axial precession

13
Axial precession This article is about the astronomical concept. For precession of the axes outside of astronomy, see Precession. For other types of astronomical precession, see Precession#Astronomy. In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an as- tronomical body’s rotational axis. In particular, it can re- fer to the gradual shift in the orientation of Earth's axis of rotation, which, similar to a wobbling top, traces out a pair of cones joined at their apices in a cycle of approx- imately 26,000 years. [1] The term “precession” typically refers only to this largest part of the motion; other changes in the alignment of Earth’s axis – nutation and polar mo- tion – are much smaller in magnitude. Earth’s precession was historically called the precession of the equinoxes, because the equinoxes moved west- ward along the ecliptic relative to the fixed stars, op- posite to the yearly motion of the Sun along the eclip- tic. This term is still used in non-technical discus- sions, that is, when detailed mathematics are absent. Historically, [2] the discovery of the precession of the equinoxes is mostly attributed to Hellenistic-era (2nd cen- tury BC) astronomer Hipparchus, although there are al- ternative suggestions claiming earlier discovery. With improvements in the ability to calculate the gravitational force between and among planets during the first half of the nineteenth century, it was recognized that the ecliptic itself moved slightly, which was named planetary pre- cession, as early as 1863, while the dominant component was named lunisolar precession. [3] Their combination was named general precession, instead of precession of the equinoxes. Lunisolar precession is caused by the gravitational forces of the Moon and Sun on Earth’s equatorial bulge, caus- ing Earth’s axis to move with respect to inertial space. Planetary precession (an advance) is due to the small an- gle 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. [4] In addition to the Moon and Sun, the other planets also cause a small movement of Earth’s axis in inertial space, making the contrast in the terms lunisolar versus planetary misleading, so in 2006 the International Astronomical Union recommended that the dominant component be renamed, the precession of the equator, and the minor component be renamed, precession of the ecliptic, but their combination is still named general precession. [5] Many references to the old terms exist in publications predating the change. 1 Precession nomenclature Etymologically, precession and procession are terms that relate to motion (derived from the Latin processio, “a marching forward, an advance”). Generally the term pro- cession is used to describe a group of objects moving for- ward, whereas, the term precession is used to describe a group of objects moving backward. The stars viewed from Earth are seen to proceed in a procession 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 move slightly retrograde, at the rate of approximately 50 arc seconds per year, a phenomenon known as the “precession of the equinox”. In describing this motion astronomers generally have shortened the term to simply “precession”. And in de- scribing 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 astro- physicists. It should be noted that the term “precession” used in astronomy generally describes the observable precession of the equinox (the stars moving retrograde across the sky), whereas the term “precession” as used in physics, generally describes a mechanical process. 2 Effects The precession of the Earth’s axis has a number of ob- servable effects. First, the positions of the south and north celestial poles appear to move in circles against the space-fixed backdrop of stars, completing one cir- cuit 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". [2] In approximately 3200 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 pre- 1

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Page 1: Axial Precession

Axial precession

This article is about the astronomical concept. Forprecession of the axes outside of astronomy, seePrecession. For other types of astronomical precession,see Precession#Astronomy.

In astronomy, axial precession is a gravity-induced,slow, and continuous change in the orientation of an as-tronomical body’s rotational axis. In particular, it can re-fer to the gradual shift in the orientation of Earth's axisof rotation, which, similar to a wobbling top, traces out apair of cones joined at their apices in a cycle of approx-imately 26,000 years.[1] The term “precession” typicallyrefers only to this largest part of themotion; other changesin the alignment of Earth’s axis – nutation and polar mo-tion – are much smaller in magnitude.Earth’s precession was historically called the precessionof the equinoxes, because the equinoxes moved west-ward along the ecliptic relative to the fixed stars, op-posite to the yearly motion of the Sun along the eclip-tic. This term is still used in non-technical discus-sions, that is, when detailed mathematics are absent.Historically,[2] the discovery of the precession of theequinoxes is mostly attributed to Hellenistic-era (2nd cen-tury BC) astronomer Hipparchus, although there are al-ternative suggestions claiming earlier discovery. Withimprovements in the ability to calculate the gravitationalforce between and among planets during the first half ofthe nineteenth century, it was recognized that the eclipticitself moved slightly, which was named planetary pre-cession, as early as 1863, while the dominant componentwas named lunisolar precession.[3] Their combinationwas named general precession, instead of precession ofthe equinoxes.Lunisolar precession is caused by the gravitational forcesof the Moon and Sun on Earth’s equatorial bulge, caus-ing Earth’s axis to move with respect to inertial space.Planetary precession (an advance) is due to the small an-gle between the gravitational force of the other planetson Earth and its orbital plane (the ecliptic), causing theplane of the ecliptic to shift slightly relative to inertialspace. Lunisolar precession is about 500 times greaterthan planetary precession.[4] In addition to the Moon andSun, the other planets also cause a small movement ofEarth’s axis in inertial space, making the contrast in theterms lunisolar versus planetary misleading, so in 2006the International Astronomical Union recommended thatthe dominant component be renamed, the precessionof the equator, and the minor component be renamed,precession of the ecliptic, but their combination is still

named general precession.[5] Many references to the oldterms exist in publications predating the change.

1 Precession nomenclature

Etymologically, precession and procession are terms thatrelate to motion (derived from the Latin processio, “amarching forward, an advance”). Generally the term pro-cession is used to describe a group of objects moving for-ward, whereas, the term precession is used to describea group of objects moving backward. The stars viewedfrom Earth are seen to proceed in a procession fromeast to west daily, due to the Earth’s diurnal motion, andyearly, due to the Earth’s revolution around the Sun. Atthe same time the stars can be observed to move slightlyretrograde, at the rate of approximately 50 arc secondsper year, a phenomenon known as the “precession of theequinox”.In describing this motion astronomers generally haveshortened the term to simply “precession”. And in de-scribing the cause of the motion physicists have also usedthe term “precession”, which has led to some confusionbetween the observable phenomenon and its cause, whichmatters because in astronomy, some precessions are realand others are apparent. This issue is further obfuscatedby the fact that many astronomers are physicists or astro-physicists.It should be noted that the term “precession” used inastronomy generally describes the observable precessionof the equinox (the stars moving retrograde across thesky), whereas the term “precession” as used in physics,generally describes a mechanical process.

