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astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In the absence of precession, the astronomical body's orbit would show axial parallelism. In particular, axial precession can refer to the gradual shift in the orientation of
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
's axis of rotation in a cycle of approximately 26,000 years.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. This is similar to the
precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In oth ...
of a spinning top, with the axis tracing out a pair of
cones A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines conn ...
joined at their apices. The term "precession" typically refers only to this largest part of the motion; other changes in the alignment of Earth's axis—
nutation Nutation () is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame ...
and
polar motion Polar motion of the Earth is the motion of the Earth's rotational axis relative to its crust. This is measured with respect to a reference frame in which the solid Earth is fixed (a so-called ''Earth-centered, Earth-fixed'' or ECEF reference ...
—are much smaller in magnitude. Earth's precession was historically called the precession of the equinoxes, because the
equinoxes A solar equinox is a moment in time when the Sun crosses the Earth's equator, which is to say, appears directly above the equator, rather than north or south of the equator. On the day of the equinox, the Sun appears to rise "due east" and set ...
moved westward along the
ecliptic The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic agains ...
relative to the fixed stars, opposite to the yearly motion of the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
along the ecliptic. Historically,Astro 101 – Precession of the Equinox
,
Western Washington University Western Washington University (WWU or Western) is a public university in Bellingham, Washington. The northernmost university in the contiguous United States, WWU was founded in 1893 as the state-funded New Whatcom Normal School, succeeding a pri ...
Planetarium, accessed 30 December 2008
the discovery of the precession of the equinoxes is usually attributed in the West to the 2nd-century-BC astronomer
Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equi ...
. 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. Their combination was named general precession, instead of precession of the equinoxes. Lunisolar precession is caused by the gravitational forces of the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
and Sun on Earth's
equatorial bulge An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere. On E ...
, 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. 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 The International Astronomical Union (IAU; french: link=yes, Union astronomique internationale, UAI) is a nongovernmental organisation with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreac ...
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. Many references to the old terms exist in publications predating the change.


Nomenclature

"
Precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In oth ...
" and "
procession A procession is an organized body of people walking in a formal or ceremonial manner. History Processions have in all peoples and at all times been a natural form of public celebration, as forming an orderly and impressive ceremony. Religious ...
" are both terms that relate to
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
. "Precession" is derived from the Latin '' praecedere'' ("to precede, to come before or earlier"), while "procession" is derived from the Latin '' 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 Diurnal motion (, ) is an astronomical term referring to the apparent motion of celestial objects (e.g. the Sun and stars) around Earth, or more precisely around the two celestial poles, over the course of one day. It is caused by Earth's ro ...
, 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". 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 Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
generally describes the observable precession of the equinox (the stars moving retrograde across the sky), whereas the term "precession" as used in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, generally describes a mechanical process.


Effects

The precession of the Earth's axis has a number of observable effects. First, the positions of the south and north
celestial pole The north and south celestial poles are the two points in the sky where Earth's axis of rotation, indefinitely extended, intersects the celestial sphere. The north and south celestial poles appear permanently directly overhead to observers a ...
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 "
north star Polaris is a star in the northern circumpolar constellation of Ursa Minor. It is designated α Ursae Minoris ( Latinized to ''Alpha Ursae Minoris'') and is commonly called the North Star or Pole Star. With an apparent magnitude tha ...
". 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 stars. 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 A solstice is an event that occurs when the Sun appears to reach its most northerly or southerly excursion relative to the celestial equator on the celestial sphere. Two solstices occur annually, around June 21 and December 21. In many countr ...
s,
equinox A solar equinox is a moment in time when the Sun crosses the Earth's equator, which is to say, appears directly above the equator, rather than north or south of the equator. On the day of the equinox, the Sun appears to rise "due east" and se ...
es, or other time defined relative to the seasons, slowly changes. For example, suppose that the Earth's orbital position is marked at the summer solstice, when the Earth's
axial tilt In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orb ...
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 A tropical year or solar year (or tropical period) is the time that the Sun takes to return to the same position in the sky of a celestial body of the Solar System such as the Earth, completing a full cycle of seasons; for example, the time f ...
, 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 26 000 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 Milankovitch cycles describe the collective effects of changes in the Earth's movements on its climate over thousands of years. The term was coined and named after Serbian geophysicist and astronomer Milutin Milanković. In the 1920s, he hypot ...
. 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 The zodiac is a belt-shaped region of the sky that extends approximately 8° north or south (as measured in celestial latitude) of the ecliptic, the apparent path of the Sun across the celestial sphere over the course of the year. The pat ...
, at the rate of about 50.3 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 (20 min 24.5 s, ~365.24219*86400/25772). The rate itself varies somewhat with time (see
Values In ethics and social sciences, value denotes the degree of importance of something or action, with the aim of determining which actions are best to do or what way is best to live (normative ethics in ethics), or to describe the significance of di ...
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 and Polar shift and equinoxes shift, below.


