Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called nutation. In physics, there are two types of precession: torque-free and torque-induced.

In astronomy, precession refers to any of several slow changes in an astronomical body's rotational or orbital parameters. An important example is the steady change in the orientation of the axis of rotation of the Earth, known as the precession of the equinoxes.

Gyroscope precession
Precession of a gyroscope


Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the angular momentum is a constant, but the angular velocity vector changes orientation with time. What makes this possible is a time-varying moment of inertia, or more precisely, a time-varying inertia matrix. The inertia matrix is composed of the moments of inertia of a body calculated with respect to separate coordinate axes (e.g. x, y, z). If an object is asymmetric about its principal axis of rotation, the moment of inertia with respect to each coordinate direction will change with time, while preserving angular momentum. The result is that the component of the angular velocities of the body about each axis will vary inversely with each axis' moment of inertia.

The torque-free precession rate of an object with an axis of symmetry, such as a disk, spinning about an axis not aligned with that axis of symmetry can be calculated as follows:[1]

where ωp is the precession rate, ωs is the spin rate about the axis of symmetry, Is is the moment of inertia about the axis of symmetry, Ip is moment of inertia about either of the other two equal perpendicular principal axes, and α is the angle between the moment of inertia direction and the symmetry axis.[2]

When an object is not perfectly solid, internal vortices will tend to damp torque-free precession, and the rotation axis will align itself with one of the inertia axes of the body.

For a generic solid object without any axis of symmetry, the evolution of the object's orientation, represented (for example) by a rotation matrix R that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internal moment of inertia tensor I0 and fixed external angular momentum L, the instantaneous angular velocity is

Precession occurs by repeatedly recalculating ω and applying a small rotation vector ω dt for the short time dt; e.g.:

for the skew-symmetric matrix [ω]×. The errors induced by finite time steps tend to increase the rotational kinetic energy:

this unphysical tendency can be counteracted by repeatedly applying a small rotation vector v perpendicular to both ω and L, noting that

Another type of torque-free precession can occur when there are multiple reference frames at work. For example, Earth is subject to local torque induced precession due to the gravity of the sun and moon acting on Earth's axis, but at the same time the solar system is moving around the galactic center. As a consequence, an accurate measurement of Earth's axial reorientation relative to objects outside the frame of the moving galaxy (such as distant quasars commonly used as precession measurement reference points) must account for a minor amount of non-local torque-free precession, due to the solar system's motion.


Torque-induced precession (gyroscopic precession) is the phenomenon in which the axis of a spinning object (e.g., a gyroscope) describes a cone in space when an external torque is applied to it. The phenomenon is commonly seen in a spinning toy top, but all rotating objects can undergo precession. If the speed of the rotation and the magnitude of the external torque are constant, the spin axis will move at right angles to the direction that would intuitively result from the external torque. In the case of a toy top, its weight is acting downwards from its center of mass and the normal force (reaction) of the ground is pushing up on it at the point of contact with the support. These two opposite forces produce a torque which causes the top to precess.

Gyroscopic precession 256x256
The response of a rotating system to an applied torque. When the device swivels, and some roll is added, the wheel tends to pitch.

The device depicted on the right (or above on mobile devices) is gimbal mounted. From inside to outside there are three axes of rotation: the hub of the wheel, the gimbal axis, and the vertical pivot.

To distinguish between the two horizontal axes, rotation around the wheel hub will be called spinning, and rotation around the gimbal axis will be called pitching. Rotation around the vertical pivot axis is called rotation.

First, imagine that the entire device is rotating around the (vertical) pivot axis. Then, spinning of the wheel (around the wheelhub) is added. Imagine the gimbal axis to be locked, so that the wheel cannot pitch. The gimbal axis has sensors, that measure whether there is a torque around the gimbal axis.

In the picture, a section of the wheel has been named dm1. At the depicted moment in time, section dm1 is at the perimeter of the rotating motion around the (vertical) pivot axis. Section dm1, therefore, has a lot of angular rotating velocity with respect to the rotation around the pivot axis, and as dm1 is forced closer to the pivot axis of the rotation (by the wheel spinning further), because of the Coriolis effect, with respect to the vertical pivot axis, dm1 tends to move in the direction of the top-left arrow in the diagram (shown at 45°) in the direction of rotation around the pivot axis.[3] Section dm2 of the wheel is moving away from the pivot axis, and so a force (again, a Coriolis force) acts in the same direction as in the case of dm1. Note that both arrows point in the same direction.

The same reasoning applies for the bottom half of the wheel, but there the arrows point in the opposite direction to that of the top arrows. Combined over the entire wheel, there is a torque around the gimbal axis when some spinning is added to rotation around a vertical axis.

