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Physics of the Sun With an emphasis on numerical modeling, Physics of the Sun: A First Course presents a quantitative examination of the physical structure of the Sun and the conditions of its extended atmosphere. It gives step-by-step instructions for calculating the numerical values of various physical quantities in different regions of the Sun. Fully updated throughout, with the latest results in solar physics, this second edition covers a wide range of topics on the Sun and stellar astrophysics, including the structure of the Sun, solar radiation, the solar atmosphere, and Sun–space interactions. It explores how the physical conditions in the visible surface of the Sun are determined by the opacity of the material in the atmosphere. It also presents the empirical properties of convection in the Sun, discusses the physical conditions that must be satisfied for nuclear reactions to occur in the core, and describes how radiation transports energy from the core outwards. This text enables a practical appreciation of the physical models of solar processes. Numerical modeling problems and step-by-step instructions are featured throughout, to empower students to calculate, using their own codes, the interior structure of different parts of the Sun and the frequencies of p-modes and g-modes. They encourage a firm grasp of the numerical values of actual physical parameters as a function of radial location in the Sun. It is an ideal introduction to solar physics for advanced undergraduate and graduate students in physics and astronomy, in addition to research professionals looking to incorporate modeling into their practices. Extensive bibliographies at the end of each chapter enable the reader to explore the latest research articles in the field. Features • Fully updated with the latest results from the spacecraft Hinode, STEREO, Solar Dynamics Observatory (SDO), Interface Region Imaging Spectrograph (IRIS), and Parker Solar Probe • Presents step-by-step explanations for calculating numerical models of the photosphere, convection zone, and radiative interior with exercises and simulation problems to test learning • Describes the structure of polytropic spheres and the acoustic power in the Sun and the process of thermal conduction in different physical conditions Physics of the Sun A First Course Second Edition Dermott J. Mullan Second edition published 2023 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487–2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2023 Dermott J. Mullan First edition published by CRC Press 2009 Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978–750–8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data A Library of Congress catalog record has been requested for this book ISBN: 978-0-367-71039-2 (hbk) ISBN: 978-0-367-72032-2 (pbk) ISBN: 978-1-003-15311-5 (ebk) DOI: 10.1201/9781003153115 Typeset in Times by Apex CoVantage, LLC Contents Preface............................................................................................................................................ xiii Author .............................................................................................................................................xix Chapter 1 The Global Parameters of the Sun ...............................................................................1 1.1 Orbital Motion of the Earth ...............................................................................1 1.2 The Astronomical Unit (AU) .............................................................................3 1.3 GM and the Mass of the Sun ...........................................................................6 1.4 Power Output of the Sun: The Solar Luminosity ..............................................6 1.5 The Radius of the Sun: R .................................................................................8 1.6 Acceleration due to Gravity at the Surface of the Sun .................................... 10 1.7 The Mean Mass Density of the Sun ................................................................ 10 1.8 Escape Speed from the Solar Surface ............................................................. 11 1.9 Effective Temperature of the Sun ....................................................................11 1.10 The Oblateness of the Sun ...............................................................................12 1.11 The Observed Rotation of the Sun’s Surface ................................................... 13 1.12 A Characteristic Frequency for Solar Oscillations Due to Gravity ................. 16 Exercises .....................................................................................................................16 References ..................................................................................................................17 Chapter 2 Radiation Flow through the Solar Atmosphere .......................................................... 19 2.1 Radiation Field in the Solar Atmosphere ........................................................ 19 2.2 Empirical Properties of the Radiant Energy from the Sun .............................24 2.3 The Radiative Transfer Equation (RTE) .........................................................27 2.4 Optical Depth and the Concept of “the Photosphere” .....................................29 2.5 Special Solutions of the RTE ...........................................................................29 2.5.1 S = Constant at All Optical Depths ....................................................30 2.5.2 S = Constant in a Slab of Finite Thickness ........................................30 2.5.3 Depth-Dependent S: Polynomial Form .............................................. 31 2.5.4 Depth-Dependent S: Exponential Form ............................................. 32 2.6 The “Eddington–Barbier” (or “Milne–Barbier–Unsöld”) Relationship ..................................................................................................... 32 2.7 Is Limb Brightening Possible? ......................................................................... 33 2.8 Is S( ) = a + b Realistic? The Gray Atmosphere ............................................ 