Rethinking Black Hole Accretion DiscsbyGreg Salvesen
B.S., University of Michigan Ann Arbor, 2009
B.S.E., University of Michigan Ann Arbor, 2009
M.S., University of Colorado Boulder, 2011

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Astrophysical and Planetary Sciences
2016This thesis entitled:
Rethinking Black Hole Accretion Discs
written by Greg Salvesen
has been approved for the Department of Astrophysical and Planetary Sciences

Prof. Mitchell C. Begelman

Prof. Philip J. Armitage

Dr. Jacob B. Simon

Prof. Jeremy Darling

Prof. J. Michael Shull

Date

The final copy of this thesis has been examined by the signatories, and we find that both the content andthe form meet acceptable presentation standards of scholarly work in the above mentioned discipline.

iii
Salvesen, Greg (Ph.D., Astrophysical and Planetary Sciences)
Rethinking Black Hole Accretion Discs
Thesis directed by Prof. Mitchell C. Begelman

Accretion discs are staples of astrophysics. Tapping into the gravitational potential energy of theaccreting material, these discs are highly efficient machines that produce copious radiation and extreme
outflows. While interesting in their own right, accretion discs also act as tools to study black holes anddirectly influence the properties of the Universe. Black hole X-ray binaries are fantastic natural laboratories
for studying accretion disc physics and black hole phenomena. Among many of the curious behaviors
exhibited by these systems are black hole state transitions — complicated cycles of dramatic brightening
and dimming. Using X-ray observations with high temporal cadence, we show that the evolution of theaccretion disc spectrum during black hole state transitions can be described by a variable disc atmosphericstructure without invoking a radially truncated disc geometry. The accretion disc spectrum can be a powerful
diagnostic for measuring black hole spin if the effects of the disc atmosphere on the emergent spectrum are
well-understood; however, properties of the disc atmosphere are largely unconstrained. Using statisticalmethods, we decompose this black hole spin measurement technique and show that modest uncertainties
regarding the disc atmosphere can lead to erroneous spin measurements. The vertical structure of thedisc is difficult to constrain due to our ignorance of the contribution to hydrostatic balance by magnetic
fields, which are fundamental to the accretion process. Observations of black hole X-ray binaries and theaccretion environments near supermassive black holes provide mounting evidence for strong magnetization.
Performing numerical simulations of accretion discs in the shearing box approximation, we impose a net
vertical magnetic flux that allows us to effectively control the level of disc magnetization. We study how
dynamo activity and the properties of turbulence driven by the magnetorotational instability depend on themagnetized state of the gas, spanning weak-to-strong disc magnetization regimes. We also demonstrate thata background poloidal magnetic flux is required to form and sustain a strongly magnetized accretion disc.
This thesis motivates the need for understanding how magnetic fields affect the observed spectrum fromblack hole accretion discs.

Dedication

To mom and dad
who inspired it,
but will never read it.

v

Acknowledgements
For better or worse — I think better — I never would have entered astronomy graduate school had it
not been for my undergraduate advisor, Jon Miller. Jon made a big school feel small and made me realizethat I could avoid real life for awhile by worrying about black holes. With the encouragement of anotherstellar mentor, Mateusz Ruszkowski, I chose Boulder, Colorado for graduate school. Jon’s parting advice
was, “Graduate school is a marathon, not a sprint.”
In my case, graduate school was an ultra marathon. I owe a big thanks to my advisors, Mitch Begelman
and Phil Armitage, who allowed me the freedom to pursue my interests during my seven year sabbaticalwhile looking out for me along the way. I was surrounded by top-notch postdocs who hugely influenced
my interests — Rubens Reis, Jake Simon, Chris Nixon, Becca Martin, and Kaitlin Kratter, you’ve all been
inspiring colleagues and great friends. To my officemates, Susanna Kohler and Eric Coughlin, you made me
feel like we were in this thing together, even though I finished last with a burro in tow. Jordan Mirocha wasalways game for a coffee or beer to talk science, while Grant Buckingham insisted on nixing the astronomy
talk and commencing the party.
I met some very bad influences through the Boulder Banditos and Rocky Mountain Runners, without
whom this thesis would have been completed two years earlier. Thank you guys for enabling my runningobsession, through which I met Cassie. Cassie, you really get me and I can’t wait for all of the adventures
we will have together. Summit is excited, too!
Of course, my family members are largely to blame for how I wound up. My dad used to give me
the change in his pocket if I could deduce the denominations from the total value and coin count. My mom
never got me that basketball hoop I wanted, forcing me to put all of my eggs in the astronomy basket withno backup plan as an NBA star. I enjoyed sharing Boulder with my brothers, Jeff and Brian, but I was here
first for the record. I love you all a lot.

Contents

Chapter

1 Introduction

1

1.1

Accretion Disc Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Vertical Disc Structure and Spectral Hardening . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.3

Magnetorotational Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.4

Local Simulations of Accretion Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.5

Black Hole X-ray Binary Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2 Spectral Hardening as a Viable Alternative to Disc Truncation in Black Hole State Transitions

22

2.1

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3

Observations and Data Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.3.1

GX 339–4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.3.2

Rossi X-ray Timing Explorer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

Multi-colour Disc Blackbody Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.4.1

The Standard MCD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.4.2

Zero-Torque Boundary Condition and Disc Vertical Structure . . . . . . . . . . . . . .

35

Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.5.1

Requirement of a Disc Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.5.2

Sensitivity of Spectral Fits to the Continuum Model . . . . . . . . . . . . . . . . . . .

40

2.4

2.5

vii

2.6

2.7

2.5.3

Disc Evolution Results from Spectral Fitting . . . . . . . . . . . . . . . . . . . . . . .

46

2.5.4

A Hot Inner Disc Can Masquerade as an Increased fcol

. . . . . . . . . . . . . . . . .

52

2.5.5

Disc Temperature and Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.5.6

Low Frequency Quasi-Periodic Oscillations . . . . . . . . . . . . . . . . . . . . . . . .

64Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

2.6.1

Implications of a Variable Colour Correction . . . . . . . . . . . . . . . . . . . . . . .

66

2.6.2

Physical Range of the Colour Correction Factor . . . . . . . . . . . . . . . . . . . . . .

74

2.6.3

Limitations of Simplistic Accretion Disc Models . . . . . . . . . . . . . . . . . . . . . .

78

Summary & Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

3 A Physical Model for State Transitions in Black Hole X–ray Binaries

89

3.1

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

3.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

3.3

Accretion picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

3.3.1

Disc evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

3.3.2

State transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

3.3.3

Disc–jet–corona coupling

98

3.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Discussion & conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4 Black Hole Spin Measurements are Sensitive to Accretion Disc Spectral Hardening

104

4.1

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.3

Disc Continuum Fitting Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.4

Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.1

Continuum Fitting Technique: Individual Parameter Uncertainties . . . . . . . . . . . 110

4.4.2

Effects of Spectral Hardening on Black Hole Spin Measurements: Model X-ray Binaries 113

4.4.3

Effects of Spectral Hardening on Black Hole Spin Measurements: Observed X-ray
Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

viii
4.5

4.6

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.5.1

Continuum Fitting Technique in Practice . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.5.2

Measuring fcol with the Continuum Fitting Technique . . . . . . . . . . . . . . . . . . 126

Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5 Accretion Disc Dynamo Activity in Local Simulations Spanning Weak-to-Strong Net Vertical Magnetic Flux Regimes

130

5.1

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.3

Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.1

Shearing Box Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.3.2

Initial Setup and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.3.3

Notation Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.4

Basic Properties of MRI Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.5

Properties of the MRI-Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.6

5.7

5.5.1

Toroidal Magnetic Field Reversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.5.2

Production and Escape of Toroidal Magnetic Field . . . . . . . . . . . . . . . . . . . . 158

5.5.3

Vertical Disc Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.6.1

MRI-Dynamo Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.6.2

Quasi-Periodic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.6.3

Spectral Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6 Strongly Magnetized Accretion Discs Require Poloidal Flux

170

6.1

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.3

Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

ix
6.3.1

Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.3.2

Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.4

Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.5

Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7 Conclusions

185

Bibliography

188

Appendix

A Probability Density Functions and Change of Variables

199

A.0.1 Flux Normalization PDF, f (Kflux ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
A.0.2 Inner Disc Radius PDF, f (rin ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
A.0.3 Black Hole Spin Parameter PDF, f (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
A.0.4 Innermost Stable Circular Orbit PDF, f (rISCO ) . . . . . . . . . . . . . . . . . . . . . 202
B Effects of Individual Parameter Uncertainties on Black Hole Spin Measurements

203

x

TablesTable

2.1

Table summarizing the key features of each GX 339–4 state transition. . . . . . . . . . . . . .

27

2.2

Results from using a pseudo-Newtonian model to fit simulated disc spectra of a Kerr black hole. 57

2.3

Best-fit parameters for all observations of GX 339–4 that required a disc spectral component.

4.1

Parameters for black hole X-ray binaries with vetted continuum fitting spin measurements. . 120

5.1

Basic information for the suite of shearing box simulations. . . . . . . . . . . . . . . . . . . . 136

5.2

Properties of the MRI turbulence, all evaluated at the disc mid-plane and time-averaged. . . 150

5.3

Properties of the MRI-dynamo for each simulation. . . . . . . . . . . . . . . . . . . . . . . . . 151

6.1

Summary of shearing box simulations with outflowing and periodic vertical boundary conditions.175

88

Figures

Figure1.1

Location of the innermost stable circular orbit as a function of black hole spin. . . . . . . . .

6

1.2

Schematic of accretion disc atmospheres with and without vertical magnetic pressure support. 10

1.3

Emergent accretion disc spectrum with and without vertical magnetic pressure support. . . .

1.4

Magnetorotational instability analog of a spring connecting two masses orbiting a central object. 13

1.5

Schematic diagram of the magnetorotational instability in an accretion disc. . . . . . . . . . .

14

1.6

The local approximation adopted by shearing box simulations of accretion discs. . . . . . . .

18

1.7

General behavior of black hole transients in the hardness-intensity and hardness-rms diagrams. 21

2.1

Hardness-intensity diagram for all state transitions of the black hole X-ray binary GX 339–4.

32

2.2

Accretion disc spectral component for selected RXTE observations of GX 339–4. . . . . . . .

41

2.3

Flux stealing effect from the disc spectral component for different Comptonization models. . .

43

2.4

Fractional flux contributed by the disc spectral component for different Comptonization models. 44

2.5

Relative errors on the normalization when systematic errors are or are not added to the data.

