12: The Construction of a Model Stellar Atmosphere
- Page ID
- 141678
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 12.1: Statement of the Basic Problem
- This page covers the development of a stellar atmosphere model, emphasizing the interplay between key state variables like pressure, temperature, and density, along with the emergent spectrum. It addresses challenges such as absorption coefficients and contrasts with blackbody radiation. The derivation of hydrostatic equilibrium and radiative transfer equations is explained, moving from basic boundary problems to integral equations.
- 12.2: Structure of the Atmosphere, Given the Radiation Field
- This page covers the modeling of stellar atmospheres, focusing on selecting atmospheric parameters and depth measures like optical depth. It emphasizes using monochromatic optical depth for simplification, and discusses deriving temperature distributions from radiative equilibrium and calculating hydrostatic equilibrium. Numerical methods are essential for solving these equations accurately.
- 12.3: Calculation of the Radiation Field of the Atmosphere
- This page covers the numerical challenges in solving the equation of radiative transfer, focusing on the importance of optical depth spacing for accuracy. It emphasizes the need for solutions across varying optical depths in stellar atmospheres and the special care required in depth truncation. The mapping of physical parameters across optical depth scales is crucial to minimize interpolation errors.
- 12.4: Correction of the Temperature Distribution and Radiative Equilibrium
- This page covers radiative equilibrium evaluation in the atmosphere, focusing on temperature correction schemes such as the Lambda Iteration and Avrett-Krook methods. It details perturbation equations for radiative transfer, addressing pure absorption and scattering effects, and emphasizes the iterative nature of correction techniques.
- 12.5: Recapitulation
- This page describes an iterative method for constructing model stellar atmospheres, beginning with an initial temperature distribution. It underscores the role of tools like Saha ionization and Boltzmann excitation in determining absorber abundances while maintaining hydrostatic and radiative equilibrium. The importance of self-consistent models based on specific interests is highlighted, along with the critical role of spectral absorption lines in understanding stellar properties.
- 12.6: Problems
- This page examines the establishment of Local Thermodynamic Equilibrium (LTE) in gray stellar atmospheres and the convergence to radiative equilibrium via Rosseland optical depth. It analyzes the linear behavior of the source function in deeper layers, considering contributions from gray opacity and radiation field randomization.
- 12.7: References and Supplemental Reading
- This page emphasizes the importance of foundational literature on stellar atmospheres, highlighting key contributions from authors like Mihalas, Kurucz, and Avrett & Krook. It advocates for examining older works to understand the underlying physics and temperature correction methods, essential for advanced studies in the field.