THE EARLY LUNAR ORBIT AND PRINCIPAL MOMENTS OF INERTIA

For over 100 years it has been suggested that the Moon may have frozen in its principal moments of inertia when it was closer to the Earth, based on the relatively high power of the lunar second-degree gravity harmonics C₂₀ and C₂₂.  However, the ratio of C₂₀, which would freeze-in predominantly rotational potential, to C₂₂, which would freeze-in tidal potential, should be 10/3 for a synchronous zero eccentricity (e) satellite, while it is currently 9.1 using unnormalized coefficients.  This problem is resolved if the Moon had a higher e in the past, or possibly a 3:2 resonance of lunar spin to mean motion [1].  For the 3:2 resonance, a magma ocean would have precluded the permanent C₂₂ lithosphere deformation required to stabilize the resonance.  Here we show that an equivalent C₂₂ moment can be achieved with large-scale density anomalies that may have existed after lunar accretion.

To determine if high e orbits are dynamically possible we have integrated e and semimajor axis (a) evolution equations from refs. [2, 3].  These equations give current values of da/dt and de/dt (4 cm yr^-1 and 1.4 x 10^-11 yr^-1) that are in good agreement with observations (2 cm yr^-1 and 1.4 x 10^-11 yr^-1).  In the case of a 3:2 resonance we find e > 0.3 is possible for a = 6-30 Earth radii for a range energy dissipation factors and Love numbers.  Mignard (1980) and Wisdom (unpublished notes) obtain similar results for the synchronous case, suggesting that past high e orbits of either synchronous rotation or 3:2 resonance may have contributed to the modern moments of inertia.

A separate solution to the high C₂₀/C₂₂ ratio can also be explored assuming a lunar history with a lower e and negligible initial C₂₂.  In this scenario the equilibrium tidally evolved spin rate for an axially symmetric Moon is slightly faster than synchronous [4], which would longitudinally average and cancel C₂₂ power during freeze-in.  Future growth of modest C₂₂ (~10^-7) through processes such as large cratering events could then drive the Moon to synchronous rotation.  We derive upper limits on the C₂₂ moment that would permit supersynchnronous rotation, and estimate the effect on the modern moments of inertia.

References: [1]  Garrick-Bethell, I. et al. (2006) Science 313, 652. [2]  Mignard, F. (1980) Moon and Planets 23, 185.  [3]  Hut, P. (1981) Astron. Astrophys. 99, 126.  [4]  Peale, S. J. & Gold, T. (1965) Nature 206, 1240.