Clusters Background Information: Virial Theorem Clusters Home

So how can we measure the mass of a galaxy cluster? We use the concepts of conservation of energy and virial equilibrium. What does this mean?

Let's think for a minute about a planet orbiting a star. There are two two major types of energy in this system: kinetic energy, or the planet's energy of motion, and potential energy, the energy provided by the gravitational field of the star. Mathematically, we can write these two quantities as:

Kinetic Energy
(where m is the mass of the planet, and v is its speed)
and potential energy
(where M is the mass of the star, r is the star-planet distance, and G is the gravitational constant)

Conservation of energy is one of the basic laws of physics and says that as the planet orbits, the total energy E=K+U does not change with time. If the planet is on an elliptical orbit, it may get closer to the star, so the potential energy will become more negative. But the planet will speed up to compensate, raising its kinetic energy and ensuring that total energy will stay constant.

The virial theorem, on the other hand, says that for any system in equilibrium, the total energy E is equal to one half the potential energy, U. So still thinking about our planet orbiting a star, we have

(1)

But we also had from conservation of energy:

(2)

If we combine (1) and (2), we can solve for M, the mass of the star, as

(3)

Now let's think about clusters of galaxies. Things are a little different here, since there is no single massive object at the center of the cluster. Instead, we have many different galaxies orbiting under their cumulative gravitational pull, as well as the pull from any other matter in the cluster. So the detailed numbers will change, but conceptually the task is the same: we want to measure the orbital speeds of galaxies in the cluster, as well as get an estimate of the size of the cluster. Then we can estimate the total mass from a variant of equation (3).