## Going Virial

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!

April 16, 2012 at 12:10 pm

This is entirely hand-waving, but I can imagine two points towards making this a less obviously invalid approximation.

First, the Wikipedia page says one can take an ‘ensemble average’ rather than a time average. We can’t take a time average in any case as we only observe instantaneous velocities: the rate of change of velocity is unmeasurably small. But if there are many bodies in the system they could be thought of as sampling (incompletely) the varying contributions to kinetic and potential energy which occur over the phase space of the bound system. That’s vague enough, but it might be made more precise by having a well-defined notion of ensemble averaging so as to control the errors inherent in incomplete sampling.

Second, which bodies to include in the system (since no cluster is an island entire in itself…) and what errors you get depending on this choice. You can start off by thinking of a bound system plus a small amount of extra matter which is either falling in from infinity (and so not described by a non-varying time average) or being ejected to infinity (ditto). Perhaps one can try to separate out this extra matter by seeing that it is faster than some position-dependent ‘escape velocity’. Anyway, the measured kinetic energies will be distorted by it disproportionately to the size of its potential energy contribution, so you can think about how inaccurate that makes the mass estimate.

In time (some number of ‘orbits’?) this extra matter will either return to infinity and become irrelevant, or become bound and thus contribute to making the estimate more accurate rather than less. The degree of ‘virialization’ might mean the relative amounts of matter which you believe can be legitimately included in a bound ensemble average — vs. not.

April 16, 2012 at 12:22 pm

I found this paper (Six Myths on the Virial Theorem for Interstellar Clouds) interesting, and there are a few citations to it that you may find useful from a galaxy cluster perspective (e.g. Davis et al 2011).

http://adsabs.harvard.edu/abs/2006MNRAS.372..443B

April 16, 2012 at 2:22 pm

To equate an ensemble average to a time average is the ergodic theorem, which is fearsomely difficult to prove even in the only special cases for which it *has* been proved; and those proofs depend on fine details of the boundaries (eg, proved by Sinai for billiard balls on a table of one shape but not another shape). Also, just how many clusters are there? Playing thermodynamics with small numbers is a dangerous game.

April 16, 2012 at 9:47 pm

Hmm, has this been addressed via, e.g. N-body simulations? Or would the initial conditions just define the outcome? I suppose if we can simulate structure formation, we can make ‘observations’ of simulated clusters and test the validity of the virial assumption under a variety of conditions.

April 17, 2012 at 11:07 am

Yes, that is the topic of the Davis et al 2011 paper I mentioned above (see also references therein). According to simulations, it seems that the clusters have excess kinetic energy and that a better accounting can be done by including the surface terms since the clusters are not isolated. Fig 8 of Hetznecker & Burkert 2006 shows the time evolution of for clusters in their simulations. Not sure how this is testable…

April 19, 2012 at 11:08 pm

Just some thoughts on the issue of virialisation.

The kind of galaxy cluster I would confidently expect to be virialised is one that shows signs of ancient evolution driven by dynamical causes. This means to me those clusters having a large fraction of elliptical galaxies and little active star formation, which I would interpret as evidence of galaxy-galaxy interactions many Gyr ago. That is, however, a very indirect indicator of virialisation.

(It might be worth noting here for the benefit of general readers that we mean that the system of galaxies is virialised with the particles of the system being the individual galaxies. I do not include here the hot intracluster gas or the dark matter.)

It is difficult to imagine a direct test of virialisation. One rather direct test might be to compare the potential and kinetic energies of the galaxies at the present time using observations: is the total potential energy = – 2 X total kinetic energy? However, this would require having information about the mass distribution of the cluster (dark matter, intergalactic gas and galaxies) to calculate the potential (this might come from gravitational lensing, or less directly from the observable components with an assumed mass-to-light ratio), and would require three-dimensional positions of the galaxies (to calculate the potential energy of each which depends on the positions within the potential). It would also require three-dimensional velocities of the individual galaxies to calculate the kinetic energy of each, something that we do not have (this is because we do not know the transverse components of the galaxies’ motions directly [for the uninitiated, we measure only the line-of-sight component of the velocity, from the Doppler shift in a galaxy’s spectrum]).

Whether clusters are virialised does matter. It affects the masses that are calculated for them: virialisation is often assumed. I did worry about this during preparations of a draft paper about poor galaxy groups I was involved with several years ago. I had thought of calculating masses on the assumption of virialisation and then again on the assumption that the groups were still in their initial collapse from a large distance. The groups were too sparsely sampled to allow me to make a serious attempt at the second (I seem to remember needing to know something about the sizes, mean density profiles and centroids which could not be calculated reliably given the small number of galaxies).