The X-ray Fundamental Plane and $L_X-T$ Relation of Clusters of Galaxies

We analyze the relations among central gas density, core radius, and temperature of X-ray clusters by plotting the observational data in the three-dimensional ($\log {\rho }_0$, $\log R$, and $\log T$) space and find that the data lie on a ‘fundamental plane’. Its existence implies that the clusters form a two-parameter family. The data on the plane still has a correlation and form a band on the plane. The observed relation $L_{\mathrm{X}}\propto T^3$ turns out to be the cross section of the band perpendicular to the major axis, while the major axis is found to describe the virial density. We discuss implications of this two-parameter family nature of X-ray clusters.