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{{Use dmy dates|date=December 2022}}
[[File:Earth precession.svg|thumb|right|Precessional movement of Earth. Earth rotates (white arrows) once a day around its rotational axis (red); this axis itself rotates slowly (white circle), completing a rotation in approximately 26,000 years<ref name=":crs-esaa-99"/>]]
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==Nomenclature==
[[File:Gyroscope precession.gif|thumb|300px|[[Precession]] of a [[gyroscope]]. In a similar way to how the force from the table generates this phenomenon of precession in the spinning gyro, the gravitational pull of the Sun and Moon on the Earth's equatorial bulge generates a very slow precession of the Earth's axis (see [[
In describing this motion astronomers generally have shortened the term to simply "precession". In describing the ''cause'' of the motion physicists have also used the term "precession", which has led to some confusion between the observable phenomenon and its cause, which matters because in astronomy, some precessions are real and others are apparent. This issue is further obfuscated by the fact that many astronomers are physicists or astrophysicists.
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==Effects==
[[File:Earth axial precession.svg|thumb|300px|The coincidence of the annual cycles of the apses (closest and further approach to the
The precession of the Earth's axis has a number of observable effects. First, the positions of the south and north [[celestial pole]]s appear to move in circles against the space-fixed backdrop of stars, completing one circuit in approximately 26,000 years. Thus, while today the star [[Polaris]] lies approximately at the north celestial pole, this will change over time, and other stars will become the "[[pole star|north star]]".<ref name="Astro 101"/> In approximately 3,200 years, the star [[Gamma Cephei]] in the Cepheus constellation will succeed Polaris for this position. The south celestial pole currently lacks a bright star to mark its position, but over time precession also will cause bright stars to become [[South Star]]s. As the celestial poles shift, there is a corresponding gradual shift in the apparent orientation of the whole star field, as viewed from a particular position on Earth.
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For identical reasons, the apparent position of the Sun relative to the backdrop of the stars at some seasonally fixed time slowly regresses a full 360° through all twelve traditional constellations of the [[zodiac]], at the rate of about 50.3 [[Arcsecond|seconds of arc]] per year, or 1 degree every 71.6 years.
At present, the rate of precession corresponds to a period of 25,772 years, so tropical year is shorter than sidereal year by 1,224.5 seconds {{nowrap|(20 min 24.5
The rate itself varies somewhat with time (see [[#Values|Values]] below), so one cannot say that in exactly 25,772 years the Earth's axis will be back to where it is now.
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==History==
===Hellenistic world===
====Hipparchus====
The discovery of precession usually is attributed to [[Hipparchus]] (190–120 BC) of [[Rhodes]] or [[İznik|Nicaea]], a [[Greek astronomy|Greek astronomer]]. According to [[Ptolemy]]'s ''[[Almagest]]'', Hipparchus measured the longitude of [[Spica]] and other bright stars. Comparing his measurements with data from his predecessors, [[Timocharis]] (320–260 BC) and [[Aristillus]] (~280 BC), he concluded that Spica had moved 2° relative to the [[September equinox|autumnal equinox]]. He also compared the lengths of the [[tropical year]] (the time it takes the Sun to return to an equinox) and the [[sidereal year]] (the time it takes the Sun to return to a fixed star), and found a slight discrepancy. Hipparchus concluded that the equinoxes were moving ("precessing") through the zodiac, and that the rate of precession was not less than 1° in a century, in other words, completing a full cycle in no more than
Virtually all of the writings of Hipparchus are lost, including his work on precession. They are mentioned by Ptolemy, who explains precession as the rotation of the [[celestial sphere]] around a motionless Earth. It is reasonable to presume that Hipparchus, similarly to Ptolemy, thought of precession in [[geocentric]] terms as a motion of the heavens, rather than of the Earth.
====Ptolemy====
The first astronomer known to have continued Hipparchus's work on precession is
====Other authors====
Most ancient authors did not mention precession and, perhaps, did not know of it. For instance, [[Proclus]] rejected precession, while [[Theon of Alexandria]], a commentator on Ptolemy in the fourth century, accepted Ptolemy's explanation. Theon also reports an alternate theory:
:
Instead of proceeding through the entire sequence of the zodiac, the equinoxes "trepidated" back and forth over an arc of 8°. The theory of [[trepidation]] is presented by Theon as an alternative to precession.
