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The Virial Theorem: Its Explanation, Proof,
Application and Limitations |

The Virial Theorem is an important theorem in mechanics. There are various corollaries of
the theorem which are sometimes also labeled *the Virial Theorem*. Those corollaries will
be dealt with in due course. In the material below the variables which are vectors will be
displayed in red.

Consider a system of n point particles indexed by i. Let r_{i}, v_{i}
and p_{i} be the position, velocity and momentum vectors, respectively, for the i-th
particle. Its mass is denoted as m_{i}.
Let F_{i} be the net force, internal and external, impinging upon the i-th particle.

- Statement of the Virial Theorem: For the n point particles bound together into a system
the time average of the kinetic energy of the particles, Σ½m
_{i}v_{i}², plus one half of the time average of ΣF_{i}·r_{i}is equal to zero.

For each particle

p

dp

Define H as Σp_{i}·r_{i}.
(Note that the dot product of two vectors is a scalar.) Then

Because (dp_{i}/dt) = F_{i}
the first term on the right in the above equation
reduces to ΣF_{i}·r_{i}.

Because p_{i}=m_{i}v_{i}
=m_{i}(dr_{i}/dt) the
second term on the right in the previous equation reduces to

This last expression is just twice the kinetic energy K of the system; i.e., 2Σ½m_{i}v_{i}².
Thus

The time average of a variable y(t) over the interval 0 to τ is defined as

Time averaging the equation of dH/dt gives

The time average of (dH/dt) is just

If the system is cyclical such that it returns to its initial state after an interval then τ can be chosen equal to the cycle period and dH/dt reduces to 0. If the system is not cyclical then for the system being bounded the limit of dH/dt as τ increases without bound is zero.

Thus

or, equivalently

K + ½ΣF

If the forces are generated as the gradients of a potential V(r_{1},…,r_{n}) then

and hence

K − ½(∂V/∂r

There is an important class of functions which are homogeneous. That is to say, if all of the arguments of the function are multiplied by a factor λ then the value of the function is multiplied by λ to some power, say m. The value m is said to be the degree of homogeneity of the function. For such functions the sum of the partial derivatives with respect to the arguments multiplied by the value of the arguments is just equal to m times the value of the function. This is called Euler's Theorem.

For forces which obey an inverse distance squared law the potential is just inversely proportional to the distance. Such potential functions are homogeneous to degree −1. For systems held together by mutual gravitation or electrostatic attraction the Virial Theorem reduces to

or, equivalently

K = − ½ V

For systems held together by mutual gravitational attraction the potential energy is negative so the kinetic energy is positive. The average total energy of the system T=K+V is given by

The relationship

also holds for the electrons in an atom, a system held together by the electrostatic force.

In atoms the electrons can change orbits going from a higher potential energy orbit to a lower one. When an electron does so the relationship that is relevant is

When an electron loses potential energy only half of it goes into increased kinetic energy. The other half of the energy loss goes into the emission of a photon.

Astronomy has made great use of the Virial Theorem as a way of measure gravitational mass.
Consider a set of n galaxies each of mass m. Let v²
be the measured time averaged squared velocity of a galaxy and
__v²__
the average of this quantity over the n galaxies.
Then the time averaged kinetic energy of the
system is n[½m__v²__].

The gravitational potential
for two galaxies separated by a distance R is then −Gm²/R, where G is the gravitational
constant. Let 1/__R__ be the cluster average of the time average of (1/R). There are n(n-1)/2 pairs of
galaxies so the time averaged potential of the system is then
−[n(n-1)/2][Gm²/R].
Then, according to the Virial Theorem,

which may be solved for m

giving

m = 2

and thus the total mass nm

of the cluster is

nm = [2

where __R__ is the reciprocal of (1/__R__).

In some clusters n is on the order of 1000 so n/(n-1) is essentially unity.

When this method has been applied to galactic clusters the gravitational mass computed is
vastly greater than the measured mass derived from the amount of light generated by the stars
in the galaxies. The difference is ascribed to *dark matter*.

It is sometimes noted that virial theorems are powerful but dangerous theorem; dangerous in the sense that that they may easily be misapplied. Their misapplication then involves stating untrue propositions as being true as a result of a virial theorem.

The conventional virial theorem applies only to point particles. If bodies are not point particles but have size and shape then some of the energy changes that take place when they interact goes into spin and perhaps distortions of the fundamental bodies.

Perhaps the most important limitation is that virial theorems do not apply to particles that are not in a bounded system. In particular
it does not apply to the particles of an explosion. Thus applying a virial theorem to galactic clusters
which are the unvirialized remnants of some primodial *Big Bang* could lead to absurdities.

(To be continued.)

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