INTRODUCTION TO ASTROPHYSICAL RADIATIVE TRANSFER Robert J. Rutten i Copyright c 1988 R.J. Rutten, Sterrekundig Instituut Utrecht, The Netherlands Reproduction in any form of any part of this work is not permitted without prior consent of the author. First edition: Autumn 1988, based on previous lectures of the third year course OTS by C. Zwaan and R.J. Rutten. Set in LaTeX by M. van der Klomp. Second edition: Autumn 1989, revised (with restoration from disk by H.M.G. Burm) and expanded (Chapter 8, with LaTeX assistance from F. van der Wolf). Third edition: Autumn 1990, revised (with contributions by A. Schadee) and ex- panded (Appendices A{C). Fourth edition: Autumn 1991, revised and expanded (Chapter 8). English translation of the fourth edition: Spring 1992, by Ruth C. Peterson, corrected by L. Strous, with assistance from C. Zwaan and the author. March 1995: fresh compilation of the fourth edition by Dan Kiselman, including revised version of Chapters 2 and 3 by Rob Rutten, and new figures Chapter 4. Autumn 2015: fresh compilation of the fourth edition by Luc Rouppe van der Voort, including corrections. ii Contents 1 Introduction 1 1.1 Why this book? . 1 1.1.1 EM radiation as a diagnostic . 1 1.1.2 EM radiation as a driver of structure formation . 2 1.2 These lecture notes . 2 1.3 Other books . 3 1.4 Main themes . 3 1.4.1 Wavelength, frequency and energy . 3 1.4.2 Spectral lines and continua . 4 1.4.2.1 Spectral lines . 4 1.4.2.2 Continua . 7 1.4.3 Collisional transitions and radiative transitions . 9 1.4.4 Photon creation, photon destruction, photon scattering and photon conversion . 11 1.4.5 Optically thin and optically thick . 12 1.4.6 Thermal and nonthermal . 13 1.5 Crucial questions . 13 2 Radiation quantities 15 2.1 Introduction: from luminosity to intensity . 15 2.2 Intensity and related quantities . 19 2.2.1 Intensity . 19 2.2.2 Mean intensity . 21 2.2.3 Flux . 22 2.2.4 Radiation density and radiation pressure . 23 2.2.5 Stokes parameters . 24 3 Transport equation 27 3.1 Introduction: emission and extinction . 27 3.2 Emission coefficient . 27 3.3 Extinction coefficient . 28 3.4 Transport equation . 30 3.5 Optical path length, optical thickness, optical depth . 31 3.6 Source function . 32 3.7 Formal solution of the transport equation . 34 3.7.1 Integral form of the transport equation . 34 3.7.2 Radiation from a homogeneous medium . 35 3.7.3 Radiation from a thick medium . 37 4 Radiation and matter in equilibrium 41 iii iv CONTENTS 4.1 Introduction: thermodynamical equilibrium . 41 4.2 Radiation in thermodynamical equilibrium . 41 4.2.1 Kirchhoff laws . 41 4.2.2 Planck law . 43 4.2.3 Related radiation laws . 43 4.2.3.1 Wien approximation . 43 4.2.3.2 Rayleigh-Jeans approximation . 43 4.2.3.3 Wien displacement law . 45 4.2.3.4 Stefan-Boltzmann law . 45 4.2.4 Radiation temperatures . 46 4.2.4.1 Brightness temperature . 46 4.2.4.2 Antenna temperature . 46 4.2.4.3 Color temperature . 47 4.2.4.4 Effective temperature . 47 4.3 Matter in thermodynamical equilibrium . 47 4.3.1 Maxwell distribution . 47 4.3.2 Boltzmann distribution . 48 4.3.3 Saha distribution . 49 4.3.4 Saha-Boltzmann partitioning . 50 5 Discrete processes 53 5.1 Introduction: bound-bound transitions . 53 5.2 Five transition processes . 54 5.2.1 Spontaneous deexcitation . 54 5.2.2 Radiative excitation . 55 5.2.3 Induced deexcitation . 56 5.2.4 Collisional excitation and collisional deexcitation . 56 5.3 Einstein relations . 