47

2.6

Relative measurements of the disc flux normalization for all observations. . . . . . . . . . . .

48

2.7

Temporal evolution of the colour correction factor when the inner disc radius was held at 6 Rg . 50

2.8

Relative measurements of fcol evolution from adopting the pow/bknpow Comptonization model. 53

2.9

Relative measurements of fcol evolution from adopting the comptt Comptonization model. . .

54

2.10 Relative measurements of fcol evolution from adopting the simpl Comptonization model. . .

55

11

2.11 Unabsorbed X-ray disc flux vs. maximum disc effective temperature assuming a constant fcol . 58

xii
2.12 Relative measurements of fcol vs. the maximum disc colour temperature. . . . . . . . . . . .

60

2.13 Unabsorbed X-ray disc flux vs. maximum disc effective temperature allowing a variable fcol . .

61

2.14 Time evolution of the inner disc radius derived from the ezdiskbb and diskpn disc models. .

622.15 Temporal evolution of the Keplerian radius derived from low-frequency QPO measurements. .

65

2.16 Coronal optical depth vs. maximum disc colour temperature. . . . . . . . . . . . . . . . . . .

70

2.17 Coronal optical depth vs. coronal temperature. . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.1

Schematic diagram of the black hole state transition model based on disc tearing. . . . . . . .

92

3.2

Column density projection for a tearing disc simulation with a disc–spin misalignment of 45◦ .

96

3.3

Column density projection showing the disc structure from a misaligned rings simulation. . .

99

4.1

Discrepancies in X-ray binary black hole spin measurements from two independent techniques. 108

4.2

Effect of individual parameter uncertainties on black hole spin measurements. . . . . . . . . . 112

4.3

Effect of fcol uncertainties on black hole spin measurements. . . . . . . . . . . . . . . . . . . . 114

4.4

Effect of adopting mildly incorrect fcol values on black hole spin measurements. . . . . . . . . 116

4.5

Effect of adopting erroneous fcol values on black hole spin measurements. . . . . . . . . . . . 119

4.6

Inner disc radius PDFs with and without marginalizing over fcol for real X-ray binaries. . . . 122

4.7

Inner disc radius PDFs with and without marginalizing over fcol for real X-ray binaries (cont.).123

5.1

Time evolution of the effective viscosity parameter for each shearing box simulation. . . . . . 142

5.2

Effective viscosity parameter vs. initial disc mid-plane plasma-β. . . . . . . . . . . . . . . . . 143

5.3

Volume renderings of the ratio of gas pressure-to-magnetic pressure at a snapshot in time. . . 144

5.4

Vertical profiles of the time-averaged plasma-β for each simulation. . . . . . . . . . . . . . . . 145

5.5

Vertical profiles of the horizontally- and time-averaged gas density fluctuations. . . . . . . . . 146

5.6

Normalized probability density function of gas density relative to its mean value. . . . . . . . 148

5.7

Space-time diagrams of the horizontally-averaged toroidal magnetic field. . . . . . . . . . . . 152

5.8

Zoomed in toroidal field space-time diagrams for a weakly and strongly magnetized simulation.153

5.9

Magnetic field structure during an MRI-dynamo reversal for a weakly magnetized simulation. 153

xiii
5.10 Magnetic field structure during an MRI-dynamo reversal for a strongly magnetized simulation. 154
5.11 Time evolution of the volume-averaged toroidal magnetic field and its power spectrum. . . . . 155
5.12 Vertical profiles of the horizontally- and time-averaged rise speed of the toroidal magnetic flux.159
5.13 Vertical profiles of the horizontally- and time-averaged gas pressure and magnetic pressure. . 163

6.1

Vertical profiles of plasma-β for simulations with outflowing and periodic boundary conditions.178

6.2

Space-time diagrams of the horizontally-averaged toroidal and radial magnetic field. . . . . . 179

6.3

Magnetic field structure for simulations with outflowing and periodic boundary conditions.

. 181

Chapter 1

Introduction

An accretion disc is an astrophysical machine that can be incredibly efficient at converting gravitational
potential energy into electromagnetic radiation (and sometimes kinetic energy in the form of outflows). This
property of accretion discs allows Galactic X-ray binaries to be discovered, which sometimes turn out toharbor a black hole accretor. The high radiative efficiency of accretion discs also led astronomers to realize
their role as the central engines of active galactic nuclei and quasars, which are among the most powerfulobjects in the Universe.
But, why would one expect an accretion disc to form around a black hole in the first place? In thecase of radial inflow of gas at low mass accretion rates onto a black hole (e.g., Bondi, 1952), gravity is the
only important force in dictating the dynamics of the flow. However, in realistic situations the accreting gas
will have non-negligible specific angular momentum, which the gas must somehow lose if it is to be accreted.
This need for accreting material to lose angular momentum, along with the small spatial scale of black
holes compared to their fuel reservoir, gives rise to accretion discs. Consider a binary system consisting of a
black hole primary and a donor star secondary that has filled its Roche lobe. As the gas from the secondary
streams toward the black hole, centrifugal forces on the material of a given angular momentum increase faster(Fcent ∝ R−3 ) than the force of gravity from the black hole (Fgrav ∝ R−2 ) (e.g., Blaes, 2014). Consequently,
the flow will be rotationally supported and a disc of gas will naturally form around the black hole. Inward
accretion of disc material toward the black hole and the liberation of gravitational binding energy can thenproceed if angular momentum is transported outward.
It would seem that identifying the angular momentum transport process that is ultimately responsible

2
for mediating energy conversion (i.e., gravitational binding energy → radiation) is of central importance inaccretion disc theory. While this is true and we will touch on this soon, it is important to highlight perhaps
the most salient feature of accretion disc theory. Namely, that the predicted spectrum emitted by the discrequires no knowledge of the details of the angular momentum transport mechanism (Shakura & Sunyaev,
1973).

1.1

Accretion Disc Spectrum
The spectrum of an accretion disc that is geometrically thin and optically thick in the direction per-pendicular to the disc mid-plane can be derived from considering mass and angular momentum conservation
within a given disc annulus (e.g., Shakura & Sunyaev, 1973; Pringle, 1981). The subsequent derivations
closely follow Frank et al. (2002). We adopt cylindrical coordinates (R, φ, z) with the disc mid-plane atz = 0 and a black hole of mass M residing at the origin. The motion of the disc material is circular orbits
around the black hole with angular velocity profile Ω (R). The surface area of one face of a disc annulus
with inner edge at radius R and outer edge at R + ∆R is,hi2
A = π (R + ∆R) − R2 ' 2πR∆R,(1.1)

where we’ve assumed the limit of small ∆R (i.e., neglecting second order terms). We assume the disc material
has a small radial drift velocity vR = dR/dt, where inward (outward) flow corresponds to negative (positive)vR by convention. The mass per unit surface area of the disc is the surface density Σ (R, t), which is obtained
locally by integrating the gas density ρ in the z-direction,
1
Σ=
2

Z +∞
ρdz,

(1.2)

−∞

A subtle but important point to mention is that both A and Σ as we define them above consider one faceof the disc. The rate of change of mass and angular momentum in the disc comes from considering the net
flow of each of these conserved quantities into/out of a given annulus.
The mass contained in an annulus is 2πR∆RΣ and the mass conservation equation in the limit ∆R → 0
is (Frank et al., 2002),
R

∂Σ
∂
+
(RΣvR ) = 0.
∂t
∂R

(1.3)

3
In a steady state (i.e., ∂/∂t = 0) this becomes,Ṁ = 2πRΣ (−vR ) ,(1.4)

where Ṁ is the mass accretion rate and we recall that vR < 0.
The angular momentum contained in an annulus is 2πR∆RΣR2 Ω. In addition to the net flow ofangular momentum through the disc annulus, the net angular momentum transport from viscous torques
G (R, t) from neighboring annuli must also be included. The torque exerted by an outer ring on an innerring at radius R is given by,
G (R) = 2πRνΣR2

dΩ
,
dR

(1.5)

where ν is the fluid viscosity. Note that if dΩ/dR < 0 then G (R) is negative, which means that inner annuli
lose angular momentum to the outer annuli and gas will spiral inwards. The net torque on an annulus ofgas between R and R + ∆R is then,G (R + ∆R) − G (R) =

∂G
∆R.
∂R

(1.6)

We suppose that the disc has a Keplerian orbital velocity profile (i.e., balance between gravity and thecentripetal force for a circular orbit), vφ = RΩK (R) where we’ve adopted Ω (R) = ΩK (R) =p
GM/R3 .

Taking all of this into consideration, the angular momentum conservation in the limit ∆R → 0 brings us
to an equation describing the temporal evolution of the surface density in a Keplerian accretion disc (Franket al., 2002),
∂Σ
3 ∂
=
∂t
R ∂R



R1/2

i
∂ h
νΣR1/2 .
∂R(1.7)

In a steady state with ∂/∂t = 0 and applying the boundary condition dΩ/dR = 0 (i.e., zero torque) at the
inner disc edge Rin , the angular momentum conservation equation 1.7 becomes,"1/2 #
Ṁ
Rin
νΣ =
1−
.
3π
R

(1.8)

We will have cause to use Equation 1.8 shortly, but we must first derive the energy loss rate from thedisc (i.e., the viscous dissipation rate). The rate of work done on the gas by the viscous torque is (Franket al., 2002),
Ω



∂G
∂
dΩ
∆R =
(GΩ) − G
dR.
∂R
∂R
dR

(1.9)

4
The first term on the right hand side is the rate at which rotational energy is convected through the discannulus. The second term −GdΩ/dR is the local rate that mechanical energy goes into heating the gas.
Therefore, viscous torques cause viscous dissipation of energy within the gas — which must ultimately beradiated away from the disc faces — at the rate GdΩ/dR. The rate of viscous dissipation per unit surface
area is then,D (R) =

Total dissipation rate (due to viscous torques) per annulus width ∆R
Disc emitting area (both faces)

G (dΩ/dR) ∆R
2 × 2πR∆R
9 GM
=
νΣ,
8 R3
=

(1.10)

where we used Equation 1.5 and a Keplerian angular velocity Ω = ΩK to arrive at the expression on thebottom line. Inserting Equation 1.8 into Equation 1.10 gives,"1/2 #
3GM Ṁ
Rin
D (R) =
1−
.
8πR3
R

(1.11)

Arriving at this dissipation rate per unit disc face area took some effort, but the implication is profound:the energy flux escaping the disc is independent of viscosity!
We are now in a position to derive the spectrum emitted by a face of an accretion disc. Assuming thatthe accretion disc is geometrically thin and optically thick disc in the z-direction, each element of a disc face4

will radiate a local blackbody spectrum with radial temperature profile T (R). Therefore, σ [T (R)] = D (R)with σ being the Stefan-Boltzmann constant and plugging in D (R) from Equation 1.11 gives,(
T (R) ="

1/2 #)1/4
3GM Ṁ
Rin
1−.
8πR3 σR(1.12)

From Equation 1.12 one can show that the maximum disc temperature occurs at R = (49/36) Rin and isgiven by,
63/2
Tmax = 7/4
7

3GM Ṁ
8πRin σ!1/4.

