Computing Invariant Manifolds and Connecting Orbits in the Circular Restricted Three Body Problem

TL;DR

This paper presents a numerical method for computing invariant manifolds and connecting orbits in the CR3BP, aiding space-mission design by identifying critical orbital connections.

Contribution

It introduces boundary value formulations combined with numerical continuation to effectively compute and analyze connecting orbits in the CR3BP.

Findings

Detected heteroclinic connections between periodic orbits.

Identified homoclinic and heteroclinic orbits relevant for mission planning.

Demonstrated effectiveness of the method in complex orbital systems.

Abstract

We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the Circular Restricted Three-Body Problem (CR3BP), which models the motion of a satellite in an Earth- Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar Vertical and Halo orbits. We compute the unstable manifolds of selected Vertical and Halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary, leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits could be of potential interest in space-mission design.