2 Effects

The precession of the Earth’s axis has a number of ob-servable effects. First, the positions of the south andnorth celestial poles appear to move in circles againstthe space-fixed backdrop of stars, completing one cir-cuit in approximately 26,000 years. Thus, while today thestar Polaris lies approximately at the north celestial pole,this will change over time, and other stars will becomethe "north star".[2] In approximately 3200 years, the starGamma Cephei in the Cepheus constellation will succeedPolaris for this position. The south celestial pole currentlylacks a bright star to mark its position, but over time pre-

1

Page 2: Axial Precession

2 3 HISTORY

2000 AD

spring summer

autumnwinter

Perihelion, 3 Jan\

\Aphelion, 4 Jul

Winter solstice21 Dec

Summer solstice21 Jun

Springequinox20 Mar Autumn

equinox22 Sep

around 7000 ADor 13000 BC

summer autumn

winterspring

around 12000 ADor 8000 BC

autumn winter

springsummer

around 17000 ADor 3000 BC

winter spring

summerautumn

The relationship between the apses, seasons and precession ofEarth’s axis, as viewed from the north. The seasons indicatedpertain to the Northern Hemisphere and are reversed for theSouthern Hemisphere. The tilt of Earth’s axis and the eccentricityof its orbit are exaggerated.

cession also will cause bright stars to become south stars.As the celestial poles shift, there is a corresponding grad-ual 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 aroundthe Sun at the solstices, equinoxes, or other time definedrelative to the seasons, slowly changes.[2] For example,suppose that the Earth’s orbital position is marked at thesummer solstice, when the Earth’s axial tilt is pointing di-rectly toward the Sun. One full orbit later, when the Sunhas returned to the same apparent position relative to thebackground stars, the Earth’s axial tilt is not now directlytoward the Sun: because of the effects of precession, itis a little way “beyond” this. In other words, the solsticeoccurred a little earlier in the orbit. Thus, the tropicalyear, measuring the cycle of seasons (for example, thetime from solstice to solstice, or equinox to equinox), isabout 20 minutes shorter than the sidereal year, which ismeasured by the Sun’s apparent position relative to thestars. Note that 20 minutes per year is approximatelyequivalent to one year per 25,772 years, so after one fullcycle of 25,772 years the positions of the seasons relativeto the orbit are “back where they started”. (Other effectsalso slowly change the shape and orientation of the Earth’sorbit, and these, in combination with precession, createvarious cycles of differing periods; see also Milankovitchcycles. The magnitude of the Earth’s tilt, as opposed tomerely its orientation, also changes slowly over time, butthis effect is not attributed directly to precession.)For identical reasons, the apparent position of the Sun rel-

ative to the backdrop of the stars at some seasonally fixedtime slowly regresses a full 360° through all twelve tra-ditional constellations of the zodiac, at the rate of about50.3 seconds of arc per year (approximately 360 degreesdivided by 25,772), or 1 degree every 71.6 years. Theconstellation or house of the zodiac in front of which theSun rises at the vernal equinox is therefore changed, andthis is described as “the age of (the zodiac sign or house),”(e.g.; The Age of Aquarius). Various trends in mytholog-ical and religious beliefs are associated with the changeof Ages and the different nature of the astrological signassociated with the zodiac constellation.[6]

For further details, see Changing pole stars and Polar shiftand equinoxes shift, below.

3 History

3.1 Hellenistic world

3.1.1 Hipparchus

Though there is still-controversial evidence thatAristarchus of Samos possessed distinct values for thesidereal and tropical years as early as c. 280 BC,[7]the discovery of precession usually is attributed toHipparchus (190–120 BC) of Rhodes or Nicaea, aGreek astronomer. According to Ptolemy's Almagest,Hipparchus measured the longitude of Spica and otherbright stars. Comparing his measurements with datafrom his predecessors, Timocharis (320–260 BC) andAristillus (~280 BC), he concluded that Spica had moved2° relative to the autumnal equinox. He also comparedthe lengths of the tropical year (the time it takes the Sunto return to an equinox) and the sidereal year (the time ittakes the Sun to return to a fixed star), and found a slightdiscrepancy. Hipparchus concluded that the equinoxeswere moving (“precessing”) through the zodiac, and thatthe rate of precession was not less than 1° in a century,in other words, completing a full cycle in no more than36000 years.Virtually all of the writings of Hipparchus are lost, in-cluding his work on precession. They are mentioned byPtolemy, who explains precession as the rotation of thecelestial sphere around a motionless Earth. It is reason-able to presume that Hipparchus, similarly to Ptolemy,thought of precession in geocentric terms as a motion ofthe heavens, rather than of the Earth.

3.1.2 Ptolemy

The first astronomer known to have continued Hip-parchus’ work on precession is Ptolemy in the secondcentury. Ptolemy measured the longitudes of Regulus,Spica, and other bright stars with a variation of Hip-parchus’ lunar method that did not require eclipses. Be-

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3.3 Indian views 3

fore sunset, he measured the longitudinal arc separatingthe Moon from the Sun. Then, after sunset, he measuredthe arc from the Moon to the star. He used Hipparchus’model to calculate the Sun’s longitude, and made correc-tions for the Moon’s motion and its parallax (Evans 1998,pp. 251–255). Ptolemy compared his own observationswith those made by Hipparchus, Menelaus of Alexandria,Timocharis, and Agrippa. He found that between Hip-parchus’ time and his own (about 265 years), the stars hadmoved 2°40', or 1° in 100 years (36” per year; the rate ac-cepted today is about 50” per year or 1° in 72 years). Healso confirmed that precession affected all fixed stars, notjust those near the ecliptic, and his cycle had the sameperiod of 36,000 years as found by Hipparchus.

