Computing the scattering map in the spatial Hill's problem

Amadeu Delshams,Josep Masdemont,Pablo Roldán

doi:10.3934/dcdsb.2008.10.455

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.

Highlights

The restricted three body problem describes the motion of an infinitesimal particle under the gravitational influence of two massive bodies called primaries that move in circular orbits around their common center of mass [29]
Hill’s problem is a simplified version of the classical restricted three body problem that is useful for studying motion in the neighborhood of the equilibrium points L1 and L2
The scattering map is a map between the normally hyperbolic invariant manifolds that describes the asymptotic behavior of heteroclinic solutions

Summary

Introduction

The restricted three body problem describes the motion of an infinitesimal particle (for instance, a spacecraft) under the gravitational influence of two massive bodies called primaries (for instance, the Sun and the Earth) that move in circular orbits around their common center of mass [29]. The scattering map is a map between the normally hyperbolic invariant manifolds that describes the asymptotic behavior of heteroclinic solutions The goal of this paper is to compute the scattering map associated to heteroclinic trajectories from the neighborhood of L1 to the neighborhood of L2 (or vice versa) in the spatial Hill’s problem. The scattering map gives the heteroclinic dynamics to the complete normally hyperbolic invariant manifold, not just to the primary invariant KAM tori used in Arnold’s mechanism. The computations of heteroclinic trajectories in this paper are done using Hill’s equations instead of RTBP equations, because they are simpler and they present a symmetry This allows us to compare results with [28] and [23]. In terms of these new variables, the equations of motion become (see [27]):

The Jacobi integral is again

They extend to global stable and unstable manifolds

Given heteroclinic channels