Editing Axial precession

{{Short description|Change of rotational axis in an astronomical body}}
{{About|the astronomical concept|precession of the axes outside of astronomy|Precession|non-axial astronomical precession|Precession#Astronomy{{!}}Astronomical precession}}
{{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"/>]]

In [[astronomy]], '''axial precession''' is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's [[Rotation around a fixed axis|rotational axis]]. In the absence of precession, the astronomical body's orbit would show [[axial parallelism]].<ref name="Lerner and Lerner 2003">{{cite book | last1=Lerner | first1=K. Lee | last2=Lerner | first2=Brenda Wilmoth | title=World of earth science | publisher=Thomson-Gale | publication-place=Farmington Hills, MI | date=2003 | isbn=0-7876-9332-4 | oclc=60695883 | page=105 and 454|quote=During revolution about the Sun, the earth's polar axis exhibits parallelism to Polaris (also known as the North Star). Although observing parallelism, the orientation of Earth's polar axis exhibits precession – a circular wobbling exhibited by gyroscopes – that results in a 28,000-year-long precessional cycle. Currently, Earth's polar axis points roughly in the direction of Polaris (the North Star). As a result of precession, over the next 11,000 years, Earth's axis will precess or wobble so that it assumes an orientation toward the star Vega.}}</ref> In particular, axial precession can refer to the gradual shift in the orientation of [[Earth]]'s axis of rotation in a cycle of approximately 26,000 years.<ref name=":crs-esaa-99">Hohenkerk, C.Y., Yallop, B.D., Smith, C.A., & Sinclair, A.T. "Celestial Reference Systems" in Seidelmann, P.K. (ed.) ''Explanatory Supplement to the Astronomical Almanac''. Sausalito: University Science Books. p. 99.</ref> This is similar to the [[precession]] of a spinning top, with the axis tracing out a pair of [[Cone (geometry)|cones]] joined at their [[Apex (geometry)|apices]]. The term "precession" typically refers only to this largest part of the motion; other changes in the alignment of Earth's axis—[[astronomical nutation|nutation]] and [[polar motion]]—are much smaller in magnitude.

Earth's precession was historically called the '''precession of the equinoxes''', because the [[Equinox (celestial coordinates)|equinoxes]] moved westward along the [[ecliptic]] relative to the [[fixed star]]s, opposite to the yearly motion of the [[Sun]] along the ecliptic. Historically,<ref name="Astro 101">[http://www.wwu.edu/depts/skywise/a101_precession.html Astro 101 – Precession of the Equinox] {{Webarchive|url=https://web.archive.org/web/20090102124621/http://www.wwu.edu/depts/skywise/a101_precession.html |date=2 January 2009 }}, [[Western Washington University]] [[Planetarium]]. Retrieved 30 December 2008</ref> 
the discovery of the precession of the equinoxes is usually attributed in the West to the 2nd-century-BC astronomer [[Hipparchus]]. With improvements in the ability to calculate the gravitational force between planets during the first half of the nineteenth century, it was recognized that the ecliptic itself moved slightly, which was named '''planetary precession''', as early as 1863, while the dominant component was named '''lunisolar precession'''.<ref>Robert Main, [https://archive.org/details/practicalandsph00maingoog/page/n226 <!-- pg=204 --> Practical and Spherical Astronomy] (Cambridge: 1863) pp.203–4.</ref> Their combination was named '''general precession''', instead of precession of the equinoxes.

Lunisolar precession is caused by the gravitational forces of the [[Moon]] and Sun on Earth's [[equatorial bulge]], causing Earth's axis to move with respect to [[inertial space]]. Planetary precession (an advance) is due to the small angle between the gravitational force of the other planets on Earth and its orbital plane (the ecliptic), causing the plane of the ecliptic to shift slightly relative to inertial space. Lunisolar precession is about 500 times greater than planetary precession.<ref name=Williams/> In addition to the Moon and Sun, the other planets also cause a small movement of Earth's axis in inertial space, making the contrast in the terms lunisolar versus planetary misleading, so in 2006 the [[International Astronomical Union]] recommended that the dominant component be renamed the '''precession of the equator''', and the minor component be renamed '''precession of the ecliptic''', but their combination is still named general precession.<ref>{{Cite web |url=http://www.iau.org/static/resolutions/IAU2006_Resol1.pdf |title=IAU 2006 Resolution B1: Adoption of the P03 Precession Theory and Definition of the Ecliptic |access-date=28 February 2009 |archive-url=https://web.archive.org/web/20111021185416/http://iau.org/static/resolutions/IAU2006_Resol1.pdf |archive-date=21 October 2011 |url-status=dead }}</ref> Many references to the old terms exist in publications predating the change.

