11.4: Curve of Growth for Gaussian Profiles
- Page ID
- 6716
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)By “gaussian profile” in the title of this section, I mean profiles that are gaussian in the optically-thin case; as soon as there are departures from optical thinness, there are also departures from the gaussian profile. Alternatively, the absorption coefficient and the optical depth are gaussian; the absorptance is not.
For an optically-thin thermally-broadened line, the optical thickness as a function of wavelength is given by
\[\label{11.4.1}\tau (x)=\tau (0)\text{exp}\left ( -\frac{x^2\ln 2}{g^2}\right ) ,\]
where the HWHM is
\[\label{11.4.2}g=\frac{V_m \lambda_0 \sqrt{\ln 2}}{c}.\]
Here \(V_\text{m}\) is the modal speed. (See Chapter 10 for details.) The line profiles, as calculated from equations 11.3.4 and \ref{11.4.1}, are shown in figure XI.2 for the following values of the optical thickness at the line centre: \(\tau (0)=\frac{1}{16},\frac{1}{8},\frac{1}{4},\frac{1}{2},1,2,4,8.\)
On combining equations 11.3.4 and \ref{11.4.1} and \ref{11.4.2}, we obtain the following expression for the equivalent width:
\[\label{11.4.3}W=2\int_0^\infty \left ( 1-\text{exp}\left \{-\tau (0)e^{-x^2\ln 2/g^2}\right \}\right )\,dx,\]
or
\[\label{11.4.4}W=\frac{2V_\text{m}\lambda_0}{c}\int_0^\infty \left ( 1-\text{exp}\left \{-\tau(0)e^{-\Delta^2}\right \}\right )\,d\Delta,\]
where
\[\label{11.4.5}\Delta =\frac{x\sqrt{\ln 2}}{g}=\frac{cx}{V_\text{m}\lambda_0}.\]
The half-width at half maximum (HWHM) of the expression \ref{11.4.1} for the optical thickness corresponds to \(\Delta = \sqrt{\ln 2} = 0.8326\). For the purposes of practical numerical integration of Equation \ref{11.4.4}, I shall integrate from \(\Delta = -5\text{ to }+5\) that is to say from \(\pm 6\) times the HWHm. One can appreciate from figure XI.2 that going outside these limits will not contribute significantly to the equivalent width. I shall calculate the equivalent width for central optical depths ranging from \(1/20 (\log \tau (0) = −1.3)\text{ to }10^5 (\log \tau (0) = 5.0)\).
What we have in figure XI.3 is a curve of growth for thermally broadened lines (or lines that, in the optically thin limit, have a gaussian profile, which could include a thermal and a microturbulent component). I have plotted this from \(\log \tau(0) = -1.3\text{ to }+5.0\); that is \(\tau(0) = 0.05\text{ to }10^5\).
\(\text{FIGURE XI.2}\)
\(\text{FIGURE XI.3}\)