Light Scattering in Inhomogeneous Atmospheres

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Edgard G. Yanovitskij Light Scattering in Inhomogeneous Atmospheres Translated by Sergij Ginsheimer and Oleg Yanovitskij With 37 Figures and 54 Tables Springer

Contents Introduction 1 1. Basic Concepts, Equations and Problems 7 1.1 Intensity of Radiation 7 1.2 Interaction of Radiation with Matter 8 1.3 Radiative Transfer Equation 13 1.4 Radiative Transfer Equation in a Stratified Medium 16 1.5 The Parallel External Flux Problem 18 1.5.1 Azimuthal Harmonics of the Radiation Intensity 21 1.5.2 Integral Equation for the Source Function 22 1.6 The Milne Problem 25 1.7 The Problem for Two-Sided Infinity 27 1.8 Radiation Flux 28 1.9 Characteristics of Radiation at the Boundaries of an Atmosphere. The Problem of Diffuse Reflection and Diffuse Transmission of Light 30 1.10 The Flux Integral and K Integral 33 1.11 Green Function and Reciprocity Relations 35 1.12 Invariance Principles 37 Part I. HOMOGENEOUS ATMOSPHERE 2. Radiation Field in an Infinite Atmosphere 41 2.1 Conservatively Scattering Atmosphere 41 2.2 General Case ' 42 2.3 Characteristic Equation and Method of Its Solution 43 2.4. Normalization Constant M 46 2.5 Radiation Field with Nearly Conservative Scattering 48 3. Semi-Infinite Medium 51 3.1 Invariance Relation for the Parallel External Flux Problem.. 51 3.2 The Milne Problem 53

X Contents 3.3 Relationship Between the Milne Problem and the Parallel External Flux Problem 54 3.4 Corollaries 56 3.5 Ambartsumian's Equation for the Reflection Coefficient and a Method for Its Solution 58 3.6 Some Integral Relations Involving Escape Functions 60 3.7 Integrals of the Transfer Equation 62 3.8 Separation of Variables. Angular Relaxation of Photons... 64 3.9 Radiation Field in Deep Atmosphere Layers 65 3.10 Doubling Formula. Radiation Field in an Atmospheric Surface Layer 67 3.11 Atmosphere with Nearly Conservative Scattering 75 3.11.1 Initial Relations 76 3.11.2 Radiation Intensity at the Boundary of an Atmosphere 78 3.11.3 Asymptotic Formulas for N and C 79 3.11.4 Radiation Intensity at an Arbitrary Optical Depth.... 80 3.11.5 Radiation Flux and K Integral 82 3.11.6 Albedo of Atmosphere 83 3.12 Q Form of the Transfer Equation and Solution to the General Problem 85 3.12.1 Function Q(/u,/io,V) and Its Physical Meaning 85 3.12.2 Q Form of the Transfer Equation 87 3.12.3 Conservative Scattering 89 3.12.4 Q Representation of the Green Function for the Transfer Equation in a Plane Atmosphere 90 3.12.5 Solution to the General Problem 91 4. Atmosphere of Finite Optical Thickness 95 4.1 Invariance Relation 95 4.2 Equation for Radiation Intensity in Medium 98 4.3 Radiation Intensity at Atmosphere Boundaries 99 4.4 Further Consequences of the Basic Invariance Relation 100 4.5 Doubling Method for Calculation of Transmission and Reflection Coefficients 101 4.6 Radiation Field in a Layer 103 4.7 Integrals of the Transfer Equation for a Layer of Finite Thickness 104 4.8 Atmosphere with Large Optical Thickness 107 4.8.1 Reflection and Transmission Coefficients and Other Quantities 108 4.8.2 Conservative Scattering 110 4.8.3 Nearly Conservative Scattering in an Optically Thick Layer Ill 4.8.4 Estimation of the Accuracy of Asymptotic Formulas.. 112

Contents XI 4.9 Illumination of the Boundary and Albedo of Atmospheres of Arbitrary Optical Thickness for Nearly Conservative Scattering 118 4.10 Algorithm for Solving the General Problem 121 4.10.1 Q Form Equation for the Green Function: Conservative Scattering 122 4.10.2 Solution of the General Problem: Conservative Scattering 124 4.10.3 Solution Algorithm for Nonconservative Scattering... 127 5. Atmosphere Above a Reflecting Surface 129 5.1 Radiation Field in Atmospheres 129 5.2 Reflection and Transmission Coefficients 131 5.3 The Case of a Lambertian Surface 131 5.4 Albedo of Atmospheres and Illumination of Surfaces 133 5.5 Optically Thick Atmosphere Above a Reflective Surface 133 5.5.1 The Milne Problem with Reflection 134 5.5.2 Radiation Field in Atmospheres 136 5.5.3 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.. 149 6.1 Formulation of the Problem 149 6.2 A Two-Layer Atmosphere 149 6.3 Choosing the Direction to Add Layers 152 6.4 Radiation Field in a Multilayer Atmosphere 153 6.5 A Semi-Infinite Multilayer Atmosphere 155 6.6 A Multilayer Atmosphere Above a Reflecting Surface... 156 6.7 The Milne Problem 157 6.8 The Milne Intensity at a Large Depth in Layer n 160 6.9 Normalization of the Solution of the Milne Problem 160 6.10 Solution of the General Problem* 162 7. Light Scattering in Two Adjacent Half-Spaces 165 7.1 Statement of the Problem and Main Equations 165 7.2 Radiation Intensity at the Boundary 166 7.3 Isotropic Scattering 167 7.4 Radiation Field for Nearly Conservative Scattering 170 7.5 Radiation Field Away from the Boundary 172

