Viscoelastic relaxation within the Moon and the phase lead of its Cassini state

Analyses of Lunar Laser Ranging data show that the spin-symmetry axis of the Moon is ahead of its expected Cassini state by an angle of $ϕ_p$ = 0.27 arcsec. This indicates the presence of one or more dissipation mechanisms acting on the lunar rotation. A combination of solid-body tides and viscous core-mantle coupling have been proposed in previous studies. Here, we investigate whether viscoelastic deformation within a solid inner core at the centre of the Moon can also account for a part of the observed phase lead angle $ϕ_p$. We build a rotational dynamic model of the Cassini state of the Moon that comprises an inner core, a fluid core and a mantle, and where solid regions are allowed to deform viscoelastically in response to an applied forcing. We show that the presence of an inner core does not change the global monthly Q of the Moon and hence, that the contribution from solid-body tides to $ϕ_p$ is largely unaffected by an inner core. However, we also show that viscoelastic deformation within the inner core, acting to realign its figure axis with that of the mantle, can contribute significantly to $ϕ_p$ through inner core-mantle gravitational coupling. We show that the contribution to $ϕ_p$ is largest when the inner core viscosity is in the range of $10^{13}$ to $10^{14}$ Pa s, when the inner core radius is large and when the free inner core nutation frequency approaches a resonance with the precession frequency of $2π/18.6$ yr$^{-1}$.

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