# Velocity dispersion

In astronomy, the velocity dispersion (σ) is the statistical dispersion of velocities about the mean velocity for a group of objects, such as an open cluster, globular cluster, galaxy, galaxy cluster, or supercluster. By measuring the radial velocities of its members, the velocity dispersion of a cluster can be estimated and used to derive the cluster's mass from the virial theorem.[1] Radial velocity is found by measuring the Doppler width of spectral lines of a collection of objects. The more radial velocities one measures, the more accurately one knows their dispersion. A central velocity dispersion refers to the σ of the interior regions of an extended object, such as a galaxy or cluster.

This relationship takes several forms in astronomy based on the object(s) being observed. For instance, the M-σ relation was found for material circling black holes, the Faber-Jackson relation for elliptical galaxies, and the Tully-Fisher relation for spiral galaxies. For example, the σ found for objects about the Milky Way's supermassive black hole (SMBH) is about 75 km/s.[2] The Andromeda Galaxy (Messier 31) hosts a SMBH about 10 times larger than our own, and has a σ ≈ 160 km/s.[2]

Groups and clusters of galaxies have a wider range of velocity dispersions than smaller objects. For example, our own poor group, the Local Group, has a σ = 61±8 km/s.[3] But rich clusters of galaxies, such as the Coma Cluster, have a σ ≈ 1,000 km/s.[4] The dwarf elliptical galaxies in Coma have their own, internal, velocity dispersion for their stars, which is a σ ≲ 80 km/s, typically.[5] Normal elliptical galaxies, by comparison, have an average σ ≈ 200 km/s.[6]

For spiral galaxies, the increase in velocity dispersion in population I stars is a gradual process which likely results from the random momentum exchanges, known as dynamical friction, between individual stars and large interstellar gas and dust clouds with masses ≳ 105 M.[7] Face-on spiral galaxies have a central σ ≲ 90 km/s; slightly more if viewed edge-on.[8]