WHAT DOES A Black Hole Look Like?

WHAT DOES A

Black Hole Look Like?

PRINCETON FRONTIERS IN PHYSICS

Abraham Loeb, How Did the First Stars andGalaxies Form?

Joshua Bloom, What Are Gamma Ray Bursts?

Charles D. Bailyn, What Does a Black Hole Look Like?

WHAT DOES A


CHAR L E S D . B A I L Y N

PRINCETON UNIVERS ITY PRESS

PR INCETON AND OXFORD

Copyright c© 2014 by Princeton University PressPublished by Princeton University Press, 41 William Street,

Princeton, New Jersey 08540In the United Kingdom: Princeton University Press, 6 Oxford Street,

Woodstock, Oxfordshire OX20 1TW

press.princeton.edu

All Rights Reserved

Library of Congress Cataloging-in-Publication Data

Bailyn, Charles D., author.What does a black hole look like? / Charles D. Bailyn.

pages cm. – (Princeton frontiers in physics)Includes index.

Summary: “Emitting no radiation or any other kind of information, blackholes mark the edge of the universe–both physically and in our scientificunderstanding. Yet astronomers have found clear evidence for the existence ofblack holes, employing the same tools and techniques used to explore othercelestial objects. In this sophisticated introduction, leading astronomer CharlesBailyn goes behind the theory and physics of black holes to describe howastronomers are observing these enigmatic objects and developing a remarkablydetailed picture of what they look like and how they interact with theirsurroundings. Accessible to undergraduates and others with some knowledge ofintroductory college-level physics, this book presents the techniques used toidentify and measure the mass and spin of celestial black holes. These keymeasurements demonstrate the existence of two kinds of black holes, those withmasses a few times that of a typical star, and those with masses comparable towhole galaxies–supermassive black holes. The book provides a detailed accountof the nature, formation, and growth of both kinds of black holes. The bookalso describes the possibility of observing theoretically predicted phenomenasuch as gravitational waves, wormholes, and Hawking radiation. A cutting-edgeintroduction to a subject that was once on the border between physics andscience fiction, this book shows how black holes are becoming routine objectsof empirical scientific study.”– Provided by publisher.

ISBN 978-0-691-14882-3 (hardback : acid-free paper) – ISBN 0-691-14882-1(hardcover : acid-free paper) 1. Black holes (Astronomy) 2. Astrophysics. I. Title.

QB843.B55B35 2014523.8′875–dc23

2014009784

British Library Cataloging-in-Publication Data is available

This book has been composed in Garamond and Helvetica Neue

Printed on acid-free paper. ∞Typeset by S R Nova Pvt Ltd, Bangalore, India

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

http://press.princeton.edu

For D.

CON T E N T S

PREFACE xi

1. Introducing Black Holes: Event Horizonsand Singularities 11.1 Escape Velocity and Event Horizons 31.2 The Metric 61.3 What Is a Black Hole? 11

2. Accretion onto a Black Hole 132.1 Spherical Accretion and the Eddington Limit 142.2 Standard Accretion Disks 172.3 Radiatively Inefficient Accretion Flows 232.4 Accretion Instabilities 242.5 Radiation Emission Mechanisms 272.6 Radiative Transfer 322.7 The α-Disk 35

3. Outflows and Jets 433.1 Superluminal Motion 453.2 Jet Physics and Magnetohydrodynamics 48

viii Contents

4. Stellar-Mass Black Holes 534.1 X-Ray Binaries 544.2 Varieties of X-Ray Binaries 584.3 X-Ray Accretion States 604.4 Compact Objects 634.5 Mass Measurements in X-Ray Binaries 684.6 Are High-Mass Compact Objects Black Holes? 734.7 Isolated Stellar-Mass Black Holes 764.8 The Chandrasekhar Limit 79

5. Supermassive Black Holes 845.1 Discovery of Quasars 855.2 Active Galaxies and Unification 885.3 Superluminal Jets and Blazars 945.4 Nonaccreting Central Black Holes 985.5 Mass Determinations for Extragalactic SMBHs 99

6. Formation and Evolution of Black Holes 1066.1 Stellar-Mass Black Holes 1076.2 Supermassive Black Holes 119

7. Do Intermediate-Mass Black Holes Exist? 1277.1 Ultraluminous X-Ray Binaries 1277.2 Black Holes in Star Clusters and Low-Mass

Galaxies 132

8. Black Hole Spin 1358.1 The Innermost Stable Circular Orbit 1378.2 Observations of the ISCO through Line

Emission 139

Contents ix

8.3 Observations of the ISCO through

Thermal Emission 1448.4 Consequences of Spin for Jets and Other

Phenomena 147

9. Detecting Black Holes throughGravitational Waves 1509.1 Gravitational Waves and Their Effects 1529.2 Binary Pulsars 1569.3 Direct Detection of Gravity Waves 1589.4 Detecting Astrophysical Signals 163

10. Black Hole Exotica 16710.1 Hawking Radiation 16710.2 Primordial Black Holes 17110.3 Wormholes 17410.4 Multiverses 176

GLOSSARY 179I NDEX 187

P R E F A C E

The goal of this book is to introduce readers to the empiri-cal study of black holes. Readers are assumed to have someknowledge of basic college-level physics and mathematics.The focus is on current understanding and research relat-ing to astrophysical manifestations of black holes ratherthan on the underlying physical theories. Nevertheless, anunderstanding of the physics is necessary to understandand interpret observations. So, I have presented descrip-tions and derivations of the physical processes in a way thatI hope will illuminate the observations and have focusedon the physical principles involved rather than the fullpresentation needed to solve detailed problems. Interestedreaders should consult the many excellent textbooks in thefield for further discussions. In particular, Bernard Schutz’sbook A First Course on Relativity; Rybicki and Lightman’sRadiative Processes in Astrophysics; Shapiro and Teukolsky’sBlack Holes, White Dwarfs and Neutron Stars: The Physicsof Compact Objects; and Frank, King, and Raine’s AccretionPower in Astrophysics are all classic texts at the upperundergraduate and introductory graduate student levelthat present the relevant physics clearly and in detail. Thefocus in this text is on the observational astrophysics—on

xii Preface

what is observed and what can be inferred from theobservations—in short, on what black holes look like.

I am very fortunate to have lived my life as part of acommunity of incisive thinkers who have helped me learnand understand the material in this book, and many otherthings besides. In particular, I am grateful to my under-graduate advisors-turned-colleagues Pierre Demarque, BobZinn, and Richard Larson, who have been teaching me andencouraging me for more years than any of us would careto admit; to my graduate advisors Peter Eggleton and JoshGrindlay, who introduced me to the wonderful world ofinteracting binary stars; to my long-time collaborators JeffMcClintock, Ron Remillard, and Jerry Orosz, who haveworked with me for two decades to ferret out the char-acteristics of stellar-mass black holes; to my many friendsand colleagues who have made Yale such a wonderful placeto work on high-energy astrophysics, including but notlimited to Meg Urry, Paolo Coppi, Andy Szymkowiak,Michelle Buxton, Ritaban Chatterjee, and Erin Bonning;and to my outstanding students-turned-colleagues whohave struggled with me to understand the many manifesta-tions of black holes, particularly Raj Jain, Dipankar Maitra,Andy Cantrell, Laura Kreidberg, Jedidah Isler, and RachelMacDonald. Finally, I must express gratitude and lovefor my learned and loving nuclear and extended family,most particularly to the gentleman to whom this book isdedicated, who taught me more about how to think thananyone else. That I have anything at all to say on this orany other scientific topic is due largely to conversationswith these and many other interlocutors, both in personand in the published literature. But any deficiencies inunderstanding or in exposition are mine alone.

WHAT DOES A


1IN T RODUC I NG B L A C K HO L E S :E V E N T HOR I Z ON S ANDS I N GU L A R I T I E S

Black holes are extraordinary objects. They exert an at-tractive force that nothing can withstand; they stop time,turn space inside out, and constitute a point of no returnbeyond which our universe comes to an end. They addressissues that have always fascinated humans—literature andphilosophy in all times and cultures explore irresistiblelures, the limits of the universe, and the nature of timeand space. In our own time and place, science has becomea dominant force both intellectually and technologically,and the scientific manifestation of these ancient themesprovides a powerful metaphor that has come to permeateour culture—black holes abound not only in specula-tive fiction but in discussions of politics, culture, andfinance, and in descriptions of our internal and publiclives.

But black holes are not just useful metaphors or remark-able constructs of theoretical physics; they actually exist.Over the past few decades, black holes have moved fromtheoretical exotica to a well-known and carefully studiedclass of astronomical objects. Extensive data archives reveal

2 Introducing Black Holes

the properties of systems containing black holes, andmany details of their behavior are known. In the currentastronomical literature, the seemingly bizarre properties ofblack holes are now taken for granted and are used as abasis for understanding a wide variety of phenomena.

The title of this book is oxymoronic. The definingproperty of black holes is that they do not emit radiation(hence “black”)—so they cannot “look like” anything atall. Nevertheless, black holes are the targets of a widevariety of observational studies. This paradox is of apiece with much of modern astrophysics, in which objectsthat cannot be observed directly are studied in detail.Cosmologists have found that more than 90% of the massenergy of the Universe is in the form of unobservable“dark matter” and “dark energy.” Thousands of planetshave been discovered orbiting stars other than our Sun,but only a tiny handful have been observed directly. So itis with black holes. They cannot be observed directly, andyet they can be studied empirically, in some detail.

My goal for this book is to describe how astronomerscarry out empirical studies of a class of objects that isintrinsically unobservable, and what we have found outabout them. I will focus on current observations andunderstanding of the astrophysical manifestations of blackholes, rather than on the underlying physical theories.There are a number of excellent textbooks on the physicalprocesses, and I will refer to them along the way. The firstthree chapters sketch some of the physics needed to under-stand and interpret the observational results. Subsequentchapters describe observed black holes, and thus providean answer to the question, What do black holes look like?

Escape Velocity & Event Horizons 3

1.1 Escape Velocity and Event Horizons

One of the basic concepts to emerge from Newton’s theoryof gravity is the escape velocity, denoted Vesc. The escapevelocity is the speed required to escape the gravitationalattraction of a spherical object. It can be shown from basicprinciples that

Vesc =√

2G M/R

where G is the gravitational constant (equal to 6.674 ×10−11 m3 kg−1 s−2), and M and R are the mass and radius,respectively, of a spherical object.1

It is a simple matter to calculate the escape velocityfor any combination of size and mass. For example,numbers approximating the size and mass of a humanbeing (1 m and 50 kg) result in an escape velocity of justover 80 µm s−1 (or about a foot per hour). While thisresult would apply precisely only to a spherical object withR = 1 m and M = 50 kg, an object of comparable massand size would have a comparable escape velocity. Becausethe resulting escape velocity is much slower than the speedsassociated with everyday life, gravitational effects betweenordinary objects (people, cars, buildings) can generally beignored. By contrast, the Earth, with a mean radius ofa bit less than 6400 km and a mass of 5.9 × 1024 kg,has an escape velocity of 11 km s−1—much faster thaneveryday speeds. So, without mechanical assistance weremain bound to the Earth.

1Technically, the escape velocity thus calculated applies to test particlesattempting to escape from the surface of the sphere.


The most conceptually straightforward description of ablack hole is an object whose escape velocity is equal to orgreater than the speed of light. Such objects had alreadybeen contemplated in the eighteenth century.2 In such acase, we can rewrite the escape velocity equation as

R ≤ Rs = 2G M/c 2,

where c is the speed of light, and Rs is the Schwarzschildradius (named after the early twentieth-century physicistKarl Schwarzschild). In the context of Newtonian physicssuch objects have no particularly striking physical qualitiesother than their small size (Rs of a mass equal to that of theEarth is only about a centimeter). Presumably, light wouldnot be able to escape from them, so they would be hard toobserve. But the fascinating physics associated with blackholes emerged only when general relativity was developed.

Nevertheless, it is amusing to play with the Newtonianconcept of black holes as objects with Ves c ≥ c andto notice how the size and density of black holes varywith their mass. The density ρ of an object is defined asmass/volume, so the density of a black hole must be

ρbh ≥ 3

32π

c 6

M2G3.

Thus the density required to form a black hole decreasesas the mass of the black hole increases—masses 108 timesthat of the Sun (which, as we will see, are common in

2The eighteenth century British philosopher John Michell is generallycredited with the first published consideration of objects with escape velocitiesgreater than c . See Gary Gibbon 1979, “The Man Who Invented Black Holes,”New Scientist 28 (June) 1101.

Escape Velocity & Event Horizons 5

the center of large galaxies) will become black holes evenif their density is no greater than that of water. Blackholes with masses comparable to those of typical stars mustattain densities comparable to those of atomic nuclei, thatis, much greater than the density of ordinary matter. Lessmassive black holes would require densities far beyond thatof any known substance.

In the general theory of relativity, the Schwarzschildradius becomes fundamentally important. Gravity is notconsidered a force in general relativity but, rather, is a con-sequence of the curvature of space-time. Mass causes space-time to curve, and this curvature affects the trajectories ofobjects. In situations where the distances between objectsare large compared with their Schwarzschild radii, thepredictions of general relativity become indistinguishablefrom those of Newtonian gravity, and all the familiarNewtonian results can be recovered. However, as distancesbetween objects approach Rs , objects begin to behavedifferently from Newtonian predictions. Indeed, the firstobservational evidence supporting general relativity camefrom slight anomalies in the orbit of Mercury, the planetclosest to the Sun. Mercury’s mean distance to the Sun isabout 20 million times the Schwarzschild radius associatedwith the mass of the Sun, so the deviations are quite small,but the orbits of the planets are known very precisely, sothe deviation was already known before Einstein developedhis theory. Closer to the Schwarzschild radius, the differ-ences between Newtonian and relativistic physics becomegreater, leading eventually to drastic qualitative differencesin behavior. These dramatic effects cannot be observed inEarth-bound laboratories, or indeed anywhere in the solar


system, because all nearby objects have radii that are manyorders of magnitude bigger than Rs . But as we will see,black holes and the dramatic physical effects associatedwith them can be found in other astronomical contexts.

For objects that fit inside their Schwarzschild radius, thespherical surface where r = Rs is often referred to as theevent horizon. This name comes about because informationfrom inside the event horizon cannot propagate to theoutside world. Consequences of events that occur at r <

Rs cannot be seen by an observer outside Rs . The interiorof the event horizon is thus causally disconnected fromthe rest of the Universe—in a sense, it is not part of ourUniverse. The behavior of matter and energy inside theevent horizon can be explored mathematically by assumingthat the equations of general relativity apply and theninterpreting the results of mathematical manipulations ofthese equations. However, the laws that lead to the equa-tions also categorically prohibit them from being testedby experiments or observations conducted by observerslocated at r > Rs . Thus from an epistemological pointof view, physics inside an event horizon is a different kindof science from physics in parts of the Universe that arecausally connected to us.

1.2 The Metric

We will not explore the details of the mathematics associ-ated with general relativity here.3 But simply looking at the

3For a good introduction at the undergraduate level, see Bernard Schutz,A First Course in General Relativity (Cambridge: Cambridge University Press,2009).

The Metric 7

form of some of the relevant equations can reveal some ofthe remarkable qualities of black holes.

Mathematically, the curvature of space-time is definedby a metric. A metric defines a line element ds , andthe separation between two space-time events is givenby the integral of ds . In general, this integral dependson the trajectory taken by the object. In the absence ofexternal forces, objects follow trajectories that minimizethe separation.4 In an uncurved space-time, this behavioris in accordance with Newton’s first law of motion, whichrequires that (in the absence of forces) objects move instraight lines (the closest distance between two points). Inrelativity, objects in a gravitational field follow a curvedtrajectory in space not because of a “gravitational force”that redirects their motion but rather because of thecurvature of space-time itself: the minimum separationbetween two space-time events follows a curve in spatialcoordinates.

A single point mass generates a space-time curvatureassociated with the so-called Schwarzschild metric:

ds 2 = −(1 − Rs /r )c 2dt2 + dr 2

1 − Rs /r+ r 2d�2.

Here space is measured in polar coordinates (r , �,where d� = sin θdθdφ), with the point mass at the origin.To be specific, dt is the time interval seen at infinity, andRs is the circumference around the black hole divided by2π . By looking at the limiting cases of this equation, wecan gain some insight into how black holes behave.

4Formally, the “proper time” is maximized.


r → ∞: When r is large, the gravitational influenceis small. In this case the spatial coordinates of themetric approximate polar coordinates of aEuclidean space, and time and space decouple. Inthis limit, the equations of general relativity reduceto the familiar results of Newtonian physics.

r → Rs : As r approaches the Schwarzschild radius,the term (1 − Rs /r ) goes to zero. The time termof the metric thus becomes zero, and the radialterm becomes infinite. Something very peculiarmust happen at r = Rs ! Indeed, as an object fallstoward the black hole, it appears to an outsideobserver that time is slowing down. That is, aclock mounted on the infalling object runs slowerthan an identical clock that remains at a largedistance from the black hole. This observedslowness applies also to the frequencies of emittedradiation. Radiation emitted near a black hole willbe observed to have lower frequencies, and thuslonger wavelengths. This effect is calledgravitational redshift and can be observed in avariety of ways (see chapter 8). However, for anobserver on the infalling object, local clocksappear to be accurate, the Universe far from theblack hole appears to speed up, and the radiationfrom distant objects appears to be blueshifted.

r < Rs : At radii smaller than Rs , the term(1 − Rs /r ) becomes negative. This means thesigns of the time and radial terms of the metric arereversed. As a consequence, radial motion can beonly unidirectional (as time is in ordinary

The Metric 9

situations), while it is possible, in principle, tomove forward and backward in time. Inside theSchwarzschild radius “time machines” are thus inprinciple possible. This property of black holes hasled to a wide variety of speculative fiction, andsome interesting physics as well. But inside theevent horizon, it is not possible to move outward,any more than it is possible to move backward intime in less exotic regions of space-time. Anyobject that finds itself inside the Schwarzschildradius of a black hole will inexorably travel towardthe center of the coordinate system at r = 0. Thusmaterial will pile up in a point of zero volume andinfinite density at the center of a black hole. Thispoint is sometimes referred to as a singularity.

There are thus a number of situations related to blackholes in which physical quantities should become infinite.At r = Rs , terms of the metric become infinite, and timestops.5 At the center of the black hole, where r = 0,the density of matter becomes infinite. The existence ofthis central singularity suggests that the physical theory islikely to be incomplete.6 In particular, there are likely to bequantum effects (which become important at small sizes)that need to be accounted for. But relativity is a continuoustheory and does not fit easily with quantum mechanics.

5The mathematical divergence of the terms of the metric at Rs = r can beavoided by an appropriate change in coordinates, but that does not change thepredicted behavior of an infalling object as observed from a long distance away.

6An example is the “ultraviolet catastrophe”’ in radiation theory, in whichclassical physics requires the radiation emitted from a blackbody to becomeinfinite at high frequencies. One of the first triumphs of quantum physics was toeliminate this infinity from the theory.


The search for a “unified theory” which combines generalrelativity and quantum mechanics is ongoing. Until thissearch is complete, physical predictions of the behaviorof matter and energy at the Schwarzschild radius or nearthe central singularity might plausibly be regarded as theresults of an incomplete theory rather than as any sortof accurate representation of reality. One of the long-term goals of observational relativistic astrophysics is toprobe this regime where current theory might encounterdifficulties. However, as we will see, there are no currentobservations that contradict general relativity.

The Schwarzschild metric is actually a special case thatapplies only to black holes with no angular momentum. Ifthe material forming the black hole has angular momen-tum (as one would expect to find in any physical object,particularly if it forms from the collapse of a much largerstructure), then a more complicated metric known as theKerr metric is the correct description of space-time. TheKerr metric has a key parameter in addition to the mass ofthe central object, namely, its angular momentum, usuallygiven in dimensionless units as a = J /(G M2/c ), where Jis the angular momentum of the object. If a ≤ 1, the Kerrmetric generates an event horizon at

RK = (G M/c 2)(1 +√

1 − a2)

(note that this reduces to the Schwarzschild radius whena = 0). Situations in which event horizons exist and a > 1are generally thought to be nonphysical. In Kerr blackholes, the central singularity takes the form of a ring, ratherthan a point, and there is an additional critical surface

What Is a Black Hole? 11

Figure 1.1. The even( horizon ami the ergosphere of a spinning black hole.

outside the event horizon called the ~rgospher~. inside of

which objects cannO( remain stationary but can escape

fro m the black hole (see figure 1.1). Exchange of energy

between particles within the ergosphere. some of which

escape (0 infinity and some of which fall into the event

horizon, allows much of the rotational energy of a Kerr

black hole (0 be extracted and transferred (0 the outside

universe.7 The observable consequences of black hole spin

will be explored further in chapter 8.

1.3 What Is a Black Hole?

So what exactly is a "black hole"? The term "black hole"

is nO( defined in a technical way and is used in differ

ent contexts to mean different things. The phrase itself

was popularized by the physicist John Archibald Wheeler

to replace the cumbersome description "gravitationally

7R. P~nr""', 1969, RivimltUi N uolJO CimmlO I (& r. I): 252.


completely collapsed object.”8 The term can be used todescribe an object whose escape velocity is greater thanthe speed of light, which leads to a quasi-Newtoniandescription of such objects. Sometimes “black hole” is usedto denote the volume inside the event horizon, that is, theregion “outside our Universe.” In this context it makessense to discuss the “size” or “density” of a black hole,since there is a nonzero radial distance (Rs in the caseof a nonspinning black hole) associated with the object.Sometimes “black hole” is used to refer specifically to thesingularity, in which case such physical quantities are notwell defined. Finally, “black hole” has become a commonlyused metaphor for anything with an inexorable pull leadingto destruction. As we will see, the assumptions aboutblack hole behavior associated with these metaphors areoften quite misleading when applied to the physical objectsthemselves.

8By his own account, Wheeler himself did not invent the term. Rather,the phrase was called out by an anonymous voice at a conference, and Wheeleradopted it then and afterwards. J. A. Wheeler, Geons, Black Holes, and QuantumFoam (New York: Norton, 2000), 296–97.

2ACCR E T I O N ON TO A B L A C K HO L E

One might think that it would be difficult to observe blackholes, given that their defining characteristic is that they donot emit light. However, their presence is clearly detectedthrough their gravitational effects on nearby objects. Inparticular, gas accreting onto a black hole generates hugeamounts of energy that create easily observable effects. Infact, accretion energy powers the most luminous objects inthe Universe and can be much more efficient at turningmass into energy than the thermonuclear processes thatpower ordinary stars like the Sun. The fusion of hydrogenatoms to make more massive nuclei generates energyequivalent to just under 1% of the mass of the hydrogengas. But accretion onto a black hole can produce energymuch more efficiently, depending on the way the gasflows onto the black hole and the way that the gastransforms kinetic energy into radiation. In this chapterwe will explore the ways that accretion produces energyand radiation.

As material falls into a gravitational potential well, en-ergy is transformed from gravitational potential energy intoother forms of energy, so that total energy is conserved. Ifall the energy is turned into kinetic energy, as in the case

14 Accretion onto a Black Hole

of a single object in free fall, then

V 2/2 − G M/D = constant,

where V is the velocity of the infalling object, M is themass of the accretor, and D is the distance between thetwo. As the object falls in, D decreases, so V must increase.This kinetic energy can in turn be converted into otherforms of energy, and thence into detectable radiation. Inparticular, an infalling stream of gas can convert kineticenergy of the individual gas particles into heat energy ofthe gas as a whole. Hot gas glows, and thus the infalling gasis a radiation source. The deeper the material falls into thepotential well, the more energy it can, in principle, pick up.Black holes have the deepest possible potential wells, so ac-cretion onto a black hole is especially energetic. Observingsuch accretion energy is one of the primary ways that astro-physicists pinpoint the locations of potential black holes.The spectrum and intensity of this radiation is governedby the geometry of the gas flow, the mass infall rate, andthe mass of the accretor. Thus careful study of the accretionradiation, and comparison with detailed models of accre-tion, can reveal not only the presence of a black hole butthe ways in which the infalling matter is influenced by thestrong relativistic effects that the black hole creates.

2.1 Spherical Accretion and the Eddington Limit

The simplest flow geometry is that of a stationary objectaccreting mass equally from all directions. Such spherically

Spherical Accretion & the Eddington Limit 15

symmetric accretion is referred to as Bondi-Hoyle accretion.As the gas falls inward it heats up and releases radiation.This radiation can be intercepted by gas farther out, whichcan retard the flow. Bondi-Hoyle accretion is thus self-limiting, in the sense that a large infall can generate enoughradiation to halt the inflow altogether. This limit is calledthe Eddington limit, and it occurs when the gravitationalforce inward is balanced by the outward force generatedby the radiation. In the case of ionized hydrogen gas,the gravitational force is dominated by the force on theprotons, which is Fin = G Mm p/r 2, where M is the massof the accreting object, and m p is the mass of the proton.The gravitational force on the electrons is negligible, sincethe mass of an electron is so much smaller than the mass ofa proton.

By contrast, the outward force generated by the ra-diation is dominated by the interaction of the radiationwith the electrons. The force on a single electron can becomputed by multiplying the radiation flux divided by thespeed of light c by a cross section which is the effectivegeometric area that the electron presents to the radiationfield. The radiation pressure is given by the intensity of theradiation field I divided by the speed of light c , and therelevant cross-sectional area is the Thomson cross sectionσT = 6.65 × 10−29 m2. For a central source of radiationexpanding spherically, the local intensity I is related tothe overall luminosity L by I = L/4πr 2, where r is thedistance to the central source, which is the same r as thatused in the calculation of the gravitational force, providedthe radiation is generated by accretion onto the centralobject. Thus the radiation force is given by σT L/(4cπr 2).


The cross section of the electrons to radiation is muchgreater than that of the protons, so in this case, theprotons can be neglected. In most situations the coulombinteractions between the protons and the electrons arestrong enough that the plasma remains neutral, and theoutward force on the electrons is balanced by the inwardforce on the protons, and G Mm p = LσT/4πc . Thesetwo forces balance when the luminosity is equal to theEddington luminosity LEdd:

LEdd = 4πG Mm pc/σT .

At this luminosity, the outward force of the radiationmatches the inward gravitational force and prevents anyadditional infall. Interestingly, the Eddington limit doesnot depend on the radius or density of the accreting objectbut only on its mass. Thus the Eddington limit can beexpressed as a function of mass as follows:

LEdd ≈ 1.2 × 1038(M/M�) ergs−1,

where M� is the mass of the Sun, and the compositionof the gas is assumed to be pure hydrogen. This limitapplies not just to energy generated by accretion—a starthat attempts to radiate at greater than its Eddington limitwill blow itself apart, so this relation provides an upperlimit on the brightness of stars generally.

There are some limitations to the derivation shown.A plasma of fully ionized hydrogen was assumed so thereare equal numbers of protons and electrons. If the com-position is different, then the mean atomic weight of theions per electron can be different, leading to a different

Standard Accretion Disks 17

numerical result. If the gas is not fully ionized, the crosssection to radiation can be different as well. For example,if a load of bricks were dumped on a black hole, the crosssection of each brick would be its geometric size, and theradiation pressure it absorbed would have to balance theinward gravitational force on the brick—although bythe time the bricks got close to the black hole, they wouldlikely be heated up and turned into an ionized plasma.

Beyond these considerations of detail, there is a moreprofound requirement on the Eddington limit, namely,that both the accretion and the radiation are assumed tobe spherically symmetric. If the infall is on only part of theobject, and radiation escapes in a different direction, thebasic assumptions behind the derivation of the Eddingtonlimit do not apply. As we will see, accretion flows ontoblack holes are not thought to be spherically symmetric—the infall is much more frequently in the form of aflattened disk. But even when the formal requirementsfor the derivation of the Eddington limit are not met, theEddington luminosity and the associated limit on massaccretion rate provide useful reference values that have sig-nificant physical consequences for accretion flows. Indeed,the vast majority of accreting black holes for which themass of the black hole is known are radiating at less thanLEdd (some possible exceptions are discussed in chapter 7).

2.2 Standard Accretion Disks

In most cases, gas accretion is not spherically symmet-ric. Infalling gas, like anything else, typically has some


rotational component and thus has some angular mo-mentum (usually denoted by J ). As the mass falls in,the distance D to the accretor decreases, and the angularvelocity Vang must increase so that the angular momentumJ ∝ D × V is conserved. The increasing angular velocitygenerates a centrifugal force that acts against the inwardgravitational force. But the gas rotates within a plane,so this outward-directed force operates only in the planeof the rotation. Consequently, the infalling gas naturallyassumes the form of a disk oriented perpendicular to theangular momentum vector J.

In principle, any object will establish a stable orbitaround a black hole, or any other central gravitationalmass, with the orbital period and eccentricity determinedby the masses of the two objects, and the total energy andangular momentum of the orbit. In Newtonian mechanics,such orbits are stable, and in the absence of other forceslike tides or additional gravitating objects, the orbitalparameters will not change in time—indeed, the planetsin our own solar system have remained in stable orbits forbillions of years. Orbits around a black hole can be justas stable as they are around less exotic objects. Whetheran orbit is “Newtonian” in its stability is not related tothe nature of the orbiting objects themselves but rather tothe size of the orbit. If the orbit is large compared with theSchwarzschild radius of the two orbiting objects, the orbitwill be Newtonian. If the Sun were suddenly replaced by ablack hole of one solar mass, the orbit of the Earth wouldnot change at all. Thus the popular impression that a blackhole necessarily sucks in everything around it can be verymisleading.


One can imagine a single atom establishing a New-tonian orbit around a black hole (or any other object).If that atom is the only orbiting object, it will remainin such an orbit indefinitely. However, if there is a largequantity of gas, each atom of which interacts with theothers, the situation becomes quite different. Whateverangular momentum the gas contains is likely to make thegas orbits align in the same plane. If the orbits of thegas are noncircular, then some gas will be moving inwardand some, outward. The gas streams will run into eachother and exchange energy and angular momentum, sothat orbits will become smoothly circular. Thus orbitinggas will naturally configure itself into a disk in which eachatom follows a circular orbit around the central body. Suchdisks are called accretion disks.

In an accretion disk, the orbital velocity at distances farfrom the black hole is determined by standard Newtonianphysics, which requires that the centrifugal accelerationV 2/D, plus any other outward forces, be balanced bythe inward gravitational pull from the accreting object(G M/D2). For infall rates well below the Eddington limit,this requirement leads to the Keplerian formula for theorbital velocity V = √

G M/D, and the familiar resultthat orbits closer to the central object must be faster.Consequently in a gas disk, this results in friction betweengas at any particular radius and the gas immediatelycloser or farther out, as the particles in the gas slide pasteach other. This friction generates heat—in formal terms,the viscosity of the gas transfers energy from the kineticenergy of the gas flow to heat energy in the gas. As theviscosity removes energy from the bulk motion of the


gas into the random motion of the gas particles it alsotransports angular momentum outward in the disk. As thebulk kinetic energy decreases, and angular momentum istransported away, the orbiting gas must move into orbitscloser and closer to the central object. Thus there is asteady flow of material through the disk toward the centralaccreting object.

Under some circumstances, the structure of accretiondisks can be simply described (see section 2.7). Assumingthat the disk is geometrically thin and optically thick, andthat the energy dissipated in the disk is radiated thermally,the energy radiated at some radius R in the disk must equalthe energy generated by the inward motion of the materialthrough the disk at that radius. Thus

σ T4 Rdr = mdr G M/R2,

where σ is the Stefan-Boltzmann contant, and m =dm/dr is the rate of mass flow inward through the disk.Thus

T ∝ R−3/4.

The total emitted spectrum of the disk is then the integralof the blackbody emission from the outer to the inner edgeof the disk (note that the inner and outer boundaries of thedisk generate special conditions). The resulting spectrumhas a cutoff similar to a single-temperature blackbody atboth the high-energy and the low-energy end but a widerpeak (due to the range of temperatures available).

The total luminosity emitted by an accretion disk de-pends on the rate at which mass is accreting, m. When the


luminosity approaches the Eddington limit, radiation pres-sure starts to contribute significantly to holding the gasup against gravity, and the thin disk–like geometry canno longer be maintained. Under these circumstance, theaccretion flow takes on other geometries. Interestingly,the temperature associated with the peak emission of thedisk near the Eddington limit scales with the mass of theaccreting object as T ∝ M−1/4, so more massive black holestend to produce cooler disks. Of course, the Eddingtonlimit does not strictly apply to nonspherical structures likedisks. Nevertheless, the assumption of a thin-disk geome-try modified by radiation pressure at luminosities close tothe Eddington limit provides reasonable explanations for asurprisingly wide range of observational data.

This approach to accretion disk structure takes the massaccretion rate as a free parameter imposed by the physicalsituation that generates the gas flow. The assumption isthat the disk will adjust itself so that the mass flow isconstant throughout the disk—if m is lower at some radiusof the disk than it is farther out, then mass will accumulateat that radius. The higher density will then generate morefriction, more energy loss, and a higher mass flow rate. Thisprocess leads, in principle, to a situation in which the localvalue of m is the same at all radii in the disk. To understandthe density distribution in the disk, one must then makesome assumption about how energy is transformed from ki-netic energy associated with the rotation flow into heat andradiation, and in particular how this energy transformationprocesses depend on the density, temperature, and verticalstructure of the disk at different distances from the blackhole. One assumption, discussed in detail in section 2.7,


is that the viscosity that creates the gas friction can bedescribed as a constant α multiplied by the sound speedand scale height of the disk. This assumption leads to acomplete physical description of the structure of the disk.Such disks are referred to as α-disks.

But this approach does not identify the physical processresponsible for the viscosity—it simply assumes a particularfunctional form for the viscosity. The absence of anynotion of the actual physical effects that are taking placemakes it all the more remarkable that α-disks really doseem to describe the overall features of real physical objects.However, as observations have improved, many detailshave emerged that require models that go beyond the α-disk approach and thus necessitate a clear understandingof how the viscosity is generated.

It turns out that the electromagnetic forces betweenindividual ions in the disk gas are much too small to gen-erate the observed flows of mass and angular momentum.Turbulence in the fluid acts to enhance viscosity, but fora long time it was not clear exactly what would drive theturbulence. Recently, it has become generally accepted thata magneto-rotational instability is most likely responsiblefor the viscosity in accretion disks. In this approach, it isassumed that a magnetic field permeates the disk. Magnetictension then connects gas orbiting at a given radius withgas slightly farther in (or out). This tension can be thoughtof as a spring that connects gas particles with other particlesat different distances from the black hole. Since the innergas is orbiting faster than the outer gas, the inner gas pullsthe outer gas along, thus transferring angular momentumfrom the inside of the disk outward, while the outer gas

Radiatively Inefficient Accretion Flows 23

slows down the inner gas, extracting energy from theorbital motion. Detailed simulations of this mechanismhave been carried out which demonstrate that it resultsin disks similar to α-disks in which 0.01 < α < 0.1,comparable to the values required by observations.

