Hamiltonian stability of spin-orbit resonances in celestial mechanics

The behaviour of 'resonances' in the spin-orbit coupling in celestial mecha nics is investigated in a conservative setting. We consider a Hamiltonian n early-integrable model describing an approximation of the spin-orbit intera ction. The continuous system is reduced to a mapping by integrating the equ ations of motion through a symplectic algorithm. We study numerically the s tability of periodic orbits associated to the above mapping by looking at t he eigenvalues of the matrix of the linearized map over the full cycle of t he periodic orbit. In particular, the value of the trace of the matrix is r elated to the stability character of the periodic orbit. We denote by epsil on*(p/q) the value of the perturbing parameter at which a given elliptic pe riodic orbit with frequency p/q becomes unstable. A plot of the critical fu nction epsilon*(p/q) versus the frequency at different orbital eccentriciti es shows significant peaks at the synchronous resonance (for low eccentrici ties) and at the synchronous and 3:2 resonances (at higher eccentricities) in good agreement with astronomical observations.