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 centrifugal 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.[1]

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.

Achernar
This illustration shows the oblate appearance of the star Achernar caused by rapid 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 portion 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.[2] However, this broadening must be carefully separated from other effects that can increase the line width.

V sin i
This star has inclination i to the line-of-sight of an observer on the Earth and rotational velocity ve at the equator.

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 , 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 .[2] This is sometimes referred to as the projected rotational velocity. In fast rotating stars polarimetry offers a method of recovering the actual velocity rather than just the rotational velocity; this technique has so far been applied only to Regulus.[3]

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.[4]

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.[5] 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.[6]

Physical effects

Equatorial bulge

Gravity tends to contract celestial bodies into a perfect sphere, the shape where all the mass is as close to the center of gravity as possible. But a rotating star is not spherical in shape, it has an equatorial bulge.

As a rotating proto-stellar disk contracts to form a star its shape becomes more and more spherical, but the contraction doesn't proceed all the way to a perfect sphere. At the poles all of the gravity acts to increase the contraction, but at the equator the effective gravity is diminished by the centrifugal force. The final shape of the star after star formation is an equilibrium shape, in the sense that the effective gravity in the equatorial region (being diminished) cannot pull the star to a more spherical shape. The rotation also gives rise to gravity darkening at the equator, as described by the von Zeipel theorem.

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.[7] Other rapidly rotating stars include Alpha Arae, Pleione, Vega and Achernar.

The break-up velocity of a star is an expression that is used to describe the case where the centrifugal force at the equator is equal to the gravitational force. For a star to be stable the rotational velocity must be below this value.[8]

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.[9][10] The first such star, other than the Sun, to have its differential rotation mapped in detail is AB Doradus.[1] [11]

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.[12][13]

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.[1]

Rotation braking

During formation

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 gains 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 centrifugal 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 in magnetic braking. The expanding wind carries away the angular momentum and slows down the rotation rate of the collapsing protostar.[14][15]

Average
rotational
velocities[16]
Stellar
class
ve
(km/s)
O5 190
B0 200
B5 210
A0 190
A5 160
F0 95
F5 25
G0 12

Most main-sequence stars with a spectral class between O5 and F5 have been found to rotate rapidly.[7][17] 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."

After formation

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

where is the angular velocity at the equator and t is the star's age.[18] This relation is named Skumanich's law after Andrew P. Skumanich who discovered it in 1972,[19][20] but which had actually been proposed much earlier by Evry Schatzman.[21] Gyrochronology is the determination of a star's age based on the rotation rate, calibrated using the Sun.[22]

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.[23]

At the end of the main sequence

Ultracool dwarfs and brown dwarfs experience faster rotation as they age, due to gravitational contraction. These objects also have magnetic fields similar to the coolest stars. However, the discovery of rapidly rotating brown dwarfs such as the T6 brown dwarf WISEPC J112254.73+255021.5[24] lends support to theoretical models that show that rotational braking by stellar winds is over 1000 times less effective at the end of the main sequence.[25]

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.[26]

Each of the members of a close binary system raises tides on the other 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 (tidal acceleration). 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.[26]

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.[27]

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.[28] (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 effective gravity is diminished in the equatorial region, thus allowing the white dwarf 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.[29]

Neutron star

Pulsar schematic
The neutron star (center) emits a beam of radiation from its magnetic poles. The beams are swept along a conic surface around the axis of rotation.

A neutron star is a highly dense remnant of a star that is primarily composed of neutrons—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 1.2 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 hundred rotations per second.

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.[30]

Black hole

A black hole is an object with a gravitational 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 all of the angular momentum that is not shed in the form of ejected gas. 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 some portion of the mass can then be ejected without falling into the black hole. When the mass is ejected, the black hole loses angular momentum (the "Penrose process").[31] The rotation rate of a black hole has been measured as high as 98.7% of the speed of light.[32]

References

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External links

AK Pictoris

AK Pictoris is a star system in the constellation Pictor. Its combined apparent magnitude is 6.182. Based on the system's parallax, it is located 69 light-years (21.3 parsecs) away. AK Pictoris is a member of the AB Doradus moving group, a group of stars with similar motions that are thought to be associated.

AK Pictoris is a binary star. Its two stars orbit each other every 217.6 years, separated by 2.004″. The primary star is a G-type star with similar properties to the Sun. The secondary star is a K-type star. The primary star is a young BY Draconis variable, a class of variable stars that derive their variability from stellar rotation. It is also known to host a debris disk, inferred from its infrared excess.

