Updating the $\sigma$-$T$ relationship for galaxy clusters
  The relationship between the X-ray determined temperature $T$ of the intracluster gas and the optical measured velocity dispersion $\sigma$ of the cluster galaxies is often believed to be not only a straightforward but also robust test for the dynamical properties of galaxy clusters. Here, we present the $\sigma$-$T$ relationship using the 94 clusters drawn from the largest sample of 149 clusters in literature, for which both $\sigma$ and $T$ are observationally determined. Employment of the doubly weighted orthogonal distance regression to our sample yields $\sigma=10^{2.47\pm0.06}T^{0.67\pm0.09}$, indicating an apparent deviation of dynamical state from that predicted by the isothermal and hydrostatic equilibrium model for galaxy clusters, though the average ratio $\beta_{spec}$ of specific energy in galaxies to that in gas is found to be in excellent agreement with unity. It shows that a nonisothermal gas distribution with a mean polytropic index of $\gamma=1.3$ can account for the reported $\sigma$-$T$ relationship, while overall clusters can still be regarded as dynamically-relaxed systems. 
Xiang-Ping Wu, Li-Zhi Fang and Wen Xu