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Axial precession
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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.
Precessional movement of Earth. Earth rotates (white arrows) once a day around its rotational axis (red); this axis itself rotates slowly (white circle), completing a rotation in approximately 26,000 years
In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In particular, it can refer 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 approximately 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 motion—are much smaller in magnitude.
Earth's precession was historically called the precession of the equinoxes, because the equinoxes moved westward along the ecliptic relative to the fixed stars, opposite to the yearly motion of the Sun along the ecliptic. This term is still used in non-technical discussions, that is, when detailed mathematics are absent. Historically,[2] the discovery of the precession of the equinoxes is usually attributed in the west to the Hellenistic-era (second-century BCE) astronomer, Hipparchus, although there are claims of its earlier discovery, such as in the Indian text, Vedanga Jyotisha, dating from 700 BC.[citation needed] 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 precession, 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, 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.[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.
Contents
[hide]
1 Precession nomenclature
2 Effects
3 History
3.1 Hellenistic world
3.1.1 Hipparchus
3.1.2 Ptolemy
3.1.3 Other authors
3.2 Alternative discovery theories
3.2.1 Babylonians
3.2.2 Maya
3.2.3 Ancient Egyptians
3.3 Indian views
3.4 Chinese astronomy
3.5 Middle Ages and Renaissance
3.6 Modern period
4 Hipparchus's discovery
5 Mithraic question
6 Changing pole stars
7 Polar shift and equinoxes shift
8 Cause
9 Equations
10 Values
11 See also
12 Notes
13 Bibliography
14 External links
Precession nomenclature[edit]
Etymologically, "precession" and "procession" are both terms that relate to motion. "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 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 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.
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.
Effects[edit]
The coincidence of the annual cycles of the apses (closest and further approach to the sun) and calendar dates (with seasons noted) at four equally spaced stages of precessionary 26,000-year-cycle. The season dates are those in the north. The tilt of Earth's axis and the eccentricity of its orbit are exaggerated. Approximate estimates. Effects of weak planetary precession on the stages shown are ignored.
The precession of the Earth's axis has a number of observable effects. First, the positions of the south and north celestial poles 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".[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 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 solstices, equinoxes, or other time defined relative to the seasons, slowly changes.[2] For example, suppose that the Earth's orbital position is marked at the summer solstice, when the Earth's axial tilt is pointing directly toward the Sun. One full orbit later, when the Sun has returned to the same apparent position relative to the background stars, the Earth's axial tilt is not now directly toward the Sun: because of the effects of precession, it is a little way "beyond" this. In other words, the solstice occurred a little earlier in the orbit. Thus, the tropical year, measuring the cycle of seasons (for example, the time from solstice to solstice, or equinox to equinox), is about 20 minutes shorter than the sidereal year, which is measured by the Sun's apparent position relative to the stars. After about 26 000 years the difference amounts to a full year, so the positions of the seasons relative to the orbit are "back where they started". (Other effects also slowly change the shape and orientation of the Earth's orbit, and these, in combination with precession, create various cycles of differing periods; see also Milankovitch cycles. The magnitude of the Earth's tilt, as opposed to merely its orientation, also changes slowly over time, but this effect is not attributed directly to precession.)
For identical reasons, the apparent position of the Sun relative to the backdrop of the stars at some seasonally fixed time slowly regresses a full 360° through all twelve traditional constellations of the zodiac, at the rate of about 50.3 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, but the rate itself varies somewhat with time (see Values below), so one cannot say that in exactly 25,772 years the earth's axis will be back to where it is now.
For further details, see Changing pole stars and Polar shift and equinoxes shift, below.
History[edit]
Hellenistic world[edit]
Hipparchus[edit]
Though there is still-controversial evidence that Aristarchus of Samos possessed distinct values for the sidereal and tropical years as early as c. 280 BC,[6] the discovery of precession usually is attributed to Hipparchus (190–120 BC) of Rhodes or Nicaea, a Greek astronomer. According to Ptolemy's Almagest, Hipparchus measured the longitude of Spica and other bright stars. Comparing his measurements with data from his predecessors, Timocharis (320–260 BC) and Aristillus (~280 BC), he concluded that Spica had moved 2° relative to the autumnal equinox. He also compared the lengths of the tropical year (the time it takes the Sun to return to an equinox) and the sidereal year (the time it takes the Sun to return to a fixed star), and found a slight discrepancy. Hipparchus concluded that the equinoxes were moving ("precessing") through the zodiac, and that the rate of precession was not less than 1° in a century, in other words, completing a full cycle in no more than 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 terms as a motion of the heavens, rather than of the Earth.
Ptolemy[edit]
The first astronomer known to have continued Hipparchus's work on precession is Ptolemy in the second century AD. Ptolemy measured the longitudes of Regulus, Spica, and other bright stars with a variation of Hipparchus's lunar method that did not require eclipses. Before sunset, he measured the longitudinal arc separating the Moon from the Sun. Then, after sunset, he measured the arc from the Moon to the star. He used Hipparchus's model to calculate the Sun's longitude, and made corrections for the Moon's motion and its parallax (Evans 1998, pp. 251–255). Ptolemy compared his own observations with those made by Hipparchus, Menelaus of Alexandria, Timocharis, and Agrippa. He found that between Hipparchus's time and his own (about 265 years), the stars had moved 2°40', or 1° in 100 years (36" per year; the rate accepted today is about 50" per year or 1° in 72 years). 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[edit]
Most ancient authors did not mention precession and, perhaps, did not know of it. For instance, Proclus rejected precession, while Theon of Alexandria, a commentator on Ptolemy in the fourth century, accepted Ptolemy's explanation. Theon also reports an alternate theory:
According to certain opinions ancient astrologers believe that from a certain epoch the solstitial signs have a motion of 8° in the order of the signs, after which they go back the same amount. . . . (Dreyer 1958, p. 204)
Instead of proceeding through the entire sequence of the zodiac, the equinoxes "trepidated" back and forth over an arc of 8°. The theory of trepidation is presented by Theon as an alternative to precession.
Alternative discovery theories[edit]
Babylonians[edit]
Various assertions have been made that other cultures discovered precession independently of Hipparchus. According to Al-Battani, the Chaldean astronomers had distinguished the tropical and sidereal year so that by approximately 330 BC, they would have been in a position to describe precession, if inaccurately, but such claims generally are regarded as unsupported.[7]
Maya[edit]
The archaeologist Susan Milbrath has speculated that the Mesoamerican Long Count calendar of "30,000 years involving the Pleiades...may have been an effort to calculate the precession of the equinox."[8] This view is held by few other professional scholars of Mayan civilization.[citation needed]
Ancient Egyptians[edit]
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 temple complex, for instance, allegedly were oriented toward the point on the horizon where certain stars rose or set at key times of the year.[citation needed] 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 at Dendera from a late (Ptolemaic) age, allegedly records precession of the equinoxes (Tompkins 1971). In any case, if the ancient Egyptians knew of precession, their knowledge is not recorded as such in any of their surviving astronomical texts.
Michael Rice 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)
Indian views[edit]
Main article: Ayanamsa
