Active Galactic Nuclei E-Book

Volker Beckmann

• Herausgeber: Wiley-VCH
• Kategorie: Lebensstil
• Sprache: Englisch
• Veröffentlichungsjahr: 2013

Seitenzahl: 759

Beschreibung

The book gives an overview on the current knowledge of the Active Galactic Nuclei phenomenon. The spectral energy distribution will be discussed, pointing out what can be observed in different wavebands and discussing different physical models.

Leseprobe

Contents

Preface

Abbreviations and Acronyms

Astronomical and Physical Constants

Color Plates

1 The Observational Picture of AGN

1.1 From Welteninseln to AGN

1.2 Broad Lines, Narrow Lines, and the Big Blue Bump

1.3 Jets and Other Outflows

1.4 X-ray Observations: Probing the Innermost Regions

1.5 Up, Up and Away: from Gamma-Rays toward the TeV Range

2 Radiative Processes

2.1 Scattering of Photons

2.2 Synchrotron Emission

3 The Central Engine

3.1 The Black Hole

3.2 Accretion Processes

3.3 Absorption Close to the Black Hole

3.4 Photoionization Modeling

3.5 Narrow and Broad-Line Regions

3.6 Reverberation Mapping: Probing the Scale of the BLR

3.7 AGN Jets: Emission, Dynamics and Morphologies

4 AGN Types and Unification

4.1 Seyfert Galaxies

4.2 Low-Luminosity AGN

4.3 Ultraluminous X-ray Sources

4.4 Ultraluminous Infrared Galaxies – ULIRGs

4.5 Radio Galaxies

4.6 Quasars

4.7 Blazars

4.8 Unification of AGN

5 AGN through the Electromagnetic Spectrum

5.1 Radio: Probing the Central Engine

5.2 Infrared: Dust Near and Far

5.3 Optical: Where It All Began

5.4 UV: The Obscured Inner Disk

5.5 X-rays: Absorption, Reflection, and Relativistically Altered Line Profiles

5.6 Gamma Rays: the Blazar-Dominated Sky

5.7 VHE: the Evolving Domain

5.8 The Whole Picture: the Spectral Energy Distribution

6 AGN Variability

6.1 Variability in Radio-Quiet AGN

6.2 Analysis Methods for Variability Studies

6.3 Variability of Radio-Loud AGN

6.4 Quasiperiodic Oscillations in AGN

6.5 Rapid Variability

7 Environment

7.1 Host Galaxies of AGN

7.2 The AGN-Starburst Connection

7.3 Merging

7.4 AGN in Clusters of Galaxies

8 Quasars and Cosmology

8.1 The Universe We Live in

8.2 AGN and the Distribution of Matteron LargeScales

9 Formation, Evolution and the Ultimate Fate of AGN

9.1 The First AGN: How Did They Form?

9.2 Tools to Study AGN Evolution

9.3 Luminosity Functions of AGN

9.4 AGN and the Cosmic X-ray Background

9.5 The Late Stages of an AGN’s Life and Reignition SMBH

10 What We Don’t Know (Yet)

10.1 The Central Engine

10.2 Environment, Interaction, and Feedback

10.3 Origin, Evolution, and Fate

10.4 Continuing the Quest

References

Index

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The Authors

Prof. Dr. Volker Beckmann

APC Laboratory, CNRS/IN2P3

Université Paris Diderot

10 rue Alice Domon et Léonie Duquet

75013 Paris

France

Dr. Chris Shrader

NASA Goddard Space Flight Center and

Universities Space Research Association

Mail Code 661

Greenbelt, MD 20771

USA

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Preface

Active galactic nuclei (AGN) are the most energetic persistent objects in the Universe. Our understanding of them is roughly summarized in Chapter 1. Such a summary will be insufficient, but the goal is to provide an overview of the topic for the newcomer in the field. In Chapter 2 we take a quick tour through the radiative processes which are common in AGN and give the relevant formulas in order to make the arguments in the book understandable. Chapter 3 then discusses our understanding of what mechanisms drive the AGN emission, and what the main elements are, such as the black hole itself, the accretion disk, the broad and narrow line regions, out flowing jets and absorbing material. The different types of AGN are discussed in Chapter 4, including an attempt to explain all different types by the most simple model possible. In Chapter 5 we take a look at AGN in different energy bands, from the radio to the gamma-ray domain, and examine the overall energy output for beamed and nonbeamed sources. In Chapter 6 we will discuss what one can learn from variability studies of AGN. Up to that point we deal with the central engine and its closest surroundings. In Chapter 7 we will then have a look at in what types of galaxies and galaxy clusters supermassive black holes reside and how the AGN is influenced by the host galaxy and how in turn the central engine might affect the star formation in the surrounding medium. In order to understand the role of AGN in the Universe, we briefly discuss the current cosmological model in Chapter 8 and show how AGN can be used as tools for cosmological studies. We then turn in Chapter 9 to the ultimate question of where quasars come from, how they might be formed, and how they evolve. This also includes the aspects of AGN density evolution in time and how this might depend on the type of AGN or the energy range we study. Chapter 10 summarizes the open issues remaining in AGN research, that is, the big questions which still lack a satisfying answer. We hope that the reader will find that stimulating – and that she or he through their own research will contribute to further progress in this thriving field of astrophysics.

The literature about AGN is full of acronyms and abbreviations. We list the most important ones and those which are used in this book starting after the Preface. Finally, the physical and astronomical constants applied in this book are listed. Throughout the book we use cgs units rather than SI, as it is still common practice in astrophysics.

As for any such undertaking as writing a text book covering a broad scientific topic, we heavily relied on the experience and publications of a large number of scientists. Many of those publications are listed in the bibliography. Some of the textbooks we had at hand when compiling this book, were Osterbrock (1989), Peterson (1997), Kembhavi and Narlikar (1999), Krolik (1999), De Young (2002), and of course Rybicki and Lightman (1986). Throughout the book we point the reader to review articles on certain topics, and give some bibliographic references for further reading. Here we have put an emphasis on recent publications over more established ones. The idea is that the reader will usually find references to earlier work in the most recent publications in a field. We would like to apologize to all the colleagues whose work we have not mentioned or have not given the proper weight in this book.

