Adapted from P. Coles, 1999, *The Routledge Critical
Dictionary of the New Cosmology*, Routledge Inc., New York. Reprinted
with the author's permission. To order this book click here:
http://www.routledge-ny.com/books.cfm?isbn=0415923549

An important result from the field of statistical
mechanics that deals with the properties of self-gravitating systems
in equilibrium. According to the theory of the **Jeans instability**,
small initial fluctuations grow by virtue of the attractive nature of
**gravity** until they become sufficiently dense to collapse. When such a
structure collapses it undergoes what is sometimes called *violent
relaxation*: the material that makes up the structure rapidly adjusts
itself so that it reaches a kind of pressure balance with the
gravitational forces. The velocities of particles inside the structure
become randomised, and the structure settles down into an equilibrium
configuration whose properties do not undergo any further change. This
process is sometimes called *virialisation*.

The virial theorem, which applies to gravitationally bound objects
of this kind, states that the total kinetic energy *T* contained in the
structure is related to the total gravitational potential energy *V* by
the equation

*T* + *V* = 0

This theorem can be applied to gravitationally bound objects such as
some kinds of **galaxy** and clusters of galaxies, and its importance lies
in the fact that it can be used to estimate the mass of the object in
question.

Because the motions of matter within a virialised structure are
random, they are characterised by some dispersion (or *variance*) around
the mean velocity. If the object is a galaxy, we can estimate the
variance of stellar motions within it by using **spectroscopy** to measure
the widths of spectral lines affected by the **Doppler shift**. If the
object is a galaxy cluster, we have to measure the **redshifts** of all
the galaxies in the cluster. The mean redshift corresponds to the mean
motion of the cluster caused by the **expansion of the Universe**; the
variance around this mean represents the **peculiar motions** of the
galaxies caused by the self-gravity of the material in the cluster. If
the variance of the velocities is written as
*v*^{2}, then the total
kinetic energy of the object is simply 1/2 *Mv*^{2}, where
*M* is the total mass.

If the object is spherical and has the physical dimension *R*, then
the total gravitational potential energy will be of the form
-*GM*^{2} / *R*,
where is a numerical factor
that measures how strongly the object's
mass is concentrated towards its centre. Note that *V* is negative
because the object is gravitationally bound. We can therefore make use
of the virial theorem to derive an expression for the mass *M* of the
object in terms of quantities which are all measurable: *M* =
*Rv*^{2} / *G*.

This illustration is very simplified, but illustrates the basic
point. More detailed analyses do not assume spherical symmetry, and
can also take into account forms of energy other than the kinetic and
gravitational energy discussed here, such as the energy associated
with gas pressure and **magnetic fields**. In rich clusters of galaxies,
for example, the galaxies are moving through a very hot gas which
emits X-ray's: the high temperature of the gas reflects the fact that
it too is in equilibrium with the gravitational field of the cluster.
Viralisation can produce gas temperatures of hundreds of millions of
degrees in this way, and this can also be used to measure the mass of
clusters (see **X-ray astronomy**). A virial analysis by Fritz
**Zwicky** of
the dynamics of the relatively nearby Coma Cluster provided the first
evidence that these objects contain significant amounts of **dark
matter**. The material in these clusters is sufficient to allow a value
of the **density parameter** of around
_{0}
0.2.

A modified version of the virial theorem, called the *cosmic virial
theorem*, applies on scales larger than individual gravitationally
bound objects like galaxy clusters: it allows us to relate the
statistics of galaxies' peculiar motions to the density
parameter. This method also usually produces an estimated value of
_{0}
0.2, indicating that dark
matter exists on cosmological scales, but
not enough to reach the critical density required for a **flat universe**.

FURTHER READING:

Tayler, R.J., *The Hidden Universe* (Wiley-Praxis, Chichester, 1995).
Coles, P. and Ellis, G.F.H., *Is the Universe Open or Closed?*
(Cambridge University Press, Cambridge, 1997).