Light Scattering in Inhomogeneous Atmospheres

Meaning.- 3.12.2 Q Form of the Transfer Equation.- 3.12.3 Conservative Scattering.- 3.12.4 Q Representation of the Green Function for the Transfer Equation in a Plane Atmosphere.- 3.12.5 Solution to the General Problem.- 4. Atmosphere of Finite Optical Thickness.- 4.1 Invariance Relation.- 4.2 Equation for Radiation Intensity in Medium.- 4.3 Radiation Intensity at Atmosphere Boundaries.- 4.4 Further Consequences of the Basic Invariance Relation.- 4.5 Doubling Method for Calculation of Transmission and Reflection Coefficients.- 4.6 Radiation Field in a Layer.- 4.7 Integrals of the Transfer Equation for a Layer of Finite Thickness.- 4.8 Atmosphere with Large Optical Thickness.- 4.8.1 Reflection and Transmission Coefficients and Other Quantities.- 4.8.2 Conservative Scattering.- 4.8.3 Nearly Conservative Scattering in an Optically Thick Layer.- 4.8.4 Estimation of the Accuracy of Asymptotic Formulas.- 4.9 Illumination of the Boundary and Albedo of Atmospheres of Arbitrary Optical Thickness for Nearly Conservative Scattering.- 4.10 Algorithm for Solving the General Problem.- 4.10.1 Q Form Equation for the Green Function: Conservative Scattering.- 4.10.2 Solution of the General Problem: Conservative Scattering.- 4.10.3 Solution Algorithm for Nonconservative Scattering.- 5. Atmosphere Above a Reflecting Surface.- 5.1 Radiation Field in Atmospheres.- 5.2 Reflection and Transmission Coefficients.- 5.3 The Case of a Lambertian Surface.- 5.4 Albedo of Atmospheres and Illumination of Surfaces.- 5.5 Optically Thick Atmosphere Above a Reflective Surface.- 5.5.1 The Milne Problem with Reflection.- 5.5.2 Radiation Field in Atmospheres.- 5.5.3 Atmosphere with Nearly Conservative Scattering.- Bibliographical Comments and Additions to Part I.- II. Multilayer Atmosphere.- 6. Parallel External Flux Problem and the Milne Problem.- 6.1 Formulation of the Problem.- 6.2 A Two-Layer Atmosphere.- 6.3 Choosing the Direction to Add Layers.- 6.4 Radiation Field in a Multilayer Atmosphere.- 6.5 A Semi-Infinite Multilayer Atmosphere.- 6.6 A Multilayer Atmosphere Above a Reflecting Surface.- 6.7 The Milne Problem.- 6.8 The Milne Intensity at a Large Depth in Layer n.- 6.9 Normalization of the Solution of the Milne Problem.- 6.10 Solution of the General Problem.- 7. Light Scattering in Two Adjacent Half-Spaces.- 7.1 Statement of the Problem and Main Equations.- 7.2 Radiation Intensity at the Boundary.- 7.3 Isotropic Scattering.- 7.4 Radiation Field for Nearly Conservative Scattering.- 7.5 Radiation Field Away from the Boundary.- 8. Atmosphere Consisting of Layers with Large Optical Thickness.- 8.1 Parallel External Flux Problem: General Discussion.- 8.2 Asymptotic Formulas for Intensity at a Boundary and for Transmission and Reflection Coefficients.- 8.3 Radiation Field in a Medium.- 8.4 Conservatively Scattering Atmosphere.- 8.5 The Milne Problem.- 8.6 Normalization of the Milne Problem.- 8.7 A Two-Layer Atmosphere: Basic Formulas.- 8.7.1 A Two-Layer Semi-Infinite Atmosphere.- 8.7.2 Optically Thick Layers.- 8.7.3 Conservative Scattering.- Bibliographical Comments and Additions to Part II.- III. Atmosphere with Continuously Varying Parameters.- 9. Diffuse Reflection and Transmission of Light by Atmospheres.- 9.1 Integro-Differential Equations for the Source Function and Reflection and Transmission Coefficients.- 9.2 Method of Truncated Atmosphere for Determining Reflection and Transmission Coefficients.- 9.3 A Semi-Infinite Atmosphere.- 10. Basic Equations Defining the Radiation Field in a Vertically Inhomogeneous Layer.- 10.1 Equation for the Radiation Intensity in a Plane Layer.