This book would not have been possible to write without the help, advice of and discussions with many colleagues. Here we would like to thank in particular Chiara Caprini (CEA/Saclay) and Olaf Wucknitz (AIfA Bonn) for their advice on the cosmology chapter, Markos Georganopoulos (UMBC) on jets and radiative processes, Knud Jahnke (MPIA Heidelberg) for many comments on the environment of AGN, Demos Kazanas (NASA/GSFC) for interesting discussions and advice, Ralf Keil for digging out the NLS1 spectrum, Dirk Lorenzen, who knows how to get a book ready for publication, Piotr Lubiński (CAMK Torun) for reading the entire manuscript, Fabio Mattana (APC Paris) for the discussion about synchrotron self-absorption, Marie-Luise Menzel for the artwork on the unified model, Pierre-Olivier Petrucci (LAOG Grenoble) who provided advice on the typology of AGN, Michael Punch (APC Paris) for improving the VHE discussion, Tapio Pursimo (IAC Canary Islands), discussing the AGN phenomenology and appearance, Fabrizio Tavecchio (OAB Merate) for valuable advice on blazars and jets, Marc Türler (ISDC Geneva) who provided input on several topics, Jane Turner (UMBC) for sharing her abundant expertise on the X-ray properties of AGN, Lisa Winter (University of Colorado) for her advice on ultraluminous X-ray sources, and Christian Wolf (Oxford University) for explaining the multiband surveys. Volker thanks his mother for pointing out that NGC4889 hosts the most massive black hole known to mankind. We also would like to mention the support of our institutions and in particular Pierre Binétruy (APC Paris), Neil Gehrels (NASA/GSFC) and Mike Corcoran (USRA) for their support and encouragement in enabling us to write this book. A very special thank you goes to Simona Soldi (CEA/Saclay), for encouraging the project from the beginning, discussing the context on a daily basis, and correcting the whole manuscript several times.

Of the team at Wiley we would like to thank Oliver Dreissigacker, for contacting us in the first place and thus initiating the AGN book project, Ulrike Fuchs, the commissioning editor, Anja Tschörtner, for helping in the early phase of the project, and Nina Stadthaus, who was a very efficient and friendly editor helping with the technical realization of the project. Finally, we would like to thank Petra Möws and the team at le-tex for their support in the proof-reading process.

Paris and Greenbelt, May 2012

Volker Beckmann, Chris R. Shrader

Abbreviations and Acronyms

Here we give the list of the most common acronyms and abbreviations used in this book and in AGN science in general. We caution that the use of abbreviations is not consistent throughout the literature.

ACF

Autocorrelation function

ADAF

Advection-dominated accretion flow

AGN

Active galactic nucleus

ALMA

Atacama Large Millimeter/submillimeter Array

ASCA

Advanced Satellite for Cosmology and Astrophysics

ATHENA

Advanced Telescope for High ENergy Astrophysics

BAL

Broad absorption line quasar

BAO

Baryon acoustic oscillations

BHXRB

Galactic black hole X-ray binary

BLR

Broad-line region

BLRG

Broad-line radio galaxy

BSC

ROSAT All-Sky Survey Bright Source Catalogue

CADIS

Calar Alto Deep Imaging Survey

CCD

Charge-coupled device

CCF

Cross-correlation function

CDF

Chandra Deep Field (X-rays)

CGRO

(Arthur Holley) Compton Gamma-Ray Observatory

CMB

Cosmic microwave background

COMBO-17

Classifying Objects by Medium-Band Observations in 17 Filters

COSMOS

Cosmic Evolution Survey

CTA

Cherenkov Telescope Array

CXB

Cosmic X-ray background

DCF

Discrete cross-correlation function

EBL

Extragalactic background light

EC

External Compton scattering model

EGRET

Energetic Gamma-Ray Experiment Telescope (CGRO)

EMSS

Einstein Observatory Extended Medium Sensitivity Survey

eROSITA

extended ROentgen Survey with an Imaging Telescope Array

EUVE

Extreme Ultraviolet Explorer

EW

Equivalent width

FIRST

Faint Images of the Radio Sky at twenty-centimeters

FR

Fanaroff and Riley

FSRQ

Flat-spectrum radio quasar

FUSE

Far Ultraviolet Spectroscopic Explorer

FWHM

Full-width at half-maximum

GALEX

Galaxy Evolution Explorer

GBH

Galactic black hole

GPS

Gigahertz-peaked spectrum source

HBL

High-frequency cutoff BL Lac object/high frequency Peaked BL Lac object

HEAO-2

EINSTEIN satellite

HES

Hamburg/ESO Survey

HPQ

Highly polarized quasar

HQS

Hamburg Quasar Survey

HR

Hardness ratio

HRI

ROSAT High-Resolution Imager

HST

Hubble Space Telescope

HzRG

High-redshift radio galaxy

IACT

Imaging Atmospheric Cherenkov Telescope

IC

Inverse Compton scattering

ICM

Intracluster medium

IBL

Intermediate BL Lac object

IDV

Intraday variability

IGM

Intergalactic medium

IMF

Initial mass function

INTEGRAL

International Gamma-Ray Astrophysics Laboratory

IPC

EINSTEIN Imaging Proportional Counter

IRAS

Infrared Astronomical Satellite

ISO

Infrared Space Observatory

IUE

International Ultraviolet Explorer

LAT

Large Area Telescope (Fermi)

LBL

Low-frequency cutoff BL Lac object/low-frequency Peaked BL Lac object

LDDE

Luminosity-dependent density evolution

LECS

Low Energy Concentrator Spectrometer (BeppoSAX)

LF

Luminosity function

LINER

Low-ionization nuclear emission-line region

LISA

Laser Interferometer Space Antenna

LMC

Large Magellanic Cloud

LOFAR

Low Frequency Array (radio)

LSST

Large Synoptic Survey Telescope (optical)

MACHO

Massive compact halo object

mas

Milliarcseconds

MECS

Medium Energy Concentrator Spectrometer

MHD

Magnetohydrodynamics

NED

NASA/IPAC Extragalactic Database

NELG

Narrow-emission-line galaxy

NLR

Narrow-line region

NLRG

Narrow-line radio galaxy

NLS1

Narrow-line Seyfert 1 galaxy

NRAO

National Radio Astronomy Observatory (USA-VA)

NVSS

NRAO VLA Sky Survey

OSSE

Oriented Scintillation Spectrometer Experiment (CGRO)

OVV

Optical violent variable blazar

PAH

Polycyclic aromatic hydrocarbon

PDS

Power Density Spectrum or Phoswich Detector System (BeppoSAX)

PLE

Pure-luminosity evolution

PSF

Point spread function

PSPC

Position Sensitive Proportional Counter (ROSAT)

