Computing the scattering map in the spatial Hill's problem

Abstract

Let $A_1$ and $A_2$ be two normally hyperbolic invariant manifolds for a flow, such that the stable manifold of $A_1$ intersects the unstable manifold of $A_2$ transversally along a manifold Γ. The scattering map from $A_2$ to $A_1$ is the map that, given an asymptotic orbit in the past, associates the corresponding asymptotic orbit in the future through a heteroclinic orbit. It was originally introduced to prove the existence of orbits of unbounded energy in a perturbed Hamiltonian problem using a geometric approach.  &nbspWe recently computed the scattering map in the planar restricted three body problem using non-perturbative techniques, and we showed that it is a (nontrivial) integrable twist map.   &nbspIn the present paper, we compute the scattering map in a problem with three degrees of freedom using also non-perturbative techniques. Specifically, we compute the scattering map between the normally hyperbolic invariantmanifolds $A_1$ and $A_2$ associated to the equilibrium points $L_1$ and $L_2$ in the spatial Hill's problem.  &nbspIn the planar problem, for each energy level (in a certain range) there is a unique Lyapunov periodic orbit around $L_{1,2}$. In the spatial problem, this periodic orbit is replaced by a three-dimensional invariant manifold practically full of invariant 2D tori. There are heteroclinic orbits between $A_1$ and $A_2$ connecting these invariant tori in rich combinations. Hence the scattering map in the spatial problem is more complicated, and it allows nontrivial transition chains.  &nbspScattering maps have application to e.g. mission design in Astrodynamics, and to the construction of diffusion orbits in the spatial Hill's problem.