Mercuryʹs capture into the 3/2 spin–orbit resonance including the effect of core–mantle friction

The rotation of Mercury is presently captured in a 3/2 spin–orbit resonance with the orbital mean motion. The capture mechanism is well understood as the result of tidal interactions with the Sun combined with planetary perturbations [Goldreich, P., Peale, S., 1966. Astron. J. 71, 425–438; Correia, A.C.M., Laskar, J., 2004. Nature 429, 848–850]. However, it is now almost certain that Mercury has a liquid core [Margot, J.L., Peale, S.J., Jurgens, R.F., Slade, M.A., Holin, I.V., 2007. Science 316, 710–714] which should induce a contribution of viscous friction at the core–mantle boundary to the spin evolution. According to Peale and Boss [Peale, S.J., Boss, A.P., 1977. J. Geophys. Res. 82, 743–749] this last effect greatly increases the chances of capture in all spin–orbit resonances, being 100% for the 2/1 resonance, and thus preventing the planet from evolving to the presently observed configuration. Here we show that for a given resonance, as the chaotic evolution of Mercuryʹs orbit can drive its eccentricity to very low values during the planetʹs history, any previous capture can be destabilized whenever the eccentricity becomes lower than a critical value. In our numerical integrations of 1000 orbits of Mercury over 4 Gyr, the spin ends 99.8% of the time captured in a spin–orbit resonance, in particular in one of the following three configurations: 5/2 (22%), 2/1 (32%) and 3/2 (26%). Although the present 3/2 spin–orbit resonance is not the most probable outcome, we also show that the capture probability in this resonance can be increased up to 55% or 73%, if the eccentricity of Mercury in the past has descended below the critical values 0.025 or 0.005, respectively.