Astrophysics of Black Holes

Astrophysics and Space Science Library 440
Cosimo Bambi Editor
Astrophysics
of Black Holes
From Fundamental Aspects
to Latest Developments
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Astrophysics of Black Holes
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Astrophysics and Space Science Library
EDITORIAL BOARD
Chairman
W.B. BURTON, National Radio Astronomy Observatory, Charlottesville,
VA, USA (bburton@nrao.edu); University of Leiden, The Netherlands
(burton@strw.leidenuniv.nl)
F. BERTOLA, University of Padua, Italy
C.J. CESARSKY, Commission for Atomic Energy, Saclay, France
P. EHRENFREUND, Leiden University, The Netherlands
O. ENGVOLD, University of Oslo, Norway
A. HECK, Strasbourg Astronomical Observatory, France
E.P.J. VAN DEN HEUVEL, University of Amsterdam, The Netherlands
V.M. KASPI, McGill University, Montreal, Canada
J.M.E. KUIJPERS, University of Nijmegen, The Netherlands
H. VAN DER LAAN, University of Utrecht, The Netherlands
P.G. MURDIN, Institute of Astronomy, Cambridge, UK
B.V. SOMOV, Astronomical Institute, Moscow State University, Russia
R.A. SUNYAEV, Space Research Institute, Moscow, Russia
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Cosimo Bambi
Editor
Astrophysics of Black Holes

From Fundamental Aspects to Latest
Developments
123
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Editor
Cosimo Bambi
Department of Physics
Fudan University
Shanghai
China
ISSN 0067-0057 ISSN 2214-7985 (electronic)

Astrophysics and Space Science Library
ISBN 978-3-662-52857-0 ISBN 978-3-662-52859-4 (eBook)
DOI 10.1007/978-3-662-52859-4
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Preface
The Fudan Winter School on Astrophysical Black Holes was held at Fudan
University, in Shanghai, from February 10 to 15, 2014. The school was organized
with the spirit to contribute to the growth of the high-energy astrophysics com-
munity in China and to promote the interaction among people working in different
places of the country. It seemed also to be a good way to make know the new
gravity and high-energy physics group at the Department of Physics of Fudan
University. Departments of physics in China have a strong tradition in condensed
matter and applied physics, while they are weak in the other fields. As the rest of the
country, they are now in a phase of quick expansion and they are trying to form
research groups working even in other areas. High-energy astrophysics seems to be
one of the most active new directions, and there are an increasing number of
students who want to work in this field. However, with the exception of a few cases,
departments of physics in China have not yet a satisfactory astrophysics program.
Fudan University is probably one of the best examples: While it is one of the top
universities in China, there is no undergraduate or postgraduate course in
astronomy/astrophysics.
The school consisted of six series of lectures given by experts in their research
field, covering basic topics in black hole astrophysics and important directions in
current research. It was mainly intended for master/Ph.D. students. The event was
attended by approximately 60 participants. At the end of the school, we considered
the possibility of organizing a contributed volume, by inviting each lecturer to write
an independent chapter. Most lecturers accepted the invitations, and the result is this
book. This volume has all the features to become a very useful reference book to
researchers and students working on astrophysical black holes.
Shanghai Cosimo Bambi

January 2016
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Contents
1 Black Hole Accretion Discs . . . . . . . . . . . . . . . . . . . . . . . . . . .... 1

Jean-Pierre Lasota
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Disc-Driving Mechanism; Viscosity . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 The a-Prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Geometrically Thin Keplerian Discs . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Disc Vertical Structure . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Disc Radial Structure . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.3 Self-gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.4 Stationary Discs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.5 Radiative Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.6 Shakura–Sunyaev Solution . . . . . . . . . . . . . . . . . . . . . 18
1.4 Disc Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 The Thermal Instability . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.2 Thermal Equilibria: The S-Curve . . . . . . . . . . . . . . . . . 21
1.4.3 Irradiation and Black Hole X-Ray Transients. . . . . . . . . 24
1.4.4 Maximum Accretion Rate and Decay Timescale . . . . . . 30
1.4.5 Comparison with Observations . . . . . . . . . . . . . . . . . . 31
1.5 Black Holes and Advection of Energy . . . . . . . . . . . . . . . . . . . 33
1.5.1 Advection-Dominated-Accretion-Flow Toy Models . . . . 34
1.6 Accretion Discs in Kerr Spacetime. . . . . . . . . . . . . . . . . . . . . . 40
1.6.1 Kerr Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.6.2 Privileged Observers . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.6.3 The Ergosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.6.4 Equatorial Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.7 Accretion Flows in the Kerr Spacetime. . . . . . . . . . . . . . . . . . . 50
1.7.1 Kinematic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.7.2 Description of Accreting Matter. . . . . . . . . . . . . . . . . . 51
1.8 Slim-Disc Equations in Kerr Geometry. . . . . . . . . . . . . . . . . . . 52
1.8.1 Mass Conservation Equation . . . . . . . . . . . . . . . . . . . . 52
1.8.2 Equation of Angular Momentum Conservation . . . . . . . 53
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viii Contents
1.8.3 Equation of Momentum Conservation. . . . . . . . . . . . . . 53

1.8.4 Equation of Energy Conservation. . . . . . . . . . . . . . . . . 54
1.8.5 Equation of Vertical Balance of Forces. . . . . . . . . . . . . 55
1.9 The Sonic Point and the Boundary Conditions . . . . . . . . . . . . . 55
1.9.1 The “No-Torque Condition” . . . . . . . . . . . . . . . . . . . . 55
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2 Transient Black Hole Binaries . . . . . . . . . . . . . . . . . ........... 61
Tomaso M. Belloni and Sara E. Motta
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2 X-ray Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2.1 Energy Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.2.2 Fast Time Variability . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.2.3 Long-Term Time Evolution. . . . . . . . . . . . . . . . . . . . . 75
2.3 Radio/IR Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3.1 Radio Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3.2 Accretion–Ejection . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.4 Winds and Outflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.4.1 Accretion Disc Winds and Atmospheres . . . . . . . . . . . . 83
2.4.2 Winds Launching Mechanism . . . . . . . . . . . . . . . . . . . 84
2.5 The Full Accretion–Ejection Picture . . . . . . . . . . . . . . . . . . . . . 85
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3 Black Hole Spin: Theory and Observation . . . . . . . . . . . . . . . . ... 99
M. Middleton
3.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.2 Useful Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.2.1 Frame-Dragging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.3 Observational Tests of Spin I—the Energy Spectral Domain . . . . 108
3.3.1 Modelling the Continuum (Disc) Spectrum . . . . . . . . . . 108
3.3.2 Modelling the Reflection Spectrum . . . . . . . . . . . . . . . 115
3.3.3 Results: BHBs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.3.4 Results: AGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.3.5 Implications: Powering of Ballistic Jets. . . . . . . . . . . . . 135
3.3.6 Implications: Retrograde Spins? . . . . . . . . . . . . . . . . . . 138
3.4 Observational Tests of Spin II—The Time Domain and
Relativistic Precession Model . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.5 Observational Tests of Spin III—The Energy–Time Domain . . . . 141
3.6 Concluding Remarks and Future Approaches . . . . . . . . . . . . . . 143
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4 Winds from Black Hole Accretion Flows: Formation and Their
Interaction with ISM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Feng Yuan
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
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4.2 Formation of Wind from a Hot Accretion Flow . . . . . . . . . . . . . 154

4.2.1 Brief History of Study of Wind from Hot
Accretion Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.2.2 Main Properties of Winds . . . . . . . . . . . . . . . . . . . . . . 157
4.2.3 Acceleration Mechanism of Wind and Disk Jet . . . . . . . 162
4.2.4 Why Do Winds Exist? . . . . . . . . . . . . . . . . . . . . . . . . 163
4.3 Interaction of Winds with Interstellar Medium:
The Formation of the Fermi Bubbles . . . . . . . . . . . . . . . . . . . . 164
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5 A Brief Review of Relativistic Gravitational Collapse . . . . . . . . . . . 169
Daniele Malafarina
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.2 Einstein’s Equations for the Collapsing Interior . . . . . . . . . . . . . 171
5.2.1 Co-moving Coordinates . . . . . . . . . . . . . . . . . . . . . . . 171
5.2.2 Misner–Sharp Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.2.3 Einstein’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.3 Matching with an Exterior Metric . . . . . . . . . . . . . . . . . . . . . . 176
5.4 Regularity, Scaling, and Energy Conditions. . . . . . . . . . . . . . . . 178
5.4.1 Regularity and Scaling . . . . . . . . . . . . . . . . . . . . . . . . 178
5.4.2 Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.4.3 Shell Crossing Singularities. . . . . . . . . . . . . . . . . . . . . 183
5.5 Trapped Surfaces and Singularities. . . . . . . . . . . . . . . . . . . . . . 183
5.6 Homogeneous Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.6.1 Apparent Horizon and Singularity . . . . . . . . . . . . . . . . 188
5.7 Inhomogeneous Dust and Collapse with Pressures . . . . . . . . . . . 188
5.8 Collapse in Astrophysics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Appendix A: General Relativity in a Nutshell. . . . . . . . . . . . . . . . . . . . 199
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Contributors
Tomaso M. Belloni INAF-Osservatorio Astronomico di Brera, Merate, Italy

Jean-Pierre Lasota Institut d’Astrophysique de Paris, CNRS et Sorbonne
Universités, UPMC Univ Paris 06, Paris, France; Nicolaus Copernicus
Astronomical Center, Warsaw, Poland
Daniele Malafarina Department of Physics and Center for Field Theory and
Particle Physics, Fudan University, Shanghai, China; Physics Department, SST,
Nazarbayev University, Astana, Kazakhstan
M. Middleton Institute of Astronomy, Cambridge, UK
Sara E. Motta Department of Physics, Astrophysics, University of Oxford,
Oxford, UK
Feng Yuan Shanghai Astronomical Observatory, Chinese Academy of Sciences,
Shanghai, China
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Chapter 1
Black Hole Accretion Discs
Jean-Pierre Lasota
Abstract This is an introduction to the models of accretion discs around black

holes. After a presentation of the nonrelativistic equations describing the structure and
evolution of geometrically thin accretion discs, we discuss their steady-state solutions
and compare them to observation. Next, we describe in detail the thermal–viscous
disc instability model and its application to dwarf novae for which it was designed
and its X-ray irradiated disc version which explains the soft X-ray transients, i.e.
outbursting black hole low-mass X-ray binaries. We then turn to the role of advection
in accretion flows onto black holes illustrating its action and importance with a toy
model describing both ADAFs and slim discs. We conclude with a presentation of
the general-relativistic formalism describing accretion discs in the Kerr spacetime.
1.1 Introduction
The author of this chapter is old enough to remember the days when even serious
astronomers doubted the existence of accretion discs and scientists snorted with
contempt at the suggestion that there might be such things as black holes; the very
possibility of their existence was rejected, and the idea of black holes was dismissed
as a fancy of eccentric theorists. Today, some 50 years later, there is no doubt about
the existence of accretion discs and black holes; both have been observed and shown
to be ubiquitous in the universe. The spectacular ALMA image of the protostellar
disc in HL Tau [46] is breathtaking, and we can soon expect to see the silhouette
of a supermassive black hole in the centre of the Galaxy in near infrared [57] or
millimetre waves [10].
J.-P. Lasota (B)

Institut d’Astrophysique de Paris, CNRS et Sorbonne Universités,
UPMC Univ Paris 06, UMR 7095, 98bis Bd Arago, 75014 Paris, France
e-mail: lasota@iap.fr
J.-P. Lasota
Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland
© Springer-Verlag Berlin Heidelberg 2016 1

C. Bambi (ed.), Astrophysics of Black Holes, Astrophysics
and Space Science Library 440, DOI 10.1007/978-3-662-52859-4_1
2 J.-P. Lasota
Understanding accretion discs around black holes is interesting in itself because of

the fascinating and complex physics involved but is also fundamental for understand-
ing the coupled evolution of galaxies and their nuclear black holes, i.e. fundamental
for the understanding the growth of structures in the universe. The chance that inflows
onto black holes are strictly radial, as assumed in many models, is slim.
The aim of the present chapter was to introduce the reader to models of accretion
discs around black holes. Because of the smallness of black holes, the sizes of their
accretion discs span several orders of magnitude: from close to the horizon up 100 000
or even 1 000 000 black hole radii. This implies, for example, that the temperature
in a disc around a stellar-mass black hole varies from 107 K, near the its surface, to
∼103 K near the disc’s outer edge at 105 black hole radii, say. Thus, studying black
hole accretion discs allows the study of physical regimes relevant also in a different
context and, inversely, the knowledge of accretion disc physics in other systems,
such as protostellar discs or cataclysmic variable stars, is useful or even necessary
for understanding the discs around black holes.
Section 1.2 contains a short discussion of the disc-driving mechanisms and intro-
duces the α-prescription used in this chapter. In Sect. 1.3 after presenting the general
framework of the geometrically thin-disc model, we discuss the properties of station-
ary solutions and the Shakura–Sunyaev solution in particular. The dwarf nova disc
instability model and its application to black hole transient sources are the subject of
Sect. 1.4. The role of advection in accretion onto black holes is presented in Sect. 1.5
with the main stress put on high accretion rate flows. Finally, Sects. 1.6 and 1.7 about
the general-relativistic version of the accretion disc equations conclude the present
chapter.
Notations and definitions

The Schwarzschild radius (radius of a non-rotating black hole) is
2GM M
RS = 2
= 2.95 × 105 cm, (1.1)
c M
where M is the mass of the gravitating body and c the speed of light.
The Eddington accretion rate is defined as
LEdd 1 4π GM 1 2π cRS −1 M
ṀEdd = = = = 1.6 × 1018 η0.1 g s−1 , (1.2)
ηc2 η cκes η κes M
where η = 0.1η0.1 is the radiative efficiency of accretion and κes the electron
scattering (Thomson) opacity.
We will often use accretion rate measured in units of Eddington accretion rate:
Ṁ
ṁ = . (1.3)
ṀEdd
1 Black Hole Accretion Discs 3
Additional reading: There are excellent general reviews of accretion disc physics,
and they can be found in references [9, 18, 26, 55].
1.2 Disc-Driving Mechanism; Viscosity
In recent years, there have been an impressive progress in understanding the physical
mechanisms that drive disc accretion. It is now obvious that the turbulence in ionized
Keplerian discs is due to the magnetorotational instability (MRI) also known as the
Balbus–Hawley mechanisms [7, 8]. However, despite these developments, numerical
simulations, even in their global 3D form, suffer still from weaknesses that make their
direct application to real accretion flows almost infeasible.
One of the most serious problems is the value of the ratio of the (vertically aver-
aged) total stress to thermal (vertically averaged) pressure
τrϕ z
α= (1.4)
Pz
which according to most MRI simulation is ∼10−3 , whereas observations of dwarf

nova decay from outburst unambiguously show that α ≈ 0.1 − 0.2 [29, 54]. Only
recently Hirose et al. [22] showed that effects of convection at temperatures ∼104 K
increase α to values ∼0.1. This might solve the problem of discrepancies between
the MRI calculated and the observed value of α [11]. One has, however, to keep in
mind that the simulations in question have been performed in the so-called shearing
box and their validity in a generic 3D case has yet to be demonstrated.
Another problem is related to cold discs such as quiescent dwarf nova discs [31]
or protostellar discs [7]. For the standard MRI to work, the degree of ionization in
a weakly magnetized, quasi-Keplerian disc must be sufficiently high to produce the
instability that leads to a breakdown of laminar flow into turbulence which is the
source of viscosity-driving accretion onto the central body. In cold discs, the ionized
fraction is very small and might be insufficient for the MRI to operate. In any case,
in such a disc non-ideal MHD effects are always important. All these problems still
await their solution.
Finally, and very relevant to the subject of this chapter, there is the question of
stability of discs in which the pressure is due to radiation and opacity to electron
scattering. According to theory, such discs should be violently (thermally) unstable,
but observations of systems presumed to be in this regime totally infirm this predic-
tion. MRI simulations not only do not solve this contradiction but rather reinforce
it [24].
4 J.-P. Lasota
1.2.1 The α-Prescription
The α-prescription [53] is a rather simplistic description of the accretion disc physics,
but before one is offered better and physically more reliable options, its simplicity
makes it the best possible choice and has been the main source of progress in describ-
ing accretion discs in various astrophysical contexts.
One keeps in mind that the accretion-driving viscosity is of magnetic origin,
but one uses an effective hydrodynamical description of the accretion flow. The
hydrodynamical stress tensor is (see e.g. [34])
∂vϕ dΩ
τrϕ = ρν =ρ , (1.5)
∂R d ln R
where ρ is the density, ν the kinematic viscosity coefficient and vϕ the azimuthal
velocity (vϕ = RΩ).
In 1973, Shakura and Sunyaev proposed the (now famous) prescription
τrϕ = αP, (1.6)
where P is the total thermal pressure and α ≤ 1. This leads to

−1
dΩ
ν= αcs2 , (1.7)
d ln R
√
where cs = P/ρ is the isothermal sound speed and ρ the density. For the Keplerian
angular velocity

GM 1/2
Ω = ΩK = (1.8)
R3
this becomes
2 2
ν= αc /ΩK . (1.9)
3 s
Using the approximate hydrostatic equilibrium Eq. (1.19), one can write this as
2
ν≈ αcs H. (1.10)
3
Multiplying the rhs of Eq. (1.5) by the ring length (2π R) and averaging over the
(total) disc height, one obtains the expression for the total torque
dΩ
T = 2π RΣνR , (1.11)
d ln R
where +∞
Σ= ρ dz. (1.12)
−∞
For a Keplerian disc

T = 3π Σν K , (1.13)
( K = R2 ΩK is the Keplerian specific angular momentum.)

The viscous heating is proportional to τrϕ (dΩ/dR) [34]. In particular, the viscous
heating rate per unit volume is
dΩ
q+ = −τrϕ , (1.14)
d ln R
which for a Keplerian disc, using Eq. (1.6), can be written as
3
q+ = αΩK P, (1.15)
2
and the viscous heating rate per unit surface is therefore
TΩ
9
Q+ = = ΣνΩK2 . (1.16)
4π R 8
(The denominator in the first rhs is 2 × 2π R taking into account the existence of two
disc surfaces.)
Additional reading: References [7, 8, 22].
1.3 Geometrically Thin Keplerian Discs
The 2D structure of geometrically thin, non-self-gravitating, axially symmetric accre-

tion discs can be split into a 1 + 1 structure corresponding to a hydrostatic vertical
configuration and radial quasi-Keplerian viscous flow. The two 1D structures are
coupled through the viscosity mechanism transporting angular momentum and pro-
viding the local release of gravitational energy.
1.3.1 Disc Vertical Structure
The vertical structure can be treated as a one-dimensional star with two essential
differences:
1. the energy sources are distributed over the whole height of the disc, while in a
star, there limited to the nucleus,
6 J.-P. Lasota
2. the gravitational acceleration increases with height because it is given by the tidal
gravity of the accretor, while in stars, it decreases as the inverse square of the
distance from the centre.
Taking these differences into account, the standard stellar structure equations (see
e.g. [48]) adapted to the description of the disc vertical structure are listed below.
• Hydrostatic equilibrium
The gravity force is counteracted by the force produced by the pressure gradient:
dP
= ρgz , (1.17)
dz
where gz is the vertical component (tidal) of the accreting body gravitational accel-
eration:
∂ GM GM z
gz = ≈ 2 . (1.18)
∂z (R + z )
2 2 1/2 R R
The second equality follows from the assumption that z R. Denoting the typical
(pressure or density) scale height by H, the condition of geometrical thinness of
the disc is H/R 1, and writing dP/dz ∼ P/H, Eq. (1.17) can be written as
H cs
≈ , (1.19)
R vK
√
where vK = GM/R is the Keplerian velocity, and we made use of Eq. (1.18).
From Eq. (1.19), it follows that
H 1
≈ =: tdyn , (1.20)
cs ΩK
where tdyn is the dynamical time.

• Mass conservation
In 1D hydrostatic equilibrium, the mass conservation equation takes the simple
form of
dς
= 2ρ. (1.21)
dz
• Energy transfer—temperature gradient

d ln T d ln P
=∇ . (1.22)
dz dz
For radiative energy transport
κR PFz
∇rad = , (1.23)
4Pr cgz
where Pr is the radiation pressure and κR the Rosseland mean opacity. From
Eqs. (1.22) and (1.23), one recovers the familiar expression for the radiative flux
16 σ T 3 ∂T 4σ ∂T 4
Fz = − =− (1.24)
3 κR ρ ∂z 3κR ρ ∂z
(Fz is positive because the temperature decreases with z so ∂T /∂z < 0).
The photosphere is at optical thickness τ 2/3 (see Eq. 1.74). The boundary con-
ditions are as follows: z = 0, Fz = 0, T = Tc , and ς = 0 at the disc midplane;
at the disc photosphere, ς = Σ and T 4 (τ = 2/3) = Teff 4
. For a detailed discus-
sion of radiative transfer, temperature stratification and boundary conditions see
Sect. 1.3.5.
In the same spirit as Eq. (1.19), one can write Eq. (1.24) as
4 σ Tc4 8 σ Tc4
Fz ≈ = , (1.25)
3 κR ρH 3 κR Σ
where Tc is the midplane (“central”) disc temperature. Using the optical depth
τ = κR ρH = (1/2)κR Σ, this can be written as
8 σ Tc4
Fz (H) ≈ = Q− , (1.26)
3 τ
(see Eq. 1.77 for a rigorous derivation of this formula).
Remark 1.1 In some references (e.g. in [18]), the numerical factor on the rhs is “4/3”
instead of “8/3”. This is due to a different definition of Σ: in our case, it is =2ρH,
whereas in [18] Σ = ρH.
In the case of convective energy transport, ∇ = ∇conv . Because convection in

discs is still not well understood (see, however, [22]), there is no obvious choice
for ∇conv . In practice, a prescription designed by Paczyński [41] for extended
stellar envelope is used [21], but this most probably does not represent very
accurately what is happening in convective accretion discs [11].
• Energy conservation
Vertical energy conservation should have the form
dFz
= q+ (z), (1.27)
dz
where q+ (z) corresponds to viscous energy dissipation per unit volume.

8 J.-P. Lasota
Remark 1.2 In contrast with accretion discs, stellar envelopes have dFz /dz = 0
The α-prescription does not allow deducing the viscous dissipation stratification (z
dependence), and it just says that the vertically averaged viscous torque is propor-
tional to pressure. Most often one assumes therefore that
3
q+ (z) = αΩK P(z), (1.28)
2
by analogy with Eq. (1.15), but such an assumption is chosen because of its simplicity
and not because of some physical motivation. In fact, MRI numerical simulations
suggest that dissipation is not stratified in the way is pressure [22].
• The vertical structure equations have to be completed by the equation of state

(EOS):
4σ 4 R
P = Pr + Pg = T + ρT , (1.29)
3c μ
where R is the gas constant and μ the mean molecular weight, and an equation
describing the mean opacity dependence on density and temperature.
1.3.2 Disc Radial Structure
• Continuity (mass conservation) equation has the form
∂Σ 1 ∂ S(R, t)
=− (RΣvr ) + , (1.30)
∂t R ∂R 2π R
where S(R, t) is the matter source (sink) term.

In the case of an accretion disc in a binary system,
∂ Ṁext (R, t)
S(R, t) = (1.31)
∂R
represents the matter brought to the disc from the Roche lobe filling/mass losing
(secondary) companion of the accreting object. Ṁext ≈ Ṁtr , where Ṁtr is the mass-
transfer rate from the companion star. Most often one assumes that the transfer
stream delivers the matter exactly at the outer disc edge, but although this assump-
tion simplifies calculations, it is contradicted by observations that suggest that the
stream overflows the disc surface(s).
• Angular momentum conservation

∂Σ 1 ∂ 1 ∂ dΩ S (R, t)
=− (RΣ vr ) + R Σν
3
+ . (1.32)
∂t R ∂R R ∂R dR 2π R
This conservation equation reflects the fact that angular momentum is transported
through the disc by a viscous stress τrϕ = RΣνdΩ/dR. Therefore, if the disc is not
considered infinite (recommended in application to real processes and systems),
there must be somewhere a sink of this transported angular momentum S (R, t). For
binary semidetached binary systems, there is both a source (angular momentum
brought in by the mass-transfer stream form the stellar companion) and a sink
(tidal interaction taking angular momentum back to the orbit). The two respective
terms in the angular momentum equation can be written as
k ∂ Ṁext Ttid (R)

Sj (R, t) = − . (1.33)
2π R ∂R 2π R
Assuming Ω = ΩK , from Eqs. (1.30) and (1.32), one can obtain an diffusion equa-
tion for the surface density Σ:

∂Σ 3 ∂ 1/2 ∂

= R νΣR 1/2
. (1.34)
∂t R ∂R ∂R
Comparing with Eq. (1.30) one sees that the radial velocity induced by the viscous
torque is
3 ∂
vr = − νΣR1/2 , (1.35)
ΣR ∂R1/2
which is an example of the general relation

ν
vvisc ∼ . (1.36)
R
Using Eq. (1.10), one can write
−2
R R2 H H
tvis := ≈ ≈ α −1 . (1.37)
vvisc ν cs R
The relation between the viscous and the dynamical times is

−2
−1 H
tvis ≈ α tdyn . (1.38)
R
In thin (H/R 1) accretion discs, the viscous time is much longer that the
dynamical time. In other words, during viscous processes, the vertical disc
structure can be considered to be in hydrostatic equilibrium.
10 J.-P. Lasota
• Energy conservation
The general form of energy conservation (thermal) equation can be written as
follows:
ds ∂s ∂s
ρT := ρT + vr = q+ − q− + q, (1.39)
dR ∂t ∂R
where s is the entropy density, q+ and q− are the viscous and radiative energy
density, respectively, and
q is the density of external and/or radially transported
energy densities. Using the first law of thermodynamics Tds = dU + PdV , one
can write
ds dU ∂vr
ρT =ρ +P , (1.40)
dt dt ∂r
where U = Tc /μ(γ − 1).
Vertically averaging, but taking T = Tc , using Eq. (1.30) and the thermodynamical
relations from Appendix (for β = 1), one obtains
∂Tc ∂Tc Tc 1 ∂(Rvr ) Q+ − Q−

Q
+ vr + =2 + , (1.41)
∂t ∂R μcP R ∂R cP Σ cP Σ
where Q+ and Q− are the heating and cooling rates per unit surface, respectively.

Q = Qout + J with Qout corresponding to energy contributions by the mass-transfer
stream and tidal torques; J(T , Σ) represent radial energy fluxes that are a more
or less ad hoc addition to the 1+1 scheme to which they do not belong since it
assumes that radial gradients (∂/∂R) of physical quantities can be neglected.
The viscous heating rate per unit surface can be written as (see Eq. 1.16)
9
Q+ = νΣΩK2 (1.42)
8
while the cooling rate over unit surface (the radiative flux) is obviously
Q− = σ Teff
4
. (1.43)
In thermal equilibrium, one has
Q+ = Q− . (1.44)
The cooling time can be easily estimated from Eq. (1.44). The energy density to be
radiated away is ∼ρcs2 (see Eqs. 1.228 and 1.232), so the energy per unit surface
is ∼Σcs2 and the cooling (thermal) time is
Σcs2 Σcs2
tth = = ∼ α −1 ΩK−1 = α −1 tdyn . (1.45)
Q− Q+
www.ebook3000.com
Since α < 1, tth > tdyn and during thermal processes, the disc can be assumed to
be in (vertical) hydrostatic equilibrium.
For geometrical thin (H/R 1) accretion discs, one has the following hierar-
chy of the characteristic times
tdyn < tth tvis . (1.46)
(This hierarchy is similar to that of characteristic times in stars: the dynamical is

shorter than the thermal (Kelvin–Helmholtz) and the thermal is much shorter than
the thermonuclear time.)
1.3.3 Self-gravity
In this chapter, we are interested in discs that are not self-gravitating, i.e. in discs
where the vertical hydrostatic equilibrium is maintained against the pull of the accret-
ing body’s tidal gravity, whereas the disc’s self-gravity can be neglected. We will see
now under what conditions this assumption is satisfied.
The equation of vertical hydrostatic equilibrium can be written as

1 dP gs
= −g = (−gz − gs ) = gz 1 + =: −gz (1 + A) , (1.47)
ρ dz gz
and therefore, self-gravity is negligible when A 1. Treating the disc as an infinite

uniform plane (i.e. assuming the surface density does not vary too much with radius),
one can write its self-gravity as gs = 2π GΣ, whereas the z-component of the gravity
provided by the central body is gz = ΩK2 z (Eq. 1.18). Therefore, evaluating A at
z = H one gets
gs 2π GΣ
AH := = . (1.48)
gz H ΩK2 H
AH is related to the so-called Toomre parameter [56]
cs Ω
QT := , (1.49)
π GΣ
widely used in the studies of gravitational stability of rotating systems, through AH ≈
QT−1 . We will therefore express the condition of negligible self-gravity (gravitational
stability) as
QT > 1. (1.50)
12 J.-P. Lasota
Using Eqs. (1.19), (1.10) and (1.57), one can write the Toomre parameter as
3cs3
QT = , (1.51)
GṀ
or as function of the midplane temperature T = 104 T4

3/2
α T4
QT ≈ 4.6 × 107 , (1.52)
m ṁ
where m = M/ M . This shows that hot ionized (T 104 ) discs become self-
gravitating for high accretor masses and high accretion rates. Discs in close binary
systems (m 30) are never self-gravitating for realistic accretion rates (ṁ < 1000,
say) and even in IMBH binaries (if they exist) (hot) discs would also be free of grav-
itational instability. Around a supermassive black hole, however, discs can become
self-gravitating quite close to the black hole. For example, when the black hole mass
is m = 108 a hot disc will become self-gravitating at R/RS ≈ 100, for ṁ ∼ 10−2 . In
general, geometrically thin, non-self-gravitating accretion discs around supermassive
black holes have very a limited radial extent.
Additional reading: References [12, 19, 20, 35, 43, 56].
1.3.4 Stationary Discs
In the case of stationary (∂/∂t = 0) discs, Eq. (1.30) can be easily integrated giving
Ṁ := 2π RΣvr , (1.53)
where the integration constant Ṁ (mass/time) is the accretion rate.

Also, the angular momentum equation (1.32) can be integrated to give
− 2π RΣvr + 2π R3 ΣνΩ
= const. (1.54)
Or, using Eq. (1.53),

− Ṁ + T = const., (1.55)
where the torque T := 2π R3 ΣνΩ

(for a Keplerian disc T = 3π R2 ΣνΩK ).
Assuming that at the inner disc radius, the torque vanishes one gets const. =
−Ṁ in , where in is the specific angular momentum at the disc inner edge. Therefore,
Ṁ( − in ) = T (1.56)
which is a simple expression of angular momentum conservation.

For Keplerian discs, one obtains an important relation between viscosity and
accretion rate 1/2
Ṁ Rin
νΣ = 1− . (1.57)
3π R
From Eqs. (1.57), (1.42) and (1.43), and the thermal equilibrium equation (1.44), it
follows that 1/2
8 σ Tc4 3 GM Ṁ Rin
σ Teff =
4
= 1− . (1.58)
3 τ 8π R3 R
This relation assumes only that the disc is Keplerian and in thermal (Q+ =
Q− ) and viscous (Ṁ = const.) equilibrium. The viscosity coefficient is absent
because of the thermal equilibrium assumption: in such a state, the emitted
radiation flux cannot contain information about the heating mechanism, and it
only says that such a mechanism exists. Steady discs do not provide information
about the viscosity operating in discs or the viscosity parameter α. To get
this information, one must consider (and observe) time-dependent states of
accretion discs.
From Eq. (1.58) one obtains a universal temperature profile for stationary
Keplerian accretion discs
Teff ∼ R−3/4 . (1.59)
For an optically thick disc, the observed temperature T ∼ Teff and T ∼ R−3/4 should
be observed if stationary, optically thick Keplerian discs exist in the universe. And
vice versa, if they are observed, this proves that such discs exist not only on paper.
In 1985, Horne and Cook [23] presented the observational proof of existence of
Keplerian discs when they observed the dwarf nova binary system ZCha during
outburst (see Fig. 1.1).
1.3.4.1 Total Luminosity
The total luminosity of a stationary, geometrically thin accretion disc, i.e. the sum
of luminosities of its two surfaces, is
Rout 1/2
3GM Ṁ Rout Rin dR
2 σ Teff 2π RdR =
4
1− . (1.60)
Rin 2 Rin R R2
For Rout → ∞, this becomes

14 J.-P. Lasota
Fig. 1.1 The observed

temperature profile of the
accretion disc of the dwarf
nova Z Cha in outburst. Near
the outburst maximum such
a disc is in
quasi-equilibrium. The
observed profile, represented
by dots (pixels), is compared
with the theoretical profiles
calculated from Eq. (1.58)
and represented by
continuous lines. Pixels with
R < 0.03RL1 correspond to
the surface of the accreting
white dwarf whose
temperature is 40 000 K. The
accretion rate in the disc is
≈10−9 M y−1 . [Fig. 6
from [23]]
1 GM Ṁ 1
Ldisc = = Lacc . (1.61)
2 Rin 2
In the disc, the radiating particles move on Keplerian orbits; hence, they retain half
of the potential energy. If the accreting body is a black hole, this leftover energy
will be lost (in this case, however, the nonrelativistic formula of Eq. 1.61 does not
apply—see Eq. 1.178). In all the other cases, the leftover energy will be released in
the boundary layer, if any, and at the surface of the accretor, from where it will be
radiated away (Fig. 1.2).
The factor “3” in the rhs of Eq. (1.58) shows that radiation by a given ring in
the accretion disc does not come only from local energy release. Indeed, in a
ring between R and R + dR only
GM ṀdR
(1.62)
2R2
is being released, while
1/2
3GM Ṁ Rin
2 × 2π R Q+ dR = 1− dR (1.63)
2R2 R
is the total energy release. Therefore, the rest


GM Ṁ 3 Rin 1/2
1− dR (1.64)
R2 2 R
must diffuse out from smaller radii. This shows that viscous energy transport
redistributes energy release in the disc.
1.3.5 Radiative Structure
Here, we will show an example of the solution for the vertical thin disc structure which
exhibits properties impossible to identify when the structure is vertically averaged.
We will also consider here an irradiated disc—such discs are present in X-ray sources.
We write the energy conservation as:
dF
= q+ (R, z), (1.65)
dz
where F is the vertical (in the z direction) radiative flux and q+ (R, z) is the viscous
heating rate per unit volume. Equation (1.65) states that an accretion disc is not in
radiative equilibrium (dF/dz = 0), contrary to a stellar atmosphere. For this equation
to be solved, the function q+ (R, z) must be known. As explained and discussed in
Sect. 1.3.1, the viscous dissipation is often written as
Fig. 1.2 The observed

temperature profile of the
accretion disc of the dwarf
nova Z Cha in quiescence.
This one of the most
misunderstood figures in
astrophysics (see text). In
quiescence, the disc is not in
equilibrium. The flat
temperature profile is exactly
what the disc instability
model predicts: in
quiescence, the disc
temperature must be
everywhere lower than the
critical temperature, but this
temperature is almost
independent of the radius
(see Eq. 1.89) [Fig. 11 from
[58]]
16 J.-P. Lasota
3
q+ (R, z) = αΩK P(z). (1.66)
2
Viscous heating of this form has important implications for the structure of optically
thin layers of accretion discs and may lead to the creation of coronae and winds.
In reality, it is an ad hoc formula inspired by Eq. (1.15). We do not know yet (see,
however, [11]) how to describe the viscous heating stratification in an accretion
disc and Eq. (1.66) just assumes that it is proportional to pressure. It is simple and
convenient, but it is not necessarily true.
When integrated over z, the rhs of Eq. (1.65) using Eq. (1.66) is equal to viscous
dissipation per unit surface:
+∞
3
F+ = αΩK Pdz, (1.67)
2 0
where F + = (1/2)Q+ because of the integration from 0 to +∞, while Q+ contains

Σ which is integrated from −∞ to +∞ (Eq. 1.12).
One can rewrite Eq. (1.65) as
dF Fvis
= −f (τ ) , (1.68)
dτ τtot
where we introduced a new variable,the optical depth dτ = −κR ρdz, κR being the
+∞
Rosseland mean opacity and τtot = 0 κR ρdz is the total optical depth. f (τ ) is
given by:
+∞
P 0 κR ρdz
f (τ ) = . (1.69)
+∞
Pdz κR ρ
0
As ρ decreases approximately exponentially, f (τ ) is the ratio of two rather well

defined scale heights, the pressure and the opacity scale heights, which are compa-
rable, so that f is of order of unity.
At the disc midplane, by symmetry, the flux must vanish: F(τtot ) = 0, whereas at
the surface, (τ = 0)
F(0) ≡ σ Teff
4
= F+. (1.70)
Equation (1.70) states that the total flux at the surface is equal to the energy dissipated
by viscosity (per unit time and unit surface). The solution of Eq. (1.68) is thus
τ
+ f (τ )dτ
F(τ ) = F 1− 0
, (1.71)
τtot
τ
where 0 tot f (τ )dτ = τtot . Given that f is of order of unity, putting f (τ ) = 1 is a
reasonable approximation. The precise form of f (τ ) is more complex and is given
by the functional dependence of the opacities on density and temperature; it is of no

importance in this example. We thus take:

+ τ
F(τ ) = F 1− . (1.72)
τtot
To obtain the temperature stratification, one has to solve the transfer equation.
Here, we use the diffusion approximation
4 σ dT 4
F(τ ) = , (1.73)
3 dτ
appropriate for the optically thick discs we are dealing with. The integration of
Eq. (1.73) is straightforward and gives:

3 τ
T 4 (τ ) − T 4 (0) = τ 1− 4
Teff . (1.74)
4 2τtot
The upper (surface) boundary condition is as follows:
1 4
T 4 (0) = T + Tirr
4
, (1.75)
2 eff
4
where Tirr is the irradiation temperature, which depends on r, the albedo, the height
at which the energy is deposited and on the shape of the disc. In Eq. (1.75), T (0)
corresponds to the emergent flux and, as mentioned above, Teff corresponds to the
4
total flux (σ Teff = Q+ ) which explains the factor 1/2 in Eq. (1.75). The temperature
stratification is thus:

3 4 τ 2
T (τ ) = Teff τ 1 −
4
+ + Tirr
4
. (1.76)
4 2τtot 3
For τtot 1, the first term on the rhs has the form familiar from the stellar atmosphere
models in the Eddington approximation.
In this case at τ = 2/3, one has T (2/3) = Teff .
Also for τtot 1, the temperature at the disc midplane is
3
Tc4 ≡ T 4 (τtot ) = τtot Teff
4
+ Tirr
4
. (1.77)
8
It is clear, therefore, that for the disc inner structure to be dominated by irradiation
and the disc to be isothermal, one must have
18 J.-P. Lasota
Firr σ Tirr
4
≡ F+ (1.78)
τtot τtot
and not just Firr F + as is sometimes assumed. The difference between the two
criteria is important in LMXBs since, for parameters of interest, τtot 102 − 103 in
the outer disc regions.
1.3.6 Shakura–Sunyaev Solution
In their seminal and famous paper, Shakura and Sunyaev [53], found power law
stationary solutions of the simplified version of the thin-disc equations presented in
Sects. 1.3.1, 1.3.2 and 1.3.4. The 8 equations for the 8 unknowns Tc , ρ, P, Σ, H, ν,
τ and cs can be written as
Σ = 2Hρ (ı)
cs R3/2
H= (ıı)
(GM)1/2

P
cs = (ııı)
ρ
RρT 4σ 4
P= + T (ıv)
μ 3c
τ (ρ, Σ, Tc ) = κR (ρ, Tc )Σ (v)
2
ν(ρ, Σ, Tc , α) =
αcs H (vı)
3
1/2
Ṁ R0
νΣ = 1− (vıı)
3π R
1/2
8 σ Tc4 3 GM Ṁ R0
= 1− . (vııı)
3 τ 8π R3 R
Equations (ı) and (ıı) correspond to vertical structure Eqs. (1.21) and (1.19), Eq. (vıı)
is the radial Eq. (1.57), while Eq. (vııı) connects vertical to radial equations. Equation
(ııı) defines the sound speed, Eq. (ıv) is the equation of state, and Eq. (vı) contains
the information about opacities. The viscosity α parametrization introduced in [53]
provides the closure of the 8 disc equations. Therefore, they can be solved for a given
set of α, M, R and Ṁ.
Fig. 1.3 Stationary accretion disc surface density profiles for 4 values of accretion rate. From top
to bottom: Ṁ = 1018 , 1017 , 1016 and 1015 gs−1 . m = 10 M , α = 0.1. The continuous line corre-
sponds to the unirradiated disc, the dotted lines to an irradiated configuration. The inner, decreasing
segments of the continuous lines correspond to Eq. (1.80). Dashed lines describe irradiated disc
equilibria (see Sect. 1.4.3) [Fig. 9 from [15]]
Power law solutions of these equations exist in physical regimes where the opacity
can be represented in the Kramers form κ = κ0 ρ n T m and one of the two pressures, gas
or radiation, dominates over the other. In [53], three regimes have been considered:
(a) Pr Pg and κes κff

(b) Pg Pr and κes κff
(c) Pg Pr and κff κes .
Regimes (a) and (b) in which opacity is dominated by electron scattering will be
discussed in Sect. 1.5. Here, we will present the solutions of regime (c), i.e. we will
assume that
Pr = 0 and κR = κff = 5 × 1024 ρTc−7/2 cm2 g−1 . (1.79)
The solution for the surface density Σ, central temperature Tc and the disc relative
height (aspect ratio) are, respectively,
−3/4
Σ = 23 α −4/5 m1/4 R10 Ṁ17 f 7/10 g cm−2 ,
7/10
(1.80)
−3/4
Tc = 5.8 × 104 α −1/5 m1/4 R10 Ṁ17 f 3/10 K,
3/10
(1.81)
H
= 2.4 × 10−2 α −1/10 m−3/8 R10 Ṁ17 f 3/20 ,
1/8 3/20
(1.82)
R
where m = M/M , R10 = R/(1010 cm), Ṁ17 = Ṁ/(1017 g s−1 ), and f = 1

−(Rin /R)1/2 .
20 J.-P. Lasota
Although for a 10 M black hole, say, Shakura–Sunyaev solutions (1.80) and

(1.81) describe discs rather far from its surface (R 104 RS ) the regime of physical
parameters it addresses; particularly, temperatures around 104 K are of great impor-
tance for the disc physics because it is where accretion discs become thermally and
viscously unstable. This instability triggers dwarf nova outbursts when the accreting
compact object is a white dwarf and (soft) X-ray transients in the case of accreting
neutron stars and black holes.
It is characteristic of the Shakura–Sunyaev solution in this regime that the three
Σ, Tc , and Teff radial profiles vary as R−3/4 . (This implies that the optical depth τ
is constant with radius—see Eq. vııı.) For high accretion rates and small radii, the
assumption of opacity dominated by free–free and bound-free absorption will break
down and the solution will cease to be valid. We will come to that later. Now, we
will consider the other disc end: large radii.
One sees in Fig. 1.3 that for given stationary solution (Ṁ = const.), the R−3/4
slope of the Σ profiles extends down only to a minimum value Σmin (R) after which
the surface density starts to increase. With the temperature dropping below 104 K,
the disc plasma recombines and there is a drastic change in opacities leading to a
thermal instability.
Additional reading: We have assumed that accretion discs are flat. This might not
be true in general because accretion discs might be warped. This has important and
sometimes unexpected consequences, see, e.g. [28, 40, 45], and references therein.
1.4 Disc Instabilities
In this section, we will present and discuss the disc thermal and the (related) viscous
instabilities. First, we will discuss in some detail the cause of the thermal instability
due to recombination.
1.4.1 The Thermal Instability
A disc is thermally stable if radiative cooling varies faster with temperature than
viscous heating. In other words,
d ln σ Teff
4
d ln Q+
> . (1.83)
d ln Tc d ln Tc
Using Eq. (1.77), one obtains

−1
4
d ln Teff Tirr 4 d ln κ
=4 1− − . (1.84)
d ln Tc Tc d ln Tc
In a gas pressure-dominated disc, Q+ ∼ ρTH ∼ ΣT ∼ Tc . The thermal instability is

due to a rapid change of opacities with temperature when hydrogen begins to recom-
bine. At high temperatures, d ln κ/d ln Tc ≈ −4 (see Eq. 1.79). In the instability
region, the temperature exponent becomes large and positive d ln κ/d ln Tc ≈ 7 − 10
and in the end cooling is decreasing with temperature. One can also see that irra-
diation by furnishing additional heat to the disc can stabilize an otherwise unstable
equilibrium solution (dashed lines in Fig. 1.3).
This thermal instability is at the origin of outbursts observed in discs around
black holes, neutron stars and white dwarfs. Systems containing the first two classes
of objects are known as soft X-ray transients (SXTs, where “soft” relates to their
X-ray spectrum), while those containing white dwarfs are called dwarf novae (despite
the name that could suggest otherwise, nova and supernova outbursts have nothing
to do with accretion disc outbursts).
1.4.2 Thermal Equilibria: The S-Curve
We will first consider thermal equilibria of an accretion disc in which heating is

due only to local turbulence, leaving the discussion of the effects of irradiation
to Sect. 1.4.3. We put therefore Tirr =
Q = 0. Such an assumption corresponds to
discs in cataclysmic variables which are the best test bed for standard accretion disc
models. The thermal equilibrium in the disc is defined by the equation Q− = Q+
(see Eq. 1.41), i.e. by
9
σ Teff
4
= νΣΩK2 (1.85)
8
(Equation 1.16). In general, ν is a function of density and temperature, and in the
following, we will use the standard α-prescription Eq. (1.9). The energy transfer
equation provides a relation between the effective and the disc midplane temperatures
so that thermal equilibria can be represented as a Teff (Σ)—relation (or equivalently a
Ṁ(Σ)-relation). In the temperature range of interest (103 Teff 105 ), this relation
forms an S on the (Σ, Teff ) plane as in Fig. 1.4. The upper, hot branch corresponds
to the Shakura–Sunyaev solution presented in Sect. 1.3.6. The two other branches
correspond to solutions for cold discs—along the middle branch convection plays a
crucial role in the energy transfer.
Each point on the (Σ, Teff ) S-curve represents an accretion disc’s thermal equilib-
rium at a given radius, i.e. a thermal equilibrium of a ring at radius R. In other words,
each point of the S-curve is a solution of the Q+ = Q− equation. Points not on the
S-curve correspond to solutions of Eq. (1.41) out of thermal equilibrium: on the left
of the equilibrium curve, cooling dominates over heating, Q+ < Q− ; on the right
heating over, cooling Q+ > Q− . It is easy to see that a positive slope of the Teff (Σ)
curve corresponds to stable solutions. Indeed, a small increase of temperature of an
equilibrium state (an upward perturbation) on the upper branch, say, will bring the
ring to a state where Q+ < Q− so it will cool down getting back to equilibrium. In
22 J.-P. Lasota
Fig. 1.4 Thermal equilibria

of a ring in an accretion discs
around a m = 1.2 white
dwarf. The distance from the
centre is 109 cm; accretion
rate 6.66 × 1016 g/s. The
solid line corresponds to
Q+ = Q− . Σmin is the
critical (minimum) surface
density for a hot stable
equilibrium; Σmax is the
maximum surface density of
a stable cold equilibrium
a similar way, an downward perturbation will provoke increased heating bringing

back the system to equilibrium.
The opposite is happening along the S-curve’s segment with negative slope as
both temperature increase and decrease lead to a runaway. The middle branch of the
S-curve corresponds therefore to thermally unstable equilibria.
A stable disc equilibrium can be represented only by a point on the lower, cold
or the upper, hot branch of the S-curve. This means that the surface density in a
stable cold state must be lower than the maximal value on the cold branch, Σmax ,
whereas the surface density in the hot stable state must be larger than the minimum
value on this branch, Σmin . Both these critical densities are functions of the viscosity
parameter α, the mass of the accreting object and the distance from the centre and
depend on the disc’s chemical composition. In the case of solar composition, the
critical surface densities are
−0.80 1.11 −0.37
Σmin (R) = 39.9 α0.1 R10 m g cm−2 (1.86)
−0.83 1.18 −0.40
Σmax (R) = 74.6 α0.1 R10 m1 g cm−2 , (1.87)
(α = 0.1α0.1 ) and the corresponding effective temperatures are (T + designates the

temperature at Σmin , T − at Σmax )
+ −0.09
Teff = 6890 R10 M10.03 K (1.88)
− −0.10
Teff = 5210 R10 M10.04 K. (1.89)
The critical effective temperatures are practically independent of the mass and radius
because they characterize the microscopic state of disc’s matter (e.g. its ionization).
On the other hand, the critical accretion rates depend very strongly on radius:
+ −0.01 2.64
Ṁcrit (R) = 8.07 × 1015 α0.1 R10 M1−0.89 g s−1 (1.90)
−
Ṁcrit (R) = 2.64 × 10 15
α0.1
0.01 2.58
R10 M1−0.85 −1
gs . (1.91)
A stationary accretion disc in which there is a ring with effective temperature

contained between the critical values of Eqs. (1.89) and (1.88) cannot be stable.
Since the effective temperature and the surface density both decrease with radius,
the stability of a disc depends on the accretion rate and the disc size (see Fig. 1.3).
For a given accretion rate, a stable disc cannot have an outer radius larger than the
value corresponding to Eq. (1.86).
A disc is stable if the rate (mass-transfer rate in a binary system) at which mass
is brought to its outer edge (R ∼ Rd ) is larger than the critical accretion rate at
+
this radius Ṁcrit (Rd ).
In general, the accretion rate and the disc size are determined by mechanisms
and conditions that are exterior to the accretion process itself. In binary systems, for
instance, the size of the disc is determined by the masses of the system’s components
and its orbital period, while the accretion rate in the disc is fixed by the rate at which
the stellar companion of the accreting object loses mass, which in turn depends on the
binary parameters and the evolutionary state of this stellar mass donor. Therefore,
the knowledge of the orbital period and the mass-transfer rate should suffice to
determine whether the accretion disc in a given interacting binary system is stable.
Such knowledge allows testing the validity of the model as we will show in the next
section.
1.4.2.1 Dwarf Nova and X-Ray Transient Outbursts
• Local view: the limit cycle
Let us first describe what is happening during outbursts with a disc’s ring. Its states
are represented by a point moving in the Σ − Teff plane as shown in Fig. 1.4 which
represents accretion disc states at R = 109 cm (the accreting body has a mass of
1.2 M ). To follow the states of a ring during the outburst, let us start with an unstable
equilibrium state on the middle, unstable branch and let us perturb it by increasing
its temperature, i.e. let us shift it upwards in the Teff (Σ) plane. As we have already
learned, points out of the S-curve correspond to solutions out of thermal equilibrium
and in the region to the right of the S-curve heating dominates over cooling. The
resulting runaway temperature increase is represented by the point moving up and
reaching (in a thermal time) a quasi-equilibrium state on the hot and stable branch. It
is only a quasi -equilibrium because the equilibrium state has been assumed to lie on
the middle branch which corresponds to a lower temperature (and lower accretion
rate—see Eq. 1.58). Trying to get to its proper equilibrium, the ring will cool down and
move towards lower temperatures and surface densities along the upper equilibrium
24 J.-P. Lasota
Fig. 1.5 Local limit cycle of

the state of disc ring at
109 cm during a dwarf nova
outbursts. The arrows show
the direction of motion of the
system in the Teff (Σ) plane.
The figure represents results
of the disc instability model
numerical simulations. As
required by the comparison
of the model with
observations, the values of
the viscosity parameter α on
the hot and cold branches are
different. [Figure adapted
from [36]]
branch (in a viscous time). But the hot branch ends at Σmin , i.e. at a temperature higher
(and surface density lower) than required, so the ring will never reach its equilibrium
state, which is not surprising since this state is unstable. Once more the ring will find
itself out of thermal equilibrium, but this time in the region where cooling dominates
over heating. Rapid (thermal time) cooling will bring it to the lower cool branch.
There, the temperature is lower than required so the point representing the ring will
move up towards Σmax where it will have to interrupt its (viscous time) journey
having reached the end of equilibrium states before getting to the right temperature.
It will find itself out of equilibrium where heating dominated over cooling so it will
move back to the upper branch.
Locally, the state of a ring performing a limit cycle on the Σ − Teff plane moves
in viscous time on the stable S-curve branches and in a thermal time between them
when the ring is out of thermal equilibrium. The states on the hot branch correspond
to outburst maximum and the subsequent decay, whereas the quiescent corresponds
to moving on the cold branch. Since the viscosity is much larger on the hot than
on the cold branch, the quiescent is much longer than the outburst phase. The full
outburst behaviour can be understood only by following the whole disc evolution
(Fig. 1.5).
1.4.3 Irradiation and Black Hole X-Ray Transients
We will present the global view of thermal–viscous disc outbursts for the case of
X-ray transients. The main difference between accretion discs in dwarf novae and in
these systems is the X-ray irradiation of the outer disc in the latter. Assuming that
the irradiating X-rays are emitted by a point source at the centre of the system, one
can write the irradiating flux as
LX
σ Tirr
4
=C with LX = η min Ṁin , ṀEdd c2 , (1.92)
4π R2
where C = 10−3 C3 , η is the radiative efficiency (which can be 0.1 for ADAFs—
see below) and Ṁin is the accretion rate at the inner disc’s edge. Since the physics
and geometry of X-ray self-irradiation in accreting black–black hole systems are still
unknown, the best we can do is to parametrize our ignorance by q constant C that
observations suggest is ∼10−3 . Of course, one should keep in mind that in reality,
C might not be a constant [17].
Because the viscous heating is ∼Ṁ/R3 , there always exists a radius Rirr for which
σ Tirr
4
> Q+ = σ Teff
4
. If Rirr < Rd , where Rd is the outer disc radius, the outer disc
emission will be dominated by reprocessed X-ray irradiation and the structure modi-
fied as shown in Sect. 1.3.5. Irradiation will also stabilize outer disc regions (Eq. 1.84
and Fig. 1.6) allowing larger discs for a given accretion rate (see Fig. 1.3).
Irradiation modifies the critical values of the hot disc parameters:
+ −0.28 −0.78 0.92
Σirr = 72.4 C−3 α0.1 R11 M1−0.19 g cm−2 (1.93)
irr,+ −0.09 0.01 −0.15
Teff = 2860 C−3 α0.1 R11 M10.09 K (1.94)
+ −0.36 0.04 2.39
Ṁirr = 2.3 × 1017 C−3 α0.1 R11 M1−0.64 −1
gs . (1.95)
As we will see in a moment, irradiation also strongly influences the shape of

outburst’s light curve.
Fig. 1.6 Example S-curves for a pure helium disc with varying irradiation temperature T irr. The
various sets of S-curves correspond to radii R = 106 , 109 and 1010 cm. For each radius, the irra-
diation temperature Tirr is 0 K, 10 000 K and 20 000 K. α = 0.16. The instable branch disappears
for high irradiation temperatures. [From [32]. Reproduced with permission from Astronomy and
Astrophysics, c ESO]
26 J.-P. Lasota
* Rise to Outburst Maximum

