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This book is aimed at studying the scattering of monochromatic radiation in plane inhomogeneous media. We are dealing with the media whose optical properties depend on a single spatial coordinate, namely of a depth. The most widely known books on radiation transfer, for instance 1. S. Chandrasekhar, Radiative Transfer, Oxford, Clarendon Press, 1950, (RT), 2. V. V. Sobolev, Light Scattering in Planetary Atmospheres, New York, Pergamon Press, 1975, (LSPA), 3. H. C. van de Hulst, Multiple Light Scattering. Tables, Formulas and - plications. Vol. 1,2, New York, Academic Press, 1980, (MLS), treat mainly the homogeneous atmospheres. However, as known, the actual atmospheres of stars and planets, basins of water, and other artificial and nat­ ural media are not homogeneous. This book deals with the model of vertically inhomogeneous atmosphere, which is closer to reality than the homogeneous models. This book is close to the aforementioned monographs in its scope of prob­ lems and style. Therefore, I guess that a preliminary knowledge of the con­ tents of these books, particularly of the book by Sobolev, would facilitate the readers' task substantially. On the other hand, all concepts, problems, and equations used in this book are considered in full in Chap. 1. So, it will be possible for those readers who do not possess the above knowledge to understand this book. A general idea about the content of the book can be gained from both the Introduction and the Table of Contents.

Inhaltsverzeichnis

Frontmatter

Introduction

Introduction

Abstract
We embark with a brief excursion into the history of the formation and development of transfer theory.
Edgard G. Yanovitskij

Basic Concepts, Equations and Problems

1. Basic Concepts, Equations and Problems

Abstract
This introductory chapter discusses the scattering of radiation in a planetary atmosphere. Its results refer equally to the whole complex of physical problems described by the similar equations. Most of the equations and problems are formulated here for the case of an inhomogeneous stratified atmosphere whose optical properties depend on one spatial variable. Thus, throughout the book we will constantly refer to the formulas contained in this chapter.
Edgard G. Yanovitskij

Homogeneous Atmosphere

Frontmatter

2. Radiation Field in an Infinite Atmosphere

Abstract
We embark this chapter with the simple problem for two-sided infinity, previously formulated in Sect. 1.7. Let us recall its formulation for a homogeneous atmosphere.
Edgard G. Yanovitskij

3. Semi-Infinite Medium

Abstract
This chapter deals with solving the parallel external flux problem, and the Milne problem. We rely on the generalized invariance principle formulated in Sect. 1.12. We also describe some concrete methods for numerical solution of these problems.
Edgard G. Yanovitskij

4. Atmosphere of Finite Optical Thickness

Abstract
The present chapter deals with the problem of finding the radiation field in a layer of an arbitrary thickness τ0, which is, in other words, the parallel external flux problem.
Edgard G. Yanovitskij

5. Atmosphere Above a Reflecting Surface

Abstract
The model of an atmosphere considered in the previous chapter is hardly realized in nature because an atmosphere is usually adjacent to a solid planetary surface, which reflects light. The percentage of the radiation reflected can be rather high — up to 80% for snow covered ground. Having been scattered in the atmosphere the reflected photons undergo further reflections. Hence, there occurs the process of multiple exchange of photons between the surface and the atmosphere.
Edgard G. Yanovitskij

Backmatter

Multilayer Atmosphere

Frontmatter

6. Parallel External Flux Problem and the Milne Problem

Abstract
The key problem studied in this chapter is to determine a radiation field in a plane multilayer atmosphere illuminated by parallel rays. The geometry of this problem is given in Fig. 6.1.
Edgard G. Yanovitskij

7. Light Scattering in Two Adjacent Half-Spaces

Abstract
An unbounded medium consisting of two homogeneous half-spaces is one of the simplest examples of a multilayer atmosphere. The study of the radiation transfer in such a medium is of considerable interest from the standpoint of the methods that can be used to derive asymptotic formulas for atmospheres consisting of homogeneous layers of large optical thickness. We will give detailed consideration to this subject in the next chapter, so this chapter bears, on the whole, auxiliary character.
Edgard G. Yanovitskij

