Semi-analytical estimates for the orbital stability of Earths 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 $J_2$ 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 Nekhoroshevs 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 Hamiltonians integrable part.