It is important to note that the torque around the gimbal axis arises without any delay; the response is instantaneous.

In the discussion above, the setup was kept unchanging by preventing pitching around the gimbal axis. In the case of a spinning toy top, when the spinning top starts tilting, gravity exerts a torque. However, instead of rolling over, the spinning top just pitches a little. This pitching motion reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity – via the pitching motion – elicits gyroscopic precession (which in turn yields a counter torque against the gravity torque) rather than causing the spinning top to fall to its side.

Precession or gyroscopic considerations have an effect on bicycle performance at high speed. Precession is also the mechanism behind gyrocompasses.

Classical (Newtonian)

The torque caused by the normal force – Fg and the weight of the top causes a change in the angular momentum L in the direction of that torque. This causes the top to precess.

Precession is the change of angular velocity and angular momentum produced by a torque. The general equation that relates the torque to the rate of change of angular momentum is:

where and are the torque and angular momentum vectors respectively.

Due to the way the torque vectors are defined, it is a vector that is perpendicular to the plane of the forces that create it. Thus it may be seen that the angular momentum vector will change perpendicular to those forces. Depending on how the forces are created, they will often rotate with the angular momentum vector, and then circular precession is created.

Under these circumstances the angular velocity of precession is given by:

where Is is the moment of inertia, ωs is the angular velocity of spin about the spin axis, m is the mass, g is the acceleration due to gravity and r is the perpendicular distance of the spin axis about the axis of precession. The torque vector originates at the center of mass. Using ω = /T, we find that the period of precession is given by:

Where Is is the moment of inertia, Ts is the period of spin about the spin axis, and τ is the torque. In general, the problem is more complicated than this, however.

There is an easy way to understand why gyroscopic precession occurs without using any mathematics. The behavior of a spinning object simply obeys laws of inertia by resisting any change in direction. A spinning object possesses a property known as rigidity in space, meaning the spin axis resists any change in orientation. It is the inertia of matter comprising the object as it resists any change in direction that provides this property. Of course, the direction this matter travels constantly changes as the object spins, but any further change in direction is resisted. If a force is applied to the surface of a spinning disc, for example, matter experiences no change in direction at the place the force was applied (or 180 degrees from that place). But 90 degrees before and 90 degrees after that place, matter is forced to change direction. This causes the object to behave as if the force was applied at those places instead. When a force is applied to anything, the object exerts an equal force back but in the opposite direction. Since no actual force was applied 90 degrees before or after, nothing prevents the reaction from taking place, and the object causes itself to move in response. A good way to visualize why this happens is to imagine the spinning object to be a large hollow doughnut filled with water, as described in the book "Thinking Physics" by Lewis Epstein. The doughnut is held still while water circulates inside it. As the force is applied, the water inside is caused to change direction 90 degrees before and after that point. The water then exerts its own force against the inner wall of the doughnut and causes the doughnut to rotate as if the force was applied 90 degrees ahead in the direction of rotation. Epstein exaggerates the vertical and horizontal motion of the water by changing the shape of the doughnut from round to square with rounded corners.

Now imagine the object to be a spinning bicycle wheel, held at both ends of its axle in the hands of a subject. The wheel is spinning clock-wise as seen from a viewer to the subject's right. Clock positions on the wheel are given relative to this viewer. As the wheel spins, the molecules comprising it are traveling exactly horizontal and to the right the instant they pass the 12-o'clock position. They then travel vertically downward the instant they pass 3 o'clock, horizontally to the left at 6 o'clock, vertically upward at 9 o’clock and horizontally to the right again at 12 o'clock. Between these positions, each molecule travels components of these directions. Now imagine the viewer applying a force to the rim of the wheel at 12 o’clock. For this example's sake, imagine the wheel tilting over when this force is applied; it tilts to the left as seen from the subject holding it at its axle. As the wheel tilts to its new position, molecules at 12 o’clock (where the force was applied) as well as those at 6 o’clock, still travel horizontally; their direction did not change as the wheel was tilting. Nor is their direction different after the wheel settles in its new position; they still move horizontally the instant they pass 12 and 6 o’clock. BUT, molecules passing 3 and 9 o’clock were forced to change direction. Those at 3 o’clock were forced to change from moving straight downward, to downward and to the right as viewed from the subject holding the wheel. Molecules passing 9 o’clock were forced to change from moving straight upward, to upward and to the left. This change in direction is resisted by the inertia of those molecules. And when they experience this change in direction, they exert an equal and opposite force in response AT THOSE LOCATIONS-3 AND 9 O’CLOCK. At 3 o’clock, where they were forced to change from moving straight down to downward and to the right, they exert their own equal and opposite reactive force to the left. At 9 o’clock, they exert their own reactive force to the right, as viewed from the subject holding the wheel. This makes the wheel as a whole react by momentarily rotating counter-clockwise as viewed from directly above. Thus, as the force was applied at 12 o’clock, the wheel behaved as if that force was applied at 3 o’clock, which is 90 degrees ahead in the direction of spin. Or, you can say it behaved as if a force from the opposite direction was applied at 9 o'clock, 90 degrees prior to the direction of spin.