History


Hellenistic world


Hipparchus

The discovery of precession usually is attributed to
Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equi ...
(190–120 BC) of
Rhodes Rhodes (; el, Ρόδος , translit=Ródos ) is the largest and the historical capital of the Dodecanese islands of Greece. Administratively, the island forms a separate municipality within the Rhodes regional unit, which is part of the S ...
or
Nicaea Nicaea, also known as Nicea or Nikaia (; ; grc-gre, Νίκαια, ) was an ancient Greek city in Bithynia, where located in northwestern Anatolia and is primarily known as the site of the First and Second Councils of Nicaea (the first and s ...
, a Greek astronomer. According to
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...
's '' Almagest'', Hipparchus measured the longitude of
Spica Spica is the brightest object in the constellation of Virgo and one of the 20 brightest stars in the night sky. It has the Bayer designation α Virginis, which is Latinised to Alpha Virginis and abbreviated Alpha Vir or α Vir. Analys ...
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 autumnal equinox. He also compared the lengths of the
tropical year A tropical year or solar year (or tropical period) is the time that the Sun takes to return to the same position in the sky of a celestial body of the Solar System such as the Earth, completing a full cycle of seasons; for example, the time f ...
(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 36000 years. 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 In astronomy, the geocentric model (also known as geocentrism, often exemplified specifically by the Ptolemaic system) is a superseded description of the Universe with Earth at the center. Under most geocentric models, the Sun, Moon, stars, an ...
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 Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...
in the second century AD. Ptolemy measured the longitudes of
Regulus Regulus is the brightest object in the constellation Leo and one of the brightest stars in the night sky. It has the Bayer designation designated α Leonis, which is Latinized to Alpha Leonis, and abbreviated Alpha Leo or α Leo. Reg ...
,
Spica Spica is the brightest object in the constellation of Virgo and one of the 20 brightest stars in the night sky. It has the Bayer designation α Virginis, which is Latinised to Alpha Virginis and abbreviated Alpha Vir or α Vir. Analys ...
, 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 (Evans 1998, pp. 251–255). Ptolemy compared his own observations with those made by Hipparchus, Menelaus of Alexandria, Timocharis, and
Agrippa Agrippa may refer to: People Antiquity * Agrippa (mythology), semi-mythological king of Alba Longa * Agrippa (astronomer), Greek astronomer from the late 1st century * Agrippa the Skeptic, Skeptic philosopher at the end of the 1st century * Agri ...
. 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 found by Hipparchus.


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 Theon of Alexandria (; grc, Θέων ὁ Ἀλεξανδρεύς;  335 – c. 405) was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's '' Elements'' and wrote commentaries on wor ...
, 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 Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī ( ar, محمد بن جابر بن سنان البتاني) ( Latinized as Albategnius, Albategni or Albatenius) (c. 858 – 929) was an astron ...
, the Chaldean astronomers had distinguished the
tropical The tropics are the regions of Earth surrounding the Equator. They are defined in latitude by the Tropic of Cancer in the Northern Hemisphere at N and the Tropic of Capricorn in the Southern Hemisphere at S. The tropics are also referred to ...
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.


