Towards an analytical theory of the third-body problem for highly elliptical orbits

TL;DR

This paper develops new analytical tools for modeling gravitational perturbations on satellites in highly elliptical orbits, overcoming limitations of traditional methods that are restricted to near-circular orbits and neglect time-dependent effects.

Contribution

It introduces a polynomial expansion of the disturbing function using Fourier series in eccentric anomaly and a novel time-dependent normalization method for the Hamiltonian.

Findings

Expanded the disturbing function into a finite polynomial with Fourier series.

Developed a time-dependent Lie transformation for Hamiltonian normalization.

Provided iterative solutions for complex differential equations involved.

Abstract

When dealing with satellites orbiting a central body on a highly elliptical orbit, it is necessary to consider the effect of gravitational perturbations due to external bodies. Indeed, these perturbations can become very important as soon as the altitude of the satellite becomes high, which is the case around the apocentre of this type of orbit. For several reasons, the traditional tools of celestial mechanics are not well adapted to the particular dynamic of highly elliptical orbits. On the one hand, analytical solutions are quite generally expanded into power series of the eccentricity and therefore limited to quasi-circular orbits [17, 25]. On the other hand, the time-dependency due to the motion of the third-body is often neglected. We propose several tools to overcome these limitations. Firstly, we have expanded the disturbing function into a finite polynomial using Fourier expansions of elliptic motion functions in multiple of the satellite's eccentric anomaly (instead of the mean anomaly) and involving Hansen-like coefficients. Next, we show how to perform a normalization of the expanded Hamiltonian by means of a time-dependent Lie transformation which aims to eliminate periodic terms. The difficulty lies in the fact that the generator of the transformation must be computed by solving a partial differential equation involving variables which are linear with time and the eccentric anomaly which is not time linear. We propose to solve this equation by means of an iterative process.