Axial precession - Knowledge

4622: 5214: 5202: 890:, 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. Observations such as these eclipses, incidentally, are the main source of data about when Hipparchus worked, since other biographical information about him is minimal. The lunar eclipses he observed, for instance, took place on 21 April 146 BC, and 21 March 135 BC. 1338: 870:). Also in VII.2, Ptolemy gives more precise observations of two stars, including Spica, and concludes that in each case a 2° 40' change occurred between 128 BC and AD 139. Hence, 1° per century or one full cycle in 36,000 years, that is, the precessional period of Hipparchus as reported by Ptolemy; cf. page 328 in Toomer's translation of Almagest, 1998 edition. He also noticed this motion in other stars. He speculated that only the stars near the zodiac shifted over time. Ptolemy called this his "first hypothesis" ( 145:. Planetary precession (an advance) is due to the small angle between the gravitational force of the other planets on Earth and its orbital plane (the ecliptic), causing the plane of the ecliptic to shift slightly relative to inertial space. Lunisolar precession is about 500 times greater than planetary precession. In addition to the Moon and Sun, the other planets also cause a small movement of Earth's axis in inertial space, making the contrast in the terms lunisolar versus planetary misleading, so in 2006 the 1309:, during most of the year the half of this bulge that is closest to the Sun is off-center, either to the north or to the south, and the far half is off-center on the opposite side. The gravitational pull on the closer half is stronger, since gravity decreases with the square of distance, so this creates a small torque on the Earth as the Sun pulls harder on one side of the Earth than the other. The axis of this torque is roughly perpendicular to the axis of the Earth's rotation so the axis of rotation 1080: 4616: 230: 438:. He found that between Hipparchus's time and his own (about 265 years), the stars had moved 2°40', or 1° in 100 years (36" per year; the rate accepted today is about 50" per year or 1° in 72 years). It is possible, however, that Ptolemy simply trusted Hipparchus' figure instead of making his own measurements. He also confirmed that precession affected all fixed stars, not just those near the ecliptic, and his cycle had the same period of 36,000 years as that of Hipparchus. 5286: 964:(three Saros cycles for the accurate eclipse prediction). Study of the Antikythera Mechanism showed that the ancients used very accurate calendars based on all the aspects of solar and lunar motion in the sky. In fact, the Lunar Mechanism which is part of the Antikythera Mechanism depicts the motion of the Moon and its phase, for a given time, using a train of four gears with a pin and slot device which gives a variable lunar velocity that is very close to 906:

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. By giving a minimum rate, he may have been allowing for errors in observation.

44: 5250: 5274: 1008: 985: 5226: 287:, 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 5262: 1354:, 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 5238: 1117:, 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). 422:, 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 956:, an ancient astronomical computer of the second century BC. The mechanism is based on a solar year, the Metonic Cycle, which is the period the Moon reappears in the same place in the sky with the same phase (full Moon appears at the same position in the sky approximately in 19 years), the Callipic cycle (which is four Metonic cycles and more accurate), the 1019:(Beta Ursae Minoris, β UMi, β Ursae Minoris), the brightest star in the bowl of the "Little Dipper", located 16 degrees from Polaris. It held that role from 1500 BC to AD 500. It was not quite as accurate in its day as Polaris is today. Today, Kochab and its neighbor Pherkad are referred to as the "Guardians of the Pole" (meaning Polaris). 1317:

the gravitational attraction. It approaches zero when they are perpendicular. For example, this happens at the equinoxes in the case of the interaction with the Sun. This can be seen to be since the near and far points are aligned with the gravitational attraction, so there is no torque due to the difference in gravitational attraction.

1088: 395:(the time it takes the Sun to return to a fixed star), and found a slight discrepancy. Hipparchus concluded that the equinoxes were moving ("precessing") through the zodiac, and that the rate of precession was not less than 1° in a century, in other words, completing a full cycle in no more than 36,000 years. 180:). This off-center push or pull causes a torque, and a torque on a spinning body results in precession. The gyro can be analyzed in its parts, and each part within the disk is trying to fall, but the rotation brings it from down to up, and the net result of all particles going through this is precession. 3056:

increases. Sufficient accuracy can be obtained over a limited time span by fitting a high enough order polynomial to observation data, rather than a necessarily imperfect dynamic numerical model. For present flight trajectory calculations of artificial satellites and spacecraft, the polynomial method

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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

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According to W. R. Ward, in about 1,500 million years, when the distance of the Moon, which is continuously increasing from tidal effects, has increased from the current 60.3 to approximately 66.5 Earth radii, resonances from planetary effects will push precession to 49,000 years at first, and then,

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The solar equation is a good representation of precession due to the Sun because Earth's orbit is close to an ellipse, being only slightly perturbed by the other planets. The lunar equation is not as good a representation of precession due to the Moon because the Moon's orbit is greatly distorted by

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The precession of Earth's axis is a very slow effect, but at the level of accuracy at which astronomers work, it does need to be taken into account on a daily basis. Although the precession and the tilt of Earth's axis (the obliquity of the ecliptic) are calculated from the same theory and are thus

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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

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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

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This average torque is perpendicular to the direction in which the rotation axis is tilted away from the ecliptic pole, so that it does not change the axial tilt itself. The magnitude of the torque from the Sun (or the Moon) varies with the angle between the Earth's spin axis direction and that of

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

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The coincidence of the annual cycles of the apses (closest and further approach to the Sun) and calendar dates (with seasons noted) at four equally spaced stages of a fictitious precessionary cycle of 20,000 years (rather than the Earth's true precessionary cycle of 26,000 years). The season dates

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has chosen the best-developed available theory. For up to a few centuries into the past and future, none of the formulas used diverge very much. For up to a few thousand years in the past and the future, most agree to some accuracy. For eras farther out, discrepancies become too large – the exact

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The constant term of this speed (5,028.796195 arcseconds per century in above equation) corresponds to one full precession circle in 25,771.57534 years (one full circle of 360 degrees divided by 50.28796195 arcseconds per year) although some other sources put the value at 25771.4 years, leaving a

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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

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The equinoxes occur where the celestial equator intersects the ecliptic (red line), that is, where the Earth's axis is perpendicular to the line connecting the centers of the Sun and Earth.The term "equinox" here refers to a point on the celestial sphere so defined, rather than the moment in time

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Rice, p. 10 "...the Precession is fundamental to an understanding of what powered the development of Egypt"; p. 56 "...in a sense Egypt as a nation-state and the king of Egypt as a living god are the products of the realisation by the Egyptians of the astronomical changes effected by the immense

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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

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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

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of about half the modern value, these resonances will not be reached until about 3,000 and 4,000 million years, respectively. However, due to the gradually increasing luminosity of the Sun, the oceans of the Earth will have vaporized before that time (about 2,100 million years from now).

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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

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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

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During revolution about the Sun, the earth's polar axis exhibits parallelism to Polaris (also known as the North Star). Although observing parallelism, the orientation of Earth's polar axis exhibits precession – a circular wobbling exhibited by gyroscopes – that results in a 28,000-year-long

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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

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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

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Michael Rice, a popular writer on Ancient Egypt, has written that Ancient Egyptians must have observed the precession, and suggested that this awareness had profound affects on their culture. Rice noted that Egyptians re-oriented temples in response to precession of associated stars.

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in −3101 and again 3,600 years later in 499. The direction changed from prograde to retrograde midway between these years at −1301 when it reached its maximum deviation of 27°, and would have remained retrograde, the same direction as modern precession, for 3600 years until 2299.

3081:) 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 3504:, p. 128). "Whether or not the ancients knew of the mechanics of the Precession before its definition by Hipparchos the Bithynian, in the second century BC is uncertain, but as dedicated watchers of the night sky they could not fail to be aware of its effects." 1813: 591:

librated 27° in both directions from the sidereal epoch. Thus the equinox moved 54° in one direction and then back 54° in the other direction. This cycle took 7200 years to complete at a rate of 54″/year. The equinox coincided with the epoch at the beginning of the

558:, one with a rate and another without a rate, and several related models of precession. Each had minor changes or corrections by various commentators. The dominant of the three was the trepidation described by the most respected Indian astronomical treatise, the 3257:

precessional cycle. Currently, Earth's polar axis points roughly in the direction of Polaris (the North Star). As a result of precession, over the next 11,000 years, Earth's axis will precess or wobble so that it assumes an orientation toward the star Vega.

2619: 198:, 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". 2889:) 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. 176:. In a similar way to how the force from the table generates this phenomenon of precession in the spinning gyro, the gravitational pull of the Sun and Moon on the Earth's equatorial bulge generates a very slow precession of the Earth's axis (see 3523:

Rice, p. 170 "to alter the orientation of a temple when the star on whose position it had originally been set moved its position as a consequence of the Precession, something which seems to have happened several times during the New

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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

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constellation. For the last 2,000 years or so, the Southern Cross has pointed to the south celestial pole. As a consequence, the constellation is difficult to view from subtropical northern latitudes, unlike in the time of the

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Tidal force on Earth due to the Moon or another celestial body. It shows both the tidal field (thick red arrows) and the gravity field (thin blue arrows) exerted on Earth's surface and center (label O) by the Moon (label

458:"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) 1995: 1321:(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 1385: 2432: 2204: 118:. With improvements in the ability to calculate the gravitational force between planets during the first half of the nineteenth century, it was recognized that the ecliptic itself moved slightly, which was named 254:

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

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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

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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

1051:, 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 874:

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.

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are those in the north. The tilt of fictitious Earth's axis and the eccentricity of its orbit are exaggerated. Approximate estimates. Effects of weak planetary precession on the stages shown are ignored.

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period). These claims remain controversial. Ancient Egyptians kept accurate calendars and recorded dates on temple walls, so it would be a simple matter for them to plot the "rough" precession rate.

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accounts for the average distance cubed of the Sun or Moon from Earth over the entire elliptical orbit, and ε (the angle between the equatorial plane and the ecliptic plane) is the maximum value of

542:, allegedly records the precession of the equinoxes. In any case, if the ancient Egyptians knew of precession, their knowledge is not recorded as such in any of their surviving astronomical texts. 1683: 945:(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 3957:

Simon, J. L.; Bretagnon, P.; Chapront, J.; Chapront-Touze, M.; Francou, G.; Laskar, J. (1994). "Numerical expressions for precession formulae and mean elements for the Moon and the planets".

2492: 737:, mentions Hipparchus's calculation of precession, and Ptolemy's value of 1 degree per 100 solar years, says that he measured precession and found it to be one degree per 66 solar years. 166: 1111:

The images at right attempt to explain the relation between the precession of the Earth's axis and the shift in the equinoxes. These images show the position of the Earth's axis on the

800:(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. 1329:

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.

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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

2527: 270:, or other time defined relative to the seasons, slowly changes. For example, suppose that the Earth's orbital position is marked at the summer solstice, when the Earth's 259:. 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. 3944:

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

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It is more difficult to find the south celestial pole in the sky at this moment, as that area is a particularly bland portion of the sky. The nominal south pole star is

4520: 613:). His trepidation consisted of an arc of 46°40′ in one direction and a return to the starting point. Half of this arc, 23°20′, was identified with the Sun's maximum 398:

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

291:. 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.) 495:

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.

