In the Dark

Here’s something a bit different. I was talking the other day with some folks here about the use of the Virial Theorem to measure masses of galaxy clusters. In case you’ve forgotten,  an important consequence of the virial theorem is that the average potential energy of an isolated system in gravitational equilibrium is equal to minus twice the average kinetic energy, i.e.

Being mathematicians they wanted to  have a precise definition of when this theorem holds, i.e. what it means for a system to be in virial equilibrium. I have to admit I was a bit stumped.

The problem is that the proof of the theorem (which you can find on the wikipedia page) involves assuming that the time-average of a scalar quantity (the virial), derived from the positions and momenta of the particles in the system, is zero. That’s fine, but the average is taken over an infinite time and most cosmic objects we apply it too are rather younger than the age of the Universe. So how accurately does it apply to, e.g., galaxy clusters? How large are the fluctuations about the mean?

Another problem is that clusters aren’t really isolated either. According to prevailing wisdom clusters sit at the intersections of filaments and sheets of dark matter from which matter continually accretes onto them, increasing their mass.

Clusters also contain a sizeable amount of substructure. Does this cast further doubt on how well actual clusters are described by the virial theorem?

I’ve heard a number of lectures and seminars about virial mass estimates of clusters but never have I heard a precise, testable definition of when it is expected to apply and how large the deviations from it are in realistic situations. I’ve taught courses in which the theorem is applied to a variety of situations, but I never looked too deeply into its foundations – which is, of course, very sloppy of me.  I tried asking a few people, and posted a question of Twitter, but didn’t get a really convincing response. Naturally, therefore, I decided to try it out on the readership of this blog….

So, please, would anyone out there please give me a precise  testable definition of what is meant by a “virialised system”  and explain how how well the virial theorem is supposed to apply to real clusters? Pointers to convincing discussions in the literature would be welcome!