Unstable and Stable Galaxy Models

Abstract

To determine the stability and instability of a given steady galaxy configuration is one of the fundamental problems in the Vlasov theory for galaxy dynamics. In this article, we study the stability of isotropic spherical symmetric galaxy models f 0(E), for which the distribution function f 0 depends on the particle energy E only. In the first part of the article, we derive the first sufficient criterion for linear instability of f 0(E) : f 0(E) is linearly unstable if the second-order operator $$A_{0} \equiv-\Delta+4\pi\int f_{0}^{\prime}(E)\{I-{\mathcal{P}}\}dv$$ has a negative direction, where $${\mathcal{P}}$$ is the projection onto the function space {g(E, L)}, L being the angular momentum [see the explicit formulae (29) and (28)]. In the second part of the article, we prove that for the important King model, the corresponding A 0 is positive definite. Such a positivity leads to the nonlinear stability of the King model under all spherically symmetric perturbations.