THE COLLISIONAL EVOLUTION OF THE
ASTEROID BELT AND ITS CONTRIBUTION TO
THE ZODIACAL CLOUD
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.
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
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.
also very grateful to Bo Gustafson and Yu-Lin Xu for the many discussions and helpful
advice through the
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-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.
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
TABLE OF CONTENTS
LIST OF TABLES .
LIST OF FIGURES .
ABSTRACT. .. .. ..
S S S S S S S S 1 iii
. . 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
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 .
Description of the Self-consistent Collisional Model
. S S 3
Verification of the Collisional Model .
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 .
* S S S S 85
* S S S S S S S S 5 85 ^
Number of Families.
Evolution of Individual Families
IRAS AND THE ASTEROIDAL CONTRIBUTION TO THE ZODIACAL
C L.JOlU Dl\ . a *
* S S 1 9
, S S 3
S S S S S S 5 3
The Ratio of Family to Non-Family Dust
6 SUMMARY .
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
* 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
. S S 5 12 1
. .*. .. 123
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
. *. 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
Magnitude-frequency distribution for catalogued mainbelt asteroids.
Absolute magnitude as a function of discovery date for all catalogued
. p a p a a a 2 1
Magnitude-frequency distribution for PLS zone I: PLS and catalogued
Magnitude-frequency distribution for PLS zone II: PLS and catalogued
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:
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.
. 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
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.
and small particle cutoffs used in our model.
Wave-like deviations imposed by a sharp particle cutoff
Size distribution resulting from gradual particle cutoff matching the
observed interplanetary dust flux (x
Collisional relaxation of a perturbation to an equilibrium size
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.
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
The evolved size distribution after 4.5
billion years using our modified
scaling law for (a) a massive initial population and (b) a small initial
. 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.
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
. Comparisons with model profiles based on
prominent Hirayama families are shown in (b), (c), and (d).
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
DANIEL DAVID DURDA
Stanley F. Dermott
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
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
which must occur during the course of collisional evolution.
ratio of the area associated
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.
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
1976). However, it has been known for some time that inter-asteroid collisions are likely
to occur over geologic time (Piotrowski
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.
contribution made by asteroids to the interplanetary dust complex has received renewed
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
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.
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.
initial mass of the
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
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
derive the size distribution of mainbelt asteroids down to ~,5 km diameter.
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.
use the model to extend
our assumptions beyond those of previous works and to shed some light on the impact
mass of the
The collisional history
providing further constraints
.- 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..
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.
Description of the Catalogued Population of Asteroids
The size-frequency distribution of the asteroids is very important in constraining
to affect the
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
we must use
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.
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.
Immediately evident is a
, or excess,
Although previous researchers have interpreted this excess as a remnant of
evolution (Hartmann and Hartmann 1968), the current interpretation is that it represents
the preferential preservation of larger asteroids effectively strengthened by gravitational
(Davis et al.
Holsapple and Housen
primarily Dohnanyi (1969, 1971), have noted from surveys of faint asteroids (discussed
is well described
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
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
no asteroids brighter than
= 7 have been discovered since
1940 the completeness limit was a magnitude fainter.
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
= 10 and H = 11 the mainbelt appears to make
a transition to a linear, power-law size distribution.
An absolute magnitude of H
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
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
(Van Houten et al.
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 surveyed the entire ecliptic nearly twice around to a width of
magnitude of nearly
plate limit for the PLS survey was about five magnitudes fainter.
To survey and detect
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
a < 2.6,
a < 3.0,
a < 3.5.
Within each zone the asteroids were
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
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
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.
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.
cases the mean of the two values given in the MDS has been assumed.
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
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.
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
to an absolute
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.
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
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
limit for the
the transition to the PLS data.
Beyond the completeness limit the observed number
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
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
magnitude bin from the catalogued population for those bins brighter than the discovery
completeness limit and from either the PLS data or catalogued population,
below the completeness
counts are determined directly from
numbers after the asteroids
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
Houten et al.