Alpha Telescopii

Alpha Telescopii, Latinized from α Telescopii, is the brightest star in the southern constellation of Telescopium, with an apparent visual magnitude of 3.5. The ancient Roman astronomer Ptolemy included it in the constellation Corona Australis, but it was moved to Telescopium when that constellation was created by French astronomer Nicolas Louis de Lacaille in the 18th century. Parallax measurements put it at a distance of 278 light-years (85 parsecs) from Earth. At that range, the visual magnitude of the star is diminished by an extinction of 0.22 due to interstellar dust.This star is much larger than the Sun, with an estimated 5.2±0.4 times the mass and 3.3±0.5 times the radius. The spectrum of the star matches a stellar classification of B3 IV, where the luminosity class of 'IV' indicates this is a subgiant star that has nearly exhausted the supply of hydrogen at its core and is evolving away from the main sequence. Alpha Telescopii is a bright star that is radiating nearly 800 times the Sun's luminosity. This energy is being emitted from the star's outer envelope at an effective temperature of around 16,700 K, giving it the characteristic blue-white hue of a B-type star.This is possibly a type of variable star known as a slowly pulsating B-type star. It has a longitudinal magnetic field with a mean strength of –233 ± 43 G. A projected stellar rotation velocity of about 14 km s−1 is considered low for a star of this type, which may indicate it is being viewed from nearly pole-on.

BB Phoenicis

BB Phoenicis is a variable star in the constellation of Phoenix. It has an average visual apparent magnitude of 6.17, being visible to the naked eye with excellent viewing conditions. From parallax measurements by the Gaia spacecraft, it is located at a distance of 448 light-years (137 parsecs) from Earth. Its absolute magnitude is calculated at 0.6.BB Phoenicis is a Delta Scuti variable, and shows stellar pulsations that cause brightness variations with an amplitude of 0.04 magnitudes. Its variability was discovered by accident in 1981, when the star was used as a comparison star for the eclipsing binary AG Phoenicis. Photometric and spectroscopic data have allowed the detection of at least 13 modes of radial and non-radial pulsations, the strongest one having a period of 0.174 days and an amplitude of 11.1 milli-magnitudes. Observations in different epochs show evidence that the pulsations modes vary in amplitude, which is common among Delta Scuti variables. Pulsation models indicate that the stellar rotation axis is inclined by 50–70° in relation to the line of sight.This star is classified as an F-type giant with a spectral type of F0/2III. It appears to be expanding after depleting all the nuclear hydrogen and leaving the main sequence. BB Phoenicis has an estimated mass of 2.25 times the solar mass and a radius of 4.7 times the solar radius. It is radiating 55 times the Sun's luminosity from its photosphere at an effective temperature of 7,200 K.

COROT-4b

COROT-4b (formerly known as COROT-Exo-4b) is an extrasolar planet orbiting the star COROT-4. It is probably in synchronous orbit with stellar rotation. It was discovered by the French COROT mission in 2008.

Catalogue of rotational velocities of the stars

Catalogue of rotational velocities of the stars is the name for catalogue of projected stellar rotation, published in 1982 by Uesugi, A. and Fukuda, I.

Contact binary

In astronomy, a contact binary is a binary star system whose component stars are so close that they touch each other or have merged to share their gaseous envelopes. A binary system whose stars share an envelope may also be called an overcontact binary. Almost all known contact binary systems are eclipsing binaries; eclipsing contact binaries are known as W Ursae Majoris variables, after their type star, W Ursae Majoris.Contact binaries are not to be confused with common envelopes. Whereas the configuration of two touching stars in a contact binary has a typical lifetime of millions to billions of years, the common envelope is a dynamically unstable phase in binary evolution that either expels the stellar envelope or merges the binary in a timescale of months to years.

Doppler imaging

Inhomogeneous structures on stellar surfaces, i.e. temperature differences, chemical composition or magnetic fields, create characteristic distortions in the spectral lines due to the Doppler effect. These distortions will move across spectral line profiles due to the stellar rotation. The technique to reconstruct these structures on the stellar surface is called Doppler-imaging, often based on the Maximum Entropy image reconstruction to find the stellar image. This technique gives the smoothest and simplest image that is consistent with observations.

To understand the magnetic field and activity of stars, studies of the Sun are not sufficient. Therefore, studies of other stars are necessary. Periodic changes in brightness have long been observed in stars which indicate cooler or brighter starspots on the surface. These spots are larger than the ones on the Sun, covering up to 20% of the star. Spots with similar size as the ones on the Sun would hardly give rise to changes in intensity. In order to understand the magnetic field structure of a star, it is not enough to know that spots exist because their location and extent are also important.

