Effect of the Earth’s compression on the physical libration of the Moon

Abstract

The effect of the Earth’s compression on the physical libration of the Moon is studied using a new vector method. The moment of gravitational forces exerted on the Moon by the oblate Earth is derived considering second order harmonics. The terms in the expression for this moment are arranged according to their order of magnitude. The contribution due to a spherically symmetric Earth proves to be greater by a factor of 1.34 × 106 than a typical term allowing for the oblateness. A linearized Euler system of equations to describe the Moon’s rotation with allowance for external gravitational forces is given. A full solution of the differential equation describing the Moon’s libration in longitude is derived. This solution includes both arbitrary and forced oscillation harmonics that we studied earlier (perturbations due to a spherically symmetric Earth and the Sun) and new harmonics due to the Earth’s compression. We posed and solved the problem of spinorbital motion considering the orientation of the Earth’s rotation axis with regard to the axes of inertia of the Moon when it is at a random point in its orbit. The rotation axes of the Earth and the Moon are shown to become coplanar with each other when the orbiting Moon has an ecliptic longitude of L☾ = 90° or L☾ = 270°. The famous Cassini’s laws describing the motion of the Moon are supplemented by the rule for coplanarity when proper rotations in the Earth-Moon system are taken into account. When we consider the effect of the Earth’s compression on the Moon’s libration in longitude, a harmonic with an amplitude of 0.03″ and period of T8 = 9.300 Julian years appears. This amplitude exceeds the most noticeable harmonic due to the Sun by a factor of nearly 2.7. The effect of the Earth’s compression on the variation in spin angular velocity of the Moon proves to be negligible.