QPO

Quasiperiodic oscillation

RASS

ROSAT All-Sky Survey

RBL

Radio-selected BL Lac object

ROSAT

Röntgensatellit

RXTE

Rossi X-ray Timing Explorer

SDSS

Sloan Digitized Sky Survey

SED

Spectral energy distribution

SF

Structure function

SIMBAD

Set of Identifications, Measurements, and Bibliography for Astronomical Data

SIS

Solid-State Imaging Spectrometer (ASCA)

SKA

Square Kilometer Array (radio telescope)

SL

Superluminal

SMBH

Supermassive black hole

SMC

Small Magellanic Cloud

SNR

Supernova remnant

SOFIA

Stratospheric Observatory for Infrared Astronomy

SRG

Spectrum-Roentgen-Gamma

SRSQ

Steep radio spectrum quasar

SSC

Synchrotron self-Compton scattering

UHBL

Ultrahigh-frequency peaked BL Lac object

UHECR

Ultrahigh-energy cosmic rays

ULIRG

Ultraluminous infrared galaxy

ULX

Ultraluminous X-ray source

VHE

Very high energy

VLA

Very Large Array (radio)

VLBA

Very Long Baseline Array (radio)

VLBI

Very Long Baseline Interferometry (radio)

VSOP

VLBI Space Observatory Programme

WFI

Wide Field Imager or Wide Field Instrument

WHIM

Warm-hot intergalactic medium

WIMP

Weakly interacting massive particles

XBL

X-ray-selected BL Lac object

XBONG

X-ray bright optically inactive galaxy

XMM

X-ray Multimirror Mission

2MASS

Two-Micron All-Sky Survey

Astronomical and Physical Constants

We list here some astronomical and physical constants which are used throughout the book.

Physical Constantsa)

Speed of light

c

2.997 924 58 × 10

10

cm s

−1

Planck constant

h

6.626 07 × 10

−27

g cm

2

s

−1

Boltzmann’s constant

k

1.380 65 × 10

−16

erg K

−1

Stefan–Boltzmann constant

σ

5.6704 × 10

−5

erg cm

−2

s

−1

K

−4

Gravitational constant

G

6.6738 × 10

−8

cm

3

g

−1

s

−2

Electron mass

m

e

9.109 383 × 10

−28

g

Proton mass

m

p

1.672 621 8 × 10

−24

g

Neutron mass

m

n

1.674 927 4 × 10

−24

g

Atomicmass unit

1 u

1.660 538 × 10

−24

g

Elementary charge

e

1.602 176 6 × 10

−19

C

Electron volt

1 eV

1.602 176 6 × 10

−12

erg

Fine-structure constant

a

7.297 352 57 × 10

−3

Wien’s displacement constant

b

0.289 777 cm K

a) Values taken from the on-line version of Mohr et al. (2008).

Astronomical Constants

Solar mass

1

M

1.989 × 10

33

g

Solar luminosity

1

L

3.839 × 10

33

erg s

−1

Astronomical unit

1 AU

1.4960 × 10

33

cm

Siderial year

1 yr

3.155 815 × 10

7

s

Light year

1 ly

9.4605 × 10

17

cm

Parsec

1 pc

3.0857 × 10

18

cm

3.2616 ly

Hubble constant

H

0

70.8 km s

−1

Mpc

−1

Color Plates

Figure 3.3 This figure illustrates a recent calibration of the M–σ relation from Gültekin et al. (2009). The colors indicate different galaxy types and the symbols (star, circle and asterisk), indicate the type of black-hole measurement.

Figure 3.4 The so called Fundamental Plane for accreting black holes with which jet outflows are also associated (Merloni et al., 2003). The observable quantities, LR and LX are proxies for the jet and disk power respectively. Objects with mass determinations from other methods were used in constructing this diagram. The tight correlation over many decades of central black hole masses suggests a commonality in the physics underlying the disk-accretion and jet launching mechanism across object sub-classes.

Figure 3.16 Examples of photoionization models of ionized plasma fitted to dispersive X-ray spectra obtained with the Chandra X-ray observatory (Kallman, 2010). The fits led to refinements in iron recombination rates and revealed multiple gas velocity components.

Figure 4.9 The FR-II type radio galaxy Cygnus A is the brightest extragalactic radio source. The picture shows the 5GHz image taken with the VLA telescope array with 0.4″ resolution (Carilli and Barthel, 1996). The AGN core is located at the bright spot at the center, the radio lobes extend out to about 50 kpc from the core, far beyond the host galaxy which is not visible in the radio domain.

Figure 4.18 Schematic representation of the connection of jet emission, inner disk radius, and spectral state in galactic black hole binaries. The top panel shows the hardnessintensity diagram, in which black holes seems to follow the paths indicated by the arrows. A flat-spectrum radio flux appears and increases with X-ray intensity in the hard state – the rith-hand vertical track of the “q” diagram. The radio emission becomes optically thin and the jet appears as the emission transitions leftwards along the upper horizontal track. The jet disappears after the source moves into the high-soft state. In the bottom panel we see the dependence of jet speed and the inner disk radius on the hardness of the spectrum. X-ray states are indicated with HS (high/soft state), HS/IS (very high and intermediate state), and LS (low/hard state). The sketches around the outside illustrate the concept of the relative contributions of jet, corona (light grey) and accretion disc (dark grey) at these different stages. Graphic from Fender et al. (2004).

Figure 5.19 SSC model fit of the spectral energy distribution of Mrk 421 based on simultaneous data during a quiescent state of the source. Graphic from Abdo et al. (2011b), reproduced by permission of the AAS.

Figure 5.22 Normalized mean SEDs for strong far-infrared (FIR) emitting quasars (a, top curve) and weak FIR quasars (b, top curve). The adjacent red SED curves show “intrinsic” AGN SEDs obtained by the subtraction of the scaled mean starburst (ULIRG) spectrum (shown as the lowest curve in black) from the mean SEDs. From Netzer et al. (2007), reproduced by permission of the AAS.

Figure 6.7 Examples for variability of the spectral energy distribution in blazars in low and high emission states. Blazars tend to shift the peak of the synchrotron and inverse Compton branch to higher frequencies, while when comparing different blazars, usually the ones with higher peak frequencies are the less luminous ones as discussed in Section 5.8.1. Graphic from Paggi et al. (2009), reproduced with permission ©ESO.

