The Basic Escape Rate of Stars from Clusters.

Abstract

In the escape of stars from an isolated cluster the basic mechanism is the relaxation of a velocity distribution as a result of stellar encounters. All estimates of the rate of escape of stars therefore contain a multiplicative factor that expresses this relaxation rate. Short of solving the entire problem of the dynamics of an isolated cluster, the best method of estimating this basic escape rate is to solve an idealized problem that represents the relaxation process without any further dynamical complications. This idealized problem is to find a velocity distribution in which encounters maintain a steady state, subject to a cutoff at the escape velocity. As stars disappear at the cutoff, the total number of stars decreases while the shape of the velocity distribution remains unchanged. The problem has been solved previously under the assumption that the stars under consideration encounter another group of stars having a Gaussian velocity distribution. The present paper removes this restriction and treats the case of a group of stars encountering each other, so that the unknown velocity distribution in the Fokker-Planck equation appears not only in the usual derivatives but also in the encounter integrals. The usual separation-of-variables technique can be used, but a scale change in the velocities must be included, in order to conserve total energy as high-speed stars disappear across the cutoff. The resulting solution gives an escape rate about 50% higher than that recently found by Spitzer and Ha~rm. The escape rate depends somewhat on the way in which energy is fed into the velocity distribution by processes such as contraction of the cluster. % higher than that recently found by Spitzer and Ha~rm. The escape rate depends somewhat on the way in which energy is fed into the velocity distribution by processes such as contraction of the cluster. % higher than that recently found by Spitzer and Ha~rm. The escape rate depends somewhat on the way in which energy is fed into the velocity distribution by processes such as contraction of the cluster. % higher than that recently found by Spitzer and Ha~rm. The escape rate depends somewhat on the way in which energy is fed into the velocity distribution by processes such as contraction of the cluster.