10: Solution of the Equation of Radiative Transfer
- Page ID
- 141662
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 10.1: Introduction
- This page highlights the importance of the radiative transfer equation in stellar atmospheres, noting that its formulation depends on the medium's geometry and conditions. It explains how the source function is affected by the radiation field, resulting in diverse solutions. Despite complexities, the notion of plane parallelism is commonly encountered, underscoring the need for in-depth analysis of the equation in this context.
- 10.2: Classical Solution to the Equation of Radiative Transfer and Integral Equations for the Source Function
- This page examines approaches to solving the equation of transfer in radiative transfer, highlighting integral equations for the source function and direct differential equations, noting their respective benefits. It details the classical solution for plane-parallel atmospheres and the implications for stellar models.
- 10.3: Gray Atmosphere
- This page covers the gray atmosphere model in radiative transfer, emphasizing its frequency-independent opacity for simplified solutions. It details numerical methods for the Schwarzschild-Milne equations and discusses the Eddington approximation and its limitations. The behavior of radiation in semi-infinite gray atmospheres is explored, particularly the effects of scattering and absorption on radiation fields.
- 10.4: Nongray Radiative Transfer
- This page explores advanced methods for solving non-gray radiative transfer equations, focusing on integral and differential approaches. Integral methods excel in polarization issues, while differential methods, particularly the Feautrier method, offer adaptability and stability. The Feautrier method uses finite difference approximations to create matrix-vector equations, improving upon earlier methods by addressing optical depth points.
- 10.5: Radiative Transport in a Spherical Atmosphere
- This page addresses the complexities of modeling stellar atmospheres, especially for red supergiants, due to limitations of the plane-parallel approximation and the need for considering atmospheric curvature. It discusses radiative transfer in spherical geometries, presenting equations and methods, such as the Feautrier method and the Eddington factor, which measure radiation isotropy.
- 10.6: Problems
- This page discusses radiative transfer in gray atmospheres, emphasizing mathematical formulations and numerical solutions. It includes integral equations for the source function, fixed-point iterations, and eigenvalue analysis within discrete ordinate methods. Key components involve the derivation of the Hopf function, self-consistency in absorbing atmospheres, and spherical coordinate transformations.
- 10.7: References and Supplemental Reading
- This page covers essential literature on radiative transfer in stellar atmospheres, featuring significant contributions like Chandrasekhar's advanced mathematical treatise and Mihalas's discussions on gray atmospheres. It stresses various approaches, particularly from the Russian school, including Sobolev's work, while underlining the importance of scattering theory and neutron diffusion in understanding radiative transfer, providing insights for both basic and advanced studies.