==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 [[#Cause|§Cause]]).  This off-center push or pull causes a torque, and a torque on a spinning body results in precession.  The gyro can be analyzed in its parts, and each part within the disk is trying to fall, but the rotation brings it from down to up, and the net result of all particles going through this is precession.]]
The term "[[Precession]]" is derived from the Latin ''[[wikt:praecedo|praecedere]]'' ("to precede, to come before or earlier"). The stars viewed from Earth are seen to proceed from east to west daily (at about 15 degrees per hour), because of the Earth's [[diurnal motion]], and yearly (at about 1 degree per day), because of the Earth's revolution around the Sun. At the same time the stars can be observed to anticipate slightly such motion, at the rate of approximately 50 arc seconds per year (1 degree per 72 years), a phenomenon known as the "precession of the equinoxes".

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.

The term "precession" used in [[astronomy]] generally describes the observable precession of the equinox (the stars moving [[Apparent retrograde motion|retrograde]] across the sky), whereas the term "precession" as used in [[physics]], generally describes a mechanical process.

==Effects==
[[File:Earth axial precession.svg|thumb|300px|The coincidence of the annual cycles of the apses (closest and further approach to the Sun) and calendar dates (with seasons noted) at four equally spaced stages of a fictitious precessionary cycle of 20,000 years (rather than the Earth's true precessionary cycle of 26,000 years). The season dates are those in the north. The tilt of fictitious Earth's axis and the eccentricity of its orbit are exaggerated. Approximate estimates. Effects of weak planetary precession on the stages shown are ignored.]]
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.

Secondly, the position of the Earth in its orbit around the Sun at the [[solstice]]s, [[equinox]]es, or other time defined relative to the seasons, slowly changes.<ref name="Astro 101"/> For example, suppose that the Earth's orbital position is marked at the summer solstice, when the Earth's [[axial tilt]] is pointing directly toward the Sun. One full orbit later, when the Sun has returned to the same apparent position relative to the background stars, the Earth's axial tilt is not now directly toward the Sun: because of the effects of precession, it is a little way "beyond" this. In other words, the solstice occurred a little ''earlier'' in the orbit. Thus, the [[tropical year]], measuring the cycle of seasons (for example, the time from solstice to solstice, or equinox to equinox), is about 20 minutes shorter than the [[sidereal year]], which is measured by the Sun's apparent position relative to the stars. After about 26&nbsp;000 years the difference amounts to a full year, so the positions of the seasons relative to the orbit are "back where they started". (Other effects also slowly change the shape and orientation of the Earth's orbit, and these, in combination with precession, create various cycles of differing periods; see also [[Milankovitch cycles]]. The magnitude of the Earth's tilt, as opposed to merely its orientation, also changes slowly over time, but this effect is not attributed directly to precession.)

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 a tropical year is shorter than a sidereal year by 1,224.5 seconds {{nowrap|(20 min 24.5 sec &asymp; (365.24219 &times; 86400) / 25772).}}

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.

For further details, see [[#Changing pole stars|Changing pole stars]] and [[#Polar shift and equinoxes shift|Polar shift and equinoxes shift]], below.

==History==
[[File:Precession table Metius.jpg|thumb|right|"Table indicating the longitude of three stars observed at different times." Compiled by [[Adriaan Metius]], 1624.]]
===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 36,000 years.<ref name=Ptolemy>{{citation |author=Ptolemy |author-link=Ptolemy |title=Ptolemy's Almagest |translator-last=Toomer |translator-first=G. J. |translator-link=Gerald J. Toomer |year=1998 |orig-year=1984 {{circa|150}} |publisher=Princeton University Press |pages=131–141, 321–340 |isbn=0-691-00260-6}}</ref>

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 Ptolemy in the second century AD. Ptolemy measured the longitudes of [[Regulus]], [[Spica]], and other bright stars with a variation of Hipparchus's lunar method that did not require eclipses. Before sunset, he measured the longitudinal arc separating the Moon from the Sun. Then, after sunset, he measured the arc from the Moon to the star. He used Hipparchus's model to calculate the Sun's longitude, and made corrections for the Moon's motion and its [[parallax]].<ref>Evans 1998, pp.&nbsp;251–255</ref> Ptolemy compared his own observations with those made by Hipparchus, [[Menelaus of Alexandria]], [[Timocharis]], and [[Agrippa (astronomer)|Agrippa]]. He found that between Hipparchus's time and his own (about 265 years), the stars had moved 2°40', or 1° in 100 years (36" per year; the rate accepted today is about 50" per year or 1° in 72 years). It is possible, however, that Ptolemy simply trusted Hipparchus' figure instead of making his own measurements. He also confirmed that precession affected all fixed stars, not just those near the ecliptic, and his cycle had the same period of 36,000 years as that of Hipparchus.<ref name=Ptolemy/>

====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:

:"According to certain opinions ancient astrologers believe that from a certain epoch the solstitial signs have a motion of 8° in the order of the signs, after which they go back the same amount. ..." (Dreyer 1958, p. 204)

