# High Energy Radiation from Black Holes: Gamma Rays, Cosmic Rays, and Neutrinos

Charles D. Dermer
Govind Menon
Pages: 568
https://www.jstor.org/stable/j.ctt7rphp

1. Front Matter
(pp. i-viii)
(pp. ix-xviii)
3. Preface
(pp. xix-xxii)
4. Chapter One Introduction
(pp. 1-13)

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 include¹$\~\;(3-10){M_\odot}$black holes found in our Galaxy, intermediate-mass$(\~\;({10^2}-{10^4}){M_\odot})$black holes observed in nearby galaxies, massive$(\~\;({10^5}-{10^7}){M_\odot})$black holes found in the centers of relatively nearby (redshift$z\;\lesssim\;1$) galaxies, and supermassive$(\~({10^8}{\kern1pt}-{\kern1pt}{10^9}){M_\odot})$black holes at all redshifts (from$z\;\ll\;1$to$z\;>\;7$). Moreover, newly formed black holes are thought to power the intense emissions from gamma-ray bursts² (GRBs) that occur in galaxies at all distance scales, from relatively nearby galaxies...

5. Chapter Two Relativistic Kinematics
(pp. 14-24)

In this chapter, the framework of Einstein’s special theory of relativity is presented. Transformations that render the spacetime interval$d{s^2}\;=\;-{c^2}d{t^2}+\;d{x^2}+d{y^2}\;+\;d{z^2}$unchanged define the category of physical theories that are Lorentz invariant. Further relativistic invariants, used to transform particle and photon distributions, are derived. The kinetic theory of reaction rates and secondary spectra occupies the final sections of this chapter.

Consider two coordinate systemsKand${K'}$in uniform relative motion, defining an inertial reference system. The reference frames are aligned along the${\hat x}$and${\hat x'}$axes, with frame${K'}$moving at speed$v\;=\;\beta c$in the positive${\hat x}$direction with respect...

6. Chapter Three Introduction to Curved Spacetime
(pp. 25-35)

After a review of aspects of special relativity relevant to the discussion in this chapter, the notions of curved spacetime and geodesic motion are introduced. Schwarzschild spacetime and gravitational redshift are considered as examples. While we do rely heavily on the topics covered in Appendix A, the covariant derivative involved in geodesic motion is independently derived here from a variational principle. Good introductions to special and general relativity can also be found in [38], [39], and [40].

In relativity there is no unique way to separate space from time. Unlike classical physics, where we have three-dimensional vectors, in special relativity...

7. Chapter Four Physical Cosmology
(pp. 36-49)

In this chapter, we forsake rigor in the interests of brevity [43]. The Robertson-Walker metric for an isotropic, homogeneous universe is derived from simple arguments. A Newtonian derivation of the motion of a gravitating fluid element leads to equations for Friedmann cosmologies relating proper-frame emission time${t_*}$to redshiftz. We deduce the luminosity and angular-diameter distances for flat cosmological models, yielding equations suitable for analyzing the statistics of cosmological black-hole sources, and for deriving the intensity from unbeamed and beamed sources.

In flat or Minkowski spacetime, a photon follows a null geodesic defined by$d{\kern 1pt} {s^2}\; = \;0$, where the invariant interval...

8. Chapter Five Radiation Physics of Relativistic Flows
(pp. 50-69)

Elementary concepts to analyze and transform radiation fields are introduced. The properties of the Planck blackbody function are derived. The results are applied to two geometries used to treat relativistic outflows from black-hole jet sources, namely, the blob geometry consisting of a relativistic spherical ball of radiating plasma, and a relativistic spherical shell geometry. The essential equivalence of the two geometries to infer properties of relativistic flows is demonstrated.

The intensity${I_\varepsilon }$is defined such that${I_\varepsilon }d\varepsilon dAdtd\Omega$is the infinitesimal energy$d\mathcal{E}$in photons with energy between$\varepsilon$and$\varepsilon \; + \;d\varepsilon$lying within solid angle element$d\Omega$that pass through area...

9. Chapter Six Compton Scattering
(pp. 70-116)

The astrophysics of the Compton scattering process for relativistic particles is treated in this chapter, specialized to relativistic electrons. After deriving the elementary Compton scattering formula, the behavior of the Compton cross section in the Thomson and Klein-Nishina regimes is examined. Compton scattering regimes are determined by the value of the invariant$\bar \varepsilon \; = \;\gamma {\kern 1pt} \varepsilon (1\; - \;{\beta _{{\text{par}}}}\mu )$, which is the photon energy in the electron rest frame ($\bar \varepsilon \; = \;\varepsilon$for electrons at rest). In the Thomson regime,$\bar \varepsilon \; \ll \;1$, and in the Klein-Nishina regime,$\bar \varepsilon \; \gg \;1$. For isotropic photon fields, analysis of the Compton energy-loss rate shows that the two regimes are characterized by the value of...

