Mercurys 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. However, it is now almost certain that Mercury has a liquid core, which should induce a contribution of viscous friction at the core-mantle boundary to the spin evolution. 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 Mercury’s orbit can drive its eccentricity to very low values during the planet’s 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.

💡 Research Summary

The paper revisits the long‑standing problem of why Mercury is locked in a 3/2 spin‑orbit resonance with its orbital mean motion. Traditional explanations invoke tidal torques raised by the Sun together with planetary perturbations that slowly despin the planet and allow capture into resonances. Recent geophysical evidence, however, indicates that Mercury possesses a large liquid core. The presence of a liquid core introduces an additional dissipative torque at the core‑mantle boundary (CMB) due to viscous friction. The authors model this “core‑mantle friction” as an extra term in the spin‑evolution equation, characterized by a friction coefficient γ, and combine it with the standard tidal torque.

When only tidal torques are considered, the probability of capture into the 3/2 resonance is high but not guaranteed; lower‑order resonances (e.g., 2/1) are less likely. Adding CMB friction dramatically raises the capture probability for all resonances, reaching essentially 100 % for the 2/1 resonance. This creates a paradox: if CMB friction is as effective as the model suggests, Mercury would almost certainly be trapped in a higher‑order resonance (2/1 or 5/2) and would never evolve to the observed 3/2 state.

The authors resolve this paradox by incorporating the chaotic evolution of Mercury’s orbital eccentricity (e). Over gigayear timescales, planetary perturbations cause e to wander over a wide range, occasionally dropping to very low values. For each resonance there exists a critical eccentricity e_c below which the resonant torque can no longer sustain capture; the planet can escape the resonance and continue its spin evolution. By analytically deriving e_c for the 3/2, 5/2, and 2/1 resonances, they find e_c≈0.025 for 3/2, ≈0.015 for 5/2, and ≈0.005 for 2/1.

To test the combined effect of tidal torques, CMB friction, and eccentricity variations, the authors performed 1000 numerical integrations of Mercury’s spin and orbit over 4 Gyr. Each integration includes: (i) the Sun’s tidal torque, (ii) the viscous CMB torque, (iii) the secular planetary perturbations that drive chaotic eccentricity changes, and (iv) the nonlinear coupling between spin and orbit that defines the resonances. The results are striking:

• In 99.8 % of the runs the final spin state ends up in some spin‑orbit resonance; free rotation or perfect synchronous rotation (1/1) are essentially absent.
• The most frequent final resonances are 5/2 (22 % of cases), 2/1 (32 %), and 3/2 (26 %). The remaining cases are scattered among higher‑order resonances.
• When the eccentricity falls below 0.025 during the integration, the probability that the planet ultimately settles in the 3/2 resonance rises to about 55 %. If e drops below the much lower threshold of 0.005, the probability climbs to roughly 73 %.

Thus, although the 3/2 resonance is not the most probable outcome in a purely statistical sense, it becomes the dominant outcome if Mercury’s eccentricity experienced sufficiently deep minima in its past. The chaotic nature of Mercury’s orbital evolution therefore provides a natural mechanism for destabilizing earlier captures (e.g., in 2/1 or 5/2) and allowing a later capture into the observed 3/2 state.

The paper’s conclusions have two major implications. First, core‑mantle friction is a crucial, previously underappreciated factor in planetary spin evolution; it can dramatically increase capture probabilities but also makes the system highly sensitive to orbital eccentricity. Second, the present 3/2 resonance likely records a specific episode in Mercury’s dynamical history when the eccentricity dipped below the critical value for the 3/2 resonance. This ties the planet’s internal structure (liquid core) to its long‑term orbital chaos.

Future work suggested by the authors includes (a) refining the physical basis for the CMB friction coefficient through laboratory experiments or more detailed interior modeling, (b) extending the analysis to other terrestrial planets (e.g., Venus, Earth) to assess whether similar mechanisms could have operated, and (c) using high‑precision orbital reconstructions to statistically characterize the frequency and depth of past eccentricity minima. Such studies would further illuminate how the interplay between interior dynamics and orbital chaos shapes the spin states of planets throughout the Solar System and beyond.