High Energy Astrophysics Is One of the Most Important and Exciting Areas of Contemporary Astronomy, Involving the Most Energetic Phenomena in the Universe

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High energy astrophysics is one of the most important and exciting areas of contemporary astronomy, involving the most energetic phenomena in the universe. The highly acclaimed first and second editions of Professor Longair’s series immediately established themselves as essential text books on high energy astrophysics. In this third edition, the subject matter is brought up to date and consolidated into a single volume covering Galactic, extragalactic and cosmological aspects of High Energy Astrophysics. The material is presented in four parts. The first provides the necessary astronomical background for understanding the context of high energy astrophysical phenomena. The second provides a thorough treatment the physical processes that govern the behaviour of particles and radiation in astrophysical environments such as interstellar gas, neutron stars, and black holes. The third part applies these tools to a wide range of high energy astrophysical phenomena in our own Galaxy, while the fourth and final part deals with extragalactic and cosmological aspects of high energy astrophysics. This book assumes that readers have some knowledge of physics and mathematics at the undergraduate level, but no prior knowledge of astronomy is required. The new third edition of the book covers all aspects of modern high energy astrophysics to the point at which the key concerns of current research can be understood.

High Energy Astrophysics

Third Edition

Malcolm S. Longair Emeritus Jacksonian Professor of Natural Philosophy, Cavendish Laboratory, University of Cambridge, Cambridge

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon´ 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org c Cambridge University Press 1981, 1992, 1994, 1997, 2000, 2002, 2004 First published 1981 Second edition in two volumes 1992, 1994, reprinted with corrections 1997, 2000, 2002, 2004 British Library cataloguing in publication data Longair, M. S. (Malcolm S.) High energy astrophysics. I. Title XXX.YYYYY ISBN 0 XXX YYYYY Z hardback Transferred to digital printing 2004 For Deborah Contents

Preface page ii Acknowledgements v Part I. Astronomical background 1

1 High energy astrophysics – an introduction 3 1.1 High energy astrophysics and modern physics and astronomy 3 1.2 The sky in diﬀerent astronomical wavebands 5 1.3 Optical waveband 7 1.4 Infrared waveband 10 1.5 Millimetre and submillimetre waveband 14 1.6 Radio waveband 19 1.7 Ultraviolet waveband 23 1.8 X-ray waveband 25 1.9 γ-ray waveband 27 1.10 Cosmic ray astrophysics 29 1.11 Other non-electromagnetic astronomies 34 1.12 Concluding remarks 36 2 The stars and stellar evolution 39 2.1 Introduction 39 2.2 Basic observations 39 2.3 Stellar structure 43 2.4 The equations of energy generation and energy transport 48 2.5 The equations of stellar structure 53 2.6 The Sun as a star 56 2.7 Evolution of high and low mass stars 66 2.8 Stellar evolution on the colour-magnitude diagram 76 2.9 Mass loss 77 2.10 Conclusion 84

vii viii Contents 3 The galaxies 85 3.1 Introduction 85 3.2 The Hubble sequence 86 3.3 The red and blue sequences 89 3.4 Further correlations among the properties of galaxies 95 3.5 The masses of galaxies 98 3.6 The luminosity function of galaxies 104 4 Clusters of galaxies 109 4.1 The morphologies of rich clusters of galaxies 109 4.2 Clusters of galaxies and isothermal gas spheres 113 4.3 The Coma cluster of galaxies 117 4.4 Mass distribution of hot gas and dark matter in clusters 119 4.5 Cooling ﬂows in clusters of galaxies 123 4.6 The Sunyaev-Zeldovich eﬀect in hot intracluster gas 125 4.7 Gravitational lensing by galaxies and clusters of galaxies 129 4.8 Dark matter in galaxies and clusters of galaxies 135 Part II. Physical processes 141

5 Ionisation losses 143 5.1 Introduction 143 5.2 Ionisation losses – non-relativistic treatment 144 5.3 The relativistic case 149 5.4 Practical forms of the ionisation loss formulae 154 5.5 Ionisation losses of electrons 158 5.6 Nuclear emulsions, plastics and meteorites 159 5.7 Dynamical friction 165 6 Radiation of accelerated charged particles and bremsstrahlung of electrons 169 6.1 Introduction 169 6.2 The radiation of accelerated charged particles 169 6.3 Bremsstrahlung 178 6.4 Non-relativistic bremsstrahlung energy loss rate 182 6.5 Thermal bremsstrahlung 183 6.6 Relativistic bremsstrahlung 189 7 The dynamics of charged particles in magnetic fields 195 7.1 A uniform static magnetic field 195 7.2 A time-varying magnetic field 197 7.3 The scattering of charged particles by irregularities in the magnetic field 202 7.4 The scattering of high energy particles by Alfven´ and hydromagnetic waves 205 7.5 The diffusion-loss equation for high energy particles 207 Contents ix 8 Synchrotron radiation 213 8.1 The total energy loss rate 213 8.2 Non-relativistic gyroradiation and cyclotron radiation 216 8.3 The spectrum of synchrotron radiation – physical arguments 219 8.4 The spectrum of synchrotron radiation - a fuller version 223 8.5 The synchrotron radiation of a power law distribution of electron energies 233 8.6 The polarisation of synchrotron radiation 235 8.7 Synchrotron self-absorption 239 8.8 Useful numerical results 244 8.9 The radio emission of the Galaxy 246 9 Interactions of high energy photons 251 9.1 Photoelectric absorption 251 9.2 Thomson and Compton scattering 255 9.3 Inverse Compton scattering 261 9.4 Comptonisation 267 9.5 The Sunyaev-Zeldovich effect 282 9.6 Synchrotron–self Compton radiation 286 9.7 Cherenkov radiation 291 9.8 Electron–positron pair production 297 9.9 Electromagnetic showers 299 9.10 Electron-positron annihilation and positron production mechanisms 302 10 Nuclear interactions 307 10.1 Nuclear interactions and high energy astrophysics 307 10.2 Spallation cross-sections 310 10.3 Nuclear emission lines 315 10.4 Cosmic rays in the atmosphere 321 11 Aspects of plasma physics and magnetohydrodynamics 327 11.1 Elementary concepts in plasma physics 327 11.2 Magnetic flux freezing 334 11.3 Shock waves 344 11.4 The Earth’s magnetosphere 349 11.5 Magnetic buoyancy 352 11.6 Reconnection of magnetic lines of force 354 Part III. High Energy Astrophysics in our Galaxy 361

12 Interstellar gas and magnetic fields 363 12.1 The interstellar medium in the life cycle of stars 363 12.2 Diagnostic tools - neutral interstellar gas 363 12.3 Ionised interstellar gas 370 12.4 Interstellar dust 378 x Contents 12.5 An overall picture of the interstellar gas 387 12.6 Star formation 393 12.7 The Galactic magnetic field 402 13 Dead stars 413 13.1 Supernovae 414 13.2 White dwarfs, neutron stars and the Chandrasekhar limit 431 13.3 White dwarfs 438 13.4 Neutron stars 439 13.5 The discovery of neutron stars 444 13.6 The galactic population of neutron stars 458 13.7 Thermal emission of neutron stars 460 13.8 Pulsar glitches 461 13.9 The pulsar magnetosphere 464 13.10 The radio and high energy emission of pulsars 467 13.11 Black holes 469 14 Accretion power in astrophysics 483 14.1 Introduction 483 14.2 Accretion - general considerations 483 14.3 Thin accretion discs 491 14.4 Thick discs and advective flows 504 14.5 Accretion in binary systems 506 14.6 Accreting binary systems 516 14.7 Black holes in X-ray binaries 531 14.8 Final thoughts 537 15 Cosmic rays 539 15.1 The energy spectra of cosmic ray protons and nuclei 539 15.2 The abundances of the elements in the cosmic rays 543 15.3 The isotropy and energy density of cosmic rays 549 15.4 Gamma ray observations of the Galaxy 550 15.5 The origin of the light elements in the cosmic rays 554 15.6 The confinement time of cosmic rays in the Galaxy and cosmic ray clocks 563 15.7 The confinement volume for cosmic rays 565 15.8 The Galactic halo 568 15.9 The highest energy cosmic rays and extensive air-showers 571 15.10 Observations of the highest energy cosmic rays 573 15.11 The isotropy of ultra-high energy cosmic rays 578 15.12 The Greisen-Kuzmin-Zatsepin (GKZ) cut-off 581 16 The origin of cosmic rays in our Galaxy 585 16.1 Introduction 585 16.2 Energy loss processes for high energy electrons 585 Contents xi 16.3 diffusion-loss equation for high energy electrons 590 16.4 Supernova remnants as sources of high energy particles 594 16.5 The minimum energy requirements for synchrotron radiation 599 16.6 Supernova remnants as sources of high energy electrons 603 16.7 The evolution of supernova remnants 604 16.8 The adiabatic loss problem and the acceleration of high energy particles 607 17 The acceleration of high energy particles 613 17.1 General principles of acceleration 613 17.2 The acceleration of particles in solar flares 614 17.3 Fermi acceleration - original version 616 17.4 Diffusive shock acceleration in strong shock waves 621 17.5 Beyond the standard model 627 17.6 The highest energy cosmic rays 635 Part IV. Extragalactic High Energy Astrophysics 637

18 Active galaxies 639 18.1 Introduction 639 18.2 Radio galaxies and high energy astrophysics 639 18.3 The quasars 641 18.4 Seyfert galaxies 648 18.5 Blazars, superluminal sources and γ-ray sources 651 18.6 Low Ionisation Nuclear Emission Regions – LINERS 653 18.7 Ultra-Luminous Infrared Galaxies – ULIRGs 654 18.8 X-ray Surveys of Active Galaxies 656 18.9 Unification Schemes for Active Galaxies 658 19 Black Holes in the Nuclei of Galaxies 667 19.1 The Properties of Black Holes 667 19.2 Elementary Considerations 669 19.3 Dynamical evidence for supermassive black holes in galactic nuclei 670 19.4 The Soltan Argument 681 19.5 Black holes and spheroid masses 683 19.6 X-ray observations of fluorescence lines in active galactic nuclei 685 19.7 The Growth of Black Holes in the Nuclei of Galaxies 693 20 The Vicinity of the Black Hole 697 20.1 The prime ingredients of active galactic nuclei 697 20.2 The continuum spectrum 697 20.3 The emission line regions – the overall picture 701 20.4 The narrow-line regions - the example of Cygnus A 702 20.5 The broad-line regions and reverberation mapping 707 20.6 The alignment effect and shock excitation of emission line regions 715 Contents i 20.7 Accretion discs about supermassive black holes 718 21 Extragalactic Radio Sources 723 21.1 Extended Radio Sources – Fanaroff-Riley types 723 21.2 The astrophysics of FR2 radio sources 730 21.3 The FR1 radio sources 738 21.4 The microquasars 740 21.5 Jet physics 742 22 Compact extragalactic sources and superluminal motions 745 22.1 Compact radio sources 745 22.2 Superluminal motions 747 22.3 Relativistic beaming 750 22.4 The superluminal source population 759 22.5 Synchro-Compton Radiation and the Inverse Compton Catastrophe 763 22.6 γ-ray Sources in Active Galactic Nuclei 764 22.7 γ-ray bursts 771 23 Cosmological aspects of high energy astrophysics 781 23.1 The cosmic evolution of galaxies and active galaxies 781 23.2 The essential theoretical tools 783 23.3 The Evolution of non-thermal sources with cosmic epoch 787 23.4 The Evolution of thermal sources with cosmic epoch 798 23.5 Mid and far-infrared number counts 807 23.6 Submillimetre Number Counts 810 23.7 The global star formation rate 813 23.8 The Old Red Galaxies 817 23.9 Putting It All Together 819 A1 Astronomical conventions and nomenclature 825 A1.1 Galactic coordinates and projections of the celestial sphere onto a plane 825 A1.2 Distances in astronomy 829 A1.3 Masses in astronomy 832 A1.4 Flux densities, luminosities, magnitudes and colours 833 A1.5 Diffraction-limited telescopes 838 A1.6 Interferometry and synthesis imaging 845 A1.7 The sensitivities of astronomical detectors 848 A1.8 Units and relativistic notation 854 References 858

9 Interactions of high energy photons

The three main processes involved in the interaction of high energy photons with atoms, nuclei and electrons are photoelectric absorption, Compton scattering and electron–positron pair production. These processes are important not only in the study of high energy astrophysical phenomena in a wide variety of different circumstances but also in the detection of high energy particles and photons. For example, photoelectric absorption is observed in the spectra of most X-ray sources at energies ε . 1 keV. Thomson and Compton scattering appear in a myriad of guises from the processes occurring in stellar interiors, to the spectra of binary X-ray sources and inverse Compton scattering figures prominently in sources in which there are intense radiation fields and high energy electrons. Pair production is bound to occur wherever there are significant fluxes of high energy γ-rays – evidence for the production of positrons by this process is provided by the detection of the 511 keV electron–positron annihilation line in our own Galaxy.

9.1 Photoelectric absorption 2 At low photon energies, ~ω mec , the dominant process by which photons interact with matter is photoelectric, or bound–free, absorption and is one of the principal sources of opacity in stellar interiors. We are principally interested here in the process in somewhat more rareﬁed plasmas. If the energies of the incident photons ε = ~ω are greater than the energy of the X-ray atomic energy level EI, an electron can be ejected from that level, the remaining energy (~ω−EI) being carried away as the kinetic energy of the ejected electron, the photoelectric eﬀect. The photon energy at which ~ω = EI corresponds to a absorption edge in the spectrum of the radiation because ejection of electrons from this energy level is impossible if the photons are of lower energy. For photons with higher energies, the cross- section for photoelectric absorption from this level decreases as roughly ω−3. Examples of the absorption cross-sections for a number of common elements are shown in Fig. 9.1 and the X-ray atomic energy levels of atoms up to iron are listed in Table 9.1. The evaluation of these cross-sections is one of the standard calculations in the quantum

251 252 Interactions of high energy photons

Fig. 9.1. Photoabsorption cross-sections of the abundant elements in the interstellar medium as a function of wavelength (Cruddace et al., 1974). theory of radiation (Heitler, 1954). For example, the analytic solution for the absorption 2 cross-section for photons with energies ~ω EI and hω mec due to the ejection of electrons from the K-shells of atoms, that is, from the 1s level, is

2 !7/2 12 3/2 5 !7/2 √ mec e m Z 1 σ = 4 2σ α4Z5 = √ e . (9.1) K T ~ω 5 6 4 ~ω 192 2π 0 ~ c

2 2 2 2 2 4 α = e /4π0~c is the fine structure constant and σT = 8πre /3 = e /6π0 mec the Thomson cross-section. This cross-section takes account of the fact that there are 2 K-shell electrons in all elements except hydrogen, both 1s electrons contributing to the opacity of the material. The absorption cross-section has a strong dependence upon the atomic number Z and so, although heavy elements are very much less abundant than hydrogen, the combination of the ω−3 dependence and the fifth-power dependence upon Z means that quite rare elements can make significant contributions to the total absorption cross-section at ultraviolet and X-ray energies. More detailed calculations of these cross-sections with appropriate Gaunt factors are given by Karzas and Latter (Karzas and Latter, 1961). These data enable the X-ray absorption coefficient for interstellar matter to be deter- 9.1 Photoelectric absorption 253

Table 9.1. The X-ray atomic energy levels for elements up to iron (Bearden and Burr, 1967).

Element Energies in electron-volts (eV) X-ray term KLI LII LIII MI MII,III MIV,V Hydrogen 13.598 Helium 24.587 Lithium 54.75 Beryllium 111.0 Boron 188.0 4.7 Carbon 283.8 6.4 Nitrogen 401.6 9.2 Oxygen 532.0 23.7 7.1 Fluorine 685.4 31 8.6 Neon 866.9 45 18.3 Sodium 1072.1 63.3 31.1 Magnesium 1305.0 89.4 51.4 Aluminium 1559.6 117.7 73.1 Silicon 1838.9 148.7 99.2 Phosphorus 2145.5 189.3 132.2 Sulphur 2472.0 229.2 164.8 Chlorine 2822.4 270.2 201.6 200.0 17.5 6.8 Argon 3202.9 320 247.3 245.2 25.3 12.4 Potassium 3607.4 377.1 296.3 293.6 33.9 17.8 Calcium 4038.1 437.8 350.0 346.4 43.7 25.4 Scandium 4492.8 500.4 406.7 402.2 53.8 32.3 6.6 Titanium 4966.4 563.7 461.5 455.5 60.3 34.6 3.7 Vanadium 5465.1 628.2 520.5 512.9 66.5 37.8 2.2 Chromium 5989.2 694.6 583.7 574.5 74.1 42.5 2.3 Manganese 6539.0 769.0 651.4 640.3 83.9 48.6 3.3 Iron 7112.0 846.1 721.1 708.1 92.9 54.0 3.6

mined. Absorption cross-sections of the forms shown in Fig. 9.1 are summed, weighted by the cosmic abundance of the different elements, 1 X σ (ε) = n σ (ε) . (9.2) e n i i H i In this computation, the K-edges, corresponding to the ejection of electrons from the 1s shell of the atom or ion, provide the dominant source of opacity. The resulting total absorption coefficient for X-rays, assuming the standard cosmic abundances of the chemical elements, is shown in Fig. 9.2, the K-edges of different elements being indicated. In low resolution X-ray spectral studies, these edges cannot be resolved individually as distinct features and a useful linear interpolation formula for the X-ray absorption coefficient, σe, 254 Interactions of high energy photons

Fig. 9.2. The eﬀective absorption cross-section per hydrogen atom for interstellar gas with typical cosmic abundances of the chemical elements. The solid line is for the gaseous component of the interstellar medium ; the dot-dashed line includes molecular hydrogen. The discontinuities in the absorption cross-section as a function of energy are associated withR the K-shell absorption edges of the elements indicated. The optical depth of the medium is τe = σe(ε)NH dl where NH is the number density of hydrogen atoms (Cruddace et al., 1974). Note that the cross-section is presented in units of cm2. For reference, 1Å ≡ 12.4 keV and 100Å ≡ 0.124 keV.

and the corresponding optical depth, τe is Z ! Z ~ω −8/3 τ (~ω) = σ N dl = 2 × 10−26 N dl , (9.3) e e H 1 keV H R where the column depth NH dl is expressed in particles per square metre and NH is the number density of hydrogen atoms in particles per cubic metre. For example, if the interstellar gas density were 106 hydrogen atoms m−3, the optical depth of the medium 9.2 Thomson and Compton scattering 255 is roughly unity for a path length of 1 kpc at 1 keV. Thus, the spectra of many X-ray sources turn over at about 1 keV because of interstellar photoelectric absorption. Because of the steep energy dependence of τe, photoelectric absorption is only important at energies ~ω 1 keV for sources with large column densities of matter between the source and the observer.

9.2 Thomson and Compton scattering In 1923, Compton discovered that the wavelength of hard X-ray radiation increases when it is scattered by stationary electrons (Compton, 1923). This was deﬁnitive proof of Ein- stein’s quantum picture of the nature of light according to which it may be considered to possess both wave-like and particle-like properties (Einstein, 1905). In the Compton scattering process, the incoming high energy photons collide with stationary electrons and transfer some of their energy and momentum to the electrons. Consequently, the scattered photons have lesser energies and momenta than before the collisions. Since the energy and momentum of the photon is proportional to frequency, E = ~ω and p = (~ω/c) ik, where ik is the unit vector in the direction of travel of the photon, the loss of energy of the photon corresponds to an increase in its wavelength. We begin with the simpler process of Thomson scattering in which the photons, or electromagnetic waves, are scattered without change of energy.

9.2.1 Thomson scattering Thomson first published the formula for what is now called the Thomson cross-section in 1906 (Thomson, 1906) and used his result to show that the number of electrons in each atom is of the same order as the element’s atomic number. He used Larmor’s formula, which we derived using Thomson’s methods in Sect. 6.2.2. We can carry out a completely classical analysis of the scattering of an unpolarised parallel beam of radiation through an angle α by a stationary electron using the radiation formula (6.6). It is assumed that the incident beam propagates in the positive z-direction (Fig. 9.3) and, without loss of generality, we can arrange the geometry of the scattering to be such that the scattering angle α lies in the x − z plane. The electric field strength of the unpolarised incident field is resolved into components of equal intensity with electric vectors in the orthogonal ix and iy directions (Fig. 9.3). The electric fields experienced by the electron in the x and y directions, Ex = Ex0 exp(iωt) and Ey = Ey0 exp(iωt) respectively, cause the electron to oscillate and the accelerations in these directions are

r¨x = eEx/me ;r ¨y = eEy/me . (9.4) We can therefore enter these accelerations into the radiation formula (6.6), which describes the angular dependence of the emitted intensity upon the polar angle θ. 256 Interactions of high energy photons

Fig. 9.3. Illustrating the geometry of the Thomson scattering of a beam of radiation by a free electron.

Treating ﬁrst the x-acceleration, (6.6) can be used with the substitution α = π/2 − θ. The intensity of radiation scattered through angle θ into the solid angle dΩ is then ! dE e2|r¨ |2 sin2 θ e4|E |2 − dΩ = x dΩ = x cos2 α dΩ . (9.5) 2 3 2 2 3 dt x 16π 0c 16π me0c

2 2 2 Taking time averages of Ex, Ex = Ex0/2. We sum over all waves contributing to the Ex- component of radiation and express the result in terms of the incident energy per unit area 2 upon the electron. The latter is given by Poynting’s theorem, Sx = (E × H) = c0Ex iz. Since the radiation is incoherent, we sum over all the time-averaged waves to ﬁnd the total P 2 intensity in the direction α from the x-component of the acceleration is S x = i c0Ex0/2, and so ! dE e4 cos2 α X e4 cos2 α − dΩ = E2 dΩ = S dΩ . (9.6) dt 2 2 3 x 2 2 2 4 x x 16π me0c i 16π me0 c

Next consider scattering of the Ey-component of the incident field. From the geometry of Fig. 9.3, the radiation in the x − z plane due to the acceleration of the electron in the y-direction corresponds to scattering through θ = 90◦ and therefore the scattered intensity in the α-direction is ! dE e4 − dΩ = S dΩ . (9.7) dt 2 2 2 4 y y 16π me0 c The total scattered radiation into dΩ is found by adding the intensities of the two inde- 9.2 Thomson and Compton scattering 257 pendent field components, ! dE e4 S − dΩ = 1 + cos2 α dΩ , (9.8) dt 2 2 2 4 2 16π me0 c where S = S x + S y and we recall that S x = S y for unpolarised radiation. We now express the scattered intensity in terms of a differential scattering cross-section dσT in direction α by the following relation, dσ (α) energy radiated per unit time per unit solid angle T = . (9.9) dΩ incident energy per unit time per unit area Since the total incident energy per unit time per unit area is S , the differential cross-section for Thomson scattering is e4 (1 + cos2 α) 3σ dσ = dΩ = T (1 + cos2 α) dΩ , (9.10) T 2 2 2 4 2 16π 16π 0 mec 2 2 which can be expressed in terms of the classical electron radius re = e /4π0mec , r2 dσ = e (1 + cos2 α) dΩ . (9.11) T 2 To find the total cross-section for scattering, we integrate over all solid angles, Z π r2 8π e4 σ = e (1 + cos2 α) 2π sin α dα = r2 = = 6.653 × 10−29 m2 . (9.12) T 2 3 e 2 2 4 0 6π0 mec This is Thomson’s famous result for the total cross-section for scattering of electromagnetic waves by stationary free electrons. It will reappear in many different guises in the course of the exposition. Let us note some of the important features of Thomson scattering.

(i) The scattering is symmetric with respect to the scattering angle α. Thus, as much radiation is scattered in the backward as in the forward direction. (ii) The scattering cross-section for 100% polarised emission can be found by integrating the scattered intensity (9.5) over all angles, ! Z   2 2  4  dE e |r¨x| 2  e  − = sin θ 2π sin θ dθ =   S x = σTS x . (9.13) dt 2 3 2 2 4 x 16π 0c 6π0 mec We find the same total cross-section for scattering as before. This should not be surprising because it does not matter how the electron is forced to oscillate. For incoherent radiation, the energy radiated is proportional to the sum of the incident intensities of the radiation field and so the only important quantity so far as the electron is concerned is the total intensity of radiation incident upon it. It does not matter how anisotropic the incident radiation field is. One convenient way of 258 Interactions of high energy photons expressing this result is to write the formula for the scattered radiation in terms to the energy density of radiation urad at the electron X X urad = ui = S i/c , (9.14) i i and hence

−(dE/dt) = σTcurad . (9.15) (iii) One distinctive feature of Thomson scattering is that the scattered radiation is polarised, even if the incident beam of radiation is unpolarised. This can be seen in- tuitively from Fig. 9.3 because all the E-vectors of the unpolarised beam lie in the x − y plane. Therefore, when the electron is observed precisely in the x − y plane, the scattered radiation is 100% polarised. On the other hand, if we look along the z- direction, we observe unpolarised radiation. If the degree of polarisation is deﬁned as I − I Π = max min , (9.16) Imax + Imin the fractional polarisation of the radiation is 1 − cos2 α Π = . (9.17) 1 + cos2 α This is therefore a means of producing polarised radiation from an initially unpolarised beam. (iv) Thomson scattering is one of the most important processes which impedes the escape of photons from any region. If the number density of photons of frequency ν is N, the rate at which energy is scattered out of the beam is d(Nhν) − = σ cNhν . dt T There is no change of energy of the photons in the scattering process and so, if there are Ne electrons per unit volume, the number density of photons decreases exponentially with distance R −dN/dt = σTcNeN , −dN/dx = σTNeN , N = N0 exp − αTNe dx . (9.18)

We can express this by stating that the optical depth τT of the medium for Thomson scattering is Z

τ = σTNe dx (9.19)

In this process, the photons are scattered in random directions and so they perform a random walk, each step corresponding to the mean free path λT of the photon −1 through the electron gas, where λT = (σTNe) . Thus, there is a very real sense 9.2 Thomson and Compton scattering 259 in which the Thomson cross-section is the physical cross-section of an electron for the scattering of electromagnetic waves.

9.2.2 Compton scattering In Thomson scattering, there is no change in the frequency of the radiation. This remains a good approximation provided the energy of the photon is much less than the rest mass 2 energy of the electron, ~ω mec . In general, as long as the energy of the photon is less 2 than mec in the centre of momentum frame of reference, the scattering may be treated as Thomson scattering, as in our treatment of inverse Compton scattering in Sect. 9.3.3. There are, however, many important cases in which the frequency change associated with the collision between the electron and the photon cannot be neglected. Let us establish some of the more important general results. Suppose the electron moves with velocity u through the laboratory frame of reference S . Let us use four-vectors to ﬁnd an elegant solution for the change in energy of the scattered photons. The momentum four-vectors of the electron and the photon before and after the collision are as follows: Before After

0 0 0 0 Electron P = [γmec, γmeu] P = [γ mec, γ meu ]

" # " 0 0 # ~ω ~ω 0 ~ω ~ω Photon K = , i K = , i 0 c c k c c k The collision conserves four-momentum and hence P + K = P0 + K0 (9.20) Now, square both sides of this four-vector equation and use the properties of the norms of the momentum four-vectors of the electron and the photon:

0 0 2 2 0 0 P · P = P · P = mec and K · K = K · K = 0 . (9.21) Therefore, (P + K)2 = (P0 + K0)2 , P · P + 2P · K + K · K = P0 · P0 + 2P0 · K0 + K0 · K0 , P · K = P0 · K0 . (9.22) Now multiply (9.20) by K0 and use the equality (9.22). P · K0 + K · K0 = P0 · K0 + K0 · K0 , P · K0 + K · K0 = P · K . (9.23) 260 Interactions of high energy photons This is the four-vector equation we seek. Let us reduce it to somewhat more familiar form by multiplying out the four-vector products. The scattering angle is given by ik ·ik0 = cos α. The angle between the incoming photon and the velocity vector of the electron is θ and the 0 0 0 0 angle between them after the collision θ . Then, cos θ = ik · u/|u| and cos θ = ik0 · u /|u |. After a little algebra, ω0 1 − (v/c) cos θ = 0 2 . (9.24) ω 1 − (v/c) cos θ + (~ω/γmec )(1 − cos α) In the traditional argument, the Compton effect is described in terms of the increase in wavelength of the photon on scattering from a stationary electron, that is, for the case v = 0, γ = 1. ω0 1 ∆λ λ0 − λ ~ω = 2 ; = = 2 (1 − cos α) . (9.25) ω 1 + (~ω/mec )(1 − cos α) λ λ mec This effect of ‘cooling’ the radiation and transferring the energy to the electron is sometimes called the recoil effect. Note, however, that (9.24) also shows more generally how energy can be exchanged between the electron and the radiation field. In the limit ~ω 2 γmec , the change in frequency of the photon is ω0 − ω ∆ω v (cos θ − cos θ0) = = . (9.26) ω ω c [1 − (v/c) cos θ0] Thus, to first order, the frequency changes are ∼ v/c. Also to first order, if the angles θ and θ0 are randomly distributed, a photon is just as likely to decrease as increase its energy. It can be shown that there is no net increase in energy of the photons to first order in v/c and it is only in second order, that is, to order v2/c2, that there is a net energy change. The Thomson cross-section is only adequate for cases in which the electron moves with 2 velocity v c or if the photon has energy ~ω mec in the centre of momentum frame of reference. If a photon of energy ~ω collides with a stationary electron, according to the analysis of Sect. 5.3.3, the centre of momentum frame moves at velocity v ~ω = 2 . (9.27) c mec + ~ω 2 Therefore, if the photons have energy ~ω & mec , we must use the proper quantum relativistic cross-section for scattering. Another case which can often arise is if the photons 2 are of low energy ~ω mec but the electron moves ultrarelativistically with γ 1. Then, the centre of momentum frame moves with a velocity close to that of the electron 2 and in this frame the energy of the photon is γ~ω. If γ~ω ∼ mec , the quantum relativistic cross-section has to be used. The relevant total cross-section is the Klein–Nishina formula: (" # ) 1 2(x + 1) 1 4 1 σ = πr2 1 − ln(2x + 1) + + − . (9.28) K−N e x x2 2 x 2(2x + 1)2 9.3 Inverse Compton scattering 261

Fig. 9.4. A schematic diagram showing the dependence of the Klein–Nishina cross-section upon photon energy.

2 2 2 where x = ~ω/mec and re = e /4π0mec is the classical electron radius. For low energy photons, x 1, this expression reduces to 8π σ = r2(1 − 2x) = σ (1 − 2x) ≈ σ . (9.29) K−N 3 e T T In the ultrarelativistic limit, γ 1, the Klein–Nishina cross-section becomes ! 1 1 σ = πr2 ln 2x + . (9.30) K−N e x 2 so that the cross-section decreases roughly as x−1 at the highest energies (Fig. 9.5). If the atom has Z electrons, the total cross-section per atom is ZσK−N. Note that scattering by nuclei can be neglected because they cause very much less scattering than electrons, 2 roughly by a factor of (me/mN) , where mN is the mass of the nucleus.

9.3 Inverse Compton scattering In inverse Compton scattering, ultrarelativistic electrons scatter low energy photons to high energies so that the photons gain energy at the expense of the kinetic energy of the electrons. The process is called inverse Compton scattering because the electrons lose energy rather than the photons. We consider the case in which the energy of the photon in the cen- 2 tre of momentum frame of reference is much less than mec and consequently the Thomson scattering cross-section can be used to describe the probability of scattering. Many of the most important results can be worked out using simple arguments (Blumen- thal and Gould, 1970; Rybicki and Lightman, 1979). The geometry of inverse Compton scattering is illustrated in Fig. 9.5 which depicts the collision between a photon and a relativistic electron as seen in the laboratory frame of reference S and in the rest frame of the 262 Interactions of high energy photons

Fig. 9.5. The geometry of inverse Compton scattering in the laboratory frame of reference S and that in which the electron is at rest S 0.

0 2 electron S . In the case in which γ~ω mec , the centre of momentum frame of reference is very closely that of the relativistic electron. If the energy of the photon is ~ω and the angle of incidence θ in S , its energy in the frame S 0 is ~ω0 = γ~ω[1 + (v/c) cos θ] , (9.31) according to the relativistic Doppler shift formula. The angle of incidence θ0 in the frame S 0 is related to θ in S by the aberration formulae sin θ cos θ + v/c sin θ0 = ; cos θ0 = . (9.32) γ[1 + (v/c) cos θ] [1 + (v/c) cos θ] 0 2 Provided ~ω mec , the Compton interaction in the rest frame of the electron is Thomson scattering and hence the energy loss rate of the electron in S 0 is the rate at which energy is reradiated by the electron. According to (9.15), this loss rate is

0 0 −(dE/dt) = σTcurad , (9.33) 0 where urad is the energy density of radiation in the rest frame of the electron. As shown in Sect. 9.2.1, it is of no importance whether or not the radiation is isotropic – the free electron accelerates in response to any incident field. Therefore, our strategy is to work out 0 0 the energy density urad in the frame S of the electron and then to use expression (9.15) to find (dE/dt)0. Using the result obtained in Sect. 6.3.1, this is also the loss rate (dE/dt) in the frame S . We give two derivations of the key result. In the first method, we consider the rate of arrival of photons at the origin of the moving frame S 0. Suppose the number density of photons in a parallel beam of radiation incident at angle θ to the x-axis is N. Then, the energy density of these photons in S is N~ω and the flux density of photons incident upon a stationary electron in S is uradc = N~ωc. To work out the flux density of the beam as 9.3 Inverse Compton scattering 263

Fig. 9.6. Illustrating the rate of arrival of photons at the observer in the laboratory frame of reference (see text). observed in the frame of reference of the stationary electron S 0, we need two things: the energy of each photon in S 0 and the rate of arrival of photons at the electron. The ﬁrst of these is given by (9.31). To ﬁnd the second factor, consider two photons which arrive at 0 0 0 0 0 the origin of S at times t1 and t2 at the angle θ to the x -axis. The coordinates of these events in S 0 are 0 0 [ct1, 0, 0, 0] and [ct2, 0, 0, 0] . The coordinates of these events in S are

0 0 0 0 [ct1, x1, 0, 0] = [γct1, γVt1, 0, 0] and [ct2, x2, 0, 0] = [γct2, γVt2, 0, 0] respectively, where we have used the inverse Lorentz transformations, ! Vx0 ct = γ ct0 + and x = γ x0 + Vt0 . (9.34) c This calculation makes the important point that the photons in the beam propagate along parallel but separate trajectories at an angle θ to the x-axis in S , as illustrated in Fig. 9.6. From the geometry of Fig. 9.6, it is apparent that the time difference when the photons arrive at a plane perpendicular to their direction of propagation in S is (x − x ) ∆t = t + 2 1 cos θ − t = (t0 − t0 )γ[1 + (v/c) cos θ] , (9.35) 2 c 1 2 1 264 Interactions of high energy photons that is, the time interval between the arrival of photons from the direction θ0 is shorter by a factor γ[1 + (v/c) cos θ] in S 0 than it is in S . Thus, the rate of arrival of photons and correspondingly the number density of photons is greater by this factor γ[1 + (v/c) cos θ] in S 0 as compared with in S . Comparison with (9.31) shows that this is exactly the same factor by which the energy of the photon has increased. Thus, as observed in S 0, the energy density of the beam is 0 2 urad = [γ(1 + (v/c) cos θ)] urad . (9.36) The second way of deriving the result (9.36) is somewhat more elegant. It uses the fact that the four-volume dt dx dy dz is invariant between any pair of inertial frames of reference. For reference frames in standard configuration, dy = dy0 and dz = dz0. Therefore, we need only consider the transformation of the differential product dt dx. According to the standard procedure for relating differential areas in different coordinate systems,

∂t ∂x

∂t0 ∂t0 dt dx = dt0 dx0 , (9.37) ∂t ∂x

∂x0 ∂x0 where the determinant is the Jacobian of the transformation between the frames S and S 0. It is straightforward to use the inverse Lorentz transformations (9.34) to show that the value of the determinant in (9.37) is unity. Therefore, the four-volume element dt dx dy dz is an invariant between inertial frames of reference. We now combine this result with other invariants to create new invariant relations between inertial frames of reference. Consider the number density of particles of energy E, n(E), moving at velocity v at an angle θ to the x-axis, as illustrated in Fig. 9.6. The number of photons in the differential three-volume dN(E) = n(E) dx dy dz is an invariant between inertial frames, where n(E) is the number density of photons of energy E. Consequently, because the four-volume element dt dx dy dz is an invariant between inertial frames of reference, so also is n(E)/dt. But, as was shown in Sect. 6.2.1, dt and E transform in the same way between reference frames and so n(E)/E is also an invariant between inertial frames. The change in energy of the photons between the frames S and S 0 is given by (9.31) and so the number density of photons increases by the same factor. We therefore recover the result (9.36) somewhat more economically. The procedure of the last two paragraphs provides a powerful tool for creating many useful relativistic invariants. For example, the differential momentum four-vector is P = [dE/c, dpx, dpy, dpz] and so dE dpx dpy dpz is an invariant volume in four-momentum space. We will return to this result in the discussion of occupation numbers in phase space in the context of Comptonisation in Sect. 9.4. Returning to (9.36), it is now a simple calculation to work out the energy density of radiation observed by the electron in its rest frame. It is assumed that the radiation field is 9.3 Inverse Compton scattering 265

0 isotropic in S and therefore the contribution to urad from the solid angle dΩ in S is 1 du0 = u γ2[1 + (v/c) cos θ]2 dΩ = u γ2[1 + (v/c) cos θ]2 sin θ dθ . (9.38) rad rad rad 2 Integrating over solid angle, Z π ! 0 2 2 1 4 2 1 urad = urad γ [1 + (v/c) cos θ] sin θ dθ = urad γ − . (9.39) 0 2 3 4 Substituting into (9.33) and using the result (6.2) that (dE/dt) = (dE/dt)0, ! dE 4 1 = σ cu γ2 − . (9.40) dt 3 T rad 4 This is the energy gained by the photon field due to the scattering of the low energy photons. We have therefore to subtract the initial energy of the low-energy photons to find the total energy gain of the photon field in S . The rate at which energy is removed from the low energy photon field is σT curad and therefore, subtracting, ! dE 4 1 4 = σ cu γ2 − − σ cu = σ cu (γ2 − 1) . dt 3 T rad 4 T rad 3 T rad Using the identity (γ2 − 1) = (v2/c2)γ2, the loss rate in its final form is

! 2 ! dE 4 v 2 = σT curad 2 γ . (9.41) dt IC 3 c 2 This is the result we have been seeking. It is exact so long as γ~ω mec . Notice the remarkable similarity of the result (9.41) to the expression (8.9) for the mean energy loss rate of the ultrarelativistic electron by synchrotron radiation

! 2 ! dE 4 v 2 = σT cumag 2 γ . (9.42) dt sync 3 c The reason for this is that the energy loss rate depends upon the electric field which accelerates the electron in its rest frame and it does not matter what the origin of that field is. In the case of synchrotron radiation, the electric field is the (u × B) field due to motion of the electron through the magnetic field whereas, in the case of inverse Compton scattering, it is the sum of the electric fields of the electromagnetic waves incident upon the electron. In the latter case, the sum of the squares of the electric field strengths appears in the formulae for incoherent radiation and so the energies of the waves add linearly (see Sect. 9.2.1). Another way of expressing this similarity between the loss processes is to consider synchrotron radiation to be the scattering of ‘virtual photons’ observed by the electron as it gyrates about the magnetic field (Jackson, 1999). The similarity of the synchrotron and inverse Compton scattering processes means that we can use the results of Sect. 8.5.1 to work out the spectrum of radiation produced by a 266 Interactions of high energy photons

Fig. 9.7. The emission spectrum of inverse Compton scattering; ν0 is the frequency of the unscattered radiation (Blumenthal and Gould, 1970). power-law distribution of electron energies. The spectral index of the scattered radiation is a = (p − 1)/2, where p is the spectral index of the electron energy spectrum. Notice that this relation is true for the intensity of radiation measured in W m−2 Hz−1. In terms of photon flux density, the spectral index would be one power of frequency, or energy, steeper aph = (p + 1)/2 The next step is to determine the spectrum of the scattered radiation. This is a somewhat lengthy, but straightforward, calculation. Because of the extreme effects of aberration, the photons which interact with the electron in the frame S 0 propagate in the negative direction along the x0-axis, the spectrum of the incident radiation being found from (9.36). They are then scattered in the moving frame with the probability distribution given by the differential Thomson cross-section (9.8). The spectrum of the scattered radiation is then transformed back into the laboratory frame of reference. The result of these calculations is given by Blumenthal and Gould for an incident isotropic photon field at a single frequency ν0 (Blumenthal and Gould, 1970). The spectral emissivity I(ν) may be written " ! 2 # 3σTc N(ν0) ν 2 ν I(ν) dν = 4 2 ν 2ν ln 2 + ν + 4γ ν0 − 2 dν , (9.43) 16γ ν0 4γ ν0 2γ ν0 where the isotropic radiation field in the laboratory frame of reference S is assumed to be monochromatic with frequency ν0; N(ν0) is the number density of photons. This spectrum is shown in Fig. 9.7. At low frequencies, the term in square brackets in (9.42) is a constant and hence the scattered radiation has a spectrum of the form I(ν) ∝ ν. It is an easy calculation to show that the maximum energy which the photon can acquire corresponds to a head-on collision in which the photon is sent back along its original path. 9.4 Comptonisation 267 The maximum energy of the photon is

2 2 2 (~ω)max = ~ωγ (1 + v/c) ≈ 4γ ~ω0 . (9.44) Another important result can be derived from (9.41), the total energy loss rate of the electron. The number of photons scattered per unit time is σT curad/~ω0 and hence the average energy of the scattered photons is 4 4 ~ω = γ2(v/c)2~ω ≈ γ2~ω . (9.45) 3 0 3 0 This result gives substance to the hand-waving argument that the photon gains typically one factor of γ in transforming into S 0 and then gains another on transforming back into S . The general result that the frequency of photons scattered by ultrarelativistic electrons 2 is ν ∼ γ ν0 is of profound importance in high energy astrophysics. There are certainly electrons with Lorentz factors γ ∼ 100 − 1000 in various types of astronomical source and consequently they scatter any low energy photons to very much higher energies. To give some examples, consider radio, infrared and optical photons scattered by electrons with γ = 1000. The scattered radiation has average frequency (or energy) roughly 106 times 9 that of the incoming photons. Radio photons with ν0 = 10 Hz become ultraviolet photons 15 12 with ν = 10 Hz (λ = 300 nm); far-infrared photons with ν0 = 3 × 10 Hz, typical of the photons seen in galaxies which are powerful far-infrared emitters, produce X-rays with 18 14 frequency 3 ×10 Hz, that is, about 10 keV; optical photons with ν0 = 4×10 Hz become γ-rays with frequency 4 × 1020 Hz, that is, about 1.6 MeV. It is apparent that the inverse Compton scattering process is an eﬀective means of creating very high energy photons. It also becomes an inevitable drain of energy for high energy electrons whenever they pass through a region in which there is a large energy density of radiation.

9.4 Comptonisation The calculations carried out in Sect. 9.2.2 demonstrate how energy can be interchanged between photons and electrons by Compton scattering in particular limiting cases. If the evolution of the spectrum of the source is dominated by Compton scattering, the process is often referred to as Comptonisation. This enormous subject is considered in much more detail by Pozdnyakov, Sobol and Sunyaev, by Liedahl and by Rybicki and Lightman (Pozd- nyakov et al., 1983; Liedahl, 1999; Rybicki and Lightman, 1979).

