Precession (astronomy)
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The precession of Earth's axis of rotation with respect to inertial space is also called the precession of the equinoxes. Like a wobbling top, the direction of the Earth's axis is changing; while today, the North Pole points roughly to Polaris, over time it will change. Because of this wobble, the position of the earth in its orbit around the sun at the moment of the equinoxes and solstices will also change.
The term precession typically refers only to the largest periodic motion. Other changes of Earth's axis are nutation and polar motion; their magnitude is very much smaller.
Currently, this annual motion is about 50.3 seconds of arc per year or 1 degree every 71.6 years. The process is slow, but cumulative. A complete precession cycle covers a period of approximately 25,765 years, the so called Great or Platonic year, during which time the equinox regresses a full 360° through all twelve constellations of the zodiac. Precessional movement is also the determining factor in the length of an astrological age.
In ancient times the precession of the equinox referred to the motion of the equinox relative to the background stars in the zodiac; this is equivalent to the modern understanding. It acted as a method of keeping time in the Great year.[citation needed]
Hipparchus is credited with discovering that the positions of the equinoxes move westward along the ecliptic compared to the fixed stars on the celestial sphere. The exact dates of his life are not known, but astronomical observations attributed to him date from 147 BC to 127 BC and were described in his writings, none of which survive to date.
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[edit] Discovery of precession
Though there is still-controversial evidence that Aristarchus of Samos possessed distinct values for the sidereal and tropical years as early as ca. 280 BC, the discovery of precession is usually attributed to Hipparchus of Rhodes or Nicaea, a Greek astronomer who was active in the 2nd century BC. Virtually all Hipparchus' writings are lost, including his work on precession. They are mentioned in Ptolemy's Almagest, where precession is explained as the rotation of the celestial sphere around a motionless Earth. It is reasonable to assume that Hipparchus, like Ptolemy, thought of precession in geocentric terms as a motion of the heavens. The first definite reference to precession as the result of a motion of the Earth's axis is Nicolaus Copernicus's De revolutionibus orbium coelestium (1543). He called precession the third motion of the earth. Over a century later it was explained in Isaac Newton's Philosophiae Naturalis Principia Mathematica (1687) to be a consequence of gravitation (Evans 1998, p. 246). However, Newton's original precession equations did not work and were revised considerably by Jean le Rond d'Alembert and subsequent scientists.
Various claims have been made that other cultures discovered precession independent of Hipparchus. At one point it was suggested that the Babylonians may have known about precession. According to al-Battani, Chaldean astronomers had distinguished the tropical and sidereal year (the value of precession is equivalent to the difference between the tropical and sidereal years). He stated that they had, around 330 BC, an estimation for the length of the sidereal year to be SK = 365 days 6 hours 11 min (= 365.258 days) with an error of (about) 2 min. It was claimed by P. Schnabel in 1923 that Kidinnu theorized about precession in 315 BC (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.) Neugebauer's work on this issue in the 1950s superseded Schnabel's (and earlier, Kugler's) theory of a Babylonian discoverer of precession.
Similar claims have been made that precession was known in Ancient Egypt prior to the time of Hipparchus. Some buildings in the Karnak temple complex, for instance, were allegedly oriented towards the point on the horizon where certain stars rose or set at key times of the year. A few centuries later, when precession made the orientations obsolete, the temples would be rebuilt. Note however that the observation that a stellar alignment has grown wrong does not necessarily mean that the Egyptians understood that the stars moved across the sky at the rate of about one degree per 72 years. 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, supposedly records precession of the equinoxes (Tompkins 1971). In any case, if the ancient Egyptians knew of precession, their knowledge is not recorded in surviving astronomical texts.
The former professor of the history of science at MIT, Giorgio de Santillana, argues in his book, Hamlet's Mill, that many ancient cultures may have known of the slow movement of the stars across the sky; the observable result of the precession of the equinox. This 700 page book, co-authored by Hertha von Dechend, makes reference to approximately 200 myths from over 30 ancient cultures that hinted at the motion of the heavens, some of which are thought to date to the neolithic period.
Identifying alignments of monuments with solar, lunar, and stellar phenomena is a major part of archaeoastronomy. Stonehenge is the most famous of many megalithic structures that indicate the direction of celestial objects at rising or setting. Precession complicates the attempt to find stellar alignments, especially for very old sites. Many archaeological sites cannot be dated exactly, making it difficult or impossible to know whether a proposed alignment would have worked when the site was founded.
Yu Xi (fourth century CE) was the first Chinese astronomer to mention precession. He estimated the rate of precession as 1° in 50 years (Pannekoek 1961, p. 92).