Figure 6.9 Multi-wavelength lightcurve of BL Lac (Marscher et al., 2011). As described in the text, patterns are beginning to emerge between the radio intensity, morphological evolution and flaring at X- and gamma-ray bands. Additional information is potentially forthcoming from comparing polarization measurements of individual jet components to optical polarization of flaring features; specifically, discrete increases in polarization are expected to coincide with gamma-ray flares. The gamma-ray peaks are sharp, and coincide with the rising phase of the radio flux. The radio peaks lag behind the gamma-ray peaks by days to weeks. The VLBA images, not shown here, indicate that the jet knots usually pass through the position corresponding to the radio core concurrently ith the gamma-ray flares (vertical arrows).

Figure 7.4 Color u–r versus galaxy mass. The larger the u–r value, the redder the galaxy. The central bulge of the spirals are similar to small elliptical galaxies. Ellipticals and spiral bulges have similar stellar mass to black hole mass ratio of the order of 0.1%. Ellipticals are located above a critical mass of Mcrit 1012M. Graphic from Cattaneo et al. (2009).

Figure 7.6 Star formation rate (SFR), black hole mass accretion rate (BHAR) and black hole mass for a simulated merging event between two galaxies containing each a super massive black hole about the size of the one in our Galaxy. For comparison, in the upper most panel also the evolution of SFR for a merger without central black holes is shown. Cases of different virial velocity are given. The dots indicate specific times in the merging process. From left to right: first passage of the two galaxies, tidal interaction just before merging, coalescence, and after the merging process has finished. In this graphic from Di Matteo et al. (2005) a feedback on the SFR is assumed in the sense that the AGN shuts off the creation of new stars, a scenario disputed for example by Debuhr et al. (2010) who argue that the AGN has an impact only on the innermost part of the bulge.

Figure 7.7 The Perseus Cluster as seen by the Chandra X-ray telescope. The center of the cluster is dominated by the emission of the Seyfert and radio galaxy NGC1275. North and south of the AGN, cavities are visible as well as a larger dark bubble in the northwest. Chandra data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center.

Figure 9.5 X-ray luminosity function as presented by Hasinger et al. (2005); reproduced with permission © ESO. Left panel: the space density of AGN as a function of redshift for several X-ray luminosity bins. The lines shown are for a pure-luminosity evolution (PLE) and for a luminosity dependent density evolution (LDDE). Right panel: emissivity of AGN as a function of redshift. The uppermost curve shows the summed emissivity of all luminosity classes considered here.

Figure 9.6 Optical luminosity function in several redshift bins derived from SDSS data. The dotted lines denote the predictions of the best-fitting double power law exponential evolution mode. From Croom et al. (2004).

Figure 9.8 Recent measurements of the cosmic X-ray background by BeppoSAX, Swift, and INTEGRAL. The continuous line shows the best fitting model derived by Türler et al. (2010). Figure courtesy of Marc Türler.

1

The Observational Picture of AGN

1.1 From Welteninseln to AGN

The dawn of extragalactic astronomy can be attributed to the year 1750, in which Thomas Wright speculated that some of the nebulae observed in the sky were not actually part of the Milky Way, but rather independent Milky Ways themselves (Wright, 1750). A few years later, Immanuel Kant introduced the term “Welteninseln” for these distant nebulae (“island universes”; Kant, 1755). It was François Arago in 1842 who first called the attention of astronomers to Kant,1) whom he calls “the Astronomer of Königsberg,” and declared that his name in that connection did not deserve the oblivion into which it had fallen (Arago and Barral, 1854). Thus the extragalactic hypothesis spread rapidly in the scientific community, although it was still not completely accepted as true. One main difficulty was the fact that some of the nebulae were actually of galactic origin, such as planetary nebulae and globular clusters. A significant step forward was the compilation of a large catalog of some 5000 nebulae assembled by William Herschel in the late eighteenth and early nineteenth century. Another advance was made by Lord Rosse, who constructed in 1845 a new 72″ telescope in Ireland, managing to distinguish individual point sources in some of the nebulae, and therefore giving further support to Kant’s and Wright’s hypothesis. Spectroscopic observations by Vesto Slipher of nebulae in the early twentieth century revealed that some of these show redshifted lines indicating they are moving relative to the Milky Way at velocities exceeding the escape velocity of our Galaxy (Slipher, 1913).

The first evidence that some galaxies were hosting some additional strongly emitting component in their center was found by Carl Seyfert in the 1940s. He obtained spectra of six galaxies, showing high-excitation nuclear emission lines superposed on a normal star-like spectrum (Seyfert, 1943). He also noticed that some galaxies showed broad emission lines, while others exhibited only narrow ones. The nature of the strong emission from the center of some galaxies remained a mystery. A common hypothesis was the assumption that a large number of stars would produce the observed features. Woltjer (1959) pointed out though that the observed concentration of the emission within the central 100 pc of the galaxies would require a mass of a few 108M. A step closer to current understanding was the idea that in the center of these galaxies resides a stellar type object of very large mass, which then would emit mainly by accretion processes of a surrounding disk of gas (Hoyle and Fowler, 1963). It was not until a year later that the idea was put forward to assume that in the center of an AGN there could lie a black hole as opposed to a hypermassive star (Salpeter, 1964; Zel’Dovich and Novikov, 1964).

The hypothesis that there might exist objects in the Universe whose gravity would be sufficient to trap even light was discussed first by John Mitchell2) in the late eighteenth century (Mitchell, 1784). Independently Pierre-Simon Laplace developed the concept of “dark stars,” speculating that the most massive stars would be invisible due to their strong gravity (Laplace, 1796). The concept of the black hole was ignored though in later years, as light was considered to be made of massless particles with no interaction with a gravitational field. When Albert Einstein (1916) formulated the general relativity theory the possible existence of black holes was shown to be a solution for the gravitational field of a point mass and of a spherical mass by Karl Schwarzschild (1916). Nevertheless, this solution to Einstein’s theory was thought to be merely hypothetical. Only when solutions had to be found to explain phenomena like AGN, and the fact that massive stars had to collapse into a black hole (Oppenheimer and Volkoff, 1939), was the existence of black holes accepted by a continuously growing fraction of the scientific community.

The idea of a supermassive black hole in the center of active galactic nuclei (Salpeter, 1964; Zel’Dovich and Novikov, 1964; Lynden-Bell, 1969) and also in the center of our own galaxy (Lynden-Bell and Rees, 1971) was a powerful model. It explained not only the large energy output based on the release of gravitational energy through accretion phenomena, but also the small size of the emitting regions and connected to it the short variability time scales of AGN. The field was now open to study the physics involved in the accretion phenomenon, to observe and explain AGN emission throughout the electromagnetic spectrum, and to study the distribution in space, the origin, the evolution and fate of these elusive objects.