Indian astrologers were aware of axial precession since before the Common Era. Although many of the astronomical texts stored in Taxila were burnt during Muslim invasion of India, the classic astronomical text Suryasiddhanta survived and contains references about ayana movements. In a later commentary on Suryasiddhanta around twelfth century, Bhāskara II[9] says: "sampāt revolves negatively 30000 times in a Kalpa of 4320 million years according to Suryasiddhanta, while Munjāla and others say ayana moves forward 199669 in a Kalpa, and one should combine the two, before ascertaining declension, ascensional difference, etc."[10] Lancelot Wilkinson translated the last of these three verses in a too concise manner to convey the full meaning, and skipped the portion combine the two which the modern Hindu commentary has brought to the fore. According to the Hindu commentary, the final value of period of precession should be obtained by combining +199669 revolutions of ayana with −30000 revolutions of sampaat, to get +169669 per Kalpa, i.e. one revolution in 25461 years, which is near the modern value of 25771 years.
Moreover, Munjāla's value gives a period of 21636 years for ayana's motion, which is the modern value of precession when anomalistic precession also is taken into account. The latter has a period of 136000 years now, but Bhāskar-II gives its value at 144000 years (30000 in a Kalpa), calling it sampāt. Bhāskar-II did not give any name of the final term after combining the negative sampāt with the positive ayana. The value he gave indicates, however, that by ayana he meant precession on account of the combined influence of orbital and anomalistic precessions, and by sampāt he meant the anomalistic period, but defined it as equinox. His language is a bit confused, which he clarified in his own Vāsanābhāshya commentary Siddhānta Shiromani,[11] by saying that Suryasiddhanta was not available and he was writing on the basis of hearsay. Bhāskar-II did not give his own opinion, he merely cited Suryasiddhanta, Munjāla, and unnamed "others".
Extant Suryasiddhanta supports the notion of trepidation within a range of ±27° at the rate of 54" per year according to traditional commentators, but Burgess opined that the original meaning must have been of a cyclical motion, for which he quoted the Suryasiddhanta mentioned by Bhāskar II.[12]
Chinese astronomy[edit]
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[edit]
In medieval Islamic astronomy, precession was known based on Ptolemy's Almagest, and by observations that refined the value.
Al-Battani, 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 on degree per 66 solar years.[13]
Subsequently, Al-Sufi mentions the same values in his Book of Fixed Stars, 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'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.[14]
Later, the Zij-i Ilkhani compiled at the Maragheh observatory sets the precession of the equinoxes at 51 arc seconds per annum, which is very close to the modern value of 50.2 arc seconds.[15]
In the Middle Ages, Islamic and Latin Christian astronomers treated "trepidation" as a motion of the fixed stars to be added to precession. This theory is commonly attributed to the Arab astronomer Thabit ibn Qurra, but the attribution has been contested in modern times. Nicolaus Copernicus published a different account of trepidation in De revolutionibus orbium coelestium (1543). This work makes the first definite reference to precession as the result of a motion of the Earth's axis. Copernicus characterized precession as the third motion of the Earth.
Modern period[edit]
Over a century later precession was explained in Isaac Newton'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 and subsequent scientists.
Hipparchus's discovery[edit]
Hipparchus gave an account of his discovery in On the Displacement of the Solsticial and Equinoctial Points (described in Almagest III.1 and VII.2). He measured the ecliptic longitude of the star Spica during lunar eclipses and found that it was about 6° west of the autumnal equinox. By comparing his own measurements with those of Timocharis of Alexandria (a contemporary of Euclid, who worked with Aristillus early in the 3rd century BC), he found that Spica's longitude had decreased by about 2° in the meantime (exact years are not mentioned in Almagest). Also in VII.2, Ptolemy gives more precise observations of two stars, including Spica and concludes that in each case a 2°:40' change occurred during 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.
Why did Hipparchus need a lunar eclipse to measure the position of a star? The equinoctial points are not marked in the sky, so he needed the Moon as a reference point. Hipparchus already had developed a way to calculate the longitude of the Sun at any moment. A lunar eclipse happens during Full moon, when the Moon is in opposition. At the midpoint of the eclipse, the Moon is 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 April 21, 146 BC, and March 21, 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 is the length of time that the Sun, as viewed from the Earth, takes to return to the same position along the ecliptic (its path among the stars on the celestial sphere). The sidereal year is the length of time that the Sun takes to return to the same position with respect to the stars of the celestial sphere. Precession causes the stars to change their longitude slightly each year, so the sidereal year is longer than the tropical year. Using observations of the equinoxes and solstices, Hipparchus found that the length of the tropical year was 365+1/4−1/300 days, or 365.24667 days (Evans 1998, p. 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 by modifying those of Meton and Callippus in On Intercalary Months and Days (now lost), as described by Ptolemy in the Almagest III.1 (Toomer 1984, p. 139). The Babylonian calendar used a cycle of 235 lunar months in 19 years since 499 BC (with only three exceptions before 380 BC), but it did not use a specified number of days. The Metonic cycle (432 BC) assigned 6,940 days to these 19 years producing an average year of 365+1/4+1/76 or 365.26316 days. The Callippic cycle (330 BC) dropped one day from four Metonic cycles (76 years) for an average year of 365+1/4 or 365.25 days. Hipparchus dropped one more day from four Callippic cycles (304 years), creating the Hipparchic cycle with an average year of 365+1/4−1/304 or 365.24671 days, which was close to his tropical year of 365+1/4−1/300 or 365.24667 days.
We find Hipparchus's mathematical signatures in the Antikythera Mechanism, an ancient astronomical computer of the second century BC. The mechanism is based on a solar year, the Metonic Cycle, which is the period the Moon reappears in the same star in the sky with the same phase (full Moon appears at the same position in the sky approximately in 19 years), the Callipic cycle (which is four Metonic cycles and more accurate), the Saros cycle and the Exeligmos 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, i.e. it takes into account the fast motion of the Moon at perigee and slower motion at apogee. This discovery proves that Hipparchus mathematics were much more advanced than Ptolemy describes in his books, as it is evident that he developed a good approximation of Kepler΄s second law.
Mithraic question[edit]
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The Mithraic Mysteries, colloquially also known as Mithraism, was a 1st–4th century neo-platonic mystery cult of the Roman god Mithras. The near-total lack of written descriptions or scripture necessitates a reconstruction of beliefs and practices from the archaeological evidence, such as that found in Mithraic temples (in modern times called mithraea), which were real or artificial "caves" representing the cosmos. Until the 1970s most scholars followed Franz Cumont in identifying Mithras as a continuation of the Persian god Mithra. Cumont's continuity hypothesis, and his concomitant theory that the astrological component was a late and unimportant accretion, is no longer followed. Today, the cult and its beliefs are recognized as a product of (Greco-)Roman thought, with an astrological component even more heavily pronounced than the already very astrology-centric Roman beliefs generally were. The details, however, are debated.
