= Single-line greenhouse deficit
{title2=$D=\pi\int E(p,q)\,\partial_pB\,dp\ge0$}
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.
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