In summary, when you apply a force to a spinning object to change the direction of its spin axis, you are not changing the direction of the matter comprising the object at the place you applied the force (nor at 180 degrees from it); matter experiences zero change in direction at those places. Matter experiences the maximum change in direction 90 degrees before and 90 degrees beyond that place, and lesser amounts closer to it. The equal and opposite reaction that occurs 90 degrees before and after then causes the object to behave as it does. This principle is demonstrated in helicopters. Helicopter controls are rigged so that inputs to them are transmitted to the rotor blades at points 90 degrees prior to and 90 degrees beyond the point at which the change in aircraft attitude is desired. The effect is dramatically felt on motorcycles. A motorcycle will suddenly lean and turn in the opposite direction the handle bars are turned.

Gyro precession causes another phenomenon for spinning objects such as the bicycle wheel in this scenario. If the subject holding the wheel removes a hand from one end of its axle, the wheel will not topple over, but will remain upright, supported at just the other end. However, it will immediately take on an additional motion; it will begin to rotate about a vertical axis, pivoting at the point of support as it continues spinning. If you allowed the wheel to continue rotating, you would have to turn your body in the same direction as the wheel rotated. If the wheel was not spinning, it would obviously topple over and fall when one hand is removed. The initial action of the wheel beginning to topple over is equivalent to applying a force to it at 12 o'clock in the direction toward the unsupported side (or a force at 6 o’clock toward the supported side). When the wheel is spinning, the sudden lack of support at one end of its axle is equivalent to this same force. So, instead of toppling over, the wheel behaves as if a continuous force is being applied to it at 3 or 9 o’clock, depending on the direction of spin and which hand was removed. This causes the wheel to begin pivoting at the one supported end of its axle while remaining upright. Although it pivots at that point, it does so only because of the fact that it is supported there; the actual axis of precessional rotation is located vertically through the wheel, passing through its center of mass. Also, this explanation does not account for the effect of variation in the speed of the spinning object; it only illustrates how the spin axis behaves due to precession. More correctly, the object behaves according to the balance of all forces based on the magnitude of the applied force, mass and rotational speed of the object. Once it is visualized why the wheel remains upright and rotates, it can easily be seen why the axis of a spinning top slowly rotates while the top spins as shown in the illustration on this page. A top behaves exactly like the bicycle wheel due to the force of gravity pulling downward. The point of contact with the surface it spins on is equivalent to the end of the axle the wheel is supported at. As the top's spin slows, the reactive force that keeps it upright due to inertia is overcome by gravity. Once the reason for gyro precession is visualized, the mathematical formulas start to make sense.

Relativistic (Einsteinian)

The special and general theories of relativity give three types of corrections to the Newtonian precession, of a gyroscope near a large mass such as Earth, described above. They are:

  • Thomas precession a special relativistic correction accounting for the observer's being in a rotating non-inertial frame.
  • de Sitter precession a general relativistic correction accounting for the Schwarzschild metric of curved space near a large non-rotating mass.
  • Lense–Thirring precession a general relativistic correction accounting for the frame dragging by the Kerr metric of curved space near a large rotating mass.


In astronomy, precession refers to any of several gravity-induced, slow and continuous changes in an astronomical body's rotational axis or orbital path. Precession of the equinoxes, perihelion precession, changes in the tilt of Earth's axis to its orbit, and the eccentricity of its orbit over tens of thousands of years are all important parts of the astronomical theory of ice ages. (See Milankovitch cycles.)

Axial precession (precession of the equinoxes)

Axial precession is the movement of the rotational axis of an astronomical body, whereby the axis slowly traces out a cone. In the case of Earth, this type of precession is also known as the precession of the equinoxes, lunisolar precession, or precession of the equator. Earth goes through one such complete precessional cycle in a period of approximately 26,000 years or 1° every 72 years, during which the positions of stars will slowly change in both equatorial coordinates and ecliptic longitude. Over this cycle, Earth's north axial pole moves from where it is now, within 1° of Polaris, in a circle around the ecliptic pole, with an angular radius of about 23.5°.

