Some effects of elasticity on lunar rotation

A general Hamiltonian for a rotating Moon in the field of the Earth is expanded in terms of parameters orienting the spin angular momentum relative to the pricipal axes of the Moon and relative to coordinate axes fixed in the orbital plane. The effects of elastic distortion are included as modifications of the moment of inertia tensor, where the magnitude of the distortion is parameterized by the Love numberk2. The principal periodic terms in the longitude of a point on the Moon due to variations of the tide caused by the Earth are shown to have amplitudes between 3?.9 × 10-3 and 1?.6 × 10-2 with a period of an anomalistic month, 3?.0 × 10-4 and 1?.2 × 10-3 with a period of one-half an anomalistic month and 2?.4 × 10-4 and 9?.6 × 10-4 with a period of one-half of a nodical month. The extremes in the amplitudes correspond to rigidities of 8 × 1011 cgs and 2 × 1011 cgs, respectively, the former rigidity being comparable to that of the Earth. Only the largest amplitude given above is comparable to that detectable by the projected precision of the laser ranging to the lunar retrorereflectors, and this amplitude corresponds to an improbably low rigidity for the Moon. A detailed derivation of the free wobble of the lunar spin axis about the axis of maximum moment of inertia is given, where it is shown that elasticity can alter the period of the free wobble of 75.3 yr by only 3 × 10-4 to 10-3 of this period. Also, the effect of elasticity on the period of free libration is completely negligible by many orders of magnitude. If the Moon's rigidity is close to that of the Earth there is no effect of elasticity on the rotation which can be measured with the laser ranging and, therefore, no elastic properties of the Moon can be determined from variations in the rotation.