 Differential rotation

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 shearing. Galaxies and protostars usually show differential rotation; and examples in our solar system include the Sun, Jupiter and Saturn.
Contents
History
Around the year 1610, Galileo Galilei 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.
The cause of differential rotation
Because of the prestellar accretion phase, and the conservation of angular momentum, 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 winds. 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 measurements of solar “pmodes” 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 v_{rot}sini, where i is the angle between the line of sight and the rotation axis, permitting the study of the rotational velocity’s lineofsight component vrot. This is calculated from Fourier transforms of the line shapes, using equation (2) below for v_{rot} 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 largescale magnetic field, through magnetohydrodynamical (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 absorptionline spectra through line broadening caused by lines being differently Dopplershifted across the stellar surface.
Solar differential rotation causes shear at the socalled tachocline. This is a region where rotation changes from differential in the convection zone to nearly solidbody 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:
Ω = Ω_{0} − ΔΩsin^{2}Ψ
(1)
where Ω_{0} is the rotation rate at the equator, and ΔΩ = (Ω_{0} − Ω_{pole}) is the difference in angular velocity between pole and equator, called the strength of the rotational shear. Ψ 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 Dopplershifted absorption lines), can be approximated as:
(3)
where θ is the colatitude (measured from the poles).
Differential rotation of the Sun
See also: Solar rotationOn 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 meanfield models or largeeddy simulations according to the approximations.
See also
 Carrington rotation
 Giovanni Cassini
 Solar nebula
 Stellar rotation
 Sunspot
Further reading
 Annu. Rev. Astron. Astrophys. 2003. 41:559643 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.unigoettingen.de/~areiners/DiffRot/interactive.htm A simulation of the effects of differential rotation on stellar absorptionline 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, 155162 (2002) 
Categories: Astronomical objects
 Astrophysics
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