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====
Archaeologist Susan Milbrath has speculated that the [[Mesoamerican Long Count calendar]] of "30,000 years involving the [[Pleiades]]...may have been an effort to calculate the precession of the equinox."<ref>Susan Milbrath, [https://web.archive.org/web/20110726181418/http://www.instituteofmayastudies.org/Milbrath2012.pdf "Just How Precise is Maya Astronomy?"], Institute of Maya Studies newsletter, December 2007.</ref> This view is held by few other professional [[Mayanist|scholars of Maya civilization]].{{citation needed|date=January 2017}}

====Ancient Egyptians====
Similarly, it is claimed the precession of the equinoxes was known in [[Ancient Egypt]], prior to the time of Hipparchus (the [[Ptolemaic Kingdom|Ptolemaic]] period). These claims remain controversial. Ancient Egyptians kept accurate calendars and recorded dates on temple walls, so it would be a simple matter for them to plot the "rough" precession rate. 

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, a popular writer on Ancient Egypt, has written that Ancient Egyptians must have observed the precession,<ref>Rice, Michael. ''Egypt's Legacy'', p.&nbsp;128). "Whether or not the ancients knew of the mechanics of the Precession before its definition by Hipparchos the Bithynian, in the second century BC is uncertain, but as dedicated watchers of the night sky they could not fail to be aware of its effects."</ref> and suggested that this awareness had profound affects on their culture.<ref>Rice, p. 10 "...the Precession is fundamental to an understanding of what powered the development of Egypt"; p. 56 "...in a sense Egypt as a nation-state and the king of Egypt as a living god are the products of the realisation by the Egyptians of the astronomical changes effected by the immense apparent movement of the heavenly bodies which the Precession implies."</ref> Rice noted that Egyptians re-oriented temples in response to precession of associated stars.<ref>Rice, p.&nbsp;170
 "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 equinox]]. The March equinox of the ''Surya Siddhanta'' librated 27° in both directions from the sidereal epoch. Thus the equinox moved 54° in one direction and then back 54° in the other direction. This cycle took 7200 years to complete at a rate of 54″/year. The equinox coincided with the epoch at the beginning of the ''[[Kali Yuga]]'' in −3101 and again 3,600 years later in 499. The direction changed from prograde to retrograde midway between these years at −1301 when it reached its maximum deviation of 27°, and would have remained retrograde, the same direction as modern precession, for 3600 years until 2299.<ref name=Surya>{{citation |author=Surya |author-link=Surya |title=Translation of Surya Siddhanta: A Textbook of Hindu Astronomy |publisher=University of Calcutta |year=1935 |orig-year=1860 |translator-last=Burgess |translator-first=Ebenezzer |editor-last=Gangooly |editor-first=Phanindralal |url=https://archive.org/details/TranslationOfTheSuryaSiddhanta/page/n169 |pages=114}}</ref><ref name=Pingree>{{citation |last=Pingree |first=David |title=Precession and trepidation in Indian astronomy before A.D. 1200 |journal=Journal for the History of Astronomy |volume=3 |pages=27–35 |year=1972|bibcode=1972JHA.....3...27P |doi=10.1177/002182867200300104 |s2cid=115947431 }}</ref>{{rp|29–30}}

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{{spaces}}revolutions in a [[Kalpa (aeon)|Kalpa]] of 4,320,000,000{{spaces}}years, which would be a rate of {{sfrac|200,000×360×3600|4,320,000,000}}{{spaces}}= 60″/year. They probably deviated from an even 200,000{{spaces}}revolutions to make the accumulated precession zero near 500. Visnucandra ({{circa|550–600}}) mentions 189,411{{spaces}}revolutions in a Kalpa or 56.8″/year. [[Bhaskara I]] ({{circa|600–680}}) mentions [1]94,110{{spaces}}revolutions in a Kalpa or 58.2″/year. [[Bhāskara II]] ({{circa|1150}}) mentions 199,699{{spaces}}revolutions in a Kalpa or 59.9″/year.<ref name=Pingree/>{{rp|32–33}}

===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.<ref>Pannekoek 1961, p.&nbsp;92</ref>

===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 work ''Zij Al-Sabi'', mentions Hipparchus's calculation of precession, and Ptolemy's value of 1 degree per 100 solar years, says that he measured precession and found it to be one degree per 66 solar years.<ref>{{Cite web
 |title=Zij Al-Sabi'
 |author=Al-Battani
 |url=http://shamela.ws/browse.php/book-452#page-132
 |access-date=30 September 2017
 |archive-url=https://web.archive.org/web/20170105192525/http://shamela.ws/browse.php/book-452#page-132
 |archive-date=5 January 2017
 |url-status=dead
 }}</ref>

