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

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 J2 problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining HJ2. (ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the ‘geolunisolar’ Hamiltonian Hgls), 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 HJ2 and Hgls models, which reflect necessary conditions for the holding of Nekhoroshev’s theorem on the exponential stability of the orbits. We find that the HJ2 model is non-convex, but satisfies a ‘three-jet’ condition, while the Hgls model restores quasi-convexity by adding lunisolar terms in the Hamiltonian’s integrable part.