Linear stability of ring systems with generalized central forces

Grupo de Mecánica Espacial and Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, Zaragoza 50009, Spain e-mail: elipe@unizar.es

Received: 4 April 2008
Accepted: 1 May 2008

Abstract

We analyze the linear stability of a system of n equal mass points uniformly distributed on a circle and moving about a single massive body placed at its center. We assume that the central body makes a generalized force on the points on the ring; in particular, we assume the force is generated by a Manev's type potential. This model represents several cases, for instance, when the central body is a spheroid or a radiating source. The problem contains 3 parameters, namely, the number n of bodies of the ring, the mass factor μ, and the radiation or oblateness coefficient ϵ. For the classical case (Newtonian forces), it has been known since the seminal work of Maxwell that the problem is unstable for . For the problem is stable when μ is within a certain interval. In this work, we determine the region () in which the problem is stable for several values of n. Unstable cases () may become stable for negative values of ϵ.