Axial precession

thumb|right|Precessional movement of Earth. Earth rotates (white arrows) about once a day around its rotational axis (red); this axis itself rotates slowly (white circle), completing a rotation in approximately 26,000 years In particular, [[Earth's axial precession is the gradual shift in the orientation of Earth's axis of rotation. It has a cycle of approximately 26,000 years, tracing out a double cone with a half-aperture of about 23.4°, an angle known as the obliquity of the ecliptic. This is similar to the precession of a spinning top. The term "precession" typically refers only to this largest part of the motion; other changes in the alignment of Earth's axis—nutation and polar motion—are much smaller in magnitude.

Earth's precession was historically called the precession of the equinoxes, because the equinoxes moved westward along the ecliptic relative to the fixed stars, opposite to the yearly motion of the Sun along the ecliptic. Historically,

the discovery of the precession of the equinoxes is usually attributed in the West to the 2nd-century-BC astronomer Hipparchus. With improvements in the ability to calculate the gravitational force between planets during the first half of the nineteenth century, it was recognized that the ecliptic itself moved slightly, which was named planetary precession, as early as 1863, while the dominant component was named lunisolar precession. Their combination was named general precession, instead of precession of the equinoxes.

Lunisolar precession is caused by the gravitational forces of the Moon and Sun on Earth's equatorial bulge, causing Earth's axis to move with respect to inertial space. Planetary precession (an advance) is due to the small angle between the gravitational force of the other planets on Earth and its orbital plane (the ecliptic), causing the plane of the ecliptic to shift slightly relative to inertial space. Lunisolar precession is about 500 times greater than planetary precession. Many references to the old terms exist in publications predating the change.

Nomenclature

thumb|300px|[[Precession of a gyroscope. 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 §Cause). 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.]]

The term "precession" as it is used in astronomy generally describes the observed but exceedingly slow 26,000 year changing location of the place where the Sun appears against the background stars when it travels northward across the equator, at the moment of the March equinox. The stars oppositely seem to be moving retrograde across the sky, relative to the false-perspective that the stars move, rather than the equinox locations moving. In physics the term "precession" is generally used slightly differently to instead describe the analogous mechanical process of spinning tops shifting their orientation to their surroundings when tipped slightly while spinning.

The term "precession" is derived from the Latin praecedere ("to precede, to come before, to be earlier"). As viewed from Earth, the stars are seen to drift slightly faster than one full revolution across the sky from east to west, at about 15 degrees per hour, due to the Earth's actual diurnal motion in the opposite direction. The stars also rotate around the Earth's axis even more slowly, once every year (at about 1 degree per day), which is a mirror reflection of the Earth's orbital motion around the Sun. At the same time the stars can be observed to move yet again more slowly in the same direction as their annual circuit of the Earth's polar axis, very slightly faster than their normal yearly motion. That later rate is approximately 50 arcseconds per year – 1 degree every 72 years or one full rotation about every 26,000 years – a phenomenon known as the "precession of the equinoxes".

In describing the motion they see, astronomers generally have shortened the term to simply "precession". Physicists also use the term "precession" to describe the cause of the motion, which has led to some confusion between the phenomenon observed in the sky and the imbalance of torques that causes tops to wobble. That distinction matters for astronomy, because some precessional motions are actual and real, whereas others are only apparent motion caused by the false perception that the background stars drift and the Earth's equinoxes are fixed, a consequence of viewing the sky from the perspective of a very slowly moving reference frame (the Earth's axis of rotation and its orbital axis). The issue is further obfuscated by the fact that modern astronomers are almost entirely physicists, and mix the astronomical and physical vocabularies, whose fine distinctions that might be more easily understood if kept separate.

Effects

thumb|300px|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 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.

The precession of the Earth's axis has a number of observable effects. First, the positions of the south and north celestial poles appear to move in circles against the space-fixed backdrop of stars, completing one circuit in approximately 26,000 years. Thus, while today the star Polaris lies approximately at the north celestial pole, this will change over time, and other stars will become the "north star".

Virtually all of the writings of Hipparchus are lost, including his work on precession. They are mentioned by Ptolemy, who explains precession as the rotation of the celestial sphere around a motionless Earth. It is reasonable to presume that Hipparchus, similarly to Ptolemy, thought of precession in geocentric terms as a motion of the heavens, rather than of the Earth.

