
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
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, 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
Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
, known as the
precession of the equinoxes
In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's Rotation around a fixed axis, rotational axis. In the absence of precession, the astronomical body's orbit would show ...
.
Torque-free or torque neglected
Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the
angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
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. , , ). 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:
where is the precession rate, is the spin rate about the axis of symmetry, is the moment of inertia about the axis of symmetry, is moment of inertia about either of the other two equal perpendicular principal axes, and is the angle between the moment of inertia direction and the symmetry axis.
When an object is not perfectly
rigid, inelastic dissipation 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 that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internal
moment of inertia tensor and fixed external angular momentum , the instantaneous angular velocity is
Precession occurs by repeatedly recalculating and applying a small
rotation vector
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
for the short time ; e.g.:
for the
skew-symmetric matrix . 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 perpendicular to both and , noting that
Torque-induced
Torque-induced precession (gyroscopic precession) is the phenomenon in which the
axis of a spinning object (e.g., a
gyroscope
A gyroscope (from Ancient Greek γῦρος ''gŷros'', "round" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining Orientation (geometry), orientation and angular velocity. It is a spinning wheel or disc in ...
) 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
In kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a non-negative scalar quantity. Intro ...
of the rotation and the
magnitude of the external torque are constant, the spin axis will move at
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s 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
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...
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.
The device depicted on the right 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 . At the depicted moment in time, section is at the
perimeter
A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimet ...
of the rotating motion around the (vertical) pivot axis. Section , therefore, has a lot of angular rotating
velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
with respect to the rotation around the pivot axis, and as 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, 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.
Section 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 . 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
A bicycle, also called a pedal cycle, bike, push-bike or cycle, is a human-powered transport, human-powered or motorized bicycle, motor-assisted, bicycle pedal, pedal-driven, single-track vehicle, with two bicycle wheel, wheels attached to a ...
performance at high speed. Precession is also the mechanism behind
gyrocompasses.
Classical (Newtonian)

Precession is the change of
angular velocity and
angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
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 the
moment of inertia, is the angular velocity of spin about the spin axis, is the mass, is the acceleration due to gravity, is the angle between the spin axis and the axis of precession and is the distance between the center of mass and the pivot. The torque vector originates at the center of mass. Using , we find that the
period of precession is given by:
Where is the
moment of inertia, is the period of spin about the spin axis, and is the
torque. In general, the problem is more complicated than this, however.
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 an object (such as a gyroscope) being accelerated along a curved path.
*
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.
The
Schwarzschild geodesics (sometimes Schwarzschild precession) is used in the prediction of the
anomalous perihelion precession of the planets, most notably for the accurate prediction of the
apsidal precession of Mercury.
Astronomy
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
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
of its orbit over tens of thousands of years are all important parts of the astronomical theory of
ice age
An ice age is a long period of reduction in the temperature of Earth's surface and atmosphere, resulting in the presence or expansion of continental and polar ice sheets and alpine glaciers. Earth's climate alternates between ice ages, and g ...
s. ''(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
Polaris is a star in the northern circumpolar constellation of Ursa Minor. It is designated α Ursae Minoris (Latinisation of names, Latinized to ''Alpha Ursae Minoris'') and is commonly called the North Star or Pole Star. With an ...
, in a circle around the
ecliptic pole, with an angular radius of about 23.5°.
The
ancient Greek astronomer Hipparchus
Hipparchus (; , ; BC) was a Ancient Greek astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hippar ...
(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°), although there is some minor dispute about whether he was. In
ancient China
The history of China spans several millennia across a wide geographical area. Each region now considered part of the Chinese world has experienced periods of unity, fracture, prosperity, and strife. Chinese civilization first emerged in the Y ...
, the
Jin-dynasty scholar-official
Yu Xi ( 307–345 AD) made a similar discovery centuries later, noting that the position of the Sun during the
winter solstice
The winter solstice, or hibernal solstice, occurs when either of Earth's geographical pole, poles reaches its maximum axial tilt, tilt away from the Sun. This happens twice yearly, once in each hemisphere (Northern Hemisphere, Northern and So ...
had drifted roughly one degree over the course of fifty years relative to the position of the stars. 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
The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
and
Sun apply torque to the equator, attempting to pull the
equatorial bulge into the plane of the
ecliptic
The ecliptic or ecliptic plane is the orbital plane of Earth's orbit, Earth around the Sun. It was a central concept in a number of ancient sciences, providing the framework for key measurements in astronomy, astrology and calendar-making.
Fr ...
, but instead causing it to precess. The torque exerted by the planets, particularly
Jupiter
Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...
, also plays a role.
Apsidal precession

The
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
s 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 nearly stationary.
Discrepancies between the observed perihelion precession rate of the planet
Mercury and that predicted by
classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
were prominent among the forms of experimental evidence leading to the acceptance of
Einstein's
Theory of Relativity
The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical ph ...
(in particular, his
General Theory of Relativity), which accurately predicted the anomalies.
Deviating from Newton's law, Einstein's theory of gravitation predicts an extra term of , which accurately gives the observed excess turning rate of 43
arcseconds every 100 years.
Nodal precession
Orbital nodes also
precess over time.
See also
*
Larmor precession
*
Nutation
*
Polar motion
Polar motion of the Earth is the motion of the Earth's rotation, Earth's rotational axis relative to its Earth's crust, crust. This is measured with respect to a reference frame in which the solid Earth is fixed (a so-called ''Earth-centered, Ea ...
*
Precession (mechanical)
*
Precession as a form of parallel transport
References
External links
*
Explanation and derivation of formula for precession of a top
{{Authority control
Earth
Dynamics (mechanics)