Worksheet 8, Problem 3 - The Virial Theorem: All About that Mass

Question: 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 physics constants.

Okay, so we are not given a lot of information here. Let's extrapolate from the information given to us to see if there is more things we can find out about this scenario. We know we are dealing with a star cluster, so let's draw one below:

A star cluster typically has thousands of stars, each with a mass, $m$. To account for the mass of the entire star cluster, we need to sum the masses of each individual star, $m_i$, such that the mass of the entire cluster is represented by: $M = \sum_0^n m_i$, where $n$ is the number of stars in the star cluster, $m_i$ is the mass of any given individual star in the star cluster, and $M$ is the total mass of the entire star cluster.

Now that we can represent the mass of the entire cluster as $M$, let's consider the energy present in the star cluster. Everything in the universe has gravitational potential energy, and everything not at absolute zero has kinetic energy. Let's try to quantify the kinetic and potential energy of this star cluster.

The kinetic energy $K$ of the star cluster is given by: $K = \frac{1}{2}M<v>^2$, where $M$ is the mass of the star cluster, and $<v>$ is the average speed of an individual star in the cluster.

The potential energy $U$ of the star cluster is given by $U = -\frac{GM^2}{R}$, where $M$ is the mass of the star cluster, $R$ is the radius of the star cluster, and $G$ is the universal gravitational constant.

Knowing the average velocity of the individual stars in the star cluster $<v>$, and the radius of the star cluster $R$, we can use the Virial Theorem, which relates the total kinetic and potential energies of a stable system like the star cluster system, to solve for the mass of the entire cluster:

\begin{align}
K &= -\frac{1}{2}U  \qquad \leftarrow \text{Virial Theorem}\\
\frac{1}{2}M<v>^2 &= -\frac{1}{2}(-\frac{GM^2}{R})\\
<v>^2 &= \frac{GM}{R}\\
M &= \frac{<v>^2 R}{G}
\end{align}

We have derived the equation to solve for the total mass $M$ of a star cluster given it's radius $R$ and the velocity of the average star in the star cluster $<v>$ using the Virial Theorem!