Nekhoroshev estimates for the orbital stability of Earth’s satellites

We provide stability estimates, obtained by implementing the Nekhoroshev theorem, in reference to the orbital motion of a small body (satellite or space debris) around the Earth. We consider a Hamiltonian model, averaged over fast angles, including the J2 geopotential term as well as third-body perturbations due to Sun and Moon. We discuss how to bring the Hamiltonian into a form suitable for the implementation of the Nekhoroshev theorem in the version given by Pöschel, (Math Z 213(1):187–216, 1993) for the ‘non-resonant’ regime. The manipulation of the Hamiltonian includes (i) averaging over fast angles, (ii) a suitable expansion around reference values for the orbit’s eccentricity and inclination, and (iii) a preliminary normalization allowing to eliminate particular terms whose existence is due to the nonzero inclination of the invariant plane of secular motions known as the ‘Laplace plane’. After bringing the Hamiltonian to a suitable form, we examine the domain of applicability of the theorem in the action space, translating the result in the space of physical elements. We find that the necessary conditions for the theorem to hold are fulfilled in some nonzero measure domains in the eccentricity and inclination plane (e, i) for a body’s orbital altitude (semimajor axis) up to about 20 000 km. For altitudes around 11 000 km, we obtain stability times of the order of several thousands of years in domains covering nearly all eccentricities and inclinations of interest in applications of the satellite problem, except for narrow zones around some so-called inclination-dependent resonances. On the other hand, the domains of Nekhoroshev stability recovered by the present method shrink in size as the semimajor axis a increases (and the corresponding Nekhoroshev times reduce to hundreds of years), while the stability domains practically all vanish for a>20000 km. We finally examine the effect on Nekhoroshev stability by adding more geopotential terms (J3 and J4) as well as the second-order terms in J2 in the Hamiltonian. We find that these terms have only a minimal effect on the domains of applicability of Nekhoroshev theorem and a moderate effect on the stability times.