2.3 Radiatively Inefficient Accretion Flows

Not all accretion flows can be described as α-disks. Ifthe various assumptions that give rise to α-disks are vi-olated, quite different accretion flow geometries can beestablished. Considerable recent research has demonstratedthe possibility of a wide variety of accretion flows. Whileall these models have not been confirmed in observedaccreting black holes, some observations that cannot beexplained by simple accretion disks have been associatedwith other kinds of accretion flows.

One important class of accretion flows is the so-calledADAFs, or advection dominated accretion flows. In anADAF, the assumption that the gravitational energy dis-sipated by the infalling matter is largely radiated whereit is generated is no longer valid. Instead, the gas carries(advects) the energy inward, resulting in a flow that movesinward relatively quickly and is not confined to a plane. Atthe inner boundary of the accretion flow the gas is likelyto have considerable internal energy that was been pickedup in the fall inward but has not yet been radiated away.In most situations, the gas flow terminates at a surface,and the energy is transformed into radiation in a boundarylayer associated with that surface. However, accretion onto


a black hole does not encounter a surface but rather anevent horizon. The internal energy of the accretion flow isnot liberated at the event horizon but can advect throughthe event horizon along with the mass that carries it. Thusone of the observational characteristics of accretion onto ablack hole is the absence of boundary layer radiation.

One situation in which ADAFs are likely to occurinvolves very low values of m, which result in low-densitygas flows. In an ionized gas of sufficiently low density, thecoupling between positively charged ions and negativelycharged electrons through coulomb interactions breaksdown, and the ions and electrons can have differenttemperatures. Since the ions are much more massive thanelectrons, most of the kinetic energy is carried by the ions.However, the electrons radiate away their energy moreefficiently. If the electrons radiate energy faster than it canbe replaced by interactions with the ions, then the electronswill be much cooler than the ions, and the ions will advecttheir energy inward as they accrete onto the central object.Such an accretion flow is necessarily optically thin, and sothe observed radiation field is not thermal, as is the case forstandard accretion disks, but rather some combination ofnonthermal processes.

2.4 Accretion Instabilities

We have assumed so far that the accretion rate and othercharacteristics of the accretion disk do not change in time.However, accretion flows are observed to vary, sometimesquite drastically. A typical light curve showing intensity

Accretion Instabilities 25

0

Inte

nsity

Time

DaysWeeks/Months

Years

Figure 2.1. X-ray binaries often display a fast rise and exponentialdecay (FRED) light curve like the one shown. In some cases sourcescan go from invisible to among the brightest sources in the skywithin a few days.

versus time for an unstable accreting stellar mass black holeis shown in figure 2.1. There are in general two causesof variability in disks and other accretion flows. The firstresults from changes in the overall mass accretion rate m.Such a change implies that the physical situation in whichthe accretor is embedded has changed. Since the energeticsof accretion depend on how much mass is being accreted,such changes result in changes in the physical conditionsand observed properties of the accretion flow.

However, accretion flows can be unstable even if theoverall accretion rate is constant, as a result of instabilitiesin the gas flow. In the same way that water can dripfrom a faucet rather than flowing continuously, the gasflow through the disk can be variable or episodic, even ifthe gas flow onto the disk is constant and continuous.One situation that has been explored in detail involves


changes in the ionization of the gas. The free electrons andions in ionized gas generate a significantly higher viscositythan neutral gas, and thus the local m will be greater whenthe gas is ionized than when it is not. If the mass enteringsome region of the disk is less than the disk can processwhen the gas is ionized, but more than moves through thedisk when the gas is neutral, then a limit cycle can be setup. In such a limit cycle, the gas density increases whenthe gas is neutral, since the mass accretion rate is greaterthan the rate at which gas is moving through that locationof the disk. As the gas accumulates, it gets denser andhotter, until ionization occurs, which greatly increases theviscosity of the disk. Then, the accumulated gas “flushes”rapidly down through that region of the disk, leading inturn to a rapid drop in density and temperature, and arecombination of the gas. The cycle then begins anew.In the α-disk formalism, such disk instabilities can bemodeled as abrupt changes in the assumed value for α.

One consequence of a change in either the overall massaccretion rate or the mass transfer rate through differentparts of the accretion flow is that the geometry of the flowcan change dramatically. Objects have been observed tochange from α-disks to ADAFs and back, and some situ-ations seem to require different kinds of flows at differentdistances from the accreting object. Such changes in theaccretion process are accompanied by significant changesin the observed spectral properties of the object, so changesin the mass accretion rate do not change merely the overallluminosity of the source but also the spectrum and otherdetailed characteristics of the emerging radiation. Suchchanges in the observed characteristics of accreting objects

Radiation Emission Mechanisms 27

Very High State

High State

Intermediate State

Low State

Quiescent State

0.5

m

0.09

0.08

0.01

Figure 2.2. Cartoon representation of accretion states in an X-raybinary, where m is given in units of the Eddington rate. Note theincreased dominance of a disk at higher accretion rates. It is nowbelieved that accretion rate alone is not the only determinant of theaccretion flow. From A. Esin, J. McClintock, and R. Narayan, 1997,Astrophys J., 489: 865.

are referred to as state transitions. One hypothezised set ofstate transitions as M changes is shown in figure 2.2.

2.5 Radiation Emission Mechanisms

Accretion flows onto a black hole cannot be mappeddirectly. The angular scale projected on the sky by the


event horizon and its environs is generally much too smallto be resolved by current instrumentation. Therefore, whatwe observe is a single point source of emission in which theradiation from the whole accretion flow is combined. Thuswe have to infer the geometry of the accretion flow fromthe spectrum and variability of the total observed emission.To do this we must understand how the energy generatedby the infall of material is transformed into radiation (theemission mechanisms) and how that radiation propagatesfrom where it is produced to where it is observed (radiativetransfer).

The simplest emission mechanism occurs when anobject is optically thick, that is to say, when the photons arerepeatedly scattered by the material so that they eventuallyemerge only from the surface of the object, and theirproperties are determined solely by the temperature of theemitting surface. Such radiation is referred to as blackbodyradiation. As noted earlier, the emitted spectrum of anoptically thick accretion disk can be simply determinedby integrating the blackbody spectrum from the hot inneredge of the accretion disk to the cooler outer edge. Such aspectrum has an exponential cutoff at wavelengths shorterthan those associated with the temperature at the inneredge of the disk, and a power-law cutoff at wavelengthslonger than those associated with the temperature at theouter edge of the disk. In between, the spectrum is flatterthan that of a single-temperature blackbody. Nevertheless,one can compute a wavelength region where most of theemission is expected to be emitted. This wavelength varieswith the mass of the accreting object as λmax ∝ M1/4. Sincethe temperature of a blackbody is inversely proportional to


the wavelength where the maximum radiation is emitted,this means that the inferred temperature for an accretiondisk varies as T ∝ M−1/4. For accreting masses similar tothose of stars, the radiation peaks in the soft (low-energy)X-rays, as is observed from the strong galactic X-raysources. This agreement of theory and observation helpedestablish the nature of the galactic sources as accretingblack holes and validated the general disk accretion hy-pothesis. For much more massive black holes, the radiationshould peak in the ultraviolet, as is also observed.

In other kinds of accretion flows, the accreting materialis often optically thin, and in this case the radiation emis-sion mechanisms can be more varied and more complex.1

Generally speaking, the emission arises from accelerationof electrons by various mechanisms; acceleration of elec-tric charges generates electromagnetic radiation, and theelectrons, being much less massive than protons or othercharged particles, are particularly prone to acceleration.The acceleration can be generated by interactions withother charged particles, by magnetic fields, or by interac-tions with photons.

Bremsstrahlung. Bremsstrahlung radiation is emitted whenan electron is deflected by the electromagnetic charge ofsome other particle. This process is common in ionizedplasmas, where there are many free electrons and also largenumbers of positively charged ions. When the electrons areunbound both before and after they interact with the ions,the radiation is sometimes referred to as free-free emission.

1See the classic textbook by G. Rybicki and A. Lightman, Radiative Processesin Astrophysics (New York: Wiley, 1979) for a full discussion.


The wavelengths and intensity of the resulting radiationdepend on the distribution and velocity of the electronsand ions, and thus on the temperature of the plasma.The specific case in which the electrons all have the sametemperature is called thermal bremsstrahlung.

Synchrotron emission. In the presence of magnetic fields,the electrons circle the magnetic field lines. Nonrelativisticelectrons generate cyclotron emission, at a frequency deter-mined by the energy of the electrons and the strength of themagnetic field. Relativistic electrons complicate the situa-tion somewhat, since the resulting radiation is beamed for-ward, rather than being emitted isotropically. The resultingradiation is referred to as synchrotron emission. In general,an emitting plasma contains electrons with a distributionof energies interacting with magnetic fields of a variety ofstrengths and directions. The resulting spectrum can oftenbe described by an intensity distribution that varies as apower law with frequency. Radio emission from accretingblack holes is often due to synchrotron emission.

Compton scattering. Compton scattering occurs when elec-trons and photons interact. Energy can be exchangedbetween the photons and electrons, resulting in dramaticchanges to the observed spectrum of photons. In par-ticular, a seed population of photons interacting with adistribution of relativistic electrons can be scattered tohigh energies. One common situation is referred to assynchrotron self-Compton or SSC emission. In this config-uration, an initial seed spectrum of low-energy radiation isgenerated by the synchrotron process, and then the same


relativistic electrons that initially generated the photonsalso generate much higher energy radiation through theCompton process.

Line emission. Emission confined to specific wavelengths(emission lines) can be created and removed by atomicand nuclear processes. When an electron falls from a highenergy level to a lower energy level, a photon of a specificenergy, and thus a specific wavelength, is released. If someprocess returns the atoms to the higher energy level, asteady stream of photons at a specific wavelength canbe produced. Several processes can repopulate the higherenergy levels, including collisions or energy input from aradiation field.

As a plasma is heated the gas becomes increasinglyionized, which changes the atomic transitions responsiblefor line emission, so the high-temperature plasmas nearblack holes have spectral lines that are quite different fromthose commonly observed in stars. A fully ionized plasmadoes not generate atomic line emission at all. The highest-energy atomic line generally observed comes from 25-times-ionized iron (that is, an iron nucleus surroundedby a single electron), which generates a hydrogen-likespectrum, but at 262 = 676 times higher energy, giventhat the energy levels of single-electron atoms scale asthe square of the nuclear charge. The equivalent of theLyman-α line in hydrogen (produced by electrons fallingfrom the second to the lowest energy level) occurs at6.4 keV in 25-times-ionized iron. X-ray telescopes caneasily observe this energy, and since iron is one of the


more commonly occurring heavy elements in astrophysicalplasmas, this line is often observed and studied. It isthe highest energy at which atomic transition is generallyobserved. However, other processes can generate lines athigher energies. Nuclear processes and radioactivity cangive rise to line emission, and annihilation of electrons andpositrons generates radiation at 511 keV.

Line emission from a disk (or from an optically thinatmosphere above a disk) has a particular shape due tothe Doppler shift generated by the motions in the disk.Such a line will be broadened by the range of Dopplershifts and will appear to have two peaks, generated bythe parts of the disk that are approaching and recedingfrom the observer. Such double-peaked emission lines arecommonly observed and are generally considered to beclear evidence for the presence of a rapidly rotating disk.

2.6 Radiative Transfer

Once the radiation has been generated, it must travelto the observer. The observed spectrum can be changedsignificantly by interactions between the radiation andmaterial it passes through. The relevant physical processesare referred to as radiative transfer. Changes in the spec-trum due to radiative transfer will naturally occur as theradiation field emerges from the source—all the processesalready described are examples of how interactions with anambient medium can affect an emerging radiation field.But the radiation field can also be absorbed by materialfar from the original source, along the line of sight to the

Radiative Transfer 33

observer. Since the amount of absorption depends on thewavelength of the radiation, such interstellar absorptioncan dramatically change the observed spectrum. Crucialsources of absorption include the following.

1) Dust. Interstellar dust particles can absorbradiation in ultraviolet, optical, and infraredwavelengths. Dust generally absorbs shorterwavelengths (bluer light) more than longerwavelengths. In the optical, such absorptionmakes the spectrum look redder, so dustabsorption is sometimes referred to as interstellarreddening. There are standard models of dust thatpredict the amount of absorption as a function ofwavelength and column density of the dust, andthese models work well under mostcircumstances.

2) Hydrogen ionization. Hydrogen is by far themost common element in the Universe, and theinterstellar medium has a significant density ofhydrogen. Most of this hydrogen is cold, with itssingle electron in the lowest energy state. Thisinterstellar hydrogen absorbs photons whoseenergy is sufficient to ionize ground-statehydrogen. Thus ultraviolet radiation below912 Å (>13.6 eV in photon energy) generallydoes not propagate successfully through thegalaxy. At much higher energies the cross sectionof hydrogen becomes smaller, so that for softX-rays at >1 keV the effect of hydrogenabsorption is greatly reduced. But the extreme


ultraviolet, where hydrogen ionization has a largeeffect, is a very difficult wavelength regime toobserve. The amount of interstellar hydrogenscales roughly with the amount of interstellardust, so optical reddening and hydrogenabsorption of ultraviolet and X-ray photons areoften correlated.

3) At the highest energy, photons with energies≥10 TeV interact with the low-energy photonsof the pervasive microwave background to createelectron-positron pairs. This process severelylimits the propagation of very high energyphotons across cosmological distances.

All these physical processes are observed in accreting blackholes, often simultaneously. Thus understanding the ra-diation spectrum generated by accreting black holes oftenrequires a superposition of thermal emission, nonthermalemission (often approximated as a power law), and lineemission, as well as a consideration of the effects ofinterstellar absorption. Often, different components of thespectrum, and different wavelength regimes, are generatedin different parts of the accretion flow. We observe thermaloptical emission from a standard accretion disk, radioemission generated by synchrotron emission, and high-energy emission from comptonization all in the same ob-ject. It is not always straightforward to correctly decomposean observed spectrum into components associated withspecific emission mechanisms. It can also be difficult toassociate specific accretion flow geometries with emissionmechanisms and the resulting observed spectrum. Ideally,

The α-Disk 35

one would create fully self-consistent models of observedspectra, taking into account the energy generated by theaccretion, all the processes that transform this energy intoobservable radiation, and all relevant radiative transfereffects. In practice, this is difficult to achieve, so often,assumptions are made about the accretion flow or theradiation mechanisms that are not fully justified by theobservations.

2.7 The α-Disk

The α-disk or thin-disk model for accretion disks was firstworked out by Shakura and Sunyaev.2 In essence, a setof assumptions about the nature of the accretion processis made that leads to a set of eight algebraic equationsin eight unknowns. These are the numbered equationsthat follow. These equations can then be solved as afunction of the distance from the accreting object R forany particular value of the accreting mass M and the massaccretion rate m. The assumptions made are by no meansalways applicable, so accretion flows can be dramaticallydifferent in nature from α-disks. Nevertheless, the α-diskformulation is useful for many applications.

The first key assumption is that all elements of thedisk travel in Keplerian circular orbits around the accretingobjects. In this case one can write the equation of circular

2N. Shakura and R. Sunyaev, 1973, Astronomy & Astrophysics 24:337. Herewe use the notation of J. Frank, A. King, and D. Raine in chapter 5 of AccretionPower in Astrophysics (Cambridge: Cambridge University Press, 3rd ed., 2002),where further comments and details of the derivations can be found.


motion Vcirc(R) = R�(R) = (G M/R)1/2, where R isthe distance of a particular gas element from the accretingobject, and � is the angular velocity of the gas. Notethat this equation also assumes that the gravity from theaccreting object dominates any gravitational effect of thecompanion star or the disk itself.

However, if this assumption were strictly true, the gaswould simply continue in stable orbits, and no accretionwould take place. The assumption neglects the viscosityof the gas. Viscosity provides a way to transfer angularmomentum and energy within the disk, allowing the gasto move from one orbit to another, and thus drives theaccretion process. One way to think about the effects ofviscosity is to consider that gas in a circular orbit at distanceR from a central gravitating object has a greater angularvelocity than gas orbiting at a greater distance. Viscositycreates tension between gas at different distances from theaccreting object, which results in the outer gas being pulledforward by the more rapidly orbiting inner gas, while theinner gas is slowed down by the outer gas. Thus angularmomentum moves from the inner gas outward, while theinner gas tends to slow down and thus spiral inward. Asthe gas moves toward the center the total energy of the gasdecreases, and the lost energy increases the internal energyof the disk itself and is ultimately radiated away. It is thisradiation that is ultimately observed.

Given the presence of viscosity, the circular orbit equa-tion is an approximation, and its use assumes that theinward drift of the gas in the radial direction vR is smallcompared with the orbital speeds, that is, vR << Vcirc.The mass transfer rate M for the gas can then be written

The α-Disk 37

M = 2π R�vR , where �(R) is the surface density ofthe gas, and thus the total mass at any given annulus is2π R�d R . As the mass moves in toward the center ofthe system, mass, angular momentum, and energy mustbe conserved. Each of these conservation laws leads to anequation that relates the mass flow through an annulusat distance R from the accreting object to vR and theconditions in the disk at that point.

In writing these three equations, we make two ad-ditional crucial assumptions. First, we assume that thedisk is in a steady state, and thus its structure does notchange with time. Second, we assume that the disk is thin,that is, that its vertical scale height H(R) is everywheresmall compared with R . More specifically, H relates themidplane density ρ(R) at any point in the disk to thesurface density � by

ρ = �/H.(1)

Furthermore, for any quantity X that varies with verticaldistance above the disk z, δX/δz ≈ X/H. A “thin” diskimplies H ≈ z << R . For example, the pressure gradientδP/δz can be written P/H. In the vertical direction, thepressure gradient must be balanced by the z component ofthe gravitational force from the central object, so that

δPδz

(1/ρ) = −G MzR3

.

If we use the equation of state

P = ρc 2s ,(2)


where c s is the sound speed, we find that

H = c s

(R3

G M

)1/2

.(3)

The last equation, the equation of state, and the definitionof � give us three of our eight equations of disk structure.

Another relevant equation can be found by setting thepressure equal to the sum of the gas pressure ρkTc /µm p ,where k is the Boltzmann constant, Tc is the centraltemperature of the disk, µ is the atomic weight of thegas, and m p is the mass of the proton; and the radiationpressure, given by (4σ/3c )T4

c , where σ is the Stefan-Boltzmann constant:

P = ρK Tc

µm p+ 4σ

3cT4

c .(4)

Conservation of mass can be simply expressed as

δ(R�vR )

δR= 0.

R� is proportional to the amount of mass in an annulus atR , and vR is the rate at which the mass is flowing towardthe interior, so this equation simply states that there can beno net flow of mass from one part of the disk to another—mass being transported inward must be balanced by massflowing in from the outside. If this were not true, thetotal amount of mass at some point in the disk would bechanging, thus violating the assumption of a steady-statedisk.

The α-Disk 39

The equations of angular momentum conservation andenergy conservation are more complex, because the viscos-ity provides a local source for these quantities. The angularmomentum of the gas at R can be written as 2π R�R2�,so conservation of angular momentum can be written

δ(2π R�R2�vR )

δR= δQ

δR,

where Q(R) is the torque exerted by the viscosity. Notethat at an annulus at R the net torque is the differencebetween the torque exerted by gas at R + δ on that atR , and that of the gas at R on that at R − δR . This iswhy the right-hand side of the equation takes the form ofa derivative of the torque. Here we ignore the constantof integration, which relates to the angular momentumtransfer between the inner edge of the accretion disk andthe accreting object, and simply set Q equal to the quantityin parentheses.

If the torque Q is caused by viscosity across neighboringgas annuli, that is, if it is a purely local effect, it must beproportional to the strength of the viscosity ν, and thechange in the angular velocity across neighboring annulid�/d R , and can be written Q(R) = Rν�R2(d�/d R).If we now assume Keplerian rotation and use the definitionof M, we find that angular momentum conservation in thedisk can be expressed as

ν� = M/(3π ).(5)

This is a reasonable result, in that it says that the masstransfer rate increases with the density of matter in the


disk (�) and with the strength of the viscosity, which isthe physical process that leads to the mass transfer in thefirst place.

Similarly, the energy deposited in the annulus at R canbe written as a luminosity L = Q�′ = 3G MM/(2R2),where we have used the expressions for Q, M, and � givenpreviously. This energy then must be radiated away fromthe disk. The radiant blackbody flux emerging from a thindisk can be written F = 4σ T4

c /3τ , where the thin-diskapproximation is used to relate the optical thickness of thedisk τ and the central temperature Tc to the temperatureappropriate for the blackbody radiation, which is thetemperature at τ = 1. The flux emerges from both sides ofthe disk, and thus over an annulus is equal to 4π R F , withF as given earlier. This expression can then be set equal tothe luminosity generated by the viscosity to yield

4σ T4c

3τ= 3G MM

8π R3.(6)

Note that this assumes that the energy is transported byradiation and that the disk is optically thick—otherwisethe assumption of blackbody radiation does not hold. Theoptical depth can be determined from the state of the gasand is generally taken to be well represented by Kramer’sopacity, in which case

τ = ρHκR = C�ρT−7/2c ,(7)

where C is a constant determined by the Gaunt factorsassociated with quantum mechanical effects and the com-position and ionization state of the gas.

The α-Disk 41

We now have a set of seven equations in the eightunknowns ρ, �, H, c s , P , Tc , τ , and ν, all of whichare functions of the mass of the accreting object M, themass accretion rate M, and the annular distance fromthe accreting source R . To close the system, we needan expression for the viscosity in terms of the otherparameters. The final assumption then is then that

ν = αc s H,(8)

where α is a constant, generally taken to be less than 1.This assumption, first used by Shakura and Sunyaev,produces a set of eight algebraic equations in eight un-knowns, which can then be solved to determine the den-sity, temperature, and emitted spectrum for every value ofR . Integration of the emission over 2π Rd R then producesa spectrum for the disk as a whole. Thus for any value of Mand M, the spectrum produced by the accretion disk can becalculated without solving any differential equations at all.

This calculation contains a large number of assump-tions, which we collect here for reference:

• The gas orbits in near-circular orbits; that is,vR << Vcirc.

• The disk is in a steady state with a mass accretionrate M that is constant in time and across thedisk.

• The disk is geometrically thin and optically thick.• The details of the inner boundary, where mass

accumulates on the accreting object, areunimportant (note that this assumption can be


relaxed and the equations rederived—see Frank,King, and Raine).

• Viscosity is local and can be approximated byν = αc s H.

Remarkably, many (but by no means all) accreting objectshave spectra that resemble those predicted by this simpletheory. This fact, along with the ease with which theemission and physical properties of these α-disks can becalculated, has led to the extensive use of α-disks as thefirst, most obvious approximation of disk structure for alarge range of research on accretion disks over the past 40years.

3OUT F LOWS AND J E T S

One somewhat unexpected feature of accretion flows isthe presence of outflows, or jets. There is strong observa-tional evidence that some fraction of the infalling materialreverses course near the accreting object and is shot outperpendicularly to the accretion disk. In some cases out-flow velocities in the jet can be close to the speed of light,and the jets can carry energy over very large distances. Inparticular, narrow collimated beams of emission are ob-served emerging from the central-most regions of galaxiesand continuing across the whole of the galaxy, depositingtheir energy hundreds of kiloparsecs away from their origin(see figure 3.1). Blobs of material are observed to moveacross the sky at relativistic speeds. These phenomena aresometimes described as jets “emerging” from a black hole.This parlance is misleading—the jets do not, and indeedcould not, emerge from inside the event horizon. Rather,some mechanism redirects the energy generated by theaccretion process into a fraction of the infalling materialand provides enough bulk kinetic energy for the materialto escape the accretion process before the material entersthe event horizon. This energy is eventually depositedinto the regions surrounding the black hole, often at large

44 Outflows & Jets

Figure 3.1. Radio image of the quasar 3C175 obtained at the VeryLarge Array (VLA) at a wavelength of 6 cm. The AGN is at thecentral bright spot, and the radio lobes are hundreds of kiloparsecsaway. In this case a one-dimensional jet connecting the AGN andthe lobe is clearly seen. Image copyright to the National RadioAstronomy Observatory (NRAO) 1996.

distances from the black hole itself. In this way the blackhole can influence its surroundings in ways other than bygravitational attraction.

Various kinds of accretion processes unrelated to blackholes also generate jets. For example, accretion in a varietyof binary stars is known to result in jets. Young stellarobjects that are still accreting gas in an accretion disk fromthe protostellar cloud that formed them also generate jets.Thus it appears that disk accretion often leads to redi-rection of material in the vertical direction. This materialtends to emerge at velocities comparable to the escape

Superluminal Motion 45

velocity of the accreting object. This suggests that the phys-ical mechanisms involved in jet production occur close tothe surface of the accreting object and are associated withthe accretion process in general rather than being specificto the nature of the particular accretor. Consequently, theexistence of jets emerging from accreting objects at closeto the speed of light implies that the accreting object itselfmust have an escape velocity close to c and thus is likely tobe a black hole.

3.1 Superluminal Motion

Jets associated with black hole accretion often appear tomove across the sky with a velocity vobs that appears to begreater than the speed of light. That is, the angular velocityof jet material across the sky ω multiplied by the distance tothe object D, can yield a value Dω = vobs > c . Such jetsare known as superluminal jets. The apparent superluminalmotion is an illusion—the true space velocity v of thematerial in the jet is necessarily less than c .

The apparent superluminal motion is a combination oftwo effects. As the jet approaches observer the light traveltime to the observer decreases, so the observed time for thejet to travel a certain distance from the source is decreasedby the change in the light travel time. This decrease iscoupled with time dilation due to special relativity (seefigure 3.2). These effects combine, such that

βobs = β sin θ

1 − β cos θ,

46 Outflows & Jets

A

BC

D ββc Δt

θ

ToObserver

Figure 3.2. Geometry of a superluminal jet. If an emitting regionmoves from A to B in time �t at a speed approaching the speed oflight (that is β = v/c ∼ 1) and the angle θ is small, its transversevelocity will appear to be greater than the speed of light.

where βobs = vobs/c , β = v/c , and θ is the angle betweenthe direction of jet propagation and the line of sight.It can easily be seen that βobs can sometimes be greaterthan unity, that is, that the observed motion of a blobemitted at relativistic speeds across the sky can appear tobe greater than the speed of light, even if β < 1. Themaximum value of βobs is obtained for cos θ = β, in whichcase βobs = β/

√1 − β2. There are many examples of

such apparently superluminal motion associated with bothgalactic and extragalactic black holes.

Special relativity can also create dramatic changes in theobserved intensity, variability, and energy of the radiationfrom the jet. The critical quantity is δ = (1 − β cos θ )−1,

Superluminal Motion 47

which ranges from unity for nonrelativistic flows to infinityfor a flow directly toward the observer at the speed oflight. In terms of δ, the observed timescales in the sourcediminish as tobs = t/δ, and so the observed frequenciesincrease as νobs = νδ. Thus the energy of the photonsin the observed frame is greater than that of those in theemitted frame. In addition to these effects, an isotropicradiation field is beamed, so that half the photons areconcentrated into a cone with opening angle θ in theforward direction, where θ is given by sin θ = 1/γ , whereγ is the Lorentz factor γ = 1/

√1 − β2.

The consequence of these effects is that the overallluminosity of the jet goes as Lobs = Lδ4, and is thusgreatly enhanced, as well as bluer and more rapidly varying,in a jet directed toward the observer compared with theluminosity of an otherwise equivalent stationary source.1

Thus sources in which the jet is beamed toward Earthprovide particularly good observations of jets, since thepower of the jet is enhanced in comparison with thatof stationary sources of emission. The class of extragalac-tic objects for which the jet emission dominates otherradiation is known as blazars—these sources emit someof the most luminous and highest energy radiation yetobserved.

1Derivations of these results from the Lorentz transformations are anexcellent exercise for students and can be found in many textbooks and lecturenotes. A useful discussion of the relationship between the textbook specialrelativity results and observations of relativistic jets is given by G. Ghiselliniin “Special Relativity at Action in the Universe” in B. Casciaro, D. Fortunato,M. Francaviglia, and A. Masiello (eds.), Recent Developments in General Relativity(Milano: Springer-Verlag, 2000), 5.

48 Outflows & Jets

3.2 Jet Physics and Magnetohydrodynamics

Some explanations of relativistic jets rely on the relativisticeffects of a rotating Kerr black hole to collimate andredirect the flow. Such explanations imply that the spinof the black hole powers the jet, but spin measurements inaccreting stellar black holes cast some empirical doubt onthis argument (see chapter 8). This class of explanations isalso presumably not relevant to the nonrelativistic jets seenin other kinds of systems.

A more widely accepted model that could apply toother accretion-powered jets is that magnetic fields in theinner accretion disk play the critical role. The study ofthe interplay of fluid flows and magnetic fields is calledmagnetohydrodynamics (MHD), the basic equations ofwhich can be derived from considerations of conservationof mass, momentum, angular momentum, and energy,in a manner similar to that for nonmagnetic fluid flows.These equations are combined with Maxwell’s equationsto generate a set of differential equations that describe boththe motion of the fluid and the topology and evolution ofthe magnetic field.

These equations can only be solved analytically in verysimplified cases. However, insight into the basic physicscan be found from considering two nondimensional num-bers. The first is the magnetic Reynolds number Rm =V L/η, where V and L are typical velocities and lengthscales associated with the flow and the magnetic fieldevolution, and η is the magnetic diffusivity. High valuesof Rm often prevail in astrophysical settings, in which casethe magnetic field is “frozen in” in the fluid. That is,

Jet Physics & Magnetohydrodynamics 49

a particular piece of the fluid is attached to a particularmagnetic field line. As the gas flows and the field evolves,the field and the flow remain attached to each other, andthe magnetic field does not diffuse across the velocity fieldof the fluid, and vice versa. This situation is sometimescalled ideal MHD.

Ideal MHD can sometimes result in a very tangledmagnetic field as a complicated fluid flow ties the field upin knots, which, in turn, dramatically decreases the lengthscale L . One way to consider L is as the length acrosswhich the magnetic field changes direction or strength bya significant amount. If the field becomes knotted, L is thelength across one strand of the knot. As the gas continuesto flow, carrying the field lines with it, the knots canbecome tighter and tighter, L decreases, and Rm decreasesalong with it. Eventually, Rm becomes sufficiently smallthat the conditions for ideal MHD no longer apply. Then,magnetic diffusion becomes important, and the magneticfield lines can slip across the fluid. This slippage resultsin magnetic reconnection, in which the field lines reattachthemselves into simpler configurations, and complex knot-ted field lines are resolved into simpler topologies. Thissimplification is accompanied by a release of the energyassociated with the magnetic field.

The other key parameter is the ratio of the gaspressure to the magnetic pressure, generally denoted byβ = nkT/(2µ0 B2), where nkT is the gas pressure, µ0 isthe magnetic permeability of the vacuum, and B is themagnetic field strength. Note that this β is not to be con-fused with the relativistic parameter β = v/c . If β � 1,the gas pressure dominates. In this case, for ideal MHD the

50 Outflows & Jets

motion of the gas is dominated by hydrodynamics, and themagnetic fields are pulled along by the movement of thegas, as described earlier. However, if β � 1, the flow isdetermined by the magnetic field, and the gas is forcedto follow the magnetic field lines, like beads on a wire.In this case, the magnetic field configuration determineswhere the gas flows. It is thought that the collimation andconfinement of the jet to a narrow stream is a result of asituation where the magnetic pressure dominates, and thefield lines form a “nozzle” confining the flow.

Thus the basic scenario for jet formation is that in thedifferentially rotating accretion disk, the magnetic fieldsare pulled along and tend to wind up, becoming ever morepowerful. At some point the magnetic field reconnects,breaking free of the disk, creating very energetic flares thatmay be the trigger for jet events. The jet is collimatedby the wound-up magnetic field, which constrains theoutward flow of material to a tight beam. As the jetpropagates away from the accreting object, hydrodynamicand magnetohydrodynamic effects generate the observedbends, twists, and knots in the flow. These ideas canin principle account for the formation, collimation, andpropagation of jets.

Each piece of this scenario can be studied using simpli-fied approximations that can be explored analytically. Butwhile this approach can verify that these ideas might workin principle, they are not sufficiently detailed to comparedirectly with the observations—computer simulations arerequired. Such simulations generally take one of twoapproaches. The first is a smooth particle hydrodynamics(SPH) approach, in which the gas is divided into small

Jet Physics & Magnetohydrodynamics 51

point masses, and the motion of each is subject to gravity,gas pressure, and magnetic pressure generated by the otherpoints. Then, the trajectories of all the points are followed,providing a simulation of the gas flow. Algorithms ofthis type are often referred to as Lagrangian approaches.The alternative Eulerian approach is to divide the spaceinto cells, each of which is assigned a gas density andtemperature, a magnetic field strength and direction, andother thermodynamic variables. Then, the computationdetermines how the properties of each cell change withtime under the influence of the various forces.

Whatever approach is used, detailed calculations of thekind necessary to compare results with the extensive obser-vations of jets are difficult to carry out. Three-dimensionaltreatments are necessary, since an inward-flowing diskmust be redirected into a perpendicular outward-flowingjet. To cover a substantial volume of space requires a largenumber of SPH points or Eulerian cells, each of whichmust be updated at each time step. A wide range of scalesizes must be treated—from microscopic magnetic eddiesto bulk flows across hundreds of kiloparsecs of space—which also increases the required number of points or cells.Formulating useful approximations of the physics is alsodifficult, in that all the various regimes are important indifferent parts of the system. Ideal MHD with high Rm

is appropriate in some cases, but diffusive reconnection isalso required to transmit energy to the jet. Gas-dominatedflows are needed in the disk to wind up the flow, whilemagnetic-dominated configurations are important to colli-mate the flow. Thus realistic simulations of jet physics canbe very complex.

52 Outflows & Jets

However, qualitative advances in the simulations haverecently become possible, leading to a new era in thestudy of jet physics. Increasing computer power has madefully three-dimensional flows tractable. At the same time,more sophisticated algorithms, some of which combine thevirtues of Eulerian and Lagrangian approaches, have beendeveloped and put into codes that take advantage of paral-lel computing hardware. For these reasons, computationalmagnetohydrodyamics has become a very active field, anda more detailed understanding of jet physics may soonbecome available.