Ptolemy

The first astronomer known to have continued Hipparchus's work on precession is Ptolemy in the second century AD. Ptolemy measured the longitudes of Regulus, Spica, and other bright stars with a variation of Hipparchus's lunar method that did not require eclipses. Before sunset, he measured the longitudinal arc separating the Moon from the Sun. Then, after sunset, he measured the arc from the Moon to the star. He used Hipparchus's model to calculate the Sun's longitude, and made corrections for the Moon's motion and its parallax. Ptolemy compared his own observations with those made by Hipparchus, Menelaus of Alexandria, Timocharis, and Agrippa. He found that between Hipparchus's time and his own (about 265 years), the stars had moved 2°40', or 1° in 100 years (36" per year; the rate accepted today is about 50" per year or 1° in 72 years). 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.

Maya

Archaeologist Susan Milbrath has speculated that the Mesoamerican Long Count calendar of "30,000 years involving the Pleiades...may have been an effort to calculate the precession of the equinox." This view is held by few other professional scholars of Maya civilization.

Ancient Egyptians

Similarly, it is claimed the precession of the equinoxes was known in Ancient Egypt, prior to the time of Hipparchus (the Ptolemaic 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.

The Dendera Zodiac, a star-map inside the Hathor temple at Dendera, 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.

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 effects on their culture. Rice noted that Egyptians re-oriented temples in response to precession of associated stars.

India

Before 1200, India had two theories of trepidation, one with a rate and another without a rate, and several related models of precession. Each had minor changes or corrections by various commentators. The dominant of the three was the trepidation described by the most respected Indian astronomical treatise, the Surya Siddhanta (3:9–12), composed but revised during the next few centuries. It used a sidereal epoch, or ayanamsa, that is still used by all Indian calendars, varying over the ecliptic longitude 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 March equinox. The March equinox of the Surya Siddhanta librated 27° in both directions from the sidereal epoch. Thus the equinox moved 54° in one direction and then back 54° in the other direction. This cycle took 7200 years to complete at a rate of 54″/year. The equinox coincided with the epoch at the beginning of the Kali Yuga in −3101 and again 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.

Another trepidation was described by Varāhamihira (). His trepidation consisted of an arc of 46°40′ in one direction and a return to the starting point. Half of this arc, 23°20′, was identified with the Sun's maximum declination on either side of the equator at the solstices. But no period was specified, thus no annual rate can be ascertained.

Middle Ages and Renaissance

In medieval Islamic astronomy, precession was known based on Ptolemy's Almagest, and by observations that refined the value.

Al-Battani, in his work Zij Al-Sabi, 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.

Subsequently, Al-Sufi, in his Book of Fixed Stars, mentions the same values that Ptolemy's value for precession is 1 degree per 100 solar years. He then quotes a different value from Zij Al Mumtahan, which was done during Al-Ma'mun's reign, of 1 degree for every 66 solar years. He also quotes the aforementioned Zij Al-Sabi 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.

Later, the Zij-i Ilkhani, compiled at the Maragheh observatory, sets the precession of the equinoxes at 51 arc seconds per annum, which is very close to the modern value of 50.2 arc seconds.

In the Middle Ages, Islamic and Latin Christian astronomers treated "trepidation" as a motion of the fixed stars to be added to precession. This theory is commonly attributed to the Arab astronomer Thabit ibn Qurra, but the attribution has been contested in modern times. Nicolaus Copernicus published a different account of trepidation in De revolutionibus orbium coelestium (1543). This work makes the first definite reference to precession as the result of a motion of the Earth's axis. Copernicus characterized precession as the third motion of the Earth.

Modern period

Over a century later, Isaac Newton in Philosophiae Naturalis Principia Mathematica (1687) explained precession as a consequence of gravitation. However, Newton's original precession equations did not work, and were revised considerably by Jean le Rond d'Alembert and subsequent scientists.

