Effect of core–mantle and tidal torques on Mercury’s spin axis orientation

Abstract

The rotational evolution of Mercury's mantle plus crust and its core under conservative and dissipative torques is important for understanding the planet's spin state. Dissipation results from tidal torques and viscous, magnetic, and topographic torques contributed by interactions between the liquid core and solid mantle. For a spherically symmetric core-mantle boundary (CMB), the system goes to an equilibrium state wherein the spin axes of the mantle and core are fixed in the frame precessing with the orbit, and in which the mantle and core are differentially rotating. This equilibrium exhibits a mantle spin axis that is offset from the Cassini state by larger amounts for weaker core-mantle coupling for all three dissipative core-mantle coupling mechanisms, and the spin axis of the core is separated farther from that of the mantle, leading to larger differential rotation. Relatively strong core-mantle coupling is necessary to bring the mantle spin axis to a position within the uncertainty in its observed position, which is close to the Cassini state defined for a completely solid Mercury. Strong core-mantle coupling means that Mercury's response is closer to that of a solid planet. Measured or inferred values of parameters in all three core-mantle coupling mechanisms for a spherically symmetric CMB appear not to accomplish this requirement. For a hydrostatic ellipsoidal CMB, pressure coupling dominates the dissipative effects on the mantle and core positions, and dissipation with pressure coupling brings the mantle spin solidly to the Cassini state. The core spin goes to a position displaced from that of the mantle by about 3.55 arcmin nearly in the plane containing the Cassini state. The core spin lags the precessing plane containing the Cassini state by an increasing angle as the core viscosity is increased. With the maximum viscosity considered of ν ∼ 15.0 cm 2 /s if the coupling is by the circulation through an Ekman boundary layer or ν ∼ 8.75 × 10 5 cm 2 /s for purely viscous coupling, the core spin lags the precessing Cassini plane by 23 arcsec, whereas the mantle spin lags by only 0.055 arcsec. Larger, non hydrostatic values of the CMB ellipticity also result in the mantle spin at the Cassini state, but the core spin is moved closer to the mantle spin. Current measurement uncertainties preclude using the mantle offset to constrain the internal core viscosity.-2 -

Consequences of a solid inner core on Mercury’s spin configuration

“…This result suggests that a smaller moment of inertia can compensate for the gravitational torque from the solid inner core and bring the mantle spin back to its observed position in the presence of the inner core. We found earlier that dissipative torques displaced the spin from the Cassini state by amounts that exceeded observational uncertainties, but that pressure torques between a liquid core and solid mantle restored the mantle spin to the nominal Cassini state position (Peale et al 2014). Here the introduction of an ellipsoidal solid inner core overrides this pressure effect and moves the mantle spin away from the Cassini state that corresponds to there being no solid inner core.…”

“…(The theoretical position of the Cassini state is that determined for a rigid planet that depends on J 2 , C 22 , and C/mR 2 c (e.g., Peale 1969Peale , 1974. The same position results for a planet with a fluid core but no solid inner core (Peale et al 2014). The measured values of the obliquity, J 2 , and C 22 are, in fact, used together with the obliquity to determine C/mR 2 c .)…”

“…This result also means that the determination of C/mR 2 c will depend on the size and shape of the inner core and that it will thereby be more uncertain than heretofore assumed. Peale et al (2014) found that as the dissipative core-mantle torques weakened, the mantle spin approached a Cassini state with a lower obliquity. The weakened dissipative torque means that the mantle is less affected by the fluid core, and the Cassini state approached is that for the smaller moment of inertia of the mantle alone.…”

“…Pressure torque between Mercury's fluid core and its mantle drive Mercury's spin axis to the Cassini state (Peale et al 2014) (hereinafter Paper 1). It thereby dominates dissipative viscous, topographic, and magnetic effects that would otherwise result in significant displacement of the observable spin axis from the Cassini state position.…”

“…The displacement vanishes if the direct gravitational torque between the mantle and inner core is set to zero. The obliquity is used to determine the moment of inertia of Mercury (e.g., Peale et al 2014). The increase in the obliquity of the mantle from gravitational torques due to the inner core means that C/mR 2 c must be reduced by an amount that increases with inner core radius to bring the mantle spin back to the observed position.…”