9.4.1 The basic physics of Comptonisation The requirement that the evolution of the spectrum be determined by Compton scattering means that the plasma must be rarefied so that other radiation processes such as bremsstrahlung do not contribute additional photons to the system. In addition, the effects of Compton- isation are important if the plasma is very hot because then the exchange of energy per 268 Interactions of high energy photons collision is greater. Examples of sources in which such conditions are found include the hot gas in the vicinity of binary X-ray sources, the hot plasmas in the nuclei of active galaxies, the hot intergalactic gas in clusters of galaxies and the early evolution of the hot primordial plasma. Let us build up a simple picture of the Comptonisation process. We restrict the dis- 2 2 cussion to the non-relativistic regime in which kTe me c and ε = ~ω mec and so the Thomson cross-section can be used for interactions between radiation and the electrons. The expression for the energy transferred to stationary electrons from the photon field (9.25) can be written in terms of the fractional change of energy of the photon per 2 collision in the limit ~ω mec , ∆ε ~ω = 2 (1 − cos α) . (9.46) ε mec In the frame of reference of the electron, the scattering is Thomson scattering and so the probability distribution of the scattered photons is symmetrical about their incident directions. Therefore, when averages are taken over the scattering angle α, opposite values of cos α cancel out and the average energy increase of the electron is

2 h∆ε/εi = ~ω/mec . (9.47)

This is the recoil effect discussed in Sect. 9.2.2. In the opposite limit in which energy is transferred from the electrons to the photon field, we can adopt the low energy limit of the energy loss rate of high energy electrons by inverse Compton scattering. The derivation of (9.41) is correct for all values of the Lorentz factor γ and hence incorporates the effects of aberration and Doppler scattering, even if these effects are small. The low energy limit of (9.41) is dE 4 v 2 = σ cu . (9.48) dt 3 T rad c

The number of photons scattered per second is σTNphot c = σTuradc/~ω, and so the average energy gain by the photons per Compton collision is * + ∆ε 4 v 2 = . (9.49) ε 3 c The average energy gain per collision is second order in v/c because the ﬁrst-order eﬀects cancel out. The net increase in energy is statistical because implicitly, in deriving (9.41), we integrated over all angles of scattering. 1 2 If the electrons have a thermal distribution of velocities at temperature Te, 2 mehv i = 3 2 kTe and hence

∆ε 4kTe = 2 . (9.50) ε mec 9.4 Comptonisation 269 As a result, the equation describing the average energy change of the photon per collision is ∆ε 4kTe − ~ω = 2 . (9.51) ε mec

There is therefore no energy transfer if ~ω = 4kTe. If 4kTe > ~ω, energy is transferred to the photons whilst if ~ω > 4kTe energy is transferred to the electrons. In the case in which the electrons are hotter than the photons, the fractional increase in 2 energy is 4kTe/mec per collision and hence we need to evaluate the number of collisions which the photon makes with electrons before they escape from the scattering region. If the region has electron density Ne and size l, the optical depth for Thomson scattering is

τe = Ne σTl . (9.52)

If τe 1, the photons undergo a random walk in escaping from the region and so the 1/2 −1 photon travels a distance l ≈ N λe in N scatterings where λe = (Ne σT) is the mean free path of the photon. Therefore, in the limit τe 1, which is necessary to alter signiﬁcantly 2 2 the energy of the photon, the number of scatterings is N = (l/λe) = τe. If τe 1, the number of scatterings is τe and hence the condition for a signiﬁcant distortion of the photon spectrum by Compton scattering is 4y & 1, where

kTe 2 y = 2 max(τe, τe) . (9.53) mec y is referred to as the Compton optical depth. Normally, the condition for Comptonisation to change signiﬁcantly the spectrum of the photons is

kTe 2 1 y = 2 τe & . (9.54) mec 4 Let us investigate how repeated scatterings change the energy of the photons. The analysis of Liedahl is rather pleasant (Liedahl, 1999). First of all, we convert (9.51) into a differential equation for the rate of change of the energy of the photons. If N is the number of scatterings, 2 dε 4kTe − ~ω ε = 2 ε = Aε − 2 , (9.55) dN mec mec 2 2 2 where A = 4kTe/mec . Setting x = ~ω/mec = ε/mec , we find dx = Ax − x2 . (9.56) dN It is straightforward to find the definite integral of this equation for a photon with initial 2 energy ε0, or equivalently x0 = ε0/mec , x x = 0 eAN . (9.57) A − x A − x0 270 Interactions of high energy photons

Initially, the photon has energy ε0 4kTe and therefore A x0. Hence, x x = 0 eAN . (9.58) A − x A 2 Furthermore, the Compton optical depth of the medium is y = (kTe/mec )N = AN/4 and so, solving (9.58) for x, we ﬁnd e4y ε = ε . (9.59) 0 ~ω 1 + 0 e4y 4kT 4y Thus, when (~ω0/4kT) e is small, the energy of the photon increases exponentially as 4y 4y ε = ε0 e . However, when the Comptonisation is strong, (~ω0/4kT) e 1, the energy of the photons saturate at ε = 4kT, as expected from (9.51). We can now work out the number of scatterings in order to approach saturation. The saturation energy is ~ω = ε = 4kTe and so let us work out the number of scatterings to attain the energy ε/2 = 2kTe. Inserting this value into (9.59) and recalling that y = (kT /m c2)N, we ﬁnd e e ! m c2 4kT N = e ln e . (9.60) 4kTe ~ω0

In the case in which the Thomson optical depth of the cloud τe is much greater than unity, 2 N ∼ τe and so " 2 !#1/2 mec 4kTe τe = ln . (9.61) 4kTe ~ω0

Liedahl gives the example of injecting optical photons with energy ~ω0 = 4 eV into a gas with temperature kTe = 10 keV. Then, there would have to be N ≈ 100 scatterings for the photons to approach a Comptonised X-ray spectrum. If this were associated with 2 random scattering within a cloud of optical depth τe, there would have to be roughly τe scatterings and so the Thomson optical depth of the region would have to be τ ≈ 10. The 2 corresponding value of the Compton optical depth is y = (kTe/mec )N ≈ 2. If the Compton optical depth y of the medium is very much greater than unity, the photon distribution approaches its equilibrium form entirely under Compton scattering. Photons are bosons and, consequently, the equilibrium spectrum is given in general by the Bose– Einstein distribution, the energy density of which is " ! # 8πhν3 hν −1 u dν = exp + µ − 1 dν . (9.62) v c3 kT µ is the chemical potential. In the case of the Planck spectrum, µ = 0 and the number and energy densities of the photons are uniquely deﬁned by a single parameter, the thermal equilibrium temperature of the matter and radiation T. If there is a mismatch between the number density of photons and the energy density of radiation, the equilibrium spectrum is 9.4 Comptonisation 271

Fig. 9.8. Illustrating the intensity spectra of Bose–Einstein distributions with different values of the dimensionless chemical potential µ. The distribution with µ = 0 is the Planck function. At energies hν µkT the distributions are similar to a Wien distribution but with intensity reduced by a factor 3 exp (−µ). At energies hν µkT, the intensity spectrum is Iν ∝ ν . In general, for large values of µ, the distribution follows closely that of a Wien distribution with intensity reduced by a factor exp (−µ). the Bose–Einstein distribution with a finite chemical potential µ. The forms of these spectra are shown in Fig. 9.8 for different values of the chemical potential µ. In the limiting case µ 1, the spectrum is the Wien distribution reduced by the factor exp(−µ), ! 8πhν3 hν u = exp (−µ) exp − . (9.63) ν c3 kT The average energy of the photons is R ∞ 3 0 x exp (−x) dx h~ωi = kTe R = 3kTe , (9.64) ∞ 2 0 x exp (−x) dx exactly the same result derived by Einstein in his great paper of 1905 in which he intro- duced the concept of light quanta (Longair, 2003). 272 Interactions of high energy photons 9.4.2 Pedagogical interlude - occupation number We now need the equation which describes how the spectrum of radiation evolves towards the Bose-Einstein distribution. In the non-relativistic limit, this equation is known as the Kompaneets equation, which is discussed in Sect. 9.4.3. It is written in terms of the occupation number of photons in phase space, because we need to include both spontaneous and induced processes in the calculation. Let us compare this approach with that involving the coefficients of emission and absorption of radiation. My favorite reference for understanding the basic physics of spontaneous and induced processes is the beautiful discussion by Feynman in Chap. 4 of Volume III of his Lectures on Physics (Feynman et al., 1965). In Sect. 4.4, he enunciates the key rule for the emission and absorption of photons, which are spin-1 bosons.

The probability that an atom will emit a photon into a particular ﬁnal state is increased by a factor (n + 1) if there are already n photons in that state.

Notice that the statement is made in terms of probabilities rather than quantum√ mechanical amplitudes – in the latter case, the amplitude would be increased by a factor n + 1. We will use probabilities in our analysis. n will turn out to be the occupation number. To derive the Planck spectrum, consider an atom which can be in two states, an upper state 2 with energy ~ω greater than the lower state 1. N1 is the number of atoms in the lower state and N2 the number in the upper state. In thermodynamic equilibrium, the ratio of the numbers of atoms in these states is given by the Boltzmann relation,

N 2 = exp (−∆E/kT) = exp (−~ω/kT) , (9.65) N1 where ∆E = ~ω and the statistical weights g2 and g1 are assumed to be the same. When a photon of energy ~ω is absorbed, the atom is excited from state 1 to state 2 and, when a photon of the same energy is emitted from state 2, the atom de-excites from state 2 to state 1. In thermodynamic equilibrium, the rates for the emission and absorption of photons between the two levels must be exactly balanced. These rates are proportional to the product of the probability of the events occurring and the number of atoms present in the appropriate state. Suppose n is the average number of photons in a given state in the phase space of the photons with energy ~ω. Then, the absorption rate of these photons by the atoms in the state 1 is N1np12, where p12 is the probability that the photon will be absorbed by an atom in state 1, which is then excited to state 2. According to the rule enunciated above by Feynman, the rate of emission of photons when the atom de-excites from state 2 to state 1 is N2(n + 1)p21. At the quantum mechanical level, the probabilities p12 and p21 are equal. This is because the matrix element for, say, the p12 transition is the complex conjugate of the transition p21 and, since the probabilities depend upon the square of the magnitude of the matrix elements, they must be equal. This is called the principle of 9.4 Comptonisation 273 jump rate symmetry. Therefore,

N1n = N2(n + 1) . (9.66)

Solving for n and using (9.65), 1 n = . (9.67) e~ω/kT − 1 The elementary volume of phase space for photons is (2π)3. There are two independent polarisations for each state and hence the number of states in the phase space volume d3k is 2 d3k/(2π)3. If the photon distribution is isotropic, the photons which lie in the frequency interval ν to ν + dν have wave vectors k which lie in a spherical shell of radius k and thickness dk and so volume d3k = 4πk2 dk. Therefore, the number of states in this volume of photon phase space is 8πk2 dk 8πν2 dν ω2 dω = = . (9.68) (2π)3 c3 c3π2 To complete the calculation, the energy density of radiation is the product of the energy of each photon, the volume of phase space in which the photons have energies in the interval ~ω to ~(ω + dω) and the occupation number of each state,

8πhν3 1 ~ω3 1 u(ν) dν = dν or u(ω) dω = dω . (9.69) c3 ehν/kT − 1 π2c3 e~ω/kT − 1 We have recovered the Planck spectrum. For our present purposes, the important relation is the general expression for the occupation number of the photons in phase space. If the energy density of isotropic radiation in the frequency interval ν to ν + dν is uν dν, the number density of photons is uν dν/hν and the mean occupation number, n(ν) or n(ω), is

u c3 u(ω) π2c3 n(ν) = ν or n(ω) = . (9.70) 8πhν3 ~ω3 There are particularly simple expressions for the occupation number of photons for the Bose–Einstein and Planck distributions,

Bose–Einstein n(ν) = [exp(x + µ) − 1]−1 ; (9.71) Planck n(ν) = [exp x − 1]−1 ,

where x = hν/kT = ~ω/kT. The occupation number n(ν) determines when it is necessary to include stimulated emission terms in the expressions for interactions of photons. If n > 1, then the eﬀects of stimulated emission cannot be neglected. For black–body radiation, this means in the Rayleigh-Jeans region of the spectrum, hν kT. Let us now rewrite the transfer equation for radiation in terms of occupation numbers. 274 Interactions of high energy photons For the case of isotropic radiation, equation (6.61) can be written dI(ω) = ~ω N A − N B ~ω I(ω) + N B ~ω I(ω) . (9.72) dx 2 21 1 12 2 21

We recall that the spontaneous emission coefficient is κν = ~ω N2A21. Notice that this equation is written in terms of the intensity of radiation integrated over 4π steradians per unit angular frequency and so is exactly equivalent to the analysis using the Einstein coefficients in Sect. 6.5.2. We now use the relations between the Einstein coefficients, ~ω3 B = B ; A = B , (9.73) 12 21 21 π2c2 21 to rewrite the transfer equation as dI(ω) ~ω3 = ~ω N B − N B ~ω I(ω) + N B ~ωI(ω) . (9.74) dx 2 π2c2 21 1 12 2 21 We now rewrite the transfer equation in terms of occupation numbers using I(ω) π2c2 n(ω) = , (9.75) ~ω3 so that dn(ω) = ~ω B {−N n(ω) + N [1 + n(ω)]} . (9.76) dx 21 1 2

This is the equation we have been seeking. Notice how the rule described by Feynman comes naturally out of an analysis of Einstein’s coeﬃcients for spontaneous and stimulated . It also illustrates how the transfer equation for radiation can be written in a remarkably compact form using occupation numbers, including both spontaneous emission and simu- lated emission and absorption. To make this clearer, let us simplify the notation of (9.76) to be similar to that used in the next section, dn = ~ω B [−N n + N (1 + n)] . (9.77) dx 21 1 2 The three terms on the right-hand side in square brackets represent stimulated absorption with the minus sign and spontaneous emission and stimulated emission with the plus sign respectively. It will be noticed that the right hand side of (9.77) is identical with (9.66) when the left hand side is set equal to zero in thermal equilibrium. The occupation number n is a Lorentz invariant. This is most easily understood by considering the invariant volume of the diﬀerential momentum four-vector for photons,

dP = [dE, dp1, dp2, dp3] ≡ ~ [dω, dk1, dk2, dk3] = ~ dK . (9.78) By exactly the same argument as given in Sect.9.3˙ for the invariant volume dt dx dy dz, it follows that the volume element in four-dimensional momentum space dω dk1 dk2 dk3 is 9.4 Comptonisation 275 invariant under Lorentz transformation. The number of photons in the element of four- dimensional phase space N is an invariant number and so N n = (9.79) dω d3k is a Lorentz invariant, recalling that n is deﬁned per unit angular frequency per unit volume of k-space.

9.4.3 The Kompaneets equation The Kompaneets equation for the evolution of the occupation number n under Compton scattering is named after the Soviet physicist Aleksander Solomonovich Kompaneets who published its derivation in 1956 (Kompaneets, 1956). In fact, the equation had been derived in the late 1940s by the combined efforts of Kompaneets, Landau, Gel’fand and Dyakov under the direction of Zeldovich as part of the Soviet atomic and hydrogen bomb programme. The derivation of the Kompaneets equation is non-trivial since it has to take account of the interchange of energy between the photons and electrons and also include induced effects which become important when the occupation number n is large. The derivations outlined by Rybicki and Lightman and by Liedahl give an excellent impression of what is involved (Rybicki and Lightman, 1979; Liedahl, 1999). Provided the fractional changes of energy per Compton interaction are small, the Boltz- mann equation can be used to describe the evolution of the photon occupation number, Z ∂n(ω) dσ = c d3 p dΩ f (p0) n(ω0)(1 + n(ω)) − f (p) n(ω)(1 + n(ω0)) . (9.80) ∂t dΩ The first terms in square brackets within the integral describes the increase in the occupation number due to photon scattering from frequency ω0 to ω. Notice that this term includes the factor (1 + n(ω)) which takes account of the fact that the photons are bosons and so, as discussed in Sect. 9.4.2, there is an increased probability of scattering by this factor if the occupation number of the final state is already n(ω). The second term in square brackets describes the loss of photons of frequency ω by Compton scattering from ω to ω0, again the stimulated term (1 + n(ω0)) being included to take account of induced scattering. In deriving the Kompaneets equation, it is assumed that the electron distribution remains Maxwellian at temperature T, so that

Ne 2 f (p) = e−p /2mekT . (9.81) 3/2 (2πme kT) The diﬀerential cross-section for Thomson scattering is given by (9.11). The change of angular frequency of the photon is given by (9.24) in the non-relativistic limit in which the 276 Interactions of high energy photons frequency changes are small. Liedahl provides a clear description of the approximations involved and the tricks which can be used to derive the ﬁnal form of the equation which is " !# ∂n 1 ∂ ∂n = x4 n + n2 + (9.82) ∂y x2 ∂x ∂x

2 where dy is the increment of Compton optical depth, dy = (kTe/mec ) σT Ne c dt, and x = ~ω/kTe. Let us analyse the meanings of the various terms in (9.82), following the presentations of Liedahl and Blandford (Liedahl, 1999; Blandford, 1990). In the process of Compton- isation, the total number of photons is conserved, although their energies are changed by Compton scattering. Consequently, the conservation equation for the total number of photons is Z Z d ∞ ∞ dn(x) ω2 n(ω) dω = 0 or x2 dx = 0 , (9.83) dt 0 0 dt and so the evolution of the photon spectrum corresponds to the conservation of a photon ‘fluid’ in phase space. There is therefore a continuity equation describing the flow of photons in phase space which can be written ∂n + ∇ · J = 0 , (9.84) ∂t where J is the ‘current’ of photons in phase space. Now, the present analysis assumes that the distribution of photons is isotropic in phase space and so we need the divergence in spherical polar coordinates with only the radial x-component of the divergence present. It follows that ∂n 1 ∂ = −∇ · J = − [x2 J(x)] . (9.85) ∂t x2 ∂x where J(x) is a scalar function of ‘radius’ x. This equation can be compared with the Kompaneets equation (9.82). It follows that ! kTe 2 2 ∂n J(x) = −NeσTc 2 x n + n + . (9.86) mec ∂x This equation enables us to understand the meanings of the various terms in the Kom- panets equation. Consider first the term

kTe 2 J(x) = −NeσTc 2 x n . (9.87) mec This term corresponds to the recoil eﬀect described by (9.47),

dω ~ω dx kTe = − 2 or = −x 2 . (9.88) ω mec x mec Just as the current or ﬂux of particles in real space is J = Nv, so the current associated with 9.4 Comptonisation 277 the drift of photons in phase space because of the recoil eﬀect is J(x) = n dx/dt. The rate of change of x is given by the number of scatterings per second times the average change in x per scattering,

dx 2 kTe = −x 2 × NeσTc . (9.89) dt mec Therefore, the ﬂux of photons J(x) is

dx kTe 2 J(x) = n = −NeσTc 2 x n , (9.90) dt mec exactly the same as (9.87). This analysis immediately enables us to understand the term in n2 in (9.86). The terms n + n2 = n(1 + n) and so the n2 term takes account of the effects of induced scattering when the occupation number n is greater than unity. The third term in (9.86) corresponds to the statistical increase of energy of the photons by Compton scattering. As discussed in Sect. 9.2.2, the ‘heating’ of the photon gas by the hotter electrons is a statistical phenomenon which corresponds to the diffusion of the photons in phase space. The equation governing this process is of the same form as that encountered in the diffusion-loss equation (7.42) which includes the statistical acceleration term. The corresponding term in the case of photon diffusion is that the current J(x) in momentum space is described by a diffusion coefficient Dx so that J(x) = Dx ∂n/∂x. In the case of the isotropic diffusion of photons in phase space, the transfer equation is ∂n 1 ∂J(x) = − . (9.91) ∂t x2 ∂x

The diffusion coefficient Dx in this case is the mean square change in x per unit time, h(∆x)2i, the same as is found in the stochastic acceleration of particles according to the diffusion loss equation (7.42). In the present case, the change of energy of the photon is given by the non-relativistic limit of (9.31), ~ω0 = ~ω(1+(v/c) cos θ), or ∆x = x(v/c) cos θ. Averaging the mean square energy change over 4π steradians, Z v2 1 1 v2 h(∆x)2i = x2 cos2 θ × sin θ dθ = x2 . (9.92) c2 2 3 c2

3 1 2 Setting 2 kTe = 2 mev , we ﬁnd the variance of ∆x in a single scattering,

2 2 kTe h(∆x) i = x 2 . (9.93) mec

There are NeσTc scatterings per unit time and so, since the variance per unit time is the sum of the separate variances, we find ! 2 2 kTe D(x) = h(∆x) i = NeσTcx 2 . (9.94) mec 278 Interactions of high energy photons Therefore the diffusion current has the form ! 2 kTe ∂n J(x) = −NeσTcx 2 , (9.95) mec ∂x exactly the same as the third term in the Kompaneets equation (9.86). It is interesting to note that the Kompaneets equation has the same formal content as (9.51), which refers to the energy of an individual photon. The importance of the Kompa- neets equation is that it describes the evolution of the spectrum of the photon field in phase space and necessarily includes induced processes. Generally, the solutions of the equation have to be found numerically, but there are a number of cases in which analytic solutions can be found. First of all, it is a useful exercise to show that the right-hand side of (9.82) is zero for a Bose–Einstein distribution for which the occupation number is n = [exp(x + µ) − 1]−1. This solution also includes the case of the Planck spectrum for which µ = 0. These solutions are found when there is time for the system to come to equilibrium under Compton scattering in the limit of very large values of y. Pozdnyakov and his colleagues provide examples of the spectra of X-ray sources for increasing values of the Compton optical depth y (Fig. 9.9) (Pozdnyakov et al., 1983). In these examples, the input photons are of very low energy and the electrons have temperature kTe = 25 keV. The Thomson scattering optical depth τ takes values between 3 and 10, so that the Comptonisation process does not reach saturation, although the beginnings of the formation of the Wien peak is seen at the largest optical depths. At smaller optical depths, the spectrum mimics very closely a power-law spectrum up to energies hν ≈ kTe, above which a roughly exponential cut-off is found. It is helpful to illustrate how these features come about. Following the presentation of Liedahl, it is convenient to modify the Kompaneets equation to take account of the source of low energy photons and the diffusion of photons out of the source region. Both terms can be taken to have similar forms to those found in the diffusion-loss equation (7.42). The rate of production of soft photons Q(x) can be written Q(x) = Q0(x) for photons with values of x ≤ x0 and Q(x) = 0 for x > x0, so that there are no initial photons with values of x > x0. The escape time from the source region is determined by the optical depth of the region to Thomson scattering τes. As discussed 2 in Sect. 9.4.1, the number of scatterings is given by the greater of τes or τes, or in terms 2 2 of the Compton optical depth by yes = (kTe/mec ) max(τes, τes). Therefore the modified Kompaneets equation can be written " !# ∂n 1 ∂ 4 2 ∂n n = 2 x n + n + + Q(x) − , (9.96) ∂y x ∂x ∂x yes where the source term Q(x) is defined as the number of photons per unit element of phase space per unit Compton optical depth. 9.4 Comptonisation 279

Fig. 9.9. The Comptonisation of low frequency photons in a spherical plasma cloud having kTe = 25 keV. τ is the Thomson scattering optical depth τ = NeσT and α is the spectral index. The solid curves are analytic solutions of the Kompaneets equation using the parameters given by the relations (9.102) and (9.103) (Pozdnyakov et al., 1983); the results of Monte Carlo simulations of the Compton scattering process are shown by the histograms and there is generally good agreement with the analytic solutions. A slightly better ﬁt to the Monte Carlo calculations is found for the cases τ = 3 and τ = 4 if the analytic formula is ﬁtted to the spectral index α found from the Monte Carlo simulations (dashed curve). These computations illustrate the development of the Wien peak for large values of the optical depth τ at energies hν ≈ kTe. 280 Interactions of high energy photons

We are interested in steady-state solutions for photon energies x x0 and so ∂n/∂y = 0 and Q(x) = 0. For our present purposes, we are interested in cases in which the photon occupation number n very small and so we can neglect the induced Compton scattering terms in n2. With these simpliﬁcations, the modiﬁed Kompaneets equation becomes " !# ∂ ∂n y x4 n + − nx2 = 0 . (9.97) es ∂x ∂x

Let us first consider the case of very large values of x = ~ω/kTe 1. Then, since the occupation number n is very small, the term nx2 in (7.97) is very much less than the first two terms. In this approximation, the first integral of (7.97) is ! ∂n constant + n = . (9.98) ∂x x4 For large values of x, the right-hand side tends to zero and so the solution for n tends to n = e−x. Converting the occupation number to an intensity of radiation using (9.75), we find ~ω3 ~ω3 e−~ω/kTe I(ω) dω = n(ω) dω = dω . (9.99) π2c2 π2c2

This is Wien’s law in the limit ~ω kTe and accounts for the exponential cut-off of the Comptonised spectrum at these high energies. For small values of x, the diffusion of photons in phase space results in heating of the photon gas and the recoil effect, which results in a loss of energy of the photons, can be neglected. In this case, the Kompaneets equation becomes " !# ∂ ∂n y x4 − nx2 = 0 . (9.100) es ∂x ∂x Inspection of (9.100) shows that power-law solutions of the form n(x) = A xm can be found. It is straightforward to find the value of m, ! 3 9 1 1/2 m = − ± + . (9.101) 2 4 yes and hence the intensity spectrum has the form I(ω) ∝ ω3+m. The positive root of (9.101) is appropriate if yes 1 and the negative root if yes 1. Thus, if the Compton optical depth yes is very large, the value of m is zero and then we recover the Wien spectrum 3 I(ω) ∝ ω in the limit ~ω kTe. If yes 1, a power-law spectrum is obtained with an exponential cut-off at high energies, the solutions having to be joined together numerically. This is an intriguing example of a power-law spectrum being created through ‘thermal’ processes rather than being ascribed to some ‘non-thermal’ radiation mechanism involving ultrarelativistic electrons. The above calculation is the simplest example of the formation of a power-law spectrum 9.4 Comptonisation 281

Fig. 9.10. The hard X-ray spectrum of the Galactic X-ray source Cygnus X-1 observed in a balloon ﬂight of the Max Planck Institute for Extraterrestrial Physics on 20 September 1977 compared with the analytic solution of the Kompaneets equation with parameters τ0 = 5, kTe = 27 keV (Sunyaev and Titarchuk, 1980).

by purely thermal processes. The predicted power-law index is sensitive to the geometry of the source. For example, Pozdnyakov and his colleagues derived an improved version of the above calculation in which the predicted spectral index is

! 3 9 1/2 m = − − + γ , (9.102) 2 4 282 Interactions of high energy photons where, for spherical geometry,

2 2 π mec γ = . (9.103) 3 2 2 τ + 3 kTe τ is the Thomson optical depth from the centre to the edge of the cloud. For a disc geometry, 2 2 π mec γ = , (9.104) 12 2 2 τ + 3 kTe where now τ is the Thomson optical depth from the centre to the surface of the disc. The theoretical curves shown by the solid lines in Fig. 9.9 have been obtained using the relations (9.102) and (9.103). In a number of hard X-ray sources, a characteristic power-law spectrum with a high energy exponential cut-off is observed. Pozdnyakov and his colleagues have fitted the hard X-ray spectrum of the source Cygnus X-1 by such a form of spectrum with the parameters given in the figure caption (Fig. 9.10). Similar spectra have been observed for a number of hard X-ray sources which contain neutron stars or black holes from observations by the XMM-Newton X-ray observatory and the INTEGRAL γ-ray observatory.

9.5 The Sunyaev-Zeldovich effect An important application of the Kompaneets equation concerns spectral distortions of the Cosmic Microwave Background Radiation if the radiation traverses extensive regions of hot ionised gas with electron temperature Te much greater than the radiation temperature Trad. Compton scattering leads to distortions of the thermal spectrum of the background radiation if there are no additional sources of photons to match the number required for a Planck spectrum. Such conditions can occur in the pre- and post-recombination phases of the standard Big Bang. There are two convenient ways of describing the degree to which the observed spectrum differs from that of a perfect black-body, both of them discussed by Zeldovich and Sunyaev in the late 1960s (for details of their work, see (Sunyaev and Zeldovich, 1980)). If there were injection of thermal energy into the intergalactic gas prior to the epoch of recombination at z ≈ 1000 and the number of photons was conserved, the spectrum would relax to an equilibrium Bose-Einstein intensity spectrum with a finite dimensionless chemical potential µ, " ! # 2hν3 hν −1 Iν = 2 exp + µ − 1 , (9.105) c kTr as discussed in Sect. 9.4.3. Such an injection of energy might have been associated with 9.5 The Sunyaev-Zeldovich effect 283 matter-antimatter annihilation or with the dissipation of primodial fluctuations and turbulence. If the heating took place after the epoch of recombination, there would not be time to set up the equilibrium distribution and the predicted spectrum is found by solving the Kompaneets equation without the terms describing the cooling of the photons, that is, ! ∂n 1 ∂ ∂n = x4 . (9.106) ∂y x2 ∂x ∂x Assuming the distortions are small, Zeldovich and Sunyaev inserted the trial solution n = (ex − 1)−1 into the right-hand side of (9.106). It is straightforward to show that ! ∆n ∆I(ω) x ex ex + 1 = = y x − 4 , (9.107) n I(ω) ex − 1 ex − 1 R 2 where the Compton optical depth y = (kTe/mec ) σTNe dl (Zeldovich and Sunyaev, 1969). The effect of Compton scattering is to shift the spectrum to higher energies with the result that the intensity of radiation in the Rayleigh-Jeans region of the spectrum, x 1, decreases while that at x 1 increases – the change-over occurs at x = 4 (Fig. 9.11). Expanding (9.107) for small values of x, the fractional decrease in intensity is ∆I(ω) = −2y . (9.108) I(ω) In this process, the total energy in the radiation spectrum increases as the photons gain energy from the hot electrons. The increase of energy in the background radiation can be found from (9.59) for small values of y. Therefore, the increase in energy density of the background radiation is ∆ε r = e4y . (9.109) εr The net result is that there is more energy in the background radiation than would be predicted from the measured temperature in the Rayleigh-Jeans region of the spectrum. Another way of expressing this result is use the fact that dI(ω) dT = RJ = −2y , (9.110) I(ω) TRJ −2y and so TRJ = e T0. Consequently, if the radiation temperature of the background radia- 4 12y tion is measured to be TRJ, the total energy density is predicted to be ε = aTRJ e . The precision with which the observed spectrum of the Cosmic Microwave Background Radiation fits a perfect black-body spectrum therefore sets strong upper limits to the values of µ and y. The precise spectral measurements made by the FIRAS instrument of the Cosmic Background Explorer result in the following limits (Page, 1997): |y| ≤ 1.5 × 10−5 , |µ| ≤ 10−4 . (9.111) 284 Interactions of high energy photons

Fig. 9.11. Illustrating the Compton scatteringR of a Planck distribution by hot electrons in the case in 2 which the Compton optical depth y = (kTe/me c )σTNe dl = 0.15. The intensity decreases in the Rayleigh–Jeans region of the spectrum and increases in the Wien region. (Sunyaev, 1980).

These limits are of astrophysical importance in the study of the physics of the intergalactic gas, as well as constraining the amount of star and metal formation which could have taken place in young galaxies. The importance of the Sunyaev-Zeldovich eﬀect in causing decrements in the Cosmic Microwave Background Radiation because of the presence of hot gas in rich clusters of galaxies has already been discussed in Sect. 4.6 and illustrated by the remarkable radio maps of clusters with redshifts up to z ≈ 1 in Fig. 4.11. These observations were made in the Rayleigh-Jeans region of the spectrum of the CosmicR Microwave Background Radia- 2 tion and so provides a measure of the quantity y = (kTe/me c )σTNe dl. In conjunction with observations of the bremsstrahlung emission of the hot intracluster gas, the physical parameters of the hot gas cloud can be determined and, as explained in Sect. 4.6, enable estimates of Hubble’s constant to be made. It is useful to make a pedagogical remark about the origin of the result dI(ω)/I(ω) = −2y. The result (9.50) shows that the average increase in energy of the photons in the 2 Compton scattering process is ∆ε/ε = 4kTe/mec and so naive application of this energy change results in the wrong answer for the amplitude of the decrement in that, since I(ω) ∝ ω2, an intensity decrement of −8y would be expected. The reason for this discrepancy is the statistical nature of the Compton scattering process. Fig. 9.12 shows the probability distribution of scattered photons in a single Compton scattering (Sunyaev, 1980). 9.5 The Sunyaev-Zeldovich eﬀect 285

Fig. 9.12. The probability distribution of photons scattered in a single Compton scattering(a) using the exact expression for Compton scattering (solid line) and (b) using the diﬀusion term in the Kom- paneets equation (dashed line) for the case in which the hot gas has temperature kTe = 5.1 keV or 2 kTe/me c = 0.01. The insert shows these distributions on a linear scale. It can be seen that the dis- 2 1/2 tributions are broad with half-widths σ ∼ (kTe/me c ) , that is, ∆ω/ω ∼ 0.1. The average increase 2 in energy of the photon is ∆~ω/ω = 4(kTe/me c ) = 0.04 (Sunyaev, 1980).

The average increase in energy is of order (v/c)2, that is, second order in v/c, compared with the breadth of the wings of the scattering function which are of order v/c. There- fore, in addition to the increase in energy due to the second-order effect in (v/c)2, we also have to take account of the scattering of photons by first order Compton scatterings. In 2 the Rayleigh–Jeans limit, in which the spectrum is Iω ∝ ω , there are more photons scattered down in energy to frequency ω than are scattered up from lower frequencies. These Doppler scatterings increase the intensity at frequency ω by an increment +6y so that the net decrement is −2y, as given by the Kompaneets equation. This digression illustrates the power of the Kompaneets equation in automatically taking account of the statistical aspects of the diffusion of photons in phase space. The spectral signature of the Sunyaev-Zeldovich effect has a distinctive form over the peak of the spectrum of the Cosmic Microwave Background Radiation and is given in the 286 Interactions of high energy photons

3 2 Fig. 9.13. Intensity change in units of 2(kT0) /(hc) , plotted against X = ~ω/kT0 for three values of 2 kTe (in keV), where θe = kTe/mec . The solid curves are calculated using the second-order correction to the Kompaneets equation, while the dashed lines are calculated from the ﬁrst-order correction. The points are the result of a Monte Carlo evaluation of the Boltzmann collision integral by Garrett and Gull (Challinor and Lasenby, 1998).

first order approximation by (9.107). The exact shape of this function has been the subject of a number of studies which take full account of special relativistic effects and expand the Kompaneets equation to higher orders in ∂n/∂x. The results of numerical solutions of the Boltzmann equation and further analytic studies are summarised by Challinor and Lasenby (Challinor and Lasenby, 1998) (Fig. 9.13). Notice that the results are presented in terms of the absolute change in intensity ∆I(ω) which tends to zero in the high and low frequency limits. This form of distortion has been measured in a number of Abell clusters in the SuZIE experiment carried out at the CalTech Submillimetre Observatory on Mauna Kea (Benson et al., 2004). Fig. 9.14 shows that the expected change in sign of the Sunyaev-Zeldovich effect on either side of the frequency ~ω/kTe = 4 has been clearly detected.

9.6 Synchrotron–self Compton radiation The physics of inverse Compton scattering was discussed in Sect. 9.3 and is an important source of high energy radiation whenever large ﬂuxes of photons and relativistic electrons occupy the same volume. The case of special interest in this Section is that in which the 9.6 Synchrotron–self Compton radiation 287

Fig. 9.14. The observed Sunyaev-Zeldovich spectrum associated with hot gas in clusters of galaxies. In each plot the solid line is the best-fit model for the spectral distortions, the dashed line is the thermal component of the Sunyaev-Zeldovich effect and the dotted line is the kinematic component (Benson et al., 2004). The kinematic component is associated with first-order Compton scattering due to the peculiar motions of the clusters

relativistic electrons which are the source of low energy photons are also responsible for scattering these photons to X- and γ-ray energies, the process known as synchrotron–self Compton radiation. A case of special importance is that in which the energy density of low 288 Interactions of high energy photons energy photons is so great that most of the energy of the electrons is lost by synchrotron– self Compton rather than by synchotron radiation. This is likely to be the source of the ultra-high energy γ-rays observed in some of the most extreme active galactic nuclei. We can derive some of the essential features of synchrotron–self Compton radiation from the formulae we have already derived. The ratio η of the rates of loss of energy of an ultrarelativistic electron by synchrotron and inverse Compton radiation in the presence of a photon energy density urad and a magnetic ﬁeld of magnetic ﬂux density B are given by the formulae (9.42) and (9.41):

(dE/dt)IC uphoton η = = 2 . (9.112) (dE/dt)sync B /2µ0 Thus, if the synchrotron radio flux density and the X- and γ-radiation from the same source region are observed, estimates of the magnetic flux density within the source region can be made, the only problems being the upper and lower limits to the electron energy spectrum and in ensuring that electrons of roughly the same energies are responsible for the radio and X-ray emission. A good example of this procedure for estimating the magnetic flux density in the hot-spot regions in the powerful double radio source Cygnus A is discussed in Sect. 22.2. The synchro–Compton catastrophe occurs if the ratio η is greater than 1. In this case, low energy radio photons produced by synchrotron radiation are scattered to X-ray energies by the same relativistic electrons. Since η is greater than 1, the energy density of the X-rays is greater than that of the radio photons and so the electrons suffer even greater energy losses by scattering these X-rays to γ-ray energies. In turn, these γ-rays have a greater energy density than the X-rays . . . and so on. It can be seen that as soon as η becomes greater than one, the energy of the electrons is lost at the very highest energies and so the radio source should be a very powerful source of X- and γ-rays. Before considering the higher order scatterings, let us study the first stage of the process for a compact source of synchrotron radiation, so compact that the radiation is self-absorbed. First of all, the energy density of radiation within a synchrotron self-absorbed radio source is estimated. As shown in Sect. 8.7, the flux density of such a source is 2kT r2 S = b Ω where Ω ≈ θ2 = and γm c2 = 3kT = 3kT . (9.113) ν λ2 D2 e e b

Ω is the solid angle subtended by the source, r its physical size and D its distance. Te is the thermal temperature equivalent to the energy of a relativistic electron with total energy 2 γmec . As explained in Sect. 8.7, for a self-absorbed source, the electron temperature of the relativistic electrons is equal to the brightness temperature Tb of the source, Te = Tb. The radio luminosity of the source in W Hz−1 is therefore 8πkT L = 4πD2S ≈ b r2 . (9.114) ν ν λ2 9.6 Synchrotron–self Compton radiation 289

Lν is the luminosity per unit bandwidth and so, to order of magnitude, the bolometric luminosity is roughly νLν. Therefore, the energy density of radiation uphoton is L ν 2kT ν u ∼ ν = b , (9.115) photon 4πr2c λ2c and the ratio η is ! 2kTbν 2 uphoton λ c 4kTeνµ0 η = 2 = 2 ! = 2 2 . (9.116) B /2µ0 B λ cB

2µ0 We now use the theory of synchrotron self-absorbed sources to express the magnetic ﬂux density B in terms of observables. Repeating the calculations carried out in Sect. 8.7,

2 2 νg ≈ ν/γ and 3kTb = 3kTe = γmec where νg = eB/2πme . (9.117) Reorganising these relations, we ﬁnd ! 2πm m c2 2 B = e e ν . (9.118) e 3kTb Therefore, the ratio of the loss rates, η, is ! (dE/dt) 81e2µ k5 η = IC = 0 νT 5 . (9.119) 2 6 11 b (dE/dt)sync π mec This is the key result. The ratio of the loss rates depends very strongly upon the brightness temperature of the radio source. Substituting the values of the constants, the critical brightness temperature for which η = 1 is

12 −1/5 Tb = Te = 10 ν9 K , (9.120) 9 where ν9 is the frequency at which the brightness temperature is measured in units of 10 Hz, that is, in GHz. According to this calculation, no compact radio source should have 12 brightness temperature greater than Tb ≈ 10 K without suﬀering catastrophic inverse Compton scattering losses, if the emission is incoherent synchrotron radiation. The most compact sources, studied by Very Long Baseline Interferometry (VLBI) at centimetre wavelengths, have brightness temperatures less than the synchrotron–self Comp- 11 ton limit, typically, the values found being Tb ≈ 10 K. These observations in themselves provide direct evidence that the radiation is the emission of relativistic electrons since the temperature of the emitting electrons must be at least 1011 K. This is not, however, the whole story. If the time-scales of variability τ of the compact radio sources are used to estimate their physical sizes, l ∼ cτ, the source regions must be considerably smaller than those inferred from the VLBI observations, and brightness temperatures exceeding 1012 K 290 Interactions of high energy photons

Fig. 9.15. Examples of the spectra of the synchrotron–self Compton radiation of compact radio sources. (a) The ‘standard’ synchrotron–self Compton spectrum of a homogeneous source with −4 −3 −3 magnetic flux density 5 × 10 T and electron number density Ne(γ) dγ = 4 γ dγ m in a spherical source of radius 2 × 1011 m. The solid line is the synchrotron radio spectrum, the small-dashed line the first scatterings and the large-dashed line the second scatterings. (b) The spectrum of the 9 10 inhomogeneous model with inner and outer radii r1 = 10 m and r2 = 10 m, within which the −4 −2 magnetic flux density varies as B = 10 (r/r1) T and the electron number density as Ne(γ) dγ = −3 −2 −3 4 γ (r/r1) dγ m for 1 ≤ γ ≤ 10 (Band and Grindlay, 1985).

are found. It is likely that relativistic beaming is the cause of this discrepancy, a topic taken up in Chap. 23. Models of synchrotron-self Compton sources are best worked out numerically and are strongly dependent upon the input assumptions. A good impression of the forms of spectra expected is provided by the computations of Band and Grindlay (Band and Grindlay, 1985), who take account of the transfer of radiation within the self-absorbed source and consider 9.7 Cherenkov radiation 291 both homogeneous and inhomogeneous cases. A number of important reﬁnements are included in their computations. Of particular importance is the use of the Klein–Nishina cross-section at relativistic energies, ~ω ≥ 0.5 MeV, rather than the Thomson cross-section for photon-electron scattering. In the ultrarelativistic limit, the cross-section tends to π2r2 1 σ = e (ln 2hν + ) , (9.121) KN hν 2 and so decreases as (~ω)−1 at high energies. Consequently, higher order scatterings result in signiﬁcantly reduced luminosities as compared with the non-relativistic calculation. Many features of such computations can be understood from the Fig. 9.15a and b. The homogeneous source has the standard form of spectrum at radio frequencies, namely, a −α power-law distribution in the optically thin spectral region Lν ∝ ν , while, in the optically 5/2 thick region, the spectrum has the self-absorbed form Lν ∝ ν . The relativistic boosting of the spectrum of the radio emission from the compact radio source is clearly seen. Both the low and high frequency spectral features of the radio source spectrum follow the relativistic ‘boosting’ relations

2 4 νg → γ νg → γ νg .... (9.122) These features are most apparent in the case of the homogeneous source. The higher order 2 scatterings for photon energies hν mec are signiﬁcantly reduced because of the use of the Klein–Nishina cross-section at high energies. In the case of the inhomogeneous source, the magnetic ﬁeld strength and number density of relativistic electrons decrease outwards as power-laws, resulting in a much broader ‘synchrotron-peak’. As a result, only one Compton scattering is apparent because of the wide range of photon energies produced by the radio source. These computations assume that the source of radiation is stationary. As will be discussed in Chap. 23, the extreme ultra-high energy γ-ray sources, which are variable over short time-scales, display many of the features expected of synchrotron-self Compton radiation, but they must also involve relativistic bulk motion of the source regions to account for their extreme properties. As a consequence, the predictions of the models are somewhat model-dependent.

9.7 Cherenkov radiation When a fast particle moves through a medium at a constant velocity v greater than the speed of light in that medium, it emits Cherenkov radiation. The process ﬁnds application in the construction of threshold detectors in which Cherenkov radiation is only emitted if the particle has velocity greater than c/n. If the particles pass through, for example, lucite or plexiglass, for which n ≈ 1.5, only those with v > 0.67c emit Cherenkov radiation which can be detected as an optical signal. Particles with extreme relativistic energies can 292 Interactions of high energy photons

Fig. 9.16. Illustrating Huygens’ construction for the wavefront of coherent radiation of a charged particle moving at constant velocity v > c/n through a medium with refractive index n. be detected in gas Cherenkov detectors in which the refractive index n of the gas is just greater than 1. A second application is in the detection of ultrahigh energy γ-rays when they enter the top of the atmosphere. The high energy γ-ray initiates an electron–photon cascade (see Sect. 9.9) and, if the electron–positron pairs acquire velocities greater than the speed of light in air, optical Cherenkov radiation is emitted which can be detected by light detectors at sea-level. The origin of the emission is best appreciated from the expressions (6.19), the Li´enard– Wiechert potentials A(r, t) and φ(r, t) which are repeated here: " # " # µ qu 1 q A(r, t) = 0 ; φ(r, t) = , (9.123) 4πr 1 − (u · iobs)/c ret 4πε0r 1 − (u · iobs)/c ret where iobs is the unit vector in the direction of observation from the moving change. In the case of a vacuum, one of the standard results of electromagnetic theory that a charged particle moving at constant velocity u does not radiate electromagnetic radiation. As shown in Sect. 6.2, in a vacuum, radiation is emitted if the particle is accelerated. In the case of a medium with a ﬁnite permittivity , or refractive index n, however, the denominators of (9.123) become

[1 − (nu · iobs)/c]ret , (9.124) where n is the refractive index of the medium. It follows that the potentials become singular along the cone for which 1 − (nu · iobs)/c = 0, that is, for cos θ = c/nv. As a result, the usual rule that only accelerated charges radiate no longer applies. The geometric representation of this process is that, because the particle moves superluminally through the medium, a ‘shock wave’ is created behind the particle. The wavefront of the radiation propagates at a ﬁxed angle with respect to the velocity vector of the particle because the wavefronts only add up coherently in this direction according to Huygens’ 9.7 Cherenkov radiation 293

Fig. 9.17. Illustrating the geometry used in the derivation of the expressions for Cherenkov radiation. construction (Fig. 9.16). The geometry of Fig. 9.16 shows that the angle of the wave vector with respect to the direction of motion of the particle is cos θ = c/nv. Let us derive the main features of Cherenkov radiation in a little more detail. Consider an electron moving along the positive x-axis at a constant velocity v. This motion corresponds to a current density J where†

J = ev δ(x − vt) δ(y) δ(z) ix . (9.125)

Now take the Fourier transform of this current density to ﬁnd the frequency components J(ω) corresponding to this motion. Z 1 J(ω) = J exp(iωt) dt , (2π)1/2 e = δ(y) δ(z) exp(iωx/v) i . (9.126) (2π)1/2 x This Fourier decomposition corresponds to representing the motion of the moving electron by a line distribution of coherently oscillating currents. Our task is to work out the coherent emission, if any, from this distribution of oscillating currents. The full treatments given in standard texts such as Jackson and Clemmow and Dougherty are quite complex (Jackson, 1999; Clemmow and Dougherty, 1969). We adopt here an approach developed by John Peacock.