Subsequently, [[Al-Sufi]], in his ''[[Book of Fixed Stars]]'', mentions the same values  that Ptolemy's value for precession is 1 degree per 100 solar years. He then quotes a different value from ''Zij Al Mumtahan'', which was done during [[Al-Ma'mun]]'s reign, of 1 degree for every 66 solar years. He also quotes the aforementioned ''Zij Al-Sabi'' of Al-Battani as adjusting coordinates for stars by 11 degrees and 10 minutes of arc to account for the difference between Al-Battani's time and Ptolemy's.<ref>
{{Cite web
 |title=Book of Fixed Stars
 |author=Al-Sufi
 |url=https://www.wdl.org/ar/item/18412/view/1/20/
}}</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, [[Isaac Newton]] in ''[[Philosophiae Naturalis Principia Mathematica]]'' (1687) explained precession as a consequence of [[gravitation]].<ref>Evans 1998, p.&nbsp;246</ref> However, Newton's original precession equations did not work, and were revised considerably by [[Jean le Rond d'Alembert]] and subsequent scientists.

==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° 40' change occurred between 128 BC and AD 139. Hence, 1° per century or one full cycle in 36,000 years, that is, the precessional period of Hipparchus as reported by Ptolemy; cf. page 328 in Toomer's translation of Almagest, 1998 edition. He also noticed this motion in other stars. He speculated that only the stars near the zodiac shifted over time. Ptolemy called this his "first hypothesis" (''Almagest'' VII.1), but did not report any later hypothesis Hipparchus might have devised. Hipparchus apparently limited his speculations, because he had only a few older observations, which were not very reliable.

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.<ref>Evans 1998, p.&nbsp;251</ref> Observations such as these eclipses, incidentally, are the main source of data about when Hipparchus worked, since other biographical information about him is minimal. The lunar eclipses he observed, for instance, took place on 21 April 146 BC, and 21 March 135 BC.<ref>Toomer 1984, p.&nbsp;135 n. 14</ref>

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 Sun, as viewed from the Earth, takes to return to the same position along the ecliptic (its path among the stars on the celestial sphere). The [[sidereal year]] is the length of time that the Sun takes to return to the same position with respect to the stars of the celestial sphere. Precession causes the stars to change their longitude slightly each year, so the sidereal year is longer than the tropical year. Using observations of the equinoxes and solstices, Hipparchus found that the length of the tropical year was 365+1/4−1/300 days, or 365.24667 days (Evans 1998, p.&nbsp;209). Comparing this with the length of the sidereal year, he calculated that the rate of precession was not less than 1° in a century. From this information, it is possible to calculate that his value for the sidereal year was 365+1/4+1/144 days.<ref>Toomer 1978, p.&nbsp;218</ref> By giving a minimum rate, he may have been allowing for errors in observation.

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.<ref>Toomer 1984, p.&nbsp;139</ref> The [[Babylonian calendar]] used a cycle of 235 lunar months in 19 years since 499 BC (with only three exceptions before 380 BC), but it did not use a specified number of days. The [[Metonic cycle]] (432 BC) assigned 6,940 days to these 19 years producing an average year of 365+1/4+1/76 or 365.26316 days. The [[Callippic cycle]] (330 BC) dropped one day from four Metonic cycles (76 years) for an average year of 365+1/4 or 365.25 days. Hipparchus dropped one more day from four Callippic cycles (304 years), creating the [[Hipparchic cycle]] with an average year of 365+1/4−1/304 or 365.24671 days, which was close to his tropical year of 365+1/4−1/300 or 365.24667 days.

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 Metonic Cycle, which is the period the Moon reappears in the same place in the sky with the same phase (full Moon appears at the same position in the sky approximately in 19 years), the Callipic cycle (which is four Metonic cycles and more accurate), the [[Saros cycle]], and the [[Exeligmos|Exeligmos cycles]] (three Saros cycles for the accurate eclipse prediction). Study of the Antikythera Mechanism showed that the ancients used very accurate calendars based on all the aspects of solar and lunar motion in the sky. In fact, the Lunar Mechanism which is part of the Antikythera Mechanism depicts the motion of the Moon and its phase, for a given time, using a train of four gears with a pin and slot device which gives a variable lunar velocity that is very close to [[Kepler's laws|Kepler's second law]]. That is, it takes into account the fast motion of the Moon at [[perigee]] and slower motion at [[apogee]].