XII Contents 8. Atmosphere Consisting of Layers with Large Optical Thickness 173 8.1 Parallel External Flux Problem: General Discussion 173 8.2 Asymptotic Formulas for Intensity at a Boundary and for Transmission and Reflection Coefficients 175 8.3 Radiation Field in a Medium 177 8.4 Conservatively Scattering Atmosphere 178 8.5 The Milne Problem 180 8.6 Normalization of the Milne Problem 182 8.7 A Two-Layer Atmosphere: Basic Formulas 184 8.7.1 A Two-Layer Semi-Infinite Atmosphere 184 8.7.2 Optically Thick Layers 185 8.7.3 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 195 9.1 Integro-Differential Equations for the Source Function and Reflection and Transmission Coefficients 195 9.2 Method of Truncated Atmosphere for Determining Reflection and Transmission Coefficients 200 9.3 A Semi-Infinite Atmosphere 205 10. Basic Equations Defining the Radiation Field in a Vertically Inhomogeneous Layer 207 10.1 Equation for the Radiation Intensity in a Plane Layer 207 10.2 Invariance Relation for a Plane Sublayer and Some of Its Corollaries 210 10.3 On Numerical Methods to Compute Radiation Field in an Inhomogeneous Atmosphere 213 10.4 An Inhomogeneous Atmosphere Above a Reflecting Surface.. 214 10.4.1 The Radiation Field in an Atmosphere 214 10.4.2 The Case of a Lambertian Surface. Reflection and Transmission Coefficients -.-: 215 10.4.3 Albedo of the Atmosphere and Illumination of the Surface 217 11. Invariance Relations and Their Corollaries for a Semi-Infinite Atmosphere 219 11.1 Invariance Relations 219

Contents XIII 11.2 Basic Equations Determining the Radiation Field 221 11.3 Two Integral Relations, Normalization of Escape Function and M Integral 224 11.4 Relationship Between the Milne Problem and the Parallel External Flux Problem 226 11.5 Some Integral Relations 228 11.6 Integrals of the Transfer Equation 230 11.7 The Concept of an Inverted Semi-Infinite Atmosphere 232 11.8 Discussion of the General Approach to the Solution of the Stated Problems 235 12. Asymptotic Properties of Radiation Fields in Inhomogeneous Atmospheres 237 12.1 Radiative Transfer in an Infinite Medium 237 12.1.1 Isotropization of Radiation. Pi Approximation 239 12.1.2 P 2 Approximation 240 12.1.3 M Integral. Relationship Between j/o( T) and yo{r).. 242 12.2 Deep Layer Regime in a Semi-Infinite Atmosphere 243 12.3 Separation of Angular Variables in the Problem of Light Scattering in an Optically Thick Layer 244 12.4 Reflection Coefficient for a Semi-Infinite Atmosphere with Nearly Conservative Scattering 248 12.5 Escape Function, Albedo of Atmospheres and Other Quantities for Small True Absorption 251 12.6 An Inhomogeneous Atmosphere with Conservative Scattering 255 12.7 Conservatively Scattering Atmosphere Above a Reflecting. Surface 257 12.8 Radiation Field in an Atmosphere with Nearly Conservative Scattering 260 12.8.1 Radiation Field in an.inverted Atmosphere and in Optically Thick Layer 262 12.8.2 Radiation Flux 263 13. Atmospheres with Exponentially Varying Characteristics. 265 13.1 Coefficient of Reflection from a Semi-Infinite Atmosphere... 265 13.2 Results of Calculations and an Estimation of Accuracy of Asymptotic Formulas for A(T) = \ l e- mr 268 13.3 Algorithm for Calculating Internal Radiation Field 273 13.4 Linear Integral Equation for Intensity of Radiation Emerging from Isotropically Scattering Semi-Infinite Atmosphere 275 14. Astrophysical, Geophysical, and Other Possible Applications of the Theory 277 14.1 Effect of Inhomogeneity of a Cloudless Earth Atmosphere on the Radiation Field 277

XIV Contents 14.2 Vertical Structure of the Venusian Atmosphere According to Data Obtained by Probes 281 14.2.1 Vertical Distribution of the Absorption Coefficient in an Atmosphere 282 14.2.2 Optical Parameters of Atmospheres in Different Spectral Regions 287 14.3 Absorption Line Formation in an Inhomogeneous Planetary Atmosphere. Basic Concepts and General Formulas 296 14.4 Absorption Line in an Optically Thick Nearly Conservatively Scattering Atmosphere 299 14.4.1 Dependence of the Observed Spectrum on the Width and Orientation of Spectrograph Slit 302 14.4.2 Profile and Equivalent Line Width for Different Models of Atmospheres 304 14.4.3 Arbitrary Model of an Atmosphere. Reduction to the Cauchy Problem 316 14.5 Effect of Inhomogeneity of Stellar Photospheres on the Continuous Spectrum 318 14.5.1 Basic Equations and Relations 318 14.5.2 Reduction to the Standard Problem 319 14.5.3 Various Methods of Solution 320 14.5.4 Isothermic Photosphere with a Density Decreasing by the Barometric Law 323 14.5.5 Asymptotic Formulas 326 14.6 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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