4STE L L A R -M A S S B L A C K HO L E S

The empirical study of black holes began in the 1960s withthe discovery of quasars (discussed in the next chapter)and the advent of X-ray astronomy. Since the atmosphereis opaque, celestial X-rays cannot be observed from theground, so X-ray astronomy did not begin until the spaceage. Once observing X-rays became possible, it quicklybecame clear that the X-ray sky is dramatically differentfrom the sky seen in optical light. Most of the optical lightin the sky comes from stars, with nuclear fusion as theunderlying power source. It is not surprising that this isthe kind of radiation that our eyes have evolved to detect,bathed as we are by radiation from a particular nearbystar. But other kinds of radiation have revealed whole newcategories of objects, powered by processes that are notimportant in the Sun, particularly accretion. Indeed, theseries of dramatic advances in astronomy from the mid-twentieth century to the present have largely been due tothe development of instruments that can detect new formsof radiation. In the case of X-ray astronomy, the brightestsources of celestial X-rays turned out to be accretion-powered binary star systems. In many of these systems,there is compelling evidence that the accreting object is ablack hole.

54 Stellar-Mass Black Holes

4.1 X-Ray Binaries

In the early 1960s, rockets with X-ray detectors beganto be launched above the Earth’s atmosphere. These in-struments were essentially rocket-mounted Geiger coun-ters that could detect X-rays coming from a particulardirection—as the instrument rotated, the detector scannedthe sky. The initial motivation was to study solar systemphenomena. In particular, the earliest X-ray observatorieswere designed to detect solar X-rays reflecting from theMoon, in hope of determining the nature of the lunarsurface in preparation for the manned missions to theMoon. It was already known that the Sun is an X-raysource, although the total X-ray luminosity is much smallerthan the total solar luminosity, which is predominantlyoptical light. The inferred brightness of other much moredistant stars was very low, so it was not expected that X-raysources from outside the solar system would be detectable.

However, it was quickly discovered that there werestrong X-ray sources that appeared in the same positionin every scan. Such spatial stability would be expectedfrom celestial sources but not from solar system objects,which one would expect to move by significant amountsrelative to the Earth, even during the short rocket flights.The observed sources were not coincident with any ofthe known nearby stars and thus had to be located atsignificant cosmic distances. The inferred luminosity of thesources was hundreds or thousands of times brighter thanthe Sun. When coincident optical stars were identified,they proved to be relatively faint. Thus it was clear thata new class of celestial sources must exist whose radiation

X-Ray Binaries 55

is predominantly in the form of X-rays, with a total lumi-nosity comparable to or greater than that of ordinary stars.

Attempts to understand this exciting new X-ray uni-verse led to the launch of more X-ray–sensitive probes,culminating in 1970 with the launch of the first orbitingX-ray observatory, the Uhuru satellite. In the years sincethen, a series of increasingly sophisticated X-ray satelliteshave been launched, culminating in the currently operatingChandra and XMM-Newton missions. The sensitivity andspatial and spectral resolution of X-ray observatories nowrival those at optical and other wavelengths, and X-rayastronomy has become a vital part of virtually all areas ofastrophysics.

Uhuru observed and cataloged hundreds of X-raysources across the sky. The Uhuru sky map is shown infigure 4.1, in galactic coordinates, in which the plane ofthe galaxy is a horizontal band across the middle of thediagram, and the direction toward the center of the galaxyis in the middle of the plot. It is immediately apparent thatthe majority of the observed sources must be galactic, sincethey are concentrated in the galactic plane and in particulartoward the galactic center.

Simply identifying the general location of these brightX-ray sources led to a determination of some of theirkey physical characteristics. In particular, the approximatedistance to the X-ray sources must be comparable to thedistance to the galactic center, where many of them clearlyreside. Since the distance toward the galactic center isknown to be around 8 kpc, the luminosity of the sourcescould be approximately determined from the observedflux F and the known distance by the inverse square law

56

eyg X-3

Cyg X-I

Stellar-Mass Black Holes

The Fourth Uhuru Cata log

""'" Virgo 30n

NGC6824

Figure 4.1. Founh and final catalog of X-ray sources from me Uhuru satellite. The sources are displayed in galactic coordinates

(with the plane of the galaxy in the equator), and the strength of

the source: is indicated by the size: of the cirde. Both gaiaccic and

extragalaccic sources are clearly detected. Imagt: from NASA, Uhuru

Miss ion.

F = L / 4][ JJl. The resulring luminosiry proved (0 be in (he range L :=:::: 1036_1038 ergs- I, dose to (he Eddington

luminosity associated with stellar masses.

I t was also dear that (he volume from which the X-rays

are emined must be very small. Two lines of argument

led (0 (his conclusion. First. the size of a thermal source

of radiation can be determined by its luminosiry and

temperature, th rough (he Stefan-Boltzmann equation

where L is (he luminosiry of a thermal source, R is the

radius of (he emining region, T is (h e (emperalUre, and

X-Ray Binaries 57

σ is the Stefan-Boltzmann constant. The temperature of ablackbody is determined by Tλmax = 2.9×10−3, where Tis the temperature (in Kelvin) and λmax is the wavelength(in meters) at which the most radiation is emitted. If theradiation is dominated by soft X-rays (λ ≈ 10−9 m),then the blackbody temperature must be in the millions.Plugging this temperature and the observed luminositiesinto the blackbody formula demonstrates that the emittingregion must be smaller than the Earth and much smallerthan most stars.

An even stronger argument for a small emitting regioncame from the very short variability timescales of theemission, which is observed to change significantly ontimescales shorter than a second. The light travel timeacross an emitting region must be smaller than the shortestobserved variability timescale, or the change in brightnesswill be smoothed out by the variation in arrival time ofphotons emitted from different parts of the source. Thusthe size of the source R must be less than the typicaltimescale over which the luminosity changes �t times thespeed of light, so R ≤ c�t . Thus a source that displayssignificant luminosity changes on timescales much shorterthan a second, as the luminous X-ray sources are observedto do, must emit its light from a region smaller than a light-second across. Again, this is a size considerably smaller thanmost kinds of stars.

The general nature of these strong galactic X-ray sourcesis now well understood. They are double star systemsconsisting of a Sun-like main sequence or giant star anda much more compact object. The size of the orbit ofthis binary star system is such that material from the


companion star can accrete onto the compact object. Asdescribed in chapter 2, such an accretion flow can generatestrong high-energy emission. The accreting mass must bevery compact to satisfy the size constraints imposed by theobservations and to ensure that the gravitational potentialwell is deep enough to generate sufficient energy to powerthe source.

4.2 Varieties of X-Ray Binaries

While the observed phenomena from X-ray binaries arevery diverse, it is clear that these systems are of twobasic types. In the “low-mass” systems, the companionstar is less massive than the accreting compact object.The gravitational force of the compact object deformsthe companion star into a teardrop shape known as aRoche lobe. The point of the Roche lobe closest to thecompact object is the inner Lagrangian point, where thegravitational forces of the two stars and the centrifugalforce generated by the orbital motion balance precisely.Any material from the companion star that crosses theLagrangian point will fall away from the companion andonto the compact object.

It is not surprising that systems of this kind generallyhave a mass ratio such that the accreting object is moremassive than the star that is losing the mass. Angularmomentum conservation ensures that mass transfer froma high-mass star to a lower-mass accreting object willbring the two objects closer together. If the mass transferis driven by Roche lobe overflow, then the rate of mass

Varieties of X-Ray Binaries 59

transfer will increase if the objects move closer together,since this will make the Roche lobe smaller and increase theamount of overflow. This can lead to a runaway situationin which an initially small mass transfer rate can increaseuntil enough of the mass-losing star is transferred that themass ratio is reversed, and the mass-accreting star becomesthe more massive of the pair.

The effect of mass transfer on the separation betweenorbiting objects can be determined by considering thetotal angular momentum of the two stars. For circularorbits, the total angular momentum J can be written asJ = m1V1 D1+m2V2 D2, where m, V , and D are the mass,velocity relative to the center of mass, and distance to thecenter of mass of the two objects, respectively. The velocityand distance of star 1 can be related to the total relativevelocity between the two objects Vt as V1 = Vt (m2/mt ),where mt is the total mass of the two stars, mt = m1 +m2.The distance of star 1 from the center of mass can berelated to the distance between the two stars Dt by D1 =Dt (m2/mt ). Similar relations apply to star 2. The relativevelocity and distance of the two stars are related throughKepler’s laws by V 2 = (Gmt/D). Combining all theserelations gives

J = G1/2m3/2t D1/2

t (m1m2/m2t ).

If the total angular momentum J is conserved, and all themass stays in the system—a situation known as conservativemass transfer—then J and mt must remain constant. Inthis case, Dt must increase if m1m2 decreases, and viceversa. The quantity m1m2 is maximized when the two


masses are equal, so mass transfer that brings the massesof the two stars closer together (that is, transfer from themore massive star to the less massive star) requires that Dt

decrease. This condition leads to runaway mass transfer,while mass transfer from the less massive to the moremassive star leads to a stable situation.

In contrast, the “high-mass” systems have companionstars that are generally more massive than the compactobjects. Such high-mass stars eject large quantities ofmatter in the form of a stellar wind. A fraction of thiswind can be captured and accreted by the compact object.Such mass transfer is not conservative, as most of themass is lost to the system, carrying angular momentumwith it, so the preceding derivation does not apply. Inthis case, a black hole can accrete matter from a higher-mass companion. These two kinds of systems are likelyto have quite different evolutionary histories and are thusfound in somewhat different circumstances. In particular,the high-mass systems tend to be found in regions ofongoing star formation—they must be quite young, sincethe lifetime of the massive companion stars is relativelyshort. In contrast, the low-mass systems can persist forbillions of years, and thus have time to travel far from theirbirthplace.

4.3 X-Ray Accretion States

Observations of X-ray binaries continued to improve asincreasingly capable X-ray observatories were launched.A key observation was that these sources are strongly

X-Ray Accretion States 61

variable, on timescales ranging from fractions of a secondto many years. In particular, many sources proved tobe transients, which rise from undetectable levels to be-come some of the brightest X-ray sources in the skyover a few days. These sources then gradually becomefainter again, until after a few weeks or months they areagain no longer observable. Recently, the most powerfulX-ray observatories have detected X-rays from transientsin quiescence, demonstrating that their X-ray flux rises byas much as eight orders of magnitude during an outburst.The quiescent state can persist for years to decades; indeed,some transients have been observed to go into outburstonly once in the history of X-ray astronomy. In quiescence,the optical light from the companion star often dominatesthe flux associated with mass accretion, and observationsof the companion provide crucial information about thesource, as described in the next section.

Nontransient X-ray binaries also show very markedchanges in behavior over short periods of time; the lumi-nosity and the spectrum can change dramatically. Spectraof X-ray binaries are sometimes dominated by thermalemission of the kind expected from an accretion disk. Suchspectra are superpositions of blackbody spectra, each ata different temperature, with higher temperatures beingassociated with the inner parts of an accretion disk. Atother times, X-ray binaries display nonthermal spectra inthe form of a power-law function of flux as a functionof wavelength. The power law typically extends to muchhigher energies than the thermal spectrum. Much of theemission in these states comes from hot gas above andaround the accretion disk. This gas, often referred to as an


accretion disk corona (ADC) is thought to be the source ofmuch of the nonthermal emission from X-ray binaries. Therelative strength of the thermal and nonthermal compo-nents of the spectra can change dramatically within a singleobject. These changes are often associated with changes inoverall brightness of the X-ray source, and with changesin the amount and kind of short-term variability. Thesechanges are referred to as state changes and presumablyresult from changes in the rate and geometry of the flowof matter onto the compact objects.

X-ray binaries also display a variety of phenomena. Theorbital period is often revealed by eclipses and other effectsof changes in viewing angle during an orbit. Variabilityon shorter timescales is also generally seen, sometimes inthe form of random “flickering” and sometimes as “quasi-periodic variations,” in which a specific timescale is fa-vored. A single object can display quasi-periodic variabilityon timescales much shorter than a second, orbital changeson timescales of hours or days, and state changes over manymonths.

As the state changes were studied it became clearthat certain characteristic combinations of effects wereoften seen together, which led to many efforts to developa taxonomy of X-ray states for X-ray binaries and tounderstand the transitions that are observed between them.It seems clear that the different X-ray states are associatedwith different kinds of accretion flows, such as disks orADAFs. Two common states are the thermal state, inwhich most of the emission comes from an accretion disk,and the hard state, in which an outer disk is observable inoptical and infrared wavelengths, while the inner accretion

Compact Objects 63

flow is from a nonthermal ADC and may also includea component from a jet or outflow (see figure 4.2).Observable radio emission, generally assumed to be froma jet, is often present in the hard state. The thermal stateis generally associated with lower levels of variability thanthe hard state. A variety of other extreme and intermediatestates have also been identified. The most appropriatedefinitions for the various states, the physics that gives riseto them, and the events that trigger transitions betweenstates are active topics of current research.

4.4 Compact Objects

Not all the compact accreting objects in X-ray binaries areblack holes. Other kinds of objects are sufficiently denseto generate X-rays by accretion. All of them are the naturalresult of stellar evolution—they are how stars end their life.Stars like the Sun are supported by gas pressure generatedby the nuclear fusion process in their core; they are in astate of hydrostatic equilibrium, in which outward pressureforces balance the inward force of gravity. Eventually,however, a star exhausts its nuclear fuel, and hydrostaticequilibrium can no longer be maintained by gas pressure.The core of the star then collapses until some other sourceof pressure restores the equilibrium. Such pressure canbe provided by the quantum mechanical effect of Fermipressure, which arises because two particles are prohibitedfrom occupying the same volume simultaneously. In starslike the Sun, the Fermi pressure of the electrons holds thestar up when it reaches a density of approximately a ton per

0.D1

10

0.1

, , , ,

10-<

10

0.1

10 Energy (keV)

100 lO-l 0.1 10 Frequen(y (Hz}

Figure 4.2. One current definition of X-ray states (from Remillard

and McClintock 2(06): me left-hand panel shows the spectrum,

and the right-hand panel shows me temporal power spectrum

(variabilicy) . Top panel: the steep power state; middle panel: the soft

thermal nate; bottom panel: me low hard scate. The differences in

spectral and temporal behavior between these states are dear. T here

are also a variety of intermediate states. From R. Remillard and

J. McClintock, 2006, Annual Rrvil!W of Astronomy 6- Astrophysics, 44: 49.

Compact Objects 65

cubic centimeter, or a size comparable to that of the Earth.Such stars are known as white dwarfs. Mass-accreting whitedwarfs are sometimes referred to as cataclysmic variablesand are responsible for nova explosions and other phenom-ena. Some classes of cataclysmic variables emit X-rays butnot in sufficient quantities to account for the strongest X-ray sources, like those identified by Uhuru in the galacticcenter.

However, not all stars end their life as a white dwarf.In the 1930s, the great astrophysicist SubrahmanyanChandrasekhar showed that white dwarfs with a massgreater than 1.4 M� cannot support themselves againstgravitational collapse—this bound is referred to as theChandrasekhar limit after its discoverer. This result wascontroversial at the time—Sir Arthur Eddington famouslyremarked that “there should be a law of Nature to preventstars from behaving in this absurd way.”1 One version ofthe derivation is given in section 4.8.

Stars that exceed the Chandrasekhar limit after theirnuclear fuel is exhausted must continue to collapse—butto do so the electrons must be eliminated, which occursthrough electron capture by the protons: e + p → n + ν,where ν is a neutrino. Essentially all the matter is trans-formed into neutrons, creating a neutron star. Neutronstars can be observed in the form of pulsars, in X-raybinaries, and in a few cases as isolated objects. The largenumber of neutrinos generated stream outward from theimplosion. The enormous energy created by the implosionblows off much of the envelope of the star in a supernova

1The Observatory, 1935, 58 (no. 729): 38.


explosion. Neutron stars achieve enormous densities (mil-lions of tons per cubic centimeter), comparable to thedensity of an atomic nucleus, and thus have extraordinarilydeep gravitational potential wells. Accreting neutron starscould generate the X-rays observed from X-ray binaries,and indeed this is what many of the X-ray binaries arethought to be.

However, there is also an upper limit on the mass ofneutron stars. They are held up by Fermi pressure of theneutrons, in much the same way as white dwarfs are heldup by pressure from the electrons. Thus in principle thereis an upper limit on the mass of a neutron star similar to theChandrasekhar limit for white dwarfs. However, this limitis modified by the strong general relativistic effects presentin neutron stars, which are only slightly bigger than theirSchwarzschild radius. As shown in section 4.8, the mass-radius relationship for objects held up by Fermi pressure isinverted, in the sense that a larger mass results in a slightdecrease in radius. Therefore, above some mass the radiusof a neutron star will be less than its Schwarzschild radius,and the object will be a black hole.

The exact upper limit on the mass of a neutron stardepends on the equation of state of neutron-dominatedmatter, that is, on the relationship between density andpressure. The correct equation of state to use for neutronstars is not completely understood—“soft” equations ofstate, which produce more compressible configurations,can result in an upper limit on the mass of a neutronstar as low as 1.5 M�, while “hard” equations of statecan result in neutron stars as massive as 2.2–2.5 M�. Thesoftest equations of state are now ruled out by observations

Compact Objects 67

of neutron stars with masses approaching 2 M�, but themass-radius relationship of neutron stars is still understudy.

However, there are some basic requirements that anyequation of state for neutron stars must satisfy. Fromthe experimental point of view, equations of state mustbe compatible with the results of direct experimentationin nuclear physics, while from a theoretical perspectiveequations of state must be causal—that is, the speed ofsound in the material must be less than the speed oflight. This minimal requirement on the equation of stateresults in a maximum mass for a neutron star of around3 M�. Thus a compact object with mass greater than threesolar masses is likely to be a black hole—it cannot be awhite dwarf, since it is above the Chandrasekhar limit, andit cannot be a neutron star, since such an object wouldcollapse to become a black hole.

In fact, there are ways in which a compact object withM > 3 M� could avoid being a black hole. For example,strong differential rotation can hold up a neutron star asmassive as six solar masses; however, the strong tangledmagnetic fields inside a black hole would damp out suchdifferential rotation in seconds. If the compact object iscomposed of exotic materials (e.g., particles containingstrange or charm quarks), then the equation of state couldbe quite different from that of a neutron star, resultingin exotic objects quite different from standard neutronsstars. Finally, it is possible that general relativity requiresmodification in the limit of strong fields, in which case theconcept of the Schwarzschild radius may not be applicable.The extraordinary nature of these alternatives reinforces


the strong probability that a massive compact object mustbe a black hole.

4.5 Mass Measurements in X-Ray Binaries

Since compact objects with M > 3M� are almost certainlyblack holes, determining the mass of the accreting object inX-ray binaries becomes an important measurement. For-tunately, standard techniques of binary star astronomy canbe used to make such mass determinations. In particular,Kepler’s laws of orbital motion can be used to find the massof orbiting objects from their velocity and orbital period.Repeated observations of the Doppler shift of an orbitingobject can be used to construct a velocity curve, in whichthe radial velocity (that is, the speed toward or away fromthe observer) of the source is plotted against time. Duringan orbit, the object first moves toward the observer, thenaway, so the velocity curve is periodic at the orbital period.For circular orbits, the velocity curve is sinusoidal. Fromsuch a sinusoidal velocity curve one can compute the massfunction f = P K 3

∗/(2πG), where P is the orbital period,and K∗ is the amplitude of the sinusoidal velocity curve(see figure 4.3; note that despite its name, f is not really afunction of mass but rather a quantity with units of mass).It is straightforward to show that

f = P K 3∗/(2πG) = Mx sin3 i/(1 + M∗/Mx )2

where M∗ is the mass of the observed star, Mx is themass of the (unobserved) compact object whose accretion

Mass Measurements in X-Ray Binaries 69

Redshifted

Period

Ampli-tude

Time

App

roac

hing

Rece

ding

Radi

al V

eloc

ity

Observer

Blueshifted

＋2

Figure 4.3. Radial velocity curve of X-ray binary with a circularorbit.

presumably generates the X-rays, and i is the angle ofinclination of the binary orbit. The importance of thisexpression is that f ≤ Mx . Thus if the velocity curve ofthe companion star can be measured, a minimum massof the compact object immediately follows solely fromconsiderations of the dynamics of the binary orbit. Suchobservations can be made if the companion star generatesmost of the optical emission from the system, which canoccur in high-mass systems in which the companion star isintrinsically very bright, or in low-mass transient systemswhen the accretion flow is in quiescence. If f > 3 M�this provides dynamical confirmation of a black hole inthe system. There are now over a dozen “dynamicallyconfirmed black hole candidates,” that is, X-ray binariesin which the mass function f has been measured to begreater than 3 M�.

But while it is true that f > 3 M� implies Mx > 3 M�,it is not necessarily the case that f <3 M� ⇒ Mx <3 M�.


To determine the mass of the compact object from themass function requires knowing the mass ratio M∗/Mx

and the orbital inclination i . Thus two additional obser-vational constraints are required beyond the mass functionto fully determine the masses and inclination of the binarysystem. These constraints are generally provided by therotational broadening of the stellar absorption lines, whichis strongly sensitive to the mass ratio, and the amplitude ofthe ellipsoidal variations, which is strongly sensitive to theorbital inclination.

The mass ratio can be constrained by determining therotational velocity of the surface of the companion star. ForRoche lobe–filling systems, the companion star is tidallylocked to the orbit, so the rotation period is always thesame as the orbital period, as is the case for the Moon in itsorbit around the Earth. If the rotational period is known,the rotational velocity at the equator depends on the size ofthe object, since 2πR/Prot = Vrot. The rotational velocityof the companion star can be measured by observing thebreadth of the spectral lines from the star—the lines willbe Doppler broadened because part of the rotating star iscoming toward the observer and the other side is receding.Since the companion star fills its Roche lobe, its size isfixed by the semimajor axis of the orbit and the massratio, since these two parameters determine the size of theRoche lobe. The semimajor axis can be calculated from theknown orbital period and the total mass, so measuring therotational broadening gives a constraint on the total massof the system and the mass ratio.

The orbital inclination can also be constrained bystandard binary star techniques. In this case, the key

Mass Measurements in X-Ray Binaries 71

observation is the change in brightness of the companionduring the course of an orbit. Such an orbital light curvegenerally displays ellipsoidal variations. These variations inbrightness are caused by the change in viewing angle ofthe companion star (see figure 4.4, top). The companionstar is tidally distorted and hence not spherical. Whenviewed from the side, it displays a larger cross section, andthus appears brighter, than when viewed end on. Thus anorbital variation in brightness is expected in which thereare two maxima of the light (when the star is viewed sideon) and two minima (when the star is viewed from thefront or the back). The amplitude of the variation dependsstrongly on the inclination, since in a face-on configurationthe companion star always displays the same side to theobserver, whereas an edge-on configuration maximizes theeffect (see figure 4.4, bottom).

This variation is roughly similar to that of a rotatingellipsoid (hence the name “ellipsoidal” variations), but thisis not precise, and it is now possible to make detailedcomputational models of the exact shape of the distendedstar. Currently used models now include both the effects ofthe unusual geometry of the star and possible contributionsto the overall flux from the accretion flow. Excellent agree-ment between data and models can be found, implyingthat the inclination is well determined. There are now closeto two dozen sources in which the mass of the compactobject is well determined and significantly greater than3 M�.

Most of the known systems with Mx > 3 M� are tran-sient low-mass systems, in which the companion star canbe observed when the accretion flow, and the luminosity

1.

1.

b 1.

o. o. 1.

1.

0 1. o. o. 1.

1.

b 1. N

o. o. 1.

1.

0 1. ~

o. o. 1.

1.

~1. o. o.

, 3

4

, , 0

9

8 , , 0

9

8 , , 0

9

8 , , 0

9

8 , , 0

9

8

Ellipsoidal Variations

Side-on Ii = 901

•

0 •

--------~

~

, , , ,

• ,

1.2

1. ,

,

Fa{e-on Ii = OJ , , ~ , , , , , ,

• • , 3 , , , , , , , k '

4

~ ~ 1.0

0.9

0.8 1.2

1.

$ 1.0

0.9

0.8 1.2

1.

'~

0 1.0 ' /\)\ , 0.9

0.8 1.2

1. ' f\J\

' f\J\

2 1.0

0.9

0.8 1.2

1.

~ 1.0

0.9

0.8

o 0.5 o orbital Pha5e

0.5

Figure 4.4.

Are High-Mass Compact Objects “Black Holes”? 73

associated with it, turns off. However there are also persis-tent high-mass systems with dynamical evidence for blackholes. Within our own galaxy, only the famous systemCygnus X-1 falls into this category, but recently, dynamicalevidence for such black holes has been discovered insystems in other galaxies. It should be remembered thatthe taxonomy of “high mass” versus “low mass” refers tothe mass of the companion star, not that of the compactobject. Interestingly, it appears that the most massivestellar black holes are found in the wind-fed “high-mass”systems. There also appears to be a significant lack of blackholes in the 3–5 M� range. There is as yet no compellingexplanation for the observed distribution of black holemasses in these systems.

4.6 Are High-Mass Compact Objects “Black Holes”?

Given the existence of a sample of “dynamically confirmedblack hole candidates,” that is, compact objects withmeasured masses above the 3 M� limit, it is important todetermine whether these objects really do behave like theblack holes predicted by general relativity. To date, theevidence suggests that they do. The most basic propertyof black holes is the absence of a surface, which has po-tentially observable consequences, as there are phenomena

�Figure 4.4. Ellipsoidal variations. Top panel: face-on and edge-on configurations; bottom panel: orbital light curves as a functionof inclination.


associated with accreting gas that falls onto a surface. Sucheffects would not be expected in accreting black holes,although they might well occur in accreting neutron stars.

An example of such surface behavior are X-ray bursts.2

When infalling gas lands on a neutron star, it mustundergo a series of fusion reactions, eventually ending upas neutron-rich material that can be absorbed into theneutron star itself. Depending on the accretion rate, someof these thermonuclear reactions may occur explosively.Explosive nucleosynthesis occurs when the accreted ma-terial reaches a critical mass. Beyond this threshold thereaction occurs explosively in all the accreted materialover a fraction of a second. Such explosions are observedin the form of bursts of X-rays, which can increase theluminosity of X-ray binaries by an order of magnitudeover a small fraction of a second. X-ray bursts have beenobserved in many X-ray binaries; however, they have neverbeen observed in any system in which the compact objectmass has been shown to be above the 3 M� limit. Theimplication is that these more massive compact objects areindeed black holes and as such, lack surfaces on which theaccreted material can accumulate. The observation of anX-ray burst from a massive compact object would consti-tute a significant challenge to the current belief that theseobjects must be black holes. But thus far, the extensivearchive of X-ray bursts comes only from systems whoseaccreting objects have a mass consistent with neutron stars.

Another consequence of the existence of a surface onan accreting object is the existence of a boundary layer at

2There are actually several kinds of X-ray bursts. I refer here to the so-called“Type 1” events.

Are High-Mass Compact Objects "Black Holes"? 75

Infaill ing Gas •

Black Hole

Boundary Layer X-Rays ,

Neutron S""

Figure 4.5. Accretion 0fi(0 a black hole versus a neutron star.

Accreting neutron stars emit boundary layer radiation, which black

holes do nor.

which the accretion Row terminates. This boundary layer

is likely to generate significant amounts of radiation, as

the kinetic energy of the infalling gas must be turned into

some other form of energy at the point where the gas lands on the accreting object (see figure 4.5.) In the case

of a classic accretion disk, in which the gas is in circular

Keplerian orbits, the kinetic energy is always equal to half

the potential energy, so half of the energy associated with

the infalling material must be dissipated in the disk itself,

and the other half m ust be associated with the termination

of the disk at the accreting surface. Thus an accreting

object with a surface would be expected to be twice as

luminous as a black hole for a given mass accretion rate.

H owever, it is hard (0 associate an observed spectrum with

a mass accretion rate, so this tes t is hard to apply. In

contrast, in some ADAF-like Rows, almost all the energy


is advected inward, and only a tiny fraction of the energyis dissipated during the infall. In these cases, accretingneutron stars would be expected to be orders of magnitudebrighter than accreting black holes. In many cases the X-ray emission from low m sources (which are expected tohave ADAF-like flows) appears to be higher for neutronstars than for dynamically confirmed black hole candidates,providing support for the idea that these objects may nothave a surface.

4.7 Isolated Stellar-Mass Black Holes

The black holes in X-ray binaries represent only a smallsubset of the stellar-mass black holes that probably existin our galaxy. Most stars with initial masses greater than≈30 M� (the exact limit is only poorly determined) willlikely retain enough matter to evolve into a black hole.While such massive stars are relatively rare, the galaxycontains around 1011 stars, so millions of black holes areexpected. Only those that happen to be in binary systemswith very specific configurations appear as X-ray binaries.The vast majority of black holes are not in such systemsand thus cannot be detected in this way. So, it is naturalto ask whether there might be other ways to identify blackholes that are not contained in X-ray binaries. Two generalschemes have been suggested, but both also depend onvery unusual interactions with other objects, and successfulidentifications have been rare.

One approach is simply to look for black holes that areaccreting mass from the diffuse interstellar medium (ISM)

Isolated Stellar-Mass Black Holes 77

rather than from a companion star. An isolated black holewould be expected to accrete from the ISM according tothe standard Bondi-Hoyle accretion process described inchapter 2. The mass accretion rate depends on the densityof the ambient medium, which in general is very low. Thusthe luminosity generated should also be very low. Indeed,the luminosity might be lower still if the accretion is in theform of an ADAF or other radiatively inefficient flow. So,such objects would generally not be expected to generatesufficient luminosity to be detected. However, the ISMis very inhomogeneous, and parts of the ISM are muchdenser than average. In particular, the molecular clouds inwhich star formation takes place have quite high densityand might generate significant accretion if a black holewere to travel through them. However such dense cloudscover quite a small fraction of the galaxy, which greatlydiminishes the number of such sources expected to be seen.Dense star-forming clouds also contain young stars thatmight generate X-rays and other high-energy radiation inother ways. The combination of these factors means thatno isolated black hole has compellingly been identified inthis way.

Another technique for identifying isolated black holesis gravitational microlensing. Gravitational lensing occurswhen a massive object is directly in the line of sight ofa luminous background object. The massive object actsmuch like an optical lens and bends the light rays fromthe background source, resulting in a magnification of thelight of the background object. Such lensing can occuronly when the two objects are very precisely aligned.Since objects in the galaxy are in constant motion, this


magnification is not expected to persist; rather, the magni-fication is expected to grow over the course of a few weeksand then to decay again. Thus a black hole passing in frontof a background star might magnify the light of the starby an observable amount for a few weeks, even if the blackhole itself emits no radiation at all.

A chance alignment precise enough to produce sig-nificant magnification of the background object is quiteunlikely. Therefore, millions of background stars must beobserved on a daily basis to provide a significant eventrate. Several large survey programs have been carried outto do this, focusing on the Galactic Bulge and the LargeMagellanic Cloud, fields in which many background starscan be observed simultaneously.3 The original goal was todetermine whether the “dark matter” in the galaxy requiredby galaxy rotation curves and other evidence could beexplained by dim stars or compact objects. Such objectsmight be unobservable directly but might reveal their pres-ence by generating a significant number of microlensingevents. The observed event rates demonstrated that suchmacroscopic objects could not, in fact, account for thedark matter, and relatively few events were observed thatwere compatible with massive (M > 3 M�) nonluminouslenses. But while stellar mass black holes clearly cannotaccount for the dark matter, the microlensing results arenevertheless compatible with a large number of isolatedblack holes in the galaxy.

3See C. Alcock et al., 2001, Nature 414:617; and A. Udalski et al., 1997,Acta Astronomica 47:319.

The Chandrasekhar Limit 79

4.8 The Chandrasekhar Limit

The structure of a star is determined by the balancebetween gravitational forces, which pull material inwardtoward the center, and pressure forces pointed outward. Instable stars, the forces balance at every point, so that thestar neither implodes nor explodes and is thus in a state ofhydrostatic equilibrium. For spherically symmetric stars, thegravitational acceleration at any point at distance R fromthe center is due only to the material located inside thatpoint and is equal to agrav = −G M(R)/R2, where M(R)is the mass contained between the center of the star and thedistance R . The minus sign denotes that the accelerationis directed inward. The pressure depends on the nature ofthe material in the star. In stars like the Sun, the pressure isgas pressure and is determined by an equation of state suchas the ideal gas law P ∝ ρT, where ρ is the density andT is the temperature. However, the pressure thus definedis exerted in all directions, whereas what is required is thenet pressure outward. This is determined by the pressuregradient, that is, the difference between the pressure justinside a given point R from the center and the pressurejust outside that point. Thus in general, a star must satisfy(1/ρ)d P/d R = −G M(R)/R2.

One of the features of an equation of state like theideal gas law is that the pressure is directly proportionalto the density. The gravitational force inward is alsorelated to the density, in that M(R) is determined byintegrating the density from the center of the star out to R .If the gas pressure is linearly dependent on the density,as would be the case if the temperature were constant


(an isothermal configuration), then it can be shown thathydrostatic equilibrium cannot be achieved—increasingd P/d R requires the density to rise quickly toward thecenter of the star, which creates a large value of M(R), andthe pressure forces cannot keep up. Therefore gas pressurecan balance gravitational force only if the temperaturealso rises toward the center of the star, providing anothercontribution to d P/d R . But if there is a temperaturegradient, basic thermodynamics requires that heat flowoutward from the hot interior. If there is a source ofenergy in the center of the star, the heat flowing outwardcan be replaced, so that the temperature gradient andhydrostatic equilibrium can be maintained. But without acentral energy source, the star evolves toward an isothermalsituation, gravity overcomes the gas pressure, and thecentral regions of the star collapses.

This collapse continues until the gas in the star becomesdegenerate. Gas is degenerate when the Pauli-Fermi exclu-sion principle, which states that two fermions cannot bein the same place at the same time, becomes an importantfactor. When the gas becomes sufficiently dense, the dis-tance between the electrons becomes sufficiently small thatthe exclusion principle creates a Fermi pressure that keepsthe electrons from coming too close to one another. ThisFermi pressure increases as a high power of the density andthus is sufficient to balance the inward gravitational forceeven for an isothermal star. White dwarfs are stars that haveexhausted their nuclear fuel and contracted to such a highdensity that Fermi pressure from the electrons suffices tomaintain hydrostatic equilibrium.


In a nonrelativistic gas (that is, a gas in which the energyarising from the momentum of the particles is much lessthan that associated with the rest mass of the particles)an approximate radius for a star held up by degeneracypressure can be derived as follows. If a star contains Nfermions and has radius R , then a typical separation xbetween individual fermions will be x ≈ (V/N)1/3 ≈R/N1/3, where V ∝ R3 is the volume of the star. Themomentum of each particle must be 1/x , and the averageFermi energy per particle must be E ≈ p2/m, where m isthe mass of the particle. In an equilibrium state, the typicalgravitational energy of each particle ≈ G M(R)m/R mustbalance the Fermi energy; since M(R) ≈ Nm, we canequate the Fermi energy and the negative gravitationalpotential energy per particle and find that

R ≈ 1/(Gm3 N)1/3.