The ancient Greek astronomer Hipparchus (c. 190–120 BC) is generally accepted to be the earliest known astronomer to recognize and assess the precession of the equinoxes at about 1° per century (which is not far from the actual value for antiquity, 1.38°),[4] although there is some minor dispute about whether he was.[5] In ancient China, the Jin-dynasty scholar-official Yu Xi (fl. 307-345 AD) made a similar discovery centuries later, noting that the position of the Sun during the winter solstice had drifted roughly one degree over the course of fifty years relative to the position of the stars.[6] The precession of Earth's axis was later explained by Newtonian physics. Being an oblate spheroid, Earth has a non-spherical shape, bulging outward at the equator. The gravitational tidal forces of the Moon and Sun apply torque to the equator, attempting to pull the equatorial bulge into the plane of the ecliptic, but instead causing it to precess. The torque exerted by the planets, particularly Jupiter, also plays a role.[7]

Precessional movement of the axis (left), precession of the equinox in relation to the distant stars (middle), and the path of the north celestial pole among the stars due to the precession. Vega is the bright star near the bottom (right).

Earth precession
Equinox path
Precession N

Apsidal precession

Precessing Kepler orbit 280frames e0.6 smaller
Apsidal precession—the orbit rotates gradually over time.

The orbits of planets around the Sun do not really follow an identical ellipse each time, but actually trace out a flower-petal shape because the major axis of each planet's elliptical orbit also precesses within its orbital plane, partly in response to perturbations in the form of the changing gravitational forces exerted by other planets. This is called perihelion precession or apsidal precession.

In the adjunct image, Earth's apsidal precession is illustrated. As the Earth travels around the Sun, its elliptical orbit rotates gradually over time. The eccentricity of its ellipse and the precession rate of its orbit are exaggerated for visualization. Most orbits in the Solar System have a much smaller eccentricity and precess at a much slower rate, making them nearly circular and stationary.

Discrepancies between the observed perihelion precession rate of the planet Mercury and that predicted by classical mechanics were prominent among the forms of experimental evidence leading to the acceptance of Einstein's Theory of Relativity (in particular, his General Theory of Relativity), which accurately predicted the anomalies.[8][9] Deviating from Newton's law, Einstein's theory of gravitation predicts an extra term of A/r4, which accurately gives the observed excess turning rate of 43″ every 100 years.

The gravitational force between the Sun and moon induces the precession in Earth's orbit, which is the major cause of the climate oscillation of Earth that has a period of 19,000 to 23,000 years. It follows that changes in Earth's orbital parameters (e.g., orbital inclination, the angle between Earth's rotation axis and its plane of orbit) is important to the study of Earth's climate, in particular to the study of past ice ages.

Nodal precession

Orbital nodes also precess over time.

See also


  1. ^ Schaub, Hanspeter (2003), Analytical Mechanics of Space Systems, AIAA, pp. 149–150, ISBN 9781600860270, retrieved 1 May 2014
  2. ^ Boal, David (2001). "Lecture 26 – Torque-free rotation – body-fixed axes" (PDF). Retrieved 2008-09-17.
  3. ^ Teodorescu, Petre P (2002). Mechanical Systems, Classical Models. Springer. p. 420.
  4. ^ Barbieri, Cesare (2007). Fundamentals of Astronomy. New York: Taylor and Francis Group. p. 71. ISBN 978-0-7503-0886-1.
  5. ^ Swerdlow, Noel (1991). On the cosmical mysteries of Mithras. Classical Philology, 86, (1991), 48-63. p. 59.
  6. ^ Sun, Kwok. (2017). Our Place in the Universe: Understanding Fundamental Astronomy from Ancient Discoveries, second edition. Cham, Switzerland: Springer. ISBN 978-3-319-54171-6, p. 120; see also Needham, Joseph; Wang, Ling. (1995) [1959]. Science and Civilization in China: Mathematics and the Sciences of the Heavens and the Earth, vol. 3, reprint edition. Cambridge: Cambridge University Press. ISBN 0-521-05801-5, p. 220.
  7. ^ Bradt, Hale (2007). Astronomy Methods. Cambridge University Press. p. 66. ISBN 978 0 521 53551 9.
  8. ^ Max Born (1924), Einstein's Theory of Relativity (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and Earth.)
  9. ^ An even larger value for a precession has been found, for a black hole in orbit around a much more massive black hole, amounting to 39 degrees each orbit.

External links

Apsidal precession

In celestial mechanics, apsidal precession is the precession (gradual rotation) of the line connecting the apsides (line of apsides) of an astronomical body's orbit. The apsides are the orbital points closest (periapsis) and farthest (apoapsis) from its primary body. The apsidal precession is the first derivative of the argument of periapsis, one of the six main orbital elements of an orbit. Apsidal precession is considered positive when the orbit's axis rotates in the same direction as the orbital motion. An apsidal period is the time interval required for an orbit to precess through 360°.

