In a recent article, the dynamics of a particle in the gravitational field of a uniform ring in the plane of the ring was examined. Parts of this work duplicate the results of previous work by West et al. in a paper published in this journal in 1998, and additional work extending those results in a paper by Tobin and West in 2006.
In the work recently published in this journal by Schumayer and Hutchinson, some interesting results and insights regarding orbital motion of a point mass under the influence of a massive filamentary ring are provided.1 Although two previous papers involving the author2,3 were not included in the list of references,1 it is gratifying to see some of those previous results validated by this new and independent examination. In particular in the first paper with West,2 it was shown that there is the possibility of stable orbits for values of Lz > 1.53Lo, where Lo = m(GMR)1/2 (determined numerically, while Schumayer and Hutchinson obtained an analytic expression). The graph in Fig. 10 of Ref. 1 shows the effective potential in the plane of the ring for multiple values of angular momentum. This is an extended version of Fig. 1 in Ref. 2, where a single case of the effective potential was shown. The value of L in Ref. 2 is equivalent to either the L3 or L4 line of Ref. 1. In discussing their Fig. 10, Schumayer and Hutchinson mention three characteristics of the effective potential that are worth noting, largely concentrating on the stable orbits found in the region outside of the local effective potential maxima. In connection with this, it is worth noting the surprising fact that orbits within the radius of a maximum in the effective potential can result in bound orbits with total energy that is positive. Particles in such an orbit are bound to, and doomed to collide with, the ring, even with an initial trajectory with a radial velocity away from the ring (hence the “provocative” title of Ref. 2).
The result for the potential along the axis of symmetry of the ring (see Fig. 5) as discussed by Schumayer and Hutchinson1 is also already known and was presented by Tobin and West in Ref. 3 (see Eq. (3) and paragraph preceding it).3
Finally, Schumayer and Hutchinson show that the circular orbits in the plane of the ring are stable against perturbations into the axial direction so that they introduce some aspects of three-dimensional motion as well (see Fig. 14 for a trajectory of such an orbit). However, they do not go on to explore the rich “zoo” of closed orbits that were found in the follow up paper by Tobin and in 2006.3
