Editing Libration (section)

== Physical libration ==
Also called real libration, as opposed to the optical libration of longitudinal, latitudinal and diurnal types, the orientation of the Moon exhibits small oscillations of the pole direction in space and rotation about the pole.

This libration can be differentiated between forced and free libration. Forced libration is caused by the forces exerted during the Moon's orbit around the Earth and the Sun, and free libration represents oscillations that occur over longer time periods.

=== Forced physical libration ===
[[File:Libration des Vollmondes.jpg|thumb|Full moon at opposite librations]]
[[Cassini's laws]] state the following:

# 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 three axes. These oscillations are called physical librations. Apart from the 1.5427° 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''.<ref>{{Cite journal|last=Eckhardt|first=Donald H.|date=1981|title=Theory of the libration of the moon|url=https://doi.org/10.1007/BF00911807|journal=The Moon and the Planets|language=en|volume=25|issue=1|pages=3–49|doi=10.1007/BF00911807|bibcode=1981M&P....25....3E|s2cid=123650603 |issn=1573-0794|url-access=subscription}}</ref> 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'' = {{val|6.31|e=-4}} and ''γ'' = {{val|2.28|e=-4}}.<ref>{{Cite journal|last1=Williams|first1=James G.|last2=Konopliv|first2=Alexander S.|last3=Boggs|first3=Dale H.|last4=Park|first4=Ryan S.|last5=Yuan|first5=Dah-Ning|last6=Lemoine|first6=Frank G.|last7=Goossens|first7=Sander|last8=Mazarico|first8=Erwan|last9=Nimmo|first9=Francis|last10=Weber|first10=Renee C.|last11=Asmar|first11=Sami W.|date=2014|title=Lunar interior properties from the GRAIL mission: Lunar Interior Properties|journal=Journal of Geophysical Research: Planets|language=en|volume=119|issue=7|pages=1546–1578|doi=10.1002/2013JE004559|s2cid=7045590 |doi-access=free}}</ref>

The placement of three retroreflectors on the Moon by the [[Lunar Laser Ranging experiment]] and two retroreflectors by [[Lunokhod programme|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.<ref>{{Cite journal |last=Calame |first=O. |date=1976 |title=Determination des librations libres de la Lune, de l'analyse des mesures de distances par laser |journal=Comptes Rendus de l'Académie des Sciences, Série B |language=fr |volume=282 |issue=5 |pages=133–135 |bibcode=1976CRASB.282..133C}}</ref><ref>{{Cite journal |last=Calame |first=O. |date=1976 |title=Free librations of the Moon determined by an analysis of laser range measurements |journal=The Moon |volume=15 |issue=3–4 |pages=343–352 |bibcode=1976Moon...15..343C |doi=10.1007/BF00562246 |s2cid=119505889 |bibcode-access=free |doi-access=free}}</ref> 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.<ref>{{Cite journal|last1=Rambaux|first1=N|last2=Williams|first2=J. G.|date=2011|title=The Moon's physical librations and determination of their free modes|journal=Celestial Mechanics and Dynamical Astronomy|volume=109|issue=1|pages=85–100|doi=10.1007/s10569-010-9314-2|bibcode=2011CeMDA.109...85R|s2cid=45209988|url=https://hal.archives-ouvertes.fr/hal-00588671/file/PEER_stage2_10.1007%252Fs10569-010-9314-2.pdf}}</ref>

The fluid core can cause a fourth mode with a period around four centuries.<ref>{{Cite journal|last1=Viswanathan|first1=V.|last2=Rambaux|first2=N|last3=Fienga|first3=A.|last4=Laskar|first4=J.|last5=Gastineau|first5=M.|date=2019|title=Observational constraint on the radius and oblateness of the lunar core-mantle boundary|url=https://doi.org/10.1029/2019GL082677|journal=Geophysical Research Letters|volume=46|issue=13|pages=7295–7303|doi=10.1029/2019GL082677|arxiv=1903.07205|bibcode=2019GeoRL..46.7295V|s2cid=119508748 }}</ref> 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.