# Virial theorem

The virial theorem states that the average kinetic energy of a system of particles whose motions are bounded is given by

[itex] \overline{K} = -\frac{1}{2} \overline{\sum_i \mathbf{F}_i \cdot \mathbf{r}_i} [itex]

where ri and Fi are the position and force vectors on the i th particle respectively.

If the force is derivable from a potential the theorem becomes,

[itex] \overline{K} = \frac{1}{2} \overline{\sum_i \nabla \mathbf{V} \cdot \mathbf{r}_i} [itex]

If V is a power-law function of r,

[itex] V= a r^{n+1} [itex]

then the virial theorem can be written as

[itex] \overline{K} = \frac{n+1}{2} \overline{V} [itex]

In particular, for the further special case of inverse square law forces (i.e. n=-2), the virial theorem states:

Equivalently:

• the time-average of the potential energy of the system is equal to twice the total energy
• the time-average of the kinetic energy of the system is equal to minus the total energy

Since the gravitational force obeys an inverse square law relation, the virial theorem is a remarkably useful simplifying result for otherwise very complex physical systems such as solar systems or galaxies, and is also applicable to a number of other similar scenarios.

The theorem is also very useful in the theory of gases and can be used to derive Boyle's Law for perfect gases.

Note that e.g. in the case of a solid or liquid celestial body, there are gravitational as well as reaction forces, so the potential of the total force does not satisfy a power-law. In the case of elastic collisions the reaction forces act only a short time and the result is not affected.

The virial theorem takes its name from the quantity known as the virial (rooted in the Latin vires, "forces"), defined as:

[itex]G = \sum_i \mathbf{r}_i \cdot \mathbf{p}_i [itex]

where ri and pi are the position and momentum vectors of the ith particle respectively.

The virial theorem can be derived by considering the properties of the virial in the limit over a long period of time.

## Unbound case

Without assuming boundedness we have the following more general properties:

• the time-average of the potential energy of the system is equal to twice the total energy minus the average time-rate of change of G.
• the time-average of the kinetic energy of the system is equal to minus the total energy plus the average time-rate of change of G.

For two simple cases:

• For a parabolic orbit the properties still apply: the averages are zero, G increases slower than t.
• For a hyperbolic orbit the time-average of the kinetic energy is equal to the total energy, without the minus, the average rate of change of G is the difference between the value and minus the value, that is twice this value.

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