Maya

The archaeologist Susan Milbrath has speculated that the
Mesoamerican Long Count calendar The Mesoamerican Long Count calendar is a non-repeating, vigesimal (base 20) and octodecimal (base 18) calendar used by several pre-Columbian Mesoamerican cultures, most notably the Maya. For this reason, it is often known as the May ...
of "30,000 years involving the
Pleiades The Pleiades (), also known as The Seven Sisters, Messier 45 and other names by different cultures, is an asterism and an open star cluster containing middle-aged, hot B-type stars in the north-west of the constellation Taurus. At a distance ...
...may have been an effort to calculate the precession of the equinox." This view is held by few other professional scholars of Mayan civilization.


Ancient Egyptians

Similar claims have been made that precession was known in Ancient Egypt during the dynastic era, prior to the time of Hipparchus ( Ptolemaic period). However, these claims remain controversial. Some buildings in the
Karnak The Karnak Temple Complex, commonly known as Karnak (, which was originally derived from ar, خورنق ''Khurnaq'' "fortified village"), comprises a vast mix of decayed temples, pylons, chapels, and other buildings near Luxor, Egypt. Constr ...
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. Nonetheless, they kept accurate calendars and if they recorded the date of the temple reconstructions it would be a fairly simple matter to plot the rough precession rate. The Dendera Zodiac, a star-map from the
Hathor temple Dendera Temple complex ( Ancient Egyptian: ''Iunet'' or ''Tantere''; the 19th-century English spelling in most sources, including Belzoni, was Tentyra; also spelled Denderah) is located about south-east of Dendera, Egypt. It is one of the best ...
at
Dendera Dendera ( ar, دَنْدَرة ''Dandarah''; grc, Τεντυρις or Τεντυρα; Bohairic cop, ⲛⲓⲧⲉⲛⲧⲱⲣⲓ, translit=Nitentōri; Sahidic cop, ⲛⲓⲧⲛⲧⲱⲣⲉ, translit=Nitntōre), also spelled ''Denderah'', ancient ...
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 wrote in his ''Egypt's Legacy'', "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." (p. 128) Rice believes that "the Precession is fundamental to an understanding of what powered the development of Egypt" (p. 10), to the extent that "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. 56). Rice says that "the evidence that the most refined astronomical observation was practised in Egypt in the third millennium BC (and probably even before that date) is clear from the precision with which the Pyramids at Giza are aligned to the cardinal points, a precision which could only have been achieved by their alignment with the stars. " (p. 31) The Egyptians also, says 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. 170)


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 but revised during the next few centuries. It used a sidereal epoch, or ayanamsa, that is still used by all Indian calendars, varying over the
ecliptic longitude The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets (except Mercury) and many small Solar System b ...
of 19°11′ to 23°51′, depending on the group consulted. This epoch causes the roughly 30 Indian calendar years to begin 23–28 days after the modern vernal equinox. The vernal 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 3600 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. Another trepidation was described by
Varāhamihira Varāhamihira ( 505 – 587), also called Varāha or Mihira, was an ancient Indian astrologer, astronomer, and polymath who lived in Ujjain (Madhya Pradesh, India). He was born at Kapitba in a Brahmin family, in the Avanti region, roughly co ...
(). 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. Several authors have described precession to be near 200,000revolutions in a
Kalpa Kalevan Pallo (KalPa) is a professional ice hockey team which competes in the Finnish Liiga. They play in Kuopio, Finland at the Olvi Areena. Team history Established in 1929 as ''Sortavalan Palloseura'' in Sortavala, the club relocated to Kuop ...
of 4,320,000,000years, which would be a rate of = 60″/year. They probably deviated from an even 200,000revolutions to make the accumulated precession zero near 500. Visnucandra () mentions 189,411revolutions in a Kalpa or 56.8″/year. Bhaskara I () mentions 4,110revolutions in a Kalpa or 58.2″/year. Bhāskara II () mentions 199,699revolutions in a Kalpa or 59.9″/year.


Chinese astronomy

Yu Xi (fourth century AD) was the first Chinese astronomer to mention precession. He estimated the rate of precession as 1° in 50 years (Pannekoek 1961, p. 92).