These counts are then scaled to
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
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
e results for
the inner zones are very interesting, as the linear portions qualitatively match very well
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
quoted in his work.
however, was performed on the cumulative distribution of the combined data from the
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
Having assigned errors to the independent points in the incremental magnitude-
portions of the distributions in each of the three
We must be cautious,
to work within the completeness
limits of the
10 is a
histogram of the number of asteroids per half-magnitude interval as derived from the
7 and 8 of Van Houten et al.
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
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
mean semimajor axis for each of the zones we calculate the adjusted completeness limits
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
the data (H
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
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
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
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.
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
sizes of the
numbered mainbelt asteroids may
if we can estimate a value for th
e albedo (See
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,
types assigned by
Tedesco et al.
(1989) when available and by
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
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
catalogued population has
magnitude data described
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
we assume that all
the asteroids in the population have the same mean
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.
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.
we have combined the data from the catalogued asteroids and the
PLS magnitude distributions to define the observed mainbelt size distribution.
the distribution is determined directly from
asteroids and IRAS-derived albedos.
The shaded band indicates the
to sampling statistics.
estimated from PLS data.
to smaller sizes.
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.
Table 1: Numbers of asteroids in three PLS zones (MDS/PLS data).
I + II + III
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
Table 2: Numbers of asteroids in three PLS zones (catalogued/PLS data).
I + II + III
.0 < a < 3.5
3: Adjusted completeness limits for PLS zones.
Semimajor Axis Zone Mean Semimajor Axis Completeness limit in H
2.0 < a < 2.6 2.43 15.3
2.6 < a < 3.0 2.75 14.6
3.0 < a < 3.5 3.17 13.8
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THE COLLISIONAL MODEL
Before describing the details
of the collisional model developed in
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
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
c m -dm,
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
The equilibrium slope index q was found to be insensitive to the fragmentation
power law 77 of the colliding bodies, provided that y
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
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.
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
there were discontinuities in the plot of the slope as a function of time at the times of
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
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
(strain-rate) dependent impact
strengths (Davis et al.
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
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
the apparent dearth
used to investigate the collisional history of asteroid families (Davis and Marzari 1993).
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.
= 1.833 and demonstrate that this value
parameters of the
Dohnanyi had thought.
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)
more realistically assumed that the
mass of the
largest fragment resulting
from a catastrophic fragmentation decreases with increasing projectile mass.
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
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
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.
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).
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.. ................................................................................................................................
total number of particles in
the asteroid belt is
The population of
was chosen as a subset
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.
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.
elements were taken to be the proper elements as computed by Milani and Knezevi6
which are more representative of the long-term orbital elements than are the
The resulting intrinsic collision rates and mean relative encounter
speeds for several representative mainbelt asteroids are given in
rate and relative encounter speed calculated
bias-free set are 2.668
x 10-18 yr1 km-2
and 5.88 km s1
672 asteroids of
collision probability for a finite-sized asteroid with diameter D is
P1 = 4'I,
/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.
D ,0,ta .1
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,
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
the energy needed
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
, or an impact strength,
. 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
from laboratory experiments to asteroid-sized bodies are reviewed by Fujiwara et al.
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
Other researchers (Farinella et al.
1982; Holsapple and Housen 1986; Housen
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
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
we take the binding energy to be that of a spherical shell of mass 1M
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,
is the energy required to disperse one half the mass of the target asteroid to infinity
projectile kinetic energy partitioned into kinetic energy of the members of the family
most consistent with
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
projectile is imparted into the kinetic energy of the
comminutional energy for these fragments amounts to some 0.1% of the impact energy.
We shall take
flE to be a parameter which may assume values of from a few to few
kinetic energy needed
fragment and disperse a target asteroid of mass M
and diameter D
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
rm i n V2
= -PD mVe2
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,
is usually the collision lifetime for bin
1. In most cases we have let
. During a single timestep the number of particles removed from bin i
is then found from the expression
with the stipulation that only an integer number of particles are allowed to be destroyed
z is rounded
small size bins this procedure gives the same results as calculated directly by Equation
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.
distributed into smaller size bins following a power-law size distribution given by
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
so that the total mass of debris equals the mass of the parent asteroid (Greenberg and
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.
