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Stellar rotation is the rotation of a star in the manner that the Sun and Earth rotate. The rotation can sometimes be determined using the timing of periodic variations or the spectrum of the star. The star's rotation causes it to take a non-spherical (oblate) shape through centrifugal force. The rotation can be different for different parts of the star (differential rotation), such as the Sun's higher rotation at its equator than its poles (a rotation period of 26 days at the equator, 38 days at the poles). A high rotation rate can produce other observed effects in a star, e.g., those of Be stars.
Variations in a star's brightness can be due to starspots, and some variation-patterns of some duration of periodic patterns int he brightness suggest such spots that are lasting for multiple rotations, which yield very precise rotation periods.
The spectrum of the star shows rotation through a characteristic type of spectral line shape, which reveals varying Doppler shifts of portions of the star that face us, due to the star's rotation: ordinary Doppler broadening is shaped by random motion of molecules, but rotational motion of the star's surface produces varying radial velocities of each portion of the star, which contributes a distinct line-shape component. The shape reveals a minimum relative speed of the star's equator due to its rotation, which is not fully determined due to any obliquity of the axis of rotation to the line of sight. Together with the star's radius, this reveals a minimum on the angular speed (in radians or rotations per unit time), i.e., the vector-magnitude of the angular velocity. (The full angular velocity vector would also require knowledge of the axis of rotation and the direction of rotation around it.) The critical velocity (or break-up velocity or critical rotation) of a star is that above which it is unstable and would break up.
Stars generally lose their rotation over time and the measured rotation is used as a proxy for the age of the star (gyrochronology). The star's initial rotation results from the rotation of the nebular material that formed the star, but is faster due to the mass drawn toward the axis. I believe that if the resulting rotation rate is very high, it is unstable such that it quickly slows to a more-stable rate, thus providing a set-point for gyrochronology.
When a star later collapses, e.g., into a white dwarf, a neutron star, or a black hole, the conservation of angular momentum (again0 causes the rotation to increase, thus the occurrence of neutron stars observed with rotational periods on the order of a second. Accretion (e.g., from a binary companion) also affects rotation, increasing it (or in some cases, decreasing it), and is generally considered the cause of the highest rotation rates observed among neutron stars, periods of a few thousandths of a second.