† Strictly speaking, we should multiply by Ne to create a current density, but Ne would cancel out when we revert to a single particle. 294 Interactions of high energy photons First, let us review some of the standard results concerning the propagation of electromagnetic waves in a medium of permittivity, or refractive index n = 1/2. It is a standard result of classical electrodynamics that the flow of electromagnetic energy through a surface dS is given by the Poynting vector flux, N · dS = (E × H) · dS. The electric and magnetic field strengths E and H are related to the electric flux density D and the magnetic flux density B by the constitutive relations

D = 0 E; B = µµ0 H . (9.127) The energy density of the electromagnetic ﬁeld in the medium is given by the standard formula Z Z u = E · dD + H · dB . (9.128)

If the medium has a constant real permittivity and permeability µ = 1, the energy density in the medium is 1 2 1 2 u = 2 0E + 2 µ0H . (9.129) 2 2 The speed of propagation of the waves is found from the dispersion relation k = 0µ0ω , −1/2 1/2 that is, c() = ω/k = (0µ0) = c/ . This demonstrates the well-known result that, in a linear medium, the refractive index n is 1/2. Another useful result is the relation between the E and B fields in the electromagnetic wave – the ratio E/B is c/1/2 = c/n. Substituting this result into the expression for the electric and magnetic field energies (9.129), it is found 2 that these are equal. Thus, the total energy density in the wave is u = 0E . Furthermore, 1/2 2 2 the Poynting vector flux E × H is 0E c = n0E c. This energy flow corresponds to 2 the energy density of radiation in the wave 0E propagating at the velocity of light in 2 the medium c/n. As is expected, N = n0E c. This is the result we have been seeking. It is similar to the formula used in Sects 6.2.2 and 6.2.3 but now the refractive index n is included in the right place. We now write down the expressions for the retarded values of the current which contributes to the vector potential at the point r (Fig. 9.17). From (6.17a), the expression for the vector potential A due to the current density J at distance r is Z Z µ J(r0, t − |r − r0|/c) µ [J] A(r) = 0 d3 r0 = 0 d3 r0 . (9.130) 4π |r − r0| 4π |r − r0| where the square brackets refer to retarded potentials. Taking the time derivative Z ∂A µ [J˙] E(r) = − = − 0 d3 r0 . (9.131) ∂t 4π |r − r0|

In the far field limit, the electric field component Er of the radiation field is perpendicular to the radial vector r and so, as indicated in Fig. 9.17, Er = E(r) × ik, that is,

Z ˙ µ0 sin θ [J] 3 0 |Er| = d r . (9.132) 4π |r − r0| 9.7 Cherenkov radiation 295

This formulaR reduces to the expression (6.5) for the radiation of a point charge by the substitution [J˙] d3 r0 = er¨. We now go through the same procedure described in Sect. 6.2.5 to evaluate the frequency spectrum of the radiation. First of all, we work out the total radiation rate by integrating the Poynting vector flux over a sphere at a large distance r. ! Z dE 2 = nc0Er dS , dt rad S Z Z nc µ2 sin2 θ ˙ 2 0 0 [J] 3 0 2 = 2 0 d r r dΩ . (9.133) Ω 16π |r − r | We now assume that the size of the emitting region is much smaller than the distance to the point of observation, L r. Therefore, we can write |r − r0| = r and then, ! Z Z dE n sin2 θ 2 ˙ 3 0 = 2 3 [J] d r dΩ . (9.134) dt rad 16π 0c Now, we take the time integral of the radiation rate to find the total radiated energy. Z ! Z Z Z ∞ dE ∞ n sin2 θ 2 ˙ 3 0 Erad = dt = 2 3 [J] d r dΩ dt . (9.135) −∞ dt rad −∞ Ω 16π 0c We use Parseval’s theorem to transform from an integral over time to one over frequency. Noting, as in Sect. 6.2.5, that we are only interested in positive frequencies, we find Z Z Z ∞ n sin2 θ 2 ˙ 3 0 Erad = 2 3 [J(ω)] d r dΩ dω . (9.136) 0 Ω 8π 0c R Let us now evaluate the volume integral [J˙(ω)] d3 r0. We take R to be the vector from the origin of the coordinate system to the observer and x to be the position vector of the current element J(ω) d3r0 from the origin. Thus, r0 = R − x. Now the waves from the current element at x propagate outwards from the emitting region at velocity c/n with phase factor exp[i(ωt − k · r0)] and therefore, relative to the origin at O, the phase factor of the waves, which we need to find the retarded value of J˙(ω), is

exp[i(ωt − k · (R − x))] = exp(−ik · R) exp[i(ωt + k · x)] . (9.137)

Therefore, evaluating [J˙(ω)], we ﬁnd Z Z ˙ 3 0 3 0 [J(ω)] d r = iω [J(ω)] d r .

Now we include the retarded component of J(ω) explicitly by including the phase factor, Z Z ˙ 3 0 3 0 [J(ω)] d r = ω exp[i(ωt + k · x)] J(ω) d r . 296 Interactions of high energy photons Using (9.126), we ﬁnd Z Z ωe ωx [J˙ (ω)] d3 r0 = exp(iωt) exp i k · x + dx , (2π)1/2 v Z ωe ωx = exp i k · x + dx . (9.138) (2π)1/2 v This is the key integral in deciding whether or not the particle radiates. If the electron propagates in a vacuum, ω/k = c and we can write the exponent

kx(cos θ + ω/kv) = kx(cos θ + c/v) . (9.139)

Since, in a vacuum, c/v > 1, this exponent is always greater than zero and hence the exponential integral over all x is always zero. This means that a particle moving at constant velocity in a vacuum does not radiate. If, however, the medium has refractive index n, ω/k = c/n and then the exponent is zero if cos θ = −c/nv. This is the origin of the Cherenkov radiation phenomenon. The radiation is only coherent along the angle θ corresponding to the Cherenkov cone derived from Huygens’ construction. We can therefore write down formally the energy spectrum by using (9.67) recalling that the radiation is only emitted at an angle cos θ = c/nv. We therefore ﬁnd from (9.136) Z Z dE nω2e2 sin2 θ ω 2 rad = 3 3 exp ikx cos θ + dx dΩ , dω Ω 16π 0c kv ! Z Z nω2e2 c2 ω 2

= 3 3 1 − 2 2 exp[ikx cos θ + dx dΩ . 16π 0c n v Ω kv We now evaluate the integral. Let us write k(cos θ + ω/kv) = α. The integral therefore becomes Z Z 2

exp(iαx) dx 2π sin θ dθ . (9.140) θ Let us take the line integral along a finite path length from −L to L. It should be noted that there is a problem in evaluating the integral of a function which only has finite value at a specific value of θ from −∞ to +∞. This is why the normal derivation involves the use of contour integration to get rid of the infinites. The integral should be taken over a small finite range of angles about θ = cos−1(c/nv) for which (cos θ + ω/kv) is close to zero. Therefore, we can integrate over all values of θ (or α) knowing that most of the integral is contributed by values of θ very close to cos−1(c/nv). Therefore, the integral becomes Z sin2 αL dα 8π . (9.141) α2 k Taking the integral over all values of α from −∞ to +∞, we find that the integral becomes 9.8 Electron–positron pair production 297 (8πc/nω)π2L. Therefore the energy per unit bandwidth is ! du ωe2 c2 = 3 1 − 2 2 L . (9.142) dω 2π0c n v We now ought to take the limit L → ∞. However, there is no need to do this since we obtain directly the energy loss rate per unit path length by dividing by 2L. Therefore, the loss rate per unit path length is ! du(ω) ωe2 c2 = 3 1 − 2 2 . (9.143) dx 4π0c n v Since the particle is moving at velocity v, the energy loss rate per unit bandwidth is ! du(ω) ωe2v c2 I(ω) = = 3 1 − 2 2 . (9.144) dt 4π0c n v Notice that the intensity of radiation depends upon the variation of the refractive index with frequency n(ω).

9.8 Electron–positron pair production 2 If the photon has energy greater than 2mec , pair production can take place in the ﬁeld of the nucleus. Pair production cannot take place in free space because momentum and energy cannot be conserved simultaneously. To demonstrate this, consider a photon of energy ~ω 2 decaying into an electron–positron pair, each of which has kinetic energy (γ − 1)mec . The best one can do to conserve both energy and momentum is if the electron–positron pair moves parallel to the original direction of the photon, then

2 Conservation of energy: energy of photon = ~ω = 2γmec .

momentum of pair = 2γmev = (~ω/c)(v/c) . But, initial momentum of photon = ~ω/c . Since v cannot be equal to c, we cannot conserve both energy and momentum in free space and this is why we need a third body, such as a nucleus, which can absorb some of the energy or momentum. Let us quote some useful results for electron–positron pair production (Chupp, 1976; Ramana Murthy and Wolfendale, 1993): Intermediate photon energies In the case of no screening, the cross-section for pho- 2 1/3 tons with energies in the range 1 ~ω/mec 1/αZ can be written " ! # 2 2 28 2~ω 218 2 −1 σpair = αre Z ln 2 − m atom . (9.145) 9 mec 27 298 Interactions of high energy photons

Fig. 9.18. The total mass absorption coeﬃcient for high energy photons in lead, indicating the contributions associated with the photoelectric absorption, Compton scattering and electron–positron pair production (Enge, 1966).

re is the classical electron radius and α the ﬁne structure constant.

Ultrarelativistic limit In the case of complete screening and for photon energies 2 1/3 ~ω/mec 1/αZ , the cross-section becomes " ! # 28 183 2 σ = αr2Z2 ln − m2 atom−1 . (9.146) pair e 9 Z1/3 27 2 In both cases, the cross-section for pair production is ∼ ασTZ . Notice also that the cross- section for the creation of pairs through interactions with electrons is very much smaller than the above values and can be neglected. Exactly as in Sect. 6.6, we deﬁne a radiation length ξpair for pair production

ξpair = ρ/Niσpair = MA/N0σpair , (9.147) where MA is the atomic mass, Ni is the number density of nuclei and N0 is Avogadro’s number. If we compare the radiation lengths for pair production and bremsstrahlung by ultrarelativistic electrons, we ﬁnd that ξpair ≈ ξbrems. This reﬂects the similarity of the Feynman diagrams for the bremsstrahlung and pair production mechanisms according to quantum electrodynamics (Leighton, 1959). 9.9 Electromagnetic showers 299

Fig. 9.19. A simple model for an electromagnetic shower.

We can now put together the three main loss processes for high energy photons – ionisation losses, Compton scattering and electron–positron pair production – to obtain the total mass absorption coefficient for X-rays and γ-rays. Fig. 9.18 shows how each of these processes contributes to the total absorption coefficient in lead. Notice that the energy range 500 keV . ~ω . 5 MeV is a complex energy range for the experimental study of photons from cosmic sources because all three processes make a significant contribution to the absorption coefficient for γ-rays. Consequently, this is a particularly difficult energy range for the design and construction of γ-ray telescopes. To make matters worse, the fluxes of photons from astrophysical sources are generally low in this energy range.

9.9 Electron–photon cascades, electromagnetic showers and the detection of ultrahigh energy γ-rays We can now understand how cascades, or showers, initiated by high energy electrons or γ-rays can come about. When, for example, a high energy photon enters the upper atmosphere, it generates an electron–positron pair, each of which in turn generates high energy photons by bremsstrahlung, each of which generates an electron–positron pair, each of which . . . and so on. Let us build a simple model of an electron–photon cascade in the following way. In the ultra-relativistic limit, the radiation lengths for pair production and bremsstrahlung are the same, as discussed in Sect. 9.8. Therefore the probability of these processes taking place 300 Interactions of high energy photons is one-half at path length ξ given by

1 exp(−ξ/ξ0) = 2 or ξ = R = ξ0 ln 2 . (9.148)

Therefore, if the cascade is initiated by a γ-ray of energy E0, after a distance of, on average R, an electron–positron pair is produced. For simplicity, it is assumed that the pair share the energy of the γ-ray, that is, E0/2 each. In the next length R, the electron and positron lose, on average, half their energy and they each radiate a photon of energy E0/4. Thus, we end up with two particles and two photons, all having energy E0/4 after distance 2R. This process is repeated as illustrated in Fig. 9.19 as the energy of the photons and particles is degraded through the atmosphere. After distance nR, the number of (photons + electrons + positrons) is 2n and their av- n 2 erage energy is E0/2 . On average, the shower consists of 3 positrons and electrons and 1 3 photons. The cascade eventually terminates when the average energy per particle drops to the critical energy Ec, below which the dominant loss process for the electrons is ionisation losses rather than bremsstrahlung. This process produces copious quantities of electron–ion pairs but they are all of very low energy. In addition, with decreasing energy, the production cross-section for pairs decreases until it becomes of the same order as that for Compton scattering and photoelectric absorption, as illustrated in Fig. 9.18. Thus, the shower reaches its maximum development when the average energy of the cascade particles is about Ec. The number of high energy photons and particles is roughly E0/Ec and the number of radiation lengths nc over which this occurs is ln(E /E ) n = 0 c . (9.149) c ln 2 At larger depths, the number of particles falls off dramatically because of ionisation losses which become catastrophic once the electrons become non-relativistic. These simple arguments give some impression of what needs to be included in a proper calculation. Appro- priate cross-sections for different energy ranges have to be used and integrations carried out over all possible products with the relevant probability distributions. Among the first calculations to illustrate these features were the pioneering efforts of Rossi and Greisen shown in Fig. 9.20a. These calculations confirm the predictions of the simple model, namely, that the initial growth is exponential, that the maximum number of particles is proportional to E0 and that after maximum development, there is a rapid attenuation of the electron flux. These computations have been considerably enhanced since these early results because of the need for precise understanding of the development of such showers which are central to the Auger ultra-high energy cosmic ray observatory and the new generation of ground- based ultra-high energy γ-ray telescopes. A more recent example of the development of an electromagnetic shower in iron is shown in Fig. 9.20b (Amsler et al., 2008). Amsler and his colleagues also provide further details about the properties of electromagnetic showers. 9.9 Electromagnetic showers 301

Fig. 9.20. (a) The total number of particles in a shower initiated by an electron of energy E0 as a function of depth through the medium measured in radiation lengths N; E0 is the critical energy (Rossi and Greisen, 1941). (b) A more recent computation of the development of an electromagnetic shower in iron (Amsler et al., 2008). 302 Interactions of high energy photons An important feature of these results is that the showers consist only of electrons, positrons and γ-rays – there are no muons, pions and other debris produced. This helps distinguish the arrival of high energy γ-rays from other types of particle. These electron–photon cascades, or electromagnetic showers, were among the ﬁrst high interactions to be detected inside cloud chambers. These showers also accompany the nuclear cascades which are considered in Chap. 10.

9.10 Electron-positron annihilation and positron production mechanisms Perhaps the most extreme form of energy loss mechanism for electrons is annihilation with their antiparticles, the positrons. Particle-antiparticle annihilation results in the production of high energy photons and, conversely, high energy photons can collide with ambient photons to produce particle-antiparticle pairs. There are several sources of positrons in astronomical environments. Perhaps the simplest is the decay of positively charged pions π+ described in Sect. 10.4. The pions are created in collisions between cosmic ray protons and nuclei and the interstellar gas, roughly equal numbers of positive, negative and neutral pions being created. Since the π0s decay into γ-rays, the flux of interstellar positrons created by this process can be estimated from the γ-ray luminosity of the interstellar gas. A second process is the decay of long-lived radioactive isotopes created by explosive nucleosynthesis in supernova explosions. For example, the β+ decay of 26Al has a mean lifetime of 1.1 × 106 years. This element is formed in supernova explosions and so is ejected into the interstellar gas where the decay results in a flux of interstellar positrons. A third process is the creation of electron–positron pairs through the collision of high energy photons with the field of a nucleus (see Sect. 9.9) . Electron-positron pair production can also take place in photon–photon collisions. The threshold for this process can be worked out using similar procedures to those used in our discussion of Compton scattering. If P1 and P2 are the momentum four-vectors of the photons before the collision,

P1 = [ε1/c, (ε1/c) i1]; P2 = [ε2/c, (ε2/c) i2] , (9.150) conservation of four-momentum requires

P1 + P2 = P3 + P4 , (9.151) where P3 and P4 are the four-vectors of the created particles. To ﬁnd the threshold for pair production, we require that the particles be created at rest and therefore

P3 = [mec, 0]; P4 = [mec, 0] . (9.152)

Squaring both sides of (9.151) and noting that P1 · P1 = P2 · P2 = 0 and that

2 2 P3 · P3 = P4 · P4 = P3 · P4 = mec , 9.10 Electron-positron annihilation and positron production mechanisms 303

Table 9.2. Threshold energies of ultra-high energy photons (ε2) which give rise to electron–positron pairs in collision with photons of diﬀerent energies (ε1)

ε1 (eV) ε2 (eV) Microwave Background Radiation 6 × 10−4 4 × 1014 Starlight 2 1011 X-ray 103 3 × 108

P1 · P1 + 2P1 · P2 + P2 · P2 = P3 · P3 + 2P3 · P4 + P4 · P4 , ε ε ε ε 2 1 2 − 1 2 cos θ = 4m2c2 , c2 c2 e 2 4 2mec ε2 = , (9.153) ε1(1 − cos θ) where θ is the angle between the incident directions of the photons. Thus, if electron– positron pairs are created, the threshold for the process occurs for head-on collisions, θ = π, and hence, 2 4 12 mec 0.26 × 10 ε2 > = eV , (9.154) ε1 ε1 where ε1 is measured in electron volts. This process thus provides not only a means for creating electron–positron pairs, for example in the vicinity of active galactic nuclei and hard X-ray sources, but also results an important source of opacity for high-energy γ-rays. Table 9.2 illustrates some of the examples we will encounter as our story unfolds. Photons with energies greater than those in the last column are expected to suffer some degree of absorption when they traverse regions containing large numbers of photons with the energies listed in the first column. The cross-section for this process for head-on collisions in the ultrarelativistic limit is 2 4 " ! # 2 mec 2¯ε σ = πre 2 ln 2 − 1 , (9.155) ε1ε2 mec 1/2 2 whereε ¯ = (ε1ε2) and re is the classical electron radius. In the regimeε ¯ ≈ mec , the cross-section is ! 1 m2c4 2 σ = πr2 1 − , (9.156) e ε¯2 (Ramana Murthy and Wolfendale, 1993). These cross-sections enable the opacity of the interstellar and intergalactic medium to be evaluated. Electron-positron annihilation can proceed in two ways. In the first case, the electrons and positrons annihilate at rest or in flight through the interaction e+ + e− → 2γ . (9.157) 304 Interactions of high energy photons

Fig. 9.21. (a) Observations of the whole sky in the 0.511 MeV electron–positron annihilation line made by the INTEGRAL γ-ray space observatory. (b) The right-hand panel shows the distribution of hard low mass X-ray binary stars. This stellar population has a distribution that matches the extent of the 511 keV map. (Courtesy of ESA, the Integral Science Team and G. Weidenspointner and his colleagues at the Max-Planck Institute for Extraterrestrial Physics, 2008).

When emitted at rest, the photons both have energy 0.511 MeV. When the particles annihilate ‘in flight’, meaning that they suffer a fast collision, there is a dispersion in the photon energies. It is a useful exercise in relativity to show that, if the positron is moving with velocity v and Lorentz factor γ, the centre of momentum frame of the collision has velocity V = γv(1 + γ) and that the energies of the pair of photons ejected in the direction of the line of flight of the positron and in the backward direction are

m c2(1 + γ) V E = e 1 ± . (9.158) 2 c

From this result, it can be seen that the photon which moves off in the direction of the incoming positron carries away most of the energy of the positron and that there is a lower 2 limit to the energy of the photon ejected in the opposite direction of mec /2. If the velocity of the positron is small, positronium atoms, that is, bound states consisting of an electron and a positron, can form by radiative recombination; 25% of the positronium 1 3 atoms form in the singlet S0 state and 75% of them in the triplet S1 state. The modes of 1 −10 decay from these states are different. The singlet S0 state has a lifetime of 1.25 × 10 s and the atom decays into two γ-rays, each with energy 0.511 MeV. The majority triplet 3 −7 S1 states have a mean lifetime of 1.5 × 10 s and three γ-rays are emitted, the maximum energy being 0.511 MeV in the centre of momentum frame. In this case, the decay of positronium results in a continuum spectrum to the low energy side of the 0.511 MeV line. If the positronium is formed from positrons and electrons with significant velocity dispersion, the line at 0.511 MeV is broadened, both because of the velocities of the particles and because of the low energy wing due to the continuum three-photon emission. This is a useful diagnostic tool in understanding the origin of the 0.511 MeV line. If the annihilations take place in a neutral medium with particle density less than 1021 m−3, positronium atoms 9.10 Electron-positron annihilation and positron production mechanisms 305 are formed. On the other hand, if the positrons collide in a gas at temperature greater than about 106 K, the annihilation takes place directly without the formation of positronium. The cross-section for electron–positron annihilation in the extreme relativistic limit is πr2 σ = e [ln 2γ − 1] . (9.159) γ For thermal electrons and positrons, the cross-section becomes πr2 σ ≈ e . (9.160) (v/c) The 0.511 MeV electron–positron annihilation line has been detected from the direction of the Galactic Centre and observations by the ESA INTEGRAL γ-ray observatory have shown that the emission is extended along the Galactic plane with a spatial distribution similar to that of hard low mass X-ray binary stars (Fig. 9.21). We will have more to say about these observation and source of positrons as the story unfolds.

17 The acceleration of high energy particles

Observations of cosmic rays and sources of non-thermal radiation indicate that the process of acceleration of high energy particles must account for the following features:

(i) The formation of a power-law energy spectrum for all types of charged particles. The energy spectrum of cosmic rays and the electron energy spectrum of non- thermal sources have the form dN(E) ∝ E−x dE, where the exponent x typically lies in the range 2 to 3. (ii) The acceleration of cosmic rays to energies E ∼ 1020 eV. (iii) In the process of acceleration, the chemical abundances of the primary cosmic rays should be similar to the cosmic abundances of the elements.

It would be helpful if we could appeal to the physics of laboratory plasmas for some guidance, but the evidence is somewhat ambivalent. On the one hand, if we want to accelerate particles to very high energies, we need to go to a great deal of trouble to ensure that the particles remain within the region of the accelerating ﬁeld, for example, in machines such as betatrons, synchrotrons, cyclotrons, and so on. Nature does not go to all this trouble to accelerate high energy particles. On the other hand, as soon as we try to build machines to store high-temperature plasmas, such as tokamaks, the conﬁgurations are usually grossly unstable and, in the instability, particles are accelerated to suprathermal energies.

17.1 General principles of acceleration The acceleration mechanisms may be classiﬁed as dynamic, hydrodynamic and electromagnetic. Often, there is no clear distinction between these because, being charged particles, they are closely tied to magnetic ﬁeld lines. In some models, the acceleration is purely dynamical, for example, in those cases where acceleration takes place through the collision of particles with clouds. Hydrodynamic models can involve the acceleration of whole layers

613 614 The acceleration of high energy particles of plasma to high velocities. The electromagnetic processes include those in which particles are accelerated by electric fields, for example, in neutral sheets, in electromagnetic or plasma waves and in the magnetospheres of neutron stars. The general expression for the acceleration of a charged particle in electric and magnetic fields is d (γmu) = e(E + u × B) . (17.1) dt In most astrophysical environments, static electric fields cannot be maintained because of the very high electrical conductivity of ionised gases – any electric field is very rapidly short-circuited by the motion of free charges (Sect. 11.1.3). Therefore, electromagnetic mechanisms of acceleration can only be associated either with non-stationary electric fields, for example, strong electromagnetic waves, or with time-varying magnetic fields. In a static magnetic field, no work is done on the particle but, if the magnetic field is time- varying, work can be done by the induced electric field, that is, the electric field E given by Maxwell’s equation curl E = −∂B/∂t. It has been suggested that phenomena such as the betatron effect might be applicable in some astrophysical environments. For example, the collapse of a cloud of ionised gas with a frozen-in magnetic field could lead to the acceleration of charged particles since they conserve their adiabatic invariants in a time-varying magnetic field (see Sect. 7.2). There is no obvious way in which this mechanism could play a role in the regions where particles are known to be accelerated to high energies, for example, in the shells of supernova remnants.

17.2 The acceleration of particles in solar flares An important example of particle acceleration by induced electric fields occurs in neutral sheets, the physics of which was discussed in some detail in the context of solar flares in Sect. 11.6. The source of energy of the flare is the reconnection of magnetic field lines and strong electric fields are generated because curl B and ∂B/∂t are not zero in these regions. The issue is how effective these processes are in creating electric fields which can accelerate charged particles to high energies. Suppose a DC electric field E is created in the reconnection region. We need the equation of motion of a charged particle in this field, taking account of the fact the particle collides with other particles in the plasma in stochastic electrostatic encounters (Sect. 11.1.2). The equation of motion of the electron is

du m = −eE − ν m v , (17.2) e dt c e where νc is the collision frequency for electrostatic collisions given by (11.17) and (11.18). 17.2 The acceleration of particles in solar ﬂares 615 For simplicity, considering the electric ﬁeld to be parallel to the velocity of the electron, dv m = eE − ν m v . (17.3) e dt c e

Adopting (11.17) for the self-collision frequency of the electrons, νc = 1/tc, we set m = me, N = ne and Z = 1 and so 4 4 e ne ln Λ e ne ln Λ νc = = . (17.4) 2π2m2v3 2 1/2 3/2 0 e 2π0 me (3 kT) In the last equality of (17.4), the mean squared velocity of the electrons is set equal to −3 3kT/me for a Maxwellian velocity distribution at temperature T. Because of the v dependence of the collision frequency upon the velocity of the electron, once the electron’s velocity becomes greater than a critical velocity vc, the effect of collisions becomes less and less important and hence the electrons are accelerated without any impediment under the influence of the electric field. This process is known as electron runaway. For velocities less than the critical velocity, the particles are not accelerated. The critical velocity is found by setting the right hand side of (17.3) equal to zero:    e3n ln Λ 1/2  e  vc =  2  . (17.5) 2π0 me E Correspondingly, for a thermal plasma, there is a critical electric field associated with this process which is known as the Dreicer field ED, 3 e ne ln Λ ED = 2 , (17.6) 6π0 kT at which electron runaway occurs for all the electrons. Writing this relation in terms of the 2 1/2 Debye length of the plasma λD = (0 kT/ne ) , e ln Λ ED = 2 . (17.7) 6π0λD This result is remarkably similar to that quoted by Heyvaerts (1981). Inserting values of the numerical constants, n ln Λ E = 2 × 10−13 e V m−1 , (17.8) D T −3 where ne is measured in electrons m and T in degrees Kelvin. 16 Adopting the values of these quantities quoted in Sect. 11.6 for solar flares, ne = 10 −3 6 −2 −1 m and T = 2 × 10 K, the Dreicer field is ED = 2 × 10 V m . The strength of the −1 electric field in the neutral sheet may be estimated by writing E ∼ vRB ≈ 0.02 V m . Thus, this process may well be important in particle acceleration in neutral sheets in solar flares. There are, however, a number of problems. First of all, there is a limit to the flow of 616 The acceleration of high energy particles charge because the beam of particles creates a magnetic field which cannot exceed the initial magnetic flux density. The result is an induced return current which tends to neutralise the effect of the electric field. Second, the beam of accelerated particles is subject to a variety of streaming instabilities. In particular, the beam may excite ion sound waves which enhance the resistivity of the plasma. If, however, only a small fraction of the electrons have velocities exceeding the critical velocity, these may be accelerated without exciting plasma instabilities. This process may also be important as part of the stochastic acceleration of particles. Heyvaerts (1981) gives a comprehensive description of the plasma instabilities which are likely to be associated with electron runaway. As discussed in Sect. 13.9, ‘gaps’ are expected in the structure of the electromagnetic fields in pulsar magnetospheres. These are regions in which induced electric fields are maintained and are potential sites for the acceleration of charged particles. The acceleration in the polar cap gaps is likely to be limited by pair-production processes, but the outer gaps, which have scale of roughly that of the light cylinder, offer the possibility of particle acceleration to high energies (Sect. 17.6).

17.3 Fermi acceleration - original version The Fermi mechanism was first proposed in 1949 as a stochastic means by which particles colliding with clouds in the interstellar medium could be accelerated to high energies (Fermi, 1949). We first consider Fermi’s original version of the theory, the problems it encounters and how it can be reincarnated in a modern guise. The analysis contains a number of features which are important for particle acceleration in general. In Sect.17.4, the modern version of first-order Fermi acceleration is described. In Fermi’s original picture, charged particles are reflected from ‘magnetic mirrors’ associated with irregularities in the Galactic magnetic field. The mirrors are assumed to move randomly with typical velocity V and Fermi showed that the particles gain energy stochastically in these reflections. If the particles remain within the acceleration region for some characteristic time τesc, a power-law distribution of particle energies is obtained. Let us repeat Fermi’s calculation in which the collision between the particle and the mirror takes place such that the angle between the initial direction of the particle and the normal to the surface of the mirror is θ (Fig. 17.1a). We carry out a relativistic analysis of the change in energy of the particle in a single collision. The mirror is taken to be infinitely massive and so its velocity is unchanged in the collision. The centre of momentum frame of reference is therefore that of the cloud moving at velocity V. The energy of the particle in this frame is ! V2 −1/2 E0 = γ (E + V p cos θ) , where γ = 1 − . (17.9) V V c2 17.3 Fermi acceleration - original version 617

Fig. 17.1. Illustrating the collision between a particle of mass m and a cloud of mass M.(a) a head- on collision; (b) a following collision. The probabilities of head-on and following collisions are proportional to the relative velocities of approach of the particle and the cloud, namely, v + V cos θ for (a) and v − V cos θ for (b). Since v ≈ c, the probabilities are proportional to 1 + (V/c) cos θ where 0 < θ < π.

The x-component of the relativistic three-momentum in the centre of momentum frame is VE p0 = p0 cos θ0 = γ p cos θ + . (17.10) x V c2 0 0 In the collision, the particle’s energy is conserved, Ebefore = Eafter, and its momentum in the 0 0 x-direction is reversed, px → −px. Therefore, transforming back to the observer’s frame, 00 0 0 E = γV (E + V px) . (17.11) 2 Substituting (17.9) and (17.10) into (17.11) and recalling that px/E = v cos θ/c , the change in energy of the particle is " # 2Vv cos θ V 2 E00 = γ2 E 1 + + . (17.12) V c2 c Expanding to second order in V/c, we find " # 2Vv cos θ V 2 ∆E = E00 − E = E + 2 . (17.13) c2 c We now average over the angle θ. Because of scattering by hydromagnetic waves or irregularities in the magnetic field, it is likely that the particle is randomly scattered in pitch angle between encounters with the clouds and we can therefore work out the mean increase in energy by averaging over a random distribution of angles θ in (17.13). There is a slightly greater probability of head-on encounters as opposed to following collisions, as illustrated in Fig. 17.1. The probability of encounters taking place at an angle of incidence θ is given by exactly the same reasoning which led to rate of arrival of photons at an angle θ in our analysis of inverse Compton scattering (see the discussion of Sect. 9.3). The only difference is that the particles move at a velocity v rather than c. For simplicity, let us consider 618 The acceleration of high energy particles the case of a relativistic particle with v ≈ c in which case the probability of a collision at angle θ is proportional to γV [1 + (V/c) cos θ]. Recalling that the probability of the angle lying in the angular range θ to θ+dθ is proportional to sin θ dθ, on averaging over all angles in the range 0 to π, the first term in (17.13) in the limit v → c becomes

Z 1 * + ! x[1 + (V/c)x] dx ∆E 2V 2 V 2 = Z−1 = , (17.14) E c 1 3 c [1 + (V/c)x] dx −1 where x = cos θ. Including the last term in (17.13), the average energy gain per collision is * + ∆E 8 V 2 = . (17.15) E 3 c This illustrates the famous result derived by Fermi that the average increase in energy is only second-order in V/c. This result leads to an exponential increase in the energy of the particle since the same fractional increase occurs per collision. If the mean-free path between clouds along a ﬁeld line is L, the time between collisions is L/(c cos φ) where φ is the pitch angle of the particle with respect to the magnetic ﬁeld direction. Averaging cos φ over the pitch angle φ, the average time between collisions is 2L/c. Therefore, the average rate of energy increase is ! dE 4 V2 = E = αE . (17.16) dt 3 cL

The particle is assumed to remain within the accelerating region for a time τesc. The resulting spectrum can be found from the diffusion-loss equation (7.41), dN ∂ N = D∇2N + [b(E)N(E)] − + Q(E) . (17.17) dt ∂E τesc We are interested in the steady-state solution in the absence of diffusion and so dN/dt = 0 and D∇2N = 0. It is also assumed that there are no sources, Q(E) = 0. The energy loss term b(E) = −dE/dt now becomes an energy gain term, b(E) = −αE. Therefore, (17.17) is reduced to d N(E) − [αEN(E)] − = 0 . (17.18) dE τesc Differentiating and rearranging this equation, ! dN(E) 1 N(E) = − 1 + , (17.19) dE ατesc E and so N(E) = constant × E−x , (17.20) 17.3 Fermi acceleration - original version 619

Fig. 17.2. Comparison of the acceleration rate and energy loss rate due to ionisation losses for a high energy particle.

−1 where x = 1 + (ατesc) . The Fermi acceleration mechanism results in a power-law energy spectrum. In Fermi’s paper of 1949, it was assumed that collisions with interstellar clouds were the main source of energy for the particles. There were, however, problems with this picture.

(i) The random velocities of interstellar clouds in the Galaxy are very small in comparison with the velocity of light, V/c ≤ 10−4. Furthermore, the mean-free path for the scattering of cosmic rays in the interstellar medium is of the order of 0.1 pc (Sect. 15.7) and so the number of collisions would amount to about a few per year, resulting in a very slow gain of energy by the particles. We might do rather better if attention was restricted to regions where there is small-scale turbulence, for example, in the shells of young supernova remnants where there is certainly a great deal of small-scale structure present and the velocities are very much greater than in the general interstellar medium. (ii) Ionisation losses hamper the acceleration of particles from low energies. The form of the ionisation loss rate as a function of kinetic energy is compared qualitatively with a uniform acceleration rate in Fig. 17.2. If the acceleration mechanism is to be eﬀective, the particles must either be injected into the acceleration region with energies greater than that corresponding to the maximum energy loss rate or else the initial acceleration process must be suﬃciently rapid to overcome the ionisation energy losses. This is known as the injection problem and is present in all acceleration mechanisms. (iii) There is nothing in the theory which tells us why the exponent x of the energy spectrum should be roughly 2.5. It would be remarkable if the mechanism of ac- 620 The acceleration of high energy particles celeration in very diverse types of source were such that the product of the characteristic escape-time τesc and the rate of energy gain, as represented by the constant α, conspired to give the same value of x.

This version of second-order Fermi acceleration disguises a key aspect of the acceleration process. The particle’s energy changes all the time stochastically and so, if particles were injected with a single energy, the energy distribution would be broadened by random encounters with interstellar clouds. On average, the root-mean-square change of energy of the particle is O(V/c), whereas the systematic increase in energy is only O(V/c)2. In the full calculation, we have to take account of the statistical nature of the acceleration process as well as the average systematic increase in energy. The problem is exactly the same as that discussed in the context of Comptonisation and the Sunyaev-Zeldovich effect in Sect. 9.5 and illustrated in Fig. 9.12. That figure shows the broadening of the photon spectrum which necessarily accompanies the second-order mean increase in energy. The full treatment which takes into account of the stochastic nature of the acceleration process and the spreading of the energy spectrum by scattering starts from the Fokker-Planck equation for the diffusion of the particles in momentum space. This approach is described in the review by Blandford and Eichler (1987) who show that that, for an isotropic distribution of scatterers, the differential energy spectral index is ! 3 4cL 1/2 1 x = 1 + 2 − . (17.21) 2 3hV iτesc 2 This result is slightly different from that found in the simple Fermi argument, except in the limit of small accelerations. The Fokker-Planck equation can be reduced to a diffusion-loss equation with an additional term which describe the diffusion of the particles in momentum space, or energy space in the case of isotropic scatterers. From (7.42),

2 dN 2 ∂ N 1 ∂ = D∇ N + [b(E)N(E)] − + Q(E) + 2 [d(E)N(E)] , (17.22) dt ∂E τesc 2 ∂E where d(E) is the mean square change in energy per unit time, d d(E) = h(∆E)2i . (17.23) dt As before, an expression for the average value of (∆E)2 can be found from (17.13). To second order in V/c, the only term which survives is (∆E)2 = 4E2(V/c)2 cos2 θ and, averaging over angle as before, Z D E V 2 1 π 4 V 2 (∆E)2 = 4E2 cos2 θ sin θ dθ = E2 . (17.24) c 2 0 3 c Thus, there is a very close relation between the mean square energy change and the average 17.4 Diffusive shock acceleration in strong shock waves 621 increase in energy per collision. Comparing (17.16) and (17.23), d(E) = −Eb(E)/2 = αE2/2, recalling that b(E) is defined to be the rate of loss of energy. We seek steady state solutions of (17.22), including the stochastic acceleration term, that is, ∂ N 1 ∂2 [b(E)N(E)] − + 2 [d(E)N(E)] = 0 . (17.25) ∂E τesc 2 ∂E Substituting for b(E) and d(E) and seeking solutions of power-law form, N ∝ E−x, the partial differential equation reduces to the quadratic equation ! 4 x2 + x − 2 + = 0 , (17.26) ατesc which has solution ! 3 16 1/2 1 x = 1 + − , (17.27) 2 9ατesc 2 exactly the result derived by Blandford and Eichler since α = (4/3)(V2/cL). In the second- order acceleration mechanism, the values of α and τesc to be used in the formulae for b(E) and d(E) are model dependent. In the modern version of second-order Fermi acceleration, the particles interact with various types of plasma wave and gain energy by being scattered stochastically by them. This process is similar to that described in Sect. 7.4 concerning the interaction of high energy particles with waves or irregularities in the interstellar magnetic field.

17.4 Diffusive shock acceleration in strong shock waves The acceleration mechanism which has dominated much astrophysical thinking since the late 1970s is associated with particle acceleration in strong shock waves, often referred to as diffusive shock acceleration. The key feature of this process is that the acceleration is first order in the shock velocity and automatically results in a power-law spectrum with energy spectral index x ≈ 2. Let us first write the essence of the original Fermi mechanism in a slightly different way. We define the constants β and P as follows: E = βE0 is the average energy of the particle after one collision and P is the probability that the particle remains within the accelerating k region after one collision. Then, after k collisions, there are N = N0P particles with k energies E = E0β . Eliminating k between these quantities, ln(N/N ) ln P 0 = , (17.28) ln(E/E0) ln β and hence ! N E ln P/ ln β = . (17.29) N0 E0 622 The acceleration of high energy particles In fact, this value of N is N(≥ E) since this is the number which reach energy E and some fraction of them continue to be accelerated to higher energies. Therefore N(E) dE = constant × E−1+(ln P/ ln β) dE . (17.30) In this reformulation, we have again recovered a power law energy distribution. The equivalence with the derivation of Fermi acceleration in Sect. 17.3 follows from equation −1 (17.11), namely, (ln P/ ln β) ≡ −(ατesc) . Evidently, β is related to α and P to τesc. In the original version of the Fermi acceleration described in Sect. 17.3, α is proportional to (V/c)2 and is referred to as second order Fermi acceleration. We would do much better if there were only head-on collisions. Inspection of (17.13) shows, that the fractional energy increase is ∆E/E ∝ 2V/c, that is, first-order in V/c and, appropriately, this is referred to as first-order Fermi acceleration. The diffusive shock acceleration process is first-order Fermi acceleration in the presence of strong shock waves and was discovered independently by a number of workers in the late 1970s. The papers by Axford, Leer and Skadron (1977), Krymsky (1977), Bell (1978) and Blandford and Ostriker (1978) stimulated an enormous amount of interest in this process in many diverse astrophysical environments. There are two different ways of tackling the problem, the first starting from the Fokker-Planck equation for the evolution of the momentum distribution of high energy particles in the vicinity of a strong shock and the second, a more physical approach in which the behaviour of individual particles is followed. Bell (1978) adopts the latter approach which we will follow here. The model involves a strong shock propagating through a diffuse medium, for example, the shock waves which propagate through the interstellar medium ahead of the supersonic shells of supernova remnants (Fig. 16.8). A flux of high energy particles is assumed to be present both in front of and behind the shock front. The particles are assumed to be propagating at speeds close to that of light and so the velocity of the shock is very much less than those of the high energy particles. The high energy particles scarcely notice the shock at all since its thickness is normally be very much smaller than the gyroradius of the high energy particle. Because of scattering by streaming instabilities or turbulent motions on either side of the shock wave, when the particles pass though the shock in either direction, they are scattered so that their velocity distribution rapidly becomes isotropic in the frame of reference of the moving fluid on either side of the shock. The physics of strong shock waves was analysed in Sect. 11.3. The material ejected in supernova explosions attains velocities of order 104 km s−1, which is very much greater than the sound and Alfven´ speeds of the interstellar medium, which are at most about 10 −1 km s . A strong shock wave travels at a highly supersonic velocity U cs, where cs is the sound speed in the ambient medium (Fig. 17.3a), the Mach number M being U/cs 1. It is often convenient to transform into the frame of reference in which the shock front is at rest and then the up-stream gas flows into the shock front at velocity v1 = U and leaves the shock with a downstream velocity v2 (Fig. 17.3b). The equation of continuity requires 17.4 Diffusive shock acceleration in strong shock waves 623

Fig. 17.3. The dynamics of high energy particles in the vicinity of a strong shock wave. (a) A strong shock wave propagating at a supersonic velocity U through stationary interstellar gas with density ρ1, pressure p1 and temperature T1. The density, pressure and temperature behind the shock are ρ2, p2 and T2 respectively. The relations between the variables on either side of the shock front are given by the relations (11.72, 11.73 and 11.74). (b) The flow of interstellar gas in the vicinity of the shock front in the reference frame in which the shock front is at rest. In this frame of reference, the ratio of the upstream to the downstream velocity is v1/v2 = (γ+1)/(γ−1). For a fully ionised plasma, γ = 5/3 and the ratio of these velocities is v1/v2 = 4 as shown in the figure. (c) The flow of gas as observed in the frame of reference in which the upstream gas is stationary and the velocity distribution of the high energy particles is isotropic. (d) The flow of gas as observed in the frame of reference in which the downstream gas is stationary and the velocity distribution of high energy particles is isotropic. mass to be conserved through the shock and so

ρ1v1 = ρ1U = ρ2v2 . (17.31)

In the case of a strong shock, ρ2/ρ1 = (γ + 1)/(γ − 1) where γ is the ratio of specific heat capacities of the gas (Sect. 11.3.1). Taking γ = 5/3 for a monatomic or fully ionised gas, ρ2/ρ1 = 4 and so v2 = (1/4)v1 (Fig. 17.3b). Now consider high energy particles ahead of the shock. Scattering ensures that the particle distribution is isotropic in the frame of reference in which the gas is at rest. It is instructive to draw diagrams illustrating the situation so far as typical high energy particles upstream and downstream of the shock are concerned. The shock advances through the medium at velocity U but the gas behind the shock travels at a velocity (3/4)U relative to the upstream gas (Fig. 17.3c). When a high energy particle crosses the shock front, it ob- tains a small increase in energy, of order ∆E/E ∼ U/c as we will show below. The particles 624 The acceleration of high energy particles are then scattered in the region behind the shock front so that their velocity distributions become isotropic with respect to that flow. Now consider the opposite process of the particle diffusing from behind the shock to the upstream region in front of the shock (Fig. 17.3d). Now the velocity distribution of the particles is isotropic behind the shock and, when they cross the shock front, they encounter gas moving towards the shock front, again with the same velocity (3/4)U. In other words, the particle undergoes exactly the same process of receiving a small increase in energy ∆E on crossing the shock from the downstream to the upstream flow as it did in travelling from upstream to downstream. This is the clever aspect of the acceleration mechanism. Every time the particle crosses the shock front it receives an increase of energy – there are never crossings in which the particles lose energy – and the increment in energy is the same going in both directions. Thus, unlike the original Fermi mechanism in which there are both head-on and following collisions, in the case of the shock front, the collisions are always head on and energy is transferred to the particles. By simple arguments, due to Bell (1978), both β and P can be determined quantitatively for a complete acceleration cycle. First, we evaluate the average increase in energy of the particle on crossing from the upstream to the downstream sides of the shock. The gas on the downstream side approaches the particle at a velocity V = (3/4)U and so, performing a Lorentz transformation, the particle’s energy when it passes into the downstream region is

0 E = γV (E + pxV) , (17.32) where the x-coordinate is taken to be perpendicular to the shock. The shock is assumed to be non-relativistic, V c, γV = 1, but that the particles are relativistic and so E = pc, px = (E/c) cos θ. Therefore, ∆E V ∆E = pV cos θ = cos θ . (17.33) E c The probability that the particles which cross the shock arrive within the angles θ to θ+dθ is proportional to sin θ dθ and the rate at which they approach the shock front is proportional to the x-component of their velocities, c cos θ. Therefore the probability of the particle crossing the shock is proportional to sin θ cos θ dθ. Normalising so that the integral of the probability distribution over all the particles approaching the shock is equal to 1, that is, those with θ in the range 0 to π/2,

p(θ) = 2 sin θ cos θ dθ . (17.34)

Therefore, the average gain in energy on crossing the shock is * + Z ∆E V π/2 2 V = 2 cos2 θ sin θ dθ = . (17.35) E c 0 3 c 17.4 Diffusive shock acceleration in strong shock waves 625 The particle’s velocity vector is randomised without energy loss by scattering in the downstream region and it then recrosses the shock, as illustrated in Fig. 17.3d when it gains another fractional increase in energy (2/3)(V/c). Therefore, in making one round trip across the shock and back again, the fractional energy increase is, on average, * + ∆E 4 V = . (17.36) E 3 c Consequently, E 4V β = = 1 + , (17.37) E0 3c in one round trip. To work out the escape probability P, we use the argument due to Bell (1978). According 1 to classical kinetic theory, the number of particles crossing the shock is 4 Nc where N is the number density of particles. This is the average number of particles crossing the shock in either direction since, as noted above, the particles scarcely notice the shock. Downstream, however, the particles are swept away, or ‘advected’, from the shock because the particles are isotropic in that frame. Referring to Fig. 17.3b, it can be seen that particles are removed 1 from the region of the shock at a rate NV = 4 NU. Thus, the fraction of the particles lost 1 1 per unit time is 4 NU/ 4 Nc = U/c. Since the shock is assumed to be non-relativistic, only a very small fraction of the particles is lost per cycle. Thus, P = 1 − (U/c). Therefore, ! U U 4V 4V U ln P = ln 1 − = − and ln β = ln 1 + = = . (17.38) c c 3c 3c c Inserting ln β and ln P into (17.32), ln P = −1 . (17.39) ln β Therefore, the differential energy spectrum of the high energy particles is N(E) dE ∝ E−2 dE . (17.40) This is the result we have been seeking. It may be objected that we have obtained a value of 2 rather than 2.5 for the exponent of the differential energy spectrum and that problem cannot be neglected. The mechanism excited a great deal of interest however because it provided an excellent physical reason why power-law energy spectra with a unique spectral index should be found in diverse astrophysical environments. The only requirements are the presence of strong shock waves and that the velocity vectors of the high energy particles are randomised on either side of the shock. There are undoubtedly strong shocks in sources of high energy particles, for example, in supernova remnants, active galactic nuclei and the extended components of extragalactic radio sources. An important feature of the model is that the particles are scattered in both the upstream and down-stream regions. In Bell’s original proposal, the particles which recross 626 The acceleration of high energy particles the shock from downstream to upstream result in bulk streaming of the relativistic particles through the unperturbed interstellar medium and, consequently, the particles are scattered by the generation of Alfven´ and hydromagnetic waves which grow to large amplitude, as described in Sect. 7.4. As the growing waves are advected into the region behind the shock, they create turbulent motions which can scatter the particles as described in Sect. 7.3. It is therefore expected that the high energy particles will be confined within some characteristic distance in front of the shock. The number of high energy particles is expected to decrease exponentially ahead of the shock wave. In consequence, the acceleration process can be modelled by the ‘box-model’ of Drury and his colleague in which the acceleration occurs within one diffusion length on either side of the shock front (Drury et al., 1999). There is an upper limit to the energy to which particles can be accelerated by this mechanism. Lagage and Cesarsky (1983) describe in detail the processes which can limit the acceleration process. The basic problem is that, although the first order acceleration mechanism is a distinct improvement upon the original Fermi mechanism, it is still not a rapid process. The particles have to diffuse back and forth across the shock wave many times and, in the case of the shells of supernova remnants, their energies increase by about one part in 100 at each crossing. The supernova blast wave is decelerated once the remnant has swept up roughly its own mass of interstellar gas and then enters the Sedov phase of evolution (Sect. 16.7). Although the acceleration mechanism continues throughout the life of a supernova remnant until it dissolves into the interstellar medium after about 105 to 106 years, most of the particle acceleration occurs during the undecelerated blast wave phase which lasts less than about 103 years, as illustrated by the computations of Lagage and Cesarsky. Assuming the magnetic flux density of the interstellar magnetic field to be 10−10 T, they estimated that the maximum upper limit to the energy of particles is about 105 GeV nucleon−1 = 1014 eV nucleon−1, their more realistic estimates corresponding to about an order of magnitude less than this value. The figure of 1014 eV corresponds roughly to a radius of gyration of a proton of about 1 pc if the magnetic flux density is B = 10−10 T (see Table 15.6). The energy spectrum of cosmic rays extends well beyond this upper limit to 1020 eV. The origin of this problem can be appreciated from an argument due to Syrovatskii for the maximum attainable energy of a particle in a magnetic field of flux density B and scale L. The first Maxwell equation, or Faraday’s law, can be used to estimate the induced electric field in the region,

∂B ∇ × E = − . (17.41) ∂t

If U is the speed of the shock, this equation can be rewritten to order of magnitude as

E B ∼ : E ∼ BU . (17.42) L L/U 17.5 Beyond the standard model 627 The energy of a particle of charge ze accelerated by the induced electric ﬁeld is then Z

Emax = zeE dx = zeBUL . (17.43)

Hence, to order of magnitude, the energy of the accelerated particle is roughly eBUL per nucleon. For the parameters adopted for young supernova remnants above, B = 10−10 T, U = 104 km s−1 and t ≈ 103 years, we recover roughly the upper limit derived by Lagage 14 and Cesarsky, Emax ∼ 10 eV. There is one further important aspect of this acceleration process. The particles are accelerated where they are needed and this enables the adiabatic loss problem (Sect. 16.8) to be overcome. In supernova remnants such as Cassiopeia A and Tycho’s supernova, the particles are accelerated in situ and the energy for accelerating them is extracted from the kinetic energy of the expanding supernova shell. As suggested in sect. 16.8, there are therefore good reasons why both the magnetic ﬁeld energy and the particle energy in shell- like supernova remnants are derived from the kinetic energy of expansion of the supernova.

