Light Scattering in Inhomogeneous Atmospheres
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1 Edgard G. Yanovitskij Light Scattering in Inhomogeneous Atmospheres Translated by Sergij Ginsheimer and Oleg Yanovitskij With 37 Figures and 54 Tables Springer
2 Contents Introduction 1 1. Basic Concepts, Equations and Problems Intensity of Radiation Interaction of Radiation with Matter Radiative Transfer Equation Radiative Transfer Equation in a Stratified Medium The Parallel External Flux Problem Azimuthal Harmonics of the Radiation Intensity Integral Equation for the Source Function The Milne Problem The Problem for Two-Sided Infinity Radiation Flux Characteristics of Radiation at the Boundaries of an Atmosphere. The Problem of Diffuse Reflection and Diffuse Transmission of Light The Flux Integral and K Integral Green Function and Reciprocity Relations Invariance Principles 37 Part I. HOMOGENEOUS ATMOSPHERE 2. Radiation Field in an Infinite Atmosphere Conservatively Scattering Atmosphere General Case ' Characteristic Equation and Method of Its Solution Normalization Constant M Radiation Field with Nearly Conservative Scattering Semi-Infinite Medium Invariance Relation for the Parallel External Flux Problem The Milne Problem 53
3 X Contents 3.3 Relationship Between the Milne Problem and the Parallel External Flux Problem Corollaries Ambartsumian's Equation for the Reflection Coefficient and a Method for Its Solution Some Integral Relations Involving Escape Functions Integrals of the Transfer Equation Separation of Variables. Angular Relaxation of Photons Radiation Field in Deep Atmosphere Layers Doubling Formula. Radiation Field in an Atmospheric Surface Layer Atmosphere with Nearly Conservative Scattering Initial Relations Radiation Intensity at the Boundary of an Atmosphere Asymptotic Formulas for N and C Radiation Intensity at an Arbitrary Optical Depth Radiation Flux and K Integral Albedo of Atmosphere Q Form of the Transfer Equation and Solution to the General Problem Function Q(/u,/io,V) and Its Physical Meaning Q Form of the Transfer Equation Conservative Scattering Q Representation of the Green Function for the Transfer Equation in a Plane Atmosphere Solution to the General Problem Atmosphere of Finite Optical Thickness Invariance Relation Equation for Radiation Intensity in Medium Radiation Intensity at Atmosphere Boundaries Further Consequences of the Basic Invariance Relation Doubling Method for Calculation of Transmission and Reflection Coefficients Radiation Field in a Layer Integrals of the Transfer Equation for a Layer of Finite Thickness Atmosphere with Large Optical Thickness Reflection and Transmission Coefficients and Other Quantities Conservative Scattering Nearly Conservative Scattering in an Optically Thick Layer Ill Estimation of the Accuracy of Asymptotic Formulas.. 112
4 Contents XI 4.9 Illumination of the Boundary and Albedo of Atmospheres of Arbitrary Optical Thickness for Nearly Conservative Scattering Algorithm for Solving the General Problem Q Form Equation for the Green Function: Conservative Scattering Solution of the General Problem: Conservative Scattering Solution Algorithm for Nonconservative Scattering Atmosphere Above a Reflecting Surface Radiation Field in Atmospheres Reflection and Transmission Coefficients The Case of a Lambertian Surface Albedo of Atmospheres and Illumination of Surfaces Optically Thick Atmosphere Above a Reflective Surface The Milne Problem with Reflection Radiation Field in Atmospheres Atmosphere with Nearly Conservative Scattering 138 Bibliographical Comments and Additions to Part I 141 Part II. MULTILAYER ATMOSPHERE 6. Parallel External Flux Problem and the Milne Problem Formulation of the Problem A Two-Layer Atmosphere Choosing the Direction to Add Layers Radiation Field in a Multilayer Atmosphere A Semi-Infinite Multilayer Atmosphere A Multilayer Atmosphere Above a Reflecting Surface The Milne Problem The Milne Intensity at a Large Depth in Layer n Normalization of the Solution of the Milne Problem Solution of the General Problem* Light Scattering in Two Adjacent Half-Spaces Statement of the Problem and Main Equations Radiation Intensity at the Boundary Isotropic Scattering Radiation Field for Nearly Conservative Scattering Radiation Field Away from the Boundary 172
