Curve of growth

This article is about the curve of growth in astronomy. For other articles related to "growth curve", see Growth curve.

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. ^[1]

Example of a curve of growth

Shape

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]