10. Chapter Seven Synchrotron Radiation
(pp. 117-159)

Relativistic charged particles emit synchrotron radiation when they are accelerated in the presence of a magnetic field. When both electric and magnetic fields are present, the electromagnetic force on a charged particle is given by the Lorentz force equation

${\vec F_{\text{L}}}\; = \;\frac{d}{{dt}}\;(\gamma m\vec \upsilon )\; = \;Q\;\left( {\vec E\; + \;\frac{1} {c}\;\vec \upsilon \; \times \;\vec B} \right).$(7.1)

Here${\vec E}$is the electric field vector,Qis the charge of the particle, and$\vec \upsilon \; = \;{{\vec \beta }_{{\text{par}}}}c$is its velocity, so${\beta _{{\text{par}}}}$is the particle beta factor.

In most astrophysical plasmas, the mobility of free charges causes the electric field to be shorted out.We assume this to be the case and take$\vec E\; = \;0$. When a particle does not experience significant...

11. Chapter Eight Binary Particle Collision Processes
(pp. 160-186)

This chapter considers binary particle collision processes, including

1. Coulomb interactions and ionization losses;

2. bremsstrahlung;

3. secondary nuclear production and spallation;

4. thermal electron-positron annihilation radiation; and

5. nuclear$\gamma {\text{ - ray}}$line radiation.

The rate of thermalization and energy transfer between ionic and leptonic species is determined in the first approximation by the rate of binary collision processes. In a magnetized plasma, collective effects may be as important as binary collisions to transfer energy, and so could increase the thermalization and energy-exchange rate. Elastic binary particle collisions between electrons, positrons, and protons involve the Møller$(e\; + \;e\; \to \;e\; + \;e)$, Bhabha$({e^ + } + {e^ - }\; \to \;{e^ + } + {e^ - })$, and Coulomb...

12. Chapter Nine Photohadronic Processes
(pp. 187-226)

In this chapter, three fundamental photohadronic processes are considered, each involving high-energy protons or ionsNwith atomic chargeZand atomic massAinteracting with target photons. They are the reactions

1.$N\; + \;\gamma \; \to \;N\; + \;\pi$(denoted$\phi {\kern 1pt} \pi$), photopion or photomeson production;

2.$N\; + \;\gamma \; \to \;N\; + \;{e^ + } + \;{e^ - }$, photopair$(\phi {\kern 1pt} e)$production; and

3.$N\; + \;\gamma \; \to \;N'\; + \;N''$, photodisintegration, for ions.

In addition, particles in intergalactic space lose energy adiabatically as the universe expands.

Charged pions formed by the photopion process decay into leptons and neutrinos, and neutral pions decay into$\gamma$rays. The secondary neutrino flavor ratio from$\phi {\kern 1pt} \pi$processes at production is

${\nu _e}:\;{\nu _\mu }\;:\;{\nu _\tau }\; = \;1\;:\;2\;:\;0,$

because the products of thedecay...

13. Chapter Ten $\gamma {\kern 1pt} \gamma$ Pair Production
(pp. 227-257)

In this chapter, the astrophysical importance of the$\gamma {\kern 1pt} \gamma \; \to \;{e^ + }{e^ - }$process that destroys$\gamma$rays and creates electron-positron pairs is considered. This reaction is the inverse of the pair annihilation process considered in section 8.4.

Topics include:

1. the$\gamma {\kern 1pt} \gamma \; \to \;{e^ + }{e^ - }$cross section${\sigma _{\gamma {\kern 1pt} \gamma }}(s)$and the$\gamma {\kern 1pt} \gamma$pair-attenuation optical depth per unit pathlength,$d{\tau _{\gamma {\kern 1pt} \gamma }}\;({\varepsilon _1}){\kern 1pt} /{\kern 1pt} dx$, for a$\gamma$ray with energy${\varepsilon _1}$;

2.$\delta {\text{ - function}}$approximation to the$\gamma {\kern 1pt} \gamma \; \to \;{e^ + }{e^ - }$cross section and an estimate of the$\gamma {\kern 1pt} \gamma$opacity${\tau _{\gamma {\kern 1pt} \gamma }}({\varepsilon _1})$;

3. calculations of the$\gamma {\kern 1pt} \gamma \; \to \;{e^ + }{e^ - }$opacity of the universe for$\~100\;{\text{GeV}}$– multi-TeV$\gamma$rays emitted from a source at redshiftzthat interact with photons...

14. Chapter Eleven Blast-Wave Physics
(pp. 258-313)

The methods of blast-wave physics, developed to explain gamma-ray burst (GRB) afterglows, can be used to model emissions from a relativistic flow that is energized through shock formation. The shocks are formed when the relativistic jetted plasma sweeps up material from the surrounding medium at an external shock, or when shocks are formed in collisions between inhomogeneous outflows in a relativistic wind. Because black holes are observed to be sources of collimated relativistic plasma outflows, as most clearly revealed by observations of jets and superluminal motion in Solar mass and supermassive black-hole systems, and also by achromatic breaks in GRB...