17.5 Beyond the standard model The simple physical model of diffusive particle acceleration described in Sect. 17.4 contains the essence of the physics of the acceleration process. That analysis has an almost ‘thermodynamic’ flavour to it in that it omits all mention of the facts that the strong shock waves are collisionless and that there are magnetic fields in the plasma which mediate the effective viscosity which transports energy and momentum through the shock wave. A full treatment begins with a Fokker-Planck type equation for the evolution of the energy spectrum of the particles, the approach taken by Axford, Leer and Skadron (1977), Krymsky (1977) and Blandford and Ostriker (1978). Since the pioneering researches of the 1970s, a huge amount of effort has been devoted to the detailed physics of diffusive particle acceleration. The motivation has been the need to understand how the prediction of the standard model N(E) ∝ E−2 is modified when the effects of the magnetic field are included in the model, as well as considering the full range of weak, strong and relativistic shocks. The shocks may be parallel, perpendicular or inclined to the magnetic field direction, all resulting is slightly different predictions for the particle energy spectrum. For our present purposes, it is sufficient to note that, for non-relativistic shocks, the same calculation performed in Sect. 17.4 can be repeated for the case r = v1/v2 < 4 and then the following relations are found for the spectrum of the particles:

−q+2 −q r = v1/v2 = %2/%1 ; q = 3r/(r − 1) ; N(E) ∝ E ; f (p) ∝ p , (17.44) where the last expression describes an isotropic particle distribution in momentum space. Steeper spectra are expected for smaller velocity or density ratios. 628 The acceleration of high energy particles The mechanism has been the subject of detailed numerical simulation. These studies have shown that diffusive particle acceleration can be remarkably effective, the conversion of kinetic energy of the shock into high energy particles amounting to about 50%. As a result, the influence of the energy density of the accelerated particles upon the structure of the shock and the stability of the flows cannot be neglected. The result also means that the process of acceleration in intrinsically non-linear. Let us first review observational evidence and theoretical developments concerning the strengths of the magnetic fields present in the shock fronts.

17.5.1 Magnetic fields in supernova shock fronts In a strong shock wave in an fully ionised gas, the increase in density behind a strong shock corresponds to a factor of (γ + 1)/(γ − 1) = 4 greater than the ambient gas density. Conse- quently, if magnetic flux freezing is applied to the ionised gas on either side of the shock wave, an increase by only a factor of four would be obtained over the interstellar magnetic field. This is a contributory factor to the problem of accelerating particles to energies greater than about 1014 eV discussed in Sect. 17.4 (Lagage and Cesarsky, 1983). Two results have suggested that the magnetic fields in the strong shocks in supernova remnants undergo a much greater flux density increase. The first piece of evidence is derived from observations of the X-ray emission from the sharp boundaries of young supernova remnants such as those observed in the images of Cassiopeia A (Fig. 13.3a) and Tycho’s supernova remnant (Fig. 16.4a). These narrow shells are interpreted as the shock waves about the supernova remnant. Vink and Laming (2003) interpreted the narrow non-thermal X-ray emission bounding Cassiopeia A as the synchrotron emission of very energy electrons accelerated in the shock itself. These very high energy electrons suffer such strong synchrotron radiation losses that they lose all their energy in the vicinity of the shock. The transient nature of some of these very narrow filaments is compelling evidence that the process of diffuse shock acceleration of ultrarelativistic electrons is indeed taking place in these shock waves. Volk¨ and his colleagues showed that the X-ray images of Cassiopeia A, Tycho and four other young supernova remnants all possess very narrow X-ray filaments which define the location of the shock wave about the supernova remnant and that these are all consistent with a similar model (Volk¨ et al., 2005). In all cases, very strong magnetic fields were inferred to be present in the shock wave, typical values being about (3 − 10) × 10−8 T. These are very much greater values than could be accounted for by compressing the interstellar magnetic field in a strong shock. The second piece is evidence is that physical mechanisms have been discovered by which such strong magnetic fields can be generated in strong shock waves. Bell and Lucek (2000; 2001) carried out numerical calculations in which they found that the magnetic flux densities behind strong shocks which were accelerating particle to high energies could be 17.5 Beyond the standard model 629 very considerably enhanced over the standard compressed values as a result of instabilities which tangled and amplified the magnetic field. In subsequent papers, Bell (2004; 2005) demonstrated that these simulations lead to a non-resonant instability driven by the cosmic ray current itself. In a simple picture, cosmic ray currents interact with loops of magnetic field resulting a j× B force which expands the loops. Since the mass density of particles on the loop decreases as the loop expands, the acceleration increases resulting in an instability which produces strong loops of magnetic field. He showed that the energy density in the magnetic field could approach the energy density in the high energy particles. The pressure of the accelerated particles can approach the ram pressure of the swept up gas and so the amplification of the magnetic flux density is of the order of the Alfven´ Mach number, of the order 100-1000. This is, however, only one of a number of mechanisms by which strong fields can be created. In the upstream region, other mechanisms include the streaming instability proposed by Bell (1978), the Drury instability (Drury and Falle, 1986), the firehose instability (Quest and Shapiro, 1996) and small-scale dynamos (Beresnyak et al., 2009). Furthermore, instabilities can also operate downstream, for example, the nonlinear Richtmeyer-Meshkov instability (Giacalone and Jokipii, 2007) and charge exchange current-driven instabilities (Ohira et al., 2009). The inference is that the magnetic flux density can be amplified to very high values by these instabilities, accounting for the strong fields now known to be present in supernova shock waves. These arguments suggest that the values of B which should be used in (17.43) could be two or three orders of magnitude greater than the values used by Lagage and Cesarsky. In consequence, there would be no problem in accelerating particles up to the knee at 1015 − 1016 eV observed in the cosmic ray spectrum.

17.5.2 Non-linear diﬀuse shock acceleration Because of the strong non-linearity of the acceleration process, numerical methods have to be employed to understand the process of acceleration in more detail. The enhance- ments needed to construct more realistic models of the process of shock acceleration are illustrated schematically in Fig. 17.4. Going round the diagram,

1 The basic picture of diﬀusive shock acceleration is shown, taken from the single particle simulations of Ballard and Heavens (1992). In fact, this diagram is for a relativistic shock, but the principles of diﬀusive shock acceleration are the same.

2 The high energy particles diﬀuse from downstream to upstream.

3 There must be some source of injection of seed particles and in this diagram it is assumed that these are thermal leakage particles.

4 The streaming of the particles gives rise to the generation of resonant Alfven´ waves, non-resonant waves and turbulence. 630 The acceleration of high energy particles

Fig. 17.4. Illustrating some of the physical processes to be included in a fully self-consistent model of diﬀusive shock acceleration (Courtesy of Professor Hyesung Kang).

5 The results of these instabilities are: (i) to enhance greatly the strength of the magnetic ﬁeld, (ii) to accelerate particles to the ‘knee’ in the cosmic ray energy spectrum, (iii) to cause further scattering of particles and the dissipation of waves and (iv) the diﬀusion of particles in physical and momentum space.

Description of various approaches to the numerical modelling of the acceleration process and of the many diverse aspects of the physics involved are given by Kang and Jones (2006). The important difference from the simple model of Sect. 17.4 is the influence of the diffusion of the high energy particles back upstream upon the structure of the shock. The resulting precursor has a roughly exponential pressure distribution ahead of the shock and the streaming of the particles results in a decrease in the velocity jump across it, as is illustrated in Fig. 17.5a. Of particular interest is the fact that Kang and her colleagues have shown that the resulting momentum distribution of the accelerated particles has self-similar form as a function of time, as illustrated in Fig. 17.5b (Kang et al., 2009). −4 The momentum spectrum of the high energy particles has the form f (p) ∝ gs(p) p −4 −2 and so if gs(p) is independent of momentum f (p) ∝ p and N(E) ∝ E . The results shown in Fig. 17.5b display a number of important features. The fact that the solution is self-similar suggests that the form of the energy spectrum is stationary as the shock 17.5 Beyond the standard model 631

Fig. 17.5. (a) Illustrating the modiﬁcation to the velocity distribution in the shock precursor due to the streaming of high energy particles back upstream through the shock. (b) The self-similar solution for the momentum distribution of the particles for a shock with Mach number M = 10 over a factor of 10 in dimensionless time (Kang et al., 2009). The momentum distribution of the high energy −4 particles is N(p) ∝ gs(p) p .

expands into the surrounding interstellar medium. The diagram shows that the self-similar spectrum is steepened relative to N(E) ∝ E−2 at intermediate energies and then flattens at higher energies. This can be understood in terms of the decreased value of the velocity, or density, ratio r = u1/u2 across the shock, which according to (17.44) leads to a steeper energy spectrum. The flattening at higher energies results from the fact that the higher energy particles have greater gyroradii and experience the acceleration associated with the larger ratio of u0/u2 rather than u1/u2. At the highest energies, the exponential decrease in the energy spectrum is associated with Lagage and Cesarsky cut-off. The predicted energy spectra for different cosmic ray species from calculations by Berezhko and Volk¨ (2007) are shown in Fig. 17.6 which displays a number of the features of the spectrum discussed above. 632 The acceleration of high energy particles

Fig. 17.6. The energy spectra of individual species ﬁtted to the results of diﬀusive shock acceleration by Berezhko and Volk¨ (2007).

An important recent addition to this story has been the detection of Hα emission from the shock wave and from the precursor region (Lee et al., 2007). Fig. 17.7a shows a recent image of the shock front in knot b of Tycho’s supernova remnant, as observed by the Hub- ble Space Telescope. The narrow emission coincides with the location of the synchrotron X-ray emission of the shock front. To the left of the narrow emission feature, diﬀuse Hα can be observed which falls oﬀ in intensity with distance from the shock front. The origin of the Hα emission is change exchange interactions between the neutral interstellar gas and the hot plasma on either side of the shock. The atomic physics of this process enables information about physical conditions in the shock front and its vicinity to be obtained. For example, it provides a direct estimate of the ratio of the electron to ion temperatures in these regions. In Fig. 17.7b, the intensity of Hα emission from the shock itself and the precursor are clearly distinguished. The Hα emission provides a new diagnostic tool 17.5 Beyond the standard model 633

Fig. 17.7. (a) The shock wave in knot b in the Tycho’s supernova remnant as observed by the HST in the Hα line of hydrogen. (b) A fit of the profile of the Hα intensity in the vicinity of knot b, showing the emission from the shock itself and from the precursor region (Both images courtesy of Dr. Jae-Joon Lee. for probing the shock and precursor regions and are consistent with the general picture developed above. The conclusions of these studies are that high energy particles and turbulent magnetic fields are the natural products of the formation of collisionless shocks in cosmic environments with a more of less universal power-law energy spectrum which is a good match to the observed spectra. About 50% of the shock kinetic energy is transferred to the cosmic rays if the strong shocks have Mach numbers Ms ≥ 30.

17.5.3 The energy spectrum of cosmic rays at and above the ‘knee’ The results of Sect. 17.5.2 provide an explanation for the origin of the energy spectrum of cosmic rays beyond the ‘knee’. Protons with energies up to 1016 − 1017 eV can be accelerated by diffusive shock acceleration in the strong magnetic fields now known to be present in the shocks present in young supernova remnants and, since the process accelerates particles to the same energy per nucleon, helium, carbon, oxygen and iron nuclei are accelerated to even greater total energies. Support for this picture is provided by observations of cosmic ray air showers by the KASCADE experiment. The Karlsruhe air-shower array is dedicated to the precise characterisation of cosmic ray showers with energies in the range 1016 − 1018 eV and uses a number of different types of detector to achieve this. The development of cosmic ray 634 The acceleration of high energy particles

Fig. 17.8. The energy spectra of diﬀerent cosmic ray species with energies in the range 1016 − 1018 eV as deduced from the Karlsruhe KASCADE experiment. The characterisation of diﬀerent species was determined by the SYBILL 2.1 simulations of the development of cosmic rays showers. Energy spectra for protons, helium nuclei and carbon are shown in the left panel and silicon and iron in the right panel based on SIBYLL simulations. The shaded bands are estimates of the systematic uncertainties in the modelling procedure (Antoni et al., 2005).

Fig. 17.9. Illustrating how the overall energy spectrum of cosmic rays could be accounted for as the superposition of the energy spectra of different species accelerated in the strong magnetic fields in supernova shock waves. The data are taken from the survey by Nagano and Watson (2000) (Courtesy of Professors Hyesung Kang and Dongsu Ryu) 17.6 The highest energy cosmic rays 635 showers is modelled by two independent codes using the best understanding of nucleon- nucleon interactions at very high energies. A variety of diagnostic tools are used to separate the showers into those initiated by protons, helium nuclei and heavier species. Fig. 17.8 shows the results of the KASCADE experiment. The knee in the energy spectrum of helium nuclei is observed at significantly higher energy than that of the protons and this trend continues for species such as carbon, silicon and iron nuclei. As the authors state in their paper (Antoni et al., 2005): ‘For both models the position of the knees in these spectra is shifted towards higher energy with increasing element number’. They also caution, however, that: ‘At present, the limiting factors of the analysis are the properties of the high energy interaction models used and not the quality or the understanding of the KASCADE data.’ Note that the separate determinations of the energy spectra for different species sum to the total energy spectrum shown in Fig. 15.19. Support for this result is provided by the HiRes 1 and 2 experiments which suggested that the composition of the cosmic rays at energies E ∼ 1018 eV is significantly heavier than that of protons alone (Fig. 15.21). Fig. 17.9 shows schematically how the overall spectrum of cosmic rays up to 1018 eV and possibly higher could be accounted for by this process.

17.6 The highest energy cosmic rays It seems highly probable that cosmic rays with energies up to E ∼ 1018 eV originate in our Galaxy and supernovae remnants are their likely source. There remain, however, the problems of accounting for the very highest energy cosmic rays with E ∼ 1019 − 1020 eV. It is convenient to introduce the diagram ﬁrst presented by Hillas (1984) which indicates potential sites for their acceleration (Fig. 17.9). The original Hillas diagram has been adapted in the light of new astrophysical understanding since 1984. The expression for the maximum energy of a cosmic ray (17.41) can be written as follows:

E E = zeBUL ; max = eBcL , (17.45) max zβ where β = U/c. If the particles have energies ∼ 1020 eV, the necessary combination of scale L and magnetic ﬂux density B is given by 3 × 1011 = BL , (17.46) zβ where L is in metres and B in tesla. This relation is plotted in Fig. 17.10. The diagram needs to be interpreted judiciously. If potential sources lie to the left of the locus (17.45), protons cannot be accelererated to 1020 eV in these objects. The diagram indicates that there are many potential sites of particle acceleration to E ∼ 1020 eV in a wide range of high energy astrophysical objects. Note, however, that the denominator of the left hand side of (17.45) contains the factor β = U/c. Therefore, the potential sites of acceleration are only properly 636 The acceleration of high energy particles

Fig. 17.10. The combinations of length scale and magnetic flux density necessary to accelerate particles to energy 1020/zβ eV where β = U/c (after Hillas (1984)). The typical magnetic flux densities found in different classes of astronomical objects on different scales are indicated by crosses. represented on the diagram if the velocities involved are U ∼ c. This may indeed be the case for pulsar magnetospheres, γ-ray bursts, active galactic nuclei and radio jets, but it unlikely to be so in the cases of the shocks in clusters of galaxies and supernova remnants. The magnetic flux densities would have to be correspondingly stronger for acceleration to 1020 eV to be feasible. The crucial issues concern the masses and charges of the highest energy cosmic rays. If they are protons, an origin in active galaxies seems the most natural explanation, given their isotropy and correlation with relatively nearby large scale structures. On the other hand, if they are iron nuclei, energies of 1020 eV can be attained even if the velocities U are significantly less than the speed of light.

22 Compact extragalactic sources and superluminal motions

The evidence of Chap. 21 shows that the huge ﬂuxes of relativistic material needed to power extended extragalactic radio sources originate close to the active galactic nuclei of the host galaxies. Direct evidence for extreme events in active galactic nuclei is provided by the superluminal motions observed in compact radio sources, by the properties of variable extragalactic γ-ray sources and by the γ-ray bursts. The extreme properties of these sources require them to be moving at highly relativistic velocities.

22.1 Compact radio sources Direct evidence for the presence of ultrarelativistic electrons in the nuclei of active galaxies is provided by very long baseline interferometric (VLBI) studies of radio quasars and BL- Lac objects at centimetre wavelengths. Combining the angular sizes of these ultra-compact 2 radio sources with their ﬂux densities S ν, the brightness temperature Tb = (λ /2k)(S ν/Ω) of the source region can be determined, where Ω is the solid angle subtended by the radio source. Observations of large samples of strong compact radio sources with structures on the scale of 1 marcsec have shown that the the maximum brightness temperatures are of the order 1011 to 1012 K, none of them exceeding the limit of 1012 K at which catastrophic synchrotron self-Compton radiation would take place, as described in Sect. 9.6 (Kellermann et al., 1998). Brightness temperatures of the order 1011 K can be compared with the thermodynamic temperature of electrons which have kinetic energies equal to their rest mass energies, 2 9 kT ≈ mec , and so T ≈ 5 × 10 K. Assuming the emission is the incoherent radiation of high energy electrons, this argument demonstrates that the emitting electrons must have ul- 2 trarelativistic energies, E mec , since particles cannot radiate at brightness temperatures greater than their thermodynamic temperatures. Generally, the angular sizes currently mea- surable for distant quasars by VLBI techniques correspond to about 1 pc compared with 8 −5 the Schwarzschild radius of a 10 M black hole of only 10 pc. Thus, the processes of jet

745 746 Compact extragalactic sources and superluminal motions

Fig. 22.1. Illustrating the evolution of the spectrum of a synchrotron self-absorbed spherical radio source in a uniform expansion in which the energies of the particles decrease adiabatically and the magnetic flux density decreases as r−2. In this example, α = 0.75, equivalent to p = 2.5. formation probably occur on scales smaller than those currently accessible by centimetre VLBI. A characteristic signature of compact radio sources is the observation of a synchrotron self-absorption cut-off at centimetre wavelengths, as indicated schematically in Fig. 8.12. The peak of the spectrum occurs roughly at the frequency at which the brightness temperature of the radio source is equal to the effective ‘thermal’ temperature of the electrons, as explained in Sect. 8.7.1. A useful reference calculation is how the spectrum of such a synchrotron self-absorbed source evolves as it expands. In the idealised case of a uniformly expanding spherical source in which the relativistic electrons lose energy adiabatically as E ∝ r−1 and the magnetic flux density decreases as r−2 because of magnetic flux freezing, it was shown in Sect. 16.8 that the flux density of the source decreases −(p−1)/2 −2p −α −(4α+2) as S ν ∝ ν r ∝ ν r , where the electron energy spectrum has the form N(E) dE ∝ E−p and α = (p − 1)/2. Supernova remnants such as Cassopaeia A decrease in flux density at roughly this rate. A similar calculation can be carried out in the optically thick regime using the same approximate methods described in Sect. 8.7.1. The flux density of the source at frequency 2 2 ν is S ν = (2kTb/λ ) Ω where Ω ∝ r is the solid angle subtended by the source. The Lorentz factor γ of the electrons radiation at this frequency is given approximately by 2 2 ν ≈ γ νg ∝ γ B, where νg = eB/2πme is the non-relativistic gyrofrequency of the electron. As the source expands, B ∝ r−2 and so, at a fixed frequency, the electrons radiating at the 2 frequency ν must have energies proportional to r. Therefore, since kTb ∝ γmec , it follows 2.5 3 that the flux density in the optically thick regime increases as S ν ∝ ν r . This form of 22.2 Superluminal motions 747 spectral evolution is illustrated schematically in Fig. 22.1 for a spectral index α = 0.75 in the optically thin region of the spectrum. The maximum of the spectrum moves to lower −(4p+6)/(p+4) frequencies as νmax ∝ r . Different assumptions can be made about the variation of the magnetic flux density and particle energy as a function of radius. For example, the particles may be re-accelerated and the magnetic field may change as a different power of r in the expansion. These cases can be developed as straightforward generalisations of the above calculation, a number of useful examples for the expansion of a young radio supernova remnant being given by Chevalier (1998). Spectra of roughly the shape shown in Fig. 22.1 are observed in compact radio sources at high radio frequencies and are interpreted as synchrotron self-absorbed radio sources. The Gigahertz Peaked Spectrum objects (GPS) are examples of such sources. Often the observed source spectra are flat with α ∼ 0 in the centimetre waveband and these are interpreted as the superposition of synchrotron self-absorbed components within the source regions.

22.2 Superluminal motions VLBI observations show that the extended radio jets observed in FR1 and 2 radio sources have their origin in the associated active galactic nuclei, as discussed in Chap. 21. The large-scale jets can be traced back more or less continuously to aligned structures on a scale of 10 pc or less, as has been observed in sources such as Cygnus A, Virgo A (M87) and 3C31. One of the great surprises of the 1970s and 1980s was that many of the bright compact radio components observed by the VLBI technique were observed to move out from their nuclei at speeds in excess of the speed of light. One of the ﬁrst examples of such superluminal motions was observed in the quasar 3C273 in which a source component moved away from the nucleus at a speed of roughly eight times the speed of light (Fig. 22.2a, top panel) (Pearson et al., 1981, 1982). With the construction of the Very Long Baseline Array and the European VLBI network, both dedicated to VLBI studies, systematic studies of compact radio sources were carried out. More recent examples of superluminal motions in the radio quasar 3C 279 and the nearby Seyfert 1 galaxy 3C 120 are shown in Fig. 22.2a (lower panel) and 22.2b respectively. With the vastly improved imaging capabilities of these arrays, it has become clear that the typical superluminal jet consists of a series of components which travel outwards from the nucleus along more or less the same axis, as demonstrated by the example of 3C 120. Characterising the nature of the population of superluminal sources has become possible with the imaging of large statistical samples of compact ﬂat spectrum radio sources selected at high frequencies. This is a highly non-trivial problem because the sources are intrinsically variable. Cohen, Kellermann and their colleagues have carried out a major survey of these sources using the VLBA, the results of which are contained in a series of papers over a ten-year period (for detailed references, see the papers cited by Cohen (2007) 748 Compact extragalactic sources and superluminal motions

(a) (b)

Fig. 22.2. (a) The superluminal radio source 3C 273 as observed by VLBI over the period July 1977 to July 1980 (top panel) (Pearson et al., 1981, 1982). At a redshift z = 0.158, the hot spot in the radio jet moved a distance of about 25 light years over the three year period of the observation. In the lower panel, the superluminal expansion of 3C 279 is shown over a six year period (courtesy of NRAO and AUI) (b) The superluminal radio source 3C 120 showing the stream of radio components moving out from the nucleus along the same axis at superluminal speeds of 4.5 to 5 times the speed of light (Marscher et al., 2002). 22.2 Superluminal motions 749 (paper IV) and Homan (2009) (Paper VII) of the series). The sample of sources is based upon a complete survey of radio sources with flat radio spectra, α ≤ 0.5, at 5 GHz. This was then extrapolated to 15 GHz, at which frequency the major imaging survey was carried out and the observations then continued over the succeeding years to determine the kinematics of all the sources in the sample. The flux density limit of the sample was about 1.5 Jy at 15 GHz (2 cm wavelength). The first paper of the series presents images of the first 132 sources and gives a good impression of the typical structures observed (Kellermann et al., 1998). Cohen and his colleagues (2007) provide a summary of many of the key points of these observations, the sample having expanded to 225 compact radio sources observed at 15 GHz through the period 1994 to 2002. For 119 of these sources, transverse velocities were determined from multi-epoch observations, enabling velocities to be determined which satisfied their ‘excellent’ or ‘good’ criteria. Of the 199 sources, 10 are galaxies, 17 BL Lac objects and 92 quasars. Some of the key properties highlighted by Cohen and his colleagues are as follows: • The transverse velocities range from mildly relativistic motion to transverse speeds up to 30 times the speed of light and monochromatic luminosities up to 3 × 1028 W Hz−1 at 15 GHz. Because of relativistic beaming, however, this is certainly an overestimate of the luminosity in the frame of the moving component which is likely to be about two orders of magnitude less, as described below. • The sources possess a one-sided ‘core-jet ’ structure in which the core is identified with the active galactic nucleus. The core component has a flatter spectrum than the component which moves superluminally away from the nucleus. The flatness of the spectrum of the core is naturally explained by high frequency synchrotron self-absorption. • The jets consist of a series of components which move along the same trajectory from the nucleus. In the example of 3C 120 shown in Fig. 22.2b, the trajectories of four source components are shown by lines, their observed velocities lying in the range 4.5 − 5.0 c. Each of these ejections originated in an absorption event in the X-ray light curve as observed by the RXTE (Marscher et al., 2002). • Although many of the trajectories are straight, there are bends and kinks in some of the jets. As we will show, many of the jets must be travelling from the nucleus at angles close to the line of sight and so there will undoubtedly the considerable fore-shortening of the actual trajectory. Right-angled kinks might in fact consist of quite small deviations from the axis of the ejection of the source components. For example, the precession observed in SS 433 seen end on would result is dramatic kinks in the jet. • The structures of the relativistically moving components often extend in the direction of the larger scale radio jets. These properties can be naturally explained in terms of what has become the standard model for superluminal sources, many of the ideas being foreshadowed in the prescient 750 Compact extragalactic sources and superluminal motions

Fig. 22.3. A typical model for the observation of superluminal motion in compact radio sources (Marscher, 1993). The observer is located at an angle θ relative to the axis of ejection of the relativistic jet. The base of the jet is identified with the compact, flat spectrum source component and that this part of the jet also moves out relativistically from the nucleus. papers of Rees of 1967 (Rees, 1967). A typical representation of the standard model is shown in Fig. 22.3 (Marscher, 1993). In this idealised picture, the jet material propagates along the axis of the beam at speed βj = vj/c. The opening angle of the beam is φ and the angle between the axis of the beam and the line of sight to the observer is θ. The ejection events may give rise to shocks within the beam which move at speed β = v/c. These shocks may be identified with the radio source components which move down the beam axis. There will undoubtedly be a shock wave at the interface between jet and the ambient interstellar medium. Just as in the case of supernova remnants, there is a difference between the speed of the jet material vjet and the speed of the shock front v, which is often referred to as the pattern speed. The term pattern speed is used more widely that just in this context. The component may not be moving with the material which is the source of the emission. For example, a search-light beam illuminating high clouds can result in the appearance of a spot on the cloud moving faster than the speed of light if the beam is rotated fast enough. This apparent motion is the source’s pattern speed. In this case, however, there is no movement of the clouds themselves. Before studying the observations in more detail, let us first work out some of the basic properties of relativistic beaming which will be used throughout the rest of this chapter.

22.3 Relativistic beaming Let us begin with the simplest model for superluminal sources, what is often referred to as the relativistic ballistic model. The ﬁrst part of the calculation is the determination of the kinematics of a relativistically moving source component, speciﬁcally the observed 22.3 Relativistic beaming 751

Fig. 22.4. illustrating the motion of a relativistically moving source component as viewed from above. The ejection from the nucleus takes place at time t0 from O. The diagram shows the con- ﬁguration at time t1. transverse speed of a component ejected at some angle θ to the line of sight at a relativistic velocity v (Fig. 22.4). The observer is located at a distance D from the source. The source component is ejected from the origin O at some time t0 and the signal from that event sets oﬀ towards the observer, where it arrives at time t = D/c later. After time t1, the component is located at a distance vt1 from the origin and so is observed at a projected distance vt1 sin θ according to the distant observer. The light signal bearing this information arrives at the observer at time D − vt cos θ t = t + 1 , (22.1) 2 1 c since the signals have to travel a shorter distance D − vt1 cos θ to reach the observer. There- fore, according to the distant observer, the transverse speed of the component is

vt1 sin θ vt1 sin θ v sin θ v⊥ = = = . (22.2) t − t vt1 cos θ v cos θ 2 t − 1 − 1 c c

The maximum observed transverse speed is found by diﬀerentiating v⊥ with respect to θ and occurs at an angle cos θ = v/c; the maximum transverse velocity is v⊥ = γv, where γ = (1 − v2/c2)−1/2 is the Lorentz factor. Thus, provided the source component moves at a speed close enough to the speed of light, apparent motions on the sky v⊥ c can be observed without violating causality and the postulates of special relativity. For example, if the source component were ejected at a speed 0.98c, transverse velocities up to γc = 5c are perfectly feasible, the case illustrated in Fig. 22.5. The diagram shows that superluminal 752 Compact extragalactic sources and superluminal motions

Fig. 22.5. The origin of superluminal motion for a source component ejected at 0.98c from the nucleus of an active galaxy at an angle 11.5◦ to the line of sight as observed from above. Because the source of the radiation moves at a speed very close to that of light, it almost catches up with the radiation it emits. velocities are observed because the source component almost catches up with the radiation it emits. Next we need to understand the effects of what is loosely referred to as ‘relativistic beaming’ upon the observed intensities of the source components. Let us consider first a classical undergraduate problem in relativity: A rocket travels towards the Sun at speed v = 0.8c. Work out the luminosity, colour, angular size and brightness of the Sun as observed from the spaceship when it crosses the orbit of the Earth. It may be assumed that the Sun radiates like a uniform disc with a black-body spectrum at temperature T0. This problem includes many of the effects found in relativistic beaming problems. Let us work out the separate effects involved in evaluating the intensity of radiation observed in the moving frame of reference. It is simplest to use four-vectors to work out the frequency shifts and aberrations. Consider the radiation from an annulus of angular width ∆θ at angle θ with respect to the centre of the Sun (Fig. 22.6). (i) The frequency shift of the radiation The frequency four-vector in the frame of the Solar System S is given in Rindler’s notation as ω ω K = 0 , k = 0 , −k cos θ, −k sin θ, 0 , (22.3) c c 0 0 where the light rays are assumed to propagate towards the observer at the orbit of the Earth, as illustrated in Fig. 22.6†. The frequency four-vector in the frame of

† Notes on the relativistic conventions used are given in Appendix A1.8.2 22.3 Relativistic beaming 753

Fig. 22.6. Illustrating the geometry of the propagation of light from the Sun to the observer in a spaceship moving radially towards the Sun at speed v.

reference of the spaceship S0 is " # " # ω0 ω0 K0 = 0 , k0 = 0 , −k0 cos θ0, −k0 sin θ0, 0 , (22.4) c c 0 0 We use the time transform to relate the ‘time’-components of the four-vectors: vx ct0 = γ ct − , (22.5) c and so 0 ω = γ (ω0 + vk0 cos θ) . (22.6)

Since k0 = ω0/c, v ν0 = γν 1 + cos θ = κν . (22.7) 0 c 0 This is the expression for the ‘blue-shift’ of the frequency of the radiation due to the motion of the spacecraft. (ii) The waveband ∆ν, in which the radiation is observed, is blue shifted by the same factor 0 ∆ν = κ∆ν0 . (22.8)

(iii) Time intervals are also diﬀerent in the stationary and moving frames. The expression for the diﬀerence can be found by comparing the periods of the waves as observed in S and S0

0 1 1 ν = 0 ; ν0 = , (22.9) T T0 754 Compact extragalactic sources and superluminal motions and so T 0 ν = 0 . (22.10) T ν0 Since the periods T and T 0 can be considered to be the times measured on clocks, the radiation emitted in the time interval ∆t is observed in the time interval ∆t0 by the observer in S0 such that ∆t0 = ∆t/κ. (iv) Solid Angles Finally, we work out how the solid angle subtended by the annulus shown in Fig. 22.6 changes between the two frames of reference. It is simplest to begin with the cosine transform, which is derived from the transformation of x0 component of the frequency four-vector: v cos θ + 0 c cos θ = v . (22.11) 1 + cos θ c Diﬀerentiating with respect to θ on both sides of this relation,

0 0 sin θ dθ sin θ dθ sin θ dθ = = . (22.12) v 2 κ2 γ2 1 + cos θ c This result has been derived for an annular solid angle with respect to the x-axis, but we can readily generalise to any solid angle since dφ0 = dφ and so sin θ dθ dφ dΩ sin θ0 dθ0 dφ0 = dΩ0 = . (22.13) κ2 κ2 Thus, the solid angle in S0 is smaller by a factor κ2 as compared with that observed in S. This is a key aspect of the derivation of the aberration formulae. Exactly the same form of beaming occurred in the derivation of the formulae for synchrotron radiation.

We can now put these results together to work out how the intensity of radiation from the region of the Sun within solid angle dΩ changes between the two frames of reference. First of all, the intensity I(ν) is deﬁned to be the power arriving at the observer per unit frequency interval per unit solid angle from the direction θ. The observer in the spacecraft observes the radiation arriving in the solid angle dΩ0 about the angle θ0 and we need to transform its other properties to those observed in S0. Let us enumerate how the factors change the observed intensity. The energy hνN(ν) received in S in the time interval ∆t, in the frequency interval ∆ν and in solid angle ∆Ω is observed in S0 as an energy hν0N(ν0) in the time interval ∆t0, in the frequency interval ∆ν0 and in solid angle ∆Ω0, where N(ν) = N(ν0) is the invariant number of photons. Therefore, the intensity observed in S0 is κ × κ × κ2 I(ν0) = I(ν) × = I(ν)κ3 . (22.14) κ 22.3 Relativistic beaming 755 Let us now apply this result to the spectrum of black-body radiation, for which

2hν3 −1 I(ν) = ehν/kT − 1 . (22.15) c2 Then, 3 3 03 2hν κ −1 2hν 0 0 −1 I(ν0) = ehν/kT − 1 = ehν /kT − 1 , (22.16) c2 c2 where T 0 = κT. In other words, the observer in S0 observes a black-body radiation spectrum with temperature T 0 = κT. Another way of deriving the same result is to write the expression for black-body radiation as 2hν3 I(ν) = n(ν) , (22.17) c2 where n(ν) is the occupation number, which we demonstrated in Sect. 9.4.2 is a Lorentz 0 invariant. Therefore, since ν = κν0, 2hν03 2hκ3ν3 I(ν0) = n(ν0) = n(ν) , (22.18) c2 c2 as before. Furthermore, n(ν) need not be of black-body form, but applies to any occupation 0 0 number and so the brightness temperature Tb also transforms as Tb(ν ) = κTb(ν). A number of useful results follow from this analysis. For example, (22.16) describes the temperature distribution of the Cosmic Microwave Background Radiation over the sky as observed from the Solar System, which is moving through the frame of reference in which the sky would be perfectly isotropic on the large scale at a velocity of about 600 km s−1. Since v/c ≈ 2 × 10−3 and γ ≈ 1, the temperature distribution is rather precisely a dipole distribution, T = T0[1 + (v/c) cos θ] with respect to the direction of motion of the Solar System through the Cosmic Microwave Background Radiation. In the example of the spacecraft travelling at v = 0.8c towards the Sun, we can illustrate a number of the features of relativistic beaming. For example, if the motion of the spacecraft is significantly relativistic, the dipole pattern is distorted and the zeros of the polar diagram are shifted in the direction of motion. If v = 0.8c, γ = 5/3 and the angle at which there is no change of temperature, corresponding to γ[1 + (v/c) cos θ] = 1, is θ = 60◦. For the case of relativistically moving source components, all we need do is determine the value of κ for the source component moving at velocity v at an angle θ with respect to the line of sight from the observer to the distant quasar as illustrated in Fig. 22.4. In this case, a straightforward calculation shows that the value of κ is 1 κ = , (22.19) v cos θ γ 1 − c 756 Compact extragalactic sources and superluminal motions where the source is moving towards the observer as illustrated in the figure. As in the above example, the observed flux density of the source is therefore L(ν ) S (ν ) = 0 × κ3 , (22.20) obs 4πD2 where νobs = κν0. In the case of superluminal sources, the spectra can often be described −α by a power-law L(ν0) ∝ ν0 and so L(ν ) S (ν ) = 0 × κ3+α . (22.21) 0 4πD2 It is interesting to compare this result with the expressions used in the case in which the sources are at cosmological distances. In that case, we have to use the appropriate distance measure D to describe how the radiation spreads out over a sphere in the curved geometry of space (see, Longair (2008)). In that case, L(ν ) S (ν ) = 0 × κ1+α , (22.22) 0 4πD2 where κ = (1 + z)−1. Notice the difference of a factor of κ2 between these formulae. The reason is that, in the cosmological case, the radiation is emitted isotropically over a sphere centred on the source, whereas in the case of the moving source component, the radiation is beamed towards the observer, the change of solid angle being given by κ−2. If the superluminal sources consisted of identical components ejected from the radio source at the same angle in opposite directions, the relative intensities of the two components would be in the ratio  v 3+α 1 + cos θ  S 1  c  =  v  . (22.23) S 2 1 − cos θ  c It is therefore expected that there should be large differences in their observed intensities. For example, adopting the largest apparent velocities for a given value of γ, cos θ = v/c, in the limit v ≈ c, S 3+α 1 = 2γ2 . (22.24) S 2 Since values of γ up to 30 are observed and α ∼ 0 − 1, it follows that the approaching component would be very much more luminous than the receding component. It is, therefore, not at all unexpected that the sources should be one-sided. The above analysis has treated the source components as identical point sources ejected in opposite directions from the nucleus. In fact, the source components are better represented by relativistic jets, as illustrated in Fig. 22.3. This slightly modifies the above analysis. Suppose the superluminal source consists of two identical uniformly emitting jets moving in opposite directions from the nucleus. Let us consider the luminosity of a 22.3 Relativistic beaming 757 section of the jet of length ∆x0 on either side of the nucleus. Then, the luminosity of these sections of the jet is L0 = ∆x0µ where µ is the luminosity per unit path length in the frame of the jet. The increment of length ∆x0 is observed in the observer’s frame as an increment ∆x = κ∆x0, as can be appreciated from the example shown in Fig. 22.5. Thus, the enhancement of the luminosity per unit path length is a factor of κ less than that given by (22.21), 2+α 2+δ 3+δ 2 S ∝ κ rather than S ∝ κ . In the same limit as in (22.24), S 1/S 2 = 2γ . Thus, the precise form of the relativistic beaming factor is model dependent. Often the relativistic beaming factor is written κδ where δ = α+n with n ∼ 2−3. It should also be noted that the external observer observes the source components at different proper times in their their own rest frames. Therefore, if the jets were time variable and evolved in exactly the same way, the intrinsic luminosities of the components observed at a fixed time in the observer’s frame would be different. We can now revisit some aspects of the jet model shown in Fig. 22.3. The jet is assumed to be relativistic right down to the base of the jet which, on compact enough scales, becomes synchrotron self-absorbed. Thus, the compact core itself is also relativistically beamed and this accounts for the frequency with which compact cores as well as relativistic jets are observed. In a few extreme blazars, it is assumed that the jet is observed more or less precisely along the line of sight, but despite the brightness of the nucleus, high dynamic range synthesis imaging has enabled the underlying double radio sources to be observed. A good example is the blazar 3C 371, which looks to all appearances like an FR2 radio source observed end-on (Fig. 18.15). Since the superluminal jets are pointing at a small angle to the line of sight, foreshort- ening effects are important in understanding the physical structure of the jets. A good example is the observed half-angle of the jet, φ in Fig. 22.3. If the jet is observed to have half-angle φobs, then the intrinsic opening half-angle is φ = φobs tan θ, where θ is the angle between the axis of the source and the line of sight. Marscher (1993) showed that, in the cases of the sources NRAO 140 and 3C 345, the intrinsic opening angles of the jets must be only φ ≈ 1.5◦ to 2◦. Thus, the relativistic beams must already be very well collimated indeed within a distance of only 1 − 10 pc from the nucleus. Bends and kinks in the jets are also strongly foreshortened and so, although the angles may look large on radio maps, when deprojected by the tan θ factor, the bends in the jets are quite small. There are similar deviations from perfect symmetry in the large-scale angular distribution of hot spots of about 7◦ − 10◦ (Best et al., 1995) and such deviations can probably encompass even the largest deviations observed in superluminal sources. While VLBI provides the bulk of the imaging evidence for superluminal motions, the jet in M87 seen in Fig. 3.2a is an example of an FR1 radio source in which relativistic motions have also been measured by optical imaging. VLA and VLBI observations showed that the radio jet moves out from the nucleus at about (0.28 ± 0.08)c, while knot A, the brightest feature in the jet, has a projected velocity of (0.52 ± 0.03)c. Optical observations by the 758 Compact extragalactic sources and superluminal motions

Fig. 22.7. The inner regions of the jet of M87 as observed by the Hubble Space Telescope through the period 1992 to 1998 (Biretta et al., 1999). The optical knots are observed to move out from the nucleus at speeds up to 6c.

