a < 2.6,


zone


a < 3.0,


zone


a < 3.5.


Within each zone the asteroids were


grouped in


half-magnitude intervals


of absolute photographic magnitude, g, and the numbers corrected for incompleteness

in the apparent magnitude cutoff and the inclination cutoff of the survey (see Kuiper et


al. 1958).


The g absolute magnitudes given by Van Houten et al. are in the standard B


band we transformed these absolute magnitudes to the H, G system by applying the


- -- rn ~ ~' 'a -a (an1 o ii OO TI, a~ 1tni c'* r.~ nrro,-'en*aI nwirv, ka<"r\ nF y, ttar^in Ac' nor


TIr








the PLS, as described by Van Houten et al.


The MDS values for the number of asteroids


per half-magnitude bin are assumed until the corrections for incompleteness approach


about 50% of the values themselves.


Where the


MDS values require correction for


incompleteness, a maximum and minimum number of asteroids is calculated based upon


two different extrapolations of the log N(mo) relation (Kuiper et al.


1958).


In these


cases the mean of the two values given in the MDS has been assumed.


The correction


factors for incompleteness in zone Il given in the MDS, however, are incorrect.


corrected values are given in Table D-I of Dohnanyi (1971).


For fainter values of H the


number of asteroids is taken from


Table


of Van Houten et al., the values given there


corrected by multiplying log N(H) by


1.38 to extend the counts to cover the asteroid


belt over all longitudes to match the coverage of the MDS.


Table


1 gives the adopted


bias-corrected number of asteroids per half-magnitude bin (H magnitudes) for each of

the three PLS zones and for the entire mainbelt as derived from the MDS and PLS data.

While the MDS, which surveyed the asteroid belt over all longitudes, is regarded


as complete


down


to an absolute


magnitude of


about g


= 9.5,


data need


to be corrected for completeness at all magnitudes as the survey covered only a few


percent of the area of the MDS.


There have been a number of discussions regarding


selection effects within the PLS and problems involved with linking up the MDS and


PLS data (cf. Kresik 1971 and Dohnanyi 1971).


We have taken a very simple approach


which indicates that the MDS and PLS data link up quite well and that any selection


effects within the PLS either cancel each other or are minor to begin with.


Figures 4,


C A^ ,^^ ^-/-. J. L ^ /^ Mk A N^ *J 1^ fi I/1 T 0 A^ a- Sk..q~l~ a-^ n,* *-. an h n. A^ nt* + n^ Sn *' i-^4 an j- a rT n n A








vertical line indicates the completeness limit for the


MDS.


beyond which correction


factors were adopted based on extrapolations of the observed trend of the number of


asteroids per mean opposition magnitude bin.


The solid vertical line indicates where


the PLS data have been adopted to extend the MDS distribution.


In each of the three


zones


completeness


limit for the


catalogued


population


roughly


coincides


the transition to the PLS data.


Beyond the completeness limit the observed number


catalogued


asteroids


per half-magnitude


bin continues to


increase (although at a


decreasing level of completeness) until the numbers fall markedly.


In each of the three


zones the data for the catalogued population merges quite smoothly with the PLS data.

This is particularly evident in zone II, where there is a significant decline in the number


of asteroids with H


11, right in the transition region between the incompleteness


corrected MDS data and the PLS data, producing an apparent discontinuity between the


two data sets.


The catalogued population, however, which is complete to about H = 11


in this zone, nicely follows the same trend, even showing the sharp upturn beyond the


completeness limit between H


= 11.25 and H


= 11.75.


With the catalogued population


making a smooth transition between the MDS and PLS data in each of the three zones,

we conclude that any selection effects which might exist within the PLS data are minor

and that there is no problem with combining the MDS data (roughly equivalent to the

current catalogued population) and PLS data as published.

The PLS Extension in Zones I, II, and Im


Having established that the PLS data may be directly used to extend our discussion




10

magnitude bin from the catalogued population for those bins brighter than the discovery


completeness limit and from either the PLS data or catalogued population,


whichever


is greater,


for the


magnitude


below the completeness


limit.


to sampling


statistics


there


a V


error


associated


each


independent


point


incremental


magnitude-frequency


diagram.


errors


catalogued


asteroid


counts are determined directly from


the raw


numbers after the asteroids


have


been


binned and counted.


For the PLS data the


errors must be determined from the


number of asteroids per magnitude interval before the counts have been corrected for


the apparent magnitude and inclination cutoffs.


The corrected counts themselves are


given


Table 5


of Van


Houten et al.


These counts are then scaled to


match


coverage of the MDS as described above. Since the errors in the PLS counts are based


on the uncorrected, unsealed counts, the PLS data points have a larger associated


error than the corrected counts themselves would indicate.


