Statistics of impacts among orbiting bodies: a Monte Carlo approach

Abstract

In this paper, we describe a method to investigate the statistics of collisions among bodies orbiting around a common central mass but not interacting with each other, like minor bodies of the Solar system. This method can be used to derive the frequency of collisions and the distribution of any dynamical parameter related to the collision circumstances. It is based on deriving the statistics of impacts from a random sampling of the orbital element space of the bodies under investigation. The fundamental approach is not new, but the mathematical framework is completely original and the final procedure is very simple and easily implementable. The motivation behind this work is to overcome all limitations due to the assumptions about the dynamical behaviour of orbits, which the methods developed so far are based on. We show all the theoretical details of the method and its practical usage, including the determination of error. We show a set of examples demonstrating the satisfactory agreement with independent approaches, when they are available. The limits and drawbacks of the methodology are also highlighted.