Ejection-Collision orbits in the symmetric collinear four–body problem

Abstract

In this paper, we consider the collinear symmetric four-body problem, where four masses m3=α, m1=1, m2=1, and m4=α, α > 0, are aligned in this order and move symmetrically about their center of mass. We introduce regularized variables to deal with binary collisions as well as McGehee coordinates to study the quadruple collision manifold for a negative value of the energy. The paper is mainly focused on orbits that eject from (or collide to) quadruple collision. The problem has two hyperbolic equilibrium points, located in the quadruple collision manifold. We use high order parametrizations of their stable/unstable manifolds to devise a numerical procedure to compute ejection-collision orbits, for any value of α. Some results from the explorations done for α=1 are presented. Furthermore, we prove the existence of ejection-direct escape orbits, which perform a unique type of binary collisions

E. Barrabés has been supported by grants MTM2016-80117-P (MINECO/FEDER,UE), and Catalan (AGAUR) grant 2017 SGR 1374. M. Ollé is partially supported by Spanish MNECO/FEDER grant MTM2015-65715 and the Catalan grant 2017SGR1049