Semi-analytical estimates for the orbital stability of Earth's satellites

TL;DR

This paper develops semi-analytical methods using high-order normal forms to estimate the long-term orbital stability of Earth's satellites under various perturbations, providing new insights into stability conditions.

Contribution

It introduces three different normal form-based estimates for satellite orbit stability, including long-term semimajor axis stability and eccentricity/inclination stability under perturbations.

Findings

Semimajor axis stability demonstrated in the $J_2$ model.

Eccentricity and inclination stability shown in the geolunisolar model.

Quasi-convexity restored by lunisolar terms in the Hamiltonian.

Abstract

Normal form stability estimates are a basic tool of Celestial Mechanics for characterizing the long-term stability of the orbits of natural and artificial bodies. Using high-order normal form constructions, we provide three different estimates for the orbital stability of point-mass satellites orbiting around the Earth. i) We demonstrate the long term stability of the semimajor axis within the framework of the $J_2$ problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $H_{J_2}$ . ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the 'geolunisolar' Hamiltonian $H_{gls}$), after a suitable reduction of the Hamiltonian to the Laplace plane. iii) We numerically examine the convexity and steepness properties of the integrable part of the secular Hamiltonian in both the $H_{J_2}$ and $H_{gls}$ models, which reflect necessary conditions for the holding of Nekhoroshev's theorem on the exponential stability of the orbits. We find that the $H_{J_2}$ model is non-convex, but satisfies a 'three-jet' condition, while the $H_{gls}$ model restores quasi-convexity by adding lunisolar terms in the Hamiltonian's integrable part.