in the previous sections through simple power-law models. [@kaiser86] analytically derived the functional shapes of the expected scaling relations under the assumption of virial equilibrium between the kinetic and thermal energy of a galaxy cluster and its gravitational potential [ a recent extension of the formalism in dependance on the ICM physics is provided in @ettori2015]. According to this model, called self-similar (SS), the cluster total mass is the only parameter that defines both thermal and dynamical properties of the ICM (e.g., @2013SSRv..177..247G and references therein).

[**The $M-{M_{\rm g}}$ relation.**]{} The total mass and the gas mass are linearly related: $$M = C_{{M_{\rm g}}} E(z)^{0} {M_{\rm g}}, 
   \label{eq:mmg}$$ without any dependence with the redshift. Here and in the following equations, $C$ is a constant.

[**The $M-T$ relation.**]{} The total mass is related to the temperature according to: $$M =  C_{T} E(z)^{-1} T^{3/2}. 
 \label{eq:mt}$$

[**The $M-{Y_{\rm X}}$ relation.**]{} From Eqs. \[eq:yx\], \[eq:mmg\], and \[eq:mt\], the self-similar scaling of ${Y_{\rm X}}$ with cluster mass is given by $$M = C_{{Y_{\rm X}}} E(z)^{-2/5} {Y_{\rm X}}^{3/5}. 
  \label{eq:myx}$$

[**The luminosity relations: $L-T$ and $L-M$.**]{} Assuming thermal bremsstrahlung emission, the luminosity can be related to the total mass by $$L \propto E(z)^2 T^{1/2} f_{gas}^2 M, \label{8}$$ 