Hydrostatic Equilibrium in Galaxy Clusters

One of the methods of mass calculation in Galaxy Clusters is to make a profile of the X-Ray gas and then (after several assumptions) assume hydrostatic equilibrium: the force pushing down on the gas (gravity) is equal to the amount of force pushing up (pressure of the plasma). What is unclear is: how do we know the gas inside the cluster isn't spinning? It seems to be that any angular momentum, however small, would invalidate the assumptions of equilibrium.

Answer

It is correct that X-ray estimates of gas mass usually make the assumption of hydrostatic equilibrium and ignore rotation. Both of these assumptions are open to challenge. Clusters are thought to have formed through a rich history of mergers that could leave them out of equilibrium or impart some angular momentum. These issues are of course being thoroughly explored, are generally well known in the literature, and are comparatively small effects compared with the "dark matter" problem (see for example Suto et al. 2013).

For the angular velocity to make a significant difference to the mass calculations then any centrifugal acceleration would need to be comparable with the gravitational acceleration.

Let's take the Coma cluster and assume a total mass of $10^{15}M_{\odot}$ within about 5 Mpc radius. Equating $$\omega^2 R = GM/R^2,$$ we find rotation is important if $\omega \sim 6\times 10^{-18}$ rad/s. Sounds not much, but this means a rotation velocity of about 1000 km/s.

Detecting such gas motions in X-ray data with relatively modest spectral resolution is difficult. One can try to look for systematic doppler shifts in ionised iron lines. As far as I can tell, a 1000 km/s rotation is just on the edge of detectablity and has been observed in the Centaurus cluster by Dupke & Bregman (2006). A source of additional support besides pressure, as could be contributed by rotation or turbulent motions would mean that the X-ray profile fitting would underestimate the total cluster mass (see for example Suto et al. 2013) . Zhang et al. (2010) compare total cluster masses found from weak lensing and from X-ray measurements. The agreement was within 10% ! However, there is a hint of additional non-thermal support for some of the clusters.

Instead of directly looking at the gas we could look to the galaxies in clusters for evidence of rotation. The velocity dispersion of the galaxies in the Coma cluster is also of order 1000 km/s (not a coincidence), but individual galaxy velocities can be measured to a precision that is a small fraction of this - easily sufficient to detect rotations of even 100 km/s, yet no definitive evidence for rotation in Coma has been found. In fact another way of thinking about this, is that the rotation will become important in the virial theorem if the systematic rotation accounts for a lot of the observed velocity dispersion.

Hwang & Lee (2007) have done a systematic search for rotation using SDSS data. They find that about 20% of Galaxy clusters have significant rotation (rotational velocities becoming comparable to the velocity dispersion). In their table 6, the re-derive the total masses of these clusters, finding that they decrease by around 20% on average. So again, the effect is small (compared with the "dark matter deficit"), but worth considering.

Edit: A new study by Smith et al. (2015) appeared today ( http://arxiv.org/abs/1511.01919 ) that compares weak lensing masses with derivations based on X-rays and hydrostatic equilibrium. The headline ratio is 0.95 +/- 0.04 indicating that departures from hydrostatic equilibrium are generally small, at least in the local universe (z<0.3). However they do point out that the ratio falls to 0.6 for higher redshift clusters, but suggest that the data quality or other measurement biases are most likely responsible.

EDIT2: Another recent paper. Maughan et al. (2015) compare X-ray-derived masses (assuming hydrostatic equilibrium) with masses derived using the "caustic technique" (this is essentially using the kinematics of the visible galaxies to determine a mass profile). They conclude that "There is no indication of any dependence of the mass ratio on the X-ray morphology of the clusters, indicating that the hydrostatic masses are not strongly systematically affected by the dynamical state of the clusters. Overall, our results favour a small value of the so-called hydrostatic bias due to non-thermal pressure sources." The paper also has a good introduction that refers to a few other recent papers in the literature.