During quiescence, the disc’s surface density, temperature and accretion rate are
everywhere (at all radii) on the cold branch, below their respective critical values
− −
Σmax (R), Teff and Ṁcrit (R). It is important to realize that in quiescence, the disc is not
steady: Ṁ = const. Matter transferred from the stellar companion accumulates in the
disc and is redistributed by viscosity. The surface density and temperature increase
(locally, this means that the solution moves up along the lower branch of the S-curve)
finally reaching their critical values. In Fig. 1.7, this happens at ∼1010 cm. The disc
parameters entering the unstable regime triggers an outburst. In the local picture,
this corresponds to leaving the lower branch of the S-curve. The next “moment” in a
thermal time is represented in the left panels of Fig. 1.7. This is when a large contrast
forms in the midplane temperature profile and when a surface-density spike is already
above the critical line. The disc is undergoing a thermal runaway at r ≈ 8 × 109 cm.
The midplane temperature rises to ∼70000 K. This raises the viscosity which leads
to an increase of the surface density, and a heating fronts start propagating inwards
and outwards in the disc as shown in Fig. 1.7. In this model, the disc is truncated at an
inner radius Rin ≈ 6 × 109 cm so the inwards propagating front quickly reaches the
inner disc radius with no observable effects. It is the outwards propagating heating
front that produces the outburst by heating up the disc and redistributing the mass
and increasing the surface density behind it because it is also a compression front.
One should stress here that two ad hoc elements must be added to the model
for it to reproduce observed outbursts of dwarf novae and X-ray transients.
• Viscosity. First, if the increase in viscosity were due only to the rise in the
temperature through the speed of sound (ν ∝ cs2 , see Eq. 1.9), the resulting
outbursts would have nothing to do with the observed ones. To reproduce
observed outbursts, one increases the value of α when a given ring of the
disc gets to the hot branch. Ratios of hot-to-cold α of the order of 4 are
used to describe dwarf nova outburst. Although in the outburst model, the
α increase is an ad hoc assumption, recent MRI simulations with physical
parameters corresponding to dwarf nova discs show an α increase induced
by the appearance of convection [22].
• Inner truncation. Second, as mentioned already, the inner disc is
assumed to be truncated in quiescence and during the rise to outburst.
Although such truncation is implied and/or required by observations, its
physical origin is still uncertain. The inner part of the accretion flow is of
course not empty but supposed to form a Ṁ = const. ADAF (see Sect. 1.5).
In our case (Fig. 1.7), the heating front reaches the outer disc radius. This corre-
sponds to the largest outbursts. Smaller-amplitude outbursts are produced when the
Fig. 1.7 The rise to outburst described in Sect. 1.4.3. The upper left panel shows Ṁin and Ṁirr
(dotted line); the bottom left panel shows the V magnitude. Each dot corresponds to one of the Σ
and Tc profiles in the right panels. The heating front propagates outwards. The disc expands during
the outburst due to the angular momentum transport of the material being accreted. At t ≈ 5.5 days,
the thin disc reaches the minimum inner disc radius of the model. The profiles close to the peak
are those of a steady-state disc (Σ ∝ Tc ∝ R−3/4 ). [From [16]. Reproduced with permission from
Astronomy and Astrophysics, c ESO]
front does not reach the outer disc regions. In an inside-out outburst1 the surface-
density spike has to propagate uphill, against the surface-density gradient because
just before the outburst Σ ∼ R1.18 —roughly parallel to the critical surface density.
Most of the mass is therefore contained in the outer disc regions. A heating front will
be able to propagate if the post-front surface density is larger than Σmin —in other
words, if it can bring successive rings of matter to the upper branch of the S-curve.
If not, a cooling front will appear just behind the Σ spike, the heating front will die
out, and the cooling front will start to propagate inwards (the heating-front will be
“reflected”).
The difficulty in inside-out fronts encounters when propagating is due to angular
momentum conservation. In order to move outwards, the Σ spike has to take with
it some angular momentum because the disc’s angular momentum increases with
radius. For this reason, inside-out front propagation induces a strong outflow. In
order for matter to be accreted, a lot of it must be sent outwards. That is why during
an inside-out dwarf nova outburst only ∼10 % of the disc’s mass is accreted onto
1 X-ray transient outbursts are always of inside-out type. In dwarf novae, both inside-out and outside-
in outbursts are observed and result from calculations [31].

28 J.-P. Lasota
the white dwarf. In X-ray transients, irradiation facilitates heating front propagation
(and disc emptying during decay—see next section).
The arrival of the heating front at the outer disc rim does not end the rise to
maximum. After the whole disc is brought to the hot state, a surface density (and
accretion rate) “excess” forms in the outer disc. The accretion rate in the inner
disc corresponds to the critical one but is much higher near the outer edge. While
irradiation keeps the disc hot, the excess diffuses inwards until the accretion rate
is roughly constant. During this last phase of the rise to outburst maximum, Ṁin
increases by a factor of 3:
+ −0.36 2.39
Ṁmax ≈ 3Ṁirr ≈ 7.0 × 1017 C−3 Rd,11 m−0.64 g s−1 . (1.96)
Irradiation has little influence on the actual vertical structure in this region and
Tc ∝ Σ ∝ R−3/4 , as in a non-irradiated steady disc. Only in the outermost disc
regions does the vertical structure becomes irradiation-dominated, i.e. isothermal.
* Decay
Figure 1.8 shows the sequel to what was described in Fig. 1.7. In general, the decay
from the outburst peak of an irradiated disc can be divided into three parts:
• First, X-ray irradiation of the outer disc inhibits cooling-front propagation. But
since the peak accretion rate is much higher than the mass-transfer rate,2 the disc
is drained by viscous accretion of matter.
• Second, the accretion rate becomes too low for the X-ray irradiation to prevent the
cooling front from propagating. The propagation speed of this front, however, is
controlled by irradiation.
• Third, irradiation plays no role and the cooling front switches off the outburst on
a local thermal timescale.
“Exponential Decay”
In Fig. 1.8 the “exponential decay” of the phase lasts until roughly day 80–100. At
the outburst peak, the accretion rate is almost exactly constant with radius; the disc is
quasi-stationary. The subsequent evolution is self-similar: the disc’s radial structure
evolves through a sequence of quasi-stationary (Ṁ(r) = const) states. Therefore,
νΣ ∼ Ṁin (t)/3π and the total mass of the disc is thus

2r
Md = 2π RΣdR ∝ Ṁin dr. (1.97)
3ν
At the outburst peak, the whole disc is wholly ionized, and except for the outermost
regions, its structure is very well represented by a Shakura–Sunyaev solution. In such
discs, as well as in irradiation dominated discs, the viscosity coefficient satisfies the
2 The peak luminosity is ∼3Ṁ + (R ), and for the disc to be unstable, the mass-transfer rate must be
irr d
+
lower than the critical rate: Ṁtr < Ṁirr (Rd ).
Fig. 1.8 Decay from outburst peak. The decay is controlled by irradiation until evaporation sets
in at t ≈ 170 days (Ṁin = Ṁevap (Rmin )). This cuts off irradiation, and the disc cools quickly. The
irradiation cut-off happens before the cooling front can propagate through most of the disc; hence,
the irradiation-controlled linear decay (t ≈ 80 − 170 days) is not very visible in the light curve.
Tirr (dotted line) is shown for the last temperature profile. [From [16]. Reproduced with permission
from Astronomy and Astrophysics, c ESO]
relation ν ∝ T ∝ Ṁ β/(1+β) . In hot Shakura–Sunyaev discs, β = 3/7 (Eq. 1.81), and

in irradiation dominated discs, β = 1/3 (Eq. 1.92). During the first decay phase, the
outer disc radius is almost constant so that using Eq. (1.97) the disc-mass evolution
can be written as:
dMd 1+β
= −Ṁin ∝ Md (1.98)
dt
β
showing that Ṁin evolves almost exponentially, as long as Ṁin can be considered as
constant (i.e. over about a decade in Ṁin ). “Exponential” decays in the DIM are only
approximately exponential.
The quasi-exponential decay is due to two effects:
1. X-ray irradiation keeps the disc ionized, preventing cooling-front propagation;
2. tidal torques keep the outer disc radius roughly constant.
“Linear” Decay
The second phase of the decay begins when a disc ring cannot remain in thermal
equilibrium. Locally, this corresponds to a fall onto the cool branch of the S-curve.
In an irradiated disc, this happens when the central object does not produce enough
30 J.-P. Lasota
X-ray flux to keep the Tirr (Rout ) above ∼104 K. A cooling front appears and propa-
gates down the disc at a speed of vfront ≈ αh cs .
In an irradiated disc, however, the transition between the hot and cold regions is
set by Tirr because a cold branch exists only for Tirr 104 K. In an irradiated disc, a
cooling front can propagate inwards only down to the radius at which Tirr ≈ 104 K,
i.e. as far as there is a cold branch to fall onto. Thus, the decay is still irradiation-
controlled. The hot region remains close to steady state, but its size shrinks Rhot ∼
1/2
Ṁin (as shown in Eq. 1.92 with Tirr (Rhot ) = const).
Thermal Decay
In the model shown in Fig. 1.8, irradiation is unimportant after t 170 − 190 days
because η becomes very small for Ṁin < 1016 g·s−1 when an ADAF forms. The
cooling front thereafter propagates freely inwards, on a thermal timescale. In this
particular case, the decrease of irradiation is caused by the onset of evaporation at
the inner edge which lowers the efficiency. In general, there is always a moment
at which Tirr becomes less than 104 K; evaporation just shortens the “linear” decay
phase.
1.4.4 Maximum Accretion Rate and Decay Timescale
Now, we will see that there are two observable properties of X-ray transients that,
when related one to the other, provide information and constraints on the physical
properties of the outbursting system. The first is the maximum accretion rate Ṁmax
(Eq. 1.96). The second is the decay time of the X-ray flux: as we have seen, disc
irradiation by the central X-rays traps the disc in the hot, high state, and only allows
a decay of Ṁ on the hot-state viscous timescale. This is
R2
t (1.99)
3ν
which using Eq. (1.9) gives
(GMR)1/2
t . (1.100)
3αcs2
Taking the critical midplane temperature Tc+ ≈ 16000 K, one gets for the decay
timescale 1/2 −1
t ≈ 32 m1/2 Rd,11 α0.2 days, (1.101)
where α0.2 = α/0.2. Eliminating R between (1.96) and (1.101) gives the accretion
rate through the disc at the start of the outburst as
Ṁ = 5.4 × 1017 m−3.03 (t30 α0.2 )4.78 g s−1 , (1.102)

with t = 30 t30 d. Assuming an efficiency of η of 10 %, the corresponding luminos-

ity is
L = 5.0 × 1037 η0.1 m−3.03 (t30 α0.2 )4.78 erg s−1 . (1.103)
1.4.5 Comparison with Observations
1.4.5.1 Sub-Eddington Outbursts
The peak luminosities of most of the soft X-ray transients are sub-Eddington. Equa-
tion (1.102) can be written using the Eddington ratio m := Ṁ/ṀEdd as
ṁ = 0.42η0.1 (α0.2 t30 )4.78 m−4.03 . (1.104)
This equation shows that the outburst peak will be sub-Eddington only if the outburst
decay time is relatively short or the accretor (black hole) mass is high; i.e. the observed
decay timescale is
−0.21 −1 0.84
t 50 η0.1 α0.2 m d, (1.105)
in good agreement with the compilation of X-ray transients outburst durations found
in [59]. This shows that the standard value of efficiency η0.1 1, and the value
α0.2 1 deduced from observations of dwarf novae, gives the correct order of mag-
nitude for the decay timescale of X-ray transients (from ≈3 days to ≈300 days). This
equation also implies that black hole transients should have longer decay timescales
than neutron star transients, all else being equal. Yan and Yu [59] find that outbursts
last on average ≈2.5 × longer in black hole transients than in neutron star transients
thus confirming this conclusion.
For sub-Eddington outbursts, Eq. (1.103) gives a useful relationship between dis-
tance D, bolometric flux F and outburst decay time t,
1/2
−1.5 η0.1
DMpc 1.0 m (α0.2 t50 )2.4 , (1.106)
F12
where D = DMpc Mpc and F=10−12 F12 erg s−1 cm−2 ; F = L/4π D2 and t = 50 t50 d.
Equation (1.106) shows that distant (D > 1Mpc) X-ray sources exhibiting vari-
ability typical of soft X-ray transients cannot contain black holes with masses superior
to stellar masses [33].
1.4.5.2 Observational Tests
Finally, one can test observationally if soft X-ray transients satisfy the necessary
condition for instability Ṁtr < Ṁcrit (Rd ), where Ṁcrit is the critical accretion rate for
32 J.-P. Lasota
either non-irradiated or irradiated discs. In Fig. 1.9, the critical accretion rates (1.90)
and (1.95) for respectively non-irradiated and irradiated disc around black holes are
plotted as Ṁ(Porb ) relation. This relation was obtained from disc radius—orbital
separation relation Rd (a) [42], where (from Kepler’s law) the orbital separation a =
2/3
3.53 × 1010 (m1 + m2 )1/3 Phr cm, where mi are the masses of the components in solar
units, and Phr is the orbital period in hours. Against these two critical lines, the actual
positions of the observed sources are marked. The mass-transfer rate being difficult
to measure, a proxy in the form of the accumulation rate
ΔE
Ṁaccum = (1.107)
trec ηc2
has been used. ΔE is the energy corresponding to the integrated X-ray luminos-
ity from during an outburst and trec the recurrence time of the outbursts. One can
see that all low-mass-X-ray-binary (LMXB) transients are in the unstable part of
the figure, as they should be if the model is correct. One can also see that all
Fig. 1.9 Mass-transfer rate as a function of the orbital period for SXTs with black holes. The
transient and persistent sources have been marked with respectively filled and open symbols. The
shaded grey areas indicated “DIM irr” and “DIM non irr” represent the separation between persistent
(above) and transient systems (below) according to the disc instability model when, respectively,
irradiation is taken into account and when it is neglected. The horizontal dashed line indicates the
Eddington accretion rate for a 10 M black hole. All the upper limits on the mass-transfer rate
are due to lower limits on the recurrence time. The upper limits on the mass-transfer rate of 4U
1957+115 and GS 1354-64 result from lower limits on the distance to the sources. The three left
closed arrows do not indicate actual upper limits on the orbital period of Cyg X-1, LMC X-1 and
LMC X-3. They emphasize that the radius of any accretion disc in these three high-mass XRBs is
likely to be smaller than the one derived from the orbital period since they likely transfer mass by a
(possibly focused) stellar wind instead of fully developed Roche lobe overflow. In the legend, the
solid horizontal line separates transient and persistent systems. (The dashed horizontal line stresses
that the persistent nature of 1E 1740.7-2942 and GRS 1758-258 is unclear.) [From [13]]
black hole LMXBs are transient. This is not true of neutron star LMXBs. Cyg X-
1 in which the stellar companion of the black hole is a massive star is observed
to be stable but according to Fig. 1.9 should be transient. This is not a problem
because in such a system matter from the high-mass companion is not transferred by
Roche-lobe overflow as in LMXBs, but lost through a stellar wind. In this case, the
Rd (a) relation used in the plot is not valid—the discs in such systems are smaller
which is marked by a left-directed arrow at the symbol marking the position of this
and two other similar objects (LMC X-1 and LMC X-3).
Additional reading: References [15, 16, 21, 31, 32].
1.5 Black Holes and Advection of Energy
Until now, we have neglected advection terms in the energy and momentum equa-
tions for stationary accretion flows. There two regimes of parameters where this
assumption is not valid, in both cases for the same reason: low radiative efficiency
when the time for radial motion towards the black hole is shorter than the radiative
cooling time. Low density (low accretion rate), hot, optically thin accretion flows are
poor coolers, and they are one of the two configurations where advection instead of
radiation is the dominant evacuation-of-energy (“cooling”) mechanism. Such opti-
cally thin flows are called ADAFs, for advection-dominated accretion flows. Also,
advection-dominated are high-luminosity flows accreting at high rates, but they are
called “slim discs” to account for their property of not being thin but still being
described as if this were not of much importance.
We shall start with optically thin flows.
• ADAFs
Advection-dominated accretion flow (ADAF) is a term describing accretion of
matter with angular momentum, in which radiation efficiency is very low. In their
applications, ADAFs are supposed to describe inflows onto compact bodies, such
as black holes or neutron stars; but very hot, optically thin flows are bad radiators
in general so that, in principle, ADAFs are possible in other contexts. Of course in
the vicinity of black holes or neutron stars, the virial (gravitational) temperature
is Tvir ≈ 5 × 1012 (RS /R) K, so that in optically thin plasmas, at such tempera-
tures, both the coupling between ions and electrons and the efficiency of radiation
processes are rather feeble. In such a situation, the thermal energy released in the
flow by the viscosity, which drives accretion by removing angular momentum, is
not going to be radiated away, but will be advected towards the compact body.
If this compact body is a black hole, the heat will be lost forever, so that advec-
tion, in this case, acts as sort of a “global” cooling mechanism. In the case of
infall onto a neutron star, the accreting matter lands on the star’s surface and the
(reprocessed) advected energy will be radiated away. There, advection may act
only as a “local” cooling mechanism. (One should keep in mind that, in general,
34 J.-P. Lasota
advection may also be responsible for heating, depending on the sign of the tem-
perature gradient—in some conditions, near the black hole, advection heats up
electrons in a two-temperature ADAF.)
In general, the role of advection in an accretion flow depends on the radiation
efficiency which in turns depends on the microscopic state of matter and on the
absence or presence of a magnetic field. If, for a given accretion rate, radiative cool-
ing is not efficient, advection is necessarily dominant, assuming that a stationary
solution is possible.
• Slim discs
At high accretion rates, discs around black holes become dominated by radiation
pressure in their inner regions, close to the black hole. At the same time, the
opacity is dominated by electron scattering. In such discs, H/R is no longer 1.
But this means that terms involving the radial velocity are no longer negligible
since vr ∼ αcs (H/R). In particular, the advective term in the energy conservation
equation vr ∂S/∂R (see Eq. 1.39) becomes important and finally, at super-Eddington
rates, dominant. When Q+ = Qadv , the accretion flow is advection dominated and
called a slim disc.
1.5.1 Advection-Dominated-Accretion-Flow Toy Models
One can illustrate the fundamental properties of ADAFs and slim discs with a simple
toy model. The advection “cooling” (per unit surface) term in the energy equation
can be written as
Ṁ 2
Qadv = c ξa (1.108)
2π R2 s
(see Eq. 1.239).
Using the (nonrelativistic) hydrostatic equilibrium equation
H cs
≈ (1.109)
R vK
one can write the advection term as

2
κes c ṁ H
Qadv = Υ ξa (1.110)
2R η R
whereas the viscous heating term can be written as

+ 3 κes c ṁ
Q =Υ , (1.111)
8 R η
where 2
cRS
Υ = . (1.112)
κes R
Since ξa ∼ 1,
2
+ H
Q adv
≈Q (1.113)
R
and, as said before, for geometrically thin discs (H/R 1), the advective term Qadv
is negligible compared to the heating term Q+ and in thermal equilibrium viscous
heating must be compensated by radiative cooling. Things are different at, very
high temperatures, when (H/R) ∼ 1. Then, the advection term is comparable to the
viscous term and cannot be neglected in the equation of thermal equilibrium. In some
cases, this term is larger than the radiative cooling term Q− and (most of) the heat
released by viscosity is advected toward the accreting body instead of being locally
radiated away as happens in geometrically thin discs.
From Eq. (1.57), one can obtain a useful expression for the square of the relative
disc height (or aspect ratio):
2 √ 1/2
H 2 ṁ RS
= (αΣ)−1 . (1.114)
R κes η R
Deriving Eq. (1.114), we used the viscosity prescription ν = (2/3)αcs2 /ΩK .

Using this equation, one can write for the advective cooling
2
−1 ṁ
Q adv
= Υ ΩK ξa (αΣ) . (1.115)
η
The thermal equilibrium (energy) equation is
Q+ = Qadv + Q− . (1.116)
The form of the radiative cooling term depends on the state of the accreting matter,
i.e. on its temperature, density and chemical composition. Let us consider two cases
of accretion flows:
• optically thick
and
• optically thin.
For the optically thick case, we will use the diffusion approximation formula
8 σ Tc4
Q− = , (1.117)
3 κR Σ
36 J.-P. Lasota
and assume κR = κes . With the help of Eq. (1.114), this can be brought to the form
1/2 2 1/2
− κes RS R ṁ
ΩK (αΣ)−1/2
3/2
Qthick = 8Υ . (1.118)
c RS η
For the optical thin case of bremsstrahlung radiation, we have
Q− = 1.24 × 1021 Hρ 2 T 1/2 (1.119)
which using Eq. (1.114) can be written

2
− R
Qthin = 3.4 × 10−6 Υ ΩK α −2 (αΣ)2 . (1.120)
RS
• In the optically thick case, we have therefore

2 1/2
ṁ R ṁ
ξa + 0.18 (αΣ)
η RS η
5/4 1/2
R ṁ
+ 2.3 (αΣ)1/2 =0 (1.121)
RS η
• In the optically thin case, the energy equation has the form
2 1/2
ṁ R ṁ
ξa + 0.18 (αΣ)
η RS η
2
R
+ 3 × 10−6 α −2 (αΣ)3 = 0 (1.122)
RS
There are two distinct types of advection-dominated accretion flows: optically thin
and optically thick. We will first deal with optically thin flows known as ADAFs.
1.5.1.1 Optically Thin Flows: ADAFs
For prescribed values α and ξa , Eq. (1.122) is a quadratic equation in (ṁ/η) whose
solutions in the form of ṁ(Σ) describe thermal equilibria at a given value of R.
Obviously, for a given Σ, this equation has at most two solutions. The solutions
form two branches on the ṁ(αΣ)—plane:
• the ADAF branch 1/2
R
ṁ = 0.53κes η ξa−1 αΣ. (1.123)
RS
and
• the radiatively cooled branch
3/2
R
ṁ = 1.9 × 10−5 η ξa−1 α −2 (αΣ)2 . (1.124)
RS
From Eqs. (1.123 and 1.124), it is clear that there exists a maximum accretion rate
for which only one solution of Eq. (1.122) exists. This implies the existence of a
maximum accretion rate at
1/2
R
ṁmax ≈ 1.7 × 10 η α 3 2
. (1.125)
RS
This is where the two branches formed by thermal equilibrium solutions on the
ṁ(αΣ)—plane meet as seen on Fig. 1.10.
Fig. 1.10 a Thermal equilibria for optically thick (The right solid S-shaped line) and optically thin
(the left solid line) accretion flows. The upper branches represent advection-dominated solution
(ADAFs). Flows above the dotted lines τ = 1 are optically thin − − −τ is the effective opti-
cal depth calculated for radiation-pressure-dominated (upper line) or gas-dominated (lower line)
configurations. It is assumed that MBH = 10 M , R = 5RS , α = 0.1 and ξa = 1. b The same for
α = 0.01 [From [4]]
38 J.-P. Lasota
The value of ṁmax depends on the cooling mechanism in the accretion flow;
the free–free cooling is not a realistic description of the emission in the vicin-
ity ((R/RS ) 103 ) of a black hole. The flow there most probably forms a two-
temperature plasma. In such a case, ṁmax ≈ 10α 2 with almost no dependence on
radius. For larger radii, ṁmax decreases with radius.
1.5.1.2 Optically Thick Flows: Slim Discs
Since the first two terms in Eq. (1.121) are the same as in (Eq. 1.122), the high,
ṁ, advection-dominated solution is the same as in the optically thin case but now
represents the
• Slim-disc branch 1/2
R
ṁ = 0.53 κes η ξa−1 αΣ. (1.123)
RS
Now, the full Eq. (1.121) is a cubic equation in ṁ1/2 , and on the ṁ(αΣ) plane,
its solution forms the two upper branches of the S-curve shown in Fig. 1.10. The
uppermost branch corresponds to slim discs, while the branch with negative slope
represents the Shakura–Sunayev solution in the regime a. (see Sect. 1.3.6), i.e.
• a radiatively cooled, radiation-pressure-dominated accretion disc
3/2
−1 R
ṁ = 160 κes η (αΣ)−1 (1.126)
RS
1.5.1.3 Thermal Instability of Radiation–Pressure-Dominated Discs
Radiation-pressure-dominated (P = Prad ) accretion discs are thermally unstable

when opacity is due to electron scattering on electrons. Indeed,
4
d ln Teff
=4 (1.127)
d ln Tc
because κR = κes = const., while in a radiation pressure-dominated disc Q+ ∼

νΣ ∼ HT 4 ∼ T 8 /Σ so
d ln Q+ 4
d ln Teff
=8 > (1.128)
d ln Tc d ln Tc
and the disc is thermally unstable. This solution is represented by the middle branch
with negative slope (see Eq. 1.126) in Fig. 1.10. The presence of this instability in
the model is one of the unsolved problems of the accretion disc theory because it
contradicts observations which do not show any unstable behaviour in the range of
luminosities where discs should be in the radiative pressure and electron-scattering
opacity domination regime.
1.5.1.4 Slim Discs and Super-Eddington Accretion
From Eqs. (1.114) and (1.126), one obtains for the disc aspect ratio

H ṁ RS
= 0.11 (1.129)
R η R
which shows that the height of a radiation dominated disc is constant with radius and
proportional to the accretion rates.
But this means that with increasing ṁ, advection becomes more and more impor-
tant (see, e.g. Eq. 1.113) and for
ṁ R
≈ 9.2 (1.130)
η RS
advection will take over radiation as the dominant cooling mechanism and the solu-
tion will represent a slim disc. Equation (1.130) can be also interpreted as giving the
transition radius between radiatively and advectively cooled disc for a given accretion
rate ṁ:
Rtrans 0.1
≈ ṁ (1.131)
RS η
Another radius of interest is the trapping radius at which the photon diffusion (escape)
time Hτ/c is equal to the viscous infall time R/vr

Hτ vr HκΣ Ṁ H ṁ
Rtrapp = = = RS . (1.132)
c c 2π RΣ R η
Notice that both Rtrans and Rtrapp are proportional to the accretion rate.
In an advection-dominated disc, the aspect ration H/R is independent of the
accretion rate: 1/4
H R
= 0.86 ξa , (1.133)
R RS
and therefore contrary to radiatively cooled discs, slim disc do not puff up with
increasing accretion rate.
Putting (1.133) into Eq. (1.132), one obtains
1/4
Rtrapp R ṁ
= 0.86 ξa−1/2 . (1.134)
R RS η
Radiation inside the trapping radius is unable to stop accretion, and since Rtrapp ∼ ṁ,
there is no limit on the accretion rate onto a black hole.
The luminosity of the toy-model slim disc can be calculated from Eqs. (1.118)
and (1.123) giving
40 J.-P. Lasota
0.1 LEdd
Q− = σ Teff
4
= , (1.135)
ξa R 2
which implies Teff ∼ 1/R1/2 . The luminosity of the slim-disc part of the accretion
flow is then
Rtrans
0.8 Rtrans
Lslim = 2 σ Teff
4
2π RdR = LEdd · ln ≈ LEdd ln ṁ, (1.136)
Rin ξa Rin
where we used Eq. (1.131).

Therefore, the total disc luminosity
Ltotal = Lthin + Lslim

Rtrans R∞
= 4π σ Teff
4
RdR + σ Teff
4
RdR ≈ LEdd (1 + ln ṁ), (1.137)
Rin Rtrans
where Lthin is the luminosity of the radiation-cooled disc for which Eq. (1.58) applies.
It is easy to see that the same luminosity formula L ≈ LEdd (1 + ln ṁ) is obtained
when one assumes mass loss from the disc resulting in a variable (with radius)
accretion rate: Ṁ ∼ R.
At very high accretion rates, the disc emission will be also strongly beamed by
the flow geometry so that observer situated in the beam of the emitting system will
infer a luminosity
1
Lsph = LEdd (1 + ln ṁ), (1.138)
b
where b is the beaming factor (see [27] for a derivation of b in the case of ultralumi-
nous X-ray sources).
Numerical simulations do not seem to correspond to this analytical solutions (see
e.g. [25, 51, 52], but they also disagree between themselves. The reasons for these
contradictions are worth investigating.
Additional reading: References [1, 3, 4, 33, 38, 49, 50, 60].
1.6 Accretion Discs in Kerr Spacetime
In this section, we will present and discuss the set equations whose solutions represent
α-accretion discs in the Kerr metric. This section is based on references [5, 30, 49]
and to be understood requires some basic knowledge of Einstein’s general relativity.
1.6.1 Kerr Black Holes
The components gij of the metric tensor with respect to the coordinates (t, x α ) are
expressible in terms of the lapse N, the components β α of the shift vector and the
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components γαβ of the spatial metric:
gij dx i dx j = −N 2 dt 2 + γαβ (dx α + β α dt)(dx β + β β dt), (1.139)
which is a modern way of writing the metric.
Remark 1.3 In this section only I will use conventions different from those
used in other parts of the chapter. First, I will use the so-called geometrical
units that are linked to the physical units for length, time and mass by
length in physical units = length in geometrical units,

1
time in physical units = length in geometrical units,
c
c2
mass in physical units = length in geometrical units.
G
(1.140)
Second, the radial coordinate will be called “r” and not “R”. This should not
confuse the reader since R is used only in the nonrelativistic context where
it denotes a radial coordinate and a radial distance, while in the relativistic
context, it only a coordinate.
1.6.1.1 General Structure, Boyer–Lindquist Coordinates
The Kerr metric in the Boyer–Lindquist (spherical) coordinates t, r, θ, ϕ corresponds

to:
ς
N=√ , β r = β θ = 0, β ϕ = −ω, (1.141)
AΔ
ς2 A2
grr = , gθθ = ς 2 , gϕϕ = 2 sin2 θ (1.142)
Δ ς
with
ς = r 2 + a2 cos2 θ, Δ = r 2 − 2Mr + a2 , (1.143)
2 2Jr 2Mar
A = r 2 + a2 − Δa2 sin2 θ, ω = = , (1.144)
A A
where M is the mass and a = J/M is the angular momentum per unit mass. In
application,s one often uses the dimensionless “angular momentum” parameter a∗ =
a/M.
42 J.-P. Lasota
Therefore in BL coordinates, the Kerr metric takes the form of
ς 2 Δ 2 A sin2 θ ς2 2
ds2 = − dt + (dϕ − ωdt) 2
+ dr + ς 2 dθ 2 . (1.145)
A ς2 Δ
The time (stationarity) and axial symmetries of the metric are expressed by two
Killing vectors
ηi = δ i(t) , ξ i = δ i(ϕ) , (1.146)
where δ i(k) is the Kronecker delta.

Remark 1.4 Using Killing vectors (1.146), one can define some useful scalar func-
tions: the angular velocity of the dragging of inertial frames ω, the gravitational
potential Φ and the gyration radius R,
η·ξ ξ ·ξ
ω=− , e−2Φ = ω2 ξ · ξ − η · η, R2 = − . (1.147)
ξ ·ξ η·η
In the Boyer–Lindquist coordinates, the scalar products of the Killing vectors are
simply given by the components of the metric,
η · η = gtt , η · ξ = gtϕ , ξ · ξ = gϕϕ , (1.148)
and therefore, quantities defined in Eq. (1.147) can be explicitly written down in
terms of the Boyer–Lindquist coordinates as:
A2 −2Φ r2Δ
R2 = , e = . (1.149)
r4Δ A
• The horizon
The black hole surface (event horizon) is at

rH = M + M 2 − a2 . (1.150)
Therefore, a horizon exists for a∗ ≤ 1 only. At the horizon, the angular velocity
of the dragging of inertial frame is equal to
a
ωH = ΩH = , (1.151)
2MrH
where ΩH is the angular velocity of the horizon, i.e. the angular velocity of the
horizon-forming light rays with respect to infinity. The horizon rotates.
The area of the horizon is given by

S = 8π MrH = 8π M M 2 − a2 . (1.152)
The extreme (maximally rotating) black hole corresponds to
a = M. (1.153)
For a > M, the Kerr solution represents a naked singularity. Such singularities would
be a great embarrassment not only because of their visibility but also because the
solution of Einstein equation in which they appear violate causality by containing
closed timelike lines. The conjecture that no naked singularity is formed through
collapse of real bodies is called the cosmic censorship hypothesis (Roger Penrose).
Remark 1.5 Rotation of astrophysical bodies

Since this is a lecture in astrophysics, let us leave for a moment the geometrical
units. They are great for calculations but usually useless for comparing their
results with observations. In the physical units,
2 2 1/2
GM GM J
rH = 2 + − . (1.154)
c c2 Mc
Therefore, the maximum angular momentum of a black hole is

2
GM 2 M
Jmax = = 8.9 × 1048 g cm2 s−1 . (1.155)
c M
This is slightly more than the angular momentum of the Sun (J = 1.63 ×
1048 g cm2 s−1 , a∗ = 0.185): the gain in velocity is almost fully compensated
by the loss in radius.
For a millisecond pulsar which is a neutron star with a mass of ∼1.4 M
and radius ∼10km, the angular momentum is
2 −1
α(x) MNS RNS PS
JNS = INS ΩS ≈ 8.6 × 1048 g cm2
0.489 1.4 M 10 km 1 ms
(1.156)
where INS ≈ α(x)MNS RNS 2
is the moment of inertia and x =
(MNS / M )(km/RNS ) the compactness parameter. For the most compact
neutron star, x ≤ 0.24 and α(x) 0.489. Therefore for neutron stars that
rotate at millisecond periods
−1 2 −1
α(x) MNS RNS PS
a∗NS ≈ 0.5 . (1.157)
0.489 1.4 M 10 km 1 ms
44 J.-P. Lasota
By definition
• the specific (per unit rest-mass) energy is
E := −η · u, (1.158)
• the specific (per unit rest-mass) angular momentum is
L := ξ · u (1.159)
and
• the specific (per unit mass-energy) angular momentum (also called geometrical
specific angular momentum) is
L ξ ·u
J := − =− (1.160)
E η·u
1.6.2 Privileged Observers
Let us consider observers privileged by the symmetries of the Kerr spacetime. The
results below apply to any spacetime with the same symmetries, e.g. the spacetime
of a stationary, rotating star. The four-velocity of a privileged observer is the linear
combination of the two Killing vectors:
u = Z (η + Ωobs ξ ) (1.161)
where the redshift factor Z is (from the normalization u · u = 1)
Z −2 = η · η + 2Ωobs η · ξ + Ωobs
2
ξ ·ξ (1.162)
Since for a = 0, the Kerr spacetime is stationary but not static; i.e. the timelike
Killing vector η is not orthogonal to the spacelike surfaces t =const. In such a
spacetime,“non-rotation” is not uniquely defined.
Stationary observers are immobile with respect to infinity; their four-velocities
are defined as
i
ustat = (ηη)−1/2 ηi (1.163)
but are locally rotating: Lstat = ξi ustat

i
= 0.
The four-velocity of a locally non-rotating observer is a unit timelike vector
orthogonal to the spacelike surfaces t =const.:

i
uZAMO = eΦ ηi + ωξ i , (1.164)
Table 1.1 Summary of properties of privileged observers

Observer Four-velocity Angular velocity with respect
to stationary observers
Stationary u = (η · η)−1/2 η Ωstat = 0
ZAMO (LNR) u = e−Φ (η + ωξ ) ΩZAMO = ω
Comoving (with matter) u = Z (η + Ωξ ) Ωcom = Ω
defines the four-velocity of the local inertial observer or ZAMO, i.e. zero-angular
momentum observers since
LZAMO = ξi uZAMO
i
= 0.
Finally, in the presence of matter forming a stationary and axisymmetric config-

uration, there are privileged observers comoving with matter (Table 1.1).
1.6.3 The Ergosphere
For ZAMOs, Ω = ω, but for stationary observers, Ω − ω = −ω. Therefore, ZAMOs

rotate with respect to infinity (but are locally non-rotating). They may exist down to
the black hole horizon, where they become null: uZAMO · uZAMO = 0.
Stationary observers immobile with respect to infinity but rotating with angular
velocity −ω with respect to ZAMOs can exist (their four-velocity must be timelike,
η · η < 0) only outside the stationarity limit whose radius is defined by η · η = 0:

rer (θ ) = M + M 2 − a2 cos2 θ . (1.165)
The stationarity limit is called the ergosphere.
1.6.4 Equatorial Plane
We will discuss now orbits in the equatorial plane, where they have the axial symme-
try. We are introducing the cylindrical vertical coordinate z = cos θ which is defined
very close to the equatorial plane, z = 0. The metric of the Kerr black hole in the
equatorial plane, accurate up to the (z/r)0 , is
r2Δ 2 A Δ
ds2 = − dt + 2 (dϕ − ωdt)2 + r 2 r 2 + dz2 , (1.166)
A r d
46 J.-P. Lasota
where now
2 2Mar
Δ = r 2 − 2Mr + a2 , A = r 2 + a2 − Δa2 , ω = , (1.167)
A
or simpler

2M A r2
ds2 = − 1 − dt 2 − 2ωdtdϕ + 2 dϕ 2 + dr 2 + dz2 . (1.168)
r r Δ
1.6.4.1 Orbits in the Equatorial Plane
The four-velocity of matter ui has components ut , uϕ , ur ,
ui = ut δ i(t) + uϕ δ i(ϕ) + ur δ i(r) . (1.169)
The angular frequency Ω with respect to a stationary observer and the angular fre-
quency Ω̃ with respect to a local inertial observer are respectively defined by
uϕ
Ω= , Ω̃ = Ω − ω, (1.170)
ut
The angular frequencies of the corotating (+) and counterrotating (−) Keplerian
orbits are
M 1/2
ΩK± = ± 3/2 , (1.171)
r ± aM 1/2
the specific energy is
r 2 − 2Mr ± a(Mr)1/2
E±
K = 1/2 , (1.172)
r r 2 − 3Mr ± 2a(Mr)1/2
and the specific angular momentum is given by

(Mr)1/2 r 2 ∓ 2a(Mr)1/2 + a2
L±
K =± 1/2 , (1.173)
r r 2 − 3Mr ± 2a(Mr)1/2
or
(Mr)1/2 r 2 ∓ 2a(Mr)1/2 + a2
JK = ± . (1.174)
r 2 − 2Mr ± a(Mr)1/2
Both J and L have a minimum at the last stable orbit, more often called ISCO
(innermost stable circular orbit).
Because of the rotation of space, there is no direct relation between angular

momentum and angular frequency but
η · ξ + Ωξ · ξ R2 Ω − ω
J= = . (1.175)
η · η + Ωη · ξ Ωω − 1
For the Schwarzschild solution (a = ω = 0)
JK = R2 ΩK , (1.176)
so a Newtonian-like relation (justifying the name “gyration radius” for R) between

angular frequency and angular momentum exists for J. No such relation exists for L.
ISCO
The minimum of the Keplerian angular momentum corresponding to the innermost
stable circular orbit (ISCO) is located at
±
rISCO = M{3 + Z2 ∓ [(3 − Z1 )(3 + Z1 + 2Z2 )]1/2 },
1/3
Z1 = 1 + 1 − a2 /M 2 (1 + a/M)1/3 + (1 − a/M)1/3 ,
1/2
Z2 = 3a2 /M 2 + Z12 . (1.177)
Binding Energy
The binding energy
Ebind = 1 − EK (1.178)
at the ISCO is
√
• 1 − √8/9 ≈ 0.06 for a = 0
• 1 − 1/3 ≈ 0.42 for a = 1.
This corresponds to the efficiencies of accretion in a geometrically thin (quasi-
Keplerian) disc around a black hole (Fig. 1.11).
For a Schwarzschild black hole, the frequency associated with the ISCO at rISCO =
6M is
M −1
νK (rISCO ) = 2197 Hz. (1.179)
M
IBCO
The binding energy of a Keplerian orbit 1 − E = 0 at the marginally bound orbit (or
IBCO: innermost bound circular orbit) is

±
rIBCO = 2M ∓ a + 2 M 2 ∓ aM. (1.180)
48 J.-P. Lasota
Fig. 1.11 Radii of

characteristic orbits in the
Kerr metric as a function of
a∗ = a/M. The innermost
stable circular orbit: rISCO ;
the marginally bound orbit
rIBCO (marked rmb ); the
photon orbit: rph ; and the
black hole horizon: rH
(marked rh ) (Courtesy of A.
Sa̧dowski)
For a non-rotating black hole, rIBCO = 4M, and the frequency associated with the
IBCO is
M −1
νK (rIBCO ) = 4037 Hz. (1.181)
M
ICO (Circular photon orbit)

The innermost circular orbit (ICO), i.e. the circular photon orbit, is at
a
± 2
rph = 2M 1 + cos cos−1 ∓ . (1.182)
3 M
For a non-rotating black hole, rph = 3M.
1.6.4.2 Epicyclic Frequencies
We will consider now consider a perturbed orbital motion in, and slightly off the
equatorial plane. In the Newtonian case, the angular frequency of such motions must
be equal to the Keplerian frequency ΩK since there is only one characteristic scale
defined by the gravitational constant G. In general relativity, the presence of two
constants G and c implies that the epicyclic frequency does not have to be equal to
ΩK .
The four-velocity for the perturbed circular motion can be written as

ui = 1, ũr , ũθ , ΩK + ũϕ , (1.183)
where ũα are the velocity perturbations.

