virial mass
/ VEER-ee-uhl /
Suppose you find a crowd of people running around inside a fenced yard, never escaping. From how fast they run and how big the yard is, you could work out, with a bit of physics, how strong the fence must be to keep them in. For a cluster of galaxies or a globular cluster there is no fence — only gravity holds it together — but the same logic works. Measure how fast the members move and how far apart they are, and you can deduce the total mass whose gravity does the holding. That deduced mass is the virial mass.
The tool behind it is the virial theorem, a result from classical mechanics. For a stable, gravitationally bound system that is neither collapsing nor flying apart, it links the average kinetic energy of motion to the gravitational potential energy: roughly, twice the kinetic energy equals minus the potential energy. Turned into a practical recipe, the mass comes out as about M is approximately (velocity dispersion squared times size) divided by the gravitational constant G. So if you can measure how fast the galaxies move (their velocity dispersion) and the cluster's radius, you get its total mass — no light required.
Virial masses are how astronomers 'weigh' clusters using motion alone, and they keep coming out far larger than the mass of everything that shines. The visible stars and the hot X-ray gas together account for only a small fraction; the rest is dark matter. The method's weakness is its key assumption — that the system is relaxed and in equilibrium. A cluster caught mid-collision (like the Bullet Cluster) violates this, which is why independent mass measurements from gravitational lensing are so valuable as a cross-check.
For a cluster with a velocity dispersion of 1,000 km/s and a radius of a few million light-years, the virial theorem yields a mass of order 10^15 solar masses — roughly a thousand trillion Suns. Yet its visible galaxies and hot gas total only a small fraction of that, so most of the virial mass must be dark.
The virial theorem turns measured speeds and sizes into a total mass — including invisible mass.
The virial theorem only applies to systems in equilibrium. Apply it to a collapsing, exploding, or colliding system and the mass you get will be wrong — a real pitfall for clusters that are still forming or merging.