Spectral line profile ($\phi (\nu )$)

A normalized spectral line profile distributes a transition's emission or absorption over frequency, with $\int \phi (\nu )d\nu =1$ and units $\mathrm{H}{\mathrm{z}}^{-1}$. Thermal motion, finite excited-state lifetime and collisions can give different contributions to the shape. Normalization lets an optically thin integrated line measure the relevant column density without requiring the detailed profile.

The integrated absorption coefficient of an isolated spectral line is its line strength. For a mass coefficient write ${\kappa }_{\nu }=S{\varphi }_{\nu }$, where $\int {\varphi }_{\nu }d\nu =1$. The units of $S$ include the chosen spectral-coordinate unit; a strength integrated over wavenumber is not numerically interchangeable with one integrated over frequency. Changing normalized width at fixed strength redistributes opacity without changing its spectral integral.

A normalized collisional line model ${\kappa }_{\nu }=(S/\pi )\Delta /[(\nu -{\nu }_0{)}^2+{\Delta }^2]$, with integrated strength $S$ and half-width $\Delta$. The centre scales as $S/\Delta$ and far wings as $S\Delta /(\nu -{\nu }_0{)}^2$. In a dilute pressure-broadened gas $\Delta$ scales approximately with pressure.

A regime where optical-depth-one transmission occurs in the far wings of a strongly absorbing line. With a uniformly mixed absorber, $\Delta ={\Delta }_{s}p/p_s$ and $C=qS{p}_s/(2\pi g{\Delta }_s)\gg 1$, the radiative-wing optical depth is $\tau \simeq qS{\Delta }_s{p}^2/[2\pi g{p}_s(\nu -{\nu }_0{)}^2]$. This condition does not make pressure broadening valid in the arbitrarily rarefied upper atmosphere.