5: Absorption, Scattering, Extinction and the Equation of Transfer
- Page ID
- 6677
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 5.1: Introduction
- This page explains the interaction of radiation with stellar atmospheres, highlighting extinction, absorption, and scattering. Extinction results from both absorption and scattering, which can be influenced by dust particles. It details how atoms absorb energy and re-emit it as light through scattering, but if excited atoms collide before re-emission, the energy transfers to kinetic energy, illustrating the shift from absorption to kinetic energy transfer.
- 5.2: Absorption
- This page explores absorption in an atmosphere, defining the linear absorption coefficient (\(\alpha\)) that measures intensity reduction over distance. It includes equations connecting intensity and \(\alpha\), addressing uniform absorption scenarios. Additionally, it introduces atomic (\(\alpha_a\)) and mass (\(\alpha_m\)) absorption coefficients, relating them to atom and mass density.
- 5.3: Scattering, Extinction and Opacity
- This page covers scattering and absorption concepts, detailing coefficients for measurement, including linear, atomic, and mass scattering coefficients (\(\sigma\)). It introduces extinction coefficients (\(\kappa\)), which represent the combined effect of absorption and scattering. The discussion addresses the applicability of these equations across various scales and frequencies, highlighting that the mass extinction coefficient is commonly referred to as opacity.
- 5.4: Optical Depth
- This page explains optical depth (\(\tau\), the measure of light absorption in a medium) and its relationship to specific intensity, which diminishes exponentially with optical depth. It defines optical depth as the integral of the extinction coefficient over distance and contrasts the logarithm bases used in stellar atmospheres versus filters. Additionally, it introduces filter density and outlines methods for calculating the dimming effect caused by clouds or filters.
- 5.5: The Equation of Transfer
- This page covers the equation of transfer, detailing how radiation interacts with an atmosphere that absorbs, scatters, and emits energy. It introduces fundamental coefficients for absorption, scattering, and emission at a specific frequency, explaining how changes in specific intensity are influenced by these factors. This forms a basic equation that highlights the balance of radiation intensity affected by absorption, scattering, and emission processes.
- 5.6: The Source Function (Die Ergiebigkeit)
- This page explores the relationship between emission and extinction coefficients, emphasizing the source function \(S\) as the ratio of the emission coefficient \(j_\nu\) to the combined extinction and scattering coefficients \(\kappa(\nu)\). It manipulates this relationship to express \(S\) as specific intensity per unit optical thickness and outlines plans to evaluate \(S\) under varying atmospheric absorption and scattering conditions.
- 5.7: A Series of Problems
- This page covers the behavior of radiative flux and specific intensity in stellar atmospheres, starting with an infinite plane radiating surface and the effects of absorption by gas. Problems progress to calculate flux at a depth \(\tau\), emphasizing the source function's role and integrating \(S(t)\) and exponential integrals. It concludes with key relationships in radiative transfer that are significant for astrophysics.
- 5.8: Source function in scattering and absorbing atmospheres
- This page explains the source function \(S_\nu\) in stellar atmospheres, highlighting its dependence on scattering and absorption. In purely scattering atmospheres, \(S_\nu\) aligns with the mean specific intensity \(J_\nu\), while in purely absorbing atmospheres, it corresponds to the black body radiance \(B_\nu\). In atmospheres with both processes, \(S_\nu\) is a weighted combination of \(B_\nu\) and \(J_\nu\), indicating the balance between absorption and scattering contributions.
- 5.9: More on the Equation of Transfer
- This page explores the radiative transfer equation for a spherical star with a shallow atmosphere, detailing the relationship between intensity, source function, and optical depth. It illustrates how mean specific intensity \(J_\nu\) increases with optical depth and derives temperature relations for the star, including surface temperature \(T_0\) and effective temperature \(T_{\text{eff}}\).