In astronomy, the curve of growth describes the equivalent width of a spectral line as a function of the column density of the material from which the spectral line is observed.
Shape The curve of growth describes the dependence of the equivalent width W {\displaystyle W} , which is an effective measure of the strength of a feature in a emission or absorption spectrum, on the column density N {\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 τ {\displaystyle \tau } of a medium that either absorbs or emits light, the strength of the feature develops non-trivially. 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 τ ≪ 1 {\displaystyle \tau \ll 1} corresponding to low N {\displaystyle N} , increasing the thickness of the medium leads to a linear increase of absorption and the equivalent line width grows linearly W ∝ N {\displaystyle W\propto N} . Once the central Gaussian part of the profile saturates, τ ≈ 1 {\displaystyle \tau \approx 1} and the Gaussian tails will lead to a less effective growth of W ∝ ln N {\displaystyle W\propto {\sqrt {\ln N}}} . Eventually, the growth will be dominated by the Lorentzian tails of the profile, which decays as ∼ 1 / x 2 {\displaystyle \sim 1/x^{2}} , producing a dependence of W ∝ N {\displaystyle W\propto {\sqrt {N}}} .
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