10: Line Profiles
- Page ID
- 6712
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Spectrum lines are not infinitesimally narrow; they have a finite width. A graph of radiance or intensity per unit wavelength (or frequency) versus wavelength (or frequency) is the line profile. There are several causes of line broadening, some internal to the atom, others external, and each produces its characteristic profile. Some types of profile, for example, have a broad core and small wings; others have a narrow core and extensive, broad wings. Analysis of the exact shape of a line profile may give us information about the physical conditions, such as temperature and pressure, in a stellar atmosphere.
- 10.1: Natural Broadening (Radiation Damping)
- This page covers the classical oscillator model of atoms, discussing electron motion in electromagnetic fields and energy loss due to radiation. It details energy absorption in classical oscillators, introducing concepts like absorptance and linear absorption coefficients. The text further explores optical properties of absorption lines, equivalent widths, and their calculations, highlighting factors such as damping constants and column density.
- 10.2: Thermal Broadening
- This page covers thermal broadening in emission and absorption lines, highlighting how atomic motion causes Doppler shifts that widen spectral lines. It introduces concepts from kinetic theory, including the Maxwell distribution of speeds and Gaussian profiles. The derivation of the full width at half maximum (FWHM) for these lines is provided, comparing Gaussian and Lorentzian profiles. The relationship between line width and the kinetic temperature of the gas is also emphasized.
- 10.3: Microturbulence
- This page explains the differences between microturbulence and macroturbulence in stellar atmospheres, focusing on microturbulence as small gas cells with Gaussian velocity distributions. It details modeling line profiles influenced by microturbulence akin to thermal broadening, provides formulas for Full Width at Half Maximum (FWHM), and discusses differentiating thermal from microturbulent broadening based on atomic mass effects.
- 10.4: Combination of Profiles
- This page covers the broadening of emission lines through mechanisms like thermal broadening and radiation damping, introducing convolution to combine various broadening profiles, including Gaussian and Lorentzian functions. It includes computations for these profiles and the Voigt profile, which merges the two. The page elaborates on the Voigt profile's integration, its area, equivalent widths, and related formulas for accuracy in spectral analysis.
- 10.5: Pressure Broadening
- This page explains pressure broadening in spectral lines, highlighting differences between giant and main sequence stars due to atomic collisions. It details how truncated sine waves produce Lorentz profiles and discusses additional broadening effects, including the Stark effect and van der Waals interactions, which contribute to asymmetric broadening in hydrogen lines.
- 10.6: Rotational Broadening
- This page explores the Doppler broadening and limb darkening of spectral lines from rotating stars, focusing on differences in light due to rotation and viewing angles. Early-type stars show greater broadening. It discusses how rotational velocity, limb darkening coefficients, and intensity distribution complicate spectral analysis. Key equations are presented, including the limb-darkening formula.
- 10.7: Instrumental Broadening
- This page explores instrumental broadening in spectroscopy, highlighting how devices like prisms and gratings can distort spectral lines. It explains that the observed spectral profile results from the convolution of the true line profile and the instrumental profile. Additionally, it discusses methods to retrieve the true profile using Borel's theorem and Fourier transforms, emphasizing practical applications in areas such as radio astronomy.
- 10.8: Other Line-Broadening Mechanisms
- This page examines the causes of line broadening in spectral lines, focusing on unresolved Zeeman splitting in white dwarf stars and its effect on polarization analysis. It describes the influence of uniform magnetic fields on the polarization of Zeeman components, along with the impacts of unresolved hyperfine structure and autoionization. The page also clarifies that while broadening does not affect the equivalent width of optically thin lines, this changes for optically thick lines.
- 10.9: Appendix A- Convolution of Gaussian and Lorentzian Functions
- This page focuses on the convolution of functions, deriving \(G(x)\) and \(L(x)\) from Lorentzian functions through integrals and algebraic techniques. It discusses the Voigt profile \(V(x)\), emphasizing efficient numerical integration methods that avoid trigonometric functions while ensuring precision, particularly near zero limits.
- 10.10: APPENDIX B- Radiation Damping as Functions of Angular Frequency, Frequency and Wavelength
- This page covers radiation damping, detailing formulas linking the absorption coefficient with angular frequency, frequency, and wavelength. It revises the absorption coefficient using effective atom numbers and discusses its dependence on frequency and wavelength, including important derived expressions. Additionally, it examines thermal broadening's effects on absorption curves, emphasizing the constancy of area in optically thin conditions.
- 10.11: APPENDIX C- Optical Thinness, Homogeneity and Thermodynamic Equilibrium
- This page explores the validity of gas absorption equations, highlighting the conditions required for their application, such as optical thinness and thermodynamic equilibrium. It discusses key concepts like the linear absorption coefficient, optical thickness, equivalent width, and central intensity in relation to gas densities. The page emphasizes the relevance of these concepts in atmospheric contexts and the significance of understanding local thermodynamic equilibrium.