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Figure 3.—Cumulative number N of the PLS asteroids with a least-squares fit to N.
where N(m) is the cumulative number of asteroids having masses of magnitude m or greater and A is a constant. A least-squares fit to the data of equation (4) gives
which, in view of uncertainties, can be regarded as identical to the theoretical result (eq. (3) of a = 1.837 obtained in Dohnanyi (1969) and found to represent very well the MDS results. (If the five objects too bright for measurement in the iris photometer employed by PLS are included, one obtains a = 1.815; i.e., an insignificant difference of 0.024 for magnitudes g P 11.) Kessler (1969) has studied the joint distribution of magnitudes, radial distance from the Sun, and heliocentric longitudes of the cataloged asteroids. It appears, from his results, that equations (4) and (5) are good representations of his overall results (NASA SP-8038, 1970). Recent work by Roosen (1970) indicates that the counterglow may be caused, almost entirely, by particles in the asteroid belt. We may therefore have direct evidence that the distribution of minor planets extends to the size range of micrometeoroids. (See Dohnanyi, 1971.) We shall, in the remainder of this paper, discuss the manner in which power-law distributions of the types in equations (3), (4), and (5) arise.
Mean Impact Velocity
When two asteroidal objects collide, the damage done to the colliding bodies depends on, besides other factors, the magnitude of the relative velocity of the two colliding objects. A statistical treatment of asteroidal collisions should, therefore, include the velocity distribution function as well as the mass distribution of the colliding masses. We shall, however, confine our attention to the influence of collisions on the mass distribution, using a mean encounter velocity. Such a simplified approach leads to a model that is mathematically tractable, as we shall see later. An alternate approach, in which the velocity distribution is modeled using Monte Carlo techniques but using an assumed mass distribution, has been given elsewhere. (See Wetherill, 1967, for a review and references.) Consider two asteroidal objects with masses M1 and M2. Using a simple molecules-in-a-box approach, kinetic theory tells us that the expected number of times these two objects collide per unit time is
where R1 and R2 are the effective radii of the two objects, V is the mean encounter velocity, and Vo is the effective volume of the asteroid belt.
Using the distribution of the inclinations and eccentricities for known asteroids, I have estimated (Dohnanyi, 1969) the rms encounter velocity with the estimated dispersion as
in agreement with Piotrowski’s (1953) estimate of 5 km/s. The distribution of encounter velocities appears to be rather broad and individual encounter velocities may vary considerably as suggested by equation (7).
Collisions at impact velocities of several kilometers per second are inelastic and result in fragmentation. Gault et al. (1963) have fired projectiles into effectively semi-infinite basalt targets at very high velocities over a range not exceeding 10 km/s and over a range of projectile kinetic energies from 10 to 10° J. The result of the impact was the production of a crater and the ejection of crushed material. The total ejected mass Me was found to be proportional to the projectile kinetic energy and the size distribution of the ejecta could be approximated by a power-law distribution.
We, therefore, choose (Dohnanyi, 1969) a comminution law of the form
where g(m; M1, M2) dm is the number of fragments in the mass range m to m + dm created when a projectile object M1 strikes a larger target object of mass M2. The factor C(M1, M2) is a function of the colliding masses and n is a constant,
for semi-infinite targets. (See Hartmann, 1969, for a survey.) Using the fact that mass is conserved during impact, it is readily shown that
where Me is the total ejected mass and M, is the upper limit to the mass of the largest fragment.
Erosive and Catastrophic Collisions
We shall presently distinguish between two different types of collisions depending on the mass M1 of the projectile compared with the mass M2 of the target. For
the target mass is effectively infinite and Gault's (Gault et al., 1963) results apply. These collisions we shall denote as erosive; clearly, during erosive collisions the projectile craters out a relatively minor amount of mass, leaving the large target mass otherwise intact. For these collisions, Me is proportional to the projectile mass M1 (Gault et al., 1963) and we write (Dohnanyi, 1969), for basalt targets,
with the impact speed v expressed in kilometers per second. (See Marcus, 1969, for a detailed discussion.) The upper limit to the mass of the largest fragment is given by
If the target mass M2 is not effectively infinite, then some projectile masses will be sufficiently large to catastrophically disrupt the target. Not much is known about the precise relationship between the target mass M2 and the smallest projectile mass M1 necessary for catastrophic disruption of M2 or about the precise nature of the catastrophic failure mode of colliding objects with arbitrary sizes, shapes, and physical composition.
Experiments (Moore and Gault, 1965) with basalt targets conducted at relatively low impact velocities in the range of 1.4 to 2 km/s imply that a target mass M2 about 50T times the projectile mass or smaller will be catastrophically disrupted. The failure mode of the spherical target consists in the separation of a spherical shell of debris leaving an approximately spherical core behind as the largest fragment.
More recent experiments (Gault and Wedekind, 1969) on finite glass targets indicate a failure mode in which, in addition to a crater having a size determined by equation (12) for semi-infinite targets, a spall fragment on the surface of the spherical target opposite the point of impact will be produced. Both glass and basalt targets are seen to have comparable failure modes; the difference is that the basalt target fails by the production of a spall engulfing most of the spherical surface of a spherical target M2, whereas the glass sphere target fails by the formation of a spall opposite the impact.
In both cases the distribution of fragments can be represented reasonably well by a formula of the form of equation (8). The total ejected mass is now given by
for catastrophic collisions, and the largest target mass M2 catastrophically disrupted by Mi will be taken as M2 = T'M1. Thus,
M2 < T'Mi (15) for catastrophic collisions, and M2 > T'Mi (16)
for erosive collisions.
The quantity T' is difficult to estimate precisely; combining results by Gault et al. (1963), Moore and Gault (1965), and Gault and Wedekind (1969), we may write
The large difference in these numbers is due mainly to the differences in the catastrophic failure modes between basalt and glass. Less energy is needed to detach a spall from a glass sphere than to detach a spherical shell of fragments from a basalt sphere.
The limit of the mass of the largest fragment for catastrophic collisions can be taken as
This formula is an idealization because for catastrophic collisions the size of the largest fragment should be approximately inversely proportional to the collisional kinetic energy. This relation defines the expected size of the largest fragment during an average catastrophic collision. For a more detailed definition of Mt we can take
where Mb is inversely proportional to M1 and No is a constant. The main effect of this refinement on the subsequent analysis (unpublished) is to add further detail without, however, altering the main conclusions. We therefore choose to retain the mathematically simpler but physically less correct definition of Mb (eq. (18)). Collecting formulas, we have
Collisions between asteroids must undoubtedly affect their mass distribution. To gain insight into this problem, we give a precise mathematical model of the evaluation of the asteroidal mass distribution under the influence of mutual inelastic collisions.
Let f(m, t) dm be the number density per unit volume of asteroids in the mass range m to m + dm at time t. Clearly, f(m, t) dm will change in time because of (1) erosion, (2) removal by catastrophic collisions of objects in this mass range, and (3) creation of fragments into this mass range by the erosive or catastrophic collisions of larger objects.
Assuming a uniform spatial distribution throughout the asteroid belt, one can write a continuity equation for the number density f(m, t):
of (m, t) of of dm = dron – + dm – Ot Ot erosion 0t catastrophic collisions of of +dm — + dron — (21) 0t catastrophic creation Ot erosive creation