Left panel: Entropy in units of K₅₀₀ versus radius in units of R₅₀₀. The three blue lines are derived from the model presented in Sect. 5 with corrections described in Sects. 6.1.1 and 6.1.2. All 3 profiles are computed assuming M₅₀₀ = 10¹⁵ M_⊙ and for the gas fraction parameters: l_m = 13.2 and l_s = −0.75. The l_n parameter is set to −1.1, −0.7 and −0.3 respectively for the top, medium and bottom curves. The region between the top and bottom curves is shaded for radii where the cooling time is longer than the time since the gas was shock heated. The red shaded region is the median entropy and intrinsic scatter from the HIGHMz sample presented in Riva et al. (2024). The black dashed line shows the self-similar approximation introduced in Voit et al. (2005). Right panel: Density in units of ρ_g, 500 versus radius in units of R₅₀₀. The three blue lines are derived from the model presented in Sect. 5 with corrections described in Sects. 6.1.1 and 6.1.2 using the same parameters for the gas fraction adopted for the entropy profile presented in the left panel. The region between the top and bottom curves is shaded for radii where the cooling time is longer than the time since the gas was shock heated. The red shaded region is the median and intrinsic scatter derived from the HIGHMz sample.