12.5: Recapitulation
- Page ID
- 141683
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Building on the results of the previous three chapters, We present in this chapter the basic approach to the construction of a model stellar atmosphere. The process is essentially an iterative one where an initial guess of the temperature distribution throughout the atmosphere yields the atmosphere's physical structure. To obtain this structure, one needs a lot of information about the dependence of the opacity on the state variables of the gas. One generally assumes that the Saha ionization and Boltzmann excitation formulas hold so that one can relatively easily calculate the abundance of each type of absorber in the atmosphere. This, then, allows for the solution of the equation of hydrostatic equilibrium and the calculation of the radiation field at all points in the atmosphere. Application of the condition of radiative equilibrium and a temperature correction procedure allow for the entire process to be iterated until a self-consistent model of the atmosphere is obtained.
What constitutes a converged, self-consistent atmosphere is not at all obvious and may depend on which properties of the model are of particular interest to the investigator. For example, it makes little sense to require 0.1 percent constancy in the radiative flux at optical depth 100 if one is interested in only the emergent flux. Conversely, if the atmosphere is to form the boundary layer for the calculation of a model stellar interior, then some care should be taken with the deep solution. If one is concerned about strong spectral lines, then considerable care should be taken with the surface solution. Since it is still relatively difficult to construct a model stellar atmosphere that exhibits both a constant flux and a zero flux derivative throughout the atmosphere at an arbitrary level of accuracy, such considerations regarding the use of the model should be weighed.
We have now completed the fundamental physics concerning the construction of models for both the inside and surface layers of a star. For normal stars, these models would give a reasonably accurate picture of the structure of these stars and the processes that take place within them. We have even included the departure of the radiation field from strict thermodynamic equilibrium that results from the escape of photons that are near the surface into free space. To the extent that the continuous opacity dominates the total stellar opacity, this will even yield a reasonably correct picture of the grosser aspects of the star's spectrum. However, as anyone who has looked at a stellar spectrum knows, the most salient feature of such a spectrum is the dark lines that cover it. These lines provide most of the information that we have about stars, from their composition to their motions. No description of stellar structure can hope to be taken seriously unless it provides some explanation of the occurrence of these lines. Therefore, for the majority of the rest of this book, we discuss the fundamentals of the formation of spectral absorption lines and the physics that yields their characteristic shape.