==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 is located about one degree from the pole, with no stars of similar brightness too close.<ref name="hipparcos">{{cite web | url=http://webviz.u-strasbg.fr/viz-bin/VizieR-5?-out.add=.&-source=I/311/hip2&recno=11739 | title=HIP 11767 | work=Hipparcos, the New Reduction | author=van Leeuwen, F. |year=2007 | access-date=1 March 2011}}</ref>
[[Image:Precession S.gif|left|thumb|upright=1.2|Precession of Earth's axis around the south ecliptical pole]]

The previous pole star was [[Beta Ursae Minoris|Kochab]] (Beta Ursae Minoris, β UMi, β Ursae Minoris), the brightest star in the bowl of the "Little Dipper", located 16 degrees from Polaris. It held that role from 1500 BC to AD 500.<ref name=StardateKochab>{{cite news|last1=Benningfield|first1=Damond|title=Kochab|url=http://stardate.org/radio/program/2015-06-14|access-date=14 June 2015|work=Stardate Magazine|publisher=University of Texas McDonald Observatory|date=14 June 2015}}</ref> It was not quite as accurate in its day as Polaris is today.<ref name=StardateKochab/> Today, Kochab and its neighbor Pherkad are referred to as the "Guardians of the Pole" (meaning Polaris).<ref name=StardateKochab/>

On the other hand, [[Thuban]] in the [[constellation]] [[Draco (constellation)|Draco]], which was the pole star in 3000 BC, is much less conspicuous at magnitude 3.67 (one-fifth as bright as Polaris); today it is invisible in [[Light pollution|light-polluted]] urban skies.

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. The nominal south pole star is [[Sigma Octantis]], which with magnitude 5.5 is barely visible to the naked eye even under ideal conditions. That will change from the 80th to the 90th centuries, however, when the south celestial pole travels through the [[False Cross]].

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 in the time of the [[ancient Greeks]].  The Southern Cross can be viewed from as far north as [[Miami]] (about 25°&nbsp;N), but only during the winter/early spring.{{clear|right}}

==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&nbsp;year cycle of precession as seen from near the Earth. The current north [[pole star]] is [[Polaris]] (top). In about 8,000&nbsp;years it will be the bright star [[Deneb]] (left), and in about 12,000&nbsp;years, [[Vega]] (left center). The Earth's rotation is not depicted to scale – in this span of time, it would actually rotate over 4&nbsp;million times.]]

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).

The rotation axis of the Earth describes, over a period of 25,700&nbsp;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&nbsp;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.The term "equinox" here refers to a point on the celestial sphere so defined, rather than the moment in time when the Sun is overhead at the Equator (though the two meanings are related). When the axis ''[[precession|precesses]]'' from one orientation to another, the equatorial plane of the Earth (indicated by the circular grid around the equator) moves. The celestial equator is just the Earth's equator projected onto the celestial sphere, so it moves as the Earth's equatorial plane moves, and the intersection with the ecliptic moves with it. The positions of the poles and equator ''on Earth'' do not change, only the orientation of the Earth against the fixed stars.

<span id="equinox_shift_rate_anchor" class="anchor"></span>
[[File:Equinox path.png|left|upright=2|thumb|Diagram showing the westward shift of the [[March equinox]] among the stars over the past 6,000&nbsp;years.]]
As seen from the {{brown|brown grid}}, 5,000&nbsp;years ago, the [[March equinox]] was close to the star [[Aldebaran]] in [[Taurus (constellation)|Taurus]]. Now, as seen from the yellow grid, it has shifted (indicated by the {{red|red arrow}}) to somewhere in the constellation of [[Pisces (constellation)|Pisces]].

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.

The precessional eras of each constellation, often known as "''Great Months''",<!--essentially the same thing as Astrological Ages--> are given, approximately, in the table below:<ref>{{cite book |last=Kaler |first=James B. |author-link=James B. Kaler |year=2002 |title=The Ever-Changing Sky: A guide to the celestial sphere |type=Reprint |page=152 |publisher=Cambridge University Press |isbn=978-0521499187 |url=https://books.google.com/books?id=KYLSMsduNqcC&pg=PA152}}</ref>

{| class=wikitable
! rowspan=2 | Constellation
! colspan=2 | Approximate year
|-
! Entering
! Exiting
|-
| Taurus
| 4500 BC
| 2000 BC
|-
| Aries
| 2000 BC
| 100 BC
|-
| Pisces
| 100 BC
| 2700
|-
| [[Age of Aquarius|Aquarius]]
| 2700
| 5300
|}

==Cause==
The precession of the equinoxes is caused by the gravitational forces of the [[Sun]] and the [[Moon]], and to a lesser extent other bodies, on the Earth. It was first explained by [[Isaac Newton]].<ref>{{Cite web|url=https://www.infoplease.com/encyclopedia/science/space/astronomy/precession-of-the-equinoxes|title=precession of the equinoxes &#124; Infoplease|website=infoplease.com}}</ref>

Axial precession is similar to the precession of a spinning top. In both cases, the applied force is due to gravity. For a spinning top, this force tends to be almost parallel to the rotation axis initially and increases as the top slows down. For a gyroscope on a stand it can approach 90 degrees. For the Earth, however, the applied forces of the Sun and the Moon are closer to perpendicular to the axis of rotation.

