Unstable and Stable Galaxy Models

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 formula (\ref{A0-radial})]. 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.