4443: 3052:, but since it is impossible for a polynomial to match a periodic function over all numbers, the difference in all such approximations will grow without bound as 1839: 882:

to measure the position of a star. Hipparchus already had developed a way to calculate the longitude of the Sun at any moment. A lunar eclipse happens during

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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

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of Al-Battani as adjusting coordinates for stars by 11 degrees and 10 minutes of arc to account for the difference between Al-Battani's time and Ptolemy's.

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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

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Lieske, J. H.; Lederle, T.; Fricke, W. (1977). "Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants".

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toward that body due to the gradient in the gravitational field. For precession, this tidal force can be grouped into two forces which only act on the

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along the ecliptic. Historically, the discovery of the precession of the equinoxes is usually attributed in the West to the 2nd-century-BC astronomer

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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

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The first astronomer known to have continued Hipparchus's work on precession is Ptolemy in the second century AD. Ptolemy measured the longitudes of

1511:{\displaystyle {\overrightarrow {T}}={\frac {3GM}{r^{3}}}(C-A)\sin \delta \cos \delta {\begin{pmatrix}\sin \alpha \\-\cos \alpha \\0\end{pmatrix}}} 583:

of 19°11′ to 23°51′, depending on the group consulted. This epoch causes the roughly 30 Indian calendar years to begin 23–28 days after the modern

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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

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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

283:, measuring the cycle of seasons (for example, the time from solstice to solstice, or equinox to equinox), is about 20 minutes shorter than the 2215: 2022: 1604:

on a line within the ecliptic plane (the intersection of Earth's equatorial plane with the ecliptic plane) directed toward the March equinox,

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At present, the rate of precession corresponds to a period of 25,772 years, so tropical year is shorter than sidereal year by 1,224.5 seconds

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Still pictures like these are only first approximations, as they do not take into account the variable speed of the precession, the variable

1033:, 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 462:

Instead of proceeding through the entire sequence of the zodiac, the equinoxes "trepidated" back and forth over an arc of 8°. The theory of

4436: 814: 2974:. It is a polynomial expression centred on the J2000 datum, empirically fitted to observational data, not on a deterministic model of the 866:

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

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beginning now because the secular decrease in precession is beginning to cross a resonance in Earth's orbit caused by the other planets.

17: 1107:(left center). The Earth's rotation is not depicted to scale – in this span of time, it would actually rotate over 4 million times. 157:, but their combination is still named general precession. Many references to the old terms exist in publications predating the change. 750:, mentions the same values that Ptolemy's value for precession is 1 degree per 100 solar years. He then quotes a different value from 194:("to precede, to come before or earlier"). The stars viewed from Earth are seen to proceed from east to west daily, due to the Earth's 2513:, which is adjusted to the observed precession because Earth's internal structure is not known with sufficient detail. If Earth were 3479: 4974: 3276: 1351: 3369: 949:

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.

5311: 4429: 4244: 775:, 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. 83:. The term "precession" typically refers only to this largest part of the motion; other changes in the alignment of Earth's axis— 3779: 1000:

is extremely well suited to mark the position of the north celestial pole, as Polaris is a moderately bright star with a visual

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waveform varying from zero at the equinoxes (0°, 180°) to 0.36495 at the solstices (90°, 270°). The value in the direction of

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When Polaris becomes the north star again around 27,800, it will then be farther away from the pole than it is now due to its

3634: 4943: 1808:{\displaystyle T_{x}={\frac {3}{2}}{\frac {GM}{a^{3}\left(1-e^{2}\right)^{\frac {3}{2}}}}(C-A)\sin \epsilon \cos \epsilon } 454:, a commentator on Ptolemy in the fourth century, accepted Ptolemy's explanation. Theon also reports an alternate theory: 3218:

Hohenkerk, C.Y., Yallop, B.D., Smith, C.A., & Sinclair, A.T. "Celestial Reference Systems" in Seidelmann, P.K. (ed.)

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show that the precessional rate has a period of about 41,000 years, the same as the obliquity of the ecliptic. That is,

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For the Earth, however, the applied forces of the Sun and the Moon are closer to perpendicular to the axis of rotation.

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In the Middle Ages, Islamic and Latin Christian astronomers treated "trepidation" as a motion of the fixed stars to be

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around a motionless Earth. It is reasonable to presume that Hipparchus, similarly to Ptolemy, thought of precession in

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itself, presently around an axis located on the plane, with longitude 174.8764°) and the proper motions of the stars.

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Various assertions have been made that other cultures discovered precession independently of Hipparchus. According to

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Simon, J. L. (1994). "Numerical expressions for precession formulae and mean elements for the Moon and the planets".

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rate and period of precession may not be computed using these polynomials even for a single whole precession period.

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Because the equinoctial points are not marked in the sky, Hipparchus needed the Moon as a reference point; he used a

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on either side of the equator at the solstices. But no period was specified, thus no annual rate can be ascertained.

511:...may have been an effort to calculate the precession of the equinox." This view is held by few other professional 3058: 2909:

adopted a new constant value in 2000, and new computation methods and polynomial expressions in 2003 and 2006; the

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The precession of the Earth's axis has a number of observable effects. First, the positions of the south and north

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In describing this motion astronomers generally have shortened the term to simply "precession". In describing the

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The longitudes of the first point of Aries, according to the two schools therefore differ by 23°′ (–) 19°11′ ...

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of 2.1 (variable), and is located about one degree from the pole, with no stars of similar brightness too close.

99: 941:(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 4767: 4273: 3535: 4948: 3284: 539: 246:

lies approximately at the north celestial pole, this will change over time, and other stars will become the "

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related one to the other, the two movements act independently of each other, moving in opposite directions.

5316: 4274:"Hipparch und die Entdeckung der Präzession. Bemerkungen zu David Ulansey, Die Ursprünge des Mithraskultes" 3148: 60: 4903: 4883: 4829: 4714: 4550: 3176: 1305:, with an equatorial diameter about 43 kilometers larger than its polar diameter. Because of the Earth's 720: 316:

below), so one cannot say that in exactly 25,772 years the Earth's axis will be back to where it is now.

4416: 4182:"Report of the International Astronomical Union Division I Working Group on Precession and the Ecliptic" 2518: 5192: 4570: 4555: 4493: 4472: 3703: 3561: 823: 214: 2614:{\displaystyle e''^{2}={\frac {\mathrm {a} ^{2}-\mathrm {c} ^{2}}{\mathrm {a} ^{2}+\mathrm {c} ^{2}}}} 4724: 4535: 4313: 1137: 30:

This article is about the astronomical concept. For precession of the axes outside of astronomy, see

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Explanatory supplement to the Astronomical ephemeris and the American ephemeris and nautical almanac

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Theoretical models may calculate the constants (coefficients) corresponding to the higher powers of

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equals 5,029.0966 arcseconds (or 1.3969713 degrees) per Julian century. Modern techniques such as

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This situation also is seen on a star map. The orientation of the south pole is moving toward the

4984: 4762: 4664: 4580: 4575: 3078: 2841: 822:. However, Newton's original precession equations did not work, and were revised considerably by 446:

Most ancient authors did not mention precession and, perhaps, did not know of it. For instance,

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is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's

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Neugebauer, O. (1950). "The Alleged Babylonian Discovery of the Precession of the Equinoxes".

3370:"IAU 2006 Resolution B1: Adoption of the P03 Precession Theory and Definition of the Ecliptic" 3321: 1141:. The direction of precession is opposite to the daily rotation of the Earth on its axis. The 5137: 3161: 2514: 1322: 1153:

when the Sun is overhead at the Equator (though the two meanings are related). When the axis

1030: 953: 84: 4365: 4306: 4237: 4141: 4116: 4039: 3970: 4964: 4878: 4704: 4565: 4361: 4302: 4233: 4196: 4137: 4112: 4072: 4035: 4022:

Laskar, J.; Robutel, P.; Joutel, F.; Gastineau, M.; Correia, A. C. M.; Levrard, B. (2004).

3966: 3684: 3591: 3376: 3333: 1125: 772: 484: 435: 4397: 4285: 3985: 3675:

Rufus, W. C. (May 1939). "The Influence of Islamic Astronomy in Europe and the Far East".

3582:

Pingree, David (1972), "Precession and trepidation in Indian astronomy before A.D. 1200",

8: 5306: 5278: 5174: 4969: 4824: 4689: 4641: 3181: 2902: 1325:. The most important term has a period of 18.6 years and an amplitude of 9.2 arcseconds. 934: 791: 746: 451: 288: 4510: 4200: 4076: 3943: 3688: 3595: 3337: 3273: 1091:

The 25,700 year cycle of precession as seen from near the Earth. The current north

5321: 5266: 5254: 5206: 5149: 4923: 4847: 4777: 4212: 3607: 3461: 3351: 3196: 3070: 1920:

for the Sun and the average maximum value for the Moon over an entire 18.6 year cycle.

1824: 1364: 1016: 1001: 910: 580: 67:. In particular, axial precession can refer to the gradual shift in the orientation of 3884: 1120:

The rotation axis of the Earth describes, over a period of 25,700 years, a small

5201: 4997: 4862: 4679: 4320:. With an appendix by Livio Catullo Stecchini. New York: Harper Colophon Books, 1971. 4216: 4084: 3846: 3713: 3707: 3611: 3416: 3355: 3247: 3237: 1545: 1355: 851: 708: 528: 384: 71:'s axis of rotation in a cycle of approximately 26,000 years. This is similar to the 64: 4063:

Ward, W. R. (1982). "Comments on the long-term stability of the earth's obliquity".

2494:

accounts for the inclination of the Moon's orbit relative to the ecliptic. The term

1600:

The three unit vectors of the torque at the center of the Earth (top to bottom) are

375:

and other bright stars. Comparing his measurements with data from his predecessors,

5230: 5142: 4888: 4867: 4819: 4782: 4595: 4369: 4342:

The Origins of the Mithraic Mysteries: Cosmology and Salvation in the Ancient World

4204: 4145: 4080: 4043: 3599: 3453: 3412: 3341: 2013: 2005: 1376: 1360: 1290:, and to a lesser extent other bodies, on the Earth. It was first explained by Sir 1113: 965: 946: 787: 399: 138: 4374: 4349: 758:'s reign, of 1 degree for every 66 solar years. He also quotes the aforementioned 4873: 4852: 4684: 4656: 4421: 4181: 4158: 4150: 4125: 4048: 4023: 4005: 3909: 3638: 3280: 1608:

on a line in the ecliptic plane directed toward the summer solstice (90° east of

1589: 1302: 1087: 1034: 942: 560: 358: 80: 76: 35: 4391:

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

3810: 1337: 1145:

was the Earth's rotation axis 5,000 years ago, when it pointed to the star

603: 5290: 5218: 5169: 4928: 4918: 4857: 4814: 4646: 4636: 4605: 4505: 4477: 4257:

Parker, Richard A. "Egyptian Astronomy, Astrology, and Calendrical Reckoning."

3836: 3661: 3603: 2874:

the Sun and neither the radius nor the eccentricity is constant over the year.

