Statistics – Computation
Scientific paper
Jan 1968
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1968jqsrt...8..193h&link_type=abstract
Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 8, Issue 1, p. 193-217.
Statistics
Computation
5
Scientific paper
A central problem in the theory of stellar atmospheres is the determination of the radiation field and distribution of atomic states throughout the atmosphere, given the temperature-pressure structure of the atmosphere, and the chemical composition and the dynamical state of the gas. This problem has a diagnostic aspect, since it can be used to obtain the external radiation field for the purpose of comparing stellar models with observations. There is also a constructive aspect, since the computation of a self-consistent model, without the LTE assumptions, must necessarily involve the problem, usually as one step in an iterative process. Here attention is limited to the radiation field in resonance lines, in which the problem is especially simple. From the considerable body of numerical solutions now available, it is possible to discern some rather general features of these problems, which can be conveniently discussed in terms of processes coupling the radiation field to the electron gas. A concept which has proven useful in understanding these solutions is that of the so-called thermalization length, defined as the optical distance between the point where a photon is created at the expense of electron kinetic energy and the point where it is converted back into kinetic energy. For example, it is seen that a rather severe limit is placed on the resolution with which any observation of the radiation field can determine the temperature-pressure structures of an atmosphere. One technical difficulty encountered in this work is that of solving the combined transfer and statistical equilibrium equations for an atmosphere with optical properties depending on depth. Mathematically, the problem is the numerical solution of a set of coupled linear differential equations with non-constant coefficient, with two-point boundary conditions. Some recent work on this problem by Dr. G. B. Rybicki and the author will be discussed, in which the linear equations are converted into a set of coupled non-linear equations with a one-point boundary condition by means of a transformation due to Rybicki and Usher.
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