Editing Libration

{{Short description|Apparent oscillation of a minor body seen from the major body it orbits}}
{{About|astronomical observations|molecular motion|libration (molecule)|orbital mechanics|libration point orbit}}
{{distinguish|Liberation (disambiguation){{!}}liberation|libation|vibration}}
{{Generalize|date=July 2020}}
[[File:Moon Phases 2019 - Northern Hemisphere - 4K.webm|thumb|The lunar phases and librations in 2019 in the Northern Hemisphere at hourly intervals, with music, titles, and supplemental graphics]]
[[File:Lunar_libration_with_phase_Oct_2007_HD.gif|alt=Over one lunar month more than half of the Moon's surface can be seen from the surface of the Earth.|thumbtime=0:02|thumb|Simulated views of the Moon over one month, demonstrating librations in [[latitude]] and [[longitude]]. Also visible are the different phases, and the variation in visual size caused by the variable distance from the Earth.]]
[[File:MoonVisibleLibration.jpg|thumb|Theoretical extent of visible lunar surface (in green) due to libration, compared to the extent of the visible lunar surface without libration (in yellow). The projection is the [[Winkel tripel projection|Winkel Tripel projection]]. [[Mare Orientale]], just outside the yellow region, is brought into visibility from Earth by libration.]]
[[File:Diurnal libration of the Moon (animated).gif|thumb|Diurnal libration of the moon as actually observed from beginning to end of a single night. The two angles are created by the different position of the observer with respect to the Moon because of the rotation of the Earth over a few hours.]]

In lunar astronomy, '''libration''' is the cyclic variation in the apparent position of the [[Moon]] that is perceived by observers on the [[Earth]] and caused by changes between the orbital and rotational planes of the moon. It causes an observer to see slightly different hemispheres of the surface at different times. It is similar in both cause and effect to the changes in the Moon's [[apparent size]] because of changes in [[lunar distance (astronomy)|distance]]. It is caused by three mechanisms detailed below, two of which cause a relatively tiny '''physical libration''' via [[tidal forces]] exerted by the Earth. Such true librations are known as well for other moons with [[locked rotation]].

The quite different phenomenon of a [[trojan asteroid]]'s movement has been called ''Trojan libration'', and ''Trojan libration point'' means [[Lagrangian point]].

==Lunar libration==
[[File:Libration longitude 1080p30 1.gif|thumb|Animation showing the changing position of the Moon due to libration, in relation to a fictitious red position on perfectly circular orbit.]]
The [[Moon]] keeps one [[wikt:hemisphere|hemisphere]] of itself facing the [[Earth]] because of [[tidal locking]]. Therefore, the first view of the [[far side of the Moon]] was not possible until the Soviet probe [[Luna 3]] reached the Moon on October 7, 1959, and further [[lunar exploration]] by the [[United States]] and the [[Soviet Union]]. This simple picture is only approximately true since over time, slightly ''more'' than half (about 59% in total) of the Moon's surface is seen from Earth because of libration.<ref>{{cite encyclopedia |encyclopedia=World Book at NASA |first=Paul D. |last=Spudis |url=http://www.nasa.gov/worldbook/moon_worldbook.html |title=Moon |date=2004 |access-date=May 27, 2010 |url-status=dead |archive-url=https://web.archive.org/web/20130703162844/http://www.nasa.gov/worldbook/moon_worldbook.html |archive-date=July 3, 2013 }}</ref>

Lunar libration arises from three changes in perspective because of the non-circular and inclined orbit, the finite size of the Earth, and the orientation of the Moon in space. The first of these is called '''optical libration''', the second '''parallax''', and the third '''physical libration'''. Each of these can be divided into two contributions.

