Two analytic relations connecting the hot gas astrophysics with the cold
  dark matter model for galaxy clusters
  Galaxy clusters are good targets for examining our understanding of
cosmology. Apart from numerical simulations and gravitational lensing, X-ray
observation is the most common and conventional way to analyze the
gravitational structures of galaxy clusters. Therefore, it is valuable to have
simple analytical relations that can connect the observed distribution of the
hot, X-ray emitting gas to the structure of the dark matter in the clusters as
derived from simulations. In this article, we apply a simple framework that can
analytically connect the hot gas empirical parameters with the standard
parameters in the cosmological cold dark matter model. We have theoretically
derived two important analytic relations, $r_s \approx \sqrt{3}r_c$ and $\rho_s
\approx 9\beta kT/8 \pi Gm_gr_c^2$, which can easily relate the dark matter
properties in galaxy clusters with the hot gas properties. This can give a
consistent picture describing gravitational astrophysics for galaxy clusters by
the hot gas and cold dark matter models.
