Differential rotation
Encyclopedia
Differential rotation is seen when different parts of a rotating object move with different angular velocities
(rates of rotation
) at different latitudes and/or depths of the body and/or in time. This indicates that the object is not solid. In fluid objects, such as accretion disks, this leads to shear
ing. Galaxies
and protostar
s usually show differential rotation; and examples in our solar system
include the Sun
, Jupiter
and Saturn
.
observed sunspots and calculated the rotation of the Sun
. In 1630, Christoph Scheiner
reported that the Sun had different rotational periods at the poles and at the equator, in good agreement with modern values.
, rotation is induced. Differential rotation is caused by convection
in stars. This is movement of mass, due to steep temperature gradients from the core outwards. This mass carries a portion of the star’s angular momentum, thus redistributing the angular velocity, possibly even far enough out for the star to lose angular velocity in stellar wind
s. Differential rotation thus depends on temperature differences in adjacent regions.
By doing helioseismological
measurements of solar “p-modes” it is possible to deduce the differential rotation. The Sun has very many acoustic modes that oscillate in the interior simultaneously, and the inversion of their frequencies can yield the rotation of the solar interior. This varies with both depth and (especially) latitude.
The broadened shapes of absorption lines in the optical spectrum depend on
, where i is the angle between the line of sight and the rotation axis, permitting the study of the rotational velocity’s line-of-sight component vrot. This is calculated from Fourier transforms of the line shapes, using equation (2) below for 
at the equator and poles. See also plot 2.
Solar differential rotation is also seen in magnetograms, images showing the strength and location of solar magnetic fields.
The inner differential rotation is one part of the mixing processes in stars, mixing the materials and the heat/energy of the stars.
Differential rotation affects stellar optical absorption-line spectra
through line broadening
caused by lines being differently Doppler-shifted across the stellar surface.
Solar differential rotation causes shear at the so-called tachocline. This is a region where rotation changes from differential in the convection zone to nearly solid-body rotation in the interior, at 0.71 solar radii from the center.
(1)
where
is the rotation rate at the equator, and 
is the difference in angular velocity between pole and equator, called the strength of the rotational shear. 
is the heliographic latitude, measured from the equator.
(2)
(3)
where θ is the co-latitude (measured from the poles).
On the Sun, the study of oscillations revealed that rotation is roughly constant within the whole radiative interior and variable with radius and latitude within the convective envelope. The Sun has an equatorial rotation speed of ~2 km/s; its differential rotation implies that the angular velocity decreases with increased latitude. The poles make one rotation every 34.3 days and the equator every 25.05 days, as measured relative to distant stars (sidereal rotation).
The highly turbulent nature of solar convection and anisotropies induced by rotation complicate the dynamics of modeling. Molecular dissipation scales on the Sun are at least six orders of magnitude smaller than the depth of the convective envelope. A direct numerical simulation of solar convection would have to resolve this entire range of scales in each of the three dimensions. Consequently, all solar differential rotation models must involve some approximations regarding momentum and heat transport by turbulent motions that are not explicitly computed. Thus, modeling approaches can be classified as either mean-field models or large-eddy simulations according to the approximations.
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
(rates of rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
) at different latitudes and/or depths of the body and/or in time. This indicates that the object is not solid. In fluid objects, such as accretion disks, this leads to shear
Shearing (physics)
Shearing in continuum mechanics refers to the occurrence of a shear strain, which is a deformation of a material substance in which parallel internal surfaces slide past one another. It is induced by a shear stress in the material...
ing. Galaxies
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
and protostar
Protostar
A protostar is a large mass that forms by contraction out of the gas of a giant molecular cloud in the interstellar medium. The protostellar phase is an early stage in the process of star formation. For a one solar-mass star it lasts about 100,000 years...
s usually show differential rotation; and examples in our solar system
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
include the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
, Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...
and Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...
.
History
Around the year 1610, Galileo GalileiGalileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...
observed sunspots and calculated the rotation of the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
. In 1630, Christoph Scheiner
Christoph Scheiner
Christoph Scheiner SJ was a Jesuit priest, physicist and astronomer in Ingolstadt....
reported that the Sun had different rotational periods at the poles and at the equator, in good agreement with modern values.
The cause of differential rotation
Because of the pre-stellar accretion phase, and the conservation of angular momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
, rotation is induced. Differential rotation is caused by convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....
in stars. This is movement of mass, due to steep temperature gradients from the core outwards. This mass carries a portion of the star’s angular momentum, thus redistributing the angular velocity, possibly even far enough out for the star to lose angular velocity in stellar wind
Stellar wind
A stellar wind is a flow of neutral or charged gas ejected from the upper atmosphere of a star. It is distinguished from the bipolar outflows characteristic of young stars by being less collimated, although stellar winds are not generally spherically symmetric.Different types of stars have...
s. Differential rotation thus depends on temperature differences in adjacent regions.
Measuring differential rotation
There are many ways to measure and calculate differential rotation in stars to see if different latitudes have different angular velocities. The most obvious being tracking spots on the stellar surface.By doing helioseismological
Helioseismology
Helioseismology is the study of the propagation of wave oscillations, particularly acoustic pressure waves, in the Sun. Unlike seismic waves on Earth, solar waves have practically no shear component . Solar pressure waves are believed to be generated by the turbulence in the convection zone near...
measurements of solar “p-modes” it is possible to deduce the differential rotation. The Sun has very many acoustic modes that oscillate in the interior simultaneously, and the inversion of their frequencies can yield the rotation of the solar interior. This varies with both depth and (especially) latitude.
