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Foundations of High-Energy Astrophysics THEORETICAL ASTROPHYSICS Fulvio Melia, Series Editor Foundations of High-Energy Astrophysics THEORETICAL ASTROPHYSICS Fulvio Melia, series editor EDITORIAL BOARD A. C. Fabian, Roy Kerr, Douglas N. C. Lin, Angela V. Olinto, Saul Rappaport Other Theoretical Astrophysics titles available from the University of Chicago Press Inner Space / Outer Space Edward Kolb, Michael Turner, Keith Olive, and David Seckel, editors (1985) Theory of Neutron Star Magnetospheres Curtis F. Michel (1990) High-Energy Radiation from Magnetized Neutron Stars Peter Meszaros (1992) Cauldron in the Cosmos Claus E. Rolf and William S. Rodney (1988) Stars as Laboratories for Fundamental Physics Georg G. Raffelt (1996) Foundations of High-Energy Astrophysics Mario Vietri THE UNIVERSITY OF CHICAGO PRESS Chicago and London MARIO VIETRI is professor of astrophysics at the Scuola Normale Superiore in Pisa, Italy. The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London C 2008 by The University of Chicago All rights reserved. Published 2008 Printed in the United States of America 17161514131211100908 12345 ISBN-13: 978-0-226-85569-1 (cloth) ISBN-10: 0-226-85569-4 (cloth) Library of Congress Cataloging-in-Publication Data Vietri, Mario. Foundations of high-energy astrophysics / Mario Vietri. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-226-85569-1 (cloth : alk. paper) ISBN-10: 0-226-85569-4 (cloth : alk. paper) 1. Astrophysics—Textbooks. 2. Infrared astronomy—Textbooks. 3. X-ray astronomy—Textbooks. 4. Gamma ray astronomy—Textbooks. 5. Particles (Nuclear physics)— Textbooks. I. Title. QB468.V54 2008 523.01976—dc22 2007030145 ∞ The paper used in this publication meets the minimum requirements of the American National Standard for Information Sciences—Permanence of paper for Printed Library Materials, ANSI Z39.48-1992. A Luisa, che aveva ragione Contents Preface xv 1 Hydrodynamics 1 1.1 The Mass Conservation Equation . .... 2 1.2 The Momentum Conservation Equation . 3 1.3TheEnergyConservationEquation...... 5 1.4 Bernoulli’s Theorem . ............ 7 1.5 The Equations of Hydrodynamics in ConservativeForm............... 8 1.6ViscousFluids................. 11 1.7SmallPerturbations.............. 15 1.8Discontinuity.................. 18 1.8.1 SurfacesofDiscontinuity....... 19 1.8.2 ShockWaves.............. 22 1.8.3 Physical Interpretation of Shock Waves.................. 23 1.8.4 Collisional and Noncollisional Shocks.................. 25 1.8.5 FormationofaShock......... 26 1.9Self-similarSolutions.............. 28 1.9.1 Self-similar Solutions of the Second Kind................... 33 1.10 Relativistic Hydrodynamics . ........ 38 1.10.1 Shock Waves in Relativistic Hydrodynamics............. 41 vii viii CONTENTS 1.10.2TheStrongExplosion......... 43 1.11 The De Laval Nozzle . ............ 47 1.12Problems.................... 53 2 Magnetohydrodynamics and Magnetic Fields 55 2.1EquationsofMotion.............. 56 2.1.1 The Limit of Ideal Magnetohydrodynamics ........ 60 2.1.2 Equations of Motion in a Conservative Form................... 62 2.2 The Force Exerted by the Magnetic Field . 64 2.3 Magnetic Flux Freezing ............ 66 2.4 Small Perturbations in a Homogeneous Medium..................... 70 2.5 Stability of Tangential Discontinuities .... 77 2.6Two-TemperatureFluids............ 81 2.7 Magnetic Buoyancy and Reconnection .... 84 2.7.1 Magnetic Buoyancy . ........ 84 2.7.2 Reconnection.............. 88 2.8ShockWaves.................. 92 2.9 Magnetic Fields in Astrophysics ........ 95 2.9.1 Observations.............. 95 2.9.2 Origin of Magnetic Fields . .... 101 2.10Problems.................... 113 3 Radiative Processes 115 3.1RadiativeTransport.............. 115 3.1.1 RadiationTransport.......... 120 3.2Low-TemperatureThermalEmission..... 124 3.3 Bremsstrahlung . ................ 128 3.4Synchrotron................... 131 3.4.1 Power Radiated by a Single Particle . 131 3.4.2 The Spectrum of a Single Particle . 134 CONTENTS ix 3.4.3 The Spectrum of a Group of Nonthermal Particles . ........ 138 3.4.4 Quantum Corrections . ........ 140 3.4.5 Self-absorption............. 142 3.4.6 Cyclotron Lines . ............ 144 3.4.7 Processes in an Intense Magnetic Field . ................ 146 3.4.8 TheRazin-TsytovichEffect...... 148 3.5ComptonProcesses............... 150 3.5.1 Physical Mechanism of the Inverse Compton................ 152 3.5.2 The Spectrum of Inverse Compton Processes................ 157 3.5.3 AbouttheComptonParameter.... 167 3.5.4 Self-synchro-Compton and Compton Limit.................. 168 3.5.5 ComptonBroadening......... 171 3.6 Relativistic Effects . ............ 172 3.6.1 Superluminal Motions . ........ 173 3.6.2 Emission Properties of Relativistic Sources . ................ 174 3.7PairCreationandAnnihilation........ 