High-Order Resonant Orbit Manifold Expansions For Mission Design In the Planar Circular Restricted 3-Body Problem

TL;DR

This paper introduces a high-order manifold expansion method for accurately computing invariant manifolds of resonant periodic orbits in the planar circular restricted 3-body problem, improving space mission trajectory design.

Contribution

It develops a parameterization method for high-precision manifold computation and algorithms for finding intersections, enhancing transfer trajectory planning in celestial mechanics.

Findings

Accurate computation of invariant manifolds using Taylor series.

Effective algorithms for intersection of stable and unstable manifolds.

Successful application to Jupiter-Europa system transfers.

Abstract

In recent years, stable and unstable manifolds of invariant objects (such as libration points and periodic orbits) have been increasingly recognized as an efficient tool for designing transfer trajectories in space missions. However, most methods currently used in mission design rely on using eigenvectors of the linearized dynamics as local approximations of the manifolds. Since such approximations are not accurate except very close to the base invariant object, this requires large amounts of numerical integration to globalize the manifolds and locate intersections. In this paper, we study hyperbolic resonant periodic orbits in the planar circular restricted 3-body problem, and transfer trajectories between them, by: 1) determining where to search for resonant periodic orbits; 2) developing and implementing a parameterization method for accurate computation of their invariant manifolds as Taylor series; and 3) developing a procedure to compute intersections of the computed stable and unstable manifolds. We develop and implement algorithms that accomplish these three goals, and demonstrate their application to the problem of transferring between resonances in the Jupiter-Europa system.