On 'spoke approximation' in the stability theory of stellar systems with highly elongated orbits in near-Keplerian potentials

In this paper, we review different approaches to studying the spectrum of eigenmodes in systems with highly elongated stellar orbits. The simplest and most natural approach suggests a replacement of the orbits by thin rotating spokes, in which rotation imitates the precession of real orbits. We show that such an obvious approach does not allow us to properly analyse the stellar systems for stability. For disc stellar systems, this does not allow us to obtain unstable modes, even in the leading order of the perturbation theory for a small parameter, characterizing the dispersion of nearly radial orbits in angular momentum. The situation is better in spherical systems, where the spectrum of eigenmodes can be determined, but in the leading order of the perturbation theory for a small parameter only. For a correct description of the stability of stellar systems, a rigorous approach should be used, based on the solution of the integral equations that are given in this paper.