Geometry of transit orbits in the periodically-perturbed restricted three-body problem

TL;DR

This paper explores the geometric structure of transit orbits in the periodically-perturbed restricted three-body problem, extending classical methods to time-periodic systems and analyzing the phase space dynamics around periodic orbits.

Contribution

It introduces a geometric framework for understanding transit orbits in periodically-perturbed three-body systems using symplectic maps and effective Hamiltonians, extending prior unperturbed analyses.

Findings

Transit and non-transit orbit structures are preserved under periodic perturbations.

The phase space geometry around periodic orbits remains consistent with the unperturbed case.

A local conservation of an effective Hamiltonian guides the orbit dynamics in the perturbed system.

Abstract

In the circular restricted three-body problem, low energy transit orbits are revealed by linearizing the governing differential equations about the collinear Lagrange points. This procedure fails when time-periodic perturbations are considered, such as perturbation due to the sun (i.e., the bicircular problem) or orbital eccentricity of the primaries. For the case of a time-periodic perturbation, the Lagrange point is replaced by a periodic orbit, equivalently viewed as a hyperbolic-elliptic fixed point of a symplectic map (the stroboscopic Poincar\'e map). Transit and non-transit orbits can be identified in the discrete map about the fixed point, in analogy with the geometric construction of Conley and McGehee about the index-1 saddle equilibrium point in the continuous dynamical system. Furthermore, though the continuous time system does not conserve the Hamiltonian energy (which is time-varying), the linearized map locally conserves a time-independent effective Hamiltonian function. We demonstrate that the phase space geometry of transit and non-transit orbits is preserved in going from the unperturbed to a periodically-perturbed situation, which carries over to the full nonlinear equations.