3.1.3 Other authors

Most ancient authors did notmention precession and, per-haps, did not know of it. Besides Ptolemy, the list in-cludes Proclus, who rejected precession, and Theon ofAlexandria, a commentator on Ptolemy in the fourth cen-tury, who accepted Ptolemy’s explanation. Theon alsoreports an alternate theory:

According to certain opinions ancient as-trologers believe that from a certain epoch thesolstitial signs have a motion of 8° in the orderof the signs, after which they go back the sameamount. . . . (Dreyer 1958, p. 204)

Instead of proceeding through the entire sequence of thezodiac, the equinoxes “trepidated” back and forth over anarc of 8°. The theory of trepidation is presented by Theonas an alternative to precession.

3.2 Alternative discovery theories

3.2.1 Babylonians

Various assertions have beenmade that other cultures dis-covered precession independently of Hipparchus. Ac-cording to Al-Battani, the Chaldean astronomers had dis-tinguished the tropical and sidereal year so that by ap-proximately 330 BC, they would have been in a positionto describe precession, if inaccurately, but such claimsgenerally are regarded as unsupported.[8]

3.2.2 Maya

There has been speculation that the Mesoamerican LongCount calendar is somehow calibrated against the preces-sion, but this view is not held by professional scholarsof Mayan civilization. Milbrath states, however, that “along cycle 30,000 years involving the Pleiades ... mayhave been an effort to calculate the precession of theequinox.”[9]

3.2.3 Ancient Egyptians

Similar claims have been made that precession wasknown in Ancient Egypt prior to the time of Hipparchus,but these claims remain controversial. Some buildings inthe Karnak temple complex, for instance, allegedly wereoriented toward the point on the horizon where certainstars rose or set at key times of the year. Nonetheless,they kept accurate calendars and if they recorded the dateof the temple reconstructions it would be a fairly simplematter to plot the rough precession rate. The DenderaZodiac, a star-map from the Hathor temple at Denderafrom a late (Ptolemaic) age, allegedly records precessionof the equinoxes (Tompkins 1971). In any case, if theancient Egyptians knew of precession, their knowledge isnot recorded as such in any of their surviving astronomi-cal texts.Michael Rice wrote in his Egypt’s Legacy, “Whether ornot the ancients knew of the mechanics of the Precessionbefore its definition by Hipparchos the Bithynian in thesecond century BC is uncertain, but as dedicated watch-ers of the night sky they could not fail to be aware ofits effects.” (p. 128) Rice believes that “the Precession isfundamental to an understanding of what powered the de-velopment of Egypt” (p. 10), to the extent that “in a senseEgypt as a nation-state and the king of Egypt as a livinggod are the products of the realisation by the Egyptians ofthe astronomical changes effected by the immense appar-ent movement of the heavenly bodies which the Preces-sion implies.” (p. 56). Rice says that “the evidence thatthe most refined astronomical observation was practisedin Egypt in the third millennium BC (and probably evenbefore that date) is clear from the precision with whichthe Pyramids at Giza are aligned to the cardinal points,a precision which could only have been achieved by theiralignment with the stars. " (p. 31) The Egyptians also,says Rice, were “to alter the orientation of a temple whenthe star on whose position it had originally been set movedits position as a consequence of the Precession, somethingwhich seems to have happened several times during theNew Kingdom.” (p. 170)

3.3 Indian views

Main article: Ayanamsa

Indian astrologers were aware of axial precession sinceCE erra. Although many of the astronomical textsstored in Taxila were burnt during Muslim invasion ofIndia, the classic astronomical text Suryasiddhanta sur-vived and contains references about ayana movements.In a later commentary on Suryasiddhanta around twelfthcentury, Bhāskara II[10] says: “sampāt revolves negatively30000 times in a Kalpa of 4320 million years accordingto Suryasiddhanta, while Munjāla and others say ayanamoves forward 199669 in a Kalpa, and one should com-bine the two, before ascertaining declension, ascensional

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4 4 HIPPARCHUS’ DISCOVERY

difference, etc.”[11] Lancelot Wilkinson translated thelast of these three verses in a too concise manner to con-vey the full meaning, and skipped the portion combine thetwo which the modern Hindu commentary has brought tothe fore. According to the Hindu commentary, the finalvalue of period of precession should be obtained by com-bining +199669 revolutions of ayana with −30000 rev-olutions of sampaat, to get +169669 per Kalpa, i.e. onerevolution in 25461 years, which is near the modern valueof 25771 years.Moreover, Munjāla’s value gives a period of 21636 yearsfor ayana’s motion, which is the modern value of preces-sion when anomalistic precession also is taken into ac-count. The latter has a period of 136000 years now,but Bhāskar-II gives its value at 144000 years (30000in a Kalpa), calling it sampāt. Bhāskar-II did not giveany name of the final term after combining the negativesampāt with the positive ayana. The value he gave in-dicates, however, that by ayana he meant precession onaccount of the combined influence of orbital and anoma-listic precessions, and by sampāt he meant the anomal-istic period, but defined it as equinox. His language isa bit confused, which he clarified in his own Vāsanāb-hāshya commentary Siddhānta Shiromani,[12] by sayingthat Suryasiddhanta was not available and he was writingon the basis of hearsay. Bhāskar-II did not give his ownopinion, he merely cited Suryasiddhanta, Munjāla, andunnamed “others”.Extant Suryasiddhanta supports the notion of trepidationwithin a range of ±27° at the rate of 54” per year accord-ing to traditional commentators, but Burgess opined thatthe original meaning must have been of a cyclical mo-tion, for which he quoted the Suryasiddhanta mentionedby Bhāskar II.[13]

3.4 Yu Xi

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

3.5 Middle Ages and Renaissance

In medieval Islamic astronomy, the Zij-i Ilkhani compiledat the Maragheh observatory set the precession of theequinoxes at 51 arc seconds per annum, which is veryclose to the modern value of 50.2 arc seconds.[14]

In the Middle Ages, Islamic and Latin Christian as-tronomers treated “trepidation” as a motion of the fixedstars to be added to precession. This theory is com-monly attributed to the Arab astronomer Thabit ibnQurra, but the attribution has been contested in moderntimes. Nicolaus Copernicus published a different accountof trepidation in De revolutionibus orbium coelestium(1543). This work makes the first definite reference to

precession as the result of a motion of the Earth’s axis.Copernicus characterized precession as the third motionof the Earth.

