### virial theorem

(average kinetic energy equals half the average negative potential energy)

The **virial theorem** is
a theorem relating the kinetic and potential energies of
a stable system as averaged over time.

2 <T> = - Sum(over k=1..N) < F(k) × r(k) >

- <
*X*> - *X* averaged over time.
- T - kinetic energy.
- F(k) - force on the kth object.
- r(k) - position of the kth object

In a gravity-bound system, this amounts to the time-averaged
kinetic energy equaling the negative of the time-averaged
potential energy of the objects.

The theorem can be used to calculate of the average total kinetic
or potential energy in complex systems: from one, you can get the
other. In astrophysics, it is used to derive physical parameters
(size, mass) of a system
(stellar cluster, galaxy, galaxy cluster)
from the velocities of the component bodies, which can be derived
from spectral-line radial velocity measurements of the
component bodies or from line broadening, if individual objects
are not resolved.

The terms **virial radius** (sometimes **virial length** or **virial scale**)
and **virial mass** are used in astrophysics
in a couple of ways in regards to a system such as a galaxy or
galaxy cluster:

- The radius and mass within that radius at which the velocity dispersion is maximal.
- The radius and mass within that radius within which virial equilibrium holds.

Such values are often defined including everything up to the radius
within which the average density (of the whole volume)
is at some chosen multiple of the
universe's average/critical density.
Multiples of 100 and 200 are commonly used.

A **virial temperature** (**T**_{Vir}) is the
temperature, e.g., of a gas, based on
the kinetic energy of the gas particles made necessary
by the virial theorem and other determined values
related to the body of gas.

(*physics*)
**Further reading:**

http://en.wikipedia.org/wiki/Virial_theorem

**Referenced by pages:**

circumgalactic medium (CGM)

cluster radius

carbon monoxide (CO)

velocity dispersion (σ)

gravitational collapse

Kelvin-Helmholtz timescale (KH timescale)

Index