Single-line greenhouse deficit ($D=\pi \int E(p,q){\partial }_{p}Bdp\ge 0$)

A single-line greenhouse deficit measures the reduction of outgoing thermal radiative flux by one isolated absorption line. Consider a plane-parallel atmosphere in local thermodynamic equilibrium, without scattering, above a blackbody surface. Let pressure $p$ increase from zero at the top to $p_s$ at the surface, and let the surface share the adjacent gas temperature $T_s$.

To specify the angular approximation, use one effective upward direction cosine $0<{\mu }_d\le 1$ and a flux closure $F_{\nu }=\pi I_{\nu }$. Define the corresponding transmission ${\mathfrak{T}}_{\nu }(p)=\exp [-{\tau }_{\nu }(p)/{\mu }_d]$. Integration by parts of the formal radiative transfer solution givesFor a sufficiently narrow absorption line centred at frequency ${\nu }_0$, replace the Planck function and its temperature derivative across the line by their values at ${\nu }_0$. With the line equivalent widththe outgoing radiative flux becomesIf temperature increases downward, $D\ge 0$. An atmospheric thermal inversion can invalidate that sign argument; absorption alone does not guarantee a positive deficit for every imposed thermal profile.

For constant absorber mass fraction $q$, constant gravity $g$ and an opacity ${\kappa }_{\nu }=S{\varphi }_{\nu }$ per absorber mass, assume $\int {\varphi }_{\nu }d\nu =1$ and constant integrated strength $S$. Then ${\tau }_{\nu }=qS{\varphi }_{\nu }p/g$. In the optically thin limit for the specified direction,This leading term is independent of the normalized shape of the absorption spectrum. The angular closure and abundance convention must accompany the coefficient; a full angular integration uses its own flux-weighted transmission.