Axial precession: Difference between revisions

{{shortShort description|Change of rotational axis in an astronomical body}}

{{aboutAbout|the astronomical concept|precession of the axes outside of astronomy|Precession|non-axial astronomical precession|Precession#Astronomy{{!}}Astronomical precession}}

[[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 [[Axial precession#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.]]

"[[Precession]]"The andterm "[[processionPrecession]]" are both terms that relate to [[motion (physics)|motion]]. "Precession" is derived from the Latin ''[[wikt:praecedo|praecedere]]'' ("to precede, to come before or earlier"), while "procession" is derived from the Latin ''[[wikt:procedo|procedere]]'' ("to march forward, to advance"). Generally the term "procession" is used to describe a group of objects moving forward. The stars viewed from Earth are seen to proceed from east to west daily, due to the Earth's [[diurnal motion]], and yearly, due to 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, a phenomenon known as the "precession of the equinoxes".

[[File:Earth axial precession.svg|thumb|300px|The coincidence of the annual cycles of the apses (closest and further approach to the sunSun) 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.]]

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 s,sec ~≈ (365.24219* × 86400) / 25772).}}

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 3600036,000 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>

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 foundthat byof Hipparchus.<ref name=Ptolemy/>

:''"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)

TheArchaeologist 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 MayanMaya civilization]].{{citation needed|date=January 2017}}

SimilarSimilarly, claimsit haveis beenclaimed made thatthe precession of the equinoxes was known in [[Ancient Egypt]] during the dynastic era, prior to the time of Hipparchus (the [[Ptolemaic Kingdom|Ptolemaic]] period). However, theseThese claims remain controversial. SomeAncient buildings in the [[Karnak]] temple complex, for instance, allegedly were oriented toward the point on the horizon where certain stars rose or set at key times of the year.{{Citation needed|date=January 2015}} Nonetheless, theyEgyptians kept accurate calendars and if they recorded thedates dateon oftemple thewalls, temple reconstructionsso it would be a fairly simple matter for them to plot the "rough" precession rate. The [[Dendera Zodiac]], a star-map from the [[Hathor temple]] at [[Dendera]] from a late (Ptolemaic) age, allegedly records precession of the equinoxes (Tompkins 1971). 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, wrotea inpopular hiswriter 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> (p.&nbsp;128)and suggested that this awareness had profound affects on their culture.<ref>Rice, believesp. that10 "...the Precession is fundamental to an understanding of what powered the development of Egypt"; (p.&nbsp;10), to the extent that56 "...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." (p.&nbsp;56).</ref> Rice saysnoted that "theEgyptians evidencere-oriented that the most refined astronomical observation was practisedtemples in Egypt in the third millennium BC (and probably even before that date) is clear from the precision with which the [[Giza pyramid complex|Pyramids at Giza]] are alignedresponse to theprecession cardinalof points, a precision which could only have been achieved by their alignment with theassociated stars. " (p.&nbsp;31) The Egyptians also, says <ref>Rice, were "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." (p.&nbsp;170)

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|vernal equinox]]. The vernalMarch 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 36003,600 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}}

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

[[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>

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']]', after mentioningmentions Hipparchus's calculation calculatingof 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

Subsequently, [[Al-Sufi]] mentions the same values, 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, asof 1 degree for every 66 solar years. He also quotes the aforementioned [[''Zij Al-Battani]]Sabi''s [[Zijof Al-Sabi']]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>

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>

Over a century later precession was explained in, [[Isaac Newton]]'s in ''[[Philosophiae Naturalis Principia Mathematica]]'' (1687), toexplained precession beas a consequence of [[gravitation]] (.<ref>Evans 1998, p.&nbsp;246).</ref> However, Newton's original precession equations did not work, however, and were revised considerably by [[Jean le Rond d'Alembert]] and subsequent scientists.

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. (henceHence, 1° per century or one full cycle in 3600036,000 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). The studyStudy of the Antikythera Mechanism provesshowed that the ancients have been usingused 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 the[[Kepler's laws|Kepler's second law]]. ofThat [[Kepler]]is, i.e. it takes into account the fast motion of the Moon at [[perigee]] and slower motion at [[apogee]]. This discovery proves that Hipparchus mathematics were much more advanced than Ptolemy describes in his books, as it is evident that he developed a good approximation of [[Kepler's laws of planetary motion#Second law|Kepler's second law]].

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 it 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>

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

[[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 94 million times.]]

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. (Note that theThe 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.

[[File:Equinox path.png|left|upright=2|thumb|Diagram showing the westward shift of the [[March equinox|vernal equinox]] among the stars over the past 6,000 years.]]

As seen from the {{brown|brown grid}}, 5,000 years ago, the [[March equinox|vernal 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]].

[[Image:FieldTidal tidalfield and gravity field.svg|thumb|Tidal force on Earth due to the Sun, Moon, or aanother planetcelestial 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>

:''α'', [[right ascension]] of the perturbing body (east from vernal [[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 vernalMarch 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.

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 edititonedition, 1842) 200.</ref>

|''ωa'' = 71.2921154959802{{E|−511}} rad/sm

|ε''e'' = 230.43928°016708634

:''p_A'' = 5,028.796195{{nnbsp}}''T'' + 1.1054348{{nnbsp}}''T''² + higher order terms, in arcseconds, with ''T'', the time in Julian centuries (that is, 36,525 days) since [[J2000|the epoch of 2000]].

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[[Solar systemSystem]]. 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,

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. Note that althoughAlthough 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}}

Note that thisThis gives an average cycle length now of 25,676 years.

According to W. R. Ward, when, 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).