Curve of growth

The curve of growth describes the dependence of the equivalent width ${\displaystyle W}$, which is an effective measure of the strength of a feature in a emission or absorption spectrum, on the column density ${\displaystyle N}$. Because the spectrum of a single spectral line has a characteristic shape, being broadened by various processes from a pure line, by increasing the optical depth ${\displaystyle \tau }$ of a medium that either absorbs or emits light, the strength of the feature develops non-trivially.^[2]

In the case of the combined natural line width, collisional broadening and thermal Doppler broadening, the spectrum can be described by a Voigt profile and the curve of growth exhibits the approximate dependencies depicted on the right. For low optical depth ${\displaystyle \tau \ll 1}$ corresponding to low ${\displaystyle N}$, increasing the thickness of the medium leads to a linear increase of absorption and the equivalent line width grows linearly ${\displaystyle W\propto N}$. Once the central Gaussian part of the profile saturates, ${\displaystyle \tau \approx 1}$ and the Gaussian tails will lead to a less effective growth of ${\displaystyle W\propto {\sqrt {\ln N}}}$. Eventually, the growth will be dominated by the Lorentzian tails of the profile, which decays as ${\displaystyle \sim 1/x^{2}}$, producing a dependence of ${\displaystyle W\propto {\sqrt {N}}}$.^[2]