Rotation of a Planet in a Three-Body System: a Non-Resonant Case

We investigate the temporal evolution of the rotation axis of a planet in a system comprised of the planet (which we call an exo-Earth), a star (an exo-Sun) and a satellite (an exo-Moon). The planet is assumed to be rigid and almost spherical, the difference between the largest and the smallest principal moments of inertia being a small parameter of the problem. The orbit of the planet around the star is a Keplerian ellipse. The orbit of the satellite is a Keplerian ellipse with a constant inclination to the ecliptic, involved in two types of slow precessional motion, nodal and apsidal. Applying time averaging over the fast variables associated with the frequencies of the motion of exo-Earth and exo-Moon, we obtain Hamilton’s equations for the evolution of the angular momentum axis of the exo-Earth. Using a canonical change of variables, we show that the equations are integrable. Assuming that the exo-Earth is axially symmetric and its symmetry and rotation axes coincide, we identify possible types of motions of the vector of angular momentum on the celestial sphere. Also, we calculate the range of the nutation angle as a function of the initial conditions. (By the range of the nutation angle we mean the difference between its maximal and minimal values.)