Spin–orbit evolution of Mercury revisited

Abstract Although it is accepted that the significant eccentricity of Mercury (0.206) favours entrapment into the 3:2 spin–orbit resonance, open are the questions of how and when the capture took place. A recent work by Makarov (Makarov, V.V. [2012]. Astrophys. J., 752, 73) has proven that trapping into this state is certain for eccentricities larger than 0.2 , provided we use a realistic tidal model based on the Darwin–Kaula expansion of the tidal torque. While in Ibid. a Mercury-like planet had its eccentricity fixed, we take into account its evolution. To that end, a family of possible histories of the eccentricity is generated, based on synthetic time evolution consistent with the expected statistics of the distribution of eccentricity. We employ a model of tidal friction, which takes into account both the rheology and self-gravitation of the planet. As opposed to the commonly used constant time lag (CTL) and constant phase lag (CPL) models, the physics-based tidal model changes dramatically the statistics of the possible final spin states. First, we discover that after only one encounter with the spin–orbit 3:2 resonance this resonance becomes the most probable end-state. Second, if a capture into this (or any other) resonance takes place, the capture becomes final, several crossings of the same state being forbidden by our model. Third, within our model the trapping of Mercury happens much faster than previously believed: for most histories, 10–20 Myr are sufficient. Fourth, even a weak laminar friction between the solid mantle and a molten core would most likely result in a capture in the 2:1 or even higher resonance, which is confirmed both semi-analytically and by limited numerical simulations. So the principal novelty of our paper is that the 3:2 end-state is more ancient than the same end-state obtained when the constant time lag model is employed. The swift capture justifies our treatment of Mercury as a homogeneous, unstratified body whose liquid core had not yet formed by the time of trapping. We also provide a critical analysis of the hypothesis by Wieczorek et al. (Wieczorek, M.A., Correia, A.C.M., Le Feuvre, M., Laskar, J., Rambaux, N. [2012]. Nat. Geosci., 5, 18–21) that the early Mercury might had been retrograde, whereafter it synchronised its spin and then accelerated it to the 3:2 resonance. Accurate processing of the available data on cratering does not support that hypothesis, while the employment of a realistic rheology invalidates a key element of the hypothesis, an intermediate pseudosynchronous state needed to spin-up to the 3:2 resonance.