Libration

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The lunar phases and librations in 2019 in the Northern Hemisphere at hourly intervals, with music, titles, and supplemental graphics
Over one lunar month more than half of the Moon's surface can be seen from the surface of the Earth.
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 Earth.
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. Mare Orientale, just outside the yellow region, is brought into visibility from Earth by libration.

In lunar astronomy, libration is the wagging or wavering of the Moon perceived by Earth-bound observers and caused by changes in their perspective. It permits 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 due to changes in 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[edit]

The Moon keeps one hemisphere of itself facing the Earth, due to 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: over time, slightly more than half (about 59% in total) of the Moon's surface is seen from Earth due to libration.[1]

Lunar libration arises from three changes in perspective due to: 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 is called parallax, and the third is physical libration. Each of these can be divided into two contributions.  

The following are the four types of lunar libration:

  • Libration in longitude results from the eccentricity of the Moon's orbit around 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.[2] It can reach 7°54′ in amplitude.[3]
  • Libration in latitude results from a slight inclination (about 6.7°) between the Moon's axis of rotation and the normal to the plane of its orbit around Earth. Its origin is analogous to how the seasons arise from Earth's revolution about the Sun. Galileo Galilei is sometimes credited with the discovery of the lunar libration in latitude in 1632,[2] although Thomas Harriot or William Gilbert might have done so before.[4] Note Cassini's laws. It can reach 6°50′ in amplitude.[3] The 6.7º depends on the orbit inclination of 5.15º and the negative equatorial tilt of 1.54º.
  • Diurnal libration is a small daily oscillation due to 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.[3] Diurnal libration is one effect of parallax, which depends on both the longitude and latitude of the site.
  • Physical libration is the oscillation of orientation in space about uniform rotation and precession. There are physical librations about all 3 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[edit]

The orientation of the Moon exhibits small oscillations of the pole direction in space and rotation about the pole.

Forced physical libration[edit]

Cassini’s laws state that:

  1. The Moon rotates uniformly about its polar axis keeping one side toward the Earth.
  2. The Moon’s equator plane is tilted with respect to the ecliptic plane and it precesses uniformly along the ecliptic plane.
  3. 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.[5] The sensitive combinations are β = (CA)/B and γ = (BA)/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 = 6.31×10−4 and γ = 2.28×10−4.[6]

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[edit]

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.[7][8] Modern values are:

  1. 1.3 seconds of arc with a 1056 day (2.9 year) period for rotation about the polar axis,
  2. a 74.6 year elliptical wobble of the pole of size 8.18 × 3.31 arcseconds, and
  3. an 81 year rotation of the pole in space that is 0.03 seconds of arc in size.[9]

The fluid core can cause a fourth mode with a period around 4 centuries.[10] 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[edit]

References[edit]

  1. ^ Spudis, Paul D. (2004). "Moon". World Book at NASA. Archived from the original on July 3, 2013. Retrieved May 27, 2010.
  2. ^ a b Jacqueline Bergeron, ed. (2013). Highlights of Astronomy: As Presented at the XXIst General Assembly of the IAU, 1991. Springer Science & Business Media. p. 521. ISBN 978-9401128285.
  3. ^ a b c Ratkowski, Rob; Foster, Jim (May 31, 2014). "Libration of the Moon". Earth Science Picture of the Day.
  4. ^ Stephen Pumfrey: Harriot’s Maps of the Moon: New Interpretations. Notes Rec. R. Soc. 63, 2009, doi:10.1098/rsnr.2008.0062.
  5. ^ Eckhardt, Donald H. (1981). "Theory of the libration of the moon". The Moon and the Planets. 25 (1): 3–49. Bibcode:1981M&P....25....3E. doi:10.1007/BF00911807. ISSN 1573-0794.
  6. ^ Williams, James G.; Konopliv, Alexander S.; Boggs, Dale H.; Park, Ryan S.; Yuan, Dah-Ning; Lemoine, Frank G.; Goossens, Sander; Mazarico, Erwan; Nimmo, Francis; Weber, Renee C.; Asmar, Sami W. (2014). "Lunar interior properties from the GRAIL mission: Lunar Interior Properties". Journal of Geophysical Research: Planets. 119 (7): 1546–1578. doi:10.1002/2013JE004559.
  7. ^ Calame, O. (1976). "Determination des librations libres de la Lune, de l'analyse des mesures de distances par laser". Comptes Rendus de l'Académie des Sciences, Série B. 282: 133–135.
  8. ^ Calame, O. (1976). "Free librations of the Moon determined by an analysis of laser range measurements". The Moon. 15 (3–4): 343–352. Bibcode:1976Moon...15..343C. doi:10.1007/BF00562246.
  9. ^ Rambaux, N; Williams, J. G. (2011). "The Moon's physical librations and determination of their free modes" (PDF). Celestial Mechanics and Dynamical Astronomy. 109: 85–100. doi:10.1007/s10569-010-9314-2.
  10. ^ Viswanathan, V.; Rambaux, N; Fienga, A.; Laskar, J.; Gastineau, M. (2019). "Observational constraint on the radius and oblateness of the lunar core-mantle boundary". Geophysical Research Letters. 46 (13): 7295–7303. arXiv:1903.07205. Bibcode:2019GeoRL..46.7295V. doi:10.1029/2019GL082677.

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