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
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.
power-law was set equal
equilibrium value of q
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
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
increased to 0.2.
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
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
(1-100 meters) of a ~-60 bin model. In Figures 19, 20, and 21 the slopes of the initial
size distributions are
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
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.
found the same behavior in his numerical solution of the fragmentation equation.
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
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--- -- -
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
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.
this series of
runs that our numerical
reproduces the results of Dohnanyi
independent impact strengths
our model produces evolved power-law size distributions with slopes essentially equal
the computer code
assumptions concerning the colliding asteroids.
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.
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.
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.
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
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.
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).
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
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
to -4.0 (Belton et al.
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
< 100 meters.
The slope of the crater distribution on Gaspra is also
consistent with the crater distribution observed in the lunar maria (Shoemaker
and the size distribution of small Earth-approaching asteroids discovered by Spacewatch
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,
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).
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.
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.
a was set equal to 1.9
for this model to produce a
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 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
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.
this population as our initial distribution, the collisional model was run for 500 million
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,
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.
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.
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
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
(1969) result that the size distribution of asteroids
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
gravitational compression lead to size-dependent impact strengths, with both increasing
and decreasing strengths
(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
In order to examine the effects of size-dependent impact strengths on the equi-
librium slope of the
we created a number of hypothetical
size-strength scaling laws.
As will be discussed in the following section,
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
dependent strengths such
(1990) nominal case,
the equilibrium slope has a higher value of q
the other hand,
is positive, an equilibrium slope
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
This could apply to the members of an individual family of a unique taxonomic
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
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:
has the same arbitrary strength scaling law as the background population of
< 0 in this case),
while the scaling law for family
2 has g'
significantly different than that of family
or the background population, even though
which are solely responsible
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
(1989) for a review of strength scaling laws).
attempts have been
to determine the strength scaling laws
first principles either analytically
(Housen and Holsapple
1990) or numerically through hydrocode studies (Ryan
However, we have taken a different approach of using the numerical collisional model
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
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
various collisional models
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.
over the volume of the asteroid we have for the effective impact strength
is the material impact strength,
p is the density,
is the diameter.
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,-
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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.
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.
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.
Under their nominal rate-dependent model the constant
in the strength regime,
where gravitational self
results of the
the parent bodies of the Koronis, Eos, and Themis asteroid families (open dots).
most recent studies,
indicate that the laboratory results are to be taken as
upper limits to the magnitude of the gravitational compression (Holsapple 1993, private
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
the initial mass of the belt is not known.
we assume the same definition as used by
Davis et al.
the mass at
the time the mean collision speed first reached the current
. 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
extremes for an initial belt mass:
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.
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
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
1-10 km-scale asteroids,
leading to a pronounced
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
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
greatest influence on
the shape of the evolved size distribution,
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.
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.
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.^ .- ^-
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.
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
After 4.5 billion years of collisional evolution we fit the "hump"
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
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
bodies in a qualitative manner.
Recent hydrocode simulations by Nolan et al.
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
strengths due to the poor conduction of tensile stress waves in the
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
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HIRAYAMA ASTEROID FAMILIES
A Brief History of Asteroid Families
The Hirayama asteroid families represent natural experiments in asteroid collisional
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),
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,
first studies, classifications of asteroids into families have been
given by many other researchers (Brouwer 1951; Arnold 1969; Lindblad and Southworth
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
families which were previously unseen), the different perturbation theories which are
used to calculate the proper elements, and the different methods used to distinguish
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
Only in the last few years have different methods lead to a convergence
in the families identified by different researchers (Zappala and Cellino
The Zappala Classification
To date, probably the most reliable and complete classification of Hirayama family
a set of
numbered asteroids whose proper elements were calculated using a second-order (in
perturbation theory (Milani and Kne2evid 1990) and checked for long-term stability by
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
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.
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
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
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
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.
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
, 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
with asteroids of
the background population, requiring larger projectiles for fragmentation.
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.
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.
In the region 2.501
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) = 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,
Zappal& et al.
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
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
for a parent of this size is approximately