Middle Ages and Renaissance

In medieval Islamic astronomy, precession was known based on Ptolemy's Almagest, and by observations that refined the value.
Al-Battani Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī ( ar, محمد بن جابر بن سنان البتاني) ( Latinized as Albategnius, Albategni or Albatenius) (c. 858 – 929) was an astron ...
, in his Zij Al-Sabi', after mentioning Hipparchus calculating 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. Subsequently, Al-Sufi mentions the same values in his
Book of Fixed Stars The ''Book of Fixed Stars'' ( ar, كتاب صور الكواكب ', literally ''The Book of the Shapes of Stars'') is an astronomical text written by Abd al-Rahman al-Sufi (Azophi) around 964. Following the translation movement in the 9th centu ...
, 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, as 1 degree for every 66 solar years. He also quotes the aforementioned
Al-Battani Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī ( ar, محمد بن جابر بن سنان البتاني) ( Latinized as Albategnius, Albategni or Albatenius) (c. 858 – 929) was an astron ...
's Zij Al-Sabi' 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. Later, the '' Zij-i Ilkhani'' compiled at the
Maragheh observatory The Maragheh observatory (Persian: رصدخانه مراغه), also spelled Maragha, Maragah, Marageh, and Maraga, was an astronomical observatory established in the mid 13th century under the patronage of the Ilkhanid Hulagu and the directorship ...
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. 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 The Arabs (singular: Arab; singular ar, عَرَبِيٌّ, DIN 31635: , , plural ar, عَرَب, DIN 31635: , Arabic pronunciation: ), also known as the Arab people, are an ethnic group mainly inhabiting the Arab world in Western Asia, ...
astronomer
Thabit ibn Qurra Thabit ( ar, ) is an Arabic name for males that means "the imperturbable one". It is sometimes spelled Thabet. People with the patronymic * Ibn Thabit, Libyan hip-hop musician * Asim ibn Thabit, companion of Muhammad * Hassan ibn Sabit (died 674 ...
, but the attribution has been contested in modern times.
Nicolaus Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, Niklas Koppernigk, german: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic canon, who formulated ...
published a different account of trepidation in ''
De revolutionibus orbium coelestium ''De revolutionibus orbium coelestium'' (English translation: ''On the Revolutions of the Heavenly Spheres'') is the seminal work on the heliocentric theory of the astronomer Nicolaus Copernicus (1473–1543) of the Polish Renaissance. The book, ...
'' (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.