Astrological age

An astrological age is a time period in astrologic theology which astrologers claim parallels major changes in the development of Earth's inhabitants, particularly relating to culture, society, and politics. There are twelve astrological ages corresponding to the twelve zodiacal signs in western astrology. Advocates believe that when one cycle of the twelve astrological ages is completed, another cycle of twelve ages begins. The length of one cycle of twelve ages is 25,860 years.Some astrologers believe that during a given age, some events are directly caused or indirectly influenced by the astrological sign associated with that age, while other astrologers believe the different astrological ages do not influence events in any way.Astrologers do not agree upon exact dates for the beginning or ending of the ages, with given dates varying hundreds of years.

Axial precession

In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In particular, it can refer to the gradual shift in the orientation of Earth's axis of rotation in a cycle of approximately 25,772 years. This is similar to the precession of a spinning-top, with the axis tracing out a pair of cones joined at their apices. The term "precession" typically refers only to this largest part of the motion; other changes in the alignment of Earth's axis—nutation and polar motion—are much smaller in magnitude.

Earth's precession was historically called the precession of the equinoxes, because the equinoxes moved westward along the ecliptic relative to the fixed stars, opposite to the yearly motion of the Sun along the ecliptic. This term is still used in non-technical discussions, that is, when detailed mathematics are absent. Historically,

the discovery of the precession of the equinoxes is usually attributed in the west to the Hellenistic-era (second-century BCE) astronomer Hipparchus, although there are claims of its earlier discovery, such as in the Indian text, Vedanga Jyotisha, dating from 700 BC.

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. In addition to the Moon and Sun, the other planets also cause a small movement of Earth's axis in inertial space, making the contrast in the terms lunisolar versus planetary misleading, so in 2006 the International Astronomical Union recommended that the dominant component be renamed the precession of the equator, and the minor component be renamed precession of the ecliptic, but their combination is still named general precession. Many references to the old terms exist in publications predating the change.

Axial tilt

In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, or, equivalently, the angle between its equatorial plane and orbital plane. It differs from orbital inclination.

At an obliquity of 0 degrees, the two axes point in the same direction; i.e., the rotational axis is perpendicular to the orbital plane. Earth's obliquity oscillates between 22.1 and 24.5 degrees on a 41,000-year cycle; Earth's mean obliquity is currently 23°26′12.4″ (or 23.43678°) and decreasing.

Over the course of an orbital period, the obliquity usually does not change considerably, and the orientation of the axis remains the same relative to the background of stars. This causes one pole to be directed more toward the Sun on one side of the orbit, and the other pole on the other side—the cause of the seasons on Earth.


In astronomy, declination (abbreviated dec; symbol δ) is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of the celestial equator, along the hour circle passing through the point in question.

The root of the word declination (Latin, declinatio) means "a bending away" or "a bending down". It comes from the same root as the words incline ("bend toward") and recline ("bend backward").In some 18th and 19th century astronomical texts, declination is given as North Pole Distance (N.P.D.), which is equivalent to 90 - (declination). For instance an object marked as declination -5 would have a NPD of 95, and a declination of -90 (the south celestial pole) would have a NPD of 180.

Equinox (celestial coordinates)

In astronomy, equinox is either of two places on the celestial sphere at which the ecliptic intersects the celestial equator. Although there are two intersections of the ecliptic with the celestial equator, by convention the equinox associated with the sun's ascending node is used as the origin of celestial coordinate systems and referred to simply as the equinox. In contrast to the common usage of spring and fall, or vernal and autumnal, equinoxes, the celestial coordinate system equinox is a direction in space rather than a moment in time.

The equinox moves because of perturbing forces, therefore in order to define a coordinate system it is necessary to specify the date for which the equinox is chosen. This date should not be confused with the epoch. Astronomical objects show real movements such as orbital and proper motions, and the epoch defines the date for which the position of an object applies. Therefore, a complete specification of the coordinates for an astronomical objects requires both the date of the equinox and of the epoch.The currently used standard equinox and epoch is J2000.0, which is January 1, 2000 at 12:00 TT. The prefix "J" indicates that it is a Julian epoch. The previous standard equinox and epoch was B1950.0, with the prefix "B" indicating it was a Besselian epoch. Before 1984 Besselian equinoxes and epochs were used. Since that time Julian equinoxes and epochs have been used.

Foucault pendulum

The Foucault pendulum (English: foo-KOH; French pronunciation: ​[fuˈko]) or Foucault's pendulum is a simple device named after French physicist Léon Foucault and conceived as an experiment to demonstrate the Earth's rotation. The pendulum was introduced in 1851 and was the first experiment to give simple, direct evidence of the earth's rotation. Today, Foucault pendulums are popular displays in science museums and universities.

