oscillation of longer time periods. Forced physical libration Cassini’s laws state that: The Moon rotates uniformly about its polar axis keeping one side toward the Earth. The Moon’s equator plane is tilted with respect to the ecliptic plane and it precesses uniformly along the ecliptic plane. The descending node of the equator on the ecliptic matches the ascending node of the orbit plane. In addition to uniform rotation and uniform precession of the equator plane, the Moon has small oscillations of orientation in space about all 3 axes. These oscillations are called physical librations. Apart from the 1.5427º tilt between equator and ecliptic, the oscillations are approximately ±100 seconds of arc in size. These oscillations can be expressed with trigonometric series that depend on the lunar moments of inertia A < B < C. The sensitive combinations are β = (C – A)/B and γ = (B – A)/C. The oscillation about the polar axis is most sensitive to γ and the 2-dimensional direction of the pole, including the 1.5427° tilt, is most sensitive to β. Consequently, accurate measurements of the physical librations provide accurate determinations of β = and γ = . The placement of 3 retroreflectors on the Moon by the Lunar Laser Ranging experiment and 2 retroreflectors by Lunokhod rovers allowed accurate measurement of the physical librations by laser ranging to the Moon. Free physical libration A free physical libration is similar to the solution of the reduced equation for linear differential equations. The periods of the free librations can be calculated, but their amplitudes must be measured. Lunar Laser Ranging provides the determinations. The two largest free librations were discovered by O. Calame. Modern values are: 1.3 seconds of arc with a 1056 day (2.9 year) period for rotation about the polar axis, a 74.6 year elliptical wobble of the pole of size 8.18 × 3.31 arcseconds, and an 81 year rotation of the pole in space that is 0.03 seconds of arc in size. The fluid core can cause a fourth mode with a period around 4 centuries. The free librations are expected to damp out in times very short compared to the age of the Moon. Consequently, their existence implies that there must be one or more stimulating mechanisms. See also Parallactic angle References External links Libration of the Moon from educational website From Stargazers to Starships Astronomy Picture of the Day: 2005 November