Modern period

Over a century later precession was explained in
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
's '' Philosophiae Naturalis Principia Mathematica'' (1687), to be a consequence of gravitation (Evans 1998, p. 246). Newton's original precession equations did not work, however, and were revised considerably by
Jean le Rond d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopéd ...
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 Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...
of the star
Spica Spica is the brightest object in the constellation of Virgo and one of the 20 brightest stars in the night sky. It has the Bayer designation α Virginis, which is Latinised to Alpha Virginis and abbreviated Alpha Vir or α Vir. Analys ...
during lunar eclipses and found that it was about 6° west of the autumnal equinox. By comparing his own measurements with those of Timocharis of Alexandria (a contemporary of
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
, 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 36000 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 The full moon is the lunar phase when the Moon appears fully illuminated from Earth's perspective. This occurs when Earth is located between the Sun and the Moon (when the ecliptic longitudes of the Sun and Moon differ by 180°). This means ...
, when the Moon is at
opposition Opposition may refer to: Arts and media * ''Opposition'' (Altars EP), 2011 EP by Christian metalcore band Altars * The Opposition (band), a London post-punk band * '' The Opposition with Jordan Klepper'', a late-night television series on Com ...
, 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 (Evans 1998, p. 251). 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 (Toomer 1984, p. 135 n. 14). Hipparchus also studied precession in ''On the Length of the Year''. Two kinds of year are relevant to understanding his work. The
tropical year A tropical year or solar year (or tropical period) is the time that the Sun takes to return to the same position in the sky of a celestial body of the Solar System such as the Earth, completing a full cycle of seasons; for example, the time f ...
is the length of time that the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
, 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. 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 (Toomer 1978, p. 218). 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 A lunisolar calendar is a calendar in many cultures, combining lunar calendars and solar calendars. The date of Lunisolar calendars therefore indicates both the Moon phase and the time of the solar year, that is the position of the Sun in the ...
by modifying those of
Meton Meton of Athens ( el, Μέτων ὁ Ἀθηναῖος; ''gen''.: Μέτωνος) was a Ancient Greece, Greek mathematician, astronomer, list of geometers, geometer, and engineer who lived in Athens in the 5th century BC. He is best known for ...
and
Callippus Callippus (; grc, Κάλλιππος; c. 370 BC – c. 300 BC) was a Greek astronomer and mathematician. Biography Callippus was born at Cyzicus, and studied under Eudoxus of Cnidus at the Academy of Plato. He also worked with Aristotle at th ...
in ''On Intercalary Months and Days'' (now lost), as described by
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...
in the ''Almagest'' III.1 (Toomer 1984, p. 139). The
Babylonian calendar The Babylonian calendar was a lunisolar calendar with years consisting of 12 lunar months, each beginning when a new crescent moon was first sighted low on the western horizon at sunset, plus an intercalary month inserted as needed by decree. Th ...
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 The Metonic cycle or enneadecaeteris (from grc, ἐννεακαιδεκαετηρίς, from ἐννεακαίδεκα, "nineteen") is a period of almost exactly 19 years after which the lunar phases recur at the same time of the year. The rec ...
(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 The Antikythera mechanism ( ) is an Ancient Greek hand-powered orrery, described as the oldest example of an analogue computer used to predict astronomical positions and eclipses decades in advance. It could also be used to track the four-yea ...
, an ancient astronomical computer of the second century BC. The mechanism is based on a solar year, the
Metonic Cycle The Metonic cycle or enneadecaeteris (from grc, ἐννεακαιδεκαετηρίς, from ἐννεακαίδεκα, "nineteen") is a period of almost exactly 19 years after which the lunar phases recur at the same time of the year. The rec ...
, 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 The saros () is a period of exactly 223 synodic months, approximately 6585.3211 days, or 18 years, 10, 11, or 12 days (depending on the number of leap years), and 8 hours, that can be used to predict eclipses of the Sun and Moon. One saros period ...
and the Exeligmos cycles (three Saros cycles for the accurate eclipse prediction). The study of the Antikythera Mechanism proves that the ancients have been using 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 second law of
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
, i.e. it takes into account the fast motion of the Moon at
perigee An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any ell ...
and slower motion at
apogee An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any ell ...
. 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 second law.


Changing pole stars

A consequence of the precession is a changing
pole star A pole star or polar star is a star, preferably bright, nearly aligned with the axis of a rotating astronomical body. Currently, Earth's pole stars are Polaris (Alpha Ursae Minoris), a bright magnitude-2 star aligned approximately with its ...
. 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
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
of 2.1 (variable), and it is located about one degree from the pole, with no stars of similar brightness too close. The previous pole star was 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. It was not quite as accurate in its day as Polaris is today. Today, Kochab and its neighbor Pherkad are referred to as the "Guardians of the Pole" (meaning Polaris). On the other hand,
Thuban Thuban (), with Bayer designation Alpha Draconis or α Draconis, is a binary star system in the northern constellation of Draco. A relatively inconspicuous star in the night sky of the Northern Hemisphere, it is historically signi ...
in the constellation Draco, which was the pole star in
3000 BC The 30th century BC was a century that lasted from the year 3000 BC to 2901 BC. Events * Before 3000 BC: An image of a deity (detail from a cong) recovered from Tomb 12 in Fanshan, Yuyao, Zhejiang, is made during the Neolithic period by the Li ...
, is much less conspicuous at magnitude 3.67 (one-fifth as bright as Polaris); today it is invisible in 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, and 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 An asterism is an observational astronomy, observed pattern or group of stars in the sky. Asterisms can be any identified pattern or group of stars, and therefore are a more general concept than the IAU designated constellations, formally defined ...
. This situation also is seen on a star map. The orientation of the south pole is moving toward the
Southern Cross Crux () is a constellation of the southern sky that is centred on four bright stars in a cross-shaped asterism commonly known as the Southern Cross. It lies on the southern end of the Milky Way's visible band. The name ''Crux'' is Latin for ...
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 Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...
. The Southern Cross can be viewed from as far north as Miami (about 25° N), but only during the winter/early spring.


