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Knowing the speed of a star's surface rotation gives us what data and information. For example, if it is a big star, is it possible to measure distance using the doppler effect? Or can we calculate how much matter it lost by looking at its spin rate?

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The question is too broad to provide extensive detail.

The rotation rate of a star can determine a star's lifetime on the main sequence. Rotation drives mixing that can draw new fuel into the fusion regions, leading to longer lifetimes. This is very important in stars with mostly radiative interiors - high mass stars. A rotation rate can therefore be crucial in estimating the evolutionary stage of a star, its age and mass.

Rotation in low-mass star's appears to determine how magnetically active they are. Rotation combined with convection drives a dynamo that produces and amplifies magnetic fields. This leads to diverse phenomena such as starspots, flares and coronal X-rays. This in turn may determine the strength of mass loss and stellar winds or the habitability of exoplanets.

Rotation can be a diagnostic of accretion processes - of gas, of other stars or even of exoplanets. Stars can be spun up or spun down by accretion depending on the circumstances.

The rotation rate of an isolated low-mass star is indicative of its age. Stars are born rotating quickly but then get spun down as they get older, as a magnetised wind carries away angular momentum. The process can be empirically calibrated using stars of known age and then applied to other stars to estimate their ages - known as gyrochronology.

Knowledge of the rotation rate of a star is needed to accurately model its spectrum. Rotation leads to the broadening of spectral features that must be separated from other broadening phenomena like gravity, granulation and microturbulence.

If you know the rotation period of a star (e.g. through the rotational modulation of starspots) then measuring the rotation rate of the stellar surface tells you how big the star is. Important, because the radius of most stars cannot be measured directly.

Rotation in non-accreting binary systems can tell us about tidal forces. Short period binary systems tend to synchronise their rotation rates with their orbital periods.

Knowing the rotation rates of some fast rotating stars allows the mapping of features on the stellar surface. This "Doppler mapping" works by tracking spectral features produced by hot or cold spots as they move from blue to red across the profile of a stellar absorption or emission line. A variant of this method - the Rossiter-McLaughlin effect - allows measurement of the inclination of an exoplanet's orbit to the stellar rotation axis.

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