Hipparchus's discovery

Hipparchus gave an account of his discovery in On the Displacement of the Solsticial and Equinoctial Points (described in Almagest III.1 and VII.2). He measured the ecliptic longitude of the star Spica during lunar eclipses and found that it was about 6° west of the autumnal equinox. By comparing his own measurements with those of Timocharis of Alexandria (a contemporary of Euclid, who worked with Aristillus early in the 3rd century BC), he found that Spica's longitude had decreased by about 2° in the meantime (exact years are not mentioned in Almagest). Also in VII.2, Ptolemy gives more precise observations of two stars, including Spica, and concludes that in each case a 2° 40' change occurred 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" (Almagest VII.1), but did not report any later hypothesis Hipparchus might have devised. Hipparchus apparently limited his speculations, because he had only a few older observations, which were not very reliable.

Because the equinoctial points are not marked in the sky, Hipparchus needed the Moon as a reference point; he used a lunar eclipse to measure the position of a star. Hipparchus already had developed a way to calculate the longitude of the Sun at any moment. A lunar eclipse happens during Full moon, when the Moon is at opposition, 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.

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. 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. The Babylonian calendar used a cycle of 235 lunar months in 19 years since 499 BC (with only three exceptions before 380 BC), but it did not use a specified number of days. The Metonic cycle (432 BC) assigned 6,940 days to these 19 years producing an average year of 365+1/4+1/76 or 365.26316 days. The Callippic cycle (330 BC) dropped one day from four Metonic cycles (76 years) for an average year of 365+1/4 or 365.25 days. Hipparchus dropped one more day from four Callippic cycles (304 years), creating the Hipparchic cycle with an average year of 365+1/4−1/304 or 365.24671 days, which was close to his tropical year of 365+1/4−1/300 or 365.24667 days.

Hipparchus's mathematical signatures are found in the Antikythera Mechanism, an ancient astronomical computer of the second century BC. The mechanism is based on a solar year, the Metonic Cycle, which is the period the Moon reappears in the same place in the sky with the same phase (full Moon appears at the same position in the sky approximately in 19 years), the Callipic cycle (which is four Metonic cycles and more accurate), the Saros cycle, and the Exeligmos cycles (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 Kepler's second law. That is, it takes into account the fast motion of the Moon at perigee and slower motion at apogee.

Changing pole stars

right|thumb|upright=1.2|Precession of Earth's axis around the north ecliptical pole

A consequence of the precession is a changing pole star. Currently Polaris is extremely well suited to mark the position of the north celestial pole, as Polaris is a moderately bright star with a visual magnitude of 2.1 (variable), and is located about one degree from the pole, with no stars of similar brightness too close.

left|thumb|upright=1.2|Precession of Earth's axis around the south ecliptical pole

The previous pole star was Kochab (Beta Ursae Minoris, β UMi, β Ursae Minoris), the brightest star in the bowl of the "Little Dipper", located 16 degrees from Polaris. It held that role from 1500 BC to AD 500. It was not quite as accurate in its day as Polaris is today.

thumb|upright=1.3|Axial and [[apsidal precession with four of the nearest northern pole stars over the course of millenia]]

{| class=wikitable

! rowspan=2 | Constellation

! colspan=2 | Approximate year

! Entering

! Exiting

| Taurus

| 4500 BC

| 2000 BC

| Aries

| 2000 BC

| 100 BC

| Pisces

| 100 BC

| 2700

| Aquarius

| 2700

| 5300

Cause

The precession of the equinoxes is caused by the gravitational forces of the Sun and the Moon, and to a lesser extent other bodies, on the Earth. It was first explained by Isaac Newton.

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

The Earth is not a perfect sphere but an oblate spheroid, with an equatorial diameter about 43 kilometers larger than its polar diameter. Because of the Earth's axial tilt, during most of the year the half of this bulge that is closest to the Sun is off-center, either to the north or to the south, and the far half is off-center on the opposite side. The gravitational pull on the closer half is stronger, since gravity decreases with the square of distance, so this creates a small torque on the Earth as the Sun pulls harder on one side of the Earth than the other. The axis of this torque is roughly perpendicular to the axis of the Earth's rotation so the axis of rotation precesses. If the Earth were a perfect sphere, there would be no precession.

This average torque is perpendicular to the direction in which the rotation axis is tilted away from the ecliptic pole, so that it does not change the axial tilt itself. The magnitude of the torque from the Sun (or the Moon) varies with the angle between the Earth's spin axis direction and that of the gravitational attraction. It approaches zero when they are perpendicular. For example, this happens at the equinoxes in the case of the interaction with the Sun. This can be seen to be since the near and far points are aligned with the gravitational attraction, so there is no torque due to the difference in gravitational attraction.