Polar shift and equinoxes shift

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 years, a small among the stars near the top of the diagram, centered on the
ecliptic The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic agains ...
north pole (the ) and with an angular radius of about 23.4°, an angle known as the ''
obliquity of the ecliptic In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and o ...
''. The direction of precession is opposite to the daily rotation of the Earth on its axis. The was the Earth's rotation axis 5,000 years ago, when it pointed to the star
Thuban Thuban (), with Bayer designation Alpha Draconis or α Draconis, is a binary star system in the northern constellation of Draco. A relatively inconspicuous star in the night sky of the Northern Hemisphere, it is historically signi ...
. 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. (Note that 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 '' 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. As seen from the , 5,000 years ago, the vernal equinox was close to the star
Aldebaran Aldebaran (Arabic: “The Follower”, "الدبران") is the brightest star in the zodiac constellation of Taurus. It has the Bayer designation α Tauri, which is Latinized to Alpha Tauri and abbreviated Alpha Tau or α Tau. Alde ...
in
Taurus Taurus is Latin for 'bull' and may refer to: * Taurus (astrology), the astrological sign * Taurus (constellation), one of the constellations of the zodiac * Taurus (mythology), one of two Greek mythological characters named Taurus * '' Bos tauru ...
. Now, as seen from the yellow grid, it has shifted (indicated by the ) to somewhere in the constellation of Pisces. Still pictures like these are only first approximations, as they do not take into account the variable speed of the precession, the variable
obliquity In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orbi ...
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''”, are given, approximately, in the table below:


Cause

The precession of the equinoxes is caused by the gravitational forces of the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
and the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
, and to a lesser extent other bodies, on the Earth. It was first explained by Sir
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
. 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 In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orb ...
, 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 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
nutation Nutation () is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame ...
. The most important term has a period of 18.6 years and an amplitude of 9.2 arcseconds. 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