(1.13)

Defining the dimensionless radius r̃ ≡ R/Rin and re-parametrizing in terms of Tmax , Equation 1.12 becomes,
T (r̃) =


1/4
77/4
−3/4
−1/2
T
r̃
1
−
r̃
.
max
63/2

(1.14)

5
The disc spectrum emitted over the area of a disc face then comes from integrating the Planck function overthe disc area (e.g., Zimmerman et al., 2005),

FE = Kflux

4πE 3
h3 c2

Z ∞
1



r̃
dr̃ erg s−1 cm−2 keV−1 ,exp [E/kT (r̃)] − 1(1.15)

where,

Kflux =RinD

2
cos (i) ,(1.16)

is the flux normalization parameter for a source distance D and disc inclination i. In some circumstances it
is useful to replace Rin with rin ≡ Rin /Rg , where Rg = GM/c2 is the gravitational radius.
Amazingly, not only is the disc spectrum independent of viscosity, but it is also fully described by
two parameters: Tmax is a shape parameter (Equation 1.13) and Kflux is a normalization constant (Equation
4.3), which itself depends on intrinsic properties of the source. Equation 1.15 is a composite of the local
blackbody spectrum emitted by each disc annulus according to a radial temperature profile. Therefore, thisaccretion disc spectral model is known as the multi-colour disc blackbody (Mitsuda et al., 1984; Makishima
et al., 1986).
The accretion disc spectrum from a supermassive black holes peaks in the ultraviolet and therefore
will be heavily attenuated by Galactic absorption. However, the spectrum from an accretion disc around astellar mass black hole peaks in the soft X-ray band (∼ 1 keV), which makes black hole X-ray binaries ideal
sources for studying accretion discs. High quality CCDs on-board X-ray observatories (e.g., XMM-Newton,
Chandra) are at best sensitive down to ∼ 0.1 keV; therefore, observers are often faced with the challenge of
relying on the Wien tail to fit the X-ray disc spectrum. In practice, X-ray data can only constrain two freeparameters, which makes the multi-colour disc blackbody solution immensely powerful.
If the black hole mass, distance to the source, and disc inclination are known independently, thenmeasuring Kflux becomes equivalent to constraining the dimensionless inner disc radius rin (e.g., Zhang et al.,
1997; McClintock et al., 2006). One might then reasonably assume that the inner edge of the accretion disc,rin , coincides with the innermost stable circular orbit (ISCO), rISCO , inside of which the gas must dynamicallyplunge into the black hole. This rin = rISCO assumption is physically motivated because the gas viscoustimescale is long relative to the free-fall timescale for gas at the ISCO; therefore, the gas interior to rISCO

6

9

8

7

rISCO

6

5

4

3

2

1
−1.0

−0.5

0.0

a

0.5

1.0

Figure 1.1: Location of the innermost stable circular orbit, rISCO , as a function of the dimensionless blackhole spin parameter, a (Bardeen et al., 1972). A prograde (retrograde) system consists of the black holeand disc angular momenta being aligned (anti-aligned). A maximally-spinning retrograde system (a = −1)
has rISCO = 9, a non-spinning black hole (a = 0) has rISCO = 6, and a maximally-spinning prograde system(a = 1) has rISCO = 1, where rISCO is given in units of gravitational radii Rg = GM/c2 and M is the blackhole mass.

7
does not have time to radiate appreciably before being accreted. Additionally, numerical simulations ofmagnetized accretion discs demonstrate that a magnetic field is unable to couple the disc material to thegas interior to the ISCO (Armitage et al., 2001).
A robust measurement of the location of the ISCO is important because it turns out that rISCO is a
monotonic function of the non-dimensional black hole spin parameter, a = cJ/ GM 2 , where J is the blackhole angular momentum (see Figure 1.1; Bardeen et al., 1972). Therefore, measuring rISCO is equivalent tomeasuring black hole spin, a. This demonstrates that accretion discs, while interesting in their own right,
also provide powerful diagnostics from which fundamental properties of black holes can be probed. How
the black hole spin measurements are affected by uncertainties in the disc vertical structure is the topic ofChapter 4.
Another well-known interesting result of accretion disc theory comes from integrating Equation 1.11
from R = Rin to Rout → ∞ over both faces of the disc to obtain the bolometric disc luminosity,
Ldisc =

GM Ṁ
.
2Rin

(1.17)

The accretion luminosity corresponding to converting all of the gravitational binding energy of the accretion
gas into radiation is Lacc = GM Ṁ /Rin (e.g., Frank et al., 2002). Therefore, half of the available accretion
power is radiated away by the disc during the accretion process and half goes into kinetic energy of the gas at
the inner disc edge. This means that for an accretion disc around a black hole with an inner edge extending
down to the ISCO, half of the available accretion energy is presumably swallowed by the hole!

1.2

Vertical Disc Structure and Spectral Hardening
The thorn in the side of the multi-colour disc blackbody model (Equation 1.15) from the previous

section is that the effects of the disc atmosphere are neglected. In other words, the details of the disc verticalstructure — temperature, density, ionization state — where the optical depth τ . 1 will modify the spectrum
emitted at the disc photosphere. Modeling the disc atmosphere quickly becomes quite complicated, with therelevant physics including free-free/bound-free opacities and electron scattering (i.e., incoherent Compton
scattering). Adopting the methodology of Ross et al. (1978, 1992) to self-consistently solve the radiative

8
transfer along with the vertical disc structure, Shimura & Takahara (1995) demonstrated that the complicatedeffects of the disc atmosphere on the emitted spectrum can be approximated by a phenomenological constantfactor. This is the spectral hardening factor fcol (also called the colour correction factor). A pure blackbody
spectrum has fcol = 1. The concept of exploiting the simplistic and practical fcol parametrization toincorporate the effects of the disc vertical structure is central to this thesis.
It is important to distinguish between the effective temperature and the colour temperature, which
are related by Tcol = fcol Teff . The effective temperature depends on the source intensity and is described by
the familiar blackbody relation for the emergent flux (e.g., Rybicki & Lightman, 1979),
4
F = σTeff.

(1.18)

When a spectrum has a blackbody shape but the incorrect normalization, the color temperature corresponds
to the blackbody that best describes the spectrum without regard to the vertical scale; therefore, Tcol depends
only on the shape of the observed spectrum (Rybicki & Lightman, 1979). The spectral hardening factor canbe incorporated into a modified multi-colour disc blackbody model, which is derived as before by replacing
each occurrence of T (R) with Tcol (R) = fcol T (R). Then, Equation 1.15 becomes,
FE = Kflux

4πE 3
h3 c2

Z ∞
1



r̃
dr̃ erg s−1 cm−2 keV−1 ,exp [E/kTcol (r̃)] − 1(1.19)

where now,
Kflux =

1
4
fcol



Rin
D

2
cos (i) ,(1.20)

The colour temperature describes the observed accretion disc spectrum, while fcol tells us by how much totranslate the spectrum in energy space in order to recover the underlying photospheric spectrum. Sophisticated disc atmospheric models tend to converge on the range 1.4 . fcol . 2.0 (e.g., Shimura & Takahara,
1995; Davis et al., 2005). However, fcol can potentially be significantly enhanced if vertical magnetic pressure
support is considered.
Introducing a magnetic pressure contribution in the disc to stave off the vertical gravity due to theblack hole means that less gas pressure support is now required in order to achieve vertical equilibrium.
This diminished gas pressure support means the vertical density gradients are less steep than they would be9without magnetic pressure support. Therefore, when magnetic support is included, the density scale heightincreases and the disc atmosphere becomes vertically extended. The location of the effective photosphere,zeff , is the important location to consider for determining the observed disc spectrum and is set by a constantvalue of the effective optical depth, τeff ' 1. Therefore, as shown schematically in Figure 1.2, an astronomer
observing a magnetically supported disc will encounter the effective photosphere at a larger disc height
and lower density compared to a disc without magnetic support. The gas density determines the relative
contributions to the total opacity from absorption (κabs ∝ ρ) and electron scattering (κes ∝ ρ0 ), with a lower
density reducing the ratio κabs /κes . Taking all of this together, a disc with magnetic pressure support has
a lower density at the effective photosphere, ρeff and, therefore, enhanced electron scattering, which acts toharden the disc spectrum, as shown in Figure 1.3. Blaes et al. (2006) used numerical simulations to showthat this hardening is modest for very weakly magnetized discs (dashed line), while the analytic argumentsof Begelman & Pringle (2007) predict substantial hardening for strongly magnetized discs (dotted line).
Although magnetic fields are undeniably fundamental to disc accretion, their potentially important
effects on the observed disc spectrum have gone largely unappreciated. Chapter 2 addresses the possibility
that the evolution of the inner disc regions during black hole state transitions is attributed to a variablevertical disc structure (as parametrized by fcol ).

1.3

Magnetorotational Instability
In a landmark paper, Balbus & Hawley (1991) identified the magnetorotational instability (MRI;

Velikhov, 1959; Chandrasekhar, 1961) as the physical mechanism that mediates angular momentum transport
in accretion discs. Accretion discs around black holes are expected to be highly ionized due to the hightemperature of the gas, especially in the inner disc regions. Therefore, if a magnetic field is present andthe fluid is a perfect conductor (i.e., resistivity η = 0), then the magnetic flux is “frozen-in” to the plasma.
In other words, the plasma and the magnetic field move together. Because magnetic fields have tension,
any displacement of a fluid element perpendicular to a magnetic field line results in an attractive force thatis proportional to the displacement. Usually, such a force is restorative and acts to stabilize the plasma;
however, it turns out that this attractive force can be destabilizing if the fluid is rotating. The MRI is a

10

Figure 1.2: Vertical magnetic pressure support causes the disc atmosphere to become vertically extended.
Consequently, the effective photosphere is encountered at a larger disc height (zeff ) and lower density (ρeff )compared to a disc without magnetic support. This lower density leads to enhanced electron scattering by
reducing the ratio of absorption-to-scattering opacities (κabs /κes ∝ ρ/ρ0 ). Therefore, an accretion disc withmagnetic pressure support (bottom panel) will have a harder spectrum than one without magnetic pressure
support (top panel).

