• sun-shadow problem
• perturbation theory
• celestial mechanics
• patched dynamics
• close encounter
• restricted three-body problem

The motion of natural and artificial celestial bodies deviates from the simple Keplerian model by perturbations of different kinds, giving rise to complex problems in the field of Celestial Mechanics. Two problems are analysed: i) the orbital evolution of Earth satellites perturbed by the solar radiation pressure, considering the effects of successive eclipses caused by the passages through the Earth's shadow; ii) the restricted three-body problem, used to study the dynamics of small bodies in heliocentric orbits perturbed by the gravitational influence of a planet. Regarding this latter problem, planetary close encounters can cause drastic variations of the heliocentric orbital elements in relatively short intervals of time. Moreover, even if no close encounters occur, the gravity of a planet can cause non-negligible effects over long (secular) timescales. Patched-dynamics models, suitable to approximate the above problems over relatively short timescales, are investigated. Concerning the secular dynamics of a small body in the restricted three-body problem, we rely on Hamiltonian perturbation theory. In particular, we examine a closed-form normalisation method, suitable for small bodies perturbed by the gravity of an external planet, i.e. a planet whose orbit is external to the orbit of the small body.