This equation has the curious result that, unlike ordinarystars, massive degenerate stars (with larger values of N)are smaller than their lower-mass counterparts. In essence,the greater gravitational force generated by a more massivestar must be countered by greater Fermi pressure, whichrequires that the particles be closer together and the totalconfiguration be smaller. As the electrons are packed incloser together, x , the typical distance between themdecreases, and the Fermi momentum increases. Eventually,the Fermi energy becomes comparable to the rest massenergy of the electrons, and the gas becomes relativistic.For relativistic gas the energy per particle is N1/3/R . This


means that the gravitational energy and the Fermi energyhave the same dependence on R , which drops out of theequilibrium equation. So instead of a relationship betweenN and R , as is found for a nonrelativistic gas, we find thatN depends only on constants:

N ≈ (1/Gm2)3/2.

Thus there can be only one value for N, and for a givenmix of atomic species, only one mass, for a fully relativisticgas. This mass is the Chandrasekhar limit—greater massescannot be sustained by Fermi pressure against gravitationalcollapse. Typical white dwarfs contain no hydrogen orhelium, since those are the elements that provide energyfrom nuclear fusion to maintain gas pressure in nonde-generate stars—it is only when the hydrogen and heliumare exhausted that a white dwarf forms in the first place.This means that white dwarfs are typically dominated byelements like carbon, nitrogen, oxygen, neon, and magne-sium, for which there are two nucleons per electron. In thiscase the Chandrasekhar limit turns out to be 1.44 M�.

We might then ask, what happens to a degener-ate configuration that happens to be greater than theChandrasekhar masses. Gravitational forces dominate,making the star contract and thus forcing the electronscloser together than is allowed by the uncertainty principle,so the star must lose the electrons altogether. Electronsand protons combine to make neutrons and neutrinos, inthe process of inverse beta decay. Thus the entire massof the star is turned into neutrons, forming a neutronstar, and huge numbers of neutrinos are released. This


transformation is observed in the form of supernovae,which often leave behind neutron stars. Neutrons are alsofermions, so in principle the same ideas apply: nonrelativis-tic configurations get smaller with increasing mass untilthe particles become relativistic, at which point a limitsimilar to the Chandrasekhar limit applies, and the starcollapses. The main difference is that because neutrons aremore massive than electrons, the equivalent Fermi energyis generated by much smaller distances between them,and thus neutron stars are much smaller and denser thanwhite dwarfs. In fact, however, general relativistic effectscome into play before the equivalent of the Chandrasekharlimit becomes relevant. When the mass of a neutron starbecomes greater than 2.2–3 M� (the exact value dependson the specifics of the neutron star equation of state), itcollapses to become a black hole.

This derivation of the Chandrasekhar limit is approxi-mate, as it assumes a constant density and thus a constantseparation between the fermions. Real degenerate starshave a density gradient, and thus the requirement of hydro-static equiliibrium must be satisfied at every radius, whichresults in differential equations for mass, density, andpressure as a function of radius that must be solved. Thiscan be done with results comparable to those describedhere.4

4See, for example, S. Shapiro and S. Teukolsky, Black Holes, White Dwarfs,and Neutron Stars: The Physics of Compact Objects (New York: Wiley, 1983).

5SUPERMAS S I V E B L A C K HO L E S

Stellar-mass black holes are clearly common consequencesof stellar evolution, but they are not the only kinds ofblack holes identified by astronomers. Much more massiveblack holes are located in the center of many, and perhapsall, galaxies, including our own. These black holes are re-ferred to as supermassive black holes, sometimes abbreviated“SMBHs.” They are responsible for a range of phenomenaoriginating from objects described as active galactic nuclei(abbreviated AGN), which were first observed in the formof quasi-stellar objects (QSOs) or quasars.1 AGN are amongthe most luminous objects in the Universe and can beobserved at great distances. The distances can be so greatthat the light travel time from the AGN to Earth is a largefraction of the age of the Universe—we are seeing AGNnot as they are today but as they were when the Universewas much younger. They are therefore often used to probethe evolution of the Universe.

1The terms QSO and quasar are sometimes used interchangeably, butsometimes QSO is used to denote objects discovered by their optical properties,whereas quasars denote those discovered in radio frequencies.

Discovery of Quasars 85

5.1 Discovery of Quasars

QSOs were identified in the 1960s as a class of very blueobjects that appear as single points of light, like stars, incontrast with galaxies and nebulae, which are extendedobjects. However, their spectra are quite different fromthose of ordinary stars. In particular, the line emission fromthe QSOs did not correspond to any known atomic transi-tions. The key to understanding QSOs was the realizationthat the spectral features could be identified provided theobjects were strongly redshifted, with redshifts as high as�λ/λ ≈ 1. This result implied that QSOs are extragalacticobjects located at cosmologically large distances whoseredshifts are caused by the expansion of the Universe. Suchlarge distances required QSOs to be the most distant andmost intrinsically bright objects yet discovered.

The cosmological distance scale for QSOs was notimmediately accepted by the full astronomical community.At the time, there was still support for the “steady statetheory” of cosmology, which requires continuous mattercreation throughout the cosmos to preserve a constant den-sity as the Universe expands. It was suggested that QSOsmight be the site of this matter creation. Soon, however,the evidence for the Big Bang and for cosmological dis-tances for QSOs became overwhelming. But for some timethe cosmological distance scale for QSOs was disputed,primarily on the grounds that high-redshift quasars wereoften seen close to low-redshift galaxies. If the galaxiesand QSOs were, in fact, part of the same object, then thehigh QSO redshifts could not be due to their distance.A few prominent astronomers, most notably Halton Arp,

86 Supermassive Black Holes

still question the cosmological distances for QSOs. But theevidence for associations between QSOs and low-redshiftobjects have become very much weaker, and holdouts likeArp are no longer taken seriously. The spatial coincidencesare now interpreted as chance superpositions. A crucialeffect is that distant QSOs can be gravitationally lensed bynearby galaxies, which makes background quasars brighterand thus easier to detect than they otherwise would be.When this effect is taken into account the number ofchance superpositions is in accord with expectations. Thereis thus no reason to doubt the distance scale of QSOs, andthe associated high luminosities.

In fact, the QSO redshifts provided a key piece ofevidence in favor of Big Bang cosmology. As surveys forthis new class of objects began to be carried out it becameapparent that there were more QSOs at high redshiftsthan expected by extrapolating the number of low-redshiftQSOs, which suggested that the population of QSOs haschanged over cosmic time. QSOs with high redshifts, andthus large distances, are seen as they were in an earlierepoch of the Universe, since the light travel time is asignificant fraction of the age of the Universe. Thus therewere apparently more QSOs at earlier times than there arenow. Such changes in the overall distribution of cosmicobjects are expected in a Big Bang cosmology, in whichthere may be changes in the bulk properties of the Universesuch as density and age, but not in the steady state theory,which as its name implies, requires that the cosmos notchange its overall nature.

It soon became clear that QSOs are closely related to thepreviously discovered class of Seyfert galaxies, the centers of

Discovery of Quasars 87

which exhibit quasar-like activity. As observational tech-niques improved, QSOs and quasars were firmly identifiedas an extreme case of the general phenomena of AGN,extreme in the sense that they are unusually luminous andthus can be seen at distances great enough that the hostgalaxy itself is invisible or difficult to detect.

Two basic features of AGN provide strong supportfor the idea that these sources are powered by accretiononto very massive black holes. First, significant variabilityis observed on timescales as short as a few days, andin some cases less than a day. This limits the size ofthe emitting region to a maximum of ≈1 light-day or≈3×1013 m. At the same time, the cosmological distancesimply luminosities as high as 1044−47 ergs−1. To avoidoutshining the Eddington limit by orders of magnituderequires a central mass Mc � 105 M�. For extreme cases,much of the emission must come from within a fewhundred Schwarzschild radii of the central mass. It wouldbe difficult to produce so much energy in a small regionwithout involving black holes in some way.

More support for the idea that AGN must be poweredby accretion onto black holes comes from their overallenergy distribution and spectra, which are remarkablysimilar to what would be expected if the X-ray binarieswere scaled up in mass by many orders of magnitude.In particular, one of the key observational characteristicsof QSOs is their blue color, and space-based observa-tions, particularly those conducted with the InternationalUltraviolet Explorer satellite, confirmed that many AGNhave high levels of ultraviolet radiation as well. Thermalemission from accreting black holes should have peak at a


wavelength λmax that scales with mass as λmax ∝ M−1/4.Thus the ultraviolet peak in thermal radiation shown bythe QSOs, sometimes called the “Big Blue Bump,” isanalogous to the thermal X-rays seen in the X-ray binaries.

5.2 Active Galaxies and Unification

QSOs and quasars are related to a variety of categories ofunusual galaxies. Many QSOs proved to be radio sourcesand thus seemed to be related to the class of radio galaxieswhich had already been identified as having extended radioemission. As noted earlier, the Seyfert galaxies, identifiedby the strong optical emission lines emanating from thecentral regions of the galaxy, appeared to be QSOs situatedwithin a host galaxy. As these phenomena were studied inmore detail it became apparent that the central regionsof galaxies showed a very wide range of phenomena,sometimes referred to as the “AGN Zoo.” A bewilderingarray of classification schemes was developed, each startingwith one or several of the various observed characteristicsdescribed next.

Radio intensity. The early observations of quasars (iden-tified as radio sources) and QSOs (identified as opticalsources) had already demonstrated that the ratio of radioemission to that in other wavebands varies widely. As moredetails of AGN began to be understood, it became clearthat about 10% of AGN are “radio loud,” with signifi-cantly more radio flux than their radio-quiet counterparts.

Active Galaxies & Unification 89

Radio morphology. Radio galaxies had been identified evenbefore quasars were discovered. Detailed maps of the brightradio emission from these sources show that the radioemission often emanates from two “lobes” surroundingthe center of the galaxy. Some of these lobes extend forhundreds of kiloparsecs from the center of the galaxy. Insome cases the center of the galaxy also emits observableradio flux, and “bridges” can be identified connecting thecentral source to the lobes. It is now generally believedthat the substantial amount of energy required to powerthe radio lobes is generated in the galactic nucleus andtransported to the lobes through collimated jets of the kinddescribed in chapter 3.

Optical spectra. Many AGN show emission lines in opti-cal and infrared wavelengths, and was the high redshiftsof these emission lines that led to the identification ofthe QSOs as objects at cosmological distances. But thestructure of these emission lines varies significantly amongAGN. Some AGN have “broad lines,” with widths corre-sponding to Doppler motions of thousands of kilometersper second. The breadth of these emission lines is generallyinterpreted as being due to a large velocity dispersion ofthe gas that produces the lines. The range of velocities ofthe emitting gas results in changes in the Doppler shiftof the lines away from the mean redshift of the AGN.Superposing emission from gas with different velocitiesresults in a broadened line. In addition having broad lines,most AGN also have much narrower lines. There arenarrow-line objects that do not exhibit broad lines, but thereverse is not generally seen, although there are some AGN


(known as BL Lac objects) in which line emission is notobserved at all.

Polarization. The radiation observed from AGN is oftenstrongly polarized, which requires that the radiation sourcebe nonthermal (in contrast with starlight) or that theradiation pass through or be reflected by some kind ofpolarizing medium.

Host galaxy characteristics. The original distinction be-tween QSOs and Seyfert galaxies rested on whether thehost galaxy could be identified and thus on the relativeluminosity of the galaxy and the nucleus. Seyferts wereoriginally identified as a subcategory of spiral galaxies, butAGN can occur in galaxies of many shapes and sizes.

Time variability. AGN are strongly variable objects. SomeAGN are known to vary in luminosity by as much asa factor of 2 in less than a day. More commonly, thetimescales of variability are longer, but some level ofvariability is observed in almost all cases.

This wide range of phenomena led to attempts toimpose order on the AGN Zoo through some kind ofunifying principle. The most successful approach to AGNunification was the suggestion that much of the diver-sity of AGN phenomena could be explained simply byviewing angle.2 As seen in figure 5.1, a generic AGNis now thought to comprise a number of components,

2A classic review of AGN unification is provided in C. M. Urry andP. Padovani, 1995, Publications of the Astronomical Society of the Pacific 107:803.


Figure 5.1. The components of an AGN are shown in canoon form. An observer looking down onto the jet sees a blazar. Looking from the side, observers see a variety of AGN, depending on how much of the central region is obscured by me torus. After C. M. Vrry and P. Padovani, 1995, Pub. ktron. Soc. Pacific, 107: 803.

in particular, the following:

• The central source. At the center of the galaxy is a black hole, of a million to a few billion solar

masses. The vast energy emined by AGN

originates in accretion onto this black hole.


• The accretion disk. Just outside the black hole isan accretion disk, which feeds the black hole withmaterial, much like in stellar mass systems. Thesource of the accreting material is not clear—itcould be interstellar gas funneled down towardthe center of the galaxy, or the remnants of starsthat have ventured too close to the black hole andbeen torn apart by tidal forces. Presumably, therate of gas accretion can vary strongly from sourceto source, leading to a wide range of luminosities,including the possibility of central black holes inwhich too little mass is being accreted for anAGN to be observed at all.

• The “broad-line region.” Outside and above theaccretion disk are clouds of gas which are ionizedby radiation from the accretion disk. As theelectrons recombine they emit observable spectrallines. Since these clouds are relatively close to theblack hole, they travel at high velocities, and thusthe emission from the lines is Doppler shifted fromthe mean value for the AGN as a whole. The com-bined emission from many of these clouds as theymove in different directions then creates broademission lines, which are seen in many AGN.

• The torus. Outside the broad-line region is a dustyopaque torus, which obscures material withinit but radiates in the infrared. This torus is createdby the outer edges of the accretion disk, wherethe temperature is low enough for dust to form,and the disk itself fragments into clumps. Sucha region is not found in X-ray binaries, because it


would lie outside the Roche lobe of the accretingobject, and the angular momentum and energyin the outer disk is transferred by tidal forces backinto the binary orbit. In AGN viewed from theside (sometimes called “Type 2 AGN”), the torusprevents the broad-line clouds from being seendirectly, thus accounting for explaining the exis-tence of AGN without broad lines. Type 1 AGN,in contrast, are viewed from the top, so their innerregions are not obscured by the torus. In someAGN, the broad lines can be seen only in polarizedlight, which suggests that light from the obscuredbroad-line region is being reradiated towardthe observer’s line of sight outside the torus.

• The “narrow-line region.” Beyond andabove the torus are more gas clouds, which againcan be ionized by radiation coming from the heartof the AGN. These clouds also emit spectral lines,but since they are farther from the central source,they move more slowly, so the lines are muchless Doppler broadened. In some nearby AGN,the narrow-line region can be seen to extendwell beyond the central regions of thegalaxy.

• The jet. As was discussed in chapter 3, somefraction of the accreting material is redirectedinto a jet moving outward perpendicularly to theaccretion disk at close to the speed of light. Thesejets can create dramatic observational phenomena.In cases when the jets are pointed towardan observer, strong relativistic boosting can


dramatically change the observed characteristicsof AGN. Such objects are called blazars.

• Radio lobes. The jets can transport large amountsof energy far away from the nucleus. Eventually,this energy is deposited into the galactic or inter-galactic gas. This energy can then power the radiolobes that are observed far from the galacticnucleus.

Thus the wide range of AGN activity is ultimately poweredby accretion onto an SMBH, whose mass can be as highas several billion solar masses. This does not mean thatall aspects of AGN phenomenology are understood—in particular, the circumstances that distinguish betweenradio-loud and radio-quiet sources are not explained bysimple unification models, and many aspects of the de-tailed physics of AGN are still poorly understood. But thebasic characteristics described here seem quite solid, andessentially all current work on AGN is being carried outwithin this framework.

5.3 Superluminal Jets and Blazars

Jets can be observed as collimated streams of materialextending outward from AGN, frequently terminating inradio lobes. Often, the jets appear to bend, implying thatthe direction of jet emission precesses over time. The jetsare often lumpy or knotty, suggesting that the jets turnon and off in specific jet events. They carry considerableamounts of energy and in some cases may dramatically

Superluminal Jets & Blazars 95

influence the interstellar medium throughout the galaxy.A feedback loop is generated as the jet energy derives fromaccretion from the galaxy and in turn influences the galaxy,heating the interstellar gas and potentially changing theaccretion process. Another key feedback process is the for-mation of new stars in gas that is gravitationally influencedby the SMBH. These AGN feedback processes can bedifficult to untangle and are the subject of considerablecurrent research.

As described in chapter 3, in some cases jets and knotsare observed to travel across the sky at speeds apparentlygreater than the speed of light. Such superluminal motionis an illusion but requires the actual space velocity of thejet to be highly relativistic. In cases in which the jet ispointed almost directly toward the observer the viewingangle results in the blazar phenomenon (see figure 5.2). Asnoted in chapter 3, relativistic effects boost the intensityand energy of the jets and make the time variability shorter.In the extreme case of blazars, radiation from the jet candominate all the other sources of radiation. Blazars thusprovide excellent labs for studying jet phenomena, sincethere is effectively no contaminating radiation from otherparts of the accretion flow.

Blazars are bright at all wavelengths, with synchrotronemission in the radio through optical, comptonized radi-ation at higher energies, and, finally, the most energeticobservable photons. Orbiting satellites, including the re-cently launched Fermi satellite, show that blazars areamong the brightest objects in the sky at photon energiesabove 10 GeV, and ground-based air shower arrays haveidentified photons of tera-electron-volt energies coming

Light-Years20 40 60 80

Tim

e (y

rs)

1992

1994

1996

1998

Figure 5.2. Repeated radio images of quasar 3C 279, showingsuperluminal motion. The radio lobe on the right appears tohave moved almost 30 light-years between 1992 and 1998, whichimplies faster-than-light speed. This is in fact an optical illusion, asexplained in chapter 3. Image courtesy of NRAO/AUI.

Superluminal Jets & Blazars 97

from these sources. This high-energy radiation is thoughtto come from Compton upscattering of the synchrotronemission generally observed in radio and infrared wave-lengths. This high-energy radiation is then observed tobe even brighter and more energetic due to the Dopplerboosting provided by the motion of the jet.

Blazars also often vary substantially on quite shorttimescales—indeed, one subclass of AGN, the opticallyviolently variable (OVV) sources, were identified from thischaracteristic. The OVVs are now understood to representa particular observational characteristic of blazars. Again,this strong observed variability is a combination of intrinsicproperties of the jets and the relativistic boosting from thejet motion. Jets are clearly unstable, both in space and intime. The events that launch the jets are highly episodic, sothat specific “jet events” can often be identified as suddenluminosity flares. In blazars, the time variability of the jetsis enhanced by the relativistic time compression, so thatstrong changes in luminosity occur on timescales of a dayor less.

One difficulty in understanding blazars is that the sameDoppler boosting that makes the jet so visible has the effectof making the other components of the AGN harder to see.The presence of the accretion disk is much less apparentin blazars than in other AGN, and many blazars have noobservable line emission at all (the BL Lac objects describedin section 5.2). Blazars thus provide wonderful laboratoriesfor studying the propagation and emission from jets, butthe connection to the underlying accretion processes thatpresumably provide the ultimate source of energy is notalways clear.


5.4 Nonaccreting Central Black Holes

Given the large observed variability in luminosity observedfrom AGN, the question naturally arises whether galaxiesthat are not observed to have AGN may neverthelesscontain central black holes which are simply accretingtoo little material to be observable. Such a non-AGNblack hole has been identified most dramatically in thecenter of our own Milky Way. The galactic center isgenerally taken to be coincident with the radio sourceSgr A*, which has also been identified as an X-ray sourcewhose characteristics are generally similar to what wouldbe expected from a scaled-up quiescent X-ray binary.There are severe difficulties in observing Sgr A* in opticalwavelengths, since dust near the galactic center blocks theoptical light from the source. However, the region can beobserved in the infrared, which allows stars near Sgr A* tobe identified.

By reobserving the region for many years, it has beenpossible to follow the motion of the innermost starsas they orbit around Sgr A*. These motions are small,and the observations require adaptive optics techniquesthat compensate for the blurring effect of the Earth’satmosphere. But such techniques have been available forover a decade, comparable to the orbital period of someof the stars. Since the distance to the galactic center isknown, the spatial scale of these orbits is known, as arethe orbital periods, so the mass of the central object canbe computed. This turns out to be around 4 million solarmasses, confined to a region comparable in size to oursolar system. The total energy emitted by Sgr A* is less

Mass of SMBHs 99

than that of the most luminous stars, so the mass-to-light ratio of this central source is enormously high. Thecombination of mass, compactness, and low luminosityleaves little doubt that this central source is a black holethat is currently in a very low accretion state. Models ofthe spectrum of Sgr A* suggest that the flow is in the formof an ADAF, as is apparently the case in most quiescentbinary systems.

Such observations are difficult to make in other galaxies.But while individual stars are hard to observe in thecenter of distant galaxies, the distribution of light can bemeasured. In many cases, this light forms a cusp risingsharply toward the center, suggesting the presence of alarge central mass. In some cases, the width of spectral linesreveals the spread of velocities of the stars contributing tothe light. Often, the velocity dispersion also rises sharplytoward the center, again implying the presence of a largeunseen central mass. So, it seems likely that many, perhapsall, non-AGN galaxies contain central black holes as well.

5.5 Mass Determinations for Extragalactic SMBHs

The mass of the black hole in the center of our galaxycan be determined directly by the motions of individualstars. However, black holes in the nuclei of other galaxiescannot be observed in this way, since the distances are toogreat for stars close to the central object to be observedindividually. However, the masses of some nearby super-massive black holes can be determined by other means.One of the most dramatic early results of the Hubble Space


Telescope (HST) was an observation of the gas aroundthe supermassive object in the nucleus of the nearby giantelliptical galaxy M87. The high spatial precision of HSTallowed measurements to be made of the Doppler shift ofgas on either side of the galactic center. The results showedthat the gas is orbiting the central object with a velocity of550 km s−1 at a typical distance of about 20 pc (6×1017 m)from the galactic center. Using the standard orbital velocityformula V = √

G M/R , the mass of the central object wasdetermined to be at least 4 billion times the mass of theSun, about a thousand times greater than the black hole inour own galaxy.

This observation did not in itself prove that the centralobject must be a black hole—the Schwarzschild radius ofthe black hole is only around 1013 m, orders of magnitudesmaller than the observed orbits of the gas. Conceivably,a very dense cluster of nonradiating objects that are notthemselves black holes might provide the mass requiredfor the observed orbits of the gas. But the precision ofthe HST measurements reduced the maximum size ofthe region containing the central mass by a large factor.This meant that any such cluster must have a very highdensity of objects, which would undergo relatively closeencounters regularly. The trajectories of the objects wouldbe significantly altered by one another during the courseof a relaxation time Tr , which can be shown to be of orderTr = σ 3/(G2 M2 N), where σ is the velocity dispersionof the cluster, M is the mass of a typical star, and N isthe number density of the stars. As the cluster relaxes,the density profile of the cluster changes such that thedensity at the core increases, and energy is removed from

Mass of SMBHs 101

the cluster by stars that escape from it altogether. After alarge number of relaxation times, the core density becomesvery high.

In the case of a putative cluster of compact objects inthe center of M87, the large density implied by the sizelimits on the cluster required by the HST results requires arelaxation time much shorter than the age of the galaxy.One would thus expect the cluster to have collapsed,possibly generating a black hole but certainly lacking thestability required to persist for any substantial length oftime. If the relaxation time is short, as must be the casein the central regions of M87, no conceivable cluster ofstarlike objects would be able to survive the large numberof collisions that would inevitably occur. Thus even if thecentral mass were contained in dim stars, those stars wouldcollide and collapse into a single dense object within a shorttime.

A similar geometric approach takes advantage of theexistence of water masers in the inner parts of the accretiondisks around central black holes in AGN. Masers are themicrowave equivalents of lasers and generate strong, sharpemission lines that can be observed with radio telescopes.Long-baseline interferometry using widely separated radioantennas, is used to obtain very high spatial precision—distinct masers are observable at separations of only a fewmilliarcseconds from the center of the galaxy, correspond-ing to linear separations of a fraction of a parsec. Theredshift and blueshift of individual masers allows com-pelling Keplerian velocity curves to be obtained, leadingto very precise measurements of the central black holemass. That the individual maser measurements trace out


a R−1/2 velocity curve indicates that the mass is central.The small volume enclosed by the masers also supportsthe idea that all the mass being measured comes from asingle black hole. In some cases the masers are seen tophysically move across the face of the galaxy over the courseof years. Equating the angular motion across the sky tothe velocity measured from the Doppler shifts permits anaccurate distance measurement also to be obtained. Thereare by now over a dozen maser measurements of black holemasses in galaxies. The exceptionally precise positionalaccuracy possible with maser measurements has proved tobe particularly useful for measuring galaxies with moderate(≈ 107 M�) mass AGN.

While these direct dynamical measurements of blackholes are generally very reliable, they can be carried outon only a small handful of AGN. Even moderately distantAGN have angular scales on the sky that preclude sepa-rating the inner parts of the disk from the center of thegalaxy. And some indication of the inclination of the diskis necessary to determine an accurate mass—in the case ofthe masers, the disk must be oriented almost edge on. Thusfor most AGN, other less direct methods must be used.

One fruitful approach is known as reverberation map-ping. This technique uses the emission features in thebroad-line region that originate when gas in the clouds isionized by radiation from the central source. An increasein the luminosity of the central source thus results in anincrease in the line emission. But there is a time delaybetween the observation of an increase in the ionizingradiation and the corresponding increase in line emission,which arises because radiation travels directly from the

Mass of SMBHs 103

central source to the observer, whereas the ionizing radi-ation must in general travel across the line of sight to getto the gas clouds of the broad-line region. Therefore thelight travel time from the central source to the broad-lineregion to the observer is longer than the light travel timefrom the central source directly to the observer. Measuringthis delay allows us, in principle, to determine the size ofthe broad-line region. But the width of the emission linesreveals the velocities of the broad-line region clouds. If theclouds are gravitationally bound to the central mass, thecombination of distance and velocity can once again beused to determine its mass.

The observations required for good reverberation map-ping measurements are demanding: the time delays aretypically weeks to months, so the central source and theline emission must be observed over long periods of time.An element of luck is also required, in that there mustbe a readily observable change in luminosity of the sourcethat generates an unambiguous response in the lines. Theideal is a single large change in luminosity that is sustainedover a time that is longer than the light travel time acrossthe broad-line region, so that the response of the wholeobservable line can be observed. Changes that are shorteror smaller become harder to observe, and disentanglingmultiple or continuous luminosity changes can be difficult.Nevertheless, a few dozen examples of mass measurementsfrom reverberation mapping exist.

These mass measurement reveal a correlation betweenluminosity and size of the broad-line region. This allowsthe luminosity of the AGN, which is relatively easy todetermine, to be used as a proxy for the size scale in systems


for which the appropriate long variability studies requiredfor reverberation mapping have not been carried out.Large surveys, in particular the Sloan Digital Sky Survey,containing many thousands of AGN spectra are available.Large numbers of AGN masses can be determined byconverting the observed luminosity to a rough size andthen using the width of the lines as a measurement ofvelocity. When this is done, a remarkable feature of AGNmasses emerges, namely, a close relationship between themass of the black hole buried at the center of the galaxyand that of the galaxy itself, often referred to as theM–σ relationship, where sigma is the velocity dispersion ofthe galaxy, and thus provides a measurement of the galaxymass.

This relationship came as a surprise, since it requiresa close bond between the origin and evolution of thecentral black hole, whose gravitational influence is quitelocal, and that of the host galaxy. One hypothesis for thisbond is hierarchical growth of galaxies. This model of galaxyevolution postulates that protogalaxies are relatively small,contain small black holes, and grow by successive mergers.When these mergers occur, the black holes at the centers ofthe merging galaxies fall to the middle of the new, biggergalaxy and themselves merge. Thus studies of the evolutionof supermassive black holes and of galaxy evolution appearto be intertwined.

This somewhat unexpected juxtaposition of two areasof astrophysics is currently prompting intense research.The evolution of galaxies is now being studied directlyby comparing observations of high-redshift galaxies, whichare observed as they were long ago due to the light

Mass of SMBHs 105

travel time from these distant systems, with observationsof nearer systems. Similarly, temporal changes in AGN-related phenomena can be studied by examining changes inthe demographics of AGN with redshift. It is now possibleto carry out detailed computer simulations of the mergerprocess for galaxies and the black holes they contain andcompare the results with the observations. And the blackhole mergers themselves may soon be observable as sourcesof gravitational wave radiation, as described in chapter 9.

6FORMA T I O N AND E VO L U T I O N O FB L A C K HO L E S

Most objects in the Universe are not black holes. Onemight wonder why not: any collection of matter withnegative total energy (including the intrinsically negativegravitational potential energy) tends to collapse due toits self-gravity. This tendency, if unchecked, will pull allmatter together until it falls within an event horizon. Sowhy shouldn’t any gravitationally bound object quicklybecome a black hole? Alternatively, if the Universe isexpanding faster than its own escape velocity, and thetotal energy of the Universe is positive, one might expectall matter to diverge into an increasingly sparse plasmaand form no objects at all. Thus there appear to be twopossible configurations for the Universe, one in which ithas negative total energy and quickly collapses into one ora few black holes, and one in which it has positive totalenergy and expands into an infinitely large, sparse, cold,unfilled void. Obviously, our Universe has managed toavoid these two extremes and is sufficiently close to zerototal energy that a variety of condensed objects can formeven within an overall expansion.

It is something of a mystery why the initial conditionsof the Universe landed in the infinitesimal range that

Stellar-Mass Black Holes 107

allows complex differentiation of matter to occur. Ofcourse, observers are unlikely to exist in universes thatcontain only a single black hole or a vast sparse void. Theidea that human existence necessitates that the Universesatisfy very particular conditions is referred to as theanthropic principle.

Within individual clumps of matter the same problemarises—negative total energy leads to collapse, positiveenergy means that the object expands, and the range ofstable objects appears to be infinitesimally small. The ex-istence of stable objects requires an equilibrium, in whichoutward forces balance gravity. This equilibrium must bestable, with chance outward motion resulting in a restoringinward force, and vice versa. These outward forces vary de-pending on the object—for small objects, up to planetarysize, gravity is sufficiently weak that material strength andchemical bonds dominate and can stop any collapse. Instars, internal gas pressure is the countervailing force, andthe stable equilibrium thus obtained is demonstrated bythe existence of many billions of stars in each of manybillions of galaxies. Given the success of our particularUniverse in forming stable individual objects, the questionis reversed: In a Universe that routinely generates stableobjects in a dizzying array of shapes and sizes, under whatcircumstances can black holes form?

6.1 Stellar-Mass Black Holes

Stellar-mass black holes are generally understood to becreated in supernova explosions that mark the end of the

108 Formation & Evolution of Black Holes

life of a massive star.1 Many supernovae create neutronstars rather than black holes, and the precise conditionsunder which black holes form are still not fully understood.If the black hole is to be detected, further events arerequired, such as the formation of a binary star system ofa kind that can be observed, and in which the existenceof a black hole can be demonstrated. Here we discuss firstthe supernova event and then the binary star evolution thatleads to a situation in which the black hole can be observed.

6.1.1 Black Holes from Supernovae

Stellar mass black holes are the final stage in the life cycleof massive stars. The key event that creates the blackhole is a supernova explosion. Prior to the supernova, thestar evolves by generating energy through fusion of lightelements into heavier elements, starting with hydrogeninto helium. In a low-mass star, this process terminateswhen the stellar core is dominated by carbon and oxygen(sometimes oxygen, magnesium, and neon) supported byFermi degeneracy pressure. Once the outer envelope ofthe star is ejected, the core remains as a white dwarf.Significant amounts of mass can be lost by the star as itevolves, so a star that ends up as a white dwarf does not

1The term supernova is used to denote two kinds of events. A Type Iasupernova occurs when a white dwarf accumulates sufficient mass to exceed theChandrasekhar limit. These supernovae have luminosities that can be preciselydetermined by their observable characteristics and are thus extremely useful ascosmological markers. They undergo explosive nucleosynthesis that blows thestar apart and thus does not leave behind a compact remnant. Here we arereferring to the “core-collapse” supernova that occurs at the end of the life of amassive star, which under most conditions does leave behind a compact remnant.


necessarily spend most of its life with a mass less thanthe Chandrasekhar limit. It is currently believed that starswith an initial mass of less than 8 M� evolve into whitedwarfs, whereas stars more massive than 8 M� undergo asupernova explosion.

In contrast with low-mass stars, the cores of massivestars become hot enough for carbon to fuse into heavierelements, which leads to a succession of fusion processesculminating in a stellar core dominated by iron. Irondoes not generate energy from fusion, so further energyproduction is not possible. The iron core can thus besupported only by Fermi degeneracy pressure. Outside thecore, material from the star continues to be fused into iron,so the degenerate core grows quickly. When the iron coreexceeds the Chandrasekhar limit, it collapses, setting offthe chain of events that results in a supernova.

To collapse, the core must lose most of its electrons,which it does through the process of electron capture—electrons and protons combine to form neutrons andneutrinos. The result is a very small core of approximatelythe Chandrasekhar mass composed largely of neutrons.The neutrinos stream outward, carrying with them muchof the energy associated with the gravitational collapse ofthe core.

The key to the formation of a black hole lies in the be-havior of the material outside the iron core. It must initiallycollapse onto the core. This collapse generates gravitationalenergy, and this additional energy is transferred into theincreasingly dense outer material. The neutrinos streamingoutward from the newly formed neutron star also interactwith the outer material, injecting energy into the plasma.


All this energy generates fusion processes in the outer ma-terial, leading rapidly to the formation of heavier elements,including elements heavier than iron. In many cases, thetotal amount of energy generated in or imparted to theregions of the star outside the iron core are sufficient togravitationally unbind that material. Effectively, the outerregions of the star “bounce” off the newly formed neutronstar and are ejected into the interstellar medium. Thisejected material radiates strongly, largely from radioactivedecay of the elements created in the supernova event itself,generating the intense emission observed from supernovae.The ejected material forms a supernova remnant, expand-ing and eventually merging into the interstellar medium.Thus most core-collapse supernovae result in a centralneutron star and an expanding supernova remnant.

In some cases, however, the outer layers may not beejected but may fall back onto the compact object at thecenter. It is this situation that gives rise to the creation ofa black hole. Such fallback of the outer regions of the staris more likely when the progenitor star immediately priorto the supernova is more massive—the energy required tounbind all the outer regions of the star is greater for moremassive stars. Thus in general terms, there is a progenitormass below which a supernova gives rise to a neutron starand above which sufficient mass falls back to create a blackhole. This mass boundary between supernovae that createneutron stars and those that create black holes is thoughtto be between 20 and 40 M�.

The precise conditions under which a black hole, ratherthan a neutron star, is born is not currently well under-stood and is a subject of considerable current research.