11

Figure 1.3: The enhanced electron scattering resulting from vertical magnetic pressure support in the discleads to a hardened disc spectrum. Numerical simulations of very weakly magnetized discs show thatthe spectral hardening is modest (dashed line) (Blaes et al., 2006). Simplified analytic models of strongly
magnetized discs predict drastic spectral hardening (dotted line) (Begelman & Pringle, 2007), but the details
remain unknown. Adapted from Blaes et al. (2006).

12consequence of this unintuitive behavior.
An intuition for how the MRI operates can be gleaned by the almost exact analogy to the mechanicalsystem shown in Figure 1.4 of a spring under tension in a differentially rotating disc. Imagine two masses
connected by a spring in a Keplerian orbit about a central object. Allowing the inner mass to orbit slightly
closer to the center than the outer mass results in the inner mass having a greater angular velocity (leftpanel), but smaller angular momentum, than the outer mass. In response to this displacement, the spring
pulls backward on the inner mass while pulling forward on the outer mass (right panel). The inner mass
therefore experiences a negative torque, causing it to lose angular momentum and move further inward
where the angular momentum of an orbit is smaller. Conversely, the outer mass experiences a positive
torque, causing it to gain angular momentum and move further outward where the angular momentum of anorbit is larger. Now, the inner (outer) mass has even higher (lower) angular momentum than it had before,
the spring stretches even more, the torques exerted on the masses by the attractive force of the spring areeven greater, the masses must be displaced even further apart, and we have a runaway process! Figure 1.5
shows a schematic of the MRI operating in an accretion disc, where a vertical magnetic field threading thedisc replaces the instructive spring analogy and fluid parcels replace the masses.
The MRI is a weak field instability, meaning that so long as a vertical magnetic field is present —however weak — an accretion disc with the appropriate differential rotation profile (e.g., Keplerian) will be
unstable to the MRI. The stability requirement for MRI assuming a Keplerian orbital velocity profile in thedisc is (Balbus & Hawley, 1998),
2
vA,z
&

where cs is the gas sound speed and vA,z =

p

6
cs ,
π2

(1.21)

Bz2 / (4πρ) is the Alfvén speed associated with the verticalmagnetic field Bz and gas density ρ. A useful parametrization for the magnetization of a plasma is the βparameter, defined as the ratio of gas pressure-to-magnetic pressure,β≡

pgas
ρc2 /γ= 2 s.
pB
B / (8π)(1.22)

Combining Equations 1.21 and 1.22 and arbitrarily choosing the adiabatic index γ = 5/3, we find that the
MRI stability requirement becomes βz . 2.0. Therefore, the MRI is a weak field instability in the sense that

13

Figure 1.4: Magnetorotational instability analog of two masses connected by a spring in Keplerian-likeorbits about a central object. The arrows on the masses in the left panel represent the angular velocity,which increases for orbits closer to the central object unlike the angular momentum, which decreases fororbits closer to the central object. The arrows on the masses in the right panel represent the attractive
force of the spring pulling backward on the inner mass and forward on the outer mass. A description of the
runaway process represented by this schematic is given in the text.

14

Figure 1.5: Schematic diagram of the magnetorotational instability in a Keplerian-like accretion disc. The
red dotted line marks the initial orbit of the two parcels of gas, shown by the translucent grey boxes. Initially,
these two fluid elements are connected by a vertical magnetic field line, shown by the dashed vertical line.
When one parcel is perturbed inward, the magnetic field is also displaced due to the frozen-in condition.
The attractive force from magnetic tension acts to negatively torque this inner parcel. This causes angularmomentum loss and the parcel moves further inward, while the outer parcel is instead positively torqued
and moves further outward due to angular momentum gain. The process repeats starting from this new
configuration where the torques exerted on the parcels by the increased magnetic tension will be more
extreme than before; therefore, the disc is dynamically unstable.

15
the vertical magnetic field must be subthermal if the disc is to be MRI-active. The tension associated witha sufficiently strong vertical magnetic field would suppress the MRI.
As we will discuss below, a dynamo process generates a dominant toroidal magnetic field in theaccretion disc. Therefore, the disc can still achieve a state of strong magnetization (i.e., β . 1) thanks tothis dominant toroidal field, while the vertical field can remain unstable to the MRI. Pessah & Psaltis (2005)
showed that magnetic tension due to the toroidal magnetic field will quench the MRI when the Alfvén speed
exceeds the geometric mean of the Keplerian speed and the gas speed: vA &√vK cs for stability. From thisrelation, we can obtain a limit on the maximal magnetic field strength that an MRI-active accretion disc canachieve. Assuming that the toroidal field is dominant (i.e., B ' Bφ ), we can rewrite the Pessah & Psaltis(2005) stability criterion as,
Bφ2 & 4πρvK cs .

(1.23)

ρ=Σ
H(1.24)

Σ=Ṁ
,
3παcs R

(1.25)

Combining this with the relations,

we obtain the maximal magnetic field strength,
9 −8/17Bφ,max ' 2.9 × 10 α

Ṁ
ṀEdd

!31/68 MM−1/2 R
Rg

−179/136
Gauss.

(1.26)

Recently, Miller et al. (2016) placed a lower limit of 103−5 Gauss on the magnitude of the magnetic fieldthat drives the accretion disc wind in the black hole X-ray binary GRS 1915+105. Adopting a typical windlaunching radius R ∼ 102−4 Rg , mass accretion rate in a disc wind-producing state Ṁ ∼ 0.1ṀEdd , blackhole mass M ∼ 10M , and α ∼ 0.1, Equation 1.26 implies a maximal magnetic field strength in the discof ∼ 104−6 Gauss. Therefore, the lower limit measured by Miller et al. (2016) is suggestive of strongly
magnetized accretion discs in black hole X-ray binaries. How strongly magnetized discs can form and persist
is the topic of Chapter 6.
We established that a black hole accretion disc is expected to be MRI-unstable, but how does thisinstability develop and produce an effective viscosity? After the initial linear growth described above, the16
MRI will enter a non-linear growth phase that causes the accretion disc to enter a self-sustaining turbulent
steady state, wherein an effective viscosity is generated that drives accretion. Therefore, in order to understand angular momentum transport in accretion discs, one must understand MRI-driven turbulence. Thisrequires numerical simulations of accretion discs with high temporal and spatial resolution.

1.4

Local Simulations of Accretion Discs
Before the MRI was identified as the generator of turbulent stresses that redistribute angular momen-

tum within an accretion disc, Shakura & Sunyaev (1973) introduced the α-prescription,wRφ = αP,(1.27)

which relates the local (vertically-averaged) stress wRφ to the local (vertically-averaged) pressure P — usuallytaken to be the total pressure. With turbulence in mind as the source of the effective viscosity, Shakura &Sunyaev (1973) suggest the relation,
ν = αHcs ,(1.28)

where the disc scale-height H roughly corresponds to the maximum turbulent eddy size and the sound speed
cs corresponds to the maximum expected velocity of turbulent motions, as shocks would likely thermalize
supersonic turbulence. Therefore, we expect α to lie somewhere between 0 (no accretion) and ∼ 1. The
practical importance of Equation 1.28 is that it allows one to solve for the radial structure of verticallyintegrated, geometrically thin accretion discs (see e.g., Frank et al., 2002). However, the validity of the
α-prescription could not be tested until the advent of sophisticated numerical simulations of accretion discs
in the local limit.
The ultimate goal of numerical simulations of MRI turbulence is to replace the phenomenological αprescription with observationally testable models rooted in real physics. While this ambitious goal remains
far off, tremendous progress has been made since the first studies of MRI turbulence by Hawley et al.
(1995). Ideally, simulations of MRI turbulence in accretion discs would capture all relevant spatial scales
from the large-scale energy injection associated with the differential rotation to the microscopic dissipation
scales associated with the plasma viscosity and resistivity. These simulations would also resolve the short17
timescales associated with wave propagation to the very long timescales associated with viscous evolution.
Of course, this is an absurdly tall order even given modern-day supercomputing resources. Instead, theapproach to numerical studies of MRI turbulence has been to devise local “shearing box” simulations andhope that the relevant physics is properly captured and converged, provided that one uses enough grid zones.
The shearing box (see Figure 1.6; Hawley et al., 1995) adopts the geometry of a Cartesian rectangular
patch inserted into the differentially rotating fluid disc. The box co-rotates locally with the background shear
flow and includes the effects of Coriolis forces, centrifugal forces, and gravity, while neglecting curvature andradial gradients. Therefore, any non-local behavior of the turbulence cannot be captured by the shearingbox (e.g., Simon et al., 2012). Boundary conditions are perfectly periodic in the azimuthal direction andshearing periodic in the radial direction, as dictated by the differential rotation profile of the disc. There ismore freedom in the choice of vertical boundary conditions, with outflowing boundary conditions now being
generally preferred over periodic boundary conditions. Next, we highlight some of the major findings fromshearing box simulations of MRI turbulence relevant to this thesis.
Shearing box simulations conserve any initial net vertical magnetic flux. A net vertical flux provides areservoir from which an MRI dynamo process can more efficiently generate toroidal field. This increases the
MRI
MRI turbulent stress wRφarising from correlated radial and azimuthal fluctuations in the magnetic field

and fluid velocity, given by,
MRI
wRφ
=



BR Bφ
ρvR δvφ −4π

,(1.29)

where δvφ is the local turbulent azimuthal velocity (i.e., the background shear velocity is subtracted off).
The expected increased stress due to increased net vertical magnetic flux is seen in shearing box simulations(e.g., Hawley et al., 1995; Pessah et al., 2007) and in local regions of global simulations (Sorathia et al.,2010). Encouragingly, shearing box simulations show that the stress from the combined Reynolds andMaxwell components of Equation 1.29 is well-described by the α-prescription of Equation 1.27, with a tightcorrelation emerging between α and the imposed net vertical magnetic flux (e.g., Blackman et al., 2008).
Including vertical stratification in shearing box simulations due to the vertical gravity of the blackhole allows for magnetic buoyancy, which is essential to the accretion disc dynamo. This dynamo mechanism
channels the accretion power liberated by the MRI into a self-sustaining, ordered toroidal magnetic field.

