3. Analysis methods
3.1. The stellar mass profile
The stellar matter content can be computed at any radius from the cluster center using the projected number density profile of galaxies, their luminosity function and a mass to light ratio for the stellar population calibrated on the observation of nearby galaxies. Most often, the density profile is fitted by the common King form:
where is the galactic core radius. The case is an approximation to the isothermal sphere, in which galaxies have reached their equilibrium distribution. The advantage of such a model is that the volume density is obtained by an analytical deprojection. However, de Vaucouleurs profiles, which are much steeper in the cluster core, provide a better approximation to the real distribution (Rhee & Latour 1991; Cirimele et al. 1997), at the same time leading to a finite total number of galaxies:
p being the projected distance to the cluster centre and r the true distance. Because this deprojection is numerically unstable, we computed it by assuming to be constant inside a grid step and then integrating analytically the denominator. The mass to light ratio applied to all clusters and groups (but the supposed fossil group RXJ 1340.6+4018 consisting of only one giant elliptical galaxy, for which we used ) is , obtained by White et al. (1993) by averaging over the Coma luminosity function the ratio from van der Marel (1991) given as a function of luminosity for bright ellipticals. Then, using the Schechter luminosity function:
the luminosity emitted by a shell of thickness dr and situated at the radius r writes as:
When no parameters for the luminosity function were found in the literature, we adopted the standard ones (Schechter 1975): and .
As a few clusters observed in X-rays do not have any available spatial galaxy distribution (or with too poor statistics), but only either a luminosity profile or even several total luminosities given at different radii, we then assumed a King profile and fitted the few points by the resulting integrated luminosity profile:
by varying simultaneously and . In addition to those cases, RXJ 1340.6+4018 was treated in a special way: we deprojected a de Vaucouleurs luminosity profile (Ponman et al. 1994).
3.2. The X-ray gas mass profile
In their pioneering work, Cavaliere & Fusco-Femiano (1976) have shown under the isothermality assumption that the X-ray gas profile is described by:
which translates to the observed X-ray surface brightness with the following simple analytical form (the so-called -model):
The slope and the core radius , which are interdependent in their adjustment to the surface brightness, are generally found to range between 0.5 and 0.8 and between 100 and 400 kpc respectively. Very often, central regions of clusters have to be excluded from the fit, due to cooling flows resulting in an emission excess. The gas mass can be inferred accurately from the knowledge of , and . Uncertainties in the gas mass are small in general, as long as it is computed inside a radius at which the emission is detected. The relationship between the electron number density and the gas mass density used here is (assuming a helium mass fraction of 24% and neglecting metals).
3.3. The binding mass profile
Mass estimation is certainly the most critical aspect of recent studies of the baryonic fraction in clusters. Clarifying this issue is one important aspect of this paper. We derived the gravitational mass in two ways:
The hydrostatic isothermal -model : First, we used the standard IHE assumption which, using spherical symmetry, translates into the mass profile:
The total mass thus depends linearly on both and . Hence, if the slope of the gas density is poorly determined (and this is the case if the instrumental sensitivity is too low to achieve a good signal to noise ratio in the outer parts of the cluster), it will have a drastic influence on the derived mass. This mass profile results in the density profile:
and in a flat density at the cluster centre. The isothermality assumption can raise doubt, since Markevitch et al. (1998) found evidence for strong temperature gradients in clusters, which may lead to IHE mass estimates smaller by 30% (Markevitch 1998). However, the reality of these gradients has recently been questioned (Irwin et al. 1999; White 2000).
The universal density profile: An alternative approach is to use the universal dark matter density profile of Navarro et al. (1995, hereafter NFW) derived from their numerical simulations:
where stands for the radius from the cluster center where the mean enclosed overdensity equals 200 (this is the virial radius) and is the critical density. It varies as near the centre, being thus much steeper than in the hydrostatic case; NFW claim this behavior fits their high resolution simulations better than a flat profile. Furthermore, contrary to the -model , the dark matter density profile obtained by NFW is independent on the shape of the gas density distribution. This will introduce a further difference. The normalization of the scaling laws ensures a relationship between temperature, virial radius and virial mass. Here, this normalization is taken from numerical simulations. Different values have been published in the literature (see for instance Evrard 1997; Evrard et al. 1996, EMN hereafter; Pen 1998; Bryan & Norman 1998, BN hereafter). Frenk et al. (1999) investigated the formation of the same cluster with various hydrodynamical numerical simulations. They found a small dispersion in the mass-temperature relationship: the rms scatter is found to be of , EMN and BN lying at the edges of the values found, representing a 4 difference. EMN provide a scaling law between and :
(in terms of comoving radius) which was used here to compute , writing:
with , where is a corrective factor to transform dark matter mass into total mass, so that is really equal to 200. Solving this equation gives . The relationship at between virial mass and temperature can then be written as:
BN did provide the following constant of normalization:
This difference is quite significant: it does correspond to a virial mass 40% higher. Using this normalization will obviously significantly change the inferred gas fraction.
© European Southern Observatory (ESO) 2000
Online publication: October 2, 2000