The resulting magnitude-


frequency diagrams for each of the PLS zones are shown in Figures


the numbers tabulated in


8, and 9 and


Table


The PLS data greatly extend the workable observed magnitude-frequency distrin-


butions for the mainbelt asteroids.


We immediately see that the inner two zones of the


mainbelt display a well defined, linear power-law distribution for the fainter asteroids,


with the prominent excess of asteroids at the brighter end of the distribution.

bution in the outer third of the belt appears somewhat less well defined. Thi


The distri-


e results for


the inner zones are very interesting, as the linear portions qualitatively match very well


Dohnanyi'


(1969,


1971) prediction of an equilibrium power-law distribution of frag-








through the MDS and PLS data, found a mass index of


1.839, in good agreement


with the theoretical expected value of q


= 1.837


quoted in his work.


His analysis,


however, was performed on the cumulative distribution of the combined data from the


three zones.


We feel that it is more appropriate to consider only incremental frequency


distributions since the data points are independent of one another and the limitations


of the data set are more readily apparent.


In this analysis we will also consider the


three zones independently to take advantage of any information that the distributions

may contain on the variation of the collisional evolution of the asteroids with location


the mainbelt.


Having assigned errors to the independent points in the incremental magnitude-


frequency


diagrams,


a weighted


least-squares


solution


can be


fit through


linear


portions of the distributions in each of the three


PLS zones.


We must be cautious,


however,


to work within the completeness


limits of the


data.


Figure


10 is a


histogram of the number of asteroids per half-magnitude interval as derived from the


data in


Tables


7 and 8 of Van Houten et al.


(1970).


These are the


1836 asteroids for


which orbits were able to be determined plus the 187 asteroids for which the computed


orbits had to be discarded.


The survey was complete to a mean photographic opposition


magnitude of approximately 19, beyond which the numbers would need to be corrected


for incompleteness.


Recognizing the uncertainties involved in trying to estimate the


degree of completeness for fainter asteroids on the photographic plates,


work within the completeness limits of the raw data set.


we prefer to


Given the completeness limit





12

mean semimajor axis for each of the zones we calculate the adjusted completeness limits


given in


Table 3.


Based on these more conservative completeness limits we may now


calculate the least squares solutions for the individual zones.

Zone I displays a distinctly linear distribution for absolute magnitudes fainter than


about H


= 11.


weighted least-squares


fit to


the data (H


= 11.25


fainter)


yields a slope of a = 0.469 0.011,


1.782 0.018 (Figure 11).


which corresponds to a mass-frequency slope of


(If we assume that all the asteroids in a semimajor axis


qz=

zone


have


the same mean albedo we may directly convert the magnitude-frequency


slope into the more commonly used mass frequency slope via q =


the slope of the magnitude-frequency data.


where a is


See Appendix B.) Zone II shows a similar,


though somewhat less distinct and shallower, linear trend beyond H


= 11.25.


A fit


through these data yields a slope of a = 0.479 0.012 (q = 1.799 0.020, Figure

In Zone III we obtain the solution a = 0.447 0.017 (q = 1.745 0.028, Figure


for magnitudes fainter than H


Dohnanyi equilibrium value of


= 10.75.


1.833.


These slopes are significantly lower than the


The weighted mean slope for the three zones


1.781 0.007, essentially equal to the well determined slope for zone I.


In addition


to the slope,


least-squares solution for each zones produces an


estimate for the intercept of the linear distribution,

number of asteroids in the population. With an esti


which is a measure of the absolute


mate of the mean albedo of asteroids


in the population, the expressions derived in Appendix B allow us to use the parameters

of the magnitude-frequency plots to quantify the size-frequency distributions for the

three zones and for the mainbelt as a whole.


*


1 + a,








The Observed Mainbelt Size Distribution


We may define the observed mainbelt size distribution that we will work with by

combining data from the catalogued population of asteroids and the least-squares fits


to the


PLS data.


The

absolute


sizes of the

brightnesses


numbered mainbelt asteroids may

if we can estimate a value for th


reconstructed


e albedo (See


from their


Appendix A).


Fortunately, an extensive set of albedos derived by IRAS is available for a great many


asteroids. A recent study by Matson et al.


(1990) demonstrates that asteroid diameters


derived using IRAS-derived albedos show no significant difference between those found


by occultation studies.


Although an even larger number of asteroids exists for which


no albedo measurements have been made, the IRAS data base is extensive enough to


allow a statistical reconstruction of their albedos.


without albedo estimates:


There are two subsets of asteroids


those for which a taxonomic classification is available, and,


larger


group,


those


which have


not been


typed.


have


used


taxonomic


types assigned by


Tedesco et al.


