Geostationary secular dynamics revisited: application to high area-to-mass ratio objects

TL;DR

This paper revisits the long-term secular dynamics of geostationary orbits using canonical perturbation theory, providing analytical insights into the behavior of high area-to-mass ratio objects and their implications for space debris management.

Contribution

It offers a new analytical framework for understanding GEO orbit dynamics, especially for high area-to-mass ratio objects, by applying higher order normal form theory to the Hamiltonian model.

Findings

The forced equilibrium corresponds to a trajectory on a lower-dimensional torus.

Analytical expressions for the trajectory near the forced equilibrium are derived.

Results are validated against numerical orbit propagation.

Abstract

The long-term dynamics of the geostationary Earth orbits (GEO) is revisited through the application of canonical perturbation theory. We consider a Hamiltonian model accounting for all major perturbations: geopotential at order and degree two, lunisolar perturbations with a realistic model for the Sun and Moon orbits, and solar radiation pressure. The long-term dynamics of the GEO region has been studied both numerically and analytically, in view of the relevance of such studies to the issue of space debris or to the disposal of GEO satellites. Past studies focused on the orbital evolution of objects around a nominal solution, hereafter called the forced equilibrium solution, which shows a particularly strong dependence on the area-to-mass ratio. Here, we i) give theoretical estimates for the long-term behavior of such orbits, and ii) we examine the nature of the forced equilibrium itself. In the lowest approximation, the forced equilibrium implies motion with a constant non-zero average `forced eccentricity', as well as a constant non-zero average inclination, otherwise known in satellite dynamics as the inclination of the invariant `Laplace plane'. Using a higher order normal form, we demonstrate that this equilibrium actually represents not a point in phase space, but a trajectory taking place on a lower-dimensional torus. We give analytical expressions for this special trajectory, and we compare our results to those found by numerical orbit propagation. We finally discuss the use of proper elements, i.e., approximate integrals of motion for the GEO orbits.