Consider a set of *n* point masses, with the *i*th mass having position vector **r**_{i} and mass *m*_{i}. Then the total scalar moment of inertia of this collection about our origin is

,

now, using the fact that , we see that, assuming the masses are constant, the time derivative

,

where **p**_{i}=*m*_{i}**v**_{i} is the momentum of the *i*th mass. Let us now define a quantity equal to half of the above:

The quantity *G* is known as the virial.

Let us now take the time derivative of the virial:

,

where *T* is the total kinetic energy of the masses, and **F**_{i} is the force on the *i*th mass.

Now, let us denote by **F**_{ij} the force exerted on mass *i* by mass *j*. If we have no external forces, then . Note also that for all *i* and *j*. (For *i*≠*j*, this is Newton’s third law of motion. For *i*=*j*, this says **F**_{ii}=0; no object exerts a force on itself.)

Then we see that

Now, suppose that the system is a stable, bound system. Then, we expect the virial to be bounded to some finite range; thus, when one takes the time average of the derivative, , one should expect it to go to zero when the averaging is over a sufficiently long time scale; thus, since , for such a system, we get

. This is known as the virial theorem.

Now, suppose that the force between the masses is a central force; the force between two masses depends only upon the distance between them, and is directed along the line between them. Then we have a potential *V*(*r*), a function of the distance only, of which the force is the negative gradient .

Then, the force on mass *i* due to mass *j* is given by

, where is the distance between mass *i* and mass *j*.

Thus,

,

and so

,

and the virial theorem becomes

.

Now, if the potential is given by a power law, , then , and . Therefore, in the power law case, , and so the virial theorem becomes

.

For systems bound by gravity (or for electrostatic systems), we have an inverse square force, so *k*=-1, and we obtain .

It was in applying the virial theorem to the Coma galaxy cluster that astronomer Fritz Zwicky inferred the existence of what we now call dark matter.

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Tags: Dark Matter, Friday Physics, physics, Virial Theorem

This entry was posted on April 16, 2010 at 12:02 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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April 16, 2010 at 7:08 am |

I’ve seen the theorem before, but that is the first time I have ever heard of it being applied to anything useful. Thanks for that final bit of information that gives it an application.