Semi-Analytical Estimates for the Orbital Stability of Earth’s Satellites

• Published in: Journal of nonlinear science Vol. 31; no. 6
• Main Authors: De Blasi, Irene, Celletti, Alessandra, Efthymiopoulos, Christos
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V
• Subjects: Analysis; Canonical forms; Celestial mechanics; Classical Mechanics; Convexity; Economic Theory/Quantitative Economics/Mathematical Methods; Estimates; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Orbital stability; Orbits; Perturbation; Satellites; Slopes; Stability analysis; Theoretical; Stability; Space debris; Normal forms; 37N05; Orbital lifetime; 34C60; 70F15; Satellite dynamics
• ISSN: 0938-8974; 1432-1467
• DOI: 10.1007/s00332-021-09738-w

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.