
The statistical spread of velocities of objects in a system, used to estimate mass.Velocity dispersion is a measure of how much the velocities of objects in a system (like stars in a galaxy, or galaxies in a cluster) vary around the average. It is a statistical quantity, typically denoted by σ, and is calculated as the standard deviation of the velocity distribution. In astrophysics, velocity dispersion is a powerful tool because it directly relates to the gravitational potential of the system through the virial theorem: the kinetic energy of the system is half its gravitational potential energy. This allows astronomers to estimate the total mass of a galaxy or cluster, including dark matter, even when most of the mass is invisible.
The concept was first applied to stellar dynamics in the early 20th century. In 1933, Fritz Zwicky used velocity dispersions of galaxies in the Coma cluster to infer the presence of dark matter, finding that the visible mass was insufficient to hold the cluster together.
For a system in dynamical equilibrium, the velocity dispersion is tied to the mass distribution. In a simple spherical system, the mass M within radius r is approximately M ≈ σ² r / G, where G is the gravitational constant. By measuring the Doppler shifts of spectral lines from stars or gas, astronomers can map the line-of-sight velocity dispersion and infer the mass profile. A higher velocity dispersion means the objects are moving faster on average, requiring more mass to keep them bound. This method is essential for detecting dark matter, as the observed velocity dispersions in galaxy clusters are far too large to be explained by visible matter alone.
Velocity dispersion is used to measure masses of galaxies and clusters, to study the dynamics of globular clusters, and to calibrate the mass-luminosity relation used in cosmological surveys. It also helps in identifying supermassive black holes by measuring the velocity dispersion of stars near galactic centers.
The velocity dispersion of stars in the Milky Way's disk is about 20-30 km/s, but in the central bulge it rises to over 100 km/s, reflecting the stronger gravitational pull there.
| r_{\rm inf} | variable | Influence radius — radius metre |
| M_{\rm BH} | variable | Black hole mass — mass kilogram |
| \sigma | variable | Velocity dispersion of stars — velocity dispersion metre per second |
| G | constant | Newtonian gravitational constant — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| M | variable | Total mass of the system — mass kilogram |
| R | variable | Characteristic radius of the system — radius metre |
| v | variable | Velocity dispersion (typical random velocity) — velocity dispersion metre per second |
| G | constant | Constant of universal gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| m | variable | Mass of a particle — mass kilogram |
| \sigma_v | variable | Velocity dispersion (standard deviation of particle velocities) — velocity dispersion metre per second |
| T | variable | Temperature — temperature kelvin |
| k | constant | Boltzmann constant, relates temperature to energy — boltzmann constant = 1.380649e-23 joule per kelvin |