(1989) when available and by


Tholen (1989,


1993


private communication) if a classification based upon an IRAS-derived albedo was not

available. For those asteroids with a taxonomic type but no IRAS-observed albedo, we

have estimated the albedo by assuming the mean value of other asteroids with the same


classification.


If no taxonomic information was available we assumed an albedo equal


to that of the IRAS-observed asteroids at the same semimajor axis.


The diameters for


the catalogued asteroids, calculated using Equation 11 of Appendix A, are then collected







distribution


asteroids


smaller


completeness


limit


catalogued population has


been derived


using the


magnitude data described


previous


section.


Linear


least-squares


solutions,


constrained


to have


same


weighted mean slope of


q = 1.781, were fit through the linear portions of the magnitude


distributions in each of the three PLS zones.


The individual distributions were then


added to determine the intercept parameter (equivalent to the brightest asteroid in the


power-law distribution) for the mainbelt as a whole.


To convert the parameters of the


magnitude-frequency distribution determined using the PLS data into a size-frequency


distribution,


we assume that all


the asteroids in the population have the same mean


albedo.


Of the


well-observed asteroids


in the


mainbelt,


that is, asteroids with


IRAS-determined albedos and measured B-V colors, we found mean albedos of 0.121,


0.105, and 0.074 in PLS zones I, II, and II, respectively.


The weighted mean albedo


for the entire


mainbelt population is 0.097


. We chose to calculate the mean albedo


based on those asteroids with diameters between 30 and 200 km, in order to avoid any


possible selection effects which might affect the smallest and largest asteroids.


With an


estimate for the mean albedo the magnitude parameters may be converted directly into

a size-frequency distribution using Equations 6 and 15 of Appendix B.


In Figure


we have combined the data from the catalogued asteroids and the


PLS magnitude distributions to define the observed mainbelt size distribution.


Down


to approximately


30 km


the distribution is determined directly from


the catalogued


asteroids and IRAS-derived albedos.


The shaded band indicates the


error associated


with


the catalogued


population due


to sampling statistics.


For diameters








estimated from PLS data.


to smaller sizes.


asteroids.


We thus use the PLS data to extend the usable size distribution


The dashed line is the best fit through the magnitude data for the small


This size distribution is very well determined and will be used in the next


chapter to place strong constraints on collisional models of the asteroids.





16
Table 1: Numbers of asteroids in three PLS zones (MDS/PLS data).


Zone I


a<2.6


N(H)


Zone II


N(H)


Zone III
)

N(H)


I + II + III


N(H)


3.25 1 1 0 2
3.75 0 1 0 1
4.25 0 0 0 0
4.75 0 0 0 0
5.25 0 2 1 3
5.75 2 1 0 3
6.25 5 4 2 11
6.75 5 4 5 14
7.25 5 15 11 31
7.75 13 20 24 57
8.25 15 39 31 5


114.5


10.25
10.75
11.25
11.75
12.25
12.75
13.25
13.75
14.25
14.75
15.25
15.75
16.25


143.93
143.93
503.75
1007.51
2254.90
4125.99
6093.04
10914.69
17151.66


287.86
791.61
551.73
1103.46
2614.73
3958.07
7532.34
6788.70
12401.97


215.89
95.95
287.86
503.75
503.75
575.72
1727.16
4941.60
5109.51
6069.05
7868.17


219.5
329.89
219.45
477.36
918.61


1439.29
1271.38
3334.37
8563.84
11322.48
17727.38
20749.91







Table 2: Numbers of asteroids in three PLS zones (catalogued/PLS data).


325


3.25
3.75
4.25
4.75
5.25
5.75
6.25
6.75
7.25
7.75
8.25
8.75
9.25
9.75
10.25
10.75
11.25
11.75
12.25
12.75
13.25
13.75
14.25
14.75
15.25
15.75
16.25


N(H)


1007.51
2254.90
4125.99
6093.04
10914.69
17151.66


Zone II




N(H)


294
791.61
551.73
1103.46
2614.73
3958.07
7532.34
6788.70
12401.97


Zone III


N(H)


503.75
503.75
575.72
1727.16
4941.60
5109.51
6069.05
7868.17


I + II + III
.0 < a < 3.5


N(H)


938.7


1570.36
1642.45
3614.62
8563.84
11322.48
17727.38
20749.91


Zone I






Table


3: Adjusted completeness limits for PLS zones.


Semimajor Axis Zone Mean Semimajor Axis Completeness limit in H
(AU)
2.0 < a < 2.6 2.43 15.3
2.6 < a < 3.0 2.75 14.6
3.0 < a < 3.5 3.17 13.8










































































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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