“Filling an important gap in a topical and fast-evolving area, this interesting book will be a valuable addition to the astrophysics literature. The scientific content is of a high quality, and includes a notable level of rigor in the derivations.”–Peter Mészáros, Pennsylvania State University

“There is definite need for a book on this topic. Dermer and Menon have gathered together a wide range of useful results and the book’s rigorous and comprehensive coverage of high-energy processes will be valuable to the advanced researcher in the field. Astronomers working on black-hole processes, jets, AGN, and gamma-ray bursts will want to have this for reference.”–Andrew Fabian, University of Cambridge

Charles D. Dermer is a theoretical astrophysicist at the U.S. Naval Research Laboratory. Govind Menon is professor of physics at Troy University.

High Energy Radiation from Black Holes

Gamma Rays, Cosmic Rays, and NeutrinosBy Charles D. Dermer Govind Menon

PRINCETON UNIVERSITY PRESS

Copyright © 2009 Princeton University Press
All right reserved.
ISBN: 978-0-691-14408-5

Contents

Preface………………………………………………………..xixChapter 1. Introduction………………………………………….1Chapter 2. Relativistic Kinematics………………………………..14Chapter 3. Introduction to Curved Spacetime………………………..25Chapter 4. Physical Cosmology…………………………………….36Chapter 5. Radiation Physics of Relativistic Flows………………….50Chapter 6. Compton Scattering…………………………………….70Chapter 7. Synchrotron Radiation………………………………….117Chapter 8. Binary Particle Collision Processes……………………..160Chapter 9. Photohadronic Processes………………………………..187Chapter 10. Pair Production………………………………………227Chapter 11. Blast-Wave Physics……………………………………258Chapter 12. Introduction to Fermi Acceleration……………………..314Chapter 13. First-Order Fermi Acceleration…………………………327Chapter 14. Second-Order Fermi Acceleration………………………..351Chapter 15. The Geometry of Spacetime……………………………..379Chapter 16. Black-Hole Electrodynamics…………………………….417Chapter 17. High-Energy Radiations from Black Holes…………………452Appendix A: Essential Tensor Calculus……………………………..473Appendix B: Mathematical Functions………………………………..488Appendix C: Solutions of the Continuity Equation……………………492Appendix D: Basics of Monte Carlo Calculations……………………..497Appendix E: Supplementary Information……………………………..499Appendix F: Glossary and Acronym List……………………………..505Bibliography……………………………………………………509Index………………………………………………………….531

Chapter One

Introduction

1.1 BLACK HOLES IN NATURE

Because of their brilliance, black holes give us a unique view of our universe at both the smallest and largest scales. Masses of candidate black holes associated with astrophysical sources include ~ (3-10)M]??] black holes found in our Galaxy, intermediate-mass (~([10.sup.2]-[10.sup.4])M]??]) black holes observed in nearby galaxies, massive (~([10.sup.5]-[10.sup.7])M]??]) black holes found in the centers of relatively nearby (redshift z [less than or equal to] 1) galaxies, and supermassive (~([10.sup.8]-[10.sup.9])M]??]) black holes at all redshifts (from z <<1 to z > 7). Moreover, newly formed black holes are thought to power the intense emissions from gamma-ray bursts (GRBs) that occur in galaxies at all distance scales, from relatively nearby galaxies within [10.sup.2] Mpc to structures at redshifts z > 8.

A supermassive black hole found at the center of an active galaxy can shine more brightly than all the stars in the galaxy. Whereas the total starlight power from a large spiral galaxy like the Milky Way may reach [approximately equal to] [10.sup.45] ergs [s.sup.-1], the total luminous powers of the nuclei of active galaxies, called active galactic nuclei (AGNs), commonly exceed ~[10.sup.44] ergs [s.sup.-1]. The very brightest AGNs can have apparent isotropic bolometric powers exceeding [10.sup.49] ergs [s.sup.-1]; apparent because the radiation may be directed into a small fraction of the full sky, so that the actual luminosity is reduced by a beaming factor related to the solid angle into which the emission is radiated. Apparent isotropic GRB powers can exceed ~[10.sup.52] ergs [s.sup.-1]. Jet sources from candidate black holes are often most luminous at [gamma]-ray energies, and a variety of phenomena supports a picture of relativistic bulk plasma outflow powered by a black-hole engine.

