The Lunar Fossil Figure in a Cassini State

The present ellipsoidal figure of the Moon requires a deformation that is significantly larger than the hydrostatic deformation in response to the present rotational and tidal potentials. This has long been explained as due to a fossil rotational and tidal deformation from a time when the Moon was closer to Earth. Previous studies constraining the orbital parameters at the time the fossil deformation was established find that high orbit eccentricities ( e ≳ 0.2) are required at this ancient time, which is difficult to reconcile with the freezing of a fossil figure owing to the expected large tidal heating. We extend previous fossil deformation studies in several ways. First, we consider the effect of removing South Pole−Aitken (SPA) contributions from the present observed deformation using a nonaxially symmetric SPA model. Second, we use the assumption of an equilibrium Cassini state as an additional constraint, which allows us to consider the fossil deformation due to nonzero obliquity self-consistently. A fossil deformation established during Cassini state 1, 2, or 4 is consistent with the SPA-corrected present deformation. However, a fossil deformation established during Cassini state 2 or 4 requires large obliquity and orbit eccentricity ( ϵ ∼ 68° and e ∼ 0.65), which are difficult to reconcile with the corresponding strong tidal heating. The most likely explanation is a fossil deformation established during Cassini state 1, with a small obliquity ( ϵ ∼ −0.2°) and an orbit eccentricity range that includes zero eccentricity (0 ≤ e ≲ 0.3).