3.6 Modern period

Over a century later precession was explained in IsaacNewton's Philosophiae Naturalis Principia Mathematica(1687), to be a consequence of gravitation (Evans 1998,p. 246). Newton’s original precession equations did notwork, however, and were revised considerably by Jean leRond d'Alembert and subsequent scientists.

4 Hipparchus’ discovery

Hipparchus gave an account of his discovery in On theDisplacement of the Solsticial and Equinoctial Points (de-scribed in Almagest III.1 and VII.2). He measured theecliptic longitude of the star Spica during lunar eclipsesand found that it was about 6° west of the autumnalequinox. By comparing his ownmeasurements with thoseof 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 about2° in the meantime (exact years are not mentioned in Al-magest). In same chapter VII.2, Ptolemy gives more pre-cise observations of two stars, including Spica and con-cludes that in each case 2°:40' change occurred during128 BC and AD 139 (hence, 1° per century or one fullcycle in 36000 years, this is the precessional period ofHipparchus as reported by Ptolemy ; cf. page 328 inToomer’s translation of Almagest, 1998 edition)) . Healso noticed this motion in other stars. He speculated thatonly the stars near the zodiac shifted over time. Ptolemycalled this his “first hypothesis” (Almagest VII.1), but didnot report any later hypothesis Hipparchus might havedevised. Hipparchus apparently limited his speculations,because he had only a few older observations, which werenot very reliable.Why did Hipparchus need a lunar eclipse to measure theposition of a star? The equinoctial points are not markedin the sky, so he needed the Moon as a reference point.Hipparchus already had developed a way to calculate thelongitude of the Sun at any moment. A lunar eclipse hap-pens during Full moon, when the Moon is in opposition.At the midpoint of the eclipse, theMoon is precisely 180°from the Sun. Hipparchus is thought to have measuredthe longitudinal arc separating Spica from the Moon.To this value, he added the calculated longitude of theSun, plus 180° for the longitude of the Moon. He didthe same procedure with Timocharis’ data (Evans 1998,p. 251). Observations such as these eclipses, inciden-tally, are the main source of data about when Hipparchusworked, since other biographical information about himis minimal. The lunar eclipses he observed, for instance,

Page 5: Axial Precession

5

took place on April 21, 146 BC, and March 21, 135 BC(Toomer 1984, p. 135 n. 14).Hipparchus also studied precession inOn the Length of theYear. Two kinds of year are relevant to understanding hiswork. The tropical year is the length of time that the Sun,as viewed from the Earth, takes to return to the same posi-tion along the ecliptic (its path among the stars on the ce-lestial sphere). The sidereal year is the length of time thatthe Sun takes to return to the same position with respectto the stars of the celestial sphere. Precession causes thestars to change their longitude slightly each year, so thesidereal year is longer than the tropical year. Using obser-vations of the equinoxes and solstices, Hipparchus foundthat the length of the tropical year was 365+1/4−1/300days, or 365.24667 days (Evans 1998, p. 209). Compar-ing this with the length of the sidereal year, he calculatedthat the rate of precession was not less than 1° in a cen-tury. From this information, it is possible to calculatethat his value for the sidereal year was 365+1/4+1/144days (Toomer 1978, p. 218). By giving a minimum ratehe may have been allowing for errors in observation.To approximate his tropical year Hipparchus created hisown lunisolar calendar by modifying those of Meton andCallippus in On Intercalary Months and Days (now lost),as described by Ptolemy in the Almagest III.1 (Toomer1984, p. 139). The Babylonian calendar used a cy-cle of 235 lunar months in 19 years since 499 BC (withonly three exceptions before 380 BC), but it did not usea specified number of days. The Metonic cycle (432BC) assigned 6,940 days to these 19 years producingan average year of 365+1/4+1/76 or 365.26316 days.The Callippic cycle (330 BC) dropped one day fromfour Metonic cycles (76 years) for an average year of365+1/4 or 365.25 days. Hipparchus dropped one moreday from four Callipic cycles (304 years), creating theHipparchic cycle with an average year of 365+1/4−1/304or 365.24671 days, which was close to his tropical yearof 365+1/4−1/300 or 365.24667 days.We find Hipparchus’ mathematical signatures in theAntikythera Mechanism, an ancient astronomical com-puter of the second century BC. The mechanism is basedon a solar year, the Metonic Cycle, which is the periodthe Moon reappears in the same star in the sky with thesame phase (full Moon appears at the same position inthe sky approximately in 19 years), the Callipic cycle(which is four Metonic cycles and more accurate), theSaros cycle and the Exeligmos cycles (three Saros cyclesfor the accurate eclipse prediction). The study of the An-tikythera Mechanism proves that the ancients have beenusing very accurate calendars based on all the aspectsof solar and lunar motion in the sky. In fact, the LunarMechanism which is part of the Antikythera Mechanismdepicts the motion of the Moon and its phase, for a giventime, using a train of four gears with a pin and slot devicewhich gives a variable lunar velocity that is very close tothe second law of Kepler, i.e. it takes into account thefast motion of the Moon at perigee and slower motion

at apogee. This discovery proves that Hipparchus mathe-matics were muchmore advanced than Ptolemy describesin his books, as it is evident that he developed a good ap-proximation of Kepler΄s second law.