As far as axial precession is concerned, one scholar of Mithraism, David Ulansey, has interpreted Mithras as a personification of the force responsible for precession (Ulansey, 1989). He argues that the cult was a religious response to Hipparchus's discovery of precession, which—from the ancient geocentric perspective—amounted to the discovery that the entire cosmos (i.e., the outermost celestial sphere of the fixed stars) was moving in a previously unknown way. His analysis is based on the so-called "tauroctony": the image of Mithras killing a bull that was located in the central place in every Mithraic temple. In the standard tauroctony, Mithras and the bull are accompanied by a dog, a snake, a raven, and a scorpion. According to Ulansey, the tauroctony is a star chart. The bull is Taurus, a constellation of the zodiac. In the astrological age that preceded the time of Hipparchus, the vernal equinox had taken place when the Sun was in the constellation of Taurus, and during that previous epoch the constellations of Canis Minor (The Dog), Hydra (The Snake), Corvus (The Raven), and Scorpius (The Scorpion)—i.e., the constellations that correspond to the animals depicted in the tauroctony—all lay on the celestial equator (the location of which is shifted by the precession) and thus had privileged positions in the sky during that epoch. Mithras himself represents the constellation Perseus, which is located directly above Taurus the Bull: the same location occupied by Mithras in the tauroctony image. Mithras' killing of the Bull, by this reasoning, represented the power possessed by this new god to shift the entire cosmic structure, turning the cosmic sphere so that the location of the spring equinox left the constellation of Taurus (a transition symbolized by the killing of the Bull), and the Dog, Snake, Raven, and Scorpion likewise lost their privileged positions on the celestial equator.
The iconography also contains two torch-bearing twins (Cautes and Cautopates) framing the bull-slaying image—one holding a torch pointing up and the other a torch pointing down. These torch-bearers are sometimes depicted with one of them (torch up) holding or associated with a Bull and a tree with leaves, and the other (torch down) holding or associated with a Scorpion and a tree with fruit. Ulansey interprets these torch-bearers as representing the spring equinox (torch up, tree with leaves, Bull) and the autumm equinox (torch down, tree with fruit, Scorpion) in Taurus and Scorpius respectively, which is where the equinoxes were located during the preceding "Age of Taurus" symbolized in the tauroctony as a whole. Thus Ulansey concludes that Mithraic iconography was an "astronomical code" whose secret was the existence of a new cosmic divinity, unknown to those outside the cult, whose fundamental attribute was his ability to shift the structure of the entire cosmos and thereby to control the astrological forces believed at that time to determine human existence, thus giving him the power to grant his devotees success during life and salvation after death (i.e., a safe journey through the planetary spheres and a subsequent immortal existence in the realm of the stars).
Changing pole stars[edit]
Precession of Earth's axis around the north ecliptical pole
A consequence of the precession is a changing pole star. Currently Polaris is extremely well suited to mark the position of the north celestial pole, as Polaris is a moderately bright star with a visual magnitude of 2.1 (variable), and it is located about one degree from the pole.[16]
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 .[17] It was not quite as accurate in its day as Polaris is today.[17] Today, Kochab and its neighbor Pherkad are referred to as the "Guardians of the Pole" (meaning Polaris).[17]
On the other hand, Thuban in the constellation Draco, which was the pole star in 3000 BC, is much less conspicuous at magnitude 3.67 (one-fifth as bright as Polaris); today it is invisible in light-polluted urban skies.
The brilliant Vega in the constellation Lyra is often touted as the best north star (it fulfilled that role around 12,000 BC and will do so again around the year 14,000); however, it never comes closer than 5° to the pole.
When Polaris becomes the north star again around 27,800, due to its proper motion it then will be farther away from the pole than it is now, while in 23,600 BC it came closer to the pole.
Precession of Earth's axis around the south ecliptical 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.
This situation also is seen on a star map. The orientation of the south pole is moving toward the Southern Cross constellation. For the last 2,000 years or so, the Southern Cross has pointed to the south celestial pole. As a consequence, the constellation is no longer visible from subtropical northern latitudes, as it was in the time of the ancient Greeks.
Polar shift and equinoxes shift[edit]
Precessional movement as seen from 'outside' the celestial sphere
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 years it will be the bright star Deneb (left), and in about 12,000 years, Vega (left center). The Earth's rotation is not depicted to scale – in this span of time, it should rotate over 9 million times.
The images above 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 circle (blue) among the stars, centered on the ecliptic north pole (the blue E) and with an angular radius of about 23.4°, an angle known as the obliquity of the ecliptic. The direction of precession is opposite to the daily rotation of the Earth on its axis. The orange axis was the Earth's rotation axis 5,000 years ago, when it pointed to the star Thuban. The yellow axis, pointing to Polaris, marks the axis now.
The equinoxes occur where the celestial equator intersects the ecliptic (red line), that is, where the Earth's axis is perpendicular to the line connecting the centers of the Sun and Earth. (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 orange grid, 5,000 years ago, the vernal equinox was close to the star Aldebaran of Taurus. Now, as seen from the yellow grid, it has shifted (indicated by the red arrow) 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 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.
Diagram showing the westward shift of the vernal equinox among the stars over the past six millennia
The precessional eras of each constellation, often known as Great Months, are approximately:[18]
Constellation
Year entering
Year exiting
Taurus 4500 BC 2000 BC
Aries 2000 BC 100 BC
Pisces 100 BC AD 2700
Cause[edit]
The precession of the equinoxes is caused by the gravitational forces of the Sun and the Moon, and to a lesser extent other bodies, on the Earth. It was first explained by Sir Isaac Newton.[19]
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. For the Earth, however, the applied forces of the Sun and the Moon are nearly perpendicular to the axis of rotation.
The Earth is not a perfect sphere but an oblate spheroid, with an equatorial diameter about 43 kilometers larger than its polar diameter. Because of the Earth's axial tilt, during most of the year the half of this bulge that is closest to the Sun is off-center, either to the north or to the south, and the far half is off-center on the opposite side. The gravitational pull on the closer half is stronger, since gravity decreases with 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 gravitational object's alignment with the Earth's spin axis and approaches zero when it is orthogonal.
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. The most important term has a period of 18.6 years and an amplitude of 9.2″.[20]
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[edit]
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 universal gravitation, whereby the gravitational force of the perturbing body on the side of Earth nearest is said to be greater than the gravitational force on the far side by an amount proportional to the difference in the cubes of the distances between the near and far sides. If the gravitational force of the perturbing body acting on the mass of the Earth as a point mass at the center of Earth ( which provides the centripetal force causing the orbital motion ) is subtracted from the gravitational force of the perturbing body everywhere on the surface of Earth, what remains may be regarded as the tidal force. This gives the paradoxical notion of a force acting away from the satellite but in reality it is simply a lesser force towards that body due to the gradient in the gravitational field. For precession, this tidal force can be grouped into two forces which only act on the equatorial bulge outside of a mean spherical radius. This couple can be decomposed into two pairs of components, one pair parallel to Earth's equatorial plane toward and away from the perturbing body which cancel each other out, and another pair parallel to Earth's rotational axis, both toward the ecliptic plane.[21] The latter pair of forces creates the following torque vector on Earth's equatorial bulge:[4]
T → = 3 G m r 3 ( C − A ) sin  δ cos  δ ( sin  α − cos  α 0 ) {\displaystyle {\overrightarrow {T}}={\frac {3Gm}{r^{3}}}(C-A)\sin \delta \cos \delta {\begin{pmatrix}\sin \alpha \\-\cos \alpha \\0\end{pmatrix}}}
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).
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 (sinδ cosδ sinα) 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 (sinδ cosδ (−cosα)) 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 = 3 2 G m a 3 ( 1 − e 2 ) 3 / 2 ( C − A ) sin  ϵ cos  ϵ {\displaystyle T_{x}={\frac {3}{2}}{\frac {Gm}{a^{3}(1-e^{2})^{3/2}}}(C-A)\sin \epsilon \cos \epsilon }
where
