Quasi-Periodic Almost-Collision Motions in the Spatial Three-Body Problem

Abstract

We deal with the spatial three-body problem in the various regimes where the Hamiltonian is split as the sum of two Keplerian systems plus a small perturbation. This is a region of the phase space $T^{{\ast}}\mathbb{R}^{6}$ where the perturbation is small [3], the so called perturbing region $\mathcal{P}_{\varepsilon,n}$. In particular, we prove the existence of quasi-periodic motions where the inner particles describe bounded near-rectilinear trajectories whereas the outer particle follows an orbit lying near the invariable plane. These motions fill in five-dimensional invariant tori. Moreover, the inner particles move in orbits either near an axis perpendicular to the invariable plane or near the invariable plane.