Resolving the Beta-Discrepancy for Clusters of Galaxies

Previous comparisons of optical and X-ray observations of clusters of galaxies have lead to the so-called ``$\beta$ - discrepancy'' that has persisted for the last decade. The standard hydrostatic-isothermal model for clusters predicts that the parameter $\beta_{spec} = \sigma_{r}^{2}/(kT/\mu m_{p})$, which describes the ratio of energy per unit mass in galaxies to that in the gas, should equal the parameter $\beta_{fit}$ (where $\rho_{gas}(r) \propto \rho_{gal}(r)^{\beta_{fit}}$) determined from the X-ray surface brightness distribution. The observations suggest an apparent discrepancy : $\beta_{spec} \sim 1.2$ (i.e., the galaxies are ``hotter'' than the gas) while $\beta_{fit} \sim 0.65$ (i.e., the gas is ``hotter'' and more extended than the galaxies). Here we show that the discrepancy is resolved when the actual observed galaxy distribution in clusters is used, $\rho_{gal}(r) \propto r^{-2.4 \pm 0.2}$, instead of the previously assumed steeper King approximation, $\rho_{gal}(r) \propto r^{-3}$. Using a large sample of clusters, we find best-fit mean values of $\beta_{spec} = 0.94 \pm 0.08$ and $\beta_{fit}^{corrected} = 1.25 \times \beta_{fit} = 0.84 \pm 0.10$. These results resolve the $\beta$ - discrepancy and provide support for the hydrostatic cluster model.