33 2.9 How Does Temperature Vary as a Function of τ? ...........................................36 2.10 Properties of the Eddington (Milne) Relation .................................................37 Exercises .....................................................................................................................37 References ..................................................................................................................38 Chapter 3 Toward a Model of the Sun: Opacity .........................................................................39 3.1 Relationship between Optical Depth and Linear Absorption Coefficient .......................................................................................................39 3.2 Two Approaches to Opacity: Atomic and Astrophysical ................................ 41 3.2.1 Energy Levels in Atomic Hydrogen ...................................................42 v vi Contents 3.3 Atomic Physics: (i) Opacity due to Hydrogen Atoms ......................................44 3.3.1 Absorption from the Ground State: Dependence on ......46 3.3.2 Absorption from Excited States: Dependence on Wavelength and T.....48 3.4 Atomic Physics: (ii) Opacity due to Negative Hydrogen Ions .........................50 3.5 Atomic Physics: (iii) Opacity due to Helium Atoms and Ions ........................ 51 3.6 Astrophysics: The Rosseland Mean Opacity ...................................................51 3.6.1 Limit of Low Density and/or High T: Electron Scattering ................. 53 3.6.2 Low T Limit .......................................................................................54 3.6.3 Higher Density: Free-Bound Absorptions..........................................54 3.6.4 Magnitude of the Largest Opacity ..................................................... 55 3.7 Power-Law Approximations to the Rosseland Mean Opacity ......................... 55 3.8 Narrow Band Opacity: Absorption Lines in the Spectrum .............................56 3.8.1 Characterizing the Properties of Absorption Lines ........................... 57 3.8.2 Heights of Formation of Different Spectral Lines .............................60 3.8.3 Shape of an Absorption Line Profile: C-Shaped Bisectors ................63 3.8.4 Shape of an Absorption Line: Magnetic Fields ..................................65 Exercises .....................................................................................................................66 References ..................................................................................................................67 Chapter 4 Toward a Model of the Sun: Properties of Ionization ................................................69 4.1 Statistical Weights of Free Electrons ...............................................................69 4.2 Saha Equation .................................................................................................. 71 4.3 Application of the Saha Equation to Hydrogen in the Sun..............................73 4.4 Application of the Saha Equation to Helium in the Sun ................................. 75 4.5 Contours of Constant Ionization: The Two Limits .......................................... 76 4.6 Application of the Saha Equation to the Negative Hydrogen Ion ................... 76 Exercises .....................................................................................................................77 References ..................................................................................................................78 Chapter 5 Computing a Model of the Sun: The Photosphere .....................................................79 5.1 Hydrostatic Equilibrium: The Scale Height ....................................................79 5.2 Sharp Edge of the Sun’s Disk ..........................................................................81 5.3 Preparing to Compute a Model of the Solar Photosphere ...............................82 5.4 Computing a Model of the Photosphere: Step by Step ....................................82 5.5 The Outcome of the Calculation .....................................................................87 5.6 Overview of the Model of the Solar Photosphere ...........................................88 Exercises .....................................................................................................................90 References ..................................................................................................................90 Chapter 6 Convection in the Sun: Empirical Properties ............................................................. 91 6.1 Nonuniform Brightness ................................................................................... 91 6.2 Granule Shapes ................................................................................................93 6.3 Upflow and Downflow Velocities in Solar Convection ...................................95 6.4 Linear Sizes of Granules .................................................................................96 6.5 Circulation Time around a Granule .................................................................98 6.6 Temperature Differences between Bright and Dark Gas ................................99 6.7 Energy Flux Carried by Convection ..............................................................100 6.7.1 Convective Energy Flux in the Photosphere ....................................100 6.7.2 Convective Energy Flux above the Photosphere? ............................ 102 Contents vii 6.7.3 Convective Energy Flux in Gas That Lies below the Photosphere ................................................................................ 102 6.8 Onset of Convection: The Schwarzschild Criterion ......................................104 6.9 Onset of Convection: Beyond the Schwarzschild Criterion .......................... 105 6.10 Numerical Value of g .................................................................................. 106 ad 6.11 Alternative Expression for g ....................................................................... 107 ad 6.12 Supergranules ................................................................................................ 108 Exercises ...................................................................................................................111 References ................................................................................................................112 Chapter 7 Computing a Model of the Sun: The Convection Zone ........................................... 115 7.1 Quantifying the Physics of Convection: Vertical Acceleration ..................... 