18

Figure 1.6: Gas density renderings showing the concept of the shearing box as a local approximation of a
global accretion disc.

19
The dynamo quasi-periodically creates toroidal field of alternating sign that buoyantly rises through the discatmosphere and escapes, which has been seen in both local (e.g., Brandenburg et al., 1995) and global (e.g.,
O’Neill et al., 2011) simulations. We emphasize that when thinking about disc magnetization, it is important
not to conflate the imposed net vertical magnetic flux and the dynamo-generated toroidal magnetic field.
How the properties of MRI turbulence and dynamo activity depend on net vertical magnetic flux is the topicof Chapter 5.
Ultimately, global simulations of accretion discs are required to test the results of the shearing box
simulations. Indeed, much progress has already been made toward understanding accretion discs and MRI
turbulence from global simulations (e.g., Fragile, 2014). However, computational limitations require that thisthesis adopts the shearing box approximation in order to perform long-duration (i.e., hundreds of orbits)
accretion disc simulations that spatially resolve MRI turbulence.

1.5Black Hole X-ray Binary Phenomenology
Accretion onto compact objects, particularly black holes, is an inherently variable process. For su-

permassive black holes that are intricately linked to galaxy growth and evolution, the long timescales foraccretion variability are observationally inaccessible. Conversely, accretion variability in black hole X-ray
binaries operates on comparatively short timescales, showcasing dramatic transitions between quiescent andluminous states. Supposing that variability in accreting supermassive black holes can be treated in a selfsimilar fashion to variability in black hole X-ray binaries (e.g., Cui, 2004) allows astronomers to circumventthe problem of observational inaccessibility and proceed to study black hole accretion physics.
The most dramatic variability observed in black hole X-ray binaries involves transitions between astate of low luminosity, where the X-ray spectrum is dominated by a non-thermal high-energy component,
and a state of high luminosity, where the X-ray spectrum is dominated by a soft thermal component. The
observational approach toward black hole X-ray binary state transitions has been to devise a powerful modelindependent phenomenology based on spectral and timing properties (e.g., Remillard & McClintock, 2006;
Belloni, 2010). General trends emerge from this phenomenology, with the most notable and robust being themysterious hysteresis cycle in the X-ray intensity vs. spectral hardness diagram (Homan et al., 2001). Shown

20
in Figure 1.7, an outburst begins with the system climbing out of a low luminosity, spectrally hard state into aspectrally soft state of high luminosity, sometimes nearing the Eddington limit. Instead of retracing this pathon the hardness-intensity diagram back down to the pre-outburst state, the system curiously decays to its
original state along a horizontal track of lower luminosity than it did during the outburst rise. The ‘q’-shaped
hysteresis path traversed in intensity vs. hardness for a full black hole state transition cycle is sometimes
affectionately referred to as the “turtle head” diagram (Fender et al., 2004). The timescale to complete
the turtle head is typically between many months and a year. The phenomenological spectral/timing states
involved in a black hole state transition cycle are marked on the hardness-intensity diagram of Figure 1.7,
which will be a useful reference while reading this thesis.
Complicating matters, no two black hole X-ray binaries display identical state transition behavior
and even multiple state transitions within the same source are far from being periodic replications of eachother. Furthermore, state transitions observed in some systems fall into a category all their own (e.g., GRS
1915+105). The hysteresis cycle is characteristic of, but not exclusive to, black hole X-ray binary transients.
State transition behavior is also observed in neutron star X-ray binaries (e.g., Muñoz-Darias et al., 2014) andultraluminous X-ray sources (e.g., Feng & Kaaret, 2006), yet whether the state transitions observed in thesedifferent systems all share a common physical origin remains unknown. A candidate physical mechanism forblack hole state transitions is proposed in Chapter 3.
My original interest in astronomy research was inspired by the enigmatic behavior displayed by blackhole X-ray binaries. In this thesis, I hope to convey to the reader the value of a research approach that
appreciates the junction of theory, numerics, observations, and phenomenology.

21

Figure 1.7: General behavior of black hole X-ray binary transients in the hardness-intensity diagram (top)and the hardness-rms diagram (bottom). Hardness is defined as the ratio of photon counts in a hard X-ray
band to those in a soft X-ray band. Labels show the locations of the phenomenological spectral/timing
states on these diagrams. The low/hard state is accompanied by a steady jet. Transitions from the hardintermediate to soft-intermediate states are often contemporaneous with a transient jet with Lorentz factor
Γ & 2. The high/soft state is accompanied by accretion disc winds, which are thought to have an equatorial
geometry (?) and to sometimes require a magnetic driving mechanism (e.g., Miller et al., 2006b). Jets anddisc winds appear to be mutually exclusive (Miller et al., 2006c). Taken from Belloni (2010).

Chapter 2

Spectral Hardening as a Viable Alternative to Disc Truncation in Black HoleState Transitions

2.1

PrefaceThis paper was published in Monthly Notices of the Royal Astronomical Society, Volume 431, Issue4, pp. 3510-3532 on June 01, 2013. The author list is: Salvesen, G., Miller, J. M., Reis, R. C., & Begelman,
M. C. The spectral evolution of the accretion disc during black hole state transitions is commonly attributed
to the radial migration of the inner edge of the disc. However, there is no clear reason to expect a geometrically thin, optically thick accretion disc around a black hole to be severely truncated. Instead, this paper
demonstrates that changes in the accretion disc atmosphere can conceivably explain the observed evolutionof the disc spectral component during black hole state transitions.

Abstract
Constraining the accretion flow geometry of black hole binaries in outburst is complicated by theinability of simplified multi-colour disc models to distinguish between changes in the inner disc radius andalterations to the emergent spectrum, parameterised by the phenomenological colour correction factor, fcol .
We analyse Rossi X-ray Timing Explorer observations of the low mass Galactic black hole X-ray binary,
GX 339–4, taken over seven epochs when the source was experiencing a state transition. The accretion
disc component is isolated using a pipeline resulting in robust detections for disc luminosities, 10−3 .
Ldisc /LEdd . 0.5. Assuming that the inner disc remains situated at the innermost stable circular orbit over23
the course of a state transition, we measure the relative degree of change in fcol required to explain the
spectral evolution of the disc component. A variable fcol that increases by a factor of ∼ 2.0 − 3.5 as thesource transitions from the high/soft state to the low/hard state can adequately explain the observed disc
spectral evolution. For the observations dominated by a disc component, the familiar scaling between the4disc luminosity and effective temperature, Ldisc ∝ Teff
, is observed; however, significant deviations from thisrelation appear when GX 339–4 is in the hard intermediate and low/hard states. Allowing for an evolving
4
fcol between spectral states, the Ldisc − Teff
law is recovered over the full range of disc luminosities, although

this depends heavily on the physically conceivable range of fcol . We demonstrate that physically reasonablechanges in fcol provide a viable description for multiple state transitions of a black hole binary without
invoking radial motion of the inner accretion disc.

2.2

Introduction
The nature of the accretion flow for black hole binaries (BHBs) undergoing state transitions remains

unclear and is an active topic of observational and theoretical pursuit. The phenomenology describing thepredominant spectral states experienced by BHBs include the high/soft state (HSS), characterised by highluminosity and a standard thin accretion disc as the dominant spectral component; the low/hard state (LHS),
characterised by low luminosity with a dominant hard power-law spectral component that is likely produced
by inverse Compton scattering of soft photons by a relativistic electron corona (Shapiro et al., 1976); andthe intermediate state (IS), which is itself further divided into the hard intermediate state (HIS) and softintermediate state (SIS) (see, e.g., Belloni 2010 for an extensive description of these phenomenological states).
Currently, the picture of a black hole accreting at a significant fraction of the Eddington mass accretionrate, ṀEdd , from a geometrically thin, optically thick disc extending to the innermost stable circular orbit(ISCO) is the accepted description of the HSS. However, there is a lack of consensus regarding a universal
geometry of the disc and innermost accretion flow in the LHS when the black hole feeds at a low fraction ofṀEdd and/or the flow has a low radiative efficiency. According to the paradigm put forth by Esin et al. (1997),
the accretion disc in the LHS is radially recessed (i.e., truncated) with the interior region being composed ofa radiatively inefficient accretion flow (RIAF) (Narayan & Yi, 1994). Alternatively, models where the inner24disc remains at the ISCO in the LHS have been proposed by invoking magnetically reconnecting flares drivinga coronal outflow (e.g., Beloborodov, 1999). Another model advocating against disc truncation is a standard
thin disc that generates a magnetically dominated corona where the amount of accretion energy liberated
in the corona increases with decreasing mass accretion rate (Merloni & Fabian, 2002). Recently, Reis et al.
(2013) proposed that the intermediate states may differ from one another only in the geometrical extent of
the X-ray corona, with the SIS being the result of a collapse of the previously extended corona as the systemmoves into the disc-dominated HSS. The notion of an accretion disc, where the magnetorotational instability
governs dissipation and angular momentum transport (Balbus & Hawley, 1991), producing a magneticallydominated corona is supported by sophisticated numerical simulations of local shearing boxes (e.g., Hirose
et al., 2006, 2009) and global discs (e.g., Beckwith et al., 2009). By varying only the mass accretion rate,
the spectral signatures of the HSS and LHS were reproduced in the global disc simulations of Schnittman
et al. (2013), without the need for radial variation of the inner accretion disc. The physical mechanism
responsible for triggering a state transition is poorly understood, making it difficult to conceive what would
drive a sudden disc recession or a drastic change in coronal activity required to explain the observed spectral
evolution; thus, lending credence to the need for a focused study of disc evolution during state transitions.
Observational studies of BHB spectra have successfully applied a multi-colour disc (MCD) blackbody
model (Mitsuda et al., 1984; Makishima et al., 1986) to the observed quasi-thermal soft X-ray spectral
component in the HSS and even in the LHS in some cases (e.g., Miller et al., 2006a; Reis et al., 2010).
Provided the observational parameters are well-constrained, the radial location of the inner disc boundary
can be measured from the best-fit MCD model parameters. However, this technique makes the criticalassumption that the deviations to the emergent disc spectrum caused by the disc vertical structure can beparametrized by a frequency-independent constant, known as the colour-correction, or spectral hardening,
factor, fcol (Shimura & Takahara, 1995).
Recently, investigations into the possibility of a variable fcol experienced a strong revival in theobservational community. Studies of black hole X-ray binary accretion discs across various luminosity statesusing data from the Rossi X-ray Timing Explorer (RXTE) (e.g., Nowak et al., 2002; Ibragimov et al.,2005; Wilms et al., 2006; Dunn et al., 2011) and the Swift observatory (Reynolds & Miller, 2013) provide

