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Astron. Astrophys. 321, 64-70 (1997)

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3. Mass distributions

In this section we will described the method we used to deduce the cluster total mass, gas mass and other associated physical parameters from the X-ray data and compare some of these parameters (e.g. velocity dispersion) with the optical data available. We will assume that the gas is in hydrostatic equilibrium and isothermal. The X-ray surface brightness given in equation  1corresponds exactly to a gas distribution of

[EQUATION]

if the gas extends all the way to infinity. Through the equation of hydrostatic equilibrium, the gas distribution is related to the cluster gravitational potential [FORMULA] as follows:

[EQUATION]

where [FORMULA] is the gas temperature, [FORMULA] is the proton mass and µ is the mean molecular weight of the gas in proton mass units. Thus the cluster gravitational potential corresponding to the gas distribution of equation  2is given by

[EQUATION]

where [FORMULA] is the central 3-D velocity dispersion and is related to [FORMULA] through [FORMULA].

Thus from Poisson's equation the total mass density is given by:

[EQUATION]

and the cluster total mass is given by:

[EQUATION]

From the X-ray spectral data, we found the best estimate emission measure and temperature assuming Raymond-Smith spectra. Thus we have the X-ray flux, [FORMULA], within a radius of [FORMULA] (i.e. 0.65 Mpc) over the Rosat energy band of [0.1,2.4] keV, from the spectral data. The X-ray surface bightness profile given in Fig. 2 gives the shape of the profile, namely [FORMULA] and [FORMULA]. The central electron density [FORMULA] can then be estimated from [FORMULA], [FORMULA] and [FORMULA] (see Table 5). The cluster total mass and gas mass within the radius of [FORMULA] (or 1.3 Mpc), i.e. the maximum extent where there is still detectable X-ray emission from the surface brightness profile, are given in Table 5. The radius of [FORMULA] (or 1.3 Mpc) is chosen such that no extrapolation would be necessary in calculating the masses from the X-ray data. The total X-ray luminosity calculated up to the same radius of 1.3 Mpc was found to be consistent with the [FORMULA] relation given by Ebeling (1993).

We compare the X-ray deduced velocity dispersion of the cluster with the measured galaxy velocity dispersion. From Jean's equation for a spherical, steady state system with isotropic velocity distribution, we can deduce the 3-D velocity dispersion [FORMULA] from the potential given in equation  4and the total mass density (equation  5) if the galaxy number distribution follows that of the total mass. Thus the 3-D velocity dispersion in this case is given by

[EQUATION]

where [FORMULA] is the 3-D radius in core radius units and [FORMULA] is the 3-D velocity dispersion at the centre. In producing the above analytic expression, we have assumed that the cluster extends to infinity and that the galaxies can be treated as test particles in the cluster potential with a distribution that follows the mass. Fig. 3 shows the galaxy number count isocontours overlaid on the X-ray image. The southern structure in the isopleth map was identified with a background group (Colless 1987). Note also the coincidence between the X-ray emission and the optical structures. Since the cluster is fairly regular, it can be circularly averaged to produce a galaxy number density distribution. We found that the galaxy density distribution thus deduced agreed well with the shape of the projected mass distribution, thus justifying the mass-follows-light assumption. The line of sight velocity dispersion [FORMULA], can be projected from the 3-D dispersion through (Merritt 1987):

[EQUATION]

where [FORMULA] is the projected mass density, [FORMULA] is the projected radius in core radius units. Therefore, the line of sight velocity dispersion [FORMULA] is given by

[EQUATION]

(see Appendix of Mellier et al 1994, but note there is a misprint). However, the measured velocity dispersion in most cases is the average of [FORMULA] within a certain radius. We derive the [FORMULA] from the above and obtain

[EQUATION]

[FIGURE] Fig. 3. An image of the Rosat PSPC image superimposed on the galaxy number counts isocontours (obtained from Colless 1987). The axes are in decimal degrees. The contour levels are (0.06, 0.12, 0.18, 0.24, 0.3, 0.36) galaxies arcmin -2.

We can thus estimate [FORMULA] from [FORMULA] and [FORMULA] and deduce [FORMULA] Mpc [FORMULA] km s -1. Note that [FORMULA] Mpc [FORMULA], thus the difference is significant and the usual assumption of a constant velocity dispersion with radius could easily produce the difference between [FORMULA] and [FORMULA] noted in many clusters (Sarazin 1988). However, in the case of A 2717, this difference is still not large enough to obtain consistency. The measured [FORMULA] of A 2717 from the galaxies within a radius of 2 Mpc was [FORMULA] km s-1 (Colless et al. 1987). Thus the measured galaxy velocity dispersion was slightly higher than that deduced from the X-ray data. As mentioned in Colless et al. (1987), there is evidence of foreground and background groups that may have contaminated the measurements and thus overestimate the velocity dispersion.

Given the electron density and gas temperature we can calculate the central cooling time, [FORMULA], given by

[EQUATION]

(Sarazin 1988) and for A 2717, it amounts to [FORMULA] Gyr which is smaller than the Hubble time ([FORMULA] Gyr). Thus in the absence of heating, cooling occurs within the centre of the cluster with a cooling radius of [FORMULA] kpc. Pending further evidence of the existence of a cooling flow, this maybe one of the few cases where a WAT is found in a cooling flow cluster (c.f. Burns 1990).

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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