1211: 1064: 1048: 535: 239: 213:

generally describes the observable precession of the equinox (the stars moving

195: 142: 4208: 711:

to mention precession. He estimated the rate of precession as 1° in 50 years.

682: 5300: 5056: 5007: 4992: 4938: 4709: 4545: 4540: 4456: 4103:

Berger, A. L. (1976). "Obliquity and precession for the last 5000000 years".

3990: 3191: 3086: 2990:

will go to very large values. In reality, more elaborate calculations on the

2882: 1593: 1282:

The precession of the equinoxes is caused by the gravitational forces of the

1184: 1172: 1079: 1041: 1027: 938: 902: 898: 879: 767: 584: 524: 492: 488: 392: 388: 284: 280: 3251: 1990:{\displaystyle {\frac {d\psi }{dt}}={\frac {T_{x}}{C\omega \sin \epsilon }}} 190: 5242: 5154: 5041: 5036: 5031: 5018: 4757: 4323: 4024:"A long-term numerical solution for the insolation quantities of the Earth" 3186: 3171: 2975: 2817:

both of which must be converted to ″/a (arcseconds/annum) by the number of

1639: 1291: 1210:

of the ecliptic, the planetary precession (which is a slow rotation of the

809: 755: 655:

revolutions to make the accumulated precession zero near 500. Visnucandra (

625: 251: 88: 1164:

do not change, only the orientation of the Earth against the fixed stars.

5164: 5159: 4804: 4787: 4719: 4699: 4585: 4467: 4405: 3287: 2970:. In any case it must be stressed that this formula is only valid over a 2427:{\displaystyle {\frac {d\psi _{L}}{dt}}={\frac {3}{2}}\left_{L}\left_{E}} 2199:{\displaystyle {\frac {d\psi _{S}}{dt}}={\frac {3}{2}}\left_{S}\left_{E}} 1579: 1347: 1052: 957: 819: 614: 555: 463: 229: 63:. In the absence of precession, the astronomical body's orbit would show 1350:

on Earth due to a perturbing body (Sun, Moon or planet) is expressed by

571:

but revised during the next few centuries. It used a sidereal epoch, or

5179: 5132: 5116: 5002: 4908: 4729: 4590: 4560: 4498: 3566:, translated by Burgess, Ebenezzer, University of Calcutta, p. 114 3514:

apparent movement of the heavenly bodies which the Precession implies."

3166: 1310: 1306: 1207: 1155: 863: 855: 730: 667: 480: 431: 426:. Ptolemy compared his own observations with those made by Hipparchus, 403: 380: 376: 346: 271: 262:

Secondly, the position of the Earth in its orbit around the Sun at the

256: 185: 169: 165: 115: 107: 72: 31: 5285: 3465: 3322:"Contribution to the Earth's Obliquity Rate, Precession, and Nutation" 2885:'s calculation at the end of the 19th century for general precession ( 1195:. Now, as seen from the yellow grid, it has shifted (indicated by the 523:

Similarly, it is claimed the precession of the equinoxes was known in

5076: 4913: 4797: 4772: 4752: 4669: 4621: 4615: 4600: 2818: 1188: 1092: 993: 961: 918: 883: 843: 594: 300: 247: 210: 173: 52: 3956: 3925:

G. Boué & J. Laskar, "Precession of a planet with a satellite",

3709:

The Edge of Objectivity: An Essay in the History of Scientific Ideas

1313:. If the Earth were a perfect sphere, there would be no precession. 5111: 5091: 4933: 4839: 4739: 4530: 4525: 3457: 3346: 1369: 1135:) and with an angular radius of about 23.4°, an angle known as the 572: 512: 508: 423: 367: 263: 103: 43: 4266:

Egypt's Legacy: The archetypes of Western civilization, 3000–30 BC

2016:. Thus the first order component of precession due to the Sun is: 354: 133:

Lunisolar precession is caused by the gravitational forces of the

5106: 5096: 5026: 4898: 4809: 4674: 3404: 2657: 2510: 1096: 1083:

Precessional movement as seen from 'outside' the celestial sphere

997: 969: 926: 850:

during lunar eclipses and found that it was about 6° west of the

741: 447: 415: 362: 267: 243: 218: 968:. That is, it takes into account the fast motion of the Moon at 897:. Two kinds of year are relevant to understanding his work. The 383:(~280 BC), he concluded that Spica had moved 2° relative to the 5101: 5051: 5046: 2833: 2825: 2660:

rounded to seven significant digits (excluding leading 1) are:

1851:

and 1/2 accounts for the average of the sine squared waveform,

1373: 1167: 1160:

ecliptic moves with it. The positions of the poles and equator

1146: 1023: 1007: 984: 973: 859: 350: 296: 3543:, Council of Scientific and Industrial Research, p. 262, 2982:

gets large enough (far in the future or far in the past), the

5081: 4893: 4792: 4747: 3563:

Translation of Surya Siddhanta: A Textbook of Hindu Astronomy

3557: 3074: 2936: 2837: 1217:

The precessional eras of each constellation, often known as "

1124:

among the stars near the top of the diagram, centered on the

1100: 1068: 914: 909:

To approximate his tropical year, Hipparchus created his own

847: 704: 419: 372: 68: 4021: 2935:, the time in Julian centuries (that is, 36,525 days) since 1619:

The value of the three sinusoidal terms in the direction of

1149:. The yellow axis, pointing to Polaris, marks the axis now. 1099:(top). In about 8,000 years it will be the bright star 1071:(about 25° N), but only during the winter/early spring. 836:

On the Displacement of the Solsticial and Equinoctial Points

620:

Several authors have described precession to be near 200,000

406:

terms as a motion of the heavens, rather than of the Earth.

391:(the time it takes the Sun to return to an equinox) and the 5086: 5071: 5061: 4515: 4452: 4350:"New precession expressions, valid for long time intervals" 4330:. Vol. 15:207–224. New York: Charles Scribner's Sons, 1978. 3236:. Farmington Hills, MI: Thomson-Gale. p. 105 and 454. 2898: 2869:= 34.723638″/a vs 34.457698″/a from Williams. 1554:, moment of inertia around any equatorial diameter of Earth 1287: 1104: 1059: 1011:

Precession of Earth's axis around the south ecliptical pole

988:

Precession of Earth's axis around the north ecliptical pole

783: 308:(20 min 24.5 sec ≈ (365.24219 × 86400) / 25772). 134: 122:, as early as 1863, while the dominant component was named 5237: 4521:

International Earth Rotation and Reference Systems Service

4247:

has a comparison of values predicted by different theories

2859:= 15.948788″/a vs 15.948870″/a from Williams 1616:

on a line directed toward the north pole of the ecliptic.

790:, but the attribution has been contested in modern times. 217:

across the sky), whereas the term "precession" as used in

5066: 3415:, Princeton University Press, pp. 131–141, 321–340, 1841:, semimajor axis of Earth's (Sun's) orbit or Moon's orbit 1283: 1067:. The Southern Cross can be viewed from as far north as 111: 3120:

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

1909:{\displaystyle a^{3}\left(1-e^{2}\right)^{\frac {3}{2}}} 1669:

The average torque of the sine wave in the direction of

4347: 3914:

Spherical and practical astronomy as applied to geodesy

2822: 1847:, eccentricity of Earth's (Sun's) orbit or Moon's orbit 1372:

plane. The latter pair of forces creates the following

651:= 60″/year. They probably deviated from an even 200,000 466:

is presented by Theon as an alternative to precession.

149:

recommended that the dominant component be renamed the

75:

of a spinning top, with the axis tracing out a pair of

3842:

The Ever-Changing Sky: A guide to the celestial sphere

3482:, Institute of Maya Studies newsletter, December 2007. 1465: 952:

Hipparchus's mathematical signatures are found in the

782:

precession. This theory is commonly attributed to the

5190: 4395: 3225: 2530: 2448: 2218: 2025: 1932: 1857: 1827: 1686: 1388: 503:

Archaeologist Susan Milbrath has speculated that the

345:

The discovery of precession usually is attributed to

3845:(Reprint). Cambridge University Press. p. 152. 324: 4223: 3533: 3069:Precession rate exhibits a secular decrease due to 4451: 3220:Explanatory Supplement to the Astronomical Almanac 2613: 2486: 2426: 2198: 1989: 1908: 1833: 1807: 1582:of the perturbing body (north or south of equator) 1510: 1074: 854:. By comparing his own measurements with those of 3891:. Jet Propulsion Laboratory/NASA. 29 October 2013 3804: 3802: 3800: 1564:, moment of inertia of Earth's equatorial bulge ( 1221:", are given, approximately, in the table below: 1044:, while in 23,600 BC it came closer to the pole. 469: 27:Change of rotational axis in an astronomical body 5298: 4348:Vondrak, J.; Capitaine, N.; Wallace, P. (2011). 4126:"Expressions for IAU 2000 precession quantities" 4001: 3999: 818:(1687) explained precession as a consequence of 794:published a different account of trepidation in 3867:"precession of the equinoxes | Infoplease" 3777: 3232:Lerner, K. Lee; Lerner, Brenda Wilmoth (2003). 1175:among the stars over the past 6,000 years. 834:Hipparchus gave an account of his discovery in 312:The rate itself varies somewhat with time (see 141:, causing Earth's axis to move with respect to 94:Earth's precession was historically called the 3986:IERS Technical Note 13 – IERS Standards (1992) 3797: 3214: 3212: 992:A consequence of the precession is a changing 727:, and by observations that refined the value. 714: 4437: 4169:The History and Practice of Ancient Astronomy 3996: 3222:. Sausalito: University Science Books. p. 99. 34:. For non-axial astronomical precession, see 4245:Precession and the Obliquity of the Ecliptic 4162:A History of Astronomy from Thales to Kepler 3808: 3231: 3127:by the small amount of +0.135052"/a between 3057:gives better accuracy. In that respect, the 2487:{\displaystyle \left(1-1.5\sin ^{2}i\right)} 1538:, geocentric distance to the perturbing body 815:Philosophiae Naturalis Principia Mathematica 221:, generally describes a mechanical process. 4189:Celestial Mechanics and Dynamical Astronomy 3269: 3267: 3265: 3209: 2893:developed an updated theory in 1976, where 842:III.1 and VII.2). He measured the ecliptic 4444: 4430: 4344:. New York: Oxford University Press, 1989. 4171:. New York: Oxford University Press, 1998. 3817:. University of Texas McDonald Observatory 3712:. Princeton University Press. p. 24. 3577: 3575: 3573: 3443: 3315: 3313: 3311: 3309: 2931:+ higher order terms, in arcseconds, with 1171:Diagram showing the westward shift of the 723:, precession was known based on Ptolemy's 250:". In approximately 3,200 years, the star 130:, instead of precession of the equinoxes. 4373: 4149: 4123: 4047: 3771: 3702: 3399: 3397: 3345: 2947:of precession is the derivative of that: 1363:outside of a mean spherical radius. This 1301:The Earth is not a perfect sphere but an 829: 4975:International Commission on Stratigraphy 3446:Journal of the American Oriental Society 3319: 3262: 1336: 1166: 1103:(left), and in about 12,000 years, 1086: 1078: 1006: 983: 320: 303:per year, or 1 degree every 71.6 years. 228: 164: 42: 4417:Forced precession and nutation of Earth 3581: 3570: 3560:(1935) , Gangooly, Phanindralal (ed.), 3537:Report of the Calendar Reform Committee 3403: 3306: 1199:) to somewhere in the constellation of 554:Before 1200, India had two theories of 527:, prior to the time of Hipparchus (the 371:, Hipparchus measured the longitude of 110:, opposite to the yearly motion of the 14: 5299: 4278:Electronic Journal of Mithraic Studies 4271: 4179: 4102: 3632: 3527: 3394: 979: 893:Hipparchus also studied precession in 696:revolutions in a Kalpa or 59.9″/year. 681:revolutions in a Kalpa or 58.2″/year. 666:revolutions in a Kalpa or 56.8″/year. 387:. He also compared the lengths of the 4425: 4292: 3950: 3916:(New York: Frederick Unger, 1969) 59. 3835: 3809:Benningfield, Damond (14 June 2015). 3674: 3556: 3550: 3480:"Just How Precise is Maya Astronomy?" 3274:Astro 101 – Precession of the Equinox 2905:allowed further refinements, and the 1352:Newton's law of universal gravitation 602:Another trepidation was described by 153:, and the minor component be renamed 4396:Bowley, Roger; Merrifield, Michael. 4062: 3977: 3584:Journal for the History of Astronomy 699: 518: 335: 3885:"Basics of Space Flight, Chapter 2" 3659: 2992:numerical model of the Solar System 2511:dynamical ellipticity or flattening 797:De revolutionibus orbium coelestium 24: 4328:Dictionary of Scientific Biography 4259:Dictionary of Scientific Biography 2624:where a is the equatorial radius ( 2598: 2583: 2569: 2554: 1592:of the perturbing body (east from 707:(fourth century AD) was the first 25: 5333: 4695:Discrete time and continuous time 4384: 3301:Practical and Spherical Astronomy 2954:= 5,028.796195 + 2.2108696  2209:whereas that due to the Moon is: 858:of Alexandria (a contemporary of 5284: 5272: 5260: 5248: 5236: 5224: 5212: 5200: 4620: 4614: 4164:. 2nd ed. New York: Dover, 1953. 3123:Precession will be greater than 3059:International Astronomical Union 2907:International Astronomical Union 1530:standard gravitational parameter 803: 632:years, which would be a rate of 505:Mesoamerican Long Count calendar 441: 147:International Astronomical Union 91:—are much smaller in magnitude. 4095: 4056: 4015: 3935: 3919: 3903: 3877: 3859: 3829: 3762: 3753: 3744: 3735: 3726: 3696: 3668: 3653: 3626: 3617: 3517: 3507: 3494: 3485: 3472: 3135:. The jump to this excess over 1548:around Earth's axis of rotation 1075:Polar shift and equinoxes shift 507:of "30,000 years involving the 325:Polar shift and equinoxes shift 206:physicists or astrophysicists. 160: 5312:Technical factors of astrology 4768:History of timekeeping devices 3750:Toomer 1984, p. 135 n. 14 3437: 3428: 3362: 3293: 3109:= 50.475838 − 27.000654  3096:= 50.475838 − 26.368583  1784: 1772: 1439: 1427: 923:On Intercalary Months and Days 474: 470:Alternative discovery theories 209:The term "precession" used in 126:. Their combination was named 13: 1: 3285:Western Washington University 3202: 2635:) and c is the polar radius ( 1379:on Earth's equatorial bulge: 686: 671: 656: 607: 565: 513:scholars of Maya civilization 340: 4354:Astronomy & Astrophysics 4318:Secrets of the Great Pyramid 4130:Astronomy & Astrophysics 4085:10.1016/0019-1035(82)90134-8 4028:Astronomy & Astrophysics 3784:Hipparcos, the New Reduction 3534:Government of India (1955), 3434:Evans 1998, pp. 251–255 3290:. Retrieved 30 December 2008 1332: 1183:, 5,000 years ago, the 925:(now lost), as described by 575:, that is still used by all 540:the Hathor temple at Dendera 299:, at the rate of about 50.3 188:" is derived from the Latin 7: 4715:Gravitational time dilation 4551:Barycentric Coordinate Time 4375:10.1051/0004-6361/201117274 3704:Gillispie, Charles Coulston 3320:Williams, James G. (1994). 3303:(Cambridge: 1863) pp.203–4. 3177:Longitude of vernal equinox 3155: 1015:The previous pole star was 826:and subsequent scientists. 715:Middle Ages and Renaissance 587:. The March equinox of the 450:rejected precession, while 313: 275:solstice occurred a little 96:precession of the equinoxes 18:Precession of the equinoxes 10: 5338: 4571:Geocentric Coordinate Time 4556:Barycentric Dynamical Time 4494:Coordinated Universal Time 4337:. London: Duckworth, 1984. 4295:Astronomy and Astrophysics 4151:10.1051/0004-6361:20031539 4105:Astronomy and Astrophysics 4049:10.1051/0004-6361:20041335 3959:Astronomy and Astrophysics 3947:(third edition, 1842) 200. 3623:Pannekoek 1961, p. 92 3604:10.1177/002182867200300104 3044:is the 41,000-year period. 2832:10″/2π) and the number of 2656:Applicable parameters for 2519:third eccentricity squared 721:medieval Islamic astronomy 409: 330: 224: 177: 155:precession of the ecliptic 29: 5125: 5016: 4983: 4957: 4838: 4738: 4725:Time-translation symmetry 4655: 4629: 4612: 4536:International Atomic Time 4486: 4463: 4209:10.1007/s10569-006-0001-2 3889:Jet Propulsion Laboratory 3073:from 59"/a to 45"/a (a = 2986:² term will dominate and 2877: 2517:the term would equal its 1229: 1226: 1138:obliquity of the ecliptic 895:On the Length of the Year 319:For further details, see 151:precession of the equator 102:moved westward along the 4410:University of Nottingham 4272:Schütz, Michael (2000). 