Table Of ContentPhysics 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
