Analysis of the geometry of the zero-velocity curves in the N-body ring problem depending on the mass ratio parameter

The purpose of the present study is the investigation of the effect of the mass ratio parameter, β, on the geometry of the zero-velocity curves, when there are N = 3, 4, . . . , 100 peripheral bodies. It is well known that there is a bifurcation value of the β parameter in the N-body ring problem that produces a change in the number of stationary solutions in the system from 5N to 3N. By examining the behavior of the critical values of the Jacobi constant that define each of the zones of stationary solutions, we have unveiled the existence of other bifurcations or critical values of β in the scenario with 5N stationary solutions, which cause different changes in the geometry of the zero-velocity curves, which in turn affect the threshold for the total opening of the curves of zero velocity. ; Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work has been partially funded by the Algerian Ministry of Higher Education and Scientific Research, who awarded a grant for PhD training abroad in 2020 to the first author, Zahra Boureghda, at the University of Alicante.