# Stellar rotation

Stellar 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.

The rotation of a star produces an equatorial bulge due to centripetal force. As stars are not solid bodies, they can also undergo differential rotation. Thus the equator of the star can rotate at a different angular velocity than the higher latitudes. These differences in the rate of rotation within a star may have a significant role in the generation of a stellar magnetic field. [cite web
last=Donati | first=Jean-François | date=November 5, 2003
url=http://www.ast.obs-mip.fr/users/donati/diffrot.html
title=Differential rotation of stars other than the Sun
publisher=Laboratoire d’Astrophysique de Toulouse
accessdate = 2007-06-24
]

The magnetic field of a star interacts with the stellar wind. As the wind moves away from the star its rate of angular velocity slows. The magnetic field of the star interacts with the wind, which applies a drag to the stellar rotation. As a result, angular momentum is transferred from the star to the wind, and over time this gradually slows the star's rate of rotation.

Measurement

Unless a star is being observed from the direction of its pole, sections of the surface have some amount of movement toward or away from the observer. The component of movement that is in the direction of the observer is called the radial velocity. For the proportion of the surface with a radial velocity component toward the observer, the radiation is shifted to a higher frequency because of Doppler shift. Likewise the region that has a component moving away from the observer is shifted to a lower frequency. When the absorption lines of a star are observed, this shift at each end of the spectrum causes the line to broaden.cite journal
author=Shajn, G.; Struve, O.
title=On the rotation of the stars
journal=Monthly Notices of the Royal Astronomical Society
year=1929 | volume=89 | pages=222–239
accessdate=2007-06-25
] However, this broadening must be carefully separated from other effects that can increase the line width.

The component of the radial velocity observed through line broadening depends on the inclination of the star's pole to the line of sight. The derived value is given as $v_e cdot sin i$, where "ve" is the rotational velocity at the equator and "i" is the inclination. However, "i" is not always known, so the result gives a minimum value for the star's rotational velocity. That is, if "i" is not a right angle, then the actual velocity is greater than $v_e cdot sin i$. This is sometimes referred to as the projected rotational velocity.

For giant stars, the atmospheric microturbulence can result in line broadening that is much larger than effects of rotational, effectively drowning out the signal. However, an alternate approach can be employed that makes use of gravitational microlensing events. These occur when a massive object passes in front of the more distant star and functions like a lens, briefly magnifying the image. The more detailed information gathered by this means allows the effects of microturbulence to be distinguished from rotation. [cite journal
last=Gould | first=Andrew
title=Measuring the Rotation Speed of Giant Stars from Gravitational Microlensing
journal=Astrophysical Journal
year=1997 | volume=483 | pages=98–102
accessdate=2007-06-28
doi=10.1086/304244
]

If a star displays magnetic surface activity such as starspots, then these features can be tracked to estimate the rotation rate. However, such features can form at locations other than equator and can migrate across latitudes over the course of their life span, so differential rotation of a star can produce varying measurements. Stellar magnetic activity is often associated with rapid rotation, so this technique can be used for measurement of such stars. [cite journal
author=Soon, W.; Frick, P.; Baliunas, S.
title=On the rotation of the stars
journal=The Astrophysical Journal
year=1999 | volume=510 | issue=2 | pages=L135–L138
accessdate=2007-06-25
doi=10.1086/311805
] Observation of starspots has shown that these features can actually vary the rotation rate of a star, as the magnetic fields modify the flow of gases in the star. [cite journal
author=Collier Cameron, A.; Donati, J.-F.
title=Doin' the twist: secular changes in the surface differential rotation on AB Doradus
journal=Monthly Notices of the Royal Astronomical Society
year=2002 | volume=329 | issue=1 | pages=L23–L27
accessdate=2007-06-25
doi=10.1046/j.1365-8711.2002.05147.x
]

Physical effects

Equatorial bulge

The rotation of a star produces a net centripetal force in a direction perpendicular to the axis. At the pole the centripetal force is zero, so the gravitational force of the star is unopposed. Around the equator, however, the centripetal force is non-zero and it partly opposes the gravitational force. In order to balance out the difference in net forces at the pole and the equator, the surface of the star bulges outward along the equator and the star forms an oblate spheroid.

