3. The and relations
3.1. The relation
We cannot directly relate the mass function to observations because we have very little information on the actual virial mass of clusters. Note that in the spherical top-hat model, the virial radius is
which extends out beyond the region of currently available mass determinations. In order to obtain a fruitful comparison of the theoretical mass function with the observations, it is therefore necessary to construct trustworthy relations between X-ray properties and cluster virial masses. This "virial" mass should be understood in the sense in which it is employed in the mass function: it is the mass contained within the region of mean contrast density . Here we consider the and relations. We shall see that the former is more reliable.
Spectroscopic studies have demonstrated that the X-ray emission is produced by thermal bremsstrahlung in an hot, optically thin intracluster plasma with a temperature of approximately . The detailed history of this gas is not well known. The presence of the iron emission line indicates that the intracluster medium (ICM) has been partially processed through the stars of the cluster galaxies. However, the large mass of the ICM, typically several times greater than the cluster stellar mass, leads one to believe that the majority of the gas is primordial in origin, since it seems difficult that the galaxies lose through winds or ram pressure stripping more than 50% of their initial mass. In addition, the measured metallicities close to third can be accounted for by a bimodial star formation model (Arnaud et al. 1992a).
We will work under the hypothesis that the cluster gas is in hydrostatic equilibrium with an isothermal temperature profile. Despite the lack of rigorous evidence, the latter point is at least consistent with the majority of data. Under these conditions we may write
where kT and are, respectively, the temperature and the density of the gas, µ is the mean molecular weight, is the proton mass and is the binding mass of the cluster. The observed X-ray surface brightness profile can be directly converted to a three dimensional density profile:
This is just the deprojection of the isothermal form known to fit the surface brightness profiles of clusters. Best fit values for are typically around 0.6 (Jones & Forman 1984). Using this value, the relation between the temperature and the virial mass can be evaluated as
This result is in reasonable agreement with the hydrodynamic simulations of Evrard (1990a, 1990b), although he obtained a constant of proportionality which is about 20 % lower. Taking this into account, the relation between mass and temperature becomes
In this expression is the cluster virial mass in units of . More recently, Evrard et al. (1996) showed that in the CDM case, this relation between mass and temperature holds with a very good accuray. This equation will allow us to transform the mass function into a temperature function which we can compare to observations.
The temperature can also be related to the initial, comoving radius containing the mass:
This last relation illustrates an important point: as the temperature is independent of the Hubble constant when scales are expressed in Mpc, a model corresponding to a given power spectrum and normalized to a scale also measured in Mpc (such as ) will produce the same cluster abundance per . Therefore, the constraints on n and b inferred from the observed abundances of clusters are independent of the value of the Hubble constant.
3.2. The relation
The bolometric X-ray luminosity of a galaxy cluster due to thermal bremsstrahlung depends strongly on the gas density profile:
Usually, one assumes that the ICM represents a constant fraction of the cluster virial mass. Adopting this hypothesis and assuming an identical radial distribution for both the gas and the dynamical mass, Kaiser (1986) derived a scaling law for the X-ray luminosity: . However, there are reasons to suspect this scaling law. As emphasized by Blanchard et al. (1992b), the total luminosity of a cluster is dominated by the mass of the gas core and the self-similar scaling applies only if the mass of the gas core scales as the virial mass. However, the formation of a core in the gas distribution is not well understood and may result from any of several physical processes, including cooling in the center of the cluster or in the smaller structures from which the cluster was built, preheating by a first generation of collapsed objects, or gas ejection from galaxies. Thus, theoretical modeling of the X-ray luminosity is dangerously uncertain. Indeed, if one combines the self-similar relation with the highly reliable theoretical relation, then one obtains a correlation whose shape is in severe conflict with local data. In addition, Blanchard & Silk (1991) and Evrard & Henry (1991) have shown that self-similar scaling produces a luminosity function which disagrees with observations for both the CDM model and for a model with a power-law power spectrum with index . An alternative to the self-similar scheme is to parametrize the luminosity-mass relation as a power law of the cluster mass and redshift, , and to consider values of the free parameters that fit the observed luminosity function. But as these authors have noted, there is an observational degeneracy between the shape of the assumed initial power spectrum and the chosen relation. Perhaps the best way to deduce the true relation is through the observed relation. This seems a more trustworthy approach if one believes that the temperature reflects the virial energy of the cluster. In the remainder of this paper, we follow this procedure and use the local relation. Edge & Stewart (1991) have found that using EXOSAT data. This relation is very close to that found by Henry & Arnaud (1991) using a compilation of data from EXOSAT, HEAO/OSO and the Einstein satellite. Additionally, Edge & Stewart (1991) have also given the correlation resulting from a minimization of the residuals in . Using the result of a double regression fit in which the slope is defined as the square-root of the product of the two individual regression slopes, we find . We normalize this relation at , the temperature at which the two regressions cross each other, to finally obtain
This expression represents the bolometric X-ray luminosity and we must correct for the fraction
actually collected in the relevant energy band [ - ].
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998