4254:. New York: Dover, 1961. 3778:van Leeuwen, F. (2007). 3768:Toomer 1984, p. 139 3759:Toomer 1978, p. 218 3326:The Astronomical Journal 1677:for the Sun or Moon is: 1277: 754:, which was done during 549: 279:in the orbit. Thus, the 4985:Astronomical chronology 4958:Archaeology and geology 4665:Absolute space and time 4581:IERS Reference Meridian 4576:International Date Line 4487:International standards 4366:2011A&A...534A..22V 4307:1994A&A...282..663S 4238:1977A&A....58....1L 4142:2003A&A...412..567C 4117:1976A&A....51..127B 4040:2004A&A...428..261L 3971:1994A&A...282..663S 3741:Evans 1998, p. 251 3732:Evans 1998, p. 246 3017:is an approximation of 824:Jean le Rond d'Alembert 498: 36:Astronomical precession 4268:, London and New York. 4264:Rice, Michael (1997), 4252:A History of Astronomy 4124:Capitaine, N. (2003). 4012:, p. 581 expression 39 3932:(2006) 312–330, p.329. 3279:2 January 2009 at the 3234:World of earth science 3089:of Julian years (Ga): 2978:. It is clear that if 2615: 2488: 2428: 2200: 1991: 1910: 1835: 1809: 1532:of the perturbing body 1512: 1343: 1187:was close to the star 1176: 1108: 1084: 1012: 989: 913:by modifying those of 886:, when the Moon is at 830:Hipparchus's discovery 808:Over a century later, 487:had distinguished the 428:Menelaus of Alexandria 235: 181: 48: 4944:Weekday determination 4830:Sundial markup schema 4180:Hilton, J.L. (2006). 3662:"Book of Fixed Stars" 3162:Astronomical nutation 2958:+ higher order terms. 2923:= 5,028.796195  2616: 2489: 2429: 2201: 1992: 1911: 1836: 1810: 1513: 1340: 1170: 1090: 1082: 1010: 987: 972:and slower motion at 954:Antikythera Mechanism 232: 168: 46: 4965:Chronological dating 4705:Theory of relativity 4566:Daylight saving time 3983:Dennis D. McCarthy, 3941:George Biddel Airy, 2528: 2446: 2216: 2023: 1930: 1855: 1825: 1684: 1386: 773:Maragheh observatory 538:, a star-map inside 485:Chaldean astronomers 124:lunisolar precession 120:planetary precession 5317:Celestial mechanics 5175:Time value of money 4970:Geologic time scale 4825:History of sundials 4690:Cosmological decade 4642:Greenwich Mean Time 4473:Orders of magnitude 4288:on 4 November 2013. 4201:2006CeMDA..94..351H 4077:1982Icar...50..444W 3689:1939PA.....47..233R 3596:1972JHA.....3...27P 3338:1994AJ....108..711W 3182:Milankovitch cycles 3139:will occur in only 2972:limited time period 2963:small uncertainty. 1022:On the other hand, 980:Changing pole stars 966:Kepler's second law 935:Babylonian calendar 792:Nicolaus Copernicus 747:Book of Fixed Stars 579:, varying over the 564:(3:9–12), composed 452:Theon of Alexandria 321:Changing pole stars 289:Milankovitch cycles 137:and Sun on Earth's 5150:Mental chronometry 4778:Marine chronometer 4630:Obsolete standards 4398:"Axial Precession" 4335:Ptolemy's Almagest 3409:Ptolemy's Almagest 3382:on 21 October 2011 3197:Apsidal precession 3113:+ 15.603265  3100:+ 21.890862  2927:+ 1.1054348  2611: 2484: 2424: 2196: 1987: 1906: 1831: 1805: 1508: 1502: 1344: 1177: 1109: 1085: 1013: 990: 911:lunisolar calendar 862:, who worked with 771:, compiled at the 709:Chinese astronomer 692:) mentions 199,699 662:) mentions 189,411 581:ecliptic longitude 236: 182: 128:general precession 49: 5188: 5187: 4998:Nuclear timescale 4680:Continuous signal 4226:Astron. Astrophys 3989:(Postscript, use 3815:Stardate Magazine 3677:Popular Astronomy 3641:on 5 January 2017 3071:tidal dissipation 2937:the epoch of 2000 2782: 2781: 2609: 2411: 2393: 2359: 2355: 2257: 2244: 2183: 2165: 2131: 2127: 2064: 2051: 1985: 1951: 1903: 1834:{\displaystyle a} 1770: 1766: 1708: 1638:for the Sun is a 1546:moment of inertia 1425: 1397: 1356:centripetal force 1275: 1274: 1230:Approximate year 1179:As seen from the 700:Chinese astronomy 677:) mentions 94,110 624:revolutions in a 519:Ancient Egyptians 379:(320–260 BC) and 336:Hellenistic world 65:axial parallelism 16:(Redirected from 5329: 5289: 5288: 5277: 5276: 5275: 5265: 5264: 5263: 5253: 5252: 5251: 5241: 5240: 5229: 5228: 5227: 5217: 5216: 5205: 5204: 5196: 4889:Dominical letter 4820:Equation of time 4783:Marine sandglass 4624: 4618: 4596:Terrestrial Time 4453:Time measurement 4446: 4439: 4432: 4423: 4422: 4413: 4379: 4377: 4340:Ulansey, David. 4310: 4289: 4284:. Archived from 4241: 4220: 4186: 4159:Dreyer, J. L. E. 4155: 4153: 4120: 4089: 4088: 4071:(2–3): 444–448. 4060: 4054: 4053: 4051: 4019: 4013: 4003: 3994: 3981: 3975: 3974: 3954: 3948: 3939: 3933: 3923: 3917: 3907: 3901: 3900: 3898: 3896: 3881: 3875: 3874: 3863: 3857: 3856: 3833: 3827: 3826: 3824: 3822: 3806: 3795: 3794: 3792: 3790: 3775: 3769: 3766: 3760: 3757: 3751: 3748: 3742: 3739: 3733: 3730: 3724: 3723: 3700: 3694: 3692: 3672: 3666: 3665: 3657: 3651: 3650: 3648: 3646: 3637:. Archived from 3630: 3624: 3621: 3615: 3614: 3579: 3568: 3567: 3554: 3548: 3547: 3542: 3531: 3525: 3521: 3515: 3511: 3505: 3498: 3492: 3489: 3483: 3478:Susan Milbrath, 3476: 3470: 3469: 3441: 3435: 3432: 3426: 3425: 3411:, translated by 3401: 3392: 3391: 3389: 3387: 3381: 3375:. Archived from 3374: 3366: 3360: 3359: 3349: 3317: 3304: 3297: 3291: 3271: 3260: 3259: 3229: 3223: 3216: 3149:tidal rhythmites 3142: 3134: 3130: 2891:Jay Henry Lieske 2847: 2831: 2812: 2798: 2777: 2732: 2722: 2693: 2683: 2663: 2662: 2652: 2645: 2643: 2640: 2634: 2632: 2629: 2620: 2618: 2617: 2612: 2610: 2608: 2607: 2606: 2601: 2592: 2591: 2586: 2579: 2578: 2577: 2572: 2563: 2562: 2557: 2550: 2545: 2544: 2543: 2508: 2493: 2491: 2490: 2485: 2483: 2479: 2472: 2471: 2433: 2431: 2430: 2425: 2423: 2422: 2417: 2413: 2412: 2407: 2396: 2394: 2389: 2378: 2370: 2369: 2364: 2360: 2358: 2357: 2356: 2348: 2346: 2342: 2341: 2340: 2319: 2318: 2308: 2307: 2303: 2296: 2295: 2265: 2258: 2250: 2245: 2243: 2235: 2234: 2233: 2220: 2205: 2203: 2202: 2197: 2195: 2194: 2189: 2185: 2184: 2179: 2168: 2166: 2161: 2150: 2142: 2141: 2136: 2132: 2130: 2129: 2128: 2120: 2118: 2114: 2113: 2112: 2091: 2090: 2080: 2072: 2065: 2057: 2052: 2050: 2042: 2041: 2040: 2027: 2014:angular momentum 2006:angular velocity 1996: 1994: 1993: 1988: 1986: 1984: 1967: 1966: 1957: 1952: 1950: 1942: 1934: 1915: 1913: 1912: 1907: 1905: 1904: 1896: 1894: 1890: 1889: 1888: 1867: 1866: 1840: 1838: 1837: 1832: 1814: 1812: 1811: 1806: 1771: 1769: 1768: 1767: 1759: 1757: 1753: 1752: 1751: 1730: 1729: 1719: 1711: 1709: 1701: 1696: 1695: 1660: 1637: 1517: 1515: 1514: 1509: 1507: 1506: 1426: 1424: 1423: 1414: 1403: 1398: 1390: 1361:equatorial bulge 1224: 1223: 1198: 1182: 1144: 1134: 1128:north pole (the 1123: 1114:celestial sphere 962:Exeligmos cycles 947:Hipparchic cycle 852:autumnal equinox 788:Thabit ibn Qurra 695: 691: 688: 680: 676: 673: 665: 661: 658: 654: 650: 647: 645: 644: 641: 638: 637:200,000×360×3600 631: 628:of 4,320,000,000 623: 612: 609: 577:Indian calendars 570: 567: 400:celestial sphere 385:autumnal equinox 359:Greek astronomer 349:(190–120 BC) of 309: 139:equatorial bulge 106:relative to the 79:joined at their 57:axial precession 21: 5337: 5336: 5332: 5331: 5330: 5328: 5327: 5326: 5297: 5296: 5295: 5283: 5273: 5271: 5261: 5259: 5249: 5247: 5235: 5225: 5223: 5211: 5199: 5191: 5189: 5184: 5121: 5012: 4979: 4953: 4834: 4734: 4685:Coordinate time 4657:Time in physics 4651: 4625: 4619: 4610: 4482: 4459: 4450: 4387: 4382: 4314:Tompkins, Peter 4184: 4098: 4093: 4092: 4061: 4057: 4020: 4016: 4004: 3997: 3982: 3978: 3955: 3951: 3940: 3936: 3924: 3920: 3910:Ivan I. Mueller 3908: 3904: 3894: 3892: 3883: 3882: 3878: 3865: 3864: 3860: 3853: 3837:Kaler, James B. 3834: 3830: 3820: 3818: 3807: 3798: 3788: 3786: 3776: 3772: 3767: 3763: 3758: 3754: 3749: 3745: 3740: 3736: 3731: 3727: 3720: 3701: 3697: 3683:(5): 233–238 . 3673: 3669: 3658: 3654: 3644: 3642: 3631: 3627: 3622: 3618: 3580: 3571: 3555: 3551: 3540: 3532: 3528: 3522: 3518: 3512: 3508: 3500:Rice, Michael. 