A fundamental question is the power source of the luminous, energetic emissions from black holes. The energy sources are essentially limited to accretion and black-hole rotation. For matter accretion, dissipation of gravitational potential energy releases [less than or equal to] 5-30% of the rest-mass energy, depending on the spin of the black hole and how the angular momentum is deposited at the disk inner edge. The diversity of black-hole spectral states suggests that a parameter at least as important as the total black-hole mass accretion rate, M, also plays a role. This could be the spin parameter of the black hole. Rather than describe emissions formed by quasi-spherical accretion or disk accretion onto black holes, which are likely to be quasi-thermal and are well reviewed in a number of books [1-3], here we concern ourselves with the high-energy radiation from black holes associated with relativistic jetted outflows. We consider whether the electromagnetic process can effectively extract energy from a rotating black hole to make luminous black-hole jet emissions, and other high-energy cosmic-ray and neutrino radiations. For this purpose, a formulation of general relativity is developed to understand black-hole electrodynamics and the Penrose process operating near black holes.

The theory of [gamma]-ray emission in black-hole sources cannot be separated from the theory of synchrotron radiation, because the same electrons that Compton-scatter soft photons to [gamma]-ray energies also radiate synchrotron photons at radio and optical frequencies. The lepton acceleration process would likewise accelerate baryons. Ultra-high-energy cosmic ray nucleons are subject to photohadronic losses, accompanied in some cases by the formation of baryonic resonances that decay into photons, leptons, neutrons, and neutrinos. To understand the underlying connections between the accelerated particles that radiate [gamma] rays and neutrinos and those particles that become cosmic rays requires knowledge of the underlying astrophysical processes.

The goal of this monograph is to provide a systematic presentation of the theory of black-hole physics and nonthermal, high-energy radiation mechanisms. We investigate the hypothesis that the most luminous and highest-energy radiations in nature are powered by rotating black holes.

The high-energy universe of nonthermal particles and radiations begins where the hot universe leaves off: when the concept of temperature breaks down. Thermal processes involve particle distributions that are described by the Maxwell-Boltzmann form. The relativistic Maxwell-Boltzmann distribution is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

[4], where the dimensionless temperature [Theta] = [k.sub.B]T/[m.sub.e][c.sup.2], [m.sub.e] is the rest mass of the electron (table 1.1), [n.sub.0] is the particle number density (particle number and energy densities are generally denoted by lower-case symbols), and [K.sub.2](x) is a modified Bessel function of the second kind of order 2 (see Appendix B). Throughout we write the total energy E of a particle of mass m in terms of the particle Lorentz factor [gamma] = E/[mc.sup.2], so that the particle’s kinetic energy is ([gamma] – 1)[mc.sup.2], and the particle’s (dimensionless) momentum p = [[beta].sub.par][gamma], where [[beta].sub.par] = [[square root of 1 – [[gamma].sup.-2].sup.3]. Because relativistic particles are mainly considered, we will frequently take the [gamma] >> 1, [[beta].sub.par] [right arrow] 1 asymptote. The particle Lorentz factor will be denoted by in the fluid frame, except when we consider problems involving multiple species of particles, where the Lorentz factors of the different species must be distinguished.

At relativistic temperatures, [Theta] >> 1, the electron-positron pair production rate increases in proportion to the square of the particle density while the pair annihilation rate declines. Pairs begin to multiply without limit above some fixed temperature [approximately equal to] 20 MeV [5,6], requiring ever-increasing energy input. Steady thermal plasmas therefore cannot exist at tens of MeV temperature and higher. Moreover, gases with temperatures [greater than or equal to] 10 MeV cannot relax to a Maxwell-Boltzmann distribution, because the bremsstrahlung energy-loss rate dominates the Coulomb thermalization collision rates [7]. (Though both are proportional to the particle number densities, bremsstrahlung energy losses increase with particle energy and elastic Coulomb processes decrease with particle energy.) Thus there is no demonstrated way for the gas to thermalize. Plasma processes are often invoked to achieve rapid thermalization on a timescale related to the inverse of the plasma frequency, but special properties of the waves are required to give a quasi-thermal particle distribution. If a system of characteristic size R containing electrons with density [n.sub.e] has Thomson depth [[tau].sub.T] = [n.sub.e] [[sigma].sub.T]R [greater than or equal to] 1, which is required to obtain large luminosities from small regions, then no steady, thermal plasmas with temperatures >> 1 MeV can exist. Allowed system luminosities and total Thomson depths, including pairs, bifurcate into a normal and pair-dominated branch below a maximum critical temperature for fixed temperature and proton Thomson depth [8]. Photons exceeding many tens of MeV, or hundreds of MeV from pionic emissions of thermalized protons, must therefore originate from nonthermal processes. The existence of cosmic rays and GeV-TeV [gamma]-ray sources attests to the importance of nonthermal processes throughout the universe.