25extensive systematic analyses of the properties of disc evolution in BHBs. The global studies of Dunn et al.
(2011) and Reynolds & Miller (2013) examined all of the available archived data for all of the well-known,
confirmed BHBs. For nearly all BHBs where an outburst was observed, Dunn et al. (2011) found thatfcol remained relatively constant in the most disc-dominated states (i.e., the HSS), closely following the4theoretical Ldisc ∝ Teff
relation between the disc luminosity and disc effective temperature. For observations

with a relatively weaker disc component, characteristic of the IS, deviations from the luminosity-temperature
relationship arose. Supposing that the inner disc radius does not change and allowing for a variable colour
4
correction factor, 1.6 . fcol . 2.6, where fcol increased as the disc fraction decreased, the Ldisc ∝ Teff
scaling

was recovered. This finding sets the stage for a thorough consideration of an increasing fcol when a BHB
transitions from the HSS to the LHS as a possible alternative to a truncated disc geometry in the LHS.
Performing a similar analysis, Reynolds & Miller (2013) speculate that fcol may attain values as high as 5 as
the BHB spectrum becomes less disc dominated. The main differences between our work and these recent
observational studies are: (1) We focus exclusively on a single source, the well-known BHB transient, GX
339–4, over multiple state transitions. (2) We systematically explore different Comptonisation models.1 (3)We introduce a novel approach for making relative measurements in order to reduce uncertainties.
The methodology of this paper is to systematically investigate to what extent variations in the discstructure and/or disc truncation can explain the observed spectral evolution for multiple state transitionsof a BHB. Most importantly, we will demonstrate that a colour correction factor that dynamically evolves
within a physically reasonable range provides an adequate alternative to a truncated disc for explaining thechanging disc spectrum.

2.3

Observations and Data Reduction
Motivated by the lack of consensus on accretion disc evolution over the course of a BHB state transi-tion, we aim to address the question: Can physically reasonable changes in the vertical disc structure alone
explain the observed evolution of disc properties during a state transition without invoking a truncated discgeometry? In light of this question, we sought a source that met the following criteria: (1) The source was
1 Reynolds & Miller (2013) also consider both phenomenological and physical Comptonisation models.

26
observed over multiple state transitions, permitting comparison of disc variations between different transitions. (2) Given that our methodology hinges on the presence of a disc, a disc component must have been
robustly detected in the LHS. (3) Conflicting claims for both a severely recessed disc and a disc extending
to the ISCO in the LHS are published for the source. (4) Variable low-frequency quasi-periodic oscillations(LFQPOs), which may be linked to the disc evolution, have been observed coincident with state transitions. (5) Reasonable constraints are known on mass, distance, and inclination; however, this is of ancillary
importance. Based on these criteria, we selected GX 339–4 as the focus of our analysis.

Startdd/mm/yy (MJD)
18/04/02 (52382)
15/02/03 (52685)
05/07/04 (53191)
17/01/07 (54117)
02/05/07 (54222)
26/03/10 (55281)
07/01/11 (55568)

End
dd/mm/yy (MJD)
23/05/02 (52417)
11/05/03 (52770)
25/08/04 (53242)
23/02/07 (54154)
06/06/07 (54257)
20/05/10 (55336)
15/02/11 (55607)
Rise
Decay
Rise
Rise
Decay
Rise
Decay

Type
33
53
30
34
46
59
28

Nobs

pow/bknpow

32
33
21
28
40
11
12

Ndisc
22
30
21
17
39
9
12

simpl

Ndisc
31
23
21
29
40
10
12

comptt

Ndisc

min
νQPO
(Hz)
0.159(1)a0.29(1)a0.306(7)a0.142(3)a0.070(3)a0.22(1)b0.9(1)c

max
νQPO
(Hz)
7.9(1)a8.8(3)a8.0(2)a6.67(2)a4.05(9)a6.7(2)b
4.5(3)b

14
9.7
8.8
13
15
9.8
2.9

max
min
RKep
/RKep

Table 2.1: Table summarising the key features of each GX 339–4 state transition. From left to right, the columns are the state transition ID, start
time of the transition, end time of the transition, type of transition, number of archived RXTE observations, number of observations requiring a disccomponent for Comptonisation models pow/bknpow, simpl, and comptt, minimum/maximum LFQPO frequency observed, and fractional change inminmax
inner disc radius implied if νQPO
and νQPO
are associated with the Keplerian frequency and assumed to trace the inner disc location. a Shaposhnikovbc& Titarchuk (2009); Motta et al. (2011); Stiele et al. (2011)

R02
D03
R04
R07
D07
R10
D11

Trans. ID

27

28
2.3.1

GX 339–4
GX 339–4, discovered by Markert et al. (1973), is a low-mass X-ray binary and recurring transienthosting a dynamically confirmed black hole with a mass M ≥ 5.8M

(Hynes et al. 2003; see also Muñoz-

Darias et al. 2008) and distance D = 8 ± 2 kpc (Zdziarski et al. 2004; see also Hynes et al. 2004). Supposingthat the inner disc inclination, i (0◦ for face on, 90◦ for edge on), can be assumed to align with the binary
inclination, then the lack of observed eclipses constrain i ≤ 60◦ (Cowley et al., 2002), while radio observations
imply i ≤ 30◦ (Gallo et al., 2004). Results from modelling the X-ray reflection features of GX 339–4 in the
SIS also suggest a lower disc inclination of i ∼ 10◦ − 30◦ (Miller et al., 2004, 2009b) and in the LHS (Reiset al., 2008). Owing to the uncertainties on the intrinsic parameters (namely, M , D, i) of GX 339–4 and thelimitations of existing X-ray data, seeking absolute measurements that depend on these parameters presents
a serious challenge.
GX 339–4 experienced multiple outbursts since its discovery, making it an appealing source for temporal evolution studies. The seven state transitions we select to study are listed in Table 2.1. The notation
convention we adopt for referring to a transition is the letter R or D, signifying a rise or decay type, respectively, followed by the year associated with the transition. A rise (decay) type indicates a transition from the
LHS (HSS) to the HSS (LHS). The intervals of each transition were selected based on previously publisheddates and the emergence/disappearance of LFQPOs (Shaposhnikov & Titarchuk, 2009; Motta et al., 2011;
Stiele et al., 2011).2

We chose to extend the transition start/end times by ∼ 5 days beyond the intervals

quoted in the literature to ensure that we capture the full transition behavior in our subsequent analysis.
Alarming discrepancies in the location of the inner accretion disc edge, particularly in the LHS,
appear in the literature based on spectral fits to X-ray observations of GX 339–4. The strongest casefor disc truncation is presented in Tomsick et al. (2009), which used fluorescent iron emission lines tofind Rin > 175 Rg (90% confidence, assuming i = 30◦ ) when GX 339–4 was in the LHS with luminosity
LX ' 1.4 × 10−3 LEdd (assuming D = 8 kpc). Claims of a truncated disc in the LHS also appear based onmodelling of direct disc emission accounting for irradiation of the inner disc (Cabanac et al., 2009). Even
2 The end date listed for transition ID GX339-D03 in Table 1 of Shaposhnikov & Titarchuk (2009) is mistyped and should
read 06/05/03 (June 05 2003).

29
in the SIS, Yamada et al. (2009) suggest a recessed disc with Rin = 5 − 32 Rg (68% confidence, assuming
i = 30◦ ). These measurements suggest that the inner portion of the disc is absent in low luminosity states,
perhaps replaced by a RIAF. In stark contrast, there are numerous claims of the inner disc radius in GX
339–4 remaining consistent with the ISCO in the LHS. Using physically motivated, sophisticated spectral
fitting techniques, the disc inner radius was determined to lie within Rin = 2.0 − 3.0 Rg in the SIS (Miller
et al., 2004) and Rin = 3.0 − 5.0 Rg in the LHS with luminosity LX ' 0.05 LEdd (assuming D = 8 kpc)
(Miller et al., 2006a). Both of these results were confirmed by a systematic reanalysis that employed detailed
disc reflection modelling (Reis et al., 2008). Using the same data set as in Miller et al. (2006a), Wilkinson &Uttley (2009) also measured a disc extending down to 4 Rg from their best-fit iron line model. Tomsick et al.
(2008) observed GX 339–4 in the LHS at luminosities LX ' 0.023 LEdd and LX ' 8 × 10−3 LEdd , finding
the disc to extend within 10 Rg and being consistent with the disc remaining fixed at ∼ 4 Rg . Reis et al.
(2010) performed a systematic study of eight BHBs (including GX 339–4) in the LHS, robustly detecting athermal component in all sources and placing a stringent upper limit of 10 Rg on the truncation radius forsix sources confirmed by both broad iron line and thermal disc modelling independently.
In order to measure black hole spin, one must associate Rin with the ISCO, located at RISCO . A
measurement of the inner disc radius is the crucial diagnostic for measuring black hole spin, as RISCO isa monotonically decreasing function of the dimensionless spin parameter, a∗ ≡ Jc/GM 2 , where J is theblack hole angular momentum (Bardeen et al., 1972). Perhaps unsurprisingly, conflicting claims regarding
the behaviour of Rin have led to discrepancies in spin measurements of GX 339–4. Spin measurements
employing the relativistically broadened iron line method (Fabian et al., 1989; Laor, 1991) suggest that
GX339–4 harbours a near-maximally spinning black hole with a∗ = 0.94 ± 0.02 (Miller et al., 2009b; Reiset al., 2008), while the thermal disc continuum fitting technique (Zhang et al., 1997) supports a low-tomoderately spinning black hole with a∗ < 0.4 (Yamada et al. 2009; see also Kolehmainen & Done 2010;
Kolehmainen et al. 2011).
LFQPOs exhibit variability simultaneous with transitions between the LHS and HSS (Rutledge et al.,1999b). Variable LFQPOs were observed in each GX 339–4 transition studied here, with the measured
frequency ranges listed in Table 2.1 and typically spanning 0.2 − 8 Hz (Belloni et al., 2005; Shaposhnikov

30& Titarchuk, 2009; Motta et al., 2011; Stiele et al., 2011; Nandi et al., 2012). The idea that LFQPOs may
originate from oscillating modes of a quasi-spherical RIAF that is interior to, or partially overlapping with,a truncated thin disc was proposed by Giannios & Spruit (2004). Long time-scale variability of the discblackbody component of GX 339–4 was associated with disc instabilities and interpreted as arising from
intrinsic variability of the disc (Wilkinson & Uttley, 2009). This suggests that the variable LFQPOs may be
intricately linked to disc truncation; however, the truncation radius cannot exceed ∼ 20 Rg if the observed
variations are limited by the viscous timescale.

