# Differential rotation in stars

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Differential rotation in stars

What is differential rotation?

*Differential rotation is when a rotating body has different angular velocities at different latitudes and/or depths of the body and/or in time. Differential rotation can be applied to any type of fluid body such as gaseous planets, stars and galaxies.

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 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.

How to measure 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 $v_\left\{rot\right\}sini$, 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 $v_\left\{rot\right\}$ 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.

An example of measured stellar differential rotation:

*Figure 1 shows images of the active star AB Dor. Part of the rotation cycle was observed continuously, showing brightness features close to the equator systematically shifting eastward between the first and second epoch.
*By cross-correlating these images a map is obtained, showing rotational shear as a function of latitude, i.e. the differential rotation.

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: "$Omega=Omega_\left\{0\right\}-DeltaOmega sin^\left\{2\right\}Psi$"(1)

where $Omega_\left\{0\right\}$ is the rotation rate at the equator, and $DeltaOmega=\left(Omega_\left\{0\right\}-Omega_\left\{pole\right\}\right)$ is the difference in angular velocuty between pole and equator, called the strength of the rotational shear. $Psi$ is the heliographic latitude, measured from the equator.
*The reciprocal of the rotational shear "$frac\left\{2pi\right\}\left\{DeltaOmega\right\}$" 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: $alpha=frac\left\{DeltaOmega\right\}\left\{Omega_\left\{0$(2)

*The Doppler rotation rate in the Sun (measured from Doppler-shifted absorption lines), can be approximated as: $frac\left\{Omega\right\}\left\{2pi\right\}\left(451.5-65.3cos^\left\{2\right\}Theta - 66.7cos^\left\{4\right\}Theta\right)nHz$(3)

where θ is the co-latitude (measured from the poles).

Differential rotation of the Sun:

*On the Sun (figure 2), 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 (figure 3), as measured relative to distant stars (sidereal rotation).

Simulations of differential 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.

Figures and Plots:

For Figure 1 click [http://www.ast.obs-mip.fr/users/donati/images/diffrot.jpg|here] .

Figure 1: AB Dor. The second half of the rotation cycle (left hand side of both the upper images) was observed continuously at both epochs (vertical ticks above each image are phases). The figures are showing brightness features close to the equator systematically shifting eastward between the first and second epoch.By cross-correlating these images a map is obtained, showing rotational shear as a function of latitude, i.e. the differential rotation.See example link for reference.

For Figure 2 click [http://www.ast.obs-mip.fr/users/donati/images/sunrot.gif|here] .

Figure 2: The Sun where red represents faster, and blue slower rotation, depicting differential rotation.See picture links for reference.

For figure 3 click [http://www.lcsd.gov.hk/CE/Museum/Space/EducationResource/Universe/framed_e/lecture/ch11/imgs/rotation.gif| here] .

Figure 3: The Sun with different angular velocities at different latitudes.See picture links for reference.

Curtesy of GONG: http://gong.nso.edu/.

Rotation Profiles:Time-averaged rotation rates, plotted as a function of radius at different latitudes within the Sun. (

For Plot 2 click [http://www.aanda.org/articles/aa/full/2002/10/aa1929/Timg85.gif|here] , or go [http://www.aanda.org/index.php?option=article&access=standard&Itemid=129&url=/articles/aa/full/2002/10/aa1929/aa1929.right.html |here] and look at "Figure 3".

Doppler measurements and Fourier transforms:The left figure shows four cases of differential rotation, each case is an absorption line profile of light spectra of a star with limb darkening $epsilon = 0.6$ and "i"=90°. The profiles are due to Doppler-shift in the line spectra. The right figure shows the Fourier transformed versions for the profiles in the right figure. The first sidelobe (sidelobes = the arcs, in this case >0.05) narrows for larger differential rotation, while its amplitude lessens. The amplitude of the second sidelobe changes slightly.

Wikipedia:
*Galileo Galilei
*Christoph Scheiner
*Sunspots
*Carrington rotation
*Stellar rotation
*The sun

A simulation of the effects of differential rotation on stellar absorption-line profiles by Ansgar Reiners:
* [http://www.astro.physik.uni-goettingen.de/~areiners/DiffRot/interactive.htm |http://www.astro.physik.uni-goettingen.de/~areiners/DiffRot/interactive.htm]

A more technical discussion of signatures in stellar line profiles:
* [http://www.aanda.org/index.php?option=article&access=standard&Itemid=129&url=/articles/aa/full/2002/10/aa1929/aa1929.right.html 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) - ]

The tachocline :
* [http://www.gong.noao.edu/ |http://www.gong.noao.edu/]

Example / Figure 2:
* [http://www.ast.obs-mip.fr/users/donati/diffrot.html |http://www.ast.obs-mip.fr/users/donati/diffrot.html]

Figure 3:
* [http://www.lcsd.gov.hk/CE/Museum/Space/EducationResource/Universe/framed_e/lecture/ch11/ch11_cnt.html |http://www.lcsd.gov.hk/CE/Museum/Space/EducationResource/Universe/framed_e/lecture/ch11/ch11_cnt.html]

Articles:Annu. Rev. Astron. Astrophys. 2003. 41:559-643 doi: 10.1146/annurev.astro.41.01152.094848 “The Internal Rotation of the Sun”.

Book:Stellar Photospheres; The Observations and Analysis of: Third Edition. Cambridge University Press, ISBN13: 9780521851862, David F. Gray, chapter 8.

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