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Use $u$ as a spectroscopic wavenumber, in the same units throughout. Write the mass absorption coefficient of the absorbing species asHere $S$ is the spectral line strength, the integrated mass absorption coefficient; it is not the line-centre coefficient. If ${\kappa }_{\nu }$ has units ${\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}$ and $\nu$ is in $\mathrm{c}{\mathrm{m}}^{-1}$, then $S$ has units ${\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}\mathrm{c}{\mathrm{m}}^{-1}$. For absorber mass density ${\rho }_a$, the volume coefficient in Beer-Lambert law is ${\rho }_a{\kappa }_{\nu }$.

Lifetime broadening. Finite lifetimes of the upper and lower states destroy optical coherence, even for an isolated molecule. Exponential coherence decay gives a Lorentzian line shape. For population decay rates ${\Gamma }_u,{\Gamma }_l$ and no extra dephasing, its wavenumber half-width is $({\Gamma }_u+{\Gamma }_l)/(4\pi c)$, with $c$ expressed in the length unit reciprocal to $\nu$. This sets a natural linewidth and can matter at extremely low collision rates; it is generally much smaller than atmospheric collisional or thermal widths.

Thermal Doppler broadening. A Maxwell-Boltzmann distribution of molecular line-of-sight velocities givesThis Gaussian distribution has a $1/e$ half-width ${\Delta }_D$, determined by absorber molecular mass $m_a$ and temperature. It is especially relevant in the rarefied upper atmosphere, where collisions are infrequent.

Pressure broadening. Collisions perturb transition frequencies and phases. In the impact regime, exponential phase decorrelation givesHere ${\Delta }_L$ is the half-width at half maximum, proportional to the coherence-destroying collision rate. It depends on perturber density, molecular collision properties, and temperature; it is often parameterized by ${\Delta }_L\propto p(T_{\ast }/T{)}^{n_T}$ for a specified broadener. It dominates much of the dense lower atmosphere. The Lorentzian line shape has algebraic $\delta {\nu }^{-2}$ wings, whereas the Doppler profile has rapidly decreasing Gaussian wings. Where both mechanisms matter, their convolution is a Voigt profile. A pure pressure-broadened profile should not be extrapolated to arbitrarily low pressure while discarding the finite Doppler core.

Take a weak line in the radiative sense: the integrated slant optical depth is small wherever absorption materially contributes, not merely a small greenhouse-gas mass mixing ratio. Normalize every local spectral line profile so that $\int {\kappa }_{\nu }(p)d\nu =S$, with the line strength independent of pressure and, as stipulated, temperature. Keep a finite Doppler or natural core, or a Voigt profile, so that a pressure-only zero-width core at the top does not invalidate a uniform optical-depth expansion. The line must also be narrow enough to use $B_{{\nu }_0}$ across it.

To first order,Thus the detailed normalized line shape drops out, even if its width varies with pressure. Substituting in the deficit in c givesThe neglected term is second order when the relevant squared optical depths are integrable and uniformly small. A very narrow saturated line need not be weak simply because $q\ll 1$. More generally the displayed first-order integrated limit follows from the finite integrated opacity even if an idealized line centre makes a pointwise expansion nonuniform; a finite physical core is the straightforward sufficient assumption.

For an explicitly perturbative temperature estimate, evaluate the bracket at $T_{\mathrm{bare}}$:At fixed $p_s,g,q,S$ and radiative conditions, increasing $\kappa =R_b/c_{p,b}$ makes every level $0<r<1$ cooler relative to the surface, enlarging the bracket and the warming. Increasing $c_{p,b}$ at fixed $R_b$ has the opposite effect. The limit $\kappa \to 0$ is isothermal and gives zero line-induced deficit. A change of background molecular mass alone need not change this leading result: for ideal gases with the same active degrees of freedom, $R_b$ and $c_{p,b}$ scale together and their ratio is unchanged. Comparing only the height-coordinate lapse rate $g/c_{p,b}$ would miss that the absorber column is naturally a pressure-coordinate quantity.