The virial theorem is a description of the dynamical state of multi-body
systems (for example stars in elliptical galaxies,
or galaxies in clusters). The theorem consists of a tensor theorem
and a simpler, scalar theorem.

The tensor virial theorem can be derived
from the collisionless Boltzmann equation - a description of point masses
interacting with and contributing to a gravitational field.
It relates the changing moment of inertia
tensor of a system to the potential and kinetic energy
tensors:

1/2 d^{2}/dt^{2} (I_{jk}) = 2T_{jk} +
P_{jk} + W_{jk} = 2K_{jk} + W_{jk}

The term on the left is the second time derivative of the inertia tensor,
which measures how the shape of the system is changing over
time. *2T + P* are the ordered and chaotic motion components of the
kinetic energy tensor *K*_{jk} (actually, twice K), and *W*_{jk} is the
potential energy tensor, the shape of the gravitational field.

When we say a system is *virialized*, the moment of inertia does not
change with time, which makes the derivative on the left-hand side equal
to zero. We're then left with the equation

0 = 2T_{jk} + P_{jk} + W_{jk} =
2K_{jk} + W_{jk}

The *scalar* virial theorem is simply the trace of the kinetic and
potential energy tensors, yielding

2K + W = 0

where *K* and *W* are now just the scalar kinetic and potential
energies of the system. The scalar part of the theorem was derived first,
in 1870 by Rudolf Clausius, the Prussian mathematician and
physicist.

The scalar virial theorem pops up in discussions of the astronomical objects mentioned above. For example, it's a nice way to determine the mass
(and mass to light ratio) of a system, based just on the
observed velocities of the objects within the system. If we assume the
kinetic energy of a system goes as the square of the velocity dispersion,
σ, we wind up with the relation

σ^{2} = GM/R

where G is the gravitational constant, *M* is the
mass of the system, and *R* is essentially radius of the system
(specifically it is the gravitational radius). If you can measure σ
(perhaps by measuring the Doppler broadening of prominent
stellar absorption lines), and can estimate the gravitational
radius (not too hard), you can determine the mass. Along those lines, the
virial theorem also leads to another important result in extragalactic
astronomy,
namely the existence of the fundamental plane of elliptical galaxies, and
the related Faber-Jackson and
Dn-sigma relations, which allow us to derive
the luminosity of an elliptical galaxy, and hence, its distance.

Sources: *Galactic Dynamics* by J. Binney and S. Tremaine, Princeton
University Press, 3rd ed (1994), and Faber, Dressler, Davies, Burstein, and
Lynden-Bell in *Nearly Normal Galaxies*, Springer-Verlag (1987).