Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem

Author： Bakker Lennard Ouyang Tiancheng Yan Duokui Simmons Skyler

Publisher： Springer Publishing Company

ISSN： 0923-2958

Source： Celestial Mechanics and Dynamical Astronomy, Vol.110, Iss.3, 2011-07, pp. : 271-290

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Abstract

We analytically prove the existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. This is an extension of our previous proof of the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar fully symmetric equal mass four-body problem. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for m between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability when 0.54 ≤ m ≤ 1, and are linearly unstable when 0 < m ≤ 0.53.