Although the above explanation involved the Sun, the same explanation holds true for any object moving around the Earth, along or close to the ecliptic, notably, the Moon. The combined action of the Sun and the Moon is called the lunisolar precession. In addition to the steady progressive motion (resulting in a full circle in about 25,700 years) the Sun and Moon also cause small periodic variations, due to their changing positions. These oscillations, in both precessional speed and axial tilt, are known as the nutation. The most important term has a period of 18.6 years and an amplitude of 9.2 arcseconds.

In addition to lunisolar precession, the actions of the other planets of the Solar System cause the whole ecliptic to rotate slowly around an axis which has an ecliptic longitude of about 174° measured on the instantaneous ecliptic. This so-called planetary precession shift amounts to a rotation of the ecliptic plane of 0.47 seconds of arc per year (more than a hundred times smaller than lunisolar precession). The sum of the two precessions is known as the general precession.

Equations

thumb|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 S).

The tidal force on Earth due to a perturbing body (Sun, Moon or planet) is expressed by Newton's law of universal gravitation, whereby the gravitational force of the perturbing body on the side of Earth nearest is said to be greater than the gravitational force on the far side by an amount proportional to the difference in the cubes of the distances between the near and far sides. If the gravitational force of the perturbing body acting on the mass of the Earth as a point mass at the center of Earth (which provides the centripetal force causing the orbital motion) is subtracted from the gravitational force of the perturbing body everywhere on the surface of Earth, what remains may be regarded as the tidal force. This gives the paradoxical notion of a force acting away from the satellite but in reality it is simply a lesser force toward that body due to the gradient in the gravitational field. For precession, this tidal force can be grouped into two forces which only act on the equatorial bulge outside of a mean spherical radius. This couple can be decomposed into two pairs of components, one pair parallel to Earth's equatorial plane toward and away from the perturbing body which cancel each other out, and another pair parallel to Earth's rotational axis, both toward the ecliptic plane. The latter pair of forces creates the following torque vector on Earth's equatorial bulge:

:<math>\overrightarrow{T} = \frac{3GM}{r^3}(C - A) \sin\delta \cos\delta \begin{pmatrix}\sin\alpha \\ -\cos\alpha \\ 0\end{pmatrix}</math>

where

:GM, standard gravitational parameter of the perturbing body

:r, geocentric distance to the perturbing body

:C, moment of inertia around Earth's axis of rotation

:A, moment of inertia around any equatorial diameter of Earth

:C − A, moment of inertia of Earth's equatorial bulge (C > A)

:δ, declination of the perturbing body (north or south of equator)

:α, right ascension of the perturbing body (east from March equinox).

The three unit vectors of the torque at the center of the Earth (top to bottom) are x on a line within the ecliptic plane (the intersection of Earth's equatorial plane with the ecliptic plane) directed toward the March equinox, y on a line in the ecliptic plane directed toward the summer solstice (90° east of x), and z on a line directed toward the north pole of the ecliptic.

The value of the three sinusoidal terms in the direction of x for the Sun is a sine squared waveform varying from zero at the equinoxes (0°, 180°) to 0.36495 at the solstices (90°, 270°). The value in the direction of y for the Sun is a sine wave varying from zero at the four equinoxes and solstices to ±0.19364 (slightly more than half of the sine squared peak) halfway between each equinox and solstice with peaks slightly skewed toward the equinoxes (43.37°(−), 136.63°(+), 223.37°(−), 316.63°(+)). Both solar waveforms have about the same peak-to-peak amplitude and the same period, half of a revolution or half of a year. The value in the direction of z is zero.

The average torque of the sine wave in the direction of y is zero for the Sun or Moon, so this component of the torque does not affect precession. The average torque of the sine squared waveform in the direction of x for the Sun or Moon is:

:<math>T_x = \frac{3}{2}\frac{GM}{a^3 \left(1 - e^2\right)^\frac{3}{2(C - A) \sin\epsilon \cos\epsilon</math>

where

:<math>a</math>, semimajor axis of Earth's (Sun's) orbit or Moon's orbit

:e, eccentricity of Earth's (Sun's) orbit or Moon's orbit

and 1/2 accounts for the average of the sine squared waveform, <math>a^3 \left(1 - e^2\right)^\frac{3}{2}</math> accounts for the average distance cubed of the Sun or Moon from Earth over the entire elliptical orbit, and ε (the angle between the equatorial plane and the ecliptic plane) is the maximum value of δ for the Sun and the average maximum value for the Moon over an entire 18.6 year cycle.