• For perturbations in the equatorial plane, the equation of motion is

∂2 ũr
+ κ2 = 0, (1.184)
∂t 2 ũϕ
where
r 2 − 6Mr ± 8aM 1/2 r 1/2 − 3a2
κ 2 = ΩK2 (1.185)
r2
is the (equatorial) epicyclic frequency. In the Schwarzschild case a = 0 this is
κ 2 = ΩK2 (1 − 6M/r) and vanishes at ISCO.In the Newtonian limit, the epicyclic
frequency equals to the Keplerian frequency κ = ΩK .
• For vertical perturbations, the equation is

∂ ∂
+ ΩK ũθ = −Ω⊥ δθ, (1.186)
∂t ∂ϕ
where the vertical epicyclic (angular) frequency is given by
r 2 − 4aM 1/2 r 1/2 − 3a2 M 2

Ω⊥2 = ΩK2 (1.187)
r2
In the Schwarzschild case (a = 0), the vertical epicyclic frequency is equal to
the Keplerian angular frequency ΩK , which is to be expected from the spherical
symmetry of this solution.
The angular velocity Ω⊥ appears also in the equation of vertical equilibrium of a
(quasi)Keplerian disc which will be discussed later (Sect. 1.8.5). Here, let us just
notice that Eq. (1.222) can be written as
∂p
= −ρe2Φ Ω⊥2 z. (1.188)
∂z
All these characteristic frequencies can be put into the form
1
Ω = f (x, a∗ ) , (1.189)
M
where x = r/M. For all relativistic frequencies, x = x(a∗ ), and therefore, they can
be written as
1
Ω = F (a∗ ) . (1.190)
M
Additional reading: Reference [2].
50 J.-P. Lasota
1.7 Accretion Flows in the Kerr Spacetime
1.7.1 Kinematic Relations
In the reference frame of the local inertial (non-rotating) observer, the four-velocity
takes the form, i
ui = γ uZAMO + v(ϕ) τ i(ϕ) + v(r) τ i(r) . (1.191)
The vectors τ i(ϕ) and τ i(r) are the unit vectors in the coordinate directions ϕ and r.
The Lorentz gamma factor γ equals,
1
γ = 2 2 . (1.192)
1 − v(ϕ) − v(r)
The relation between the Boyer–Lindquist and the physical velocity component in
the azimuthal direction is,
v(ϕ) = R̃Ω̃, (1.193)
which justifies the name of R̃—gyration radius. It is convenient to use the (rescaled)
radial velocity component V defined by the formula,
V
√ = γ v(r) = ur grr
1/2
. (1.194)
1− V2
The Lorentz gamma factor may then be written as,

1 1
γ2 = , (1.195)
1 − Ω̃ 2 R̃2 1 − V2
which allows writing a simple expression for V in terms of the velocity components
measured in the frame of the local inertial observer,
v(r) v(r)
V = 2 = . (1.196)
1 − v(ϕ) 1 − R̃2 Ω̃ 2
Thus, V is the radial velocity of the fluid as measured by an observer corotating with
the fluid at fixed r.
Although a different quantity could have been chosen as the definition of the
“radial velocity”, only V has directly three very convenient properties, all guaranteed
by its definition:
• (i) everywhere in the flow |V | ≤ 1,
• (ii) on the horizon |V | = 1,
• (iii) at the sonic point |V | ≈ cs ,

where cs is the local sound speed.
To see that property (i) holds, let us define
Ṽ 2 = ur ur = ur ur grr ≥ 0. (1.197)
Then, one has

V 2 = Ṽ 2 /(1 + Ṽ 2 ) ≤ 1. (1.198)

Writing V = r 2 ur ur /(r 2 ur ur + Δ) demonstrates property (ii) since |V | = 1
independent of the value of r 2 ur ur .
For the proof of property (iii) of V see [4].
Other possible choices of the “radial velocity” such as u = |ur | are not that con-
venient.
1.7.2 Description of Accreting Matter
The stress–energy tensor T ik of the matter in the disc is given by
T ik = (ε + p) ui uk + p gik + S ik + uk qi + ui qk , (1.199)
where ε is the total energy density, p is the pressure,
Sik = νρσik , (1.200)
is the viscous stress tensor, ρ is the rest mass density, and qi is the radiative energy
flux. In the last equation, ν is the kinematic viscosity coefficient and σik is the shear
tensor of the velocity field. From the first law of thermodynamic, it follows that
ε+p
dε = dρ + ρTdS, (1.201)
ρ
where T is the temperature and S is the entropy per unit mass. Note that in the
physical units, ε = ρc2 + Π , where Π is the internal energy. For nonrelativistic
fluids, Π ρc2 and p ρc2 , and therefore
ε + p ≈ c2 ρ. (1.202)
We shall use this approximation (in geometrical units ε + p ≈ ρ) in all our calcu-
lations. This approximation does not automatically ensure that the sound speed is
below c, and one should check this a posteriori when models are constructed. We
write the first law of thermodynamics in the form:
52 J.-P. Lasota

1
dU = −p d + TdS, (1.203)
ρ
where U = Π/ρ.
1.8 Slim-Disc Equations in Kerr Geometry
General-relativistic effects play an important role in the physics of thin (H/r 1)

accretion discs close to the black hole, but they determine the properties of slim
(H/r 1) discs. We will derive the slim-disc equation, and before discussing their
properties, we will say few words about thin discs.
It is convenient to write the final form of all the slim-disc equations at the equa-
torial plane, z = 0. Only these equations which do not refer to the vertical structure
could be derived directly from the quantities at the equatorial plane with no further
approximations. All other equations are approximated—either by expansion in terms
of the relative disc thickness H/r, or by vertical averaging.
1.8.1 Mass Conservation Equation
From general equation of mass conservation,
∇ i (ρui ) = 0, (1.204)
and definition of the surface density Σ,

+H(r)
Σ= ρ(r, z)dz ≈ 2Hρ, (1.205)
−H(r)
we derive the mass conservation equation,
V
Ṁ = −2π Δ1/2 Σ √ . (1.206)
1 − V2
In the Newtonian limit, the mass conservation equation is as follows:
Ṁ = −2π Σ vr . (1.207)
1.8.2 Equation of Angular Momentum Conservation
From the general form of the angular momentum conservation,

∇k T ki ξi = 0, (1.208)
we derive, after some algebra,

3/2 Δ γ dΩ
1/2 3
Ṁ dL 1 d
+ ΣνA − F − L = 0, (1.209)
2π r dr r dr r4 dr
where F − = 2qz is the vertical flux of radiation and

A3/2
L ≡ −(uξ ) = −uϕ = γ Ω̃, (1.210)
r 3 Δ1/2
is the specific (per unit mass) angular momentum. The term F − L represents angular
momentum losses through radiation. Although it was always fully recognized that
angular momentum may be lost this way, it has been argued that this term must
be very small. Rejection of this term enormously simplifies numerical calculations,
because with F − L = 0 Eq. (1.209) can be trivially integrated,
Ṁ Δ1/2 γ 3 dΩ T
(L − L0 ) = −ΣνA3/2 4
≡ , (1.211)
2π r dr 2π
where L0 is the specific angular momentum of matter at the horizon (Δ = 0). In the
numerical scheme for integrating the slim Kerr equations (with F − L assumed to be
zero), the quantity L0 plays an important role: it is the eigenvalue of the solutions
that passes regularly through the sonic point. The rhs of Eq. (1.211) T represent the
viscous torque transporting angular momentum.
In the Newtonian case, a geometrically thin disc is Keplerian Ω ≈ ΩK (see

Eq. (1.214)), LK = K = R2 ΩK , and Eq. (1.211) takes the familiar form of

Ṁ 0
νΣ = 1− ,
3π K
(see Eq. 1.57).
1.8.3 Equation of Momentum Conservation
From the r-component of the equation ∇i T ik = 0, one derives

54 J.-P. Lasota
V dV A 1 dP
= − , (1.212)
1 − V dr
2 r Σ dr
where P = 2Hp is the vertically integrated pressure and
MA (Ω − Ωk+ )(Ω − Ωk− )

A =− + − . (1.213)
r 3 ΔΩk Ωk 1 − Ω̃ 2 R̃2
Note that in Eq. (1.212), the viscous term has been neglected.
The Newtonian limit of Eq. (1.212) is
dvr 2 c2
vr − Ω − ΩK2 r + s = 0 (1.214)
dr r
For a thin disc: H/r ≈ cs2 /rΩK 1, Eq. (1.214), is simply Ω ≈ ΩK ; i.e. a thin
Newtonian disc is Keplerian. The thickness of the disc depends on the efficiency
of radiative processes: efficient radiative cooling implies a low speed of sound.
1.8.4 Equation of Energy Conservation
From the general form of the energy conservation,

∇i T ik ηk = 0, (1.215)
and the first law of thermodynamics,

1 ∂ε ∂ε
T= , p=ρ − ε, (1.216)
ρ ∂S ρ ∂ρ S
the energy equation can be written in general as
Qadv = Q+ − Q− , (1.217)
where 2
+ A2 dΩ
Q = νΣ 6 γ 4 (1.218)
r dr
is the surface viscous heat generation rate, Q− is the radiative cooling flux (both
surfaces) which is discussed in Sect. 1.5, and Qadv is the advective cooling rate due
to the radial motion of the gas. It is expressed as
ΣV Δ1/2 dS Ṁ dS
Qadv = √ T ≡− T . (1.219)
1−V 2 r dr 2π r dr
In stationary accretion flows, advection is important only in the inner regions

close to the compact accretor. In the rest of the flow, the energy equation is just
Q+ = Q− . (1.220)
In the newtonian limit of Eqs. (1.218) and (1.211), one obtains

+ 3 GM Ṁ 0
Q = 1− . (1.58)
8π R3
1.8.5 Equation of Vertical Balance of Forces
The equation of vertical balance is obtained by projecting the conservation equation

onto the θ direction
hθi ∇k Tik = 0, where hθi = δθi − ui uθ (1.221)
and neglecting the terms O 3 (cos θ ). For a nonrelativistic fluid, this leads to

dP L2 − a2 E2 − 1
= −ρgz z = −ρ z. (1.222)
dz r4
In the newtonian limit, Eq. (1.222) becomes
dP K 2
= −ρ 4 z (see Eq. 1.18).
dz r
1.9 The Sonic Point and the Boundary Conditions
1.9.1 The “No-Torque Condition”
There have been a lot of discussion about the inner boundary condition in an accretion
disc. The usual reasoning is that for a thin disc, the inner boundary is at ISCO and
56 J.-P. Lasota
since it is where circular orbits end the boundary condition should be simply that
the “viscous” torque vanishes (there is no orbit below the ISCO to interact with).
Several authors have challenged this conclusion, but a very simple argument by
Bohdan Paczyński [44] shows the fallacy of these challenges.
Using Eq. (1.206), one obtains from Eq. (1.211)
A3/2 γ 2 1 dΩ
v(r) = ν . (1.223)
r 4 L − L0 dr
Next, from the viscosity prescription ν ≈ αH 2 Ω, and taking for simplicity the
nonrelativistic approximation (this does not affect the validity of the argument but
allows skipping irrelevant in this context multiplicative factors), one can write
2
dΩ Ω H
vr ≈ α H 2 ≈ α H2 ≈ α vϕ , (1.224)
− 0 dr − 0 r R − 0
where vϕ = RΩ. Although we have dropped the GR terms, the Eq. (1.224) does not
assume that the radial velocity is small; i.e. this equation holds within the disc as
well as within the stream below the ISCO.
Far out in the disc, where 0 , one obtains the standard formula (see Eq. 1.36)
2
H
vr ≈ α vϕ , R Rin . (1.225)
R
The flow crosses the black hole surface at the speed of light, and since it is subsonic
in the disc, it must somewhere become transonic, i.e. to go through a sonic point,
close to disc’s inner edge.
At the sonic point, we have vr = cs ≈ (H/R)vϕ , and the Eq. (1.224) becomes:
vr Hin in
=1≈α , R = Rin (1.226)
cs Rin in − 0
If the disc is thin, i.e. Hin /Rin 1, and the viscosity is small, i.e. α 1, then
Eq. (1.226) implies that ( in − 0 )/ in 1; i.e. the specific angular momentum at
the sonic point is almost equal to the asymptotic angular momentum at the horizon.
In a steady-state disc, the torque T has to satisfy the equation of angular momentum
conservation (1.211), which can be written as
T = Ṁ ( − 0 ) , Tin = Ṁ ( in − 0 ) . (1.227)
Thus, it is clear that for a thin, low viscosity disc, the ‘no-torque inner boundary
condition’ (Tin ≈ 0) is an excellent approximation following from angular
momentum conservation.
However, if the disc and the stream are thick, i.e. H/r ∼ 1, and the viscosity is
high, i.e. α ∼ 1, then the angular momentum varies also in the stream in accordance
with the simple reasoning presented above. However, the no-stress condition at the
disc inner edge might be not satisfied.
Additional reading: Reference [6].
Acknowledgments I am grateful to Cosimo Bambi for having invited me to teach at the 2014 Fudan
Winter School in Shanghai. Discussions with and advice of Marek Abramowicz, Tal Alexander,
Omer Blaes and Olek Sa̧dowski were of great help. I thank the Nella and Leon Benoziyo Center for
Astrophysics at the Weizmann Institute for its hospitality in December 2014/January 2015 when
parts of these lectures were written. This work has been supported in part by the French Space Agency
CNES and by the Polish NCN grants DEC-2012/04/A/ST9/00083, UMO-2013/08/A/ST9/00795
and UMO-2015/19/B/ST9/01099.
Appendix
Thermodynamical Relations
The equation of state can be expressed in the form:
R R B2
P = Pr + ρTi + ρTe + , (1.228)
μi μe 24π
where Pr is the radiation pressure, R is the gas constant, μi and μe are the mean
molecular weights of ions and electrons respectively, Ti , and Te are ion and electron
temperatures, a is the radiation constant (not to be confused with the dimensionless
angular momentum a in the Kerr metric), and B is the intensity of a isotropically tan-
gled magnetic field, includes the radiation, gas and magnetic pressures. The radiation
pressure Pr , the gas pressure Pg and the magnetic pressure Pm correspond respec-
tively to the first term, the second and third terms, and the last term in Eq. (1.228).
The mean molecular weights of ions and electrons can be well approximated by:
4 2
μi ≈ , μe ≈ , (1.229)
4X + Y 1+X
where X is the relative mass abundance of hydrogen and Y that of helium. We may
define a temperature as
Ti Te
T =μ + , (1.230)
μi μe
where −1
1 1 2
μ= + ≈ (1.231)
μi μe 1 + 3X + 1/2Y
58 J.-P. Lasota
is the mean molecular weight. In the case of a one-temperature gas (Ti = Te ), one
has T = Ti = Te . For an optically thick gas, Pr = (4σ /3c)Tr4 .
For the frozen-in magnetic field pressure Pm ∼ B2 ∼ ρ 4/3 , therefore, we may
write the internal energy as
4σ 4 RT
U= T + + eo ρ 1/3 , (1.232)
ρc r μmu (γg − 1)
where eo is a constant (Pm = 1/3eo ρ 4/3 ) and γg is the ratio of the specific heats of
the gas. We define
Pg Pg 4 − βm
β= , βm = , β∗ = β. (1.233)
p Pg + Pm 3βm
From Eqs. (1.228) and (1.232), one obtains the following formulae (see, e.g. “Cox
and Giuli” 2004) for the specific heat at constant volume:

R 12(1 − β/βm )(γg − 1) + β 4 − 3β ∗ P
cV = = (1.234)
μ(γg − 1) β Γ3 − 1 ρT
and the adiabatic indices:

(4 − 3β ∗ )(γg − 1)
Γ3 − 1 = (1.235)
12(1 − β/βm )(γg − 1) + β
Γ1 = β ∗ + (4 − 3β ∗ )(Γ3 − 1). (1.236)
The ratio of specific heats is γ = cp /cV = Γ1 /β. For β = βm , we have Γ3 = γg

and Γ1 = (4 − β)/3 + β(γg − 1). For an equipartition magnetic field (β = 0.5), one
gets Γ1 = 1.5 and for β = 0.95, Γ1 = 1.65 (here, we have used γg = 5/3). One
expects βm ∼ 0.5 − 1. Since

dS d ln T d ln Σ d ln H
T = cV − (Γ3 − 1) − , (1.237)
dR dR dR dR
the advective flux is written in the form:
Ṁ P
Qadv = ξa (1.238)
2π R2 ρ
where
4 − 3β ∗ d ln T ∗ d ln Σ
ξa = − + (4 − 3β ) . (1.239)
Γ3 − 1 d ln R d ln R
The term ∝ d ln H/d ln R has been neglected. Since no rigorous vertical averaging
procedure exists, the presence or not of the d ln H/d ln R—type terms in this (and
other) equation may be decided only by comparison with 2D calculations.
The formulae derived in this section are valid for the optically thin case τ = 0 if
one assumes β = βm .
Reference: Weiss, A., Hillebrandt, W., Thomas, H.-C., and Ritter, H. 2004, Cox
and Giuli’s Principles of Stellar Structure, Cambridge, UK: Princeton Publishing
Associates Ltd, 2004.
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Chapter 2
Transient Black Hole Binaries
Tomaso M. Belloni and Sara E. Motta
Abstract The last two decades have seen a great improvement in our understand-
ing of the complex phenomenology observed in transient black hole binary systems,
especially thanks to the activity of the Rossi X-ray Timing Explorer satellite, com-
plemented by observations from many other X-ray observatories and ground-based
radio, optical and infrared facilities. Accretion alone cannot describe accurately the
intricate behaviour associated with black hole transients, and it is now clear that the
role played by different kinds of (often massive) outflows seen at different phases
of the outburst evolution of these systems is as fundamental as the one played by
the accretion process itself. The spectral-timing states originally identified in the X-
rays and fundamentally based on the observed effect of accretion have acquired new
importance as they now allow to describe within a coherent picture the phenomenol-
ogy observed at other wavelength, where the effects of ejection processes are most
evident. With a particular focus on the phenomenology seen in the X-ray band, we
review the current state of the art of our knowledge of black hole transients, describ-
ing the accretion–ejection connection at play during outbursts through the evolution
of the observed spectral-timing properties. Although we mainly concentrate on the
observational aspects of the global X-ray transient picture, we also provide physical
insight to it by reviewing (when available) the theoretical explanations and models
proposed to explain the observed phenomenology.
2.1 Introduction
Black hole (BH) binaries (hereafter BHBs) are binary systems consisting of a non-
collapsed star and a black hole. They provide a unique laboratory for the study
not only of accretion of matter onto a compact object, but also of the strongly
T.M. Belloni (B)

INAF - Osservatorio Astronomico di Brera, via E. Bianchi 46, I-23807 Merate, Italy
e-mail: tomaso.belloni@brera.inaf.it
S.E. Motta
Department of Physics, Astrophysics, University of Oxford,
Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
e-mail: sara.motta@physics.ox.ac.uk
62 T.M. Belloni and S.E. Motta
curved spacetime in the vicinity of a black hole, where the effects of general
relativity in a strong gravitational field cannot be ignored and can be studied
(see [137]).
The first BHB, Cygnus X-1, was discovered in the early years of X-ray astronomy
with instruments on board sounding rockets. It became the first candidate for systems
hosting a black hole when optical observations revealed its binary nature and allowed
the estimate of the mass of the compact object (see Chap. 3 in this book). Its X-ray
properties included strong variability on short timescales and a very hard energy
spectrum. Cyg X-1 is a persistent system, which we know now are rather rare.
Most BHBs are transient, difficult to discover in the absence of a wide-field X-ray
instrument. The first such system, A 0620-00, was discovered in 1975 with the Ariel
V satellite [47] and in 1986. It became a much stronger candidate for hosting a
black hole, with a minimum mass of 3.2 M for the compact object [93]. A few
additional persistent systems were identified, notably the two in the LMC and the
galactic source GX 339-4, but it was only with the Japanese satellite Ginga that
new transients were discovered and followed, thanks to the availability of an all-sky
monitor and of a large-area detector. Systems such as GX 339-4 and Cyg X-1 were
known to undergo transitions in their flux, spectral and timing properties, which had
led to the definition of a high (flux) and low state (see, e.g. [169]). The detailed study
of several observations, in particular of the transient GS 1124-683 (also known as
Nova Muscae 1991), GX 339-4 and Cyg X-1 led to the identification of complex
spectral/timing properties that in turn led to the definition of additional states (see
[17, 19, 107, 110]). In particular, the first spectral studies with accretion disc models
opened the way to techniques to measure the inner radius of the disc. In the timing
domain, quasi-periodic oscillations (QPOs) were discovered. These features yield
discrete frequencies that constitute a very direct measurement of timescales in the
system, which are associated with the accretion phenomenon and/or to effects of
general relativity. A full review of the observational status after Ginga can be found
in [168]), where essentially all known information on BHBs at the time is recorded.
The situation changed dramatically starting from the second half of the nineties. At
the end of 1995, the Rossi X-ray Timing Explorer (RXTE) was launched and opened
a new window onto phenomena of fast variability from neutron star and black hole
binaries. Its All-Sky Monitor (ASM) allowed the timely discovery of outbursts of
old and new transients and the large-area proportional counter array (PCA), operated
in a very flexible way, accumulated high signal data for timing analysis and spectral
analysis at moderate spectral resolution. A few years later, the availability of high-
spectral-resolution instruments on board Chandra and XMM-Newton opened the way
to perform detailed spectral studies, in particular on relativistically skewed emission
lines. Overall, there was a real explosion of information and now, after the demise
of RXTE and with the availability of further missions such as INTEGRAL, Swift,
Suzaku and NuSTAR, it is possible to combine archival data and new observations to
obtain new insight on the emission properties of BHBs. This allows us to understand
much better the process of accretion onto a black hole and new exciting results are
possible that were not even imaginable only a few years ago. The long-term evolution
of the X-ray emission, in particular for transients, is now clear in its phenomenology
2 Transient Black Hole Binaries 63
and provides a solid framework upon which to study detailed spectral and variability
properties. Direct effects of general relativity are studied and measured through the
analysis of broadband spectra, emission lines and QPOs (see below and Chap. 3), the
properties of continuum emission are interpreted with physical models and global
models for different states are being proposed.
At the same time, the discovery of relativistic jets from BHBs [105, 106] (see also
[50, 55]) opened a new path to the study of accretion. The difference in timescale
and count statistics with active galactic nuclei, where jets were known since much
earlier, allows the study of the connection between the accretion onto the compact
object and ejection of relativistic jets, introducing a new window to understand the
physics of accretion and to unveil the mechanism responsible for the ejections (see
[50, 55]). Moreover, in the past few years the existence of powerful non-relativistic
winds has been discovered in a number of objects (see [50]). These winds appear to
be flowing in a direction parallel to the accretion disc, as opposed to jets, and to be
mutually exclusive with the jets. The full picture of an accreting black hole is now
much different from that of only a couple of decades ago, when observations in the
X-ray band alone and only at low-energy resolution made us miss very important
components (jets and winds) which can be energetically very important. Jets and
outflows are the subject of Sect. 2.3.
In addition to the comparison with active galactic nuclei, the supermassive coun-
terparts of stellar mass accreting black holes, the strong similarities between the
BHBs and other systems at subgalactic scale are being studied and exploited. Neutron
star binaries share many properties in the X-ray band and, when their magnetic field
is sufficiently low, also in the radio, where relativistic jets have been observed [102,
119]. Binaries in external galaxies are also accessible with current instrumentation,
although naturally at lower count rates, and in particular the enigmatic ultraluminous
X-ray sources (ULXs) are being studied in order to understand whether they contain
intermediate-mass black holes or are super-Eddington counterparts of our galactic
objects (see [53]). In this respect, a detailed comparison with BHBs is instrumental
to unveil this mystery.
In this review, we aim at presenting the current state of the art of BHB research,
concentrating on the properties of accretion and ejection, while the Chap. 3 addresses
the issue of measurement of physical quantities of the central object. The sheer
amount of information available today makes it impossible to cover all aspects.
Many specific and additional details can also be found, e.g., in [24, 49, 50, 85].
2.2 X-ray Emission
In BHBs, the X-ray emission originates from the inner regions of the accretion flow
and, possibly, from the base of the relativistic jets [90]. Even restricting ourselves
to the two original source states, low/hard and high/soft, the details of the emission
appear to be very complex (see, e.g. [62]). Both the energy spectra and the fast
variability contain multiple components, which vary in a correlated fashion. While the
details of these properties are given in the next sections, it is important to understand
the regularities that exist in the time evolution of the observables. Before the mid-
1990s, only a few sources were available and the coverage was not sufficient to study
these aspects in sufficient detail; only with RXTE, it has been possible to find a
general scheme for the characterization of source states. Even before then, it was
clear that a simple classification of the observed properties on the basis of energy
spectra alone was not possible and fast variability had to be included in the picture.
In this sense, the notion of “spectral states” does not carry any meaning.
While a few BHBs are persistent sources, like the archetypal system Cyg X-1
(always accreting at a high rate and emitting luminosities above 1037 erg/s) most of
them are of transient nature. They spend most of the time in a low accretion regime
(L X < 1033 erg/s), where observations are still limited by the low number of counts
(see [132] and references therein). With a recurrence period that varies between
several months and decades, the accretion rate onto the central objects increases by
orders of magnitude and the sources go into outburst for a time that can range from
a few days to, more commonly, several months (one peculiar object, GRS 1915+105
is at present active since 23 years). Their X-ray luminosity increases, peaks and then
decreases and can roughly be adopted as a proxy for accretion rate, while the detailed
properties of the energy spectra and fast variability change, at times in a very abrupt
way. It was only at the beginning of the last decade, however, that a coherent picture
emerged, which can be applied to most systems [65, 69].
Outbursts of different systems and even multiple outbursts from the same object
have time evolutions which can differ considerably (see Belloni [20]). However, when
the evolution of an outburst is represented in a hardness–intensity diagram (HID),
strong regularities emerge. A HID is equivalent to a hardness–magnitude diagram in
the optical band: on the abscissa is the ratio of counts in two separate bands (hard/soft),
which gives a rough indication of the hardness of the energy spectrum, and on the
ordinate the total count rate over a broad energy band, a proxy for luminosity and
accretion rate [20, 24, 66, 69]. As such, the diagram is source dependent (because
of interstellar absorption) and instrument dependent, but it is extremely useful to
follow BHB outbursts. An extension to an independent form of the diagram (with
flux ratio between the components and total flux respectively) has been proposed
[46]: this has the advantage of containing physical quantities, but the disadvantage
of being insensitive to small changes when one component dominates, besides being
of course model-dependent. An example of a HID based on RXTE/PCA data for
the best known system, GX 339-4, is shown in the top left panel of Fig. 2.1. Here,
four outbursts are plotted (2002, 2004, 2007, 2010). The general evolution is the
same: a “q”-shaped diagram travelled counterclockwise from the bottom right (faint
and hard). Quiescence cannot be included as the RXTE/PCA instrument was not
sensitive to low-flux observations, but we know that the hard branch softens as flux
decreases (see [7] and references therein). From the HID, one can easily identify the
two historical states as the two “vertical” branches. The hard branch extending to
the stem of the “q” corresponds to the low/hard state (LHS), which is observed at
the start and at the end of an outburst only, never in the middle. The branch is not
really vertical as the logarithmic axis suggests, but there is a marked softening as
the source brightens (see, e.g. [112]). The left branch corresponds to the high/soft
state (HSS). The scatter of the points is magnified by the log scale and contains
some excursions back to other states (see below) as well as intrinsic variability of
the hard spectral component. The transition between these two states takes place at
two different flux levels. At high flux, the source moves from LHS to HSS and at
low flux it returns to the LHS, completing a hysteresis cycle (originally identified
by [109], see also [85, 88]). Two additional diagrams have been proven useful for
following the evolution of an outburst and identify source states. The first is the HRD
(hardness–rms diagram, bottom left panel in Fig. 2.1), where the Y-axis contains the
fractional rms variability integrated over a broad range of frequencies (see [20]).
Remarkably, no hysteresis is observed in this diagram: at each hardness corresponds
a single value of fractional rms, regardless of the flux. The second is the RID (rms–
intensity diagram, right panel in Fig. 2.1), the third combination of the same three
observables. Here the hysteresis, as expected, is clearly present and the location of
specific states is rather precise (see [120]). The central part of the diagram identifies
two additional states, the hard intermediate state (HIMS) and soft intermediate state
(SIMS). The transition between the states corresponds to precise values in hardness,
obviously source and instrument dependent. The identification of these thresholds
is based on the properties of fast variability and/or changes in the multi-wavelength
relations. For a precise determination of the different states, we refer the reader to
[24, 120], here we include a brief summary:
• Low/Hard State (LHS). The LHS has only been observed in the first and last
stages of an outburst. In some cases, the first RXTE/PCA observations found a
source already in a softer state, but the initial LHS might have been too fast to
observe. It corresponds to the right branch in the HID and to the straight diagonal
line in the RID (see Fig. 2.1). It is characterized by large variability (around 40 %
fractional rms in the case of GX 339-4) and a hard spectrum (see below). In
transients, it is only observed at the start and end of outbursts (although at times
the start LHS is missed altogether) and is the most common state for the persistent
system Cyg X-1). The variability is in the form of broadband noise made of a
few components whose characteristic frequencies increase with luminosity (and
decreasing hardness).
• Hard Intermediate State (HIMS). In the HID, this state corresponds to a large part
of the area between the LHS and the HSS, covering the horizontal tracks, both at
high and low flux. This state appears after the initial LHS and reappears before
the source goes back to the LHS at the end of the outburst. In addition, secondary
transitions to and from it can be observed (see below). The softening compared to
the LHS is due to two effects: the appearance in the observational range of flux
from the thermal disc and the steepening of the hard component (see below and
[112]). The fast variability is an extension of that of the LHS, with characteristic
frequencies increasing and total fractional rms decreasing (see the HRD and the
RID in Fig. 2.1). Type-C QPOs are present (see below).
• Soft Intermediate State (SIMS). The energy spectrum is slightly softer than the
HIMS, putting this state to the left of the HIMS in the HID. In the HID and RID
Fig. 2.1 The three main diagrams for the representation of the X-ray evolution of black hole binaries
for three outbursts of GX 339-4 as observed with RossiXTE. Top left Hardness–intensity diagram
(HID), top right rms–intensity diagram (RID) and bottom left hardness–rms diagram (HRD)
these points are not immediately identifiable, but can be seen in the HRD (bottom
panel in Fig. 2.1) as a cloud of points at a lower rms than the main branch (around
hardness 0.2). While the energy spectrum below 10 keV is very similar to that of
the softer HIMS points, at high energies the spectrum unlike the HIMS does not
show a significant high-energy cut-off [112]. The identifying feature of this state
is the disappearance of the band-limited noise components in the power density
spectrum, replaced by a weaker power law component (hence, the lower fractional
rms) and the appearance of a marked type-B QPO.
• High–Soft State (HSS). The spectrum is soft, dominated by an optically thick
accretion disc, variability is low. Occasional low-frequency QPOs can be detected,
identified with type-C (see [113]). This state is reached from the intermediate states
and left through the intermediate states.
Although for different sources diagrams can look different in the HID in Fig. 2.1,
the sequence of states from quiescence to quiescence is the following: LHS—
HIMS—SIMS—HSS—(minor transitions to and from HIMS and SIMS)—HSS—
SIMS—HIMS—LHS. Some transient remained in the LHS throughout the out-
burst, a few showed failed transitions in the sense that the LHS–HIMS sequence
was not followed by a transition to the softer states (SIMS and HSS). The few bright
persistent sources show a reduced number of states. Cyg X-1 is found mostly in
the LHS with transition to the HIMS and possibly all the way to the HSS. LMC
X-1 is always in the HSS. LMC X-3 is mostly in the HSS, with brief transitions
to the LHS (no strong evidence of intermediate states).
The three diagrams presented above and the state classification are a firm basis
upon which to base detailed spectral and timing analysis. One important transition
is the HIMS–SIMS one, which appears to be associated to the ejection of relativistic
ballistic jets (the crossing of the “jet line”, see below).
2.2.1 Energy Spectra
2.2.1.1 The Truncated Disc Model
Long-term X-ray light curves, X-ray spectra, the rapid X-ray variability and the radio
jet behaviour have been shown to be consistent with the so-called truncated disc
model. According to this model, at low luminosities a Shakura–Sunyaev optically
thick, geometrically thin accretion disc is truncated at a certain (variable) radius and
coexists with a hot, optically thin, geometrically thick accretion flow, which replaces
the region between the inner edge of the disc and the innermost stable orbit. Neutron
stars are also consistent with the same description [119], but with an additional
component due to their surface, giving implicit evidence for the event horizon in
black holes.
At low luminosities, the optically thick, geometrically thin disc is truncated at
very large radii, being replaced (probably through evaporation, see e.g. [99]) in the
inner regions by a hot, inner flow which might also act as the launching site of the jet.
Only a few photons from the disc illuminate the flow at this stage; therefore, Compton
cooling of the electrons is rather inefficient compared to the heating coming from
collisions with protons. The ratio of power in the electrons to that in the seed photons
illuminating them—Lh /Ls —is the major parameter (together with the optical depth
of the plasma) which determines the shape of a thermal Comptonization spectrum
(e.g. [64]). Physically, Lh /Ls sets the energy balance between heating and cooling
and, hence, the electron temperature. In the hard state, the relative lack of seed
photons illuminating the hot inner flow produces hard thermal Comptonized spectra
(with Lh /Ls >> 1), roughly characterized by a power law in the 5–20 keV band with
photon index 1.5 < Γ < 2 (where the photon spectrum N(E) ∝ E −Γ ). Hard spectra
of this kind are typical of the LHS.
As the disc moves progressively inwards, it increasingly extends underneath the
hot inner flow so that there are more seed photons intercepted by the flow, decreasing
Lh /Ls . Therefore, the decrease in disc truncation radius leads to softer spectra, as well
as higher frequencies in the power spectra and a faster jet. This results in spectra
that are a combination of the hard spectral component described above and the soft
spectral component typical of the soft state (see below), i.e. a standard geometrically
thin, optically thick accretion disc with a progressively smaller inner radius. These
spectra are observed during the (usually) short-lived HIMS and SIMS.
When the truncation radius reaches the innermost stable orbit, the hot flow is
thought to collapse into a Shakura–Sunyaev disc and dramatic changes in both the
spectral and time domains are seen. This includes a significant decrease in radio flux,
as well as the major hard-to-soft spectral transition seen in BHs. The dramatic increase
in disc flux due to the presence of the inner disc marks the hard–soft-state transition
[48] and also means that any remaining electrons which gain energy outside of the
optically thick disc material are subject to much stronger Compton cooling (Lh /Ls
is now ≤1). This results in much softer Comptonized spectra. Thus, the soft state is
characterized by a strong disc and soft tail, roughly described by a power law index
of photon index Γ ≥ 2, extending out beyond 500 keV [61]. This tail, differently
from the hard tail observed in the LHS, is not produced by thermal Comptonization.
In order to extend to 500 keV and beyond, the spectrum should be produced in a re-
gion with rather small optical depth and high temperature. However, these conditions
would result in a bumpy spectrum, with individual Compton scattering orders sepa-
rated, in contrast with the observed smooth power law-like tail. Such a spectrum can
be instead produced by Compton scattering on a non-thermal electron population,
where the index is set predominantly by the shape of the electron distribution rather
than Lh /Ls . Soft spectra such as the ones described here are typically seen during the
HSS.
As noted in [44], even though there are no observations which unambiguously
conflict with the truncated disc models, there exist a tremendous amount of data
which can be fit within this geometry (which includes, besides energy spectra, rapid
variability characteristics and jet properties). While the truncated disc model is in-
deed currently a very simplified version of what must be a more complex reality,
nonetheless the range of data it can qualitatively explain gives confidence that it
captures the essence of the main spectral states.
2.2.1.2 Alternative Geometries
Alternative geometries that include an untruncated disc and (mostly) isotropic source
emission have significant problems in matching the observed features of the hard-
state spectra.
• The shape of the spectral continuum observed in the hard-state rules out slab
corona models as the spectra all peak at high energies. These spectra are possible
only if the luminosity in seed photons within the X-ray region is less than that
in the hot electrons (i.e. Lh /Ls >> 5), while in a slab geometry only quite small
Lh /Ls can be obtained (a disc extending underneath an isotropically radiating hot
electron region would intercept around half the Comptonized emission in a slab
corona geometry, see [64]).
• A patchy corona allows part of the reprocessed flux to escape without reilluminat-
ing the hot electron region, so it can produce the required hard spectra. A patchy
corona also allows the reflected flux to escape along with the reprocessed flux,
resulting in a strong reflection spectrum for very hard spectra, in direct conflict
with the observations (see e.g. [89]).
• Models where the X-rays are produced directly in the jet were proposed by [91].
These produce the hard X-rays by synchrotron emission from the high-energy ex-
tension of the same non-thermal electron distribution which gives rise to the radio
emission. However, the observed shape of the high-energy cut-off in the hard state
is very sharp and cannot be easily reproduced by synchrotron models [183]. How-
ever, there are now composite models where the X-rays are from Comptonization
by thermal electrons at the base of the jet (which resides in a hot flow at the centre
of a truncated disc), while the radio is from non-thermal electrons accelerated
up the jet [92]. This model practically converges onto the truncated disc model,
though with some additional weak beaming of the hard X-rays.
• One last alternative to the truncated disc model is represented by the magnetized
accretion–ejection model of [54], which has a disc-inner jet structure similar to that
by [92], though here the inner, optically thick disc is still present down to the last
stable orbit, but with properties very different from the standard Shakura–Sunyaev
disc. The transition radius between this jet-dominated disc and the standard ac-
cretion disc is variable, producing the range of behaviour seen in the hard state
spectra in a similar way to the truncated disc model.
From what has been said above, all currently viable models for the hard state
converge on a geometry where the standard disc extends down only to some radius
larger than the last stable orbit, with the properties of the flow abruptly changing at
this point.
2.2.2 Fast Time Variability
Fast time variability is an important characteristic of BHBs and a key ingredient for
understanding the physical processes in these systems. Fast (aperiodic and quasi-
periodic) variability is generally studied through the inspection of power density
spectra (PDS; [172]). Most of the power spectral components in the PDS of BHBs
are broad and can take the form of a wide power distribution over several decades of
frequency or of a more localized peak (quasi-periodic oscillations, QPOs).
QPOs were discovered several decades ago in the X-ray flux emitted from ac-
creting neutron stars and have since been observed in many BHB systems (see, e.g.
[116]). It is now clear that QPOs are a common characteristic of accreting BHs and
they have been observed also in neutron stars (NS) binaries (e.g. [14, 67, 172]), in
cataclysmic variables (see e.g. [129]), in the so-called ultraluminous X-ray sources
(possibly hosting intermediate-mass BHs or super-Eddington accreting NS, e.g. [8,
165]) and even in active galactic nuclei (AGNs, e.g. [60, 101]).
2.2.2.1 Low-Frequency QPOs
Low-frequency QPOs (LFQPOs) with frequencies ranging from a few MHz to

∼30 Hz are a common feature in almost all transient BHBs and were already found
in several sources with Ginga and divided into different classes (see, e.g. [108] for
the case of GX 339-4 and [167] for the case of GS 1124-68). Observations performed
with the Rossi X-ray Timing Explorer (RXTE) have led to an extraordinary progress
in our knowledge of the properties of variability in BHBs (see [24, 143, 174]): it was
only after RXTE was launched that LFQPOs were detected in most observed BHBs
(see [173].
Three main types of LFQPOs, dubbed types A, B and C, originally identified in
the PDS of XTE J1550-564 (see [69, 142, 178]), have been seen in several sources.
The different types of QPOs are currently identified on the basis of their intrinsic
properties (mainly centroid frequency and width, but energy dependence and phase
lags as well), of the underlying broadband noise components (noise shape and total
variability level) and of the relations among these quantities.
• Type-A QPOs (Fig. 2.2, top panel) are characterized by a weak (few per cent rms)
and broad (ν/Δν ≤3) peak around 6-8 Hz. Neither a subharmonic nor a second
harmonic are usually present (possibly because of the width of the fundamental
peak), whereas a very low-amplitude red noise is associated with these QPOs.
Originally, these LFQPOs were dubbed type-A-II by [69]. LFQPOs dubbed type-
A-I [178] were strong, broad and associated with a very low-amplitude red noise.
A shoulder on the right-hand side of this QPO was clearly visible and interpreted as
a very broadened second harmonic peak. [35] showed that this type-A-I LFQPOs
should be classified as a type-B QPOs. Type-A QPOs usually appear in the HSS,
just after the transition from the HIMS.
• Type-B QPOs (Fig. 2.2, middle panel) are characterized by a relatively strong (4 %
rms) and narrow (ν/Δν ≥6) peak, which is found in a narrow range of centroid
frequencies around 6 Hz or 1–3 Hz [114]. A weak red noise (few per cent rms or
less) is detected at very low frequencies (≤0.1 Hz). A weak second harmonic is
often present, sometimes together with a subharmonic peak. In a few cases, the
subharmonic peak is higher and narrower, resulting in a cathedral-like QPO shape
(see [34]). Rapid transitions in which type-B LFQPOs appear/disappear are often
observed in some sources (e.g. [123]). These transitions are difficult to resolve, as
they take place on a timescale shorter than a few seconds. The presence of type-B
QPOs essentially defines the SIMS.
• Type-C QPOs (Fig. 2.2, bottom panel) are characterized by a strong (upto 20 %
rms), narrow (ν/Δν ≥10) and variable peak (its centroid frequency and intensity
varying by several per cent in a few days; see, e.g. [116]) at frequencies 0.1–15 Hz,
superimposed onto a flat-top noise that steepens above a frequency comparable
to that of the QPO. A subharmonic and a second harmonic peak are often seen,
and sometimes even a third harmonic peak. The total (QPO plus flat-top noise)
fractional rms variability can be as high as 40 %. The frequency of the type-C
QPOs correlates both with the flat-top noise break-frequency ([179] and with the
characteristic frequency of some broad components seen in the PDS at higher
frequency (>20 Hz, see [138]). Type-C QPOs are usually observed during the
bright end of the LHS and during the HIMS. In some sources (see e.g. [113, 116]),
type-C QPOs can be seen also in the HSS, where they show frequencies that can
reach ∼30 Hz.
www.ebook3000.com
Fig. 2.2 Examples of type-A, type-B and type-C QPOs from our GX 339-4 observations. The
contribution of the Poisson noise was not subtracted. Adapted from [114]
2.2.2.2 Models for LFQPOs
Despite LFQPOs being known for several decades, their origin is still not understood
and there is no consensus about their physical nature. However, the study of LFQPOs
provides an indirect way to explore the accretion flow around black holes (and neutron
stars).
The existing models that attempt to explain the origin of LFQPOs are generally
based on two different mechanisms: instabilities and geometrical effects. In the latter
case, the physical process typically invoked is precession.
Titarchuk and Fiorito [170] proposed the so-called transition layer model, where
type-C QPOs are the result of viscous magneto-acoustic oscillations of a spherical
bounded transition layer, formed by matter from the accretion disc adjusting to the
sub-Keplerian boundary conditions near the central compact object.
Cabanac et al. [32] proposed a model to explain simultaneously type-C QPOs
and the associated broadband noise. Magneto-acoustic waves propagating within the
corona makes it oscillate, causing a modulation in the efficiency of the Comptoniza-
tion process on the embedded photons. This should produce both the type-C QPOs
(thanks to a resonance effect) and the noise that comes with them.
Tagger and Pellat [166] proposed a model based on the accretion–ejection in-
stability (AEI), according to which a spiral density wave in the disc, driven by
magnetic stresses, becomes unstable by exchanging angular momentum with a

Rossby vortex. This instability forms low azimuthal wavenumbers, standing spi-
ral patterns which would be the origin of LFQPOs. Varnière and Tagger [175,
176] suggested that all the tree types of QPOs (A, B and C) can be produced
through the AEI in three different regimes: non-relativistic (type-C), relativistic
(type-A, where the AEI coexists with the Rossby wave instability (RWI), see [166])
and during the transition between the two regimes (type-B QPOs).
Stella and Vietri [158] proposed the so-called relativistic precession model (RPM)
to explain the origin and the behaviour of a type of LFQPO (the so-called horizontal-
branch oscillation) and of two high-frequency QPOs (the so-called kHz QPOs) in
NS X-ray binaries, as the result of the nodal precession, periastron precession and
Keplerian motion, respectively, of a self-luminous blob of material in the accretion
flow around the compact object. This model was later extended to BHs [159]. Motta
et al. [117] recently showed that the RPM provides a good explanation for both
type-C QPOs and high-frequency QPO (see below) in at least two BH systems.
Ingram et al. [70] proposed a model based on the relativistic precession as pre-
dicted by the theory of general relativity that attempts to explain type-C QPOs and
their associated noise. This model requires a cool optically thick, geometrically thin
accretion disc [150] truncated at some radius, filled by a hot, geometrically thick
accretion flow. This geometry is known as truncated disc model [48, 135]. In this
framework, type-C QPOs arise from the Lense–Thirring precession of a radially ex-
tended section of the hot inner flow that modulates the X-ray flux through a combi-
nation of self-occultation, projected area and relativistic effects that become stronger
with inclination (see [70]). The broadband noise associated with type-C QPOs, in-
stead, would arise from variations in mass accretion rate from the outer regions of
the accretion flow that propagate towards the central compact object, modulating the
variations from the inner regions and, consequently, modulating also the radiation in
an inclination-independent manner (see [71]).
2.2.2.3 High-Frequency QPOs
Among the most important discoveries of RXTE is the detection of the so-called
kHz QPOs in neutron star binaries (see [174]). This result opened a window onto
high-frequency phenomena in BHBs. The first observations of the very bright system
GRS 1915+105 led to the discovery of a transient oscillation at ∼67 Hz [111], the
first high-frequency QPO (HFQPO) in a BHB. Since then, sixteen years of RXTE
observations have yielded only a handful of detections in other sources, although
GRS 1915+105 seems to be an exception, with a remarkably high number of detected
high-frequency QPOs (see e.g. [25]).
The properties of the few confirmed HFQPOs [25] can be summarized as follow:
• They appear only in observations at high-flux/accretion rate. This is at least partly

due to a selection effect, but not all high-flux observations lead to the detection of
Power spectral density

1.92
Power spectral density

1.9
1.88
1
10
1.86
1.84
200 300 400 500 600 700800

Frequency (Hz)
−1 0 1
10 10 10
Frequency (Hz)
Fig. 2.3 Power spectrum of GRO J1655-40 displaying three simultaneous QPO peaks: the type-C
at ∼18 Hz, upper and lower high-frequency QPO at ∼300 and ∼450 Hz, respectively (Fig. 2 from
[115])
a HFQPO, all else being equal, indicating that the properties of these oscillations
can vary substantially even when all other observables do not change.
• They can be observed as single or double peaks. Only one source, GRS J1655-
40 (see Fig. 2.3, showed two clear simultaneous peaks [115, 163], while all the
others only showed single peaks, sometimes at different frequencies (see Tab. 1
in [26]. In XTE J1550-564, the two detected peaks [142] have been detected
simultaneously after averaging a number of observations, but the lower one with
a 2.3 σ significance [103]. Méndez et al. [97], on the basis of their phase lags,
suggested that the two detected peaks might be the same physical signal at two
different frequencies. H 1743-322 showed a clear HFQPO with a weak second
simultaneous peak [66]. A systematic analysis of the data from GRS 1915+105
[22] led to the detection of 51 HFQPOs, most of which at a centroid frequency
between 63 and 71 Hz. All detections corresponded to a very limited range in
spectral parameters, as measured through hardness ratios. Additional peaks at
27, 34 and 41 Hz were discovered by [16, 22, 164]. The most recent HFQPO
discovered, in IGR J17091-3624, is consistent with the average frequency of the
67 Hz QPO in GRS 1915 + 105 [3].
• Typical fractional rms for HFQPOs is 0.5–6 % increasing steeply with energy, in
the case of GRS 1915 + 105 reaching more than 19 % at 20–40 keV (see right
panel in Fig. 6 of [111]). Quality factors Q1 are around 5 for the lower peak and
10 for the upper. In GRS 1915 + 105, a typical Q of ∼20 is observed, but values
as low as 5 and as high as 30 are observed.
• Time lags of HFQPOs have been studied for four sources [97]. The lag spectra of
the 67 Hz QPO in GRS 1915 + 105 and IGR J17091-3624 and of the 450 Hz QPO
1Q is defined as the ratio between the centroid frequency and FWHM of the QPO peak.
in GRO J1655-40 are hard (hard photon variations lag soft photon variations),
while those of the 35 Hz QPO in GRS 1915 + 105 are soft. The 300 Hz QPO in
GRO J1655-40 and both HFQPOs in XTE J1550-564 are consistent with zero
(suggesting that the two HFQPOs in XTE J1550-564 are the same feature seen at
different frequencies).
• For three sources, GRO J1655-40, XTE J1550-564 and XTE J1743-322, the two
observed frequencies are close to being in a 3:2 ratio [142, 143, 163], which has
led to a family of models, known as resonance models (see e.g. [1]). For GRS
1915 + 105, the 67 and 41 Hz QPOs, observed simultaneously, are roughly in 5:3
ratio. The 27 Hz would correspond to 2 in this sequence.
2.2.2.4 Models for HFQPOs
Many models have been proposed to describe HFQPOs of BHBs, all involving in
some form the predictions of the theory of general relativity.
The relativistic precession model (RPM) was originally proposed by [158, 160]
to explain the origin and the behaviour of the LFQPO and kHz QPOs in NS X-
ray binaries and later extended to BHs [115, 117, 159]. The RPM associates three
types of QPOs observable in the PDS of accreting compact objects to a combination
of the fundamental frequencies of particle motion. The nodal precession frequency
(or Lense–Thirring frequency) is associated with type-C QPOs LFQPOs, while the
periastron precession frequency and the orbital frequency are associated with the
lower and upper HFQPO, respectively (or to the lower and upper kHz QPO in the
case of NSs). The relativistic precession model has been proposed in two other
versions. In [31], it is assumed that radiation is modulated by the vertical oscillations
of a slightly eccentric fluid slender torus formed close to the ISCO. Stuchlik and
collaborators proposed a further version of the relativistic precession model that has
been studied in many papers by this group. Here, the model is related to the warped
disc oscillations discussed by [75] (see below).
The warped disc model proposed by [75, 76] states that the HFQPOs are resonantly
excited by specific disc deformation warps. The model was generalized to include
precession of the warped disc in [77] and spin-induced perturbations were included
in [78].
Abramowicz and Kluźniak [1] and Kluzniak and Abramowicz [79] introduced the
nonlinear resonance model, which was later studied extensively by them as well as
by other authors. This model is based on the assumption that nonlinear 1:2, 1:3 or 2:3
resonance between the orbital and radial epicyclic motion could produce the HFQPOs
observed in both BH and NS binaries. Later on, [2] proposed another version of this
model, called the Keplerian nonlinear resonance model, where the resonance occurs
between the radial epicyclic frequency and the orbital frequency instead of between
the radial epicyclic frequency and the vertical frequency. These resonance models
successfully explain black hole QPOs with frequency ratio consistent with 2:3 or
1:2 (see Sect. 3.2). As a given resonance condition is verified only at a fixed radius
in the disc, the QPO frequencies are expected to remain constant, or jump from one
resonance to another.
2.2.3 Long-Term Time Evolution
After having defined and discussed the source states in terms of their spectral and
timing properties along the HID/HRD/RID diagrams, it is useful to examine the
evolution of sources along the HID (and in parallel the other diagrams). A short
description can be found in [49], together with an animation.
BHBs spend most of the time in a “quiescent” state, where the accretion rate
reaching the central regions of the accretion flow and the black hole is very low,
typically lower than 10−5 L Edd [131, 132], indicated in Fig. 2.4 as QS but actually
located below the extension of the Y-axis. Here, the energy spectra indicate that
the spectrum deviates from that of the LHS, which hardens as flux decreases. Going
down to quiescence, at a level around 10−2 L Edd the measured power law index starts
increasing from 1.5–1.6 until leveling off around 2 at 10−5 L Edd [131, 181]).
The branch from A to B in Fig. 2.4 corresponds to the LHS and is travelled on
timescales which can be as long as months but also so short that the LHS is not
observed in pointed observations made on the day following the first alert. We do not
Fig. 2.4 Generic HID and HRD of a black hole binary outburst. The letters refer to main locations
described in the text (Fig. 1 from [85])
know directly that the A–B branch was traversed, but we have never seen evidence of
the contrary. There are a few cases of systems which never leave the LHS and return
to quiescence after having reached a peak (see, e.g. [30] and references therein), not
necessarily at low flux [126]). Along this vertical branch, the characteristic frequen-
cies of the strong noise components observed in the PDS increase, while the total
fractional rms variability decreases.
Then at B the HIMS is entered. The precise time of the transition can be identified
with the changes of timing properties (see [13] and [120]) or through a change in
the low-energy properties as was observed in GX 339-4 [66]. The duration of the
HIMS, when the source softens from B to the SIMS transition, can be less than
a day upto two weeks. A few sources, after entering the HIMS, did not proceed to a
further transition and returned to the LHS and then to quiescence, failing to reach a
full transition [28, 33, 155]. The characteristic frequencies continue to increase and
the total fractional rms to decrease, while an evident type-C QPO appears, also with
increasing frequency as the source softens.
If the outburst does not fail, a transition to the SIMS is made. While the energy
spectrum below ∼10 keV is similar to that of the softest HIMS observations, with
a hard-component photon index around 2.4, the SIMS is characterized by marked
differences in the PDS. In particular, the disappearing of the type-C QPO and the
appearing of a type-B QPO at a different frequency are relatively easy to observe.
This transition has been seen to take place on a timescale of a few seconds [34].
Multiple back and forth transitions have also been observed, on timescales of days
to weeks but also down to minutes [13, 34, 123]. The evolution of the high-energy
(>10 keV) spectral component also changes abruptly here. Around, but not exactly
coincident to, this transition, fast discrete relativistic jets are launched, observed
either as resolved moving radio spots or as bright radio flares (see [51]). Although it
would be tempting to causally associate the disappearance of timing components in
the PDS to the ejection of the plasma responsible for them, it has been shown that in
some cases the ejection starts a few days before the transition [52].
The position of the further transition to the HSS is not easy to identify, as the
presence of weak type-A QPOs is not always easy to ascertain. The HSS, after
possible back transition to the previous intermediate states, is rather stable and can
last for months. The luminosity, which in many systems peaks in the HIMS/SIMS
(but see below), decreases, most likely because of a more or less steady decrease in
mass accretion rate. A low level of aperiodic time variability is detected, in the form
of a power law component in the PDS, with a total fractional rms around 1–2 %.
At a luminosity level well below that of the early HIMS, a new transition takes
place (at D). From here on, time is reversed, first the SIMS and then the HIMS are
observed, after which the LHS is reached, after which the outburst ends and the
source goes back to its quiescent level. However, notice that at low luminosity, the
thermal disc contribution is softer, which means that HID points at the same hardness
do not correspond to the same energy spectrum. Indeed, the photon index of the hard
component at the return SIMS is around 2.1, comparable to that at the start of the
first HIMS (point B) [161].
The transition between LHS–HIMS–SIMS–HSS at high flux does not necessarily

