CONSTRUCTION OF LIBRATIONAL INVARIANT TORI IN THE SPIN-ORBIT PROBLEM

We investigate the stability of the synchronous spin-orbit resonance. In particular we construct invariant librational tori trapping periodi c orbits in finite regions of phase space. We first introduce a mathem atical model describing a simplification of the physical situation. Th e corresponding Hamiltonian function has the form H(y, x, t) = (y(2)/2 ) + epsilon V(x, t), where V is a trigonometric polynomial in x, t and epsilon is the ''perturbing parameter'' representing the equatorial o blateness of the satellite. We perform some symplectic changes of vari ables in order to reduce the initial Hamiltonian to a form which suita bly describes librational tori. We then apply Birkhoff normalization p rocedure in order to reduce the size of the perturbation. Finally the application of KAM theory allows to prove the existence of librational tori around the synchronous periodic orbit. Two concrete applications are considered: the Moon-Earth and the Rhea-Saturn systems. In the fi rst case one gets the existence of trapping orbits for values of the p erturbing oblateness parameter far from the real physical value by a f actor similar to 5. In the Rhea-Saturn case we construct the trapping tori for values of the parameters consistent with the astronomical mea surements.