===Alternative discovery theories===
====Babylonians====
Various assertions have been made that other cultures discovered precession independently of Hipparchus. According to [[Al-Battani]], the [[Babylonian astronomy|Chaldean astronomers]] had distinguished the [[tropical year|tropical]] and [[sidereal year]] so that by approximately 330 BC, they would have been in a position to describe precession, if inaccurately, but such claims generally are regarded as unsupported.<ref>{{Cite journal |jstor = 595428|title = The Alleged Babylonian Discovery of the Precession of the Equinoxes|journal = Journal of the American Oriental Society|volume = 70|issue = 1|pages = 1–8|last1 = Neugebauer|first1 = O.|year = 1950|doi = 10.2307/595428}}</ref>
====Maya====
====Ancient Egyptians====
The [[Dendera Zodiac]], a star-map inside [[Dendera Temple complex#Hathor temple|the Hathor temple at Dendera]], allegedly records the precession of the equinoxes.<ref>Tompkins, 1971</ref> In any case, if the ancient Egyptians knew of precession, their knowledge is not recorded as such in any of their surviving astronomical texts.
Michael Rice,
"to alter the orientation of a temple when the star on whose position it had originally been set moved its position as a consequence of the Precession, something which seems to have happened several times during the New Kingdom."</ref>
===India===
Before 1200, India had two theories of [[trepidation]], one with a rate and another without a rate, and several related models of precession. Each had minor changes or corrections by various commentators. The dominant of the three was the trepidation described by the most respected Indian astronomical treatise, the ''[[Surya Siddhanta]]'' (3:9–12), composed {{circa|400}} but revised during the next few centuries. It used a sidereal epoch, or [[ayanamsa]], that is still used by all [[Indian national calendar|Indian calendar]]s, varying over the [[ecliptic longitude]] of 19°11′ to 23°51′, depending on the group consulted.<ref name=Reform>{{citation |author=Government of India |title=Report of the Calendar Reform Committee |publisher=Council of Scientific and Industrial Research |year=1955 |page=262 |url=https://dspace.gipe.ac.in/xmlui/bitstream/handle/10973/39692/GIPE-043972.pdf |quote=The longitudes of the first point of Aries, according to the two schools therefore differ by 23°[51]′ (–) 19°11′ ... [Upper limit was increased by 42′ of accumulated precession 1950–2000.]}}</ref> This epoch causes the roughly 30 Indian calendar years to begin 23–28 days after the modern [[March
Another trepidation was described by [[Varāhamihira]] ({{circa|550}}). His trepidation consisted of an arc of 46°40′ in one direction and a return to the starting point. Half of this arc, 23°20′, was identified with the Sun's maximum [[declination]] on either side of the equator at the solstices. But no period was specified, thus no annual rate can be ascertained.<ref name=Pingree/>{{rp|27–28}}
Several authors have described precession to be near 200,000{{
===Chinese astronomy===
[[Yu Xi]] (fourth century AD) was the first [[Chinese astronomy|Chinese astronomer]] to mention precession. He estimated the rate of precession as 1° in 50 years
===Middle Ages and Renaissance===
In [[Astronomy in medieval Islam|medieval Islamic astronomy]], precession was known based on Ptolemy's ''Almagest'', and by observations that refined the value.
