Tidal dissipation in the lunar magma ocean and its effect on the early evolution of the Earth-Moon system

The present-day inclination of the Moon reflects the entire history of its thermal and orbital evolution. The Moon likely possessed a global magma ocean following the Moon-forming impact. In this work, we develop a coupled thermal-orbital evolution model that takes into account obliquity tidal heating in the lunar magma ocean. Dissipation in the magma ocean is so effective that it results in rapid inclination damping at semi-major axes beyond about 20 Earth radii (RE), because of the increase in lunar obliquity as the so-called Cassini state transition at ≈30 RE is approached. There is thus a “speed limit” on how fast the Moon can evolve outwards while maintaining its inclination: if it reaches 20 RE before the magma ocean solidifies, any early lunar inclination cannot be maintained. We find that for magma ocean lifetimes of 10 Myr or more, the Earth's tidal quality factor Q must have been >300 to maintain primordial inclination, implying an early Earth 1-2 orders of magnitude less dissipative than at present. On the other hand, if tidal dissipation on the early Earth was stronger, our model implies rapid damping of the lunar inclination and requires subsequent late excitation of the lunar orbit after the crystallization of the lunar magma ocean.