13.4: Problems
- Page ID
- 141690
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Find equivalent expressions for the asymptotic behavior of the residual intensity and flux as given by equations (13.2.33) through (13.2.37) for the case where the radiative transfer equation is solved by the Chandrasekhar two-stream approximation.
- Find expressions for the residual intensity and flux of a Schuster-Schwarzschild atmosphere if the equation of transfer is to be solved by using the Chandrasekhar nth-order approximation.
- Show that the expression for \(f_\nu\) can be used to obtain the expression for \(\mathrm{r}_\nu\) for the Milne-Eddington atmosphere.
- Find an expression for \(\mathrm{r}_\nu\) in a Milne-Eddington atmosphere where the source function in the continuum is given by \[B(\tau)=B^{(0)}+B^{(1)} \tau+B^{(2)} \tau^2\nonumber\]
- Derive expressions for a Milne-Eddington type of model atmosphere of finite optical depth \(\tau_0\) for \(f_\nu(\mu)\) and \(\mathrm{r}_\nu\). Assume the atmosphere is illuminated from below by a uniform isotropic intensity \(\mathrm{I_0}\).
- Compare the results of Problem 5 to the results obtained for
- the semi-infinite Milne-Eddington atmosphere for strong absorption lines and strong scattering lines and
- the Schuster-Schwarzschild atmosphere for strong scattering lines and weak scattering lines.
- If the law of limb-darkening for lines formed in a Schuster-Schwarzschild atmosphere can be written as \[I_{\nu_0}(\mu, 0)=a+b \mu\nonumber\] find \(a\) and \(b\) in terms of the continuum flux and the optical depth of the line \(\tau_0\).
- Find an expression for the residual flux from a Schuster-Schwarzschild atmosphere if the continuum photospheric intensity has the form \[I_c(\mu, 0)=a+b \mu\nonumber\]
- Use a model atmosphere code to generate \(\mathrm{r}_\nu\) for the spectral line of your choice, and compare the results to those of a Milne-Eddington atmosphere with the same effective temperature. Clearly state all assumptions and approximations that you make.