The Earth is not a perfect sphere but an [[oblate spheroid]], with an equatorial diameter about 43 kilometers larger than its polar diameter. Because of the Earth's [[axial tilt]], during most of the year the half of this bulge that is closest to the Sun is off-center, either to the north or to the south, and the far half is off-center on the opposite side. The gravitational pull on the closer half is stronger, since gravity decreases with the square of distance, so this creates a small torque on the Earth as the Sun pulls harder on one side of the Earth than the other. The axis of this torque is roughly perpendicular to the axis of the Earth's rotation so the axis of rotation [[precession|precesses]]. If the Earth were a perfect sphere, there would be no precession.

This average torque is perpendicular to the direction in which the rotation axis is tilted away from the ecliptic pole, so that it does not change the axial tilt itself. The magnitude of the torque from the Sun (or the Moon) varies with the angle between the Earth's spin axis direction and that of the gravitational attraction. It approaches zero when they are perpendicular. For example, this happens at the equinoxes in the case of the interaction with the Sun. This can be seen to be since the near and far points are aligned with the gravitational attraction, so there is no torque due to the difference in gravitational attraction.

Although the above explanation involved the Sun, the same explanation holds true for any object moving around the Earth, along or close to the ecliptic, notably, the Moon. The combined action of the Sun and the Moon is called the lunisolar precession. In addition to the steady progressive motion (resulting in a full circle in about 25,700 years) the Sun and Moon also cause small periodic variations, due to their changing positions. These oscillations, in both precessional speed and axial tilt, are known as the [[astronomical nutation|nutation]]. The most important term has a period of 18.6 years and an amplitude of 9.2 arcseconds.<ref>{{Cite web|url = http://www2.jpl.nasa.gov/basics/bsf2-1.php#nutation|title = Basics of Space Flight, Chapter 2|date = 29 October 2013|access-date = 26 March 2015|website = Jet Propulsion Laboratory|publisher = Jet Propulsion Laboratory/NASA}}</ref>

In addition to lunisolar precession, the actions of the other planets of the Solar System cause the whole ecliptic to rotate slowly around an axis which has an ecliptic longitude of about 174° measured on the instantaneous ecliptic. This so-called planetary precession shift amounts to a rotation of the ecliptic plane of 0.47 seconds of arc per year (more than a hundred times smaller than lunisolar precession). The sum of the two precessions is known as the general precession.

==Equations==
[[Image:Tidal field and gravity field.svg|thumb|Tidal force on Earth due to the Moon or another celestial body. It shows both the tidal field (thick red arrows) and the gravity field (thin blue arrows) exerted on Earth's surface and center (label O) by the Moon (label S).]]

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>

where

:''GM'', [[standard gravitational parameter]] of the perturbing body
:''r'', geocentric distance to the perturbing body
:''C'', [[moment of inertia]] around Earth's axis of rotation
:''A'', moment of inertia around any equatorial diameter of Earth
:''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 [[March equinox]]).

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 March equinox, '''y''' on a line in the ecliptic plane directed toward the summer solstice (90° east of '''x'''), and '''z''' on a line directed toward the north pole of the ecliptic.

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.

The average torque of the sine wave in the direction of '''y''' is zero for the Sun or Moon, so this component of the torque does not affect precession. The average torque of the sine squared waveform in the direction of '''x''' for the Sun or Moon is:

:<math>T_x = \frac{3}{2}\frac{GM}{a^3 \left(1 - e^2\right)^\frac{3}{2}}(C - A) \sin\epsilon \cos\epsilon</math>

where

:<math>a</math>, semimajor axis of Earth's (Sun's) orbit or Moon's orbit
:''e'', eccentricity of Earth's (Sun's) orbit or Moon's orbit

and 1/2 accounts for the average of the sine squared waveform, <math>a^3 \left(1 - e^2\right)^\frac{3}{2}</math> accounts for the average distance cubed of the Sun or Moon from Earth over the entire elliptical orbit,<ref>G. Boué & J. Laskar, "Precession of a planet with a satellite", ''Icarus'' '''185''' (2006) 312–330, p.329.</ref> and ε (the angle between the equatorial plane and the ecliptic plane) is the maximum value of ''δ'' for the Sun and the average maximum value for the Moon over an entire 18.6 year cycle.

Precession is:

:<math>\frac{d\psi}{dt} = \frac{T_x}{C\omega\sin\epsilon}</math>

where ''ω'' is Earth's [[angular velocity]] and ''Cω'' is Earth's [[angular momentum]]. Thus the first order component of precession due to the Sun is:<ref name=Williams/>

:<math>\frac{d\psi_S}{dt} = \frac{3}{2}\left[\frac{GM}{a^3 \left(1 - e^2\right)^\frac{3}{2}}\right]_S \left[\frac{C - A}{C}\frac{\cos\epsilon}{\omega}\right]_E</math>

whereas that due to the Moon is:

:<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 edition, 1842) 200.</ref>

:<math>e''^2 = \frac{\mathrm{a}^2 - \mathrm{c}^2}{\mathrm{a}^2 + \mathrm{c}^2}</math>

where a is the equatorial radius ({{val|6378137|u=m}}) and c is the polar radius ({{val|6356752|u=m}}), so {{nowrap|1=''e''<sup>2</sup> = 0.003358481}}.