5 Mithraic question

The Mithraic Mysteries, colloquially also known asMithraism, was a 1st-4th century neo-platonic, mysterycult of the Roman god Mithras. The near-total lackof written descriptions or scripture necessitates a recon-struction of beliefs and practices from the archaeologicalevidence, such as that found in Mithraic temples (in mod-ern times called mithraea), which were real or artificial“caves” representing the cosmos. Until the 1970s mostscholars followed Franz Cumont in identifying Mithrasas a continuation of the Persian god Mithra. Cumont’scontinuity hypothesis, and his concomitant theory that theastrological component was a late and unimportant accre-tion, is no longer followed. Today, the cult and its beliefsare recognized as a product of (Greco-)Roman thought,with an astrological component even more heavily pro-nounced than the already very astrology-centric Romanbeliefs generally were. The details, however, are debated.As far as axial precession is concerned, one scholarof Mithraism, David Ulansey, has interpreted Mithras(Mithras Sol Invictus – the unconquerable sun) as a sec-ond sun or star that is responsible for precession. He sug-gests the cult may have been inspired by Hipparchus’ dis-covery of precession. Part of his analysis is based on thetauroctony, an image of Mithras sacrificing a bull, foundin most of the temples. According to Ulansey, the tau-roctony is a star chart. Mithras is a second sun or hyper-cosmic sun and/or the constellation Perseus, and the bullis Taurus, a constellation of the zodiac. In an earlierastrological age, the vernal equinox had taken place whenthe Sun was in Taurus. The tauroctony, by this reason-ing, commemorated Mithras-Perseus ending the “Age ofTaurus” (about 2000 BC based on the Vernal Equinox –or about 11,500 BC based on the Autumnal Equinox).The iconography also contains two torch bearing boys(Cautes and Cautopates) on each side of the zodiac.Ulansey, and Walter Cruttenden in his book Lost Star ofMyth and Time, interpret these to mean ages of growthand decay, or enlightenment and darkness; primal el-ements of the cosmic progression. Thus Mithraism isthought to have something to do with the changing ageswithin the precession cycle or Great Year (Plato’s termfor one complete precession of the equinox).Ulansey’s theory has not received much support fromMithraic scholars or from scholars of astrological beliefs.

Page 6: Axial Precession

6 7 POLAR SHIFT AND EQUINOXES SHIFT

Precession of Earth’s axis around the north ecliptical pole

Precession of Earth’s axis around the south ecliptical pole

6 Changing pole stars

A consequence of the precession is a changing pole star.Currently Polaris is extremely well suited to mark the po-sition of the north celestial pole, as Polaris is a moderatelybright star with a visual magnitude of 2.1 (variable), andit is located about one degree from the pole.[15]

On the other hand, Thuban in the constellation Draco,which was the pole star in 3000 BC, is much less conspic-uous at magnitude 3.67 (one-fifth as bright as Polaris);today it is invisible in light-polluted urban skies.The brilliant Vega in the constellation Lyra is often toutedas the best north star (it fulfilled that role around 12,000BC and will do so again around the year 14,000); how-ever, it never comes closer than 5° to the pole.

When Polaris becomes the north star again around27,800, due to its proper motion it then will be fartheraway from the pole than it is now, while in 23,600 BC itcame closer to the pole.It is more difficult to find the south celestial pole in the skyat this moment, as that area is a particularly bland portionof the sky, and the nominal south pole star is Sigma Oc-tantis, which with magnitude 5.5 is barely visible to thenaked eye even under ideal conditions. That will changefrom the 80th to the 90th centuries, however, when thesouth celestial pole travels through the False Cross.This situation also is seen on a star map. The orientationof the south pole is moving toward the Southern Crossconstellation. For the last 2,000 years or so, the SouthernCross has pointed to the south pole. By consequence, theconstellation is no longer visible from subtropical north-ern latitudes, as it was in the time of the ancient Greeks.

7 Polar shift and equinoxes shift

Precessional movement as seen from 'outside' the celestial sphere

The images above attempt to explain the relation betweenthe precession of the Earth’s axis and the shift in theequinoxes. These images show the position of the Earth’saxis on the celestial sphere, a fictitious sphere which placesthe stars according to their position as seen from Earth,regardless of their actual distance. The first image showsthe celestial sphere from the outside, with the constel-lations in mirror image. The second image shows theperspective of a near-Earth position as seen through avery wide angle lens (from which the apparent distortionarises).The rotation axis of the Earth describes, over a periodof 25,700 years, a small circle (blue) among the stars,centered on the ecliptic north pole (the blue E) and withan angular radius of about 23.4°, an angle known as the

Page 7: Axial Precession

7

The 26,000-year cycle of precession as seen from near the Earth.The current north pole star is Polaris (top). In about 8,000 yearsit 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 should rotate over 9 million times.

obliquity of the ecliptic. The direction of precession isopposite to the daily rotation of the Earth on its axis.The orange axis was the Earth’s rotation axis 5,000 yearsago, when it pointed to the star Thuban. The yellow axis,pointing to Polaris, marks the axis now.The equinoxes occur where the celestial equator inter-sects the ecliptic (red line), that is, where the Earth’saxis is perpendicular to the line connecting the centersof the Sun and Earth. (Note that the term “equinox” hererefers to a point on the celestial sphere so defined, ratherthan the moment in time when the Sun is overhead at theEquator, though the two meanings are related.) When theaxis precesses from one orientation to another, the equa-torial plane of the Earth (indicated by the circular gridaround the equator) moves. The celestial equator is justthe Earth’s equator projected onto the celestial sphere, soit moves as the Earth’s equatorial plane moves, and the in-tersection with the ecliptic moves with it. The positionsof the poles and equator on Earth do not change, only theorientation of the Earth against the fixed stars.As seen from the orange grid, 5,000 years ago, the vernalequinox was close to the star Aldebaran of Taurus. Now,as seen from the yellow grid, it has shifted (indicatedby the red arrow) to somewhere in the constellation ofPisces.Still pictures like these are only first approximations, asthey do not take into account the variable speed of theprecession, the variable obliquity of the ecliptic, the plan-etary precession (which is a slow rotation of the eclipticplane itself, presently around an axis located on the plane,with longitude 174°.8764) and the proper motions of thestars.