Wide Field Camera of the Hubble Space Telescope showed motions of the knots in the jet in the range 0.5 − 1c, while the outermost knot has a velocity of only 0.1c (Biretta et al., 1995). More recently, Biretta and his colleagues carried out optical imaging of features in the jet very close to the nucleus and found large superluminal velocities (Fig. 22.7). There are 10 superluminal features within the first 6 arcsec of the jet, eight of them having apparent speeds in the range 4c−6c (Biretta et al., 1999). These observations are consistent with the picture of FR1 radio galaxies discussed in Sect. 21.3, in which it was asserted that such sources have relativistic jets close to the nucleus which are decelerated to mildly relativistic velocities on their outward passage through the host galaxy . 22.4 The superluminal source population 759 22.4 The superluminal source population Understanding the nature of the superluminal source phenomenon in compact extragalactic radio sources requires the systematic study of complete samples of such sources over a long time interval. As discussed in Sect. 22.2, this has now been achieved through the major surveys undertaken by Cohen, Kellermann and their colleagues (Cohen et al., 2007). From the flux density limited survey described in that Section, they assembled a statistical sample of 119 superluminal core-jet sources, most of which had been observed over a ten-year period and for which ‘excellent’ or ‘good’ structures had been determined. The maximum observed velocity βapp = vapp/c for each source is plotted against observed luminosity L in Fig. 22.8. The symbols distinguish between quasars, BL-Lacs objects and galaxies, as described in the figure caption. It is a non-trivial exercise to work out all the selection effects which determine which sources appear on Fig. 22.8 and even more of a challenge to interpret what the diagram is telling us. First of all, the sample is flux-density limited but the luminosities which appear on the diagram have been greatly enhanced from their intrinsic values by the effects of relativistic beaming, which depend upon both the Lorentz factors γ of the sources and the angles θ of their axes to the line of sight – both of these are unknown for any given source. Second, there are limits to the observable values of the proper motions of the source components. If they were too great, they would not be detected in the 10 years of the survey. The line K in Fig. 22.8a indicates an upper limit of 4 marcsec year−1. Sources in the hatched area to the left of K would not be observable in the flux density limited sample. To intepret the diagram, we need the pair of equations (22.2), (22.17) and the expression for the observed luminosity of the superluminal source: v sin θ 1 β c = v = ; κ = ; L = κδ L , (22.25) app ⊥ v cos θ v cos θ obs 0 1 − γ 1 − c c where δ = α + n and n ∼ 2 − 3. The solid line on Fig. 22.8a shows the locus corresponding 25 −1 to an intrinsic luminosity L0 = 1.0 × 10 W Hz and a Lorentz factor γ = 32 as the angle θ is varied from 90◦ at the left hand end of the line to 0◦ at the right hand end. It can be seen that this locus provides a bounding curve within which all the superluminal sources are observed. Sources with these intrinsic properties are expected to lie along this locus, but the probability of the extreme high and low values being observed is very small, because they correspond to small solid angles on the sphere. Cohen and his colleagues carried out a number of simulations to determine within what ranges of observed luminosities there would be a 92% chance that the source is observed, the truncation criteria being that the cumulative probabilities should lie between 4% at low luminosities and 96% at high luminosities. In Fig. 22.8b, the same curve with L0 = 1.0 × 1025 W Hz−1 and a Lorentz factor γ = 32 is shown, but with these truncation values 760 Compact extragalactic sources and superluminal motions

Fig. 22.8. (a) A plot of apparent transverse speed, βapp = vapp/c, against apparent luminosity, L, for the fastest component in each of 119 core-jet sources in the 2 cm VLBA survey (Cohen et al., 2007). 25 −1 The aspect curve is the locus of (βapp, L) for sources with γ = 32 and L0 = 1.0 × 10 W Hz , as θ −1 varies. Curve K is an observational limit for S VLBA, med = 0.5 Jy and µ = 4 mas yr ; sources in the hatched region are usually unobservable. The horizontal lines are the minimum values of redshift, −1 zmin(βapp), for which the angular velocity is below the limit µ = 4 mas yr . The vertical lines are the maximum values of redshift, zmax(L), for which the ﬂux density is greater than the limit, S = 0.5 Jy. Open circles are quasars, full circles are BL Lac objects and triangles are galaxies. (b) The same data as in Fig. 22.8, but with lines indicating the loci of sources with ﬁxed values of β and L for values of θ within which there is a 92% chance of the sources being observed. 22.4 The superluminal source population 761 included. It can be seen that the most likely values are quite closely clustered about the maximum observed luminosity and maximum apparent velocity – this rule holds good for the other truncation curves illustrated in Fig 22.8b. The reason for this is that the maximum enhancement takes place for angles close to the value cos θ = v/c at which the greatest value of v⊥ is observed. These curves, referred to as aspect curves by Cohen and his colleagues, are useful tools in interpreting the distribution of sources in Fig. 22.8. For example, there is a real limit to the values of apparent luminosity at high values of apparent transverse speed, which is not the result of observational selection. Cohen and his colleagues (2007) draw the following conclusions from their analysis:

• About half of the sources with βapp > 4 in a ﬂux density limited sample have values of γ within 20% of the value of βapp. 25 −1 • The aspect curve with L0 = 1.0 × 10 W Hz and a Lorentz factor γ = 32 forms an envelope within which all the sources in the ﬂux density limited sample lie. From probability arguments, the maximum Lorentz factor is about 30 and the peak intrinsic 26 −1 luminosity L0 = 1.0 × 10 W Hz .

• There is an observed correlation between βapp and L for the jets of quasars. High values of βapp are only found in radio jets with high values of L. This implies a corresponding correlation between γ and L0. In other words, high values of γ must preferentially exist in jets with high values of L0. • The exponent n for a typical source must be less than 3 or else the highly luminous jets with the fastest superluminal speeds would have intrinsic luminosities comparable to those of slow nearby galaxies which are observed to have only mildly relativistic jets. If n = 2, the largest values of γ would be assocaited with intrinsically much more luminous jets. This value of n is consistent with the assumption that the superluminal sources are continuous jets with n ∼ 2.

• According to probability arguments, there are too many low-speed (βapp < 3) quasars in the sample. Cohen and his colleagues argue that some of these may have pattern speeds substantially lower than their beam speeds. • The galaxies have a distribution of Lorentz factors up to γ = 7. Three show strong superluminal motion, but most are only mildly relativistic. They argue that these are not ‘oﬀ-axis’ versions of the powerful quasars. • The pattern and beam speeds must be approximately equal for the fastest components in many sources.

It is a major theoretical challenge to account for the formation of jets moving at ultrarelativistic speeds from the nuclei of active galaxies. An attractive possibility is the development of the electromagnetic model of Blandford and Znajek illustrated in Fig. 21.14a. In a series of papers, Lovelace and his colleagues have developed the model taking account of the twisting up of the magnetic ﬁeld lines which are tied to the accretion disc about the 762 Compact extragalactic sources and superluminal motions

Fig. 22.9. A sketch of the magnetic field configuration of a Poynting jet from Lovelace and Ro- manova (2003). The lower part of the figure shows the initial dipole-like magnetic field threading the accretion disk which rotates at the angular rate Ω(r). The upper part of the figure shows the jet at some time later when the head of the jet is at a distance Z(t). At the head of the jet there is pressure balance between the electromagnetic stress of the jet and the ram pressure of the ambient medium of density %ext. black hole - a brief summary of their results is given in the paper by Lovelace and Ro- manova (2003). As they show, the twisting of the magnetic field threading a differentially rotating accretion disk extracts angular momentum and energy from the disk magnetically. If there is significant mass flux in the outflow, the energy and angular momentum are carried outwards hydrodynamically by both the matter and the electromagnetic field. If the mass flux is negligible, energy and angular momentum are carried predominantly by the electromagnetic field, what are referred to as Poynting outflows. The structure of the Poynt- ing outflow is illustrated schematically in Fig. 22.9. Lovelace and Romanova show that the speed of advance of the jet is given by the expression ! ! 10 1/3 B 1/3 106 m−3 1/6 γ ≈ 8 0 , (22.26) < 0.1 T next

2 where < = r0/rg 1, r0 is the radius of the O-point of the magnetic ﬁeld, rg = GM/c , and B0 is the magnetic ﬂux density at the center of the disk. As expressed by Lovelace and Romanova, 22.5 Synchro-Compton Radiation and the Inverse Compton Catastrophe 763

As the ‘twist’, as measured by Ω0t, increases, a high twist field configuration appears with a different topology. A plasmoid consisting of toroidal flux detaches from the disk and propagates outward. The plasmoid is bounded by a poloidal field line which has an X-point above the O-point on the disk. The occurrence of the X-point requires that there be at least a small amount of dissipation in the evolution from the poloidal dipole field and the Poynting jet configuration.

A key feature of the model is that the mass density within the jet should be low. In the picture described by Lovelace and Romanova, the mass in the jet consists of electron- positron pairs. The attraction of the model is that it results in highly relativistic collimated ﬂows which originate in the inner regions of the accretion disc.

22.5 Synchro-Compton Radiation and the Inverse Compton Catastrophe Synchro-Compton radiation and the inverse Compton catastrophe were analysed in Sect. 9.6 where it was shown that, if the brightness temperature of the radio source exceeds 1012 K, the radio source loses energy catastrophically by synchrotron self-Compton radiation. The sample of core-jet sources discussed in Sects. 22.2 and 22.4 have been analysed by Ko- valev and his colleagues to determine the maximum brightness temperatures within their compact cores (Kovalev et al., 2005). The analysis involved determining the visibility functions for the compact cores in the sources, both along and perpendicular to the axis of jet directions. Not all the sources were resolved in both directions and so, in a number of cases, the measurements provide lower limits to the brightness temperatures. The median value of the observed brightness temperatures are about 1012 K with the values extending up to at least 5 × 1013 K. Fig. 22.10 shows a plot of the apparent velocities of the superluminal cores plotted against their maximum brightness temperatures. The largest brightness temperatures tend to be associated with the largest values of apparent velocity. It is natural to associate the largest brightness temperatures with the effects of relativistic beaming. The solid line in Fig. 22.10 is the relation Tb, max = κmaxTint ≈ βmaxTint derived 11 in Sect. 22.3, the intrinsic brightness temperature being taken to be Tint = 10 K. Thus, brightness temperatures greater than 1012 K can be accommodated assuming the cores are relativistically beamed. The same argument reappears in a slightly different guise from studies of the time variability of the source components. The most compact blazars show variability on a time- scales ∆t less than a day and so, according to the causality argument, the dimensions of the source would have to be less than c ∆t. If, however, the source component is moving towards the observer at a relativistic velocity, the time scale in the frame of reference of the source is ∆t0 = κ∆t and so the causality constraint is relaxed. An additional factor of κ2 therefore appears in the expression for the brightness temperature deduced from the flux density and time variability of the source,

3 Tb, obs = κ Tint . (22.27) 764 Compact extragalactic sources and superluminal motions

Fig. 22.10. Apparent velocity, βmax plotted against maximum core brightness temperature, Tb, max for the most compact components in core-jet superluminal sources for which ‘excellent’ or ‘good’ structures had been determined. Values of βmax are taken for the fastest components of these sources. Lower limits to the brightness temperatures are plotted as arrows. The plotted curve is Tb, max = 11 βmaxTint, where the intrinsic brightness temperature is taken to be Tint = 10 K.

It should be emphasised that exact form of the relativistic beaming ‘correction’ is somewhat model-dependent, but the sense of the changes can be appreciated from Fig. 22.5. The same type of reasoning reappears in studies of the highly variable γ-ray sources. Fig. 22.11 shows the histogram of the brightness temperature distributions of two large samples of core-jet sources, the grey-scale coding distinguishing between those sources which display strong inter-day variability, weak inter-day variability and no inter-day variability (Kovalev et al., 2005). It is apparent that the strong variable sources are those which have high brightness temperatures as observed by VLBI, but their distrbution is not so diﬀerent from the majority of sources in the sample. On the other hand, Kovalev and his colleagues ﬁnd that the interday variable sources are more compact and have more core dominant structure on sub-milliarcsecond scales than the non-variable sources.

22.6 γ-ray Sources in Active Galactic Nuclei One of the key discoveries of the Compton Gamma-Ray Observatory (CGRO) was the observation of ultraluminous, extragalactic γ-ray sources at energies ε ≥ 100 MeV. Fig. 22.12 shows the distribution of point γ-ray sources from the revised catalogue of EGRET γ-ray sources of Casandjian and Grenier (2008) in a Hammer-Aitoﬀ projection. There are 188 22.6 γ-ray Sources in Active Galactic Nuclei 765

Fig. 22.11. Distributions of the maximum brightness temperature Tb, max for interday variable (IDV) selected and non-selected sources. The gray-scale coding shows the distributions for source which display strong inter-day variability, weak variability and no variability (Kovalev et al., 2005). sources in the revised catalogue derived from the complete dataset obtained during the lifetime of the Compton Gamma-ray Observatory. The figure caption shows that the bright sources comprise a variety of high energy astrophysical objects, including γ-ray pulsars, blazars, flat spectrum radio sources and supernova remnants; for 87 of the sources, identifications have not been made. A striking feature of the early observations made by the CGRO was that 24 of the brightest extragalactic sources were all associated with radio- loud, flat-spectrum radio sources, many of which exhibited superluminal motions (Michel- son, 1994). Fig. 22.12 confirms that the majority of the extragalactic sources are associated with blazars, many of which exhibit superluminal motions. These findings have been confirmed and greatly enhanced with the launch of the Fermi Gamma-ray Space Telescope. Observations from the first three months of observation showed that the telescope has achieved much greater sensitivity than the CGRO with improved angular resolution. During this period, 132 bright sources were detected at |b| > 10◦ at better than the 10σ sensitivity level. 106 of these sources form the Large Area Telescope Bright Active Galactic Nuclei (AGN) Sample (LBAS). There are only two radio galaxies Centaurus A and NGC 1275 but 104 blazars including 57 flat spectrum radio quasars, 42 BL Lac objects, and 5 blazars with uncertain classification (Abdo et al., 2009b). All the blazars are variable γ-ray sources and are detected as radio sources. Kovalev and his colleagues (2009) monitored the radio properties of this sample with the VLBA almost contemporaneously and found that the γ-ray flux density is strongly correlated with the compact radio flux density. In particular, the γ-ray detected sources in the radio-selected complete sample generally have higher compact radio flux densities and their parsec-scale 766 Compact extragalactic sources and superluminal motions

Fig. 22.12. The revised EGRET source catalogue in Galactic coordinates (Casandjian and Grenier, 2008). Identiﬁed pulsars are shown as black squares; other ATNF pulsars as open squares; blazar candidates as black diamonds; other ﬂat-spectrum radio-sources as open diamonds; supernova remnants as stars; no counterpart as crosses.

cores have higher brightness temperatures than the jets in non-detected objects. Kovalev and his colleagues inferred that the jets of bright γ-ray active galactic nuclei are associated with the more extreme superluminal sources. The radio jets are found to be in a more active radio state within several months of their detection as strong γ-ray sources. These results are consistent with a jet model in which the γ-ray and radio emission originate from the same collimated beam, as illustrated in Fig. 22.3. The spectra of the powerful γ-ray sources show that most of the energy liberated by these objects is emitted in the γ-ray region of the spectrum. The γ-ray luminosities can be written in terms of a beaming factor f , which describes the fraction of the celestial sphere over which the source emits its observed intensity. Typically, the γ-ray luminosities amount to about 1041 f W, with a few examples up to ten times greater than this value. These are enormous luminosities, if the sources were to radiate isotropically, that is, if f ∼ 1. Fermi Large Area Telescope observations of the quasar 3C 454.3 provide a good example of the properties of these strong γ-ray sources (Abdo et al., 2009a). Over the period July 7-October 6 2008, the γ-ray emission at energies greater than 100 MeV was highly variable (Fig. 22.13a) Fig. 22.10b shows the non-thermal γ-ray spectrum of the source, while Fig. 22.10c shows the energy spectrum of the source from radio to γ-ray energies. 3C 454.3 is a well know superluminal source with relativistic beaming factor κ ≈ 25. It is apparent that the γ-ray emission originates in the same relativistic jets which give rise to the observation superluminal velocities. 22.6 γ-ray Sources in Active Galactic Nuclei 767

Fig. 22.13. The γ-ray properties of the blazar 3C454.3 showing (a) the variability of the γ-ray source, (b) the γ-ray spectrum and (c) the broad-band energy spectrum from radio to γ-ray energies (Abdo et al., 2009a)).

A further argument concerns the enormous energy densities in γ-rays which must be present in these compact variable sources. The sources are also intense hard X-ray emitters and so the γ-rays lose energy in photon-photon collisions, leading to the creation of electron-positron pairs. The result of this process is the formation of what Blandford (1994) calls a ‘γ-ray photosphere’ about the nucleus – the γ-rays we observe cannot originate from within this region. A key role is played by the compactness parameter which determines whether or not the source is opaque for γγ collisions for electron-positron pair production. For simplicity, 2 consider the ﬂux of γ-rays at the threshold for electron-positron pair production, ε ∼ mec . −1 The mean free path for γγ collisions is λ = (Nγσ) , where Nγ is the number density of 2 2 −15 photons with energies ε = hν ∼ mec and σ ≈ mere , re = 2.8 × 10 m being the classical electron radius. If the source has luminosity Lγ and radius r, the number density of photons 768 Compact extragalactic sources and superluminal motions within the source region is

Lγ N = . (22.28) γ 4πr2cε The condition for the source to be opaque is r ≈ λ, that is,

2 2 4πr cmec Lνσ r ≈ , or 3 ∼ 1 . (22.29) Lνσ 4πmec r The compactness factor C is deﬁned to be the quantity

Lνσ C = 3 . (22.30) 4πmec r Sometimes the compactness parameter is defined without the factor of 4π in the denominator. If C is much greater than unity, the γ-rays are all degraded by electron-positron pair production, resulting in a huge flux of electrons and positrons within the source region. Consequently, the source would no longer be a hard γ-ray source. The significance of the compactness parameter can be appreciated for a γ-ray source with isotropic lumi- 41 nosity Lγ ∼ 10 W which is observed to vary on the time scale of a day. Inserting these values into (22.30), C ∼ 103 1 and so there is a problem in understanding why such sources should exist. Relativistic beaming again comes to the rescue. Because of relativistic beaming, the observed luminosity is enhanced by a factor κ2+α and, in addition, because the time-scale of variability appears on the denominator of (22.27), the observed value is shorter by a factor κ and so the compactness parameter is increased by relativistic beaming by a factor of roughly κ4+α. Since α ≈ 1, C ∝ κ5. Notice again that the exact dependence upon κ is model dependent. Therefore, if γ ≥ 4, in the frame of the source components themselves, the value of the compactness parameter can be reduced below the critical value C = 1, resolving the problem of the compactness parameter. From their more detailed analysis, Abdo and his colleagues (2009a) infer that, taking C ∝ κ4 the relativistic beaming factor has to be greater than κ = 8. The favoured model is one in which the source of the γ-ray emission is the same relativistic jet which is responsible for the radio jet but, because of the compactness of the source, the emission probably originates much closer to the nucleus, in the regions in which the radio core is synchrotron self-absorbed, the region labelled ‘core’ as in Fig. 22.3. Blandford (1994) argues that the most likely region in which the γ-rays are produced is at the point where the relativistic jet has just emerged from the γ-ray photosphere, which can be taken to be the region at which C ≈ 1. The emission process is probably some version of inverse Compton scattering in which low energy photons are boosted to enormous energies by the high energy electrons present in the relativistic beam. The source of the photons which are scattered to high energies is a matter of speculation but various possibilities have been suggested. For example, the photons may be created in the inner regions of an accretion disc about a supermassive black hole. Alternatively, the photons from the accretion disc may 22.6 γ-ray Sources in Active Galactic Nuclei 769

Fig. 22.14. The spectral energy distribution of the blazar PKS 2155-304 observed by the Fermi Gamma-ray Space Telescope, the HESS atmospheric Cherenkov array , the ATOM telescope and the RXTE and Swift observatories derived from simulataneous observations through the period 25 August and 6 September 2008 (Aharonian et al., 2009). The solid line is a one zone synchro-self Compton model due to Band and Grindley (1985). The very high energy region of the spectral energy distribution is absorbed because of photon-photon interactions with extragalactic background radiation.

be scattered into the beam by Thompson scattering by free electrons in the environment of the nucleus. A third possibility is that the soft photons are created within the relativistic jet itself. The synchro-self Compton scattering models can give a reasonable ﬁt to the extremely broadband observations made by combining observations made with the Fermi Gamma- Ray Space Telescope, the HESS atmospheric Cherenkov Array and telescopes operating at X-ray and optical wavelengths. An example of this is shown in Fig. 22.14 of the blazar PKS 2155-304 (Aharonian et al., 2009). The broad band spectral energy distribution shows two broad peaks which are interpreted in terms of the models developed by Band and Grindley (1985), examples of which were discussed in Sect. 9.6 and illustrated in Fig. 9.15. To match the observations, it is assumed that both radiation peaks are relativistically beamed with a bulk beaming factor κ = 32. The radius of the emitting region is 0.05 pc and so the radiation originates from the inner regions of the jet. At the very highest TeV energies, absorption 770 Compact extragalactic sources and superluminal motions

Fig. 22.15. The integral flux density for energies ε ≥ 200 GeV observed from PKS 2155-304 on MJD 53944 versus time as observed by the HESS Cherenkov array telescope (Aharonian et al., 2007a). The horizontal line represents the corresponding flux density observed from the Crab Nebula. Five separate bursts have been identified. by γγ interactions of the γ-rays with the extragalactic optical and infrared background radiation result in an exponential cut-off to the observed spectrum. The observations discussed above referred to the blazar PKS 2155-304 in its quiescent state but, during periods of activity, it is the most highly variable TeV γ-ray source yet observed. The outburst observed by the HESS telescope array in July 2006 was the most dramatic seen from any TeV γ-ray source (Aharonian et al., 2007a). The light curve of the outburst of July 2006 reached two orders of magnitude greater flux densities than those observed in the ‘quiescent’ phase and was an order of magnitude greater than the flux density of the Crab Nebula (Fig. 22.15). Variability was observed on the timescales of a few minutes, the best measured individual flare having a rise-time of only 173 ± 23 s. Such rapid variability would require an extremely large Doppler factor. Aharonian and his colleagues argue that the host galaxy is a luminous galaxy and so is likely to host a 9 black hole with mass M ∼ 10 M . According to (14.8), the time scale associated with the Schwarzschild radius of such a black hole is 104 s. Consequently, a relativistic beaming factor of κ ≥ 60 would be required to account for the rapid changes in intensity. A similar limit would be required to account for the extreme values of the compactness parameter. A survey of recent observations of extragalactic TeV γ-ray sources detected by ground- based γ-ray Cherenkov telescopes is given by Hinton (2009), who provides a useful list of currently active telescopes as well as a list of 19 active galactic nuclei from which TeV γ-rays have been detected. Most of the sources have redshifts z . 0.2 because of the cut-off due to γγ interactions with the extragalactic background radiation. Evidence that the cut-off is indeed due to this process is provided by the steepening of the TeV γ- 22.7 γ-ray bursts 771 ray spectra at redshifts z & 0.1, although there is considerable dispersion in the spectral indices. The sensitivity of the detector arrays has, however, improving steadily. The recent detection of the flat spectrum radio quasar 3C279 at a redshift z = 0.536 by the MAGIC collaboration represents a major advance (Berger et al., 2009). These observations can be used to set significant limits to the extragalactic background intensity in the optical and infrared regions of the spectrum as illustrated by the analyses of Aharonian and his colleagues (Aharonian et al., 2006).

22.7 γ-ray bursts 22.7.1 The discovery of γ-ray bursts The discovery of γ-ray bursts was reported in a paper by Klebesadel, Strong and Olson of the Los Alamos National Laboratory in 1973, six years after the ﬁrst of these events had been discovered by the Vela surveillance satellites (Klebesadel et al., 1973). The delay in publication was not caused by security considerations, but rather because the authors wished to be certain that the γ-ray bursts were genuinely astronomical phenomena. The bursts lasted from a fraction of a second to several minutes (Fig. 22.15) and during which time the γ-ray bursts were the most luminous objects in the γ-ray sky. The Compton Gamma-ray Observatory (CGRO) included a detector system speciﬁcally designed to de- tect and locate γ-ray bursts, the Burst and Transient Source Experiment (BATSE). During the 1990s, this instrument detected 2704 bursts and established a number of their key properties:

• The γ-ray bursts are uniformly distributed over the sky (Fig. 22.16). • The number counts of the bursts show departures from those expected of a uniform Euclidean distribution of sources, suggesting a cosmological distribution of the sources. • The distribution of γ-ray burst durations is bimodal: bursts which last less than 2 seconds being termed ‘short’ and those of greater duration ‘long’ (Fig. 22.17).

The γ-ray bursts were of such short duration and the angular resolution of the BATSE instrument only a few degrees that there was not time nor adequate positional accuracy to make secure identiﬁcations of the sources. The solution to the problem was stimulated by a theoretical paper by Mesz´ aros´ and Rees (1993). They argued that, if the γ-ray bursts were extragalactic phenomena, the energy densities during the event would be so extreme that a relativistic shock wave would be created in which electrons would be accelerated to very high energies. These electrons would emit synchrotron radiation with a power- law spectrum, consistent with the observed spectra of the bursts. Although the intense γ-ray emission lasts only a short time at γ-ray energies, the emission would be expected to last much longer at lower energies – an afterglow was predicted which would appear successively in the X-ray, optical, infrared and radio wavebands. 772 Compact extragalactic sources and superluminal motions

Fig. 22.16. The distribution of 2704 γ-ray burst over the sky as observed by the BATSE experiment of the CGRO (Courtesy of NASA, G.J. Fishman and the CGRO Science Team).

Fig. 22.17. The distribution of γ-ray burst durations from the bursts observed by the BATSE experiment of the CGRO showing the bimodality between the short and long bursts (Courtesy of NASA, G.J. Fishman and the CGRO Science Team). 22.7 γ-ray bursts 773 Searches for the afterglows of γ-ray bursts were carried out by Italian-Dutch BeppoSAX satellite. In 1997, the X-ray telescope was pointed at the position of the γ-ray burst GRB 970228 within 8 hours of the event having taken place and its X-ray afterglow discovered (Costa et al., 1997). Subsequently, afterglows were observed throughout the electromagnetic spectrum. These observations enabled a precise position for the burst to be determined and the association of this burst with a very faint galaxy established by observations with the Hubble Space Telescope (Sahu et al., 1997). This observation conﬁrmed that the γ-ray bursts form a population of extragalactic objects. Within four years, afterglows from over 40 γ-ray bursts had been detected and 30 of these were associated with a distant host galaxy. A supernova-γ-ray burst association was suggested by the coincidence of the SN1998bw with the γ-ray burst GBR980425 by Galama and his colleagues (Galama et al., 1998). This association was established with certainty for the γ-ray burst GRB 030329 observed on 29 March 2003 which was observed optically and showed the characteristic broad lines of an extremely energetic supernova were observed within days of the event (Hjorth et al., 2003).

The pace of discovery has increased greatly with observations from the HETE-2, SWIFT and FERMI γ-ray telescopes which have detector systems designed speciﬁcally to enable very rapid follow-up of γ-ray bursts to be made. SWIFT, in particular, has greatly increased the numbers of γ-ray bursts for which secure identiﬁcations have been made, both for the long and short bursts. The feature of SWIFT which made this possible is the automatic repointing of the telescope when a γ-ray burst occurs so that within two minutes of the event occurring the onboard X-ray and optical-UV telescope can observed it. SWIFT has measured about 100 γ-ray bursts per year Since its launch in 2005, 380 bursts had been detected by December 2008, for 126 of which redshifts had been measured. Most of the redshift measurements are for ‘long’ bursts, but 33 short bursts have been measured and for eight of these redshifts have been measured.

The γ-ray bursts are so luminous that they can be identiﬁed in extremely distant galaxies. The gamma-ray burst GRB 090423, discovered by the SWIFT γ-ray telescope, has +0.07 the largest redshift of any known object, z = 8.26 −0.08 (Tanvir et al., 2009). This redshift can be compared with those of the most distant spectroscopically conﬁrmed galaxy (z = 6.96) and quasar (z = 6.43). Cosmologically, these objects therefore provide probes of the reionisation era when the intergalactic gas was heated and reionised by ultraviolet radiation from the earliest generations of stars in galaxies. As expressed by Gehrels and his colleagues (2009) in their comprehensive review, ‘(γ-ray bursts) are now rapidly becoming powerful tools to study the detailed properties of the galaxies in which they are embedded and of the Universe in general’. 774 Compact extragalactic sources and superluminal motions

Fig. 22.18. Illustrating a number of the properties of γ-ray bursts, distinguishing between the long (L) and short (S) bursts (Gehrels et al., 2009). UL stands for under-luminous. The meanings of the various quantities are explained in the text.

22.7.2 The properties of γ-ray bursts With the greatly expanded sets of high quality observations from the present generation of γ-ray telescopes, a vast amount of detailed information is now available. The principal distinction remains between the long and short bursts, as is vividly demonstrated by the correlation diagrams shown in Fig. 22.17 from the review by Gehrels and his colleagues (2009). The main features of these diagrams are as follows: • At cosmological distances, the energy releases of the γ-ray bursts are enormous. It is conventional to write the energy emitted as E = Eiso(Ω/4π), where Ω is the solid angle within which the energy is emitted. If the radiation were to be emitted isotropically, Ω = 4π and so E = Eiso. The diagrams make the point that the most energetic γ-ray 47 bursts have energies E = Eiso up to about 10 J, corresponding to the rest mass energy 22.7 γ-ray bursts 775

of 0.5 M . However, the radiation is likely to be strongly beamed and so the energy requirement can be very signiﬁcantly reduced.

• The overall time-scale of the bursts are parameterised by the quantity T90 which is defined to be the time within which 90% of the energy is liberated. The short bursts are significantly less energetic than the long bursts. The conventional picture is that the bimodality is associated with different types of progenitors for the long and short bursts. • There is a difference in the γ-ray spectra of the short and long bursts, the short bursts having somewhat hard (flatter) spectra than the long bursts. • The rise times of the long bursts can be as short as 100 ms and of the short bursts as short as 10 ms. The decay times are somewhat longer than the rise times. • The variability or ‘spikiness’ of the burst is correlated with the energy or peak luminosity of the burst. • For the long bursts, there is a time-lag between the burst being observed at high and low γ-ray energies. For the SWIFT observations, the energy bands are defined to be 50-100 keV (high energy) and 15-25 keV (low energy) channels. In contrast, no such lags are observed for the short bursts.

In addition to these differences, the host galaxies are different. The long bursts are associated with star-forming galaxies, with typical luminosities L ≈ 0.1 L∗ and metallicities only about 10% of the solar value – these values are significantly smaller than galaxies from the Sloan Digital Sky Survey at the same redshift. The association of the long bursts with core-collapse supernovae has been convincingly demonstrated in many cases. The statistics of short bursts are much smaller than those of the long bursts but there is a clear correlation with regions of low star formation, either in elliptical galaxies or in a region of low star formation rate within galaxies. The luminosities of the elliptical host galaxies correspond to L ∼ L∗ galaxies. The afterglows have played a particularly important role in understanding the nature of the γ-ray bursts. Before the SWIFT observations began to appear in 2005, the afterglows were typically observed about 1000 seconds after the burst and the spectrum in the X-ray, optical and radio wavebands consisted of power-law spectrum which steepened at later times. The SWIFT observations enabled the afterglows to be observed much sooner after the burst and somewhat complicated this picture. Examples of the afterglows of long and short bursts are shown in Fig. 22.19. After about 103 seconds, the power-law decline of the X-ray afterglow is observed and in a number of cases an increased decline rate is observed after about 105 seconds, as inferred from the earlier observations. The new features are the very rapid decline in the first 100 seconds followed by a very slow decline, typically from about 100 to 1000 seconds. In addition, X-ray flares are observed in both the long and short bursts. As might be expected, the wealth of data from γ-ray telescopes such as SWIFT have revolutionised understanding of nature of the γ-ray bursts, but also led to an appreciation 776 Compact extragalactic sources and superluminal motions

Fig. 22.19. Examples of the time evolution of the X-ray afterglows of representative examples of long and short γ-ray bursts (Gehrels et al., 2009). of the diversity of phenomena associated with the bursts. The above broad-brush picture is adequate for our present purposes, but there are many interesting variants on these correlations which are thoroughly described by Gehrels and his colleagues (2009).

22.7.3 The physics of γ-ray bursts We can use many the tools developed in this and previous chapters to understand the physics relevant for the γ-ray bursts. A review of many aspects of the theory of γ-ray bursts has been provided by Mesz´ aros´ (2002). According to the standard causality argument, a time-scale of the variability of the order of milliseconds indicates that the energy must originate from regions less than about 3 × 105 km in size. Hence the bursts must involve stellar mass objects and huge energy releases, typically 1046 J if the emission were isotropic. The resulting energy density in γ-rays within the source region would then be so great that the same problems afflicting the extreme γ-ray luminosities of blasars and quasars, the degradation of the γ-rays by photon-photon interactions (Sect. 22.6), apply even more severely. From the energy and scale of the emitting region, it follows that the source region must expand relativistically and so, just as in the extreme extragalactic γ-ray sources, the γ-ray bursts must involve relativistic bulk motions. The values of the relativistic beaming factors necessary to reduce the compactness factor below the critical value are κ ∼ 102 − 103, greater even that those necessary for the extragalactic γ-ray sources and those observed by radio VLBI. In the simplest picture, the source is taken to be a relativistic fire-ball, meaning a highly 22.7 γ-ray bursts 777 relativistically expanding sphere which heats the surrounding gas and drives a relativistic shock wave into it (Mesz´ aros´ and Rees, 1993). If the sphere is optically thick, the radiation would be thermalised and a thermal spectrum observed. The observed spectra of the γ- ray bursts are however of non-thermal form, often with power-law spectra of the form N(ε) ∝ ε−α, where α ∼ 2 − 3 at energies greater than about 0.1–1 MeV. Therefore the radiation must originate from optically thin regions later in the expansion. The relativistic shock wave then provides the means of accelerating electrons to very high energies by the first-order Fermi acceleration mechanism discussed in some detail in Sects. 17.4 and 17.5. The structure of the expanding sphere resembles the structure shown in Fig. 16.10a. The relativistically expanding fireball drives an external shock into the ambient medium and there is a contact discontinuity between the material of the fireball and the shocked gas. As the sphere decelerates, an internal reverse shock is formed which is driven back towards the origin and reheats the expanding sphere. The difference as compared with Fig. 16.10a is that the external shock is relativistic and the reverse shock can be relativistic as well. All the arguments concerning the formation of a power-law spectrum of accelerated electrons and the growth of the magnetic flux density towards equipartition discussed in Sects. 17.4. and 17.5 reappear, but now in relativistic guise. Prior to the launch of the SWIFT satellite, the characteristic evolution of synchrotron spectrum of such a sphere, the simplest version of which was discussed in Sect. 22.1, provided a reasonable match to the observed spectrum and time evolution of the afterglows in the X-ray, optical and radio wavebands. Examples of such spectra are shown in Fig. 22.20 (Granot and Sari, 2002). The origin of these spectra can be understood on the basis of the analysis of the properties of synchrotron radiation and the modifications to the spectra due to synchrotron losses discussed in Chaps. 8 and 16. For example, in the case of spectrum 1, the standard optically thin synchrotron spectrum is observed in region G between the frequency νm, which corresponds to the lower limit of the electron energy spectrum, and νc at which the lifetimes of the electrons equals the synchrotron loss time. The power law at frequencies greater than νc in region H results from the assumption of the continuous acceleration under the influence of dominant synchrotron losses (Fig. 16.1a). The 1/3 power law Lν ∝ ν in region D corresponds to the low-frequency tail of the spectrum of a synchrotron radiation of a single electron. The Rayleigh-Jeans spectrum in region B is the region of self-absorption for the radiation of the lowest energy electrons in the energy spectrum. In spectrum 2, there is a region A in which synchrotron self-absorption takes 2.5 place with νm νsa, resulting in the standard relation Iν ∝ ν . The other spectra correspond to different orderings of the various ‘critical’ frequencies, as discussed by Granot and Sari (2002). The reason for displaying these results is that it turned out that the simplest form of spectrum, example 1 in Fig. 22.20, could account for the spectral shape and time variability of the burst afterglows in the early observations and consequently could provide information about the physical properties of the relativistic fireball. Relativistic beaming alleviates the problems of γγ annihilation, but does not solve the 778 Compact extragalactic sources and superluminal motions

Fig. 22.20. Examples of different broad band synchrotron spectra from a relativistic blast wave, in which electrons are accelerated with a power law energy distribution N(E) dE ∝ E−p dE. The thin solid line shows the asymptotic power law segments and their points of intersection, where various break frequencies are expected. The different orderings of the various break frequencies are discussed by Granot and Sari (2002). energy problem. The fireball need not, however, be isotropic. In the case in which the energy of the γ-ray burst is emitted in a narrow collimated beam, the energy requirements can be very significantly reduced. In the case of a relativistic jet of opening angle θ, the same overall picture with external and internal shocks and a contact discontinuity illustrated in Fig. 21.7b applies, provided the opening angle θ of the jet is greater than γ−1, where γ is the Lorentz factor associated with the bulk motion of the emitting region and the observer is located within the solid angle of the jet. As the jet sweeps up the surround- 22.7 γ-ray bursts 779

Fig. 22.21. Illustrating the evolution of a γ-ray burst on increasing physical scales (Gehrels et al., 2009).

ing gas and decelerates, however, its Lorentz factor decreases until θ < γ−1. Up till that point, the decrease in luminosity has been partly compensated by the fact that more of the jet is observed as the relativistic beaming allows more of the jet to be observed. As γ continues to decreases when θ < γ−1, the emission from the whole jet is observed and the observed flux density decreases more rapidly with time. This change can explain the break in the afterglow light curve observed after about 105 seconds. In addition, the jet may expand sideways as it decelerates causing yet more rapid decline of the afterglow. If the jet model is adopted, the total energy requirements of the γ-ray bursts can be evaluated and it was found that, rather than a dispersion in intrinsic luminosities from about 1044 to 1047 J if the radiation is isotropic, there is only about an order of magnitude spread about the value 8 × 1043 J, a value typical of the energy released in the core collapse of a supernova explosion. This picture of the evolution of the afterglows of γ-ray bursts needs some revision in the light of the SWIFT observations. The full implications of these observations are discussed by Granot (2008) who describes clearly the deficiencies of the model, in particular, the need to account for the initial steep decline of the X-ray light curve and its subsequent flattening. He also points out the problem that, although a break may be observed in the X-ray light curve, this is often not observed in the optical light curve. This observation is contrary to the expectations of the decelerating jet model in which the break in the light curves should be independent of wavelength. He suggests various solutions to these problems. The association of the long bursts with core-collapse supernovae in which such energy releases are observed makes a compelling picture. A montage of images for the plausible evolution of γ-ray bursts is provided by the simulations-cartoons presented by Gehrels 780 Compact extragalactic sources and superluminal motions and his colleagues (2009) (Fig. 22.21). There is only direct observational evidence for the later stages in the evolution, but the overall picture is consistent with the types of physical process which are inferred to take place in core collapse supernovae. There are many similarities with the physical processes involved in the release of gravitational energy and the formation of relativistic jets in extragalactic radio sources discussed earlier in this chapter. The one big difference is that jet formation, collimation and the expansion into the surrounding medium take place very rapidly indeed within the collapsing envelope of the collapsing star. The diagrams make the important point that the γ-rays only become observable when the jets reach scales of order 107 km. If the relativistic beaming factor is of the order κ = 103, time variations of the γ-ray burst would be expected on time-scales as short as 30 µs. The appearance of the afterglows only occur significantly later in the evolution of the relativistic jet as it is decelerated on passing through the stellar envelope and sweeping up mass. A popular picture is to relate the γ-ray bursts to the collapse of a very massive stars, such as the Wolf-Rayet stars, the cores of which would collapse to form a Kerr black hole and then to associate the relativistic jet with the presence of electric and magnetic fields in the vicinity of the hole, as illustrated in Fig. 22.19 and discussed in Sect. 22.4. Axisymmetric collapse of the outer envelope would continue onto the collapsed core. This picture is often referred to as a collapsar model for γ-ray bursts. The short bursts are much less luminous that the long bursts and belong to different stellar populations. The very short time scales of variability suggest that neutron stars and black holes are involved but, because they are not associated with star forming regions in galaxies, they cannot be young stellar systems. An appealing possibility is that they are associated with the merger of binary neutron stars, or of a neutron star and a black hole, or of two black holes. Many of these possibilities are discussed in the review by Mesz´ aros´ (Mesz´ aros,´ 2002). If this interpretation is correct, the statistics of the short γ-ray bursts provide estimates of the numbers of stellar binary mergers which might be detected by the gravitational wave detectors.

