Worksheet 8, Problem 3: Star Cluster

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

Let's start again by drawing out our system:

Let's start by recalling the virial theorem: \( K = - \frac{1}{2} U \). Now, let's write out the full expressions for both Kinetic Energy, K, and Potential Energy, U. \[ K = \frac{1}{2} M v^2 \] \[ U = -G \frac{M^2}{R} \] In these expressions, M is equal to the total mass of all stars in the system, and v is the average velocity of the stars. Now, relating our two energies using the virial theorem, we get: \[ \frac{1}{2} M v^2 = \frac{1}{2} G \frac{M^2}{R} \] Solving for M, we get: \[ M = \frac{Rv^2}{G} \]

Acknowledgements: Johnathan Budd & Willie Pirc Team EE