Physics of the Solar Corona An Introduction with Problems and Solutions Markus J Aschwanden Physics of the Solar Corona An Introduction with Problems and Solutions Published in association with Praxis Publishing Chichester, UK Dr Markus J Aschwanden Lockheed Martin Advanced Technology Center Solar and Astrophysics Laboratory Palo Alto California USA SPRINGER–PRAXIS BOOKS IN ASTRONOMY AND PLANETARY SCIENCES SUBJECT ADVISORY EDITOR: Dr. Philippe Blondel, C.Geol., F.G.S., Ph.D., M.Sc., Senior Scientist, Department of Physics, University of Bath, Bath, UK; John Mason B.Sc., M.Sc., Ph.D. ISBN 3-540-30765-6 Springer-Verlag Berlin Heidelberg New York Springer is part of Springer-Science + Business Media (springeronline.com) Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available from the Internet at http://dnb.ddb.de Library of Congress Control Number: 2005937065 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. # Praxis Publishing Ltd, Chichester, UK, 2005 Printed in Germany Reprinted with corrections, problems and solutions and issued in paperback 2006 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Jim Wilkie Completed in LaTex: EDV-Beratung, Germany Printed on acid-free paper To Susanna Contents Preface xix Preface to 2nd Edition xxii 1 Introduction 1 1.1HistoryofSolarCoronaObservations................. 1 1.2NomenclatureofCoronalPhenomena................. 8 1.3TheSolarMagneticCycle....................... 14 1.4MagneticFieldoftheSolarCorona.................. 17 1.5GeometricConceptsoftheSolarCorona ............... 20 1.6DensityStructureoftheSolarCorona................. 23 1.7TemperatureStructureoftheSolarCorona .............. 26 1.8 Plasma-β ParameteroftheSolarCorona ............... 28 1.9 Chemical Composition of the Solar Corona . ............ 30 1.10RadiationSpectrumoftheSolarCorona................ 32 1.11Summary................................ 36 2 Thermal Radiation 37 2.1RadiationTransferandObservedBrightness ............. 37 2.2Black-BodyThermalEmission..................... 39 2.3 Thermal Bremsstrahlung (Free−FreeEmission)............ 42 2.4AtomicEnergyLevels......................... 47 2.5 Atomic Transition Probabilities . ................ 50 2.6IonizationandRecombinationProcesses ............... 52 2.7 Ionization Equilibrium and Saha Equation . ............ 55 2.8EmissionLineSpectroscopy...................... 57 2.9RadiativeLossFunction........................ 61 2.10FirstIonizationPotential(FIP)Effect................. 63 2.11Summary................................ 66 3 Hydrostatics 67 3.1HydrostaticScaleHeight........................ 67 3.2HydrostaticWeightingBias ...................... 71 3.3 Multi-Hydrostatic Corona . ....................... 76 viii CONTENTS 3.4LoopGeometryandInclination..................... 83 3.4.1 Vertical Semi-Circular Loops ................. 83 3.4.2 Inclined Semi-Circular Loops ................. 84 3.4.3 Diverging Semi-Circular Loops . ............. 85 3.4.4 CoordinateTransformations.................. 86 Image(orObserver’s)CoordinateSystem........... 87 HeliographicCoordinateSystem................ 88 LoopPlaneCoordinateSystem................. 88 Column Depth of Loops with Constant Cross Section . 89 3.5LoopLine-of-SightIntegration..................... 89 3.5.1 BrightSingleLoop....................... 90 3.5.2 FaintSingleLoop........................ 92 3.5.3 Statistical Distribution of Loops . ............. 92 3.6HydrostaticSolutionsandScalingLaws................ 93 3.6.1 UniformHeatingandRTVScalingLaws........... 94 3.6.2 Nonuniform Heating and Gravity . ............. 96 3.6.3 Analytical Approximations of Hydrostatic Solutions . 96 3.7HeatingFunction............................ 99 3.8 Instrumental Response Functions . ................. 108 3.9 Observations of Hydrostatic Loops . ................. 109 3.10HydrostaticDEMDistributions .................... 113 3.11Summary................................ 116 4 Hydrodynamics 117 4.1 Hydrodynamic Equations ........................ 117 4.2 Steady-Flow and Siphon-Flow Solutions . ............. 122 4.3 Thermal Stability of Loops . ................. 127 4.3.1 Radiative Loss Instability . ................. 128 4.3.2 Heating Scale Instability . ................. 133 4.4 Observations of Flows in Coronal Loops . ............. 135 4.5 Observations of Cooling Loops . ................. 139 4.5.1 CoolingDelays......................... 