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Kuz’min, Vadim, 582 Blain, Andrew, 812 Bland-Hawthorn, Jonathan, 803 Abell, George, 109, 111 Blandford, Roger, 276, 611, 620–622, 627, Abraham, Robert, 805 761, 767, 768 Abramovitz, Milton, 227, 231 Blumenthal, George, 179, 191, 266 Adams, Fred, 384, 401 Bondi, Hermann, 483 Aharonian, Felix, 598, 770, 771 Bothe, Walter, 31 Amsler, Claude, 300 Boyle, Brian, 647, 792 Anderson, Carl, 31, 32, 178 Bracessi, Alessandro, 643 Arnaud, Monique, 121 Brandt, W., 794, 795 Arnett, David, 422 Bruzual, Gustavo, 798, 799, 813, 814, 817 Arzoumanian, Zaven, 459 Burbidge, Geoffrey, 603 Auger, Pierre, 31, 571 Butler, Clifford, 32 Axford, Ian, 611, 622, 627 Cotˆ e,´ Patrick, 104 Babbedge, T., 809, 810 Calabretta, Mark, 827 Backer, Donald, 456, 457 Camenzind, Max, 441, 442 Bahcall, John, 61, 63, 65 Cameron, Alistair, 67, 544 Bahcall, Neta, 116 Cannon, Annie, 41 Balbus, Steven, 496 Cappelluti, Nico, 798 Ballard, K.R., 629 Carter, Brandon, 474 Band, David, 290 Caswell, James, 248 Barger, Amy, 815 Cavaliere, Alfonso, 121 Beck, Rainer, 568 Cesarsky, Catherine, 205, 207, 568, 626, 627, Beckwith, Steven, 806, 807 629, 631 Becquerel, Henri, 159 Challinor, Anthony, 127, 286 Bekefi, George, 213 Chandrasekhar, Subrahmanyan, 331, 469, 474, Bell(-Burnell), Jocelyn, 20, 444 496 Bell, Anthony, 611, 622, 624, 625, 628, 629 Charlot, Stephane,´ 798, 799, 817 Bennett, Charles, 17 Chwolson, O., 129 Berezhko, Evgeny, 631, 632 Cimatti, Andrea, 806, 818 Berezinsky, Venyamin, 583, 584 Clayton, Donald, 422 Bethe, Hans, 63, 179, 182, 191 Clemmow, Phillip, 293 Bignami, Giovanni, 218 Colless, Matthew, 119 Binney, James, 117, 167 Compton, Arthur, 255 Blaauw, Adriaan, 826 Cordes, James, 459 Blackett, Patrick, 32 Cowie, Lennox, 806, 813, 815

891 892 NAME INDEX

Cox, Donald, 389 Genzel, Reinhard, 681 Ghez, Andrea, 679 Damon, Paul, 326 Gilli, R., 658, 794, 797 Davies, Rodney, 410 Ginzburg Davis, Leverett, 407 Vitaly, 579 Davis, Raymond, 34, 61 Ginzburg, Vitali L., 213, 239, 242 de Boer, Klaas, 568 Glazebrook, Karl, 818 de Vaucouleurs, Gerard,´ 85, 86 Gold, Thomas, 444 Dermer, Charles, 552 Goldreich, Peter, 464, 466 Deubner, Franz-Ludwig, 57 Goldsmith, Donald, 392, 393 Diehl, Roland, 315 Gough, Douglas, 57 Dirac, Adrian, 32 Gould, Robert, 169, 179, 191, 266 Djorgovski, George, 97 Graham Smith, Francis, 444, 454, 457, 460, Dougherty, John, 293 461, 464, 468 Draine, Bruce, 383, 405–407, 409, 410 Green, David, 609 Dressler, Alan, 92, 97 Green, Richard, 644, 647, 792 Drury, Luke, 626 Greenstein, Jesse, 407 Dunlop, James, 790, 818 Greisen, Eric, 827 Dunn, Andrew, 119 Greisen, Kenneth, 300, 582 Dyakov, Sergei, 275 Grindlay, Jonathan, 290 Gull, Stephen, 609 Eastman, Ronald, 430 Gunn, James, 117, 118, 139, 459 Efstathiou, George, 138 Eichler, David, 620 Habing, Harm, 392, 393 Einstein, Albert, 129, 188, 255, 271 Hall, John, 405 Ellis, George, 473 Hasinger, Gunther,¨ 528, 794, 795 Ellis, Richard, 131, 800, 805, 806 Hauser, Michael, 813 Ellis, Simon, 803 Hawking, Stephen, 473, 478 Emslie, Gordon, 353, 355 Hawkins, Michael, 646 Ewen, Harold, 21 Hawley, John, 496 Ezer, D., 67 Haynes, Martha, 95, 96 Heavens, Alan, 629 Faber, Sandra, 96, 97 Heitler, Wilhelm, 179, 182, 191 Fabian, Andrew, 123, 125 Hess, Victor, 30 Fan, Xiaohui, 793 Hewish, Antony, 20, 444 Felten, James, 106, 107, 787 Hewitt, Jacqueline, 136 Ferland, Gary, 371, 375 Heyvaerts, Jean, 615, 616 Fermi, Enrico, 154, 616, 618, 619 Hildebrand, Roger, 383 Feynman, Richard, 272, 274, 463 Hillas, Michael, 549, 578, 579, 635, 636 Field, George, 392, 393, 569 Hillebrandt, Wolfgang, 416, 418, 419 Fisher, Richard, 96 Hiltner, William, 405 Fitzpatrick, Richard, 327, 328 Holloway, Nigel, 468 Forbes, Terry, 355, 358, 360 Hoyle, Fred, 483 Ford, Vincent, 407 Hubble, Edwin, 85, 86, 830 Fort, Bernard, 133 Hulse, Russell, 454 Frank, Juhan, 491, 496, 497, 501, 502, 504, Harm,¨ Richard, 334 505, 513, 518, 520, 525, 719, 721, 722 Frolov, Valery, 469 Jackson, John, 293 Jackson, John D., 151, 154 Garcia-Munoz, M., 561 Jackson, Robert, 96 Gavazzi, Raphael,¨ 135 Janka, Hans-Thomas, 419, 420, 422, 424, 459 Gel’fand, Israil, 275 Jansky, Karl, 19 NAME INDEX 893

Jenkins, Edward, 568 Leibundgut, Bruno, 416 Jokipii, Randy, 204, 205 Lequeuex, James, 97 Jones, Thomas, 630 Liedahl, Duane, 267, 269, 270, 275, 276, 278 Julian, William, 464, 466 Lifshitz, Evgenii, 344, 346, 348 Lightman, Alan, 213, 223, 231, 267, 275 Kang, Hyesung, 630, 634 Lilly, Simon, 813, 818 Karttunen, Hannu, 39 Lingenfelter, Richard, 319 Karzas, William, 185, 252 Liu, Q.Z., 529 Kauﬀmann, Guinevere, 93, 94 Lizano, Susana, 401 Kembhavi, Ajit, 639 Lochner, James, 489 Kennicutt, Robert, 96, 395 Longair, Malcolm, 812 Kent, Stephen, 117, 118 Lorimer, Duncan, 444 Kerr, Roy, 474 Lotz, Jennifer, 806 King, Andrew, 491, 492, 497, 501, 502, 504, Lucek, S.G., 628 505, 513, 518, 521, 525, 719, 722 Lyne, Andrew, 444, 454, 457, 460, 461, 464, King, Ivan, 116 468 Kippenhahn, Rudolf, 432, 435, 436 Lyttleton, Raymond, 483 Kippenhahn, Rudolph, 39, 54, 67, 70 Kirshner, Robert, 430 Mesz´ aros,´ Peter, 771, 776, 780 Kneib, Jean-Pierre, 131 MacDonald, Douglas, 478 Koch, H. William, 179, 191 Mack, Julian, 606 Kochanek, Christopher, 131 Madau, Piero, 813, 815 Kolb, Rocky, 138 Majewski, Steven, 646 Kolhorster,¨ Werner, 30, 31 Malin, David, 425 Kompaneets, Aleksander, 275 Manchester, Richard, 448 Koo, David, 645 Marsh, Thomas, 517 Kormendy, John, 85 Mathewson, Donald, 407 Kramer, Michael, 444 Mayor, Michel, 74 Krause, Oliver, 417 McCarthy, Patrick, 817, 818 Krolik, Julian, 639 McCaughrean, Mark, 397 Kron, Richard, 645 McClintock, Jeﬀrey, 531, 532, 534 Krymsky, Germogen, 611, 622, 627 McCray, Richard, 430 Kulsrud, Russell, 207, 327, 338, 340, 355, Mellier, Yannick, 133 360, 567 Mellinger, Axel, 7–9, 827 Menjo, Hiroaki, 326 Lacy, Mark, 818 Merritt, David, 118, 119 Lagache, Guilaine, 809, 810 Mestel, Leon, 464, 467 Lagage, P.O., 626, 627, 629, 631 Metcalfe, Nigel, 800 Laming, Martin, 628 Mewaldt, Richard, 562 Landau, Lev, 275, 344, 346, 348, 493, 498 Meyer, Peter, 543 Larmor, Joseph, 464 Michell, John, 469, 471 Lasenby, Anthony, 127, 286 Millikan, Robert, 31 Lasota, Jean-Pierre, 525 Mirabel, Felix, 482 Latter, Richard, 185, 252 Misner, Charles, 474, 475 Lawson, K.D., 594 Motz, J., 179, 191 Lazarian, Alexander, 409 Murray, Andrew, 825 Le Roux, Edouard, 228 Mushotzky, Richard, 489, 490 Lear, Egil, 611, 622, 627 Leavitt, Henrietta, 830 Nagano, M., 634 Lee, Jae-Joon), 633 Narlikar, Jayant, 639 Leger, Alain, 382, 383 Neddermeyer, Seth, 32 Legg, , 239 Neininger, N., 402 894 NAME INDEX

Newman, Ted, 474 Roberts, Morton, 95, 96 Niemeyer, Jens, 416, 418, 419 Robson, Ian, 639 Nikolic, Bojan, 380 Rochester, George, 32 Northrop, Theodore, 201 Rodrigues, Juan, 482 Novikov, Igor, 469 Rossi, Bruno, 300 Rowan-Robinson, Michael, 789, 830, 831 Occhialini, Giuseppe, 32 Rubin, Vera, 101 Oort, Jan, 20 Ruderman, Malvin, 468 Orosz, Jerome, 482 Rybicki, George, 213, 223, 231, 267, 275 Osmer, Patrick, 645 Ryu, Dongsu., 634 Osterbrock, Donald, 371, 375 Ostriker, Jeremiah, 102, 459, 611, 622, 627 Sersic,´ Jose´ Luis, 91 Sajina, Anna, 807 Pacholczyk, Andrej, 213 Salaman, M. H., 583 Pacini, Franco, 444, 464 Sandage, Allan, 85, 97 Pagel, Bernard, 543 Savage, Blair, 568 Palla, Francesco, 393 Sazonov, V. N., 239 Panagia, Nino, 430 Scheuer, Peter, 232 Parker, Eugene, 354, 355, 569, 570 Schmidt, Brian, 430 Peacock, John, 293, 790 Schmidt, Maarten, 394, 644, 645, 647, 789, Pearce, W. P, 207 792, 793 Pearce, William, 567 Schneider, Peter, 131, 135 Peebles, James, 102 Schroeder, Daniel, 843 Pengelly, R.M., 372 Schwarzschild, Karl, 469 Penrose, Roger, 473, 477 Schwarzschild, Martin, 104 Penzias, Arno, 15 Sedov, Leonid, 606 Peterson, Bradley, 639 Petschek, Harry, 358 Shakura, Nicolai, 496, 501 Pogson, Norman, 833 Shapiro, Paul, 569 Pozdnyakov, L., 267, 278, 281, 282 Shapiro, Stuart, 437, 440, 462, 473, 475 Pratt, Gabriel, 121 Shu, Frank, 384, 393, 401 Price, Richard, 478 Silberberg, Rein, 310, 313, 314 Priest, Eric, 353–355, 358, 360 Simpson, John, 544, 545, 558 Pringle, James, 492 Skadron, George, 611, 622, 627 Pryce, M.H.L, 468 Skilling, John, 568 Puget, Jean-Loup, 382, 383, 813 Skobeltsyn, Dmitri, 30, 32 Purcell, Edward, 21 Smart, William, 825 Smith, Barham, 389 Queloz, Didier, 74 Sobol, I., 267 Spinrad, Hyron, 818 Rontgen,¨ Wilhelm, 159 Spitzer, Lyman, 327, 331, 333, 334 Raine, Derek, 491, 497, 501, 502, 504, 505, Springel, Volker, 820 513, 518, 521, 719, 722 Stahler, Steven, 393 Raizer, Yuri, 344 Starrﬁeld, Sumner, 521, 522 Ramaty, Reuven, 319 Stecker, Floyd, 552, 583, 584 Rees, Martin, 771 Stegun, Irene, 227, 231 Reimer, Paula, 326 Steidel, Charles, 813 Remillard, Ronald, 531, 532, 534 Steigman, Gary, 136 Rest, Armin, 417 Strong, Andrew, 191, 552, 553 Reynolds, R.J., 372 Sunyaev, Rashid, 165, 267, 282, 283, 496, Richards, Gordon, 793 501 Richer, John, 392 Sutherland, Ralph, 185 NAME INDEX 895

Sweet, Peter, 355 Wentzel, Donat, 205 Syrovatskii, Sergei, 579, 626 Westfold, Kevin, 228, 239 Syrovatskii, Sergei I., 213, 239, 242 Wheeler, John, 475 White, Simon, 813 Tandberg-Hanssen, Einar, 353 Wilkinson, Peter, 136 Tayler, Roger, 39, 52, 53, 543 Willingale, Richard, 421 Taylor, Geoﬀrey, 606 Wilson, Robert, 15 Taylor, Joseph, 454 Wolfendale, Arnold, 550 Tesla, Nikola, 178 Woon, David, 367 Teukolsky, Saul, 437, 440, 462, 473, 475 Woosley, Stan, 418–420, 422–424, 459, 521 Thomson, J. J., 170, 171 Thomson, John Joseph (J.J.), 255 Yukawa, Hideki, 32 Thorne, Kip, 475, 478 Tremaine, Scott, 117, 139, 167 Zatsepin, Georgiy, 582 Trodden, Mark, 139 Zavlin, Vyacheslav, 460 Tsao, Chen-Hsiang, 310, 313, 314 Zeldovich, Yakov, 275, 282, 283, 344 Tully, Brent, 96 Zwicky, Fritz, 111 Turner, Michael, 138

Ulrich, Roger, 61

Volk,¨ Heinz, 628, 631, 632 Veron´ Phillippe, 580 Veron-Cetty´ Marie-Paule, 580 van de Hulst, Henk, 20 van den Bergh, Sidney, 85, 805 van den Heuvel, Edward, 515 van der Kris, Michiel, 528 VandenBerg, Donald, 44 Velikhov, E.P., 496 Vink, Jacco, 628 Visvanathan, N., 97

Waddington, Ian, 792 Wall, Jasper, 788 Wambsganss, Joachim, 131 Wandel, Amri, 489, 490 Wang, Wei-Hao, 815 Warren, Stephen, 645 Wasson, John, 162 Watson, Alan, 634 Wdowczyk, Jerzy, 550 Weaver, Thomas, 418 Webber, William, 542, 562 Weber, Joseph, 35 Weber, William, 313 Webster, Adrian, 568, 569, 594 Wefel, John, 563 Weigert, Alfred, 39, 54, 67, 432, 435, 436 Weisskopf, Victor, 442 Object Index

14016+2610, 385 mass-luminosity ratio of, 119 1E1207.4-5209 (isolated neutron star), 218 X-ray image of, 119 3C 273, 491, 641, 642, 698, 699, 748 Crab Nebula, 25, 26, 28, 213 3C 58, 414 Crab Nebula (M1, NGC1952), 414 3C273, 21, 26, 28 continuous injection of energy into, 448 47 Tucanae (47 Tuc), 43, 44, 73, 76, 81, 457 energy requirements of, 448 4C 21.53, 456, 457 pulsar in, 444, 447, 448, 456, 457, 461, 4U 1700-37, 452 463, 465, 467 51 Peg, 74 Cyg X-1, 26 Cyg X-3, 26 A 0620-00, 531 Cygnus A, 26, 288, 534, 603, 848 A0535+262, 514, 515 minimum energy requirements for, 603 Abell 478, 123 Cygnus Loop, 596, 607 Abell 1413, 121 Cygnus X-1, 452 Abell 2218, 110, 131 hard X-ray spectrum of, 281, 282 Antennae, 88 Cygnus X-1 (Cyg X-1), 479, 482, 489, 531 Cygnus X-2 (Cyg X-2), 527, 528 B1937+21 (millisecond pulsar), 456 B2334+61, 461 Becklin-Neugabauer (B–N) object, 397, 398 η-Carinae, 79, 80 Betelgeuse, 845 mass loss rate of, 79 EXO 033319-2554.2, 218 Cartwheel galaxy, 88 Cassiopaeia A (Cas A), 317, 318, 365, 415, G1.9+0.3, 609 421, 422, 603, 604, 608, 610 G1.9+0.3 (supernova remnant), 415 as a type IIb supernova, 421 Galactic Centre, 8, 679 equipartition magnetic ﬂux density, 603 supermassive black hole in, 13 kinetic energy of optical ﬁlaments, 603 ‘Geminga’, 461 minimum energy requirements for synchrotron Gliese 229B radiation, 603 brown dwarf companion of Gliese 229, 73, Cassiopeia A (Cas A), 595, 602–604, 607, 74 608, 627, 628, 640 spectrum similar to that of Jupiter, 74 Type IIb supernova, 596 GRB 030329, 773 Centaurus X-3 (Cen X-3), 450 GRO 1744-28, 527 Coma cluster of galaxies (Abell 1656), 26, GRO J1655-40, 532 117–120 GRO J1744-28, 514 core radius of, 116 GRS 1915+105, 481 mass of, 118, 832 mass function for, 481

896 OBJECT INDEX 897

GRS1915+105, 481 NGC 4839, 119, 120 Guitar Nebula, 459 NGC 5195, 87 GX5-1, 527, 528 NGC 5506, 489 NGC 7023, 382 HD 209458, 75 North Pole star, 8 HD 93131 (Wolf-Rayet star), 79 Her X-1 (pulsating X-ray binary), 218, 443 Ophiuchus molecular cloud, 385 Hercules X-1 (Her X-1), 450, 451 Orion giant molecular clouds, 396, 397 discovery records of, 451 Orion Molecular Cloud, 14 modulation of light curve, 527 Orion Molecular Cloud A, 396 precession of rotation axis of neutron star, Orion Nebula, 14, 18, 396, 397 527 luminous far–infrared sources in, 397–398 unpinning of crust and magentic ﬁeld from Orion star cluster, 73, 76, 78 neutron superﬂuid, 527 Orion, constellation of, 396, 397 HH211, 398, 399 HH30, 398, 399 Perseus cluster of galaxies, 26, 125 HH34, 398, 399 Lyα and Lyβ emission lines of highly ionised Homunculus Nebula (η-Carinae), 80 iron, Fe+25 in, 185, 186 HT Cas, 521 X-ray bremsstrahlung of hot intracluster gas total eclipse observed in, 521 in, 185 X-ray spectrum of, 185 Kepler’s supernova, 414 Plough or Great Bear, 8 Kleinmann-Low Nebula, 397, 398 PSR 1919+21 (CP 1919), 444, 445 discovery records of, 445 Large Magellanic Cloud (LMC), 8, 9, 13, 29, PSR B0540-69, 447 35, 415, 425, 426 PSR B0656+14, 461 LMC X-1, 482 PSR B1055-52, 461 LMC X-3, 482, 531, 534 PSR B1509-58, 447 PSR B1913+16 (binary pulsar), 454, 455 M1, see Crab Nebula (M1, NGC 1952) change of period due to emission of gravi- M3, 44 tational waves, 454 M31, Andromeda Nebula, 101, 107 tests of general relativity and, 454 M33, 480, 481 PSR J0538+2817, 461 M33 X-7 (X-ray binary), 480 PSR J0737-3039, 454–456 M49 (NGC 4472), 104, 105 change of period due to emission of gravi- M51, 402, 404 tational waves, 456 M51 (NGC 5194), 87 PSR J1119-6127, 447 M67, 44 M81, 827 RX J0852.0 − 4622, 597, 598 M87 (NGC 4486), 25, 87, 104 inverse Compton scattering model for spec- non-thermal optical jet of, 87 trum of, 598 Magellanic Clouds, 370 neutral pion decay model for spectrum of, MCG -6-30-15, 489, 669, 685, 687 598 Milky Way, 8, 9 RX J1713.7 − 3946 (G347.3 − 0.5), 597 MXB 1730-335 (rapid burster), 527 Sersic´ 159-03, 123, 124 NGC 1300, 87, 88 Sagittarius A, 826 NGC 2362, 44 Sagittarius A∗ (Sgr A∗), 679–681 NGC 253, 568 Sanduleak −69 202, 35, 425, 426 radio halo of, 569 Sco X-1, 26 NGC 2787, 87 Scorpius X-1 (Sco X-1), 527 NGC 4258, 681 Small Magellanic Cloud (SMC), 8, 9, 13 898 OBJECT INDEX

SN 1006, 326, 414 SN 1054, see Crab Nebula (M1, NGC 1952) SN 1181, 414 SN 1572, see Tycho’s supernova SN 1604, see Kepler’s supernova SN 1987A, 35, 81, 307, 317, 415, 425–430 SU Aur, 385

Taurus molecular cloud, 385 Trapezium stars in Orion Nebula, 397 Tycho’s supernova, 414, 417 remnant of, 417 Type Ia, 596 typing of by light echos, 417 Tycho’s supernova remnant, 595, 596, 607, 609, 627, 628, 632, 633

U Gem, 517, 519

Vega (α-Lyrae), 833, 834 Vela supernova remnant, 28 pulsar in, 444, 461, 463 Vela X-1, 452 Virgo cluster of galaxies, 26, 110 VSSG 23, 385

X1822-371, 523 XBT0748-676, 523 XTE J1650-500, 536

Z Cha, 517, 518 Index

Abell Catalogues of rich clusters of galaxies for protons, helium nuclei, carbon, sili- selection criteria of clusters in, 111 con and iron, 634 space density of clusters in, 111 features of their properties to be explained, Abell clusters of galaxies, 109–112 613 spatial correlations with clusters and galax- Fermi acceleration - original version, 621 ies, 111 first order Fermi acceleration, 597 aberration formula, relativistic, 262 general principles of acceleration, 613–614 absolute magnitude, 836 dynamic, 613, 614 electromagnetic, 613–614 absorption coefficient χν, 186 for thermal bremsstrahlung, 186, 187 hydrodynamic, 613, 614 corrected for stimulated emission, 188 highest energy cosmic rays, 635–636 uncorrected for stimulated emission, 188 Hillas diagram, 635, 636 absorption edges at X-ray energies, 251–255 in solar flares, 614–616 magnetic fields in supernova shock fronts, K-edges, 253 628–629 accelerated charged particles neutral sheets and, 614 polar diagram of radiation of, 171 importance of induced return currents, acceleration four-vector, 177 616 acceleration of an electron in the electrostatic importance of streaming instabilities, 616 field of a proton or nucleus, 179 non-linear diffuse shock acceleration, 629– acceleration of high energy particles, 613–636 633 acceleration of charged particle in electric reconnection of magnetic field lines and, and magnetic fields, 614 614 acceleration of charged particle in time-varying accretion magnetic fields, 614 as an energy source for X-ray sources, 25 beyond the standard model, 627–635 in X-ray binary systems, 450 critical velocity vc for electrostatic acceler- maximum energy release for Schwarzschild ation, 615 black hole, 450 diffusive shock acceleration accretion columns in magnetic cataclysmic vari- acceleration to same energy per nucleon, ables, 519–521 633 emission of far ultraviolet and soft X-ray diffusive shock acceleration in strong shock emission from, 520 waves, 621–627 shock fronts in, 520 electron runaway for electrostatic acceler- temperature of shocked gas in, 520 ation, 615, 616 accretion disc, thick, 477, 478 energy spectrum of cosmic rays at and above accretion discs, see thin accretion discs the ‘knee’, 633–635 boundary layer

899 900 INDEX

emission from, in cataclysmic variables, X-ray surveys of active galaxies, 656–658 518 adaptive optics, 7, 844 luminosity of, 499 adiabatic changes in classical mechanics, 200 boundary layer at inner edge of, 497 adiabatic expansion energy flow in, 499 of a magnetic field, 337 formation of, 485 adiabatic invariance, 203 luminosities of, 498, 499 principle of, 199 outward transfer of angular momentum, 485 adiabatic invariants, 570 viscous dissipation of energy in, 485 conservation of, in a time-varying magnetic accretion luminosity, 484 field, 614 Accretion Power in Astrophysics (Frank, King adiabatic loss problem and the acceleration of and Raine, 491 high energy particles, 607–611 accretion power in astrophysics, 483–537 adiabatic losses, 587–588 accreting binary systems, 516–531 condition for importance of, 588 accretion in binary systems, 506–515 instantaneous dynamics of the expansion black holes in X-ray binaries, 531–537 and, 588 Eddington limiting luminosity, 486–488 non-relativisitic efficiency of the accretion process, 483– in an expanding flow, 587 486 in terms of momentum, 588 efficiency of the accretion process, ξ, 484 velocity distribution inside expanding sphere, for neutron stars, 484 588 for white dwarfs, 484 adiabatic motion of charged particle, 146 onto black holes, 484 Adiabatic Motion of Charged Particles (Northrop), general considerations, 483–491 201 thick discs and advective flows, 504–505 Advanced Camera for Surveys (ACS), 135, thin accretion discs, 491–504 801 accretion radius, see capture, or accretion, ra- advection, 718–720 dius advective transport of mass and energy, 504– action integral in Lagrangian mechanics, 200, 505 201 supermassive black holes and, 505 Active Galactic Nuclei (Krolik), 639 through black hole horizon, 505 Active Galactic Nuclei (Robson), 639 Airy diffraction pattern, 841, 843, 844 active galactic nuclei Airy disc, 841, 844 ‘blue-bump’ component in spectra of, 491 Aitoff projection, 828 ratio of black hole to spheroid masses, 819, Akeno Giant Air Shower Array (AGASA), 821 571, 575 Type 1, 657, 794 Alfven´ and hydromagnetic waves in interstel- unobscured, 657, 794 lar medium, 205–207 active galaxies, 639–665 damping by neutral particles, 207 blazars, superluminal sources and γ-ray sources, damping rate of, 207 651–653 energy density of, 206 Low Ionisation Nuclear Emission Regions growth rate of, 206 – LINERS, 653 momentum density of, 206 quasars, 641–647 Alfven´ radius, 511–513 radio galaxies and high energy astrophysics, for white dwarfs, 512 639–640 Alfven´ speed, 206, 207, 352, 354, 357, 358, Seyfert galaxies, 648–651 567, 568, 622 Ultra-Luminous Infrared Galaxies – ULIRGs, alignment effect in radio galaxies, 716, 717 654–656 alignment of interstellar grains, 405–407 unification schemes for active galaxies, 658– magnetic field parallel to optical polarisa- 665 tion, 407 INDEX 901

physical mechanisms for, 406 astronomical units, 829 Barnett effect, 406 astroparticle physics, 140 paramagnetic dissipation, 407 Astrophysical Flows (Pringle and King), 492 Rowland effect, 406 Astrophysics of Gaseous Nebulae and Active suprathermal processes, 406 Galactic Nuclei, The (Osterbrock and Fer- α-discs, 496, 501, 502 land), 371 AM Herculis binaries, 218 asymptotic giant branch stars, 316 ambipolar diffusion, 341 Atacama Large Millimetre Array (ALMA), Ampere’s` theorem, 355 15 Anglo Australian Telescope 2dF galaxy sur- atmosphere as convertor for cosmic rays, 549 vey, 107, 111, 112 atmospheric turbulence and astronomical see- Anglo-Australian telescope, 644 ing, 844, 845 Anglo-Australian Telescope 2dF survey (AAT atomic binding energy, 147 2dF), 4, 85, 89, 105, 106 Auger air-shower array, 34 angular cyclotron frequency, 196 Auger ultra-high energy cosmic ray observa- angular diameter distance, 130 tory, 300 in general relativity, 470–472 aurorae, 351 angular frequency ω0 of electron in atom, 147 green and red lines of oxygen in, 351 angular gyrofrequency, 196, 201, 207 auroral zone, 351 non-relativistic, 221 Australia Telescope Compact Array, 429, 430 relativistic, 222, 233 Australia Telescope National Facility (ATNF), angular momentum transport by viscosity, 497, 23 498 Avogadro’s number, 192, 298 angular plasma frequency, 486 angular resolving power, 838 Baade-Wesselink method, 430, 830 anomalous resistivity of a plasma, 359 background intensities in ground-based ob- anomalous X-ray pulsars, 458 servations, 835, 850 location on P − P˙ diagram, 458 background radiation antenna temperature, 854 submillimetre and far-infrared minimum detectable, 854 contribution of active galactic nuclei, 815 aperture grading, 844 main contributors to, 816 aperture synthesis, 845–848 ultraviolet Earth-rotation, 848 decrease at large redshifts, 814, 815 principles of, 23 decrease in intensity from z = 1 to present apodisation, 844 epoch, 815 Apollo 12 and the Surveyor satellite, 161 Baksan Neutrino Observatory in the northern Apollo 14, 162 Caucasus mountains, 63 Apollo 17, 161 Balmer absorption line index HδA as age in- apparent magnitude dicator, 94 bolometric, 834 Balmer break, or discontinuity, Dn(4000) as definition of, 833 age indicator, 94 Archimedean spiral, 343 Balmer continuum absoption, 837 Ariel-V satellite, 794 Balmer decrement, 372 ASCA Large Sky Survey, 794 Balmer series of hydrogen, 491, 642, 649, ASCA X-ray observatory, 597, 660 699, 703, 708 associated Legendre functions, 58 BATSE instrument of the Compton Gamma- asteroids ray Observatory, 514 asteroid belt and, 162 Beckwith, Steven, 814 parent bodies of meteorites, 162 BeppoSAX gamma-ray telescope, 317, 318 astronomical seeing, 844, 845 Bernoulli’s equation, 346 astronomical unit (AU), 829 beryllium-10 10Be cosmic ray clock, 564 902 INDEX

production ratio of, 564 angular momentum on last stable orbit, 500 Bessel functions J0(z), J1(z), 841 condition of matter to fall into, 499 betatron, 213 dragging of inertial frames, 475 Bethe-Bloch formula, 153–154, 178, 183 electrodynamics of, 478 bias parameter, 133 ergospheres of rotating, 477 BIMA Millimetre Array, 127, 128 evaporation of, 478 binary pulsars, 454–456 extraction of rotational energy from rotat- binary star systems, 506–509 ing, 477 close, 506 ‘flickering’ of the X-ray intensity and, 488, contact binaries, 506 489 common envelope of, 506 general case in general relativity, 474–479 mass-luminosity relation for, 507 Hawking radiation from very low mass, 478 equipotential surfaces in the rotating sys- horizon of rotating black hole, 475 tem, 506, 507 in binary X-ray sources, 25 evolution of stars in, 507, 508 in X-ray binaries, 531–537 feeding the accretion disc, 509–511 different active states of, 531, 532 Roche lobe overflow, 509, 510 disc fraction f , 531 stellar mass loss and, 510 hard state, 531, 534 stellar mass loss and winds, 509 hard state and presence of radio jets, 534 mass transfer in, 507, 508 hot corona in, 535, 537 non-contact close binaries, 507 iron fluorescence lines, 532, 536–537 periods of, 506 luminosity of accretion disc, 533, 534 role of magnetic fields, 511–515 power density spectrum of the variabil- accretion columns and, 512 ity, 531 magnetic pressure of, 511 quasi-periodic oscillations (QPOs), 535– mass flow onto magnetic poles, 511, 512 536 ram pressure and, 511 steep power-law, 531 spectroscopic binaries, 506 steep power-law state, 534–535 statistics of, 506 temperature distribution in accretion disc symbiotic stars and, 509, 516 about, 533 visual binaries, 506 thermal state, 531, 533 binary X-ray sources, 479 last stable orbit, 488, 491, 499, 500, 505, bipolar outflows, 386, 398 536, 537, 668, 688–692, 718 magnetic fields in, 398 last stable orbit about, 485 model for, 398, 400 evidence for, 533 molecular beams in, 398 light-travel time across the last stable orbit, origin of, 401 488 magnetic fields and, 402 magnetic fields threading, 477, 478 polarisation observations of, 398 mass estimates for, in binary X-ray sys- similarity to extragalactic radio sources, 398, tems, 479, 480 402 masses from time variability, 490 BL-Lacertae objects (BL-Lac objects), 20 maximally rotating, 476 Black hole physics : basic concepts and new maximum angular momenta of, 475 developments (Frolov and Novikov), 469 maximum energy release of Schwarzschild, Black hole X-ray binaries, 532 476, 485 black hole X-ray binaries observational evidence for, 479–482 quasi-periodic oscillations (QPOs), 531, 532, primordial, 478 534–536 resistivity of, 478 increase in frequency with luminosity, 535 Schwarzschild, 476 black holes, 84, 136, 413, 469–482 sketches of, in binary X-ray systems, 482 accretion luminosity of, 500 specific angular momentum of, 499 INDEX 903

static radius about rotating, 475 low frequency spectrum of, 181, 182 supermassive, 136, 782 non-relativistic energy loss rate, 182–183 epoch of maximum quasar activity, 792 non-thermal, 318 non-thermal radiation processes and, 782 relativistic, 189–194 temperature of, 478 average energy of photons emitted in, 194 temperature of gas at last stable orbit, 491 catastrophic losses in the atmosphere, 193 Black Holes – the Membrane Paradigm, 478 collision parameters bmax and bmin for, black holes in the nuclei of galaxies, 667–696 189–191 black holes in X-ray binaries and active galac- correction for electron-electron interac- tic nuclei, 488–491 tions, 191 black-body radiation critical energy, 193 intensity of, 240, 853 in terms of the photon number flux den- Rayleigh-Jeans limit of, 240, 241 sity, 193 spectrum of, 5 radiation length X0 for, 191 blazars, superluminal sources and γ-ray sources, radiation lengths for hydrogen, air and 651–653 lead, 193 blue sequence or blue cloud, 89 the low energy γ-ray emission of the in- avoid regions of rich clustering, 93 terstellar medium and, 194 properties of galaxies of, 89 total energy loss rate, 191–193 Bohr model of the atom, 147 total stopping power for different mate- Bohr radius of the hydrogen atom, 190 rials, 193 bolometric absolute magnitude, 836 total stopping power in different materi- of Sun, 836 als, 191 bolometric flux density, 836 spectral emissivity of, 121 bolometric luminosity of Sun, 836 spectrum of, 121, 129 Boltzmann distribution, 187, 328, 364 thermal, 183–189 Boltzmann equation, 275, 286 absorption, 186–189 evolution of photon occupation number and, Gaunt factors for, 184 275 Gaunt factors for radio wavelengths, 184 Boltzmann relation, 272 Gaunt factors for X-ray energies, 184 Bose–Einstein distribution mass of gas and dark matter in clusters as a solution of Kompaneets equation, 278 of galaxies and, 186 Bose-Einstein spectrum, 282 spectral emissivity of, 183–186 Bose-Einstein statistics, 442 spectral emissivity of, in terms of pho- bound-free absorption, see photoelectric ab- ton number per unit energy inter- sorption val, 185 bound-free absorption in stars, 52 total energy loss rate of, 184 bow shock, 350 bremsstrahlung absorption in stars, 52 density enhancement across, 351 bremsstrahlung, X-ray, 431 Boyer-Lindquist coordinates, 474 bressstrahlung braking index, 445, 447 collision parameters for, 587 energy loss from pulsars and, 445 brightness temperature Tb, 240, 241, 853 for magnetic dipole radiation, 447 Rayleigh-Jeans approximation for, 853 magnetic breaking and, 445 brown dwarfs, 73, 74, 136 ‘braking radiation’, see bremsstrahlung discovered in 2MASS survey, 74 bremsstrahlung, 109, 119, 127, 145, 169, 178– discovered in Pleiades, Orion and ρ Ophi- 194, 330, 831 uchus clusters, 74 collision parameters bmax and bmin for, 179, discovered in Sloan Digital Sky Survey, 74 182 Buckingham Π theorem, 606 cooling rate, 123 BUGS experiment of Ariel-VI mission, 546, emissivity of, 129 547 904 INDEX buoyancy luminosity–period relation for, 830 hot has bubble in the Galactic plane and, Cerro Tololo InterAmerican Observatory, 13 569 ChaMP study, 794 Chandra Deep Fields, 794 Calar Alto telescope, 596 Chandra X-ray Observatory, 27, 124, 125, 414, CalTech Submillimetre Observatory, 286 421, 429, 595, 596, 723, 732, 781, 794 CalTech Submillimetre Observatory (CSO), Chandrasekhar limit, 419, 435–443 15, 127 Chandrasekhar mass, 417–419, 438, 508 Canada-France Redshift Survey, 813 for neutron stars, 437 canonical coordinates, 200 for white dwarfs, 437 canonical momentum, 200 charge exchange current-driven instabilities, of particle in a magnetic field, 200 629 capture, or accretion, radius, 529, 530 charge-coupled devices (CCD), 7, 850 carbon burning, 71 charged particles in magnetic fields carbon monoxide (CO) map of Galaxy, 18 dynamics in time-varying field, 197–201 carbon recombination lines in gaseous nebu- adiabatic invariant approach to, 200–201 lae physical approach to, 197–200 very high order transitions, 373 dynamics of, 195–207 carbon-nitrogen-oxygen (CNO) cycle, 49, 50, in static uniform field, 195–197, 213– 53, 55, 69, 80, 81, 521, 522 214 Cassegrain telescope, 843, 845 scattering by Alfven´ and hydromagnetic waves, cataclysmic variables, 419, 483, 488, 509, 511, 205–207 516–522 scattering by irregularities in the field, 202– accretion columns in magnetic, 519–521 205 AM Herculis stars, 516, 517 spiral motion in uniform field, 196 classical novae, 516, 521, 522 chemical potential µ, 270, 271, 282 event rate in our Galaxy and M31, 521 Cherenkov radiation, 291–297 thermonuclear runaway and, 516, 522 ‘shock wave’ interpretation of, 292 dwarf novae, 516, 518, 519, 521 Cherenkov cone, 296 eclipse mapping of, 517, 518 condition for, 293 temperature mapping, 517, 518 energy loss rate per unit bandwidth, 297 intermediate polars, 516 circumstellar disc, 386 magnetic, 516 Classical Electrodynamics (Jackson), 141 novae-like stars, 516 classical electron radius, 257 polars, 516, 517 classical novae, see cataclysmic variables, clas- recurrent novae, 516 sical novae strong emitters in the 1 − 10 keV X-ray clusters of galaxies, 109–140 wavebands, 519 Bautz-Morgan classification, 112, 113 strong emitters in the EUV and soft X-ray central concentration of galaxies and cen- wavebands, 519 tral profile, 112, 113 strong winds from, 521 central mass density of, 116 symbiotic stars, 516 core radius, 113, 115 celestial equator, 825, 827 crossing time, 118 celestial hemisphere dark matter in, 135–140 north, 827 galaxy content, 112, 113 south, 827 isothermal gas spheres and, 113–117 celestial sphere, 828 mass segregation, 112, 113 Central limit theorem, 577 morphologies of, 109–112 central limit theorem, 787, 849 most massive galaxies in, 167 Gaussian statistics and, 849 regular, intermediate and irregular, 111–113 Cepheid variables, 830 symmetry, 112, 113 INDEX 905

X-ray emission of, 27 probability distribution for a single scatter- clusters of stars, 42 ing, 285 coincidence counting, 31 recoil effect, 260 collision frequency for electrostatic collisions Compton scattering optical depth, 126 between particles νc, 614 Comptonisation, 267–282, 535, 620 collision of high energy particle with station- basic physics of, 267–271 ary electron Bose-Einstein distribution duration of, 146 formation of, 270, 271 limits to collision parameters, 145–148 with finite chemical potential µ, 271 relativistic case, 152–153 Compton optical depth, 269, 270 non-relativistic treatment, 144–145 condition for significant distortions of pho- maximum energy loss, 144 ton spectrum, 269 relativistic treatment, 152–153 examples of astrophysical applications, 268 collision parameter, 144 interchange of energy between matter and collision parameters radiation and, 268–270 maximum and minimum for electrostatic differential equation for, 269 scattering, 331 Kompaneets equation, 275–282 collision time of particles in a plasma, 330 number of scatterings to approach satura- collisionless plasma, 335 tion, 270 colour index occupation number, 272–275 definition of, 836 recoil effect in, 268 colour-colour diagram for stars, 837 computer simulations colour-magnitude diagram for stars, see Hertzsprung- hydrodynamical simulations of galaxy col- Russell diagram lisions, 820 colours origin and evolution of cosmic structures definition of, 836 and, 820 comoving radial distance coordinate, 786 concentration index C = (R90/R50), 94 comoving volume conductivity of a plasma, 145, 464 variation with redshift, 784 cone diagram of the distribution of galaxies, compact extragalactic sources and superlumi- 111 nal motions, 745–780 conservation of angular momentum, 471, 484 compact HII region conservation of energy for Newtonian grav- spectrum at radio wavelengths of, 188, 189 ity, 471, 472, 484 compact radio sources conservation of energy in general relativity, relativistic bulk motion in, 291 472, 484 variability of, 289 conservation of mass, 53 Compton effect, 260 convective transport of energy in stars, 50, 54 Compton Gamma-ray Observatory (CGRO), condition for, 50, 66 316–318, 320 in main sequence stars, 67 EGRET instrument of, 28, 29, 551, 552 in pre-main sequence stars, 67 Compton optical depth, 269, 270, 276, 280, in red giant stars, 67 283, 284 of different masses on the main sequence, Compton scattering, 125, 259–261 68 average energy change of photon in, 269 convective transport of mass and energy, 505 average energy gain of photons by, 268 cooling flows in clusters of galaxies, 123–125 derivation of formulae for, 259–261 Cooper pairs, 442 exchange of energy between electrons and coordinate systems and projections used in radiation field, 260 astronomy, 8 inverse, 782 coordinate systems in astronomy, 825–829 Klein–Nishina cross-section, 260–261 coordinate time in general relativity, 470 906 INDEX coordinate-space diagram for diffusion-loss equa- confinement time in the Galaxy and cosmic tion, 209 ray clocks, 563–565 Copernicus satellite, 368, 369 confinement volume for, 565–568 coronal loops, 338, 339, 342, 353 cosmic ray clocks coronal mass ejection events, 344 observed abundances of radioactive spal- coronene, 383 lation products, 565, 566 COS-B satellite, 28 radioactive spallation products and, 563, Cosmic Background Explorer, 410 564 Cosmic Background Explorer (COBE), 16, 17 differential energy spectra for different species, DIRBE instrument of, 12–14 540–542 The Cosmic Century: A History of Astrophysics differential energy spectra for electrons, 542– and Cosmology (Longair), ii–iv, 3, 5, 53, 543 213 differential energy spectrum of The Cosmic Century: A History of Astrophysics ‘ankle’ in, 539, 540 and Cosmology (Longair), 639 ‘bump’ in, 575 Cosmic Microwave Background Radiation, 15– ‘knee’ in, 539, 540, 575, 578, 629, 630, 18, 109, 125, 126, 410, 589, 755, 831 633–635 cosmological fluctuations in, 17 overall, 539, 540 dipole component of, 15–17 energy density of, 390 distortions from a perfect black-body spec- energy spectra of cosmic ray protons and trum, 282 nuclei, 539–543 due to energy injection after the recom- solar modulation and, 539 bination era, 283 energy spectrum of, 33 due to energy injection prior to the re- escape time τe from our vicinity in the Galaxy, combination era, 282 563, 566 limits to, 284 derived from mean spallation path length, motion of the Solar System relative to, 16, 563 17 derived from radioactive spallation prod- radiation temperature of, 15 ucts, 565 Sunyaev-Zeldovich distortions of, 125–127, escape time from the Galaxy, 604 286 Galactic halo, 571 cosmic ray astrophysics, 29–34 Galactic halo and, 568 extensive air showers, 31 Greisen-Kuzmin-Zatsepin (GKZ) cut-off, 581– from space and from the ground, 32–34 584 history of, 29–32 gyroradius of, 549 ionisation of the atmosphere with increas- high energy electrons and the Galactic ra- ing altitude, 30, 31 dio emission, 585, 593 cosmic ray clocks, 548 high energy protons and the Galactic γ-ray cosmic ray electrons emission, 585 spectrum of, 233, 247, 248 highest energy cosmic rays and extensive cosmic ray protons and nuclei air-showers, 571–573 solar modulation of, 204, 205 isotopic abundance anomalies in cosmic rays, 539–584 origin of, 563 abundances of the elements in the cosmic isotopic abundances of, 547–549 rays, 543–549 differences from cosmic abundances, 548 air shower technique at energies E & 1015 isotropy and energy density of, 549–550 eV, 539, 541 isotropy of, 566 antiparticles in the flux of, 543 as a function of energy, 549 chemical abundances of, 544–547 Compton-Getting effect, 567 features of, 545–546 diffusion model for, 566–568 chemical composition of, 34 leaky box model for, 566 INDEX 907

net stream of cosmic rays and, 549 nucleonic cascades, 321–323 predicted anisotropy in diffusion model, path-length for nuclear interaction, 323 567 pion production, 321–323 underground muons and, 549 radioactive nuclei produced by cosmic rays isotropy of ultra-high energy cosmic rays, in the atmosphere, 324–326 578–581 formation of carbon-14 14C, 324 local energy density of, 550 formation of tritium 3H, 324 compared with other local energy densi- residence time in atmosphere, 325 ties, 550 secondary fluxes of relativistic electrons, mean free path for scattering in the inter- 323 stellar medium, 567 vertical fluxes of, 324 mean spallation path length of 50 kg m−2, vertical fluxes of at different heights, 323 565 cosmic rays, solar observation of the highest energy, 573–578 underabundances correlated with first ion- origin of the light elements in, 554–563 isation potential, 559 abundance differences compared with cos- cosmic star formation rate mic abundances, 554 maximum at redshifts z ∼ 1 − 2, 819 overall statistics for, 539 problem of dust extinction, 815 particle detectors in space observatories, 539 submillimetre determinations of residence time in the Galaxy, 579 as a function of redshift, 816 dependence upon particle energy, 579 cosmic web, 111, 112 Solar System abundances of the elements, Cosmic-Ray Isotope Spectrometer (CRIS) of 543–545 the Advanced Composition Explorer (ACE), source abundances compared with the lo- 565, 566 cal Galactic abundances, 558–560 cosmological aspects of high energy astrophysics, correlation with first ionisation potential, 781–823 560 cosmological distance ladder, 831, 832 first ionisation potential and, 558 Coulomb’s law of electrostatics, 172 spectrum of, 233 relativistic transformation of, 149–151 streaming instability due to excitation of counts of active galaxies Alfven´ and hydromagnetic waves, 567 evolution in infrared waveband, 809 transfer equation for light nuclei, 554–561 counts of extragalactic radio sources, 787– ultra-high energy, 34 789 anisotropies in distribution of, 34 excess of faint radio sources, 787 cutoff at very high energies, 34 evidence for strong cosmological evolu- variations in the chemical composition of tion, 789 cosmic rays with energy, 561–563 counts of galaxies, 800–807 boron-to-carbon ratio, 561, 562 excess of faint blue galaxies, 801, 804 chromium-to-iron ratio, 561, 562 and starburst galaxies, 806 cosmic rays and the discovery of new parti- nature of, 804–807 cles, 31–32 fluctuations in, due to large-scale cluster- charged and neutral kaons, 32 ing, 800, 803, 804 mesotron, 32 for irregular/peculiar/merger systems, 805 muon, 32 for spheroidal and spiral galaxies, 805 pion, 32 in infrared K-band, 804 positron, 32 in infrared K-waveband, 806 strange particles, 32 in U, B, R, I and K wavebands, 803 cosmic rays in the atmosphere, 321–326 problems of determining, 800 electromagnetic cascades, 321–323 counts of galaxies and active galaxies, pre- formation rate and half-life of 14C, 324 dicted, 783 formation rate and half-life of 3H, 325 at submillimetre wavelengths, 810–812 908 INDEX

normalised differential counts, 810, 811 crossing time τcr of star or galaxy in a cluster, Euclidean, 783 166, 167 differential, 783 Cryogenic Dark Matter Search (CDMS II), integral, 783 140 for standard world models, 783–800 current sheets, 354 for sources with power-law spectra, 784 curvature of space-time in general relativity, infrared counts for galaxies, 798 470 normalised, differential, 784 curvature radiation, 222, 468 optical counts for galaxies, 798–800 cusp catatrophe, 133 slopes of integral and differential, 784 cyclotron absorption features, 443 counts of infrared and submillimetre sources cyclotron absorption features in accreting X- convergence of, at mid-infrared wavelengths, ray sources, 217–219 810 magentic field estimates from, 218 IRAS galaxies cyclotron absorption features in isolated neu- excess of faint sources, 807 tron star, 218 Spitzer Survey magentic field estimates from, 219 excess of faint sources, 807 cyclotron radiation, 216–219 submillimetre wavelengths, 812 broadening of spectral lines of, 217 excess of faint sources, 812 circular polarisation of, 217, 218 counts of infrared sources, 807 geometry of magnetic field configuration counts of mid and far-infrared sources, 810 and, 218 counts of radio quiet quasars, 646, 647, 792 magentic field estimates from, 218 counts of submillimetre sources, 810–813 harmonics of gyrofrequency, 216, 217 counts of X-ray sources, 794–798 mildly relativistic, 216 evidence for evolution of the source popu- cyclotron radius of charged particle in mag- lations, 794 netic field, 196 hard X-ray energies, 2 − 10 keV, 657, 794, 796 damping of Alfven´ and hydromagnetic waves, history of, 794 567 problems of interpretation, 794 dark energy, 417 soft X-ray energies, 0.5 − 2 keV, 794, 796 dark matter, 135–140 and the integrated X-ray background emis- astrophysical and experimental limits, 139– sion, 794 140 Crab Nebula (M1, NGC1952), 447, 599, 640, axions and, 138 705, 770 baryonic, 135–138 pulsar in, 414 black holes and, 136 critial density for degenerate gas, 433 limits to mass density from gravitational critical angular frequency for synchrotron ra- lensing, 136 diation, 229, 230 brown dwarfs and baryonic, 136 critical brightness temperature, 289, 290 clusters of galaxies and, 109, 119 critical cosmological density, 140 forms of, 135–140 in neutrinos with finite mass, 138 in clusters of galaxies, 119, 121 critical density in early type galaxies, 135 for common ions, 373, 375 in galaxies, 85, 102 critical density for stabilisation in degenerate and stability of disc galaxies, 102 neutron gas, 440 MACHOs and, 138 critical density for star formation, 820 gravitational microlensing and, 136 critical Fermi momentum pF, 440 masses of dark matter particles, 140 critical frequency for synchrotron radiation, neutrinos with finite rest mass and, 138 222, 232–234, 240, 245 non-baryonic, 138–139 crossing time, 118 searches for dark matter particles, 140 INDEX 909

