Statistical Approach to the Three-Body Problem


Paper:	Statistical Approach to the Three-Body Problem
Volume:	316, Order and Chaos in Stellar and Planetary Systems
Page:	45
Authors:	Valtonen, M.; Myllari, A.; Orlov, V.; Rubinov, A.
Abstract:	Strongly bound triple systems often evolve in a way that can be described as a deterministic chaos. Therefore the most useful description of the final state of the three-body evolution in many situations is purely statistical. One may assume that the phase space of initial configurations is uniformly covered in some sense, and that in the end of the dynamical evolution, when the triple has broken up into a binary and a single escaping body, the systems are again uniformly distributed in the allowable phase space. In this paper we study the validity of this assumption. Dynamical evolution of 100,000 equal-mass triple systems is investigated. The systems possess net angular momentum which is quantified by the parameter w = -L₀²E₀/G²m₀⁵, where G is the gravitational constant, m₀ is the mass of a body, and L₀ and E₀ are the angular momentum and the total energy of the triple system. We consider the values of w = 0.1, 1, 2, 4, 6 which covers the range of angular momenta for strongly interacting, initially bound systems. For each w, 20,000 triple systems are studied. The initial coordinates and velocities of the components are chosen in two different ways: the first one assumes a hierarchical structure initially, the second one does not. The evolution of each triple system is calculated until either the escape of one of the bodies occurs or the time exceeds 1000 mean crossing times of the system. The statistical escape theory is based on the assumption of ergodicity, i.e., that the only information on the initial conditions remaining at the time of the escape of the third body is contained in the conserved total energy, total angular momentum and the mass values. The distributions of various quantities are derived from the allowable phase space volumes. We consider as an example the distributions of escape angle predicted by theory and found from numerical simulations. Those are in agreement. The escape directions are preferentially perpendicular to the total angular momentum vector, the more so the greater is the angular momentum.