The structure of the star immediately prior to collapseand, in particular, its chemical composition and rotationas a function of radius are critical to the outcome ofthe supernova. However, the evolution of massive stars isimperfectly understood, in part because the crucial role ofmass loss from stellar winds is not well determined. Thepresence of a binary companion star can often also havea significant effect on the structure of the presupernovaprogenitor, particularly if mass is exchanged between thetwo binary components. Even if the progenitor structurewere perfectly understood, however, the results of thesupernova event would be hard to determine. Detailedcomputer models are used to predict the outcome of super-nova events. Such computations must include the effects ofenergy transport by neutrinos and a wide variety of nuclearfusion and fission processes, all in the context of a processin which crucial events take place on scales as small as a fewkilometers near the neutron star surface and as large as theouter regions of the expanding supernova remnant, whichcan start at the outer edge of the progenitor star, hundredsof millions of kilometers from the core, and expand fromthere. Recently there have been enormous improvementsin computer hardware and significant progress in thesophistication of the programs used to model supernovaevents. Nevertheless, details of the relationship betweenthe initial mass and structure of a massive star, its massand structure immediately prior to the supernova event,and the consequences of the supernova in terms of theexpanding supernova remnant and the nature and theprecise mass of the compact remnant left behind are stillpoorly understood.


6.1.2 Binary Star Evolution

To be observed as an X-ray binary, not only must a blackhole be created in a supernova event but subsequent evolu-tion must result in a companion star providing a streamof matter to the black hole so that significant numbersof X-rays can be observed. The details of binary starevolution must therefore also be considered in determiningthe number and kind of observable stellar-mass black holes.

For high-mass X-ray binaries, the situation is relativelystraightforward. These are systems in which a black hole isaccreting from a star more massive than itself that emits astrong stellar wind. This kind of situation evolves naturallyfrom a binary consisting of two massive stars. The moremassive star (the primary star) evolves faster and thusundergoes a supernova explosion first. Some fraction of themass of the primary is lost to the system, while the restcollapses into a compact object. For circular orbits, V 2 =G M/D, so the kinetic energy (proportional to V 2/2) ishalf that of the gravitational binding energy (proportionalto G M/D). Thus if half of the mass is lost to the system,the binding energy and the kinetic energy are the same;if more than half of the mass is lost, the kinetic energy isgreater than the binding energy, the total energy is positive,and the system becomes unbound. Even if the systemremains bound, the mass lost from the supernova candramatically change the parameters of the binary system.

However, there is another crucial phenomenon thatmust be understood to determine the parameters of thebinary that emerges from the supernova, namely, the massloss and the neutrino flow from the supernova is likely to


be asymmetric, which results in a “kick” being imparted tothe nascent black hole as more material is ejected from oneside of the supernova than from the other. The observeddistributions and velocities of known neutron stars andblack holes demonstrate that such kicks do indeed exist andcan be as much as hundreds of kilometers per second. If thekick is in the right direction, a binary that would otherwisebe unbound can remain intact, while a binary that wouldotherwise survive can be disrupted. But in any case, thekick will certainly alter the orbital period and eccentricityof the binary.

Provided the binary remains intact after the supernovaevent, with an orbital period of less than a year or so,then the high stellar winds emitted by the secondary starwill essentially guarantee the creation of an observableX-ray binary. Such a high-mass X-ray binary will berelatively short-lived (only a few million years), since thesecondary star will also evolve quickly and will undergo asupernova of its own. Thus they will not have time to movefar from their initial birthplace, and indeed high-massX-ray binaries in our own and nearby galaxies are observedto be in or very near regions of current high-mass starformation. If the kick is in an appropriate direction, thissecond supernova can result in the formation of a binarysystem containing two compact objects. Double neutronstar binaries in which one object is a radio pulsar areobserved and will be discussed further in chapter 9 inthe context of the gravitational wave radiation they emit.Binaries consisting of a black hole and a neutron star, orof two black holes, have not yet been definitively observedbut are expected to exist in large numbers.


The evolution of low-mass X-ray binaries (those inwhich the companion star is less massive than the compactobject and transfers mass through Roche lobe overflow) ismore complicated than that of high-mass systems. Sincethe companion star is relatively low mass, it is almost inevi-table that the system will lose more than half its total massin the initial supernova explosion. Thus the binary willremain bound only if the kick has a fairly precise size anddirection. Because low-mass stars emit very modest stellarwinds, if the binary does remain intact, the object will havequite low X-ray emission, and thus be difficult to observe,unless the binary orbit is short enough that the companionstar fills its Roche lobe—this will require an orbital periodof less than a day for typical main sequence stars.

A number of effects tend to extract energy from thebinary orbit and thus reduce the period. Of these, magneticbraking may well be the strongest effect. Magnetic brakingoccurs when material emitted from a rotating star is forcedby the star’s magnetic field to corotate with the star. Thisis an example of a magnetic-dominated fluid flow in whichthe gas is required to flow along the magnetic field lines. Asthe material moves outward its angular momentum mustincrease, because the angular velocity remains the same,but the distance from the central source increases. Thisangular momentum is extracted from the star, slowing itsrotation. In a close binary system, tidal effects force therotation of the star into synchronicity with the orbit, sothe loss of angular momentum is coupled into the orbit,thus driving the stars closer together and decreasing theorbital period.


Alternatively, the companion star may come into con-tact with the compact object when it evolves into a giant,the star which can fill the Roche lobe of orbits with periodsof up to several hundred days. In this case the angularmomentum of the orbit is maintained, so the mass transferis conservative, and the binary gradually widens as mass istransferred from the lower-mass giant to the higher-masscompact object. However the continuing evolution of thegiant causes it to increase in size as well, so the mass transferis maintained. X-ray binaries are observed with both mainsequence and giant companions, so both of these effectsmust be relevant to the creation of low-mass black holebinaries.

Conservative mass transfer requires that mass transferfrom a low-mass to a high-mass object should increasethe orbital period, If the mass transfer is to continue, thesystem must either continue to lose angular momentum(for main sequence companions), or the star must continueto increase in size (for giant companions). The rate ofmass transfer is determined by the rate of the angularmomentum loss or the size increase of the giant. Theseprocesses have long timescales, up to billions of years.Thus low-mass X-ray binaries are much longer-lived thantheir high-mass counterparts and are often found in oldstellar populations with very little current star formation.Interestingly, in galaxies like the Milky Way, there appearto be roughly equal numbers of high-mass and low-masssystems, implying that the longer lifetimes of the low-mass systems are compensated for by the more stringentconditions of their formation.


6.1.3 Binary Evolution in Star Clusters

A particular situation arises in dense star clusters. Herethe evolution of binary stars can be dramatically changedby close encounters between the closely packed clusterstars. If a single star passes close to or within the orbitof a binary star, several outcomes can lead to systems thatcould not occur under ordinary circumstances. The resultsof encounters between binaries and single stars dependstrongly on the total energy of the three-body systemincluding the binary and the incoming single star. If thenegative energy of the binary exceeds the kinetic energyof the single star, then the total energy will be negative,and the binary is referred to as a hard binary, whereasif the total energy of the three-body system is positive,the binary is described as soft. Possible outcomes of closeencounters between binary stars and single stars include thefollowing:

• The orbit of the binary star can be altered. Thisalteration can be minor or quite drastic,depending on the details of the encounter. Ingeneral, encounters between single stars and hardbinaries tend to cause the binary to become moretightly bound—that is, hard binaries tend tobecome harder. In this case, the single star leaveswith more energy than it had originally, providinga source of energy to the dynamics of the clusteras a whole. This is true only in a statistical sense,that is, the average effect of such encounters is to


make hard binaries harder, but any particularencounter can change the binary parameters inmany different ways. But the net effect in globularclusters is to bring tight binaries closer together,which can lead to the initiation of mass transfersooner than would otherwise be the case. Incontrast, loosely bound binaries, in which therelative velocity of the binary components issmaller than the velocity of the incoming star,tend to end up more loosely bound than beforethe encounter.

• The binary can be completely disrupted; that is, thethree stars can end up independent of one another.This can occur only if the total energy of the systemis positive, and thus this outcome requires that thebinary be soft. Since soft binaries become generallysofter and eventually become disrupted, and hardbinaries become harder, this tends to create asituation in which all binaries in a cluster are tightlybound, with short orbital periods, in contrast withbinaries in the field (outside of clusters), whichshow a wide range of orbital periods.

• The single star can be exchanged into the binary.In such an exchange encounter, the incomingsingle star ends up as part of the binary system,and one member of the binary is ejected. Ingeneral, exchanges tend to result in retention ofthe more massive star of the binary system andejection of a lower-mass star. In systems wherebinaries consist of low-mass main sequence or


giant stars and a large number of massive compactremnants, exchange encounters tend to result inthe creation of large numbers of binariescontaining massive compact objects. This effect islikely to be important in explaining thedisproportionate number of such binaries in densestar clusters.

Binary stars can also encounter other binary stars, inwhich case the range of outcomes is very broad. But thegeneral trends are the same as for encounters betweensingle stars and binaries: hard binaries harden; soft binariessoften and are often disrupted, and tight binaries contain-ing the most massive stars tend to be produced.

All these effects are amplified by the process of masssegregation. Once the cluster has gone through at leastone relaxation time, the massive stars in a cluster tend tosink to the middle of the cluster, while lower-mass starsmigrate toward the outer regions of the cluster and aresometimes expelled from the cluster altogether. Since hardbinaries segregate in the same way as single stars with massequal to the total of the two components of the binary,mass segregation tends to create a dense core in the clusterconsisting of binaries and the most massive stars, whichare often compact objects. Since the cluster core is dense,there are frequent interactions between the binaries andthe compact objects, further increasing the tendency forbinaries including compact objects to form.

These effects explain why dense star clusters containmany more X-ray binaries per unit mass than the generalgalactic field. It is curious, however, that all the luminous

Supermassive Black Holes 119

X-ray binaries known in galactic globular clusters exhibitX-ray bursts and thus must contain neutron stars ratherthan black holes. This observation has generated somediscussion of the fate of black holes in a cluster in whichmost stars are below 1 M�, as would be the case in an oldstar cluster. It is clear that the massive remnants wouldundergo significant mass segregation, effectively forminga cluster of their own at the very center of the cluster.How such a subcluster of massive objects evolves is thesubject of much current debate—one possibility is thatthey tend to kick each other out of the cluster, leavingperhaps only a single object or a single massive binarybehind.

6.2 Supermassive Black Holes

In contrast with stellar-mass black hole formation, there isno obvious route to creation of a black hole of ≥106 M�directly from collapsing interstellar gas. It seems unlikelythat a single object of such a mass could form withoutfragmenting into individual stars. Thus most discussions ofthe origin and evolution of supermassive black holes positan initial “seed” black hole of relatively low mass, whichthen grows over time. While this general scenario seemsplausible, the details are not yet clear. In particular, becausesupermassive black holes are observed at high redshifts,and thus at relatively early times in the evolution of theUniverse, this process must progress surprisingly quicklyin at least some cases.


6.2.1 Seed Black Holes

It is generally assumed that the seed black holes that sub-sequently evolved into supermassive black holes must haveoriginated as stellar-mass black holes in the earliest gen-eration of star formation. This assumption presents somedifficulties, as the growth mechanisms postulated nextcannot create supermassive black holes out of ≈10 M�black holes quickly enough to explain the AGN observedat high redshifts. However, the very first generation ofstars is likely to have been very different from the starsobserved today. In particular, the first generation of starswas presumably created out of a mixture of hydrogenand helium, with no heavier elements at all, since theelements heavier than helium could not be created in theearly universe but, rather, required nuclear processes insidestars to come into being and supernova explosions to bedistributed into the interstellar medium.

Stars without heavy elements are not currently ob-served, so this first generation of stars has apparentlydisappeared. However, theoretical models of the behaviorof such stars suggest that the absence of even trace amountsof heavy elements changes the nature of the stars quite dra-matically. In particular, much of the opacity and cooling instars and star-forming regions is generated by incompletelyionized heavy elements. In the absence of such ions, themass of the fragments into which a collapsing gas cloudforms is larger than it might otherwise be. Stellar windsare also harder to drive, so more of the mass is retained asthe star evolves. Finally, the eventual supernova of suchan object at the completion of nuclear burning might


plausibly lead to near-complete collapse, leaving behinda black hole much more massive than ones created byordinary stars, perhaps in some cases with a mass as high as103 M�. This scenario also has the appealing feature thatthe large mass of the first generation of stars explains whyno such stars are currently observed, since high-mass starsevolve and die on timescales much shorter than the age ofthe Universe.

6.2.2 Growth of Supermassive Black Holes

As discussed in the preceding section, it is plausible thatthere might be a significant number of black holes withmasses of up to 103 M� that were created very shortly afterthe first generation of stars was formed. However, blackholes of this mass are not observed (see chapter 8) but,rather, must grow by factors of thousands to create thesupermassive black holes observed back to very early timesin AGN. While the details of the growth of supermassiveblack holes are still the subject of intense research anddebate, two basic processes are generally assumed to beresponsible.

The first growth mechanism is simply accretion. It isknown that these black holes are accreting gas, since it isthrough the accretion process that they are observed. If asteady supply of gas can be provided to the black hole, itsmass will inevitably grow. However, accretion cannot ingeneral occur faster than the Eddington limit will allow—if mass is accreted too quickly, the luminosity will exceedLEdd, and the accretion will be halted by radiation pressure.


Since LEdd ∝ M, the allowed mass accretion rate is aconstant proportion of the mass, leading to exponentialgrowth with an e-folding timescale of

τgrowth = ησTc

4πGm p≈ 5 × 107years,

where η is an efficiency factor for turning accreted massinto radiation, usually taken to be about 10%. It might bethought that exponential growth could account for evenvery large black hole masses, but in fact the number ofavailable e-folding times is quite limited. Bright quasarshave been observed at redshifts greater than 6, whichimplies distances so great that the Universe was <109 yearsold when the quasars were observed. This is also about thetime when the first stars were formed, which producedthe seed black holes at the end of their evolution. Thusthere may be 10 or fewer e-folding times between thecreation of the seeds and the existence of bright quasarsthat seem to require 109 M� SMBHs. Since e 10 ≈ 2×104,it may be hard to grow a black hole from 10 M�, or even103–109 M�, with Eddington-limited accretion.

The source of the gas that powers AGN is also notwholly clear. There is considerable interstellar gas in mostyoung galaxies, but it is not easy to extract enough angularmomentum to allow a substantial amount of the gas tofall into the small event horizon of a black hole. Torquesexerted by nonspherical mass distributions in the center ofgalaxies (referred to as “bars”) may help this process, butit is not clear that it can provide enough gas to grow theblack holes quickly enough. Another source of gas could


be tidally disrupted stars. When a star ventures too closeto a massive black hole, it is torn apart by tidal forces andstretched out into a stream of gas. Such a stream is accretedonto the black hole relatively quickly. However, as with thedirect accretion of gas, ensuring a steady supply of stars tothe black hole can be problematic.

The other basic idea relating to black hole growth ismerger of black holes. This concept has gained currencysince the discovery of the M-sigma relationship, whichdemonstrates that the mass of the central black hole isproportional to the mass of the spherical component ofthe galaxy that contains it. Galaxies are thought to growby mergers as small condensations of gas and stars collideand create increasingly larger structures (see figure 6.1).Thus it seems natural to ask whether the black holes theycontain might also merge. If the galaxies and black holesgrow together, then the close relationship between themcould be explained.

However, the processes through which the central blackholes in merging galaxies will themselves merge are alsonot well understood. In general, the galaxies will notcollide precisely head-on, so the two central black holeswill start out in a fairly wide orbit around each other.They will gradually spiral inward, by transferring energyfrom their orbit to the stars and gas they encounter—this is essentially an extreme example of the process ofmass segregation that occurs in star clusters. However, thisprocess depletes the inner regions of the galaxy of matterand is thus self-limiting. If the two black holes are broughtsufficiently close to each other, they will merge throughthe emission of gravitational radiation, an event that could


Figure 6.1. Image of colliding galaxy SDSS J1254+0846 in opticaland X-ray light. The outer parts of the galaxy, observed in opticallight, are tidally distorted. Each galaxy contains an AGN, observedin X-rays, which are the bright sources in the center. Both the galaxyand the black holes they contain are likely to merge over the next fewhundred million years. X-ray image NASA/CXC/SAO. P. Greenet al., Optical image (Carnegie Obs/Magellan/W. Baade Telescope.J.S. Mulchaey et al.).

potentially be observed by gravitational wave detectorscurrently under development. Unfortunately, according tosimple calculations, the interaction of orbiting black holeswith stars and gas in typical galaxies does not appear tobring them close enough for gravitational radiation to take


them the rest of the way to merger. This “final-parsecproblem” is a current subject of considerable research.

One interesting test of whether gas accretion or blackhole mergers is the dominant process would be to comparethe total luminosity of AGN in the Universe with thetotal mass of AGN.2 If most of the mass of the SMBHsis due to accretion, then the mass accretion responsiblefor the observed luminosity must be equal to the currenttotal mass of the AGN. However, if there is more massthan can be accounted for by the observed accretionluminosity, then some “quiet” mode of growth mightbe required. Black hole mergers constitute such a quietgrowth process, since they emit much of their energyin the form of gravitational waves, which are currentlynot detected (but see chapter 9). There are significantobservational and theoretical problems with carrying outthis test. Observationally, one must include all energyemitted by AGN, including radiation at all wavelengthsand bulk energy carried out in jets. Theoretically, onemust understand the efficiency of the accretion, that is,how much energy is generated by the accretion of a givenamount of mass. These complexities are not yet whollyresolved.

Thus our understanding of the creation of supermas-sive black holes is currently incomplete. Observationalapproaches to the problem include determining the lumi-nosity of AGN as a function of cosmic time by detailedsurveys at all redshifts; searching for the first quasars andstars at the highest redshifts to try to identify black holes

2This approach was pioneered by A. Soltan (1982) in an article in MonthlyNotices of the Royal Astronomical Society 200:115.


in the initial or early stages of their growth; searchingfor binary black holes in merging galaxies; and creatinggravitational wave detectors that might one day detect themerger events themselves. Theoretical work is being doneon the formation and evolution of the earliest generationof stars; on the processes by which gas might be fed togrowing black holes in the centers of galaxies; and onprocesses that might bring merging black holes across thefinal parsec of their separation so that mergers can takeplace.

7DO I N T E RMED I A T E -M A S S B L A C KHO L E S E X I S T ?

There is powerful empirical evidence for two classes ofblack holes, namely, the stellar-mass black holes, withmasses a few times that of the Sun, and the supermassiveblack holes at the centers of galaxies. The considerable gapin mass between these two categories naturally prompts thequestion whether black holes might exist at other massscales. In recent years two lines of evidence have beenpresented in support of the idea that black holes withmasses intermediate between stellar mass and supermassivemight exist, that is, with masses of 102–105 M�. Suchsources are referred to as intermediate-mass black holes(IMBHs). In both cases the results are currently stillambiguous, and much debated.

7.1 Ultraluminous X-Ray Binaries

As X-ray astronomy evolved, one of the capabilities thatimproved significantly was the spatial resolution of thevarious orbiting observatories, that is, the ability of anobservation to distinguish between closely neighboring

128 Do Intermediate-Mass Black Holes Exist?

objects. Improved X-ray optics and detector technologyhas improved spatial resolution of X-ray observatories tothe point that the spatial resolution in soft X-rays of theChandra satellite is almost as good as that of the HubbleSpace Telescope in optical bandwidths.

One scientific goal for which the improved spatialresolution has been particularly important is the studyof X-ray binaries in galaxies other than our own. X-raybinaries are bright, so simply detecting them in othergalaxies in the nearby universe is straightforward. Howeverthere are many such objects in a sizable galaxy (we havehundreds in our own), and they are generally concentratedtoward the central regions, or in the regions of the greatestongoing star formation. Those parts of galaxies are alsohome to other strong X-ray sources, including AGN andsupernovae remnants. Thus studying the populations ofX-ray binaries in other galaxies requires distinguishingmany closely packed sources from each other and thusrequires the highest obtainable spatial resolution.

Maps of nearby galaxies made with Chandra and otherX-ray observatories have provided lists of hundreds ofX-ray binaries outside the Milky Way. The demographicsof the X-ray binary population vary from one galaxy toanother. One particular category of X-ray binary that hasbeen identified in other galaxies, but not in our own, is theultraluminous X-ray sources (ULXs), whose luminosities aresignificantly larger than the Eddington limit for a 10 M�object (see figure 7.1).

When the first ULXs were observed, there was somequestion about whether the large observed fluxes em-anated from an individual X-ray binary. And, indeed, as

Ultraluminous X-Ray Binaries 129

Figure 7.1. X-ray image of galaxy M66. The point sources areX-ray binaries superposed on a diffuse galactic background. Thebrightest sources are considerably brighter than the Eddington limitof a 10 M� accreting object and so are considered ultraluminoussources. NASA/CXC/Ohio State Univ. C. Grier et al.

observations were refined, some sources were identified asAGN, or as superpositions of several individual sources,or as foreground or background sources coincidentallylocated within the galaxy under study. But some ULXscould not be explained in this way. Such sources are notcoincident with the center of the galaxy (as an AGNwould be) and have spectra that are hard to explain fromforeground or background sources. A key step forward wasthe identification of large luminosity changes over timefrom some of these sources—if the flux came from severalsuperposed sources, luminosity changes of more than afactor of 2, would not be expected since that would requirethe individual sources to be able to arrange to vary in


concert. Thus at least some of the ULXs do appear to beindividual sources that have luminosities of >1040 ergs−1,well above the Eddington limit for stellar-mass blackholes.

The natural suggestion for these ULXs is that theyrepresent accretion onto objects significantly more massivethan the accreting black holes observed in our galaxy. Sincethe Eddington limit scales with mass, the ULXs couldremain within the Eddington limit if the accreting sourcescontained intermediate-mass rather than stellar-mass blackholes. This hypothesis was strengthened by the discoverythat the X-ray flux from ULXs is dominated by soft (low-energy) photons. Radiation that is thermal in origin wouldindicate lower temperatures from the accretion disk, inaccordance with the expectation that T ∝ M−1/4 foraccreting black holes.

However, detailed studies of the X-ray spectra of theULXs do not agree with the expectations of the IMBH hy-pothesis. If the ULXs were intermediate-mass black holes,they would presumably be accreting at sub-Eddingtonrates, and thus the X-ray spectra should show states similarto those of the known galactic X-ray binaries, except scaledappropriately for the difference in mass of the black hole.However, at least some of the ULXs have spectral featuresthat are different from anything else seen in the galaxy. Inparticular, some of the best-observed ULXs show an excessof flux in low-energy X-rays, even when the T ∝ M−1/4

law is accounted for, and also a sharp drop in flux at higherenergies unlike anything else seen in galactic sources.1

1J. Gladstone, T. Roberts, and C. Done, 2009, Monthly Notices of the RoyalAstronomical Society 39:1836.

Ultraluminous X-Ray Binaries 131

This suggests that the accretion flow in ULXs is notanalogous to what is observed in the galactic X-ray binaries.

The alternative hypothesis is that the ULXs are exam-ples of super-Eddington accretion onto stellar-mass blackholes. Super-Eddington accretion requires that one ormore of the assumptions that underlie the calculation ofthe Eddington limit be false. In particular, the flow isunlikely to be spherically symmetric, so from some viewingangles luminosity might be seen well beyond the nominalEddington limit. The flow is also likely to become opticallythick, which might result in emergence of observed radia-tion at lower energies, which could explain the observedspectral anomalies. Thus super-Eddington accretion ontoa stellar-mass black hole may be a viable explanation forthe ULXs.

A compelling resolution to the ULX conundrum islikely to require direct determination of the mass of theaccreting object. This could in principle be done bydetermining the orbital period and velocity of the com-panion star. In practice, this is quite difficult: the sourcesare located in external galaxies, and thus the companionstars are extremely faint. At the same time the accretionluminosity is by definition large, so the optical/infraredflux from the companion star is likely to be masked bythat of the accretion. However, such measurements maynot be out of the question if the companion is a brightearly-type giant or supergiant, as has been suggested forsome sources. If such a source goes into a low state inwhich the companion outshines the accretion flow, theorbital motion of the companion might be measured, andthe existence of an intermediate-mass black hole could beestablished or refuted.


7.2 Black Holes in Star Clusters and Low-MassGalaxies

The putative intermediate-mass black holes in ULXs arefueled by a companion star and are thus similar to X-raybinaries. One can also imagine intermediate-mass blackholes at the center of low-mass galaxies and star clusters,which would thus be low-mass analogs to the supermassiveblack holes in AGN. The mass distribution of black holesat the center of galaxies is likely to extend well below106 M�. Observations in nearby galaxies seem to supportthe presence of black holes at the low end of the AGNmass distribution, although there is some evidence that theM–σ relationship changes at the low-mass end. But therelevant observations are hard to make in very low-massgalaxies, so the low-mass extension of the SMBHs has beenhard to characterize.

If the M–σ relation extends to globular star clusters,the largest of which contain 106−7 M� in stars, thenthese systems might be expected to contain central blackholes that are intermediate between the stellar-mass andsupermassive black holes. Such black holes would bedistinct from the compact objects contained in X-raybinaries in these star clusters, as they would presumablynot have evolved as members of a binary star systemand would be located at the exact dynamic center of thecluster. Since there is no evidence of X-ray sources withLx � 1038 ergs−1 in these systems, such central IMBHsapparently do not accrete significant amounts of matter.This is not surprising, since globular clusters are knownto contain only very small amounts of interstellar gas. It

Black Holes in Star Clusters 133

is possible that newly developed sensitive radio telescopesmay be able to detect such accretion, and, indeed, onecluster in the nearby Andromeda galaxy seems to showunexpectedly large radio flux. Barring direct evidence ofaccretion, demonstrating the existence of IMBHs wouldrequire seeing the gravitational effects of the IMBH on thedynamics of observable stars near the center of the cluster.

Some evidence of this kind has been claimed, in theform of cusps of unexpectedly high density and velocitynear the center of some clusters. These cusps are analogousto the density peaks at the center of galaxies that revealthe presence of nonaccreting central black holes. The outeredge of the cusp occurs when the mass of stars interior tothat point is roughly equal to the mass of the IMBH. Thusthe total mass of stars exhibiting a significant influencefrom the IMBH cannot be greater than the mass of theIMBH itself. An IMBH with M ≈ 103 M�, for example,would influence no more than the central 103 stars in thecluster if the typical mass of a star is close to that of theSun. Since some of these stars are quite faint, there may beonly a handful of easily observable stars that would showthe effects of the IMBH. These effects may be quite subtle,and there is disagreement in the scientific community asto whether the claimed evidence for IMBHs in globularclusters might be explained by random fluctuations in thedistributions of the small number of stars, expected to beinfluenced by an IMBH. For example, in the largest clusterin our galaxy, Omega Centauri, ground-based observationsseem to show a cusp of the kind expected to be producedby an IMBH, whereas space-based observations do not.One possible resolution to this and other discrepancies lies


in ambiguities in determining the location of the centerof the cluster; if the cluster center is determined by thedistribution of light, it tends to be unduly influenced bythe location of the brightest few stars near the center. If themeasured center of light is superposed on such a bright star,the light distribution will naturally have a cusp, regardlessof whether there is a central IMBH or not.

To firmly establish (or refute) the presence of IMBHs inglobular clusters will require very precise measurements ofas many stars as possible. Two approaches may be helpful:first, repeated images by the Hubble Space Telescope canreveal small motions across the sky by many stars in thecentral region of clusters. But even in this case, the limitson the numbers of stars influenced by the IMBHs maypreclude conclusive results. Alternatively, the motion ofradio pulsars may be very precisely determined by changesin the observed pulsation periods—if such a pulsar is closeenough to an IMBH, the changes in the motion of thepulsar may reveal the presence of a large compact mass.This approach would require that a pulsar exist in just theright position near the central black hole and is thus alsounlikely to provide definitive results. Thus it seems likelythat the existence of IMBHs in star clusters will remainunproven in the foreseeable future.

8BLACK HO L E S P I N

Black holes are among the simplest objects in the Universe.Simplicity and complexity can be defined by the numberof parameters required to completely specify the propertiesof an object. This number is very large for such things aspeople, planets, stars, and galaxies—the chemical compo-sition, pressure, and temperature all need to be definedat every point within the object. But a black hole can becompletely defined by a mere three parameters: its overallmass, charge, and spin. The nature and distribution ofmaterial inside the event horizon has no impact on theobservable properties of the black hole.

The previous chapters discussed ways in which themass of black holes is determined and the evolutionaryscenarios that lead to black holes with masses similar tothose observed. Charged black holes are not likely tooccur in astrophysical situations, since a black hole withnonzero charge would attract oppositely charged particlesand would quickly achieve neutrality. But observed blackholes are expected to have nonzero angular momentum.In particular, since black holes are created by the collapse

136 Black Hole Spin

of large amounts of material, any initial spin will beincreased during the collapse due to angular momentumconservation. So the spin of a black hole is likely to besignificant.

As noted in chapter 1, the spin is usually described asa nondimensional parameter a = J /(G M2/c ), which canrange from zero (a nonspinning black hole) to 1 (a situa-tion described as “maximally spinning”). The parameter aenters into the Kerr metric, which describes a black holewith a ring singularity at the center, surrounded by anevent horizon, surrounded by an ergosphere. Between theevent horizon and the ergosphere, objects cannot remain atrest but must rotate along with the black hole; nevertheless,they can escape to infinity. If pairs of particles are producedin the ergosphere, with one particle plunging into the eventhorizon and the other escaping to infinity, the rotationalenergy of the black hole can be extracted. This mechanismis known as the Penrose process. Up to 29% of the totalenergy of a maximally rotating black hole can be extractedby this process if the entire spin of the black hole isremoved.

The differences in space-time between a nonspinningSchwarzschild black hole and a Kerr black hole of the samemass have potentially observable effects. Thus we can hopeto measure the spin by observing effects which should bedifferent depending on the value of a . The most obviousof these differences is the position of the innermost stablecircular orbit (ISCO), which has a significant effect on theinner edge of an accretion disk. It is through determinationof the physical size of the ISCO that the spins of blackholes are determined.

The Innermost Stable Circular Orbit 137

8.1 The Innermost Stable Circular Orbit

Accretion disks are composed of gas orbiting in concen-tric circular orbits around the accreting object. Viscosityextracts energy from the disk and transports angular mo-mentum through it, which has the effect of transferringmass toward the inside of the disk. This process continuesuntil the accreting matter reaches the inner edge of thedisk. Three things can truncate the inner disk. The diskcan reach all the way down to the surface of the accretingobject, at which point the accreting gas must settle on thatsurface. The disk can also terminate due to the presenceof strong magnetic fields anchored in the surface of theaccreting object. When the magnetic pressure becomessufficiently great, the gas must flow along the magneticfield lines directly to the surface of the accreting object.These fields change the balance of forces and thus disruptthe orbits of the gas. But black holes have no surface, andso neither of these effects will terminate the disk.

Instead, in the case of accretion black holes the proper-ties of space-time itself bound the inner edge of the disk.Sufficiently close to a black hole, stable circular orbits areno longer possible. For a nonspinning black hole describedby the Schwarzschild metric, this innermost stable circularorbit, or ISCO, occurs at r = 3Rs . Between the ISCOand the event horizon, material can follow trajectories thatdo not lead inside the event horizon, but those trajectoriescannot remain at a fixed distance from the black hole, socircular orbits are impossible.

The same is true of rotating black holes, described bythe Kerr metric. However, the distance of the ISCO from

138 Black Hole Spin

0 =0 D = 1

Figure 8.1 . Illustration of che innermost stable circular disk

(lSeO) for a nonspinning and maximally spinning prograde black hole. The accretion disk terminates at the IseD.

the singularity varies with the spin of the black hole. As the

spin parameter a goes from 0 to 1 the distance of the lsea from the center of the black hole d ecreases from 6GM/ c2

to 1 GM/ c2• Thus the accretion disk extends farther in for

a spinning black hole than for a nonspinning black hole

as shown in figure 8 .1. This resull applies to simarions in which the disk is (mating in the same direction as

the black hole. Relativistic effects force the inner pans of

the disk to orbit in a plane perpendicular to the angular momentum vector of the black hole. However, the d isk

can be prograde or retrograde: that is, the disk can be

rotating in the same direction as the black hole or in

the opposite direction, respectively. The prograde situation resul ts in a small ISCQ, whereas retrograde rotation creates

a much larger ISCO. Either way, the temperature of the

inner disk is quite different for a disk with a ::::: I than

for a nonspinning disk. This difference can be observed

because the inner edge of the disk is the honest pan of the

disk, so the location of the high-temperature cutoff in the

Observations of the ISCO: Line Emission 139

spectrum will change if the ISCO is in a different position.Another potentially observable effect is that light from theinnermost parts of an accretion disk that extends closer tothe black hole will be more gravitationally redshifted.

There is some debate over the correct way to modelthe effects of the ISCO. The simplest approach is simplyto assume that the material vanishes down the black holeas soon as it reaches the ISCO. But this is obviously notquite right—the density of the material doubtless dropsconsiderably at the ISCO, but the material still has tomake its way to the event horizon before disappearingfrom sight, so the density and emission must in fact becontinuous across the ISCO. It may be that the drop isquite precipitous, but some dynamical models suggest thatmaterial inside the ISCO still contributes significantly tothe observed luminosity of the disk. The best way to modelthe flow from the ISCO to the event horizon is a subjectof considerable dispute, leading to strong disagreements onthe reliability of measurements of black hole spin.

8.2 Observations of the ISCO through Line Emission

The spin of a black hole can be determined by measuringthe size of the ISCO relative to Rs . There are two primarymeasurement methods, both of which have been carriedout in some cases. But there is considerable controversyover whether either of these methods is reliable, a con-troversy that has been heightened by the quite differentanswers obtained in the very few cases in which bothmethods have been applied.

140 Black Hole Spin

The first approach to measuring black hole spininvolves observing spectroscopic emission lines from theinner parts of the accretion disk. Spectral lines are gener-ated by atomic or nuclear processes and consist of spikesof emission at a specific wavelength or, equivalently, at aspecific energy or frequency. Accretion disks sometimesdisplay such lines. Highly ionized iron, in particular, hasa number of spectral features in the X-ray range. Sinceiron is the most massive nucleus found in abundance inastronomical plasmas, these lines are the highest-energyatomic lines generally observed and have been seen inaccreting stellar-mass and supermassive black holes. Aparticularly useful and important line is a fluorescence lineat an energy near 6.4 keV, which generally thought to beexcited in the disk by photons emitted by the accretiondisk corona. Thus this line is thought to track the gas inthe disk.