[[Al-Battani]], in his
|title=Zij Al-Sabi'
|author=Al-Battani
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}}</ref>
Subsequently, [[Al-Sufi]]
{{Cite web
|title=Book of Fixed Stars
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}}</ref>
Later, the ''[[Zij-i Ilkhani]]'', compiled at the [[Maragheh observatory]], sets the precession of the equinoxes at 51 arc seconds per annum, which is very close to the modern value of 50.2 arc seconds.<ref>{{Cite journal |title=The Influence of Islamic Astronomy in Europe and the Far East |last=Rufus |first=W. C. |journal=Popular Astronomy |volume=47 |issue=5 |date=May 1939 |pages=233–238 [236]
|bibcode = 1939PA.....47..233R}}.</ref> In the Middle Ages, Islamic and Latin Christian astronomers treated "trepidation" as a motion of the fixed stars to be ''added to'' precession. This theory is commonly attributed to the [[Arab]] astronomer [[Thabit ibn Qurra]], but the attribution has been contested in modern times. [[Nicolaus Copernicus]] published a different account of trepidation in ''[[De revolutionibus orbium coelestium]]'' (1543). This work makes the first definite reference to precession as the result of a motion of the Earth's axis. Copernicus characterized precession as the third motion of the Earth.<ref>{{cite book |last=Gillispie |first=Charles Coulston |author-link=Charles Coulston Gillispie|title=The Edge of Objectivity: An Essay in the History of Scientific Ideas |year=1960 |publisher=Princeton University Press |isbn=0-691-02350-6 |url=https://archive.org/details/edgeofobjectivit00char |page=24}}</ref>
===Modern period===
Over a century later
==Hipparchus's discovery==
Hipparchus gave an account of his discovery in ''On the Displacement of the Solsticial and Equinoctial Points'' (described in ''Almagest'' III.1 and VII.2). He measured the ecliptic [[longitude]] of the star [[Spica]] during lunar eclipses and found that it was about 6° west of the [[September equinox|autumnal equinox]]. By comparing his own measurements with those of [[Timocharis]] of Alexandria (a contemporary of [[Euclid]], who worked with [[Aristillus]] early in the 3rd century BC), he found that Spica's longitude had decreased by about 2° in the meantime (exact years are not mentioned in ''Almagest''). Also in VII.2, Ptolemy gives more precise observations of two stars, including Spica, and concludes that in each case a 2°
Because the equinoctial points are not marked in the sky, Hipparchus needed the Moon as a reference point; he used a [[lunar eclipse]] to measure the position of a star. Hipparchus already had developed a way to calculate the longitude of the Sun at any moment. A lunar eclipse happens during [[Full moon]], when the Moon is at [[Opposition (astronomy)|opposition]], precisely 180° from the Sun. Hipparchus is thought to have measured the longitudinal arc separating Spica from the Moon. To this value, he added the calculated longitude of the Sun, plus 180° for the longitude of the Moon. He did the same procedure with Timocharis' data
Hipparchus also studied precession in ''On the Length of the Year''. Two kinds of year are relevant to understanding his work. The [[tropical year]] is the length of time that the
To approximate his tropical year, Hipparchus created his own [[lunisolar calendar]] by modifying those of [[Meton]] and [[Callippus]] in ''On Intercalary Months and Days'' (now lost), as described by [[Ptolemy]] in the ''Almagest'' III.1
Hipparchus's mathematical signatures are found in the [[Antikythera Mechanism]], an ancient astronomical computer of the second century BC. The mechanism is based on a solar year, the
==Changing pole stars==
[[Image:Precession N.gif|right|thumb|upright=1.2|Precession of Earth's axis around the north ecliptical pole]]
A consequence of the precession is a changing [[pole star]]. Currently [[Polaris]] is extremely well suited to mark the position of the north celestial pole, as Polaris is a moderately bright star with a visual [[apparent magnitude|magnitude]] of 2.1 (variable), and
[[Image:Precession S.gif|left|thumb|upright=1.2|Precession of Earth's axis around the south ecliptical pole]]
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When Polaris becomes the north star again around 27,800, it will then be farther away from the pole than it is now due to its [[proper motion]], while in 23,600 BC it came closer to the pole.
