4: Flux, Specific Intensity and other Astrophysical Terms
- Page ID
- 6670
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.1: Introduction
- This page presents key definitions and symbols relevant to stellar atmosphere theory, highlighting differences from standard radiation terminology. It discusses the "horizontal" surface model applicable to shallow stellar atmospheres, emphasizing its limitations for stars with deeper atmospheres while noting its relevance for the Sun's sharp limb and rapid opacity.
- 4.2: Luminosity
- This page discusses the differences between standard and astrophysical terms, focusing on "flux" and "intensity." It explains that "radiant flux" in standard contexts equates to "luminosity" in astronomy, represented by \(L\). Luminosity is measured in watts, but astronomers frequently reference it using solar luminosity, approximately \(3.85 \times 10^{26} \text{watts}\), to describe the brightness of stars.
- 4.3: Specific Intensity
- This page explains the transition in terminology from "radiance" to "specific intensity" in stellar atmosphere theory, clarifying that "specific" does not refer to "per unit mass" in this context. Instead, "specific intensity" measures brightness from a radiating surface, akin to radiance, and indicates observed watts per square meter per steradian in different directions within a stellar atmosphere. The author will consistently use "specific intensity" in future discussions.
- 4.4: Flux
- This page introduces the concept of "flux" in stellar atmosphere theory, defining upward (\(F_+\)) and downward (\(F_-\)) flux based on radiant energy at a horizontal surface, with net upward flux as \(F = F_+ - F_−\).
- 4.5: Mean Specific Intensity
- This page explores specific intensity in stellar atmospheres, emphasizing its directional dependence and noting that intensity is stronger when observed downwards than upwards. It defines the mean specific intensity \(J\) as the average intensity over a sphere, highlighting that at the star's center, this intensity becomes isotropic, leading to \(J\) equaling \(I\) in all directions.
- 4.6: Radiation Pressure
- This page explores the implications of equation 1.18.5 concerning atmospheric radiation, emphasizing non-isotropic radiation and the importance of radiation density in integrals. It introduces the equation for radiation pressure based on specific intensity and notes that isotropic radiation yields a defined value, while the radiation flux is null due to the odd power of cosine, reinforcing core physical principles governing radiation behavior.
- 4.7: Other Integrals
- This page covers key integrals \(H\) and \(K\) related to stellar atmospheres, which measure intensity \(I\) over solid angles, with \(H\) considering cosine weighting and \(K\) involving cosine squared. Notably, \(H\) equals zero for isotropic emissions, and \(K\) has ties to pressure \(P\).
- 4.8: Emission Coefficient
- This page explains the emission coefficient \(j\), representing intensity emitted per unit volume of gas, measured in \(\text{W sr}^{-1} \ \text{m}^{-3}\). It distinguishes between the volume emission coefficient and the mass emission coefficient \(j_m\), which refers to intensity per unit mass, highlighting the relationship \(j = \rho j_m\). The page notes a common confusion among authors regarding the distinction between volume and mass emission coefficients.