Random Gravitational Encounters and the Evolution of Spherical Systems. I. Method

A modified Monte Carlo method is described for following the dynamical evolution of a spherical stellar system. In this method, designed for use with a digital coinputer, the trajectory of each star is computed numerically through time on the assumption that the gravitational potential is spherically symmetric. This procedure, used earlier by Campbell and Henon, becomes exact if each star is taken to represent a spherically symmetric shell composed of many stars, all characterized by the same values of the mass, of the distance r from the center, of the radial velocity v , and of the transverse velocity V6 however, the orientation of the orbital planes is assumed to be random for the stars at each location in the shell. To take into account irregular variations in the gravitational field, produced by encounters with passing stars, the velocity of each star is perturbed at intervals of time (At)p. The perturbations are chosen to give the appropriate diffusion coefficients for stars within each small range in r. To expedite the computations, the diffusion coefficients are simplified, with ((Avi))/v taken to be constant, with ((Avj)2) set equal to 2((AviI)2) and also assumed constant, and with these constant values set equal to the exact values of these coefficients for V2 equal to the meah square velocity. In the case of an isotropic velocity distribution in which all velocities are initially equal, this method of computing the effect of gravitational encounters is shown to be in satisfactory agreement with a full numerical solution of the Fokker-Planck equation. For reliable results, (At)p must not exceed about one-fifth of the relaxation time.