Precession is:

:<math>\frac{d\psi}{dt} = \frac{T_x}{C\omega\sin\epsilon}</math>

where ω is Earth's angular velocity and Cω is Earth's angular momentum. Thus the first order component of precession due to the Sun is:

:<math>e^2 = \frac{\mathrm{a}^2 - \mathrm{c}^2}{\mathrm{a}^2 + \mathrm{c}^2}</math>

where a is the equatorial radius () and c is the polar radius (), so .

Applicable parameters for J2000.0 rounded to seven significant digits (excluding leading 1) are:

{| class=wikitable

! Sun !! Moon !! Earth

|GM = 1.3271244 m3/s2

|GM = 4.902799 m3/s2

|(C − A)/C = 0.003273763

|a = 3.833978 m

|a = 1.4959802 m

|e = 0.05554553

|e = 0.016708634

|i = 5.156690°

|ε = 23.43928°

|ω = 7.292115 rad/s

which yield

:dψS/dt = 2.450183 /s

:dψL/dt = 5.334529 /s

both of which must be converted to ″/a (arcseconds/annum) by the number of arcseconds in 2π radians (1.296″/2π) and the number of seconds in one annus (a Julian year) (3.15576s/a):

:dψS/dt = 15.948788″/a   vs   15.948870″/a from Williams

:pA = 5,028.796195T + 1.1054348T2 + higher order terms, in arcseconds, with T, the time in Julian centuries (that is, 36,525 days) since the epoch of 2000.

The rate of precession is the derivative of that:

:p = 5,028.796195 + 2.2108696T + higher order terms.

The constant term of this speed (5,028.796195 arcseconds per century in above equation) corresponds to one full precession circle in 25,771.57534 years (one full circle of 360 degrees divided by 50.28796195 arcseconds per year)

:p = 50.475838 − 26.368583T + 21.890862T2

:p = 50.475838 − 27.000654T + 15.603265T2

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

Precession will be greater than p by the small amount of +0.135052"/a between and . The jump to this excess over p will occur in only beginning now because the secular decrease in precession is beginning to cross a resonance in Earth's orbit caused by the other planets.

According to 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, when the Moon reaches 68 Earth radii in about 2,000 million years, to 69,000 years. This will be associated with wild swings in the obliquity of the ecliptic as well. Ward, however, used the abnormally large modern value for tidal dissipation. Using the 620-million year average provided by tidal rhythmites of about half the modern value, these resonances will not be reached until about 3,000 and 4,000 million years, respectively. However, due to the gradually increasing luminosity of the Sun, the oceans of the Earth will have vaporized before that time (about 2,100 million years from now).

References

Bibliography

Dreyer, J. L. E. A History of Astronomy from Thales to Kepler. 2nd ed. New York: Dover, 1953.
Evans, James. The History and Practice of Ancient Astronomy. New York: Oxford University Press, 1998.
Explanatory supplement to the Astronomical ephemeris and the American ephemeris and nautical almanac
Precession and the Obliquity of the Ecliptic has a comparison of values predicted by different theories
Pannekoek, A. A History of Astronomy. New York: Dover, 1961.
Parker, Richard A. "Egyptian Astronomy, Astrology, and Calendrical Reckoning." Dictionary of Scientific Biography 15:706–727.
Rice, Michael (1997), Egypt's Legacy: The archetypes of Western civilization, 3000–30 BC, London and New York.
Tompkins, Peter. Secrets of the Great Pyramid. With an appendix by Livio Catullo Stecchini. New York: Harper Colophon Books, 1971.
Toomer, G. J. "Hipparchus." Dictionary of Scientific Biography. Vol. 15:207–224. New York: Charles Scribner's Sons, 1978.
Toomer, G. J. Ptolemy's Almagest. London: Duckworth, 1984.
Ulansey, David. The Origins of the Mithraic Mysteries: Cosmology and Salvation in the Ancient World. New York: Oxford University Press, 1989.

External links

D'Alembert and Euler's Debate on the Solution of the Precession of the Equinoxes
Forced precession and nutation of Earth