Steady-State Size Distributions for Collisional Populations: Analytical Solution with Size-Dependent Strength

TL;DR

This paper derives an analytical expression for the size distribution of collisional populations considering size-dependent material strength, extending to cases with different strength laws for small and large bodies, and validates with simulations.

Contribution

It provides a simple analytical solution for the size distribution power-law index accounting for size-dependent strength and transitions between different strength regimes.

Findings

Analytical expression for size distribution index with size-dependent strength.

Transition effects cause a wave in the size distribution.

Results match numerical simulations well.

Abstract

The steady-state population of bodies resulting from a collisional cascade depends on how material strength varies with size. We find a simple expression for the power-law index of the population, given a power law that describes how material strength varies with size. This result is extended to the case relevant for the asteroid belt and Kuiper belt, in which the material strength is described by 2 separate power laws--one for small bodies and one for larger bodies. We find that the power-law index of the small body population is unaffected by the strength law for the large bodies, and vice versa. Simple analytical expressions describe a wave that is superimposed on the large body population because of the transition between the two power laws describing the strength. These analytical results yield excellent agreement with a numerical simulation of collisional evolution. These results will help to interpret observations of the asteroids and KBOs, and constrain the strength properties of those objects.