On librational motion of spacecraft

The paper deals with the three-dimensional librational motion of a spacecraft. It is shown that there exist periodic or quasi-periodic motions as well as chaotic trajectories merged together in the phase space. The quasi-periodic motion is investigated by reduction of the system to its normal form for circular and eccentric orbits. The normalizing transformation is singular at the resonances: the system close to such resonances is understood studying the ‘close to resonance hamiltonian’, whose stationary points give rise to periodic solutions. Surrounding any stable periodic trajectory, there are zones of quasi-periodic motion; however if the width of such zones is larger than the distance between two neighbouring resonances, a chaotic region appears both for nearly circular and eccentric orbits. The restriction to the planar case makes it possible to give a perturbative computation of the Poincaré map as well as the analytic calculation of key parameters, like the Chirikov parameter and the Melnikov function, which are analytical tests for chaos. Some final remarks on the interaction between attitude librations and orbital motion are given.