Effect of an Ellipsoidal Solid Inner Core on Mercury's Obliquity

Abstract:

The gravitational torque on Mercury's solid mantle from a solid inner core displaces the spin axis from the Cassini state when the second--degree shapes of the mantle and inner core are misaligned. Dissipation brings the spins of the inner core, outer fluid core, and mantle to stationary equilibrium positions in the frame of the precessing orbit, where such misalignment is sustained. The equilibrium spin axes of the mantle, fluid core, and inner core all lie in the plane determined by the orbit normal and the Laplace plane normal and precess with the orbit. The fluid and inner core spins have $\sim 4$ arcmin higher obliquities than the mantle spin, which is itself displaced from the Cassini state toward higher obliquity by an angle that exceeds the 5 arcsec uncertainty in the observed spin axis position if a hydrostatic inner core size exceeds $\sim 0.35$ Mercury radii. The equilibrium mantle obliquity increases with the inner core size. Rather than placing an upper bound on the inner core size, this result means that the determination of the obliquity of the Cassini state and the determination of $C/MR^2$ therefrom are incomplete, where $C,\,M,\,{\rm and}\,R$ are Mercury's polar moment of inertia, mass and radius respectively. The dependence of the mantle obliquity on the inner core size and shape as well as $C/MR^2$ and the second degree coefficients in the expansion of Mercury's gravitational field $J_2\,{\rm and}\,C_{22}$ means our determination of $C/MR^2=0.346$ from only the latter three parameters is more uncertain than previously estimated, since the inner core size and shape remain unknown. The precise value of $C/MR^2$ is a crucial constraint on Mercury's internal structure.