coincide to the maximum in accretion rate, as it is assumed in the previous description.
Indeed, in a few cases the accretion rate continued to increase after the HSS is reached.
If this happens, in the HID the source moves up from C moving to the right (see [20,
113]). Both the energy spectrum and the PDS become rather complex [45, 113]).
There are sources for which the diagram appears more complex than this (see
examples in [20, 52]), as well as sources which are observed when already in a
bright HSS. The latter must have come from quiescence and the data do not exclude
that the LHS–HIMS–SIMS branches were followed, only on a much shorter time
scale, of the order of the day.
The HID path outlined in Fig. 2.4 shows a clear evidence of hysteresis, as the
return path is different from the forward path. In other words, the luminosity (or
accretion rate) level at which the hard-to-soft transition takes place is higher than
that of the reverse transition [88]. For the prototypical source GX 339-4, which had
several outbursts, there is evidence that the higher the first level, the lower the second.
A correlation has been found between the waiting time from the previous outburst
and the hard X-ray peak, which corresponds to the hard-to-soft transition level (see
[182]), but this does not specifically address the issue of hysteresis. The same effect
is observed in neutron star transients [88, 119] and implies that luminosity (and
hence accretion rate) cannot be an absolute proxy for state. In other words, LHS and
HSS can be observed over the same range of luminosities, although the transition
from hard to soft does take place at the highest LHS flux (see also [88]). While most
generic interpretations for the observed hysteresis invoke the presence of a second
parameter in addition to accretion rate, since accretion rate determines the movement
along the diagram, but the LHS–HSS transition can take place at different accretion
rate levels, [85] associate the hysteresis with the system’s memory, in the same way
that magnetic hysteresis works. However, what sets the transition level and whether
a transition will take place or not is still an open problem. It is interesting to note
that while the upper path can take place at very different luminosities, the range
spanned by the lower path is much more reduced, around a few percentage of the
Eddington luminosity (see [74, 88]). Adding complication, [118] showed that there
is a systematic difference in the shape of the HID between high- and low-inclination
sources.
A similar hysteresis diagram has been observed in a white dwarf binary, the dwarf
nova SS Cyg, using optical emission as soft band and X-ray emission as hard band for
the production of the HID [80]. A radio flare was also detected in correspondence to
the hard-to-soft transition. Although the comparison is interesting, it has to be noted
that in a white dwarf binary the optical/UV emission originates from the accretion
disc and the X-ray emission from the boundary layer between the disc and surface. In
general sense, what we are observing from these systems is a transition from optically
thin to optically thick emission when the density has increased and a reverse transition
when the density has decreased, an effect which is also valid for the boundary layer of
a dwarf nova (see [84]). The details of the physics of the systems and the transitions
do not necessarily need to be the same.
As described in Sect. 2.2.1 and highlighted in more detail by [85], the general
outburst evolution (including the basic diagrams, energy spectra and fast time vari-
ability) at zero level can be interpreted as a simple evolution of the transition radius
within the truncated disc model (see also Sect. 2.5).
2.3 Radio/IR Emission
While the process of accretion onto stellar mass black holes has been studied, mainly
in the X-rays, since the beginning of X-ray astronomy, it is only relatively recently
that observations at longer wavelengths, notably infrared and radio, have led to the
discovery of additional processes such as the ejection of relativistic jets and of wind
outflows. It is now clear that without a broadband perspective the complete and
complex picture of an accreting compact object in a binary system is impossible to
understand. Below, we review the main observational points that in the past decade
have led to a major change in perspective.
2.3.1 Radio Jets
Although radio emission from black hole binaries had been detected a long time
before (see, e.g. Tananbaum et al. 1971), it was only in the early nineties that co-
ordinated radio campaigns were started, following the discovery of relativistic jets
in the radio band. The current observational picture is rather clear and it correlates
with the position of the source in the HID (see above). Figure 2.5 shows a sketch of
a HID with an indication of the different types of outflow observed.
from below A to B BHBs spend most of their time in the quiescent state (well
below point A in Fig. 2.5), accessible only by the most powerful focusing X-
ray telescopes. The X-ray luminosity is < 1033 erg/s. Weak flat spectrum radio
emission has been detected in this state. As the source brightens in the LHS, the
radio luminosity also increases, while the radio spectrum remains flat. In this state,
a compact jet has been resolved in a few cases in the radio band (see e.g. [162]).
The radio spectrum is flat, consistent with self-absorbed synchrotron emission and
extends upto the near infrared, while the polarization level is low. The X-ray flux
and radio flux show a strong positive nonlinear correlation (see Sect. 2.3.2.1). The
observational data are interpreted with the presence of a compact jet emitting in
the radio through synchrotron and moving outwards with moderately relativistic
speed (Γ < 2, see [51]).
from B to C Point B marks the start of the HIMS. In addition to changes in the rapid
variability (total fractional rms), a good marker for this transition is the breakdown
of the correlation between the radio-IR flux and X-ray flux (see below and [51,
66]). As the transition to the SIMS is approached, the radio emission generally
Fig. 2.5 Schematic evolution of a black hole binary along the HID (top panel). The three bottom
panels show the configuration of the system along the top branch: where red represents radio jets
and yellow wind outflows. From [49]. Reprinted with permission from AAAS
decreases, with oscillations and small flares [104], while there is indication that
the radio spectrum starts steepening [51]. All data indicate that the inner part of
the jet undergoes changes in its physical properties.
from C to D As outlined above, the transition between the HIMS and SIMS is
marked by rapid changes in the properties of fast variability, in particular a drop
in fractional rms and a switch between the type-C and type-B QPOs. Around
(but not precisely in correspondence to) the transition, the jet properties change
drastically. One or more large-amplitude flares are observed in the radio (see [52]).
When major superluminal ejections are observed, the ejection time can also be
traced back to a time close to the transition. Unfortunately, the lack of precise
correspondence of the time of jet ejection and that of HIMS–SIMS transition
indicates that there cannot be a causal connection. There is indication that these
jets have a larger Lorentz factor than the steady jets in the LHS and HIMS, which
led to the idea that what is observed is due to internal shocks caused by faster jets
hitting the slower components (see [51]).
from D to E This branch marks the HSS. Until now no radio emission that can be
attributed to the central source (and not from ejecta) has been detected down to
upper limits of >300 times that of LHS sources at the same X-ray flux (see [147]).
Outflows in the form of winds are observed (see 2.4). During this state accretion
rate decreases on a long timescale (weeks to months), leading to the vertical
track in the HID. However, at high flux there can be multiple transitions to the
SIMS/HIMS and back, as measured through the changes in timing properties [13,
20, 34, 49]. Although coverage at lower energies is sparse, the observations are
consistent with the presence of weaker radio flares corresponding to the transitions
(see e.g. [29, 34]).
from E to F On the return branch from soft to hard, as we have seen above the
reverse track is followed, through the SIMS and the HIMS, on a timescale com-
parable to that of the upper track. Observations of several transitions from multiple
systems have shown that the compact radio jet is re-formed not in correspondence
with the SIMS–HIMS transition, but when the system has reached the LHS, a de-
lay of several days (see [73, 74] and references therein). At the time of radio
reappearance, secondary maxima have been seen in the optical-infrared band,
which have been attributed to direct jet emission, although alternative models
exist [74].
2.3.2 Accretion–Ejection
2.3.2.1 Correlations
The radio and optical/IR emission from BHBs is attributed to jets (for outflows, see
Sect. 2.4) and is closely connected to the accretion properties as measured at higher
energies. In particular, the low-energy flux from the compact jet in the LHS shows a
strong nonlinear correlation with X-ray flux, modelled with a power law with index
α ∼0.6–0.7 (Fig. 2.6, see [39, 58], first discovered in GX 339-4 [40] then extended to
other systems [57]). Ignoring the points which appear to follow a different correlation
(see below) the scatter in the relation is rather small. Under the assumption that the X-
ray flux originates from accretion and therefore is not subject to relativistic beaming
like the radio emission from a jet, from the scatter around the correlation [51] have
estimated the jet Lorentz factor to be Γ ∼1–2. More recently, the correlation has
been extended to BHBs in quiescence, notably down to very low fluxes for the first
known system A 0620-00 ([56], see also [59]). All stellar mass black holes have
likely similar masses (a few to several solar masses). After a correction for mass
is added, it was found that the correlation can be extended to active galactic nuclei
([98], see also [132]), spanning 15 orders of magnitude in X-ray flux [56].
Fig. 2.6 Correlation between the radio and X-ray luminosity for BHBs in the LHS and in quiescence
(Fig. 9 from [39])
However, as more multi-wavelength observations became available, the situation

has become more complex. A group of sources are found to follow a different cor-
relation, limited to high fluxes and below the main one (often referred to as the
“radio-quiet” branch, see [58]). Figure 2.6 shows the clear presence of two branches:
the radio-quiet one is steeper, fitted with α ∼1 [58]. Recently, the statistical sig-
nificance of the presence of two separate correlations has been questioned by [59].
The system H 1743-322 was observed to move from one correlation to the other:
during outburst decay, it started on the radio-quiet branch, then as radio luminosity
decreased X-ray luminosity stalled until the system reached the radio-loud branch
[42]. This clearly indicates that the difference between the two branches cannot be
associated with fundamental parameters of the different systems, which would not
change during the outburst (see [154]). Coriat et al. [42] explored different possibil-
ities and concluded that the transition between the two branches could be due to a
transition from a radiatively efficient to a radiatively inefficient accretion flow, which
would make the “radio-quiet” branch an “X-ray loud” one (see [82]). To complicate
matters even further, [148] found that during its 2011 outburst MAXI J1836-194
followed a path in the X-radio plane which was significantly steeper and connected
the two existing branches. Although the correlation appears to be less universal than
previously assumed, it is clear that its interpretation can give important insights
on both accretion and ejection. For instance, the evaporation/condensation model by
[100] would explain the second, X-ray loud, branch as due to the presence of an inner
optically thick accretion disc which would increase the soft photon input available
to Comptonization. As this inner disc cannot work at low accretion rate, this inter-
pretation also explains why no lower-branch points are observed at low X-ray fluxes
and the switch of H 1743-322. At or after the LHS–HIMS transition, the correla-
tion breaks down (see also below), as the energy spectrum becomes more and more
dominated by an optically thick accretion disc and the radio emission also shows
non-monotonic variations (see [51]). Interestingly, neutron star LMXBs also show a
correlation between the radio and X-ray flux, with three important differences: (1)
all NS systems are more radio-quiet at a given X-ray luminosity; (2) the correlation
is steeper and (3) the radio emission is not suppressed when the source transits to the
soft states [102]. This also points to the presence of a radiatively inefficient regime
of accretion for BHBs.
At shorter wavelengths, IR and optical, the situation is complicated by the fact
that the accretion disc also contributes to the flux and the jet contribution has to
be estimated. In the prototypical system GX 339-4, in the LHS a clear positive
correlation between the X-ray and IR flux was found, which terminated abruptly
when the source entered the HIMS and the relative contribution between the jet
and disc are expected to change [66]. Additional observations have shown that the
correlation extends for four orders of magnitude and evidence for a break around
10−3 L Edd has been found [41]. Recently, a complete analysis of data from 33 systems,
both hosting black holes and neutron stars, has been published [146]. For BHBs, they
found a correlation with power law index 0.6 extending from quiescence to bright
LHS, similar to that observed in the radio.
2.4 Winds and Outflows
Over the last couple of decades we have witnessed the discovery of a multitude of
highly ionized absorbers in high-resolution X-ray spectra from both BH and NS X-
ray binaries. The first detections were obtained thanks to ASCA on the BH binaries
GRO J1655-40 [171] and GRS 1915+105 [83]. Narrow absorption lines in the spectra
of these systems identified as Fe XXV and Fe XXVI indicated the first of a myriad of
discoveries of photo-ionized plasmas in LMXBs that followed the launch of X-ray
observatories such as Chandra, XMM-Newton and Suzaku.
After the first detections, it was soon clear that the photo-ionized absorbers in the
form of winds or atmospheres could be ubiquitous to all X-ray binaries (e.g. [127]),
but only recently it has become increasingly clear that the presence of these plasmas
could be key to our understanding of these systems. It has been suggested that the
amount of mass that leaves the system once the plasmas are outflowing can be of
the order of or significantly higher than the mass accretion rate transferred through
the accretion disc (e.g. [133]), with crucial consequences on the accretion-outflow
equilibrium at play in these systems.
Much of the recent work on the accretion/ejection connection at work during the
outburst of black hole x-ray transients has focused on the X-ray/radio correlations
(see, e.g. [51, 52]). It is now established that transient sources emerge from quies-
cence entering the low/hard state (see [24]), where a steady radio jet is ubiquitously
seen. Then, they rise in luminosity until the hard-to-soft transition takes place. During
this transition strong relativistic ejections are often seen—observed as bright radio
flares—and connected to the disappearance of the steady radio jets.
Strong disc winds have been predominantly observed in the thermal-dominated
high/soft state of BH transients (and most recently in NS systems, see [134]). Con-
versely, the presence of winds has been excluded by observations in the low/hard
state, where no signature of wind-like outflows has been found so far with high level
of confidence. Therefore, it seems natural to conclude that in the soft state the radio
jets and relativistic ejections seem to be replaced by highly ionized accretion disc
winds, that may play a crucial role in the physics of accretion and ejection around
compact object. For instance, the effects of the winds on the entire system could be
key in the outburst evolution of transient systems, influencing or even triggering the
return to the hard state.
2.4.1 Accretion Disc Winds and Atmospheres
To date, several LMXBs, both containing BHs and NSs (see e.g. [43, 133]), have
shown narrow absorption lines, more often than not blueshifted (i.e. indicating out-
flowing material) that have been interpreted as a consequence of the presence of
highly ionized material local to the source, opposed to the absorption due to the in-
terstellar medium. Most of these sources are observed at a relatively high inclination
angle (60–70◦ ), pointing to a distribution of the ionizing plasma close to the accretion
disc, with an equatorial or flared geometry.
While the relative depth of the absorption lines detected in the high-resolution
spectra of LMXBs allows to determine the column density and the ionization state of
the plasma responsible for the lines, their blueshift with respect to their theoretical
wavelength provides information on the relative velocity of the plasma. The ioniza-
tion degree of the plasma is described through the ionization parameter, defined as
ξ = L/nr 2 , where L is the luminosity of the ionizing source, n is the electron density
and r is the distance between the ionizing source and the plasma.
The column density of the ionized plasma detected in LMXBs ranges between
1021 cm−2 and 1024 cm−2 and there is no obvious difference between the densities
measured for BHs and NSs systems. The vast majority of the systems for which
log(ξ ) has been measured, show high ionization degrees, with log(ξ ) > 3. However,
MAXI J1305-70 has shown both high and low ionization degrees, and GX 339-4
(see [151]) has shown only a low ionization degree plasma. Interestingly, among
the sources known to show ionization plasmas, GX 339-4 is possibly the one at the
lowest inclination: this suggests the possibility that there could be a stratification
of the ionized plasma as a function of inclination above the accretion disc. On the
other hand, GX 339-4 is affected by relatively low interstellar absorption; thus, the
detection of low ionization plasma could be affected by a significant observational
bias.
Even though the majority of the ionized plasmas detected so far in LMXBs is
characterized by significant intrinsic velocities, for a few sources there is no evidence
of an intrinsic plasma velocity. This difference justifies the classification of ionized
plasmas in disc winds/outflows and disc atmospheres. When a given system shows
absorption lines with significant blueshift and/or a P-Cygni profile, the ionized plasma
is classified as outflow. If, instead, a given system does not show a significant blueshift
in the absorption line, then the ionized plasma takes the name of atmosphere. Clearly,
the most important difference between the systems shown winds/outflows and the
ones showing atmospheres is that only the former loose mass, with a rate that can be
even significantly larger than the mass accretion rate (see, e.g. [87, 122, 133]).
Among the sources for which ionized plasmas has been detected, all BH LMXBs
shows relatively fast outflows, while only about ∼30 % of NS LMXBs have shown
clear winds (see [134]). This difference between NSs and BHS could be due to a
significant difference in size of the system, i.e. strong winds are more easily pro-
duced in long orbital period (i.e. large systems, and, consequently, large discs). This
hypothesis is supported by the fact that the NS systems producing winds are those
with orbital periods comparable to the BH ones.
2.4.2 Winds Launching Mechanism
Three main mechanisms have been identified as possible responsibles for the launch-
ing of fast winds from the accretion disc of accreting sources: thermal pressure, ra-
diative pressure and magnetic pressure. The mechanism normally invoked to explain
the strong winds detected in the radio-free soft states of BH and NS X-ray binaries
is the thermal pressure, even though it is important to bear in mind that the dominant
wind launching mechanism could change as the system evolves.
2.4.2.1 Thermally Driven Winds
Thermal pressure or Compton heated winds should arise in systems where the ac-
cretion disc is irradiated by the central regions of the accretion flow, such as in X-ray
binaries or in quasars. The disc gas can be heated to temperatures exceeding 107
K mostly through the Compton process, partially evaporating and forming a corona
above the disc. This gas, depending on the thermal velocity exceeding or not the
local escape velocity, can be either emitted as a thermal wind or stay bound to the
disc, forming an atmosphere [12]. The radial extent of this corona only depends on
the mass of the central compact object and on the Compton temperature, while it
is independent on the luminosity. It has been shown [12] that due to disc rotation a
wind can be launched via thermal pressure at radii larger than ∼0.1 rC , where rC is
the Compton radius, i.e. the radius where the escape velocity equals the isothermal
sound speed at the Compton temperature TC . For very luminous systems (where the
radiation force due to electrons must be taken into account), [136] found that strong
winds can be launched starting from radii as small as 0.01 r I C .
2.4.2.2 Radiatively Driven Winds
Radiation-driven winds might arise from an accretion disc when radiative accelera-
tion occurs due to the transfer of momentum from the photon field to the corona that
leaves the disc in the form of an outflow. Assuming that the wind has optical depth τe
higher than 1 and completely surrounds a source of radiation with luminosity Lbol ,
the multiple scattering of photons within the wind will lead to a wind momentum that
exceed the photon momentum of the primary emission. In most sources, however,
the material constituting the wind appears to be far too ionized to allow an effec-
tive momentum transfer and a consequent launching mechanism. In other words, the
momentum-flux in radiatively driven winds cannot exceed that of the radiation field,
even considering the effects of radiation of free electrons (see [144]).
2.4.2.3 Magnetically Driven Winds
Disc winds can also be driven by magnetic forces. These winds are expected to be
centrifugally accelerated down open, rotating, poloidal magnetic fields anchored in
the disc (see [27]). Being these winds accelerated by the effect of magnetic torques
from magnetic fields embedded in the accretion disc, there must be an intimate
connection between the mass loss in the wind and the accretion onto the black hole.
According to the theory of MHD winds, material is centrifugally lifted off the disc
at a certain launching radius R L and is continuously accelerated by magnetic forces
until the flow becomes super-Alfvenic at a radius R A > R L . Accurate models suggest
that the ratio R A /R L ranges between 2 and 3 (see [139]). Reynolds [144] has shown
that MHD torques would be able to produce Compton thick winds only if (1) the
accretion rate is a significant fraction of the Eddington rate, (2) the radiative efficiency
is low and (3) the Alfven radius is very close to the launching radius (R A ∼R L ). Thus,
MHD driving could become a viable explanation for disc winds only in a very limited
number of cases.
2.5 The Full Accretion–Ejection Picture
Despite the number of complications and additional details, the overall picture is
now much more clear than before the RXTE data. Not all transient sources behave
in the fundamental diagrams as neatly as the one shown in Fig. 2.1, but the general
pattern appears to be followed by most sources and this regularity points towards
fundamental aspects of accretion onto black holes (see [52]). Indeed, a comparison
with different classes of related sources shows that this must be the case.
Persistent BHBs. A few persistent systems are known in our galaxy and the Mag-
ellanic Clouds. Some of them appear locked in a single state. LMC X-1 has always
been observed in the HSS (see [145]), as the bright galactic source 4U 1957 + 11
[124]. The second BHB in the LMC, LMC X-3, was only observed in the HSS until
with the extensive coverage of RXTE brief transitions to and from the LHS were
discovered [152]. The source GRS 1758-258 in the galactic centre region is mostly
in the LHS, but shows sporadic transitions to the HSS (see [157]). The bright source
4U 1755-33 was very bright and in the HSS until 1996, when it went into quiescence
(see [5]), but unfortunately the decay into quiescence was not covered by observa-
tions. The brightest and best known Cyg X-1, the first black hole candidate, is usually
found in the LHS and makes rather frequent transitions to the HIMS, to rarely reach
the HSS when radio emission is observed to drop (see [63] and references therein).
A HID from the RXTE observations of Cyg X-1 is shown in [20]. Overall, none
of these systems shows a behaviour inconsistent with the above picture, although
clearly none of them shows a full “transient” cycle. In particular, the LHS–HIMS
and HMIS–LHS transitions of Cyg X-1 do not show any sign of hysteresis. Inter-
estingly, corresponding to one of the softest events of Cyg X-1, a radio flare was
observed, compatible with a jet ejection in correspondence to a transition to the HSS
[180].
A case of its own is represented by GRS 1915 + 105, which has started an outburst
in 1992 and is still active at the time of writing. Its behaviour is very different from all
other systems, although a few sources have been found to match it rather precisely
for some limited time [4, 9]. The original idea was that this peculiar behaviour
is connected to a very high accretion rate, which would put the source above the
standard “q” diagram (see [20] for an HID of state-C, i.e. hard, intervals of GRS
1915 + 105). However, recently the same type of structured variability has been
found in the Rapid Burster, a neutron star binary, at luminosities well below the
Eddington limit, casting doubts on this interpretation [9]. At any rate, the short-term
variability of GRS 1915 + 105 is not different from those observed in other BHBs:
during state-C, whose energy spectrum corresponds to a LHS/HIMS, the PDS is a
typical LHS/HIMS [20, 140, 141], during softer and brighter states (called A and B,
see [15] the PDS is similar to that of “anomalous” states at high accretion rate see
in other BHBs [20, 140]. HFQPOs are observed only in a very narrow region of the
HID [23]. Evidence for a type-B QPO during fast transition has also been presented
[153]. What is peculiar here is the structure of alternating states.
Neutron star binaries. Neutron star low-mass X-ray binaries show many similar-
ities in their emission properties with BHBs. Their detailed X-ray energy spectra are
rather different as the component of direct emission from the surface of the com-
pact object and the boundary layer is very bright. However, it is now clear that the
properties of fast time variability can be strongly connected to those of BHBs (see
[18, 35, 125, 138, 179]. The first evidence of a hysteresis pattern in the HID was
presented from the low-luminosity persistent system 4U 1636-53 [14]. “q” diagrams
were shown for the outbursts of Aql X-1 [20, 81]. More recently, a full analysis of
Fig. 2.7 RID diagram with

the regions occupied by
BHBs and different classes
of NS LMXBs (Z: Z sources,
BA: bright atoll sources, the
grey “q” corresponds to atoll
sources; Fig. 10 from [119]).
The X-axis in this diagram is
fractional rms variability,
which correlates almost
linearly with hardness,
which means a HID would
look almost identical
RXTE data has shown that the observed pattern is very similar and a strong con-
nection has been drawn [119]. As the difference between the classes of neutron star
LMXBs has been finally attributed to accretion rate levels (see [68], a complete dia-
gram for NS and BH binaries could be produced (see Fig. 2.7). A strong connection
with the BHBs was found in the radio emission of Aql X-1, completing the picture
[81]. The radio-X correlation for NS binaries lies below that of BHBs, suggesting
a difference in the emission efficiency between the two classes of systems in their
hard state [102].
ULX. In the case of ultraluminous X-ray sources (ULX), the comparison is not
simple, given the lower statistics and sparser coverage of observations, not to mention
the fact that there is no agreement on the nature of the systems (see [53] for a
review). There are similarities in the timing properties, but there is no consensus on
the interpretation of the few QPOs detected in ULX in terms of those in BHBs (see,
e.g. [121, 128, 165]). The energy spectra are even more complex and only basic
comparisons can be made, not always in agreement with each other [21, 156]. State
transitions have been observed (see [72] and references therein), but only with few
observations. When standard diagrams have been produced for a sample of ULX, the
results were also complex and difficult to interpret [130]. In the case of the hyper-
luminous source ESO 243-49 HLX-1, a number of XMM-Newton and Chandra
observations were used to produce HID and HRD, showing compatibility with the
BHB diagrams [149]. Radio flares have been observed from this source, although the
lack of high-SNR X-ray coverage prevents a precise association with source states
[177].
AGN. The association between the AGN properties and BHB states is also com-
plex. While Quasars and Blazars are thought to be jet-dominated systems and Seyfert
II AGN suffer of large absorption, a comparison can be attempted with Seyfert I sys-
tems. On the spectral side, the energy distribution in X-rays is similar to that of
galactic systems in the LHS, although for AGN the thermal accretion disc compo-
nent is not in the “standard” X-ray band. Indeed, the production of a HID from RXTE
data of Sy I objects shows that almost all of them lie on the LHS branch. However,
analysis of variability, which in the case of AGN must be observed upto time scales
of years, shows that their PDS is more similar to that of HSS binaries (see, e.g. [94,
96]). Two systems stand out both spectrally and in timing properties: Ton S180 and
Ark 564. When placed on a standard HID, renormalized for flux differences, these
object lie in an intermediate-hardness area [20]. The exact connection to the BHB
diagram is not easy as the horizontal branches are caused by both a softening of the
LHS component and the appearance of a thermal disc component, absent in the AGN
2–10 keV spectra, but the position is certainly intermediate. On the timing side, these
two objects also show properties which can be identified of more intermediate states
[6, 94, 95].
This general behaviour must be understood in terms of basic parameters. Interpre-
tations of the overall evolution in the HID have been proposed [10, 11, 88]. However,
the correlated properties of the emission of relativistic jets and winds from the system
must be included in order to reach a global understanding. This of course complicates
the problem and only basic attempts have been made [49, 51]. A zero-level approach
has been proposed recently, which in addition to the X-ray properties incorporates
a model for the production of poloidal magnetic fields, which can be a crucial in-
gredient for the production of jets [85]. The “cosmic battery” scenario (see [36–38,
86]) can be the link between the properties of the accretion flow and generation of
relativistic jets, which is worth exploring further.
2.6 Conclusions
Thanks to the availability of high-quality data from past and current high-energy
missions, our knowledge and understanding of black hole transients has increased
significantly in the past two decades. Instrumental to this advancements have been
observations at other wavelengths, which have allowed us to study physical compo-
nents, such as relativistic jets and outflows, that were completely ignored before. It is
now clear that studying the emission over a very broad range of energies is the only
way to properly characterize the physical properties of these objects. For the study
of fast X-ray time variability, the gap left by the demise of RossiXTE has just been
filled by the launch of the Indian satellite Astrosat, which at the time of writing is still
in the performance verification phase. Theoretical modelling of observational data
is now turning to the interpretation of joint spectral-timing properties, a crucial step
to study rapidly varying phenomena. The field is evolving very rapidly: however,
while the amount of available details make the full picture more complicated, the
basic patterns are converging towards a solid set of observational points that must
be at the base of all theoretical models. Above, we have focused onto those patterns
with the aim of outlining general properties of black hole transients. This is by no
means complete (see Middleton’s chapter on detailed spectral models), but it should
constitute a solid starting point upon which layers of complexity can later be laid.
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Chapter 3
Black Hole Spin: Theory and Observation
M. Middleton
Abstract In the standard paradigm, astrophysical black holes can be described

solely by their mass and angular momentum—commonly referred to as ‘spin’—
resulting from the process of their birth and subsequent growth via accretion. Whilst
the mass has a standard Newtonian interpretation, the spin does not, with the effect
of nonzero spin leaving an indelible imprint on the space-time closest to the black
hole. As a consequence of relativistic frame-dragging, particle orbits are affected
both in terms of stability and precession, which impacts on the emission character-
istics of accreting black holes both stellar mass in black hole binaries (BHBs) and
supermassive in active galactic nuclei (AGN). Over the last 30 years, techniques have
been developed that take into account these changes to estimate the spin which can
then be used to understand the birth and growth of black holes and potentially the
powering of astrophysical jets. In this chapter, we provide a broad overview of both
the theoretical effects of spin, the means by which it can be estimated and the results
of ongoing campaigns.
3.1 Preface
In the generally accepted model of Einstein’s General Relativity (GR—although

modified forms of GR cannot yet be ruled out), black holes (BHs) are defined by
only their mass, charge and angular momentum, hereafter referred to as spin. In an
astrophysical setting, the charge will soon neutralise (or else be radiated away during
the formation process in accordance with Price’s theorem) and so BHs can be entirely
defined by only their mass and spin—this is the often-touted ‘no-hair’ theorem.
Whilst the effect of mass is relatively benign from the standpoint of observation
and theory (generally acting only to scale the energetics, e.g. Shakura and Sunyaev
[301]—and accretion-related timescales—e.g. McHardy et al. [195]), the spin has
much more to offer in terms of revealing how BHs formed and grew, how accretion
M. Middleton (B)
Institute of Astronomy, Madingley Road, Cambridge, UK
e-mail: mjm@ast.cam.ac.uk

100 M. Middleton
operates in the regime of strong gravity and how the most powerful ejections of
material in the universe may be powered.
What follows should not be viewed as exhaustive but a summary of the state of the
art. In Sect. 3.2, we will review the necessary theory to make sense of the observa-
tions and methods that have been applied in attempts to measure BH spin in various
systems under various assumptions. In Sect. 3.3, we will discuss the traditional tech-
niques used to measure the spin in BHBs and AGN and the implications of these
measurements (with special attention paid to the launching of astrophysical jets).
Sections 3.4 and 3.5 discuss new techniques which incorporate the time domain, and
in Sect. 3.6, we conclude with some final remarks about the future prospects for the
field.
3.2 Useful Theory
Before we discuss how methods to estimate the spin have developed and been applied
to observation, it is important to understand how a spinning BH affects the space-
time in which it is embedded as the outcome dictates our approach to studying BHs.
Below, we present the formalisms which govern the nature of space-time around such
a BH and the behaviour of test particles in close proximity (at large distances, this
tends towards a Newtonian description); we stress that a deep working knowledge of
GR is not necessary to appreciate what follows, with only those formulae considered
relevant for our later discussions being presented (though for the more experienced
reader, we suggest the review of Abramowicz and Fragile [1]).
The solution to Einstein’s field equations for a spherically symmetric, nonrotating
massive body was discovered by Karl Schwarzschild in 1916 for which the metric,
which describes the geometry of empty space-time (a ‘manifold’), is named, whilst
the generalisation to a rotating (uncharged) BH is known as the Kerr metric after Roy
Kerr, who discovered the solution in 1963 (see his explanation in Kerr [150]). The
difference between the two solutions results from the inclusion of the BH angular
momentum (J) which, as we will see, has a significant impact on the orbit of test
particles and the behaviour of infalling material. The metric is usually presented
in Boyer–Lindquist coordinates (t, r, θ , φ) which can be interpreted as spherical
polar coordinates and is related to cartesian coordinates via the following standard
transforms: √
x = √r 2 + a2 sin θ cos φ
y = r 2 + a2 sin θ sin φ
z = r cos θ
Here, a is the BH-specific angular momentum (a =|J| /M) although in literature

discussing observation, is often expressed in its dimensionless form, a∗ :
|J|c
a∗ = (3.1)
GM 2
3 Black Hole Spin: Theory and Observation 101
where M is the BH mass in solar units and G and c are the usual constants.
In flat (Minkowski) space-time, a line element of length ds is simply given by
ds2 = −(cdt)2 + dx 2 + dy2 + dz2 ; however, in the Kerr space-time (in natural units
of G = c = 1), this becomes:

2Mr ρ2 2 2Mra2
ds2 = − 1 − 2 dt 2 + dr + ρ 2 dθ 2 + r 2 + a2 + sin2 θ sin2 θ dφ 2
ρ Δ ρ2
4Mra 2
− sin θ dφdt (3.2)
ρ2
where
ρ 2 = r 2 + a2 cos2 θ (3.3)
Δ = r 2 − 2Mr + a2 (3.4)
Using Einstein’s notation, this can be written in terms of the covariant metric
tensor, gμν , where ds2 = gμν dx μ dx ν and gμν in vector form is:
⎛ ⎞
gtt 0 0 gtφ
⎜ 0 grr 0 0 ⎟
⎜ ⎟
⎝ 0 0 gθθ 0 ⎠
gtφ 0 0 gφφ
By inspection, the components of the metric in Eq. 3.2 are then:

2Mr
gtt = − 1 − 2 (3.5)
ρ
−4Mr
gtφ = a sin2 θ (3.6)
ρ2

2Mra2
gφφ = r 2 + a2 + sin 2
θ sin2 θ (3.7)
ρ2
ρ2
grr = (3.8)
Δ
gθθ = ρ 2 (3.9)
and setting a = 0 returns the metric for a nonspinning, Schwarzschild BH.

Whist the above may seem mathematically daunting, the components of the metric
play a vital role in allowing us to predict the effect of nonzero spin. In the Schwarz-
schild metric, there are two important radii to consider: the first is the position of
102 M. Middleton
the static surface (also called a ‘null hypersurface’) of the event horizon, where an
observer in a distant reference frame would observe a body, travelling at the speed of
light radially away from the BH to be stationary. In this case (i.e. for a nonspinning
BH), the event horizon is located at the Schwarzschild radius (which can be easily
derived in the Newtonian case with the escape velocity set equal to the speed of light),
Rs = 2GM/c2 (=2Rg or 2M in natural units). The second important radius we need
to consider is the position of the marginally stable orbit more commonly referred to
as the innermost stable circular orbit (ISCO), inside of which stable orbits are not
possible and any accreted material takes a laminar plunge to the event horizon on a
dynamical timescale. The position of the ISCO can be found from considering the
radial equation of geodesic motion using the proper time (i.e. that measured by a
clock at rest) τ :
d2r M
2
+ 2 − (1 − 3M/r) uφ2 /r 3 = 0 (3.10)
dτ r
where uφ is the specific angular momentum. If we consider circular orbits, d 2 r/dτ 2

= 0 so uφ2 = Mr 2 /(r − 3M), the specific angular momentum has a minimum at r =
6M. Physically, this is distinct from the Newtonian case where angular momentum
decreases monotonically down to the central object. At radii within the ISCO, circular
orbits are no longer possible (see the discussion in Abramowicz and Fragile [1]) and
the material undergoes radial free fall to the event horizon and the singularity beyond.
Nonzero values of the spin indicate a Kerr BH, with a positive value correspond-
ing to the BH rotating in the same direction as the orbiting particles around it, i.e.
prograde. Conversely, negative spin values indicate that the orbits are oriented in
the opposite direction, i.e. retrograde to the BH spin. Based on the third law of BH
thermodynamics which states that a BH cannot have zero surface gravity [17], the
spin must have natural limiting values of −1 and 1 (at which point the surface gravity
is zero). An additional constraint arises from the consideration of the Kerr solution
and by setting Eq. 3.4 equal to zero: Δ = r 2 − 2Mr + a2 = 0, which is a coordinate
singularity in Eq. 3.2. It can be readily seen that there are no real solutions when
a2 > M 2 , which implies that in such a case, there is no horizon and no BH, leading
to a ‘naked singularity’ which is forbidden (due to paradoxes); so once again, we
find a limiting value of |a| < M or (as a∗ = a/M), |a∗ | < 1 as before.
Assuming that, at the point of BH formation, the spin is less than maximal (in the
case of stellar mass BHs, this is eminently sensible, given that angular momentum
may be lost in the supernova explosion), accretion of matter in the prograde direction
will both increase the mass and ‘spin-up’ the BH, analogous to the situation of
accreting neutron stars (e.g. Bisnovatyi-Kogan and Komberg [26]). Considering only
the accretion of matter, the BH’s growth follows Bardeen’s law [17]; however, as
discussed in Thorne [325], the effect of radiation is important. If one were to ignore
the radiation from the disc, then in principle the spin could reach a limiting value
of 1, however, as pointed out by Bardeen et al. [17], the capture cross section for
photons with oppositely aligned momentum is larger than when aligned and so these
photons will act to buffer against the spin reaching unity. More accurately, above
a∗ = 0.90, radiation effects cannot be ignored and lead to a deviation in evolution

away from Bardeen’s law such that the limiting, ‘maximal’ value is reached at a∗
= 0.998 for prograde spin or −0.998 in the retrograde case (and changes only very
little depending on the nature of the illumination: [325]). It is important to note that
this maximal value does not account for the effect of torques which are expected to
result from magnetic fields threading the plunging region which may act to reduce
the maximum spin that can be achieved [97].
It is important to note that whilst accretion must inevitably change the spin
(unless maximal already), we cannot yet observe this on any human timescale. From
Bardeen’s law, it can be seen that it would take more than the mass of the BH itself
to be accreted to change the spin from 0 to 1; given typical mass transfer rates in
BHBs via Roche Lobe overflow of 10−6 –10−7 M /year and outburst duty cycles of
∼1 % (e.g. Fragos et al. [92]), it is clear that we will have to wait around the lifetime
of the binary itself (∼ billions of years) to see such a substantial change in the spin
of stellar mass BHs (and equally for supermassive BHs—SMBHs—in AGN where
the duty cycle is thought to be similar, but mass loss via winds could be substantial).
Smaller changes are, however, possible on smaller fractions of the binary’s lifetime,
depending on the starting mass and spin of the black hole (see Fragos and McClin-
tock [93]); however, for changes in the spin that occur on observable timescales, we
presently—and for the foreseeable future—will lack the ability to detect them via
the methods we will discuss in the forthcoming sections of this chapter.
3.2.1 Frame-Dragging
A key consequence of having a spinning BH—in terms of observational

implications—is the concept of relativistic frame-dragging. As a result of the BH’s
nonzero angular momentum, space-time moves (is frame-dragged) in the direction of
the spin in its vicinity, thereby imparting energy to an orbiting test particle. This can
be seen directly when we consider an observer, with (contravariant) four velocity uμ
(= dx μ /dτ ) who falls into the BH with zero angular momentum or L = uφ = 0. This is
the usual definition of a zero angular momentum observer (ZAMO). The contravari-
ant component of the velocity is nonzero (except as r → 0), so uφ = gφt ut = 0. The
angular velocity of the ZAMO is as follows:
uφ
dφ
dφ
Ω= = dτ
dt
= = 0 (3.11)
dt dτ
ut
Ω can be computed from uφ = 0 = gφφ uφ + gφt ut which gives:
uφ gφt
Ω= =− (3.12)
ut gφφ
104 M. Middleton
Substituting Eqs. 3.5 and 3.6 for the components of the metric leads to:
2Mar
Ω= (3.13)
(r 2 + a 2 )2− a2 Δ sin2 θ
By substituting Eq. 3.4, we can see that (r 2 + a2 )2 > a2 (r 2 − 2Mr + a2 ) sin2 θ and
so Ω/Ma > 0. Therefore, as a result of the nonzero BH spin, a ZAMO is forced to
co-rotate (frame-dragged) in the direction of its rotation (as the angular velocity has
the same sign as the angular momentum).
As a consequence of frame-dragging, whilst the Schwarzschild metric is only
singular (in Boyer–Lindquist coordinates) at the static surface of the event horizon,
solutions are singular across two null hypersurfaces in the Kerr metric. The new
position of the event horizon is found where grr , the radial component of the metric
(Eq. 3.2) tends to infinity. By setting 1/grr = 0, we can see that this is the same as
the solution to the coordinate singularity at Δ = 0 = r 2 − 2Mr + a2 . Solving the
quadratic leads to the solution for the radius of the horizon:

rH = M ± M 2 − a2 (3.14)
The positive solution of this equation defines the event horizon (as radii below this
are forced to travel faster than the speed of light). As we can see, for nonzero spin,
the position of the event horizon can be within the Schwarzschild radius. The second
hypersurface occurs when gtt changes sign or, from Eqs. 3.4 and 3.5:

2Mr
gtt = 0 = − 1 − 2 → 0 = r 2 − 2Mr + a2 cos2 θ (3.15)
r + a2 cos2 θ
which has the solutions:

rE = M ± M 2 − a2 cos2 θ (3.16)
The surface at rE+ is referred to as the ‘ergosphere’√

and the region between√ this and
the event horizon, the ergoregion. It is easy to see that M 2 − a2 cos2 θ > M 2 − a2
for all co-latitudes (θ ) except at the poles where the position of the event horizon and
ergosphere meets (see Fig. 3.1).
As rE+ > rH+ (when θ = 0 and θ = π ), an observer within the ergosphere can
still be in causal contact with the outside universe; this is plotted in Fig. 3.2. Within
the ergosphere, it is not possible for a physical observer to remain at rest, and from
calculating the effect of orbits within this region, it was discovered that negative
energy (retrograde) trajectories/orbits are possible (see, e.g. Penrose [258]; Bardeen
and Press [18]). Should an orbiting body fragment within the ergosphere, then the
total energy of those fragments not induced into negative energy orbits will be greater,
having effectively tapped the energy (angular momentum) of the BH. This tapping
of the BH’s spin is called the Penrose effect [258] and the magnetic field analog,
Fig. 3.1 2D positions of the

event horizon (Eq. 3.14:
dashed line) and ergosphere
(Eq. 3.16: solid line) as a
function of BH spin (across
polar coordinate
space—where the x-axis is
the radial distance from the
BH). The hypersurfaces meet
at the poles (co-latitude of 0
degrees) with the ergosphere
‘pinched’ down as the spin
increases
the Blandford–Znajek effect [29], which we shall discuss in Sect. 3.3.5 in relation to
powering ‘superluminal’ ejections.
As stable orbits are possible closer to the BH for prograde spin [18], frame-
dragging changes the position of the ISCO which is a well-defined, monotonic func-
tion of a/M [18] and is shown in Fig. 3.2:

risco = M 3 + Z2 ∓ [(3 − Z1 ) (3 + Z1 + 2Z2 )]1/2 (3.17)
where
1/3
Z1 = 1 + 1 − a2 /M 2 [(1 + a/M)1/3 + (1 − a/M)1/3 ] (3.18)
1/2
Z2 = 3a2 /M 2 + Z12 (3.19)
where ∓ is used the top sign refers to a treatment where the spin is prograde and the
bottom sign to where the spin is retrograde.
As is shown in Fig. 3.2, for a > 0, the position of the ISCO lies within that for
a Schwarzschild BH (6 M), reaching a minimum at 1.235 M (for a∗ = 0.998), and
is pushed further out in the case of a < 0 (retrograde spin) towards a maximum of
9 M. As we shall see in the following sections, this correspondence between the spin
and location of the ISCO allows us to construct models to estimate the spin from
observation.
106 M. Middleton
Fig. 3.2 Position of the

respective hypersurfaces in
the Kerr metric as a function
of BH spin (Eqs. 3.14, 3.16
and 3.17), RH , the event
horizon, RE , the ergosphere
(shown for θ = π/2, i.e. in
the plane of an accretion
disc) and Risco , the innermost
stable circular orbit. The top
right inset shows an enlarged
version covering the highest
spins where Risco sits inside
the ergosphere
It is worth noting as a point of general interest that in addition to the changes

to the positions of the hypersurfaces discussed above, the singularity itself can no
longer be point-like but must take the form of a ring (we will not discuss the effects
of ring singularities further but point the interested reader to Burko and Ori [36]).
Additionally, although we have discussed the effect of frame-dragging on test parti-
cles, electromagnetic fields (generated either in the flow via the Magnetorotational
instability (MRI): [14] or in the local environment to the BH) will also be affected
and can lead to reconnection events and particle acceleration (e.g. Karas et al. [148]).
3.2.1.1 Relativistic Precession
There are two further implications of a spinning BH, resulting from the effect of
frame-dragging: Lense–Thirring precession and the Bardeen–Peterson effect, both
of which may provide important observational diagnostics of the spin and region of
strong gravity close to the BH.
Lense–Thirring precession (also called the Lense–Thirring effect: [169]) describes
the behaviour of orbiting and vertically displaced motion in proximity to a rotating
massive body. For this reason, it is relevant not only for Kerr BHs but more generally
for satellites of astrophysical bodies such as stars and planets. Due to frame-dragging,
the orbital motion undergoes precession leading to epicyclic oscillations about peri-
apsis (position of closest approach) and the ecliptic as long as the orbit is vertically
misaligned with the rotation axis of the rotating body.
Should the accretion disc be misaligned with the BH spin axis (for instance due to
a supernova kick, see Brandt and Podsiadlowski [31]), then the precession of orbits
due to the Lense–Thirring effect produces a torque. If this torque is larger than the
viscous torques in the disc, a fluid (nonsolid body) inner disc will align perpendicular
to the BH spin axis, whilst beyond the warp radius the disc aligns with the binary
orbit; this is the Bardeen–Petterson effect [19]. Should the torques not be dissipated
(though see discussions by Armitage and Natarajan [10]; Marković and Lamb [185]),
the whole of the inner disc can precess leading to important effects on the emergent
spectrum and jets (see Sect. 3.4). Such precession may lead to clear signatures in the
time domain as well as the energy domain [130], and as we will discuss later, can
provide a diagnostic for the BH spin.
The manner in which the disc warp created by the Bardeen–Peterson effect prop-
agates depends on the nature of the disc: if it is thin or the viscosity high, then the
warp diffuses (due to viscous torques as discussed by Pringle [261]), whilst if the
disc is thick or viscosity low, the warp propagates as a wave [253]. King et al. [151]
discuss the general case (for both viscosity cases and for the full range of disc tilts and
misalignment) of how the torque between the BH and disc as a result of precession
can lead to co-alignment or counteralignment of the BH/disc system. The authors
conclude that alignment depends on the detailed properties of the disc, namely how
the warp is propagated, although, in general, on short timescales, it is possible that
the disc tries to misalign with the hole (essentially spinning the hole down), whilst
on long timescales there is a tendency towards co-alignment (and spin-up). Making
comparisons to observation, Maccarone [177] reports that the jet angle in the BHBs,
GRO J1655-40 and SAX J 1819–2525, is misaligned with respect to (i.e. not perpen-
dicular to) the binary orbit. Assuming the jet angle is tied to the inner disc and the
spin of the BH, this may indicate that the misalignment resulted from the formation
process (e.g. Brandt and Podsiadlowski [31]) and the time required for alignment is
potentially a significant fraction of the binary lifetime [187, 309]. King et al. [151]
note that in such systems, the angular momentum of the BH is much larger than
that of the disc and so the crucial timescale is that on which tidal forces can transfer
angular momentum from the binary orbit to the disc—on timescales shorter than
this, counteraligned discs may be possible.
In the case of AGN, the picture is less clear as the timescales for accretion-driven
changes (and alignment) are considerably longer (typically scaling with mass), whilst
the means by which material reaches the inner sub-pc disc is still debated. Should
material fall through the galactic disc, then the angular momentum is expected to
be in a single direction and the BH’s spin axis should appear aligned with the host
galaxy’s stellar disc (assuming that the growth is driven by accretion rather than
via BH–BH mergers). Instead, should material with a range of angular momenta
be accreted (e.g. via condensed filaments: [236])—a situation often referred to as
‘chaotic’ accretion—then it is possible that the disc-BH system will be initially
misaligned with respect to the inflowing material (which is assumed to be misaligned
with the host galaxy’s stellar disc) and then co-align—as a consequence, the sub-
pc system does not need to be aligned with the galactic plane. Observations of jets
108 M. Middleton
(see Hopkins et al. [125]) and the inclination of the sub-pc disc determined from
reflection spectroscopy (Middleton et al. [202]), indicates that many AGN-galaxy
systems do indeed appear misaligned which likely points towards a recent chaotic
accretion history.
3.3 Observational Tests of Spin I—the Energy Spectral

Domain
The effect of frame-dragging and the change in the position of the ISCO with spin
(Fig. 3.2) have led to the development of methods by which the BH spin can be
estimated. We purposely use the word estimated here to signify the uncertainty
inherent in characterising such an intrinsic yet complex property, relying on models
which are themselves based on caveats and assumptions. However, this should not
be read as a criticism of efforts both past and ongoing to better estimate and constrain
the spin, rather that the reliability of a chosen method should be evaluated against
the backdrop of systematic uncertainties.
3.3.1 Modelling the Continuum (Disc) Spectrum

As mentioned elsewhere in this compilation, transient (predominantly low-mass
companion) BHBs undergo outburst cycles regulated by disc instabilities (e.g. Lasota
[168]) evolving in brightness and spectral shape (see McClintock and Remillard [192]
and Done et al. [69] for reviews). Towards the peak of the outburst, the spectrum
becomes increasingly dominated by emission originating from the accretion disc.
Under the assumption that the inner disc radius (Rin ) sits at the ISCO (as seen in
GRMHD simulations: [257, 298]—and see Zhu et al. [345] for the effect of the
plunging region—and evidenced in the observational study of LMC X-3 by Steiner
et al. [307]), we can show that the properties (namely the temperature and luminosity)
are related to the spin through the following formulae (for a rigourous discussion, we
point the reader to Frank et al. [94]). We point out that the following derivations are
meant only to illustrate the case using the simplest, nonrelativistic treatment (we dis-
cuss the relativistic disc modelling in Sect. 3.3.1.1). The presence of viscous torques
on the differential (Keplerian) orbits leads to dissipation of mechanical energy with
the torque defined as:
tφ (R) = 2π RvΣR2 (3.20)
where v is the kinematic viscosity, Σ is the surface density and Ω is the radial
gradient of angular momentum (dΩ/dr).
Although v becomes unimportant in terms of calculating the emission profile
(as we shall soon see), its form is relatively important historically as it can also be
parameterised as:
v = αcs H (3.21)
where cs is the sound speed, H is the height of the disc from the mid-plane and α
is the viscosity parameter that underpins the α-prescription of Shakura and Sunyaev
[301]. This formula assumes that turbulence drives the viscosity and results from a
consideration of the typical size of a turbulent eddy (which must be less than the disc
scale height, H/R) and the assumption that the turbulent velocity is not supersonic.
As a consequence, we would expect α < 1. As Frank et al. [94] point out, this is not
a physical statement as the true nature of the viscosity is unknown (although they
also point out that magnetic stress would also lead to α < 1).
Irrespective of the nature of the viscosity, the amount of mechanical heat loss
is given by t φ (R)Ω dR and is dissipated across both sides of the disc (2×2π RdR),
giving a heat loss per unit area (D(R)) of:
tφ Ω vΣ
D(R) = = (RΩ )2 (3.22)
4π R 2
Setting Ω to be Keplerian (i.e. differential rotation: Ωk = (GM/R3 )1/2 ), gives:

