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 $$J_2$$ J 2 problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $${\mathcal {H}}_{J_2}$$ H J 2 . (ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the ‘geolunisolar’ Hamiltonian $${\mathcal {H}}_\mathrm{gls}$$ 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 $${\mathcal {H}}_{J_2}$$ H J 2 and $${\mathcal {H}}_\mathrm{gls}$$ 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 $${\mathcal {H}}_{J_2}$$ H J 2 model is non-convex, but satisfies a ‘three-jet’ condition, while the $${\mathcal {H}}_\mathrm{gls}$$ H gls model restores quasi-convexity by adding lunisolar terms in the Hamiltonian’s integrable part.