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.

Consider a plane-parallel atmosphere in hydrostatic equilibrium, with constant gravitational acceleration $g$ and a uniformly mixed absorber of mass fraction $q$. Let its mass absorption coefficient be the Lorentz line shape ${\kappa }_{\nu }=(S/\pi )\Delta /[(\nu -{\nu }_0{)}^2+{\Delta }^2]$, with pressure-independent integrated strength $S$ and half-width $\Delta (p)={\Delta }_{s}p/p_s$. This assumes a fixed-temperature pressure broadening model over the pressures used.

For vertical propagation from the top to pressure $p$, optical depth is ${\tau }_{\nu }=(q/g){\int }_0^p{\kappa }_{\nu }(p^{\prime })d{p}^{\prime }$. Integration givesThe strong-line limit is $C\gg 1$. At optical depth one, $|\nu -{\nu }_0|=\Delta (p)/\sqrt{e^{1/C}-1}\simeq \Delta (p)\sqrt{C}$, so the escaping frequencies lie in the far wings. Expanding the logarithm there recovers ${\tau }_{\nu }\simeq qS{\Delta }_s{p}^2/[2\pi g{p}_s(\nu -{\nu }_0{)}^2]$. This identifies both the dimensionless control parameter and the physical approximation; pressure broadening still requires a suitable collisional regime.