The
tidal force The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...
on Earth due to a perturbing body (Sun, Moon or planet) is expressed by
Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
, 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 An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere. On E ...
outside of a mean spherical radius. This 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 The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic agains ...
plane. The latter pair of forces creates the following
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
on Earth's equatorial bulge: :\overrightarrow = \frac(C - A) \sin\delta \cos\delta \begin\sin\alpha \\ -\cos\alpha \\ 0\end 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 vernal
equinox A solar equinox is a moment in time when the Sun crosses the Earth's equator, which is to say, appears directly above the equator, rather than north or south of the equator. On the day of the equinox, the Sun appears to rise "due east" and se ...
). 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 vernal 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 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 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: :T_x = \frac\frac(C - A) \sin\epsilon \cos\epsilon where :a, 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, a^3 \left(1 - e^2\right)^\frac accounts for the average distance cubed of the Sun or Moon from Earth over the entire elliptical orbit, 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: :\frac = \frac where ''ω'' is Earth's angular velocity and ''Cω'' is Earth's
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
. Thus the first order component of precession due to the Sun is: :\frac = \frac\left frac\rightS \left frac\frac\rightE whereas that due to the Moon is: :\frac = \frac\left frac\rightL \left frac\frac\rightE 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 \left(1 - 1.5\sin^2 i\right) accounts for the inclination of the Moon's orbit relative to the ecliptic. The term is Earth's 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 homogeneous the term would equal its third eccentricity squared, :e''^2 = \frac where a is the equatorial radius () and c is the polar radius (), so . Applicable parameters for J2000.0 rounded to seven significant digits (excluding leading 1) are:Dennis D. McCarthy,
IERS Technical Note 13 – IERS Standards (1992)
' (Postscript, us
XConvert
.
which yield :''dψS/dt'' = 2.450183 /s :''dψL/dt'' = 5.334529 /s both of which must be converted to ″/a (arcseconds/annum) by the number of arcseconds in 2 π
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s (1.296″/2π) and the number of seconds in one
annum A year or annus is the orbital period of a planetary body, for example, the Earth, moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the h ...
(a Julian year) (3.15576s/a): :''dψS/dt'' = 15.948788″/a   vs   15.948870″/a from Williams :''dψL/dt'' = 34.723638″/a   vs   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 Simon Newcomb (March 12, 1835 – July 11, 1909) was a Canadian–American astronomer, applied mathematician, and autodidactic polymath. He served as Professor of Mathematics in the United States Navy and at Johns Hopkins University. Born in N ...
'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 Very-long-baseline interferometry (VLBI) is a type of astronomical interferometry used in radio astronomy. In VLBI a signal from an astronomical radio source, such as a quasar, is collected at multiple radio telescopes on Earth or in space. T ...
and LLR allowed further refinements, and the
International Astronomical Union The International Astronomical Union (IAU; french: link=yes, Union astronomique internationale, UAI) is a nongovernmental organisation with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreac ...
adopted a new constant value in 2000, and new computation methods and polynomial expressions in 2003 and 2006; the accumulated precession is:N. Capitaine ''et al.'' 2003
p. 581 expression 39
:''pA'' = 5,028.796195''T'' + 1.1054348''T''2 + higher order terms, in arcseconds, with ''T'', the time in Julian centuries (that is, 36,525 days) since the epoch of 2000. The rate of precession is the derivative of that: :''p'' = 5,028.796195 + 2.2108696''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) 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''2 + … 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. For present flight trajectory calculations of artificial satellites and spacecraft, the polynomial method gives better accuracy. In that respect, the
International Astronomical Union The International Astronomical Union (IAU; french: link=yes, Union astronomique internationale, UAI) is a nongovernmental organisation with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreac ...
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. Note that 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. Precession rate exhibits a secular decrease due to tidal dissipation from 59"/a to 45"/a (a =
annum A year or annus is the orbital period of a planetary body, for example, the Earth, moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the h ...
= 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
billion Billion is a word for a large number, and it has two distinct definitions: *1,000,000,000, i.e. one thousand million, or (ten to the ninth power), as defined on the short scale. This is its only current meaning in English. * 1,000,000,000,000, i. ...
s of Julian years (Ga): :''p'' = 50.475838 − 26.368583''T'' + 21.890862''T''2 :''p'' = 50.475838 − 27.000654''T'' + 15.603265''T''2 Note that this gives an average cycle length now of 25,676 years. Precession will be greater than ''p'' by the small amount of +0.135052"/a between and . The jump to this excess over ''p'' will occur in only 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 Ward, when, in about 1,500 million years, 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 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. Using the 620-million year average provided by 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 million years from now).


See also

* Age of Aquarius *
Astrological age An astrological age is a time period in astrological theory which astrologers say, parallels major changes in the development of Earth's inhabitants, particularly relating to culture, society, and politics. There are twelve astrological ages corr ...
*
Astronomical nutation Astronomical nutation is a phenomenon which causes the orientation of the axis of rotation of a spinning astronomical object to vary over time. It is caused by the gravitational forces of other nearby bodies acting upon the spinning object. Al ...
*
Axial tilt In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orb ...
* Euler angles * Longitude of vernal equinox *
Milankovitch cycles Milankovitch cycles describe the collective effects of changes in the Earth's movements on its climate over thousands of years. The term was coined and named after Serbian geophysicist and astronomer Milutin Milanković. In the 1920s, he hypot ...
* Sidereal year


References


Bibliography

* * * Dreyer, J. L. E. ''A History of Astronomy from Thales to Kepler''. 2nd ed. New York: Dover, 1953. * Evans, James. ''The History and Practice of Ancient Astronomy''. New York: Oxford University Press, 1998. * ''Explanatory supplement to the Astronomical ephemeris and the American ephemeris and nautical almanac'' * *
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. * * * Tompkins, Peter. ''Secrets of the Great Pyramid''. With an appendix by Livio Catullo Stecchini. New York: Harper Colophon Books, 1971. * 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. * *


External links


D'Alembert and Euler's Debate on the Solution of the Precession of the Equinoxes
*

{{DEFAULTSORT:Axial Precession (Astronomy) Precession Technical factors of astrology Celestial mechanics