8. Atmosphere Consisting of Layers with Large Optical Thickness

Abstract
The equations derived in Chap. 6 are valid for layers of arbitrary optical thickness. However, when τ j (j = 1,2,…,n) increases (especially if it occurs simultaneously in all layers), it becomes more difficult to solve these equations. In order to describe the internal radiation field in detail, one needs to take an enormous number of equations and, consequently, to store a lot of information on computer. Therefore, it is important that one can derive simple asymptotic formulas for the quantities that describe the radiation field in an atmosphere consisting of layers with large optical thickness τ j ≫ 1). Accuracy of these formulas grows with increasing τ j . It turns out to be possible to express analytically these quantities in terms of functions describing the radiation field in the corresponding semi-infinite layers and intensities at interlayer boundaries. The present chapter deals with such asymptotic formulas.
Edgard G. Yanovitskij

Backmatter

Atmosphere with Continuously Varying Parameters

Frontmatter

9. Diffuse Reflection and Transmission of Light by Atmospheres

Abstract
From now on we commence a systematical study of radiation fields in an inhomogeneous atmosphere whose optical properties depend on only one spatial coordinate, namely, optical depth. In the beginning we will consider the simpler problem of how to determine the transmission and reflection coefficients for a plane layer of inhomogeneous atmosphere. So, in other words, we will be interested in the intensity of the radiation diffusely transmitted and reflected by the medium (see Sect. 1.9). This problem is one of the most important in astrophysics. Observing celestial bodies, we mostly deal with either the intensity of the radiation diffusely reflected by their surface (planets, gaseous and dust nebula) or the intensity of the radiation that diffuses through their atmosphere from a source in the deep layers of the medium (stars). For a homogeneous atmosphere, both problems are known to be closely connected (see Sect. 3.3). It is important to note here that the problem of determination of the reflection and transmission coefficients can be solved without the main problem (i.e., the determination of the radiation field in the atmosphere) being solved.
Edgard G. Yanovitskij

10. Basic Equations Defining the Radiation Field in a Vertically Inhomogeneous Layer

Abstract
In Chap. 9 we considered the problem of calculating the reflection and transmission coefficients of a plane-parallel layer. In other words, the problem of finding the intensity of the radiation at the boundaries of the layer was tackled.
Edgard G. Yanovitskij

11. Invariance Relations and Their Corollaries for a Semi-Infinite Atmosphere

Abstract
The prime objective of this chapter is to derive all the most important rigorous equations and formulas relating to the parallel external flux problem and to the Milne problem. The derivation is based on the generalized invariance principle and the concept of a truncated inhomogeneous atmosphere. In the final section of the chapter, a new important concept — the inverted semi-infinite atmosphere — is introduced, and a general approach to the solution of the main problems is discussed.
Edgard G. Yanovitskij

12. Asymptotic Properties of Radiation Fields in Inhomogeneous Atmospheres

Abstract
The fact that the number of variables and parameters is large is known to be one of the major causes of calculating difficulties in the transfer theory. For an inhomogeneous medium, the situation is even more difficult because of additional parameters characterizing the inhomogeneity of the atmosphere. Therefore, there is a practical necessity to study the cases in which the number of parameters can be reduced.
Edgard G. Yanovitskij

13. Atmospheres with Exponentially Varying Characteristics

Abstract
It has already been said that no restrictions have so far been imposed on the way the functions λ(τ) and χ*τ) depend on optical depth. Moreover, it turns out that even if we consider particular forms of these functions, which at the same time admit considerable generality, it is possible to derive several relatively simple equations and relations. The significance of these equations and relations is due to the following reasons. First, for real inhomogeneous atmospheres, the dependence of optical properties on depth can, more or less accurately, be described by a concrete function. Second, one can design a simple algorithm for the numerical solution of these equations. Thus, all the above points allow us both to estimate the influence of inhomogeneity of actual media on the radiation field, and provide a testing tool for more complicated methods of solution.
Edgard G. Yanovitskij

14. Astrophysical, Geophysical, and Other Possible Applications of the Theory

Abstract
We now in fact enter the closing chapter. It’s aim is to show how the previously elaborated methods can be applied to various problems in astrophysics, geophysics, and other areas of science. My concern of “other areas” looks superficial, for the following reason. Working in astrophysics and, partly, geophysics, I have gained, I suppose, sufficient experience in resolving the problems arising therein. For this reason, I am cautious of entering other fields. However, one may expect that these methods can be utilized somewhere else.
Edgard G. Yanovitskij

Backmatter

Backmatter

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