The Electromagnetic Spectrum

During the past ~75 years, the cosmic electromagnetic spectrum outside the visual domain has been explored using telescopes with different limiting sensitivities and imaging capabilities over the range of frequencies v (Hz) extending from [approximately equal to] 30 kHz to v [greater than or equal to] [10.sup.27] Hz (i.e., photon energies E [greater than or equal to] 10 TeV); see figure 1.1. Absorption by interplanetary plasma limits observations from space-based radio detectors below [approximately equal to] 30 kHz, and the Earth’s ionosphere hampers observations of cosmic radio emission below [approximately equal to] 10-30 MHz frequencies. The sky is transparent at higher radio frequencies and in the optical waveband. Space research overcomes the limitations of the obscuring atmosphere in many wavelength ranges.

Ultraviolet and X-ray fluxes from compact sources are often attenuated by photoelectric absorption from intervening neutral or ionized matter, including absorption by Milky Way gas. At TeV energies, the universe becomes opaque to pair production attenuation of very high-energy photons ([gamma]) on cosmic infrared photons ([gamma]’). This process, represented by the reaction [gamma][gamma]’ [right arrow] [e.sup.+][e.sup.-], prevents us from seeing extragalactic sources of multi-TeV radiation at distances [greater than or equal to] 1 Gpc (redshifts z [greater than or equal to] 0.2). The highest-energy cosmic photon yet detected was at [approximately equal to] 90 TeV from a Milagro source [9]. At [greater than or equal to] 100 TeV energies and higher, telescopes yet lack sufficient sensitivity to detect cosmic sources or the plane of our Galaxy.

It is convenient to express the total energy [E.sub.[gamma]] of a photon in units of the electron rest-mass energy [m.sub.e][c.sup.2]. The dimensionless energy of a photon with frequency v is denoted by

[member of] = [E.sub.[gamma]]/[m.sub.e][c.sup.2] = hv/[m.sub.e][c.sup.2]. (1.2)

The units of [m.sub.e][c.sup.2] give the energy units (table 1.1) of [E.sub.[gamma]]. The term [member of] in eq. (1.2) can also denote the dimensionless energy of highly relativistic ([gamma] >> 1) particles, such as cosmic rays or neutrinos. At nonrelativistic ([gamma] – 1 <<1) and transrelativistic ([gamma] – 1 [approximately equal to] 1) particle energies, the total energy, kinetic energy, and momentum must be more carefully related than in the ultrarelativistic regime, where p = [square root of [[gamma].sup.2] – 1] [right arrow] [gamma] when [gamma] >> 1. Figure 1.1 illustrates the energy scale of astronomical photon observations in Hz, eV, and dimensionless units.

The Astroparticle Spectrum

Cosmic rays consist of nonthermal protons, ions, electrons, positrons, photons, and antiprotons. Intensities of various cosmic-ray ionic components, multiplied by [E.sup.2.75], are shown in the left panel in figure 1.2. The dashed line gives the approximation [I.sub.p](E) = 2.2[E.sup.2.75] protons/([cm.sup.2] s GeV sr) multiplied by [E.sup.2.75] (therefore a straight line in this plot) to the demodulated flux of cosmic-ray protons, valid from E [less than or equal to] 1 GeV to [approximately equal to] [10.sup.14] eV. Note that [alpha]-particles and heavier nuclei make an increasing contribution to to the cosmic-ray flux near the knee of the cosmic ray spectrum at [approximately equal to] 3 PeV.