177 3.8 Cosmological Attenuations . ........ 181 3.8.1 Protons................. 181 3.8.2 Photons................. 186 3.9Problems.................... 188 4 Nonthermal Particles 191 4.1 The Classic Theory of Acceleration . .... 192 4.1.1 Acceleration . ............ 193 4.1.2 Injection................. 206 4.2 Constraints on the Maximum Energy...................... 208 x CONTENTS 4.3 More Details in the Newtonian Limit....................... 213 4.3.1 From the Vlasov Equation to the Convection-Scattering Equation .... 215 4.3.2 Scattering in the Angle of Motion in a MediumatRest............ 217 4.3.3 Scattering and Convection in a MediuminMotion........... 219 4.4GeneralDiscussion............... 226 4.4.1 An Equation for f ........... 227 4.4.2 The Small Pitch Angle Scattering Limit.................. 232 4.4.3 Distributions of Probability Pu and Pd 238 4.4.4 The Particles’ Spectrum ........ 241 4.4.5 The Equations for Pu and Pd ..... 243 4.4.6 Results................. 247 4.5 The Unipolar Inductor . ............ 250 4.6Problems.................... 254 5 Spherical Flows: Accretion and Explosion 255 5.1 Accretion from Cold Matter . ........ 256 5.2 Accretion from Hot Matter . ........ 260 5.2.1 TheCriticalPoint........... 265 5.3TheIntermediateCase............. 267 5.4 Doubts about the Bondi Accretion Rate . 268 5.5TheEddingtonLuminosity.......... 270 5.6 The Efficiency of Spherical Accretion . .... 275 5.7ExplosiveMotions............... 277 5.7.1 Supernovae............... 278 5.7.2 Gamma Ray Bursts . ........ 286 5.8Problems.................... 300 6 Disk Accretion I 303 6.1QualitativeIntroduction............ 305 6.2FundamentalEquations............ 307 CONTENTS xi 6.3 Special Relations ................ 310 6.4 The α Prescription............... 315 6.5EquationsfortheStructureofDisks..... 318 6.6TheStandardSolution............. 322 6.7 The Origin of Torque . ............ 326 6.8 Disk Stability . ................ 331 6.8.1 TimeScales............... 331 6.8.2 Instability ................ 332 6.9 Lense-Thirring Precession . ........ 337 6.10Problems.................... 345 7 Disk Accretion II 347 7.1OtherDiskModels............... 347 7.1.1 The Origin of Particles . ........ 349 7.1.2 Dynamic Peculiarities of Pair Plasmas . ................ 351 7.1.3 The Pair Plasma without Input of ExternalPhotons............ 352 7.1.4 The Pair Plasma with Input of ExternalPhotons............ 362 7.2 Thick Accretion Disks . ............ 366 7.2.1 SomeGeneralProperties....... 368 7.2.2 The Inapplicability of the EddingtonLimit............ 371 7.2.3 PolytropicModels........... 374 7.2.4 PropertiesofThickDisks....... 376 7.3 Nondissipative Accretion Flows ........ 379 7.4FurtherDevelopmentsoftheTheory..... 386 7.4.1 General-Relativistic Corrections . 387 7.4.2 The Fate of Angular Momentum at LargeRadii............... 387 7.5 Accretion Disks on Magnetized Objects..................... 393 7.5.1 The Alfv´enRadius........... 394 xii CONTENTS 7.5.2 Interaction between the Disk and the Magnetosphere . ........ 401 7.5.3 Accretion Columns . ........ 403 7.6 Boundary Layers ................ 414 7.7Problems.................... 417 8 Electrodynamics of Compact Objects 419 8.1TheGold-PaciniMechanism.......... 420 8.2 The Magnetospheres Surrounding Pulsars . 422 8.2.1 Quasi-Neutral or Charge-Separated Plasma? . ................ 427 8.2.2 The Goldreich and Julian Magnetosphere . ............ 429 8.2.3 ThePulsarEquation.......... 432 8.2.4 TheSolution.............. 443 8.2.5 The Transport of Angular Momentum 444 8.2.6 Discussion................ 448 8.3 The Blandford-Znajek Model . ........ 450 8.3.1 The Magnetic Field of a Black Hole . 450 8.3.2 TheBlackHoleEquation....... 456 8.3.3 The Transport of Energy and of Angular Momentum . ........ 470 8.3.4 AQualitativeDiscussion........ 473 8.3.5 A Simplified Discussion of Total Energetics................ 475 8.4TheGenerationofCharges.......... 479 8.5Disk-JetCoupling............... 482 8.5.1 The Lovelace-Blandford Model .... 484 8.5.2 A Special Solution . ........ 485 8.5.3 Discussion................ 490 8.5.4 A Model Including Inertial Effects . 494 8.5.5 A Special Solution . ........ 500 8.5.6 Results................. 503 8.5.7 A Brief Summary ............ 508 8.6Problems.................... 510 CONTENTS xiii Appendix A Propagation of Electromagnetic Waves 513 B Orbits Around Black Holes 519 B.1Problem..................... 527 C Useful Formulae 529 C.1VectorIdentities................ 529 C.2 Cylindrical Coordinates ............ 529 C.3 Spherical Coordinates . ............ 530 Bibliography 533 Index 543 Preface When we bring a new scientific text to the attention of col- leagues and students, it is necessary to provide a justification for this novelty. The main reason that drove me to write this book is the desire to present all the basic notions
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