It is more difficult to find the south celestial pole in the sky at this moment, as that area is a particularly bland portion of the sky
This situation also is seen on a star map. The orientation of the south pole is moving toward the [[Crux|Southern Cross]] constellation. For the last 2,000 years or so, the Southern Cross has pointed to the south celestial pole. As a consequence, the constellation is difficult to view from subtropical northern latitudes, unlike
==Polar shift and equinoxes shift==
[[Image:Outside view of precession.jpg|thumb|upright=1.3|Precessional movement as seen from 'outside' the celestial sphere]]
[[File:Precession animation small new.gif|right|thumb|upright=1.3|The 25,700 year cycle of precession as seen from near the Earth. The current north [[pole star]] is [[Polaris]] (top). In about 8,000 years it will be the bright star [[Deneb]] (left), and in about 12,000 years, [[Vega]] (left center). The Earth's rotation is not depicted to scale – in this span of time, it would actually rotate over
The images at right attempt to explain the relation between the precession of the Earth's axis and the shift in the equinoxes. These images show the position of the Earth's axis on the ''[[celestial sphere]]'', a fictitious sphere which places the stars according to their position as seen from Earth, regardless of their actual distance. The first image shows the celestial sphere from the outside, with the constellations in mirror image. The second image shows the perspective of a near-Earth position as seen through a very wide angle lens (from which the apparent distortion arises).
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The rotation axis of the Earth describes, over a period of 25,700 years, a small {{blue|blue circle}} among the stars near the top of the diagram, centered on the [[ecliptic coordinates|ecliptic]] north pole (the {{blue|blue letter '''E'''}}) and with an angular radius of about 23.4°, an angle known as the ''[[obliquity of the ecliptic]]''. The direction of precession is opposite to the daily rotation of the Earth on its axis. The {{brown|brown axis}} was the Earth's rotation axis 5,000 years ago, when it pointed to the star [[Thuban]]. The yellow axis, pointing to Polaris, marks the axis now.
The equinoxes occur where the celestial equator intersects the ecliptic (red line), that is, where the Earth's axis is perpendicular to the line connecting the centers of the Sun and Earth.
[[File:Equinox path.png|left|upright=2|thumb|Diagram showing the westward shift of the [[March
As seen from the {{brown|brown grid}}, 5,000 years ago, the [[March
Still pictures like these are only first approximations, as they do not take into account the variable speed of the precession, the variable [[Axial tilt|obliquity]] of the ecliptic, the planetary precession (which is a slow rotation of the [[ecliptic plane]] itself, presently around an axis located on the plane, with longitude 174.8764°) and the proper motions of the stars.
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==Equations==
[[Image:
The [[tidal force]] on Earth due to a perturbing body (Sun, Moon or planet) is expressed by [[Newton's law of universal gravitation]], whereby the gravitational force of the perturbing body on the side of Earth nearest is said to be greater than the gravitational force on the far side by an amount proportional to the difference in the cubes of the distances between the near and far sides. If the gravitational force of the perturbing body acting on the mass of the Earth as a point mass at the center of Earth (which provides the [[centripetal force]] causing the orbital motion) is subtracted from the gravitational force of the perturbing body everywhere on the surface of Earth, what remains may be regarded as the tidal force. This gives the paradoxical notion of a force acting away from the satellite but in reality it is simply a lesser force toward that body due to the gradient in the gravitational field. For precession, this tidal force can be grouped into two forces which only act on the [[equatorial bulge]] outside of a mean spherical radius. This [[couple (mechanics)|couple]] can be decomposed into two pairs of components, one pair parallel to Earth's equatorial plane toward and away from the perturbing body which cancel each other out, and another pair parallel to Earth's rotational axis, both toward the [[ecliptic]] plane.<ref>[[Ivan I. Mueller]], ''Spherical and practical astronomy as applied to geodesy'' (New York: Frederick Unger, 1969) 59.</ref> The latter pair of forces creates the following [[torque]] [[Euclidean vector|vector]] on Earth's equatorial bulge:<ref name="Williams">{{Cite journal |last1=Williams |first1=James G. |year=1994 |title=Contribution to the Earth's Obliquity Rate, Precession, and Nutation |url=https://articles.adsabs.harvard.edu/pdf/1994AJ....108..711W |journal=The Astronomical Journal |volume=108 |pages=711 |bibcode=1994AJ....108..711W |doi=10.1086/117108|s2cid=122370108 |doi-access=free }}</ref>
:<math>\overrightarrow{T} = \frac{3GM}{r^3}(C - A) \sin\delta \cos\delta \begin{pmatrix}\sin\alpha \\ -\cos\alpha \\ 0\end{pmatrix}</math>
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:''C'' − ''A'', moment of inertia of Earth's equatorial bulge (''C'' > ''A'')
:''δ'', [[declination]] of the perturbing body (north or south of equator)
:''α'', [[right ascension]] of the perturbing body (east from
The three unit vectors of the torque at the center of the Earth (top to bottom) are '''x''' on a line within the ecliptic plane (the intersection of Earth's equatorial plane with the ecliptic plane) directed toward the
The value of the three sinusoidal terms in the direction of '''x''' {{nowrap|(sin''δ'' cos''δ'' sin''α'')}} for the Sun is a [[sine squared]] waveform varying from zero at the equinoxes (0°, 180°) to 0.36495 at the solstices (90°, 270°). The value in the direction of '''y''' {{nowrap|(sin''δ'' cos''δ'' (−cos''α''))}} for the Sun is a sine wave varying from zero at the four equinoxes and solstices to ±0.19364 (slightly more than half of the sine squared peak) halfway between each equinox and solstice with peaks slightly skewed toward the equinoxes (43.37°(−), 136.63°(+), 223.37°(−), 316.63°(+)). Both solar waveforms have about the same peak-to-peak amplitude and the same period, half of a revolution or half of a year. The value in the direction of '''z''' is zero.