9 GM
D(R) = vΣ 3 (3.23)
8 R
From conservation of mass and angular momentum, and assuming zero torque at
the innermost edge of the disc (i.e. Ω = 0 at R = Rin ; see Krolik [161]; Gammie [96]
and Balbus [13] for issues associated with this assumption), it can be seen that:
1/2
Ṁ Rin
vΣ = 1− (3.24)
3π R
Combining Eqs. 3.23 and 3.24 leads to the formula for the dissipation of energy
per unit area more commonly seen in the literature:
1/2
3GM Ṁ Rin
D(R) = 1− (3.25)
8π R3 R
which importantly demonstrates that the heating is independent of viscosity (either

related to the sum of radiation and gas pressure in the disc: [301], or via magnetic
stresses caused by the MRI: [14]). Assuming the disc to be fully optically thick to
the emergent thermal radiation, we should expect local emission (i.e. at each radius)
to be a blackbody (Planck’s distribution) with a peak, effective temperature, Teff
according to Stefan–Boltzmann’s law:
D(R) = σSB Teff

4
(3.26)
110 M. Middleton
or
1/2 1/4
3GM Ṁ Rin
Teff = 1− (3.27)
8π R3 σSB R
Thus, in this nonrelativistic approximation, the temperature of the thermal emission

is in principle related to the position of the inner radius and is therefore a diagnostic
of the spin.
The luminosity that emerges as a result of the process of accretion through the
disc can be approximated by:
GM Ṁ
L= (3.28)
2R
where R is the position of the inner edge of the flow and the factor of 2 in the
denominator results from the virialisation of the system (i.e. half of the potential
energy is radiated, whilst the remaining half is converted into kinetic energy and lost
to the BH). This luminosity can also be parameterised as the conversion of rest mass
to energy given by:
L = ηṀc2 (3.29)
where η is the radiative efficiency. By equating the two formulae above, it is clear
that in the simplest picture, the radiative efficiency is a function of the position of the
inner edge and therefore the spin, ranging from ∼8–40 % for zero through to maximal
(prograde) spin. This is the simplified Newtonian case and is a good approximation
to the actual efficiency as a function of spin which goes as:
η = 1 − (RISCO − 2M ± A1 ) (RISCO − 3M ± 2A1 )−1/2 (3.30)

√
where A1 = a M/RISCO . We note that the above equations assume that the mass
falling onto the BH is only converted into radiation. This is demonstrably not the case
as powerful winds and jets are ubiquitous to accretion flows; however, this remains
an important illustrative point and a useful theoretical framework for discussing BH
accretion discs.
In practice, the emission spectrum from the accretion disc is a convolution of the
thermal emission from all radii or a ‘multicolour’ disc blackbody (e.g. Mitsuda et al.
[223]; Makishima et al. [181]). In addition to this deviation from a blackbody, a fur-
ther complication arises due to the effect of opacity which determines how deep into
the disc atmosphere we observe, i.e. the position of the ‘effective photosphere’, τeff .
The two ‘competing’ forms of opacity are electron scattering (κT ) and absorption
via both free-free (κff ) and via metal edges/bound-free transitions (κbf ). Whilst κT is
independent of temperature and density, both forms of absorption
√ opacity scale as
ρ ∗ T −7/2 (Kramer’s law). Thus, the position of τeff ≈ τT κabs /κT = 1 is a function
of temperature/frequency (where κabs is the sum of the contributions to the absorption

opacity). At higher frequencies, we can see further into the disc as the absorption
opacity is lower; as there is a negative, vertical temperature gradient through the
disc, when τeff = 1 is further into the disc, we observe a larger offset in temperature
compared to the surface. Such effects lead to the requirement of a colour correc-
tion/spectral hardening factor, fcol where the observed temperature of a blackbody at
a given radius is Tcol = fcol ∗ Teff and the intensity at a given frequency (Iν ):
1
Iν = 4
Bν (Tcol ) (3.31)
fcol
where Bν is the Planck function and fcol can be roughly parameterised by the ratio
(i.e. relative importance) of the competing opacities in the disc:
1/4
κtot
fcol ∼ (3.32)
κabs
where κtot = κabs + κT and fcol reaches saturation [56] at:

1/9
fcol ∼ 72 keV /Teff (3.33)
The simplest and most widely adopted disc model, diskbb (for use in the spectral
fitting package xspec: [12] or ISIS: [126]), takes a value of fcol = 1.7 [304], assuming
electron scattering dominates the opacity (see, e.g. Ebisuzaki et al. [75]). Although
density and temperature (giving the relative balance of opacities) and therefore fcol
are unlikely to be constant across the disc (e.g. Gierliński and Done [110]), this
model is commonly used to describe the thermal emission seen in BHBs and neutron
star binaries and can be used to provide a crude estimate of Rin and therefore the BH
spin.
3.3.1.1 Beyond the Simple Picture
A more accurate picture of accretion in the framework of GR was developed by

Novikov and Thorne [246] and Page and Thorne [252], assuming a razor-thin disc
and zero-torque inner boundary condition (and can be seen as the relativistic analog
to the Shakura–Sunyaev disc), more commonly referred to as the general relativistic
accretion disc (GRAD) model. Building upon this relativistic framework, models
are now available that include the full ‘suite’ of relativistic corrections (Doppler
boosting and gravitational redshift), the effect of returning radiation and importantly
nonzero inner boundary conditions (i.e. tφ (Rin ) = 0). This last point is hotly debated
as magnetic fields crossing the ISCO may connect the disc to the BH or plunging
region and thereby provide a torque (see discussions by Paczyński [250]; Armitage
et al. [11]; Hawley and Krolik [119]; Afshordi and Paczyński [3]; Li [170]). One of
the most widely used of the GRAD models is kerrbb [171] which includes a grid
112 M. Middleton
of spectra created via ‘ray-tracing’ in the Kerr metric. The method of ray-tracing
is a well-established and reliable means of mapping photon paths in a given metric
and can be seen as a way to effectively visualise emission from the accretion flow
(e.g. Cunningham and Bardeen [47]; Cunningham [48]; Rauch and Blandford [266];
Fanton et al. [86]; Čadež et al. [38]; Müller and Camenzind [232]; Schnittman and
Bertschinger [296]; Dexter and Agol [62]). In practice, the disc ‘image’ seen by an
observer in some observer-system geometry is broken into a number of small ele-
ments and photon paths are traced back to the disc. By assuming a local flux density
profile at each location in the disc and by incorporating relativistic effects (Doppler
boosting and gravitational redshift), the final spectrum can be reconstructed by sum-
ming over the disc elements the paths intercept. In addition to direct illumination,
ray-tracing can also track photons which return to the disc from the far side due
to gravitational light-bending, leading to a change in the locally emitted flux. The
grids in the kerrbb model allow for a range of spin, inclination and BH mass whilst
assuming a standard disc structure with constant fcol (= 1.7, although this value can
be changed in the model) and allows for limb-darkening (e.g. Svoboda et al. [319]).
In determining fcol in the above models, bound-free absorption has been ignored;
however, it can dominate over free-free opacity and lead to changes in fcol with
an increased likelihood that photons are instead ‘destroyed’ rather than propagated
and scattered. At the time of writing, the only disc model which incorporates metal
edges is bhspec [57, 58]. This model describes a GRAD [246] but unlike kerrbb
calculates the disc spectrum by including a relativistic transfer function in place of
ray-tracing. The transfer function provides the integration kernel in calculating the
disc emission and contains information regarding the Doppler boost due to rotation
and gravitational light-bending (see Cunningham [48, 49]; Laor [167]; Speith et al.
[306]; Agol [4]; Agol and Krolik [5]; Dovčiak et al. [71]).
Davis et al. [57] create the bhspec model tables following the methods laid out
in Hubeny et al. [127, 128] (and references therein) adopting the tlusty stellar
atmosphere code [129] to solve the equations for the vertical structure and angular
dependence of the radiative transfer. In addition to including bound-free opacities
(assuming ground state populations), the model fully accounts for Comptonisation
of the escaping radiation. The authors find that bound-free absorption does indeed
play an important role in determining fcol with typical values between 1.5 and 1.6
(lower than found by Shimura and Takahara [304] and Merloni et al. [199]), with
the actual value depending on the mass accretion rate and spin (and weakly on the
adopted value of the α parameter or stress prescription). As with kerrbb, bhspec
accounts for the angular dependence of limb-darkening.
The hybrid code, kerrbb2, combines the ability of bhspec to self-consistently
determine spectral hardening values with the ability of kerrbb to account for return-
ing radiation. To do this, tables of spectral hardening values are precomputed (pub-
licly released for RXTE only) by fitting spectra generated with bhspec with kerrbb
at fixed sets of input values (see McClintock and Remillard [192] for an example of
its use).
In the case of the above models, it is important to have reliable estimates of the
system parameters, notably the inclination which sets the amount of limb-darkening
and Doppler boosting (and visible area from which the flux is emitted), the BH mass,
which shifts the peak temperature, sets the Eddington limit (and in extreme cases can
influence fcol through changing the density) and the distance which sets the luminosity
(and therefore Eddington ratio). As the luminosity scales as the inverse square of the
distance, the resulting spin values are highly sensitive to this parameter and obtaining
an accurate distance measurement (for instance via radio parallax measurements: see
Miller-Jones [215]) is critical.
As we will discuss in Sect. 3.3.3, the above models have been widely utilised in
estimating the spins for BHBs. However, whilst lauding their successes, it is also
important to consider the limitations of such models. There are a number of key
assumptions that go into both kerrbb and bhspec including the assumption that the
discs are steady-state (i.e. time independent) and that the flows are radiatively effi-
cient. The first assumption breaks down should winds be driven from the innermost
disc by thermal, MHD processes or radiative pressure (e.g. Ponti et al. [259]) as the
mass accretion rate is then a function of time and radius. The second assumption is
likely to be invalid at very low mass accretion rates where the flow is low density
and hot material is carried into the BH before it can radiate, i.e. advected [238, 239];
such flows are referred to as advection-dominated accretion flows (ADAFs) or radia-
tively inefficient accretion flows (RIAFs). Such ADAFs/RIAFs also appear at very
high mass accretion rates where the scale height of the disc is very large and so pho-
tons are effectively trapped as the diffusion timescale is longer than the viscous infall
timescale [2]. The effect of advection at high accretion rates is to stabilise the disc (i.e.
removes heat from the flow) with a change in the emission profile from T ∝ R−3/4
(see Eq. 3.27) to T ∝ R−1/2 . Nonrelativistic models (with multicolour discs) with
a free emissivity profile are available (e.g. diskpbb: [120, 163, 164, 216, 337]);
however, until recently, these did not include the complex calculations necessary to
describe a physically motivated disc atmosphere and relativistic transfer/ray-tracing
important in the Kerr metric.
The model slimbb ([293, 316] utilises ray-tracing and the radial and vertical
profile solutions given in Sadowski et al. [293, 294]. This model once again does
not account for mass loss in a wind but accounts for three key components in high
mass accretion rate ADAFs, the radial advection of heat and subsequent change in
the emissivity, the position of the inner edge (which can move from the marginally
stable to marginally bound orbit: [2]) and the location of the effective photosphere. A
later version of this code, slimbh [317], incorporates the tlusty stellar atmosphere
code directly and so is closer in nature to bhspec.
Unlike the case for high mass accretion rate ADAFs, the emission from low accre-
tion rate ADAF/RIAFs may arise from synchrotron cooling in radiatively inefficient
jets [88] or strong outflowing winds [28], Bremsstrahlung emission or Compton scat-
tering in the plasma. Whilst the exact nature of the emission is still debated (although
correlations between radio and X-ray luminosity may tend to favour emission from
a jet, e.g. Corbel et al. [45]; Gallo et al. [95]), it is clear that the optically thick disc
is not present, and therefore, such accretion states are not presently used to diagnose
the spin.
114 M. Middleton
There is also a more fundamental assumption that goes into models that derive
from the Novikov–Thorne [246] prescription: real accretion discs will have a finite
thickness and will not behave as if razor thin. Paczyński [250] and Afshordi and
Paczyński [3] argue for a monotonically decreasing deviation with decreasing scale
height for small α. This assertion was later confirmed by calculations [298] and so
in these limits, the GRAD models should be reliable. However, this does not account
for the presence of magnetic fields in the disc which are expected to be generated via
a dynamo effect and give rise to magnetic stresses and angular momentum transport
[14]. Recent 3D GRMHD simulations [243, 244, 257, 299] estimate that both the
luminosity and stress in the inner regions differ substantially (by up to 20 %: [244])
to that expected in the Novikov–Thorne prescription. Both Kulkarni et al. [166] and
Zhu et al. [345] explore how this might affect estimates for the spin derived from
the use of codes such as kerrbb and bhspec. The former obtain the disc flux profile
for a series of spin values resulting from the 3D GRMHD thin disc simulations
of Penna et al. [257], setting fcol = 1.7 (as with kerrbb), arguing that the extra
sophistication of calculating the position of the effective photosphere is unnecessary
in this instance. The spectra themselves are then determined via ray-tracing (see
the discussion on kerrbb) without returning radiation but taking into account limb-
darkening. Zhu et al. [345] perform a similar set of GRHMD simulations but, by
including tlusty and radiative transfer, also include the distortion to the spectrum
as a result of spectral hardening (as described above). Both sets of authors find that
unlike Novikov–Thorne discs, emission can be seen to originate from within the
ISCO due to a combination of advection at high accretion rates (as discussed above:
[295]) and nonzero inner torque, resulting from a finite disc thickness and leading to
increased viscous dissipation at radii close to the ISCO. The combination of these
two effects leads to the emission peaking at smaller radii than Novikov–Thorne discs
giving both a higher peak temperature and larger emitted flux (and which extends
inside the plunging region: [345]). By fitting the simulated spectra with kerrbb
[166] and bhspec [345], the spin is found to be systematically overestimated as a
result of the increased disc brightness. Crucially, the typical error resulting from
the use of the Novikov–Thorne profile is far less than the errors associated with the
system parameters of mass, inclination and distance (see Gou et al. [113]; Steiner
et al. [307]); the spin values derived from GRAD-based models can therefore be
treated as representative where the errors on the system parameters dominate.
As an addendum, it is also possible that the disc emission profile differs dramati-
cally from all of those mentioned thus far. Novikov–Thorne (and Shakura–Sunyaev)
discs are both thermally [302] and viscously [173] unstable, whilst MHD simulations
carried out by Hirose et al. [121] find that small patches of the disc can be thermally
stable yet viscously unstable. As a result (and motivated by both spectra and variabil-
ity arguments), Dextor and Agol [63] proposed a toy model for an inhomogeneous
disc (ID) which consists of zones which undergo independent fluctuations driven by
radiation pressure instabilities and leads to random walks in the effective tempera-
ture at that radius. As Dexter and Quataert [64] point out, this model can explain the
spectral and variability properties of the soft states in BHBs. Importantly, should this
model be an accurate depiction of the disc and its emission, there are implications
for our ability to measure the spin from fits to the continuum. The ID model assumes
a magnitude of temperature fluctuations, σT (not to be confused with the Thompson
cross section); increasing this value leads to a higher characteristic temperature at
the inner edge (see Eq. 3.27). As a consequence, deriving the position of the ISCO
from the temperature using the Novikov–Thorne disc profile could be misleading,
resulting in systematic overestimates for the spin (as with the consideration of a
finite thickness disc discussed above). Dexter and Quataert [64] quantify the likely
effect, finding that the impact depends on the model used to account for the hard tail
accompanying the disc emission [192]. In practice, they determine that the errors
can be far larger than statistical uncertainty or systematic uncertainty resulting from
not considering emission from within the plunging region [166] except in the most
disc-dominated states (with a disc fraction 0.95 or σT 0.15).
3.3.2 Modelling the Reflection Spectrum
A hard tail of emission ubiquitously accompanies the thermal disc emission in BHBs
(and AGN—see, e.g. Jin et al. [138, 139]) and must originate by inverse Compton-
isation in a corona of thermal/nonthermal plasma in some, as yet, undetermined
geometry (e.g. Liang and Price [172]; Haardt et al. [118]). The exact spectral shape
(and properties such as variability) of this component is discussed in detail elsewhere
in this compilation but can very broadly (though not always accurately) be described
by a power law which breaks at the peak temperature of the thermal electron distri-
bution.
In all models, the corona producing the power law emission has a larger scale
height than the disc, and as a consequence, the disc will subtend some solid angle to
the upscattered seed photons which will re-illuminate the disc down to a typical scat-
tering surface at τeff = 1 (e.g. Guilbert and Rees [117]; Lightman and White [174];
George and Fabian [107]). The resulting ‘reflection’ spectrum is composed of scat-
tered re-emission (i.e. Compton scattering in the surface layers), bound-free edges,
bound-bound absorption and emission lines. The three most important components
of the spectrum are the Fe Kα edge, the Fe Kα fluorescent line(s) and the Compton
downscattered hump. The details of these components are discussed in detail in the
review of Fabian et al. [85] with only a brief overview provided here.
As Fe is cosmically abundant and the fluorescent yield (the probability that a
fluorescent line is produced following photoelectric absorption) of neutral elements
scales as Z 4 , emission from Fe is expected to be of great importance. Moderate
energy (a few keV) X-ray photons produced via inverse Compton scattering in the
corona are energetic enough to remove the inner K (1s) shell electron (leading to
a sharp photoelectric edge at 7.1 keV). As long as electrons are available in the
L (2s) shell, one will drop to fill the K shell gap and release a photon at either
6.404 keV (Kα1 ) or via spin–orbit interaction, a secondary transition at 6.391 keV
(Kα2 ). The probability of this occurrence is only 34 %, whilst the most favoured
(66 % probability) outcome is Auger de-excitation where the production of a photon
116 M. Middleton
via the L→K shell transition is absorbed by a bound electron which is subsequently
expelled. These energies assume that the Fe is neutral; however, illumination of the
disc will act to increase the ionisation state of the reflecting material (e.g. Ross and
Fabian [283]; Ross et al. [284]), parameterised as ξ = L/nr 2 , where n is the electron
density and r is the distance from the ionising source of luminosity L [322]. An
increase in ionisation state increases the electron binding energy and the energy of
the edge and Fe K fluorescent doublet accordingly; however, the lines only emerge
significantly above 6.4 keV for Fe XVII and above. The fluorescent yield will also
(weakly) depend on ionisation state, and when Fe is Li- through to H-like, Auger
de-excitation is no longer possible as two L shell electrons are required. Instead,
photoelectric recombination can lead to line emission (with a high fluorescent yield)
at ∼6.8 keV.
Although Fe K is the strongest feature of the reflected emission, other metal
transitions (e.g. Ni Kα ) also contribute to the overall picture through line emission
and absorption edges. Thus, the precise details of the reflected emission clearly
depend on the elemental abundances and ionisation state of the illuminated material
and have been discussed in detail by Matt et al. [189, 190] who consider illumination
of a constant density slab. They find four distinct regimes of increasing ξ , ranging
from ‘cold’ reflection for ξ < 100 ergs/cm/s (where reflection around Fe K resembles
that expected from neutral material and the absorption edge is saturated and weak)
and terminating at ξ > 5000 ergs/cm/s where there is no absorption edge or line.
In between these regimes, the strength of the line is dependent on the number of
electrons available in the L shell as described above, with a weak line and moderate
edge produced by ionised species of Fe up to Fe XXIII (where the Auger effect
may still take place) and a stronger ‘hot’ line by species above FeXXIII (where the
Auger effect no longer takes place and line emission is a result of recombination). In
the latter case, the edge appears stronger due to increased flux below the edge as a
result of diminishing opacity. One of the most successful models which accounts for
illumination onto a semi-infinite slab of optically thick material in the atmosphere
of an accretion disc is reflion and its later incarnation, reflionx [284, 285]. These
models, based on the work of Ross et al. [287] and Ross [286], fully incorporate the
radiative transfer of continuum X-rays (using the Fokker–Planck diffusion equation),
line emission and Comptonisation (using the modified Kompaneets operator) and
thus were an important step forward from earlier models which, whilst not including
line emission or intrinsic emission from inside the gas, were the first to incorporate
Green’s functions to describe the scattering of photons by electrons in cold gas (e.g.
pexrav: [179]).
The important effects in creating the reflection spectrum discussed so far make no
mention of the effect of spin, but this has a significant impact on the emergent spec-
trum for a number of reasons. Principally, the Fe Kα emission line (s) originates from
illuminated disc radii which rotate in circular Keplerian orbits (although deviations
are expected to scale with (H/R)2 ). The observed reflection spectrum is a compos-
ite of emission from the receding Doppler red-shifted and approaching blue-shifted
sides which leads to separation in line energies and a classic ‘double-horned’ profile;
as these radii approach the BH, the radial velocity increases which leads to a greater
separation. As the orbiting material near the BH is mildly relativistic, beaming of

the emission (resulting from relativistic aberration and time dilation) leads to a flux
change of a factor D3 where D is the Doppler factor:
D = {Γ [1 − β cos(i)]}−1 (3.34)
Γ is the Lorentz factor, β = v/c, v is the approaching or receding velocity and i is the
inclination of the observer from the rotation axis. As a result of relativistic beaming,
flux from the approaching, blue-shifted side is Doppler boosted, whilst the receding
side is Doppler deboosted (see, e.g. Fabian et al. [82, 85]; Stella [312]). Accom-
panying these Newtonian and special relativistic effects are the special relativistic
effect of transverse Doppler shift and the general relativistic effect of gravitational
redshift, both of which act to reduce the observed energy of the line at each radius.
It can be seen that the combination of these effects leads to a heavily skewed and
broadened line, where the blue wing depends heavily on the inclination (Eq. 3.34)
and the shape of the red wing is dominated by the position of the inner edge and
can therefore be used as a proxy for the BH spin (e.g. Laor [167]). In Fig. 3.3, we
demonstrate these effects on a fluorescence line originating from two annuli in the
Fig. 3.3 Distortion of an emission line (using the model relline: [54]) for two annuli in the
accretion disc seen at 45 degrees about a BH with a∗ = 0.998. The red line is from an annulus
20–22 Rg from the BH and is broadened into the characteristic ‘double-horned’ profile by Doppler
shifting. The blue and red ‘wings’ of this line are then increased and decreased in flux, respectively,
through the effect of Doppler boosting (relativistic beaming: see Eq. 3.34). The blue line comes
from an annulus at the ISCO (1.3-2 Rg from the BH) where the red wing gives a measure of the
position of the ISCO; the entire line is shifted to lower energies due to the combination of transverse
Doppler shift and gravitational redshift. In reality, we see a blend of lines from all radii
118 M. Middleton
disc. Importantly, unlike the case with continuum fitting, the method of fitting the
relativistically broadened Fe line is independent of BH mass. In addition, the tech-
nique is insensitive to the distance to the source and the inclination is a ubiquitous
parameter of the spectral models (determined from the shape of the blue line and
Fe Kα absorption edge) and so can be estimated concurrently with the spin. As a
consequence, the major source of error when determining the spin is the systematic
uncertainty within the model itself. Of importance when determining the emergent
reflection spectrum is the modelling of the primary illuminating continuum which
may be more complex than a simple power law but can be probed through the use
of an expanded bandpass (e.g. the NuSTAR observations of Cyg X-1: [255]) as well
as through the application of advanced spectral timing techniques (see the review of
Uttley et al. [327]). Additionally, the radial and vertical dependent density structure
of the optically thick disc and the geometry of the corona/disc system (which sets
the emissivity, e.g. Ghisellini et al. [108]) are important considerations for accurate
modelling of the reflected emission. The following subsections discuss these issues
in turn.
3.3.2.1 Emissivity and Geometry
An important ingredient of the reflection spectrum is the ‘emissivity’ () of the flux
from the disc which is proportional to the radial dependence of the illuminating
radiation onto the disc (where the irradiation goes as I(R) ∝ R− ). For an irradiating
point source in flat space-time, sitting above the disc, the emissivity goes as the
product of the inverse square law and the cosine of the normal to the plane of the disc
(see the discussion in Wilkins and Fabian [339]) or, for a source, h above the disc,
goes as (R2 + h2 )−1 ∗ cos θ (where cos θ is just h/ (R2 + h2 )). Thus, at small
radii (R h), this would tend to a flat profile but at large radii (around R = h) will
instead tend to R−3 which is that of the disc emission in Eq. 3.27. This discounts the
effect of relativity however, and the effect of light-bending close to the BH can have
a significant impact by focusing more of the radiation onto smaller radii.
The impact of light-bending naturally depends on the location of the corona and
the geometry of the disc–corona system. One of the most popular geometries is that
of the ‘lamp-post’ (see Fig. 3.4) where the corona sits on the BH spin axis some
height above the BH; in a physical sense, this would then be associated with the
base of a jet (see Markoff and Nowak [184]; Dauser et al. [53]; Wilkins and Fabian
[339]; Wilkins and Gallo [340]). The result of light-bending in this geometry can
lead to highly anisotropic illuminations and is predicted to produce a radial profile
that is a twice-broken power law, with very steep emissivities in the most inner
regions then flattening before tending to constant emissivity at large radii [188,
217]. Naturally, as the profile is a function of the light-bending, it is a function of
the source height above the disc. For a decrease in height, the emissivity profile is
steepened due to an increased amount of light-bending, i.e. a larger value of [218,
273, 339], which drops the observed flux from the corona and increases the amount
of reflection, the ratio of which is the ‘reflected fraction’. In addition, the position of
Fig. 3.4 Lamp-post geometry showing the X-ray source (blue) at some height h above the BH (on
the spin axis) as expected for the base of a jet. The X-ray photon paths are bent by the strong gravity
and illuminate the disc, leading to a steeper emissivity profile at small radii than a simple flat profile
(which breaks to R−3 at large disc radii) otherwise expected from illumination in flat space-time
the break to a flatter emissivity changes as does the extent of the region over which
this is predicted to hold [339]. Although reflection models without ray-tracing (or
incorporating a relativistic transfer integration kernel) cannot account for this effect
directly, convolution models which take into account the effect of photon orbits in
the Kerr metric have been developed and can be used to alter the emissivity profile
accordingly. The most commonly used models (besides relconv which we will
mention shortly in the context of next-generation models) are kdblur/kdblur2
which are based on the relativistic transfer function of Laor [167] and kerrconv
which uses an analytical prescription for relativistic beaming and a pared down
transfer function for ease of computing [32].
The geometry of the illuminating source remains a fundamental issue in accre-
tion physics in general but has especially important relevance for reflection models
(although as we will discuss, there are time domain methods now in use to help con-
strain this) as this sets the possible emissivity profiles. It may well be lamp-post-like
in the picture where the optically thin electron population is associated with the base
of a jet (and, if the component is analogous to that in BHBs, probably composed of
a thermal electron population). In the case where the scattering is not in a jet base,
other geometries must be considered. Wilkins and Fabian [339] explored a series
of potential geometries for the corona, notably testing not only for position of the
illuminating source but also for the source’s extent which had not been previously
considered from a theoretical perspective. They calculated emissivity profiles from
ray-tracing using GPUs (graphics processing units) and included a full treatment
of the relativistic effects including the reduction in effective disc area (which they
find goes as the classical disc area divided by the redshift: see their Fig. 3.2) and
blue-shifting of photons onto the disc. The geometries considered include an axial
source (lamp-post geometry) both stationary and moving away from the BH (i.e.
a jet), an orbiting source (which can be seen as a single element in a ring) and an
extended disc-like source. In the case of the ring source, the act of moving it further
from its position above the BH leads to a somewhat shallower emissivity profile in
the central regions with a flatter profile after the first break and extending to larger
radii, dependent on the radial position of the ring. As there are several arguments
120 M. Middleton
for an extended corona from both variability studies (e.g. Churazov et al. [44]) and
more directly from gravitational microlensing (which indicates an extension of up to
a few Rg : [52]), an isotropic point source is unlikely to be representative, and to
this end, the study of an extended source is extremely valuable. Wilkins and Fabian
[339] describe such a scenario as the sum of points spread over a vertical (giving a
source of finite thickness) and radial extent (i.e. a disc above the disc). The source
is assumed to have a constant luminosity across its extent and is optically thin to
the radiation it has produced (and so does not interact with itself); as expected, the
resulting emissivity profile is a combination of the breaks and indices from the con-
sideration of point sources (although may well be further complicated if magnetic
reconnection occurs at distinct radii in the disc, e.g. Sochora et al. [305]). Wilkins
and Fabian [339] explore the effect of changing the spin on the emissivity, finding
that the increased disc area resulting from a higher spin leads to a steeper emissivity
for an axial or ring geometry (with [339] showing that the emissivity is only greatly
steepened in these arrangements for a∗ ≥ 0.8).
Finally, in addition to the issues associated with the as-yet-unknown coronal geom-
etry, a further simplifying assumption commonly made is that the coronal plasma
itself is uniform, being of a single temperature and density. This is unlikely to be
the case (see, e.g. Parker et al. [255]) and in future can be tested using eclipses to
determine the radial dependence of the coronal properties (see Sect. 3.5).
3.3.2.2 Disc Vertical Density Profile
As with estimating the BH spin via the disc emission (see previous section), the
reflection spectrum also depends on the vertical structure of the disc; in the former,
the important consideration was the position of the effective optical depth (which, as
we discussed, can change depending on the relative dominance of opacities). In the
case of reflection spectra, the important consideration is the ionisation state of the
material in which the majority of the reflection takes place (i.e. one effective optical
depth). Shakura and Sunyaev [301] point out that in radiation pressure-dominated
discs of large optical depth and heated by viscous dissipation, the density in the
vertical direction is roughly constant; this has led to models for both AGN [190,
283, 350] and BHBs [284, 288] incorporating this simplifying assumption of a
constant vertical density structure in the disc (or a Gaussian distribution: [284]).
This assumption has been questioned by Nayakshin et al. [237] who point out that
the illuminating X-ray heating of the outer disc layers (where reflection occurs) could
be orders of magnitude greater than the viscous heating in the disc at such height
above the mid-plane. The disc material is expected to be thermally unstable when the
supporting gas pressure in the illuminated atmosphere falls below a fraction of the
downward radiation pressure provided by the illuminating continuum [157, 162, 265,
289, 290] which leads to a large jump in temperature. Nayakshin et al. [237] argue
that a self-consistent determination of the density gradient is therefore necessary
and determine the density profile from the condition of hydrostatic equilibrium,
simultaneously solving the equations of ionisation, energy balance and radiative
transfer, finding constant density (or Gaussian distributions) to be broadly unphysical.

Their calculations instead suggest that the structure of the illuminated layer of the
disc is a two-phase structure formed of an ionised skin where, in the case of ‘hard’
illuminating flux, Fe is completely ionised, and below this layer, the material is cold,
i.e. weakly ionised. The strength of the reflection features is then a function of the
optical depth of the top layer (which does not imprint features due to being totally
ionised), with smaller optical depths of the skin leading to less of an impact on the
emergent reflection spectrum from the cold layers beneath (as such layers are reached
more readily). This in turn depends on the strength of the illumination and is therefore
intrinsically connected to the emissivity profile discussed above. As pointed out by
Fabian and Ross [84], simulations of discs supported by magnetic pressure (e.g. Blaes
et al. [27]) add a complicating factor, as this can potentially reduce the density and
increase the impact of photoionisation. Indeed, should magnetic pressure dominate
the hydrostatic support, then the discontinuous vertical profile resulting from the
thermal instability may not exist, although large-scale inhomogeneities could be
present and these could have an impact on the emergent reflection spectrum [16, 27].
As the impact of magnetic fields on the disc structure remains an open question, so
too does the nature of hydrostatic balance.
3.3.2.3 Effect of the Plunging Region
A standard assumption in modelling the emission from the BH accretion flow is

that truncation at the ISCO is final with no radiation arriving to the observer from
the ballistically infalling material in the plunging region. Zhu et al. [345] studied
this low-density region through 3D MHD simulations, discovering that only a small
amount of thermal emission was produced and was unlikely to distort spin values
obtained via the continuum fitting method (Sect. 3.3.1).
In order to determine the effect of including the plunging region on the reflected
spectrum (as first discussed in Reynolds and Begelman [273]), Reynolds and Fabian
[272] performed high-resolution 3D MHD simulations (thereby incorporating MRI-
driven turbulence: [14, 15]) of the geometrically thin accretion disc close to the
ISCO. Their simulation demonstrates that some Fe Kα emission can originate from
the plunging region which introduces an additional systematic error for models which
terminate at the ISCO. As a notable caveat to this effect, at high mass accretion rates
the density in this region drops and the material may become ionised to the point
where there are no longer any notable reflection features (with the exception of the
Compton hump). The authors estimate the uncertainty introduced by the inclusion
of reflection from within the plunging region (see their Fig. 3.5), finding it to be
an overestimate of the true spin (as for the inclusion of emission from this region
in continuum fitting, e.g. Zhu et al. [345]) with a larger error for slowly spinning
BHs, and to be relatively insensitive to uncertainty in the inclination (as long as the
inclination can be constrained from modelling the Fe Kα profile). The uncertainty
grows when the scale height is large just outside of the ISCO; whilst further work
is needed to establish the full impact of this, Reynolds [271] notes that inclusion of
122 M. Middleton
Fig. 3.5 Schematic of a wind showing a gap (assumed to have been formed via Rayleigh–Taylor or
similar instabilities), rotating and providing a changing view through to the inner disc regions (the
emission from which is determined from ray-tracing: [62]). Analogous to the situation of eclipses
by Compton-thick clouds [281], the emission seen by the observer as a function of time is dependent
on the inclination and spin (as well as disc structure). By incorporating the additional lever arm of
variability, Doppler tomography provides a means to obtain tighter constraints on the spin than is
possible through the use of the time-averaged spectrum alone [211]
such uncertainties can relax the constraints for otherwise maximal spinning BHs to
>0.9.
3.3.2.4 Next-Generation Models
The most recent codes (developed principally by Thomas Dauser, Javier García and
colleagues) provide important updates to existing models. In the case of nonrelativis-
tic reflection (i.e. no light-bending effects), xillver [99–101] includes xstar [142]
to solve for the ionisation state of the disc atmosphere and makes use of the most
updated, accurate and complete atomic database for atomic transitions. A series of
codes have also been developed which include a relativistic transfer function from
ray-tracing which allows the emissivity of the irradiation to be determined (although
the disc is still assumed to be ‘thin’ as in the Novikov–Thorne prescription). These
models include relline [54], which determines lines emission for spins ranging from
maximal retrograde through to maximal prograde and incorporates limb-darkening,
limb brightening and is similar in output to previous models including kerrdisk and
kyrline [71] but is evaluated over a finer energy grid than laor. This model has
been adapted to act as a convolution kernel (relconv) for the entire spectrum which,
when combined with xillver, allows the entire reflection spectrum to be calculated
with a prescribed emissivity law (relxill: [53]). Using relxill, Dauser et al. [55]
showed that by considering the additional spin dependence of the reflected fraction
(which most models do not account for), it becomes possible to place increasingly
stringent constraints on the spin by discounting unphysical solutions (see also Parker
et al. [254]), although consideration of distant, neutral reflection may complicate mat-
ters somewhat. In addition to improvements in determining the emissivity, relxill

also allows the thermal cut-off (at energies often well beyond the detector bandpass)
to be determined [103] whilst calculating the angular dependence of the reflection
spectrum (approximated in the past by a convolution of the angle-averaged reflection
spectrum with a relativistic kernel) in a fully self-consistent manner. The latter has
only a minor impact on measurements of the spin and inner disc inclination (although
the constraints on both parameters improve) but can have a major impact on the esti-
mated Fe abundance [102]. Although an undoubted step forward, this model still
relies on several caveats that are important to consider; the radial and vertical density
profile is assumed constant, as is the ionisation state of the reflecting material. Fur-
ther iterations of the relxill1 model have begun to deal with these assumptions, by
considering a power law-like radial density structure, multizone ionisation structure
or with the ionisation gradient calculated self-consistently from irradiation.
3.3.2.5 The Geometry Through Reverberation
As the geometry of the disc–corona system is potentially important for reliably

estimating the spin, we will very briefly discuss the role of reverberation as the most
promising method for placing tight (spectrally independent) constraints.
It was realised early on (Fabian et al. [82], Stella [312]) that the physical separation
of the corona and the accretion disc will, from the perspective of an observer at
infinity, lead to a light-travel time delay between the emission arriving directly from
the corona and that reflected from the disc (see Matt and Perola [191]; Campana and
Stella [39]; Fabian et al. [85]); a simple schematic showing this for the lamp-post
geometry is given in Fig. 3.4. This ‘reverberation’ off the disc can be studied in the
time domain from the cross-correlation function (e.g. Gandhi et al. [98]) between a
band containing the intrinsic hard emission (from the corona) and a band dominated
by reflection. For ease of evaluating the lag across multiple frequencies, reverberation
is more commonly studied in the Fourier domain via ‘phase lags’. As the phase lag
is a function of the geometry, it can provide important insights into the nature of
the accretion flow and has implications for how we measure the spin. In practice,
obtaining the phase lag requires evaluating the components of the cross-spectrum
[247, 331] which in turn are found from the frequency-dependent complex ordinates
of the Fourier-transformed light curve. Here, we present only the most basic of
descriptions; for a detailed review of Fourier analysis, we point the reader to van der
Klis [329], and for a detailed review of reverberation and phase lags, we recommend
that of Uttley et al. [327].
The light curve in each energy band (e.g. those dominated by either the primary
or reflected emission) is given by x(t) and y(t) with their Fourier transforms X(ν)
and Y (ν). An alternative way of displaying these is X(ν) = A(ν)ei and Y (ν) =
B(ν)ei+φ where φ is the phase lag between them. By defining the cross-spectrum as
1 www.sternwarte.uni-erlangen.de/~dauser/research/relxill/.
124 M. Middleton
C(ν) = X ∗ (ν)Y (ν) (Nowak and Vaughan [245]), the phase can then be determined
from:
(ν)
φ(ν) = arg[C(ν)] = tan−1 (3.35)
(ν)
where and are the real and imaginary Fourier coefficients of the cross-spectrum
at each frequency. The details of how to estimate the phase in practice (i.e. taking
averages over segments and normalising) are discussed in Uttley et al. [327]. The
important question is how do we go from the phase as measured from the light curve
to the geometry? Each light curve can be related via an impulse response to an input
light curve, called a ‘driving signal’, s(t), such that:
∞
x(t) = h(t − τ )s(τ )dτ (3.36)
−∞
where τ is the time lag (φ/2π ν). In Fourier space, this is equivalent to saying X(ν)
= H(ν)S(ν) (and similarly, Y (ν) = G(ν)S(ν)), where H(ν) (or equally G(ν)) is the
Fourier transform of h(t − τ ) and is called the transfer function. As described in
Uttley et al. [327], the impulse response (and therefore the geometry) is related to
the phase lag in Eq. 3.35 via the cross-spectrum:
C(ν) = H ∗ (ν)S ∗ (ν)G(ν)S(ν) = |S(ν)|2 H ∗ (ν)G(ν) (3.37)
Thus, the cross-spectrum of two light curves contains the cross-spectrum of the
transfer functions (H ∗ (ν)G(ν)) and the power spectrum of the driving signal. As
the power spectrum of the driving signal (|S(ν)|2 ) is by definition real-valued, it has
no effect on the phase, and so, given an input relativistic transfer function from the
geometries described in Sect. 3.3.2.1, it therefore becomes possible to compare the
expected phase lag to observations and thereby begin to constrain potential geome-
tries (e.g. Cackett et al. [37]). In so doing, such analyses also hold the promise of
better understanding contributory factors in measuring the spin.
As reverberation in BHBs occurs on very fast timescales, the number of photon
counts per light crossing time is small; as such, observations of reverberation due
to reflection have to date focused mainly on AGN (where conversely, the number
of photon counts per light crossing time is substantial). The first tentative hints of
a signal were noticed by McHardy et al. [196] with the first significant (≥5-σ )
detection of a reverberation lag due to reflection made by Fabian et al. [83] from
XMM–Newton observations of the AGN, 1H0707-495 (see also Zoghbi et al. [346]).
Due to the remarkably strong Fe L emission in this source (which is noted for having
supersolar abundance of Fe), the authors were able to take a hard band without strong
contributions from reflection and detect the signature of reverberation as a ‘soft lag’,
i.e. the soft emission lagging the hard. Notably, the strong Fe Kα emission seen in
the energy lag spectra (e.g. Zoghbi et al. [347, 348]; Kara et al. [143, 145, 146]) and
its continuation to higher energies mapping out the Compton hump (through the use
of NuSTAR data: [144, 349]) on the frequencies of the reverberation lag all indicate
that its origin lies in relativistic reflection.
Following the initial discovery of reverberation, a soft lag has now been detected
(or hinted at) in several AGN (e.g. Emmanoulopoulos et al. [77]) with the detection
of a significant trend of frequency/amplitude with mass [60]. This would require that
the coronal geometry be the same throughout, although in at least one AGN (IRAS
13224-3809: [147]) the soft lag is observed to change in frequency and amplitude;
this does not necessarily invalidate a correlation, but understanding the stability and
evolution of the corona–disc geometry is clearly of great importance.
Work is now turning towards reconstructing the spin-dependent geometry (see
Sect. 3.3.2.1 and Wilkins et al. [340] for a recent study of this) through the use of
impulse response functions (e.g. [274]) and direct modelling of the phase lags. Young
and Reynolds [342] and more recently Cackett et al. [37] and Emmanoulopoulos
et al. [78] simulate the effect of reverberation assuming a lamp-post geometry and
obtain the associated impulse response functions for a variety of physical parameters
(either via precalculated transfer functions in the case of the former or via direct
ray-tracing in the case of the latter). The major effect on the frequency-dependent
lag results from changing the mass or vertical displacement of the axial illuminator
and is relatively insensitive to the inclination and spin. The energy profile of the
lag, however, is far more sensitive, with the profile of Fe Kα —found to be strong at
frequencies corresponding to the soft lag—changing with spin in a similar manner
to its time-averaged counterpart. Cackett et al. [37] compare predicted frequency-
lag and energy-lag spectra to the real data of NGC 4151 and constrain the physical
parameters including the spin (found to be maximal), inclination and source height.
In a similar approach, Emmanoulopoulos et al. [78] perform the first sample analysis,
modelling the frequency-lag spectra in 12 AGN using general relativistic impulse
response functions, finding a consistent source height across the sample and a possible
bimodality of spin values (which might indicate SMBH growth via mergers, e.g.
Volonteri et al. [332]). In order to reliably estimate the spin, these techniques typically
rely on high-quality data with high energy resolution; whilst not widely available
at present, the arrival of next-generation missions such as ESA’s Athena will allow
such methods to be used to their full potential. In addition, Athena’s location at
the L2 point will allow uninterrupted observations which, in turn, will allow lower
frequencies to be explored, important for larger mass AGN where the reverberation
signal is expected to be found (e.g. De Marco et al. [60]).
3.3.3 Results: BHBs
Here, we will discuss the results of campaigns to estimate the spin of stellar mass
BHs from applying the methods described in Sects. 3.3.1 and 3.3.2. Due to their
success, the number of sources for which spin measurements have been obtained
is constantly growing and we apologise for any results which are therefore absent.
Whilst we briefly touch upon the relevance of measuring the BH spin for the purposes
126 M. Middleton
of probing the launching of astrophysical jets, obtaining an understanding of the spin

distribution is important in its own right as this provides a view of the natal spin which
is set during the supernova process (as insufficient mass can been accreted onto the
BH to change the spin by a considerable amount: see the discussion of Reynolds
[271] and Sect. 3.2).
3.3.3.1 Continuum Fitting
As discussed elsewhere in this compilation, the outburst of BHBs (with low-mass

companion stars) follows a predictable path through X-ray spectra, variability and
multiwavelength properties [89, 90]. The luminosity of a source is usually parame-
terised as a ratio to the Eddington limit for spherical accretion, given by:
4π GMmp
LEdd = (3.38)
σT
where σT is the Thompson cross section and mp is the proton mass. This is frequently
referred to by its numerical approximation (and under the assumption that the material
is entirely ionised Hydrogen), LEdd = 1.26 × 1038 MBH /M .
On the rise to outburst, at low-to-moderate mass accretion rates (typically <70 %
of the Eddington limit: [73]), the spectrum is dominated by a hard tail of emission
resulting from inverse Compton scattering by an optically thin, thermal population of
electrons (thus, the spectrum is referred to as being in a hard state). This emission is
accompanied by synchrotron emission extending across several decades in frequency
and is associated with a low bulk Lorentz factor (Γ ) jet. At the highest mass accretion
rates, the spectrum begins to soften and becomes increasingly dominated by emission
from the accretion disc, passing through the intermediate states and then into the soft
state (we note that there are accompanying changes in the variability properties [24],
but these are not of relevance for the discussion here).
The soft state (also called the thermal dominant state) is characterised by a strong
disc component and a nonthermal (rather than thermal) tail of emission to high
energies which is only of the order of a few per cent or less of the total flux in the
X-ray bandpass (e.g. McClintock and Remillard [192]; Remillard and McClintock
[270]). As the vast majority of the emission originates from the disc (with very little
energy being liberated in a corona), the spin can in principle be determined from
the application of a suitable model for the disc emission (see Sect. 3.3.1). However,
a condition of applying this technique relies on the inner edge being located at the
ISCO (as expected from GRMHD simulations: [257, 299]). As the luminosity of
the disc component for a fixed emitting area (i.e. fixed inner edge) is expected to
follow a T 4 dependence (see Eq. 3.27), this provides a means by which to test the
consistency of the position of the ISCO. Determining the position and stability of
the ISCO via this approach has been attempted by a number of authors (e.g. Ebisawa
et al. [74]; Muno et al. [233]; Kubota and Done [165]; Kubota and Makishima [164];
Steiner et al. [307]). In particular, Gierliński and Done [110] use the expected relation
for the integrated disc luminosity as a function of the maximum observed colour
temperature [111] considering a pseudo-Newtonian potential (Paczyńsky and Wiita
[251]) and corrections to the observed flux due to inclination and GR effects [48,
344] to obtain the predicted form of the relation between Eddington ratio (L/LEdd )
and temperature. Through the use of RXTE data (which covers a nominal 3–20 keV
energy range) for a number of well-known BHBs, Gierliński and Done [110] showed
that the disc emission in the soft state is consistent with a fixed inner edge for a range
of Eddington ratios, with deviations at the very highest and lowest values. As the
predicted relation depends on GR corrections and therefore the spin, Gierliński and
Done [109] were able to use these plots to indicate that the spin in the case of XTE
J1550-564 is non-maximal.
The first attempts to directly constrain the spin from modelling the disc emission
(and indeed pioneering the field of continuum fitting) were carried out by Zhang
et al. [344]. The authors utilised values for the peak temperature and flux for a
number of BHBs available from the literature (e.g. Dotani et al. [70]; Belloni et al.
[22]) to obtain the position of the inner edge after accounting for relativistic effects.
Following the development of new models (namely kerrbb and bhspec), there has
been a steady and substantial increase in spin measurements obtained via this method,
with 12 in total covering persistent and transient sources both within our Galaxy and
in nearby galaxies. McClintock et al. [194] provide a review of the continuum fitting
method and its application up to 2013, and we highly recommend this as an excellent
overview.
As mentioned in Sect. 3.3.1.1, of critical importance in all attempts to constrain
the spin via the continuum fitting method, is the accuracy with which the system
parameters—the inclination, distance and BH mass—can be determined as the uncer-
tainty on these dominates over typical model systematics of ∼5 %. As noted by Orosz
et al. [248], having an accurate distance to the system is key to reducing uncertainty
on the mass estimate. For those systems in nearby galaxies, cosmic distance lad-
ders can be used and result in ∼a few per cent uncertainty, substantially better than
distance measurements for Galactic sources [141], although these are being sub-
stantially improved through radio parallax measurements (see Miller-Jones [215]).
Once the distance is known, the mass can be estimated from modelling the orbital
dynamics of the system using eclipsing light curve (ELC) models (e.g. Orosz and
Hauschildt [249]; Orosz et al. [248]) and requires the radial velocity of the companion
star derived from line spectroscopy. Such modelling also determines the inclination
of the system; however, the question remains as to whether the system inclination is
representative of the inclination of the inner disc which is key for determining the
spin via the continuum fitting method. As discussed in Sect. 3.2.1.1, misalignment of
the BH spin axis with that of the binary orbit can lead to relativistic precession and
the Bardeen–Peterson effect which aligns the inner regions with the BH spin axis,
whilst the outer regions align with the binary plane. There may already be evidence
in support of this scenario; the inclination of the jet, which is expected to be the
same as the BH spin axis, appears misaligned with the binary orbit in the BHBs,
GRO J1655-40 ([114, 122] and also [219] for possible issues associated with accu-
rately determining such physical properties) and SAX J 1819–2525 (also known as
128 M. Middleton
V4641 Sgr: [177, 183]). As an interesting aside, it is possible that the ubiquitous
low-frequency quasi-periodic oscillations (LFQPOs) seen in the variability power
spectra of BHBs when in the hard through to the intermediate states (see the review
of [23]) are associated with precession of a low-density flow in the inner regions due
to misalignment [132, 133]. Incorporating the effect of reflection of the radiation
from the precessing regions leads to unambiguous, observational tests for misalign-
ment which, if confirmed, would have implications for our ability to measure the
spin reliably (see Ingram and Done [130]). We will return to the observational effect
of precession in Sect. 3.4.
In obtaining the spin via the continuum method, a further selection criterion is
often applied to the X-ray spectra. As discussed in McClintock et al. [193], above
∼30 % of the Eddington limit, the spectra may deviate from those expected from a
simple thin disc possibly due to the creation of an inner, optically thick, radiation-
pressure-dominated corona/slim disc due to the high mass accretion rates [2, 205,
206, 326] or a region supported by magnetic pressure [317]. It is assumed that either
disc truncation or simply a cooling of the disc photons by the corona leads to a lower
disc temperature than should be expected from the innermost edge of the disc and
a deviation away from the expected L ∝ T 4 relation [109]. Such deviations in disc
structure can in principle lead to the spin being underestimated and has been pro-
posed to explain the difference in spin results for the extreme BHB, GRS 1915+105
with both moderate [209] and maximal [192] values claimed. However, the actual
luminosity at which this distortion appears in GRS 1915+105 is model dependent and
requires careful treatment [209]; as a result, the spin of this unusual source remains
somewhat contentious. In general, the point at which radiation pressure starts to
affect the structure of the flow must be related to the spin (which sets the radiative
efficiency—see Eq. 3.30) and so the demarcation at 30 % of the Eddington limit trans-
lates into differing mass accretion rates for different sources. Should the structure
of the disc be affected instead by something coupled to the mass accretion rate, the
change in the flow could conceivably occur at a different Eddington ratio; thus, to
ensure rigour, selection criteria should ideally be determined on an object-by-object
basis (e.g. Steiner et al. [307]).
An important consideration for spectral studies of BHBs (in general but in partic-
ular in using the spectrum to estimate the spin) is that whilst their proximity leads to
high fluxes, these can lead to severe issues for CCD spectrometers (see [158] for a
discussion of these effects on XMM-Newton data) including photon pile-up (where
the arrival of multiple photons is read as a single event), charge transfer inefficiency
(CTI, where a loss of charge occurs during CCD read-out) and X-ray loading (where
very bright sources contaminate the ‘offset map’—analogous to a ‘dark frame’ in
optical instruments). Such effects can readily distort the spectrum, and therefore,
care must be taken to ensure that their impact on estimating the spin is understood.
Finally, it is important to note that mass-loaded winds are launched from the
accretion disc (perhaps as a result of radiation pressure, e.g. Proga and Kallman
[262]; thermal reprocessing in the outer disc, e.g. Begelman et al. [21], or MHD
driving, e.g. Neilsen and Homan [241]) and become stronger as the source becomes
dominated by thermal disc emission [259]. Although these winds are highly ionised
(as the source is X-ray bright) and are unlikely to present an obstacle for studying
the continuum (although see the following section for their impact on AGN spectra),
the models applied in order to determine the spin (see Sect. 3.3.1.1) are at present
only steady-state and do not take this mass loss into account—the effect of mass loss
on the measurement of the spin at this time is therefore unknown.
In Table 3.1, we present the available BH spin values for BHBs (both Galactic
and extra-galactic) along with the system parameters and the model used (useful in
the light of the assumptions that underpin each model as discussed above). The vast
majority of these values are reported in the recent review by Miller and Miller [214],
and for the sake of brevity, we direct the interested reader there for the individual ref-
erences for the spin values and system parameters, although we note, where possible,
updated values and any additional sources.
Notably, all bar two of the BHBs in Table 3.1 is Galactic (or located in the large
Magellanic cloud), and as such, the issue of precision in the distance measurement
is important. In the case of the source in M31 [203, 205], the distance is known
to within a few per cent, and as this is relatively large, the source flux (which is
high in such soft states) does not pose an issue for CCD detectors. Thus, such bright
(yet relatively nearby) extra-galactic sources offer a means to expand our sample of
BHBs with spin measurements which, as we discuss in Sect. 3.3.5, may be extremely
important for understanding how astrophysical jets are launched.
3.3.3.2 Reflection Fitting
As mentioned in the previous section, due to the proximity of Galactic BHBs, their X-
ray flux is typically very high; unsurprisingly, these provided the very first discovery
of a broad Fe Kα line through an EXOSAT observation of Cygnus X-1 [20, 82].
As the disc emission does not need to be isolated, even higher Eddington fraction
observations than those typically selected for the continuum fitting method may be
utilised (although should advection occur at such higher rates, then particle orbits
may be affected, e.g. Narayan and Yi [238]). Due to the source brightness, CCD
detector issues may once again become important, but where these distorting effects
can be reliably ignored or corrected (see Miller et al. [212] for a discussion of pile-up
on Fe Kα line measurements), CCD spectroscopy of BHBs offers the opportunity
to study the reflection spectrum in remarkable detail and thereby well constrain the
spin.
As discussed in Sect. 3.3.2, modelling of the reflection spectrum is less sensitive
to uncertainties in system parameters such as the distance and BH mass (whilst the
inclination is a free parameter in the models), and as such, the method is in principle
more robust than modelling the disc emission. However, an important consideration
when modelling the reflection spectrum in BHBs is the modelling of the continuum
which, in the bandpass of most detectors, contains a significant contribution from the
disc due to the BH mass (see Eq. 3.27). In addition, the nature of the Comptonised
emission remains a point of debate, with suggestions from broadband spectroscopy
extending to high energies, that it may be a combination of more than one component
130 M. Middleton
Table 3.1 Table of BHB spins

