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

For a black surface thermally coupled to a dry convective atmosphere, define $E(p,q)=\int [1-{\mathfrak{T}}_{\nu }(p)]d\nu$, the line equivalent width between pressure $p$ and space. If the line is sufficiently narrow to evaluate the Planck function at its centre, the outgoing flux is $\sigma T_s^4-D$. Integration by parts of the formal radiative transfer solution gives the displayed deficit, nonnegative because temperature and the Planck function increase downward. In the weak-line limit with constant integrated strength, $E\simeq qSp/(g{\mu }_d)$ and the leading deficit is independent of normalized line shape.