Dynamics of a pair of spherical gravitating shells

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Abstract
The dynamical N body problem for a system of mass points interacting solely through gravitational forces is not integrable. The difficulties which arise in constructing accurate numerical codes for simulating the motion over long time scales are legend. Thus, in order to test their theories, astronomers and astrophysicists resort to simpler, one-dimensional models which avoid the problems of binary formation, escape, and the singularity of the inverse square force law. To date, the most frequently employed “test’’ model consists of a system of parallel mass sheets moving perpendicular to their surface. While this system avoids all of the above problems, the time scale for reaching equilibrium is extremely long and probably arises from the system’s weak ergodic properties, which become manifest even in the three sheet system. Here we consider a different one-dimensional gravitating system consisting of nonrotating concentric mass shells. For the case of two shells we investigate the structure of the phase space by studying the stability of periodic trajectories. By employing an event driven algorithm, we are able to directly investigate the influence of the singularity without having to resort to regularization of the force. Although stable structures are present at every energy, we find that the ergodic properties of this system are more robust than its planar counterpart.

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