Download Differential rotation of the Sun

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; examples in the
Solar System include the Sun, Jupiter and Saturn.
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 pre-stellar 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 "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 vrotsin(i), 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 vrot 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 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.
Calculating differential rotation
For observed sunspots, the differential rotation can be calculated as:
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:
The Doppler rotation rate in the Sun (measured from Doppler-shifted absorption
lines), can be approximated as:
where θ is the co-latitude (measured from the poles).
Differential rotation of the Sun
Internal rotation in the Sun, showing differential rotation in the outer convective region and
almost uniform rotation in the central radiative region.
See also: Solar rotation
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.