Although the photons associated with a particular lineare emitted at the same energy, they are not all observedat that energy. This phenomenon is called line broadening;that is the emission created by a spectral line is spread overa range of energies. If the spectrum of the source is plotted,the bump associated with the line has a particular shapedetermined by the line broadening (see figure 8.2). Thephysical mechanisms that give rise to the broadening canbe determined by measuring the shape of the line.

In the case of line emission from an accretion disk,the Doppler shift results in considerable line broadening.Different parts of the disk move at different speeds relativeto the observer, so the Doppler shift changes the ob-served energy of the photons differently at different points.


Line

flux

FE (k

eV c

m–2

s–1

keV

–1)

02

x 10

–44

x 10

–46

x 10

–48

x 10

–4

Energy (keV)

4 6 8 10

Figure 8.2. Iron line from the AGN MCG-6-30-15 obtained withthe XMM satellite. Note the extended red wing, caused by gravita-tional redshift of the inner disk. The extent of the line is limited bythe innermost edge of the accretion disk and can thus be used todetermine the position of the ISCO. Source: Fabian et al. “A LongHard Look at MCG-6-30-15 with XXM-Newton.” MNRAS, 2002,volume 335, L1-L5. By permission of Oxford University Press.

When the disk as a whole is observed, the line is broadenedbecause the velocity of the emitting gas varies acrossthe disk. The internal motions of the disk generate aparticular kind of line broadening, resulting in double-peaked emission—one peak corresponds to emission fromthe parts of the disk moving toward the observer, and theother, from those parts of the disk moving away from theobserver.

But disks in strong gravitational fields display anothereffect, namely, broadening from gravitational redshift. The

142 Black Hole Spin

gravitational redshift changes the energy of photons dueto the strength of the gravitational field from which aphoton emerges. Like any other entity, a photon will loseenergy as it climbs out of a gravitational potential well.In massive objects, this causes the object to slow downas it recedes from a large mass. But photons must bydefinition move at the speed of light, so they lose energy bychanging wavelength rather than by changing speed. Whenthe gravitating field is strong, as in an accretion disk near ablack hole, the relativistic expression for the gravitationalredshift for a Schwarzschild black hole as measured byan observer at infinite distance from the source can beexpressed as

z = 1/√

1 − Rs /r − 1,

where z is the observed redshift, and r is the distance fromthe massive central object to the source of emission.

The combination of Doppler shift and gravitationalredshift produces a very particular line shape for lineemission from an accretion disk near a black hole. Thegravitational redshift moves the line to lower energies, butthis effect is more pronounced for photons emerging fromthe inner parts of the accretion disk, which are nearest theblack hole. This is also the part of the accretion disk inwhich the gas moves most rapidly, so the wings of the lineprofile are the most affected by the gravitational redshift.The high-energy wing moves back toward the center ofthe line, since the positive shift from the Doppler effect iscountered by the gravitational redshift. But the low-energywing becomes even more pronounced, as the Doppler shift


and the gravitational redshift work in tandem. Thus theeffect of the gravitational redshift is to make the line shapestrongly asymmetric, stretched toward the low-energy end.

The amount of this stretching is determined by thestrength of the gravitational field, that is, from the radialposition of the emitting material relative to Rs . This iswhere the connection between line shape and the locationof the ISCO, and thus the spin of the black hole, ismade. For nonspinning black holes, the ISCO is relativelylarge, and thus the effects of gravitational redshift cannever become too big. In contrast, the disk of a maximallyspinning black hole extends much farther in, and thusthe effects of gravitational redshift are much greater. Thusa rapidly spinning black hole will have lines with muchgreater asymmetry than a nonspinning black hole, so thespin of the black hole can be determined by measuring theshape of the lines.

There are a number of difficulties with this procedurein practice. Most important, the amount of line emissiongenerated by a gas is determined by the physical condi-tions of the gas—in particular, the chemical composition,temperature, and pressure. The temperature and pressurevary greatly as a function of radius in an accretion disk,and thus the amount of line emission generated at eachposition in the accretion disk varies. One could imagine adisk in which the inner parts simply do not generate lineemission, for example, if it is hot enough that the ironis completely ionized, and single-electron iron atoms arevery few. The line shapes created by such a disk mightmimic those of a disk with a much larger ISCO. Anotherproblem is that accretion flows generate large amounts of

144 Black Hole Spin

line emission only in certain X-ray states, and those X-raystates may not be those in which the flow is in the formof a standard accretion disk. Thus the Doppler broadeningmay not be as expected from a disk. These effects can inprinciple be modeled, but the models are complex and notfully developed, so they introduce considerable uncertaintyinto the spin measurements. Nevertheless, studies of emis-sion line shapes have produced spin measurements for anumber of stellar and supermassive black holes.

8.3 Observations of the ISCO throughThermal Emission

A different approach to determining the location of theISCO is to observe the continuous thermal emission fromthe hot gas in the inner parts of the disk. Specifically,a standard α-disk is expected to emit radiation that ap-proximates blackbody radiation from a series of concentricrings, each of which has a different temperature. Thetemperature rises toward the inner edge of the disk by anamount that can readily be calculated. At the ISCO, therise in temperature is abruptly terminated, so the highesttemperature seen in the integrated spectrum of the diskcorresponds to the temperature at the ISCO.

Blackbody radiation follows the Stefan-Boltzmannrelationship L = σ R2T4, which relates the luminosity,temperature, and size of the emitting region. The highesttemperature of the disk can be determined by the high-energy cutoff of the observed spectrum. So, if the luminos-ity can be determined, the size of the emitting region can

Observations of the ISCO: Thermal Emission 145

be calculated. For the highest temperature, that size willpresumably be that of a ring of emission situated at theinner edge of the disk. The perimeter of that ring revealsthe size of the ISCO. If the mass of the black hole, andthus Rs , can be separately determined, then the size of theISCO in terms of Rs can be measured, and thus the spinof the black hole can be calculated.

Determining the luminosity of the disk emission re-quires knowing the distance to the black hole, so thatthe observed flux can be converted into an intrinsic lu-minosity. For supermassive black holes, the distance canbe determined by measuring the redshift of the AGNor its host galaxy. But supermassive black holes presentother difficulties to using this approach, as discussed later.Determining the distance to stellar-mass black holes ismore difficult but can be done in a number of importantcases.

Two approaches have been used to determine accuratedistances to stellar black hole systems. The most directmethod is to measure the parallax of the system. A parallaxmeasurement is made by determining the tiny shift in ob-served position that occurs because the vantage point fromEarth changes as the Earth orbits the Sun. The definition ofa parsec, the basic unit of astronomical distance, is that atthis distance the parallax shift is equal to one second of arc(hence par-sec). The importance of the parallax method toastronomy is demonstrated by its definition as the standardunit of distance; parallax measurements are the gold stan-dard on which all other astronomical distances are based.Unfortunately, it is quite difficult to measure tiny fractionsof an arcsecond, so most astronomical objects beyond a few

146 Black Hole Spin

hundred parsecs of distance, including most X-ray binaries,cannot be measured in this way. However, radio parallaxeshave recently been used to determine the distances of a fewX-ray binaries. Combining radio observations from manystations across the Earth can lead to positions accurateto better than a milliarcsecond, which corresponds to adistance of a kiloparsec. The distances of several of theclosest X-ray binaries have been measured in this way.

But parallax measurements cannot be made for mostX-ray binaries. Nevertheless, some of their distances aremoderately secure. In transient systems, the accuratelyknown orbital parameters (period, mass function, massratio, and inclination) lead to a reliable distance determina-tion. Information about the size of the Roche lobe filled bythe companion star is combined with the temperature de-termined by its spectral type. Again, the Stefan-Boltzmannrelation is used, this time to calculate the luminosity of thecompanion star. The distance can be determined by com-paring the luminosity with the observed flux. This distanceis subject to all the uncertainties associated with the orbitalparameters and also requires accounting for any interstellarabsorption along the line of sight. The calculation of thedistance is in most cases the limiting factor in measuringthe spin of the black hole from continuum emissions.

Key requirements for this method are that the accretiondisk extends to the ISCO and that observed emissionis dominated by thermal radiation from the inner disk.Equivalently, if the accretion flow is in the form of astandard α-disk, this is likely to be true but is certainly notalways true. In X-ray binaries, these measurements can bemade when the inner disk is a significant component of the

Consequences of Spin for Jets 147

flux, typically in the soft X-ray state. In AGN, the situationis even more difficult, since the thermal emission from theinner disk is typically in the far ultraviolet, a wavelengthrange in which photons do not propagate through thegalaxy. Therefore, this method has been used only forX-ray binaries. Even then, questions abound: Does the diskreally extend all the way to the ISCO? Are the black holemass and the distance well measured? Does the materialinside the ISCO contribute significantly to the emission?One hopeful sign for this method is that in some sources,the size of the ISCO has been measured many times withthe same result, even though the X-ray flux and spectrumwere quite different. This consistency strongly suggeststhat the potential problems arising from an understandingof the accretion flow itself are not significant: however, thebinary parameters must still be accurately determined.

8.4 Consequences of Spin for Jets and OtherPhenomena

A great deal of effort has gone into trying to measure thespin of black holes. One might ask why. Mass determi-nations are crucial to establishing that an object is indeeda black hole, and in understanding how black holes aremade. But the spin measurements assume that the systembeing observed contains a black hole and so cannot be usedto confirm the nature of the system.

One possible consequence of black hole spin is thecreation and collimation of the relativistic jets observed inmany AGN and some X-ray binaries. That jets in these

148 Black Hole Spin

two kinds of systems are similar (up to the scaling bymass of the black hole) suggests that some deep physicsassociated with accretion onto black holes must be invokedto account for them. One of the earliest models wasthat of Blandford and Znajek, who suggested that framedragging from a rapidly spinning black hole could createthe necessary jet power.1 This model would imply that thepower in the jet should be at least loosely correlated withthe spin of the black hole. An alternative mechanism, nowmore widely accepted, proposed that the jets are collimatedby winding up the magnetic field of the disk, in whichcase the spin of the black hole is much less important.Recent results from the thermal disk method appear tofavor the magnetic model: while the X-ray binary withthe strongest observed jet does indeed appear to have aspin parameter close to unity, another source which isbelieved to have a strong jet has the lowest observed spinparameter (a < 0.5 at high confidence, and consistentwith zero). Taken at face value, this result would likelyrule out the Blandford-Znajek mechanism, but it shouldbe noted that the jet in this system was observed onlyindirectly and that the usual caveats about the accuracyof the spin measurements apply. It seems likely that morenumerous and more accurate measurements of the spin ofblack holes will provide a definitive ruling on whether theBlandford-Znajek mechanism is in fact responsible for theproduction of the observed superluminal jets.

Another situation in which the spin of the black holeis important is in the details of colliding or merging black

1R. Blandford and R. Znajek, 1977, Monthly Notices of the Royal AstronomicalSociety, 179: 433.

Consequences of Spin for Jets 149

holes. When the event horizons of two black holes comeinto contact, the black holes rapidly merge. Energy carriedaway in the form of gravitational waves is potentiallyobservable, as described in the next chapter. The mergerprocess will proceed differently depending on the spin—and the orientation of the spin—of the black holes, andso the expected behavior at the moment of merger, whenmost of the gravitational waves are produced, dependscritically on the spin of the individual black holes. Thespin of the merger product is the vector sum of the orbitalangular momentum of the orbit and that of the twooriginal black holes, less whatever angular momentum isshed in the merger. Supermassive black holes may be builtup by gas accretion or by successive mergers of blackholes, whose spins should be randomly aligned. If theprimary growth mechanism is many small mergers, thespins should cancel out, whereas if the growth is due toonly a few mergers, or by mergers with very lopsided massratios, or by gas accretion, a black hole could have a verysignificant spin. Thus observations of spin in black holesare closely connected with the merger process that resultsin supermassive black holes.

In general terms, spinning black holes raise issues thatare not present in standard Schwarzschild black holes. Inparticular, the region of space between the ergosphere andthe event horizon can produce a variety of very strangephenomena, some of which are potentially observable.If black holes are used to test and explore the moreextreme predictions of general relativity, examples withwell-determined nonzero spin seem likely to produce themost interesting results.

9DET EC T I N G B L A C K HO L E ST H ROUGH GRA V I T A T I O N A L WA V E S

The vast majority of our information concerning thecosmos arrives in the form of photons (electromagneticradiation). Each photon can be characterized by fournumbers, namely, the energy and arrival time of thephoton, and two numbers characterizing its position onthe sky.1 So, our information about the Universe can beimagined as being contained in a very long table, each entryof which consists of four numbers, as well as the accuracyof each determination.

Viewed in this light, the enormous progress in ob-servational astrophysics over the past half-century can beviewed as advances in the range and resolution of ourdetermination of these numbers. The range in energydetectable has been of particular importance. A centuryago we could observe only optical photons. The radio,X-ray, infrared, ultraviolet, microwave, and gamma-rayranges were successively opened, so that now observatories

1In fact, additional information is carried by the polarization of the radiation,which in some cases is extremely useful. But special observational apparatus isrequired to detect polarization, and most celestial radiation is unpolarized.

Black Holes & Gravitational Waves 151

can detect essentially all photon energies capable of prop-agating through interstellar and intergalactic space. Eachnew wavelength regime revealed a qualitatively differentUniverse. At the same time, new technologies dramaticallyimproved the temporal, spatial, and spectral precision ofour measurements of photons.

While we can expect further improvements in ourability to detect and measure electromagnetic radiation, itis possible that the next great advances in observationalastrophysics will come from the detection of other kindsof information altogether. A number of such nonelectro-magnetic “messengers” are known. Cosmic rays, whichare charged particles (electrons, protons, alpha particles,and others), have been studied for some time. They aredifficult to analyze, however, because the trajectories ofcharged particles are changed by the presence of magneticfields, so the specific origin of a particular cosmic ray isoften hard to determine. Nevertheless, observations of theintensity and energy of cosmic rays has led to insight intohigh-energy cosmic events. Neutrinos from celestial eventshave also been observed, most notably from the Sun andfrom the supernova that was observed in 1987 in the LargeMagellanic Cloud. Neutrino astrophysics is also an activefield, but the very low detectability of neutrinos seemslikely to restrict observations to a few objects that have theright combination of intensity and proximity.

Currently, there is great excitement about the possi-bility of directly detecting an entirely new “celestial mes-senger,” namely, gravitational radiation. The existence ofgravitational waves is a prediction of general relativity, andcurrent technology has put us very close to being able to

152 Black Holes & Gravitational Waves

detect them directly. The strongest sources of gravitationalradiation are expected to be merging black holes. Since weexpect such mergers to occur, both between stellar-massand supermassive black holes, the detection of gravitationalradiation would provide a new way not only to exploregravitational physics but also to look for and to studycelestial black holes.

9.1 Gravitational Waves and Their Effects

General relativity holds that the presence of mass warpsthe space and time around it. When a mass moves, thespace-time distortion moves with it. If the mass changesits motion (that is to say, accelerates), this creates a wigglein the space-time distortion that propagates away from themass at the speed of light. This wiggle takes the form ofa gravitational wave. The generation and propagation ofgravitational waves is in some ways analogous to that ofelectromagnetic waves, which are created by the acceler-ation of a charged particle. However, in the case of anelectromagnetic wave, quantum mechanics describes howthe wave can also be analyzed as a particle (a photon).In the absence of a good theory of quantum gravity,consideration of “gravitons” is more problematic.

One common form of acceleration occurs when themasses in question are in orbit around each other. Forcircular orbits, there is a constant amount of acceleration ina smoothly changing direction, which in turn produces asteady flow of gravitational waves (see figure 9.1). Thesewaves carry energy with them; this energy is extracted

Gravitational Waves & Their Effects 153

Figure 9.1. Gravicational waves propagating outward from black holes in orbit.

from the binary orbit. Thus the existence of gravitational

waves is an intrinsically non-Newtonian process, in that

it reduces the total energy of the orbit, causing the two

objects to spiral inward rather than to remain in a per

manent stable orbit with a constant period, as Newtonian

theory requires. The orbital period will decrease due to

gravitational radiation as

where M\ and M2 are the masses of the binary components

measured in solar masses, and the orbital period P is

measured in seconds. The form of the mass term on the

right-hand side means that systems in which the masses are

roughly equal decay more quickly, while the high negative

power of P means that binary stars with orbits of a few

days or more do not exhibit measurable effects. Another

consequence of this equation is that as the period decreases,


the rate of the period change increases, creating a runawaycondition—that is, an ever-decreasing orbital period thatdecreases more and more quickly. By integrating thechange in P over time, one can compute how long it willtake for the binary period to formally approach zero. Thisin-spiral time T0 represents a maximum time after whichthe two orbiting objects will merge and can be written

T0 = 2.33 × 10−3 (M1 + M2)1/3

M1 M2P 8/3years.

Orbiting objects much larger than their Schwarzschildradius will merge sooner, since the binary separation willstill be large when their surfaces touch. Other ways toextract energy from a binary orbit most notably includetidal effects, which can become very large for objects whoseradius is comparable to the orbital separation. But fororbiting black holes T0 is a good measure of the timeuntil merger. For a binary system containing two objectsof 5 M� in an orbital period of 10 hours—comparable tomany observed X-ray binaries—the merger time is onlyabout 10 million years, which is much less than the ageof the Universe. As we will see, there are observed systemswhose parameters are such that we expect such mergers totake place.

The luminosity of the gravitational wave radiationemitted by a black hole merger event is immense. There isa “natural” gravitational wave luminosity that can be con-structed by combining the fundamental constants G and cin a form with units of luminosity, namely, L0 = c 5/G =3.6 × 1059 erg s−1, which is 26 orders of magnitude

Gravitational Waves & Their Effects 155

greater than that of the Sun. This value is greater than theelectromagnetic luminosity of the entire Universe. L0 isroughly the luminosity of the gravitational wave radiationemitted by a system in which the velocities are stronglyrelativistic (V ≈ c ), and the spatial scales are comparableto the Schwarzschild radius. Thus L0 is an upper limit tothe gravitational wave radiation emitted by any particularevent, since c and Rs limit the velocity and size of anyphysical object, and the luminosity falls off quite rapidlyas V becomes less than c , and R becomes greater thanRs . However, as a black hole binary system merges, thevelocity and scale of the two objects approach Rs and c , sojust before the merger the luminosity in gravitational wavesbecomes very large.

It is interesting to note that this maximum luminosityis not dependent on the mass of the objects, just on theirvelocity and the size relative to the Schwarzschild radius.So, merging supermassive black holes approach the samelimiting luminosity as merging stellar-mass black holes. Atfirst this seems counterintuitive—surely the more massivesystems should emit more energy than the lower-masssystems. But it must be remembered that the timescalesfor the more massive systems are greater than those for thelower-mass systems. The orbital period just before mergerscales with the Schwarzschild radii and thus with the massof the black holes. Thus more massive systems emit at highL for longer than the lower-mass systems do, so the totalenergy emitted is correspondingly greater, even though theinstantaneous maximum luminosity is the same.

By the same token, the frequency of the gravitationalradiation depends on the mass of the merging objects.


The frequency of the radiation when L approaches itsmaximum is related to the inverse of the orbital period atthe last stable orbit. Since the orbital velocity is the samefraction of the speed of light at this point, the period islinearly proportional to the circumference of the orbit andthus to Rs and to the mass of the black hole. For mergingstellar-mass black holes, frequencies are in the kilohertzrange. Similar phenomena are expected, for supermassiveblack holes but with frequencies at or below a millihertz.Thus the best range for seeking these kinds of events isseveral orders of magnitude on each side of 1 Hz.

9.2 Binary Pulsars

While gravitational waves have not yet been observeddirectly, one class of systems in which their consequencescan be readily observed is the binary pulsars. Pulsars aremagnetized spinning neutron stars. When the magneticpole of the pulsar points toward the observer (once perrotation), a “pulse” of radiation is observed. These pulseshave been observed and timed to great accuracy by radioobservatories since the late 1960s, and more recently inother wavelengths as well. The short periods of the pulsars(typically a few seconds or less) indicate that the rotationalperiods and thus the size of the objects themselves must besmall, much smaller than the categories of stars that wereknown at the time. Indeed, pulsars were the first directevidence for the existence of neutron stars.

In the 1970s, several binary pulsars were discoveredwhose changing velocity curves determined from the

Binary Pulsars 157

Doppler shifts of the pulsation period clearly indicated thatthe pulsar was in a binary orbit of a few tens of hours withanother neutron star. The tiny radius of the neutron starsmeans that tidal effects are negligible, so the relativisticeffects on the orbits could readily be seen. In particular,the decay of the orbit due to gravitational radiation isclearly apparent in several systems. The discoverers ofthe first binary pulsar, Russell Hulse and Joseph Taylor,were awarded the Nobel Prize in physics for this thrillingdemonstration of what had previously been a theoreticalconcept.2

Given the high precision of pulsar measurements, anumber of relativistic effects could be observed, includingthe precession of the orbital periastron (whose presencein the orbit of Mercury was one of the first empiri-cal confirmations of general relativity), the gravitationalredshift due to the changing distance between the twosources, and the expected decrease in the orbital period dueto gravitational radiation. These observations completelyconstrained the parameters of the orbit and the mass ofthe orbiting objects, which could be determined to greatprecision, and precisely predicted second-order relativisticeffects such as the Shapiro time delay. These second-ordereffects have since been observed and are measured to beconsistent with the predictions of general relativity. Theseobservations of the orbital motion of the binary pulsarsnow provide severe constraints on possible non-Einsteinian

2Hulse and Taylor’s accounts of their discovery, recounted in theirNobel Prize lectures (which can be found on the Nobel Prize sitehttp://www.nobelprizes.org), provide a wonderful description of the interplaybetween technical expertise, deep physical insight, and good fortune in physics.


gravitational theories. But while these systems provide verypowerful indirect evidence for the existence of gravitationalwave radiation, a yet more stringent test would be thedirect detection of gravitational waves, in the manner thatelectromagnetic radiation is detected by devices rangingfrom the human eye to photographic film to digitizedsilicon devices.

9.3 Direct Detection of Gravity Waves

Attempts to detect gravitational waves directly have a longand somewhat checkered history. The basic approach is toobserve small changes in space-time caused by the passageof a gravitational wave by observing the minute resultingchanges in size of an object as the wave passes through it,which requires extraordinarily precise size measurements.For most expected astrophysical sources of gravitationalwaves, the precision must be �L/L ≈ 10−22 or better:that is, the expected change in size or separation will beonly 10−22 of the size or separation of the objects beingmeasured. Given that the size scale of an atomic nucleusis a few femtometers (1 fm = 10−15 m) it is remarkablethat such experiments can even be contemplated—for a1 m object, the measurement precision would have to bebetter than a millionth of a single nucleus!

The first attempts at detecting gravitational waves usedlarge bars of aluminum. The effect of gravitational wavesat the resonance frequency of the bars would be amplified,thus allowing measurements of a change in length. Startingin the late 1960s, Joseph Weber claimed positive results

Direct Detection of Gravity Waves 159

using this method. However, the results could not be du-plicated in other laboratories, and in any case the claimedamplitude of the observed effects was much larger than theexpected size of the astrophysical events thought to causethem. These results are therefore generally discounted.

Elaborate efforts are currently under way to detect grav-itational waves using the technique of laser interferometry,in which light emitted by a laser is combined with lightfrom that same laser that has traveled down a differentpath. The basic design is that the laser emission is sentdown two different paths at right angles, then reflectedand recombined.3 Depending on the precise distancesof the paths, the peaks and troughs of the light waveseither combine constructively, and the light is amplified,or destructively, in which case the light is dimmed. If thelength of one of the arms changes even slightly due toa gravitational wave passing through the apparatus, thestrength of the recombined signal will change significantly.Changes of length of a small fraction of a wavelength ofthe light can be detected by monitoring the strength of thecombined light beam.

The frequency range of the gravitational waves observ-able from such an interferometer is related to the inverseof the travel time of the beam. So, for deca- and kilohertzfrequencies, as expected from merging stellar mass blackholes, the light travel time should be a fraction of asecond, and thus the length of the light path would ideallybe hundreds or thousands of kilometers. For mergers of

3This is basically the same arrangement as the famous Michelson-Morleyexperiment, which demonstrated that the speed of light is the same regardless ofthe motion of the light source and detector.


SMBHs, timescales of hundreds or thousands of secondswould be desirable, and thus beam lengths of millions ofkilometers would be required.

This approach is now being implemented in the LaserInterferometer Gravitational Wave Observatory (LIGO),which is an experiment being carried out at two sites, onein Washington state and one in Louisiana. At both sites,a laser shines in two directions down a several-kilometers-long track to thermally and vibrationally isolated mirrors(see figure 9.2). Tiny changes in length in the two di-rections are recorded. Having two widely separated sitesprovides confirmation that any observed signal is celestialin origin, and any time delay between the effects at the twosites provides information on the direction to the source ofthe radiation, since gravity waves must propagate at thespeed of light.

LIGO is a very delicate experiment, and a large teamof scientists is working to reduce all the sources of noisein the system to levels where expected astrophysical sig-nals could be detected. Key limitations on the precisionof the measurements include seismic vibrations, thermalvibrations in the mirror and laser apparatus, and the abilityto measure tiny changes in the strength of the combinedlight beam. LIGO is currently sensitive to displacementsnear 10−18 m. Given the kilometer size of the experiment,sensitivity is thus near �L/L = 10−21, tantalizingly closeto gravitational wave amplitudes expected from astronom-ical sources, but no detections have yet been confirmed.Work is currently underway on Advanced LIGO, whichshould improve the sensitivity by more than a factor of 10,allowing frequent detections.

Direct Detection of Gravity Waves

Be,m Splitter

Photodetector

light Bounces Many TImes between the Mirrors on Each Arm

Re<:yding MirrOf

161

Mirror

Figure 9.2. Schematic of LIGO setup. Laser light is sent down two

long corridors and reflected to the start. When recombined, the

interference patterns can reveal changes in the relative length

traveled of a smal l fraction of the wavelength of the light. Extra

precision is obtained by repeated reflections using the recycling

mirror. After diagram from UGO website (www.ligo.caltech.edu) .

Ground-based interferometers are limited by the vibra

tions fro m the Earth due ro human-caused seismic activity.

These limitations apply particularly strongly to frequencies

ofless than 1 H z, so low-frequency gravitational waves, like

those from merging supermassive black holes, are unlikely

ever to be detectable fro m ground-based experiments. This


Sun

Earth

RelativeOrbit ofSpacecraft20°

60°

5 × 106 km

1 AU

Venus Mercury

Figure 9.3. Planned orbits of the three LISA satellites. AfterFigure 5 from P. Aufmuth and K. Danzmann, 2005, New Journalof Physics, 7: 202.

drawback has prompted interest in creating a space-basedgravitational wave observatory. In particular, a projectcalled LISA (the Laser Interferometer Space Antenna)has been developed by a collaboration between NASAand the European Space Agency and is currently beingreviewed prior to authorization (see figure 9.3). LISA willfly three spacecraft in an equilateral triangle several millionkilometers on a side. Inside each spacecraft will be a free-flying, and thus vibrationally shielded, set of mirrors anda laser. Each satellite will constantly monitor distancesto both of the others. The large distances and carefulshielding of the mirror and laser apparatuses from all noisesources should allow for great precision and sensitivity atthe relatively low frequencies associated with mergers ofsupermassive black holes. Some technology developmentis still required before this mission can be flown, but theprimary limitation now appears to be financial—if full

Detecting Astrophysical Signals 163

Dis

tort

ion

Time

Figure 9.4. Intensity as a function of time for a chirp. The charac-teristic of a chirp is that both the frequency and the amplitude ofthe signal increase exponentially with time.

funding were available, estimates are that LISA could flywithin a decade or so.

9.4 Detecting Astrophysical Signals

As the binary spirals together ever more rapidly the grav-itational radiation takes the form of a chirp, which isthe term used for an oscillating signal whose frequencyand amplitude both increase exponentially (see figure 9.4).Just before the merger, the gravitational waves reach amaximum strength, with a period equal to the orbitalperiod just prior to the merger. In the case of neutronstars, which have a radius only somewhat bigger than theirSchwarzschild radius, this frequency is of order ≈103 Hz,corresponding to an orbital period of about a millisecond.


While these high-frequency gravitational waves occur foronly a fraction of a second, the amplitude is so highthat such an event would be observable by a device likeAdvanced LIGO over hundreds of megaparsecs. Our owngalaxy is known to contain a few binary pulsars, eachof which has a decay timescale of a few million years,suggesting that such an event will be generated everymillion years or so. But since the events may be observablein millions of galaxies, we would expect to see one at leastonce per year. Thus known systems will necessarily gen-erate observable gravitational wave events quite regularly.It is this expectation that justifies the construction of thecurrent generation of gravitational wave observatories.

While neutron star mergers generate strong gravita-tional radiation signatures, mergers of black holes are evenstronger. Two black holes in orbit release gravitationalradiation, and the chirp continues to rise in both frequencyand amplitude until the event horizons are in contact. Atthat point the event horizons merge, creating a single blackhole, and a last intense burst of gravitational wave radiationcarries off energy and angular momentum until the mergerremnant settles into the configuration of a single Kerrblack hole. It seems likely that stellar-mass systems resultin mergers between black holes or between black holes andneutron stars, in addition to neutron star systems like thebinary pulsars. But the clear existence of double neutronstar systems that must evolve to mergers demonstratesthat events with frequencies of ≈103 Hz almost certainlyoccur.

But supermassive black holes can also merge—andindeed they must if current ideas about the evolution of

Detecting Astrophysical Signals 165

galaxies and the SMBH at their center are correct. Asnoted in chapter 6, SMBHs can grow through repeatedmergers of smaller black holes. These mergers result inhuge bursts of gravitational radiation, but with muchlower frequencies than mergers of stellar-mass objects. Thefrequency of gravitational waves from a black hole mergerare comparable to the orbital frequency near the eventhorizon. Observations of such events will be a direct test ofhierarchical galaxy formation models and thus importantto cosmology as well as to relativistic astrophysics. But asdiscussed earlier while these frequencies are accessible fromplanned space missions like LISA, they are unlikely to beobserved from the ground.

Observation of any kind of black hole merger eventrequires that the gravitational merger signal—the chirp—be extracted from data containing sources of noise muchlarger than the signal itself. Fortunately, the repetitivenature of the wave signal as it passes over the detector willallow a very faint signal to be extracted, and sophisticatedtiming analysis allows signals with a known time signatureto be identified with high precision. In the early stages ofthe merger, a classic chirp is expected, whose functionalform is easy to define. However when the two eventhorizons approach each other, the pattern becomes morecomplex. But this is the moment when the signal isstrongest, so it is of great importance to be able toaccurately predict the expected pattern of the incomingradiation, which can be done by careful computationalmodeling of the space-time associated with the merger.The field of numerical relativity has advanced greatly inrecent years, allowing ever more accurate computations


of the expected wave patterns. These patterns depend onthe initial configuration of the merging black holes—theirorbit, their relative masses, and the strength and directionof the black hole spins. Thus libraries of expected patternsare being computed for comparison with the data that willeventually be obtained.

10BLACK HO L E E XO T I C A

Black holes are a staple of science fiction, where theireffects are used as plot devices in a variety of ways. Butthe effects evoked by science fiction writers are generallynot those observed by astronomers in the ways that aredescribed in this book. Rather, they are extrapolationsfrom the predicted behavior of black holes that have notyet been observed. But in principle these exotic behaviorsare observable, and as science fiction correctly points out,they could have dramatic effects on people’s lives underthe right circumstances. In this final chapter we exploresome of these predicted effects and the possibility that theymight someday be explored in fact as well as in fiction.

10.1 Hawking Radiation

Relativity is in some senses a very classical theory. It isdeterministic; that is, if the initial conditions of a systemare precisely known, its further behavior can in principle beprecisely predicted. It is also continuous, in that the theoryremains intact if one extrapolates distances to zero. Relativ-ity shares these characteristics with Newtonian theory, but

168 Black Hole Exotica

the situation is very different in quantum mechanics. It isthese differences that make a theory of quantum gravity sodifficult (thus far impossible) to construct. However, onesituation in which the interaction between quantum me-chanics and relativity has been explored with some successinvolves Hawking radiation, a process through which blackholes are expected to emit energy and ultimately evaporate.

Such a process runs counter to our basic ideas aboutblack holes. Black holes, after all, are supposed to involveone-way processes, in which an object gains mass andenergy but cannot lose them or emit anything. But space-time as derived from a metric is a continuous constructand by its nature does not allow for quantum effects. Nev-ertheless, those quantum effects must occur, particularlyin the immediate vicinity of an event horizon. The eventhorizon around a Schwarzschild black hole, for example,is an infinitely thin spherical surface. But “infinitely thin”is a phrase that leaps out as demonstrating the need for adeeper theory. In particular, phenomena that occur so closeto an event horizon that quantum uncertainties extendacross the horizon clearly cannot be adequately describedby relativity theory alone.

One important quantum phenomenon is the produc-tion of virtual pairs in a vacuum. In this process, aparticle and its antiparticle are spontaneously produced butrecombine quickly enough to conserve mass and energywithin the quantum uncertainties. Predictions of particleinteractions carried out using Feynman diagrams and otheranalytic tools require the creation of virtual pairs to explaina variety of observed phenomena. Hawking radiation ariseswhen a virtual pair is created in the near vicinity of an

Hawking Radiation 169

event horizon. If one of the newly created particles crossesthe event horizon, it cannot reemerge to recombine with itspartner. The remaining particle can then travel away fromthe event horizon. When viewed from a distance, the blackhole appears to be emitting a stream of particles. Theseparticles carry off mass and energy from the black hole,which eventually evaporates.

One way to think about Hawking radiation is throughthe simple approximation that a black hole emits black-body radiation with the peak wavelength of order thesize of the Schwarzschild radius. This heuristic approachleads to results that reveal the basic features of Hawkingradiation, and their implications for the observability andeffects of this phenomenon.

The wavelength of maximum emission of a blackbodycan be determined by differentiating the blackbody spec-trum as a function of wavelength λ and setting the resultequal to zero. The result is an expression for λmax as afunction of blackbody temperature T: λmax = hc/CkT,where C is a numerical constant defined by Ce C/(e C −1) = 5, or C = 4.965.... In dimensional units

λmaxT ≈ 2.9 × 10−3

where λmax is measured in meters and T in kelvins. Settingλmax equal to the Schwarzschild radius than gives T ≈2.9 × 10−3c 2/2GM in standard mks units.

Given this temperature, we can determine the black-body luminosity of the black hole using the standardformula L = σ4π R2T4, where the relevant radius is againthe Schwarzschild radius. Since Rs ∝ M, and T ∝ 1/M,


we find L ∝ M−2. Thus low-mass black holes will radiatefar more than massive black holes.