a {\displaystyle 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 ( 1 − e 2 ) 3 / 2 {\displaystyle a^{3}(1-e^{2})^{3/2}} accounts for the average distance cubed of the Sun or Moon from Earth over the entire elliptical orbit,[22] and ϵ {\displaystyle \epsilon \,\!} (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:
d ψ d t = T x C ω sin  ϵ {\displaystyle {\frac {d\psi }{dt}}={\frac {T_{x}}{C\omega \sin \epsilon }}}
where ω is Earth's angular velocity and Cω is Earth's angular momentum. Thus the first order component of precession due to the Sun is:[4]
d ψ S d t = 3 2 [ G m a 3 ( 1 − e 2 ) 3 / 2 ] S [ ( C − A ) C cos  ϵ ω ] E {\displaystyle {\frac {d\psi _{S}}{dt}}={\frac {3}{2}}\left[{\frac {Gm}{a^{3}(1-e^{2})^{3/2}}}\right]_{S}\left[{\frac {(C-A)}{C}}{\frac {\cos \epsilon }{\omega }}\right]_{E}}
whereas that due to the Moon is:
d ψ L d t = 3 2 [ G m ( 1 − 1.5 sin 2  i ) a 3 ( 1 − e 2 ) 3 / 2 ] L [ ( C − A ) C cos  ϵ ω ] E {\displaystyle {\frac {d\psi _{L}}{dt}}={\frac {3}{2}}\left[{\frac {Gm(1-1.5\sin ^{2}i)}{a^{3}(1-e^{2})^{3/2}}}\right]_{L}\left[{\frac {(C-A)}{C}}{\frac {\cos \epsilon }{\omega }}\right]_{E}}
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 ( 1 − 1.5 sin 2  i ) {\displaystyle (1-1.5\sin ^{2}i)} accounts for the inclination of the Moon's orbit relative to the ecliptic. The term (C−A)/C 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,[23]
e ″ 2 = a 2 − c 2 a 2 + c 2 {\displaystyle e''^{2}={\frac {\mathrm {a} ^{2}-\mathrm {c} ^{2}}{\mathrm {a} ^{2}+\mathrm {c} ^{2}}}}
where a is the equatorial radius (6378137 m) and c is the polar radius (6356752 m), so e2 = 0.003358481.
Applicable parameters for J2000.0 rounded to seven significant digits (excluding leading 1) are:[24][25]
Sun
Moon
Earth
Gm = 1.3271244×1020 m3/s2 Gm = 4.902799×1012 m3/s2 (C − A)/C = 0.003273763
a = 1.4959802×1011 m a = 3.833978×108 m ω = 7.292115×10−5 rad/s
e = 0.016708634 e = 0.05554553 ϵ {\displaystyle \epsilon \,\!} = 23.43928°
i= 5.156690°
which yield
dψS/dt = 2.450183×10−12 /s
dψL/dt = 5.334529×10−12 /s
both of which must be converted to "/a (arcseconds/annum) by the number of arcseconds in 2π radians (1.296×106"/2π) and the number of seconds in one annum (a Julian year) (3.15576×107s/a):
dψS/dt = 15.948788"/a vs 15.948870"/a from Williams[4]
dψL/dt = 34.723638"/a vs 34.457698"/a from Williams.
The solar equation is a good representation of precession due 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[edit]
Simon Newcomb's calculation at the end of the 19th century for general precession (p) in longitude gave a value of 5,025.64 arcseconds per tropical century, and was the generally accepted value until artificial satellites delivered more accurate observations and electronic computers allowed more elaborate models to be calculated. Lieske developed an updated theory in 1976, where p equals 5,029.0966 arcseconds per Julian century. Modern techniques such as VLBI and LLR allowed further refinements, and the International Astronomical Union adopted a new constant value in 2000, and new computation methods and polynomial expressions in 2003 and 2006; the accumulated precession is:[26]
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 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 with 5,028.796195 arcseconds per century)[26] 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 T² 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 constants have a period of about 41,000 years, the same as the obliquity of the ecliptic. Note that the constants mentioned here are the linear and all higher terms of the formula above, not the precession itself. That is,
p = A + BT + CT2 + …
is an approximation of
p = a + b sin (2πT/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 (finite) polynomial to match a periodic function over all numbers, the difference in all such approximations will grow without bound as T increases. However, greater 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. So for present flight trajectory calculations of artificial satellites and spacecraft, the polynomial method gives better accuracy. In that respect, the International Astronomical Union chose the best-developed available theory. For up to a few centuries in the past and the future, all formulas do not 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 thus, are related to each other, the two movements act independently of each other, moving in opposite directions.
Precession exhibits a secular decrease due to tidal dissipation from 59"/a to 45"/a (a = annum = 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 billions of Julian years (Ga):[27]
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 jump to this excess over p+ will occur in only 20 Ma beginning now because the secular decrease in precession is beginning to cross a resonance in Earth's orbit caused by the other planets.
According to 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, unless the atmosphere is artificially shielded, the oceans of the Earth will have vaporized before that time (about 2,100 million years from now).
See also[edit]
Astrological age
Age of Aquarius
Axial tilt
Euler angles
Longitude of vernal equinox
Milankovitch cycles
Astronomical nutation
Sidereal year
Notes[edit]
Jump up ^ 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.
^ Jump up to: a b c Astro 101 – Precession of the Equinox, Western Washington University Planetarium, accessed 30 December 2008
Jump up ^ Robert Main, Practical and Spherical Astronomy (Cambridge: 1863) pp.203–4.
^ Jump up to: a b c d James G. Williams, "Contributions to the Earth's obliquity rate, precession, and nutation", Astronomical Journal 108 (1994) 711–724, pp.712&716. All equations are from Williams.
Jump up ^ IAU 2006 Resolution B1: Adoption of the P03 Precession Theory and Definition of the Ecliptic
Jump up ^ Dennis Rawlins, Continued fraction decipherment: the Aristarchan ancestry of Hipparchos' yearlength & precession DIO (1999) 30–42.
Jump up ^ Neugebauer, O. "The Alleged Babylonian Discovery of the Precession of the Equinoxes", Journal of the American Oriental Society, Vol. 70, No. 1. (Jan. – Mar., 1950), pp. 1–8.
Jump up ^ Susan Milbrath, "Just How Precise is Maya Astronomy?", Institute of Maya Studies newsletter, December 2007.
Jump up ^ Siddhānta-shiromani, Golādhyāya, section-VI, verses 17–19
Jump up ^ Translation of the Surya Siddhānta by Pundit Bāpu Deva Sāstri and of the Siddhānta Siromani by the Late Lancelot Wilkinson revised by Pundit Bāpu Deva Sāstri, printed by C B Lewis at Baptist Mission Press, Calcutta, 1861; Siddhānta Shiromani Hindu commentary by Pt Satyadeva Sharmā, Chowkhambā Surbhārati Prakāshan, Varanasi, India.
Jump up ^ Vāsanābhāshya commentary Siddhānta Shiromani (published by Chowkhamba)
Jump up ^ cf. Suryasiddhanta, commentary by E. Burgess, ch.iii, verses 9–12.
Jump up ^ Al-Battani. "Zij Al-Sabi'".
Jump up ^ Al-Sufi. "Book of Fixed Stars".
Jump up ^ Rufus, W. C. (May 1939). "The Influence of Islamic Astronomy in Europe and the Far East". Popular Astronomy. 47 (5): 233–238 [236]. Bibcode:1939PA.....47..233R. .
Jump up ^ van Leeuwen, F. (2007). "HIP 11767". Hipparcos, the New Reduction. Retrieved 2011-03-01.
^ Jump up to: a b c Benningfield, Damond (14 June 2015). "Kochab". Stardate Magazine. University of Texas McDonald Observatory. Retrieved 14 June 2015.
Jump up ^ Kaler, James B. (2002). The ever-changing sky: a guide to the celestial sphere (Reprint). Cambridge University Press. p. 152. ISBN 978-0521499187.
Jump up ^ The Columbia Electronic Encyclopedia, 6th ed., 2007
Jump up ^ "Basics of Space Flight, Chapter 2". Jet Propulsion Laboratory. Jet Propulsion Laboratory/NASA. 2013-10-29. Retrieved 2015-03-26.
Jump up ^ Ivan I. Mueller, Spherical and practical astronomy as applied to geodesy (New York: Frederick Unger, 1969) 59.
Jump up ^ G. Boué & J. Laskar, "Precession of a planet with a satellite", Icarus 185 (2006) 312–330, p.329.
Jump up ^ George Biddel Airy, Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics (third edititon, 1842) 200.
Jump up ^ J.L. Simon et al., "Numerical expressions for precession formulae and mean elements for the Moon and the planets", Astronomy and Astrophyics 282 (1994) 663–683.
Jump up ^ Dennis D. McCarthy, IERS Technical Note 13 – IERS Standards (1992) (Postscript, use PS2PDF).
^ Jump up to: a b N. Capitaine et al. 2003, p. 581 expression 39
Jump up ^ J. Laskar et al., "A long-term numerical solution for the insolation quantities of the Earth Archived May 29, 2008, at the Wayback Machine.", Astronomy and Astrophysics 428 (2004) 261–285, pp.276 & 278.
Bibliography[edit]
Berger, A. L. (1976). "Obliquity & precession for the last 5 000 000 years". Astronomy and Astrophysics. 51. pp. 127–135. Bibcode:1976A&A....51..127B.
Capitaine, N. (2003). "Expressions for IAU 2000 precession quantities". Astronomy and Astrophysics. 412: 567–586. Bibcode:2003A&A...412..567C. doi:10.1051/0004-6361:20031539.
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
Hilton, J.L. (2006). "Report of the International Astronomical Union Division I Working Group on Precession and the Ecliptic" (PDF). Celestial Mechanics and Dynamical Astronomy. 94. pp. 351–367. Bibcode:2006CeMDA..94..351H. doi:10.1007/s10569-006-0001-2.
