A test of galaxy cluster fundamental plane for the X-COP sample

We test the galaxy cluster fundamental plane using the X-COP sample of 12 clusters. The fundamental plane is given by the relation $T_X \propto M_s^{\alpha} r_s^{\beta}$, where $T_X$, $M_s$, and $r_s$ correspond to the gas temperature, NFW halo mass, and scale radius, respectively. We did this analysis using two different temperatures: the error-weighted temperature in $(50-500)h^{-1}$ kpc as well as the mass-weighted temperature in the same range. With both these temperatures, we find a very tight fundamental plane with dispersion of about 0.02 dex. The best-fit values for $\alpha$ and $\beta$ are in-between those expected from virial equilibrium and self-similarity solution for secondary infall and collapse, with $\alpha$ being closer to the virial expectation. Our best-fit values are also consistent with a recent re-analyses of the fundamental plane for the CLASH sample, after excluding the hottest clusters.