The energy that generates this luminosity is extractedfrom the mass of the black hole, which can be convertedinto energy using the familiar expression E = Mc 2. ThusHawking radiation reduces the mass of the black hole,which makes it radiate more intensely. So a low-massblack hole will undergo a runaway and proceed rapidlyto complete evaporation. Since luminosity is defined byL = d E/dt , the time to evaporation tevap will be

tevap =∫ 0

Mc 2(1/L)d E ∝ M3.

More sophisticated calculations than the simple modelpresented here show that

tevap = 5120πG2

�c 4M3,

or in dimensional units

tevap = 2 × 1067(M/M�)3years,

where M� is the mass of the Sun. Thus stellar-massblack holes will take far longer to evaporate than the ageof the Universe (currently estimated to be 1.4 × 1010

years), and SMBHs will take orders of magnitude longerthan that. These enormous timescales show how tiny theeffects of Hawking radiation would be on black holes ofthe kinds that we know exist—indeed, the accretion of asingle optical photon every few years would counteract theeffects of Hawking radiation for a stellar-mass black hole

Primordial Black Holes 171

altogether. Thus Hawking radiation remains a theoreticalprediction, not yet observable in any known situation.

We can also calculate the mass for which a black holewould evaporate over the current age of the universe, whichproves to be Mevap ≈ 2×1011 kg, or much less than stellar-mass or supermassive black holes. Since the evaporationtime scales with a high power of M, most black holes withM < Mevap will have evaporated, while black holes withM > Mevap will still exist.

Another consequence of the strong dependence ofHawking radiation on M is that the final stages of blackhole evaporation will proceed very rapidly. Thus one mightexpect to see a burst of radiation associated with theevaporation of a black hole, and if a class of primordialblack holes with M ≈ Mevap existed, one might expectto see occasional bursts of radiation associated with thedemise of these.

10.2 Primordial Black Holes

The preceding formulas show that there is a sharp cutoffbetween black holes with a mass of less than ≈1011 kg,which would evaporate over the course of the universe,and those with a higher mass, which would still exist. Thestrong dependence on mass means that black holes forwhich Hawking radiation is significant will radiate at in-creasing rates and proceed exponentially toward completeevaporation, while black holes a few orders of magnitudemore massive will show little effect over cosmic time.Therefore the effects of Hawking radiation are relevant


only for black holes with M < 1011 kg ≈ 10−19 M�.Standard evolution of stars and galaxies cannot lead toblack holes this small, since pressure forces and chemicalbonds will more than suffice to hold up objects of therequired low mass against gravity. However, it is possiblethat such small black holes could have been created in theearly Universe.

In its earliest stages, the Universe was much denserand hotter than it is now. Tiny fluctuations might thenhave given rise to objects that collapsed inside their eventhorizon. The density and temperature of the plasma isso great that ordinary matter is replaced by more exoticparticles and antiparticles, which may not be able to holdsuch objects up against collapse. A number of specificmoments in cosmic history have been suggested when suchprimordial black holes might have been created, and whatmasses they might have. Such hypotheses are hard to testobservationally, since low-mass black holes will have longsince evaporated, and isolated high-mass black holes arehard to detect.

However, black holes with evaporation times compara-ble to the age of the Universe might be readily observable.If such black holes were created in the early Universe, theymight be completing their evaporation right now. SinceHawking radiation is a runaway effect, increasing in inten-sity as the black hole becomes less massive, an evaporationevent will end in a sudden large increase in radiation,whose intensity might briefly be very high and thus observ-able. This effect was one of the more exotic explanations ofthe sudden γ -ray burst events that are observed through-out the Universe, but there is now much more compelling

Primordial Black Holes 173

evidence that these are associated with particular kinds ofcollapsing stars.1 Currently, astrophysicists are taking a dif-ferent approach, using the absence of observable Hawkingradiation events to set limits on the number of primordialblack holes that could have formed in the early Universe,which in turn provides information on physical processesassociated with the first moments of the Big Bang.

Some limits can be set on the formation of black holes atother masses. The radiation generated by the evaporationof low-mass black holes at much earlier times in cosmichistory might have affected observable phenomena likethe cosmic microwave background. No such effects areobserved, and that limits the number of primordial blackholes that could have been evaporating at some stages ofthe evolution of the Universe. Gravitational microlensingobservations limit the number of more massive black holeswhich currently exist. It is clear that in most mass rangesthere are too few primordial black holes to account forthe observed dark matter. Nevertheless, direct clues to thepossible creation, existence, and evaporation of primordialblack holes are hard to come by, and significant popula-tions of these black holes could be hidden in the Universe.

One possible way to explore the creation of primordialblack holes is through particle physics experiments thatreproduce (very briefly) the conditions that were present inthe early Universe. The possibility that black holes mightbe produced in particle accelerators has received consid-erable press attention—given the popular impression ofblack holes as objects that can swallow up everything

1See J. Bloom, What Are Gamma-Ray Bursts? (Princeton, NJ: PrincetonUniversity Press, 2011).


nearby, one can understand why there might be someconcern! Fortunately however, this concern is misplaced.Events as energetic as those in any particle acceleratoroccur frequently at the top of the atmosphere, whenenergetic cosmic rays encounter atoms in the ionosphere.This process has been going on for billions of years, andobviously the Earth has not yet been swallowed by ablack hole. Furthermore, any black hole that could beformed would be of very low mass and thus would havehigh Hawking radiation. Very small black holes wouldevaporate essentially immediately and be transformed intostreams of particles similar to those that created them. Forexample, a 1 kg black hole would evaporate in less than10−16 second. Furthermore, any black hole in which theHawking radiation is greater than the Eddington limitwould not be able to accrete mass. Since the Hawking radi-ation of a low-mass black hole is high, while the Eddingtonlimit is low, low-mass black holes of the kind that mightbe created in particle experiments cannot accrete under anycircumstances.

10.3 Wormholes

One of the most enticing possible effects associated withblack holes is that they might form wormholes throughwhich widely separated parts of the Universe can be closelyconnected. This is perhaps the most common use of blackholes in science fiction, since it provides an escape fromthe central difficulty of space opera, namely, that that lighttravel times between most celestial objects are vastly greater

Wormholes 175

than a human lifetime.2 The basic idea is that two singu-larities at different points of space-time might be joinedby a “bridge” or “throat” which takes a different paththrough space-time than would ordinarily be followed. Ifthis connection is shorter than the ordinary space-timetrajectory, then mass and energy might appear to travelfrom one black hole to the other at faster-than-light speeds.Alternatively, it is possible that the connection might allowchanges in the time coordinate, creating a time machine.

To use the wormhole as a practical means of transporta-tion, matter and energy would have to be able to enter oneof the black holes and emerge from the other. This wouldrequire that the singularity not be surrounded by an eventhorizon but, rather, that it be a “naked” singularity. Inprinciple, such naked singularities can exist, but in general,they are not stable. To hold the black hole open requiresmatter with negative energy density, for which there is nocurrent evidence. Even if such exotic matter did exist, thereis no conceivable sequence of events that would lead to thecreation of a wormhole. Penrose has proposed the “cosmiccensorship” theory, which essentially says that singularitiesare not permitted to be naked. Extensive computationalsimulations of the gravitational collapse of various con-figurations of ordinary and exotic matter have producedambiguous outcomes, but no convincing demonstration ofa real situation in which a naked singularity would result.

2The invention of the concept of wormholes provides an especially intimateconnection between science fiction and science fact. In his book Black Holesand Time Warps: Einstein’s Outrageous Legacy (New York: Norton, 1994), KipThorne provides a charming account of how some of the key ideas behindwormholes were developed at the behest of Carl Sagan for use in Sagan’s novelContact.


If a wormhole did exist, what might it look like? Herethe science fiction representations (e.g., Star Trek) departcompletely from reality. The manifestation of a wormholewould presumably be that interstellar matter accreted byone black hole would appear to emanate from another.Thus one end of the wormhole would be a “white hole” inwhich matter and energy was being extruded. This wouldlook quite different from a black hole emitting Hawkingradiation, since the rate at which mass/energy was emittedwould depend on how much was being accreted at the farend of the wormhole, rather than the mass of the blackhole itself. Given the wide range of radiating astrophysicalobjects, it would be difficult to establish that a particularobserved object was in fact a wormhole. Many other seem-ingly much more plausible explanations would need to beeliminated. It might be even harder to identify the locationof the other end of the wormhole, since if these objectsexist at all, there is no reason for their two ends to haveany particular relationship to each other. Nevertheless,the possibility of wormholes being discovered and perhapsbeing technologically exploited by a sufficiently advancedcivilization remains, even though there is currently notheoretical demonstration that they can form and noempirical evidence that suggests that they exist.

10.4 Multiverses

One final suggestion that might be contemplated is thata separate universe, perhaps with natural laws differentfrom our own, might exist inside the event horizon of a

Multiverses 177

black hole. This is one version of the multiverse concept,in which a variety of universes with a variety of charac-teristics exist, of which our Universe is only a particularexample. Much of the attraction of this concept lies inhow it explains the peculiar friendliness of our Universeto complexity and thus to life. As noted in chapter 5,most values of the total energy of the Universe result ineither a single black hole or a dispersed lifeless void, butour Universe lives in the infinitesimally small range thatallows for many individual objects to exist. Many othernatural laws and values of constants also appear to requiresuch fine tuning for life to exist. One way to address theseseeming coincidences is to postulate that many universeswith all sorts of laws exist, and that most of them are infact quite simple, but that we necessarily exist in a Universethat allows complexity.

One version of this idea that has a particularly closelink to black holes is described by Lee Smolin in hisbook The Life of the Cosmos.3 Smolin postulates that thecomplexity and fine tuning we see in our physical universeis created in much the same way that complexity arisesin biological systems, namely, through natural selection.If a new universe is birthed within each black hole, thena universe that creates lots of black holes will create morenumerous progeny and thus be “fitter” in the evolutionarysense than a universe that produces zero black holes orone black hole. If each child universe has characteristicssimilar to its parent, then universes that produce the largestpossible number of black holes will soon be the dominant

3L. Smolin (Oxford: Oxford University Press, 1997).


species. Smolin suggests that this may be testable: if onesimulates a large number of possible universes, it shouldturn out that ones that actually exist are among the onesthat produce the most black holes.

There are some problems with this formulation. Inparticular, it isn’t clear what rules would be used togenerate the small but nonzero changes between parentand child universes that are required for the evolutionof the multiverse. Nor is it clear that our understandingof cosmic processes is sufficient to accurately determinehow many black holes a universe with a particular set ofcharacteristics may have. Nevertheless, it is fascinating toimagine that the existence of a universe conducive to lifemight be due to the existence of black holes, and that atleast one example of the resulting life-forms might beginto explore and understand how that happened.

G LO S S A R Y

Accretion Disk (2.2, 2.7): A thin rotating disk of gas surroundinga compact object. Viscous forces within the disk lead to gradualaccretion of the gas onto the compact object.

Accretion Disk Corona (4.3): Hot optically thin gas above anaccretion disk. It generates significant X-ray emission in someX-ray states of X-ray binaries.

Active Galactic Nuclei (chapter 5): Accreting supermassive blackholes in the center of a galaxy.

ADAF (2.3): Advection Dominated Accretion Flow. One kindof radiatively inefficient accretion flow, in which most of theenergy in the accretion flow is advected onto the central object.The flow is generally more radial than in an accretion disk.

AGN (see Active Galactic Nuclei)

α-Disk (2.7): A simple accretion disk model originally developedby Shakura and Sunyaev in which the viscosity is parameterizedby a constant α multiplied by the height of the accretion diskand the sound speed.

Anthropic Principle (6.0): The idea that human existence neces-sitates that the Universe satisfy very particular conditions.

Binary Pulsar (9.2): A rotating magnetized neutron star in abinary orbit with another neutron star. Binary pulsars show clear

180 Glossary

but indirect evidence for gravitational wave radiation and areused as tests of general relativity.

Blazar (5.3): AGN in which a relativistic jet is directed towardthe observe. The Doppler boosting of the jet results in unusuallybright, energetic, and variable jet emission.

Bondi-Hoyle Accretion (2.1): Spherically symmetric accretion.It is generally not realized in physical situations, since angularmomentum is generally significant in accretion flows.

Boundary Layer (4.6): Material accreted onto the surface of awhite dwarf or neutron star. Radiation from a boundary layershould not be observed from an accreting black hole.

Bremsstrahlung (2.5): Radiation emitted when electrons aredeflected by the electromagnetic charge of some other particles.

Cataclysmic Variable (4.4): A generic term applied to an accret-ing white dwarf.

Chandrasekhar Limit (4.4, 4.8): The maximum mass of a whitedwarf. If the Chandrasekhar limit is exceeded, the star cannot besupported by Fermi pressure of electrons.

Chirp (9.4): A signal whose frequency and amplitude increaseexponentially with time. The gravitational waves emitted bymerging black holes are expected to be in the form of a chirp.

Compton Scattering (2.5): Emission mechanism due to interac-tion of electrons and photons, resulting in dramatic changes tothe observed photon spectrum.

Eddington Limit (2.1): An upper limit on the accretion rateimposed by radiation pressure from the accretion luminosity.

Ellipsoidal Variation (4.5): An orbital change in the flux observedfrom the companion star of an X-ray binary due to the nonspher-ical shape of the star.

Glossary 181

Equation of State (4.8): The relationship between density andpressure in a gas.

Ergosphere (1.2, 8): The region between the event horizon andthe Schwarzschild radius of a rotating black hole.

Escape Velocity (1.1): The speed required to escape the gravita-tional attraction of a spherical object: Vesc = √

2G M/R .

Event Horizon (1.1): The boundary beyond which informationand radiation cannot travel to a distant observer. For a nonro-tating black hole, the event horizon occurs at the Schwarzschildradius.

Fermi Pressure (4.8): Pressure that arises because two particles(fermions) are prohibited from occupying the same volumesimultaneously.

Globular Cluster (6.1.3): A dense cluster of stars. X-ray binariesare unusually prevalent in globular clusters, presumably due todynamic interaction between compact objects and binaries andother stars. Globular clusters are sometimes thought to harborintermediate mass black holes.

Gravitational Redshift (1.2, 8.2): The change in wavelengthof an observed photon relative to its rest wavelength: z =1/

√1 = Rs /r − 1.

Gravitational Microlensing (4.7): The bending of light rays froma visible star by an unseen star in the line of sight, resulting in amagnification of the visible star. Used to detect isolated stellar-mass black holes.

Gravitational Wave Radiation (chapter 9): Radiation emitted bythe acceleration of masses.

Hard Binary (6.1.3): A binary star whose binding energy isgreater than the kinetic energy of an incoming third object.

182 Glossary

Interactions between hard binaries and other objects tend toharden the binary further.

Hawking Radiation (10.1): Emission from a black hole due toquantum mechanical effects.

Hydrostatic Equilibrium (4.8): The balance of gravitational forcesinward with pressure forces outward.

Innermost Stable Circular Orbit (ISCO) (7.1): The closest closedorbit to a black hole. The distance from the event horizondepends on the spin of the black hole, and thus the location ofthe ISCO is used to measure black hole spin.

Intermediate-Mass Black Hole (chapter 7): A black hole moremassive than a star, but less massive than a supermassive blackhole. The existence of IMBHs is disputed.

Laser Interferometry (9.3): A method for very precise distancemeasurements used in attempts to directly observe gravitationalwaves.

Line Emission (2.5): Emitted radiation that is concentrated in asmall range of wavelengths.

M-σ Relation (5.5): An observed correlation between the mass ofa supermassive black hole at the center of galaxy and the mass ofthe stars in the central portion of the galaxy.

Magnetic Braking (6.1.2): A mechanism for loss of angularmomentum from a rotating or orbiting body, in which theangular momentum is transferred to corotating material throughmagnetic fields.

Magnetohydrodynamics (3.2): The study of fluid flows in mag-netic fields.

Magnetic Reynolds Number (3.2): A nondimensional numberreflecting the importance of fluid flow compared with magneticdissipation: Rm = V L/η.

Glossary 183

Mass Function (4.5): A quantity with units of mass computablefrom the observed radial velocity curve of binary stars. The massof the unobserved component of the binary system must begreater than or equal to the mass function. The large value ofmass function of some X-ray binaries provides evidence that thecompact objects cannot be neutron stars and are most likelyblack holes.

Mass Segregation (6.1.3): The tendency for more massive bod-ies to fall to the center of a star cluster over a relaxation time orlonger.

Metric (1.2): A line element that defines the separation betweenspace-time events. The Schwarzschild metric is used for non-rotating black holes, and the Kerr metric for rotating blackholes.

Multiverse (10.4): The concept that there may be multipleuniverses, each of which may have different laws of physics.

Neutron Star (4.4): A star made primarily of neutrons, supportedby Fermi pressure of the neutrons.

Numerical Relativity (9.4). Techniques for computer simulationsof relativistic situations; actively pursued in exploring the ex-pected signals from gravitational waves from black hole mergerevents.

Parallax (8.3): A method of determining distance based on theslight changes in an object’s apparent location in the sky due tothe motion of the Earth around the Sun.

Penrose Process (1.2, 8.4): The process by which energy can beextracted from the spin of a black hole.

Quasar (5.1): A very bright AGN: sometimes used to denotebright AGN detected in radio wavelengths.

184 Glossary

QSO (5.1): Quasi-Stellar Object. A very bright AGN sometimesused to denote bright AGN detected in optical wavelengths.

Radiative Transfer (2.6): The process of transmission of radi-ation through gas or plasma, which may affect the observedspectrum of the radiation.

Radio Galaxies (5.2): A class of galaxies that emit unusually largeamounts of radio emission.

Relaxation Time (5.5): The time required for a cluster of objectsto exchange significant amounts of energy through gravitationalinteractions.

Reverberation Mapping (5.5): A method of determining themass of an SMBH by comparing the delay between luminositychanges in the accretion disk and the broad-line region with theluminosity of the AGN.

Schwarzschild Radius (1.1): The radius of the event horizon ofa nonrotating black hole: Rs = 2G M/c 2.

Singularity (1.1): The point or ring at the center of a black holewhere the density of matter is formally infinite.

Sgr A* (5.4): The radio source at the center of the Milky Waygalaxy. Measurements of the motions of stars near Sgr A* havedemonstrated that the galactic center harbors a black hole with amass about 3 million times that of the Sun.

SMBH (chapter 5): Supermassive Black Hole. A black hole witha mass of 106 M� or greater generally found in the center ofAGN and other large galaxies.

Superluminal Jets (3.1): Jets emitted by AGN (and occasionallyX-ray binaries) that appear to propagate faster than the speed oflight. This is an optical illusion created by the effects of specialrelativity.

Glossary 185

Supernova (6.1): An explosion caused by the collapse of thecenter of a massive or accreting star.

Synchrotron Emission (2.5): Radiation emitted due to accelera-tion of electrons by magnetic fields.

Uhuru Satellite (4.1): The first orbiting X-ray observatory,launched in 1970.

Ultraluminous X-Ray Binaries (ULXs) (7.1): Extragalactic X-raybinaries that radiate at significantly greater than the Eddingtonlimit for an object of a few tens of solar mass.

Unification (5.2): The idea that many different observable phe-nomena in AGN are due to the effects of orientation to theobserver.

White Dwarf (4.4): A star held up by Fermi pressure of theelectrons. The final stage of stellar evolution for stars with a finalmass of less than the Chandrasekhar limit.

Wormhole (10.3): A pair of black holes joined by several differ-ent paths through space-time. To be stable, the throat of thewormhole must be surrounded by matter with negative energydensity. There is currently no empirical evidence for wormholesor for matter with negative energy density.

Velocity Curve (4.5): A plot of radial velocity against time foran orbiting star. The period and amplitude of the velocity curvehelp determine the mass of the compact objects in X-ray binaries.

X-Ray Binaries (4.2): Binary stars in which accretion onto acompact object generates large numbers of observable X-rays.High-mass and low-mass X-ray binaries refer to the mass of themass-losing star relative to that of the accreting compact object.

X-Ray Burst (4.6): A brief X-ray flare observed in X-ray binariesthat occurs explosively in a fraction of a second. It is thought to

186 Glossary

arise from thermonuclear ignition of material accreted onto thesurface of a neutron star. X-ray bursts should therefore not beobserved from black holes, and indeed they are not.

X-Ray States (4.3): Different configurations of spectra and tim-ing characteristics of X-ray binaries.

I N D E X

absorption, 20, 33–34, 70, 146accretion disk corona (ADC),

62–63, 140accretion disks: α-disks and,

22–23, 26, 35–42, 144, 146;emission spectra and, 28;innermost stable circular orbit(ISCO) and, 136–47;intermediate-mass black holesand, 130; jets and, 43–44, 48,50; spin and, 136–44, 146;standard, 17–24, 34;stellar-mass black holes and,61–62, 75; superluminalblack holes and, 91–94;supermassive black holes and,92–94, 96, 97, 101;temperature of, 29

accretor, 14, 18, 25, 45active galactic nuclei (AGN):

accretion and, 44, 87–88;blazars and, 94–97; BL Lacobjects and, 90, 97;Eddington limit and, 121–22;feedback processes of, 95;

formation and, 120–25,128–29, 132; host galaxycharacteristics and, 90;International UltravioletExplorer satellite and, 87–88;luminosity and, 84, 125;narrow-line region and, 89,93, 96; nonaccreting centralblack holes and, 98–99;optically violently variable(OVV) sources and, 97;optical spectra and, 89–90;quasars and, 44, 84–90; radiointensity and, 88; redshiftand, 105, 141, 145;reverberation mapping and,102–4; seed black holes and,120–21; spin and, 141, 145,147; superluminal jets and,45–47, 94–97; supermassiveblack holes and, 84, 87–105,121–26; time variability and,90–94; ultraluminous X-raysources (ULXs) and, 129;water masers and, 101–2

188 Index

advection dominated accretionflow (ADAF), 23–24, 26, 62,75–77, 99

α-disks: accretion and, 22–23,26, 35–42; spin and, 144, 146

alpha particles, 151angular momentum: accretion

and, 18–20, 22, 36–37, 39;conservation laws and, 37–39,48, 58, 136; conservativemass transfer and, 59–60,115; event horizons and, 10;formation and, 114–15, 122;gravitational waves and, 164;jets and, 48; Schwarzschildradius and, 10; spin and,135–38, 149; stellar-massblack holes and, 58–60;supermassive black holes and,93

angular velocity, 18, 36, 39, 45,114

anthropic principle, 107atmosphere, 32, 53–54, 98, 174atomic weight, 16, 38atoms: accretion and, 13, 16, 19,

31–32, 38; cosmic rays and,151, 174; electrons, 15–16(see also electrons); fission and,111; fusion and, 13, 49, 53,63, 74, 82, 108–11;neutrinos, 65, 82, 109–13,151; neutrons, 65–66, 82–83,109; nucleus, 5, 13, 84, 158;protons, 15–16, 29, 38, 65,82, 109, 151; spectra and,

140 (see also spectra); spinand, 140, 143; stellar-massblack holes and, 66, 82;supermassive black holes and,85; thermonuclear processesand, 13, 74

Big Bang, 85–86, 173Big Blue Bump, 88binaries: accretion and, 25, 27;

evolution and, 108, 112–19;exchange encounters and,117–18; formation and, 108,111–19, 126; gravitationalwaves and, 153–58, 163–64;hard, 116–18;intermediate-mass black holesand, 127–32; jets and, 44;pulsars and, 156–58, 164;soft, 117–18; spin and,146–48; stellar-mass blackholes and, 53–76;supermassive black holes and,87–88, 92, 98–99;supernovae and, 112–14;ultraluminous X-ray sources(ULXs) and, 127–32; X-ray,53 (see also X-ray binaries)

binding energy, 112blackbody radiation, 9n6;

accretion and, 20, 28–29, 40;Hawking radiation and,169–70; Stefan-Boltzmannconstant and, 144–46;stellar-mass black holes and,57, 61

Index 189

Blandford-Znajek mechanism,148

blazars, 47, 94–97BL Lac objects, 90, 97blueshift, 8, 69, 101Bondi-Hoyle accretion, 15, 77boundary layer, 23–24,

74–75Bremsstrahlung radiation,

29–30broad-line regions, 89, 91,

92–93, 102–3

carbon, 82, 108–9cataclysmic variable, 65Chandrasekhar, Subrahmanyan,

65Chandrasekhar limit: formation

and, 108n1, 109; stellar-massblack holes and, 65–67,79–83; supernovae and,108n1, 109

Chandra X-Ray Observatory,59, 128

charge: alpha particles and, 151;ions and, 24 (see also ions);negative, 24 (see alsoelectrons); neutral, 135;positive, 24, 29, 106–7, 112,116–17, 142

chemical composition: gasesand, 143; gravity and, 172;line emission and, 143;stellar-mass black holes and,107, 111, 135

chirp, 163–65

clouds, 44, 77–78, 92–93,102–3, 120, 151

compact objects: Chandrasekharlimit and, 66–67; Fermipressure and, 63–64, 66,80–82; formation and, 110,112–15, 118;intermediate-mass black holesand, 132; stellar-mass blackholes and, 57–58, 60, 62–76,78; supermassive black holesand, 101

Compton scattering, 30–31, 34,95, 97

conservation laws, 37–39, 48,58, 136

conservative mass transfer,59–61, 115

cosmic censorship, 175cosmic rays, 151, 174cosmological distances: accretion

and, 34; high energy photonsand, 34; quasars and, 85–87,89

coulomb interactions, 16, 24cyclotron emission, 30Cygnus X-1, 73

dark energy, 2dark matter, 2, 78, 173degeneration, 80–83, 109density: accretion and, 16, 21,

24, 26, 33, 37, 39–41;Eulerian algorithm and, 51;intermediate-mass black holesand, 133; mass and, 4–5;

190 Index

density (continued.) plasma and,172; singularities and, 9–12,136–38, 175; spin and, 139;stellar-mass black holes and,63–64, 66, 77, 79–80, 83;supermassive black holes and,85–86, 100–1; volume and,12; wormholes and, 175

Doppler shifts: accretion and,32; gravitational waves and,157; spin and, 140–44;stellar-mass black holes and,68, 70; supermassive blackholes and, 89, 92–93, 97,100, 102

dust, 33–34, 92–93, 98

Earth, 57, 65; active galacticnuclei (AGN) and, 84;atmosphere of, 54, 98, 174;celestial observations from, 5,98, 145–46, 161; danger tofrom black holes, 174; escapevelocity of, 3; jets and, 47;Moon and, 54, 70; orbit of,18, 145; parallax and,145–46; Schwarzschild radiusand, 4; X-ray detectors and,54

eccentricity, 18, 113Eddington, Arthur, 65Eddington limit: accretion and,

14–17, 19, 21, 27, 121–22;active galactic nuclei (AGN)and, 121–22; formation and,121–22; intermediate-mass

black holes and, 128–31;luminosity and, 16, 21, 56,87; primordial black holesand, 174; quasars and, 87;stars and, 16; stellar-massblack holes and, 56;supermassive black holes and,87; symmetry and, 17;ultraluminous X-ray sources(ULXs) and, 130–31

Eddington luminosity, 16–17,56

efficiency: accretion and, 13,23–24, 122, 125; formationand, 122, 125; stellar-massblack holes and, 77

e-folding timescale, 122Einstein, Albert, 5. See also

relativityelectromagnetism, 22, 29,

150–52, 155, 158electron capture, 65, 109electrons: accretion and, 15–16,

24, 26, 29–34;Bremsstrahlung and, 29–30;cyclotron emission and, 30;Fermi pressure and, 63–64,66, 80–82; formation and,109; free, 26, 29; free-freeemission and, 29; gravity and,15, 151; spin and, 143;stellar-mass black holes and,63–66, 80–83; supermassiveblack holes and, 92, 96–97

ellipsoidal variation, 70–71, 72emission, 2, 8; absorption and,

Index 191

20, 33–34, 70, 146; accretionand, 13, 20–21, 27–32, 34,41–42; active galactic nuclei(AGN) and, 87 (see also activegalactic nuclei (AGN));blackbody radiation and, 9n6,20, 28–29, 40, 57, 61, 144,169–70; blazars and, 47,94–97; BL Lac objects and,90, 97; broad-line regionsand, 89, 91, 92–93, 102–3;chemical composition and,143; Compton scattering and,30–31, 34, 95, 97; cyclotron,30; formation and, 112–14,123, 125; free-free, 29;gravitational waves and,154–55, 159–62; Hawkingradiation and, 167–74, 176;innermost stable circular orbit(ISCO) and, 139–47;International UltravioletExplorer satellite and, 87–88;jets and, 43–52, 63, 89,93–97, 125, 147–49; laser,101, 159–62; line, 31–32, 34,85, 90, 97, 102–3, 139–44;mechanisms of, 27–32, 34;narrow-line region and, 89,91, 93; nonthermal, 24, 34,61–63, 90; photons and,28–34, 47, 57, 96, 130,140–42, 147, 150–52, 170;spin and, 139–47; stellar-massblack holes and, 56–58,61–63, 65, 69, 75, 76, 78;

superluminal jets and, 45–47,94–97; supermassive blackholes and, 85, 87–90, 91–93,95–98, 101–3; synchrotron,30–31, 34, 95–97; thermal,34, 61, 87–88, 144–47

energy: accretion and, 13–14,16, 19–21, 23–24, 28–40;angular momentum and,114–15, 122 (see also angularmomentum); binding, 112;blazars and, 47, 94–97;conservation laws and, 37–39,48, 58, 136; dark, 2; E =Mc2 and, 170; efficiency and,13, 23–24, 77, 122, 125;escape velocity and, 3–4, 12,45, 106; event horizons and,6, 10–11; Fermi, 81–83;fission and, 111; formationand, 106–12, 114, 116–17,123–25; fusion and, 13, 49,53, 63, 74, 82, 108–11;gravitational potential welland, 13–14, 58, 66, 142;gravitational waves and,150–55, 164; Hawkingradiation and, 168–70; heat,14–15, 17, 19–21, 31, 80, 95;intermediate-mass black holesand, 130; jets and, 43–52;Keplerian laws and, 19, 35,39, 59, 68, 75, 101; kinetic,13–14, 19–22, 24, 43, 75,112, 116; motion and, 11,19–20 (see also motion);

192 Index

energy (continued.) multiversesand, 177; negative, 24, 81,106–7, 116, 153, 175;positive, 24, 29, 106–7, 112,116–17, 142; potential, 13,75, 81, 106; range ofdetecting, 150–51;singularities and, 9–12,136–38, 175; spin and,136–37, 140–44, 149;stellar-mass black holes and,58, 64, 65, 75–77, 80–83;supermassive black holes and,87, 89, 92–100; supernovaeand, 107–11 (see alsosupernovae); thermonuclearprocesses and, 13, 74;viscosity and, 19, 22, 26, 36,39–42, 137; wormholes and,175–76; zero total, 106

equation of state, 37–38, 66–67,79, 83

equations: accretion, 16, 20,37–41; conservation ofangular momentum, 39;conservation of mass, 38;Eddington limit, 16;Eddington luminosity, 16;e-folding timescale, 122;escape velocity, 3–4;exponential growth, 122;formation, 122; gravitationalpotential well, 14;gravitational waves, 153–54;Hawking radiation, 170; jets,45; Kerr metric, 10; Kramer’sopacity, 40; luminosity

generated by viscosity, 40;midplane density, 37; orbitalvelocity, 100; pressuregradient, 37; Schwarzschildmetric, 7; spin, 136;Stefan-Boltzmann constantand, 20, 38, 56–57, 144,146; stellar-mass black holes,58–59, 68, 81–82;supermassive black holes, 122;viscosity, 41

equilibrium: Chandrasekharlimit and, 79–82; formationand, 107; gravity and, 107;hydrostatic, 63, 79–80;stellar-mass black holes and,79–82

ergosphere, 11, 136, 149escape velocity, 3–4, 12, 45,

106Eulerian algorithm, 51–52European Space Agency,

162event horizons: angular

momentum and, 10; energyand, 6, 10–11; ergosphereand, 11, 136, 149; gravityand, 3, 5, 7–8, 11–12; matterand, 5–6, 9–10; the metricand, 6–11; motion and, 7–8;Newtonian physics and, 3–5,7–8, 12; radiation and, 8;rotation and, 11;Schwarzschild radius and,4–10; space-time and, 5, 7,9–10; ultraviolet catastropheand, 9n6

Index 193

evolution: Big Bang and, 85–86,173; binaries and, 108,112–19; compact objects and,63; earliest stars and, 126;hierarchical galaxy growthand, 104; high-mass systemsand, 60; magnetic field and,48; massive stars and, 111;multiverses and, 177–78;primordial black holes and,172–73; redshift studies and,104–5; seed black holes and,122; star clusters and,116–19; stellar-mass blackholes and, 84, 108, 112–19;supermassive black holes and,164–65

exchange encounters,117–18

fast rise and exponential decay(FRED), 25

Fermi energy, 81–83fermions, 80–83Fermi pressure, 63–64, 66,

80–82, 108–9Fermi satellite, 95fission, 111formation: accretion and,

121–23, 125; active galacticnuclei (AGN) and, 120–25,128–29, 132; angularmomentum and, 114–15,122; anthropic principle and,107; binaries and, 108,111–19, 126; Chandrasekharlimit and, 108n1, 109;

compact objects and, 110,112–15, 118; Eddington limitand, 121–22; efficiency and,122, 125; e-folding timescaleand, 122; electrons and, 109;emission and, 112–14, 123,125; energy and, 106–12,114, 116–17, 123–25;equilibrium and, 107;exchange encounters and,117–18; galaxies and, 107,113, 115, 122–26; gas and,107, 114, 119–26; gravityand, 106–7, 109–10, 112–13,123–26; hydrogen and, 108,120; initial conditions ofUniverse and, 106–7; ionsand, 120; luminosity and,121–22, 125; magnetism and,114; matter and, 106–7, 112,123; mergers and, 123–26;motion and, 107; orbits and,112–17, 123–24; plasma and,106, 109; pressure and,107–9, 121; protons and,109; radiation and, 113,121–25; rotation and, 111,114; singularities and, 9–12,136–38, 175; stars and,107–26; stellar-mass blackholes and, 107–19;supermassive black holes and,119–26; supernovae and,107–14, 120; symmetry and,113; velocity and, 106, 114,117; X-rays and, 112–15,118–19, 124

