Birth of resonances in the spin-orbit problem of Celestial Mechanics

The behaviour of resonances in the spin-orbit coupling in Celestial Mechanics is investigated. We introduce a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. A parametric representation of periodic orbits is presented. We provide explicit formulae to compute the Taylor series expansion in the perturbing parameter of the function describing this parametrization. Then we compute approximately the radius of convergence providing an indication of the stability of the periodic orbit. This quantity is used to describe the different probabilities of capture into resonance. In particular, we notice that for low values of the orbital eccentricity the only significative resonance is the synchronous one. Higher order resonances (including 1:2, 3:2, 2:1) appear only as the orbital eccentricity is increased.