1.2 Broad Lines, Narrow Lines, and the Big Blue Bump

The first notably distinct observational characteristic of AGN was the presence of emission lines with widths upwards of 1000 km s−1 and far in excess of any known class of objects. Furthermore, the centers of these broad emission lines did not correspond to the laboratory wavelengths of any known atomic species and certainly not to the well known hydrogen Balmer series or other common lines known to be of astrophysical origin. This dilemma was resolved in the 1960s leading to the basic AGN paradigm described in the previous section of a distant and highly luminous object powered by a massive, accreting black hole. The deep gravitational potential of the black hole was responsible for the dynamical broadening of the observed lines and for radiatively efficient accretion leading to the extreme luminosities. The line identification dilemma was solved with the realization that the distances involved were of such magnitude that the cosmological expansion of the Universe redshifted atomic emission lines to the observed values including some high-ionization UV lines were.

Fast forwarding ahead several decades, it became evident that these broad emission line spectra could be exploited as a diagnostic of the physical conditions in the environment ambient to the central black hole. As we discuss in some detail later in the text, correlated variability of line and continuum emission components have been applied to “reverberation mapping” analyses leading to constraints on the broad-line emission region size and on the mass of the central black hole, for example Peterson and Horne (2004); Bentz et al. (2009a). This led to a dramatic revision of our basic understanding of AGN. Additionally, this knowledge has been used to cross-calibrate alternative black hole mass estimation methods and to better constrain physical models of the broad-line emission media as virialized gas clouds in photoionization equilibrium with the central engine radiation field.

Another distinguishing observational feature of some AGN is the presence of narrow, nonvariable forbidden emission lines. The similarity to nebular line emission in our galaxy was noted and some of the atomic physics and computational formalism developed and to study those objects was employed (Osterbrock, 1989). There were some significant differences between the galactic nebulae and the AGN as well. In particular, the AGN narrow lines required a broad-band ionizing continuum extending far bluewards of the stellar radiation fields responsible for photoionizing the galactic nebulae.

At present, with an improved understanding of the narrow-line region morphologies and thermodynamics, insight into the gas and dust distributions of the central regions of AGN can be gained. The dynamics of the inner AGN region can also be probed in cases using the narrow lines. For example, approximately 1% of low-redshift (z 0.3) optically selected type 2 AGNs exhibit a double-peaked [O III] narrow-line profile in spatially resolved spectra Shen et al. (2011). These types features have been interpreted in the context of kinematics, such as biconical outflows (Kraemer et al., 2008) or rotation of the narrow-line region about the central black hole, or to the relative motion of two distinct NLRs in an ongoing AGN merger event.

Another observational aspect of AGN that was evident in early observations was that the continuum spectral distribution was very distinct from an integrated stellar continuum characteristic of normal galaxies (Oke and Sargent, 1968). Observationally, AGN were comparatively very blue. In fact, radio-quiet AGN would later be identified and cataloged primarily by performing multicolor imaging of sky regions, sorting the results in color-color plots, and performing spectroscopic follow-ups on the blue excess subpopulation thus identified (Green et al., 1986, it is now known that this approach omits redder objects that are picked up in X-ray surveys).

The blue colors were due to both the fact that the continuum emission extended into the UV and beyond and that structure was often seen in the blue continua – the so-called “big blue bump” (Richstone and Schmidt, 1980). The big blue bump spectral component was a positive flux excess relative to an underlying power-law continuum. It exhibited curvature that suggested a thermal origin. This was interpreted as the first observational evidence for the presence of an accretion disk (Malkan and Sargent, 1982a), thus lending support to the basic paradigm of Lynden-Bell (1969). The quest to further corroborate this basic paradigm and to gain a deeper fundamental understanding of putative AGN accretion disks was the driver behind many subsequent observational campaigns and theoretical efforts during the decades which followed.

1.3 Jets and Other Outflows

The sixteenth century French astronomer Charles Messier published a catalog comprising 103 spatially extended or nebular objects. Among the most prominent, roughly spherical, examples in his catalog was object number 87, thus its designation as M87. In 1918 the American astronomer Heber Curtis noted that the presence of a “curious straight ray” that protruded from the nebula and apparently traced back to its nucleus (Curtis, 1918). As noted, there was at that time still disagreement as to whether or not the nebulae were external to or contained within our galaxy, but as that issue was soon resolved and it became clear that M87 was a giant elliptical galaxy, we consider Heber’s observation to be the first documented example of an AGN jet.

Figure 1.1 Schematic representation of our understanding of the AGN phenomenon and its main components. Note that this is a simplified view and not to scale. Graphic courtesy of Marie-Luise Menzel.

There were relatively few of examples of these highly collimated, bipolar outflows or jets from the centers of AGN until the observational techniques of radio interferometry matured in the decades subsequent to the Second World War. There are now AGN subclasses, which are believed to be related to each other within the context of the unification scenarios described in subsequent chapters. The physics underlying the launching and propagation of these AGN jets – their remarkable energetics, enormous size and plethora of associated phenomenology (hot spots, knots, bends) – remains enigmatic in many regards and its pursuit may be considered a “holy grail” of modern astrophysics.

Our general understanding of the main components of an AGN is shown in Figure 1.1. The schematic representation distinguishes between sources which display a jet and are therefore bright in the radio band, and those which do not show strong radio dominance and in which case one assumes no or weak jet emission.

1.4 X-ray Observations: Probing the Innermost Regions

During the same decade of the 1960s when the basic AGN paradigm was being developed, the first cosmic X-ray source, known as Scorpios X-1, was discovered using a rocket-borne detector (Giacconi et al., 1962). It quickly became evident that X-ray emission was characteristic of compact, accretion-powered sources associated with galactic binaries. With the realization that the deep gravitational wells of massive black holes were likely the source for the extreme energetics exhibited by quasars, the generation of X-rays seemed natural. However, the rocket-borne experiments had limited capabilities and the first significant breakthrough came with the launch of the Uhuru satellite (also known as SAS-A) in 1970. A catalog of sources detected with Uhuru ultimately included nearly 340 objects, mostly galactic binaries, but also including about a dozen AGN (Forman et al., 1978).

