## 2. Projection effect of aspherical clustersIn 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 The other important observable towards clusters is the SZ effect, which is given by: where 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, 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 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 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 |