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Astron. Astrophys. 339, 623-628 (1998)

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3. Hubble constant and gas fraction

Table 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 [FORMULA] 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 [FORMULA] 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 [FORMULA], [FORMULA], and H0 = 50 [FORMULA] km s-1 Mpc-1. In order to facilitate comparison between the gas mass fractions measured at various radii, we scaled them to the [FORMULA] radius based on relations presented by Evrard (1997). The [FORMULA] radius has been shown to be a good approximation to the outer hydrostatic boundary of galaxy clusters (Evrard, Metzler, Navarro 1996). We list the derived cluster gas mass fraction at the [FORMULA] radius in Table 1.


Table 1. SZ Effect/X-ray [FORMULA] Measurements and X-ray Gas Mass Fractions.

In Fig. 1, we show the calculated [FORMULA] against [FORMULA] values for each of the clusters. As shown, the gas fraction measurements have a broad distribution with a scatter of [FORMULA] 40% from the mean value. A similar broadening of the Hubble constant, from 30 to 70 km s-1 Mpc-1 with a mean of [FORMULA] 50 km s-1 Mpc-1 is observed. The correlation is negative, and suggest that clusters with high gas mass fraction measurements produces Hubble constant values at the low end of the distribution, while the opposite is seen for clusters with high gas mass fraction. The solid line in Fig. 1 is the best-fit relation between h and [FORMULA] assuming [FORMULA]. For values in Table 1, the best-fit line, when the slope between h and [FORMULA] is allowed to vary, scales as [FORMULA], which is fully consistent with the expected relation. Since the current SZ cluster sample is small, a careful study of a complete sample of galaxy clusters are need to fully justify the projection effects between SZ and X-ray derived Hubble constant and gas mass fractions values. We derived a similar negative correlation between h and [FORMULA] when the cluster gas mass fraction is measured from SZ. For example, Myers et al. (1997) derived a gas mass fraction of ([FORMULA]) [FORMULA] for A2256, which is at the low end of the gas mass fraction values, while a gas mass fraction of ([FORMULA]) [FORMULA] was derived for A478, which is the cluster at the high end. We note here that, as we discuss later, the SZ derived gas mass fractions scale with h as only [FORMULA], while X-ray derived gas mass fractions, which are presented in Table 1, scale with h as [FORMULA]. In comparison, the gas mass fractions derived from SZ and X-ray observations may be affected similar to the measurements based on only the X-ray data. Additional probes of the total mass are the gravitational lensing measurements and the optical virial analysis of internal galaxy velocity dispersion measurements. In the present SZ/X-ray sample, A2218 (Kneib et al. 1995) and A2163 (Squires et al. 1997) have lensing mass measurements. In both these clusters total virial masses when measured using X-ray gas temperature, agrees with the weak lensing mass measurements at large radii, and since these two clusters are not the ones which are primarily responsible for the observed negative correlation, we cannot state the effect of lensing mass measurements on the above data. Also, in the present SZ cluster sample, A2256 and A2142 (Girardi et al. 1998), and Cl0016+16 (Carlberg et al. 1997) have measured total masses from optical virial analysis. These virial masses are in good agreement with X-ray masses, allowing an independent robust measurement of the total mass (Girardi et al. 1998).

[FIGURE] Fig. 1. The observed gas mass fraction of galaxy clusters and the SZ/X-ray Hubble constant. The vertical dashed line is the mean value of the gas mass fraction. The solid line is the best-fit relation between the [FORMULA] values and the [FORMULA] values, assuming [FORMULA]. This line is favored at [FORMULA] 2 [FORMULA] confidence over a constant [FORMULA].

Finally, there is a slight possibility that the observed broad distribution and negative correlation in [FORMULA] and [FORMULA] is not really present. The negative correlation is only present at a level of [FORMULA] 2 [FORMULA], assuming that the errors in h and [FORMULA] are independent. The removal of either one of the clusters at high or low end reduces the negative correlation, decreasing the significance of the observed correlation. However, both the Hubble constant and, possibly, the gas mass fraction is expected to be constant, suggesting that a point, or a region when considering errors in [FORMULA] and [FORMULA], is preferred. We rule out the possibility that both [FORMULA] and [FORMULA] are constants in the present data with a confidence greater than 95%.

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 [FORMULA] and [FORMULA].

It has been suggested that cluster gas clumping may overestimate [FORMULA] from the true value. As reviewed in Evrard (1997), cluster gas clumping also overestimates [FORMULA], suggesting that if gas clumping is responsible for the observed trend, a positive correlation should be present. The nonisothermality underestimates [FORMULA] by as much as 25% (e.g. Roettiger et al. 1997). To explain the distribution of [FORMULA] 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 [FORMULA] measurements as measured using [FORMULA]-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 [FORMULA] and [FORMULA], 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 [FORMULA], and its distribution.

The third possibility is the cluster asphericity. The effect of cluster projection on [FORMULA] was first suggested by Birkinshaw et al. (1991), who showed that the derived values for [FORMULA] 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 [FORMULA] may be offset as much as [FORMULA] 27% (e.g., Holzapfel et al. 1997). The present [FORMULA] 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 [FORMULA] 40%. The [FORMULA] distribution is similar to what has been seen in Cen (1997). It is more likely that the projection effects are causing the distribution of [FORMULA] and [FORMULA] 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.

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Online publication: October 21, 1998