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:<math>\frac{d\psi_L}{dt} = \frac{3}{2}\left[\frac{GM\left(1 - 1.5\sin^2 i\right)}{a^3 \left(1 - e^2\right)^\frac{3}{2}}\right]_L \left[\frac{C - A}{C}\frac{\cos\epsilon}{\omega}\right]_E</math>
where ''i'' is the angle between the plane of the Moon's orbit and the ecliptic plane. In these two equations, the Sun's parameters are within square brackets labeled S, the Moon's parameters are within square brackets labeled L, and the Earth's parameters are within square brackets labeled E. The term <math>\left(1 - 1.5\sin^2 i\right)</math> accounts for the inclination of the Moon's orbit relative to the ecliptic. The term {{nowrap|(''C'' − ''A'')/''C''}} is Earth's [[geodesy|dynamical ellipticity or flattening]], which is adjusted to the observed precession because Earth's internal structure is not known with sufficient detail. If Earth were [[Homogeneity (physics)|homogeneous]] the term would equal its [[angular eccentricity#Eccentricity|third eccentricity squared]],<ref>George Biddel Airy, ''[https://archive.org/details/mathematicaltra06airygoog Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics]'' (third
:<math>e''^2 = \frac{\mathrm{a}^2 - \mathrm{c}^2}{\mathrm{a}^2 + \mathrm{c}^2}</math>
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|(''C'' − ''A'')/''C'' = 0.003273763
|-
|
|''a'' = 1.4959802{{E|11}} m▼
|''a'' = 3.833978{{E|8}} m
|''
|-
|
|''e'' = 0.05554553
|
|-
|
|''i'' = 5.156690°
|ε = 23.43928°
|-
|
|
|}
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[[Simon Newcomb]]'s calculation at the end of the 19th century for general precession (''p'') in longitude gave a value of 5,025.64 arcseconds per tropical century, and was the generally accepted value until artificial satellites delivered more accurate observations and electronic computers allowed more elaborate models to be calculated. [[Jay Henry Lieske]] developed an updated theory in 1976, where ''p'' equals 5,029.0966 arcseconds (or 1.3969713 degrees) per Julian century. Modern techniques such as [[VLBI]] and [[Lunar laser ranging|LLR]] allowed further refinements, and the [[International Astronomical Union]] adopted a new constant value in 2000, and new computation methods and polynomial expressions in 2003 and 2006; the '''accumulated''' precession is:<ref name=Capitaine2003>[http://syrte.obspm.fr/iau2006/aa03_412_P03.pdf N. Capitaine ''et al.'' 2003], p. 581 expression 39</ref>
:''p<sub>A</sub>'' = 5,028.796195{{nnbsp}}''T'' + 1.1054348{{nnbsp}}''T''<sup>2</sup> + higher order terms, in arcseconds, with ''T'', the time in Julian centuries (that is, 36,525 days) since [[J2000|the epoch of 2000]].