15. Chapter Twelve Introduction to Fermi Acceleration
(pp. 314-326)

The subject of particle acceleration is integral to our understanding of energetic nonthermal radiations. Our focus on Fermi acceleration does not preclude the existence of other accelerators based on magnetic reconnection, wakefield acceleration driven by the ponderomotive force of a radiation pulse, shear flows (which can, but need not be, a Fermi mechanism), Weibel or other plasma instability mechanisms, etc. Electrodynamic acceleration, as discussed in relation to neutron stars, pulsars, and magnetars [368], can also operate near black holes, and is studied in chapter 16. Because of its highly developed theoretical basis, and its ability to produce nonthermal power-law particle...

16. Chapter Thirteen First-Order Fermi Acceleration
(pp. 327-350)

The importance of first-order shock Fermi acceleration in driving ideas about particle acceleration and the origin of the cosmic rays demands a fuller review than a single chapter can provide, but here we present basic results about the first-order Fermi acceleration mechanism that are essential knowledge for deeper study.

We begin by deriving the hydrodynamic relation between Mach number$\mathcal{M}$and compression ratio$\chi$. The derivation of the convection-diffusion equation for nonrelativistic shocks and nonrelativistic particles is outlined, and the momentum representation of the particle spectrum is extended to relativistic energies. The shock spectral index in nonrelativistic shock acceleration is...

17. Chapter Fourteen Second-Order Fermi Acceleration
(pp. 351-378)

The mean energy gain per cycle in second-order Fermi acceleration was derived in section 12.4. Revisiting this point, let us consider a test particle that changes energy by first entering a magnetized cloud, scattering around until isotropized, and then leaving the cloud, as shown in figure 14.1. If${\gamma _1}$is the initial Lorentz factor of the particle, then its Lorentz factor in the cloud is${{\gamma '}_1}\; = \;\Gamma \,{\gamma _1}\;(1\; - \;\beta {\kern 1pt} {\beta _1}{\mu _1})$, where arccos${\mu _1}$is the angle between the cloud direction of motion and the direction of the particle. For isotropic elastic scattering, the Lorentz factor${{\gamma '}_2}$of the scattered particle is the same as...

18. Chapter Fifteen The Geometry of Spacetime
(pp. 379-416)

The presence of a massive object (compact or otherwise) deforms the ambient Minkowski geometry of spacetime in accordance with the Einstein equation. The gravitational field is described by the metric tensor${g_{\mu {\kern 1pt} \nu }}$of the manifold. On any region of the manifold we can place a coordinate system$(t,\;{x^1},\;{x^2},\;{x^3})$such thattis a timelike coordinate, and${x^i}$are spacelike for$i\; = \;1,\;2,\;3$. The metric tensor can then be written in the form

$g\; = \;{g_{tt}}d{\kern 1pt} t\; \otimes \;d{\kern 1pt} t\; + \;{g_{ti}}d{\kern 1pt} t\; \otimes \;d{\kern 1pt} {x^i} + \;{g_{it}}d{\kern 1pt} {x^i}\; \otimes \;d{\kern 1pt} t\; + \;{g_{i\,j}}d{\kern 1pt} {x^i} \otimes \;d{\kern 1pt} {x^j}.$(15.1)

Here${g_{\mu {\kern 1pt} \nu }}\; = \;{g_{\nu {\kern 1pt} \mu }}$, and sincetis a timelike coordinate,${g_{tt}}\; < \;0$. At every point of our manifold, the tangent space is isomorphic to Minkowski space. In particular, the...

19. Chapter Sixteen Black-Hole Electrodynamics
(pp. 417-451)

We have seen in chapter 15 that energy and angular momentum can be extracted from rotating black holes. Extraction of energy using particles does not seem to be a very efficient process in an astrophysical setting [459]. Energetic jets that emanate from supermassive black holes probably rely on electromagnetic fields and currents for their existence. To undertake a study of these matters we shall begin by looking at the equations of electrodynamics in a curved background whose metric is written in the form given by eq. (15.13). Just as we did with the metric, we shall rewrite the usual covariant...

20. Chapter Seventeen High-Energy Radiations from Black Holes
(pp. 452-472)

Energy from a black hole is generated by one of three mechanisms: (i) the release of gravitational potential energy by accretion of matter onto a black hole; (ii) dissipation of rotational energy stored in black-hole spin; and (iii) Hawking radiation from low-mass, evaporating black holes. For case (i), the energy of accreting matter is surely the source of the luminous UV and X-rays detected from radio-quiet Seyfert AGNs, reaching to$\approx \,{10^{46}}\;{\rm{ergs}}\;{{\rm{s}}^{ - 1}}$, and X-ray binaries in the Galaxy as bright as$\approx {10^{39}}\;{\rm{ergs}}\;{{\rm{s}}^{ - 1}}$. For case (iii), no evidence for evaporating black holes was found with EGRET [458], and it would be a...

21. Appendix A: Essential Tensor Calculus
(pp. 473-487)
22. Appendix B: Mathematical Functions
(pp. 488-491)
23. Appendix C: Solutions of the Continuity Equation
(pp. 492-496)
24. Appendix D: Basics of Monte Carlo Calculations
(pp. 497-498)
25. Appendix E: Supplementary Information
(pp. 499-504)
26. Appendix F: Glossary and Acronym List
(pp. 505-508)
27. References
(pp. 509-530)
28. Index
(pp. 531-538)