3499: 3495: 3490: 3486: 3477: 3473: 3442: 3438: 3433: 3429: 3423: 3402: 3395: 3385: 3383: 3379: 3372: 3368: 3367: 3363: 3318: 3307: 3298: 3294: 3281:Wayback Machine 3272: 3263: 3244: 3230: 3226: 3217: 3210: 3205: 3158: 3140: 3132: 3128: 2921: 2913:precession is: 2880: 2866: 2856: 2845: 2829: 2810: 2806: 2796: 2792: 2775: 2730: 2720: 2691: 2681: 2647: 2641: 2638: 2636: 2630: 2627: 2625: 2602: 2597: 2596: 2587: 2582: 2581: 2580: 2573: 2568: 2567: 2558: 2553: 2552: 2551: 2549: 2539: 2535: 2531: 2529: 2526: 2525: 2495: 2467: 2463: 2453: 2449: 2447: 2444: 2443: 2418: 2397: 2395: 2379: 2377: 2376: 2372: 2371: 2365: 2347: 2336: 2332: 2325: 2321: 2320: 2314: 2310: 2309: 2291: 2287: 2277: 2273: 2266: 2264: 2260: 2259: 2249: 2236: 2229: 2225: 2221: 2219: 2217: 2214: 2213: 2190: 2169: 2167: 2151: 2149: 2148: 2144: 2143: 2137: 2119: 2108: 2104: 2097: 2093: 2092: 2086: 2082: 2081: 2073: 2071: 2067: 2066: 2056: 2043: 2036: 2032: 2028: 2026: 2024: 2021: 2020: 1968: 1962: 1958: 1956: 1943: 1935: 1933: 1931: 1928: 1927: 1923:Precession is: 1895: 1884: 1880: 1873: 1869: 1868: 1862: 1858: 1856: 1853: 1852: 1826: 1823: 1822: 1758: 1747: 1743: 1736: 1732: 1731: 1725: 1721: 1720: 1712: 1710: 1700: 1691: 1687: 1685: 1682: 1681: 1646: 1623: 1590:right ascension 1501: 1500: 1494: 1493: 1478: 1477: 1461: 1460: 1419: 1415: 1404: 1402: 1389: 1387: 1384: 1383: 1335: 1303:oblate spheroid 1280: 1196: 1180: 1142: 1129: 1121: 1077: 982: 943:Callippic cycle 832: 806: 752:Zij Al Mumtahan 717: 702: 693: 689: 678: 674: 663: 659: 652: 648: 642: 639: 636: 635: 633: 629: 621: 610: 589:Surya Siddhanta 568: 561:Surya Siddhanta 552: 521: 501: 477: 472: 444: 412: 361:. According to 343: 338: 333: 307: 240:celestial poles 227: 163: 61:rotational axis 39: 28: 23: 22: 15: 12: 11: 5: 5335: 5325: 5324: 5319: 5314: 5309: 5294: 5293: 5281: 5269: 5257: 5245: 5233: 5221: 5209: 5186: 5185: 5183: 5182: 5177: 5172: 5170:Time metrology 5167: 5162: 5157: 5152: 5147: 5146: 5145: 5135: 5129: 5127: 5126:Related topics 5123: 5122: 5120: 5119: 5114: 5109: 5104: 5099: 5094: 5089: 5084: 5079: 5074: 5069: 5064: 5059: 5054: 5049: 5044: 5039: 5034: 5029: 5023: 5021: 5014: 5013: 5011: 5010: 5005: 5000: 4995: 4989: 4987: 4981: 4980: 4978: 4977: 4972: 4967: 4961: 4959: 4955: 4954: 4952: 4951: 4946: 4941: 4936: 4931: 4926: 4921: 4916: 4911: 4906: 4901: 4896: 4891: 4886: 4881: 4876: 4871: 4865: 4860: 4855: 4850: 4844: 4842: 4836: 4835: 4833: 4832: 4827: 4822: 4817: 4815:Dialing scales 4812: 4807: 4802: 4801: 4800: 4790: 4785: 4780: 4775: 4770: 4765: 4760: 4755: 4750: 4744: 4742: 4736: 4735: 4733: 4732: 4727: 4722: 4717: 4712: 4707: 4702: 4697: 4692: 4687: 4682: 4677: 4672: 4667: 4661: 4659: 4653: 4652: 4650: 4649: 4647:Prime meridian 4644: 4639: 4637:Ephemeris time 4633: 4631: 4627: 4626: 4613: 4611: 4609: 4608: 4606:180th meridian 4603: 4598: 4593: 4588: 4583: 4578: 4573: 4568: 4563: 4558: 4553: 4548: 4543: 4538: 4533: 4528: 4523: 4518: 4513: 4508: 4503: 4502: 4501: 4490: 4488: 4484: 4483: 4481: 4480: 4475: 4470: 4464: 4461: 4460: 4449: 4448: 4441: 4434: 4426: 4420: 4419: 4414: 4393: 4386: 4385:External links 4383: 4381: 4380: 4345: 4338: 4333:Toomer, G. J. 4331: 4326:"Hipparchus." 4321: 4311: 4290: 4269: 4262: 4255: 4250:Pannekoek, A. 4248: 4242: 4221: 4195:(3): 351–367. 4177: 4172: 4167:Evans, James. 4165: 4156: 4136:(2): 567–586. 4121: 4111:(1): 127–135. 4099: 4097: 4094: 4091: 4090: 4055: 4014: 3995: 3976: 3949: 3934: 3918: 3902: 3876: 3871:infoplease.com 3858: 3852:978-0521499187 3851: 3828: 3796: 3770: 3761: 3752: 3743: 3734: 3725: 3718: 3695: 3667: 3652: 3635:"Zij Al-Sabi'" 3625: 3616: 3569: 3549: 3526: 3516: 3506: 3502:Egypt's Legacy 3493: 3491:Tompkins, 1971 3484: 3471: 3458:10.2307/595428 3436: 3427: 3421: 3393: 3361: 3347:10.1086/117108 3305: 3292: 3261: 3242: 3224: 3207: 3206: 3204: 3201: 3200: 3199: 3194: 3189: 3184: 3179: 3174: 3169: 3164: 3157: 3154: 3118: 3117: 3104: 3046: 3045: 3015: 3014: 2960: 2959: 2941: 2940: 2919: 2879: 2876: 2871: 2870: 2864: 2860: 2854: 2815: 2814: 2804: 2800: 2790: 2780: 2779: 2769: 2767: 2764: 2763: 2762:ε = 23.43928° 2760: 2754: 2751: 2750: 2749:= 0.016708634 2744: 2738: 2735: 2734: 2724: 2714: 2711: 2710: 2709:= 0.003273763 2695: 2685: 2674: 2673: 2670: 2667: 2622: 2621: 2605: 2600: 2595: 2590: 2585: 2576: 2571: 2566: 2561: 2556: 2548: 2542: 2538: 2534: 2482: 2478: 2475: 2470: 2466: 2462: 2459: 2456: 2452: 2435: 2434: 2421: 2416: 2410: 2406: 2403: 2400: 2392: 2388: 2385: 2382: 2375: 2368: 2363: 2354: 2351: 2345: 2339: 2335: 2331: 2328: 2324: 2317: 2313: 2306: 2302: 2299: 2294: 2290: 2286: 2283: 2280: 2276: 2272: 2269: 2263: 2256: 2253: 2248: 2242: 2239: 2232: 2228: 2224: 2207: 2206: 2193: 2188: 2182: 2178: 2175: 2172: 2164: 2160: 2157: 2154: 2147: 2140: 2135: 2126: 2123: 2117: 2111: 2107: 2103: 2100: 2096: 2089: 2085: 2079: 2076: 2070: 2063: 2060: 2055: 2049: 2046: 2039: 2035: 2031: 1998: 1997: 1983: 1980: 1977: 1974: 1971: 1965: 1961: 1955: 1949: 1946: 1941: 1938: 1902: 1899: 1893: 1887: 1883: 1879: 1876: 1872: 1865: 1861: 1849: 1848: 1842: 1830: 1816: 1815: 1804: 1801: 1798: 1795: 1792: 1789: 1786: 1783: 1780: 1777: 1774: 1765: 1762: 1756: 1750: 1746: 1742: 1739: 1735: 1728: 1724: 1718: 1715: 1707: 1704: 1699: 1694: 1690: 1598: 1597: 1583: 1573: 1555: 1549: 1539: 1533: 1519: 1518: 1505: 1499: 1496: 1495: 1492: 1489: 1486: 1483: 1480: 1479: 1476: 1473: 1470: 1467: 1466: 1464: 1459: 1456: 1453: 1450: 1447: 1444: 1441: 1438: 1435: 1432: 1429: 1422: 1418: 1413: 1410: 1407: 1401: 1396: 1393: 1334: 1331: 1279: 1276: 1273: 1272: 1269: 1266: 1262: 1261: 1258: 1255: 1251: 1250: 1247: 1244: 1240: 1239: 1236: 1232: 1231: 1228: 1227:Constellation 1212:ecliptic plane 1076: 1073: 1065:ancient Greeks 1060:Southern Cross 1049:Sigma Octantis 1035:light-polluted 981: 978: 838:(described in 831: 828: 805: 802: 740:Subsequently, 733:, in his work 716: 713: 701: 698: 675: 600–680 660: 550–600 551: 548: 536:Dendera Zodiac 520: 517: 500: 497: 476: 473: 471: 468: 460: 459: 443: 440: 411: 408: 342: 339: 337: 334: 332: 329: 301:seconds of arc 226: 223: 196:diurnal motion 162: 159: 143:inertial space 98:, because the 26: 9: 6: 4: 3: 2: 5334: 5323: 5320: 5318: 5315: 5313: 5310: 5308: 5305: 5304: 5302: 5292: 5287: 5282: 5280: 5270: 5268: 5258: 5256: 5246: 5244: 5239: 5234: 5232: 5222: 5220: 5215: 5210: 5208: 5203: 5198: 5197: 5194: 5181: 5178: 5176: 5173: 5171: 5168: 5166: 5163: 5161: 5158: 5156: 5153: 5151: 5148: 5144: 5141: 5140: 5139: 5136: 5134: 5131: 5130: 5128: 5124: 5118: 5115: 5113: 5110: 5108: 5105: 5103: 5100: 5098: 5095: 5093: 5090: 5088: 5085: 5083: 5080: 5078: 5075: 5073: 5070: 5068: 5065: 5063: 5060: 5058: 5055: 5053: 5050: 5048: 5045: 5043: 5040: 5038: 5035: 5033: 5030: 5028: 5025: 5024: 5022: 5020: 5019:units of time 5015: 5009: 5008:Sidereal time 5006: 5004: 5001: 4999: 4996: 4994: 4993:Galactic year 4991: 4990: 4988: 4986: 4982: 4976: 4973: 4971: 4968: 4966: 4963: 4962: 4960: 4956: 4950: 4949:Weekday names 4947: 4945: 4942: 4940: 4939:Tropical year 4937: 4935: 4932: 4930: 4927: 4925: 4922: 4920: 4917: 4915: 4912: 4910: 4907: 4905: 4904:Intercalation 4902: 4900: 4897: 4895: 4892: 4890: 4887: 4885: 4882: 4880: 4877: 4875: 4872: 4870:(lunar Hijri) 4869: 4866: 4864: 4861: 4859: 4856: 4854: 4851: 4849: 4846: 4845: 4843: 4841: 4837: 4831: 4828: 4826: 4823: 4821: 4818: 4816: 4813: 4811: 4808: 4806: 4803: 4799: 4796: 4795: 4794: 4791: 4789: 4786: 4784: 4781: 4779: 4776: 4774: 4771: 4769: 4766: 4764: 4761: 4759: 4756: 4754: 4751: 4749: 4746: 4745: 4743: 4741: 4737: 4731: 4728: 4726: 4723: 4721: 4718: 4716: 4713: 4711: 4710:Time dilation 4708: 4706: 4703: 4701: 4698: 4696: 4693: 4691: 4688: 4686: 4683: 4681: 4678: 4676: 4673: 4671: 4668: 4666: 4663: 4662: 4660: 4658: 4654: 4648: 4645: 4643: 4640: 4638: 4635: 4634: 4632: 4628: 4623: 4617: 4607: 4604: 4602: 4599: 4597: 4594: 4592: 4589: 4587: 4584: 4582: 4579: 4577: 4574: 4572: 4569: 4567: 4564: 4562: 4559: 4557: 4554: 4552: 4549: 4547: 4546:24-hour clock 4544: 4542: 4541:12-hour clock 4539: 4537: 4534: 4532: 4529: 4527: 4524: 4522: 4519: 4517: 4514: 4512: 4509: 4507: 4504: 4500: 4497: 4496: 4495: 4492: 4491: 4489: 4485: 4479: 4476: 4474: 4471: 4469: 4466: 4465: 4462: 4458: 4454: 4447: 4442: 4440: 4435: 4433: 4428: 4427: 4424: 4418: 4415: 4411: 4407: 4403: 4402:Sixty Symbols 4399: 4394: 4392: 4389: 4388: 4376: 4371: 4367: 4363: 4359: 4355: 4351: 4346: 4343: 4339: 4336: 4332: 4329: 4325: 4324:Toomer, G. J. 4322: 4319: 4315: 4312: 4308: 4304: 4300: 4296: 4291: 4287: 4283: 4280:(in German). 