Nonlinear librations of distant retrograde orbits: a perturbative approach -- The Hill problem case

TL;DR

This paper introduces a new perturbation method for analyzing nonlinear librations of distant retrograde orbits in the Hill problem, providing simpler low-order solutions and enabling higher-order calculations involving special functions.

Contribution

It presents an alternative perturbation approach that yields simple trigonometric solutions at low order and facilitates higher-order solutions with special functions for distant retrograde orbits.

Findings

Developed a simple low-order analytical solution using trigonometric functions.

Enabled computation of higher-order solutions involving elliptic functions.

Improved accuracy over previous qualitative descriptions of distant retrograde orbit dynamics.

Abstract

The non-integrability of the Hill problem makes that its global dynamics must be necessarily approached numerically. However, the analytical approach is feasible in the computation of relevant solutions. In particular, the nonlinear dynamics of the Hill problem close to the origin, and the libration point dynamics have been thoroughly investigated by perturbation methods. Out of the Hill sphere, the analytical approach is also feasible, at least in the case of distant retrograde orbits. Previous analytical investigations of this last case succeeded in the qualitative description of the dynamics, but they commonly failed in providing accurate results. This is a consequence of the essential dependance of the dynamics on elliptic functions, a fact that makes to progress in the perturbation approach beyond the lower orders of the solution really difficult. We propose an alternative perturbation approach that allows us to provide a very simple low order analytical solution in trigonometric functions, on the one hand, and, while still depending on special functions, to compute higher orders of the solution, on the other.