Lieske, J. H.; Lederle, T.; Fricke, W. (1977). "Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants". Astron. Astrophys. 58. pp. 1–16. Bibcode:1977A&A....58....1L.
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.
Schütz, Michael (2000). "Hipparch und die Entdeckung der Präzession. Bemerkungen zu David Ulansey, Die Ursprünge des Mithraskultes". Electronic Journal of Mithraic Studies. [permanent dead link] (in German) archived version
Simon, J. L. (1994). "Numerical expressions for precession formulae and mean elements for the Moon and the planets". Astronomy and Astrophysics. 282. pp. 663–683. Bibcode:1994A&A...282..663S.
Tomkins, 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.
Vondrak, J.; Capitaine, N.; Wallace, P. (2011). "New precession equations, valid for long time intervals". Astron. Astrophys. 534. p. A22. Bibcode:2011A&A...534A..22V. doi:10.1051/0004-6361/201117274.
Ward, W. R. (1982). "Comments on the long-term stability of the earth's obliquity". Icarus. 50. pp. 444–448. Bibcode:1982Icar...50..444W. doi:10.1016/0019-1035(82)90134-8.
External links[edit]
Wikisource has the text of The New Student's Reference Work article Precession.
D'Alembert and Euler's Debate on the Solution of the Precession of the Equinoxes
Bowley, Roger; Merrifield, Michael. "Axial Precession". Sixty Symbols. Brady Haran for the University of Nottingham.
Forced precession and nutation of Earth
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Economic model
From Wikipedia, the free encyclopedia
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This article is about theoretical modelling. For the overall economic structure of a society, see Economic system.
A diagram of the IS/LM model
In economics, a model is a theoretical construct representing economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified, often mathematical, framework designed to illustrate complex processes. Frequently, economic models posit structural parameters.[1] A model may have various exogenous variables, and those variables may change to create various responses by economic variables. Methodological uses of models include investigation, theorizing, and fitting theories to the world.[2]
Contents
[hide]
1 Overview
2 Types of models
3 Problems with economic models
4 History
5 Tests of macroeconomic predictions
5.1 Comparison with models in other sciences
5.2 The effects of deterministic chaos on economic models
5.3 The critique of hubris in planning
6 Examples of economic models
7 See also
8 Notes
9 References
10 External links
Overview[edit]
In general terms, economic models have two functions: first as a simplification of and abstraction from observed data, and second as a means of selection of data based on a paradigm of econometric study.
Simplification is particularly important for economics given the enormous complexity of economic processes.[3] This complexity can be attributed to the diversity of factors that determine economic activity; these factors include: individual and cooperative decision processes, resource limitations, environmental and geographical constraints, institutional and legal requirements and purely random fluctuations. Economists therefore must make a reasoned choice of which variables and which relationships between these variables are relevant and which ways of analyzing and presenting this information are useful.
Selection is important because the nature of an economic model will often determine what facts will be looked at, and how they will be compiled. For example, inflation is a general economic concept, but to measure inflation requires a model of behavior, so that an economist can differentiate between real changes in price, and changes in price that are to be attributed to inflation.
In addition to their professional academic interest, the use of models include:
Forecasting economic activity in a way in which conclusions are logically related to assumptions;
Proposing economic policy to modify future economic activity;
Presenting reasoned arguments to politically justify economic policy at the national level, to explain and influence company strategy at the level of the firm, or to provide intelligent advice for household economic decisions at the level of households.
Planning and allocation, in the case of centrally planned economies, and on a smaller scale in logistics and management of businesses.
In finance predictive models have been used since the 1980s for trading (investment, and speculation), for example emerging market bonds were often traded based on economic models predicting the growth of the developing nation issuing them. Since the 1990s many long-term risk management models have incorporated economic relationships between simulated variables in an attempt to detect high-exposure future scenarios (often through a Monte Carlo method).
A model establishes an argumentative framework for applying logic and mathematics that can be independently discussed and tested and that can be applied in various instances. Policies and arguments that rely on economic models have a clear basis for soundness, namely the validity of the supporting model.
Economic models in current use do not pretend to be theories of everything economic; any such pretensions would immediately be thwarted by computational infeasibility and the incompleteness or lack of theories for various types of economic behavior. Therefore, conclusions drawn from models will be approximate representations of economic facts. However, properly constructed models can remove extraneous information and isolate useful approximations of key relationships. In this way more can be understood about the relationships in question than by trying to understand the entire economic process.
The details of model construction vary with type of model and its application, but a generic process can be identified. Generally any modelling process has two steps: generating a model, then checking the model for accuracy (sometimes called diagnostics). The diagnostic step is important because a model is only useful to the extent that it accurately mirrors the relationships that it purports to describe. Creating and diagnosing a model is frequently an iterative process in which the model is modified (and hopefully improved) with each iteration of diagnosis and respecification. Once a satisfactory model is found, it should be double checked by applying it to a different data set.
Types of models[edit]
According to whether all the model variables are deterministic, economic models can be classified as stochastic or non-stochastic models; according to whether all the variables are quantitative, economic models are classified as discrete or continuous choice model; according to the model's intended purpose/function, it can be classified as quantitative or qualitative; according to the model's ambit, it can be classified as a general equilibrium model, a partial equilibrium model, or even a non-equilibrium model; according to the economic agent's characteristics, models can be classified as rational agent models, representative agent models etc.
Stochastic models are formulated using stochastic processes. They model economically observable values over time. Most of econometrics is based on statistics to formulate and test hypotheses about these processes or estimate parameters for them. A widely used bargaining class of simple econometric models popularized by Tinbergen and later Wold are autoregressive models, in which the stochastic process satisfies some relation between current and past values. Examples of these are autoregressive moving average models and related ones such as autoregressive conditional heteroskedasticity (ARCH) and GARCH models for the modelling of heteroskedasticity.
Non-stochastic models may be purely qualitative (for example, relating to social choice theory) or quantitative (involving rationalization of financial variables, for example with hyperbolic coordinates, and/or specific forms of functional relationships between variables). In some cases economic predictions in a coincidence of a model merely assert the direction of movement of economic variables, and so the functional relationships are used only stoical in a qualitative sense: for example, if the price of an item increases, then the demand for that item will decrease. For such models, economists often use two-dimensional graphs instead of functions.
Qualitative models – although almost all economic models involve some form of mathematical or quantitative analysis, qualitative models are occasionally used. One example is qualitative scenario planning in which possible future events are played out. Another example is non-numerical decision tree analysis. Qualitative models often suffer from lack of precision.
At a more practical level, quantitative modelling is applied to many areas of economics and several methodologies have evolved more or less independently of each other. As a result, no overall model taxonomy is naturally available. We can nonetheless provide a few examples that illustrate some particularly relevant points of model construction.
