Transparent-gas addition at fixed absorber mass ($T_s=T_e{D}^{R_a/c_{pa}}$)

In a well-mixed convective column with fixed absorber mass and constant mass opacity, adding a perfectly transparent gas leaves total optical depth $D$ unchanged. The absolute effective emission pressure scales with total surface pressure, but their ratio remains $D$. In the thick-column emitting-level model, $T_s=T_e{D}^{R_a/c_{pa}}$; added gas warms the surface only if its specific-gas-constant to heat-capacity ratio exceeds that of the original gas. Pressure broadening would violate the fixed-opacity assumption and must be treated separately.