Diagram showing the westward shift of the vernal equinoxamong the stars over the past six millennia

The precessional eras of each constellation, often knownas Great Months, are approximately:[16]

8 Cause

The precession of the equinoxes is caused by the gravi-tational forces of the Sun and the Moon, and to a lesserextent other bodies, on the Earth. It was first explainedby Sir Isaac Newton.[17]

Axial precession is similar to the precession of a spinningtop. In both cases, the applied force is due to gravity. Fora spinning top, this force tends to be almost parallel to therotation axis. For the Earth, however, the applied forcesof the Sun and the Moon are nearly perpendicular to theaxis of rotation.The Earth is not a perfect sphere but an oblate spheroid,with an equatorial diameter about 43 kilometers largerthan its polar diameter. Because of the Earth’s axial tilt,during most of the year the half of this bulge that is clos-est to the Sun is off-center, either to the north or to thesouth, and the far half is off-center on the opposite side.The gravitational pull on the closer half is stronger, sincegravity decreases with distance, so this creates a smalltorque on the Earth as the Sun pulls harder on one sideof the Earth than the other. The axis of this torque isroughly perpendicular to the axis of the Earth’s rotationso the axis of rotation precesses. If the Earth were a per-fect sphere, there would be no precession.This average torque is perpendicular to the direction inwhich the rotation axis is tilted away from the eclipticpole, so that it does not change the axial tilt itself. Themagnitude of the torque from the Sun (or the Moon)varies with the gravitational object’s alignment with theEarth’s spin axis and approaches zero when it is orthogo-nal.Although the above explanation involved the Sun, thesame explanation holds true for any object moving aroundthe Earth, along or close to the ecliptic, notably, theMoon. The combined action of the Sun and the Moon iscalled the lunisolar precession. In addition to the steady

Page 8: Axial Precession

8 9 EQUATIONS

progressive motion (resulting in a full circle in about25,700 years) the Sun and Moon also cause small peri-odic variations, due to their changing positions. Theseoscillations, in both precessional speed and axial tilt, areknown as the nutation. The most important term has aperiod of 18.6 years and an amplitude of 9.2″.[18]

In addition to lunisolar precession, the actions of the otherplanets of the Solar System cause the whole ecliptic torotate slowly around an axis which has an ecliptic longi-tude of about 174° measured on the instantaneous eclip-tic. This so-called planetary precession shift amounts toa rotation of the ecliptic plane of 0.47 seconds of arc peryear (more than a hundred times smaller than lunisolarprecession). The sum of the two precessions is known asthe general precession.

9 Equations

Tidal force on Earth due to the Sun, Moon, or a planet

The tidal force on Earth due to a perturbing body (Sun,Moon or planet) is expressed by Newton’s law of uni-versal gravitation, whereby the gravitational force of theperturbing body on the side of Earth nearest is said tobe greater than the gravitational force on the far side byan amount proportional to the cube of the distance be-tween the near and far sides. If the gravitational force ofthe perturbing body at the center of Earth (which pro-vides the centripetal force causing the orbital motion) issubtracted from the gravitational force of the perturbingbody everywhere on the surface of Earth, only the tidalforce remains. For precession, this tidal force takes theform of two forces which only act on the equatorial bulgeoutside of a pole-to-pole sphere. This couple can be de-composed into two pairs of components, one pair parallelto Earth’s equatorial plane toward and away from the per-turbing body which cancel each other out, and anotherpair parallel to Earth’s rotational axis, both toward theecliptic plane.[19] The latter pair of forces creates the fol-lowing torque vector on Earth’s equatorial bulge:[4]

−→T =

3Gm

r3(C −A) sin δ cos δ

sinα− cosα

0

where

Gm = standard gravitational parameter of theperturbing bodyr = geocentric distance to the perturbing bodyC = moment of inertia around Earth’s axis ofrotationA = moment of inertia around any equatorialdiameter of EarthC −A =moment of inertia of Earth’s equatorialbulge (C > A)δ = declination of the perturbing body (northor south of equator)α = right ascension of the perturbing body (eastfrom vernal equinox).

The three unit vectors of the torque at the center of theEarth (top to bottom) are x on a line within the eclipticplane (the intersection of Earth’s equatorial plane withthe ecliptic plane) directed toward the vernal equinox, yon a line in the ecliptic plane directed toward the summersolstice (90° east of x), and z on a line directed towardthe north pole of the ecliptic.The value of the three sinusoidal terms in the directionof x (sinδ cosδ sinα) for the Sun is a sine squared wave-form varying from zero at the equinoxes (0°, 180°) to0.36495 at the solstices (90°, 270°). The value in thedirection of y (sinδ cosδ (−cosα)) for the Sun is a sinewave varying from zero at the four equinoxes and solsticesto ±0.19364 (slightly more than half of the sine squaredpeak) halfway between each equinox and solstice withpeaks slightly skewed toward the equinoxes (43.37°(−),136.63°(+), 223.37°(−), 316.63°(+)). Both solar wave-forms have about the same peak-to-peak amplitude andthe 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 ofy is zero for the Sun or Moon, so this component of thetorque does not affect precession. The average torque ofthe sine squared waveform in the direction of x for theSun or Moon is:

Tx =3

2

Gm

a3(1− e2)3/2(C −A) sin ϵ cos ϵ

where

a

e = eccentricity of Earth’s (Sun’s) orbit orMoon’s orbit

and 1/2 accounts for the average of the sine squared wave-form, a3(1 − e2)3/2 accounts for the average distancecubed of the Sun or Moon from Earth over the entire el-liptical orbit,[20] and ϵ (the angle between the equatorial

Page 9: Axial Precession

9

plane and the ecliptic plane) is the maximum value of δfor the Sun and the average maximum value for theMoonover an entire 18.6 year cycle.Precession is:

dt=

TxCω sin ϵ

where ω is Earth’s angular velocity and Cω is Earth’sangular momentum. Thus the first order component ofprecession due to the Sun is:[4]

dψS

dt=

3

2

[Gm

a3(1− e2)3/2

]S

[(C −A)

C

cos ϵω

]E

whereas that due to the Moon is:

dψL

dt=

3

2

[Gm(1− 1.5 sin2 i)a3(1− e2)3/2

]L

[(C −A)

