Random Gravitational Encounters and the Evolution of Spherical Systems. III. Halo

Accumulation of stars in the halo of a spherical stellar system, together with escape of stars from the system, result from stellar encounters within a dense central core; the resultant halo structure is studied both analytically and numerically for an isolated system. On the assumption that conditions in the central core remain constant with time, a simple steady-state solution is obtained for the outermost halo, or "fringe," defined as the region from which stars may escape after a few additional orbits through the central core of the system. In this steady-state solution the density in phase space,J(E, J), obtained fmm a simple integral equation for the fringe region, is found to vary nearly linearly with E, for E less than the energy of escape E E and J are the energy and orbital angular momentum of a star per unit mass. The flux of stars into the fringe, which about equals the rate of escape from the system, is regarded as an arbitrary constant, Fj, for stars of each J-value; Fj is determined by the evolution of the system as a whole rather than by conditions within the halo. In the particular case of an isolated system, where the escape energy is zero, this steady state cannot be reached for E arbitrarily close to zero since the total number of the stars in the halo would become infinite, but is reached more and more closely as N, the number of stars in the system, increases. The density distribution in this steady state, which should be very nearly reached in practice for N sufficiently large, varies as ' out to the radius r where the escaping stars give a density varying as 1/r2. Ionte Carlo runs for somewhat more realistic but isolated clusters confirm that the simple steady-state solution gives a reasonable approximation to the general structure of the halo.