Fluctuations in Isothermal Spheres

Abstract

Isolated isothermal spheres of N gravitationally interacting points with equal mass are believed to be stable when density contrasts do not exceed 709. That stability limit does, however, not take into consideration fluctuations of temperature near the onset of instability. These are important when N is finite. Here we correlate {\it global mean quadratic temperature fluctuations} with onset of instability. We show that such fluctuations trigger instability when the density contrast reaches a value near $709\cdot\exp(-3.3N^{-1/3})$. These lower values of limiting density contrasts are significantly smaller than 709 when N is not very big and this suggests (i) that numerical calculations with small N may not reflect correctly the onset of core collapse in clusters with big N and (ii) that a greater number of globular clusters than is normally believed may already be in an advanced stage of core collapse because most of observed globular clusters whose parameters fit quasi-isothermal configurations are close to marginal stability.