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.