2.3.2

Rossi X-ray Timing Explorer
The monitoring campaign of the Rossi X-ray Timing Explorer (RXTE) (Bradt et al., 1993; Swank,1999) observed GX 339–4 transitioning between states in 2002, 2003, 2004, 2007, 2010, and 2011, obtaining
well-sampled (in time) observations spanning each transition before being decommissioned in early 2012.
We examined 283 RXTE observations of GX 339–4 covering seven time intervals, during which the source
experienced state transitions (see Table 2.1). The RXTE standard products, comprised of the source andbackground spectra along with instrument response matrices, were obtained from the High Energy Astrophysics Science Archive Research Center data archive.3

The Proportional Counter Array (PCA) (Jahoda

et al., 1996) and High Energy X-ray Timing Experiment (HEXTE) (Rothschild et al., 1998) standard source
spectra were analysed jointly. The PCA energy spectrum (Standard 2 mode) is built from summing alllayers from various Proportional Counter Units (PCUs). The PCU-1, PCU-2, and PCU-3 are used in all
observations prior to 25 December 2006 (i.e., R02, D03, R04), while only the PCU-2 and PCU-3 are usedin observations thereafter (i.e., R07, D07, R10, D11) due to a propane loss in the top layer of PCU-1.4
For observations corresponding to transitions R02, D03, R04, R07, and D07, we used the HEXTE Cluster B
spectrum, but could not use these data for R10 and D11 because the HEXTE Cluster B was permanently leftin an off-source pointing position from 29 March 2010 onward.5 The HEXTE Cluster A data for transitions
R10 and D11 are affected by strong residuals, resulting from difficultly in determining the appropriate con3 http://heasarc.gsfc.nasa.gov/cgi-bin/W3Browse/w3browse.pl
4 http://heasarc.nasa.gov/docs/xte/recipes/stdprod guide.html
5 http://heasarc.nasa.gov/docs/xte/xhp new.html

31
tribution of the background spectra. Consequently, we excluded all HEXTE data while analysing transitions
R10 and D11, using PCA data only for these observations. This resulted in many observations with a strong
soft component, which is indicative of a disc, being excised from our sample due to poor spectral modellingof the high-energy continuum (see §2.5.1).
While the time sampling capability of the RXTE satellite makes it well-suited for a study of BHB statetransitions, which are a highly time-dependent phenomenon, RXTE has important limitations. Determining
the spectral properties of the disc component for a given observation is made difficult by the limited lowenergy coverage of RXTE, which only extends down to ∼ 3 keV, while the disc spectrum peaks at ∼ 1
keV. Especially in the HIS and LHS, the diminished disc component, if present, is difficult to characterise
due to a combination of its weakness and the dominance of a Comptonised component. Figure 2.1 presentsa hardness-intensity diagram (HID) of all 283 RXTE observations of GX 339–4 considered in this work.
Observations that do not statistically require a disc component (see §2.5.1) are indicated by grey crosses,
while the various glyphs indicate the transition associated with the observation. The plethora of IS and HSS
observations (i.e., low-to-moderate hardnesses) deemed to not require a disc comes from the R10 and D11
data sets, where the absence of HEXTE data prevented acceptable fits to disc-dominated spectra. Notably,
Figure 2.1 shows that the set of observations considered in the subsequent analysis span all spectral states,
which indicates that we are studying the complete evolution of state transitions in GX 339–4.
For a given observation, hardness and intensity are calculated from the PCA data as follows. The
distribution of counts across all channels is obtained from both the standard PCA source spectral file andbackground spectral file. The background counts are then subtracted from the source counts channel-bychannel, yielding the counts in each channel associated with GX 339–4. The hardness is then defined as theratio of GX 339–4 counts in channels 15 - 24 (6.3 - 10.5 keV) to those in channels 8 - 14 (3.8 - 6.3 keV)
(Belloni et al., 2005). A count rate is obtained by summing the counts across all channels and dividing by
the exposure time. Good time intervals (GTIs) are identified from the GTI file included in the standard
products. The average number of PCUs that are turned on during the GTIs is determined from the filter
file included in the standard products. The intensity is then defined as the count rate divided by the average
number of active PCUs during the observation GTIs.

32

Fdisc / Ftotal
0.05

0.1

0.2

0.5

1

PCA Intensity (cnts s-1 PCU-1)

1000

100

10
R02
D03
R04
R07
D07
R10
D11
1
0.1

0.2

0.5

1

Hardness Ratio
Figure 2.1: Hardness-intensity diagram for all observations of GX 339–4 considered in this work generated
from the results of the pow/bkn model (see §2.5.2). The disc fraction for each observation is given by theglyph colour and is an indicator of the spectral state. Data points marked by a grey cross correspond toobservations that were not deemed to statistically require a disc component. The key in the bottom left
identifies the glyph associated with each transition ID. See §2.3.2 for a description of how the HID was
generated. A wide range of hardness is considered in this work.

33
To justify using the χ2 fitting statistic for deriving parameter constraints, both the PCA and HEXTE
spectra were binned to require a minimum of 20 counts per energy bin using the FTOOL grppha. A systematic
error of 0.6% was added to the PCA spectrum using grppha; however, the PCA spectra without systematic
errors added are used when deriving relative measurements (see §2.5.3.1). All spectral fits are performed withXSPEC version 12.7.1 (Arnaud, 1996). The PCA and HEXTE spectra are fitted over the 2.8 − 25 keV and
20−100 keV energy ranges, respectively. The response energy range over which spectral models are calculated
is extended to 0.1 keV and 100 keV using the energies extend command in XSPEC with 50 logarithmic
steps in each direction. All observation times correspond to the midpoint of the PCA observation. Quoted
errors from spectral fits are 1σ uncertainties, unless specified otherwise.

2.4

Multi-colour Disc Blackbody Models
The accretion disc in BHBs manifests itself as a soft, quasi-blackbody component with a typicalpeak inner disc temperature of ∼ 1 keV (e.g., Fender & Belloni, 2004). While the accepted description
for the origin of the soft thermal component in the HSS is a geometrically thin, optically thick accretion
disc (Shakura & Sunyaev, 1973) extending down to the ISCO, no such consensus has been reached for thenature of the innermost flow in the HIS or LHS. Appealing to spectral fits of the thermal continuum withmulti-colour disc (MCD) blackbody models to isolate the disc component, one can measure the temporal
evolution of the accretion disc over the course of a state transition.

2.4.1

The Standard MCD Model
The most commonly adopted disc model is the MCD blackbody model (diskbb in XSPEC) (Mitsuda

et al., 1984; Makishima et al., 1986). The attractiveness of the MCD model is owed to its legacy, applicability,
and simplicity. However, the familiarity and broad acceptance of diskbb often mask subtle features of themodel, possibly leading to unintended misuse. Here, we dissect the diskbb model in order to elucidate its
underlying assumptions and expose its limitations. Assuming a geometrically thin, optically thick accretion
disc that locally emits a blackbody spectrum in concentric annuli, the colour temperature for a MCD model

34
at a given radius is described as,Tcol (r) = T∗rRin

−3/4 "


1−β

r
,
Rin

−1/2 #1/4
,

(2.1)

where Rin is the radius of the inner disc edge and β describes the boundary condition at this location.
For discs in the HSS, Rin is commonly associated with RISCO . A zero-torque boundary condition at Rin
corresponds to β = 1, while β = 0 allows for non-zero stresses at the inner disc and is the choice adopted
in diskbb (Zhang et al., 1997; Gierliński et al., 1999; Zimmerman et al., 2005). The leading factor, whichroughly describes the colour temperature at the inner disc, is given by,T∗ = fcol

3GM Ṁ
3
8πσRin

!1/4
,

(2.2)

where G is the gravitational constant, σ is the Stefan-Boltzmann constant, M is the compact object mass,Ṁ is the mass accretion rate, and fcol is the colour correction factor (Shimura & Takahara, 1995), also
referred to as the spectral hardening factor. For a zero-torque condition at Rin (i.e., β = 1 in Equation
2.1), T∗ is related to the maximum temperature attained in the disc by Tcol,max = 0.488T∗ , which occurs at
r = (49/36)Rin . Since the disc temperature does not reach T∗ at any radius, Tcol,max is a more physicallymeaningful characterisation of the blackbody temperature associated with the spectrum. For a thin discthat is torqued at the inner boundary, as is the case for diskbb, Tcol,max coincides with T∗ because thetemperature profile grows asymptotically with decreasing r (see Figure 1 of Zimmerman et al. 2005). Oneof the two free parameters of the diskbb model is T∗ , implicitly assuming fcol = 1. The ‘normalisation’ isthe other free parameter of diskbb and is given by,
Kbb = 100

1
4
fcol



Rin,kmDkpc

2
cos i,(2.3)

where, again, fcol = 1, Dkpc is the distance to the source in kiloparsecs, Rin,km is the inner disc radius
in kilometers, and i is the inner disc inclination. Inner disc radii are commonly derived from Kbb by
measuring intrinsic system properties and imposing a constant fcol (not necessarily unity as assumed by
diskbb), regardless of the source luminosity state (e.g., Zdziarski et al., 2004). Remarkably, the simple
two-parameter diskbb model fully describes the accretion disc spectrum under the appropriate assumptionsof a non-zero torque inner disc boundary condition and a fixed colour correction factor for characterising the35
overall deviation from a blackbody spectrum.