An accounting model is one based on the premise that for every credit there is a debit. More symbolically, an accounting model expresses some principle of conservation in the form
algebraic sum of inflows = sinks − sources
This principle is certainly true for money and it is the basis for national income accounting. Accounting models are true by convention, that is any experimental failure to confirm them, would be attributed to fraud, arithmetic error or an extraneous injection (or destruction) of cash, which we would interpret as showing the experiment was conducted improperly.
Optimality and constrained optimization models – Other examples of quantitative models are based on principles such as profit or utility maximization. An example of such a model is given by the comparative statics of taxation on the profit-maximizing firm. The profit of a firm is given by
π ( x , t ) = x p ( x ) − C ( x ) − t x {\displaystyle \pi (x,t)=xp(x)-C(x)-tx\quad }
where p ( x ) {\displaystyle p(x)} is the price that a product commands in the market if it is supplied at the rate x {\displaystyle x} , x p ( x ) {\displaystyle xp(x)} is the revenue obtained from selling the product, C ( x ) {\displaystyle C(x)} is the cost of bringing the product to market at the rate x {\displaystyle x} , and t {\displaystyle t} is the tax that the firm must pay per unit of the product sold.
The profit maximization assumption states that a firm will produce at the output rate x if that rate maximizes the firm's profit. Using differential calculus we can obtain conditions on x under which this holds. The first order maximization condition for x is
∂ π ( x , t ) ∂ x = ∂ ( x p ( x ) − C ( x ) ) ∂ x − t = 0 {\displaystyle {\frac {\partial \pi (x,t)}{\partial x}}={\frac {\partial (xp(x)-C(x))}{\partial x}}-t=0}
Regarding x as an implicitly defined function of t by this equation (see implicit function theorem), one concludes that the derivative of x with respect to t has the same sign as
∂ 2 ( x p ( x ) − C ( x ) ) ∂ 2 x = ∂ 2 π ( x , t ) ∂ x 2 , {\displaystyle {\frac {\partial ^{2}(xp(x)-C(x))}{\partial ^{2}x}}={\partial ^{2}\pi (x,t) \over \partial x^{2}},}
which is negative if the second order conditions for a local maximum are satisfied.
Thus the profit maximization model predicts something about the effect of taxation on output, namely that output decreases with increased taxation. If the predictions of the model fail, we conclude that the profit maximization hypothesis was false; this should lead to alternate theories of the firm, for example based on bounded rationality.
Borrowing a notion apparently first used in economics by Paul Samuelson, this model of taxation and the predicted dependency of output on the tax rate, illustrates an operationally meaningful theorem; that is one requiring some economically meaningful assumption that is falsifiable under certain conditions.
Aggregate models. Macroeconomics needs to deal with aggregate quantities such as output, the price level, the interest rate and so on. Now real output is actually a vector of goods and services, such as cars, passenger airplanes, computers, food items, secretarial services, home repair services etc. Similarly price is the vector of individual prices of goods and services. Models in which the vector nature of the quantities is maintained are used in practice, for example Leontief input-output models are of this kind. However, for the most part, these models are computationally much harder to deal with and harder to use as tools for qualitative analysis. For this reason, macroeconomic models usually lump together different variables into a single quantity such as output or price. Moreover, quantitative relationships between these aggregate variables are often parts of important macroeconomic theories. This process of aggregation and functional dependency between various aggregates usually is interpreted statistically and validated by econometrics. For instance, one ingredient of the Keynesian model is a functional relationship between consumption and national income: C = C(Y). This relationship plays an important role in Keynesian analysis.
Problems with economic models[edit]
Most economic models rest on a number of assumptions that are not entirely realistic. For example, agents are often assumed to have perfect information, and markets are often assumed to clear without friction. Or, the model may omit issues that are important to the question being considered, such as externalities. Any analysis of the results of an economic model must therefore consider the extent to which these results may be compromised by inaccuracies in these assumptions, and a large literature has grown up discussing problems with economic models, or at least asserting that their results are unreliable.
History[edit]
One of the major problems addressed by economic models has been understanding economic growth. An early attempt to provide a technique to approach this came from the French physiocratic school in the Eighteenth century. Among these economists, François Quesnay should be noted, particularly for his development and use of tables he called Tableaux économiques. These tables have in fact been interpreted in more modern terminology as a Leontiev model, see the Phillips reference below.
All through the 18th century (that is, well before the founding of modern political economy, conventionally marked by Adam Smith's 1776 Wealth of Nations) simple probabilistic models were used to understand the economics of insurance. This was a natural extrapolation of the theory of gambling, and played an important role both in the development of probability theory itself and in the development of actuarial science. Many of the giants of 18th century mathematics contributed to this field. Around 1730, De Moivre addressed some of these problems in the 3rd edition of The Doctrine of Chances. Even earlier (1709), Nicolas Bernoulli studies problems related to savings and interest in the Ars Conjectandi. In 1730, Daniel Bernoulli studied "moral probability" in his book Mensura Sortis, where he introduced what would today be called "logarithmic utility of money" and applied it to gambling and insurance problems, including a solution of the paradoxical Saint Petersburg problem. All of these developments were summarized by Laplace in his Analytical Theory of Probabilities (1812). Clearly, by the time David Ricardo came along he had a lot of well-established math to draw from.
Tests of macroeconomic predictions[edit]
In the late 1980s the Brookings Institution compared 12 leading macroeconomic models available at the time. They compared the models' predictions for how the economy would respond to specific economic shocks (allowing the models to control for all the variability in the real world; this was a test of model vs. model, not a test against the actual outcome). Although the models simplified the world and started from a stable, known common parameters the various models gave significantly different answers. For instance, in calculating the impact of a monetary loosening on output some models estimated a 3% change in GDP after one year, and one gave almost no change, with the rest spread between.[4]
Partly as a result of such experiments, modern central bankers no longer have as much confidence that it is possible to 'fine-tune' the economy as they had in the 1960s and early 1970s. Modern policy makers tend to use a less activist approach, explicitly because they lack confidence that their models will actually predict where the economy is going, or the effect of any shock upon it. The new, more humble, approach sees danger in dramatic policy changes based on model predictions, because of several practical and theoretical limitations in current macroeconomic models; in addition to the theoretical pitfalls, (listed above) some problems specific to aggregate modelling are:
Limitations in model construction caused by difficulties in understanding the underlying mechanisms of the real economy. (Hence the profusion of separate models.)
The law of unintended consequences, on elements of the real economy not yet included in the model.
The time lag in both receiving data and the reaction of economic variables to policy makers attempts to 'steer' them (mostly through monetary policy) in the direction that central bankers want them to move. Milton Friedman has vigorously argued that these lags are so long and unpredictably variable that effective management of the macroeconomy is impossible.
The difficulty in correctly specifying all of the parameters (through econometric measurements) even if the structural model and data were perfect.
The fact that all the model's relationships and coefficients are stochastic, so that the error term becomes very large quickly, and the available snapshot of the input parameters is already out of date.