Gyrochronology

Gyrochronology is a method for estimating the age of a low-mass star like the Sun from its rotation period. The term is derived from the Greek words gyros, chronos and logos, roughly translated as rotation, age, and study respectively. It was coined in 2003 by Sydney Barnes to describe the associated procedure for deriving stellar ages, and developed extensively in empirical form in 2007.The technique builds on an insight of Andrew Skumanich,

who realized that another measure of stellar rotation (v sin i) declined steadily with stellar age. Gyrochronology uses the rotation period P of the star instead of the doubly ambiguous v sin i, which depends on the unknown inclination of the star's axis of rotation, i. In particular, the technique accounts for the substantial mass dependence of stellar rotation, as exemplified by early rotation-period work on the Hyades open cluster. These two improvements are largely responsible for the precision in the ages provided by gyrochronology. The associated age estimate for a star is known as the gyrochronological age.

The basic idea underlying gyrochronology is that the rotation period P, of a main-sequence cool star is a deterministic function of its age t and its mass M (or a suitable proxy such as color). The detailed dependencies of rotation are such that the periods converge rapidly to a certain function of age and mass, mathematically denoted by P = P (t, M), even though stars have a range of allowed initial periods. Consequently, cool stars do not occupy the entire 3-dimensional parameter space of (mass, age, period), but instead define a 2-dimensional surface in this space. Therefore, measuring two of these variables yields the third. Of these quantities, the mass (or a proxy such as color) and the rotation period are the easier variables to measure, providing access to the star's age, otherwise difficult to obtain.

Defining a star as "Sun-like" is very difficult, because to be Sun-like the star should have a mass, radius, age, temperature metallicity, and spectral type that is similar to the Sun's. Measuring most of these factors is difficult, and determining the age of a star is extremely difficult, so astronomers tend to ignore it when deciding if a star is Sun-like or not. However, this is not ideal, because the Sun, and all stars change over time. If a star's rotation period is less than 25 days, the star can be determined as being younger than the Sun, if the rotation rate is longer, the star can be determined as being older than the Sun.The relationship between rotation and age was initially discovered by Soren Meibom and colleagues by measuring the period of rotation of stars in a billion-year-old cluster. Because the ages of the stars were already known, the researchers could discover a relationship between a star's age and its rotation period. A study of 30 cool stars in the 2.5-billion-year-old cluster NGC 6819 allowed to estimate the age–period relationship for older stars. Using these results, the ages of a large number of cool galactic field stars can be derived with 10% precision.

Kepler-102

Kepler-102 is a star in the constellation of Lyra. It has five known exoplanets. Kepler-102 is less luminous than the Sun.

Kepler-17

Kepler-17 is main-sequence yellow dwarf star. This star is known to host one exoplanet, Kepler-17b, in orbit around it.

Kepler-65

Kepler-65 is a star slightly more massive than the Sun and has at least three planets.

Kepler-66

Kepler-66 is a star with slightly more mass than the Sun in the NGC 6811 open cluster in the Cygnus constellation. It has one confirmed planet, slightly smaller than Neptune, announced in 2013.

Kepler-67

Kepler-67 is a star with slightly less mass than the Sun in the NGC 6811 open cluster in the Cygnus constellation and has one confirmed planet, slightly smaller than Neptune, announced in 2013.

NGC 681

NGC 681 (also known as the Little Sombrero Galaxy) is an intermediate spiral galaxy in the constellation of Cetus, located approximately 66.5 million light-years from Earth. The name Little Sombrero Galaxy is a reference to a much larger and earlier observed sombrero-like galaxy designated M104, or the Sombrero Galaxy.

Otto Struve

Not to be confused with his grandfather Otto Wilhelm von Struve (1819–1905)Otto Struve (August 12, 1897 – April 6, 1963) was a Russian-American astronomer. In Russian, his name is sometimes given as Otto Lyudvigovich Struve (Отто Людвигович Струве); however, he spent most of his life and his entire scientific career in the United States. Otto was the descendant of famous astronomers of the Struve family; he was the son of Ludwig Struve, grandson of Otto Wilhelm von Struve and great-grandson of Friedrich Georg Wilhelm von Struve. He was also the nephew of Karl Hermann Struve.With more than 900 journal articles and books, Struve was one of the most distinguished and prolific astronomers of the mid-20th century. He served as director of Yerkes, McDonald, Leuschner and National Radio Astronomy Observatories and is credited with raising worldwide prestige and building schools of talented scientists at Yerkes and McDonald observatories. In particular, he hired Subrahmanyan Chandrasekhar and Gerhard Herzberg who later became Nobel Prize winners. Struve's research was mostly focused on binary and variable stars, stellar rotation and interstellar matter. He was one of the few eminent astronomers in the pre-Space Age era to publicly express a belief that extraterrestrial intelligence was abundant, and so was an early advocate of the search for extraterrestrial life.