Continuum fitting Reflection
fitting
Source Mass (M ) Inclination Distance a∗ Model a∗
(degrees) (kpc)
+0.12
Cygnus X-1 14.8 ±1.0 27.1 ± 0.8 1.86 −0.11 ≥ 0.95 K2 >0.97a
+0.58 +0.37
XTE J1550-564 9.10 ± 0.61 74.7 ± 3.8 4.38−0.31 0.34 −0.34 K2 0.55 ± 0.22
XTE J1650-500 0.79 ± 0.01
XTE J1652-453 0.45 ± 0.02
XTE J1752-223 0.52 ± 0.11
XTE J1908+094 0.75 ± 0.09
A 0620-00 6.61 ± 0.25 51.0 ± 0.9 1.06 ± 0.12 ± 0.19 K2
0.12
4U 1543-475 9.4 ± 1.0 20.7 ± 1.5 7.5 ± 1.0 0.8 ± 0.1 K 0.3 ± 0.1
+0.005
4U 1630-472 0.985−0.014
MAXI 0.88 ± 0.03
J1836-194
GRO J1655-40 6.30 ± 0.27 70.2 ± 1.2 3.2 ± 0.2 0.7 ± 0.1 K 0.98 ± 0.01
+0.05
GS 1124-683 7.24 ± 0.70 54.0 ± 1.5 5.89 ± −0.24−0.64 K
0.26
GX 339-4 >0.97b
GRS 1915+105 14.0 ± 4.4 66 ± 2 11.0 ≥0.95 K2 0.98 ± 0.01
∼0.7c B
GRS 1739-278 0.8 ± 0.2 d
+0.2
SAX 0.6 −0.4
J1711.6-3608
Swift 0.76 +011
−0.15
J1753.5-0127
Swift ≤ −0.32
J1910.2-0546
+0.05 +0.02
LMC X-1 10.91 ± 36.4 ± 2.0 48.1 ± 2.2 0.92−0.07 K2 0.97−0.13
1.54
+0.20 e
LMC X-3 6.95 ± 0.33 69.6 ± 0.6 48.1 ± 2.2 0.25−0.29 K2
M31 ULX-2 ∼10 <60 772 ± 44 <−0.17f B
M33 X-7 15.65 ± 74.6 ± 1.0 840 ± 20 0.84±0.05 K
1.45
Notes: Spin values from continuum fitting (and system parameters used for the model fits) and from
reflection fitting (see [214] for individual references) with errors typically quoted at 1–2σ (with the
exception of the upper limit for the spin of M31 ULX-2 which is at 3σ ). The differing levels of
quoted precision for the values is a result of the individual studies. The model used in the continuum
fitting is either kerrbb: K, kerrbb2: K2 or bhspec: B. Updated or additional values are indicated
by: a Parker et al. [255], b Ludlam et al. [176], c Middleton et al. ([209], as GRS 1915+105 is extreme
and spectral modelling is inherently degenerate, we include this value for completeness: see also
McClintock and Remillard [192] for additional discussion), d Miller and Miller [214], e Steiner et al.
[311], f Middleton et al. [203]
due to a two-temperature plasma or a combination of thermal and nonthermal elec-

tron populations (e.g. Parker et al. [255]). When considering the spin value obtained
via this technique, one must therefore take into account errors resulting from the com-
bination of detector effects and model uncertainty (e.g. the radial profile discussed in
Reynolds and Fabian [272]) and also any uncertainty in the continuum onto which the
reflection spectrum is imprinted. Naturally, as the count rate is typically extremely
high, these will dominate over statistical errors. As a final point, it is possible that
the Fe Kα line may be broadened due to scattering in the disc atmosphere which
can confuse measurements of the spin; however, as Steiner et al. [308] find in the
case of XTE J1550-564, the effect—whilst noticeable—is unlikely to dominate over
Doppler and GR effects.
At the time of writing, 17 BHBs (out of a total Galactic population of ∼30: [116])
have measurements for the spin from reflection fitting and these are shown in Table 3.1
(alongside those from continuum fitting). As discussed in detail by Reynolds [271],
there is general concordance with notable exceptions being 4U 1543-475 and GRO
J1655-40.
The distribution of spin values for stellar mass BHs provides insight into the natal
supernova process by which they are formed (see [153]). As remarked upon by Miller
and Miller [214], the distribution of values from the two methods is similar (although
the sample sizes do not yet allow for Gaussian-distributed statistics) with spin values
at least two orders of magnitude higher than in the case of neutron stars (where the
spin can be accurately determined from pulse periods) and demands vastly different
means of acquiring angular momentum during formation.
3.3.4 Results: AGN
Whilst the spin distribution of BHBs is considered relevant for understanding their
formation process [153], so too is it the case that the spin distribution for AGN
provides information as to the growth of SMBHs (e.g. Moderski and Sikora [224];
Madau and Quataert [178]; Volonteri et al. [333]; King and Pringle [152]; Fanidakis
et al. [87]). If the growth progressed via accretion of gas, for example driven by
minor mergers with satellite galaxies, then the infalling material will have a range of
angular momenta relative to the BH spin (see, e.g. Nayakshin et al. [236]) giving a
range of SMBH spins in the local universe. Conversely, accretion of material through
the galactic disc with a fixed direction of angular momentum is likely to produce
high spins (even if the BH is initially misaligned: [152]). As accretion is inherently
connected to the production of highly energetic outflows in the form of winds and
jets, the strength of which is potentially related to the BH spin (via the Blandford–
Znajek mechanism—see Sect. 3.3.5—and the radiative efficiency in the disc—see
Eq. 3.30), understanding how SMBHs grew and how their spin evolved is of broad
importance for our understanding of the larger scale structure of the universe due to
the interactions of outflows with the host galaxies (see the review of feedback by
Fabian [81]).
132 M. Middleton
In the following sections, we discuss the campaigns to determine the spin of AGN
excluding those accreting at very low (quiescent levels) such a Sgr A*, although we
note that spin estimates via GRMHD simulations and SED fitting for this important
SMBH have generally favoured moderate-to-high (but not maximal) spin values (e.g.
Mościbrodzka et al. [227]; Dibi et al. [66]; Drappeau et al. [72]) and will be further
constrained when the event horizon telescope is fully operational.
3.3.4.1 Continuum Fitting
From inspection of Eq. 3.27, it is readily apparent that the peak disc temperature
scales inversely with the BH mass. For typical SMBH masses of >106 M (e.g.
[341]), at sub-Eddington rates, the disc will peak in the extreme UV [182] which is
heavily absorbed by the ISM. Whilst substantial emission emerges at higher energies,
providing the illuminating continuum for reflection, direct fitting to the disc emission
to obtain the spin has not been viewed as a promising technique. In addition, for
accurate spin measurements to be obtained from the continuum, it is important that
the system parameters (i.e. inclination, mass and distance: see Sect. 3.3.1) are known
to a high degree of precision. This can be relatively challenging; whilst the distance
is known to far greater accuracy than is usually available for Galactic sources (and
motivates the study of extra-galactic BHBs—[203]), the mass and inclination are
harder to measure, with typical errors on the mass via reverberation of 0.5 dex.
A subclass of AGN—specifically some narrow-line Seyfert 1s (NLS1s)—appear
to show a very hot disc component (e.g. Middleton et al. [210], Jin et al. [138, 139],
Terashima et al. [324]) which may extend into the soft X-ray band and contribute to
the ubiquitous ‘soft excess’ [50, 110] directly or via Comptonisation [67]. Recent
progress in modelling the AGN inflow—most notably by applying a more rigourous
treatment for the opacity balance (see Sect. 3.3.1)—has led to the development of
the model optxagnf which incorporates approximate radiative transfer, somewhat
analogous to bhspec [67]. This model takes in the mass accretion rate which can be
determined from the optical flux where the mass is reasonably well estimated and
conserves energy extracted from the accretion process in creating an optically thick
thermal Compton component and the higher energy tail [59]. Done et al. [68] applied
this new model with a relativistic convolution kernel (kerrconv) to the spectrum
of the bright NLS1, PG 1244+026. The spectrum of this source is similar to other
extreme, soft NLS1s (e.g. RE J1034+396: [204]), appearing to have high (close to
the Eddington limit) mass accretion rates with a very weak tail of hard emission
and a strong soft excess. Whilst high Eddington accretion rates are not generally
considered to be an appropriate regime to apply standard continuum fitting methods
for BHBs (see Sect. 3.3.1), analogous behaviours of the AGN disc have not been
established (due to the timescales of variability). As this model conserves the energy
produced via accretion (and is therefore related to the spin) in creating the regions
of different optical depth, the limits on the spin imposed by this model are therefore
of interest (although as with more standard models does not yet take into account
energy lost in the creation of a wind or jet). In the case of PG 1244+026, Done et al.
[68] find that the spin must be low (and rule out maximal spin). Such an approach
is likely to be useful for those lower mass AGN where the broadband (optical to X-
ray) disc emission can be well modelled, i.e. relatively unobscured by gas and dust.
Whilst these appear relatively rare (see Middleton et al. [210]), the future missions
of eROSITA and Athena are expected to find many more, allowing this approach to
be more widely used and thoroughly tested.
3.3.4.2 Reflection Fitting
The vast majority of spin measurements for AGN have come from fitting reflection
models to their spectra. By AGN in this context, we are generally referring to Seyfert
1 AGN (including NLS1s and QSOs) as the unified model [8, 9] would suggest
that these are viewed at low-to-moderate inclinations and so are not obscured by the
molecular torus (which is the origin of Compton-thick AGN where Fe Kα emission
is seen but originates from the torus and so is not a measure of the BH spin). The
review of Reynolds [271] provides details of the approaches discussed already as
well as a ‘cookbook’ for obtaining estimates for the BH spin from the reflection
spectrum, and we direct the interested reader and practical observer here.
The Fe Kα line is known to be almost ubiquitous in AGN [235, 275] as a direct
result of X-ray illumination of optically thick (not fully ionised) material and was
first discovered by Tanaka et al. [321] when the bright AGN, MCG-6-30-15, was
observed by ASCA with the strong, broad line found to require maximal spin (see
also Iwasawa et al. [135]; Dabrowski [51]; Reynolds and Begelman [273] Young et al.
[343]). Since the inception of the field, the use of the reflection spectrum (the whole
of which is pivotal for accurate spin determination: [343]) has been widespread,
finding an application in probes of higher redshift, lensed QSOs [269, 335] as well
as in novel techniques to utilise the time dependence of the emission ([277]—see
Sect. 3.5).
There are a number of important considerations when fitting the reflection spec-
trum of AGN which are not relevant in the application of this method to BHBs. The
optical/UV line ratios of Seyfert AGN (e.g. Warner et al. [336]; Nagao et al. [234])
imply that the metal abundance of the gas is supersolar (e.g. Zoghbi et al. [346];
Reynolds et al. [276]) which plays an important role in the strength of the Fe Kα line;
it is therefore important to allow any applied models to extend beyond solar metal-
licities. Unlike accretion discs around stellar mass BHs, the discs around SMBHs
do not contribute in a sizeable way to the X-ray continuum (with the exception of
the very brightest, lowest mass AGN: [67, 139, 210]) giving a much cleaner view of
the reflection spectrum. However, the environment of the AGN is less ‘clean’ than in
BHBs: winds are expected to be ubiquitous given the UV radiation field and resonant
line opacity [263] and will not be as ionised as those from BHBs. These winds can
therefore result in a distorting imprint around the Fe Kα line and possible degeneracy
in the spectral fitting of AGN (e.g. Patrick et al. [256]; Brenneman et al. [34]; Mid-
dleton et al. [202]). This degeneracy can be broken in two separate ways: the first is
through the use of an observable bandpass which extends to higher energies, where
134 M. Middleton
differences between absorption and reflection differ dramatically (as demonstrated

in the case of NGC 1365 through use of NuSTAR data: [278]). The second means
to break the spectral degeneracy is through the use of techniques which utilise the
source variabliity (e.g. the cross-spectrum: Nowak and Vaughan [245]) which can
effectively isolate reverberation of the primary continuum [327]. The latter technique
has clearly demonstrated that the lags at high frequencies (i.e. those originating from
the most compact regions of the corona) contain a signature of the broad Fe Kα line
[146, 347] and Compton hump [143, 144], confirming the strong contribution of
reflection to the spectrum.
A further important consideration when applying the reflection fitting method to
AGN is the presence of the soft excess. As the name implies, this is an excess of
flux seen below 2 keV once a power law fit to the 2–10 keV band is extrapolated
backwards and has no obvious counterpart in BHB spectra. The soft excess is both
smooth and is present across a large number of AGN across a range in mass, peaking
at ∼0.5 keV in each; this raises problems for its interpretation as a continuum-only
component should not produce the same peak temperature across a range of masses
without some fine-tuning of the radiative process [109]. Instead, atomic transitions
associated with OVII/OVIII and the Fe M UTA naturally produce features in this
energy range and could be seen in absorption and emission via reflection or in an
outflow. However, to produce such an excess of flux without seeing sharp features
requires velocity broadening in either situation. Crummy et al. [46] and Middleton
et al. [210] studied a sample of AGN, finding that both reflection and absorption can
produce statistically indistinguishable fits (across the XMM-Newton bandpass) with
the spin tending towards large values and the outflow velocity of the wind tending
to being moderately relativistic. A third possibility is the combination of reflection,
outflow and, in the case of the brightest, low mass AGN, some Compton component
of the disc [67, 140]. Understanding the origin of the soft excess is considered to
be extremely important as, in cases where the inclination of the disc is unknown or
cannot be constrained in the reflection model, the spin can be driven by the need
to smear the atomic features at soft energies. Once again, applying methods which
utilise the time domain can assist in understanding the origin of this component and
has shown that for some sources, the likely origin is in partially ionised reflection
(notably 1H0707-495 [83, 346]), whilst in others it would appear that the soft excess
is dominated by a continuum component associated with Compton upscattering of
UV seed photons [140].
In the case where the inclination is well constrained by the model (and therefore
the result does not rely on the soft excess) and the whole reflection spectrum is
considered (allowing for super-solar abundances of metals), the uncertainty on the
spin value once again depends on the continuum (including the effect of absorption)
and the model being used, whilst at typical AGN fluxes, detector issues which can
be challenging for observations of BHBs become less troublesome.
In general, SMBH spins determined via reflection (see, for example the sample
studies of Walton et al. [334] and Patrick et al. [256]) show a tendency towards
high spin (as remarked upon by Reynolds [271]). As discussed by Brenneman et al.
[33] and Walton et al. [334], this distribution may be a result of selection effects; a
higher spin results in higher radiative efficiency (see Eq. 3.30) and a brighter AGN;
conversely, this may be indicating coherent rather than chaotic SMBH growth which
would result in systematically high spins (see Fanidakis et al. [87] although see
Nayakshin et al. [236] for counterarguments).
3.3.5 Implications: Powering of Ballistic Jets
As remarked upon in Sect. 3.2.1 of this chapter, the effect of frame-dragging will
have important consequences for the transfer of energy from the BH to an orbiting
test particle. Here, we briefly describe the Penrose process [258]; [18] and how it
can be linked via the Blandford–Znajek mechanism to the powering of relativistic
ejections.
3.3.5.1 The Penrose Effect and Blandford–Znajek Mechanism
The mechanical Penrose effect occurs as a result of ‘negative energy orbits’, i.e.
where the energy required to send a body to infinity is larger than its rest mass
energy. In the notation of GR, this is ut < 0 and can be shown to only be possible
for uφ > 0 and:
(uφ )2 (−gtt ) > Δ sin2 θ (3.39)
where Δ is given in Eq. 3.4. As the right-hand side of the inequality is positive, we
find that this is only satisfied if gtt < 0, i.e. within the ergosphere. It can then be shown
that for a particle to have negative energy, i.e. uφ > 0, orbits need to be retrograde
relative to the BH spin. The Penrose process can then be described as a body entering
the ergosphere and breaking apart; one part is induced into a retrograde orbit and
the other escapes to infinity after having gained energy (equal to the negative energy
captured by the BH) from the rotation of the BH. The efficiency of this process, i.e.
the ratio of maximum energy out to that going in, is of order 20 % (Chandrasekhar
[43], see also the review of Brito et al. [35]).
Whilst the mechanism described above is unlikely to have a direct impact on the
observational appearance of BHs, when coupled to magnetic fields, it may present a
viable mechanism for powering collimated, relativistic outflows in the form of jets.
Poloidal magnetic fields are expected to grow in the accretion flow via the MRI:
[14] and propagate down to the vicinity of the BH. Where these field lines thread
the ergosphere, they are forced to rotate with the matter, inducing a force on the
coupled charged plasma (Lorentz force) which will lead to acceleration of material
at relativistic speeds along the rotation axis of the BH in the form of jets (which are
then collimated via magnetic confinement). This is a highly simplified description
of the Blandford–Znajek (BZ) mechanism (or effect: [29]) where the power that can
be extracted Pjet ∝ a2 as long as the spin is not large. A more accurate relationship
136 M. Middleton
that covers the whole range of possible spins has been derived by Tchekhovskoy and
McKinney [323] to be Pjet ∝ (MΩH )2 where ΩH is the BH angular frequency which
(in natural units) is given by:
a
ΩH = √ (3.40)
2M(1 + 1 − a2 )
Measuring both the spin and jet power in a reliable way can therefore provide
insights into the launching of jets by confirming the BZ effect (or an effect with a
similar form) or by ruling it out.
3.3.5.2 Testing for the Blandford–Znajek Effect
As briefly mentioned in Sect. 3.3.3, BHBs launch jets at low bulk Lorentz factors
(Γ < 2) when accreting at low to high rates and when accompanied by an X-ray
spectrum dominated by a hard thermal tail of emission. The emission from the jet
extends from low (MHz) frequencies up to optical/IR and is typically a flat power
law (S ∝ Eγ where γ ≈ 0) across the intervening several decades in frequency. The
spectrum results from highly (ultrarelativistic) electrons spiralling around magnetic
field lines and cooling via synchrotron radiation. The flat spectrum is a result of
viewing the emission from spatially extended regions, with the low-frequency radio
emission originating further from the launching point (i.e. where the particle Lorentz
factor is lower). A break at low frequencies occurs when the lowest energy electron
population becomes optically thick to their radiation (and are self-absorbed), whilst a
high-frequency break occurs when the most energetic electrons are optically thin to
their radiation (in reality, these breaks occur throughout the spectrum and merely
convolve to give the observed spectrum). Although the spectrum evolves (notably
the position of the high-frequency break moves as the electrons cool, e.g. Russell
et al. [291]), as the emission is constant, the jet is termed ‘steady’.
The jet changes dramatically in nature when the spectrum softens into the interme-
diate and then soft states, gaining a much higher bulk Lorentz factor (Γ >2) and tak-
ing the form of discrete ejecta (e.g. Mirabel and Rodríguez [220]; Fender et al. [90])
which cool via synchrotron radiation with a high-frequency spectral break evolving
with the expansion of the ejecta (see van der Laan [330]; Kellermann and Owen [149];
Hjellming [123]; Hjellming Johnston [124]). These ejections were first observed in
the Galactic centre ‘Annihilator’, 1E 1740.7-2942 [221], and subsequently detected
in the remarkable BHB, GRS 1915+105, which was the first Galactic source where
superluminal jets (appearing as such due to their highly relativistic velocities and
orientation close to the line-of-sight) were identified; due to their resemblance to the
superluminal ejections from radio loud quasars, these sources were dubbed ‘micro-
quasars’ [222]. Such discrete ejections are thought to be ubiquitous, with all BHBs
showing (or expected to show) what are sometimes referred to as ‘ballistic’ jets.
Typically, the power in the steady jet is determined from the established correlation
with the radio synchrotron luminosity [30, 136, 159], whilst the power in the ballistic
jet has traditionally been estimated from the monochromatic radio luminosity which
Steiner et al. [310] have shown to be approximately linearly correlated with the
mechanical (bulk kinetic) jet power (see their Appendix). To obtain the intrinsic
luminosity, the effect of Doppler boosting (which acts on both the flux and energy
of any breaks in the synchrotron spectrum) has to be accounted for. In practice, the
boosting factor (D3−γ : see Eq. 3.34) is determined from the inclination (although,
as discussed in Sect. 3.2.1.1, one has to careful as to which inclination—inner or
outer disc—is being used) for a range of bulk Lorentz factors and assuming a typical
spectral index. By correlating this deboosted jet power against the spin derived from
the spectrum using the continuum fitting or reflection fitting methods, the likely
impact of the BZ (or a similar) mechanism can then be tested.
Fender et al. [91] demonstrate that there is no obvious correlation between the
steady jet power and the spin for a sample of BHBs, effectively ruling out the BZ
effect as the dominant mechanism for the launching of the slower jet. To investi-
gate the powering of the faster, ballistic jet, Narayan and McClintock [240] and
Steiner et al. [310] selected the five Galactic BHBs which are thought to reach their
Eddington limit and therefore act as ‘standard candles’, thereby removing any mass
accretion rate-dependent effects; the resulting presence of a correlation between jet
power and spin has been claimed as strong support of the BZ effect. Russell et al.
[292] have since questioned the selection criteria arguing that the inclusion of other
sources disagrees with the presence of a correlation; instead, the driving factor is
claimed to be the mass accretion rate (with similar arguments proposed for AGN:
[155, 156]). Whilst the debate is ongoing, it is abundantly clear that a larger sample of
BHBs is required in order to fully evaluate the presence of a correlation. The applica-
tion of these techniques to nearby extra-galactic BHBs (which can be studied reliably
in the X-ray and radio: [205]) is in its infancy but is the only means by which this can
be accomplished [203]. In addition, the introduction of new methods for measuring
the spin (as discussed in the following sections) will provide important cross-checks
for existing methods and provide increased confidence in any conclusions which rest
upon its measurement.
As a final remark, Garofalo et al. [104] note that the BH spin itself may have
a distorting influence on the nature of any correlation between jet power and spin
through the quenching of the jet by winds which become stronger with increasing
source brightness (e.g. [242]). This assumes that the winds are radiatively driven
(either thermal via reprocessing or radiation pressure powered via scattering) such
that a higher spin—which leads to a higher efficiency in conversion of rest mass
to energy via Eq. 3.30—more readily powers winds and could in principle shut off
the jet at lower mass accretion rates (under the assumption that the wind launching
is not dependent on the mass accretion rate-dependent structure of the disc itself).
Whilst the exact mechanism for jet quenching is not yet fully understood, certainly
the coupled interaction of inflow and outflow and the spin dependence is an important
consideration for understanding the evolution of accreting BHs of all masses (e.g.
Kovács et al. [160]).
138 M. Middleton
3.3.6 Implications: Retrograde Spins?
In a very small number of cases, only three BHBs [203, 226, 268] to date,
significantly retrograde (a∗ < 0) spin has been reported. It is possible that such
sources could launch particularly powerful jets should magnetic flux be swept from
the plunging region onto the BH, with the amount of flux being dependent on the
size of the ‘gap’ between the ISCO and BH. As this is larger for retrograde spin
(RISCO = 9Rg for a∗ = −0.998 (see Fig. 3.2), the magnetic flux trapped on the BH
can therefore be enhanced [105, 106] although simulations incorporating the effect
of magnetic field saturation [323] dispute this ‘gap paradigm model’ and arrive at
the opposite conclusion.
Irrespective of the impact, we must ask the question, how can retrograde spin
practically occur in those systems where it has been reported and how likely is it?
Co-alignment of disc and BH (which occurs through the action of viscous torques
transferred via the Lense–Thirring and Bardeen–Peterson effects, e.g. King et al.
[151]) is expected to take at least several per cent of the binary’s lifetime, and so,
assuming that the measurement of retrograde spin is genuine (and not an artefact of
inner disc truncation due to some as-yet-unknown process), retrograde spin is likely
to be an indication of the formation process (i.e. an anisotropic supernova kick: [31]),
the result of wind-fed accretion, (which can produce counteraligned inflows, e.g. GX
1+4: [41] and Cyg X-1: [303, 344]) or from the tidal capture of a star [80]. In this
last case, after the BH has formed whilst in a globular cluster, it is expelled by the
natal kick, with a subsequent stellar capture producing a retrograde orbit. Whether
retrograde spin could form in AGN is not clear but could presumably result from
minor mergers with material carrying counteraligned angular momentum accreted
onto the SMBH on short timescales (as on long timescales enough material can be
accreted to co-align the system). Notably in AGN, the entire sub-pc disc may be
misaligned with the galaxy as a result of a recent minor merger, and indeed, there is
growing evidence for this from jet launching angles and the location of the molecular
torus (see Hopkins et al. [125]).
3.4 Observational Tests of Spin II—The Time Domain and

Relativistic Precession Model
Traditional methods of determining the spin have proven to be highly illuminating;

however, methods which do not rely solely on the time-averaged spectrum have the
advantage of being able to provide a semi-independent measure of the spin and test
traditional models (and our understanding of GR).
As we discussed in Sect. 3.2.1.1, the effect of relativistic frame-dragging leads
to precession of orbits in the innermost regions of the accretion flow due to the
Lense–Thirring effect. Such precession in turn leads to epicyclic oscillations about
the vertical ‘nodes’ (where orbits of a test particle out of the equatorial plane meets
that of the ecliptic) and precession of periastron passage. When combined with the
frequency of Keplerian orbit, these three frequencies form the relativistic precession
model (or RPM: [313–315]). As long as the nature of the plasma in the accretion disc
is not so dense as to dampen the oscillations, in practice and under the assumption
that these particle orbits can leave an imprint on the source flux we could expect each
of these frequencies to leave a trace of coherent power in the light curve of accreting
black hole systems (both BHBs and AGN). These could naturally be associated with
the quasi-periodic oscillations (QPOs) detected as narrow peaks in the power density
spectrum (PDS) and seen in BHBs at low frequencies (e.g. Wijnands et al. [338];
Casella et al. [40]; Belloni et al. [24]; Motta et al. [229]), occasionally (and only
detected so far in 5 BHBs) at high frequencies (e.g. Morgan et al. [225]; Remillard
and McClintock [270]; Méndez et al. [198]) and recently discovered in AGN [6, 7,
112, 204, 207]. The AGN QPOs appear to be analogous to the high-frequency QPOs
(HFQPOs) of BHBs [208] and, in both sets of systems, appear in a 3:2 harmonic
ratio implying a common physical origin (e.g. Dexter and Blaes [65]).
In the RPM, the two HFQPO frequencies are associated with the orbital frequency,
νφ , and the periastron precession frequency, νper , which in turn is given by the differ-
ence between the orbital and radial epicyclic frequencies: νper = νφ − νr . The much
slower, vertical (Lense–Thirring) precession would instead be associated with the
low-frequency QPO (LFQPO: [130, 132]) and is given by νlt = νφ − νθ (where νθ is
the vertical epicyclic frequency). The three QPO frequencies are connected both to
one another and to the BH mass and spin through the following relations [18, 200]:
β 1
νφ = ± (3.41)
M r ±a
3/2

6 8a 3a2
νper = νφ 1 − 1 − ± 3/2 − 2 (3.42)
r r r

4a 3a2
νlt = νφ 1 − 1 ∓ 3/2 + 2 (3.43)
r r
where β = c3 /(2π GM ), r is the radius in units of Rg and M is in units of solar

mass. Where ± or ∓ are given, the top sign refers to a treatment where the spin is
prograde and the bottom sign to where the spin is retrograde.
The above set of equations lead to a set of simultaneous equations and in turn to
the following formula for the spin (see Ingram and Motta [131] for a derivation):

r 3/2 6
a=± Λ+Φ −2+ (3.44)
4 r
where:
νper 2
Φ = 1− (3.45)
νφ
140 M. Middleton
and

νper 2
Λ= 1− (3.46)
νlt
Although the solutions to the equations of the RPM do not immediately tell us
whether the spin is prograde or retrograde, Ingram and Motta [131] point out that
this can be identified from the highest frequency reached by the LFQPO (which if it
extends to within the ISCO for a = −|a|, then it implies prograde spin).
Critically, the application of the RPM in determining the spin relies on the simul-
taneous presence of LF and HFQPOs in the light curve (as their frequencies are
indicative of the radius at which they are generated), although it is not vital that all
three be present as Ingram and Motta [131] demonstrate semi-analytically. Recently,
the RPM method has been applied to two BHBs: GRO J1655-40 [228], the only
BHB to date where all three QPOs have been detected simultaneously [229] and
XTE J1550-654 [231] where only the LFQPO and one of the HFQPOs have been
detected simultaneously. The resulting spin for XTE J1550-654 was found to be
consistent with estimates from the reflection fitting and continuum method [308],
whereas the value obtained for GRO J1655-40 via the RPM is inconsistent with
those obtained via spectral means [213, 267, 300]. The reason for the discrepancy
in the spin for the latter source may be due to the misalignment of the BH spin and
orbital axis by >15◦ [114] which are likely to be closely aligned in XTE J1550-654
[309]. Thus, whilst the RPM is insensitive to the inclination (as the frequencies are
independent of the inclination unlike the measured strengths of the QPO: [230]), this
could present problems for measurements of the spin via spectroscopic methods (see
the discussions in Sect. 3.3).
Finally, it should be pointed out that mechanisms to explain the origins of the
QPOs besides the RPM have also been proposed (e.g. Esin et al. [79]; Tagger and
Pellat [320]), with little direct observational progress made in distinguishing the
correct interpretation. However, a key discriminator for the origin of the LFQPO
as Lense–Thirring precession is the QPO phase-resolved emission (see Sect. 3.3.2).
As described in Ingram and Done [130], the geometrically thin, optically thick disc
truncates (possibly due to disc evaporation, e.g. Meyer and Meyer-Hofmeister [201];
Liu et al. [175]) and the inner, lower density region (in this picture, the location of
the Compton upscattering of seed disc photons) precesses as a solid body as a result
of the Lense–Thirring effect (see Ingram et al. [133]). As the inner region precesses,
various disc azimuths are subjected to changing illumination; in turn, this leads
to changes in the observed reflected emission (Fe lines and Compton hump) as a
result of the various Doppler shifts and boosting. Should these predicted changes be
observed by long observations with present instruments or using high throughput,
high-time-resolution instruments such as the LAXPC onboard the recently launched
ASTROSAT (see Ingram et al. [134] for details of the arithmetic approaches and a
possible detection of modulation already seen in RXTE data), it will likely represent a
‘smoking gun’ for the RPM (and may likewise rule out such an origin if the predicted
variations are not observed).
3.5 Observational Tests of Spin III—The Energy–Time

Domain
As opposed to the previously described methods which use either the energy or time
domain, an approach which combines the two, promises to provide the largest lever
arm for estimating the BH spin. One such method is ‘Doppler tomography’ and
relies upon a changing view of the regions of the accretion disc due to an eclipse
by an orbiting body. This technique was first applied to the study of white dwarfs in
mapping the accretion disc via emission lines (e.g. Marsh and Horne [186]), and a
small number of authors have since developed it as a tool for the study of AGN spin
and the effects of GR.
In the majority of AGN (with masses typically >106 M : [341]), the disc is out of
the X-ray bandpass (although the hottest tail of the disc and/or a Compton upscattered
component may enter at soft energies for the lowest mass and highest mass accretion
rate sources: [210]; Jin et al. [138]; [67], see the discussion in Sect. 3.3.1), whilst
the primary Compton scattered emission and its reflected component dominate the
emission (e.g. Fabian [83]). Should an orbiting body pass across our line-of-sight, an
eclipse results and leads to changes in the spectrum as a function of time that allows
a test of the nature of the inner regions, e.g. the radial temperature dependence of
the corona and BH spin (see McKernan and Yaqoob [197]).
Variability due to obscuration by cold material is relatively common on long
timescales in AGN (see [279]) and on shorter timescales by Compton-thin material
[25, 76, 264, 277, 280]. However, obscuration by Compton-thick (i.e. τ > 1 →
nH σT >1 → nH > 1024 cm−2 ) material on observable (<100 s of ks) timescales—
which leads to the simplest form of eclipse—is relatively rare, although at least one
such event has been observed in NGC 1365 [282].
Using the model of Dovčiak et al. [71], which calculates the line emission from
different parts of the disc separately, Risaliti et al. [281] simulate the effects of an
eclipse under the assumption that the obscuring source is a Compton-thick broad-
line region cloud and completely covers the source. The result is a profound shift
in the shape of the Fe Kα line as a function of time as the approaching blue-shifted
side of the disc is covered followed by the retreating red-shifted side (i.e. a situation
where the cloud is co-rotating with the disc). As Risaliti et al. [281] point out, the
major effect is not a shift in the line profile but in flux due to Doppler boosting. As
a corollary, the high-energy continuum emission, i.e. the Compton hump, will also
rise and fall in flux, correlated with changes in the emission line (Fig. 3.1 of [281]).
Risaliti et al. [281] point out that this would constitute a case of a ‘perfect’ eclipse as
the obscuration is complete and the eclipse assumed to have sharp, linear edges. The
authors perform simulations which show that present observatories have the required
throughput to detect predicted changes for Compton-thick eclipses and can provide
independent confirmation of relativistic effects in shaping the Fe Kα line. Although
the more common eclipses by Compton-thin material (typically a few 1023 cm−2 :
[180, 282]) have less of an impact on the reflection spectrum, Risaliti et al. [281]
142 M. Middleton
show that future observatories (for instance ESA’s Athena) will still be able to detect
changes associated with the passage of such material.
Although not modelled explicitly, the change in Doppler boosting from either
side of the disc is dependent on both the inclination to the source and rotational
velocity (see Eq. 3.34). The latter is of course dependent on the location of the ISCO
and therefore the spin. Doppler tomography therefore not only provides a means
to independently verify the origin of the emission in relativistic material but also a
measurement of the spin. There are of course qualifiers and caveats to this approach,
and Risaliti et al. [281] point out that variations in the intrinsic emission can lead to
a distorting effect as can an eclipse that covers the illuminating source in a different
manner to the reflector.
An analogous situation to that described by Risaliti et al. [281] can be applied
to the disc emission directly when eclipses take place in AGN of lower mass. Such
low mass AGN are preferentially detected at the highest mass accretion rates [115],
where, analogous to the spectra of BHBs [138, 139], the spectrum is dominated
by the disc with a weak, flat power-law tail of emission to high energies. In such
sources, reflection features are therefore expected to be weak, restricting the means
by which the spin can be measured. However, at the apparent high accretion rates
associated with low-mass AGN, powerful winds are expected to be driven from
the disc, which itself may have grown in scale height (e.g. Shakura and Sunyaev
[301]; King [154]; Poutanen et al. [260]). As such winds are likely to rotate in
an approximately Keplerian manner (with deviations from this expected to scale
as (H/R)2 ), any inhomogeneities due to radiative hydrodynamic instabilities in the
surface of the wind material [263] will lead to gaps through to the inner regions
which also rotate. Should our sight line to the source intercept one of these gaps,
we can obtain a view of the approaching, blueshifted side of the disc and then the
retreating redshifted side as the gap orbits (see Fig. 3.5). As with the model presented
by Risaliti et al. [277], this form of Doppler tomography is highly sensitive to the
spin and inclination and can therefore be used to provide independent constraints.
Notably, unlike studies of the reflection, the disc emission is expected to be stable
over the timescales of an observation and so is not likely to be affected by intrinsic
variability that can have a distorting influence.
Using the ray-tracing code geokerr [62], Middleton and Ingram [211] create a
model to describe the orbit of a gap in a Compton-thick wind and apply it to the
case of the low-mass AGN RX J1301.9+2747 [318] which shows long-lived flaring
behaviour (Dewangan et al. [61]) inconsistent with the usual origin of rapid variability
in AGN (i.e. viscous—see [328]—and/or thermal). The authors instead argue that
the variability is due to gaps in the Compton-thick wind crossing our line-of-sight
(Fig. 3.5). From fitting the model across multiple phases simultaneously, the authors
find the spin to be very low irrespective of several caveats (e.g. errors on the mass and
temperature profile in the disc) although once again important assumptions remain
including the unknown structure of the disc seen through the gaps (the ray-tracing
assumes a Novikov–Thorne disc). Importantly, the combination of the time and
energy domains leads to stronger constraints on the spin than can be obtained from
traditional methods, demonstrating the power of Doppler tomography as a method
to probe AGN accretion and the region of strong gravity.
3.6 Concluding Remarks and Future Approaches
In this chapter we have discussed the core theory that is useful for an appreciation of
the role BH spin plays, notably the effect of precession and the Penrose process due
to frame-dragging and the changing position of the ISCO. As a result of the latter’s
effect on the emergent radiation (be it direct or reflected), the community has been
provided with a means to measure the BH spin in both AGN and BHBs.
Campaigns over the last 10 years have started to allow the spin distributions of
BHs to be probed, allowing progress to be made in understanding their formation.
However, many questions remain open and as yet unanswered: What role does the
spin play in the launching of ballistic jets? Is there a bias in the spin measurements of
AGN or is the spin genuinely high in most local Seyferts? Is retrograde spin common
or vanishingly rare? How reliable are our present set of techniques?
The first three questions can only be addressed by expanding our sample of sources
for which we have accurate spin measurements. This will no doubt be possible in
the forthcoming years when new, highly sensitive X-ray satellites including Athena
and eROSITA (and potentially LOFT or a descendent) become available. These will
provide the deepest views of the X-ray universe, providing access to not only the
spins of local sources but also, in the case of Athena, the cosmic evolution of the
AGN spin distribution (which in turn probes the growth mechanism of SMBHs: [87,
333]). The photon-rich spectra that Athena will obtain will not only provide high-
precision spin measurements but potentially even test for deviations from the Kerr
metric (e.g. Jiang et al. [137]).
As we have discussed in Sect. 3.3.5, to rigourously probe the BZ effect (in BHBs—
the analogy to AGN jets is still not clear) requires a much larger sample of sources
with reliable measures of both BH spin and jet power—for this, we must look to
nearby galaxies. Such an approach has been proven to be feasible with current
instrumentation [203] and in future will benefit from the introduction of both high-
throughout X-ray instruments and the next generation of radio telescopes (SKA and
pathfinders), for which the discovery of radio transients is a core aim (e.g. the Thun-
derKAT campaign).
Finally, to test the reliability of our techniques requires the use of the time domain
in an independent (RPM) or complimentary (Doppler tomography) fashion and look-
ing to the future, the use of X-ray polarimetry and gravitational wave interferometry.
Expanding briefly on the latter two techniques, as explained in Schnittman and Kro-
lik [297], the effect of returning radiation from the accretion disc leads to scattering
which is not appreciable at low energies (i.e. further out in the disc) and leads to
horizontal polarisation [42], whilst at higher energies the increased scatter to the
observer results in vertical polarisation [5]. The amount of the latter is dependent on
144 M. Middleton
the position of the ISCO and therefore the spin. This is expected to be an extremely
powerful technique and is the focus of a number of proposed (e.g. PRAXyS and
X-Calibur) and accepted (XIPE) missions. The impact of gravitational wave inter-
ferometry on the field of BH spin measurements has been discussed in detail in the
recent review by Miller and Miller [214] to which we point the interested reader; in
essence, should a BH–NS or BH–BH binary be found and if the BH spin is high and
misaligned with the orbital axis (due to LT precession: see Sect. 3.2.1), then there
can be a considerable impact on the gravitational waveform.
In conclusion, the future looks extremely bright for the field of BH spin determi-
nation and, in years to come, will allow the most detailed understanding of the most
extreme objects in the universe.
Acknowledgments The author gratefully acknowledges the assistance of Chris Reynolds, Javier
Garcia and Jack Steiner in proofreading and offering valuable suggestions.
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Chapter 4
Winds from Black Hole Accretion Flows:
Formation and Their Interaction with ISM
Feng Yuan
Abstract Black hole hot accretion flows occur in the regime of relatively low accre-
tion rates and are operating in the nuclei of most of the galaxies in the universe. In
this chapter, I will review one of the most important progresses in recent years in
this field, which is about the wind or outflow. This progress is mainly attributed to
the rapid development of numerical simulations of accretion flows, combined with
observations on, e.g., Sgr A*, the supermassive black hole in the Galactic center. The
following topics will be covered: theoretically why do we believe strong winds exist;
where and how are they produced and accelerated; what are their main properties
such as mass flux and terminal velocity; the comparison of the properties between
wind and “disk-jet”; the main observational evidences for wind in Sgr A*; and one
observational manifestation of the interaction between wind and interstellar medium,
namely the formation of the Fermi bubbles in the Galactic center.
4.1 Introduction
Black hole accretion is a fundamental physical process in the universe. It is the

standard model for the central engine of active galactic nuclei (AGNs), and also
plays a central role in the study of black hole X-ray binaries, Gamma-ray bursts,
and tidal disruption events. According to the temperature of the accretion flow, the
accretion models can be divided into two classes, namely cold and hot. The standard
thin disk model belongs to the cold disk, since the temperature of the gas is far
below the virial value Shakura and Sunyaev [45] (see reviews by [1, 6, 20, 40]). The
disk is geometrically thin but optically thick and radiates multi-temperature black
body spectrum. The radiative efficiency is high, ∼0.1, independent of the accretion
rate. The model has been successfully applied to luminous sources such as luminous
AGNs and black hole X-ray binaries in the thermal state.
F. Yuan (B)
Shanghai Astronomical Observatory, Chinese Academy of Sciences,
80 Nandan Road, Shanghai 200030, China
e-mail: fyuan@shao.ac.cn

154 F. Yuan
In addition to the cold disk model, the accretion equations have another set of
solutions, i.e., the hot accretion flow. The most well-known and pioneer hot accretion
flow model is the advection-dominated accretion flow [2, 37, 38]. This model applies
when the mass accretion rate is below ∼(0.1 − 0.3)α 2 ṀEdd , with ṀEdd ≡ 10LEdd /c2
is defined as the Eddington accretion rate and α is the viscous parameter. Above this
critical rate but below ∼α ṀEdd , we have another hot accretion flow model called
luminous hot accretion flows [54]. In contrast to the cold disk, the temperature of
the hot accretion flow is much higher, close to the virial value. The flow is thus
geometrically thick. The radiative efficiency of hot accretion flow is very low when
the accretion rate is low, but quickly increase with the increasing accretion rates [53].
Hot accretion flow is operating in the nuclei of perhaps most galaxies in the universe,
and the quiescent and hard states of black hole X-ray binaries. Yuan and Narayan
[58] present a comprehensive review of various aspects of the model, including the
one-dimensional and multi-dimensional dynamics, radiation, jet formation, wind,
and the applications in Sgr A*, low-luminosity AGNs, black hole X-ray binaries,
and AGN feedback.
For a long time, the mass accretion rate of accretion flow, both cold and hot, is
assumed to be a constant of radius. This implies that all the gas available at the outer
boundary of the accretion flow will be accreted into the black hole horizon, only
except at the innermost region very close to the black hole where a jet is formed.
However, in recent years, we have realized the existence of strong wind from black
holt accretion flow in both observational and theoretical aspects. We now understand
that most of the gas available at the outer boundary will be lost in wind rather than
accreted into the black hole horizon. In fact, the study of wind has become a hot
topic in the field of black hole accretion. The reason is twofold. The first is that this
is crucial for our understanding of the dynamics of accretion. The second reason
is that this can help us to understand observations on wind. The third reason is
related to AGN feedback. AGN feedback is now believed to play an important role
in galaxy formation and evolution [18, 27, 28]. There are two kinds of medium for
the interaction between an AGN and the interstellar medium, i.e., photons and matter.
The latter includes wind and jet. So to understand AGN feedback, we need to first
understand the main properties of wind.
In this chapter, I will review our current understanding of the various aspects of
wind from hot accretion flows.
4.2 Formation of Wind from a Hot Accretion Flow
The study of wind from hot accretion flows started from about 15 years ago. This
is young compared with the history of the study of wind from a thin disk, which
started more than twenty years ago (e.g., [41]). However, since in many cases we
do not need to take into account radiation, and the hot accretion flow is technically
much easier to simulate than a thin disk, we have a better understanding to the wind
launched from hot accretion flows.
4 Winds from Black Hole Accretion Flows … 155
4.2.1 Brief History of Study of Wind from Hot Accretion