The field progressed rapidly during the 1970s, with detectors having larger collecting area, increased spectral coverage and improved spectral resolution, for example the Ariel-5 satellite launched in 1975 and the OSO-7 (1974), OSO-8 (1975) and HEAO-1 (1977) (see, e.g., Tucker and Giacconi, 1985). This led to the characterization of AGN as a class of X-ray sources and to the first detection of the iron Kα line emission from an extragalactic source. The biggest breakthrough however came later in that decade with the launch of the Einstein Observatory (originally called HEAO-2). This was the first true orbiting X-ray telescope, in that it utilized a concentric array of grazing incidence mirrors to focus ~ keV photons onto its focal plane detectors. The resulting images provided, in addition to vastly improved spatial localization of sources on the sky, a large leap in sensitivity since the source and celestial background could be effectively separated.

It had become clear that X-ray emission was a common property of different subclasses of active galaxies with the X-ray flux comprising a significant fraction (about 5–40%) of the bolometric emission from such objects (Ward et al., 1987). Rapid variability was also found to be a prominent feature of the X-ray emission with kilosecond timescale X-ray flux variations seen in local Seyfert galaxies. This imposed new and increasingly stringent constraints on the size of the X-ray emission region and strongly supported the idea that it occurs very close to the active nucleus (e.g., Pounds et al., 1986). The origin of X-rays from close to the central black hole means that X-ray data offer a chance to study the immediate environs of supermassive black holes and the poorly understood accretion process that fuels them. Although the angular scale of the X-ray emission region is too small to image with current instrumentation, timing analysis and spectroscopy offered methods to probe these regions indirectly.

Specific spectral signatures were attributed to characteristics of the gas inflow and outflow near the central most regions in AGN. The X-ray observations also provided signatures of reprocessing of radiation in material within approximative distance of hundreds of gravitational radii and thus the potential for discerning signatures of the accretion disk at even smaller radii. Features such as the weak, broad emission lines due to low-ionization states of iron as well as other structured deviations from simple power laws had been identified in the spectra of AGN. In 1991, George and Fabian (1991) offered an interpretation of these features in terms of X-irradiation of relatively cold, dense gas in the vicinity of the central black hole. The emergent spectrum then consists of direct radiation from the central source plus a scattered or “reflected” spectrum that includes imprinted photoabsorption, fluorescent emission and Compton scattering from matter within the surrounding accretion flow. This basic idea has withstood the scrutiny of improved observational data and has become a tenant of the AGN paradigm.

X-ray observations of AGN are also being applied to address issues of fundamental black hole physics. The shapes of line profiles have also been applied to models which in principle allow one to infer an intrinsic property of the central black hole, namely its intrinsic angular momentum or spin (e.g., Brenneman and Reynolds, 2009). The basic idea is that the asymmetry of a line profile produced in the inner AGN accretion disk depends in a predictable manner on the shape of the gravitational potential which in turn depends on the black hole spin.

In the chapters that follow, we discuss these issues in further detail highlighting a number of results from the current astrophysics literature. We also speculate on the possibilities offered by future orbiting X-ray observatories, which are currently under discussion.

1.5 Up, Up and Away: from Gamma-Rays toward the TeV Range

The impact of gamma-ray astronomy on AGN research did not emerge as rapidly as did X-ray astronomy, although the fields were initiated more or less concurrently with 1960s rocket flights followed by satellite-borne experiments in the 1970s. The reasons for this are several-fold. There are fewer gamma-ray photons than lower energy photons emitted even though the overall energy budget for some AGN may be dominated by the gamma rays. There are substantial instrumental and celestial backgrounds at gamma-ray energies that need to be understood and modeled or subtracted. Gamma-ray detectors tend to be more massive for a given effective collection area than X-ray detectors and gamma rays cannot be focused. Additionally, it became apparent that only the radio-loud AGN, which comprise ~ 5% of the overall population are prolific emitters of gamma radiation. We should note here that the term “gamma rays” encompasses a huge swath of the electromagnetic spectrum. Here we will designate photons with energies above ~ 100 keV as gamma rays. AGN have been detected at ~ TeV, thus we are considering over 7 decades in our discussion of gamma-ray studies. The energy range above about 100 MeV has, somewhat surprisingly, provided the richest bounty of results as we will further discuss in later chapters.

In the 1970s the ESA mission COS-B, along with NASA’s SAS-2, provided the first detailed views of the Universe in gamma rays. COS-B, launched in August 1975, was originally projected to last two years, but it operated successfully for nearly seven. It made the first gamma-ray measurement of an AGN, that being 3C 273 (Swanenburg et al., 1978). However, it was not until nearly 20 yr later with the launch of the Compton Gamma-Ray Observatory (CGRO) that additional gammaray detections were made, starting with the discovery in 1991 of bright gamma-ray emission from 3C 279 (Hartman et al., 1992). New results came quickly after that leading ultimately to the identification of some 70 high-latitude CGRO gamma-ray sources with radio-loud AGN. Specifically, BL Lac objects and flat-spectrum radio quasars (FSRQs), known collectively as blazars, comprised the entire gamma-ray sample. It was also clear that the radiative output of the blazars was typically dominated by the gamma rays. The gamma-ray emission was also found to be variable on time scales less than a day.

These observations had several immediate implications for physical models. The emission had to emanate from a compact region. For example, a factor of 2 flux variation limits, approximately, the size r of a stationary, isotropic emitter to where δtvar is the variation time scale. The implications from the early CGRO results, which by this line of reasoning necessitated a very compact emission region, were problematic in any scenario in which the gamma-ray production involves such a stationary isotropic source. The problem involved the transparency of a compact region such as inferred here. If X-rays are produced cospatially with the gamma rays, attenuation of the gamma rays due to the process γγ → e+e− for which the cross-section for attenuation of ~ 100 MeV gamma rays is in the X-ray range ~ keV X-ray range. The inferred gamma-ray opacity from the CGRO observations would exceed unity in many instances. Either the radiating particles were strongly beamed or the emitting plasma was undergoing bulk relativistic motion. Thus beaming was very strongly implied.

Models that had been previously favored to explain the radio-to-optical continua in these objects, for example Blandford and Konigl (1979), implied that we are viewing nearly along an axis of a relativistic plasma jet ejected from near the central black hole, involving nonthermal synchrotron emission as we will discuss in later chapters. An extension of this scenario invoking a distinct second spectral component was now clearly required. The basic idea was that gamma rays emitted by blazars are produced by the same population of electrons that produced the synchrotron emission via Compton scattering of ambient low-energy photons. The ambient photon field could be the synchrotron photons themselves (e.g., Maraschi et al., 1992) or from an external source such as the accretion disk or broad-line clouds (e.g., Dermer et al., 1992).

