Peculiarities in the gravitational field of a filamentary ring

The gravitational field of a massive, filamentary ring is considered. We provide an analytic expression for the gravitational potential and demonstrate that the exact gravitational potential and its gradient—the gravitational force-field—is not central. Hence, it is a good candidate to discuss the difference between the concepts of center of mass and center of gravity. We focus on other consequences of reduced symmetry, e.g., only the z-component of the angular momentum is conserved. However, the remnant high symmetry of this system also ensures that there are special classes of motions which are restricted to invariant subspaces; thus, depending on the initial condition, the dynamics of a point particle is integrable. We also show that periodic orbits in the equatorial plane external to the ring are possible, but only if the angular momentum is above a threshold value. In this case, the orbits are stable.