Flows
One of the most important pioneer works in this aspect is Stone, Pringle and Begelman
[7] (see also [23, 24]). They performed the first global hydrodynamical numerical
simulation of hot accretion flow and calculated the following time-averaged radial
profiles of inflow and outflow rates,
π
Ṁin (r) = 2π r 2 ρ min(vr , 0) sin θ dθ , (4.1)
0 tφ
π
Ṁout (r) = 2π r 2
ρ max(vr , 0) sin θ dθ , (4.2)
0 tφ
where the angle brackets represent time averages (and also average over the azimuthal
angle φ in the case of 3D simulations). Note that the order of doing time-average
and the integral will make significant differences. Note also that the outflow rate
calculated by Eq. (4.2) does not necessarily represent the mass flux of “real outflow”,
because the positive radial velocity may just come from the turbulent motion of the
accretion flow. The most important result they obtained is that the inflow rate based
on Eq. (4.1) follows a power-law function of radius,
s
r
Ṁin (r) = Ṁin (rout ) . (4.3)
rout
Here Ṁin (rout ) is the mass inflow rate at the outer boundary rout . The dynamical range
of this simulation is not large, spanning less than two orders of magnitude in radius.
The results were later confirmed by simulations with a much larger radial dynamical
range of four orders of magnitude [56]. Moreover, MHD simulations yield very
similar results (e.g., [26, 39, 46]). In almost all cases, we typically have s ∼ 0.5 − 1
(see review in [56]).
The predicted inward decrease of accretion rate has soon been confirmed by two
observations, both are on Sgr A*. One is the detection of radio polarization at a level
of 2 − 9 % (e.g., [5, 12, 31]). Such high polarization requires that the mass accretion
rate close to the black hole horizon must be within a certain range, which is two
orders of magnitude lower than the Bondi rate obtained from Chandra observations
[43]. The other observational evidence is from the Chandra observation of the iron
emission lines originated from the hot accretion flow [52]. The modeling to the Kα
lines indicates a flat radial density profile around the Bondi radius, confirming that
the mass accretion rate decreases with decreasing radius. This is because, if the mass
accretion rate were a constant of radius, the density profile would be much steeper
(Fig. 4.1).
Now the question is, what is the reason of the inward decrease of mass accretion
rate? Two competing models have been proposed. In the adiabatic inflow–outflow
156 F. Yuan
Fig. 4.1 Radial profiles of

mass inflow rate Ṁin , mass
outflow rate Ṁout (Eqs. 4.1
and 4.2), and net mass
accretion rate Ṁnet defined as
their difference. Top Results
from a two-dimensional
Newtonian HD simulation of
a hot accretion flow (Taken
from Stone et al. [47]). Solid,
dashed, and dotted lines
correspond to Ṁin , Ṁout , and
Ṁnet , respectively. Bottom
Solid lines indicate
equivalent results from a
three-dimensional GRMHD
simulation of a hot accretion
flow around a non spinning
black hole (Taken from Yuan
and Narayan [58]). Dashed
lines indicate results for a
different kind of time
averaging, as described in
the text. The true mass
outflow rate is ∼60 % of the
solid green lines [59]
solution (ADIOS), the inward decrease of mass accretion rate is due to the mass
lost in the wind [7, 8, 11]. The other model is the convection-dominated accretion
flow (CDAF) model. In this model, the accretion flow is assumed to be convectively
unstable.1 The inward decrease of accretion rate is explained as more and more gas
is locked in convective eddies during accretion [3, 25, 35, 42]. For a long time, it is
unclear which scenario is physical.
To investigate this problem, numerical simulations have been performed [29, 36,
56]. Both Yuan, Bu and Wu [56] and Narayan et al. [36] found that in the presence
of magnetic field, the hot accretion flow is convectively stable. This indicates that
the CDAF model may not apply, leaving outflow/wind as the only possible solution.
The fundamental question is, how strong the wind is. Narayan et al. [36] calculated
the outflow rate based on Eq. (4.2), except that they move the tφ average inside
the integral. The advantage of this approach is that it eliminates contributions from
1 As we will introduce in the next paragraph, this assumption is likely true only when the magnetic
field is not included in the analysis of the stability.
turbulent motion. The disadvantage is that, as shown in Yuan et al. [59] and also
described in the present paper later, it also eliminates significant mass flux of real
outflow. Consequently, they found substantially lower outflow rate than Eq. (4.2). In
fact, only upper limit was reported in Narayan et al. [36] since the outflow rate was
found to not converge with time.
On the other hand, [56] obtained a different result and showed that the mass flux of
outflow should be large, i.e., being a significant fraction of that described by Eq. (4.2).
This conclusion is mainly based on the following argument. That is, if the main
contributor of Eq. (4.2) is turbulence, we would expect that the properties of inflow
and outflow, such as angular momentum and temperature, should be roughly same;
while they found that they are quite different. For an example, the specific angular
momentum of outflow is much higher than that of the inflow. The hydrodynamical
simulations by [29] also found strong outflow as Yuan, Bu and Wu [56].
To obtain the mass flux of wind, the crucial point is how to discriminate the
real wind and turbulent motion. Yuan et al. [59] finally solved this issue by using a
“trajectory” approach, combined with the data of three dimensional GRMHD numer-
ical simulation of accretion. Different from the streamline analysis often adopted in
accretion literature, this approach can provide the trajectory of each “virtual test
particle” in the accretion flow and thus directly show whether the flow is turbulent
outflow or real outflow. Using this approach, they found that the mass flux of wind is
quite strong. In fact, the mass flux of winds is ∼60 % of that calculated by Eq. (4.2)
(see Sect. 2.2 for details). The rather weak outflow rate obtained in Narayan et al.
[36] is because outflow is intrinsically instantaneous. The outflow stream can wander
around in 3D space thus will be cancelled if the time-average is performed first. Their
work indicates that the mass lost via the wind is the reason for the inward decrease
of the accretion rate (Eq. 4.3). In the following we summarize the main results they
have obtained.
4.2.2 Main Properties of Winds
The trajectories of some virtual test particles obtained in Yuan et al. [59] are shown in
Fig. 4.2. We can see that in the main body of the accretion flow around the equatorial
plane, it is inflow, and the motion is quite turbulent. Winds are evident in the coro-
nal region and their motion is much less turbulent. Below we summarize their main
properties, including the mass flux, poloidal speed, fluxes of energy and momentum.
Mass Flux
Figure 4.3 shows various mass flow rates as a function of radius. Black and blue solid
lines show the inflow and outflow rates calculated following Eqs. (4.1 and 4.2) while
the blue solid line shows their difference. The red and blue dashed lines show the
mass fluxes of wind and real inflow (i.e., excluding the turbulent inflow) at a certain
time of the simulation. The mass flux of wind can be described by
158 F. Yuan
Fig. 4.2 Lagrangian

trajectories of the “test
particles” originating from r
= 80rg (the black circle in
the figure) in the 2D (r − θ)
plane. Winds are evident in
the coronal region. The
inflow concentrates within
the main disk body around
the equatorial plane, and
their motion is turbulent.
Taken from Yuan et al. [59]
s
r
Ṁwind (r) = ṀBH , s ≈ 1, (4.4)
20rs
where ṀBH is the mass accretion rate at the black hole horizon and rs ≡ 2GM/c2
is the Schwarzschild radius. Comparing it with the outflow rate Ṁout calculated by
Eq. (4.2), we find that the mass flux of wind is Ṁwind ∼ 60 %Ṁout . This confirms
the conclusion obtained by Yuan, Bu and Wu [56]. Also shown in the figure are the
mass fluxes of the wind calculated following the Narayan et al. [36] method, which
is much weaker, equal to ṀBH only until 50rs .
How large can the value of r in Eq. (4.4) be? This is important since this will deter-
mine the total wind flux. [14, 15] studied this problem by using two dimensional
hydrodynamical and MHD numerical simulations. They focus on the large radius of
the accretion flow. At that region, in addition to the central black hole, the nuclear
star cluster also contributes to the gravitational potential thus needs to be taken into
account. The velocity dispersion of stars is assumed to be a constant and the grav-
itational potential of the nuclear star cluster φ ∝ σ 2 ln(r), where σ is the velocity
dispersion of stars and r is the distance from the center of the galaxy. It is found
that when the gravity is dominated by the nuclear star cluster, i.e., r > rA ≡ GM/σ 2 ,
Fig. 4.3 Various mass flow rates as a function of radius. Black and red solid lines show the total
inflow and outflow rates calculated following Eqs. (4.1–4.2), while the blue solid line denotes their
difference. The red and blue dashed lines denote the mass flux of the wind and real inflow calculated
at a certain time. The red dot-dashed line shows the mass flux of the disk jet. For comparison, the
red dotted line shows the mass flux of wind calculated following the method in Narayan et al. [36].
Taken from Yuan et al. [59]
wind launched from the accretion flow almost disappear. This result indicates that
there exists an upper limit of the value of r in Eq. (4.4), which is r < rA . Since σ is
close to the sound speed of the accreting gas around the Bondi radius, the value of
rA is close to the Bondi radius, which is the outer boundary of the accretion flow.
Poloidal Velocity of Wind and Disk Jet
We are mainly interested in the poloidal speed of wind. This is because, at large
radius, the poloidal speed of wind is the dominant component since the magnetic
field becomes subdominant. Figure 4.4 shows the angular distribution of the poloidal
speed of wind at radius r = 160rg . The results at the other radius are similar. We can
see that the poloidal speed as a function of θ has a sharp jump at θ ∼ 15◦ away from
the rotation axis. The poloidal speed of outflow close to the axis is >0.3c, much
larger than that away from the axis, which is <0.05c. The outflow within θ < 15◦
to the axis is thus naturally identified as the “disk jet”, while the outflow out of this
range is identified as wind. Note that the simulation data are for a nonrotating black
hole, so the presence of the disk jet is irrelevant to black hole spin, although the spin
of the black hole may strengthen the disk jet. The disk jet originates from the inner
region of the disk and is powered by the rotation energy of the accretion flow (see
more discussion below). This is different from the Blandford–Znajek jet originated
from the black hole horizon, which is powered by the spin energy of the black hole
[10]. Other main differences between the two types of jet are that the disk jet is
sub-relativistic and matter dominated, while the Blandford–Znajek jet is relativistic
and Poynting flux dominated (see [58] for a summary).
160 F. Yuan
Fig. 4.4 Poloidal speed (in

units of speed of light) of
wind as a function of θ at
r = 160rg and various φ.
The values of φ are denoted
by the color of the lines. We
can see that close to the axis,
the poloidal speed is much
larger than in other regions.
We identify this part as the
disk jet. Taken from Yuan et
al. [59]
The mass flux-weighted poloidal speed of wind at radius r is described by
vp,wind ≈ 0.2vk (r) (4.5)
On the one hand, the wind can be launched from any radius r. On the other hand, we
see in Fig. 4.3 that the wind mass flux increases rapidly with radius. Therefore, this
result primarily reflects wind launched close to radius r.
An important question is the evolution of the poloidal speed of wind along their
trajectories. For disk jet, their poloidal speed significantly increases; while for wind,
the poloidal speed roughly keeps constant or slightly increases along the trajectory.
If there were no acceleration during the propagation of wind, the poloidal speed
would decrease outward because of the gravity. Therefore, this result implies that the
winds are accelerated along the trajectory. The acceleration forces will be discussed
in Sect. 2.3. In summary, the terminal speed of wind originated from radius r can be
described by Yuan et al. [59] (see also [57])
vp,term ≈ (0.2 − 0.4)vk (r) (4.6)
The above behavior of poloidal velocity of wind implies that the Bernoulli parameter
of wind Be is not constant but increases. Note that only for strictly steady and invis-
cid flow Be is a constant along the trajectory, while a real accretion flow is always
turbulent.
Fluxes of Energy and Momentum
The fluxes of energy and momentum of wind and disk jet are calculated as follows,

1
Ėjet(wind) (r) = ρ(r, θ, φ)vp3 (r, θ, φ)r 2 sin(θ )dθ dφ, (4.7)
2
Fig. 4.5 The radial profile

of the energy fluxes of wind
(red solid) and disk jet (blue
dashed) for a non-rotating
black hole. Taken from [59]

Ṗjet(wind) (r) = ρ(r, θ, φ)vp2 (r, θ, φ)r 2 sin(θ )dθ dφ. (4.8)
The integration over θ for wind and the disk-jet is bounded by θ ≈ 15◦ according to
discussions above. Figure 4.5 shows the energy flux of wind and disk jet as a function
of radius. We see that the energy flux of the wind is >3 times stronger than that of the
disk jet, while the contrast in the momentum flux between wind and jet is even larger
[59]. This result is mainly because of the low density in the disk jet. Obviously, it
implies the importance of wind in comparison with disk jet in the context of AGN
feedback. Of course, this is for a non-rotating black hole. In the case of a spinning
black hole, the relative importance of a jet in comparison with winds will become
stronger ([44]2 ; Yuan et al. in preparation). For r > 40rg , we have
1 1
Ėwind (r) ≈ Ṁwind (r)vp,wind
2
(r) ≈ ṀBH c2 . (4.9)
2 1000
This result indicates that the energy flux at large radius is roughly saturated, consistent
with Fig. 4.5. The main reason energy flux saturates is s = 1 in Eq. (4.4).
The energy flux obtained in Eq. (4.9) is in good agreement with that required in
large scale AGN feedback simulations (e.g., [16]). In these works, AGN feedback
is involved to heat the inter-cluster medium to compensate for rapid cooling rate in
the systems (i.e., the cooling flow problem). It was found that to be consistent with
observations of both isolated galaxies and galaxy clusters, the required “mechan-
ical feedback efficiency”, defined as ε ≡ Ėwind /ṀBH c2 , must be in the range of
∼10−3 − 10−4 . The result shown in Eq. (4.9) provides a natural explanation for
this required value of ε.
In accretion systems with extremely low mass accretion rates such as Sgr A*, the
density of the accretion flow is very low thus the mean free path of particles may
2 In the calculations of many wind properties presented in Sadowski et al. [44] such as the mass flux
of wind, they do the time-average first to the velocity field. Since wind is instantaneous, their result
should be regarded as the lower limit.
162 F. Yuan
be large compared with the typical length-scale of the system. In this case, thermal
conduction will play an important role. Bu, Wu and Yuan [13] have studied the effect
of thermal conduction on the properties of winds. They find that the mass flux of
wind slightly increases due to the presence of thermal conduction, while the energy
flux of wind increases by a factor of ∼10 mainly because of the significant increase
of wind velocity.
4.2.3 Acceleration Mechanism of Wind and Disk Jet
What are the mechanisms to accelerate the wind against the gravity? This can be
studied by analyzing the forces in the wind and disk jet region [32, 59]. Figure 4.6
shows the results for three representative points in the disk jet, wind and main disk
regions. For the wind, the main driving forces are the centrifugal force and the gradi-
ent of magnetic and gas pressure. From the figure we notice that the gradient of the
magnetic pressure is “downward”, pointing toward the positive θ direction. This sur-
prising result reflects the strong fluctuation of the accretion flow. If we choose another
time or another location to do the force analysis, we very likely find that the gradi-
ent of the magnetic pressure becomes “upward”. The direction of the gas pressure
gradient also strongly fluctuates with time and location. Therefore the acceleration
is not a continuous process but stochastic [32, 59]. But statistically, the gradients of
both the gas and magnetic pressure are pointing along the positive r direction thus
are helpful to the acceleration of wind. Their magnitudes are also comparable to the
centrifugal force, as shown by Fig. 4.6. The centrifugal force is important because
the specific angular momentum of wind is large. For example, Yuan, Bu and Wu
[56] find that the specific angular momentum of outflow is significantly larger than
that of inflow. In that work, “outflow” includes both wind and turbulent outflow.
Figure 4.7 shows the radial profile of the angular momentum of wind. We can see
that the angular momentum of wind is actually super-Keplerian. Such a result is very
likely because of the angular momentum transport by the magnetic field, i.e., the
wind gas is somehow forced to co-rotate with the magnetic field line rooted in the
main body of accretion flow, as described in the model of Blandford and Payne [9].
In summary, the mechanism of wind acceleration is similar to the Blandford and
Payne [9] model in the sense that the magneto-centrifugal force play an important
role. But different from the Blandford and Payne [9] mechanism, here we don’t have
a large scale ordered poloidal field and the other forces such as the gradient of gas
pressure also play important roles.
For the disk jet, the dominant driving force is the gradient of the magnetic pressure.
This is consistent with the “magnetic tower” model proposed by Lynden–Bell [30].
Fig. 4.6 Force analysis at

three representative locations
corresponding to the disk jet,
wind and the main body of
the accretion disk. The
arrows indicate force
direction, while length
represents force magnitude.
Taken from Yuan et al. [59].
See also Moller and
Sadowski [32]
Fig. 4.7 The radial profiles

of flux-weighted angular 1.2
momentum (in unit of
flux weighted l l_k
Keplerian angular 1.1

momentum) of wind (solid
blue), turbulent outflow
1.0
(solid black), real inflow
(dashed blue), and turbulent
inflow (dashed blue). The 0.9
angular momentum of wind
is super-Keplerian 0.8
20 30 50 70 100
R/Rs
4.2.4 Why Do Winds Exist?
So far our discussion focuses only on whether strong wind exists in hot accretion
flows and what are their properties. But why does strong wind exist or why some
inflowing gas turn around and become outflow? In addition to above-mentioned
works, this question is also addressed in Begelman [11] and [21]. For example, [21]
argue that the energy equilibrium never can be reached in hot accretion flows unless
wind is present and take away some energy. Based on Sect. 2.3, it looks that the wind
is produced because of both “energy” and “momentum” reasons. The gradient of gas
pressure corresponds to “energy driven”. Angular momentum is transported from
one fluid element to another perhaps by magnetic field, which makes the centrifugal
force large thus the originally inflowing fluid is inclined to become into wind. The
magneto-centrifugal force corresponds to “momentum-driven”.
164 F. Yuan
4.3 Interaction of Winds with Interstellar Medium:

The Formation of the Fermi Bubbles
Using the Fermi-LAT, two giant gamma-ray Bubbles has been discovered, located
above and below the Galactic plane [48]. The bubbles extend to ∼50◦ above and
below the Galactic plane, and the width is ∼40◦ in longitude. Several models have
been proposed to explain the formation of the Fermi bubbles. They often invoke the
interaction between a jet (e.g., [22]) or wind and the interstellar medium (see [34] for
a brief review). In different models, while the jet comes from the accretion flow, the
wind can either come from the cold or hot accretion flows around the black hole [33,
34, 60, 61], or star formation [17]. In both the jet and accretion wind models, they
have to assume that the luminosity of Sgr A* was several orders of magnitude higher
in the last. There seems to be many observational evidences for it (see review in
[50]). In the jet model, they must assume that the jet is perpendicular to the Galactic
plane in order to explain the morphology of the bubbles which are perpendicular to
the Galactic plane. As pointed out by Zubovas et al. [61], this is a strong assumption
since statistical studies to other galaxies indicate that this is usually not the case. In
addition, the mass lost rate in the jet is also assumed to be very large, close to or even
larger than Eddington.
Mou et al. [33, 34] propose that the Fermi bubbles are produced by the interaction
between the wind launched from the hot accretion flow around Sgr A* and the
interstellar medium. The mass accretion rate of the accretion flow is determined
by other independent observations [50], which is ∼10−2 ṀEdd , thus is not a free
parameter. This rate is still well in the regime of the hot accretion flow [58] thus strong
winds are expected, as described in the previous sections. The properties of wind
such as its mass flux, terminal velocity, and angular distribution are all determined
by the small-scale MHD numerical studies [59] thus are again not free parameters.
In addition to thermal particles, Mou et al. [34] assume that relativistic electrons and
protons also exist in the wind. These particles are likely produced by, e.g., magnetic
reconnection process occurred in the coronal region of the accretion flow (e.g., [55]).
These relativistic protons (also called cosmic ray protons; CRp) will collide with
the thermal protons in the ISM and produce neutral pions; these poins will decay
and produce gamma-ray photons of the Fermi bubbles. So in order to obtain the
Gamma-ray radiation and compare it with observations, the main task is to calculate
the spatial distribution of CRp. This is achieved in Mou et al. [34] by treating CRp
as another kind of fluid and solving the time-dependent two-dimensional two-fluid
equations using the simulation code ZEUS.
Figure 4.8 shows the simulated morphology of the bubble. The left and right plots
show the number density and temperature of the thermal gas, respectively. The purple
line in the figure denotes the contact discontinuity (CD) between the wind and the
ISM. Near the CD, the CRp pressure is comparable to the thermal pressure of the
shocked ISM, so it expels some thermal gas away from the CD, leaving a zone with
density somewhat lower than the “typical” density of the shocked ISM. We call it a
“permeated zone”. The CRp and thermal protons are well mixed in this zone thus
Fig. 4.8 X−Z sectional views of the simulation results. Coordinates are in units of kpc. Left number
density distribution of thermal electrons (ne ) in units of cm−3 . Velocity vectors are also plotted, with
the color bar at the top of the plot denoting the value of velocity in units of km s−1 . Right temperature
in units of Kelvin. The dashed lines in these three maps denotes the contact discontinuity (CD).
Taken from [34]
p−p collision is frequent. Therefore, most of the gamma-ray radiation is produced

from this zone. We can see that the shape of this zone is roughly consistent with the
observed shape of the Bubbles. The central molecular zone located in the galactic
center plays an important role in collimating the wind and the formation of the bubble
morphology. One caveat exists in their model. That is, although a jet should also exist
for accretion rate of 10−2 ṀEdd , they assume that the interaction of the jet and ISM
can be neglected since the jet may just pierce through the ISM in a narrow channel
without depositing much energy in the ISM. This assumption seems to be reasonable
as shown by the numerical simulation of Vernaleo and Reynolds [51].
The magnetic field in the shocked ISM is roughly parallel to the CD. However, the
alignment is not perfect, which allows some CRp diffuse into the shocked ISM region
and form the “permeated zone”. But under such a kind of magnetic field configuration,
CRs cannot diffuse too far away from the CD. Therefore the morphology of the
bubbles is determined by the CD, and this is also the reason why the edges of the
Fermi bubbles look sharp.
The processes considered in Mou et al. [34] for the production of gamma-ray
emission include: (1) the production of neutral pions by the collision between thermal
protons and CRp, which will further decay and produce gamma-ray; (2) the p − p
collision will also produce some charged pions which will generate second-order
leptons. These leptons will scatter with the seed photons and produce gamma rays;
(3) In the processes of CRp production such as magnetic reconnection in the disk
corona, some CRe will also be produced. These electrons can also produce gamma-
ray photons by scattering with seed photons.
Figure 4.9 shows the calculated spectral energy distribution, together with the
observational data. The model can fit the data well. In addition to the spectrum, the
166 F. Yuan
Fig. 4.9 The gamma-ray

spectral energy distribution
calculated based on run A.
The rectangles with error
bars show the latest
observational results of
Ackermann et al. [4]. The
solid line is the sum of the
dashed (for the π 0 decays),
dotted (for IC process of the
secondary leptons generated
in hadronic reaction), and
dot-dashed (for IC of the
primary electrons) lines.
Taken from Mou et al. [34]
model can also successfully explain the other main observational features of the Fermi
bubbles: (1) limb-brightened surface brightness; (2) the width of the boundary; (3)
the temperature of the shocked ISM just outside of the CD [49]; (4) the bulk motion
velocity of the gas at the edge of the Fermi bubbles [19].
4.4 Summary
Numerical simulations have shown that the mass accretion rate decreases inward
toward the black hole (Fig. 4.1). This theoretical result has been confirmed by obser-
vations. To understand this result, two models have been proposed, namely convec-
tion and outflow (or wind). Recent theoretical works have shown that it is the mass
lost in the wind that results in the inward decrease of accretion rate (Fig. 4.2). Based
on the MHD simulation data, we can follow the trajectories of the virtual particles
thus precisely measure the various properties of wind, such as the mass flux (Fig. 4.3
and Eq. 4.4), poloidal speed (Fig. 4.4 and Eqs. 4.5 and 4.6.), the fluxes of energy and
momentum (Fig. 4.5 and Eqs. 4.7 and 4.8). It is found that The Bernoulli parameter
is usually not a constant and the poloidal speed of wind roughly keeps constant or
increases along their trajectories. The acceleration forces are the centrifugal force,
and the gradient of the gas and magnetic pressure (Fig. 4.6). It is therefore a combi-
nation of the magneto-centrifugal mechanism similar to that proposed by Blandford
and Payne [9] and the thermal mechanism.
One interesting result is that the angular distribution of the poloidal speed of
wind has a sharp jump at ∼θ = 15◦ . Close to the axis, the speed is as high as 0.3c
while beyond this region the speed is about ten time lower. The former is therefore
identified as the “disk jet”. The comparison between this kind of jet and the Blandford
and Znajek jet and more importantly, which kmind of jet is physically associated with
the observed “real” jet is interesting topics to be studied in the future.
The wind from hot accretion flow is difficult to be detected by the usual absorption
line approach, because the temperature of wind gas is too high thus fully ionized. But
the interaction of such wind with the interstellar medium may have some observa-
tional consequence and this process may result in the formation of the Fermi bubbles
observed in the Galactic center by the Fermi telescope. This idea has been worked
out in Mou et al. [33, 34] and the model has successfully explained the key observa-
tions such as the morphology and Gamma ray spectrum. The advantage of this model
compared with the other models of the Fermi bubbles is that the key parameters of the
model, i.e., the mass accretion rate in the accretion flow and the properties of wind
are not free parameters. The former is adopted from the other independent observa-
tional constrains, while the properties of wind come from the MHD simulation of
hot accretion flow as described in this review.
Acknowledgments This project was supported in part by the National Basic Research Program of
China (973 Program, grant 2014CB845800), the Strategic Priority Research Program The Emer-
gence of Cosmological Structures of CAS (grant XDB09000000), the Natural Science Foundation
of China (grants 11133005 and 11573051), and the CAS/SAFEA International Partnership Program
for Creative Research Teams.
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Chapter 5
A Brief Review of Relativistic Gravitational
Collapse
Daniele Malafarina
Abstract We review here the basic setup to describe complete gravitational collapse
of massive bodies within the general theory of relativity. We derive Einstein’s equa-
tions describing collapse and solve them in some simple well-known toy models. We
study the final outcome of collapse and the quantities that describe the formation of
trapped surfaces and of the central singularity.
5.1 Introduction
General relativity became an essential part of the curricula of astrophysicists nearly

50 years after it was first developed by Albert Einstein, when new ultra-dense objects
such as pulsars and highly energetic phenomena such as quasars were discovered.
By 1963, it was clear that general relativity was necessary to understand those phe-
nomena and that gravitational collapse played a crucial role for many astrophysical
scenarios. Nevertheless, the seeds of our modern understanding of stellar collapse
have deeper roots. It was Chandrasekhar, back in 1931, who used special relativity
to evaluate the pressure necessary to overcome the electron degeneracy in a star and
derived the famous upper mass limit for a stable white dwarf. He wrote: “...the life
history of a star of small mass must be essentially different from the life history of
a star of large mass. For a star of small mass the natural white dwarf stage is an
initial step towards complete extinction. A star of large mass cannot pass into the
white dwarf stage, and one is left speculating on other possibilities” [8]. The “other
possibilities” to which Chandrasekhar referred today are called neutron stars and
black holes.
D. Malafarina (B)
Department of Physics and Center for Field Theory and Particle Physics,
Fudan University, 220 Handan Road, Shanghai 200433, China
e-mail: daniele.malafarina@nu.edu.kz
D. Malafarina
Physics Department, SST, Nazarbayev University, 53 Kabanbay Batyr Avenue,
Astana 010000, Kazakhstan

170 D. Malafarina
In 1939, Oppenheimer and Volkov performed a similar calculation using neutrons

instead of electrons and general relativity instead of special relativity [39]. They con-
cluded that any body with a large enough mass would not be able to sustain its own
gravity and undergo complete collapse, although at the time it was not clear what the
final state would be. They wrote: “...actual stellar matter after the exhaustion of ther-
monuclear sources of energy will, if massive enough, contract indefinitely, although
more and more slowly, never reaching equilibrium.” Soon after, Oppenheimer and
Snyder and independently Datt developed the first exact solution of Einstein’s equa-
tions describing a collapsing spherical cloud of non-interacting particles [11, 38].
The end state of such collapse model is a Schwarzschild black hole.
The existence of black holes as astrophysical objects was just a conjecture fifty
years ago, while today is almost unanimously accepted by the community of astro-
physicists. Nevertheless, the theoretical paradigm upon which the whole theory of
black hole formation relies is not much different from the original Oppenheimer–
Snyder–Datt (OSD) collapse model. And while our physical knowledge of astro-
physical phenomena has progressed enormously in the last fifty years, our theoret-
ical understanding of how black holes form is still very much rooted in simple toy
models such as OSD. The reason for this relies mostly in the immense difficulty
that one encounters when trying to solve analytically Einstein’s equation in more
general and physically relevant cases. Also, the fact that we do not know much about
the behavior of matter in the strong field regime contributes in making our present
theoretical understanding very limited.
On the other hand, from an experimental perspective, new missions and obser-
vatories are due to come online in the near future, and there is great hope that they
will produce, among other things, an enormous amount of data on gravitational col-
lapse and black hole formation. Astrophysicists of tomorrow will be able to rely on
photons, neutrinos, and gravitational waves in order to study and understand what
happens at the end of the life cycle of a star. This is usually called multimessen-
ger astronomy. One of the key questions they will have to address is whether white
dwarves, neutron stars, and black holes are the only possible objects that are left
after a star dies. At present, it is natural to ask whether it is possible that there exists
some yet unknown state of matter beyond the neutron degeneracy limit and capable
of producing stable ultra-dense remnants. The purpose of this chapter is to pave the
way for astrophysicists toward a broad theoretical understanding of the processes
that lead to the formation of black holes. We do so by reviewing the paradigm for
gravitational collapse in general relativity (GR) and the most fundamental analytical
results that were obtained in the field.
The chapter is structured as follows: In Sect. 5.2, we derive the set of differential
equations that are used to describe collapse. In Sect. 5.3, we discuss how the collaps-
ing “star” can be matched to a vacuum exterior. Section 5.4 is devoted to the discus-
sion of the conditions for the model to be physically viable. In Sect. 5.5, the apparent
horizon and the singularity curve are defined. Section 5.6 presents the simplest solu-
tion for the homogenous dust collapse model, while in Sect. 5.7 inhomogeneous dust
and homogeneous perfect fluid models are briefly outlined. Section 5.8 is devoted to
5 A Brief Review of Relativistic Gravitational Collapse 171
discussing how the mathematical models can be useful for astrophysics, and finally,
in Sect. 5.9, some possible future directions of investigation are discussed.
5.2 Einstein’s Equations for the Collapsing Interior
Stars are supported in equilibrium by the balance of the gravitational attraction that
pulls inward and the push outward generated by nuclear reactions happening at their
center. When a star exhausts the nuclear fuel that was keeping it stable, it implodes
under its own weight. At this point, the future evolution of the star depends on its
mass. If a star is sufficiently massive, then there is no known force in nature capable of
halting collapse. These stars end their lives forming black holes. In order to be able to
describe the final stages of collapse, where the gravitational field becomes extremely
large, we must use general relativity. Therefore, it is useful to begin our discussion
by understanding what is a black hole in general relativity. The simplest and most
intuitive definition of a black hole is that of a space-time singularity surrounded by
an event horizon. Clearly, there are two elements that are crucial to our definition
of a black hole: the singularity and the event horizon. The event horizon acts like
a two-dimensional one-way membrane that lets particles and light enter while not
letting anything exit. The singularity, strictly speaking, is not a part of the space-
time, and it is the boundary that marks the geodesic incompleteness of all paths
for particles that enter the horizon. The horizon for a non-rotating Schwarzschild
black hole is located at a radius RSch = 2G MS /c2 , where MS is the black hole’s
mass, G is the Newton’s constant, and c is the speed of light. The singularity is
ideally “located” at the center of symmetry of the system. Intuitively, we can see that
the black hole forms once enough mass is concentrated within a sphere of a small
enough radius. Once matter is trapped inside the horizon, it can only fall toward the
singularity (if there is no rotation). In principle, the equations of general relativity
can be very difficult to solve; therefore, in order to describe the process by which all
the matter in the star falls within the horizon radius thus forming a black hole, we
need a mathematical framework that is simple enough to solve the equations but that
still retains the most important physical features. In the following, we will neglect all
the physical processes that happen in the cloud except gravity, we will assume that
the cloud is perfectly spherically symmetric and not rotating, and we will assume
that the exterior of the cloud is vacuum. Also, we shall consider here only extremely
simplified fluid models to describe the state of matter of the collapsing star. Finally,
it is custom to make use of natural units, thus setting G = c = 1.
5.2.1 Co-moving Coordinates
Our aim is to solve the system of Einstein’s equations for a spherical collapsing
matter cloud. As it is well known, Einstein’s equations possess two distinct parts that
172 D. Malafarina
both require some assumptions in order to allow us to find physically meaningful

solutions. On the left-hand side, we have the geometrical part of the set of equations
that is given by the Einstein tensor. This is determined once we know the metric
for the space-time. As said, we will consider here only spherically symmetric, non-
rotating space-times. A space-time is said to be spherically symmetric if the metric
remains invariant under the group of spatial rotations S O(3). This means that we
can define the two-dimensional metric induced on the unit two-sphere as
dΩ 2 = dθ 2 + sin2 θ dφ 2 (5.1)
and define a function R for which 4π R 2 represents the area of each two-sphere in
the space-time. The full four-dimensional metric then can be written as
ds 2 = g AB d x A d x B + R(x A , x B )2 dΩ 2 , (5.2)
with A, B = 0, 1. We can then introduce the coordinates t = x0 and r = x1 that

diagonalize the two-dimensional part of the metric g AB and write the most general
spherically symmetric line element in the simple form
ds 2 = −e2λ dt 2 + e2ψ dr 2 + R 2 dΩ 2 (5.3)
where the functions λ, ψ, and R do not depend on the coordinates θ and φ. In the
following, we will use a dot to express derivatives with respect to t and a prime to
express derivatives with respect to r , thus writing Ẋ = d X/dt, X = d X/dr . The
above coordinate system for which the metric is diagonal is called co-moving because
one can think of the labels t and r as “attached” to each collapsing particle. Then,
the functions λ, ψ, and R depend only on r and t. In this reference frame, the fluid
is instantaneously at rest and its four-velocity u μ is u t = e−λ , u r = u θ = u φ = 0.
In order to describe the collapse of a spherically symmetric massive object such
as a star, we need to solve Einstein’s equations for a space-time described by (5.3)
coupled to an energy momentum tensor describing a realistic fluid source. The energy
momentum tensor is the right-hand side of Einstein’s equations and for a fluid source
in the co-moving frame can be written as
⎛ ⎞
ρ 0 0 0
⎜0 pr 0 0 ⎟
T μν =⎜
⎝0
⎟
0 pθ 0 ⎠
0 0 0 pθ
A perfect fluid is an idealized fluid where no shear stresses, no viscosity, and no

heat conduction are present. It can be characterized by its mass density and isotropic
pressures alone. Isotropic pressure means that the radial pressure equals the tangential
pressure, and this implies pr = pθ = p. The energy density is the energy per unit
volume of the fluid in the local rest frame with four-velocity u μ and the energy
momentum tensor for a perfect fluid can then be written as
T μν = (ρ + p)u μ u ν + pg μν . (5.4)
Note that in general relativity, the pressure contributes to the gravitational field, and
therefore, the total gravitational energy need not be conserved. Nevertheless, the
baryon number is conserved. For a gas of non-interacting particles, the so-called
dust, the pressure vanishes and we can set p = 0. This is the simplest fluid model
that can be considered.
5.2.2 Misner–Sharp Mass
The metric (5.3) can be used to describe static sources in equilibrium in the case
when λ, ψ, and R do not depend on t. These are static objects with non-vanishing
energy momentum. In this case, the area radius R can be used as a radial coordinate
setting R = r . The simplest interior solution of this kind is given by the constant
density Schwarzschild interior and was found by Schwarzschild himself together
with the more famous vacuum solution (see, e.g., [47]). For the constant density
interior, one sets ρ = const. and solves Einstein’s equations that take the form of the
famous Tolman–Oppenheimer–Volkov equation [44], to find p(r ). Then, the object’s
boundary rb is determined by the condition that p(rb ) = 0. For a metric describing
a static interior case, we can define a function m(r ) such that

2ψ(r ) 2m(r ) −1
grr = e = 1− . (5.5)
r
It is easy to see that at the boundary of the static object, the function m(r ) must
become equal to the Schwarzschild parameter MS that describes the total mass of the
star, and therefore, we can interpret m(r ) as describing the amount of matter enclosed
within the radius r . We can generalize the above expression in the dynamical case
by introducing a function U (r, t) as
d R(r, t)
U = uμ = e−λ Ṙ, (5.6)
dxμ
then, we get

2m(r, t) −1 2
grr = e 2ψ(r,t)
= 1+U −
2
R . (5.7)
R
which reduces to the static case for R = r , so that R = 1 and Ṙ = 0. The Misner–
Sharp mass F(r, t) is then defined from 1 − F/R = gμν ∇ μ R∇ ν R and it is given by
F(r, t) = 2m(r, t) = R(1 − e−2ψ R 2 + e−2λ Ṙ 2 ). (5.8)

174 D. Malafarina
In analogy with what was said before, we can interpret the Misner–Sharp mass as
describing the amount of matter enclosed within the radius r at the time t [35].
5.2.3 Einstein’s Equations
Einstein’s equations couple the space-time geometry given by the metric gμν appear-
ing in the Einstein’s tensor G μν for the line element (5.3) to the matter content of the
collapsing cloud given by the energy momentum tensor (5.4). Einstein’s equations
take the usual form
1
G μν = Rμν − gμν R = Tμν , (5.9)
2
where Rμν and R are the Ricci tensor and Ricci scalar and where we have absorbed
the constant factor 8π k into the definition of Tμν . Then, Einstein’s tensor for the
collapsing system is given by
F 2 Ṙe−2λ

G tt = − 2
+
Ṙ − Ṙλ − ψ̇ R , (5.10)
R R RR
−2ψ

Ḟ 2R e
G rr = − 2 − Ṙ − Ṙλ − ψ̇ R , (5.11)
R Ṙ R Ṙ
2e−2λ

G r = −e
t
Gt =
2ψ−2λ r
Ṙ − Ṙλ − ψ̇ R , (5.12)
R
φ e−2ψ

G θθ = G φ = (λ + λ2 − λ ψ )R + R + R λ − R ψ +
R
e−2λ

− (ψ̈ + ψ̇ 2 − λ̇ψ̇)R + R̈ + Ṙ ψ̇ − Ṙ λ̇ . (5.13)
R
These equations need to be supplemented with one more equation coming from the
conservation of energy momentum that in general relativity comes as a consequence
of the fact that the connection is metric and which can be written as
∇μ Tνμ = 0. (5.14)
Then, in the simple case of pressureless (i.e., dust) collapse, and by making use of
the definition of the Misner–Sharp mass given in Eq. (5.8), the first two equations of
the above system simplify to
F
ρ = −G tt = , (5.15)
R2 R
Ḟ
p = 0 = G rr = − 2 , (5.16)
R Ṙ
the third and fourth combine to give
Ṙ = Ṙλ + ψ̇ R = 0, (5.17)
and the conservation of energy momentum (5.14) becomes
ρλ = 0. (5.18)
From Eq. (5.16), we see that for dust, we must have Ḟ = 0 which implies F =
F(r ). This shows that the amount of matter enclosed within the co-moving radius
r does not change with time. In other words, during collapse, there is no inflow or
outflow of matter across any co-moving shell r . From Eq. (5.18), since the energy
density is nonzero, we see that we must have λ = 0, which implies λ = λ(t). Now,
we can define a new co-moving time coordinate t˜ by rescaling in such a way that
d t˜
= eλ , (5.19)
dt
and therefore obtain
2
dt
d t˜ = −d t˜ .
2 2
− e dt = −e
2λ 2 2λ
(5.20)
d t˜
This means that there is always the gauge freedom to fix the co-moving time t such
that λ = 0, and in the following, we shall take t as such a gauge. Finally, Eq. (5.17)
can be written as
Ṙ
= ψ̇, (5.21)
R
from which we get
R = e g(r )+ψ . (5.22)
We call f (r ) = e2g(r ) − 1 and the Misner–Sharp mass equation (5.8) can be rewritten
in the form of the equation of motion of the system as
F(r )
Ṙ 2 = + f (r ). (5.23)
R
Once Eq. (5.23) is solved to give R(r, t), the whole system of Einstein’s equations is
solved. The metric becomes
R 2
ds 2 = −dt 2 + dr 2 + R 2 dΩ 2 . (5.24)
1+ f
176 D. Malafarina
This is the well-known Lemaitre–Tolman–Bondi (LTB) space-time [6, 33, 43]. We

see that the whole problem allows for two free functions of r , namely F and f , to be
specified at will. As said, the function F can be thought of as representing the matter
profile within the radius r , while from the above line element, the function f can
be thought of as an energy profile describing the spatial curvature of the space-time.
Then, provided that F and f are sufficiently regular, a unique solution of the equation
of motion (5.23) exists for each regular initial condition Ri = R(r, ti ).
5.3 Matching with an Exterior Metric
The metric given in Eq. (5.24) describes the dynamical collapse of a dust sphere.
This can be thought of as describing the final stages of the life of a star, provided that
gravity prevails on all other forces (thus allowing us to neglect any effect coming
from the microphysics of the collapsing fluid) and that the collapsing cloud has a
boundary. In the co-moving frame, this boundary can be identified with the surface
given by the co-moving radius r = rb . We consider the exterior of the collapsing
dust cloud to be static and vacuum. Then, Birkhoff’s theorem implies that it must
be a portion of the Schwarzschild space-time. The exterior Schwarzschild solution
can readily be derived from the metric (5.3) by including the further assumptions
of staticity and vanishing of energy momentum tensor. A space-time is said to be
stationary if it possess a time-like Killing vector, ∂t . In such a case, the metric (5.3)
becomes invariant under translations in t, and this is reflected in the metric functions
λ, ψ, and R that do not depend on t. Further, a space-time is said to be static if the
time-like Killing vector is orthogonal to the hypersurfaces of constant t. If we further
impose that the energy momentum tensor is that of vacuum, namely Tμν = 0, we
find that the only static spherically symmetric vacuum solution of Einstein’s field
equations can be written in the form

2MS 2MS −1 2
ds = − 1 −
2
dts2 + 1− drs + rs2 dΩ 2 . (5.25)
rs rs
which is the well-known Schwarzschild line element expressed in Schwarzschild

coordinates {ts , rs , θ, φ}. As said, the Schwarzschild metric has a singularity at the
center rs = 0. One way to determine the presence of singularities in solutions of Ein-
stein’s field equations is by inspecting curvature invariants looking for divergences.
The Kretschmann scalar is one of these invariants and it is defined starting from the
Riemann tensor as K = Rμνσ δ Rμνσ δ . The Kretschmann scalar for Schwarzschild is
4MS 2
K = 12 , (5.26)
rs6
from which we see that the null surface rs = 2MS is not singular; in fact, it is the
event horizon [15], and the singularity is located at rs = 0.
The global solution for the collapsing star is obtained by matching the collapsing
interior given by the metric (5.24) to the vacuum exterior across a shrinking bound-
ary surface Σ. Mathematically, “matching” means that the induced metric on the
boundary hypersurface Σ must be the same on both sides. Also, the rate of change
of the unit normal to Σ must be the same on both sides [23]. Let us label the interior
metric with (−) and the exterior metric with (+). Then, the two line elements can be
written as
±
ds±2 = gμν d x±μ d x±ν . (5.27)
The boundary hypersurface is implicitly defined on each side by Φ ± (x±μ (y a )) = 0,

and the induced metric on Σ can be written as
dsΣ2 = γab dy a dy b , (5.28)

μ
where a = 1, 2, 3. Now define the three basis 4-vectors tangent to Σ as e(a) =
± ± μ ν
∂ x μ /∂ y a . Then, the condition that the induced metric γab = gμν e(a) e(b) agrees on
+ −
both sides is simply γab = γab . Define the unit normal to Σ as

∂Φ ∂Φ −1/2 ∂Φ
n μ = g ρσ ρ σ . (5.29)
∂x ∂x ∂xμ
The extrinsic curvature (or second fundamental form) is defined as
K ab = gμν n μ ∇a e(b)
ν
. (5.30)
Then,
μ
± ∂ x± ∂ x±ν
K ab = ∇μ n ν , (5.31)
∂ ya ∂ yb
and continuity of K across Σ is given by

+ −
K ab = K ab . (5.32)
In the case of spherical collapse of dust, we have that the boundary hypersurface, in
the exterior, with coordinates {x + } = {ts , rs , θ, φ}, is given by
Φ + = rs − Rb (ts ) = 0, (5.33)
and in the interior, with coordinates {x − } = {t, r, θ, φ}, is given by
Φ − = r − rb = 0. (5.34)
178 D. Malafarina
So that the induced metric on the boundary is
dsΣ2 = −dt 2 + Rb (t)2 dΩ 2 . (5.35)
The Schwarzschild time ts can be written as a function of the co-moving time t from

dt 2MS 2MS −1 d Rb 2
= 1− − 1− , (5.36)
dts Rb Rb dts
and the matching conditions for the continuity of the metric become
Rb (ts ) = R(rb , ts (t)), (5.37)

F(rb ) = 2MS . (5.38)
Finally, continuity of K ab follows identically from the matching conditions. There-

fore, we see that the Misner–Sharp mass can be interpreted as the mass enclosed
within the co-moving radius r and that on the boundary, it becomes proportional to
the Schwarzschild mass MS . Also, we see that the area function R(r, t) in the interior
at the boundary becomes the shrinking area radius in the Schwarzschild portion of
the space-time. For more general collapse model, a matching to a suitable exterior
space-time can also be defined (see, e.g., [13, 14, 27]).
5.4 Regularity, Scaling, and Energy Conditions
In order for the model to be physically acceptable, we need to choose an initial

configuration that satisfies several conditions. The most important ones are regularity,
which corresponds to requiring that the initial matter profiles do not present any
singularities and are well behaved and the usual energy conditions, which in the dust
case can be expressed via positivity of the energy density. Another requirement that
is often imposed on the model is the absence of shell crossing singularities. These are
caustics like singularities that are due to the overlap of infalling shells. Shell crossing
singularities can possibly be removed by a suitable redefinition of the coordinates
and generally do not represent a breakdown of the model.
5.4.1 Regularity and Scaling
We now investigate regularity of the matter profiles and the condition for avoidance
of singularities at the initial time. In order to study these properties, we first need to
express the area radius R, the Misner–Sharp F mass, and the velocity profile f in an
appropriate gauge. As mentioned before, the Kretschmann scalar constitutes a valid
tool to investigate the occurrence of singularities. In the case of the metric (5.24),
this becomes
F 2 F F F2
K = 12 4 2
− 32 5 + 48 6 . (5.39)
R R R R R
We note here that with the present choice of the metric functions, one may be induced
to think that the central curve R = 0 is always singular, including at the initial time.
Nevertheless, this is not a physical singularity, as can be easily verified by evaluating
the energy density, which turns out to be finite at the initial time. We notice then
that there is a gauge degree of freedom in the scaling of R, namely in the way R is
“measured” at the initial time that can be used to remove the above ambiguity. In
fact, we can always choose arbitrarily the initial value of R. In the following, we
choose this initial scaling condition as
R(r, ti ) = r. (5.40)
From the choice of the initial data for R given in Eq. (5.40), we see that the gauge
freedom allows us to define a scaling function a(r, t) from the area function R(r, t)
as
R = ra(r, t). (5.41)
Now, the scaling factor a is an a-dimensional quantity such that

• at the initial time, we have a(r, ti ) = 1,
• at the time of formation of the singularity tsing , we have a(r, tsing ) = 0,
• collapse is given by ȧ < 0.
For dust, using Einstein’s equation (5.15), the above scaling implies that the initial
density must satisfy the following condition:
F
ρ(r, ti ) = ρi (r ) = > 0. (5.42)
r2
Therefore, in order to avoid having ρ diverging at r = 0 at the initial time, we must
impose a regularity condition on the Misner–Sharp mass. This is given by
F(r ) = r 3 M(r ), (5.43)
with M(r ) non-diverging and sufficiently regular in the interval [0, rb ]. Generally, we
assume that the function M(r ) can be written as a polynomial expansion in the vicinity
of r = 0. In general, a physically viable density profile should be non-increasing
radially outwards and therefore we must impose that the first non-vanishing term in
the polynomial expansion of M is vanishing or negative in r = 0. It is reasonable to
suppose that M ≤ 0 near the center. With the above scaling, the density becomes
180 D. Malafarina
3M + r M
ρ= . (5.44)
a 2 (a + ra )
If we add the further requirement that ρ must not present any cusps at r = 0, we
must impose that M (r ) vanishes in r = 0, and therefore, we must require M ≤ 0
near r = 0. Note that in the simplest case of homogeneous dust, ρ does not depend
on r and so M(r ) must be constant M0 . Then,
3M0
ρ(t) = , (5.45)
a3
and as a consequence, the scale factor also does not depend on r . With this choice
of the scaling factor, it is easy to verify that the central density diverges only at
the singularity. Also, we see that the Kretschmann scalar in the homogeneous case
reduces to
M02
K = 60 , (5.46)
a6
and it is regular at the initial time, its value being Ki = 60M02 . In general, for inho-
mogeneous dust, we have
(3M + r M )2 M(3M + r M ) M2
K = 12 − 32 + 48 . (5.47)
a 4 (a + ra )2 a 5 (a + ra ) a6
Note that by writing K in terms of M and a, we avoid the problem of divergence

along the central line. In the new scaling along r = 0, we see that K diverges only for
a = 0, thus showing the occurrence of the singularity. The curve tsing (r ) for which
a(r, tsing ) = 0 is the singularity curve which describes the time at which the shell
r becomes singular. As a consequence of the fact that in the homogeneous case, a
depends only on t, we see that for homogeneous dust, the singularity occurs at the
same time tsing for every co-moving shell r .
From the equation of motion (5.23), we see that at the initial time, the velocity of
the infalling particles is given by

F
Ṙi = − + f. (5.48)
r
Given the fact that the choice of the free function F corresponds to fixing the initial
density profile from the above equation, we see that fixing f corresponds to deter-
mining the initial velocity profile for the particles in the cloud. Now, by making use
of the scaling above, we can rewrite Eq. (5.48) as

f
ȧi = − M + , (5.49)
r2
from which we see that in order to have a finite initial velocity at all radii, we must
set a scaling for f as well. We shall take
f (r ) = r 2 b(r ), (5.50)
with b(r ) a sufficiently regular function (again which can be given as a polynomial
expansion near r = 0). To summarize, at the initial time, we have the freedom to
specify three functions of r as follows:
• Choose an initial condition for the scaling R(r, ti ) = Ri (r ) or equivalently set the
value of a(r, ti ).
• Choose a mass function F(r ), or equivalently M(r ), which implies an initial
density ρi = F /r 2 .
• Choose a velocity function f (r ), or equivalently b(r ), which implies the initial
condition for the velocity Ṙ(r, ti ).
Then, the system is fully determined and the equation of motion can be written as