structure of, 133 diffusion time for magnetic field in a plasma, WIMPs and, 138 340, 342, 355 dark matter haloes of galaxies and clusters, diffusion-loss equation, 278, 586, 594, 608 133 escape time, 278 Davis–Greenstein alignment mechanism, 407 including radioactive decay, 564–565 de Broglie wavelength, 307 solution of, 564 de Vaucouleurs r1/4 law for surface bright- including term for diffusion of particles in ness, 91, 116 momentum space, 620 dead stars, 413–482 source term, 278 Debye length, 327–329, 331, 486, 615 statistical acceleration and, 277 as shielding distance in plasma, 328, 329 steady-state solutions for particle accelera- decay of binary orbits and gravitational waves, tion, 618 35 diffusion-loss equation for high energy elec- declination (Dec or δ), 825, 826 trons, 590–594 deflagrations, 419 distortions of the injection energy spectrum, degeneracy pressure, 431 590–591 condition for use of relativistic equation of steady-state solutions for, 590–591 state, 433, 434 under synchrotron and inverse Compton conditions under which important, 431, 433, losses, 591, 592 434 time evolution degenerate gas under synchrotron and inverse Compton non-relativistic equation of state for, 432, losses, 591, 592 433 diffusion-loss equation for high energy parti- relativistic equation of state for, 433 cles, 207–211 of electrons, 433 coordinate-space approach, 209–211 of neutrons, 434 elementary approach, 208–209 including spallation gains and losses, catas- degenerate stars trophic loss of particles, radioactive internal structure of, 431–434 decay, 210 dendrochronology, see tree-ring-dating diffusive shock acceleration, 621–635 density parameter advection of particles from shock acceler- for stars in the Universe, 108 ation region, 625 in baryons, 108 average energy gain on crossing shock wave, density wave theory of spiral structure, 388, 624 504 average energy gain per cycle of accelera- forcing mechanisms and, 389 tion, 625 depolarisation of polarised radio signals, see box-model of, 626 Faraday depolarisation differential power-law energy spectrum from, detection of ultrahigh energy γ-rays, 292 625 detonations, 419 efficiency of conversion of kinetic energy deuterium formation in stars, 49, 60 to particle energy, 628, 633 neutrinos from, 49 first-order Fermi acceleration, 622 differential rotation of Galactic disc, 387 for shock waves with different compression diffraction optics, 838 ratios, 627, 631 diffraction-limited telescopes, 838–845 fraction of particles lost per cycle, 625 diffusion coefficient D, scalar, 208 non-linear features of, 628 diffusion coefficients for high energy parti- overcoming the adiabatic loss problem, 627 cles in fluctuations in magnetic field, 204 scattering by Alfven´ and hydromagnetic waves, physical model for, 204–205 626 diffusion of charged particles, 329–332 scattering by streaming instabilities or tur- mean free path and, 330 bulent motions, 622, 623 910 INDEX

symmetry of acceleration process in cross- mass distribution in, 135 ing of shock front, 624 Earth’s magnetosphere, see magnetosphere of upper limit to particle energy, 626 Earth diffusive shock acceleration in strong shocks, eclipsing X-ray binary with stellar mass black 616 hole, 481 diffusive velocity, 357 eddies, turbulent, 496 dimensional analysis, 606 Eddington limiting luminosity, 79, 486–488, dipole radiation, 173, 220, 221 490, 491, 495, 502–504, 511, 512, 515, Dirac δ-function, 329 521, 527, 535, 669, 670, 693, 696, 718, Dirac delta function, 176, 181 720, 721, 734 Fourier transform of, 181 Eddington luminosity, 452 dispersion measure, 459 effective aperture of an antenna, 849 dispersion measure of pulsars, 375–377 effective area of the antenna Ae, 854 displacement four-vector, 149, 169, 170 effective temperature Teff , 39, 53 displacement four-vector R, 856 effective temperature of relativistic gas, 240 dissipation time-scale, 355 Effelsberg 100-m radio telescope, 404 distance indicators, 830 Effelsberg 100-metre radio telescope, 569 problems of using, 830 Eichler, David, 621 distance measure D, 784 8-10 metre optical-infrared telescopes, 7 distances in astronomy, 10, 829–832 Einstein angle, 130, 133 astrophysical methods, 830, 832 Einstein coefficients Baade-Wesselink method, 830 relations between, 274 gravitational lenses, 832 Einstein coefficients for absorption and spon- Sunyaev-Zeldovich effect, 831 taneous and induced emission, 242 using distance indicators, 830 Einstein coefficients for spontaneous and stim- Doppler shift formula, relativistic, 262 ulated emission and stimulated absorp- Dreicer field, 615 tion, 187 ‘drop-out’ galaxies, 806, 814 Einstein radius, 130, 133 Drude model, see electrical conductivity of a Einstein ring, 129–131 fully ionised plasma, Drude model for Einstein X-ray Observatory, 27, 78, 123 Drury instability, 629 Einstein-de Sitter world model, 800 dust, see interstellar dust electric dipole moment, 172 dust emission in millimetre and submillime- electrical conductivity of a fully ionised plasma, tre waveband, 17 333–334, 338, 354, 355 dust extinction of galaxy spectral energy dis- dissipation of energy and, 354 tribution, 807, 809 Drude model for, 330, 333 dust shells about giant stars, 83 including effect of electron–electron colli- dwarf novae, see cataclysmic variables, dwarf sions, 334 novae Lorentz approximation for, 333 dynamical friction, 143, 165–167 electrical conductivity of metals clusters of galaxies and, 167 typical values for, 334 galaxies and, 167 electromagnetic field globular clusters and, 167 energy density of, in a medium, 294 limits to collision parameters and, 166 Electromagnetic Processes (Gould), 141 regular clusters of galaxies and, 167 electromagnetic showers, 292, 299–302 dynamical timescale of a star, 46 critical energy Ec for, 300, 301 dynamics of a charged particle in a magnetic degradation of energy by ionisation losses field at low energies, 300 with small scale fluctuations, 203 degradation of energy through the atmosphere, 300 early-type galaxies, 87, 89 properties of, 300, 302 INDEX 911

simple model for, 299 in terms of elementary atomic processes, total number of particles with depth through 188 a medium, 301 emissivity of telescope in infrared waveband, electromotive force, 334–336 850 electromotive force E, 198 encircled energy fraction, 842, 843 electron cyclotron radiation features in the X- for circular mirror with central hole, 846 ray spectra, 511 with varying wavefront errors, 843 electron degeneracy pressure, 76, 84, 413 end-points of stellar evolution, 84 electron scattering, see Thomson scattering, energies 502, 503, 721 useful conversion formulae, 855 electron–photon cascades, see electromagnetic useful conversion formulae for hot bodies, showers 855 electron–positron annihilation line from the useful conversion formulae for photons, 855 direction of the Galactic Centre, 304, 305 energy densities in high energy particles and electron–positron pair production, 297–299 magnetic fields in the Galaxy cross-section at intermediate photon ener- roughly equality of, 570, 571 gies, 297 energy density of starlight in our Galaxy, 589 cross-section in the ultrarelativistic limit, energy dissipation rate by viscous forces, 498 298 energy generation, 53 impossibility in free space, 297 energy loss processes for high energy elec- radiation length for, 298 trons, 585–590 similarity of radiation length for bremsstrahlung adiabatic losses, 587–588, 590, 591 and for, 298, 299 bremsstrahlung, 586–587, 590, 591 electron-positron annihilation, 302–305 inverse Compton scattering, 589–591, 593, annihilation at rest or in flight, 303 594 broadening of γ-ray lines, 304 ionisation losses, 586, 590, 591, 593 cross-section for in extreme relativistic limit, 305 maximum lifetime of high energy electrons anywhere in the Universe, 589 thermal electrons and positrons, 305 maximum and minimum energies of pho- relative imporance of inverse Compton scat- tons, 304 tering and synchrotron losses, 589 of positronium atoms, 304 synchrotron radiation, 542, 588, 590, 591, electron-positron annihilation line, 318, 320 593, 594 electron-positron, or electromagnetic, cascades, under interstellar conditions, 590 571 energy transport in stars, 50–53 electrostatic encounters between charged par- enthalpy per unit mass, 346 ticles, 614 for perfect gas, 346 elliptical galaxies entropy of plasma as triaxial systems, 103, 104 conservation of, 338 stability of, 104 equation of continuity, 622 constancy of velocity dispersion with ra- equation of hydrostatic support, 113, 492 dius, 104 equation of mass conservation, 113 dark matter in, 104 equation of state of stellar material, 44, 53, flattening not due to rotation, 103, 104 58 mass-to-luminosity ratios for, 103 equations of stellar structure, 53–56, 435 rotation along minor axis, 103 equidistant azimuthal polar projection, 7, 827, rotation of, 103 828 variation of ellipticity with radius, 103 equivalent current loop for particle gyrating emission from the Earth’s atmosphere Tatm, in magnetic field, 197–199 854 equivalent noise temperature Tn, 852 emissivity κν, 186 EROS project, 138 912 INDEX escape time of high energy particles in the explosive nucleosynthesis, 73 Galaxy, 592, 593 extensive air-showers, 323, 571 ESO-SERC Southern Sky Survey, 110 extinction Euler’s constant, 184 role of absorption and scattering, 378–381 evolution of active galaxies with cosmic epoch, extinction by interstellar dust grains, 8, 10 787–798 extinction coefficient, 10 extragalactic radio sources, 787–792 extinction curves, 378 ‘luminosity evolution’, 790 extinction efficiency Q, 380, 381 cut-off of strong evolution beyond red- extragalactic radio sources, 723–744 shift z ∼ 2 − 3, 791, 792 luminosity function of for sources with steep and flat radio spec- evolution with cosmic epoch, 789–792 tra, 790 Extreme Ultraviolet Explorer (EUVE), 24 luminosity-dependent density evolution, 791, 792 Faber-Jackson relation and fundamental plane, 96–97 radio quiet quasars, 792–793 and the distances of galaxies, 97 evolution of galaxies and active galaxies with Faraday depolarisation, 377–378 cosmic epoch, 781–813 Faraday rotation of linearly polarised radio mid and far-infrared number counts, 807– signals, 376–378, 402–404 810 direction of magnetic field and, 377 active galaxies, 782, 787–798 estimates of the Galactic magnetic field and, extragalactic radio sources, 787–792 377 radio quiet quasars, 792–793 refractive indices of elliptically polarised X-ray clusters of galaxies, 798 waves, 376 X-ray sources, 794–798 right- and left-handed elliptically polarised brief history of evidence for, 781–782 waves, 376 co-evolution of stellar and black hole prop- rotation measure, 377 erties of galaxies, 782 Fermi acceleration - original version, 616– counts of galaxies, 800–807 622, 624, 626 counts of galaxies and active galaxies, 783 as a second-order process, 618 euclidean source counts, 783 average energy gain per collision, 618 for standard world models, 800 average rate of energy gain, 618 submillimetre counts of dusty galaxies, broadening of energy spectrum by random 810–812 colllisions, 620 counts of galaxies and active galaxies, pre- characteristic loss time τesc from accelera- dicted tion region, 616 for standard world models, 783 exponential increase in energy, 618 isubmillimetre number counts, 810 formation of a power-law energy spectrum, stellar and gaseous components of galax- 619 ies, 782 head-on and following collisions, 617 submillimetre number counts, 813 probabilities of, 617 V/Vmax or luminosity-volume test, 786–787 injection problem, 619 excess of faint blue galaxies ionisation losses and, 619 irregular nature of, 806 magnetic mirrors and, 616 exoplanets, 74–76 modern version including interactions with Doppler technique for discovering, 74 plasma waves, 621 highly elliptical orbits of, 75 problems with, 619–620 Jupiter mass planets very close to parent injection problem, 619 stars, 75 ionisation losses and, 619 occultation technique for discovering, 75 slow increase in energy of particles, 619 problems of accounting for orbits of, 75 why a standard power spectrum?, 619, EXOSAT X-ray telescope, 489, 523, 527 620 INDEX 913

random scattering between cloud collisions, volume element of as a Lorentz invariant, 617 274 relation between mean square energy change four-frequency K, 857 and increase in energy per collision, four-momentum P, 857 621 four-velocity U, 857 second order Fermi acceleration, 622 Fourier analysis, 845–848 stochastic energy gains and, 616 addition theorem, 845 Fermi energy, 440 convolution theorem, 846 Fermi momentum of a degenerate Fermi gas, Fourier transform, 232 433 Fourier transformation, 178, 179, 203, 221– Fermi-Dirac distribution, 433 223 Fermi-Thomas model of the atom, 190 Fraunhofer diffraction, 839 filling factor, 372 for circular aperture, 840–842 fine structure constant, 147, 183, 252 encircled energy fraction, 843 fine structure constant in electrodynamics, 438 for circular mirror or lens, 840 Finkelstein coordinates, 473 for rectangular slit, 839 FIRAS instrument of the Cosmic Background for single slit, 839 Explorer, 283 free-fall velocity in general relativity, 471 firehose instability, 629 free-free emission, see bremsstrahlung frequencies of galaxies of different types, 803 first adiabatic invariant, 199 in different galactic environments, 800 first point in Aries, 825 fully convective stars first-order Fermi acceleration, see diffusive shock stability criterion for, 66 acceleration in strong shocks fundamental plane, 96–97 fluctuations in black body radiation, 852 fluctuations in the Cosmic Microwave Back- Galactic Astronomy (Binney and Merrifield), ground Radiation, 820 86 fluctuations per mode in black body radiation, galactic centre, 826, 828 852, 853 galactic coordinates, 826, 827 Fluid mechanics (Landau and Lifshitz), 344 Galactic Dynamics (Binney and Tremaine), flux densities, luminosity, magnitudes and colours 86 in astronomy, 833–837 galactic equator, 826, 827 flux density, 834 Galactic halo, 568 bolometric, 834, 836 galactic fountains and, 569 definition of, 833 observed in UV absorption lines of CIV Fokker–Planck equation for stars in clusters, and SiIV, 568 116 radio, 568 Fokker-Planck equation for the diffusion of Galactic halo and, 571 the particles in momentum space, 620, galactic latitude (b), 826 622, 627 galactic longitude (l), 826, 828 fold catastrophe, 133 Galactic magnetic field, 248, 402–411 forbidden transitions in spectra of gaseous neb- aligned with local spiral arm, 407 ulae, 372–375 Faraday rotation in the interstellar medium, critical density for, 375 402–404 metastable energy levels and, 374 large fluctuations in, 402, 403 radiative de-excitation and, 374 large-scale order in, 402, 403, 407 Formation of Stars, The (Stahler and Palla), magnetic flux density from pulsar rotation 393 measures, 249 48-inch Schmidt Telescope Palomar Sky Sur- mean magnetic flux density, 411 vey (POSS), 109 North Polar Spur and, 407 four-dimensional momentum space optical polarisation of starlight, 404–407 914 INDEX

radio emission of spinning dust grains and, statistics of galaxies belonging to red and 408–410 blue sequences, 95 radio emission of the Galaxy and, 411 surface brightness distributions of, 91, 94 summary of the information on, 411 discs, 91, 92 Zeeman splitting of 21-cm line radiation spheroids or bulges, 91, 92 and, 410 ‘tuning fork’ diagram for, 86 galactic north pole, 826–828 Galaxies in the Universe: an Introduction (Sparke Galactic radio emission and Gallagher), 86 spectrum of, 593 galaxies, old red, 806 break in, 593, 594 Galaxy variations with location in Galaxy, 594 dark matter in, 387 galactic south pole, 827, 828 radio emission of, 246–249 galaxies, 85–108 radio disc, 247 barred spiral SB, 86 thickness of radio disc, 248 collisions between, 88 rotation curve of, 387 colour-luminosity relation for elliptical galax- Galaxy Formation (Longair), ii, iii, 86, 111, ies, 97 130, 140, 832 elliptical E, 86 galaxy formation ellipticities of, 86 collisions between galaxies and, 819 further correlations among the properties ‘down-sizing’, 819 of galaxies, 95–98 feedback mechanisms and, 819 Faber-Jackson relation and fundamental growth of supermassive black holes and, plane, 96–97 819 mass-metallicity relation for galaxies, 97– Lyman-break galaxies 98 multicolour technique for discovering, 813 Tully-Fisher relation for spiral galaxies, observed global star formation rate, 813 96 star-forming galaxies and, 813 gas phase metallicity-luminosity relation for starburst galaxies at large redshift, 813 late-type galaxies, 98 star and element formation and, 816 Hubble sequence, 86–88 star formation and, 819 interactions between, 109 Galaxy, mass of, 832 irregular, 86 GALLEX solar neutrino experiment, 34, 63 large-scale distribution of γ-ray background emission holes and voids in, 111 Comptonisation and, 798 sheets and ﬁlaments in, 111 spectrum of, 794 lenticular or S0, 87 γ-ray bursts, 419 luminosity density of the Universe due to, γ-ray bursts, 28 107 γ-ray luminosity of the Galaxy, 552 luminosity function of, 104–108 γ-ray observations of the Galaxy, 539, 550– mass-metallicity relation for galaxies, 98 554 mass-to-luminosity ratios for spiral galax- distribution of cosmic rays in the Galaxy ies, 102 and, 550–554 masses of, 98–104 pion production and, 551 mean number density in the Universe, 107 cross-section for, 551 mean separation in the Universe, 107 relativistic bremsstrahlung of ultrarelativis- normal spiral S, 86 tic electrons, 552, 553 passive evolution of, 799 spectrum of γ-rays produced by inverse Comp- importance of red giant branch, 798 ton scattering of starlight, 552–554 peculiar, 88 spectrum of γ-rays produced from neutral red and blue sequences, 89–95 pion decay, 551, 552 ring, 88 γ-ray waveband, 27–29, 551 INDEX 915

γ-ray line emission, 28 cro (Schneider, Kochanek and Wambs- Cherenkov detection technique, 29 ganss), 131 detectors for, 28 gravitational relaxation time τr, 165–167 gauge selection in electrodynamics, 175 grazing incidence optics, 23, 25 Gaunt factor, 121, 123, 184 Great Observatories Origins Deep Survey (GOODS), bremsstrahlung, 586 802, 803 frequency averaged, 185 Great Observatories Origins Deep Surveys (GOODS), Gaunt factors, 331, 334 815 for diffusion of particles in a plasma, 332 grey-body spectrum, 807, 809 for electrical conductivity of plasma, 332 Grotrian diagram, 374 Gauss’s theorem for Newtonian gravity, 102, group velocity, 375 131 guiding centre, 214, 229 Gaussian point-spread function, 844 guiding centre motion, 197, 199, 203, 204 Gemini Deep Deep Survey, 817, 818 gyration radius of cosmic ray protons in mag- Gemini North telescope, 480 netic field, 549 general relativity, 469–473 gyrofrequency, 375, 376 dynamics of particles about point mass, 471 non-relativistic, 216, 222, 229, 230 GEO600 gravitational wave experiment, 36 relativistic, 213, 216, 222, 232, 236 geodesic distance, 470 gyrofrequency νg, 196 giant branch, 41, 43, 44, 55, 69, 77, 81, 82 non-relativistic, 197 giant molecular clouds, 341, 396–398 gyroradius, 203 formation of, 388 and rigidity R, 204 in spiral density waves, 389 gyroradius of charged particle in magnetic field, percolation processes and, 389 196 supernova explosions and, 389 gyroscopic precession about rotating black holes, globular clusters, 44, 76 475 in M49, 104, 105 in M49 and M87, 104 Hammer-Aitoff projection, 8, 9, 12, 14, 16, oldest, 77 19, 24, 29, 828 typical parameters for, 167 Handbook of Space Astronomy and Astrophysics Goldstake solar neutrino experiment, 34 (Zombeck), 141 GOLF experiment of the ESA SOHO mis- hard low mass X-ray binary stars, 304, 305 sion, 58, 59 Harvard spectral classification system, 41, 42 Gran Sasso Laboratory, in Central Italy, 63 spectra of main sequence stars in, 42 grating telescope, 848 Hawking radiation, 478 gravitational deflection of light rays, 132, 133 Hawking, Stephen, 474 by the Sun, 129 Hayashi tracks, 66–69, 71 collision parameter for, 129 for fully convective stars, 67 gravitational fine structure constant, 438 HEAO-1 X-ray Observatory, 794 gravitational lensing, 109, 832 HEAO-A2 experiment, 185 gravitational lensing by galaxies and clusters HEAO-C satellite, 28 of galaxies, 129–135 HEAO-C2 cosmic ray experiment, 545 caustics and cusps in, 133 HEAO-C3 cosmic ray experiment, 546, 547 cluster masses from, 133 heat diffusion equation, 50 critical surface density for, 132 Heisenberg Uncertainty Principle, 147, 152 distortion of background images by, 133 Heisenberg’s Uncertainty Principle, 307 galaxy-galaxy imaging and, 135 Heisenberg’s uncertainty principle, 369, 431– necessary conditions for, 131 433 gravitational lensing, strong, 135 helioseismology, 56–60 gravitational lensing, weak, 133 acoustic or p-modes, 58 Gravitational Lensing: Strong, Weak and Mi- gravity or g-modes, 58 916 INDEX

probing the structure of the Sun and, 58–60 Gaisser-Hillas function, 572, 576 helium GKZ cut-off at 5×1019 eV for protons, 581 primordial nucleosynthesis of, 60 Greisen-Kuzmin-Zatsepin (GKZ) cut-off, 581– cosmic abundance of, 43 584 helium burning, 71, 81 gyroradius of, 578–580 helium flash, 81, 438 protons and iron nuclei, 579 helium shell burning, 71 high energy neutrino production and, 581 Herbig-Haro (HH) objects, 398, 399 HiRes experiments, 572, 575–578, 582–584, Hertzsprung-Russell diagram, 39–44, 55, 66, 635 70, 71, 73, 76–78, 81–83 largest measured energies, 573 for clusters of stars, 42 maximum distance from which they origi- for white dwarfs, 439 nated for protons, 581 theorist’s, 76, 77 modification factor for, 582, 583 HESS Cherenkov γ-ray array, 29 photo-pair production dip, 582 HESS Cherenkov γ-ray telescope, 597, 769, Pierre Auger Observatory, 572, 573, 575– 770 580, 583, 584 Hess ultra γ-radiation, 30 potential sites for acceleration of, 635, 636 hierarchical models of galaxy formation scattering of, by magnetic irregularities, 579 old red galaxies and, 818 slant path length, 571, 572, 576 High Energy Astrophysics (Longair), ii spread in depth through atmosphere of max- high energy astrophysics imum shower development RMS(Xmax), definition of, 3 576–578, 584 high energy astrophysics and modern physics upper limit to particle energy, 635 and astronomy, 3–5 Hipparcos astrometric satellite, 40, 830 high energy electron energy spectrum in the Homestake gold-mine, South Dakota, 61 local interstellar medium, 592–594 homologous stellar models, 53–54, 76 high mass X-ray binaries, 529–531 energy generation rates, 53 capture, or accretion, radius for stellar wind, inadequacies of, 54 529, 530 horizontal branch, 43, 44, 77, 81 O and B stars in, 529 evolution of stars on, 81 properties of, 529 mass loss and, 43 X-ray luminosity due to accretion, 530 hot gas in clusters of galaxies highest energy cosmic rays, 571–584 absence of cool gas in, 124 superposition principle, 577 associated with heating by radio lobes, charges and masses of, 636 124 chemical composition of, 576–578, 580 models to explain the, 124 correlation with nearby galaxies and active abundance of iron, 122 galaxies, 579–581 characteristic cooling time for, 123 Cosmic Microwave Background Radiation cooling flows and, 123–125 and, 574, 581, 583 cooling flows in, 123 cut-off due to photonuclear interactions for mass inflow rates of, 123 nuclei, 583–584 cooling time of, 123 Monte Carlo calculations for, 584 distribution of, 119–125 depth through atmosphere of maximum shower iron line FeXXVI from, 119 development Xmax, 571, 576–578 sound waves in, 125 detection by emission of fluorescent radia- HR diagram for stars, see Hertzsprung-Russell tion, 572 diagram differential energy spectrum of, 573–578 Hubble Deep Field, 781, 801, 805, 806, 813 distortion of spectrum due to photo-pair pro- Hubble sequence of galaxies, 85–88, 95 duction, 582, 583 correlations along, 95–98 fly’s eye telescopes, 572 luminosity function of HII regions, 96 INDEX 917

neutral hydrogen, 95 initial mass function for stars, 393–396 star formation rates and, 96 Miller and Scalo, 394 total surface density and surface density Salpeter, 394 of neutral hydrogen, 95 inner Lagrangian point, 507, 509–511, 517, Hubble Space Telescope, 21, 79, 82, 87, 131, 525, 531 135, 398, 414, 421, 429, 430, 632, 655, instability strip, 81 660, 673, 675, 716, 725, 758, 773, 781, instantaneous rest frame, 173 801, 804, 843, 845, 846, 851 INTEGRAL γ-ray observatory, 28, 218, 282, Hubble Ultra Deep Field, 834 304, 305 Hubble Ultra Deep Field (HUDF), 781, 801– INTEGRAL Gamma-ray Observatory, 316, 317, 803, 806, 813–815 320 Hubble’s constant, 832 intensity from Sunyaev-Zeldovich effect, 129 definition of, 833 Hubble Space Telescope Key project, 832 intensity of radiation Iν, 186 measured from power spectrum of fluctua- interactions of high energy photons, 251–305 tions in Cosmic Microwave Background intercloud medium, 372 Radiation, 832 intercombination lines, 374 Huygen’s principle, 838 interferometer Huygens’ construction, 292, 293, 296 eight-element Hydrodynamics (Lamb), 57 power polar diagram of, 847 hydrogen four-element ionisation potential of, 47 power polar diagram of, 847, 848 hydrogen recombination lines from warm component of the interstellar gas, 372 grating telescope, 847, 848 hydrogen recombination lines in gaseous neb- two-element, 846 ulae, 372–373 power polar diagram of, 847 Hβ line of the Balmer series, 372 interferometry and synthesis imaging, 845– very high order transitions, 372–373 848 hydrogen shell burning, 71 internal energy per unit mass, 346 hydromagnetic waves, circularly polarised, 206 International Linear Collider (ILC), 139 hydrostatic equilibrium, 43, 53, 119, 120 International Ultraviolet Explorer (IUE), 23, hydrostatic support, equation of, 45 78, 430, 699, 709, 711 International Ultraviolet Observatory (IUE), IMB solar neutrino experiment, 35 369 IMP-7 mission, 565 interplanetary magnetic field, 332 IMP-8 mission, 565 interstellar chemistry, 22, 158, 368 interstellar dust, 378–386 impedance of free space Z0, 173 Infrared Astronomical Satellite (IRAS), 11, 217.5 nm absorption feature, 381 ∗ 13, 83 associated with π → π transitions, 381 infrared cirrus, 14 condensed matter physics of, 406 infrared luminosity function of galaxies cross-section for scattering and absorption, evolution of, 807, 809 381 Infrared Space Observatory (ISO), 807 diffuse interstellar bands and, 379, 381 infrared waveband, 10–14 electric charging of, 406 all-sky images in, 12–14 coupling of grains and neutral particles emission by dust grains at thermal infrared through, 406 wavelengths, 13 electric dipole moments of, 409 near infrared wavelengths in, 11 emissivity of heated, 383 observing it, 10–11 extinction law for, 378–379 thermal infrared wavelengths in, 11 formation of interstellar molecules and, 381 wavelength windows in, 11 graphite grains and, 381 918 INDEX

heavy elements in the interstellar medium X-ray absorption, 370 and, 378 heating mechanisms, 389–390 Mie theory of scattering and absorption, 380, collisions of old supernova shells, 389 381 cosmic rays and, 390 obscuration in our Galaxy and, 378 ionisation rate by ionisation losses, 390 optical depth for, 378 other types of, 390 ratio of total to selective absorption RV , 379 supernova explosions, 389 reddening, 379 ultraviolet radiation of hot stars, 390 reradiation of heated, 383 high velocity clouds, 569 star formation and, 383 hot component of, 389, 390 rotation frequency–grain radius relation overall picture of, 387–393 for different phases of the interstellar medium, large scale dynamics, 387–389 408, 409 overall state of, 392–393 rotation speed of dust grains, 405–406, 409 thermal instabilities, 393 rotation speed of PAH molecules, 409 condition for, 392–393 selective absorption, 379 interstellar gas and magnetic fields, 363–411 shielding of molecules by, 391, 397 interstellar medium silicate absorption feature of, 379 phases of, 392 silicate absorption features at 9.7 and 18 cold neutral medium, 392 µm, 381 coronal gas, 392, 393 size parameter, 380, 381 diffuse clouds, 392 small grains in, 382 giant molecular clouds, 392 transient heating of, 382 HI clouds, 392 sublimation temperature of, 378, 384 water ice feature at 3.1 µm, 381 intercloud medium, 392 wide range of grain sizes present in, 381 two-phase model for, 393 interstellar gas interstellar medium and the life cycle of stars, average properties of, 363 363 cooling mechanisms, 391 interstellar molecules, 21 bound-bound or bound-free emission, 391 hydroxyl molecule OH, 22 bremsstrahlung, 391 line emission of, 18 in gaseous nebulae, 391 Introduction to Active Galactic Nuclei (Peter- interstellar dust emission, 391 son), 639 low lying energy levels of common ele- An introduction to the ionosphere and mag- ments, 391 netosphere (Ratcliffe), 334 molecular line emission, 391 invariant four-volume, 264 diagnostic tools invariant four-volume in four-momentum space, 21-cm line emission and absorption, 363– 264, 274, 275 365 inverse β-decay, 421, 441 column depth from X-ray absorption, 370 inverse β-decay process, 440 dispersion measure of pulsars, 375–376 inverse Compton radiation, 534 Faraday rotation of linearly polarised ra- inverse Compton scattering, 261–268, 286, 617 dio signals, 376–378 average energy of scattered photons, 267 ionised interstellar gas, 370–378 derivation of formulae for, 261–267 molecular radio lines, 365–368 energy density of radiation in moving frame, neutral interstellar gas, 363–370 262–264 optical and ultraviolet absorption lines, geometry of, 261, 262 369–370 maximum energy of scattered photon, 267 permitted and forbidden transitions in gaseous of radio, infrared and optical photons, 267 nebula, 371–375 similarity of loss rate to synchrotron loss- thermal bremsstrahlung, 370–371 rate, 265, 288 INDEX 919

spectral index of radiation of a power-law ISAAC infrared spectrograph of the ESO VLT, distribution of electron energies, 266 481 spectrum of scattered radiation by a single isothermal carbon core, 71 electron, 266 isothermal gas spheres, 113–117, 132 total energy loss rate of, 265 core radius, 115 inverse square laws of electrostatics and grav- projected, 115 itation, 165 singular, 132, 133 ion-acoustic instability, 359 structural index, or structural length, 114 ionisation loss formula tidal radius of, 116 adapted for radiation damage density, 160 truncated, 116 ionisation losses, 321, 330, 390 isothermal helium core, 71 ionisation losses of electrons, 158, 159 Jacobian for transformation between inertial maximum energy transfer per interaction, frames of reference, 264 159 James Clerk Maxwell Telescope, 812 ionisation losses of protons and nuclei, 143– James Clerk Maxwell Telescope (JCMT), 15 167 James Webb Space Telescope (JWST), 11 average energy loss per unit path length, Jansky (Jy), 833 145 Jeans’ instability, 400–401 Bethe-Bloch formula, 153–154 fragmentation within giant molecular clouds, cancer therapy and, 158 401 density effect, 154 in molecular clouds, 401 effects of polarisation of medium, 154 time-scale for collapse, 401 energy spectrum of ejected electrons, 148 Jeans’ length, 401 heating of giant molecular clouds and, 143, Jeans’ mass, 401 158 Jet Propulsion Laboratory, 427, 428 lower limit bmin to collision parameters Jupiter, mass of, 74, 832 classical limit, 146 quantum limit, 147 K-correction, 783, 784 mean energy loss rates in different materi- K-corrections, 801 als, 155, 156 K20 sample of galaxies, 806 minimum loss rate, 154, 155 K20 Survey, 817 non-relativistic treatment, 144–148 very red galaxies in, 818 nuclear emulsions and, 159–160 Kamiokande neutrino experiment, 426 number of ion-electron pairs produced, 158 Kamiokande solar neutrino experiment, 34, 35 particle detectors and, 143 KASCADE experiment practical forms of the ionisation loss for- Karlsruhe air-shower array, 633–635 mulae, 154–158 SYBILL 2.1 simulations, 634, 635 range R and, 155, 157 Keck 10-metre telescope, 805 relativistic treatment, 149–154 Kelvin–Helmholtz instability, 351 stopping power, 155 Kelvin-Helmholz time-scale, 48 straggling, 158 Kennicutt, Robert, 820 upper limit bmax to collision parameters, 145 Kepler’s third law of planetary motion, 102, ionisation potential I, 147, 148, 155 387 IRAM millimetre interferometer, Plateau de Keplerian orbits, 492, 497, 498, 689 Bure, 398 Keplerian velocities, 498 IRAS galaxies, 810, 811 Keplerian velocity, 495, 505, 513, 515 IRAS infrared observatory, 781, 807 Kerr black holes, 474 IRAS space mission, 386 maximum energy release, 476, 477, 485 Irvine-Michigan-Brookhaven (IMB) neutrino Kerr metric, 474, 475 experiment, 426 kinetic energy per nucleon, 198 920 INDEX

ionisation losses and, 154 laser emission, 853 King profiles for mass distribution in clusters, Laser Interferometer Gravitational-Wave Ob- 116, 117 servatory (LIGO), 36 Kirchhoff’s law, 371 last stable circular orbit, 472, 473, 476, 477 Klein–Nishina cross-section, 260–261, 291 late-type galaxies, 87, 89 Kompaneets equation, 126, 272, 275–283, 285 ‘leaky box’ model, 592 ‘current’ of photons in phase space and, Lectures on Physics, Vol. III (Feynman), 272 276 Leiden–Berkeley Deep Survey (LBDS), 791 diffusion coefficient for photons in phase Lexan polycarbonate, 161 space, 277 Lienard-Weichert´ potentials, 221, 223 diffusion of photons in phase space and, doppler shift factor in, 223 277–278 Lienard–Wiechert´ potentials, 176, 292 expansion to higher orders in ∂n/∂x, 286 lifetimes of stars of different mass, 394 formation of power-law spectrum by ther- Lifshitz, Evgenii, 493, 498 mal processes, 280 light echo technique, 421, 596 formation of Wien peak, 278–280 light, L, group of elements, 555 induced scattering and, 275 light-year, definition, 829 Monte Carlo solutions of, 279 line blanketing, 97 photon conservation and, 276 Lockman Hole survey field, 794 power-law solutions of, 280–282 long period variables, 82 recoil effect and, 276, 277, 280 Lorentz factor, 152, 154, 159, 195, 198, 323, recoil effect and induced scattering, 277 471, 549, 564, 576, 577, 581, 582, 586 solutions for ~ω kT, 280 Lorentz factor γ, 213, 232, 245, 856 spectra of X-ray sources and solutions of, Lorentz force, 195, 200, 465 278–282 Lorentz gauge, 175 Kramers opacity, 502, 503 Lorentz transformation, 624 Kruskal coordinates, 473 inverse, 153 Kuiper Airborne Observatory, 15, 391 Lorentz transformations, 170, 855 lack holes inverse, 263 mass from kinematics of nuclear gas clouds, Lorentz transforms for electric and magnetic 490 fields, 149–151, 214–215 Lagrangian formulation of classical dynam- Low Ionisation Nuclear Emission Regions – ics, 200 LINERS, 653 Lagrangian mechanics low mass X-ray binaries, 522–529, 535 action integral in, 200 accretion disc corona, 524 ΛCDM model of galaxy and structure forma- dependence of observed properties on an- tion, 819–823 gle of inclination, 523 problems with eclipses and ‘dips’ in X-ray light curves, accounting for Faber-Jackson relation and 522, 523 Tully-Fisher relation, 821 quasi-periodic oscillations (QPOs), 527–529 Lane–Emden equation, 114 models for, 528–529 Lane-Emden equation, 435, 436 thick absorbing screen about accretion disc, Laplace’s equation, 464 524, 525 Large Hadron Collider (LHC), 139 X-ray burst sources (bursters), 525 Large Magellanic Cloud, 137 comparison of accretion and thermonu- distance from observations of SN 1987A, clear runaway luminosities, 525 430 properties of, 525 Larmor formula, 408, 410 Type I, 525–527 Larmor’s formula, 173 Type II, 527 Larmor’s formula for radiation of accelerated X-ray colour-colour diagram for, 528 electron, 255 luminosity INDEX 921

definition of, 836 slowly varying, 197, 199, 200 luminosity classes for stars, 42, 43 magnetic fields in our Galaxy, see Galactic luminosity density of starlight in the Universe, magnetic field 106–107 magnetic fields in supernova shock fronts, 628– luminosity function of galaxies, 104–108, 787 629 for red and blue galaxies, 105, 106 Bell-Lucek instability and, 628, 629 luminosity density of starlight and, 106– equipartition with energy in high energy par- 107 ticles, 629 mass-to-luminosity ratio generation of strong, 628 for the Universe as a whole, 107 instability mechanisms for field amplifica- Schechter, 105, 106 tion, 629 break absolute magnitude M∗, 106 X-ray emission from supernova shock fronts break luminosity L∗, 105 and, 628 written in absolute magnitudes, 106 magnetic fluctuations generated by Alfven´ and useful statistics about galaxies, 107–108 hydromagnetic waves in interstellar medium, luminosity function of stars, 393–394 205–207 luminosity indicators for stars, 42, 43 physical model for, 205 luminosity-temperature diagram for stars, see magnetic flux freezing, 334–344, 443, 464, Hertzsprung-Russell diagram 504, 570, 628 luminous infrared galaxies change of magnetic flux density with den- evolution of, 809 sity of plasma, 340 N1-015 diffusion time for magnetic field in a plasma spectral energy distribution of, 807 and, 340 luminous infrared galaxy N1-015 magnetohydrodynamic approach, 337–342 spectral energy distribution of, 809 physical approach, 334–337 Lundquist number, 357 similarity to adiabatic motion of charged Lyman continuum absorption, 24 particles, 201 Lyman limit for hydrogen, 24 magnetic lines of force Faraday’s concept of, 352 Mach number, 347 magnetic flux freezing and, 334 Mach number, M, 493, 495, 606, 622, 631, magnetic lines of force, reconnection of, see 633, 734 reconnection of magnetic lines of force Alfven,´ 629 magnetic mirroring, 199 MACHO project, 136, 137 in the Earth’s radiation belts, 199 MACHOs, 136 magnetic moment µ of the current loop, 198 mean mass of, 137 Magnetic reconnection (Priest and Forbes), 360 Magellanic Clouds, 487 magnetic reconnection, see reconnection of magnetars, 458 magnetic lines of force magnetic fields in, 458 magnetic Reynolds’ number, 341, 342 magnetic buoyancy, 352–354 ‘longitudinal’, 357 formation of magnetic loops and, 354 Lundquist number and, 357 in a plane-parallel stratified atmosphere, 353 magnetic rigidity, see rigidity R magnetic dipole radiation, 462, 464, 465 magnetic tubes of force, 352 pulsars and, 446 magnetoactive medium, 376 magnetic field fluctuations in interplanetary magnetohydrodynamics medium aspects of, 327–360 power spectrum of, 202, 203, 205 derivation from microscopic description of magnetic field, energy density of, 336 particle dynamics in a magnetic field, magnetic field, Galactic, 248 201 magnetic fields equations of, 337–338 in extragalactic radio sources, 20 entropy equation, 338 922 INDEX

equation of continuity, 337 of spiral galaxies, 107 force equation, 337 masses in astronomy, 10, 832 Maxwell’s equation, 337 masses of galaxies, 98–104 Ohm’s law, 337 masses of elliptical galaxies, 102–104 magnetopause, 350 from widths of absorption lines, 103, 105 magnetosheath, 351 rotation curves of spiral galaxies, 101–102 magnetosphere of Earth, 349–352, 354 virial theorem for galaxies and clusters, 99– collisionless plasmas in, 351 101 collisionless shocks in, 352 massive galaxies in clusters, 821 plasma wave interactions in, 352 Mathematical theory of black holes (Chan- magnetopause, 350 drasekhar), 469 magnetotail, 351 Mauna Kea Observatory, Hawaii, 15 neutral sheet, 351 Maxwell’s equations, 614 plasma sheet, 351 Maxwell’s equations in free space, 174 shock wave discontinuity in, 351 Maxwellian velocity distribution, 184 magnetotail, 351, 354 McGill pulsar group, 458 magnitude mean free path absolute, definition of, 836 for pitch angle scattering, 205 apparent, definition of, 833 of particle in a plasma, 330 bolometric absolute, definition of, 836 of a proton in the interplanetary medium, bolometric apparent, definition of, 834 332 magnitudes in optical and infrared astronomy, mean free path λ, 495 833 mean free time between particle collisions in main sequence, 41–44, 49, 54–56, 67–70, 76, a plasma, 333 78, 79, 837 Medium Deep Survey of galaxies (MDS), 805 main sequence termination point, 42, 76, 81 medium, M, group of elements, 555, 564 main-sequence lifetimes of sun and stars, 54– metallicity Z, 76, 77, 97 56 metals, 43 Mariner 4, 202 meteorites, 161–164 maser action of interstellar molecules, 22 constancy of the flux of cosmic rays and, maser emission, 853 164 mass absorption coefficient for high energy exposure ages of, 164 photons, 298, 299 fossil tracks of cosmic rays and, 162–163 mass conservation, equation of, 45 Galactic cosmic rays and, 162, 164 mass function, 479 history of Solar System and, 161–164 mass function of a binary system, 454, 480 Solar cosmic rays and, 162, 164 mass loss Mg2 index, 97 formation of white dwarfs and, 413 correlation with (B - V) colour, 97 quiescent, 78 microquasar, 481 rates of for massive stars, 79 Mie theory of scattering and absorption, 380, mass-luminosity ratio 381 early-type galaxies, 135 Mikheyev-Smirnov-Wolfenstein (MSW) effect, mass-luminosity relation for main sequence 63, 65 stars, 41, 54, 56, 76 Milky Way, 828 mass-luminosity relation for stars, 394, 799 Millennium galaxy catalogue, 91, 92, 107 mass-metallicity relation for galaxies, 97–98 Millennium Simulation, 820–822 mass-to-luminosity ratio first quasar candidate, 822 for the Universe as a whole, 107 Miller and Scalo initial mass function, 394 for Universe as a whole, 107 millimetre and submillimetre waveband, 14– for visible parts of galaxies, 108 18 of elliptical galaxies, 107 all-sky images in, 15–18 INDEX 923