An extreme example of an equatorial bulge is found on the star Regulus A (α Leonis A). The equator of this star has a measured rotational velocity of 317 ± 3 km/s. This corresponds to a rotation period of 15.9 hours, which is 86% of the velocity at which the star would break apart. The equatorial radius of this star is 32% larger than polar radius.cite journal
author=McAlister, H. A., ten Brummelaar, T. A., et al.
title=First Results from the CHARA Array. I. An Interferometric and Spectroscopic Study of the Fast Rotator Alpha Leonis (Regulus).
journal=The Astrophysical Journal
year=2005 | volume=628 | pages=439–452
doi=10.1086/430730
] Other rapidly rotating stars include Alpha Arae, Vega and Achernar.

The break-up velocity of a star is an expression that is used to describe the case where the centripetal force at the equator is equal to the gravitational force. For a star to be stable the rotational velocity must be below this value. [cite conference
author=Hardorp, J.; Strittmatter, P. A.
title =Rotation and Evolution of be Stars
booktitle=Proceedings of IAU Colloq. 4
pages=48
publisher=Gordon and Breach Science Publishers
date=September 8-11, 1969
location=Ohio State University, Columbus, Ohio
accessdate = 2007-06-26
]

Differential rotation

Surface differential rotation is observed on stars such as the Sun when the angular velocity varies with latitude. Typically the angular velocity decreases with increasing latitude. However the reverse has also been observed, such as on the star designated HD 31993.) [cite journal
author=Kitchatinov, L. L.; Rüdiger, G.
title=Anti-solar differential rotation
journal=Astronomische Nachrichten
year=2004 | volume=325 | issue=6 | pages=496–500
accessdate=2007-06-27
doi=10.1002/asna.200410297
] [cite journal
author=Ruediger, G.; von Rekowski, B.; Donahue, R. A.; Baliunas, S. L.
title=Differential Rotation and Meridional Flow for Fast-rotating Solar-Type Stars
journal=Astrophysical Journal
year=1998 | volume=494 | issue=2 | pages=691–699
url=http://www.journals.uchicago.edu/doi/full/10.1086/305216
accessdate=2007-06-27
doi=10.1086/305216
] The first such star, other than the Sun, to have its differential rotation mapped in detail is AB Doradus. [cite journal
author=Donati, J.-F.; Collier Cameron, A.
title=Differential rotation and magnetic polarity patterns on AB Doradus
journal=Monthly Notices of the Royal Astronomical Society
year=1997 | volume=291 | issue=1 | pages=1–19
accessdate=2007-07-03
] cite web
last=Donati | first=Jean-François | date=November 5, 2003
url=http://www.ast.obs-mip.fr/users/donati/diffrot.html
title=Differential rotation of stars other than the Sun
publisher=Laboratoire d’Astrophysique de Toulouse
accessdate = 2007-06-24
]

The underlying mechanism that causes differential rotation is turbulent convection inside a star. Convective motion carries energy toward the surface through the mass movement of plasma. This mass of plasma carries a portion of the angular velocity of the star. When turbulence occurs through shear and rotation, the angular momentum can become redistributed to different latitudes through meridional flow. [cite web
last=Korab | first=Holly | date=June 25, 1997
url=http://access.ncsa.uiuc.edu/Stories/97Stories/WOODward.html
title=NCSA Access: 3D Star Simulation
publisher=National Center for Supercomputing Applications
accessdate=2007-06-27
] [cite journal
author=Küker, M.; Rüdiger, G.
title=Differential rotation on the lower main sequence
journal=Astronomische Nachrichten
year=2004 | volume=326 | issue=3 | pages=265–268
accessdate=2007-06-27
doi=10.1002/asna.200410387
]

The interfaces between regions with sharp differences in rotation are believed to be efficient sites for the dynamo processes that generate the stellar magnetic field. There is also a complex interaction between a star's rotation distribution and its magnetic field, with the conversion of magnetic energy into kinetic energy modifying the velocity distribution.

Rotation braking

Stars are believed to form as the result of a collapse of a low-temperature cloud of gas and dust. As the cloud collapses, conservation of angular momentum causes any small net rotation of the cloud to increase, forcing the material into a rotating disk. At the dense center of this disk a protostar forms, which gaining heat from the gravitational energy of the collapse.

