The Stability of a One Dimensional Gravitational Cluster

Recent observations reporting 60 globular clusters or faint galaxies orbiting the Milky Way (J.D. Simon and M. Geha, Physics Today, November 2021) raise the issue of the stability of globular clusters.   We begin a program of studying the evaporation of stars from clusters. To  start modestly, we consider a simple one-dimensional gravitational cluster. Our analytic approach will use the first two terms of a six-term time evolution equation for the one-particle distribution function   (A. Muriel, Results in Physics, 2016) using the gravitational pair potential in one dimension. The resulting expressions are interpreted to be the Vlassov collisionless regime. The density of the cluster as a function of position and time is calculated  by integration over momentum. We will display the time evolution of the  density for various initial conditions and comment on the stability of the cluster. With insight from this one dimensional model, we outline the analysis for a 3-d globular cluster with spherical symmetry, effectively one dimensional as well.