4279: 4275: 4270: 4267: 4263: 4260: 4256: 4253: 4249: 4246: 4243: 4239: 4235: 4231: 4227: 4222: 4218: 4214: 4210: 4206: 4202: 4198: 4194: 4190: 4183: 4178: 4176: 4173: 4170: 4166: 4163: 4160: 4157: 4152: 4147: 4143: 4139: 4135: 4131: 4127: 4122: 4118: 4114: 4110: 4106: 4101: 4100: 4086: 4082: 4078: 4074: 4070: 4066: 4059: 4050: 4045: 4041: 4037: 4033: 4029: 4025: 4018: 4011: 4009: 4006:N. Capitaine 4002: 4000: 3992: 3988: 3987: 3980: 3972: 3968: 3964: 3960: 3953: 3946: 3945: 3938: 3931: 3928: 3922: 3915: 3911: 3906: 3890: 3886: 3880: 3872: 3868: 3862: 3854: 3848: 3844: 3843: 3838: 3832: 3816: 3812: 3805: 3803: 3801: 3785: 3781: 3774: 3765: 3756: 3747: 3738: 3729: 3721: 3719:0-691-02350-6 3715: 3711: 3710: 3705: 3699: 3690: 3686: 3682: 3678: 3671: 3663: 3656: 3640: 3636: 3629: 3620: 3613: 3609: 3605: 3601: 3597: 3593: 3589: 3585: 3578: 3576: 3574: 3565: 3564: 3559: 3553: 3546: 3539: 3538: 3530: 3520: 3510: 3503: 3497: 3488: 3481: 3475: 3467: 3463: 3459: 3455: 3451: 3447: 3440: 3431: 3424: 3422:0-691-00260-6 3418: 3414: 3413:Toomer, G. J. 3410: 3406: 3400: 3398: 3378: 3371: 3365: 3357: 3353: 3348: 3343: 3339: 3335: 3331: 3327: 3323: 3316: 3314: 3312: 3310: 3302: 3299:Robert Main, 3296: 3289: 3286: 3282: 3278: 3275: 3270: 3268: 3266: 3258: 3253: 3249: 3245: 3243:0-7876-9332-4 3239: 3235: 3228: 3221: 3215: 3213: 3208: 3198: 3195: 3193: 3192:Sidereal year 3190: 3188: 3185: 3183: 3180: 3178: 3175: 3173: 3170: 3168: 3165: 3163: 3160: 3159: 3153: 3150: 3144: 3138: 3126: 3121: 3116: 3112: 3108: 3105: 3103: 3099: 3095: 3092: 3091: 3090: 3088: 3084: 3080: 3076: 3072: 3067: 3063: 3060: 3055: 3051: 3043: 3039: 3035: 3031: 3027: 3023: 3020: 3019: 3018: 3012: 3008: 3004: 3000: 2997: 2996: 2995: 2993: 2989: 2985: 2981: 2977: 2973: 2969: 2964: 2957: 2953: 2950: 2949: 2948: 2946: 2938: 2934: 2930: 2926: 2922: 2916: 2915: 2914: 2912: 2908: 2904: 2900: 2896: 2892: 2888: 2884: 2883:Simon Newcomb 2875: 2868: 2861: 2858: 2851: 2850: 2849: 2843: 2839: 2835: 2827: 2824: 2820: 2808: 2801: 2794: 2787: 2786: 2785: 2773: 2770: 2768: 2766: 2765: 2761: 2758: 2755: 2753: 2752: 2748: 2745: 2743:= 0.05554553 2742: 2739: 2737: 2736: 2728: 2725: 2718: 2715: 2713: 2712: 2708: 2704: 2700: 2696: 2689: 2686: 2679: 2676: 2675: 2671: 2668: 2665: 2664: 2661: 2659: 2654: 2651:= 0.003358481 2650: 2603: 2593: 2588: 2574: 2564: 2559: 2546: 2540: 2536: 2532: 2524: 2523: 2522: 2520: 2516: 2512: 2507: 2503: 2499: 2480: 2476: 2473: 2468: 2464: 2460: 2457: 2454: 2450: 2440: 2419: 2414: 2408: 2404: 2401: 2398: 2390: 2386: 2383: 2380: 2373: 2366: 2361: 2352: 2349: 2343: 2337: 2333: 2329: 2326: 2322: 2315: 2311: 2304: 2300: 2297: 2292: 2288: 2284: 2281: 2278: 2274: 2270: 2267: 2261: 2254: 2251: 2246: 2240: 2237: 2230: 2226: 2222: 2212: 2211: 2210: 2191: 2186: 2180: 2176: 2173: 2170: 2162: 2158: 2155: 2152: 2145: 2138: 2133: 2124: 2121: 2115: 2109: 2105: 2101: 2098: 2094: 2087: 2083: 2077: 2074: 2068: 2061: 2058: 2053: 2047: 2044: 2037: 2033: 2029: 2019: 2018: 2017: 2015: 2011: 2007: 2003: 1981: 1978: 1975: 1972: 1969: 1963: 1959: 1953: 1947: 1944: 1939: 1936: 1926: 1925: 1924: 1921: 1919: 1900: 1897: 1891: 1885: 1881: 1877: 1874: 1870: 1863: 1859: 1846: 1843: 1828: 1821: 1820: 1819: 1802: 1799: 1796: 1793: 1790: 1787: 1781: 1778: 1775: 1763: 1760: 1754: 1748: 1744: 1740: 1737: 1733: 1726: 1722: 1716: 1713: 1705: 1702: 1697: 1692: 1688: 1680: 1679: 1678: 1676: 1672: 1667: 1665: 1658: 1654: 1650: 1645: 1641: 1635: 1631: 1627: 1622: 1617: 1615: 1611: 1607: 1603: 1595: 1594:March equinox 1591: 1587: 1584: 1581: 1577: 1574: 1571: 1567: 1563: 1559: 1556: 1553: 1550: 1547: 1543: 1540: 1537: 1534: 1531: 1527: 1524: 1523: 1522: 1503: 1497: 1490: 1487: 1484: 1481: 1474: 1471: 1468: 1462: 1457: 1454: 1451: 1448: 1445: 1442: 1436: 1433: 1430: 1420: 1416: 1411: 1408: 1405: 1399: 1394: 1391: 1382: 1381: 1380: 1378: 1375: 1371: 1366: 1362: 1357: 1353: 1349: 1339: 1330: 1326: 1324: 1318: 1314: 1312: 1308: 1304: 1299: 1295: 1293: 1289: 1285: 1270: 1267: 1264: 1263: 1259: 1256: 1253: 1252: 1248: 1245: 1242: 1241: 1237: 1234: 1233: 1225: 1222: 1220: 1215: 1213: 1209: 1204: 1202: 1194: 1190: 1186: 1185:March equinox 1174: 1173:March equinox 1169: 1165: 1163: 1158: 1157: 1150: 1148: 1140: 1139: 1133: 1127: 1118: 1116: 1115: 1106: 1102: 1098: 1094: 1089: 1081: 1072: 1070: 1066: 1061: 1056: 1054: 1050: 1045: 1043: 1042:proper motion 1038: 1037:urban skies. 1036: 1032: 1029: 1028:constellation 1025: 1020: 1018: 1009: 1005: 1003: 999: 995: 986: 977: 975: 971: 967: 963: 959: 955: 950: 948: 944: 940: 939:Metonic cycle 936: 932: 928: 924: 920: 916: 912: 907: 904: 903:sidereal year 900: 899:tropical year 896: 891: 889: 885: 881: 880:lunar eclipse 876: 873: 869: 865: 861: 857: 853: 849: 845: 841: 837: 827: 825: 821: 817: 816: 811: 804:Modern period 801: 799: 798: 793: 789: 785: 781: 776: 774: 770: 769: 768:Zij-i Ilkhani 763: 761: 757: 753: 749: 748: 743: 738: 736: 732: 728: 726: 722: 712: 710: 706: 697: 684: 669: 643:4,320,000,000 627: 618: 616: 605: 600: 597: 596: 590: 586: 585:March equinox 582: 578: 574: 563: 562: 557: 547: 543: 541: 537: 532: 530: 526: 525:Ancient Egypt 516: 514: 510: 506: 496: 494: 493:sidereal year 490: 486: 482: 467: 465: 457: 456: 455: 453: 449: 442:Other authors 439: 437: 433: 429: 425: 421: 417: 407: 405: 401: 396: 394: 393:sidereal year 390: 389:tropical year 386: 382: 378: 374: 370: 369: 364: 360: 356: 352: 348: 328: 326: 322: 317: 315: 310: 304: 302: 298: 292: 290: 286: 285:sidereal year 282: 281:tropical year 278: 273: 269: 265: 260: 258: 253: 249: 245: 241: 231: 222: 220: 216: 212: 207: 204: 199: 197: 193: 192: 187: 179: 175: 171: 167: 158: 156: 152: 148: 144: 140: 136: 131: 129: 125: 121: 117: 113: 109: 105: 101: 97: 92: 90: 86: 82: 78: 74: 70: 66: 62: 58: 54: 45: 41: 37: 33: 19: 5279:Solar System 5155:Decimal time 4884:Astronomical 4763:Complication 4758:Atomic clock 4401: 4357: 4353: 4341: 4334: 4327: 4317: 4298: 4294: 4286:the original 4281: 4277: 4265: 4258: 4251: 4229: 4225: 4192: 4188: 4174: 4168: 4161: 4133: 4129: 4108: 4104: 4096:Bibliography 4068: 4064: 4058: 4031: 4027: 4017: 4007: 3984: 3979: 3962: 3958: 3952: 3942: 3937: 3929: 3926: 3921: 3913: 3905: 3893:. Retrieved 3888: 3879: 3870: 3861: 3841: 3831: 3819:. Retrieved 3814: 3787:. Retrieved 3783: 3773: 3764: 3755: 3746: 3737: 3728: 3708: 3698: 3680: 3676: 3670: 3655: 3645:30 September 3643:. Retrieved 3639:the original 3633:Al-Battani. 3628: 3619: 3587: 3583: 3562: 3552: 3544: 3536: 3529: 3519: 3509: 3501: 3496: 3487: 3474: 3449: 3445: 3439: 3430: 3408: 3384:. Retrieved 3377:the original 3364: 3329: 3325: 3295: 3255: 3233: 3227: 3219: 3187:Polar motion 3172:Euler angles 3145: 3136: 3124: 3122: 3119: 3114: 3110: 3106: 3101: 3097: 3093: 3082: 3068: 3064: 3053: 3049: 3047: 3041: 3037: 3033: 3029: 3025: 3021: 3016: 3010: 3006: 3002: 2998: 2987: 2983: 2979: 2976:Solar System 2971: 2967: 2965: 2961: 2955: 2951: 2944: 2942: 2932: 2928: 2924: 2917: 2910: 2894: 2886: 2881: 2872: 2862: 2852: 2816: 2802: 2788: 2784:which yield 2783: 2771: 2759:= 5.156690° 2756: 2746: 2740: 2726: 2716: 2706: 2702: 2698: 2687: 2677: 2655: 2648: 2623: 2505: 2501: 2497: 2438: 2436: 2208: 2009: 2001: 1999: 1922: 1917: 1850: 1844: 1817: 1674: 1670: 1668: 1663: 1656: 1652: 1648: 1643: 1640:sine squared 1633: 1629: 1625: 1620: 1618: 1613: 1609: 1605: 1601: 1599: 1585: 1575: 1569: 1565: 1561: 1557: 1551: 1541: 1535: 1525: 1520: 1345: 1327: 1319: 1315: 1300: 1296: 1292:Isaac Newton 1281: 1219:Great Months 1218: 1216: 1205: 1178: 1161: 1154: 1151: 1136: 1131: 1130:blue letter 1119: 1112: 1110: 1057: 1046: 1039: 1021: 1014: 996:. Currently 991: 951: 930: 922: 908: 894: 892: 877: 871: 867: 846:of the star 839: 835: 833: 813: 810:Isaac Newton 807: 795: 779: 777: 766: 764: 759: 751: 745: 739: 734: 729: 724: 718: 703: 619: 604:Varāhamihira 601: 593: 588: 559: 553: 544: 533: 522: 502: 478: 461: 445: 413: 397: 366: 344: 318: 311: 305: 293: 276: 261: 252:Gamma Cephei 237: 208: 202: 200: 189: 183: 161:Nomenclature 154: 150: 132: 127: 123: 119: 95: 93: 89:polar motion 56: 50: 40: 5267:Outer space 5255:Spaceflight 5207:Mathematics 5165:System time 5160:Metric time 4879:Solar Hijri 4805:Water clock 4788:Radio clock 4720:Time domain 4700:Proper time 4586:Leap second 4468:Chronometry 4406:Brady Haran 4301:: 663–683. 