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

1. Introduction

Over the last few years, studies on galaxy clusters using X-ray emission observations have been a source of a tremendous increase in the literature, especially those using Sunyaev-Zel'dovich (SZ) effect. The SZ effect (Sunyaev & Zel'dovich 1972, Rephaeli 1995, Birkinshaw 1999) is one of the major sources of secondary anisotropies of the Cosmic Microwave Background (CMB) arising from inverse Compton scattering of the microwave photons by hot electrons in clusters of galaxies.

Many different works have been developed during recent years leading to the use of this effect for studies of cosmology (Bernstein & Dodelson 1990; Aghanim, de Luca, Bouchet et al. 1997; Barbosa, Bartlett, Blanchard et al. 1996; Bartlett, Blanchard & Barbosa 1998; Cooray 1998). Observations in the millimetre and submillimetre wavebands (Perrenod & Lada 1979; Chase et al. 1987; Silverberg et al. 1997) give important information on the characteristics of clusters of galaxies. For example, by combining the SZ intensity change and the X-ray emission observations, and solving for the number density distribution of electrons responsible for both these effects (after assuming a certain geometrical shape), the angular diameter distance to galaxy clusters can be derived. Assuming a cosmological model, this leads to an estimate of the Hubble constant (Holzapfel, Arnaud, Ade et al., 1997 for Abell 2163; Birkinshaw & Hughes, 1994 for Abell 2218).

The SZ effect thus offers the possibility to put important constraints on the cosmological models. For this reason, different projects to measure the SZ effect are under way for example the MITO instrument (De Petris, Aquilina, Canonico et al., 1996) or (the longer term) the Planck mission (ESA report 1997).

The SZ effect is difficult to measure accurately, since systematic errors can be significant. For instance, Inagaki, Suginohara & Suto (1995) made an analysis of the reliability of the Hubble constant determination based on the SZ effect.

An additional effect arises if the cluster has a peculiar velocity (kinematic effect). Several papers discussed the influence of the kinematic effect on the measurement of the thermal SZ effect (De Luca, Désert & Puget, 1995; Audit & Simmons, 1999 for transverse clusters velocities). Note that the kinematic effect can thus be used to infer the peculiar velocity of clusters of galaxies, if the value of the Hubble constant is known (Rephaeli & Lahav, 1991; Haehnelt & Tegmark, 1996). Another possible distortion on the SZ effect is due to gravitational lensing (Blain, 1998; Roettiger et al., 1997).

The extension and the geometry of hot gas distribution in clusters of galaxies is also an important source of systematic errors in the SZ effect. Cooray (1998) showed that projection effects of clusters can affect the calculations of the Hubble constant and the gas mass fraction. Recently, Sulkanen (1999) showed that galaxy cluster shapes can produce systematic errors in measured via the SZ effect. It is thus necessary to know, at least approximately, the shape of the clusters, for instance, if they are oblate or prolate (Cooray 2000, Hughes & Birkinshaw 1998) or have a more general geometry.

The -model (Cavaliere & Fusco-Femiano 1976) is widely used in X-ray astronomy to parametrize the gas density profile in clusters of galaxies by fitting their surface brightness profile. Nevertheless, fitting an aspherical distribution with a spherical -model can lead to an important inaccuracy (see Inagaki, Suginohara & Suto 1995).

The aim of this paper is to investigate the influence of the shape and the finite extension of an ellipsoidal cluster gas distribution on the SZ effect, and to discuss the possible errors induced in the inferred value for . The paper is organized as follows:

In Sect. 2 we present the calculations of the SZ effect and the X-ray surface brightness for an ellipsoidal shape with an isothermal profile and a finite cluster extension. Details of the calculations are reported in two appendices.

Sect. 3 is then devoted to a quantitative discussion of the incidence of these effects on the SZ measurements, in particular, the finite extension and the geometry (prolate and oblate) of the cluster. The influence of a polytropic thermal profile on the SZ measurements is also considered.

The discussion and conclusion are given in Sect. 4.

© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
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