Escape in Hill's Problem

Abstract

This didactic paper is motivated by the problem of understanding how stars escape from globular star clusters. One formulation of this problem is known, in dynamical astronomy, as Hill's problem. Originally intended as a model for the motion of the moon around the earth with perturbations by the sun, with simple modifications it also serves as a model for the motion of a star in a star cluster with perturbations by the galaxy. The paper includes introductory sections on the derivation of the equations of motion of Hill's problem, their elementary properties, and extensions to deal with non-point masses and non-circular orbits. We then show how the rate of escape may be calculated numerically and estimated theoretically, and discuss how this simple picture is modified if the stars in a cluster are also undergoing two-body relaxation. Finally we introduce some established ideas for obtaining the distribution of escape times.