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