11: Curve of Growth
- Page ID
- 6719
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)It will be part of the aim of this chapter to predict the curve of growth for gaussian and lorentzian profiles, and also for Voigt profile for different Gauss/Lorentz ratios. The curve of growth (die Wachstumskurve) is a graph showing how the equivalent width of an absorption line, or the radiance of an emission line, increases with the number of atoms producing the line.
- 11.1: Introduction to Curve of Growth
- This page explains the curve of growth, detailing how equivalent width of absorption and emission lines varies with atomic column density in gases. In optically thin conditions, this width increases linearly but eventually stabilizes before rising again due to line wing contributions. The impact of line profiles, specifically Gaussian and Lorentzian, on this relationship is discussed.
- 11.2: A Review of Some Terms
- This page covers foundational concepts in radiation absorption by gases, detailing the absorption coefficient, absorptance, optical thickness, and central depth. It explores the impact of a gas slab on radiation intensity and the mathematical relationships between specific intensity, absorption, and gas properties.
- 11.3: Theory of the Curve of Growth
- This page covers emergent radiance from a homogeneous gas slab under a continuum source, defining radiance per unit wavelength, optical thickness, and the equivalent width (W) for absorption features. It provides mathematical expressions for calculating W based on optical thickness and continuum radiance, and contrasts historical integral approximation methods with current computer-aided numerical integration techniques.
- 11.4: Curve of Growth for Gaussian Profiles
- This page covers Gaussian profiles in optically-thin conditions, noting that absorption coefficients and optical depths may show Gaussian behavior while absorptance does not. It explains the optical thickness function and Half-Width at Half Maximum (HWHM) related to thermally-broadened lines.
- 11.5: Curve of Growth for Lorentzian Profiles
- This page covers the optical depth related to radiation damping and its effect on spectral lines, emphasizing the relationship between optical thickness and wavelength. It explains half-width at half-maximum (HWHM), details how central optical thickness influences equivalent width, and presents the equivalent width formula in multiple forms.
- 11.6: Curve of Growth for Voigt Profiles
- This page details the construction of growth curves for Voigt profiles, concentrating on the Lorentzian to Gaussian half-width ratio. It presents equations for equivalent width against optical thickness, highlighting the significance of the Gaussian fraction. The findings reveal that at low optical thickness, equivalent width is stable despite Gaussian fraction variations, while divergence occurs at higher optical thickness levels, particularly with non-Gaussian profiles (below \(k_G \approx 0.
- 11.7: Observational Curve of Growth
- This page explains the analysis of spectral lines' equivalent width as optical thickness varies, focusing on stellar atmospheres. It details the construction of growth curves to derive properties such as half-widths related to temperature and pressure, particularly for iron group elements. Both theoretical and practical challenges in obtaining parameters are discussed, alongside methods for aligning growth curves from different spectral lines for comparison with theory.
- 11.8: Interpreting an Optically Thick Profile
- This page discusses the analysis of optically thick spectral lines, focusing on determining parameters \(g\) and \(l\) from the curve of growth with unresolved line profiles. It examines the thermally broadened emission line, noting its deviation from Gaussian shape due to optical thickness. Methodologies include comparing line profiles with known strength ratios and measuring half-widths to recover \(g\) using derived relationships from line intensity equations.
- 11.9: APPENDIX A- Evaluation of the Voigt Curve of Growth Integral
- This page examines the evaluation of equivalent width using dimensionless variables, simplifying integral formulations and transforming lengths into new units. It highlights challenges with numerical integration, especially for large parameters, and suggests effective strategies for managing step sizes. Additionally, it analyzes the characteristics of the profile and details the approach to integration using Voigt and Lorentzian profiles based on defined limits.