Modern economic models incorporate the reaction of the public and market to the policy maker's actions (through game theory), and this feedback is included in modern models (following the rational expectations revolution and Robert Lucas, Jr.'s Lucas critique of non-microfounded) models. If the response to the decision maker's actions (and their credibility) must be included in the model then it becomes much harder to influence some of the variables simulated.
Comparison with models in other sciences[edit]
Complex systems specialist and mathematician David Orrell wrote on this issue in his book Apollo's Arrow and explained that the weather, human health and economics use similar methods of prediction (mathematical models). Their systems—the atmosphere, the human body and the economy—also have similar levels of complexity. He found that forecasts fail because the models suffer from two problems : (i) they cannot capture the full detail of the underlying system, so rely on approximate equations; (ii) they are sensitive to small changes in the exact form of these equations. This is because complex systems like the economy or the climate consist of a delicate balance of opposing forces, so a slight imbalance in their representation has big effects. Thus, predictions of things like economic recessions are still highly inaccurate, despite the use of enormous models running on fast computers.[5]
The effects of deterministic chaos on economic models[edit]
Economic and meteorological simulations may share a fundamental limit to their predictive powers: chaos. Although the modern mathematical work on chaotic systems began in the 1970s the danger of chaos had been identified and defined in Econometrica as early as 1958:
"Good theorising consists to a large extent in avoiding assumptions....(with the property that)....a small change in what is posited will seriously affect the conclusions."
(William Baumol, Econometrica, 26 see: Economics on the Edge of Chaos).
It is straightforward to design economic models susceptible to butterfly effects of initial-condition sensitivity.[6][7]
However, the econometric research program to identify which variables are chaotic (if any) has largely concluded that aggregate macroeconomic variables probably do not behave chaotically. This would mean that refinements to the models could ultimately produce reliable long-term forecasts. However the validity of this conclusion has generated two challenges:
In 2004 Philip Mirowski challenged this view and those who hold it, saying that chaos in economics is suffering from a biased "crusade" against it by neo-classical economics in order to preserve their mathematical models.
The variables in finance may well be subject to chaos. Also in 2004, the University of Canterbury study Economics on the Edge of Chaos concludes that after noise is removed from S&P 500 returns, evidence of deterministic chaos is found.
More recently, chaos (or the butterfly effect) has been identified as less significant than previously thought to explain prediction errors. Rather, the predictive power of economics and meteorology would mostly be limited by the models themselves and the nature of their underlying systems (see Comparison with models in other sciences above).
The critique of hubris in planning[edit]
A key strand of free market economic thinking is that the market's "invisible hand" guides an economy to prosperity more efficiently than central planning using an economic model. One reason, emphasized by Friedrich Hayek, is the claim that many of the true forces shaping the economy can never be captured in a single plan. This is an argument that cannot be made through a conventional (mathematical) economic model, because it says that there are critical systemic-elements that will always be omitted from any top-down analysis of the economy.[8]
Examples of economic models[edit]
Cobb–Douglas model of production
Solow–Swan model of economic growth
Lucas islands model of money supply
Heckscher–Ohlin model of international trade
Black–Scholes model of option pricing
Gordon-Loeb model of information security investment[9]
See also[edit]
Economic methodology
Computational economics
Agent-based computational economics
Notes[edit]
Jump up ^ Moffatt, Mike. (2008) About.com Structural Parameters Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.
Jump up ^ Mary S. Morgan, 2008 "models," The New Palgrave Dictionary of Economics, 2nd Edition, Abstract.
Vivian Walsh 1987. "models and theory," The New Palgrave: A Dictionary of Economics, v. 3, pp. 482–83.
Jump up ^ Friedman, M. (1953). "The Methodology of Positive Economics". Essays in Positive Economics. Chicago: University of Chicago Press.
Jump up ^ Frankel, Jeffrey A. (May 1986). "The Sources of Disagreement Among International Macro Models and Implications for Policy Coordination" (PDF). NBER Working Paper. Retrieved 23 January 2012.
Jump up ^ http://www.postpythagorean.com/FAQ.html
Jump up ^ Paul Wilmott on his early research in finance: "I quickly dropped... chaos theory (as) it was too easy to construct ‘toy models’ that looked plausible but were useless in practice." Wilmott, Paul (2009), Frequently Asked Questions in Quantitative Finance, John Wiley and Sons, p. 227
Jump up ^ Kuchta, Steve (2004), Nonlinearity and Chaos in Macroeconomics and Financial Markets (PDF), University of Connecticut
Jump up ^ Hayek, Friedrich (September 1945), "The Use of Knowledge in Society", American Economic Review, 35 (4): 519–30, JSTOR 1809376.
Jump up ^ Gordon, Lawrence; Loeb, Martin (November 2002). "The Economics of Information Security Investment". ACM Transactions on Information and System Security. 5 (4): 438–457. doi:10.1145/581271.581274.
References[edit]
Baumol, William & Blinder, Alan (1982), Economics: Principles and Policy (2nd ed.), New York: Harcourt Brace Jovanovich, ISBN 0-15-518839-9 .
Caldwell, Bruce (1994), Beyond Positivism: Economic Methodology in the Twentieth Century (Revised ed.), New York: Routledge, ISBN 0-415-10911-6 .
Holcombe, R. (1989), Economic Models and Methodology, New York: Greenwood Press, ISBN 0-313-26679-4 . Defines model by analogy with maps, an idea borrowed from Baumol and Blinder. Discusses deduction within models, and logical derivation of one model from another. Chapter 9 compares the neoclassical school and the Austrian School, in particular in relation to falsifiability.
Lange, Oskar (1945), "The Scope and Method of Economics", Review of Economic Studies, The Review of Economic Studies Ltd., 13 (1): 19–32, doi:10.2307/2296113, JSTOR 2296113 . One of the earliest studies on methodology of economics, analysing the postulate of rationality.
de Marchi, N. B. & Blaug, M. (1991), Appraising Economic Theories: Studies in the Methodology of Research Programs, Brookfield, VT: Edward Elgar, ISBN 1-85278-515-2 . A series of essays and papers analysing questions about how (and whether) models and theories in economics are empirically verified and the current status of positivism in economics.
Morishima, Michio (1976), The Economic Theory of Modern Society, New York: Cambridge University Press, ISBN 0-521-21088-7 . A thorough discussion of many quantitative models used in modern economic theory. Also a careful discussion of aggregation.
Orrell, David (2007), Apollo's Arrow: The Science of Prediction and the Future of Everything, Toronto: Harper Collins Canada, ISBN 0-00-200740-1 .
Phillips, Almarin (1955), "The Tableau Économique as a Simple Leontief Model", Quarterly Journal of Economics, The MIT Press, 69 (1): 137–44, doi:10.2307/1884854, JSTOR 1884854 .
Samuelson, Paul A. (1948), "The Simple Mathematics of Income Determination", in Metzler, Lloyd A., Income, Employment and Public Policy; essays in honor of Alvin Hansen, New York: W. W. Norton .
Samuelson, Paul A. (1983), Foundations of Economic Analysis (Enlarged ed.), Cambridge: Harvard University Press, ISBN 0-674-31301-1 . This is a classic book carefully discussing comparative statics in microeconomics, though some dynamics is studied as well as some macroeconomic theory. This should not be confused with Samuelson's popular textbook.
Tinbergen, Jan (1939), Statistical Testing of Business Cycle Theories, Geneva: League of Nations .
Walsh, Vivian (1987), "Models and theory", The New Palgrave: A Dictionary of Economics, 3, New York: Stockton Press, pp. 482–83, ISBN 0-935859-10-1 .
Wold, H. (1938), A Study in the Analysis of Stationary Time Series, Stockholm: Almqvist and Wicksell .
Wold, H. & Jureen, L. (1953), Demand Analysis: A Study in Econometrics, New York: Wiley .
External links[edit]
Wikiquote has quotations related to: Economic model
Wikimedia Commons has media related to Economic models.
R. Frigg and S. Hartmann, Models in Science. Entry in the Stanford Encyclopedia of Philosophy.
H. Varian How to build a model in your spare time The author makes several unexpected suggestions: Look for a model in the real world, not in journals. Look at the literature later, not sooner.
Elmer G. Wiens: Classical & Keynesian AD-AS Model – An on-line, interactive model of the Canadian Economy.
IFs Economic Sub-Model [1]: Online Global Model
Economic attractor
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/wiki/Second_derivative_test
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Second derivative test

