The dynamics of the spin–spin problem in Celestial Mechanics

This work investigates different models of rotational dynamics of two rigid bodies with the shape of an ellipsoid, moving under their gravitational influence. The focus of this study is on their behavior, their linear stability, and numerical investigation of the main resonances. We assume that the spin axes of the two bodies are perpendicular to the orbital plane and coinciding with the direction of their shortest physical axis. In the basic approximation, we assume that the orbits of the centers of mass are Keplerian and we retain the lowest order of the potential, according to which the rotational motions of the two bodies are decoupled, the so-called spin–orbit problem. When considering highest order approximation of the potential, the rotational motions become coupled giving rise to the so-called spin–spin problem. Finally, we release the assumption that the orbit is Keplerian, which implies that the rotational dynamics is coupled to the variation of the orbital elements. The resulting system is called the full spin–spin problem. We also consider the above models under the assumption that one or both bodies are non rigid; the dissipative effect is modeled by a linear function of the rotational velocity, depending on some dissipative and drift coefficients.

We consider three main resonances, namely the (1:1,1:1), (3:2,3:2), (1:1,3:2) resonances and we start by analyzing the linear stability of the equilibria in the conservative and dissipative settings (after averaging and keeping only the resonant angle), showing that the stability depends on the value of the orbital eccentricity. By a numerical integration of the equations of motion, we compare the spin–orbit and spin–spin problems to highlight the influence of the coupling term of the potential in the conservative and dissipative case. We conclude by investigating the effect of the variation of the orbit on the rotational dynamics, showing that higher order resonant islands, that appeared in the Keplerian case, are destroyed in the full problem.