The ionization fraction of hydrogen gas undergoing adiabatic expansion, plotted as a function of temperature (in units of 10³ K). The top and bottom panels correspond to entropies per baryon of S = 36.0 and S = 45.0, respectively (corresponding to ΔS = 24.0 and ΔS = 33.0). These entropies correspond to polytropic envelope masses of 1 M_⊙ and 10 M_⊙ (see Equation (24) for more details). The solid blue lines show the ionization fraction given by the Saha equation (Equation (4)), while the black dashed lines show the ionization fraction given by the approximation of the high-entropy limit: in the regime T > 2T_ion, the gas is completely ionized (η = 1), in the regime T < T_ion the gas is non-ionized (η = 0), and between them the ionization fraction depends linearly on the temperature, i.e., η = T/T_ion − 1. The maximal mismatch between the Saha equation and the high-entropy approximation, which scales as ΔS⁻¹, reaches 40% for S = 36.0 and 33% for S = 45.0.