2. Projection effect of aspherical clusters
In order to study the effect of aspherical clusters in present SZ and X-ray Hubble constant, we extend the work of Fabricant et al. (1984) to calculate the X-ray surface brightness and the SZ temperature change produced by clusters with ellipsoidal geometries. Independent of the cluster shape, the X-ray surface brightness towards a clusters is given by:
where . In order to model the electron number density profile within clusters, we consider the -model, which can be written as:
where and are coordinates of the ellipsoid axes, while and are the observed semi-major and semi-minor axes. To simplify the calculations, we assume that the symmetry axis is at an inclination angle to the line of sight along the observer, which we take to be the z-axis. Following Fabricant et al. (1984, Appendix A), we integrate along the z-axis to derive:
The other important observable towards clusters is the SZ effect, which is given by:
is the frequency dependence with , (Fixsen et al. 1994) and is the cross section for Thomson scattering. The integral is performed along the line of sight through the cluster. As with the X-ray surface brightness, we consider the same ellipsoidal shape to evaluate the observed SZ temperature change. Again by integrating along the line of sight, z-axis, we derive:
The Hubble constant is usually derived by combining the X-ray brightness and the SZ temperature change to eliminate the central number density . By this combination, one can derive the observed length of one of the axis, e.g.:
where Z is the scale factor first introduced in Birkinshaw et al. (1991), which can now be written as:
When the symmetry axis of the cluster is along the line of sight (), then which is directly related to the observed cluster ellipticity, while when the cluster is spherical (), , and no effects due to projection is present in the data. In Eq. 7, we know from SZ and X-ray observations all the quantities except the scale factor Z. Therefore, the length of the cluster along the line of sight can be known up to a multiplicative factor. The Hubble constant is derived based on the angular diameter distance to the cluster, , using an assumed cosmological model, and the observed size of the axis, , used to calculate the distance in Eq. 7. The derived Hubble constant can be written as:
Based on observed ellipticities of galaxy clusters, we can estimate the expected error in the Hubble constant. Using X-ray emission from a sample of clusters, Mohr et al. (1995) showed that the median ellipticity is 0.25. This suggest that the ratio is 0.7 if clusters are intrinsically prolate or 1.5 if clusters oblate. Therefore, ignoring the effects due to inclination, the Hubble constant as measured from SZ and X-ray observations of an individual cluster can be offseted as much as 30% to 50%, based on a spherical model of clusters where asphericity is ignored. Here, we have assumed that clusters are ellipsoids. The derived scale factor in Eq. 8, as well as the numerical values, are likely to be different if clusters are biaxial or triaxial. Recently, Zaroubi et al. (1998) studied the projection effects of biaxial clusters and determined , where is the inclination angle. The observational evidence which suggest clusters are biaxial is limited. For ellipsoidal clusters, we have determined that h varies with both the inclination angle and the sizes of semi-major and semi-minor axes. For triaxial clusters, it is likely that h will vary with all three rotation angles and the length scales of the three axes that define the cluster. In a future paper, we plan to study the projection effects of triaxial clusters; for the purpose of this paper, we will only consider ellipsoids.
Apart from the Hubble constant, the projection effects are also present in the total gas mass derived from the X-ray emission observations with , and the total mass based on the virial theorem using X-ray temperature as . Then, the gas mass fraction can be written as . Since and the , we expect the and to exhibit a negative correlation, if both measurements are affected by the projection effect.
© European Southern Observatory (ESO) 1998
Online publication: October 21, 1998