At the lowest energies of the nonthermal particle spectrum, Solar energetic particles accompanying Solar flares and coronal mass ejections have been measured with energies ranging from keV/nucleon to GeV/nucleon. Cosmic rays from outside our Solar system range in energy from directly detected ~GeV/nucleon cosmic-ray ions, at lower energies, to air shower and fluorescence events with energies exceeding [greater than or equal to] [10.sup.20] eV at high energies. For energies [less than or equal to] few GeV/nucleon, the cosmic-ray flux varies with the phase of the 11-year Solar sunspot cycle. This Solar modulation of the low-energy cosmic-ray flux results from adiabatic losses of the cosmic rays as they penetrate the heliosphere and diffuse through the outflowing Solar wind.

The heliosphere does not significantly perturb trajectories of cosmic rays with E [greater than or equal to] 10 GeV/nucleon, and these cosmic rays sample the interstellar cosmic-ray flux. In a magnetic field with intensity B, the Larmor radius of a particle with mass m, atomic mass A, and charge Z is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.3)

Particles with [r.sub.L] >> R will escape from or penetrate into a region of size R with mean magnetic field B, and will do this on the light crossing timescale R/c. Cosmic-ray protons with energies E [greater than or equal to] [10.sup.17] eV, well above the knee energy at [approximately equal to] 3 PeV, have Larmor radii comparable to the [approximately equal to] 180 pc thickness of the Galaxy’s disk. At such energies, cosmic-ray protons originating in the disk of the Galaxy will escape into the halo of the Galaxy and escape to intergalactic space. If formed beyond the Galaxy, such energetic cosmic rays can pass through the Galactic halo and into the Galaxy’s disk.

The intensity of all cosmic-ray particles, irrespective of type, multiplied by [E.sup.2.5], is shown for E >> [10.sup.14] eV cosmic rays in the right panel of figure 1.2. Features of the high-energy spectrum are labeled. At the highest particle energies, extending from ultrahigh energies (UHE; [greater than or equal to] [10.sup.18] eV) to super-GZK energies ([greater than or equal to] [10.sup.20] eV), the different signatures of the air showers formed by a photon and a particle are not clear cut, and depend on the particle physics model used to interpret air show data [12]. The Auger collaboration report [13] that the photon fraction of the UHECR (ultrahigh-energy cosmic ray) flux at E [congruent to] [10.sup.19] eV is [less than or equal to] 2%. The Auger result [14] on clustering of arrival directions of [greater than or equal to] 6 x [10.sup.19] eV UHECRs rules out a Galactic origin, and the large gyroradii and near isotropy of cosmic rays with E [less than or equal to] 4 x [10.sup.19] eV also suggest an extragalactic origin. The energy of the transition from a Galactic disk or halo component to an extragalactic component, assuming there is a definite transition energy, could be at the second knee or at the ankle, and is an open question.

High-energy neutrinos make up another important component of the astroparticle spectrum. The high-energy nonthermal neutrino window is being opened in this decade with observations from ~TeV to [10.sup.20] eV. Various estimates [16] suggest that detection of 100 TeV-100 PeV neutrinos from cosmic sources is probable with neutrino telescopes at the km scale, such as the IceCube experiment at the South Pole or a northern hemisphere neutrino telescope in the Mediterranean Sea. High-energy neutrino observations will be crucial to identify cosmic-ray accelerators and buried nonthermal sources.

The Gravitational-Wave Spectrum

The gravitational-wave spectrum offers a new window into the universe of high-energy activity. Distortions in spacetime induced by extreme events involving, for example, the coalescence of Solar mass neutron stars and black holes, induce minute strains on underground bars like those in the Laser Interferometer Gravitational Wave Observatory (LIGO). Relative displacements of the three-detector space-based array that forms the proposed Laser Interferometer Space Antenna (LISA) will be used to monitor coalescence of massive black holes. LIGO is now taking data at its design sensitivity, and LISA may be developed as a NASA mission in the 2010-2020 time frame or later. Template gravitational waveform studies of merging compact objects and coalescing black holes will be a crucial tool to compare with data from these detectors. This subject, though of potentially great importance to black-hole physics, will not be treated here (see, e.g., Ref. [17]).

(Continues…)

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