4261:15:706–727. 4034:: 261–285. 3780:"HIP 11767" 3386:28 February 3288:Planetarium 3079:Julian year 2911:accumulated 2842:Julian year 2729:= 1.4959802 2680:= 1.3271244 2515:homogeneous 2509:is Earth's 2012:is Earth's 2004:is Earth's 1580:declination 1348:tidal force 1122:blue circle 1053:False Cross 958:Saros cycle 933:III.1. The 820:gravitation 786:astronomer 765:Later, the 760:Zij Al-Sabi 735:Zij Al-Sabi 690: 1150 683:Bhāskara II 615:declination 556:trepidation 475:Babylonians 464:trepidation 257:South Stars 108:fixed stars 5307:Precession 5301:Categories 5180:Timekeeper 5133:Chronology 5117:Millennium 5003:Precession 4909:Julian day 4730:T-symmetry 4591:Solar time 4561:Civil time 3452:(1): 1–8. 3203:References 3167:Axial tilt 2844:) (3.15576 2819:arcseconds 2809:= 5.334529 2795:= 2.450183 2774:= 7.292115 2719:= 3.833978 2690:= 4.902799 1307:axial tilt 1181:brown grid 1143:brown axis 960:, and the 888:opposition 864:Aristillus 856:Timocharis 731:Al-Battani 668:Bhaskara I 611: 550 569: 400 481:Al-Battani 432:Timocharis 404:geocentric 381:Aristillus 377:Timocharis 347:Hipparchus 341:Hipparchus 272:axial tilt 248:north star 215:retrograde 191:praecedere 186:Precession 184:The term " 170:Precession 116:Hipparchus 73:precession 32:Precession 5322:Equinoxes 5231:Astronomy 5077:Fortnight 4924:Lunisolar 4914:Leap year 4848:Gregorian 4798:stopwatch 4773:Hourglass 4753:Astrarium 4670:Spacetime 4601:Time zone 4478:Metrology 4457:standards 4217:122358401 3660:Al-Sufi. 3612:115947431 3590:: 27–35, 3524:Kingdom." 3407:(1998) , 3356:122370108 3040:), where 2778:10 rad/s 2565:− 2474:⁡ 2458:− 2409:ω 2405:ϵ 2402:⁡ 2384:− 2330:− 2298:⁡ 2282:− 2227:ψ 2181:ω 2177:ϵ 2174:⁡ 2156:− 2102:− 2034:ψ 1982:ϵ 1979:⁡ 1973:ω 1940:ψ 1878:− 1803:ϵ 1800:⁡ 1794:ϵ 1791:⁡ 1779:− 1741:− 1666:is zero. 1491:α 1488:⁡ 1482:− 1475:α 1472:⁡ 1458:δ 1455:⁡ 1449:δ 1446:⁡ 1434:− 1395:→ 1333:Equations 1311:precesses 1235:Entering 1208:obliquity 1197:red arrow 1189:Aldebaran 1156:precesses 1093:pole star 1002:magnitude 994:pole star 919:Callippus 884:Full moon 844:longitude 756:Al-Ma'mun 744:, in his 595:Kali Yuga 529:Ptolemaic 327:, below. 268:equinoxes 264:solstices 211:astronomy 174:gyroscope 100:equinoxes 53:astronomy 5138:Duration 5112:Saeculum 5092:Olympiad 4934:Solstice 4863:Holocene 4840:Calendar 4740:Horology 4531:ISO 8601 4526:ISO 31-1 4408:for the 4232:: 1–16. 3991:XConvert 3895:26 March 3839:(2002). 3811:"Kochab" 3706:(1960). 3277:Archived 3252:60695883 3156:See also 3087:billions 2848:10s/a): 2537:″ 1370:ecliptic 1323:nutation 1286:and the 1271:2700 AD 1257:2000 BC 1249:2000 BC 1246:4500 BC 1238:Exiting 1162:on Earth 1126:ecliptic 931:Almagest 872:Almagest 868:Almagest 840:Almagest 780:added to 725:Almagest 573:ayanamsa 509:Pleiades 489:tropical 424:parallax 368:Almagest 104:ecliptic 85:nutation 5291:Science 5219:Physics 5193:Portals 5107:Century 5097:Lustrum 5027:Instant 4899:Equinox 4868:Islamic 4810:Sundial 4675:Chronon 4362:Bibcode 4360:: A22. 4303:Bibcode 4234:Bibcode 4197:Bibcode 4138:Bibcode 4113:Bibcode 4073:Bibcode 4036:Bibcode 3967:Bibcode 3965:: 663. 3821:14 June 3789:1 March 3685:Bibcode 3592:Bibcode 3405:Ptolemy 3334:Bibcode 3332:: 711. 3133:+130 Ma 3032:sin (2π 2836:in one 2834:seconds 2826:radians 2694:10 m/s 2684:10 m/s 2658:J2000.0 2644: m 2633: m 1612:), and 1268:100 BC 1265:Pisces 1260:100 BC 1243:Taurus 1097:Polaris 1026:in the 998:Polaris 970:perigee 929:in the 927:Ptolemy 742:Al-Sufi 646:⁠ 634:⁠ 448:Proclus 436:Agrippa 416:Regulus 410:Ptolemy 363:Ptolemy 331:History 277:earlier 244:Polaris 225:Effects 219:physics 5102:Decade 5057:Moment 5052:Minute 5047:Second 5017:Other 4874:Julian 4853:Hebrew 4499:offset 4215: 4065:Icarus 4008:et al. 3927:Icarus 3849: 3716: 3610: 3466:595428 3464: 3419: 3354: 3250: 3240: 3129:+30 Ma 3085:is in 2878:Values 2828:(1.296 2672:Earth 2646:), so 2437:where 2000:where 1818:where 1521:where 1377:vector 1374:torque 1365:couple 1254:Aries 1201:Pisces 1193:Taurus 1147:Thuban 1024:Thuban 1017:Kochab 974:apogee 860:Euclid 694: 679: 664: 653: 649: 630: 622: 483:, the 434:, and 355:Nicaea 351:Rhodes 314:Values 297:zodiac 178:§Cause 81:apices 5243:Stars 5143:music 5082:Month 5042:Jiffy 5037:Shake 5032:Flick 4929:Solar 4919:Lunar 4894:Epact 4858:Hindu 4793:Watch 4748:Clock 4213:S2CID 4185:(PDF) 3608:S2CID 3558:Surya 3541:(PDF) 3462:JSTOR 3380:(PDF) 3373:(PDF) 3352:S2CID 3141:20 Ma 3075:annum 2838:annum 2813:10 /s 2799:10 /s 2733:10 m 2723:10 m 1655:(−cos 1568:> 1278:Cause 1101:Deneb 1069:Miami 1031:Draco 915:Meton 848:Spica 705:Yu Xi 626:Kalpa 550:India 420:Spica 373:Spica 203:cause 172:of a 77:cones 69:Earth 5087:Year 5072:Week 5062:Hour 4516:DUT1 4455:and 4010:2003 3897:2015 3847:ISBN 3823:2015 3791:2011 3714:ISBN 3647:2017 3417:ISBN 3388:2009 3248:OCLC 3238:ISBN 3131:and 2945:rate 2943:The 2901:and 2899:VLBI 2821:in 2 2669:Moon 2008:and 1647:(sin 1624:(sin 1346:The 1288:Moon 1105:Vega 917:and 784:Arab 534:The 499:Maya 491:and 357:, a 323:and 135:Moon 87:and 5067:Day 4370:doi 4358:534 4299:282 4205:doi 4146:doi 4134:412 4081:doi 4044:doi 4032:428 3963:282 3930:185 3600:doi 3454:doi 3342:doi 3330:108 3013:+ … 2903:LLR 2867:/dt 2857:/dt 2840:(a 2807:/dt 2793:/dt 2666:Sun 2642:752 2639:356 2631:137 2628:378 2465:sin 2461:1.5 2399:cos 2289:sin 2285:1.5 2171:cos 1976:sin 1797:cos 1788:sin 1651:cos 1632:sin 1628:cos 1485:cos 1469:sin 1452:cos 1443:sin 1342:S). 1284:Sun 1191:in 1095:is 921:in 812:in 719:In 365:'s 353:or 112:Sun 51:In 5303:: 4511:ΔT 4506:UT 4404:. 4400:. 4368:. 4356:. 4352:. 4316:. 4297:. 4276:. 4230:58 4228:. 4211:. 4203:. 4193:94 4191:. 4187:. 4144:. 4132:. 4128:. 4109:51 4107:. 4079:. 4069:50 4067:. 4042:. 4030:. 4026:. 3998:^ 3993:). 3961:. 3912:, 3887:. 3869:. 3813:. 3799:^ 3782:. 3681:47 3679:. 3606:, 3598:, 3586:, 3572:^ 3460:. 3450:70 3448:. 3396:^ 3350:. 3340:. 3328:. 3324:. 3308:^ 3283:, 3264:^ 3254:. 3246:. 3211:^ 3077:= 3028:+ 3024:= 3011:CT 3009:+ 3007:BT 3005:+ 3001:= 2863:dψ 2853:dψ 2803:dψ 2789:dψ 2705:)/ 2701:− 2688:GM 2678:GM 2653:. 2521:, 2504:)/ 2500:− 2010:Cω 1659:)) 1596:). 1588:, 1578:, 1560:− 1544:, 1528:, 1526:GM 1294:. 1203:. 1055:. 976:. 687:c. 672:c. 657:c. 608:c. 566:c. 515:. 430:, 418:, 266:, 55:, 5195:: 4445:e 4438:t 4431:v 4412:. 4378:. 4372:: 4364:: 4309:. 4305:: 4282:1 4240:. 4236:: 4219:. 4207:: 4199:: 4154:. 4148:: 4140:: 4119:. 4115:: 4087:. 4083:: 4075:: 4052:. 4046:: 4038:: 3973:. 3969:: 3899:. 3873:. 3855:. 3825:. 3793:. 3722:. 3693:. 3691:. 3687:: 3664:. 3649:. 3602:: 3594:: 3588:3 3468:. 3456:: 3390:. 3358:. 3344:: 3336:: 3137:p 3125:p 3115:T 3111:T 3107:p 3102:T 3098:T 3094:p 3083:T 3054:T 3050:T 3042:P 3038:P 3036:/ 3034:T 3030:b 3026:a 3022:p 3003:A 2999:p 2988:p 2984:T 2980:T 2968:T 2956:T 2952:p 2939:. 2933:T 2929:T 2925:T 2920:A 2918:p 2895:p 2887:p 2865:L 2855:S 2846:× 2830:× 2823:π 2811:× 2805:L 2797:× 2791:S 2776:× 2772:ω 2757:i 2747:e 2741:e 2731:× 2727:a 2721:× 2717:a 2707:C 2703:A 2699:C 2697:( 2692:× 2682:× 2649:e 2637:6 2626:6 2604:2 2599:c 2594:+ 2589:2 2584:a 2575:2 2570:c 2560:2 2555:a 2547:= 2541:2 2533:e 2506:C 2502:A 2498:C 2496:( 2481:) 2477:i 2469:2 2455:1 2451:( 2439:i 2420:E 2415:] 2391:C 2387:A 2381:C 2374:[ 2367:L 2362:] 2353:2 2350:3 2344:) 2338:2 2334:e 2327:1 2323:( 2316:3 2312:a 2305:) 2301:i 2293:2 2279:1 2275:( 2271:M 2268:G 2262:[ 2255:2 2252:3 2247:= 2241:t 2238:d 2231:L 2223:d 2192:E 2187:] 2163:C 2159:A 2153:C 2146:[ 2139:S 2134:] 2125:2 2122:3 2116:) 2110:2 2106:e 2099:1 2095:( 2088:3 2084:a 2078:M 2075:G 2069:[ 2062:2 2059:3 2054:= 2048:t 2045:d 2038:S 2030:d 2002:ω 1970:C 1964:x 1960:T 1954:= 1948:t 1945:d 1937:d 1918:δ 1901:2 1898:3 1892:) 1886:2 1882:e 1875:1 1871:( 1864:3 1860:a 1845:e 1829:a 1785:) 1782:A 1776:C 1773:( 1764:2 1761:3 1755:) 1749:2 1745:e 1738:1 1734:( 1727:3 1723:a 1717:M 1714:G 1706:2 1703:3 1698:= 1693:x 1689:T 1675:x 1671:y 1664:z 1657:α 1653:δ 1649:δ 1644:y 1636:) 1634:α 1630:δ 1626:δ 1621:x 1614:z 1610:x 1606:y 1602:x 1586:α 1576:δ 1572:) 1570:A 1566:C 1562:A 1558:C 1552:A 1542:C 1536:r 1504:) 1498:0 1463:( 1440:) 1437:A 1431:C 1428:( 1421:3 1417:r 1412:M 1409:G 1406:3 1400:= 1392:T 1132:E 685:( 670:( 640:/ 606:( 38:. 20:)

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