Planetary long periodic terms in Mercury’s rotation: a two dimensional adiabatic approach

Abstract

The averaged spin-orbit resonant motion of Mercury is considered, with e the orbital eccentricity, and io the orbital inclination introduced as very slow functions of time, given by any secular planetary theory. The basis is our Hamiltonian approach (D’Hoedt, S., Lemaitre, A.: Celest. Mech. Dyn. Astron. 89:267–283, 2004) in which Mercury is considered as a rigid body. The model is based on two degrees of freedom; the first one is linked to the 3:2 resonant spin-orbit motion, and the second one to the commensurability of the rotational and orbital nodes. Mercury is assumed to be very close to the Cassini equilibrium of the model. To follow the motion of rotation close to this equilibrium, which varies with respect to time through e and io, we use the adiabatic invariant theory, extended to two degrees of freedom. We calculate the corrections (remaining functions) introduced by the time dependence of e and io in the three steps necessary to characterize the frequencies at the equilibrium. The conclusion is that Mercury follows the Cassini equilibrium (stays in the Cassini forced state), in an adiabatic behavior: the area around the equilibrium does not change by more than \({\varepsilon}\) for times smaller than \({\frac{1}{\varepsilon}}\) . The role of the inclination and the eccentricity can be dissociated and measured in each step of the canonical transformation.