Phi2 Pavonis

Phi2 Pavonis (φ2 Pav, φ2 Pavonis) is a solitary star in the southern constellation of Pavo (the Peacock). It is faintly visible to the naked eye with an apparent visual magnitude of +5.10. Based upon an annual parallax shift of 40.55 mas as seen from Earth, it is located 80.4 light years from the Sun. At that distance, the visual magnitude is diminished by an extinction factor of 0.07 due to interstellar dust. It is a member of the thin disk population.This is a yellow-white hued G-type main sequence star with a stellar classification of G0 V Fe-0.8 CH-0.5. This notation indicates the surface abundance of iron and cyanogen are below normal for this class of star. It is around 5.7 billion years old and is spinning with a period of around 28 days. It has an estimated 1.09 times the mass of the Sun and is 1.86 times the Sun's radius. The star is radiating 3.39 times the solar luminosity from its photosphere at an effective temperature of 6,091 K.This system was in 1991 a test case for the Zeta Herculis moving group, of low metallicity stars with 5 billion years of age. This group includes besides Zeta Herculis: δ Trianguli, ζ Reticuli, 1 Hydrae, Gl 456, Gl 678, and Gl 9079.In 1998, using the European Southern Telescope in Chile, a planet was announced to be orbiting the star. This team retracted this claim in 2002, but found a different periodicity of 7 days possibly due to stellar rotation.

Robert Kraft (astronomer)

Robert Paul "Bob" Kraft (June 16, 1927 – May 26, 2015) was an American astronomer. He performed pioneering work on Cepheid variables, stellar rotation, novae, and the chemical evolution of the Milky Way. His name is also associated with the Kraft break: the abrupt change in the average rotation rate of main sequence stars around spectral type F8.

Wolf 1061

Wolf 1061 (also known as HIP 80824 and V2306 Ophiuchi) is an M class red dwarf star located about 14.1 light years away in the constellation Ophiuchus. It is the 36th closest known star system to the Sun and has a relatively high proper motion of 1.2 seconds of arc per year. Like many red dwarfs, it most likely has a long rotation period of more than 100 days, although it is difficult to measure accurately. Wolf 1061 does not have any unusual spectroscopic features. The star was first cataloged in 1919 by German astronomer Max Wolf when he published a list of dim stars that had high proper motions. Wolf 1061's name originates from this list. The star has a stellar rotation period of 89.3±1.8~ days. A seven years study found no evidence of photometric transits and confirms the radial velocity signals are not due to stellar activity. The habitable zone estimate for the system lies between approximately 0.1 and 0.2 AU from the star.

Zeeman–Doppler imaging

In astrophysics, Zeeman–Doppler imaging is a tomographic technique dedicated to the cartography of stellar magnetic fields, as well as surface brightness and temperature distributions.

This method makes use of the ability of magnetic fields to polarize the light emitted (or absorbed) in spectral lines formed in the stellar atmosphere (the Zeeman effect). The periodic modulation of Zeeman signatures during the stellar rotation is employed to make an iterative reconstruction of the vectorial magnetic field at stellar surface.

The method was first proposed by Marsh and Horne in 1988, as a way to interpret the emission line variations of cataclysmic variable stars. This techniques is based on the principle of maximum entropy image reconstruction; it yields the simplest magnetic field geometry (as a spherical harmonics expansion) among the various solutions compatible with the data.This technique is the first to enable the reconstruction of the vectorial magnetic geometry of stars similar to the Sun. It is now offering the opportunity to undertake systematic studies of stellar magnetism and is also yielding information on the geometry of large arches that magnetic fields are able to develop above stellar surfaces. To collect the observations related to Zeeman-Doppler Imaging, astronomers use stellar spectropolarimeters like ESPaDOnS at CFHT on Mauna Kea (Hawaii), HARPSpol at the ESO's 3.6m telescope (La Silla Observatory, Chile), as well as NARVAL at Bernard Lyot Telescope (Pic du Midi de Bigorre, France).

The technique is very reliable, as the reconstruction of the magnetic field maps with different algorithms yield almost identical results, even with poorly sampled data sets.

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