141 4.5.2 Iron Abundance and Filling Factors . ............. 142 4.5.3 Scaling Law of Cooling Loops ................. 145 4.5.4 Catastrophic Cooling Phase . ................. 145 4.6 Observations of Non-Hydrostatic Loops . ............. 148 4.6.1 TRACEObservations...................... 150 4.6.2 TheoreticalModels....................... 153 4.7 Hydrodynamic Numerical Simulations of Loops . ......... 155 4.8 Hydrodynamics of the Transition Region . ............. 159 4.9 Hydrodynamics of Coronal Holes . ................. 167 4.10 Hydrodynamics of the Solar Wind . ................. 172 4.11Summary................................ 173 CONTENTS ix 5 Magnetic Fields 175 5.1ElectromagneticEquations....................... 176 5.1.1 Maxwell’sEquations...................... 176 5.1.2 Amp`ere’sLaw......................... 176 5.1.3 Ohm’sLaw........................... 177 5.1.4 Induction Equation ....................... 177 5.2PotentialFields............................. 178 5.2.1 UnipolarField......................... 179 5.2.2 DipoleField.......................... 180 5.2.3 Potential-Field Calculation Methods . ............ 182 Green’s Function Methods . ................ 182 Eigenfunction Expansion Methods . ............ 184 5.3Force-FreeFields............................ 186 5.3.1 LinearForce-FreeFields.................... 187 5.3.2 ShearedArcade......................... 188 5.3.3 Nonlinear Force-Free Field Calculation Methods . .... 190 TheVerticalIntegrationMethod................ 190 The Boundary Integral Method . ................ 192 TheEulerPotentialMethod.................. 193 FullMHDMethod....................... 194 Potential Vector Grad−RubinMethod............. 195 Evolutionary Methods . ................ 195 5.4MagneticFieldinActiveRegionCorona ............... 196 5.4.1 3D Stereoscopy of Active Region Loops ............ 196 5.4.2 Alfv´enVelocityinActiveRegions............... 202 5.4.3 Non-Potentiality of Soft X-Ray Loops . ............ 205 5.4.4 The Width Variation of Coronal Loops . ............ 209 5.5MagneticHelicity............................ 213 5.5.1 UniformlyTwistedCylindricalForce-FreeFluxtubes..... 215 5.5.2 Observations of Sigmoid Loops ................ 217 5.5.3 ConservationofHelicity.................... 218 5.6MagneticNullpointsandSeparators.................. 220 5.6.1 Topological Definitions . ................ 220 5.6.2 ObservationsofCoronalNullpoints.............. 222 5.7 Magnetic Field Measurements in Radio ................ 224 5.7.1 Magnetic Fields Measured from Free-Free Emission . .... 224 5.7.2 GyroresonanceEmission.................... 227 5.7.3 GyroresonanceStereoscopy.................. 230 5.7.4 Non-Potential Field Modeling of Gyroresonance Emission . 233 5.8 Magnetic Field in the Transition Region ................ 234 5.8.1 TheMagneticCanopyStructure................ 235 5.8.2 Force-FreenessoftheChromosphericMagneticField..... 237 5.9Summary................................ 239 x CONTENTS 6 Magneto-Hydrodynamics (MHD) 241 6.1MHDEquations............................. 241 6.1.1 ParticleConservation...................... 242 6.1.2 MomentumorForceEquation................. 243 6.1.3 IdealMHD........................... 244 6.1.4 EnergyEquation........................ 245 6.1.5 ResistiveMHD......................... 246 6.2 MHD of Coronal Loops . ........................ 247 6.2.1 Magneto-StaticsinVerticalFluxtubes............. 247 6.2.2 LorentzForcenearMagneticNullpoints............ 249 6.2.3 LorentzForceinCurvedFluxtubes.............. 252 6.2.4 DynamicsofTwistedFluxtubes................ 253 6.2.5 MHDSimulationsofEmergingFluxtubes........... 257 6.2.6 MHD Simulations of Coronal Loops . ............. 259 6.3 MHD Instabilities in Coronal Loops . ................. 263 6.3.1 Rayleigh−Taylor Instability . ................. 264 6.3.2 Kruskal−Schwarzschild Instability . ............. 265 6.3.3 Kelvin−Helmholtz Instability ................. 265 6.3.4 Ballooning Instability . ................. 266 6.3.5 Convective Thermal Instability ................. 266 6.3.6 Radiatively-Driven Thermal Instability . ......... 266 6.3.7 Heating Scale-Height Instability . ............. 268 6.3.8 Resistive Instabilities . ................. 268 6.3.9 Kink Instability (m=1) . ................. 269 6.3.10 Sausage Instability (m=0) . ................. 270 6.4MHDofQuiescentFilamentsandProminences............ 270 6.4.1 MagneticFieldConfiguration................