An Estimate on the Lunar Figure

In 1799 Laplace discovered that the three principal moments of the Moon are not in equilibrium with the Moon's current orbital and rotational state. Some authors suggested that the Moon may carry a fossil figure. More than 3 billion years ago, the liquid Moon was closer to the Earth and revolved faster. Then the Moon migrated outwards and its rotation slowed down. During the early stage of this migration, the Moon was continually subjected to tidal and rotational stretching and formed into an ellipsoid. Subsequently the Moon cooled down and solidified quickly. Eventually, the solid Moon's lithosphere was stable and as a result we may see the very early lunar figure.In this paper, by using the lunar libration parameters and the sphericalharmonic gravity coefficients, the lengths of three radii a, b and c of the ellipsoid and the Moon's figure in tidal equilibrium are calculated. We have obtained three main conclusions as follows.(1) In the very early stage the Moon may be very close to the Earth, then it cooled down and solidified about 3 billion years ago. The present Moon is a fossil figure.(2) We demonstrate in this paper that the tidal deformation of liquid Moon was 1.934 times larger than that of the equilibrium tide. So a liquid Love number of hf = 1.934 must be taken into account when calculating the true lunar figure using the lunar spherical-harmonic gravity coefficients.(3) According to the differences between a, b and c, a Moon-Earth distance (1.7455 × 108 m) and a lunar spin period (3.652 day) at the moment of its coagulation are determined. The lunar orbital period then is calculated (8.34 d). Finally we argue that the Moon was close to a 2:1 spin-orbit resonance when it froze.