[edit] Hipparchus' discovery
Hipparchus gave an account of his discovery in On the Displacement of the Solsticial and Equinoctial Points (described in Almagest III.1 and VII.2). He measured the ecliptic longitude 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 about 150 years. 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 had already 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 like 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 Callipic 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. The three Greek cycles were never used to regulate any civil calendar—they only appear in the Almagest in an astronomical context.
[edit] Mithraic question
Mithraism was a mystery religion or school based on the worship of the god Mithras. Many underground temples were built in the Roman Empire from about the 1st century BC to the 5th century CE. Understanding Mithraism has been made difficult by the near-total lack of written descriptions or scripture; the teachings must be reconstructed from iconography found in mithraea (a mithraeum was a cave or underground meeting place that often contained bas reliefs of Mithras, the zodiac and associated symbols). Until the 1970s most scholars followed Franz Cumont in identifying Mithras with the Persian god Mithra. Cumont's thesis was re-examined in 1971, and Mithras is now believed to be a syncretic deity only slightly influenced by Persian religion.
Mithraism is now recognized as having pronounced astrological elements, but the details are debated. One scholar of Mithraism, David Ulansey, has interpreted Mithras (Mithras Sol Invictus - the unconquerable sun) as a second sun or star that is responsible for precession. He suggests the cult may have been inspired by Hipparchus' discovery of precession. Part of his analysis is based on the tauroctony an image of Mithras sacrificing a bull, found in most of the temples. According to Ulansey, the tauroctony is a star chart. Mithras is a second sun or hyper-cosmic sun and or a constellation Perseus, and the bull is Taurus, a constellation of the zodiac. In an earlier astrological age, the vernal equinox had taken place when the Sun was in Taurus. The tauroctony, by this reasoning, commemorated Mithras-Perseus ending the "Age of Taurus" (about 2000 BC based on the Vernal Equinox - or about 11,500 BC based on the Autumnal Equinox).
The iconography also contains two torch bearing boys (Cautes and Cautopetes) on each side of the zodiac. Ulansey, and Walter Cruttenden in his book Lost Star of Myth and Time, interpret these to mean ages of growth and decay, or enlightenment and darkness; primal elements of the cosmic progression. Thus Mithraism is thought to have something to do with the changing ages within the precession cycle or Great Year (Plato's term for one complete precession of the equinox).
[edit] Later studies of precession
The first astronomer known to have continued Hipparchus' work on precession is Ptolemy in the 2nd century. Ptolemy measured the longitudes of Regulus, Spica, and other bright stars with a variation of Hipparchus' 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' 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' 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.
Most ancient authors did not mention precession and perhaps did not know of it. Besides Ptolemy, the list includes Proclus, who rejected precession, and Theon of Alexandria, a commentator on Ptolemy in the 4th century, who 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. In the Middle Ages, Islamic and Latin Christian astronomers treated it 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).
[edit] Changing pole stars
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 within a half degree of the pole.
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 12000 BC and will do so again around the year AD 14000), however it never comes closer than 5° to the pole.
When Polaris becomes the north star again around 27800 AD, due to its proper motion it then will be farther away from the pole than it is now, while in 23600 BC it came closer to the pole.
It is more difficult to find the south celestial pole in the sky at this moment, as that area is a particularly bland portion of the sky, and the nominal south pole star is Sigma Octantis, which with magnitude 5.5 is barely visible to the naked eye even under ideal conditions. That will change from the eightieth to the ninetieth 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 nicely pointed to the south pole. By consequence, the constellation is no longer visible from subtropical northern latitudes, as it was in the time of the ancient Greeks.
[edit] Polar shift and equinoxes shift
The figures to the right attempt to explain the relation between the precession of the Earth's axis and the shift in the equinoxes. These figures 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 figure shows the perspective of a near-Earth position as seen through a very wide angle lens (from which the apparent distortion).
The rotation axis of the Earth describes, over a period of 25,700 years, a small circle (blue) among the stars, centered around 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. 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 (whose center lies on a circle about 6° away from the poles) and the proper motions of the stars.
[edit] Explanation
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.
In popular science books, precession is often explained with the example of a spinning top. While the physical effect is the same, some crucial details differ. For a spinning top, gravity causes the top to wobble, which in turn causes precession. The applied force in this case is 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.
The figure below explains how this process works. The Earth is given as a perfect sphere with the mass of the bulge approximated by a blue torus around its equator. The green arrows indicate the gravitational forces from the Sun on some extreme points. These tangential forces create a torque (orange), and this torque, added to the rotation (magenta), shifts the rotational axis to a slightly new position (yellow). Over time, the axis precesses along the white circle, which is centered around the ecliptic pole.
This torque is always in the same direction, 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 less than 20 seconds of arc.
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 planetary precession shift is only 0.47 seconds of arc per year (more than a hundred times smaller than lunisolar precession), and takes place along the instantaneous equator. The sum of the two precessions is known as the general precession.
