Dynamics of rotating triple systems: statistical escape theory versus numerical simulations

Abstract

Dynamical evolution of 120 000 equal-mass rotating triple systems is investigated. The system rotation is described by the parameter w = -L 2 0 E 0 /G 2 m 5 0 , where G is the gravitational constant, m 0 is the mass of a body, and L 0 and E 0 are the angular momentum and the total energy of the triple system, respectively. We consider the values of w = 0.005, 0.1, 1, 2, 4 and 6. For each w, 20 000 triple systems are studied. The initial coordinates and velocities of the components are randomly chosen. The initial data are chosen in two different ways: the first one assumes a hierarchical structure initially and 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 orbital parameters of the final binary and the escaper are recorded for each run. We compare the results of numerical simulations and predictions of a statistical escape theory. The statistical theory is based on the assumption of ergodicity, that is, 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. The distribution of binary energy agrees with earlier results by Heggie, with the angular momentum dependence of Mikkola & Valtonen being added. The eccentricities are distributed in general accordance with Monaghan's work, while the triple systems break up like in radioactive decay, as was previously found by Valtonen & Aarseth. The escape directions are preferentially perpendicular to the total angular momentum vector; the more so, the greater the angular momentum. The escape-angle distributions are derived from the statistical theory and are found to be in agreement with the numerical data. The relative orientations and magnitudes of the binary and third-body angular momenta are also explained based on the statistical theory.