As the collapse continues, the rotation rate can increase to the point where the accreting protostar can break up due to centripetal force at the equator. Thus the rotation rate must be braked during the first 100,000 years to avoid this scenario. One possible explanation for the braking is the interaction of the protostar's magnetic field with the stellar wind. The expanding wind carries away the angular momentum and slows down the rotation rate of the collapsing protostar. [cite journal
author=Ferreira, J.; Pelletier, G.; Appl, S.
title=Reconnection X-winds: spin-down of low-mass protostars
journal=Monthly Notices of the Royal Astronomical Society
year=2000 | volume=312 | pages=387–397
accessdate=2007-06-26
doi=10.1046/j.1365-8711.2000.03215.x
] [cite news
first=Terry | last=Devitt
title=What Puts The Brakes On Madly Spinning Stars?
date=January 31, 2001
url=http://www.news.wisc.edu/5732
accessdate=2007-06-27
]

Most main sequence stars with a spectral class between O5 and F5 have been found to rotate rapidly. [cite conference
author=Peterson, Deane M. "et al"
title =Resolving the effects of rotation in early type stars
booktitle=New Frontiers in Stellar Interferometry, Proceedings of SPIE Volume 5491
pages=65
publisher=The International Society for Optical Engineering
year=2004
location=Bellingham, Washington, USA
accessdate=2007-06-25
] For stars in this range, the measured rotation velocity increases with mass. This increase in rotation peaks among young, massive B-class stars. As the expected life span of a star decreases with increasing mass, this can be explained as a decline in rotational velocity with age.

For main sequence stars, the decline in rotation can be approximated by a mathematical relation:

:$v_e propto t^\left\{-frac\left\{1\right\}\left\{2$

where "ve" is the angular velocity at the equator and "t" is the star's age. [cite book
first=Jean-Louis | last=Tassoul | year=1972
title=Stellar Rotation | location=Cambridge, MA
publisher=Cambridge University Press
url=http://assets.cambridge.org/97805217/72181/sample/9780521772181ws.pdf
accessdate=2007-06-26
id=ISBN 0521772184
] This relation is named "Skumanich's law" after Andrew P. Skumanich who discovered it in 1972. [cite journal
first=Andrew P. | last=Skumanich
title=Time Scales for CA II Emission Decay, Rotational Braking, and Lithium Depletion
journal=The Astrophysical Journal
year=1972 | volume=171 | pages=565
doi=10.1086/151310
] Gyrochronology is the determination of a star's age based on the rotation rate. [cite news
author=Staff
title=Gyrochronology
publisher=Astrobiology Magazine
date=April 29, 2007
url=http://www.astrobio.net/news/article2315.html
accessdate=2007-06-26
]

Stars slowly lose mass by the emission of a stellar wind from the photosphere. The star's magnetic field exerts a torque on the ejected matter, resulting in a steady transfer of angular momentum away from the star. Stars with a rate of rotation greater than 15 km/s also exhibit more rapid mass loss, and consequently a faster rate of rotation decay. Thus as the rotation of a star is slowed because of braking, there is a decrease in rate of loss of angular momentum. Under these conditions, stars gradually approach, but never quite reach, a condition of zero rotation. [cite journal
last = Nariai | first = Kyoji
title=Mass Loss from Coronae and Its Effect upon Stellar Rotation
journal=Astrophysics and Space Science
year=1969 | volume=3 | pages=150–159
accessdate=2007-06-27
doi=10.1007/BF00649601
]

Close binary systems

A close binary star system occurs when two stars orbit each other with an average separation that is of the same order of magnitude as their diameters. At these distances, more complex interactions can occur, such as tidal effects, transfer of mass and even collisions. Tidal interactions in a close binary system can result in modification of the orbital and rotational parameters. The total angular momentum of the system is conserved, but the angular momentum can be transferred between the orbital periods and the rotation rates.cite journal
last=Hut | first=P.
title=Tidal evolution in close binary systems
journal=Astronomy and Astrophysics
year=1999 | volume=99 | issue=1 | pages=126–140
accessdate=2007-06-07
]

Each of the members of a close binary system raise tides on the companion star through gravitational interaction. However the bulges can be slightly misaligned with respect to the direction of gravitational attraction. Thus the force of gravity produces a torque component on the bulge, resulting in the transfer of angular momentum. This causes the system to steadily evolve, although it can approach a stable equilibrium. The effect can be more complex in cases where the axis of rotation is not perpendicular to the orbital plane.

For contact or semi-detached binaries, the transfer of mass from a star to its companion can also result in a significant transfer of angular momentum. The accreting companion can spin up to the point where it reaches its critical rotation rate and begins losing mass along the equator. [cite web
author=Weaver, D.; Nicholson, M. | date=December 4, 1997
url=http://hubblesite.org/newscenter/archive/releases/1997/39/text/
title=One Star's Loss is Another's Gain: Hubble Captures Brief Moment in Life of Lively Duo
publisher=NASA Hubble
accessdate=2007-07-03
]

Degenerate stars

After a star has finished generating energy through thermonuclear fusion, it evolves into a more compact, degenerate state. During this process the dimensions of the star are significantly reduced, which can result in a corresponding increase in angular velocity.

