Worksheet 8, Problem 3: Using Virial Theorem to find the Mass of a Cluster

If the average speed of a star in a cluster of thousands of stars is $\langle v \rangle$, give an expression for the total mass of the cluster in terms of $\langle v \rangle$, the cluster radius R, and the relevant physical constraints.

Globular Cluster M80
Image Credit: http://www.nightskynation.com/pics/stars-Globular-Cluster-M80.jpg

We know that a cluster of stars is virialized, which means that the total kinetic energy of the system must be equal to half of the potential energy:

$K=-\frac{1}{2}U$

The total potential energy, U, in the cluster is given by:

$U= -\dfrac{GM^2}{R}$

And the kinetic energy:

$K=\frac{1}{2}M\langle v \rangle ^2$

Putting these into the equation for Virial Theorem, we get:

$\frac{1}{2}M\langle v \rangle ^2 =-\frac{1}{2}\left(-\dfrac{GM^2}{R}\right)$

Finally, solving for the mass, we get the expression:

$M=\dfrac{R\langle v \rangle ^2}{G}$

I would like to thank Charles Law and Daniel Chen for their help in solving this problem.