Three-body scattering near triple collision or expansion

Abstract

This article studies the statistics of three-body scattering in the proximity of triple collision and triple expansion. It begins with the one-dimensional collinear problem, looking at the orbits close to triple collision/expansion on the McGehee manifold (McGehee 1974). The time spent in the neighbourhood of triple collision/expansion determines the energy of the resulting binary, and asymptotic predictions can be made about the energy distribution. Similar results can be obtained for the probability of ionization. These results can also be found by examining the Siegel exponents (e.g. Siegel & Moser 1971) for motion in the vicinity of triple collision/expansion. They are illustrated by a series of numerical integrations. The study is then extended to two and three dimensions and unequal masses. There are two possible configurations close to triple collision/expansion: the equilateral and the collinear configurations. As in one dimension, a series of analytic predictions are made, although the computation of the Siegel exponents for the collinear configuration must be carried out numerically. The results are compared with numerical data that has already been presented in papers by Szebehely (1974), Alexander (1986) and Hut & Bahcall (1983).