we sum up only the hot gas component[^5] of all multi-phase particles not containing more than 10% in cold mass and, therefore, no star-forming particles.

[**Temperatures.**]{} We consider both the mass-weighted temperature and the spectroscopic-like temperature [@2004MNRAS.354...10M]. The mass-weighted temperature is provided by: $${T_{\rm mw}}=\frac{\sum_i m_i T_i}{\sum_i m_i},
      \label{eq:tmw}$$ where $m_i$ and $T_i$ are the hot-gas mass and temperature of the $i^{th}$ gas particle. The spectroscopic-like temperature is introduced to ease the comparison with X-ray observations and the formula was derived considering the non-flat response of the instruments on board of Chandra and XMM-Newton: $${T_{\rm sl}}= \frac{\sum_i \rho_i m_i T_i^{0.25}}{\sum_i\rho_i m_i T_i^{-0.75}}, 
\label{eq:tsl}$$ where $\rho_i$ is the particle gas density [@2004MNRAS.354...10M see also @vikh2006]. For this computation, we used only particles emitting in the X-ray band with $T_i> 0.5$ keV.

[**The parameter $Y_{X}$.**]{} As previously said, this parameter is equivalent to the product of gas mass and the global temperature within $R_{500}$ and it is a powerful proxy for the total thermal content of the ICM because it is almost insensitive to the physical processes included in simulations [@2010ApJ...715.1508S; @2011MNRAS.416..801F; @2012ApJ...758...74B; @sembolini.etal.2014] and to the dynamical status of the clusters [@rasia.etal.2011; @kay.etal.2012]. $Y_X$ is derived from X-ray observations, therefore we 