Motion around a Monopole + Ring system: I. Stability of Equatorial Circular Orbits vs Regularity of Three-dimensional Motion

We study the motion of test particles around a center of attraction represented by a monopole (with and without spheroidal deformation) surrounded by a ring, given as a superposition of Morgan & Morgan discs. We deal with two kinds of bounded orbits: (i) Equatorial circular orbits and (ii) general three-dimensional orbits. The first case provides a method to perform a linear stability analysis of these structures by studying the behavior of vertical and epicyclic frequencies as functions of the mass ratio, the size of the ring and/or the quadrupolar deformation. In the second case, we study the influence of these parameters in the regularity or chaoticity of motion. We find that there is a close connection between linear stability (or unstability) of equatorial circular orbits and regularity (or chaoticity) of the three-dimensional motion.