Instability in spherical stellar systems

Abstract

In this paper we show that a spectrum of purely growing modes exists in models of spherical stellar systems which are strongly peaked towards radial orbits. This spectrum has an accumulation point at <it>ω</it> = 0 if the distribution function is unbounded for zero angular momentum orbits. We show that no stability criterion can be found of the form of a critical anisotropy parameter, e.g. 2M<it>T</it><inf></inf>/<it>T</it><inf></inf> and derive a straightforward eigenvalue problem which yields a sufficient condition for the absence of this low growth rate spectrum. We also study this spectrum for some generalized polytropic models.