# Theory of the libration of the moon

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## Abstract

In 1693, Jean Dominique Cassini disclosed his finding that the rotational motion of the Moon could be neatly described by the superposition of two uniform motions, a prograde rotation of the Moon about its polar axis and a retrograde precession of the Moon's equator along the ecliptic. The description of these motions is now called Cassini's laws. The theoretical explanation of Cassini's laws shows that physical librations with amplitudes less than 0″.5, as seen from the Earth, must also exist. Until 1970, the physical librations were just marginally discernible, and the dynamical theory was developed to a level far superior to the quality of the observations.

In 1970 the resolution of libration observations jumped by a factor of 10^{4} over earlier techniques, and existing theories became inadequate for analyzing the observations. This paper presents a new semianalytic libration theory that is of use for analyzing observations. In this development the Moon is assumed to be either rigid, elastic or anelastic, and its gravity potential is represented through its fourth-degree harmonics. The Moon is considered to be moving about the Earth in an orbit that is perturbed by the Sun (the ALE of Deprit, Henrard and Rom), and by the planets and the figures of the Earth and Moon (from the ILE, principally derived by Brown). The direct effects of the rotation of the plane of the ecliptic and the figure of the Earth are also considered. Tables for physical libration variables are tabulated which are truncated at 0″.010.

### Glossary of principal terms

*A, B, C*lunar principal moments of inertia

*a, a′*mean Earth-Moon and Earth-Sun distances

*B*lunar latitude

*C*lunar moment of inertia matrix

*C*_{nm},*S*_{nm}coefficients of spherical harmonic (

*n*= degree,*m*= order) expansion of lunar gravity potential*e, e′*eccentricities of lunar and solar orbits

*G*gravitational constant

*I*mean inclination of lunar equator to ecliptic

*i*inclination of lunar orbit to ecliptic

*J*_{2}dynamical form-factor for Earth

*k*potential disturbance Love number of Moon

*L*lunar angular momentum

*L*(or^{*}*L), L′*mean longitudes of Moon and Sun

*l, l′, F, D*Delaunay arguments

*M, m, m′*masses of Moon, Earth and Sun

*n*mean rate of lunar orbit

- N
_{2}, N_{3}, N_{4} second, third and fourth degree torques induced by Earth on Moon

- N′
_{2} second-degree torque induced by Sun on Moon

*p*general precession rate

- p
selenographic unit vector toward pole of ecliptic

*q*_{1},*q*_{2}*p*_{1}cos*F−p*_{2}sin*F, p*_{1}sin*F+p*_{2}cos*F**R*mean radius of Moon

*R*_{⊕}equatorial radius of Earth

*r, r′*Earth-Moon and Earth-Sun distances

*s*center equation and solar inequalities in lunar longitude

*t*ephemeris time

- u, u′
selenographic unit vectors toward Earth and Sun

*U*_{ijk},*U*_{ij}, etc.*u*_{i}*u*_{j}*u*_{k},*u*_{i}*u*_{j}, etc.*v*longitude of Earth from descending node of Moon's equator

*V*_{1}*V*_{2}gravitational potential of the Moon and its second degree component

*W*_{1},*W*_{2}centrifugal and tidal potentials of the Moon

- α, β, γ
*(C−B)/A, (C−A)B*and*(B−A)/C*- \(\bar \gamma \)
resonant value of γ for 2

*l*−2*F*argument- θ
inclination of lunar equator to ecliptic

- λ
correction factor for applying Kepler's third law to lunar orbit

- μ
\( - \dot \Omega /n\)

- ν
_{1},ν_{2},ν_{3} resonant angular rates for p and τ

- Π, π
longitude of ecliptic rotation axis and rotation rate

- ϱ, σ, τ
lunar physical librations in inclination, node and longitude

- ϕ, ψ
Euler angles which, along with θ, define orientation of Moon

- ω
lunar rotation vector

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