12.3: Calculation of the Radiation Field of the Atmosphere
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)All Chapter 10 was devoted to solving the equation of radiative transfer, so there is no need to repeat the specifics here. However, some numerical aspects of those solutions require comment. As even the casual reader of Chapter 10 will notice, the general solution of the equation of radiative transfer is fraught with some formidable numerical difficulties. Not the least of these is ensuring the numerical accuracy of the results. Whether one chooses to solve the integrodifferential equation for the specific intensity or the integral equation for the source function of the radiation field, the spacing of the optical depth points at which the solution is to be obtained is crucial for determining the accuracy of that solution.
However, to obtain the radiative flux and source function at a sufficient number of frequencies to accurately evaluate the condition of radiative equilibrium, we will have to solve the equation of transfer repeatedly over optical depths whose range varies widely. For example, the opacity of a normal stellar atmosphere at frequencies greater than the Lyman limit of hydrogen will be enormously greater than for frequencies in the Balmer continuum for any given physical depth in the atmosphere. Hence, the corresponding optical depth will be very much larger. Since practical realities require that any radiative transfer solution for a semi-infinite atmosphere be truncated at a finite physical depth, we can expect that the monochromatic optical depth corresponding to that physical depth will vary greatly with frequency. However, we must ensure that the radiative transfer solutions, which are calculated at a finite set of depth points that yet span a wide range of optical depths, have sufficient accuracy to facilitate the accurate calculation of radiative equilibrium.
The normal method of accomplishing this is to carry out the numerical solution of the equation of radiative transfer at a predetermined set of optical depths \(\tau_{\mathrm{ri}}\), chosen to ensure the accuracy of the solution. All physical parameters required for that solution are interpolated from the monochromatic optical depths corresponding to the reference depths onto the set of optical depths to be used for the radiative transfer solution. Mathematically, this amounts to mapping these parameters from the \(\tau_\nu\) space on which they are defined onto the \(\tau_{\mathrm{r}}\) space in which the equation of radiative transfer will be solved. In many cases, the points in the \(\tau_{\mathrm{r}}\) space can be chosen to be the same points as those used for the reference depth scale \(\tau_0\), but occasionally they may be a separate set of points. In this case, a further mapping of the radiative transfer solution from the \(\tau_\mathrm{r}\) space to the \(\tau_0\) space must be carried out. The primary reason for this convoluted procedure is to separate the numerical errors into two well-defined categories - those arising from the interpolation and those arising from the solution of the equation of transfer. The latter are generally more difficult to estimate and so are controlled by carrying out the solution over a set of optical depths for which the numerical stability of the radiative transfer solution is well understood. The errors introduced by the interpolation from one optical depth scale to another are generally easier to control. However, a rapid and accurate mapping algorithm must be available. Such an algorithm is contained in the current version of the atmosphere code known as ATLAS2.
For the general overall accuracy of the calculation, we require that the most accurate solutions be obtained at those frequencies for which the majority of the radiative flux flows through the atmosphere. If the reference set of optical depth points is chosen to correspond closely to the monochromatic optical depths at those frequencies, then the interpolation errors incurred from the mapping procedures will be minimal. Frequencies at which the atmosphere is very much optically thicker than the reference optical depths will tend to carry less flux simply because the radiation can escape more easily at the more transparent frequencies. Hence, the frequencies at which the atmosphere is relatively optically thick, and for which the interpolation errors of the mapping can be expected to be the greatest, will make a correspondingly smaller contribution to the total flux and to the conditions of radiative equilibrium. However, one must be careful that, at the frequencies for which the atmosphere is most transparent, a sufficient number of optical depth points are chosen to ensure that the maximum optical depth is optically remote from the surface. In practice, this generally means that \(\tau_\nu \gg 10\).