115 7.2 Vertical Velocities and Length-Scales ........................................................... 116 7.3 Mixing Length Theory (MLT) of Convection .............................................. 117 7.4 Temperature Excesses Associated with MLT Convection ............................ 118 7.5 MLT Convective Flux in the Photosphere ..................................................... 119 7.6 MLT Convective Flux below the Photosphere .............................................. 119 7.7 Adiabatic and Nonadiabatic Processes .......................................................... 120 7.8 Computing a Model of the Convection Zone: Step by Step .......................... 122 7.9 Overview of Our Model of the Convection Zone .......................................... 123 Exercises ...................................................................................................................125 References ................................................................................................................125 Chapter 8 Radiative Transfer in the Deep Interior of the Sun .................................................. 127 8.1 Thermal Conductivity for Photons................................................................ 127 8.2 Flux of Radiant Energy at Radius r............................................................... 129 8.3 Base of the Convection Zone ......................................................................... 130 8.4 Temperature Gradient in Terms of Luminosity ............................................. 131 8.5 Temperature Gradient in Terms of Pressure ................................................. 131 8.6 Integrating the Temperature Equation ........................................................... 132 Exercise ....................................................................................................................133 References ................................................................................................................133 Chapter 9 Computing a Mechanical Model of the Sun: The Radiative Interior ...................................................................................................................... 135 9.1 Computational Procedure: Step by Step ........................................................ 135 9.2 Overview of Our Model of the Sun’s Radiative Interior ............................... 137 9.3 Photons in the Sun: How Long before They Escape? ................................... 139 9.4 A Particular Global Property of the Solar Model ......................................... 140 9.5 Does the Material in the Sun Obey the Perfect Gas Law? ............................ 141 9.6 Summary of Our (Simplified) Solar Model ................................................... 142 Exercises ...................................................................................................................143 References ................................................................................................................143 Chapter 10 Polytropes ................................................................................................................. 145 10.1 Power-Law Behavior ..................................................................................... 145 10.2 Polytropic Gas Spheres .................................................................................. 146 viii Contents 10.3 Lane–Emden Equation: Dimensional Form .................................................. 147 10.4 Lane–Emden Equation: Dimensionless Form ............................................... 148 10.5 Boundary Conditions for the Lane–Emden Equation ...................................149 10.6 Analytic Solutions of the Lane–Emden Equation .........................................149 10.6.1 Polytrope n = 0 ................................................................................. 150 10.6.2 Polytrope n = 1 ................................................................................. 150 10.6.3 Polytrope n = 5 ................................................................................. 150 10.7 Are Polytropes in Any Way Relevant for “Real Stars”? ............................... 151 10.8 Calculating a Polytropic Model: Step by Step ............................................... 152 10.9 Central Condensation of a Polytrope .............................................................154 Exercises ...................................................................................................................155 References ................................................................................................................155 Chapter 11 Energy Generation in the Sun .................................................................................. 157 11.1 The pp‑I Cycle of Nuclear Reactions ............................................................ 158 11.2 Reaction Rates in the Sun .............................................................................. 160 11.3 Proton Collision Rates in the Sun ................................................................. 160 11.4 Conditions Required for Nuclear Reactions in the Sun................................. 162 11.4.1 Nuclear Forces: Short-Range............................................................ 162 11.4.2 Classical Physics: The “Coulomb Gap” ........................................... 164 11.4.3 Quantum Physics: Bridging the “Coulomb Gap” ............................. 165 11.4.4 Center of the Sun: Thermal Protons Bridge the Coulomb Gap ................................................................................... 166 11.4.5 Other Stars: Bridging the Coulomb Gap ..........................................167 11.4.6 Inside the Nuclear Radius ................................................................ 167 11.5 Rates of Thermonuclear Reactions: Two Contributing Factors .................... 167 11.5.1 Bridging the Coulomb Gap: “Quantum Tunneling” ........................ 168 11.5.2 Post-Tunneling Processes ................................................................. 169 11.5.3 Probability of pp‑I Cycle in the Solar Core: Reactions (a) and (b) ......................................................................... 171 11.6 Temperature Dependence of Thermonuclear Reaction Rates ....................... 171 11.7 Rate of Reaction (c) in the pp‑I cycle ............................................................ 