Instability in spherical stellar systems

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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 ω = 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. 2MT﬩/T﬩ 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.

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