Abstract
A study of the gravitational interaction of a unit point mass m, and a very thin ring of radius R and mass M (M ≫ m) is presented. The point mass moves in a plane that is perpendicular to that containing the ring. The potential due to the ring is similar in geometry to that of the two-centre problem with two large point masses, if it were possible for the positions of such point masses to be fixed. The scope of the paper is restricted almost exclusively to closed orbits (trajectories), and these display a surprisingly wide range of patterns. A characterization plot, based on the initial conditions of the trajectories, is used to illustrate and aid in the categorization of the trajectories into ‘families’. Figures showing an example member of 11 distinct families and one example trajectory in which the point mass is not restricted to motion in a plane are included. The period of a family of closed trajectories, roughly egg shaped and bound to a particular section of the ring, is found to depend almost linearly on the radius of the orbit, as expected.
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