Random Gravitational Encounters and the Evolution of Spherical Systems. II. Models

The evolution of several model spherical star systems is followed numerically in time, using a new method for computing the velocity perturbations produced by stellar encounters. The integrations extend over 10-tOO relaxation times at the center of the system. In every case a contracting, nearly isothermal, core of high star density develops at the center of the system, while stars in elongated orbits accumulate in a surrounding extended halo; the contraction as a whole is far from homologous. These results show clearly how the tendency to establish a Maxwellian velocity distribution produces a gradual evolution through two effects: (a) the stars diffuse in space toward xero energy, populating the halo in elongated orbits with some escaping from the system altogether, and (b) stars heavier than the average lose energy to the lighter stars and fall toward the center of the system. In the model systems with a single mass component, the loss of energy to escaping stars is comparable with the loss of energy to those which accumulate in the halo; the relative importance of escaping stars would presumably be even greater if the tidal force of the Galaxy, which sets an upper limit on the apocenter of a bound stellar orbit, were considered, or if the occurrence of relatively close encounters, with large resultant velocity changes, were taken into . The particular initial conditions assumed might also modify this ratio. In systems with several mass components the contraction of the core of heavier stars is in rough agreement with simple theory. If 10 percent of the system's mass is in stars with S times the mass of the lighter stars, the density in the central core increases to about 50 Ph, where Ph is the mean density for the inner half of the system's mass, during a time interval equal to , where 4i is the dynamical relaxation time at the density Ph. Calculations have also been carried through for the mass distribution anticipated in old stellar systems such as globular clusters, in which stars of 0.625, 0.25, and 0.1 Alo represent one-half, one-third, and onesixth the total mass, respectively; in this model the central density increases to 150 Ph in 2 t,h, and the lighter stars move outward. Since t? h equals about 8 X 10 years for a typical cluster, pronounced mass stratification should be present for the lower-mass stars in at least some of these systems. The structure and evolution of all these models depend only on M4 h, with no strong dependence on N, the total number of stars in the system, over the limited range of values, between about 10' and I0 , assumed in the calculations.