Rushing to equilibrium: a simple model for the collisional evolution of asteroids

A simple mathematical model for the evolution of a system of collisionally interacting bodies—such as the asteroid population—consists of two coupled, nonlinear, first-order differential equations for the abundances of “small” and “big” bodies. The model easily allows us to recover Dohnanyi's value for the exponent of the equilibrium mass distribution. Moreover, the model shows that any initial value for the ratio of “big” to “small” bodies rapidly relaxes to the equilibrium ratio, corresponding to the exponent, and that integrating the evolution equations backward in time—an attractive possibility to investigate the mass distribution of primordial planetesimals—leads to strong numerical instability.