Numerical analysis of periodic solutions and bifurcations in the planetary annulus problem

This paper discusses the dynamics of particles orbiting planetary rings under a general central potential. Starting with the mathematical description of the dynamical system, we analyze the motion of a particle with infinitesimal mass as attracted by a central body surrounded by a homogeneous circular annular disk. Throughout the paper we carry out an analytic search of the most relevant equilibria solutions and, based on that, we investigate numerically the stability matrix of the system to find stability inequalities. In this way, we describe the in-plane and out-of-plane motion by means of the numerical continuation of a wide number of uni-parametric families of planar and spatial periodic orbits. We present a description of the main families of periodic orbits encountered, their bifurcations and linear stability. With the aim of reproducing a more realistic scenario, we analyse different mass proportions between the annulus and the central body, we consider an oblate planet and we also include a composition of rings in the dynamical model.