Applicable parameters for [[J2000.0]] rounded to seven significant digits (excluding leading 1) are:<ref name=Simon>{{Cite journal |bibcode = 1994A&A...282..663S|title = Numerical expressions for precession formulae and mean elements for the Moon and the planets|journal = Astronomy and Astrophysics|volume = 282|pages = 663|last1 = Simon|first1 = J. L.|last2 = Bretagnon|first2 = P.|last3 = Chapront|first3 = J.|last4 = Chapront-Touze|first4 = M.|last5 = Francou|first5 = G.|last6 = Laskar|first6 = J.|year = 1994}}</ref><ref name=IERS>Dennis D. McCarthy, ''[http://ilrs.gsfc.nasa.gov/docs/iers_1996_conventions.ps IERS Technical Note 13 – IERS Standards (1992)]'' (Postscript, use [https://www.xconvert.com/convert-ps-to-pdf XConvert]).</ref>
{| class=wikitable
! Sun !! Moon !! Earth
|-
|''GM'' = 1.3271244{{E|20}} m<sup>3</sup>/s<sup>2</sup>
|''GM'' = 4.902799{{E|12}} m<sup>3</sup>/s<sup>2</sup>
|(''C'' − ''A'')/''C'' = 0.003273763
|-
|
|''a'' = 3.833978{{E|8}} m
|''a'' = 1.4959802{{E|11}} m
|-
|
|''e'' = 0.05554553
|''e'' = 0.016708634
|-
|
|''i'' = 5.156690°
|ε = 23.43928°
|-
|
|
|''ω'' = 7.292115{{E|−5}} rad/s
|}

which yield
:''dψ<sub>S</sub>/dt'' = 2.450183{{E|−12}} /s
:''dψ<sub>L</sub>/dt'' = 5.334529{{E|−12}} /s

both of which must be converted to ″/a (arcseconds/annum) by the number of [[arcsecond]]s in 2[[Pi|π]] [[radian]]s (1.296{{E|6}}″/2π) and the number of [[second]]s in one [[annus]] (a [[Julian year (astronomy)|Julian year]]) (3.15576{{E|7}}s/a):
:''dψ<sub>S</sub>/dt'' = 15.948788″/a &nbsp; vs &nbsp; 15.948870″/a from Williams<ref name=Williams/>
:''dψ<sub>L</sub>/dt'' = 34.723638″/a &nbsp; vs &nbsp; 34.457698″/a from Williams.

The solar equation is a good representation of precession due to the Sun because Earth's orbit is close to an ellipse, being only slightly perturbed by the other planets. The lunar equation is not as good a representation of precession due to the Moon because the Moon's orbit is greatly distorted by the Sun and neither the radius nor the eccentricity is constant over the year.

==Values==
[[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:

:''p'' = 5,028.796195 + 2.2108696{{nnbsp}}''T'' + higher order terms.

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 [[Solar System]]. It is clear that if ''T'' gets large enough (far in the future or far in the past), the ''T''² term will dominate and ''p'' will go to very large values. In reality, more elaborate calculations on the [[numerical model of the Solar System]] show that the precessional rate has a period of about 41,000 years, the same as the obliquity of the ecliptic. That is,
:''p'' = ''A'' + ''BT'' + ''CT''<sup>2</sup> + …
is an approximation of
:''p'' = ''a'' + ''b'' sin (2π''T''/''P''), where ''P'' is the 41,000-year period.

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. Although the precession and the tilt of Earth's axis (the obliquity of the ecliptic) are calculated from the same theory and are thus related one to the other, the two movements act independently of each other, moving in opposite directions.{{clarify|date=January 2022}}

Precession rate exhibits a secular decrease due to [[tidal acceleration|tidal dissipation]] from 59"/a to 45"/a (a = [[annus]] = [[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>
:''p''{{sup|−}} = 50.475838 − 26.368583{{nnbsp}}''T'' + 21.890862{{nnbsp}}''T''<sup>2</sup>
:''p''{{sup|+}} = 50.475838 − 27.000654{{nnbsp}}''T'' + 15.603265{{nnbsp}}''T''<sup>2</sup>

This gives an average cycle length now of 25,676 years.

