Behavior of the binary collision in a planar restricted ([formula omitted])-body problem

Abstract

We consider the planar restricted (N+1)-body problem, where the primaries are moving in a central configuration. It is verified that, when the energy approaches minus infinity, the infinitesimal mass m1 is arbitrarily close to a primary. We use Levi-Civita and McGehee coordinates to regularize the binary collision in this setting. A canonical transformation is constructed in such a way that it transforms the equations into the form of a perturbed resonant pair of harmonic oscillators where the perturbation parameter is the reciprocal of the energy. We first prove the existence of four transversal ejection–collision orbits. After that, we carry out the construction of the annulus mapping and verify the conditions of the Moser Invariant Curve Theorem; we are able to show the existence of long periodic solutions for the restricted (N+1)-body problem. We also prove the existence of quasi-periodic solutions close to the binary collision. The first result implies, via the KAM theorem, the existence of an uncountable number of invariant punctured tori in the corresponding energy surface for certain intervals of values of the Jacobi constant.This work grew from an attempt to carry over the methods used to study the restricted three-body problem for high values of the Jacobian constant by Conley (1963, 1968) [3,18]. Chenciner [4] and Chenciner and Llibre (1988) [5] applied their techniques to a more general restricted problem. Our goal in this paper is to give a generalization of Conley’s results (Conley, 1968 [18]). In addition, we show that the Hill terms (the terms of sixth order) that appear in this study have the same nature but with different coefficients than those in the mentioned papers. This fact allows us to present some differences with respect to known results. Thus, we point out conditions on the relative equilibrium of the N-body problem in order to overcome the apparent difficulties.