Statistical properties of encounters among asteroids: a new, general purpose, formalism

The statistical properties of asteroid mutual encounters have been studied by several authors, with the main purpose of estimating collisional rates (and thus mean collisional lifetimes) and the distribution of encounter velocities. In this paper we present a new approach, conceptually not really different with respect to the classical ones, but implemented with a rather different mathematical formalism and consequently more flexible. When a comparison is possible our results are very similar to those obtained by means of other techniques. We exploited the peculiar flexible features of the present formalism to study-in a quantitative way-what happens when special dynamical conditions occur, such as a clustering of longitudes of perihelia (as in the so-called Kresak effect) or of the longitudes of the sample around the longitude (variable in time) of Jupiter, as in the case of Trojans. These dynamical situations have never been explored in the past using statistical approaches, and the development of the present one can avoid the use of heavy N-body integrations. Concerning the Trojan asteroids, the results of our analysis, although discussed here in a simplified version, are satisfactorily compared with those emerging from a detailed numerical integration of the orbits (Marzari et al., 1996, Icarus 119, 192-201). Finally, we used our approach to analyze the statistical properties of impacts among very large samples of objects with a moderate amount of computer time, thanks to the numerical algorithm, based on a Monte Carlo technique of integration. We have tested this numerical procedure by comparing our results with previous ones published in the literature; we find an amazing agreement with the more standard and refined numerical methods.