194 Index

Frank, J., 35n2, 42friction, 19, 21–22fusion, 13, 49, 53, 63, 74, 82,

108–11

Galactic Bulge, 78galaxies, 5, 33; active galactic

nuclei (AGN) and, 44, 84,87–105, 120–22, 124, 125,128–29, 132, 141, 145, 147;centers of, 43, 84, 98–101,104, 122, 132–33; constantmotion of, 77–78; CygnusX-1 and, 73; dark matter and,2, 78, 173; dense clouds and,77; elliptical, 100;equilibrium and, 107;formation mechanisms and,107, 113, 115, 122–26;gravitational waves and,164–65; hierarchical growthof, 104; host characteristicsand, 90; intermediate-massblack holes and, 127, 132–34;isolated black holes and, 76;jets and, 43, 95; low-mass,132–34; Milky Way, 73, 98,100, 115, 128, 164;nonaccreting central blackholes and, 98–99; plane of,55, 56; primordial black holesand, 172; quasars and, 85–87;radio, 88–90; redshift and,85; relaxation time and, 101;Seyfert, 86–88, 90; spin and,135, 145, 147; stellar-mass

black holes and, 55, 56, 73,76–78; supermassive blackholes and, 84–95, 98–102,104–5; unification and,88–94; water masers and,101–2; X-rays and, 55

gamma rays, 150, 172–73gas: accretion and, 13–26, 31,

36–41; chemical compositionand, 143; degeneration and,80–83, 109; formation and,107, 114, 119–26; frictionand, 19, 21–22; Gaunt factorsand, 40; ideal, 79; infalling,14, 17–18, 74–75;intermediate-mass black holesand, 132–33; jets and, 44,49–51; nonrelativistic, 81;rotation of, 18; spin and, 137,140–44, 149; stellar-massblack holes and, 61–63,74–75, 79–82; supermassiveblack holes and, 89, 92–95,100–3; viscosity and, 19, 22,26, 36, 39–42, 137

Gaunt factors, 40Geiger counters, 54giant stars, 57, 115, 117–18,

131Gibbon, Gary, 4n2globular clusters: evolution and,

116–19; intermediate-massblack holes and, 132–34;low-mass galaxies and,132–34; stellar-mass blackholes and, 116–19;

Index 195

supermassive black holes and,100–1, 123

gravitational microlensing,77–78, 173

gravitational potential well,13–14, 58, 66, 142

gravitational redshift, 8, 141–43,157

gravitational waves: angularmomentum and, 164; binariesand, 153–58, 163–64; binarypulsars and, 156–58; chirpand, 163–65; direct detectionof, 158–63; Doppler shiftsand, 157; effects of, 152–56;electrons and, 151; emissionand, 154–55, 159–62; energyand, 150–55, 164; galaxiesand, 164–65; interferometryand, 159–64; LIGO and,160, 161, 164; luminosityand, 154–55; magnetism and,150–52, 155–56, 158; massand, 152–66; mergers and,152, 154–55, 159–60,162–65; orbits and, 152–57,162, 163–66; photons and,142, 150–52; protons and,151; quantum effects and,152; radiation and, 150–60,163–65; relativity and,151–52, 157, 165–66;resonance frequency and,158–59; rotation and, 156;Schwarzschild radius and,154–55, 163; Shapiro time

delay and, 157–58;space-time and, 152, 158,165; spectra and, 151; starsand, 153, 156–58, 161,163–64; supernovae and, 151;velocity and, 155–56; X-raysand, 150, 154

gravity: accretion and, 13–19,21, 23, 36–37; bindingenergy and, 112; chemicalcomposition and, 172;electrons and, 15, 151;equilibrium and, 107; escapevelocity and, 3–4, 12, 45,106; event horizons and, 3, 5,7–8, 11–12; formation and,106–7, 109–10, 112–13,123–26; Hawking radiationand, 168; infalling objectsand, 8, 9n5, 14–19, 23, 28,43, 74–76; intermediate-massblack holes and, 133; jets and,44, 51; Newtonian physicsand, 3, 5, 153; planets and,107; potential energy and, 13,75, 81, 106; primordial blackholes and, 172–73; quantum,152, 168; spin and, 139,141–43, 149; stellar-massblack holes and, 58, 63,65–66, 77–82, 80;supermassive black holes and,86, 95, 103–5; supernovaeand, 107–14 (see alsosupernovae); wormholes and,174–76

196 Index

hard binaries, 116–18Hawking radiation: luminosity

and, 169–70; mass and,168–71; multiverses and, 176;Newtonian physics and, 167;primordial black holes and,171–74; quantum effects and,168; relativity and, 167–68;Schwarzschild radius and,168–70; virtual pairs and,168–69

heat: accretion and, 14–15, 17,19–21, 31; stellar-mass blackholes and, 80; supermassiveblack holes and, 95

helium, 82, 108, 120hierarchical growth, 104Hubble Space Telescope (HST),

99–101, 128, 134Hulse, Russell,

157hydrogen: accretion and, 13,

15–16, 31, 33–34; formationand, 108, 120; fusion and, 13;ionized, 15–16, 33–34;spectra and, 31; stellar-massblack holes and, 82

hydrostatic equilibrium, 63,79–80

ideal MHD, 49–51, 51infalling gas, 14, 17–18, 74–75infalling objects, 8, 9n5, 14–19,

23, 28, 43, 74–76infrared light, 33, 62, 89, 92,

97–98, 131, 150

innermost stable circular orbit(ISCO): Kerr metric and,137; line emission and,139–47; spin and, 136–47;Stefan-Boltzmann constantand, 144–46; thermalemission and, 144–47

interferometry, 101, 159–64intermediate-mass black holes:

accretion and, 130–33;binaries and, 127–32; clustersand, 132–34; compact objectsand, 132; density and, 133;Eddington limit and, 128–31;energy and, 130; galaxies and,127, 132–34; gas and,132–33; gravity and, 133;light and, 134; low-massgalaxies and, 132–34; matterand, 132; motion and, 131,134; orbits and, 127, 131;photons and, 130; radiationand, 130–31; spectra and,129–31; stars and, 128,131–34; supernovae and, 128;symmetry and, 131;temperature and, 130;ultraluminous X-ray sources(ULXs) and, 127–32; velocityand, 131, 133

International Ultraviolet Explorersatellite, 87–88

interstellar reddening, 33–34ions: accretion and, 15–17, 22,

24, 26, 29–31, 33–34, 40,140, 143; atomic weight and,

Index 197

16; formation and, 120; freeelectrons and, 26, 29; Gauntfactors and, 40; hydrogenand, 15–16, 33–34;supermassive black holes and,92–93, 102–3

iron, 31, 109–10, 140–43isothermal configurations, 80

jets: angular momentum and,48; binary stars and, 44;Blandford-Znajek mechanismand, 148; Earth and, 47;energy and, 43, 46–51;Eulerian algorithm and,51–52; galaxies and, 43; gasand, 44, 49–51; gravity and,44, 51; Lagrangian algorithmand, 51–52; luminosity and,47; magnetism and, 48–51;magnetohydrodynamics and,48–52; mass and, 48, 51;motion and, 45–51; opticallyviolently variable (OVV)sources and, 97; photons and,47, 147; physics of, 48–52;pressure and, 49–51; radiationand, 46–47; relativity and,45–46; speed of light and, 43,45–47; spin and, 147–49;superluminal, 45–47, 93–97;temperature and, 51; velocityand, 45–47, 49

Keplerian laws, 19, 35, 39, 59,68, 75, 101

Kerr metric, 10, 136–37

kinetic energy, 13–14, 19–22,24, 43, 75, 112, 116

King, A., 35n2, 42knots, 49–50, 95Kramer’s opacity, 40

Lagrangian algorithm, 51–52,58

Large Magellanic Cloud, 78,151

Laser InterferometerGravitational WaveObservatory (LIGO), 160,161, 164

Laser Interferometer SpaceAntenna (LISA), 162–63, 165

lasers, 101, 159–64Life of the Cosmos, The (Smolin),

177light: accretion and, 13, 24–25,

33; Big Blue Bump and, 88;blackbody radiation and, 9n6,20, 28–29, 40, 57, 61, 144,169–70; blazars and, 47,94–97; BL Lac objects and,90, 97; blueshift and, 8, 69,101; dark matter and, 2, 78,173; emission and, 13 (see alsoemission); gravitationallensing and, 77–78; infrared,33, 62, 89, 93, 97–98, 131,150; interferometry and, 101,159–64; intermediate-massblack holes and, 134;International UltravioletExplorer satellite and, 87–88;

198 Index

light (continued.) jets and, 45;lasers and, 101, 159–64;luminosity and, 127–31 (seealso luminosity);Michelson-Morleyexperiment and, 159n3;microlensing and, 77–78,173; optical, 53–54, 61, 98,124; photons, 28–34, 47, 57,96, 130, 140–42, 147,150–52, 170; redshift and, 8,69, 85–86, 89, 101, 104–5,119–20, 122, 125, 139,141–43, 145, 157; spectraand, 20 (see also spectra);speed of, 4, 12, 15, 43,45–47, 57, 67, 94–95, 142,152, 156, 159n3, 160; spinand, 139; stellar-mass blackholes and, 53–54, 57, 61, 71,73, 77–78; supermassiveblack holes and, 84–87, 90,93, 98–99, 103–5; travel timeand, 45, 57, 84, 86, 103–5,159, 174; ultraviolet, 9n6, 29,33–34, 87–88, 147, 150;wormholes and, 174–76;X-rays and, 124 (see alsoX-rays)

light curves, 24–25, 71,73

light-years, 96line emission: accretion and,

31–32, 34; innermost stablecircular orbit (ISCO) and,139–44; spin and, 139–44;

supermassive black holes and,85, 90, 97, 102–3

low-mass systems, 58, 60,71–72, 108–9, 114–15

luminosity, 127–31; accretionand, 15–17, 20–21, 26, 40;active galactic nuclei (AGN)and, 84, 125; blazars and, 47,94–97; E = Mc2 and, 170;Eddington, 16–17, 56;formation and, 121–22, 125;gravitational waves and,154–55; Hawking radiationand, 169–70;intermediate-mass black holesand, 129–31; jets and, 47;quasars and, 122, 125; spinand, 139, 144–46;stellar-mass black holes and,54–57, 61, 71–72, 74, 77;Sun and, 155; superluminaljets and, 45–47, 94–97;supermassive black holes and,90, 97–99, 102–4;ultraluminous X-ray sources(ULXs) and, 127–32

M87, 100–1magnesium, 82, 108magnetic braking, 114magnetic reconnection, 49magnetic Reynolds number,

48–49magnetism: accretion and,

20–22, 24, 26, 28–31, 38,40–41; binary pulsars and,

Index 199

156–58, 164;electromagnetism and, 22, 29,150–52, 155, 158; formationand, 114; gravitational wavesand, 150–52, 155–56, 158;jets and, 48–51; spin and,137, 148; stellar-mass blackholes and, 67

magnetohydrodynamics(MHD), 48–52

main sequence stars, 57,114–15, 117

“Man Who Invented BlackHoles, The” (Gibbon), 4n2

masers, 101–2mass: accretion and, 13–29,

35–41; accretors and, 14, 18,25, 45; conservation laws and,37–39, 48, 58, 136;conservative mass transferand, 59–60, 115; dark matterand, 2, 78, 173; density and,4–5; escape velocity and, 3–4,12, 45, 106; exchangeencounters and, 117–18;formation of black holes and,106–26; gravitational wavesand, 152–66; Hawkingradiation and, 168–71;high-mass systems and, 58,60, 69, 73–76, 112–15, 121,172; intermediate-mass blackholes and, 127–34; jets and,48, 51; Kerr metric and, 10;low-mass systems and, 58, 60,71–72, 108–9, 114–15;

Newtonian physics and, 3–4;point, 7, 51; primordial blackholes and, 171–74;Schwarzschild radius and, 7,10, 66, 87, 100; single point,7; singularities and, 9–12,136–38, 175; spin and,135–37, 140, 142, 144–49;stars and, 5; stellar-mass blackholes and, 53 (see alsostellar-mass black holes);supermassive black holes and,84 (see also supermassive blackholes); supernovae and,107–14 (see also supernovae);thermonuclear processes and,13; typical stars and, 5;wormholes and, 174–76;X-ray binaries and, 68–73

mass accretion rate, 17, 21,25–26, 35, 41, 75–77, 122

mass function, 68–70, 146mass segregation, 118–19, 123mass transfer: angular

momentum and, 58–60, 115;circular orbit equation and,36–37; conservative, 59–61,115; density and, 39–40; flowgeometry and, 26; viscosityand, 36–37, 39–40

matter: accretion and, 14, 23,39–40, 137; dark, 2, 78, 173;density and, 5 (see alsodensity); electrons, 15–16 (seealso electrons); event horizonsand, 5–6, 9–10;

200 Index

matter (continued.) formationand, 106–7, 112, 123; gasand, 137, 140–44, 149 (seealso gas); gravity and, 13 (seealso gravity);intermediate-mass black holesand, 132; neutrinos, 65, 82,109–13, 151; neutrons,65–66, 82–83, 109; plasmaand, 16–17, 29–32, 106, 109,140, 172; primordial blackholes and, 172–73; protons,15–16, 29, 38, 65, 82, 109,151; stellar-mass black holesand, 60, 62, 65–66, 76, 78;supermassive black holes and,85; wormholes and, 175–76

Mercury, 5, 157mergers: formation and,

123–26; gravitational wavesand, 152, 154–55, 159–60,162–65; repeated, 165;supermassive black holes and,104–5, 159–60, 164–65

metric: event horizons and,6–11; Kerr, 10, 136–37;mathematical divergence and,9n5; Schwarzschild, 7, 10,137; space-time and, 6–11,136–37, 168

Michelson-Morley experiment,159n3

microlensing, 77–78, 173microwaves, 34, 101, 150, 173Milky Way, 73, 98, 100, 115,

128, 164

Moon, 54, 70motion: accretion and, 19–20,

23, 32, 36; angularmomentum and, 10 (see alsoangular momentum);blueshift and, 8, 69, 101;Doppler shifts and, 32, 68,70, 89, 92–93, 97, 100, 102,140–44, 157; escape velocityand, 3–4, 12, 45, 106; eventhorizons and, 7–8; formationand, 107; gravity and, 7 (seealso gravity); infalling objectsand, 8, 9n5, 14–19, 23, 28,43, 74–76; intermediate-massblack holes and, 131, 134; jetsand, 45–51; Keplerian lawsand, 19, 35, 39, 59, 68, 75,101; Michelson-Morleyexperiment and, 159n3;Newtonian physics and, 7;orbital, 55 (see also orbits);redshift and, 8, 69, 85–86,89, 101, 104–5, 119–20, 122,125, 139, 141–43, 145, 157;rotation, 11, 18, 21–22, 39,54, 67, 70, 78, 111, 114, 136,138, 156; spin and, 141, 152;stellar-mass black holes and,58, 68, 77–78; superluminal,45–47, 95, 96 ; supermassiveblack holes and, 89, 95, 96,97–99, 102

M-σ relationship, 104,132

multiverses, 176–78

Index 201

narrow-line region, 89, 91, 93NASA, 162negative energy, 24, 81, 106–7,

116, 153, 175neon, 82, 108neutrinos, 65, 82, 109–13, 151neutrons, 65–66, 82–83, 109neutron stars: Chandrasekhar

limit and, 66–67; equation ofstate and, 66–67; enormousdensities of, 66; formationand, 65, 108–11, 113, 119;gravitational waves and,156–57, 163–64;Schwarzschild radius and,66; stellar-mass black holesand, 65–67, 74–76, 82–83;upper limit of mass of,66–67

Newtonian physics: accretionand, 18–19; escape velocityand, 3–4, 12, 45, 106; eventhorizons and, 3–5, 7–8, 12;gravity and, 5, 153; Hawkingradiation and, 167; orbitsand, 18–19

nitrogen, 82Nobel Prize, 157nonaccreting central black holes,

98–99nonthermal emission, 24, 34,

61–63, 90numerical relativity, 165–66

optically violently variable(OVV) sources, 97

orbital period: accretion and, 54;formation and, 113–15, 117;gravitational waves and,153–57, 163;intermediate-mass black holesand, 131; stellar-mass blackholes and, 62, 68, 70;supermassive black holes and,98

orbits: accretion and, 18–20,22–23, 35–36, 41; angularmomentum and, 149 (see alsoangular momentum); circular,19, 35–36, 41, 59, 68, 69,112, 136–47, 152; eccentric,18, 113; formation and,112–17, 123–24;gravitational waves and,152–57, 162, 163–66;innermost stable circular(ISCO), 136–47;intermediate-mass black holesand, 127, 131; Keplerian lawsand, 19, 35, 39, 59, 68, 75,101; Newtonian physics and,18–19; planets and, 2, 5;Schwarzschild radius and,18, 154; spin and, 136–39;stable, 18, 36, 153, 156;stellar-mass black holes and,55–59, 62, 68–71, 72, 73,75; supermassive blackholes and, 93, 96, 98, 100;velocity and, 19, 100,156

oxygen, 82, 108

202 Index

parallax, 145–46Pauli-Fermi exclusion principle,

80Penrose, Roger, 175Penrose process, 136photons: accretion and, 28–34,

140; Compton scattering and,30–31, 34, 95, 97;gravitational waves and, 142,150–52; Hawking radiationand, 170; informationreceived through, 150;intermediate-mass black holesand, 130; jets and, 47, 147;light and, 28–34, 47, 57, 95,130, 140–42, 147, 150–52,170; redshift and, 142 (see alsoredshift); stellar-mass blackholes and, 57; supermassiveblack holes and, 96–97

planets: extrasolar, 2; gravityand, 107; precise orbits of, 5,18; spin and, 135

plasma: accretion and, 16–17,29–32; density of, 172;formation and, 106, 109; ironand, 140; radiative transferand, 28, 32–35; supernovaeexplosions and, 106, 109;temperature and, 172

polarization, 90, 93, 150n1positive energy, 24, 29, 106–7,

112, 116–17, 142potential energy, 13, 75, 81, 106pressure: accretion and, 15, 17,

21, 37–38; Fermi, 63–64, 66,

80–82, 108–9; formationand, 107–9, 121; gas, 38,49–51, 63, 79–80, 82, 107;jets and, 49–51; primordialblack holes and, 172;radiation, 15, 17, 21, 38,121–22; spin and, 135, 137,143; stellar-mass black holesand, 63, 66, 79–83

primary stars, 112primordial black holes, 171–74protons: accretion and, 15–16,

29, 38; formation and, 109;gravitational waves and, 151;stellar-mass black holes and,65, 82

pulsars, 65, 113, 134, 156–58,164

quantum effects: accretion and,40; Gaunt factors and, 40;general relativity and, 10;gravity and, 152, 168;Hawking radiation and, 168;small scales and, 9;stellar-mass black holes and,63; ultraviolet catastropheand, 9n6; unified theory and,10; virtual pairs and, 168–69

quasars: active galactic nuclei(AGN) and, 44, 84–90; BigBlue Bump and, 88; BL Lacobjects and, 90; cosmologicaldistances and, 85–87, 89;cosmological distance scalefor, 85–86; differences from

Index 203

quasi-stellar objects (QSOs),84n1; discovery of, 53,85–88; Eddington limit and,87; host galaxy characteristicsand, 90; intermediate-massblack holes and, 122, 125;International UltravioletExplorer satellite and, 87–88;radio intensity and, 88;redshift and, 86; Seyfertgalaxies and, 86–88, 90;spectra and, 85; stellar-massblack holes and, 53;supermassive black holes and,84–90, 91; unification and,88–94

radiation: accretion and, 13–17,21, 27–40; blackbody, 9n6,20, 28–29, 40, 57, 61, 144,169–70; boundary layer and,23–24, 74–75;Bremsstrahlung, 29–30;emission and, 2, 20–21,27–32, 34, 41–42 (see alsoemission); event horizons and,8; formation and, 113,121–25; gravitational wavesand, 150–60, 163–65;Hawking, 167–74, 176;inefficient flows and, 23–24;intermediate-mass black holesand, 130–31; jets and, 43–52,63, 89, 93–97, 125, 147–49;light and, 87 (see also light);polarization and, 90, 93,

150n1; pressure and, 15, 17,21, 38, 121–22; primordialblack holes and, 171–74; spinand, 144, 146; stellar-massblack holes and, 53–57, 75,77–78; supermassive blackholes and, 87–97, 102–3,105; ultraviolet catastropheand, 9n6; wormholes and,176

radiative transfer, 28, 32–35radio galaxies, 88–89radio intensity, 88radio morphology, 89radio telescopes, 101, 133, 150Raine, D., 35n2, 42redshift, 69; active galactic

nuclei (AGN) and, 105, 141,145; formation and, 119–20,122, 125; gravitational, 8,141–43, 157; quasars and, 86;spin and, 139, 141–43, 145;supermassive black holes and,85–86, 89, 101, 104–5

relativity: curvature ofspace-time and, 5, 7;E = Mc2 and, 170; general,4–8, 10, 67, 73, 149–52, 157;gravitational waves and,151–52, 157, 165–66;Hawking radiation and,167–68; initial conditionsand, 167; jets and, 45–46;numerical, 165–66; quantummechanics and, 10;Schwarzschild radius and,

204 Index

relativity (continued.) 4–10, 18,66–68, 87, 100, 136–37, 142,149, 154–55, 163–64,168–70; special, 45–46,47n1; spin and, 149;stellar-mass black holes and,67, 73; unified theory and, 10

relaxation time, 100–1, 118resonance frequency, 158–59reverberation mapping, 102–4Roche lobe, 58–59, 70, 93,

114–15, 146rotation: accretion and, 18,

21–22, 39; ergosphere and,11, 136, 149; event horizonsand, 11; formation and, 111,114; gas and, 18; gravitationalwaves and, 156; spin and,136, 138; stellar-mass blackholes and, 67, 70, 78

Schwarzschild, Karl, 4Schwarzschild metric, 7, 10, 137Schwarzschild radius: accretion

and, 18; angular momentumand, 10; ergosphere and, 11,136, 149; escape velocity and,4; event horizons and, 4–10;gravitational waves and,154–55, 163; Hawkingradiation and, 168–70; massand, 7, 10, 87; Newtonianphysics and, 8; orbits and, 18,154; relativity and, 5, 10,67–68; space-time curvatureand, 7–9; spin and, 136–37,

142, 149; stellar-mass blackholes and, 66–68;supermassive black holes and,66, 87, 100

Seyfert galaxies, 86–88, 90Sgr A, 98–99Shakura, N., 35, 41Shapiro time delay, 157–58singularities, 9–12, 136–38, 175Sloan Digital Sky Survey, 104Smolin, Lee, 177smooth particle hydrodynamics

(SPH), 50–51soft binaries, 117–18space-time: curvature of, 5, 7;

event horizons and, 5, 7,9–10; gravitational waves and,152, 158, 165; light-yearsand, 91; the metric and,6–11, 136–37, 168;singularities and, 9–12,136–38, 175; spin and,136–37; vacuum and, 49,168; wormholes and, 174–76

spatial resolution, 127–28special relativity, 45–46, 47n1spectra: absorption and, 20,

33–34, 70, 146; accretionand, 14, 20, 26, 28, 30–35,41–42; active galactic nuclei(AGN) and, 89–90; Big BlueBump and, 88; blackbodyradiation and, 9n6, 20,28–29, 40, 57, 61, 144,169–70; blazars and, 47,94–97; BL Lac objects and,

Index 205

90, 97; blueshift and, 8, 69,101; broad-line regions and,89, 91, 92–93, 102–3;emission and, 20–21, 27–32,34, 41–42 (see also emission);gravitational waves and, 151;Hawking radiation and,167–74, 176; hydrogen and,31; intermediate-mass blackholes and, 129–31; interstellarreddening and, 33–34; ironand, 31; narrow-line regionand, 89, 91, 93; nonthermal,24, 34, 61–63, 90; optical,89–90; quasars and, 85;redshift and, 8, 69, 85–86,89, 101, 104–5, 119–20, 122,125, 139, 141–43, 145, 157;spin and, 139–40, 144,146–47; state changes and,62; stellar-mass black holesand, 55, 61–62, 64, 70, 75;supermassive black holes and,85, 87, 89–90, 92–93, 99,104; X-rays and, 25, 27, 29,31–34, 150, 154 (see alsoX-rays)

spin: accretion and, 136–49;active galactic nuclei (AGN)and, 141, 145, 147; α-disksand, 144, 146; angularmomentum and, 135–38,149; atoms and, 140, 143;binaries and, 146–48; blackholes and, 11, 48, 135–49,166; Blandford-Znajek

mechanism and, 148; densityand, 139; Doppler shifts and,140–44; electrons and, 143;emission and, 139–47; energyand, 136–37, 140–44, 149;ergosphere and, 136, 149;galaxies and, 135, 145, 147;gas and, 137, 140–44, 149;gravity and, 139, 141–43,149; innermost stable circularorbit (ISCO) and, 136–47;jets and, 147–49; Kerr metricand, 136–37; light and, 139;luminosity and, 139, 144–46;magnetism and, 137, 148;magnetized neutron stars and,156; mass and, 135–37, 140,142, 144–49; maximallyspinning condition and, 136,138, 143; mergers and,148–49; motion and, 141,152; orbits and, 136–39;Penrose process and, 136;planets and, 135; pressureand, 135, 137, 143; radiationand, 144, 146; relativity and,149; rotation and, 136, 138;Schwarzschild radius and,136–37, 142, 149; space-timeand, 136–37; spectra and,139–40, 144, 146–47;symmetry and, 143;temperature and, 135, 138,143–46; velocity and, 141;X-rays and, 140, 144,146–48

206 Index

stars: accretion and, 16, 21, 29,31, 36; binary, 44, 53, 57, 68,70–71, 108, 112–18, 132,153; companion, 36, 58,60–61, 69–73, 77, 111–12,114–15, 131–32, 146;degeneration and, 80–83,109; Eddington limit and, 16;exchange encounters and,117–18; formation and,107–26; giant, 57, 115,117–18, 131; globular clustersand, 117, 119, 132–34;gravitational waves and, 153,156–58, 161, 163–64;high-mass systems and, 58,60, 69, 73–76, 112–15, 121,172; intermediate-mass blackholes and, 128, 131–34; jetsand, 44; low-mass, 58, 60,71–72, 108–9, 114–15; mainsequence, 57, 114–15, 117;mass and, 5; orbits and, 2 (seealso orbits); primary, 112;primordial black holes and,172–73; spin and, 135, 146;stellar-mass black holes and,53–83; Sun, 13 (see also Sun);supergiant, 131; supermassiveblack holes and, 85, 88, 90,92–93, 95, 98–101;supernovae and, 65–66, 83,107–14, 120–21, 128, 151;symmetry and, 79; velocitycurves and, 68–69, 101–2,156–57; white dwarfs,

65–67, 80, 82–83,108–9

Star Trek, 176state changes, 62Stefan-Boltzmann constant, 20,

38, 56–57, 144–46stellar-mass black holes:

accretion and, 53, 58, 60–63,68–69, 71, 74–77; angularmomentum and, 58–60;atoms and, 66, 82; binariesand, 53–76; Chandrasekharlimit and, 65–67, 79–83;chemical composition and,107, 111, 135; clusters and,116–19; compact objects and,57–58, 60, 62–76, 78; densityand, 63–64, 66, 77, 79–80,83; Doppler shifts and, 68,70; Eddington limit and, 56;efficiency and, 77; electronsand, 63–66, 80–83; emissionand, 56–58, 61–63, 65, 69,75, 76, 78; energy and, 58,64, 65, 75–77, 80–83;equilibrium and, 79–82;evolution and, 84, 108,112–19; formation of,107–19; galaxies and, 55, 56,73, 76–78; gas and, 61–63,74–75, 79–82; gravity and,58, 65–66, 77–82; heat and,80; high-mass compactobjects and, 73–76; hydrogenand, 82; isolated, 76–78; lightand, 53–54, 57, 61, 71, 73,

Index 207

77–78; luminosity and,54–57, 61, 71–72, 74, 77;magnetism and, 67; matterand, 60, 62, 65–66, 76, 78;motion and, 58, 68, 77–78;orbits and, 55–59, 62, 68–71,72, 73, 75, 93, 96, 98, 100;photons and, 57; protons and,65, 82; quantum effects and,63; quasars and, 53; radiationand, 53–57, 75, 77–78;relativity and, 67, 73; rotationand, 67, 70, 78;Schwarzschild radius and,67–68; spectra and, 55,61–62, 64, 70, 75, 84; starsand, 53–83; supernovae and,65–66, 83, 107–11;temperature and, 56–57, 61,79–80; velocity and, 59,68–70; X-rays and, 53–66,68–69, 74–77

stellar winds, 60, 111–14, 120Sun: accretion and, 53; density

of, 4–5; Earth’s orbit and, 18,145; Eddington limit and, 16;extrasolar planets and, 2;Fermi pressure and, 63; gaspressure and, 63, 79;luminosity and, 155;main-sequence stars and, 57;mass of, 16, 100, 127, 133,170; neutrinos and, 151;Schwarzschild radius and, 5;thermonuclear processes of,13, 63; X-rays and, 54

Sunyaev, R., 35, 41supergiant stars, 131superluminal jets, 45–47, 91,

94–97, 148supermassive black holes:

accretion and, 87, 92–101;active galactic nuclei (AGN)and, 84, 87–105; angularmomentum and, 93; atomsand, 85; binaries and, 87–88,93, 98–99; clusters and,100–1, 123; compact objectsand, 101; Doppler shifts and,89, 92–93, 97, 100, 102;Eddington limit and, 87;electrons and, 92, 96–97;emission and, 85, 87–90,92–93, 95–98, 101–3; energyand, 87, 89, 92–100;evolution and, 164–65;exchange encounters and,117–18; formation of,119–26; galaxies and, 84–95,98–102, 104–5; gas and, 89,92–95, 100–3; gravity and,63, 80, 86, 95, 103–5; growthof, 121–26; Hawkingradiation and, 170–71; heatand, 95; ions and, 92–93,102–3; light and, 84–87, 90,93, 98–99, 103–5; luminosityand, 90, 97–99, 102–4; massand, 85–86, 99–105; matterand, 85; mergers and, 104–5,159–60, 164–65; motionand, 89, 91, 95, 97–99, 102;

208 Index

supermassive black holes(continued.) nonaccretingcentral black holes and,98–99; optically violentlyvariable (OVV) sources and,97; orbital period and, 98;photons and, 96–97; pressureand, 63, 66, 79–83; quasarsand, 84–90, 91; radiationand, 87–97, 102–3, 105;radio intensity and, 88;reverberation mapping and,102–4; Schwarzschild radiusand, 87, 100; seed black holesand, 120–21; spectra and, 85,87, 89–90, 92–93, 99, 104;stars and, 85, 88, 90, 92–93,95, 98–101; superluminal jetsand, 45–47, 94–97;supernovae and, 120;temperature and, 93; timevariability and, 90–94;unification and, 88–94;velocity and, 89, 95, 99–104;water masers and, 101–2;X-rays and, 87–88, 93, 98

supernovae: binary stars and,112–14; black hole formationand, 107–14; Chandrasekharlimit and, 108n1, 109;core-collapse, 108n1; Fermipressure and, 108–9; fusionand, 108–11; gravitationalwaves and, 151;intermediate-mass black holesand, 128; neutrinos and,

109–13; Newton’s theory ofgravity and, 3; plasma and,106, 109; seed black holesand, 120–21; stellar-massblack holes and, 65–66, 83,107–11; supermassive blackholes and, 120; white dwarfsand, 109

superposition, 34, 61, 86, 129symmetry: Bondi-Hoyle

accretion and, 15; Eddingtonlimit and, 17; formation and,113; infalling gas and, 17;intermediate-mass black holesand, 131; spin and, 143; starsand, 79

synchrotron emission, 30–31,34, 97

synchrotron self-Compton(SSC) emission, 30–31

Taylor, Joseph, 157telescopes: Hubble Space

Telescope (HST), 99–101,128, 134; parallax and,145–46; radio, 101, 133, 150;spatial resolution and,127–28; X-ray, 31

temperature: accretion and,20–21, 24, 26, 28–31, 38,40–41; blackbody radiationand, 9n6, 20, 28–29, 40, 57,61, 144, 169–70; Hawkingradiation and, 169;intermediate-mass black holesand, 130; jets and, 51; plasma

Index 209

and, 172; primordial blackholes and, 172; spin and, 135,138, 143–46; stellar-massblack holes and, 56–57, 61,79–80; supermassive blackholes and, 93; thermonuclearprocesses and, 13, 74

thermal Bremsstrahlung, 30thermal emission, 34, 61,

87–88, 144–47thermonuclear processes, 13, 74torus, 92–93, 96transients, 61, 69, 71, 146

Uhuru satellite, 55–56, 65ultraluminous X-ray sources

(ULXs), 127–32ultraviolet catastrophe, 9n6ultraviolet light, 9n6, 29, 33–34,

87–88, 147, 150unification (of AGN), 88–94unified theory, 10

vacuum, 49, 168velocity: accretion and, 14,

18–19, 30, 36, 39, 45;angular, 18, 36, 39, 45, 114;Doppler shifts and, 32, 68,70, 89, 92–93, 97, 100, 102,140–44, 157; escape, 3–4, 12,45, 106; formation and, 106,114, 117; gravitational wavesand, 155–56; infalling objectsand, 14; intermediate-massblack holes and, 131, 133; jetsand, 45–47, 49; Keplerian

laws and, 19, 35, 39, 59, 68,75, 101; orbital, 19, 100, 156;relative, 59, 117; spin and,141; stellar-mass black holesand, 59, 68–70; superluminalmotion and, 45–47;supermassive black holes and,89, 95, 99–104

velocity curves, 68–69, 101–2,156–57

virtual pairs, 168–69viscosity, 19, 22, 26, 36, 39–42,

137

water masers, 101–2Weber, Joseph, 158–59Wheeler, John Archibald, 11–12white dwarfs, 65–67, 80, 82–83,

108–9wormholes, 174–76

XMM-Newton, 55, 141X-ray binaries: accretion and,

25, 27, 60–63; compactobjects and, 57–58, 60,62–76, 78; formation and,112–15, 118–19; FRED lightcurve and, 25; gravitationalwaves and, 154;intermediate-mass black holesand, 127–32; massmeasurements and, 68–73;Roche lobe and, 58–59, 70,93, 114–15, 146; spin and,146–48; stellar-mass blackholes and, 54–63, 65–66,

210 Index

X-ray binaries (continued.)68–74, 76; supermassive blackholes and, 87–88, 93, 98;ultraluminous X-ray sources(ULXs) and, 127–32; varietiesof, 58–60; velocity curvesand, 68–69, 101–2, 156–57

X-ray bursts, 74, 119X-rays: accretion and, 25, 27,

29, 31–34, 60–63; Chandramission and, 55; formationand, 112–15, 118–19, 124;Geiger counters and, 54;gravitational waves and, 150,154; intermediate-mass black

holes and, 127–32; Moonand, 54; spatial resolutionand, 127–28; spin and, 140,144, 146–48; state changesand, 62; stellar-mass blackholes and, 53–66, 68–69,74–77; Sun and, 54;supermassive black holes and,87–88, 93, 98; transients and,61, 69, 71, 146; Uhurusatellite and, 55;ultraluminous X-ray sources(ULXs) and, 127–32; vs.optical light, 53

X-ray states, 62, 64, 144, 147

Documents

WHAT DOES A Black Hole Look Like?