The '''rate''' of precession is the derivative of that:
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The constant term of this speed (5,028.796195 arcseconds per century in above equation) corresponds to one full precession circle in 25,771.57534 years (one full circle of 360 degrees divided by 50.28796195 arcseconds per year)<ref name=Capitaine2003/> although some other sources put the value at 25771.4 years, leaving a small uncertainty.
The precession rate is not a constant, but is (at the moment) slowly increasing over time, as indicated by the linear (and higher order) terms in ''T''. In any case it must be stressed that this formula is only valid over a ''limited time period''. It is a polynomial expression centred on the J2000 datum, empirically fitted to observational data, not on a deterministic model of the
:''p'' = ''A'' + ''BT'' + ''CT''<sup>2</sup> + …
is an approximation of
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Theoretical models may calculate the constants (coefficients) corresponding to the higher powers of ''T'', but since it is impossible for a polynomial to match a periodic function over all numbers, the difference in all such approximations will grow without bound as ''T'' increases. Sufficient accuracy can be obtained over a limited time span by fitting a high enough order polynomial to observation data, rather than a necessarily imperfect dynamic numerical model.{{clarify|date=January 2022}} For present flight trajectory calculations of artificial satellites and spacecraft, the polynomial method gives better accuracy. In that respect, the [[International Astronomical Union]] has chosen the best-developed available theory. For up to a few centuries into the past and future, none of the formulas used diverge very much. For up to a few thousand years in the past and the future, most agree to some accuracy. For eras farther out, discrepancies become too large – the exact rate and period of precession may not be computed using these polynomials even for a single whole precession period.
The precession of Earth's axis is a very slow effect, but at the level of accuracy at which astronomers work, it does need to be taken into account on a daily basis.
Precession rate exhibits a secular decrease due to [[tidal acceleration|tidal dissipation]] from 59"/a to 45"/a (a = [[annum]] = [[Julian year (astronomy)|Julian year]]) during the 500 million year period centered on the present. After short-term fluctuations (tens of thousands of years) are averaged out, the long-term trend can be approximated by the following polynomials for negative and positive time from the present in "/a, where ''T'' is in [[1,000,000,000|billion]]s of Julian years (Ga):<ref>{{Cite journal |doi = 10.1051/0004-6361:20041335|title = A long-term numerical solution for the insolation quantities of the Earth|journal = Astronomy & Astrophysics|volume = 428|pages = 261–285|year = 2004|last1 = Laskar|first1 = J.|last2 = Robutel|first2 = P.|last3 = Joutel|first3 = F.|last4 = Gastineau|first4 = M.|last5 = Correia|first5 = A. C. M.|last6 = Levrard|first6 = B.|bibcode = 2004A&A...428..261L|doi-access = free}}</ref>
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:''p''{{sup|+}} = 50.475838 − 27.000654{{nnbsp}}''T'' + 15.603265{{nnbsp}}''T''<sup>2</sup>
Precession will be greater than ''p''{{sup|+}} by the small amount of +0.135052"/a between {{nowrap|+30 Ma}} and {{nowrap|+130 Ma}}. The jump to this excess over ''p''{{sup|+}} will occur in only {{nowrap|20 Ma}} beginning now because the secular decrease in precession is beginning to cross a resonance in Earth's orbit caused by the other planets.
According to W. R. Ward
==See also==
* [[Astronomical nutation]]
* [[Axial tilt]]
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* [[Polar motion]]
* [[Sidereal year]]
* [[Apsidal precession]]
==References==
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* Ulansey, David. ''The Origins of the Mithraic Mysteries: Cosmology and Salvation in the Ancient World''. New York: Oxford University Press, 1989.
* {{cite journal|first1=J. |last1=Vondrak | first2=N. | last2=Capitaine | first3=P. | last3=Wallace |title= New precession expressions, valid for long time intervals| journal = Astronomy & Astrophysics| year=2011 | volume=534 | page=A22 | doi=10.1051/0004-6361/201117274|bibcode = 2011A&A...534A..22V | doi-access=free }}
{{refend}}
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[[Category:Technical factors of astrology]]
[[Category:Celestial mechanics]]
[[Category:Equinoxes]]
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