Editing Virial theorem (section)

=== Time averaging ===

The average of this derivative over a duration {{mvar|τ}} is defined as
<math display="block">
 \left\langle \frac{dG}{dt} \right\rangle_\tau = \frac{1}{\tau} \int_0^\tau \frac{dG}{dt} \,dt = \frac{1}{\tau} \int_{G(0)}^{G(\tau)} \,dG = \frac{G(\tau) - G(0)}{\tau},
</math>
from which we obtain the exact equation
<math display="block">
 \left\langle \frac{dG}{dt} \right\rangle_\tau =
 2 \langle T \rangle_\tau + \sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle_\tau.
</math>

The '''virial theorem''' states that if {{math|1={{angbr|''dG''/''dt''}}{{sub|''τ''}} = 0}}, then
<math display="block">
 2 \langle T \rangle_\tau = -\sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle_\tau.
</math>

There are many reasons why the average of the time derivative might vanish. One often-cited reason applies to stably bound systems, that is, to systems that hang together forever and whose parameters are finite. In this case, velocities and coordinates of the particles of the system have upper and lower limits, so that {{math|''G''<sup>bound</sup>}} is bounded between two extremes, {{math|''G''<sub>min</sub>}} and {{math|''G''<sub>max</sub>}}, and the average goes to zero in the limit of infinite {{mvar|τ}}:
<math display="block">
 \lim_{\tau \to \infty} \left| \left\langle \frac{dG^{\text{bound}}}{dt} \right\rangle_\tau \right| =
 \lim_{\tau \to \infty} \left| \frac{G(\tau) - G(0)}{\tau} \right| \le
 \lim_{\tau \to \infty} \frac{G_\max - G_\min}{\tau} = 0.
</math>

Even if the average of the time derivative of {{mvar|G}} is only approximately zero, the virial theorem holds to the same degree of approximation.

For power-law forces with an exponent {{mvar|n}}, the general equation holds:
<math display="block">
 \langle T \rangle_\tau = -\frac12 \sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle_\tau
 = \frac{n}{2} \langle V_\text{TOT} \rangle_\tau.
</math>

For [[gravitation]]al attraction, {{math|''n'' {{=}} −1}}, and the average kinetic energy equals half of the average negative potential energy:
<math display="block">
 \langle T \rangle_\tau = -\frac12 \langle V_\text{TOT} \rangle_\tau.
</math>

This general result is useful for complex gravitating systems such as [[planetary system]]s or [[galaxy|galaxies]].

A simple application of the virial theorem concerns [[galaxy clusters]]. If a region of space is unusually full of galaxies, it is safe to assume that they have been together for a long time, and the virial theorem can be applied. [[Doppler effect]] measurements give lower bounds for their relative velocities, and the virial theorem gives a lower bound for the total mass of the cluster, including any dark matter.

If the [[Ergodicity|ergodic hypothesis]] holds for the system under consideration, the averaging need not be taken over time; an [[ensemble average]] can also be taken, with equivalent results.