C

cos ϵω

]E

where i is the angle between the plane of the Moon’s or-bit and the ecliptic plane. In these two equations, theSun’s parameters are within square brackets labeled S,theMoon’s parameters are within square brackets labeledL, and the Earth’s parameters are within square bracketslabeled E. The term (1 − 1.5 sin2 i) accounts for the in-clination of the Moon’s orbit relative to the ecliptic. Theterm (C−A)/C is Earth’s dynamical ellipticity or flatten-ing, which is adjusted to the observed precession becauseEarth’s internal structure is not known with sufficient de-tail. If Earth were homogeneous the term would equal itsthird eccentricity squared,[21]

e′′2 =a2 − c2a2 + c2

where a is the equatorial radius (6378137 m) and c is thepolar radius (6356752 m), so e2 = 0.003358481.Applicable parameters for J2000.0 rounded to seven sig-nificant digits (excluding leading 1) are:[22][23]

which yield

dψS/dt = 2.450183×10−12 /sdψL/dt = 5.334529×10−12 /s

both of which must be converted to "/a (arcsec-onds/annum) by the number of arcseconds in 2π radians(1.296×106"/2π) and the number of seconds in oneannum (a Julian year) (3.15576×107s/a):

dψS/dt = 15.948788"/a vs 15.948870"/a fromWilliams[4]

dψL/dt = 34.723638"/a vs 34.457698"/a fromWilliams.

The solar equation is a good representation of precessiondue the Sun because Earth’s orbit is close to an ellipse,being only slightly perturbed by the other planets. Thelunar equation is not as good a representation of preces-sion due to the Moon because its orbit is greatly distortedby the Sun.

10 Values

Simon Newcomb's calculation at the end of the 19th cen-tury for general precession (p) in longitude gave a valueof 5,025.64 arcseconds per tropical century, and was thegenerally accepted value until artificial satellites deliveredmore accurate observations and electronic computers al-lowed more elaborate models to be calculated. Lieskedeveloped an updated theory in 1976, where p equals5,029.0966 arcseconds per Julian century. Modern tech-niques such as VLBI and LLR allowed further refine-ments, and the International Astronomical Union adopteda new constant value in 2000, and new computationmeth-ods and polynomial expressions in 2003 and 2006; theaccumulated precession is:[24]

pA = 5,028.796195×T + 1.1054348×T2 +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 or-der terms.

The constant term of this speed (5,028.796195 arcsec-onds per century in above equation) corresponds to onefull precession circle in 25,771.57534 years (one full cir-cle of 360 degrees divided with 5,028.796195 arcsecondsper century)[24] although some other sources put the valueat 25771.4 years, leaving a small uncertainty.The precession rate is not a constant, but is (at the mo-ment) slowly increasing over time, as indicated by thelinear (and higher order) terms in T. In any case it mustbe stressed that this formula is only valid over a limitedtime period. It is clear that if T gets large enough (far inthe future or far in the past), the T² term will dominateand p will go to very large values. In reality, more elab-orate calculations on the numerical model of the SolarSystem show that the precessional constants have a pe-riod of about 41,000 years, the same as the obliquity ofthe ecliptic. Note that the constants mentioned here arethe linear and all higher terms of the formula above, notthe precession itself. That is,

p = A + BT + CT2 + …

is an approximation of

Page 10: Axial Precession

10 12 NOTES

p = a + b sin (2πT/P), where P is the 410-century period.

Theoretical models may calculate the proper constants(coefficients) corresponding to the higher powers of T,but since it is impossible for a (finite) polynomial tomatcha periodic function over all numbers, the error in allsuch approximations will grow without bound as T in-creases. In that respect, the International AstronomicalUnion chose the best-developed available theory. For upto a few centuries in the past and the future, all formulasdo not diverge very much. For up to a few thousand yearsin the past and the future, most agree to some accuracy.For eras farther out, discrepancies become too large – theexact rate and period of precession may not be computedusing these polynomials even for a single whole preces-sion period.The precession of Earth’s axis is a very slow effect, but atthe level of accuracy at which astronomers work, it doesneed to be taken into account on a daily basis. Note thatalthough the precession and the tilt of Earth’s axis (theobliquity of the ecliptic) are calculated from the sametheory and thus, are related to each other, the two move-ments act independently of each other, moving in mutu-ally perpendicular directions.Precession exhibits a secular decrease due to tidal dissipa-tion from 59"/a to 45"/a (a = annum = Julian year) duringthe 500 million year period centered on the present. Af-ter short-term fluctuations (tens of thousands of years) areaveraged out, the long-term trend can be approximated bythe following polynomials for negative and positive timefrom the present in "/a, where T is in milliards of Julianyears (Ga):[25]

p− = 50.475838 − 26.368583T + 21.890862T2

p+ = 50.475838 − 27.000654T + 15.603265T2

Precession will be greater than p+ by the small amount of+0.135052"/a between +30 Ma and +130 Ma. The jumpto this excess over p+ will occur in only 20 Ma beginningnow because the secular decrease in precession is begin-ning to cross a resonance in Earth’s orbit caused by theother planets.According to Ward, when, in about 1,500 million years,the distance of the Moon, which is continuously increas-ing from tidal effects, has increased from the current 60.3to approximately 66.5 Earth radii, resonances from plan-etary effects will push precession to 49,000 years at first,and then, when the Moon reaches 68 Earth radii in about2,000 million years, to 69,000 years. This will be as-sociated with wild swings in the obliquity of the eclip-tic as well. Ward, however, used the abnormally largemodern value for tidal dissipation. Using the 620-millionyear average provided by tidal rhythmites of about halfthe modern value, these resonances will not be reacheduntil about 3,000 and 4,000 million years, respectively.

However, due to the gradually increasing luminosity ofthe Sun, the oceans of the Earth will have vaporized longbefore that time (about 2,100 million years from now).

11 See also• Age of Aquarius• Axial tilt• Euler angles• Longitude of vernal equinox• Milankovitch cycles• Nutation• Sidereal year

12 Notes[1] 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 Al-manac. Sausalito: University Science Books. p. 99.

[2] Astro 101 – Precession of the Equinox,WesternWashing-ton University Planetarium, accessed 30 December 2008

[3] Robert Main, Practical and Spherical Astronomy (Cam-bridge: 1863) pp.203–4.

[4] James G.Williams, "Contributions to the Earth’s obliquityrate, precession, and nutation", Astronomical Journal 108(1994) 711–724, pp.712&716. All equations are fromWilliams.