White dwarf

A white dwarf is a star that consists of material that is the by-product of thermonuclear fusion during the earlier part of its life, but lacks the mass to burn those more massive elements. It is a compact body that is supported by a quantum mechanical effect known as electron degeneracy pressure that will not allow the star to collapse any further. Generally most white dwarfs have a low rate of rotation, most likely as the result of rotational braking or by shedding angular momentum when the progenitor star lost its outer envelope. [cite book
author=Willson, L. A.; Stalio, R. | year=1990
title=Angular Momentum and Mass Loss for Hot Stars
edition=1st edition | pages=pp. 315-16 | publisher=Springer
id=ISBN 0792308816
] (See planetary nebula.)

A slow-rotating white dwarf star can not exceed the Chandrasekhar limit of 1.44 solar masses without collapsing to form a neutron star or exploding as a Type Ia supernova. Once the white dwarf reaches this mass, such as by accretion or collision, the gravitational force would exceed the pressure exerted by the electrons. If the white dwarf is rotating rapidly, however, the centripetal force at the equator can counter-balance the gravitational force and allow the mass to exceed the Chandrasekhar limit. Such rapid rotation can occur, for example, as a result of mass accretion that results in a transfer of angular momentum. [cite journal
author=Yoon, S.-C.; Langer, N.
title=Presupernova evolution of accreting white dwarfs with rotation
journal=Astronomy and Astrophysics
year=2004 | volume=419 | pages=623–644
accessdate=2007-07-03
doi=10.1051/0004-6361:20035822
]

Neutron star

A neutron star is a highly dense remnant of a star that is primarily composed of neutrons&mdash;a particle that is found in most atomic nuclei and has no net electrical charge. The mass of a neutron star is in the range of 1.35 to 2.1 times the mass of the Sun. As a result of the collapse, a newly-formed neutron star can have a very rapid rate of rotation; on the order of a thousand rotations per second. [cite web
author=Lochner, J.; Gibb, M. | date=December, 2006
url=http://imagine.gsfc.nasa.gov/docs/science/know_l1/pulsars.html
title=Neutron Stars and Pulsars
publisher=NASA
accessdate=2007-06-27
]

Pulsars are rotating neutron stars that have a magnetic field. A narrow beam of electromagnetic radiation is emitted from the poles of rotating pulsars. If the beam sweeps past the direction of the Solar System then the pulsar will produce a periodic pulse that can be detected from the Earth. The energy radiated by the magnetic field gradually slows down the rotation rate, so that older pulsars can require as long as several seconds between each pulse. [cite web
last = Lorimer | first = D. R. | date =August 28, 1998
url=http://relativity.livingreviews.org/Articles/lrr-1998-10/
title=Binary and Millisecond Pulsars
publisher=Max-Planck-Gesellschaft
accessdate=2007-06-27
]

Black hole

A black hole is an object with a gravitation field that is sufficiently powerful that it can prevent light from escaping. When they are formed from the collapse of a rotating mass, they retain angular momentum. This rotation causes the space within an oblate spheroid-shaped volume, called the "ergosphere", to be dragged around with the black hole. Mass falling into this volume gains energy by this process and can then be ejected without falling into the black hole. When the mass is ejected, the black hole loses angular momentum. [cite journal
last = Begelman | first = Mitchell C.
title=Evidence for Black Holes
journal=Science | year=2003 | volume=300
issue=5627 | pages=1898–1903
url=http://www.sciencemag.org/cgi/content/full/300/5627/1898
accessdate=2007-06-26
doi=10.1126/science.1085334
] The rotation rate of a black hole has been measured as high as 98.7% of the speed of light. [cite news
first=Lee | last=Tune
title=Spin of Supermassive Black Holes Measured for First Time
publisher=University of Maryland Newsdesk
date=May 29, 2007
url=http://www.newsdesk.umd.edu/scitech/release.cfm?ArticleID=1447
accessdate=2007-06-25
]

References

*cite web
author=Staff | date=February 28, 2006
url=http://saturn.ethz.ch/research/stars/starspots/results/results_nf.html
title=Stellar Spots and Cyclic Activity: Detailed Results
publisher=ETH Zentrum | accessdate=2007-06-26

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