Geodetic effect

The geodetic effect (also known as geodetic precession, de Sitter precession or de Sitter effect) represents the effect of the curvature of spacetime, predicted by general relativity, on a vector carried along with an orbiting body. For example, the vector could be the angular momentum of a gyroscope orbiting the Earth, as carried out by the Gravity Probe B experiment. The geodetic effect was first predicted by Willem de Sitter in 1916, who provided relativistic corrections to the Earth–Moon system's motion. De Sitter's work was extended in 1918 by Jan Schouten and in 1920 by Adriaan Fokker. It can also be applied to a particular secular precession of astronomical orbits, equivalent to the rotation of the Laplace–Runge–Lenz vector.The term geodetic effect has two slightly different meanings as the moving body may be spinning or non-spinning. Non-spinning bodies move in geodesics, whereas spinning bodies move in slightly different orbits.The difference between de Sitter precession and Lense–Thirring precession (frame dragging) is that the de Sitter effect is due simply to the presence of a central mass, whereas Lense–Thirring precession is due to the rotation of the central mass. The total precession is calculated by combining the de Sitter precession with the Lense–Thirring precession.

Incremental dating

Incremental dating techniques allow the construction of year-by-year annual chronologies, which can be temporally fixed (i.e., linked to the present day and thus calendar or sidereal time) or floating.

Archaeologists use tree-ring dating (dendrochronology) to determine the age of old pieces of wood. Trees usually add growth rings on a yearly basis, with the spacing of rings being wider in high growth years and narrower in low growth years. Patterns in tree-ring growth can be used to establish the age of old wood samples, and also give some hints to local climatic conditions. This technique is useful to about 9,000 years ago for samples from the western United States using overlapping tree-ring series from living and dead wood.

The Earth's orbital motions (inclination of the earth's axis on its orbit with respect to the sun, gyroscopic precession of the earth's axis every 26,000 years; free precession every 440 days, precession of earth orbit and orbital variations such as perihelion precession every 19,000 and 23,000 years) leave traces visible in the geological record. These changes provide a long-term sequence of climatic events, recorded as changes in the thickness of sediment layers (known as "varve analysis"—the term "varve" means a layer or layers of sediment. Typically, varve refers to lake or glacial sediment), as temperature induced changes in the isotopic ratios for oxygen isotopes in sediments, and in the relative abundance of fossils. Because these can be calibrated reliably over a period of 40 million years this provides an alternate verification to radiometric dating in cases where sufficient record exists to provide a reliable trace.Polarity reversals in the Earth's magnetic field have also been used to determine geologic time. Periodically, the magnetic field of the earth reverses leaving a magnetic signal in volcanic and sedimentary rocks. This signal can be detected and sequences recorded, and in the case of volcanic rocks, tied to radiometric dates.

Another technique used by archaeologists is to inspect the depth of penetration of water vapor into chipped obsidian (volcanic glass) artifacts. The water vapor creates a "hydration rind" in the obsidian, and so this approach is known as "hydration dating" or "obsidian dating", and is useful for determining dates as far back as 200,000 years.

Larmor precession

In physics, Larmor precession (named after Joseph Larmor) is the precession of the magnetic moment of an object about an external magnetic field. Objects with a magnetic moment also have angular momentum and effective internal electric current proportional to their angular momentum; these include electrons, protons, other fermions, many atomic and nuclear systems, as well as classical macroscopic systems. The external magnetic field exerts a torque on the magnetic moment,


where is the torque, is the magnetic dipole moment, is the angular momentum vector, is the external magnetic field, symbolizes the cross product, and is the gyromagnetic ratio which gives the proportionality constant between the magnetic moment and the angular momentum. The phenomenon is similar to the precession of a tilted classical gyroscope in an external torque-exerting gravitational field.

Lense–Thirring precession

In general relativity, Lense–Thirring precession or the Lense–Thirring effect (named after Josef Lense and Hans Thirring) is a relativistic correction to the precession of a gyroscope near a large rotating mass such as the Earth. It is a gravitomagnetic frame-dragging effect. It is a prediction of general relativity consisting of secular precessions of the longitude of the ascending node and the argument of pericenter of a test particle freely orbiting a central spinning mass endowed with angular momentum .

The difference between de Sitter precession and the Lense–Thirring effect is that the de Sitter effect is due simply to the presence of a central mass, whereas the Lense–Thirring effect is due to the rotation of the central mass. The total precession is calculated by combining the de Sitter precession with the Lense–Thirring precession.

According to a recent historical analysis by Pfister, the effect should be renamed as Einstein–Thirring–Lense effect.

Lunar precession

Precession is the change in orientation of a rotational axis with respect to a reference plane. The orbit of the Moon undergoes two important types of precessional motion: apsidal and nodal. The axis of the Moon also experiences precession.