5 XII Contents 8. Atmosphere Consisting of Layers with Large Optical Thickness Parallel External Flux Problem: General Discussion Asymptotic Formulas for Intensity at a Boundary and for Transmission and Reflection Coefficients Radiation Field in a Medium Conservatively Scattering Atmosphere The Milne Problem Normalization of the Milne Problem A Two-Layer Atmosphere: Basic Formulas A Two-Layer Semi-Infinite Atmosphere Optically Thick Layers Conservative Scattering 187 Bibliographical Comments and Additions to Part II 189 Part III. ATMOSPHERE WITH CONTINUOUSLY VARYING PARAMETERS 9. Diffuse Reflection and Transmission of Light by Atmospheres Integro-Differential Equations for the Source Function and Reflection and Transmission Coefficients Method of Truncated Atmosphere for Determining Reflection and Transmission Coefficients A Semi-Infinite Atmosphere Basic Equations Defining the Radiation Field in a Vertically Inhomogeneous Layer Equation for the Radiation Intensity in a Plane Layer Invariance Relation for a Plane Sublayer and Some of Its Corollaries On Numerical Methods to Compute Radiation Field in an Inhomogeneous Atmosphere An Inhomogeneous Atmosphere Above a Reflecting Surface The Radiation Field in an Atmosphere The Case of a Lambertian Surface. Reflection and Transmission Coefficients -.-: Albedo of the Atmosphere and Illumination of the Surface Invariance Relations and Their Corollaries for a Semi-Infinite Atmosphere Invariance Relations 219
6 Contents XIII 11.2 Basic Equations Determining the Radiation Field Two Integral Relations, Normalization of Escape Function and M Integral Relationship Between the Milne Problem and the Parallel External Flux Problem Some Integral Relations Integrals of the Transfer Equation The Concept of an Inverted Semi-Infinite Atmosphere Discussion of the General Approach to the Solution of the Stated Problems Asymptotic Properties of Radiation Fields in Inhomogeneous Atmospheres Radiative Transfer in an Infinite Medium Isotropization of Radiation. Pi Approximation P 2 Approximation M Integral. Relationship Between j/o( T) and yo{r) Deep Layer Regime in a Semi-Infinite Atmosphere Separation of Angular Variables in the Problem of Light Scattering in an Optically Thick Layer Reflection Coefficient for a Semi-Infinite Atmosphere with Nearly Conservative Scattering Escape Function, Albedo of Atmospheres and Other Quantities for Small True Absorption An Inhomogeneous Atmosphere with Conservative Scattering Conservatively Scattering Atmosphere Above a Reflecting. Surface Radiation Field in an Atmosphere with Nearly Conservative Scattering Radiation Field in an.inverted Atmosphere and in Optically Thick Layer Radiation Flux Atmospheres with Exponentially Varying Characteristics Coefficient of Reflection from a Semi-Infinite Atmosphere Results of Calculations and an Estimation of Accuracy of Asymptotic Formulas for A(T) = \ l e- mr Algorithm for Calculating Internal Radiation Field Linear Integral Equation for Intensity of Radiation Emerging from Isotropically Scattering Semi-Infinite Atmosphere Astrophysical, Geophysical, and Other Possible Applications of the Theory Effect of Inhomogeneity of a Cloudless Earth Atmosphere on the Radiation Field 277
7 XIV Contents 14.2 Vertical Structure of the Venusian Atmosphere According to Data Obtained by Probes Vertical Distribution of the Absorption Coefficient in an Atmosphere Optical Parameters of Atmospheres in Different Spectral Regions Absorption Line Formation in an Inhomogeneous Planetary Atmosphere. Basic Concepts and General Formulas Absorption Line in an Optically Thick Nearly Conservatively Scattering Atmosphere Dependence of the Observed Spectrum on the Width and Orientation of Spectrograph Slit Profile and Equivalent Line Width for Different Models of Atmospheres Arbitrary Model of an Atmosphere. Reduction to the Cauchy Problem Effect of Inhomogeneity of Stellar Photospheres on the Continuous Spectrum Basic Equations and Relations Reduction to the Standard Problem Various Methods of Solution Isothermic Photosphere with a Density Decreasing by the Barometric Law Asymptotic Formulas Other Fields of Possible Application of the Theory. 328 Bibliographical Comments and Additions to Part III 331 Appendix. Tables of Some Functions and Constants 335 References i 355 Index 369
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