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/wiki/Local_maximum
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Local maximum

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/wiki/Vector_(geometric)
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Vector (geometric)

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/wiki/Good_(accounting)
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Good (accounting)

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/wiki/Input-output_model
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Input-output model

values: Edit section: Problems with economic models

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/wiki/Wealth_of_Nations
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Wealth of Nations

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/wiki/Nicolas_Bernoulli
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Nicolas Bernoulli

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/wiki/Macroeconomic_models
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Macroeconomic models

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/wiki/Model_(economics)#Pitfalls
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Model (economics)

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/wiki/Unintended_consequence
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Unintended consequence

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/wiki/Robert_Lucas,_Jr.
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Robert Lucas, Jr.

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/wiki/Time_inconsistency
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Time inconsistency

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/wiki/Complex_systems
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Complex systems

values: Edit section: The effects of deterministic chaos on economic models

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/wiki/Chaotic_systems
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Chaotic systems

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/wiki/Neo-classical_economics
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Neo-classical economics

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/wiki/Command_economy
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Command economy

values: Edit section: Examples of economic models

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/wiki/Gordon-Loeb_Model
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Gordon-Loeb Model

values: The New Palgrave Dictionary of Economics

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http://www.nber.org/papers/w1925.pdf

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http://www.postpythagorean.com/FAQ.html

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//www.jstor.org/stable/1809376

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//doi.org/10.1145%2F581271.581274

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/wiki/Oskar_Lange
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Oskar Lange

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//doi.org/10.2307%2F2296113

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//www.jstor.org/stable/2296113

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//doi.org/10.2307%2F1884854

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//www.jstor.org/stable/1884854

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/wiki/Paul_A._Samuelson
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Paul A. Samuelson

values: The New Palgrave: A Dictionary of Economics

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http://www.ifs.du.edu/ifs

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http://www.bentamari.com/attractors

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