Curve of growth explained

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]

Shape

The curve of growth describes the dependence of the equivalent width

, which is an effective measure of the strength of a feature in a emission or absorption spectrum, on the column density

.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

\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

\tau\ll1

corresponding to low

, increasing the thickness of the medium leads to a linear increase of absorption and the equivalent line width grows linearly

W\proptoN

. Once the central Gaussian part of the profile saturates,

\tau ≈ 1

and the Gaussian tails will lead to a less effective growth of

W\propto\sqrt{lnN}

. Eventually, the growth will be dominated by the Lorentzian tails of the profile, which decays as

\sim1/x²

, producing a dependence of

W\propto\sqrt{N}

.^[2]

Notes and References

Web site: The curve of growth . Michael Richmond.
Book: Bartelmann, Matthias . Theoretical Astrophysics : An Introduction . Heidelberg University Publishing . 2021 . 978-3-96822-029-1 . 10.17885/heiup.822 . 93.