Shortly after the CGRO results began to emerge, another major discovery followed from ground-based Cherenkov gamma-ray telescopes, which measured gamma rays in the ~ TeV range. Blazars such as Markarian 421 (Punch et al., 1992) and Markarian 501 (Quinn et al., 1996) were detected during a high-amplitude variability episode. These discoveries established this subclass of AGN as emitters over ~ 20 decades of the electromagnetic spectrum. As such, they were a striking example of the value, indeed the necessity, inherent in the multiwavelength approach to studying AGN. The high-energy gamma-ray observations also fit in naturally with the synchrotron plus Comptonization model scenarios. They also had other potentially significant implications, not only on the blazar AGN themselves, but on the gamma-ray transparency of the universe and thus in turn the background radiation fields to the cosmic star-formation history.

In the two decades since these discoveries, gamma-ray studies of AGN have expanded enormously. The Fermi Gamma-Ray Space Telescope, launched in 2008, has cataloged approximately 900 gamma-ray AGN. Advances in ground-based Cherenkov telescope facilities, as well as in detection and analysis methodologies, has produced a similar order-of-magnitude increase in the TeV gamma-ray sample. Multiwavelength campaigns have begun to reveal how jet formation and propagation may be correlated with the gamma-ray flux variations. Clearly, gamma-ray astronomy will continue to be a vital component of our quest to better understand the AGN phenomenology for the foreseeable future.

1) Arago wrote: “Kant condensait ses idés dans le moindre nombre de mots possible, quand il appelait la Voie lactée le Monde des Mondes.”

2) Mitchell wrote: “If the semidiameter of a sphere of the same density as the Sun were to exceed that of the Sun in the proportion of 500 to 1, a body falling from an infinite height towards it would have acquired at its surface greater velocity than that of light, and consequently supposing light to be attracted by the same force in proportion to its vis inertiae, with other bodies, all light emitted from such a body would be made to return towards it by its own proper gravity.”

2

Radiative Processes

A basic understanding of the interaction between photons and particles and of particles with other particles or fields is essential for the understanding of emission and absorption processes. In AGN, we observe a large variety of processes. Synchrotron emission plays a key role in AGN jets, their accretion disk seems to emit thermal radiation, and the highest photon energies are reached through inverse Compton processes, for example in the jet or in a plasma close to the accretion disk. In this chapter we will give a brief overview of the physics underlying these processes. This shall be by no means a comprehensive review, and we refer the reader for example to Rybicki and Lightman (1986) for a more detailed discussion.

2.1 Scattering of Photons

Electromagnetic radiation produced in any kind of physical process may not travel undisturbed to the observer. In many cases, photons will be scattered on particles, losing or gaining energy on their way. This has important effects on our interpretation of an observed photon spectrum.

2.1.1 Thomson Scattering

Thomson scattering describes the nonrelativistic case of an interaction between an electromagnetic wave and a free charged particle. The effect was first described by Sir Joseph John Thomson, who discovered the electron when studying cathode rays in the late nineteenth century. The process can be understood as elastic or coherent scattering, as the photon and the particle will have the same energy after the interaction as before. For this process the energy E of the photon has to be much smaller than the rest energy of the particle:

(2.1)

with ν being the frequency of the photon and m the mass of the particle. Another requirement for Thomson scattering is that the particle must be moving at nonrelativistic speed (v c). In the classical view of this process, the incoming photon is absorbed by the particle with charge q, which is set into motion and then re-emits a photon of the same energy.

(2.2)

This is symmetric with respect to the angle θ, thus the amount of radiation scattered in the forward and backward direction is equal. The total cross-section is then given by

(2.3)

In the case of an electron, this gives a Thomson cross-section of σT 6.652 × 10–25 cm2. The cross-section for a photon scattering on a proton is a factor of (mp/me)2 3.4 × 106 smaller.

Since in the classical view of this process, the electron has no preferred orientation, the cross-section is independent of the polarization of the incoming electromagnetic wave. The polarization of the scattered radiation depends, however, on the polarization of the incoming photon wave. Unpolarized radiation becomes linearly polarized in the Thomson scattering process with the degree of polarization being

(2.4)

Therefore, polarization of the observed emission can be a sign that the emergent radiation has been scattered.

Thomson scattering is important in many astrophysical sources. Any photon which will be produced inside a plasma can be Thomson scattered before escaping in the direction of the observer. The chance for the single photon to be Thomson scattered and how many of the photons will be scattered out of or into the line of sight is quantified in terms of the optical depth τ of the plasma:

(2.5)

Here ne is the electron density, and dx is the differential line element. The mean free path λT of the photon, that is, the mean distance traveled between scatterings will thus be λT (σTne)–1.

2.1.2 Compton Scattering

Thomson scattering describes the interaction between the photon and the charged particle as an elastic scattering process. In this treatment, the incoming and out-going photon have the same energy. In a quantum-mechanical treatment of the problem, the photon carries momentum as well as energy, and there will be an energy transfer from the photon to the electron, thus leading to a recoil of the electron. The larger the change of direction of the photon, the larger the energy transfer toward the electron will be.

Another condition for Thomson scattering is the low energy of the photon in comparison with the charged particle. If this requirement is dropped and the photon can have an energy comparable to the rest mass of the charged particle, it is necessary to consider quantum mechanical effects. This will reduce the effective cross-section relative to the Thomson case.

(2.6)

The resulting energy E2 of the photon after the Compton scattering is

(2.7)

with E1 being the energy of the photon before the scattering. Thus, in the limit of E1mc2 this results in the case described by Thomson scattering, where no energy is transferred to the electron.

