Size Distribution of Collisionally Evolved Asteroidal Populations: Analytical Solution for Self-Similar Collision Cascades

Abstract

Collision cascades play a prominent role in processing the present asteroid belt, as well as formative planetary systems. As a first step toward understanding how this mechanism operates, an analytical solution has been obtained for the steady-state size distribution of a self-similar collisional fragmentation cascade. It is found that this corresponds to a power law for the differential mass distribution dn = Cm-αdm , where the exponent α is very nearly 11/6, equivalent to a differential radius distribution dn = C *r-3.5dr. This is in agreement with the earlier conclusion of J. S. Dohnanyi (1969, J. Geophys. Res. 74, 2531-2554). The work of Dohnanyi has been extended considerably to include a full treatment of the simultaneous occurrence of cratering and catastrophic fragmentation. A more physically realistic model of catastrophic fragmentation is used in which the size of the largest fragment decreases with projectile mass. The average steady-state value of the exponent α (1.8333) is found to be extremely insensitive to the assumed physical parameters of the fragmentation process, such as the strength of the target, the threshold for catastrophic fragmentation, the relative contribution of cratering and catastrophic fragmentation, the size and number of the largest fragments, and the size distribution of the fragments produced by individual fragmentation events. The robust nature of the result is a consequence of the steady-state solution being primarily the consequence of geometrical rather than mechanical considerations.