Hydrostatic equilibrium of a porous intracluster medium: implications for mass fraction and X-ray luminosity

The presence of dilute hot cavities in the intracluster medium (ICM) at the cores of clusters of galaxies changes the relation between gas temperature and its X-ray emission properties. Using the hydrostatic equations of a porous medium we solve for the ICM density for a given temperature as a function of the filling factor of dilute bubbles. We find that at a given temperature, the core X-ray luminosity increases with the filling factor. If the frequency of AGNs in clusters were higher in the past, then the filling factor could correspondingly be significant, with implications for the cluster scaling relations at high redshifts. This is especially important for the core properties, including the temperature-luminosity ($L_X-T$) relation and estimates of the core gas mass. The results imply an epoch-dependent sensitivity of the $L_X-T$ relation in the core to the porosity of the ICM. Detection of such an effect would give new insights into AGN feedback.