Physics Friday 115

Consider a set of n point masses, with the ith mass having position vector ri and mass mi. 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 pi=mivi is the momentum of the ith 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 Fi is the force on the ith mass.

Now, let us denote by Fij 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 ij, this is Newton’s third law of motion. For i=j, this says Fii=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.