The cross-section for the high-energy limit of Thomson scattering is described by the Klein–Nishina formula derived by applying quantum electrodynamics. Here, the differential cross-section is described by

(2.8)

with

(2.9)

In this notation, the energy of the incoming photon E1 has been expressed in units of the rest mass energy of the charged particle:

(2.10)

The total cross-section is then

(2.11)

A useful approximation for photon energies lower than E1 200 keV is

(2.12)

Again, when 1 and thus the energy of the incoming photon is small compared to the rest mass energy of the particle, this reduces to the Thomson cross-section. For energies Emc2 the cross-section is approximately

(2.13)

The Klein–Nishina effect results in a reduction of the Compton scattering cross-section at high photon energies. In other words, at the highest frequencies Compton scattering becomes less efficient. To illustrate this effect we show in Figure 2.1 the cross-section in units of the Thomson value as a function of the photon energy. Only above a few keV (v 1018 Hz) does the Klein–Nishina effect become important. At 350 keV the cross-section is reduced to 1/2σT and it falls below 10% of the Thomson cross-section at ~ 7 MeV. This effect has important consequences regarding the interpretation of AGN spectral energy distributions as we will see in subsequent chapters.

2.1.3 Inverse Compton Scattering

In the case of Compton scattering the photon loses energy which is transferred to the electron. In cases where the electron is moving at relativistic speeds, the Compton process can lead to the opposite effect: the low-frequency photon can gain in energy through the so-called inverse Compton effect. Because in this case the condition hvmec2/γ is fulfilled, one can apply the Thomson scattering cross-section.

Assume that the laboratory frame in which we observe the scattering event is L, and the frame of the relativistic electron with Lorentz factor γ is L′ in which it is at rest before the event. The Lorentz factor is defined as

(2.14)

In the electron’s frame L′ the energy of the photon is much smaller than the rest energy of the electron: hv′mc2. Thus, the center of momentum is close to that of the relativistic electron. Following the relativistic Doppler shift formula the energy of the photon in L′ is given by

(2.15)

Note that the angle θ is the angle between the direction of the photon and of the incoming electron, traveling at speed υe, in the laboratory frame L. In the electron’s frame, this angle appears smaller:

(2.16)

Because the photon energy is small compared to the electron rest energy, we can treat the event as Thomson scattering. As this is an elastic scattering process, the energy of the photon in the electron’s frame does not change, that is, . If we transform this expression back into the laboratory frame, we get

(2.17)

thus the photon gains energy in proportion to the square of the Lorentz factor γ of the electron. For relativistic electrons this means that a photon in the radio domain can be up-scattered to the optical or even X-ray range in the inverse Compton process. The maximum energy gain of the photon is determined by the energy conservation of the process as seen in the laboratory frame:

(2.18)

Thus, the maximum change in photon frequency is

(2.19)

The total power (in other words, the luminosity) of the inverse Compton process will naturally depend on the density of photons nph available for the scattering. Thus, the luminosity LIC of the inverse Compton component will be

(2.20)

In other words, the luminosity will be proportional to the energy density of the photon field:

(2.21)

We will not present the exact derivation of the luminosity of the single Compton scattering, which can be studied in Rybicki and Lightman (1986), but instead point out some general considerations.

The resulting LIC will depend on the cross-section for the inverse Compton process. We recall that this scattering can be treated as elastic, and thus the Thomson cross-section σT applies. The single electron will “see” per unit time an incoming energy from the photons, which is the product of the speed of the incoming photons c, the cross-section of the process σT, and the energy density of the photon field Uph:

(2.22)

In addition we have to consider the relativistic Doppler shift, and thus the resulting luminosity is

(2.23)

In cases where the condition hv′ mc2 is not fulfilled in the electron’s frame, the assumption of Thomson scattering is no longer valid. Thus, for high photon energies Klein–Nishina effects have to be considered, which involves a dependence on photon energy and scattering angle. Following for example Blumenthal and Gould (1970), the resulting luminosity for the inverse Compton process is

(2.24)

The first term in the brackets gives the luminosity considering Thomson scattering. Since in the Klein–Nishina realm the cross-section is reduced, the second term in the brackets reduces LIC depending on the average photon energy 〈E1〉 in the photon field. The larger 〈E1〉, the more the luminosity of the inverse Compton process will decrease. For very high photon energies, the luminosity can be negative and the photon will lose energy to the electron.

In astrophysical sources, one has to deal with not only a single electron but with electron distributions. Thus, in order to determine the luminosity of the inverse Compton process LIC one has to integrate over the single scattering described by Eq. (2.23), applying an energy distribution for the electrons. An assumption that is often valid for a distribution of electron energies is that of a simple power law:

(2.25)

with N(E) being the number of electrons at a given energy E, p the slope of the power law, and kE the normalization. If we assume that the electrons are highly relativistic, that is, that υe/c 1, then we can integrate Eq. (2.23) and derive the luminosity of the inverse Compton radiation for the electron distribution. Assuming that the electrons have a minimum Lorentz factor γmin and a maximum of γmax, this results in:

(2.26)

2.1.4 Thermal Bremsstrahlung

Another important mechanism in astrophysics for electrons losing energy, is bremsstrahlung (from German “breaking radiation”). Here, the electron interacts with another free charged particle, for example an ion. As both particles are not bound before and after the scattering process, this is also referred to as free–free emission. The interaction leads to a change in the momentum of the particles involved, and through this acceleration of charges to radiation. This type of scattering takes place for example in hot, but nonrelativistic plasmas. The most prominent source of extragalactic bremsstrahlung is the intracluster gas which is bound in the gravitational wells of galaxy clusters with temperatures of T 107–108 K (Section 7.4). Interaction of two particles with the same mass, like electron–electron or positron–positron scattering does not produce dipole radiation, and the quadrupole term is comparably low. Only in the relativistic case can this produce a significant amount of radiation.

Thus, in extragalactic sources the most important bremsstrahlung is produced in interactions of electrons with ions of charge Ze. As the ion is much more massive than the electron, we can consider the scattering event with the ion being at rest, and the electron being the only particle which undergoes acceleration. One can further assume a scattering event which does not result in a large change in the electron’s direction, and consider only small-angle scattering. In a classical approach to the problem, the acceleration will depend on how close the electron is passing by the ion. If d is the instantaneous distance between the electron and the ion, then the acceleration experienced by the electron is

(2.27)

where b is the minimum value of d, often referred to as the impact parameter. The luminosity as a function of the distance is then (see, e.g., De Young, 2002):

(2.28)

with a being the acceleration of the electron. As the radiated power depends strongly on the distance L α d–4, only the time when the two particles near the point of closest approach will be relevant. We can thus assume that the important acceleration takes place when the distance is The longer the electron is close to the ion, the larger the effect will be, thus, the acceleration will be stronger for slower electrons. In the nonrelativistic case the time during which the electron is at distance is simply

(2.29)

with v being the velocity of the electron relative to the ion. The energy radiated away through bremsstrahlung per electron will then be:

(2.30)

(2.31)