Milankovitch cycles

Milankovitch cycles describe the collective effects of changes in the Earth's movements on its climate over thousands of years. The term is named for Serbian geophysicist and astronomer Milutin Milanković. In the 1920s, he hypothesized that variations in eccentricity, axial tilt, and precession of the Earth's orbit resulted in cyclical variation in the solar radiation reaching the Earth, and that this orbital forcing strongly influenced climatic patterns on Earth.

Similar astronomical hypotheses had been advanced in the 19th century by Joseph Adhemar, James Croll and others, but verification was difficult because there was no reliably dated evidence, and because it was unclear which periods were important.

Now, materials on Earth that have been unchanged for millennia (obtained via ice, rock, and deep ocean cores) are being studied to indicate the history of Earth's climate. Though they are consistent with the Milankovitch hypothesis, there are still several observations that the hypothesis does not explain.

Pole star

A pole star or polar star is a star, preferably bright, closely aligned to the axis of rotation of an astronomical object.

With regard to planet Earth, the pole star refers to Polaris (Alpha Ursae Minoris), a magnitude 2 star aligned approximately with its northern axis, and a pre-eminent star in celestial navigation.

Tests of general relativity

Tests of general relativity serve to establish observational evidence for the theory of general relativity. The first three tests, proposed by Einstein in 1915, concerned the "anomalous" precession of the perihelion of Mercury, the bending of light in gravitational fields, and the gravitational redshift. The precession of Mercury was already known; experiments showing light bending in line with the predictions of general relativity was found in 1919, with increasing precision measurements done in subsequent tests, and astrophysical measurement of the gravitational redshift was claimed to be measured in 1925, although measurements sensitive enough to actually confirm the theory were not done until 1954. A program of more accurate tests starting in 1959 tested the various predictions of general relativity with a further degree of accuracy in the weak gravitational field limit, severely limiting possible deviations from the theory.

In the 1970s, additional tests began to be made, starting with Irwin Shapiro's measurement of the relativistic time delay in radar signal travel time near the sun. Beginning in 1974, Hulse, Taylor and others have studied the behaviour of binary pulsars experiencing much stronger gravitational fields than those found in the Solar System. Both in the weak field limit (as in the Solar System) and with the stronger fields present in systems of binary pulsars the predictions of general relativity have been extremely well tested locally.

In February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from a black hole merger. This discovery, along with additional detections announced in June 2016 and June 2017, tested general relativity in the very strong field limit, observing to date no deviations from theory.

Thomas precession

In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion.

For a given inertial frame, if a second frame is Lorentz-boosted relative to it, and a third boosted relative to the second, but non-colinear with the first boost, then the Lorentz transformation between the first and third frames involves a combined boost and rotation, known as the "Wigner rotation" or "Thomas rotation". For accelerated motion, the accelerated frame has an inertial frame at every instant. Two boosts a small time interval (as measured in the lab frame) apart leads to a Wigner rotation after the second boost. In the limit the time interval tends to zero, the accelerated frame will rotate at every instant, so the accelerated frame rotates with an angular velocity.

The precession can be understood geometrically as a consequence of the fact that the space of velocities in relativity is hyperbolic, and so parallel transport of a vector (the gyroscope's angular velocity) around a circle (its linear velocity) leaves it pointing in a different direction, or understood algebraically as being a result of the non-commutativity of Lorentz transformations. Thomas precession gives a correction to the spin–orbit interaction in quantum mechanics, which takes into account the relativistic time dilation between the electron and the nucleus of an atom.

Thomas precession is a kinematic effect in the flat spacetime of special relativity. In the curved spacetime of general relativity, Thomas precession combines with a geometric effect to produce de Sitter precession. Although Thomas precession (net rotation after a trajectory that returns to its initial velocity) is a purely kinematic effect, it only occurs in curvilinear motion and therefore cannot be observed independently of some external force causing the curvilinear motion such as that caused by an electromagnetic field, a gravitational field or a mechanical force, so Thomas precession is usually accompanied by dynamical effects.If the system experiences no external torque, e.g., in external scalar fields, its spin dynamics is determined only by the Thomas precession. A single discrete Thomas rotation (as opposed to the series of infinitesimal rotations that add up to the Thomas precession) is present in situations anytime there are three or more inertial frames in non-collinear motion, as can be seen using Lorentz transformations.

Trepidation (astronomy)

Trepidation (from Lat. trepidus, "trepidatious"), in now-obsolete medieval theories of astronomy, refers to hypothetical oscillation in the precession of the equinoxes. The theory was popular from the 9th to the 16th centuries.

