Integrability of close encounters in the spatial restricted three-body problem

TL;DR

This paper extends Levi-Civita's technique for integrating close encounters to the spatial restricted three-body problem, using Kustaanheimo-Stiefel regularization to construct a complete integral near collisions.

Contribution

It introduces a method to handle close encounters in the spatial case, expanding previous planar techniques with a convergent series solution.

Findings

Constructed a complete integral of the Hamilton-Jacobi equation near collisions.

Extended Levi-Civita's planar approach to three-dimensional space.

Provided a convergent series solution for the spatial restricted three-body problem.

Abstract

We extend to the spatial case a technique of integration of the close encounters formulated by Tullio Levi-Civita for the planar restricted three-body problem. We consider the Hamiltonian introduced in the Kustaanheimo-Stiefel regularization and construct a complete integral of the related Hamilton-Jacobi equation by means of a series convergent in a neighbourhood of the collisions with the primary or secondary body.