M(r )
ȧ(r, t) = − + b(r ). (5.51)
a(r, t)
By solving the above equation for a, we completely solve the system of Einstein’s
equations. As said, homogeneous dust collapse is given by ρ = ρ(t) = 3M0 /a 3 .
Therefore, from the above, it follows that homogeneous dust collapse can be obtained
from the following requirements:
• a = a(t)
• M(r ) = M0 = const.
• b(r ) = k = const.
In this case, we can give a precise interpretation of the velocity profile if we imagine
a dust cloud that extends to infinity. We can think at the constant k as representing the
initial velocity of particles at spatial infinity, and we can characterize the geometry
of the space-time based on the sign of k in the following way:
• k = 0 marginally bound collapse, corresponding to a flat geometry. Shells at radial
infinity begin collapse with zero initial velocity.
• k > 0 unbound collapse, corresponding to a hyperbolic geometry. Shells at radial
infinity have positive initial velocity.
• k < 0 bound collapse, corresponding to an elliptic geometry. Shells at radial infin-
ity have negative initial velocity.
Note that if one wishes to have zero initial velocity Ṙi = 0 for particles in the col-
lapsing cloud with boundary, then the only possible choice is that of bound collapse
with M0 = −k.
182 D. Malafarina
5.4.2 Energy Conditions
Einstein’s equations are often regarded as made of two different parts. The “golden”
half, more elegant, is the left-hand side that contains the Einstein tensor and therefore
encodes the information about the geometry of the space-time. The “wooden” half
is the right-hand side that contains the energy momentum tensor and in principle
should describe the physical properties of matter. As a matter of fact, it is generally
practically impossible to fully describe all the properties of the matter fields in the
energy momentum tensor, and therefore, one usually resorts to simplifications and
averaged properties that are valid for macroscopic fields. Nevertheless, we must keep
in mind that the behavior of matter under very strong gravitational fields is not known
at present, and therefore, the description of classical macroscopic fluids that is valid
in the weak field may not be enough when the curvature becomes very high. To sim-
plify things, one usually imposes that certain inequalities be satisfied by the energy
momentum tensor in order for the same to be considered physically viable [18]. The
first and most commonly used inequality is the weak energy condition (w.e.c.). To
satisfy the w.e.c., the energy momentum tensor must be given in such a way that
Tμν V μ V ν ≥ 0 for any time-like (and null) vector V μ . This means that the energy
density must be nonnegative in any reference frame. The energy momentum tensor
for a fluid made of massive particles, with respect to some orthonormal basis, can
always be written as T μν = diag{ρ, p1 , p2 , p3 }. Then, the weak energy conditions
in the co-moving frame can be written as
ρ ≥ 0 ρ + pi ≥ 0 with i = 1, 2, 3. (5.52)
This is the less demanding of the energy requirements. The weak energy condition
allows for violations of the conservation of baryon number as new particles can be
created. Still, more stringent conditions can be imposed. If one desires to impose that
the total amount of mass in the space-time is conserved, then one must impose the
dominant energy condition (d.e.c.) which states that for every time-like vector V μ ,
the energy momentum tensor must satisfy both Tμν V μ V ν ≥ 0 and Tμν V μ being a
null or time-like vector. This means that not only the energy density is nonnegative
in any frame, but also the flow of ρ must be locally not space-like. As a consequence,
we get that in an orthonormal reference frame, the energy density must be greater
than the pressures. Namely,
ρ ≥ 0 , −ρ ≤ pi ≤ ρ with i = 1, 2, 3. (5.53)
Note that if we define the speed of sound waves within the fluid travelling in the
direction of pi as vi = dpi /dρ (i = 1, 2, 3), then the d.e.c. does not allow for the
speed of sound to be greater that the speed of light. It is a very reasonable assumption
that is not implemented by the w.e.c.. A fluid that satisfies the d.e.c. obviously satisfies
also the w.e.c.. Finally, let us briefly mention a third energy condition that can be
imposed and that is not directly related to the previous two. This is the strong energy
condition (s.e.c.), and for a perfect fluid in the co-moving frame, it is equivalent to
requiring
ρ ≥ 0 , ρ + p ≥ 0 , ρ + 3 p ≥ 0. (5.54)
In the following, we shall always require that the fluid satisfies the w.e.c. and when-
ever possible that it satisfies the d.e.c. as well.
5.4.3 Shell Crossing Singularities
From Eq. (5.47), we see that the Kretschmann scalar diverges when the central sin-
gularity forms, namely when a = 0, but also when R = 0 if M = 0. In this case,
we speak of the occurrence of shell crossing singularities. These are true curvature
singularities that arise from overlapping radial shells. At the shell crossing singular-
ity, the radial geodesic distance between shells with radial coordinate r and r + dr
vanishes. These singularities are equivalent to caustics in wave propagation, and it is
reasonable to assume that the space-time can be extended through the singularity by
a suitable redefinition of the coordinates. This can also be seen from the fact that shell
crossing singularities are gravitationally weak, meaning that geodesics reaching the
singularity are not squeezed into a line (as is the case of the central singularity), and
thus, observers at the shell crossing are not crushed (see, e.g., [21, 22, 32, 49, 50]).
Nevertheless, in any collapse model, a condition that can be required is the absence
of shell crossing singularities. In order to avoid shell crossing singularities, we can
either impose R = 0 or choose M(r ) in such a way that M(r ) = 0 when R = 0 so
that M /R < ∞. During collapse, the mass function M is generally assumed to be
positive; therefore, requiring the absence of shell crossing singularities is equivalent
to requiring that R is not vanishing. Note that since R = a + ra in a neighborhood
of the center, the condition can always be satisfied if a = 0.
5.5 Trapped Surfaces and Singularities
In the Schwarzschild space-time, the surface rs = 2MS , known as the event horizon,
is “...a perfect unidirectional membrane: causal influences can cross it in only one
direction” [15]. The event horizon is the boundary of the region where light rays can
not escape to infinity. At the horizon, the time-like Killing vector is null and outgoing
null geodesics have zero radial velocity. Nevertheless, the event horizon is not a very
useful concept for practical (i.e., astrophysical) purposes. In fact, the event horizon
is a global property of the space-time which does not depend on the observer and its
determination requires the knowledge of the entire future history of the space-time.
What we need in order to be able to make experiments is a local approach to the
definition of trapped surfaces that allows us to define when a co-moving observer
184 D. Malafarina
that is collapsing with the cloud becomes causally disconnected from the outside
universe. If matter is present, as in the case of the LTB metric given in Eq. (5.24), the
event horizon is not the only possible horizon that can be defined and it is not the most
useful concept to investigate the physics that occurs as the black hole forms. We want
to know how and when the black hole forms during gravitational collapse. When light
will be trapped by the gravitational field? The exterior region will become a black
hole solution once the boundary surface Rb (t) passes the Schwarzschild radius. What
about the interior? Each collapsing shell will become causally disconnected from the
outer universe at some point and will eventually fall into the central singularity. If
we want to track the formation of the horizon inside the matter region, first we need
to know what we mean by trapped surface in the interior.
Given a 3 + 1 slicing of the space-time, consider the three-dimensional space-like
slice. Then, a “trapped surface” is defined as a smooth closed two-surface in the slice
whose future-pointing outgoing null geodesics have negative expansion. This means
that all light rays, all null geodesics, emanating from the surface are pointing inward.
The “trapped region” in the slice is then defined as the union of all trapped surfaces,
and the “apparent horizon” is the outer boundary of the trapped region [7, 20, 40].
One intuitive way to understand the difference between the apparent horizon and
the event horizon is to note that the event horizon is the surface at which any light
ray directed outward can be initially outgoing and eventually become ingoing, thus
falling back inwards at some later time, while the apparent horizon is the surface
for which all light rays directed outwards are ingoing, thus directed inward at the
time when they are emitted. In vacuum, the two surfaces coincide, and therefore,
the apparent horizon and the event horizon in the Schwarzschild space-time are the
same. Still, when matter is present, they can be different, as is the case for the LTB
metric.
The apparent horizon in general need not be a null surface and it always lies inside
the event horizon. Nevertheless, it is the apparent horizon that determines the trapped
region in the collapsing cloud. It is a local property of the space-time and is observer
dependent, and therefore, it can be experimentally tested, while the event horizon
may be undetectable. To understand this, imagine the situation of a thin spherical
shell separating a vacuum Minkowski interior from a vacuum Schwarzschild exterior.
The shell may be time-like or light-like. Let the shell have total mass MS and collapse
under its own gravity (see Fig. 5.1). As the shell collapses, an event horizon will form
at R = 0 at the time t = t0 . The event horizon curve will expand to larger radii and
eventually match the Schwarzschild radius R = 2MS in the exterior at t = tSch . An
observer living at a fixed radial coordinate R = R1 inside the Minkowski region will
experience the event horizon passing through him at the time t1 but will not have
any way to detect it. This shows how in principle we can have event horizons where
we would not expect and why local experiment cannot detect the presence of an
event horizon. For this reason, the apparent horizon is a more useful tool to study the
trapped region that develops during the formation of a black hole.
For the spherical dust collapse model, the apparent horizon is the surface for
which the surface R(r, t) = const. is null. This means
tSch
t1
t0 Minkowski Schwarzschild
R
R1 2Ms
Fig. 5.1 Collapse of a thin spherical null shell (thick line) separating a flat vacuum interior from
a Schwarzschild exterior. The angular coordinates θ and φ are suppressed, so every radius R
corresponds to a spherical surface. The thin line represents the event horizon, which forms at the
center of symmetry of the system at the time t0 and expands toward bigger radii. As the shell crosses
the Schwarzschild radius 2MS , the horizon settles to the usual event horizon of a static black hole.
Observers living in the interior at a fixed radius R1 would fall inside the trapped region at the time
t1 but would have no way of detecting the horizon that is passing through them
g μν (∂μ R)(∂ν R) = 1 + f − e−2λ Ṙ 2 = 0. (5.55)
From the definition of the Misner–Sharp mass in Eq. (5.8), we get that the condition
for the formation of trapped surfaces can be expressed as
F r2M
1− =1− = 0. (5.56)
R a
This condition can be viewed as the implicit definition of the curve tah (r ) for which
a(r, tah (r )) = r 2 M(r ). (5.57)
The above curve is called the apparent horizon curve and describes the co-moving
time t at which the co-moving shell r becomes trapped.
As we have seen, the other important curve to describe the formation of the black
hole at the end of collapse is the singularity curve tsing (r ). This is the curve that
describes the co-moving time t at which the co-moving shell r becomes singular.
This curve represents the limit of the space-time manifold, and all geodesics inside
the trapped region must terminate at the singularity. From the condition of formation
of the singularity, we see that the curve tsing (r ) is given implicitly by
186 D. Malafarina
a(r, tsing (r )) = 0. (5.58)
We have already seen that in the case of homogeneous dust, we must have tsing =
const., which means that all shells become singular at the same co-moving time.
5.6 Homogeneous Solutions
The equation of motion (5.51) for homogeneous collapse is written as

M0
ȧ = − + k. (5.59)
a
We can characterize the geometry depending on the sign of the free parameter k by
introducing the following change of coordinates
⎧
⎨ sinh χ
⎪ if k = +1, “hyperbolic” region,
r = Sk (χ ) = χ if k = 0, “flat” region,
⎪
⎩
sin χ if k = −1, “elliptic” region.
We can then write the Oppenheimer–Snyder metric in a unified form as

ds 2 = −dt 2 + a(t)2 dχ 2 + Sk (χ )2 dΩ 2 , (5.60)
and the solution of the equation of motion is given in parametric form by

⎧
2k 3/2 (t−ts )
⎪
⎪
M0
(cosh η − 1) with sinh η − η = if k > 0,
⎪
⎨
2k

M0
2/3
a(t) = 3M0 (t−ts )
if k = 0,
⎪
⎪ 2
⎪
⎩ 2(−k)3/2 (t−ts )
M0
−2k
(1 − cos η) with η − sin η = M0
if k < 0,
On the other hand, one can always solve the equation of motion (5.59) to find t (a).
• In the flat region given by k = 0, the equation of motion is easily integrated to give
3
2a 2
t (a) = − √ + tsing .
3 M0
• In the hyperbolic region, given by k > 0, we define X = M0 /k and we get

⎛ ⎞
a ⎝X 1 X
t (a) = √ tanh−1 − + 1⎠ + tsing .
k a X
+1 a
a
• In the elliptic region, given by k < 0, we define X = −M0 /k and we get

⎛ ⎞
a ⎝ X X 1 ⎠ + tsing .
t (a) = √ − 1 − tan−1
−k a a X
−1 a
Finally, for homogeneous dust, the metric (5.3) can also be written as
a2
ds 2 = −dt 2 + dr 2 + r 2 a 2 dΩ 2 , (5.61)
1 + r 2k
from which we see that the metric in the interior is just the time reversal of a dust
Friedmann–Robertson–Walker cosmological solution.
Marginally bound case is a particularly simple case given by √k = 0. Then, the ini-
tial velocity of collapse is nonzero,
√ as can be seen from ȧi = − M0 , and the equation
of motion is simply ȧ = − M0 /a for which the solution is obtained immediately
as (see Fig. 5.2)
2/3
3
a(t) = 1 − M0 t . (5.62)
2
t
2Ms
tsing
t0
Rb (t)
Interior Exterior
R
0
Fig. 5.2 Schematic view of homogeneous dust collapse. At the time ti = 0, no singularities are
present. The boundary curve Rb (t) separates the interior from the vacuum exterior. The cloud
collapses as t increases, and at the time t0 , the horizon forms at the boundary. In the exterior region,
the horizon is the Schwarzschild radius. In the interior, the apparent horizon propagates inward and
reaches the center of symmetry at the time of formation of the singularity tsing . For t > tsing , the
space-time has settled to the usual Schwarzschild solution
188 D. Malafarina
5.6.1 Apparent Horizon and Singularity
From the condition for formation of trapped surfaces given in Eq. (5.56), we obtain
the curve tah (r ) describing the time at which the shell r crosses the apparent horizon:
⎧ √
⎪
⎪ t (r ) + F
tanh −1 f
− Ff 1 + f , for k > 0,
⎪
⎪ sing 3 1+ f
⎨ f 2
tah (r ) = tsing (r ) − 23 F(r ) = 3√2M0 − 23 r 3 M0 , for k = 0,

⎪
⎪
⎪
⎪ √
⎩ tsing (r ) + F 3 tan−1 − 1+f f − Ff 1 + f , for k > 0.
(− f ) 2
Now, if we look for simplicity at the apparent horizon for marginally bound homo-
geneous dust model, we see that it forms
√ initially at the boundary of the collapsing
cloud at the time t0 = tah (rb ) = 2/3 M0 − 2rb3 M0 /3 and then propagates inward
toward the center. The time t0 is the same time at which the event horizon forms
in the exterior spacetime. For t > t0 , the apparent horizon√ curve moves to smaller
radii reaching the center at the time tsing = tah (0) = 2/3 M0 , which is the time of
formation of the singularity. Inside the trapped region, all geodesics terminate at the
singularity; therefore, an observer on the boundary, once thishas passed the horizon,
falls toward the singularity in a finite time of the order of Rb3 /G M. An observer
at infinity sees the boundary approaching the horizon becoming infinitely redshifted
and indefinitely slow [36].
The singularity is reached once the density diverges. From the above, we have
seen that the shell focusing strong curvature singularity corresponds to a = 0. In the
homogeneous dust collapse case, this gives
⎧ √
⎪
⎪ √1 X tan −1 √ 1
− X − 1 , for k > 0,
⎪ −k
⎪ X −1
⎨ 3
tsing = 2r√2 = √2 , for k = 0,
⎪
⎪ 3 F
√ 3 M 0
⎪
⎪
⎩√ 1
X + 1 − X tanh −1 √ 1
, for k > 0.
k X +1
The singularity is simultaneous, and all shells fall into the singularity at the same
co-moving time.
5.7 Inhomogeneous Dust and Collapse with Pressures
The easiest way to extend the homogeneous dust collapse model is to introduce
inhomogeneities in ρ. Inhomogeneous models have been widely considered in cos-
mology which are obtained by a time reversal of collapse models (see [5, 31] and
references therein). This means considering ρ = ρ(r, t), with ρ radially decreasing
in order for the matter profile to be physically realistic. This describes a dust cloud
that initially has higher density at the center. Following the same procedure as in
homogeneous collapse, we can evaluate Einstein’s equations as
F
ρ= , (5.63)
R2 R
Ḟ = 0, (5.64)
λ = 0, (5.65)
Ṙ
ψ̇ = . (5.66)
R
The main difference with the homogeneous case is that now we will have M(r ),
b(r ), and a(r, t). Then, it is worth asking how the boundary, the trapped surfaces,
and singularity are affected by the presence of inhomogeneities. Does a black hole
still form at the end of collapse? Yes, the singularity theorems by Hawking and
Penrose tell us that once the trapped surfaces form, the formation of the singularity
is inevitable. Eventually, all matter falls into the central singularity and we are left
with a Schwarzschild black hole [19]. Nevertheless, if we ask whether we get a
picture of collapse qualitatively similar to the OS model, then the answer is not
always in the affirmative. In fact, the way in which the apparent horizon and the
singularity curve develop depends on the form of the density and velocity profiles
and some important differences with the OS model may arise. The most striking of
these differences is that in the inhomogeneous dust collapse, there is the possibility
for the central singularity to be “naked” (i.e., not covered by a horizon) at the instant
of formation (see, e.g., [9, 12, 26, 37, 48]).
In general, if M = M(r ), we have
F 3M + r M
ρ= = . (5.67)
R2 R a 2 (a + ra )
If we consider M(r ) as a polynomial expansion near r = 0, we can take
M(r ) = M0 + M1r + M2 r 2 + · · · , (5.68)
and the condition for the energy density to be radially decreasing outward is given by
M1 ≤ 0. If we also wish to impose that the density does not present cusps at the center,
we may impose M1 = 0, and then, the condition for ρ to be decreasing becomes
M2 ≤ 0. This is consistent with the choice of density profiles in astrophysical models
that generally present only quadratic terms in r . Similar to the homogeneous case,
the value a = 0 signals the appearance of the shell focusing singularity. On the
other hand, now we need to make sure that shell crossing singularities, given by
R = a + ra = 0, do not occur before the formation of the singularity. For simplicity,
let us consider the marginally bound case. Then, R = 0 implies
190 D. Malafarina
3√ rt M
1− M − √ = 0, (5.69)
2 2 M
which can be used to obtain the time at which the shell r develops a shell crossing
singularity as
√
2 M
tsc (r ) = . (5.70)
3M + r M
For homogeneous dust, then tsc = tsing and we do not have shell crossing singularities
during collapse. Similarly, in the inhomogeneous case with M < 0, we have tsc (r ) ≥
ts (r ), and therefore, no shell crossing singularities occur before the formation of the
central singularity. From this, we see that the physical requirement of a radially
decreasing density profile is compatible with the condition for avoidance of shell
crossing singularities.
The solution for inhomogeneous dust collapse can be obtained form the one for
homogeneous dust by replacing M0 and k with M(r ) and b(r ). Again, let us consider
for simplicity the solution for marginally bound collapse. This is given by
2/3
3
a(r, t) = 1 − M(r )t . (5.71)
2
We immediately see that now each shell collapses with a different scale factor and
a different velocity. As a consequence, each shell becomes singular at a different
time. The apparent horizon curve is also affected, as now it does not necessarily
form initially at the boundary. The singularity curve and apparent horizon curve are
explicitly given by
2
tsing (r ) = √ , (5.72)
3 M(r )
2
tah (r ) = tsing (r ) − r 3 M, (5.73)
3
and near r = 0, they have the same behavior up until the third order in r ,
2 M1 r
tsing (r ) = √ − 3/2
+ ··· , (5.74)
3 M0 3M0
2 M1 r
tah (r ) = √ − 3/2
+ ··· . (5.75)
3 M0 3M0
Note that near the center, the apparent horizon curve is increasing and the central line
r = 0 becomes singular and trapped at the same time. This suggests the possibility
for the existence of geodesics that originate at the central singularity and are not
trapped inside the horizon as (see Fig. 5.3). Due to the lack of pressures in the model,
t
2Ms
t1
t0
Rb (t)
Interior Exterior
R
0
Fig. 5.3 Schematic view of inhomogeneous dust collapse. At the time ti = 0, no singularities
are present. The boundary curve Rb (t) separates the interior from the vacuum exterior. The cloud
collapses as t increases, and at the time t0 , the horizon forms at the center of the cloud. The
singularity forms at the same time. Null geodesic can originate from the singularity and reach
distant observers. In the interior, the apparent horizon propagates outward and reaches the boundary
at the time t1 > t0 . Once all the matter falls into the singularity, the space-time settles to the usual
Schwarzschild solution
the boundary of the star rb can be chosen arbitrarily. This is a mathematical artifact
of the dust solution, and in the case with pressures, the boundary would have to be
set at the radius where p vanishes. Therefore, if the boundary is chosen in such a
way that tah is always increasing, it is possible to find null geodesics that originate at
ts (0) and reach observers at infinity. Outgoing radial null geodesics tγ (r ) are given
by
dtγ
= R, (5.76)
dr
and it can be proven that there are null geodesics coming out of the first instant of
the central singularity ts (0) and reaching the boundary [9, 12, 26, 37, 48]. Naked
singularities are found in many solutions of Einstein’s equations and can be very
different from one another. The question is if they can form from physically realistic
processes. How much these models rely on the assumptions? What outcome will
come from more realistic models? We shall shortly discuss these issues in the next
sections (for a more detailed discussion, see, e.g., [28]).
Analytically, we cannot deal with rotation or departures from spherical symmetry.
In fact, when it comes to rotation at present, we do not possess an analytical solu-
tion that describes an interior for the Kerr space-time that matches smoothly to the
exterior. Nevertheless, some indications on how general is the scenario described
192 D. Malafarina
in the dust collapse model can be obtained by considering collapse of a fluid

source with pressures. For a fluid source, the energy momentum tensor is given
by Tμν = diag{ρ, pr , pθ , pθ } and we can write Einstein’s equations and energy
momentum conservation
F
ρ= , (5.77)
R2 R
Ḟ
pr = − 2 , (5.78)
R Ṙ
pθ − pr R pr
λ =2 − , (5.79)
ρ + pr R ρ + pr
Ṙ
Ġ = 2λ G , with G = R 2 e−2ψ . (5.80)
R
Then, it is easy to see that the system of equations to be solved becomes much more
complicated with respect to the dust case. Together with the Misner–Sharp mass
definition, the system has five equations and seven unknown functions. Specifying
equations of state for pr and pθ then closes the system.
In this case, the Misner–Sharp mass need not be conserved during the evolution,
and therefore, there can be an inflow or outflow of matter across each shell r as
collapse progresses. As a consequence, the matching with the exterior space-time
need not be done with the Schwarzschild solution. Requiring the boundary condition
pr (rb , t) = 0 implies that F(rb , t) is conserved during collapse and the exterior is
Schwarzschild, and on the other hand, the condition that pr vanishes at the bound-
ary translates in a variable boundary surface rb (t). It can be shown that matching
with the Vaidya solution describing ingoing or outgoing null dust can be done in
certain cases and matching with a generalized Vaidya solution is always possible
[13, 14, 27].
The simplest model with pressures that can be considered is that of a homogeneous
perfect fluid with linear equation of state. Then, requiring that the fluid be perfect
implies pr = pθ = p, while homogeneity implies p = p(t) and ρ = ρ(t). Finally,
a linear equation of state relates p to ρ via
p = γρ, (5.81)
with γ being a constant. The presence of the linear equation of state closes the system,
and Einstein’s equations can be fully integrated in this case. The third Einstein’s
equation (5.79) becomes again λ = 0, and the fourth equation gives again G =
1 + kr 2 . Then, the equation of motion is again written as
M
ȧ 2 = + k, (5.82)
a
where now M = M(t) is to be determined from the equation of state that together
with Eqs. (5.77) and (5.78) gives
3γ M
Ṁ = − . (5.83)
a
Integrating the above equation, we get
M0
M(t) = . (5.84)
a 3γ
Einstein’s equations (5.77) and (5.78) imply that the density is ρ = 3M/a 3 and the
pressure is p = − Ṁ/a 2 . To have a positive pressure, M must decrease in time. If
we require a constant co-moving boundary r = rb , then the exterior metric cannot
be a portion of the Schwarzschild space-time and some mass must be radiated away
in the exterior region. This can be easily seen from the fact that M(t) implies that
the total mass within rb changes with time, and therefore, there must be an outflow
of matter from the co-moving boundary. As said, we can always match to a non-
vacuum solution describing a radiating null fluid or we can require the matching to
be performed at a surface rb (t).
5.8 Collapse in Astrophysics
We studied here some simple analytical toy models that describe the complete grav-
itational collapse of a spherical matter cloud made of non-interacting particles in
general relativity (GR). If we assume as a first approximation that these models can
be used to describe the most relevant features of the collapse of the core of a massive
star, we see that a black hole must inevitably form as the final product of collapse. At
the time t0 , the horizon forms as the boundary of the star crosses the threshold of the
Schwarzschild radius, and at the time tsing > t0 , the singularity forms at the center
of symmetry of the system. In the end, we are left with a Schwarzschild black hole.
The Oppenheimer–Snyder–Datt (OSD) model is very simple and relies on many
simplifying assumptions that while on the one hand allow us to solve the equations
analytically thus finding a global solution, on the other hand make for a scenario that
is not very realistic. The OSD model can be seen as the bridge between mathematical
black holes and astrophysical black holes in the sense that it is a simplified mathe-
matical description of a dynamical phenomena that nevertheless captures the most
essential features. The main assumptions in the model are spherical symmetry, no
rotation, homogeneous density, and no pressures. A real star will have some small
but non-vanishing quadrupole moment, it will have angular momentum, it will be
composed of several kinds of gases with pressures and different equations of state,
and its density will not be homogeneous. Therefore, it is reasonable to ask how gen-
eral is the picture obtained in the OSD model and how much the collapse of a real
star will depart from our mathematical idealization.
What happens to singularity and horizon once we introduce inhomogeneities, pres-
sures, rotation, and asymmetries in the model? Gravitational collapse is a dynamical
194 D. Malafarina
process, and the structure and evolution of the horizon in a realistic stellar interior
is not well understood. We still do not have any analytical model of formation of a
Kerr black hole. In fact, we do not even have any analytical solution describing a
viable interior metric for the Kerr solution. Nevertheless, the evidence for the exis-
tence of black holes is now almost universally accepted, and most people believe
that the process that leads to their formation can be roughly described via the OSD
model. Still, the question of whether the black hole candidates that we observe in the
universe are well described by the Schwarzschild and Kerr metrics is still open. As is
the question of whether every collapsing star that is massive enough must inevitably
form a black hole as the final end state. In order to study more realistic models, one
needs to give up the hope to solve Einstein’s equations analytically and resort to
numerical simulations. Fully general relativistic simulations have been done in the
past years to study (among other things) gravitational core collapse with rotation and
magnetic fields, black hole mergers and black hole neutron star mergers, gravitational
wave production, recoil from black hole mergers, production of jets, and gamma ray
bursts (see, e.g., [24, 25, 42] and references therein).
Numerical simulations have improved dramatically over the last decade. Never-
theless, there are still no fully satisfactory simulations of supernovae explosions that
lead to the formation of a black hole. One reason resides in the fact that many ele-
ments of classical and quantum physics come into play during the last stages of the
life of a star. Describing accurately such scenarios is an enormous task that requires
very expansive computations on the most advanced supercomputers. Further to this,
numerical simulations must assume that GR holds unchanged at all energy scales and
therefore are limited by our lack of knowledge of gravity in the strong field. Stellar
evolution and black hole formation still present a lot of open questions, and the pos-
sibility exists that black holes are not the only possible final outcome of collapse of
very massive stars. For these reasons, despite the increasing amount of observational
evidence for the existence of black holes, it is useful to keep an eye open for other,
more exotic, possibilities (see, e.g., [10, 16, 34, 45, 46] and references therein).
5.9 Concluding Remarks
Simple analytical models of general relativistic collapse show that a black hole can
form as the end state of the life of a massive star. From a mathematical point of view,
a black hole is a space-time singularity covered by an event horizon. The curvature
singularity at the end of collapse is indicated by the divergence of the scalar K ,
and approaching the singularity matter reaches infinite density in a finite co-moving
time. In some sense, the singularity at the end of collapse is analogous to the infinite
density obtained in Newtonian collapse and can be viewed as a limit of the model
rather than a physical feature of the system. Singularities are found in many solutions
of Einstein’s equations. The question is whether they can form from physically real-
istic processes and how we should interpret their appearance. We may think that GR
works well in the strong field regime and nothing can prevent complete collapse from
happening. In this case, one has to accept that singularities are there and they may be
causally connected to the outside universe. On the other hand, we can believe that GR
works well in the strong field, but other effects arise either preventing the formation
of singularities or hiding them from view. Or we can think that GR needs modifica-
tions in the strong field regime due perhaps to quantum effects. These modifications
would then affect the space-time near the formation of classical singularities, thus
removing them. The first attitude, although legitimate, is not very common. A sin-
gularity in any classical theory such as classical mechanics or electromagnetism is
located somewhere in time and space and it does not affect the future predictability
of space-time itself. On the other hand in GR, a singularity is not a part of the space-
time. The distribution of matter determines the properties of space and time, and the
occurrence of singularities translates in geodesic incompleteness and has important
consequences for the causal structure of the space-time itself. For this reason, most
people believe that singularities must not occur in the real universe. The second atti-
tude can be summarized by the words of Roger Penrose [41]: “...does there exist a
‘cosmic censor’ who forbids the appearance of naked singularities, clothing each
one in an absolute event horizon?” This is the famous cosmic censorship conjecture
(CCC). At present, there exist counterexamples to the CCC, like the inhomogeneous
dust collapse model, but their physical relevance is not entirely clear. On the other
hand, it is highly plausible that GR is not enough to describe what happens in the
strong field regime. One needs to account for microphysics or for modifications to
GR possibly due to quantum effects. This third attitude is a view that was already
suggested by Wheeler who saw singularities as possible probes for new physics.
If the occurrence of singularities at the end of collapse signals a breakdown of
the fluid model approximation or a breakdown of GR itself, then what could be a
better mathematical framework to describe the last stages of the life of a star? Is there
any viable model for collapse that does not originate a singularity? The singularity
theorems by Hawking and Penrose tell us that if GR is the ultimate ingredient that we
need to use to describe collapse and if matter satisfies the usual energy conditions,
then a singularity must necessarily form [19]. More precisely, provided that some
energy condition is satisfied, the space-time is globally hyperbolic, and a trapped
region develops at some point, a singularity must always form. Therefore, in order
to develop non-singular model of collapse, one needs to modify GR in some way.
Several attempts have been made over the years, and the general scenario that is
arising is that singularities may be removed by quantum gravitational effects. Matter
in the strong field regime may violate standard energy conditions, and the complete
collapse to a black hole may be replaced by a bouncing scenario in which collapsing
matter re-expands after reaching a minimal size. The expansion phase may take the
form of an explosive event, and it may leave behind an exotic compact remnant (see,
e.g., [3, 4, 17] and references therein).
These compact remnants may be less massive, more dense, and smaller than a
neutron star and they would not possess an event horizon. Several types of exotic
compact objects have been investigated, and their observational properties are of
great interest for future astrophysical observations (see, e.g., [1, 2, 29, 30]). Given
the small number of astrophysical black hole candidates observed so far and the
196 D. Malafarina
peculiar features that theoretical compact objects may possess, it is reasonable to

suppose that their observation may pose a great challenge for future astrophysics.
Nevertheless, if some departure from the black hole paradigm will be observed in
the future, this may open a window onto new areas of physics where gravitation and
quantum mechanics merge.
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Appendix A
General Relativity in a Nutshell
The aim of this appendix was to provide the reader not familiar with general
relativity a basic theoretical background on some fundamental concepts. It is not
an introduction to the theory of general relativity, but is hopefully enough to under-
stand the chapters in this volume. A key concept is the existence of an innermost
stable circular orbit around a black hole, which has no Newtonian counterpart. Since
we are interested in astrophysics, here we always assume that the spacetime has 4
dimensions (1 + 3). Units in which G N = c = 1 are used, and therefore, all the
length and timescales are set by the black hole mass M. The conversion factors are

M M
M = 1.477 km, M = 4.925 µs. (A.1)
M M
A.1 Geodesic Equations
In classical mechanics, the principle of least action plays a very important role and
it can be used to obtain in an elegant way the equations of motion for a system when
its action is known. In the case of a free point-like particle, the Lagrangian is simply
given by the kinetic energy of the particle
1 2 1 dxi dx j
L= mv = mgi j , (A.2)
2 2 dt dt
where m is the mass of the particle, gi j is the metric tensor, {x i } are the coordinates,
and t is the time. Here, we use the Einstein convention of summation over repeated
indices; that is,
dxi dx j dxi dx j
gi j = gi j . (A.3)
dt dt i, j
dt dt

and Space Science Library 440, DOI 10.1007/978-3-662-52859-4
200 Appendix A: General Relativity in a Nutshell
For instance, in Cartesian coordinates {x, y, z}, we have

2 2 2
dx dy dz
v2 = + + , (A.4)
dt dt dt
which means that the only nonvanishing coefficients of the metric tensor are gx x =
g yy = gzz = 1, while all the off-diagonal ones are zero. In the case of spherical
coordinates {r, θ, φ}, we have
2 2 2
dr dθ dφ
v2 = + r2 + r 2 sin2 θ , (A.5)
dt dt dt
so grr = 1, gθθ = r 2 , gφφ = r 2 sin2 θ , and all the other coefficients vanish. It is easy
to see that if we want to go from a coordinate system {x i } to another one {x i }, the
metric tensor changes as
∂xa ∂xb
gi j → gi j = gab . (A.6)
∂ x i ∂ x j
Infinitesimal displacements {d x i } change in the opposite way; that is,
∂ x i a
d x i → d x i = dx , (A.7)
∂xa
and for this reason we write lower indices for gi j and upper index for d x i .
For what follows, it is more convenient to define the action of a free point-like
particle as proportional to the length of its path

S=m ds = m L dλ, (A.8)
γ γ
where m is the particle mass, ds is the line element,

L= gi j ẋ i ẋ j , (A.9)
λ is an affine parameter that parameterizes the particle path γ (λ) and the dot˙indicates
the derivative with respect to λ. From the principle of least action, we find the Euler–
Lagrange equations
d ∂L ∂L
− i = 0. (A.10)
dλ ∂ ẋ i ∂x
It is easy to see that Eqs. (A.2) and (A.9) provide the same equations of motion. If
we plug Eq. (A.9) into (A.10), we obtain the geodesic equations
Appendix A: General Relativity in a Nutshell 201
ẍ i + Γ jki ẋ j ẋ k = 0, (A.11)
where Γ jki are the Christoffel symbols

1 il ∂glk ∂g jl ∂g jk
Γ jki = g + − . (A.12)
2 ∂x j ∂xk ∂ xl
In Cartesian coordinates, all the Christoffel symbols vanish, and therefore, the geo-
desic equations simply reduce to ẍ = ÿ = z̈ = 0.
In special relativity, time and space are not independent entities and the line
element of the spacetime ds is given, respectively, in Cartesian and spherical coor-
dinates, by
ds 2 = −dt 2 + d x 2 + dy 2 + dz 2 , (A.13)
ds 2 = −dt 2 + dr 2 + r 2 dθ 2 + r 2 sin2 θ dφ 2 . (A.14)
Thanks to the transformation rules (A.6) and (A.7) the line element is an invariant;
that is, it is independent of the choice of the coordinates. We can thus define the
following coordinate independent types of trajectories:
ds 2 < 0 timelike trajectories,

ds 2 = 0 lightlike trajectories,
ds 2 > 0 spacelike trajectories. (A.15)
In particular, massless particles such as photons will follow light-like trajectories with
ds 2 = 0. In the case of massive particles, it is convenient to use their “proper time”
τ , i.e., the time measured in the rest-frame of the particle, as the affine parameter λ.
Since ds 2 is an invariant, dτ 2 = −ds 2 , because the coordinate system is anchored on
the particle and therefore, there is no motion along the spatial directions. With this
choice of the affine parameter, ds 2 = −1. Since we are now considering a spacetime
in 1+3 dimensions, it is common to use Greek letters μ, ν, ρ, . . . to denote spacetime
indices, for instance, gμν , with μ = 0, 1, 2, and 3, where 0 stands for the temporal
component, while 1, 2, and 3 for the spacial components. Latin letter i, j, k, . . . are
used for the space components and therefore can assume the values 1, 2, and 3.
In Newtonian mechanics, the motion of a test particle in a gravitational field can
be described by adding the correct gravitational potential to the Lagrangian of the
free particle. A key point in general relativity is that the gravitational field can be
absorbed into the metric tensor gμν : In other words, we have still a free particle, but
now it lives in a curved spacetime and follows the geodesics of that spacetime. It
is useful to see how we can recover the Newtonian limit. In Cartesian coordinates,
the metric tensor of the flat spacetime of special relativity is usually denoted by ημν ,
where [see Eq. (A.13)]
||ημν || = diag(−1, 1, 1, 1). (A.16)

The Newtonian limit should be recovered by requiring that: (i) The gravitational
field is weak, (ii) the gravitational field is stationary, and (iii) the motion of the
particle is nonrelativistic. These three conditions are given, respectively, by
gμν = ημν + h μν with |h μν | 1, (A.17)

∂gμν
= 0, (A.18)
∂t
dt dxi
. (A.19)
dλ dλ
Within these approximations, the geodesic equations reduce to
2
d2xμ dt 1 μν ∂h tt
+ Γttμ =0 with Γttμ = η . (A.20)
dλ2 dλ 2 ∂xν
After a simple integration, we find
d2xi 1 ∂h tt
=− . (A.21)
dt 2 2 ∂xi
If we compare Eq. (A.21) with the Newtonian formula m ẍ = −m∇Φ, where Φ is
the Newtonian gravitational potential, and we require that the spacetime is flat at
infinity, we find
gtt = −(1 + 2Φ). (A.22)
A.2 Einstein Equations
In the previous section, we have discussed the motion of test particles in a given back-
ground metric gμν . In general relativity, the latter takes into account the gravitational
field as well, and therefore, it is determined by the matter distribution. The Einstein
equations relate the spacetime geometry (on the left side) to the matter content (on
the right side):
G μν = 8π Tμν , (A.23)
where G μν is the Einstein tensor
1
G μν = Rμν + gμν R, (A.24)
2
Rμν and R are, respectively, the Ricci tensor and the scalar curvature
ρ ρ
∂Γμν ∂Γμν σ ρ σ ρ
Rμν = − + Γμν Γσρ − Γμρ Γνσ , (A.25)
∂xρ ∂xν
μν
R = g Rμν , (A.26)
and Tμν is the matter energy-momentum tensor. The factor 8π is just to recover the
correct Newtonian limit. If the matter content is known, we can plug its energy-
momentum tensor on the right-hand side of the Einstein equations and find the
corresponding metric tensor (modulo a choice of coordinates). However, this job is far
from being trivial, mainly because of the nonlinear nature of the Einstein equations.
Analytical solutions are thus known only in the case of special symmetries.
To find the Newtonian limit, we assume the approximations (A.17) and (A.18),
as well as that in our coordinate system all the components of the matter energy-
momentum tensor are negligible, except the tt one, which describes the energy
density and reduces to the matter density in the Newtonian limit, so that
Ttt = ρ Tμν = 0 for μ = ν = t. (A.27)
After some calculations, we find
1
Rtt = Δh tt = 8πρ, (A.28)
2
where Δ is the Laplace operator. The Poisson equation of Newtonian gravity is
recovered by replacing h tt with 2Φ, where Φ is the Newtonian gravitational potential,
as found in the previous section.
A.3 Schwarzschild Solution
The only spherically symmetric solution of the vacuum Einstein equations G μν = 0

is the Schwarzschild metric (Birkhoff’s theorem). Such a solution describes the
exterior gravitational field of any spherically symmetric source, which therefore
may also have a nonvanishing radial motion. The line element is

2M 2M −1 2
ds 2 = − 1 − dt 2 + 1 − dr + r 2 dθ 2 + r 2 sin2 θ dφ 2 , (A.29)
r r
where M is a free parameter to be related to the gravitational mass of the object. For
M/r 1, the correct Newtonian limit is recovered. In the case of a star, this metric is
valid in the exterior, for radii r > R , where R is the radial coordinate of the surface
of the star. In the case of a black hole, there is no R . r = 2M is the radius of the
event horizon, where the above coordinate system is ill defined. Since grr → +∞
for r → 2M, from the geodesic equations we find that any massless and massive
particle in the region r ≤ 2M is trapped and cannot escape to infinity. The event
horizon is the boundary of the black hole and it can be seen as a one-way membrane:
particles can cross the event horizon from the outside, and thus be swallowed by the
black hole, but once inside they cannot come back to the outside region.
The event horizon is a region with quite peculiar properties. For instance, the
relation between the temporal coordinate t and the proper time of a static observer
at (r, θ, φ) is

2M 1/2
dτ = 1 − dt < dt. (A.30)
r
The observer proper time τ is thus slower than the coordinate time t (corresponding
to the proper time of a static observer at infinity) as a consequence of the gravitational
field, and the observer proper time is frozen out (with respect to the distant observer)
for r = 2M.
Let us now consider two static observers, say A and B, with coordinates, respec-
tively, (r A , θ, φ) and (r B , θ, φ). The observer A emits an electromagnetic signal that
he/she measures to have frequency ν A and to last for a time Δτ A . The number of
wavefronts is thus n = ν A Δτ A . The signal arrives at the position of the observer B,
who measures a frequency ν B for a time Δτ B . As the number of wavefronts is an
invariant, we have ν A /ν B = Δτ B /Δτ A . For a light ray, ds 2 = 0, and therefore, since
A and B have the same θ and φ coordinates, along the photon path we have

2M 2M −1 2
1− dt = 1 −
2
dr . (A.31)
r r
The time interval with respect to the coordinate system that the first wavefront takes
to go from the observer A to the observer B is
rB
dr
t B1 − t A1 = . (A.32)
rA 1 − 2M/r
Since it is independent of the coordinate t, the last wavefront takes the same time;
that is, t B1 − t A1 = t Bn − t An and therefore, t An − t A1 = t Bn − t B1 . The proper time measured
by the observer A is instead
1/2
t An
2M 2M 1/2 n
Δτ A = 1− dt = 1 − t A − t A1 , (A.33)
t A1 rA rA
and the same for B, with the index B replacing the index A in Eq. (A.33). We can
then find the relation between ν A and ν B

νA 2M 1/2 2M −1/2
= 1− 1− . (A.34)
νB rB rA
Even in this case, we can see that these coordinates are not suitable to describe what
happens at r = 2M.
In astrophysical scenarios, one usually needs to compute the trajectories of parti-
cles and photons in this background. In the case of an accretion disk, we may imagine
that the particles of the gas follow nearly geodesic circular orbit in the equatorial
plane θ = π/2. Since the metric coefficients in Eq. (A.29) are independent of the
coordinates t and φ, there are two constants of motions, associated, respectively, with
the energy and the angular momentum of the particle

d ∂L 2M
=0 ⇒ 1− t˙ = E = const., (A.35)
dλ ∂ t˙ r
d ∂L
=0 ⇒ r 2 φ̇ = L = const., (A.36)
dλ ∂ φ̇
For a massive test particle ds 2 = gμν ẋ μ ẋ ν = −1 and we have
1 2 E2 − 1 M L2 2M L 2
ṙ = + − 2 + . (A.37)
2 2 r r r3
Equation (A.37) can be seen as the equation of motion of a test particle in Newtonian
mechanics under the effect of an effective potential. The first term on the right-hand
side is just a constant. The second term, M/r , is the standard gravitational potential of
a spherically symmetric mass in the Newtonian theory and it introduces an attractive
force (the term is positive). The third term, L 2 /r 2 , is the standard centrifugal potential
and introduces an effective repulsive force (the term is negative). The forth term,
2M L 2 /r 3 , is something absent in the Newtonian theory: It becomes dominant at
very small radii (it scales as 1/r 3 ) and it is attractive. Such a term introduces some
novel properties, like the fact that circular orbits exist only for radii larger than a
critical one, say r ≥ rc , and that stable circular orbits exist only for radii larger than
another critical value, r ≥ rISCO . These two facts can be understood by noticing
that the term 2M L 2 /r 3 makes the gravitational force very strong at small radii, and
therefore, any particle must fall onto the black hole. More details will be given in the
next section.
A.4 Kerr Solution
The only rotating uncharged black hole solution of 4-dimensional general relativity
is described by the Kerr metric, which depends only on the black hole mass M and the
black hole spin angular momentum J . This is the statement of the “no-hair” theorem,
which is called in this way to mean that a black hole has no distinguishing features
(no hair), except for its mass and spin angular momentum. An astrophysical black
hole is supposed to be well described by the Kerr solution. In general relativity, initial
deviations from the Kerr metric are quickly radiated away through the emission of
gravitational waves. The equilibrium electric charge is soon reached, because of the
highly ionized host environment, and too small for macroscopic objects to affect the
geometry of the spacetime. The effect of the mass of the accretion disk is completely
negligible in most cases.
In Boyer–Lindquist coordinates, the line element is

2Mr 4a Mr sin2 θ
ds = − 1 −
2
dt 2 − dtdφ
Σ Σ

Σ 2a 2 Mr sin2 θ
+ dr 2 + Σdθ 2 + r 2 + a 2 + sin2 θ dφ 2 , (A.38)
Δ Σ
where a = J/M, Σ = r 2 + a 2 cos2 θ and Δ = r 2 − 2Mr + a 2 . The radius of the

event horizon is defined by the larger root of Δ = 0, where grr diverges, and it is

rH = M + M 2 − a2. (A.39)
Let us note that the event horizon exists only for |a| ≤ M. For |a| > M, there is
no horizon and the Kerr metric describes the spacetime of a naked singularity. A
number of arguments (instability of the spacetime, apparent impossibility to create a
Kerr naked singularity from a Kerr black hole, etc.) suggest that the Kerr metric with
|a| > M has no astrophysical applications. An important new property of the Kerr
metric with respect to the Schwarzschild solution is the existence of the ergosphere,
where gtt changes sign and static particles are not allowed (everything must rotate).
The outer boundary of the ergosphere is

rE = M + M 2 − a 2 cos2 θ . (A.40)
As in the Schwarzschild case, the metric coefficients are independent of the coor-
dinates t and φ, and therefore, we have two constants of motion, associated, respec-
tively, with the energy E and the axial component of the angular momentum L z . We
can exploit this fact to write t˙ and φ̇ as
Egφφ + L z gtφ Egtφ + L z gtt

t˙ = , φ̇ = − . (A.41)
2
gtφ − gtt gφφ 2
gtφ − gtt gφφ
From gμν ẋ μ ẋ ν = −1, we can write
grr ṙ 2 + gθθ θ̇ 2 = Veff (r, θ ), (A.42)
where the effective potential Veff is given by
E 2 gφφ + 2E L z gtφ + L 2z gtt

Veff = − 1. (A.43)
2
gtφ − gtt gφφ
Circular motion on the equatorial plane plays an important role in the description
of accretion disks because, even if there is an initial misalignment, the disks are
forced to adjust on the equatorial plane (Bardeen–Petterson effect). Circular orbits
on the equatorial plane are located at the zeros and the turning points of the effective
potential: ṙ = θ̇ = 0, which implies Veff = 0, and r̈ = θ̈ = 0, requiring, respectively,
∂r Veff = 0 and ∂θ Veff = 0. From these conditions, one can obtain the values of E
and L z
r 3/2 − 2Mr 1/2 ± a M 1/2

E= √ , (A.44)
r 3/4 r 3/2 − 3Mr 1/2 ± 2a M 1/2

M 1/2 r 2 ∓ 2a M 1/2 r 1/2 + a 2
Lz = ± √ . (A.45)
r 3/4 r 3/2 − 3Mr 1/2 ± 2a M 1/2
The orbits are stable under small perturbations if ∂r2 Veff ≤ 0 and ∂θ2 Veff ≤ 0. In Kerr
spacetime, the second condition is always satisfied, so one can deduce the radius of
the innermost stable circular orbit (ISCO) from ∂r2 Veff = 0. After some passages,
one finds the ISCO radius

rISCO = 3M + Z 2 ∓ (3M − Z 1 )(3M + Z 1 + 2Z 2 ), (A.46)
2
2 1/3

Z1 = M + M − a (M + a) + (M − a)
1/3 1/3
,

Z 2 = 3a 2 + Z 12 , (A.47)
The ISCO radius is rISCO = 6 M for a non-rotating black hole and decreases
(increases) as the spin parameter increases for a corotating (counter-rotating) disk,
to rISCO = M (rISCO = 9 M) for a maximally rotating Kerr black hole with a = M.

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Astrophysics and Space Science Library 440

Cosimo Bambi Editor

More information about this series at http://www.springer.com/series/5664

Astrophysics of Black Holes

ISSN 0067-0057 ISSN 2214-7985 (electronic)

Library of Congress Control Number: 2016943408

© Springer-Verlag Berlin Heidelberg 2016

Printed on acid-free paper

This Springer imprint is published by Springer Nature

Shanghai Cosimo Bambi

1 Black Hole Accretion Discs . . . . . . . . . . . . . . . . . . . . . . . . . . .... 1

1.8.3 Equation of Momentum Conservation. . . . . . . . . . . . . . 53

4.2 Formation of Wind from a Hot Accretion Flow . . . . . . . . . . . . . 154

Appendix A: General Relativity in a Nutshell. . . . . . . . . . . . . . . . . . . . 199

Tomaso M. Belloni INAF-Osservatorio Astronomico di Brera, Merate, Italy

Abstract This is an introduction to the models of accretion discs around black

J.-P. Lasota (B)

© Springer-Verlag Berlin Heidelberg 2016 1

Understanding accretion discs around black holes is interesting in itself because of

Notations and definitions

1.2 Disc-Driving Mechanism; Viscosity

which according to most MRI simulation is ∼10−3 , whereas observations of dwarf

1.2.1 The α-Prescription

τrϕ = αP, (1.6)

where P is the total thermal pressure and α ≤ 1. This leads to

For a Keplerian disc

( K = R2 ΩK is the Keplerian specific angular momentum.)

1.3 Geometrically Thin Keplerian Discs

The 2D structure of geometrically thin, non-self-gravitating, axially symmetric accre-

1.3.1 Disc Vertical Structure

where tdyn is the dynamical time.

• Energy transfer—temperature gradient

For radiative energy transport

In the case of convective energy transport, ∇ = ∇conv . Because convection in

where q+ (z) corresponds to viscous energy dissipation per unit volume.

• The vertical structure equations have to be completed by the equation of state

1.3.2 Disc Radial Structure

• Continuity (mass conservation) equation has the form

where S(R, t) is the matter source (sink) term.

k ∂ Ṁext Ttid (R)

which is an example of the general relation

The relation between the viscous and the dynamical times is

∂Tc ∂Tc Tc 1 ∂(Rvr ) Q+ − Q−

In thermal equilibrium, one has

tdyn < tth tvis . (1.46)

(This hierarchy is similar to that of characteristic times in stars: the dynamical is

and therefore, self-gravity is negligible when A 1. Treating the disc as an infinite

AH is related to the so-called Toomre parameter [56]

or as function of the midplane temperature T = 104 T4

1.3.4 Stationary Discs

where the integration constant Ṁ (mass/time) is the accretion rate.

Or, using Eq. (1.53),

where the torque T := 2π R3 ΣνΩ

which is a simple expression of angular momentum conservation.

1.3.4.1 Total Luminosity

For Rout → ∞, this becomes

∂Tc ∂Tc Tc 1 ∂(Rvr ) Q+ − Q−

Also for τtot 1, the temperature at the disc midplane is

(a) Pr Pg and κes κff