Outgoing thermal flux for a pure H₂O atmosphere on an Earth gravity planet (𝑔 = 9.81 m/s²) based on the analytic gray atmosphere and full RT calculations under different assumptions. The results of the full RT calculations in a vertically extended atmosphere (Full RT) with heat capacity varying with temperature, c_P(T), are used for benchmarking (solid lines). The models are compared for the same surface temperatures and pressures, where different surface pressures are indicated by variations in color. Left panel: Results of full RT calculations with temperature-dependent c_P (solid lines), full RT with constant c_P (dashed lines), and the gray atmosphere model (dash-dotted lines). Right panel: Results of the corrected gray atmosphere (dash-dotted lines) compared to the full RT.