Analytical theory for highly elliptical orbits including time-dependent perturbations

TL;DR

This paper develops an advanced analytical method for accurately modeling highly elliptical orbits with time-dependent perturbations, overcoming limitations of traditional theories that are restricted to near-circular orbits and neglect third-body effects.

Contribution

It introduces a novel approach combining polynomial expansion of third-body perturbations with a Hamiltonian normalization technique to handle high eccentricity and time-dependent effects.

Findings

Enables long-term propagation of highly elliptical orbits with high accuracy

Maintains the classical J2 solution unchanged

Provides a practical tool for mission analysis involving highly elliptical orbits

Abstract

Traditional analytical theories of celestial mechanics are not well-adapted when dealing with highly elliptical orbits. On the one hand, analytical solutions are quite generally expanded into power series of the eccentricity and so limited to quasi-circular orbits. On the other hand, the time-dependency due to the motion of the third body (e.g. Moon and Sun) is almost always neglected. We propose several tools to overcome these limitations. Firstly, we have expanded the third-body disturbing function into a finite polynomial using Fourier series in multiple of the satellite's eccentric anomaly (instead of the mean anomaly) and involving Hansen-like coefficients. Next, by combining the classical Brouwer-von Zeipel procedure and the time-dependent Lie-Deprit transforms, we have performed a normalization of the expanded Hamiltonian in order to eliminate all the periodic terms. One of the benefits is that the original Brouwer solution for J2 is not modified. The main difficulty lies in the fact that the generating functions of the transformation must be computed by solving a partial differential equation, involving derivatives with respect to the mean anomaly, which appears implicitly in the perturbation. We present a method to solve this equation by means of an iterative process. Finally we have obtained an analytical tool useful for the mission analysis, allowing to propagate the osculating motion of objects on highly elliptical orbits (e>0.6) over long periods efficiently with very high accuracy, or to determine initial elements or mean elements. Comparisons between the complete solution and the numerical simulations will be presented.