A simple proof of dynamical stability for a class of spherical clusters

Abstract

This paper examines the dynamical stability of stationary, collisionless, spherically symmetric Newtonian clusters for which the population f<SUB>0</SUB> of stars in phase space varies monotonically as a function of the energy E<SUB>0</SUB>. A pulsation equation is obtained for arbitrary linearized perturbations which, for some cases of physical interest, admits a symmetric operator formulation and, hence, an energy principle. This energy principle provides a simple proof that, at least when ∂f<SUB>0</SUB>/∂E<SUB>0</SUB> &lt; 0, (1) all these clusters are stable to radial perturbations, and (2) isotropic clusters are stable to all perturbations.