- 10.2 Invariance Relation for a Plane Sublayer and Some of Its Corollaries.- 10.3 On Numerical Methods to Compute Radiation Field in an Inhomogeneous Atmosphere.- 10.4 An Inhomogeneous Atmosphere Above a Reflecting Surface.- 10.4.1 The Radiation Field in an Atmosphere.- 10.4.2 The Case of a Lambertian Surface. Reflection and Transmission Coefficients.- 10.4.3 Albedo of the Atmosphere and Illumination of the Surface.- 11. Invariance Relations and Their Corollaries for a Semi-Infinite Atmosphere.- 11.1 Invariance Relations.- 11.2 Basic Equations Determining the Radiation Field.- 11.3 Two Integral Relations, Normalization of Escape Function and M Integral.- 11.4 Relationship Between the Milne Problem and the Parallel External Flux Problem.- 11.5 Some Integral Relations.- 11.6 Integrals of the Transfer Equation.- 11.7 The Concept of an Inverted Semi-Infinite Atmosphere.- 11.8 Discussion of the General Approach to the Solution of the Stated Problems.- 12. Asymptotic Properties of Radiation Fields in Inhomogeneous Atmospheres.- 12.1 Radiative Transfer in an Infinite Medium.- 12.1.1 Isotropization of Radiation. P1 Approximation.- 12.1.2 P2 Approximation.- 12.1.3 M Integral. Relationship Between y0(-?) and y0(?).- 12.2 Deep Layer Regime in a Semi-Infinite Atmosphere.- 12.3 Separation of Angular Variables in the Problem of Light Scattering in an Optically Thick Layer.- 12.4 Reflection Coefficient for a Semi-Infinite Atmosphere with Nearly Conservative Scattering.- 12.5 Escape Function, Albedo of Atmospheres and Other Quantities for Small True Absorption.- 12.6 An Inhomogeneous Atmosphere with Conservative Scattering.- 12.7 Conservatively Scattering Atmosphere Above a Reflecting Surface.- 12.8 Radiation Field in an Atmosphere with Nearly Conservative Scattering.- 12.8.1 Radiation Field in an Inverted Atmosphere and in Optically Thick Layer.- 12.8.2 Radiation Flux.- 13. Atmospheres with Exponentially Varying Characteristics.- 13.1 Coefficient of Reflection from a Semi-Infinite Atmosphere.- 13.2 Results of Calculations and an Estimation of Accuracy of Asymptotic Formulas for ?(?) = ?1e-m?.- 13.3 Algorithm for Calculating Internal Radiation Field.- 13.4 Linear Integral Equation for Intensity of Radiation Emerging from Isotropically Scattering Semi-Infinite Atmosphere.- 14. Astrophysical, Geophysical, and Other Possible Applications of the Theory.- 14.1 Effect of Inhomogeneity of a Cloudless Earth Atmosphere on the Radiation Field.- 14.2 Vertical Structure of the Venusian Atmosphere According to Data Obtained by Probes.- 14.2.1 Vertical Distribution of the Absorption Coefficient in an Atmosphere.- 14.2.2 Optical Parameters of Atmospheres in Different Spectral Regions.- 14.3 Absorption Line Formation in an Inhomogeneous Planetary Atmosphere. Basic Concepts and General Formulas.- 14.4 Absorption Line in an Optically Thick Nearly Conservatively Scattering Atmosphere.- 14.4.1 Dependence of the Observed Spectrum on the Width and Orientation of Spectrograph Slit.- 14.4.2 Profile and Equivalent Line Width for Different Models of Atmospheres.- 14.4.3 Arbitrary Model of an Atmosphere. Reduction to the Cauchy Problem.- 14.5 Effect of Inhomogeneity of Stellar Photospheres on the Continuous Spectrum.- 14.5.1 Basic Equations and Relations.- 14.5.2 Reduction to the Standard Problem.- 14.5.3 Various Methods of Solution.- 14.5.4 Isothermic Photosphere with a Density Decreasing by the Barometric Law.- 14.5.5 Asymptotic Formulas.- 14.6 Other Fields of Possible Application of the Theory.- Bibliographical Comments and Additions to Part III.- Appendix. Tables of Some Functions and Constants..- References.