Families of symmetric exchange orbits in the planar \((1+2n)\)-body problem
The paper deals with a particular \((2n+1)\)-body planar configuration in which a single massive body rests at the origin and \(n\) pairs of nearly infinitesimal bodies (called ``satellites'' in the paper) orbiting around it. Both the even-indexed and odd-indexed satellites form a regular \(n\)-gon at all times during the orbit. At an initial time, these polygons are the same size, forming \(n\) isoceles triangles with these satellite pairs and the origin forming the vertices. At some future time, the configuration arises again after a rotation by a fixed angle. The paper focuses on the interplay between adjacent pairs of satellites as the configuration evolves. This is achieved by framing the problem as a suitable boundary value problem analyzed with the software package AUTO. The authors Use Maxwell's homographic ring solutions as an initial seed for the problem (after re-writing the solution in their established coordinate system). The final third of the paper describes the results of this analysis for \(n=2,3,\dots,6\) in both a fixed and rotating reference frame. Interestingly, in the \(n = 6\) case, the curves describing the orbit of the even- and odd-indexed pairs coincide. The analytic results established in the paper are not too surprising. The majority of the interesting work arises in the numerical portion. There are a few grammatical and mathematical errors throughout, though the authors' general intention is generally understood. For instance, as written, in equation (9) the conditions at time \(t = T_0\) appear to correspond with a collision of two bodies, although it is obvious from the remainder of the paper that this is not their intent.
scientific article; zbMATH DE number 7358628