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, in about 1,500 million years, when the distance of the Moon, which is continuously increasing from tidal effects, has increased from the current 60.3 to approximately 66.5 Earth radii, resonances from planetary effects will push precession to 49,000 years at first, and then, when the Moon reaches 68 Earth radii in about 2,000&nbsp;million years, to 69,000 years. This will be associated with wild swings in the obliquity of the ecliptic as well. Ward, however, used the abnormally large modern value for tidal dissipation.<ref>{{cite journal| first1=W. R.|last1= Ward|year=1982 |title=Comments on the long-term stability of the earth's obliquity |journal=Icarus|volume=50|issue= 2–3|pages=444–448 | bibcode=1982Icar...50..444W|doi = 10.1016/0019-1035(82)90134-8 }}</ref> Using the 620-million year average provided by [[tidal acceleration#Historical evidence|tidal rhythmites]] of about half the modern value, these resonances will not be reached until about 3,000 and 4,000 million years, respectively. However, due to the gradually increasing luminosity of the Sun, the oceans of the Earth will have vaporized before that time (about 2,100&nbsp;million years from now).

==See also==
* [[Astronomical nutation]]
* [[Axial tilt]]
* [[Euler angles]]
* [[Longitude of vernal equinox]]
* [[Milankovitch cycles]]
* [[Polar motion]]
* [[Sidereal year]]
* [[Apsidal precession]]

==References==
{{Reflist}}

===Bibliography===
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* [[J. L. E. Dreyer|Dreyer, J. L. E.]] ''A History of Astronomy from Thales to Kepler''. 2nd ed. New York: Dover, 1953.
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* {{cite journal|first1=J.L.|last1= Hilton |year=2006 | url=http://syrte.obspm.fr/iau2006/cm06_94_PEWG.pdf |title=Report of the International Astronomical Union Division I Working Group on Precession and the Ecliptic |journal=Celestial Mechanics and Dynamical Astronomy|volume=94|issue= 3 |pages=351–367|bibcode=2006CeMDA..94..351H |doi=10.1007/s10569-006-0001-2|s2cid= 122358401 }}
* {{cite journal |first1=J. H.|last1= Lieske |first2=T. |last2=Lederle | first3=W. |last3=Fricke |year=1977|bibcode=1977A&A....58....1L | title=Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants |journal=Astron. Astrophys.|volume=58|pages=1–16}}
* [https://web.archive.org/web/20060615160717/http://www.tenspheres.com/researches/precession.htm Precession and the Obliquity of the Ecliptic] has a comparison of values predicted by different theories
* Pannekoek, A. ''A History of Astronomy''. New York: Dover, 1961.
* Parker, Richard A. "Egyptian Astronomy, Astrology, and Calendrical Reckoning." ''Dictionary of Scientific Biography'' 15:706–727.
* Rice, Michael (1997), ''Egypt's Legacy: The archetypes of Western civilization, 3000–30 BC'', London and New York.
* {{cite journal|last1=Schütz|first1=Michael|title=Hipparch und die Entdeckung der Präzession. Bemerkungen zu David Ulansey, Die Ursprünge des Mithraskultes|journal=Electronic Journal of Mithraic Studies|url=http://www.uhu.es/ejms/Papers/Volume1Papers/ulansey.doc|year=2000|volume=1|language=de|archive-url=https://web.archive.org/web/20131104120349/http://www.uhu.es/ejms/papers.htm|archive-date=4 November 2013}}
* {{cite journal|first1=J. L. |last1=Simon |year=1994|bibcode=1994A&A...282..663S |
title=Numerical expressions for precession formulae and mean elements for the Moon and the planets
|journal=Astronomy and Astrophysics|volume=282|pages=663–683}}
* [[The Secret Life of Plants|Tompkins, Peter]]. ''Secrets of the Great Pyramid''. With an appendix by Livio Catullo Stecchini. New York: Harper Colophon Books, 1971.
* [[G. J. Toomer|Toomer, G. J.]] "Hipparchus." ''Dictionary of Scientific Biography''. Vol. 15:207–224. New York: Charles Scribner's Sons, 1978.
* Toomer, G. J. ''Ptolemy's Almagest''. London: Duckworth, 1984.
* 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}}

==External links==
{{NSRW Poster|Precession}}
* [https://web.archive.org/web/20070715044302/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=962&bodyId=1147 D'Alembert and Euler's Debate on the Solution of the Precession of the Equinoxes]
* {{cite web|last=Bowley|first=Roger|title=Axial Precession|url=http://www.sixtysymbols.com/videos/axial_precession.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|author2=Merrifield, Michael}}
* [http://farside.ph.utexas.edu/teaching/celestial/Celestial/node74.html Forced precession and nutation of Earth]

{{Time measurement and standards}}
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{{DEFAULTSORT:Axial Precession (Astronomy)}}
[[Category:Precession]]
[[Category:Technical factors of astrology]]
[[Category:Celestial mechanics]]
[[Category:Equinoxes]]