Curve of growth

The curve of growth describes the dependence of the equivalent width $W$, which is an effective measure of the strength of a feature in a emission or absorption spectrum, on the column density $N$.

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 $\backslash tau$ of a medium that either absorbs or emits light, the strength of the feature develops non-trivially.{{cite book | author = Bartelmann, Matthias | title = Theoretical Astrophysics : An Introduction | publisher = Heidelberg University Publishing | year = 2021 | isbn = 978-3-96822-029-1 | doi = 10.17885/heiup.822 | url = https://heiup.uni-heidelberg.de/catalog/book/822 | page = 93}}

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 $\backslash tau\; \backslash ll\; 1$ corresponding to low $N$, increasing the thickness of the medium leads to a linear increase of absorption and the equivalent line width grows linearly $W\; \backslash propto\; N$. Once the central Gaussian part of the profile saturates, $\backslash tau\backslash approx\; 1$ and the Gaussian tails will lead to a less effective growth of $W\; \backslash propto\; \backslash sqrt\{\backslash ln\; N\}$. Eventually, the growth will be dominated by the Lorentzian tails of the profile, which decays as $\backslash sim\; 1/x^2$, producing a dependence of $W\backslash propto\; \backslash sqrt\{N\}$.

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Category:Spectroscopy