## 3. Hubble constant and gas fractionTable 1 lists the Hubble constant values that have so far been
obtained from SZ observations (Cooray et al. 1998b, see also Hughes
1997). These values have been calculated under the assumption of a
spherical gas distribution with a profile for
the electron number density and an isothermal atmosphere. For the same
clusters, we compiled a list of gas mass fraction measurements using
X-ray, SZ, and gravitational lensing observations. Most of the
clusters in Table 1 have been analyzed by Allen & Fabian (1998),
where they included cooling flow corrections to the X-ray luminosity
and the gas temperature. For the two clusters (A2256 & Cl0016+16)
for which measurements are available, but not
analyzed in Allen & Fabian (1998), we used the results from Buote
& Canizares (1996) and Neumann & Böhringer (1996),
respectively. The gas mass fractions in Allen & Fabian (1998) have
been calculated to a radius of 500 kpc, while for the A2256 and
Cl0016+16, they have been calculated to different radii, and also
under different cosmological models. Using the angular diameter
distance dependence on the gas mass fraction measurements with
redshift (Cooray 1998), we converted all the gas mass fraction
measurements to a cosmology of ,
, and H
In Fig. 1, we show the calculated against
values for each of the clusters. As shown, the
gas fraction measurements have a broad distribution with a scatter of
40% from the mean value. A similar broadening
of the Hubble constant, from 30 to 70 km s
Finally, there is a slight possibility that the observed broad
distribution and negative correlation in and
is not really present. The negative correlation
is only present at a level of 2
, assuming that the errors in ## 3.1. Evidence for a projection effect?Usually, the broad distribution of the SZ and X-ray Hubble constants has been explained in literature based on the expected systematic effects. The systematic effects in the gas mass fraction measurements are reviewed in Evrard (1997) and Cooray (1998). We briefly discuss these systematic uncertainties in the context of their combined effects on and . It has been suggested that cluster gas clumping may overestimate from the true value. As reviewed in Evrard (1997), cluster gas clumping also overestimates , suggesting that if gas clumping is responsible for the observed trend, a positive correlation should be present. The nonisothermality underestimates by as much as 25% (e.g. Roettiger et al. 1997). To explain the distribution of values, the cluster temperature profile from one cluster to another is expected to be different. However, Markevitch et al. (1997) showed the similarity between temperature profiles of 30 clusters based on ASCA data (including A478, A2142 & A2256 in present sample). Since SZ and X-ray structural fits weigh the gas distribution differently, even a similar temperature profile between clusters can be expected to cause the change in the Hubble constant from one cluster to another. Another result from the Markevitch et al. (1997) study is that the measurements as measured using -models and standard isothermal assumption is underestimated. The similarity of cluster temperature profiles also suggests that the gas mass fractions are affected by changes in temperature from one cluster to another. It is likely that the present isothermal assumption has underestimated both and , and that temperature profiles are responsible for the observed behavior. A large sample of clusters, perhaps the same cluster sample studied by Markevitch et al. (1997), should be studied in SZ to determine the exact effect of radial temperature profiles on , and its distribution. The third possibility is the cluster asphericity. The effect of cluster projection on was first suggested by Birkinshaw et al. (1991), who showed that the derived values for can be offset by as much as a factor of 2 if the line of sight along the cluster is different by the same amount. The present cluster isophotal ellipticities suggest that may be offset as much as 27% (e.g., Holzapfel et al. 1997). The present distribution is suggestive of this behavior. Cen (1997), using numerical simulations, studied the effects of cluster projection on gas mass fraction measurements, and suggested differences of the order 40%. The distribution is similar to what has been seen in Cen (1997). It is more likely that the projection effects are causing the distribution of and values, unless a systematic effect still not seen in numerical simulations is physically present in galaxy clusters. Such effects could come from effects due to variations in the temperature profiles from one cluster to another. For the rest of the discussion, we assume that the present values are affected by projection effects, rather than temperature profiles. © European Southern Observatory (ESO) 1998 Online publication: October 21, 1998 |