Cosmic Microwave Background Radiation shielding by dust, 367 in, 15 types of, 368 molecular lines in, 15 unsaturated, 368 observing in, 14–15 vibrational transitions of, 367 millisecond pulsars, 443, 456–457 momentum four-vector, 152, 153, 169, 170 as members of old galactic populations, 459 for electrons, 259 discovery of, 457 for photons, 259, 302 distribution in the Galaxy, 459 momentum impulse, 144, 145, 151, 181, 330 in globular clusters, 457 morphological classification of galaxies, 85– limit to spin-up of, 457 88 location on P − P˙ diagram, 457 morphologies of clusters of galaxies, 109–112 magnetic fields in, 457 Mount Hopkins Observatory, Arizona, 13 space velocities of, 459 muons spin-up of, 457 decay of, 322 minimum energy requirements for synchrotron mean lifetime of, 323 radiation, 599–603 test of relativistic time dilation and length as order of magnitude estimates, 602 contraction, 323 equipartition and, 601–602 underground limitations of, 601–602 measurements of intensity and isotropy filling factor and, 602 of cosmic rays from, 323 minimum energy density in particles and magnetic field and, 602 negative pressure equation of state, 473 minimum total energy, 601 neutral hydrogen simplified estimate for, 602 21-cm line emission of, 20 Minkowski metric, 469 21-cm map of the Galaxy, 21, 22 MKK spectral classification system, 42 distribution in the Galaxy, 387–388 modified Bessel functions of order 2/3 and neutral sheets, 351, 355, 356, 358 1/3, 227 thickness of, 357 asymptotic expressions for, 231 Neutrino Astrophysics (Bahcall), 61 modified Bessel functions of order zero and neutrino interactions one, 180 charged current interaction (CC), 64, 65 asymptotic values of, 180 elastic scattering (ES), 64, 65 molecular hydrogen neutral current interaction (NC), 64, 65 distribution in the Galaxy, 388 neutrino oscillations, 63, 138 molecular radio lines, 365–368 and solar neutrinos, 63–65 list of molecules identified by their, 365– in atmospheric µ neutrinos, 65 367 in terrestrial neutrino experiments, 66 maser action and, 365 neutrinos molecular doubling processes and, 367 laboratory limits to the masses of, 138 rotational ladder of, 367, 368 laboratory limits to the number of species molecules in the interstellar gas, 365–368 of, 140 acetylenic, 368 neutrinos, solar, see solar neutrinos, 307 CO as a tracer of molecular hydrogen, 368 neutron capture γ-ray line at 2.223 MeV, 318, discovery of, 365 320 electronic transitions of, 365, 367 neutron degeneracy pressure, 84, 413 glycine, 368 neutron drip, 440 in large redshift quasars, 368 neutron production by (γ, n) interaction, 326 list of known species, 365–367 neutron star, 341 molecular hydrogen, 368 binding energy of, 414, 599 rotational transitions of, 367 diffusion time for magnetic field from, 341 self-shielding, 368 magnetic flux density of, 341 924 INDEX neutron stars, 20, 76, 84, 413, 439–443 non-relativistic gyroradiation, 216–219 cooling by neutrino emission, 460 energy loss rate by, 216 discovery of, 444–458 linear and circular polarisation of, 216, 235 ellipticity due to rotation, 463 non-thermal emission, 213 galactic population of, 458–460 non-thermal radiation, 5 in X-ray binary systems, 450–454 non-thermal sources internal structure of, 439–443 spectra of, 233 core region, 442 norm of four-vector, 177 inner crust, 442 norm of the four-vector, 856 inverse β-decay process and, 440 north celestial pole (NCP), 825–827 neutron liquid phase, 442 North Polar Spur, 247, 407 outer crust, 441 northern celestial hemisphere, 826 superconductivity and superfluidity and, novae, 419, 521–522 442–443 nuclear γ-ray lines surface layers, 441 important, 319, 320 zones within, with increasing density, 441– nuclear cascades, 302 442 number of particle produce in, 309 magnetic dipole moment of, 446 nuclear deflagration, 522 mass estimates in binary star systems, 455, nuclear emission lines, 315–320 456, 479, 481 44Ti from supernova remnant Cas A, 317, predicted surface temperatures, 460 318 rotating neutron superfluid and, 463 asymptotic giant branch stars, 316 pinning of quantised vortices in, 464 collisional excitation of nuclei, 318–320 quantisation of angular momentum, 463 cross-sections for, 318, 319 rotation of, 443 in the interstellar medium, 318, 320, 321 rotational break-up speed, 443 decay of radioactive isotopes, 315–317 scale height of atmospheres of, 460 astrophysically important examples of, thermal emission of, 460–461 315, 316 ‘truly isolated’, 461 conditions for observability of, 315–316 X-ray emission from surfaces of, 460, 461 diffuse 26Al and 60Fe emission from the Galaxy, neutronisation, 440 316, 317 Newton’s law of gravity, 832 diffuse 26Al emission from the Galaxy, 317 Newton’s laws of motion, 832 explosive nucleosynthesis and, 316 Nobeyama Radio Observatory, 45 m millime- supernovae and, 316 tre telescope, 396 Wolf-Rayet stars and, 316 noise power, 853 nuclear emulsions, 159, 160 fluctuations of, 853 development of, 159 in thermal equilibrium, 852 discovery of elementary particle and, 160 noise temperature silver bromide crystals, AgBr, 159 of the receiving system Tsys, 854 nuclear energy generation rates in stars, 48– non-electromagnetic astronomies, 34–36 50 astroparticle physics, 36 nuclear interaction cross-section, 307, 308 gravitational waves, 35–36 nuclear interactions, 307–326 neutrino astrophysics, 34–35 cosmic rays in the atmosphere, 321–326 non-linear diffuse shock acceleration, 629–633 high energy astrophysics and, 307–310 percursor region in, 630–633 multiple scattering within nuclei, 308 electron and ion temperatures in, 632 neutron production in, 308 Hα emission due to charge exchange in- nuclear emission lines, 315–320 teractions, 632 nucleus–nucleus collision, 309–310 Hα emission from, 632, 633 pion and strange particle production in, 308, self-similar solutions for, 630, 631 309 INDEX 925

spallation cross-sections, 310–315 star-forming galaxies at large redshift and, spallation fragments, 308 818 nuclear interactions in the atmosphere stellar masses as a function of redshift, 817 mean free path for, 309 ‘onion-skin’ chemical structure of massive stars, nuclear interactions of high energy particles 72 with nuclei of atoms and molecules, 143 opacity κ of stellar material, 50–54 nucleonic cascade, 577 optical and infrared wavebands used in ground- nucleonic cascades, 571 based astronomy, 835 particle detectors and, 323 optical and ultraviolet absorption lines, 369– total ionisation as a measure of energy of 370 primary cosmic ray, 323 curve of growth of, 369 nucleosynthesis D lines of sodium, 369 explosive, 422–424 deficit of heavy element in the interstellar late stages of, in massive stars, 421 gas, 370 shell burning, 419 deuterium and, 370 steady-state, 422–424 Doppler broadening of, 369 steady-state hydrostatic, 422 equivalent width of, 369 synthesis of isotopes and, 422, 423 H and K lines of calcium, 369 Nucleosynthesis and chemical evolution of galax- highly ionised OVI and, 370 ies (Pagel), 543 molecular hydrogen and, 369 nucleosynthesis in massive stars, 413 radiation or natural broadening of, 369 nucleosynthesis, primordial, 136, 547 optical depth for radiation τ, 188 nucleus, radius of, 307 optical polarisation of starlight, 404–407 Nyquist’s theorem, 852 alignment of interstellar grains and, 405– 407 OB supergiant mass-loss rates, 529, 531 as a function of galactic coordinates, 407 observability of the sky in different astronom- degree of polarisation correlated with ex- ical wavebands, 5, 6 tinction, 405 observations in cosmology for Friedman world maximum degree of, 404 models parallel to minor axis of grains, 405 flux density–redshift relations optical waveband, 7–10 for starburst galaxies in the submillime- all-sky images in, 7–10 tre waveband, 810, 811 observing in, 7 occupation number, 264, 272–275 Orbital Astronomical Observatories II (OAO- as a Lorentz invariant, 274 II), 369 for Bose-Einstein distribution, 273 orbital migration, 75 for Planck distribution, 273 origin of cosmic rays in our Galaxy, 585–611 in Rayleigh-Jeans region of Planck spec- The origin of the chemical elements (Tayler), trum, 273 543 mean, 273 OSO III satellite, 28 spontaneous and induced processes and, 272 overdensity of galaxies about any galaxy, 92, OH/IR stars, 82 93 old red galaxies, 817–819 OVRO Millimetre Array, 127, 128 constancy of masses with redshift, 818 oxygen burning, 71 early formation of stellar populations of, 819 P(D) distribution, 787, 794 evidence for old stellar populations in, 818 P-Cygni profiles, 78–81, 521 evolving stellar mass density and, 817 mass outflows and, 78, 79 massive, 817–819 p-p chain, see proton-proton chain (p-p chain) need for early starbursts, 818 paleogeomagnetic studies of Earth’s magnetic selected in the K waveband, 817 field, 326 926 INDEX

Palomar 200-inch telescope, 7, 645 cross-sections for, 251–255 parallax-second (or parsec, pc), definition, 829 spectra of X-ray sources at energies ε ∼ 1 parallaxes, 829 keV, 251–255 Parker instability, 569–571 photoelectric effect, 251 buoyancy and, 570 photoionisation Parker’s loops, 570 cross-section for hydrogen atoms, 371 Parkes Radio Telescope, 448 photon loss processes Parkes Selected Region (PSR) sample, 791 summary of absorption coefficient for, 299 Parseval’s theorem, 177–178, 295 photon–photon collisions, 302 applied to magnetic field fluctuations in in- as a source of opacity for high-energy γ- terplanetary medium, 203 rays, 303 spectral distribution of the radiation of an cross-section for 2 accelerated electron and, 177–178 in the regimeε ¯ ≈ mec , 303 partial spallation cross-sections, see spallation in ultrarelativistic limit, 303 cross-sections, partial theshold energy for electron–positron pair particle-antiparticle annihilation, 302 production, 302–303 Pauli exclusion principle, 139 examples of, 303 P˙ versus P diagram for pulsars, 448–450 photonuclear interactions for nuclei ‘graveyard’ region of, 450 disintegration of the nuclei and, 583 as an evolutionary sequence for pulsars, 449 giant dipole resonce for, 583 death line for pulsars on, 450, 457 physical constants in SI units, 854, 856 pulsar equivalent of the Hertzsprung-Russell The physics of fully ionised gases (Spitzer), diagram, 448 327, 331 peculiar galaxies, 88 The physics of plasmas (Fitzpatrick), 327 strong gravitational encounters or collisions, Physics of shock waves and high-temperature 88 hydrodynamic phenomena (Zeldovich and tails associated with prograde encounter be- Raizer), 344 tween galaxies, 88 Pioneer missions of NASA, 344 Penrose process, 477 pionisation, see cosmic rays in the atmosphere, perchloroethylene C2Cl4, 61 pion production perfect gas law, 120 pions permitted and forbidden transitions in gaseous decay of, 322 nebula, 371–375 mean lifetime of, 322 excitation by ultraviolet emission of hot stars, pitch angle, 213, 215, 217, 220, 222, 224, 371 227–230, 234–236, 245 photoexcitation and photoionisation, 371 isotropic distribution of, 215, 235 temperature of emitting regions, Tgas ≈ T?, pitch angle θ, 196, 203 371, 390, 391 pitch angle scattering, 205 permittivity physical model for, 204 relation to refractive index n, 294 Planck distribution, 383, 500 Perseus cluster of galaxies, 821 Planck spectrum, 5, 270, 272, 273 Petrosian r-band luminosity, 94 as a solution of Kompaneets equation, 278 Petschek mechanism of magnetic reconnec- Planck spectrum of black-body radiation, 186, tion, 358 187 phase space plane of the ecliptic, 825, 826 elementary volume of, 273 planetary nebulae, 82–83 photo-pair production, 581–583 central stars of, 83 photo-pion production, 581, 582 evolutionary tracks for, 83 threshold for, 581 images of, 82 photodisintegration of iron nuclei, 421 mass loss and, 82 photoelectric absorption, 251–255 mass loss events and the structures of, 83 INDEX 927 plasma frequency, 327–329, 375, 376 protostars, 341, 383–386 angular, 327 accretion shock in, 384 plasma physics as intense far infrared sources, 384 aspects of, 327–360 diffusion time for magnetic field from, 341 elementary concepts in, 327–334 dust photosphere of, 384 Plasma physics for astrophysics (Kulsrud), 327, hydrostatic core of, 383, 384 360 release of binding energy by far infrared plasma sheet, 351 emission, 384 Plummer model for elliptical galaxies, 116 spectra of, 384, 385 point-spread function, 844 structure of, 383–384 Poisson statistics, 849 pulsar glitches, 447, 461–464 Poisson’s equation changes in moment of inertia of neutron in electrostatics, 329 star and, 461 Poisson’s equation for gravity, 116, 401 glitch function, 462 polars, 218 healing parameter, 462 polycyclic aromatic hydrocarbon (PAH) molecules, migration of quantised vortices and, 464 381–383 spin-up following, 462 planes of hexagonal benzene rings and, 383 time constant of τc, 462, 463 unidentified infrared lines and, 382 starquakes and, 463 polycyclic aromatic hydrocarbon molecules two-component model of interior of neu- (PAHs), 807, 809 tron star and, 461, 463 polytropes, 435 pulsar magnetosphere, 464–467 polytropic index, 435 beaming of radio emission and, 466 positron production mechanisms, 302–305 closed field lines in, 465 positronium atoms, 304 corotation radius, 465 positrons critical field line, 466 sources of, 302 dominated by electromagnetic forces, 465 creation of electron–positron pairs, 302 inevitability of fully conducting plasma sur- decay of π+ pions, 302 rounding, 465 decay of long-lived radioactive isotopes, inner and outer acceleration gaps, 467–469, 302 616 Poynting vector, 172 limited by pair production processes, 616 Poynting vector flux, 294, 295 radio emission from, 468 Poynting’s theorem, 256 light cylinder, 465, 466 primordial nucleosynthesis, 108 magnetic field distribution in, 465, 466 principle of detailed balance, 186, 239 open field lines in, 465, 466 Principle of Equivalence, 470 polar cap regions of, 465, 466 principle of jump rate symmetry, 272, 273 potential differences in, 467 Principles of Optics (Born and Wolf), 838 sparks in, 468 probability density function, 849 space charge distribution in, 465 projected surface brightness, 121 strong electric fields in, 465 projections of celestial sphere onto a plane, zero charge cones, 465, 466, 468 825–829 pulsars, 376 proper acceleration, 173, 177 ages from braking index, 447 proper time ages of, 447 in general relativity, 470–472 anomalous X-ray, 458 in special relativity, 470 as magnetised rotating neutron stars, 464 proton decay, 426 as rotating, magnetised neutron stars, 444, proton-proton chain (p-p chain), 49, 53, 55, 446 56, 60, 63 as spiral arm populations, 459 pp1, pp2 and pp3 branches, 60, 61 associated with supernova remnants, 459 928 INDEX

binary, 454–456 radiation damage, 143, 160–164 dispersion measures of, 459 in plastics and meteorites, 160–164 high velocities of polymers and, 160 asymmetric supernova explosions and, 509 radiation damage density J, 160, 161 disruption of binary systems and, 509 radiation losses of accelerated charged parti- incoherent infrared, optical and X-ray emis- cles sion from outer gap regions, 468 non-relativistic, 170–176 kinematic ages of, 459 relativistic, 176–177 large velocities due asymmetric collapse of radiation of accelerated charged particles, 169– core-collapse supernovae, 459 178 large velocities due to disruption of close from Maxwell’s equations, 174–176 binaries, 459 J.J. Thomson’s treatment, 170–173 luminosity function of, 460 loss rate of, 214 magnetic flux density from slow-down rate, polar diagram of, 173 448–450 polarisation of, 173 millisecond, 443, 456–457 properties of, 173 normal radio, 444–450 relativistic invariants, 169–170 P˙ versus P diagram, 448–450 total energy loss rate, 169, 170, 190 radio emission of, 443 total energy loss rate of, 173 rate of loss of rotational energy, 448 radiation pressure in stars, 51, 54 rates of formation of, 460 Radiation Processes in Astrophysics (Rybicki scale-height in the Galaxy, 459 and Lightman), 141 space densities of, 460 radiation resistance, 853 space velocities of, 459 radiation-driven pulsational instability for mas- spin-up, 449, 513, 515 sive stars, 79 by accretion, 515 radiative transport of energy in stars, 50–52 maximum rotation rate and, 515 radio background emission from our Galaxy spin-up rate–luminosity relation, 514, 515 Tgal, 854 timing noise, 461 radio emission of spinning dust grains, 408– 410 quantisation of rotational angular momentum, Galactic background radiation and, 409– 367 410 quantum theory of gravity, 473 radio emission of the Galaxy, 246–249 quark matter, 442 comparison with predicted emission using quasar spectra local energy spectrum of cosmic ray Lyman-α emission line, 643–645 electrons, 248 prominent emission lines in, 644 determination of radio spectrum and radio quasars, 641–647 emissivity, 247–248 3CR sample of radio, 789 radio emissivity of, 248 discovery of, 641–642 in direction of opaque regions of ionised early formation of supermassive black holes hydrogen, 248 in, 819 radio spectrum of, 247 low-luminosity, 647, 792 radio galaxies radio quiet, see radio quiet quasars 3CR sample of, 789 ratio of black hole to spheroid masses, 819, discovery of, 19 821 energy requirements of, 19 techniques for discovering radio-quiet, 642– evidence for old stellar populations in, 818 658 radio galaxies and high energy astrophysics, Quasars and Active Galactic Nuclei – an In- 639–640 troduction (Kembhavi and Narlikar), 639 radio galaxy 0902+34, 818 quasi-stellar radio sources, 20 radio galaxy LBDS 53W069, 818 INDEX 929 radio galaxy LBDS 53W091, 818 formation of elements beyond the iron peak radio map of the sky, 246 and, 424 radio pulsars, 20, 84 proto-neutron stars and, 424 brightness temperatures of, 467 ratio of specific heat capacities coherent radiation of, 467 for non-relativistic degenerate gas, 432 maser emission of, 467 Rayleigh criterion for resolving two point sources, radio emission of, 467–469 843 curvature radiation and, 468 Rayleigh scattering, 380, 381 electron/death line and, 468 Rayleigh–Taylor instability, 419 electron/positron–photon cascades and, Rayleigh-Jeans limit, 240 468 Rayleigh-Jeans region of Planck spectrum, 189 radio quiet quasars Rayleigh-Jeans spectrum, 501, 777 counts of, 646, 647, 792 Rayleigh-Taylor instability, 607, 609, 610 cut-off at large redshifts, 645, 793 recoil effect, 260, 268, 280 definition of complete samples of, 647 reconnection of magnetic lines of force, 351, completeness of, 647 354–360 dispersion prism-grating techniques, 645 diffusive time-scale, 358 multi-colour photometric technique, 644– formation of current filaments by tearing 645 mode instabilities, 359 searches for ‘i-band drop-outs’, 645, 793 ohmic losses, 355 searches for Lyman-α and CIV emission Petschek mechanism, 358 lines, 645–646 reconnection velocity, 357, 358 searches for variability of, 646–647 self-consistency of models for, 360 ultraviolet excess technique, 643–644 Sweet–Parker mechanism for, 355–358 discovery of, 792 rectangular one-dimensional aperture evolution of, with cosmic epoch, 645, 792 diffraction pattern of, 840 luminosity function of red and blue sequences of galaxies, 89–95 ‘luminosity evolution’ of, 792 colour and absolute magnitude, 89–91 evolution of, 647, 792, 793 effect of galaxy environment, 92–93 radio waveband, 19–23 mean stellar age and concentration index neutral hydrogen and molecular line astron- C, 93–95 omy, 20–22 Sersic´ index and colour, 91–92 observing the sky in, 22–23 statistics of galaxies belonging to, 95 origin of high energy astrophysics and, 19– red sequence, 89 20 preferentially found in rich cluster environ- Radiocarbon, 325 ments, 93 radiocarbon dating, 324–326 properties of galaxies of, 89 calibration of, 325 Red-Sequence Cluster Survey, 135 impact of nuclear test explosions, 326 redshift influence of Earth’s magnetic field, 326 gravitational, 471 using coral samples, 326 reduced mass of molecule, 367 using tree ring dating, 325 reference frames in standard configuration, 148, detection of supernovae in tree-ring data, 149 326 Reissner-Nordstrøm metric, 474 radionuclides, 307 relativistic aberration, 220 radius of curvature formulae, 220, 221 instantaneous, 222, 224, 225, 229 relativistic beaming range R of high energy particles, 155 synchrotron radiation and, 220–222, 226, in different materials, 157 232 rapid or r-process, 72, 424 relativistic invariants, 169–170 930 INDEX

total energy loss rate, 169, 170, 177, 190, Sagittarius A∗ (Sgr A∗), 826 215 supermassive black hole in, 826, 827 relativistic jets Salpeter initial mass function, 394, 800, 813, heating of the interstellar gas by, 821 814 relativistic kinetic energy, 857 SAS-2 γ-ray satellite, 552 relativistic length contraction, 182 SAS-2 satellite, 28 relativistic Maxwellian distribution, 240 scalar potential φ, 174 relativistic plasma, 20 scale height of an atmosphere, 353 relativistic three-momentum, 197, 200, 857 scattering of charged particles by irregulari- relativistic total energy, 857 ties in magnetic field, 202–205 relativistic transformation of an inverse square scattering of high energy particles by Alfven´ law Coulomb field, 149–151 and hydromagnetic waves, 205–207, 567 Relativity: Special, General and Cosmologi- Schonberg–Chandrasekhar¨ limit, 54, 66 cal (Rindler), 855 Schonberg-Chandrasekhar¨ limit, 799 relaxation time of particles in a plasma, 331– Schechter luminosity function for galaxies 333 turnover luminosity L∗, 818 retarded potentials, 294 Schmidt law of star formation, see Schmidt- retarded time, 176, 223 Kennicutt law Reuven Ramaty High Energy Solar Spectro- Schwarzschild black hole, 536 scopic Imager (RHESSI), 318, 320 Schwarzschild metric, 469–475 Reynolds number R, 494, 495 Schwarzschild radius, 20, 450, 469, 471, 473, Richtmeyer-Meshkov instability, nonlinear , 475, 484, 487, 488, 667–669, 681, 688, 629 745, 770 right ascension (RA or α), 825, 826 coordinate singularity in, 473 rigidity R, 197, 198, 204 physical singularity at r = 0, 473 rigidity, magnetic, 579 X-ray emission from the vicinity of, 26 Rindler, Wolfgang, 473 SCUBA submillimetre bolometer array, 15, ring galaxies, 88 812 Roche lobe, 502, 506–510, 524 second-order Fermi acceleration, see Fermi overflow, 509, 516, 529, 531 acceleration - original version Roche lobe overflow, 80 Sedov expansion phase of supernova remnant, ROSAT All Sky Survey, 597 605–607, 626 ROSAT satellite dynamics of, 606 ultraviolet Wide Field Camera of, 389 seeing disc, 844 ROSAT Wide Field Camera, 24 variation in size with wavelength, 844 ROSAT X-ray Observatory, 26, 27, 123, 461, self-collision time of particles in a plasma, 597, 657, 781, 794 331 Rosseland Mean Opacity, 52 semi-analytic models of galaxy formation, 819– Rosseland mean opacity, 502 823 Rossi X-ray Timing Explorer (RXTE), 458, as ‘experimental computational astrophysics’, 489, 527, 531, 533, 715 820 rotation curves of spiral galaxies, 101–102 dust extinction and, 821 constancy at large radial distances, 101 formation of massive black holes in the nu- rotation measure, 377, 402–404 clei of galaxies and, 821 as a function of galactic latitude, 402, 403 formation of supermassive black holes and, pulsar, 403, 404 823 RR-Lyrae variable stars, 81 in centres of rich clusters, 823 masses of, 81 heating of the intergalactic gas in clusters and, 821 Sersic´ index n, 91–93 nuclear starbursts and, 820, 821 SAGE solar neutrino experiment, 34 objectives of, 820 INDEX 931

quasars at large redshifts and, 822 background-limited, 851 radiative cooling and star formation, 820 dependence on size of telescope for differ- spectrophotometric properties of galaxies ent limiting cases, 851 and, 821 detector-noise-limited, 851 supernova explosions and, 820 in the presence of other sources of noise, the epoch of maximum quasar activity and, 850 823 background signal from sky and telescope, two-point correlation function for galaxies 850 and dark current in the detector, 850 for different luminosities and colours, 821 read-out noise, 850 semi-forbidden transitions in spectra of gaseous photon statistics and, 850 nebulae, 374–375 photon statistics limit, 851 example of C III], 374 signature of metric, 856 intercombination lines, 374 silicate dust grains sensitivities of astronomical detectors, 848– complex dielectric constant for, 381 854 silicon burning, 72 in the photon limit, 849–851 single line spectroscopic binaries, 480 in the wave limit, 852–854 single-line spectroscopic binaries, 453 increasing signal-to-noise by increasing singularity theorems of Penrose and Hawk- bandwidth and integration time, 853 ing, 473 optical and infrared detectors, 849–851 sky in different astronomical wavebands, 5 radio and millimetre-wave receivers, 852– Sloan Digital Sky Survey (SDSS), 4, 7, 85, 854 89, 90, 92, 93, 98, 107, 108, 111, 645, Seyfert galaxies, 647–651, 792 646, 782, 793 counts of, 647, 792 slow or s-process, 72, 73 Shapiro time delay, 456 Smithsonian SubMillimetre Array (SMA), 15 shear stress, 493, 494 Smoluchowski’s envelope, 114 shock waves, 344–349, 389, 622 soft γ-ray repeaters, 458 basic properties of plane, 345–348 soft X-ray emission from the Galactic plane, collisionless, 352, 627 391 conservation relations for, 345 Solar and Heliospheric Observatory (SOHO) energy flux conservation, 346 of ESA, 58, 59 mass conservation, 345, 346 solar flares, 353 momentum flux conservation, 346 γ-ray spectrum of, 318, 320 explosions and, 344 solar luminosity, 40 heating of gas to high temperatures and, Solar magnetohydrodynamics (Priest), 353 348 solar mass, 40, 832 Mach number, 347 Solar Maximum Mission, 427, 428 oblique, 349 solar modulation, 204, 205, 247, 390, 549, passage of streamlines through, 350 552 role of atomic or molecular viscosities, 348 cosmic ray energy spectrum and, 539 shock conditions, 346 diffusive-convective model for, 540 speed of sound and, 344, 345 electron energy spectrum and, 542 stand-off distance, 349 solar neutrino experiment, 61 strong, 347–348, 622, 623 solar neutrino problem, 61 supersonic piston, 348–349 solar neutrinos, 60–66 shock waves, collisionless, 633 and neutrino oscillations, 63, 65 SI (Systeme` International) system of units, chlorine detector for, 60, 61 141 GALLEX solar neutrino experiment, 63 sidelobes, 844 gallium experiments, 63 signal-to-noise ratio Kamiokande II experiment, 62 932 INDEX

SAGE experiment, 63 features of, 313–315 Sudbury Neutrino Observatory (SNO), On- methods of determining tario, Canada, 64, 65 experimental, 312 SuperKamiokande experiment, 62, 63 Monte Carlo techniques, 312 solar radius, 40 semi-empirical relations, 312 solar seismology, see helioseismology partial, 310–315, 554, 555 Solar System abundances of the elements, 543– total inelastic, 310, 313 545 weighted average total inelastic, 556 local interstellar abundances, 543, 544 spallation fragments, 308 meteoritic abundances, 543 relativistic, 309, 310 solar wind, 20, 78, 332, 342–344, 349, 465, special relativity 540, 567 notation, 855–857 high-speed winds observed in, 343, 344 specific angular momentum, 471, 473 magnetic field in, 342 specific angular momentum in general rela- magnetosphere and the, 349–352 tivity, 474 drag forces associated with, 351 specific heat capacities γSH rotating garden sprinkler and trajectories of ratio of, for perfect gas, 240 particles in, 342 ratio of, for relativistic gas, 240 spiral configuration of magnetic field in, specific heat capacity per unit mass, 338 343 specific volume, 346 out to 20–25 AU, 344 speckles, 845 typical properties of, 342 spectral energy distribution of galaxies Soudan Cryogenic Dark Matter Search (CDMS), evolution of, 798–800 36 in far infrared and submillimetre wavebands, source function, 244 810, 811 south celestial pole (SCP), 825, 826 spectral index, 234, 239 South Pole Observatory, 15 spectrum of radiation of an arbitrarily moving Soviet atomic and hydrogen bomb programme, electron, 223–224 275 spin temperature for 21-cm absorption, 365 spallation, 162–164, 307, 542, 546, 547, 554 spinning dust grains formation of L from M elements, 556 radio emission of, 408–410 formation of radioactive isotopes in mete- spiral arm tracers in the Galaxy, 388 orites and, 164 spiral galaxies formation of rare isotopes in meteorites and, dark matter haloes, 102 162, 163 radio haloes of, 568 path length distribution for, 557–558 Spitzer infrared space observatory, 781 characteristic escape time τe, 557 Spitzer infrared space telescope, 807 dependence upon cosmic ray energy, 561 Spitzer Space Infrared Observatory, 596 exponential, 557 Spitzer Space Telescope, 11 Gaussian, 557, 558 spontaneous and induced processes, 272, 274 isotopic anomalies, 563 rules for, 272, 274 leaky box model for, 558 stability of disc galaxies and dark matter haloes, truncated exponential, 558 102 path length through interstellar gas for, 555 stand-off distance, 349, 350 production of 3He from 4He, 556 stand-off shock waves, 344, 349 products of, of iron nuclei, 556, 557 standard wavebands used in ground-based as- radioactive products, 547 tronomy, 834, 835 slab model for, 555, 557, 558 star clusters spallation cross-sections, 310–315, 554 ages of, 76 dependence upon particle energy, 313–315 open, 76 for iron nuclei, 315 star formation, 84, 393–402 INDEX 933

angular momentum problem, 400 statistical weights g1, g2, 187 energy problem, 400 Stefan-Boltzmann law, 39, 41, 51, 500, 830 initial mass function, 393–396 stellar coronae, 78 issues in the theory of, 400–402 stellar evolution Jeans’ instability, 400–401 formation of isothermal core, 66 magnetic field problem, 400 growth of central helium core, 68 model for, 384–386 hydrogen-shell burning, 66 pre-main sequence evolutionary tracks, 78 lower mass limit of 0.08 M for hydrogen pre-main sequence stars and, 385, 386 burning stars, 73 regions of, 396–398 of a 1.3 M star, 55 Schmidt-Kennicutt law, 393–396 of a 15 M star, 420 star formation in molecular clouds core collapse of, 421 radiation of heated dust grains at infrared formation of iron core, 421 wavelengths as a signature of, 383 neutrino luminosity of, 420, 421 role of interstellar dust grains, 383 of a 5 M star, 69, 70 star-forming galaxies, 383 Stellar Magnetism (Mestel), 464 spectrum of, 806 stellar structure starburst galaxies, 809 equations of, 43–56 ultraviolet spectra of, 813, 814 stellar winds, 78 starquakes, 463 stellar X-ray sources in Magellanic Clouds stars and stellar evolution, 39–84 X-ray luminosities of, 487 basic observations of, 39–43 Stokes’ theorem, 355 equations of energy generation and energy stopping power, 155 transport, 48–52 straggling, 158 equations of hydrostatic support and mass structural index, or structural length, 114 conservation, 45–46 sub-dwarf (sdB) stars, 644 equations of stellar structure, 53–56 sub-dwarf stars, 43 evolution of high and low mass stars, 66– 76 sub-giant stars, 43 high mass stars, 69–73 Subaru 8-metre telescope, 814 low mass stars, 69, 73–76 Subaru 8.2-metre telescope, 417 Hayashi tracks, 66–69 Subaru Deep Surveys, 814 helioseismology and the internal structure submillimetre background radiation, 812 of the sun, 56–60 Sudbury Neutrino Observatory (SNO), 35 importance of mass loss, 79, 80 Sun mass loss, 77–84 central pressure of, 46 horizontal branch, 81 corona of, 342 overall mass loss rate, 83–84 main sequence lifetime of, 56 P-Cygni profiles and Wolf-Rayet stars, minimum central temperature of, 47 78–81 origin of magnetic field of, 60 planetary nebulae, 82–83 properties of, 40 observations of solar neutrinos, 60–66 rotation of, 342 stellar evolution on the colour-magnitude speed of sound as a function of radius, 58– diagram, 76–77 60 stellar structure, 43–48 sunspots, 353 sun as a star, 56–66 Sunyaev-Zeldovich effect, 282–286, 620 virial theorem for stars, 46–48 hot gas in clusters of galaxies and, 284– statistical equation of motion for mean drift 286 velocity of electrons in electric field, 333 Hubble’s constant and, 284 statistical equilibrium, 100, 832 in Rayleigh-Jeans region of the spectrum, statistical weights, 272 283 934 INDEX

increase in energy density of the background contact discontinuity in, 605, 606, 609, radiation, 283 610 physical nature of result in Rayleigh-Jeans four stages of evolution of, 604–607 spectral region, 284 intense X-ray emission from, 606, 607 Sunyaev-Zeldovich effect in hot intracluster late cooling by optical emission, 607 gas, 109, 125–129, 831 reheating by reverse shock in, 607 distance estimates using, 129 reverse shock in, 605, 606, 610 in the Rayleigh-Jeans limit, 126 snowplough phase of, 607 shape of spectral distortions, 285, 286 strong shocks in, 595 spectral shape of distortions, 126 supersonic expansion of, 606 spectral signature of, 126 undecelerated expansion phase, 604 SuZIE experiment and measurement of shape X-ray bremsstrahlung of, 595 of spectral distortions, 286 X-ray emission of, 595 SuZIE experiment and measurement of spec- supernova remnants as sources of high energy tral shape of distortions, 127 electrons, 603–604 super-Alfvenic´ motions, 352 supernova remnants as sources of high energy superCDMS experiment, 140 particles, 594–599 superclustering of galaxies, 111, 112 γ-ray observations of shell-like, 597–599 superconductivity and superfluidity in neutron radio observations of, 595 stars, 442–443 radio observations of shell-like, 596 supergiant stars, 43 supernova SN 1987A, 425–430 superluminal motions, 567 44Ti energy source for light curve, 317, 429 in microquasars, 481 56Co γ-ray emission lines from, 317, 427, supernova light curve 428 radioactive origin of, 317 57Co energy source for light curve, 429 supernova rate in our Galaxy, 604 [CoII] and [NiII] infrared emission lines supernova remnants, 414, 430–431 from, 427, 428 as X-ray sources, 431 collision of envelope with mass-loss ring, acceleration of the high energy particles and 429, 430 the generation of strong magnetic fields observed at optical, X-ray and radio wave- in, 609 lengths, 430 acceleration of the high energy particles in, distance estimate from [OIII] ring, 430 610 dust formation in, 429 adiabatic loss problem, 607–611 light curve of, 425–427, 429 adiabatic losses and decrease in radio lu- limits to neutrino mass, 427 minosity of, 608 neutrino luminosity of, 427 adiabatic losses for relativistic gas, 607 neutrinos from, 307 as supersonically expanding spheres of hot progenitor of, 425, 426 gas, 610 rings of [OIII] about, 429 evolution of, 604–607 stellar mass-loss prior to, 426, 429, 430 filled-centre, or Crab-like, 594 Type IIP, 425 γ-ray properties of, 594 supernova units (SNu), 417 generation of strong magnetic fields in, 609– supernovae, 84, 414–431 610 as sources of high energy particles, 431 local energy density of high energy parti- classification of, 415 cles due to, 604 core-collapse, 419–422 radio properties of, 594–596 bounce mechanism for, 422 synchrotron radiation and, 595–596 formation of neutron stars and black holes Sedov expansion phase of, 605–607, 626 and, 419–422 shell-like, 594, 604 inhomogeneous expulsion of outer lay- adiabatic expansion of, 604, 605 ers, 422 INDEX 935

kinetic energy release, 421 relation between spectral index a and spec- neutrinos from, 427 tral index of electron spectrum p, 234 historical, 414–416 self-absorption of, 239–244 kinetic energy of ejected material, 430 absorption coefficient for, 242–244 light curves, 416 physical arguments, 239–241 rates of occurrence, 417 spectral emissivity of Type Ia, 416–419, 508 in orthogonal polarisations, 229 Type II, 509 maximum of, 230, 600 Types I and II, 415 total, 229 differences between, 415 spectrum of a power law distribution of elec- typology, 414–416 tron energies, 233–235 supersonic motion, see shock waves emissivity per unit volume, 235 supersonic piston, 348–349, 606 full analysis, 234 location of shock front relative to, 349 physical arguments, 234 supernovae and the, 349, 606 spectrum of, detailed analysis, 223–233 Sutherland, Peter, 468 algebra of, 225–229 Sweet–Parker mechanism of magnetic recon- of radiation of an arbitrarily moving elec- nection, 355–358 tron, 223–224 dissipation rate of, 356, 357 results of, 229–233 effects of gas pressure, 356 system of coordinates for, 224–225 energy released in, 358 spectrum of, physical arguments, 219–222 SWIFT satellite, 458, 773, 775, 777, 779 spontaneous transition probability for, 242 SWIRE Legacy sample, 809 total energy loss rate, 213–215, 231 symbiotic star, 419 useful numerical results, 244–246 symbiotic stars, 502 absorption coefficient χν for a random synchro–Compton catastrophe, 288–290 magnetic field, 246 synchro-Compton catastrophe, 590 critical frequency νc, 245 synchro-Compton radiation, 590 emission spectrum of a single electron, synchrotron radiation, 213–249, 411, 448, 534, 245 782 radiation spectrum of a power-law elec- as the relativistic limit of cyclotron radia- tron energy distribution, 245 tion, 217 total energy loss rate, 244 asymptotic expression for emissivity at high synchrotron radiation as the scattering of vir- and low frequencies, 231 tual photons, 265 critical angular frequency for, 229, 230 synchrotron radiation facilities, 230 critical frequency for, 222, 232–234, 240, synchrotron self-absorption, 239–244, 288, 289 245 absorption coefficient for, 242–244 high frequency limit, 231–233 for a randomly oriented magnetic field, map of Galactic, 19 243 minimum energy requirements for, 585, 599– two-level system for, 242 603 evidence for presence of relativistic elec- polarisation of, 235–239 trons, 241 circular, 239 evidence for relativistic electron in com- fractional, for power-law electron energy pact radio sources, 289 spectrum, 238–239 physical arguments, 239–241 spectra of orthogonal polarisations I⊥ and polarisation of synchrotron self-absorbed source, Ik, 236 244 system of coordinates for, 225 spectrum of radio source exhibiting, 241 total energy in polarisations I⊥ and Ik, synchrotron–self Compton radiation, 286–291 237 models of sources of, 290–291 radio emission of the Galaxy and, 246–249 homogeneous, 291 936 INDEX

inhomogeneous, 291 emission spectrum of, 501, 503 synchro–Compton catastrophe, 288–290 emission spectrum of, in optically thin re- Systeme` International d’Unites´ (SI units), 854– gion, 504 855 formation of hot spot, 511, 517, 518 instabilities in, 521 TAMA gravitational wave experiment, 36 opacity of material of, 502 tearing mode instabilities, 359 role of viscosity - the α parameter, 493– technetium Tc, 73 496 telluric absorption, 10 structure of thin discs, 496–499 temperature-frequency relation for black-body temperature distribution in, 500–501 radiation, 6 transition from dominance of gas pressure term diagram for doubly ionised oxygen OIII, to radiation pressure, 502, 503 373, 374 Third Reference Catalogue of Bright Galax- The physics of solar flares (Tandberg-Hanssen ies (de Vaucouleurs et al), 95 and Emslie), 353 Thomson and Compton scattering, 255–261 Theoretical Concepts in Physics (Longair), ii, Thomson cross-section, 52, 252, 486 175, 469, 852, 857 Thomson scattering, 52, 54, 255–486 Theoretical Concepts in Physics, 606 degree of polarisation of, 258 thermal bremsstrahlung, see bremsstrahlung, derivation of formulae for, 255–259 thermal, 370–371, 519 differential cross-section for, 257, 266, 275 absorption in plane of Galaxy, 371 Eddington limiting luminosity and, 452 absorption, at radio wavelengths, 371 for 100% polarised emission, 257 at soft X-ray energies from hot component geometry of, 256 of interstellar gas, 371 mean free path for, 258 at X-ray energies from intracluster gas, 371 optical depth for, 258, 269, 278, 279, 282 and FeXXV emission lines, 371 polarisation of, 258 at X-ray energies from supernova remnants properties of, 257–259 and FeXXV emission lines, 371 total scattering cross-section for, 257 thermal energy per unit mass, 47 Thomson scattering cross-section σT, 170, 215 thermal instabilities, 392–393 Thorne, Kip, 474 thermal paradox for stars, 48 3CR radio galaxies, 716, 717, 789, 818 thermal time-scale for stars, 48 early formation of bulk of stellar popula- thermalisation time for stellar systems, 103 tion, 819 thermalisation time of particles in a plasma, stellar masses of large redshift, 818 331–332 3CR radio sources, 789, 791 thermonuclear runaway, 516, 522, 525 threshold detectors, 291 stabilisation at about 108 K, 522 tidal radius of cluster of galaxies, 114 thick accretion discs, 504–505 tidal radius of galaxies, 133, 135 funnels along polar directions, 504, 505 time dilation formula Newtonian rotationally supported vorticity- in general relativity, 471 free torus, 504, 505 time to reach a given signal-to-noise ratio, 851 stability of, 504 tokamaks, 354 thin accretion discs, 491–504 torque, magnetic, 513, 515 about black holes, 499–500 torque, viscous, 494, 496, 513 inner boundary condition, 499, 500 total depth of the atmosphere in kg m−2, 193 conditions for, 492–493 transfer equation for radiation, 186, 188, 244 continuum spectra of, 518 in terms of Einstein coefficients for spon- detailed models of, 501–504 taneous and induced emission, 274 Doppler tomography of, 517, 519 in terms of occupation numbers, 273, 274 emission spectra of, 500–501 Transition Region and Coronal Explorer (TRACE) in black body approximation, 500 spacecraft of NASA, 339 INDEX 937 transmission function G(z) of one-dimensional UHURU X-ray observatory, 450, 529, 656 aperture, 838 UK Schmidt Telescope, 644 Fourier transform of, 839 ultra-high energy γ-ray telescopes, 300 Gaussian, 844 Ultra-Luminous Infrared Galaxies – ULIRGs, transmission function Tν of astronomical fil- 654–656 ters, 834, 835 ultrahigh energy γ-rays transparency of the atmosphere as a function detection of, 299–302 of wavelength, 5, 6 ultraluminous infrared galaxies (ULIRGs), 811 in the infrared and submillimetre wavebands, ultraviolet waveband, 23–24 10, 15 extreme ultraviolet (EUV), 23 windows in the submillimetre waveband, observations of resonance lines of common 15 elements, 23 tree-ring dating, 325 Ulysses mission of the European Space Agency Trinity atomic bomb test, 606 and NASA, 343, 344 triple-α process, 71 unification schemes for active galaxies, 658– Tully-Fisher relation for spiral galaxies, 96 665 and the distances of galaxies, 96 polarisation studies of Seyfert 1 and Seyfert infrared, 96 2 galaxies and, 658–660 turbulence radio quasars and radio galaxies, 660–665 magnetohydrodynamic, 496 unstable stars, 82 21-cm line emission and absorption, 363–365 useful statistics about galaxies, 107–108 absorption coefficient for, 364–365 V/V or luminosity-volume test, 786–787 column density of neutral hydrogen from, max banded, 787 364 for radio quiet quasars, 792 emissivity of, 364 space distribution of galaxies, quasars ac- kinematics of gas in galaxies from, 364 tive galaxies, 786–787 spin temperature for, 365 van den Heuvel, Edward, 449 spontaneous transition probability of, 363 variance, 849 2dF galaxy redshift survey, 644, 782 of sum of two quantities, 849 2dF quasar redshift survey, 644, 646, 647, total, as the sum of variances, 850 792, 793 vector potential A, 174, 176, 200, 232 Two Micron All Sky Survey (2MASS), 13, Vela satellites, 28 108 velocity cone, 236, 239 2MRS catalogue, 580 velocity of Solar System through the Cosmic two-colour diagram for stars, 837 Microwave Background Radiation, 16 Type Ia supernovae, 416–419, 522 velocity-distance relation for galaxies, 830 as standard candles, 417 vernal equinox, 825 evolution of colours of, 418 Very Large Array (VLA), 23, 136, 569, 595, 56 formation of Ni, 418 596, 664, 848 light curves Very large Array (VLA), 609 56 radioactive decay of Co, 418 Very Long Baseline Interferometry (VLBI), light curves of, 416 23, 241, 289 luminosity-width relation, 416, 417 vicinity of the black hole, 697–722 most luminous supernovae, 417 violent interstellar medium, 392 rates of occurrence, 417 supernova explosions and, 392 redshift-distance relation and, 417 VIRGO gravitational wave experiment, 36 thermonuclear explosions and, 415, 417, virial theorem, 113, 437 418 virial theorem for galaxies and clusters, 99– type Ia supernovae, 521–522 102, 104, 832 problems of application to observed sys- UHURU X-ray Observatory, 25, 26, 794 tems, 100, 101 938 INDEX virial theorem for stars, 46–48, 98 Wolf-Rayet star, 508, 509, 563, 780 viscosity, 494, 495, 502, 719 Wolf-Rayet stars, 78–81, 316 effective, in shock fronts, 627 mass loss rates of, 80 kinematic, 485, 720 WC and WN types, 80 turbulent, 504, 511 W± and Z0 bosons, 140 viscosity parameter α, 501, 504, 521, 722 viscosity, dynamic, 496, 498 X mass fraction of hydrogen, 77 viscosity, dynamic or shear η, 493, 495 X-ray absorption, 370 viscosity, kinematic ν, 494 X-ray absorption coefficient of interstellar mat- viscosity, turbulent, 495, 496, 501 ter, 252, 254 visible matter in the Universe column depth for, 254 integrated luminosity of, 107 optical depth and, 254 VLT Antu telescope, ISAAC infrared camera X-ray atomic energy levels, 253 of, 397 X-ray background at soft X-ray energies, 26, VLT Kueyen Telescope, FORS2 instrument 27 of, 398, 399 X-ray background emission voids in the distribution of galaxies, 800 at hard X-ray energies Voyager missions of NASA, 344 nature of sources contributing to, 657, 796 W UMa-type binaries, 506 at soft X-ray energies wave impedance Z0, 852 nature of sources contributing to, 794 wavebands used in ground-based astronomy, contributions of different classes of source 835 to, 797 wavefront errors, 843, 844 spectrum of, 794 Wheeler, John, 474 X-ray binaries, 509 white dwarf, 340 X-ray binary systems, 450–454, 502, 522– diffusion time for magnetic field from, 341, 537 342 accretion in, 452 magnetic flux density of, 341 estimate of mass of neutron stars in, 452– white dwarf stars, 41, 43, 76, 83, 84 454 white dwarfs, 413, 438–439, 483, 644 estimate of masses of neutron stars and black carbon-oxygen, 417, 418 holes in, 453 cooling curves for, 439 high mass X-ray binaries, 529–531 cooling times for, 439 high-mass X-ray binaries, 452 Hertzsprung-Russell diagram for, 439 late 0 or early B type companion stars, ultimate fate of, 439 452 with carbon-oxygen cores, 438 low mass X-ray binaries, 522–529 white dwarfs, neutron stars and the Chandrasekhar Galactic Bulge sources, 522 limit, 431–438 in globular clusters, 522 Wien distribution, 271 low-mass X-ray binaries, 452 average energy of photons of, 271 in globular clusters, 452 Wien’s displacement law, 5 luminosities of, 452 Wien’s law, 280 soft X-ray transients, 452 Wilkinson Microwave Anisotropy Probe (WMAP), symbiotic X-ray binaries, 452 17 typical temperature of, 450 WIMPs X-ray bursters, 452 astrophysical limits to the masses of, 139 X-ray pulsars, 452 candidates for, 138 X-ray emission from supernova shock fronts laboratory detection of, 140 synchrotron radiation and, 628 laboratory limits to the masses of, 140 X-ray emission of intracluster gas, 109 suppression mechanisms for, 140 X-ray nova, 531 INDEX 939

X-ray pulsars, 25, 479 X-ray source spectra and solutions of the Kom- paneets equation, 278–282 X-ray sources binary pulsating as rotating magnetised neutron stars, 488 Comptonisation and, 657, 797, 798 evolution of luminosity function of, with cosmic epoch, 796 populations at soft and hard X-ray energies, 657, 794, 796 strongly absorbed, 657, 797 template spectra for strongly absorbed sources, 658, 797 X-ray surveys of active galaxies, 656–658 X-ray waveband, 25–27 detectors for, 25 observing in, 25 sky in, 25–27 XMM-Newton X-ray Observatory, 27, 119– 121, 123, 219, 282, 421, 461, 687, 692, 781, 794

Y mass fraction of helium, 77 Yanasak, N.E., 565 year 2000.0 coordinate system, 825, 826 Yerkes spectral classiﬁcation system, 42

Z mass fraction of ‘metals’, 77 Zeeman splitting of 21-cm line radiation, 410 circularly polarised components of, 410 of OH absorption lines, 410, 411 zenith equidistant projection, 7, 827, 828