The following are the three types of lunar libration:
* '''Optical libration''', the combined libration of longitudinal and latitudinal libration produces a movement of the sub-Earth point and a wobbling view between the temporarily visible parts of the Moon, during a lunar orbit. This is not to be confused with the change of the Moon's [[apparent size]] because of the changing [[Lunar distance (astronomy)|distance between the Moon and the Earth]] during the [[Orbit of the Moon|Moon's elliptic orbit]], or with the change of [[positional angle]] because of the change in the position of the Moon's tilted axis, or with the observed swinging motion of the Moon because of the relative position of the Earth's tilted axis during an orbit of the Moon.<ref name="SVS 2021">{{cite web | last=SVS | first=NASA's | title=SVS: Moon Phase and Libration, 2020 | website=Home - NASA Scientific Visualization Studio | date=2021-08-18 | url=https://svs.gsfc.nasa.gov/4768 | access-date=2022-06-01}}</ref>
** '''Libration in [[Ecliptic coordinate system#Spherical coordinates|longitude]]''' results from the [[orbital eccentricity|eccentricity]] of the [[orbit of the Moon]] around the Earth; the Moon's rotation sometimes leads and sometimes lags its orbital position. The lunar libration in longitude was discovered by [[Johannes Hevelius]] in 1648.<ref name="Bergeron">{{cite book |editor=Jacqueline Bergeron |title=Highlights of Astronomy: As Presented at the XXIst General Assembly of the IAU, 1991 |publisher=Springer Science & Business Media |year=2013 |page=521 |isbn=978-9401128285}}</ref> It can reach 7°54′ in [[amplitude]].<ref name="Ratkowski">{{cite web |first1=Rob |last1=Ratkowski |first2=Jim |last2=Foster |url=https://epod.usra.edu/blog/2014/05/libration-of-the-moon.html |work=Earth Science Picture of the Day |date=May 31, 2014 |title=Libration of the Moon}}</ref> Longitudinal libration allows an observer on Earth to view at times further into the Moon's west and east respectively at different phases of the Moon's orbit.<ref name="SVS 2021"/>
** '''Libration in [[Ecliptic coordinate system#Spherical coordinates|latitude]]''' results from the Moon's [[axial tilt]] (about 6.7°) between its rotation axis and orbital axis around Earth. This is analogous to how Earth's seasons [[Season#Axial tilt|arise]] from [[Ecliptic#Obliquity of the ecliptic|its axial tilt]] (about 23.4°) between its rotation axis and orbital axis about the Sun. [[Galileo Galilei]] is sometimes credited with the discovery of the lunar libration in latitude in 1632<ref name="Bergeron" /> although [[Thomas Harriot]] or [[William Gilbert (astronomer)|William Gilbert]] might have done so before.<ref>Stephen Pumfrey: ''Harriot's Maps of the Moon: New Interpretations.'' Notes Rec. R. Soc. 63, 2009, [[doi:10.1098/rsnr.2008.0062]].</ref> Note [[Cassini's laws]]. It can reach 6°50′ in amplitude.<ref name="Ratkowski" />  The 6.7° depends on the orbit inclination of 5.15° and the negative equatorial tilt of 1.54°. Latitudinal libration allows an observer on Earth to view beyond the Moon's north pole and south pole at different phases of the Moon's orbit.<ref name="SVS 2021"/>
* '''Parallax libration''' depends on both the longitude and latitude of the location on Earth from which the Moon is observed.
** '''[[Diurnal motion|Diurnal]] libration''' is the small daily libration and oscillation from [[Earth's rotation]], which carries an observer first to one side and then to the other side of the straight line joining Earth's and the Moon's centers, allowing the observer to look first around one side of the Moon and then around the other—since the observer is on Earth's surface, not at its center. It reaches less than 1° in amplitude.<ref name="Ratkowski" />
*'''Physical libration''' is the oscillation of orientation in space about uniform rotation and precession. There are physical librations about all three axes. The sizes are roughly 100 seconds of arc. As seen from the Earth, this amounts to less than 1 second of arc. '''Forced physical librations''' can be predicted given the orbit and shape of the Moon. The periods of '''free physical librations''' can also be predicted, but their amplitudes and phases cannot be predicted.

== 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.

==See also==
* [[Parallactic angle]]

==References==
{{reflist}}
	
{{refbegin}}
* {{cite journal | author=J. Derral Mulholland, Eric C. Silverberg | title=Measurement of Physical Librations Using Laser Retroreflectors | journal=Earth, Moon, and Planets | date=1972 | volume=4 | issue=1–2 | pages=155–159 | bibcode=1972Moon....4..155M | doi=10.1007/BF00562923 | s2cid=122988376 }}
* {{cite book | last=Moore | first=Sir Patrick | author-link=Patrick Moore | others=Foreword by [[Arnold Wolfendale|Sir Arnold Wolfendale]] | title=Philip's Atlas of the Universe | publisher=Philip's | date=2003 | isbn=978-0-540-08707-5 | oclc=51966591 }}
{{refend}}

== External links ==
{{wiktionary | libration}}
* [http://www.phy6.org/stargaze/Smoon4.htm Libration of the Moon] from educational website [http://www.phy6.org/stargaze/Sintro.htm From Stargazers to Starships]
* [http://antwrp.gsfc.nasa.gov/apod/ap051113.html Astronomy Picture of the Day: 2005 November 13] – time-lapse animation of the Moon through one complete cycle, hosted by NASA
* [http://www.pixheaven.net/photo_us.php?nom=0505-0704 Libration: 2 years in 2 seconds] – 24 full moon pictures taken over two years, compiled in an animation (linked on page) showing the Moon's libration and variations in apparent diameter
* [http://users.pandora.be/lunar/jpeg/libratieboek.pdf Observing the Lunar Libration Zones]

{{The Moon}}

[[Category:Dynamics of the Solar System]]
[[Category:Orbit of the Moon]]
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