173 Exercises ...................................................................................................................173 References ................................................................................................................174 Chapter 12 Neutrinos from the Sun ............................................................................................ 175 12.1 Generation and Propagation of Solar Neutrinos............................................ 175 12.2 Fluxes of pp-I Solar Neutrinos at the Earth’s Orbit ...................................... 177 12.3 Neutrinos from Reactions Other than pp-I .................................................... 177 12.3.1 pp-II and pp-III Chains .................................................................... 178 12.3.2 Other Reactions That Occur in the Sun ........................................... 180 12.4 Detecting Solar Neutrinos on Earth .............................................................. 181 12.4.1 Chlorine Detector ............................................................................. 181 12.4.2 Cherenkov Emission ......................................................................... 182 12.4.3 Gallium Detectors ............................................................................ 183 12.4.4 Heavy Water Detector ...................................................................... 183 12.5 Solution of the Solar Neutrino Problem ........................................................ 185 Exercises ...................................................................................................................186 References ................................................................................................................186 Contents ix Chapter 13 Oscillations in the Sun: The Observations ...............................................................187 13.1 Variability in Time Only................................................................................ 188 13.2 Variability in Space and Time ....................................................................... 191 13.3 Radial Order of a Mode ................................................................................. 194 13.4 Which p-Modes Have the Largest Amplitudes? ........................................... 195 13.5 Trapped and Untrapped Modes ..................................................................... 195 13.5.1 Vertically Propagating Waves in a Stratified Atmosphere ...............196 13.5.2 Simplest Case: The Isothermal Atmosphere .................................... 197 13.5.3 Critical Frequency and the “Cut-Off” Period .................................. 199 13.5.4 Physical Basis for a Cut-Off Period ................................................. 199 13.5.5 Numerical Value of the Cut-Off Period ........................................... 199 13.6 Waves Propagating in a Non‑Vertical Direction ...........................................200 13.7 Long-Period Oscillations in the Sun ............................................................. 201 13.8 p‑mode Frequencies and the Sunspot Cycle ..................................................202 Exercises ...................................................................................................................202 References ................................................................................................................203 Chapter 14 Oscillations in the Sun: Theory ...............................................................................205 14.1 Small Oscillations: Deriving the Equations ..................................................205 14.2 Conversion to Dimensionless Variables ........................................................208 14.3 Overview of the Equations ............................................................................209 14.4 The Simplest Exercise: p‑Mode Solutions for the Polytrope n = 1 ............... 210 14.4.1 Procedure for Computation .............................................................. 211 14.4.2 Comments on the p‑mode Results: Patterns in the Eigenfrequencies .............................................................................. 213 14.4.3 Eigenfunctions .................................................................................. 214 14.5 What About g-Modes? ................................................................................... 216 14.6 Asymptotic Behavior of the Oscillation Equations ....................................... 218 14.6.1 p-modes ............................................................................................ 218 14.6.2 g-modes ............................................................................................220 14.7 Depth of Penetration of p-modes beneath the Surface of the Sun ................ 221 14.8 Why Are Certain p-Modes Excited More than Others in the Sun? ..............223 14.8.1 Depths Where p‑Modes Are Excited ...............................................223 14.8.2 Properties of Convection at the Excitation Depth ............................223 14.9 Using p-Modes to Test a Solar Model ...........................................................225 14.9.1 Global Sound Propagation ................................................................225 14.9.2 Radial Profile of the Sound Speed ...................................................225 14.9.3 The Sun’s Rotation ...........................................................................226 14.10 r-Modes in the Sun ........................................................................................229 Exercises ...................................................................................................................231 References ................................................................................................................231 Chapter 15 The Chromosphere ...................................................................................................233 15.1 Definition of the Chromosphere ....................................................................234 15.2 Linear Thickness of the Chromosphere ........................................................236 15.3 Observing the Chromosphere on the Solar Disk ...........................................236 15.4 Supergranules Observed in the H Line ....................................................... 238 15.5 The Two Principal Components of the Chromosphere .................................240