Steepest descent technique and stellar equilibrium statistical mechanics. III Stability of various ensembles

The stability of mean fields in stellar systems under various constraints has been analyzed by a steepest descent technique applied to the relevant thermodynamic potential. Necessary and sufficient conditions of stability with respect to arbitrary perturbations about mean field solutions have been obtained. Instabilities with respect to spherically symmetric perturbations are associated with a change of sign of thermodynamic quantities. With a fixed center of mass, the clusters are stable with respect to arbitrary nonspherical perturbations. Numerical results are given.