Non-Power Law Behavior in Fragmentation Cascades

TL;DR

This paper investigates deviations from the traditional power law in particle mass distributions resulting from fragmentation cascades, deriving analytic corrections and revealing persistent wave patterns in the distribution.

Contribution

It introduces a modified steady-state solution accounting for non-proportional largest fragment masses and uncovers wave phenomena in mass distributions independent of cutoffs.

Findings

Correction factor significantly affects extrapolations over many mass orders

Analytic solutions for the correction factor are derived and validated numerically

Waves can persist in mass distributions without cutoffs or strength law breaks

Abstract

Collisions resulting in fragmentation are important in shaping the mass spectrum of minor bodies in the asteroid belt, the Kuiper belt, and debris disks. Models of fragmentation cascades typically find that in steady-state, the solution for the particle mass distribution is a power law in the mass. However, previous studies have typically assumed that the mass of the largest fragment produced in a collision with just enough energy to shatter the target and disperse half its mass to infinity is directly proportional to the target mass. We show that if this assumption is not satisfied, then the power law solution for the steady-state particle mass distribution is modified by a multiplicative factor, which is a slowly varying function of the mass. We derive analytic solutions for this correction factor and confirm our results numerically. We find that this correction factor proves important when extrapolating over many orders of magnitude in mass, such as when inferring the number of large objects in a system based on infrared observations. In the course of our work, we have also discovered an unrelated type of non-power law behavior: waves can persist in the mass distribution of objects even in the absence of upper or lower cutoffs to the mass distribution or breaks in the strength law.