Editing Libration (section)

=== 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.