Astron. Astrophys. 363, 415-424 (2000)

4. Discussion and conclusions

We have calculated the relative error caused by assumptions regarding finite extension, a polytropic temperature profile, ellipsoidal geometry and projection effects, on the measurements of the X-ray surface brightness, the SZ temperature decrement and the determination of the Hubble constant. Although the X-ray data have improved dramatically in the last decade, it is still difficult to determine the internal structure of clusters from X-ray imaging alone, because such images supply only projected temperature and surface brightness information, without further indications of the internal gas dynamics. Nevertheless, recent observations show indirectly that many clusters are still dynamically evolving (see Mohr et al. 1995).

Cooray (2000) has discussed intrinsic cluster shape, in particular considering axisymmetric models such as oblate and prolate ellipsoids, using the Mohr et al. (1995) cluster sample. Their study shows that clusters do indeed have aspherical profiles, which are more likely described as prolate rather than oblate ellipsoids. Nevertheless, Mohr et al. (1995) remarked that they cannot rule out the possibility that clusters are intrinsically triaxial.

Pierre et al. (1996) studied with ROSAT the rich lensing cluster Abell 2390 and determined its gas and matter content. They found that on large scales the X-ray distribution has an elliptical shape with an axis ratio of minor to major half axis of . Using our results we see that this corresponds to a relative error in the y parameter of up to 10%, depending on the line of sight and the shape of the cluster (prolate or oblate, see Fig. 4). The surface brightness measurements lead to errors of up to 25% (see Fig. 5) and thus the Hubble constant is overestimated by about 23% (see Fig. 6).

An unresolved temperature gradient in the gas affects the gas profile and thus the total mass derived assuming hydrostatic equilibrium. If such a gradient is present, the true temperature in the central region may be higher than the emission-weighted temperature generally used. As an example, Grego et al. (2000) observed in Abell 370 a slow decline of the temperature with radius. The temperature falls to half its central value within 6-10 core radii. This temperature profile can be approximately described by a gas with a polytropic index of , which in itself is already an important modification with respect to an isothermal profile and could lead to a relative error of 37% in the evaluation of the Hubble constant (see Fig. 3).

Furthermore, the optical and X-ray observations of this cluster show a possible bimodal mass distribution. Thus, the combined temperature and geometry effects must be taken into account to obtain reliable values for such parameters as the gas and total matter content. A similar polytropic index () has also been found for Abell 3562 (Ettori et al. 2000).

Cooling flows in galaxy clusters can substantially change the temperature profiles, especially in the inner regions. Schlickeiser (1991) and Majumdar & Nath (2000) have investigated the changes induced by a cooling flow in the temperature and density profiles, and their implications on the SZ effect. We notice that for a polytropic distribution, the density profile can still be well approximated by a profile, whereas for cooling flow solutions the density becomes quite different. For example, Vikhlinin et al. (1999) showed that outside the cooling flow region, the -model describes the observed surface brightness closely, but not precisely. In this context, Majumdar & Nath (2000) found that the presence of a cooling flow in a cluster can lead to an overestimation of the Hubble constant determined from the SZ decrement.

Recently, Mauskopf, Ade, Allen et al. (2000) determined the Hubble constant from X-ray measurements obtained of the cluster Abell 1835 with ROSAT and from the corresponding millimetric observations of the SZ effect with the Sunyaev-Zel'dovich Infrared Experiment (Suzie) multifrequency array receiver. Assuming an infinitely extended, spherical gas distribution with an isothermal equation of state, characterized by , keV and cm^-3, they found a value of km s^-1 Mpc^-1 for the Hubble constant. In Fig. 9 we show the influence of geometry and of assumptions of finite extension on this result using the same input parameters. Fig. 9a shows that for a spherical geometry, displays a strong dependence on the cluster extension. Fig. 9b gives the value of assuming an infinite extended ellipsoid shaped cluster (instead of a spherical geometry), as a function of its axis ratio .

Fig. 9a and b. The Hubble constant derived from the data of Mauskopf et al. (2000). (a) shows the influence of finite extension, while (b) gives the value of assuming an axisymmetric ellipsoidal geometry. In the latter case, oblate or prolate geometry give the same value of when taking a line of sight through the cluster center, as is assumed here.

In summary, we see that it is crucial to know the shape of a cluster and its temperature profile. For this problem, the new X-ray satellites have the necessary spatial and spectral resolution to remove the effects of contaminating sources in the field and to measure the spatial variation of the cluster temperature. In this context it will be better for future studies to focus on nearby cluster samples, which are less subject to observational selection effects (as mentioned by Roettiger et al., (1997)).

© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
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