The broadened shapes of absorption lines in the optical spectrum depend on
, where i is the angle between the line of sight and the rotation axis, permitting the study of the rotational velocity’s line-of-sight component vrot. This is calculated from Fourier transforms of the line shapes, using equation (2) below for at the equator and poles. See also plot 2.Solar differential rotation is also seen in magnetograms, images showing the strength and location of solar magnetic fields.
Effects of differential rotation
Gradients in angular rotation caused by angular momentum redistribution within the convective layers of a star are expected to be a main driver for generating the large-scale magnetic field, through magneto-hydrodynamical (dynamo) mechanisms in the outer envelopes. The interface between these two regions is where angular rotation gradients are strongest and thus where dynamo processes are expected to be most efficient.The inner differential rotation is one part of the mixing processes in stars, mixing the materials and the heat/energy of the stars.
Differential rotation affects stellar optical absorption-line spectra
Line Spectra
LINE SPECTRA is a Canadian indie rock band from Montreal. The band is Diana Lalla on guitars and bass pedalboard, Vanessa Morelli on vocals, keyboard and synthesizer and Kathy Perlini on drums, glockenspiel and percussion.-History:...
through line broadening
Doppler broadening
In atomic physics, Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting particles result in different shifts, the cumulative effect of which is the line broadening.The...
caused by lines being differently Doppler-shifted across the stellar surface.
Solar differential rotation causes shear at the so-called tachocline. This is a region where rotation changes from differential in the convection zone to nearly solid-body rotation in the interior, at 0.71 solar radii from the center.
Calculating differential rotation
For observed sunspots, the differential rotation can be calculated as:(1)
where
is the rotation rate at the equator, and is the difference in angular velocity between pole and equator, called the strength of the rotational shear. is the heliographic latitude, measured from the equator.
- The reciprocal of the rotational shear
is the lap time, i.e. the time it takes for the equator to do a full lap more than the poles. - The relative differential rotation rate is the ratio of the rotational shear to the equatorial velocity:
(2)
- The Doppler rotation rate in the Sun (measured from Doppler-shifted absorption lines), can be approximated as:
(3)
where θ is the co-latitude (measured from the poles).
Differential rotation of the Sun
On the Sun, the study of oscillations revealed that rotation is roughly constant within the whole radiative interior and variable with radius and latitude within the convective envelope. The Sun has an equatorial rotation speed of ~2 km/s; its differential rotation implies that the angular velocity decreases with increased latitude. The poles make one rotation every 34.3 days and the equator every 25.05 days, as measured relative to distant stars (sidereal rotation).
The highly turbulent nature of solar convection and anisotropies induced by rotation complicate the dynamics of modeling. Molecular dissipation scales on the Sun are at least six orders of magnitude smaller than the depth of the convective envelope. A direct numerical simulation of solar convection would have to resolve this entire range of scales in each of the three dimensions. Consequently, all solar differential rotation models must involve some approximations regarding momentum and heat transport by turbulent motions that are not explicitly computed. Thus, modeling approaches can be classified as either mean-field models or large-eddy simulations according to the approximations.
See also
- Carrington rotationCarrington rotationThe Carrington rotation of the Sun is a system for comparing locations on the Sun over a period of time, allowing the following of sunspot groups or reappearance of eruptions at a later time....
- Giovanni CassiniGiovanni Domenico CassiniThis article is about the Italian-born astronomer. For his French-born great-grandson, see Jean-Dominique Cassini.Giovanni Domenico Cassini was an Italian/French mathematician, astronomer, engineer, and astrologer...
- Solar nebulaSolar nebulaIn cosmogony, the nebular hypothesis is the most widely accepted model explaining the formation and evolution of the Solar System. There is evidence that it was first proposed in 1734 by Emanuel Swedenborg. Originally applied only to our own Solar System, this method of planetary system formation...
- Stellar rotationStellar rotationStellar rotation is the angular motion of a star about its axis. The rate of rotation can be measured from the spectrum of the star, or by timing the movements of active features on the surface....
- SunspotSunspotSunspots are temporary phenomena on the photosphere of the Sun that appear visibly as dark spots compared to surrounding regions. They are caused by intense magnetic activity, which inhibits convection by an effect comparable to the eddy current brake, forming areas of reduced surface temperature....
Further reading
- Annu. Rev. Astron. Astrophys. 2003. 41:559-643 doi: 10.1146/annurev.astro.41.01152.094848 “The Internal Rotation of the Sun”
- David F. Gray, Stellar Photospheres; The Observations and Analysis of: Third Edition, chapter 8, Cambridge University Press, ISBN 9780521851862
External links
- http://www.astro.physik.uni-goettingen.de/~areiners/DiffRot/interactive.htm A simulation of the effects of differential rotation on stellar absorption-line profiles by Ansgar Reiners
- A. Reiners & J. H. M. M. Schmitt: On the feasibility of the detection of differential rotation in stellar absorption profiles, Astronomy & Astrophysics 384, 155-162 (2002) -