[5] IAU 2006 Resolution B1: Adoption of the P03 PrecessionTheory and Definition of the Ecliptic

[6] “A Complete Tour of the Astrological Ages”. RobertOhotto, Intuitive Astrologer. Retrieved 2013-12-17.

[7] Dennis Rawlins, Continued fraction decipherment: theAristarchan ancestry of Hipparchos’ yearlength & preces-sion DIO (1999) 30–42.

[8] Neugebauer, O. “The Alleged Babylonian Discovery ofthe Precession of the Equinoxes”, Journal of the Ameri-can Oriental Society, Vol. 70, No. 1. (Jan. – Mar., 1950),pp. 1–8.

[9] Milbrath, S. “Just How Precise is Maya Astronomy?", In-stitute of Maya Studies newsletter, December 2007.

[10] Siddhānta-shiromani, Golādhyāya, section-VI, verses 17–19

[11] Translation of the Surya Siddhānta by Pundit Bāpu DevaSāstri and of the Siddhānta Siromani by the Late LancelotWilkinson revised by Pundit Bāpu Deva Sāstri, printedby C B Lewis at Baptist Mission Press, Calcutta, 1861;Siddhānta Shiromani Hindu commentary by Pt SatyadevaSharmā, Chowkhambā Surbhārati Prakāshan, Varanasi,India.

Page 11: Axial Precession

11

[12] Vāsanābhāshya commentary Siddhānta Shiromani (pub-lished by Chowkhamba)

[13] cf. Suryasiddhanta, commentary by E. Burgess, ch.iii,verses 9-12.

[14] Rufus, W. C. (May 1939). “The Influence of Islamic As-tronomy in Europe and the Far East”. Popular Astronomy47 (5): 233–238 [236]. Bibcode:1939PA.....47..233R..

[15] van Leeuwen, F. (2007). “HIP 11767”. Hipparcos, theNew Reduction. Retrieved 2011-03-01.

[16] Kaler, James B. (2002). The ever-changing sky: a guide tothe celestial sphere (Reprint). CambridgeUniversity Press.p. 152. ISBN 978-0521499187.

[17] The Columbia Electronic Encyclopedia, 6th ed., 2007

[18] “Basics of Space Flight, Chapter 2”. Jet Propulsion Lab-oratory. Jet Propulsion Laboratory/NASA. 2013-10-29.Retrieved 2015-03-26.

[19] Ivan I. Mueller, Spherical and practical astronomy as ap-plied to geodesy (New York: Frederick Unger, 1969) 59.

[20] G. Boué & J. Laskar, “Precession of a planet with a satel-lite”, Icarus 185 (2006) 312–330, p.329.

[21] George Biddel Airy,Mathematical tracts on the lunar andplanetary theories, the figure of the earth, precession andnutation, the calculus of variations, and the undulatory the-ory of optics (third edititon, 1842) 200.

[22] J.L. Simon et al., "Numerical expressions for precessionformulae and mean elements for the Moon and the plan-ets", Astronomy and Astrophyics 282 (1994) 663–683.

[23] Dennis D.McCarthy, IERS Technical Note 13 – IERS Stan-dards (1992) (Postscript, use PS2PDF).

[24] N. Capitaine et al. 2003, p. 581 expression 39

[25] J. Laskar et al., "A long-term numerical solution for theinsolation quantities of the Earth", Astronomy and Astro-physics 428 (2004) 261–285, pp.276 & 278.

13 Bibliography

• A.L. Berger (1976), “Obliquity & precession for thelast 5 million years”, Astronomy & astrophysics 51,127

• N. Capitaine et al. (2003), “Expressions for IAU2000 precession quantities” 685KB, Astronomy &Astrophysics 412, 567–586.

• Dreyer, J. L. E.. A History of Astronomy fromThales to Kepler. 2nd ed. New York: Dover, 1953.

• Evans, James. The History and Practice of AncientAstronomy. New York: Oxford University Press,1998.

• Explanatory supplement to the Astronomicalephemeris and the American ephemeris and nauticalalmanac

• J.L. Hilton et al. (2006), "Report of the In-ternational Astronomical Union Division I Work-ing Group on Precession and the Ecliptic" (pdf,174KB). Celestial Mechanics and Dynamical As-tronomy (2006) 94: 351..367

• J.H. Lieske et al. (1977), "Expressions for the Pre-cession Quantities Based upon the IAU (1976) Sys-tem of Astronomical Constants". Astronomy & As-trophysics 58, 1..16

• Precession and the Obliquity of the Ecliptic has acomparison of values predicted by different theories

• Pannekoek, A. A History of Astronomy. New York:Dover, 1961.

• Parker, Richard A. “Egyptian Astronomy, Astrol-ogy, and Calendrical Reckoning.” Dictionary of Sci-entific Biography 15:706–727.

• Rice, Michael (1997), Egypt’s Legacy: Thearchetypes of Western civilization, 3000–30 BC,London and New York.

• Schütz, Michael: Hipparch und die Entdeckungder Präzession. Bemerkungen zu David Ulansey,Die Ursprünge des Mithraskultes, in: ejms = Elec-tronic Journal of Mithraic Studies, www.uhu.es/ejms/Papers/Volume1Papers/ulansey.doc

• J.L. Simon et al. (1994), "Numerical expressionsfor precession formulae and mean elements for theMoon and the planets", Astronomy & Astrophysics282, 663..683

• Tomkins, Peter. Secrets of the Great Pyramid. Withan appendix by Livio Catullo Stecchini. New York:Harper Colophon Books, 1971.

• Toomer, G. J. “Hipparchus.”Dictionary of ScientificBiography. Vol. 15:207–224. New York: CharlesScribner’s Sons, 1978.

• Toomer, G. J. Ptolemy’s Almagest. London: Duck-worth, 1984.

• Ulansey, David. The Origins of the Mithraic Myster-ies: Cosmology and Salvation in the Ancient World.New York: Oxford University Press, 1989.

• W.R. Ward (1982), “Comments on the long-termstability of the earth’s obliquity”, Icarus 50, 444

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