The origin of the theory of trepidation comes from the Small Commentary to the Handy Tables written by Theon of Alexandria in the 4th century CE. In precession, the equinoxes appear to move slowly through the ecliptic, completing a revolution in approximately 25,800 years (according to modern astronomers). Theon states that certain (unnamed) ancient astrologers believed that the precession, rather than being a steady unending motion, instead reverses direction every 640 years. The equinoxes, in this theory, move through the ecliptic at the rate of 1 degree in 80 years over a span of 8 degrees, after which they suddenly reverse direction and travel back over the same 8 degrees. Theon describes but did not endorse this theory.

A more sophisticated version of this theory was adopted in the 9th century to explain a variation which Islamic astronomers incorrectly believed was affecting the rate of precession. This version of trepidation is described in De motu octavae sphaerae (On the Motion of the Eighth Sphere), a Latin translation of a lost Arabic original. The book is attributed to the Arab astronomer Thābit ibn Qurra, but this model has also been attributed to Ibn al-Adami and to Thabit's grandson, Ibrahim ibn Sinan. In this trepidation model, the oscillation is added to the equinoxes as they precess. The oscillation occurred over a period of 7000 years, added to the eighth (or ninth) sphere of the Ptolemaic system. "Thabit's" trepidation model was used in the Alfonsine Tables, which assigned a period of 49,000 years to precession. This version of trepidation dominated Latin astronomy in the later Middle Ages.

Islamic astronomers described other models of trepidation. In the West, an alternative to De motu octavae sphaerae was part of the theory of the motion of the Earth published by Nicolaus Copernicus in De revolutionibus orbium coelestium (1543). Copernicus' version of trepidation combined the oscillation of the equinoxes (now known to be a spurious motion) with a change in the obliquity of the ecliptic (axial tilt), acknowledged today as an authentic motion of the Earth's axis.

Trepidation was the mainstay of Hindu astronomy and was used to compute ayanamsha for converting sidereal to tropical longitudes. The third chapter of the Suryasiddhanta, verses 9-10, provides the method for computing it, which E. Burgess interprets as 27 degree trepidation in either direction over a full period of 7200 years, at an annual rate of 54 seconds. This is nearly the same as the Arab period of about 7000 years. The zero date according to the Suryasiddhanta was 499 AD, after which trepidation is forward in the same direction as modern equinoctial precession. For the period before 1301 BCE, Suryasiddhantic trepidation would be opposite in sign to equinoctial precession. For the period 1301 BCE to 2299 AD, equinoctial precession and Suryasiddhantic precession would have the same direction and sign, only differing in magnitude. Brahma Siddhanta, Soma Siddhanta and Narada Purana describe exactly the same theory and magnitude of trepidation as in Suryasiddhanta, and some other Puranas also provide concise references to precession, esp Vayu purana and Matsya Purana.

True north

True north (also called geodetic north) is the direction along Earth's surface towards the geographic North Pole or True North Pole.

Geodetic north differs from magnetic north (the direction a compass points toward the Magnetic North Pole), and from grid north (the direction northwards along the grid lines of a map projection). Geodetic true north also differs very slightly from astronomical true north (typically by a few arcseconds) because the local gravity may not point at the exact rotational axis of Earth.

The direction of astronomical true north is marked in the skies by the north celestial pole. This is within about 1° of the position of Polaris, so that the star would appear to trace a tiny circle in the sky each sidereal day. Due to the axial precession of Earth, true north rotates in an arc with respect to the stars that takes approximately 25,000 years to complete. Around 2100–02, Polaris will make its closest approach to the celestial north pole (extrapolated from recent Earth precession). The visible star nearest the north celestial pole 5,000 years ago was Thuban.On maps published by the United States Geological Survey (USGS) and the United States Armed Forces, true north is marked with a line terminating in a five-pointed star. The east and west edges of the USGS topographic quadrangle maps of the United States are meridians of longitude, thus indicating true north (so they are not exactly parallel). Maps issued by the United Kingdom Ordnance Survey contain a diagram showing the difference between true north, grid north, and magnetic north at a point on the sheet; the edges of the map are likely to follow grid directions rather than true, and the map will thus be truly rectangular/square.


The zodiac is an area of the sky that extends approximately 8° north or south (as measured in celestial latitude) of the ecliptic, the apparent path of the Sun across the celestial sphere over the course of the year. The paths of the Moon and visible planets are also within the belt of the zodiac.In Western astrology, and formerly astronomy, the zodiac is divided into twelve signs, each occupying 30° of celestial longitude and roughly corresponding to the constellations Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius and Pisces.The twelve astrological signs form a celestial coordinate system, or more specifically an ecliptic coordinate system, which takes the ecliptic as the origin of latitude and the Sun's position at vernal equinox as the origin of longitude.

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