Linear stability in an extended ring system

The planar n + 1 ring body problem consists of n bodies of equal mass m uniformly distributed around a central body of mass m(0). The bodies are rotating on its own plane about its center of mass with a constant angular velocity. Since Maxwell introduced the problem to understand the stability of Saturn's rings, many authors have studied and extended the problem. In particular, we proved that if forces that are functions of the mutual distances are considered the n-gon is a central conf guration. Examples of this kind are the quasi-homogeneous potentials. In a previous work we analyzed the linear stability of a system where the potential of the central body is a Manev's type potential. By introducing a perturbation parameter (epsilon(0)) to the Newtonian potential associated with the central primary, we showed that unstable cases for the unperturbed problem, for n <= 6, may become stable for some values of the perturbation. The purpose of this paper is to show that it is possible to increase the range of values of the mass parameter (mu = m/m(0)) and the parameter epsilon(0) in order to render a stable conf guration. In order to get it, we introduce a second perturbation term (with parameter epsilon(1)) to the Newtonian potential of the bodies in the ring. We show some results for the problem with n = 7.