9: The Flow of Radiation Through the Atmosphere
- Page ID
- 141654
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 9.1: Introduction
- This page highlights the significance of a star's atmosphere in comprehending its structure, evolution, and appearance. It notes that much of our understanding of stars is derived from their surfaces and emphasizes the atmosphere's role in influencing stellar evolution through radiation loss. The page details the modeling process of a star's atmosphere, stressing the need to balance radiation flow with atmospheric structure to create a self-consistent model aligned with observational data.
- 9.2: Basic Assumptions for the Stellar Atmosphere
- This page explains thermodynamic equilibrium in stellar interiors, focusing on gas temperature and the distinction between photon and particle behavior. It introduces local thermodynamic equilibrium (LTE) in stellar atmospheres and discusses spectral characteristics and assumptions for normal atmospheres.
- 9.3: Equation of Radiative Transfer
- This page covers the flow of radiation through stellar outer layers, employing the Boltzmann transport equation to derive the radiative transfer equation. It discusses the specific intensity and the "creation" rate of photons, linking it to scattering and absorption processes. The concept of optical depth is introduced, measuring where significant photon interactions occur, alongside the source function \(S_\nu\) that reflects local radiation contributions.
- 9.4: Moments of the Radiation Field
- This page explores averaging techniques for analyzing radiation fields in a plane-parallel atmosphere, introducing key concepts like mean intensity, flux, and radiation pressure while maintaining photon momentum distribution. It emphasizes different flux definitions for clarity. Additionally, it examines the radiation pressure tensor \(K_\nu\) in spherical coordinates, deriving expressions under isotropic conditions and establishing key relationships with pressure.
- 9.5: Moments of the Equation of Radiative Transfer
- This page covers the application of moment equations in radiative transfer, focusing on the Boltzmann transport equation and the derivation of the radiative equilibrium equation under isotropic scattering. It discusses energy conservation in scattering, the diffusion approximation, and the Eddington approximation for boundary conditions in stellar atmospheres.
- 9.6: Problems
- This page explores specific intensity and radiative transfer across different geometries, detailing its constancy in empty space and methods for calculating intensity and radiative flux from an emitting disk. It derives relevant equations for spherical and plane-parallel geometries, incorporating diagonal tensors and emphasizing crucial assumptions for the derivations.
- 9.7: Supplemental Reading
- This page presents foundational texts on stellar atmospheres, focusing on radiative transfer and scattering. It suggests chapters from Mihalas for definitions, Sobolev for scattering issues, and Mustel for a comprehensive overview. The page also cites Eddington's classic paper, which offers historical perspectives and physical insights into stellar radiative equilibrium, underscoring the enduring significance of these astrophysical topics.