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Polar motion
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==Principle== In the absence of external torques, the vector of the [[angular momentum]] '''M''' of a rotating system remains constant and is directed toward a fixed point in space. If the earth were perfectly symmetrical and rigid, '''M''' would remain aligned with its axis of symmetry, which would also be its [[axis of rotation]]. In the case of the Earth, it is almost identical with its axis of rotation, with the discrepancy due to shifts of mass on the planet's surface. The vector of the [[figure axis]] '''F''' of the system (or maximum principal axis, the axis which yields the largest value of moment of inertia) wobbles around '''M'''. This motion is called [[Euler]]'s [[free nutation]]. For a rigid Earth which is an oblate [[spheroid]] to a good approximation, the figure axis '''F''' would be its geometric axis defined by the geographic north and south pole, and identical with the axis of its polar moment of inertia. The Euler period of free nutation is (1) {{pad|4em}} τ<sub>E</sub> = 1/ν<sub>E</sub> = A/(C − A) sidereal days ≈ 307 sidereal days ≈ 0.84 sidereal years {{nowrap|ν<sub>E</sub> {{=}} 1.19}} is the normalized Euler frequency (in units of reciprocal years), {{nowrap|C {{=}} 8.04 × 10<sup>37</sup> kg m<sup>2</sup>}} is the polar moment of inertia of the Earth, A is its mean equatorial moment of inertia, and {{nowrap|C − A {{=}} 2.61 × 10<sup>35</sup> kg m<sup>2</sup>}}.<ref name=Lambeck/><ref>{{cite book|last1=Munk|first1=Walter H.|last2=MacDonald|first2=Gordon J.F.|title=The Rotation of the Earth A Geophysical Discussion.|date=2009|publisher=Cambridge University Press|location=Cambridge|isbn=978-0521104067|edition=Digitally printed}}</ref> The observed angle between the figure axis of the Earth '''F''' and its angular momentum '''M''' is a few hundred [[milliarcseconds]] (mas). This rotation can be interpreted as a linear [[displacement (physics)|displacement]] of either [[geographical pole]] amounting to several meters on the surface of the Earth: 100 mas [[Subtended angle|subtends]] an [[arc length]] of 3.082 m, when converted to radians and multiplied by the [[Earth's figure|Earth's]] [[polar radius]] (6,356,752.3 m). Using the geometric axis as the primary axis of a new body-fixed coordinate system, one arrives at the Euler equation of a gyroscope describing the apparent motion of the rotation axis about the geometric axis of the Earth. This is the so-called polar motion.<ref name=MoritzMueller>{{cite book|last1=Moritz|first1=Helmut|last2=Mueller|first2=Ivan Istvan|title=Earth rotation: theory and observation|date=1987|publisher=Ungar}}</ref> Observations show that the figure axis exhibits an annual wobble forced by surface mass displacement via atmospheric and/or ocean dynamics, while the free nutation is much larger than the Euler period and of the order of 435 to 445 sidereal days. This observed free nutation is called [[Chandler wobble]]. There exist, in addition, polar motions with smaller periods of the order of decades.<ref>{{cite journal|last1=Gross|first1=Richard S.|last2=Lindqwister|first2=Ulf J.|title=Atmospheric excitation of polar motion during the GIG '91 Measurement Campaign|journal=Geophysical Research Letters|date=4 May 1992|volume=19|issue=9|pages=849–852|doi=10.1029/92GL00935|bibcode=1992GeoRL..19..849G}}</ref> Finally, a secular polar drift of about 0.10{{nbsp}}m per year in the direction of 80° west has been observed which is due to mass redistribution within the Earth's interior by continental drift, and/or slow motions within mantle and core which gives rise to changes of the moment of inertia.<ref name=MoritzMueller/> The annual variation was discovered by Karl Friedrich Küstner in 1885 by exact measurements of the variation of the latitude of stars, while [[Seth Carlo Chandler|S.C. Chandler]] found the free nutation in 1891.<ref name=MoritzMueller/> Both periods superpose, giving rise to a [[beat frequency]] with a period of about 5 to 8 years (see Figure 1). This polar motion should not be confused with the changing direction of the [[Earth's rotation axis]] relative to the stars with different periods, caused mostly by the torques on the [[Geoid]] due to the gravitational attraction of the Moon and Sun. They are also called [[nutation]]s, except for the slowest, which is the [[precession of the equinoxes]].
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