Moon’s Physical Libration

This article calls what I once thought was the Moon’s main libration phenomenon, ‘physical libration’ and notes that it was classically considered but then neglected as unobservably small. The Wikipedia’s article on libration omits physical libration. The recent corner reflectors left on the Moon are now used to measure this small phenomenon; it is small but has been detected and measured for the Moon.

Suppose that

the Moon’s orbit about the Earth were perfectly circular,
the Moon’s rotational axis were perpendicular to its orbital plane,
the Earth’s and Moon’s rotation were the same as the Moon’s orbital period.

With these assumptions the distance between some point on Earth and some point on the Moon does not vary with time. The forces are constant and any tidal stress within either body is constant. There is no tidal dissipation in this case.

With these assumptions the three librations described by Wikipedia each vanish. Physical libration is yet another possible periodic Moon motion compatible with these assumptions and that would allow us to see more than 1/2 of the Moon from a fixed point on Earth. Physical libration could be initiated from this configuration by imparting a small additional angular momentum to the Moon about its center and some axis perpendicular to the line between the bodies. The Moon is an ellipsoid with its longest axis pointing towards the Earth and its shortest North and South. This is due to the tidal force of the Earth on the Moon. The additional rotation would soon cause the ellipsoidal axis not to go thru the Earth and this would cause a restoring torque on the Moon. This would lead to an oscillation to which the term ‘physical libration’ presumably applies. The Moon assumed this shape as it ceased to periodically display its entire equator to the Earth. Before that a tide presumably circulated on the Moon as it does today on Earth. Perhaps this shape arose as the Moon last froze and the tides there ceased to wash around the Moon. Physical libration was large at first.

The period of physical libration is inversely proportional to the square root of the tidal force. None of the other librations are like this. I recall that this libration period is several months.

Current moon reflector ranging.

This article proposes that physical libration is the energy source for Enceladeus. See also Nutation.

More Abstractly

Consider two fluid bodies orbiting about each other. The orbits are circular. The three angular velocities are the same and constant. If the momentum is sufficient each body will be an ellipsoid, to good approximation, with three different axes. The largest axis will be in common with the two bodies. The smallest axis will be perpendicular to the plane of the orbit. Distance between points in this configuration is constant; there is no dissipation.

Suppose both bodies freeze without changing shape. We then impart a small additional angular momentum about the North-South axis to the smaller body. A tidal torque will cause the angular momentum of the smaller body to oscillate so that its major axis oscillates about the line joining the bodies. This is called ‘physical libration’. If we add a small angular momentum about the intermediate axis, the tidal torque on the small body due to the inequality of its major and minor axes causes a faster oscillation. Either of these oscillations lead to varying elastic forces within the bodies and are likely to cause the librations to die out.