57 5.4 Emission coefficient and extinction coefficient . 58 5.5 Source function . 60 6 Continuous processes 63 6.1 Introduction: four types of interaction . 63 6.2 Radiation from an accelerating charge . 63 6.3 Electrons and electric fields . 65 6.3.1 Free-free transitions . 65 6.3.2 Bound-free transitions . 67 6.3.3 H− transitions . 69 6.4 Electrons and photons . 70 6.4.1 Elastic scattering . 70 6.4.1.1 Rayleigh scattering . 72 6.4.1.2 Resonant scattering . 73 6.4.1.3 Thomson scattering . 74 6.4.2 Inelastic scattering . 74 6.4.2.1 Compton scattering . 74 6.4.2.2 Inverse Compton scattering . 75 6.5 Electrons and magnetic fields . 77 6.5.1 Cyclotron radiation . 77 6.5.2 Synchrotron radiation . 78 6.6 Collective processes . 79 CONTENTS v 6.6.1 Dust and droplets . 79 6.6.2 Cherenkov radiation . 79 6.6.3 Plasma cutoff . 80 6.6.4 Faraday rotation . 80 6.6.5 Razin cutoff . 80 6.7 Nuclear reactions . 81 6.7.1 Fusion and fission reactions . 81 6.7.2 Pair annihilation and pair creation . 81 7 Radiative transfer 83 7.1 Introduction: different types of equilibrium . 83 7.1.1 Thermodynamical Equilibrium (TE) . 83 7.1.2 Local Thermodynamical Equilibrium (LTE) . 83 7.1.3 Statistical Equilibrium (SE) . 84 7.1.4 Non-Local Thermodynamical Equilibrium (NLTE) . 85 7.2 Radiative transfer in LTE . 86 7.2.1 Radiation from a thin LTE slab . 87 7.2.2 Radiation within a thick LTE medium: the Rosseland approx- imation . 87 7.2.3 Radiation from a thick LTE medium . 88 7.3 Radiative transfer with photon scattering . 91 7.3.1 Pure scattering . 92 7.3.2 Absorption and scattering in a two-level atom . 93 7.3.3 Effective optical thickness . 96 7.3.4 Radiation from a thin scattering slab . 97 7.3.5 Radiation within a thick scattering inhomogeneous medium: the Eddington approximation . 97 7.3.6 Radiation from a thick scattering medium . 98 7.4 Radiative transfer with photon conversion . 101 8 Applications 103 8.1 Introduction: between thick and thin . 103 8.2 Spectra from stellar photospheres . 103 8.2.1 Continua from the Sun . 103 8.2.1.1 Extinction coefficient . 103 8.2.1.2 Height of formation . 104 8.2.1.3 Variation of emergent intensity and temperature stratification . 106 8.2.1.4 Center-limb variation . 108 8.2.2 Spectral lines from the Sun . 109 8.2.2.1 Extinction coefficient . 109 8.2.2.2 Height of formation . 109 8.2.2.3 The Na I D lines . 110 8.2.2.4 The Ca II K line . 111 8.2.2.5 Emergent intensity and temperature stratification . 113 8.2.2.6 Center-limb variation . 114 8.2.2.7 Outside the limb . 115 8.2.3 Spectra of stellar photospheres . 115 8.3 Stellar envelopes . 116 8.3.1 Stellar coronae . 116 vi CONTENTS 8.3.1.1 The solar corona in X-rays . 120 8.3.1.2 The solar corona in the visible . 121 8.3.1.3 The solar corona in radio waves . 124 8.3.2 Stellar winds . 126 8.3.3 Planetary nebulae . 129 8.3.3.1 Zanstra mechanism . 130 8.3.3.2 Fluorescence . 132 8.3.3.3 Forbidden lines . 132 8.3.3.4 Radio emission . 133 A Tables and term diagrams 135 B Formulae 149 References 85 Chapter 1 Introduction 1.1 Why this book? By \radiation" we are referring here exclusively to electromagnetic (EM) radiation. This radiation is of interest from both a diagnostic and an energetic standpoint. 1.1.1 EM radiation as a diagnostic Practically all astrophysical data which reach us are encoded in the EM spectrum; it is \the astronomer's treasure" (Pannekoek), a rich source of (diagnostic) information in that: { all objects emit EM waves, i.e. photons, and so are observable provided that.