[edit] History
Precession causes the cycle of seasons (tropical year) to be about 20.4 minutes less than the time for the Earth to return to the same position with respect to the stars. This results in a slow change (one day every 71 calendar years) in the position of the Sun with respect to the stars at an equinox.
Hipparchus estimated the Earth's precession around 130 BC, adding his own observations to those of Babylonian astronomers in the preceding centuries. In particular, they measured the distance of stars such as Spica to the Moon and the Sun during lunar eclipses, and because he could compute the distance of the Moon and the Sun from the equinox at these moments, he noticed that Spica and other stars appeared to have moved over the centuries.
It remains controversial as to whether the ancient Egyptians knew of the Precession or not. 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) Following Carl Gustav Jung, 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. This fact alone makes Jung's belief in the Egyptians' knowledge of the Precession a good deal less speculative than once it seemed." (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) see also Royal Arch and the Precession of the Equinoxes
[edit] Anomalistic precession
Because of gravitational disturbances by the other planets, the shape and orientation of Earth's orbit are not fixed, and the apsides (that is, perihelion and aphelion) slowly move with respect to a fixed frame of reference. Therefore the anomalistic year is slightly longer than the sidereal year. It takes about 112,000 years for the ellipse to revolve once relative to the fixed stars.
Because the anomalistic year is longer than the sidereal year while the tropical year (which calendars attempt to track) is shorter, the two forms of precession add. It takes about 21,000 years for the ellipse to revolve once relative to the vernal equinox, that is, for the perihelion to return to the same date (given a calendar that tracks the seasons perfectly). The dates of perihelion and of aphelion advance each year on this cycle, an average of 1 day per 58 years.
This interaction between the anomalistic and tropical cycle is important in the long-term climate variations on Earth, called the Milankovitch cycles. An equivalent is also known on Mars.
The figure to the right illustrates the effects of precession on the northern hemisphere seasons, relative to perihelion and aphelion.
Notice in the above figure that the areas swept during a specific season changes through time. Orbital mechanics require that the length of the seasons be proportional to the swept areas of the seasonal quadrants, so when the orbital eccentricity is extreme, the seasons on the far side of the orbit may be substantially longer in duration.
[edit] Values
Simon Newcomb's calculation at the end of the nineteenth century for general precession (known as 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:
pA = 5,028.796195×T + 1.1054348×T² + higher order terms,
in arcseconds per Julian century, 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 corresponds to one full precession circle in 25,772 years.
The precession rate is not a constant, but (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 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 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 + Bsin (2πT/P), where P is the 410-century period.
Theoretical models may calculate the proper constants (coefficients) corresponding to the higher powers of T, but since it is impossible for a polynomial to match a periodic function over all numbers, the error in all such approximations will grow without bound as T increases. 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, 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 mutually perpendicular directions.
Over longer time periods, that is, millions of years, it appears that precession is quasiperiodic at around 25,700 years, however, it will not remain so. According to Ward, when the distance of the Moon, which is continuously increasing from tidal effects, will have gone from the current 60.3 to approximately 66.5 Earth radii in about 1,500 million years, 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. Long before that time (about 2,100 million years from now), due to the increasing luminosity of the Sun, however, the oceans of the Earth will have boiled away, which will alter tidal effects significantly.
[edit] References
- Explanatory supplement to the Astronomical ephemeris and the American ephemeris and nautical almanac
- Precession and the Obliquity of the Ecliptic has a comparison of values predicted by different theories
- A.L. Berger (1976), "Obliquity & precession for the last 5 million years", Astronomy & astrophysics 51, 127
- J.H. Lieske et al. (1977), "Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants". Astronomy & Astrophysics 58, 1..16
- W.R. Ward (1982), "Comments on the long-term stability of the earth's obliquity", Icarus 50, 444
- J.L. Simon et al. (1994), "Numerical expressions for precession formulae and mean elements for the Moon and the planets", Astronomy & Astrophysics 282, 663..683
- N. Capitaine et al. (2003), "Expressions for IAU 2000 precession quantities", Astronomy & Astrophysics 412, 567..586
- J.L. Hilton et al. (2006), "Report of the International Astronomical Union Division I Working Group on Precession and the Ecliptic" (pdf, 174KB). Celestial Mechanics and Dynamical Astronomy (2006) 94: 351..367
- Rice, Michael (1997), Egypt's Legacy: The archetypes of Western civilization, 3000-30 BC, London and New York.
- 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.
- 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.
- 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.
- Schütz, Michael: Hipparch und die Entdeckung der Präzession. Bemerkungen zu David Ulansey, Die Ursprünge des Mithraskultes, in: ejms = Electronic Journal of Mithraic Studies, www.uhu.es/ejms/Papers/Volume1Papers/ulansey.doc
