From infinite to finite time stability in Celestial Mechanics and Astrodynamics

Time scales in Celestial Mechanics and Astrodynamics vary considerably, from a few hours for the motion of Earth's artificial satellites to millions of years for planetary dynamics. Hence, the time scales on which one needs to investigate the stability of celestial objects are different. Therefore, the methods of study are themselves different and might lead to specific definitions of stability, either in the sense of bounds on the initial conditions or rather in the sense of confinement in a given region of the phase space. In this work we concentrate on three different methods: perturbation theory, Nekhoroshev's theorem, KAM theory. All theories are constructive in the sense that they provide explicit algorithms to give estimates on the parameters of the system and on the stability time. Perturbation theory gives results on finite time scales, Nekhoroshev's theorem provides stability results on exponentially long times, KAM theory ensures the confinement between invariant tori in low-dimensional systems.We recall the basic ingredients of each theory, starting with KAM theory, then presenting Nekhoroshev's theorem and finally introducing perturbation theory. We provide examples of stability results for some objects of the Solar system. Precisely, we consider the stability of the rotational motion of the Moon (or other planetary satellites) within the spin-orbit model by means of KAM theory, we analyze the stability of asteroids, also in the triangular equilibrium Lagrangian points, using Nekhoroshev's theorem, we study the Earth's satellite dynamics through perturbation theory.