2.4.2

Zero-Torque Boundary Condition and Disc Vertical Structure
If the inner disc radius can be identified with the ISCO, a zero-torque boundary condition becomes

attractive from the physical consideration that the gas viscous inspiral timescale is long relative to thefree-fall timescale for gas at the ISCO; therefore, the gas interior to Rin does not have time to radiate
appreciably before being accreted (Shakura & Sunyaev, 1973; Novikov & Thorne, 1973; Page & Thorne, 1974;
Abramowicz & Kato, 1989; Frank et al., 1992; Afshordi & Paczyński, 2003). In principle, for a geometricallythin disc in the presence of a magnetic field, non-negligible torque resulting from magnetic stresses may
be present at the ISCO (Krolik, 1999; Gammie, 1999; Agol & Krolik, 2000; Afshordi & Paczyński, 2003).
However, magnetohydrodynamic three-dimensional simulations of global thin accretion discs around black
holes demonstrate that a weak magnetic field is unable to couple the gas within the plunging region to
the disc, despite large fluctuating magnetic stresses inside the ISCO, in accordance with the zero-torqueboundary condition (Armitage et al., 2001; Reynolds & Armitage, 2001). Similar simulations exploring
initially purely poloidal and toroidal field topologies instead observe significant stress at the ISCO andcontinuing deep within the plunging region, refuting the zero-torque assumption (Hawley & Krolik, 2001,
2002). A difficulty with allowing for a non-zero torque in disc models is that there is no obvious choice for themagnitude of the torque, necessitating the introduction of a free parameter. The zero-torque choice is both
physically motivated and is not arbitrary; however, the presence of significant torque on the inner disc may
be important in real systems depending on highly unknown factors such as the disc thickness, dissipation
properties, and magnetisation.
The underlying physics responsible for altering the locally emitted spectrum of an optically thick discfrom a pure blackbody are parameterised into the colour correction factor. The result of this approximationis the colour-corrected, or ‘diluted’, blackbody,Iν =14 Bν (fcol Teff ) ,
fcol

(2.4)

where Iν is the specific intensity, Bν is the Planck function, Teff = Tcol /fcol is the effective temperature,

36
and fcol = 1 refers to a pure blackbody spectrum. In reality, deviations in the emergent disc spectrum
from a canonical blackbody arise from frequency-dependent opacities determined from the disc verticalstructure, which is neglected in the standard thin disc treatment. The degree of spectral modification
will be governed by the combined effects of electron scattering (i.e., Comptonisation in the disc), freefree emission/absorption, and bound-free absorption, resulting in a departure from local thermodynamic
equilibrium and a hardening of the spectrum (Felten & Rees, 1972; Shimura & Takahara, 1995; Zavlin
et al., 1996; Rutledge et al., 1999a; McClintock et al., 2004). A surrounding corona acting as a depository
for some fraction of the disc accretion power provides yet another means for altering the emergent disc
spectrum (Svensson & Zdziarski, 1994; Merloni et al., 2000). Furthermore, magnetic pressure support likely
contributes significantly to hydrostatic balance in the disc photosphere and may act to vertically extend
the disc atmosphere, producing a harder spectrum (Blaes et al., 2006; Begelman & Pringle, 2007). The
phenomenological colour-correction prescription avoids all of these complications by neglecting any frequency
dependence, only permitting translational (i.e., frequency-independent) hardening and attenuation of the
Planck function. Provided that this model yields a satisfactory fit to the observed disc spectrum, one
presumes that fcol , which is assumed to be constant for all radial locations in the disc, provides an adequate
description of the disc vertical structure in an average sense.
The two disc models we appeal to in this work differ from diskbb in that they both adopt a zerotorque inner boundary condition and allow for the possibility of a variable colour correction factor. The
ezdiskbb model (Zimmerman et al., 2005) is the analog of diskbb, but with the zero-torque boundary
condition enforced. The free parameters are the maximum disc colour temperature, Tcol,max defined above,and the normalisation, given by
Kez = 100

1
4
fcol



Rin,kmDkpc

2
cos i,(2.5)

where fcol is not specified. We emphasize that changes in Kez do not necessarily imply changes in thedisc inner radius, due to the parameter degeneracy between Rin and fcol . The diskpn model (Gierliński
et al., 1999), along with the zero-torque boundary condition, incorporates a pseudo-Newtonian potential(Paczyńsky & Wiita, 1980) to more accurately calculate the radial temperature distribution in the disc at
the expense of introducing the inner disc radius as a free parameter. The two remaining free parameters are37
Tcol,max and the normalisation,
Kpn =where MM

1



4
fcol

MM
Dkpc

2
cos i,(2.6)

is the compact object mass in solar mass units. In all of the MCD model normalisations,fcol enters to the fourth power. Therefore, asking how much fcol could conceivably change during a statetransition is not unreasonable, as only small changes in fcol are required to explain the observed spectral
evolution, which may not necessitate disc truncation.
The disc models described up to this point neglect many important physical considerations. We
wish to emphasize that sophisticated, physically motivated models are available in XSPEC that account forirradiation of the disc (diskir; Gierliński et al., 2008), general relativistic effects around a Schwarzschild(grad; Ebisawa et al., 1991) and Kerr black hole (kerrbb; Li et al., 2005), and detailed modelling of thedisc vertical structure around a Kerr black hole (bhspec; Davis et al., 2005). Realistic disc models comewith the expense of introducing additional parameters. The limited quality of X-ray observations and thedesire to limit the number of model parameters forces the use of over-simplified disc models, with the hopethat they can adequately capture the most important aspects of the accretion physics. The purpose of thesubsequent analysis is to demonstrate that simplistic disc models produce good spectral fits but they cannot
be used to make reliable claims regarding the behaviour of the inner disc radius.

2.5

Analysis and Results
In what follows, we exploit the timing and monitoring capabilities of RXTE, which provide multi-

ple sequential spectra of moderate resolution tracking the transition between spectral states, to study theevolution of the accretion disc in GX 339–4. The ultimate goal of our spectral analysis is to isolate thedisc component in order to measure relative changes in disc properties during transitions over the rise (LHS
→ HSS) and decay (HSS → LHS) stages of an outburst. We show that allowing for physically reasonablechanges in fcol provides an alternative explanation to disc truncation for the observed disc spectral evolution
during state transitions. Table 2.3 provides a sample of best-fit parameters for the fits used in this work.

38
2.5.1

Requirement of a Disc Component
We wish to verify the presence of a thermal disc component with adequate confidence using phe-nomenological models that have proven successful in characterising the general features of BHB X-ray spectra. To determine whether or not a disc component is required in a given RXTE spectrum, we adopt a
methodology based on χ2 statistics that uses phenomenological models to fit numerous observations in various spectral states, as motivated by previous works (Dunn et al., 2008, 2010, 2011; Reynolds & Miller,2013). The PCA and HEXTE data are fit jointly, allowing an energy-independent constant factor to vary
for the HEXTE data to account for any normalisation offset relative to the PCA and linking all parameters
between the PCA and HEXTE spectra. The spectral models are modified by photoelectric absorption using
the phabs model with a fixed equivalent neutral hydrogen column density of NH = 5.7 × 1021 cm−2 (Miller
et al., 2009b). Both Galactic and intrinsic absorption are folded into the phabs model, leading to a potential
ambiguity as to whether low-energy spectral evolution is due to changes in the intrinsic absorption or toevolution of the source. Miller et al. (2009a) fitted high-resolution BHB spectra (including GX 339–4) overa wide range in luminosity and spectral states, demonstrating that the low-energy spectral evolution is aconsequence of the evolution of the source itself, rather than a variable neutral photoelectric absorption.
Notably, Cabanac et al. (2009) present the case for possible variations in NH during outbursts of GX 339–4;
however, the restricted low-energy coverage of RXTE does not permit NH to be a free parameter in our
spectral fits.
As a first attempt, the spectra are fitted with a simple model consisting of an absorbed power-law
and MCD blackbody (i.e., phabs×(powerlaw+diskbb) in XSPEC) to model the continuum and accretion
disc, respectively. In many observations, this simple prescription does not yield an adequate fit due to thepresence of a broad Fe Kα emission line. To check for the requirement of an iron line component, a Gaussianline profile (gauss in XSPEC) is added to the model at fixed centroid energy of 6.4 keV with the width, σ, andnormalisation, Kline , as free parameters. An F -test is inappropriate for determining whether the inclusion ofa spectral line is a statistically significant improvement (Protassov et al., 2002). The alternative diagnostics
outlined by Protassov et al. (2002) for testing for the presence of a line are not easily incorporated into

39
the spectral fitting framework used here. We are concerned with obtaining an acceptable fit in order todetermine the presence of a disc component and do not wish to make any strong statements regarding FeKα emission; therefore, we adopt the line detection procedure of Dunn et al. (2008). In order for a line tobe considered significant, two criteria must be met: (1) An F -test must determine that the line is significant
at ≥ 3σ (F -statistic probability P ≤ 0.0027). (2) Given the 1σ uncertainty in the line normalisation, σKline ,obtained with the XSPEC error command, we require Kline ≥ 3σKline . This procedure for line determination
is more conservative than appealing to an F -test alone, though we note that some detected line features may
be erroneous.
We next test for the presence of a high energy break in the spectrum by replacing the power-law
component with the broken power-law model, bknpower, and repeating the aforementioned fitting procedure.
If an F -test determines that the broken power-law model is a significant improvement over the power-law
model at ≥ 3σ confidence, then we proceed with the broken power-law model imposing the parameter ranges
for the spectral indices and break energy of 1 ≤ Γ ≤ 4 and Eb ≥ 10 keV, respectively. In instances where aline component was deemed necessary in the power-law model but not the broken power-law model, or viceversa, we select the best fitting model determined by the lowest reduced χ2 -statistic, χ2ν ≡ χ2 /ν, where νis the number of degrees of freedom in the model. For cases when the best-fit converges to the parameter
limits of the bknpower model, the pipeline reverts to the powerlaw model for that observation. If the best-fit
powerlaw model also runs into a parameter bound, the observation is discarded.
At this point in the phenomenological fitting pipeline, for a fit to be considered acceptable for use in
subsequent analysis, we require χ2ν ≤ 2. Often, a poor fit to the high-energy spectrum is the culprit for failure
to meet this criterion. For instance, a poor fit may result from the inability of simplistic, unphysical models
to properly fit a reflection component or a complicated spectrum arising from a dynamic corona. Since we areusing phenomenological models to characterize the soft disc component and are not interested in a detailed
characterization of the high-energy spectrum, if the best-fit model to the combined PCA/HEXTE data gives
χ2ν > 2, then we repeat the above fitting procedure on the PCA data alone in search of an acceptable fit. In
these cases where only the PCA data are being fitted, we use the powerlaw model only because any spectralenergy break is expected to lie beyond the high-energy cutoff of the PCA, rendering the use of the bknpower

40
model inappropriate.
The disc component (diskbb) is then removed from the best-fit model and the spectrum is re-fitted.
An F -test is performed to determine whether or not the inclusion of the disc improved the fit by ≥ 3σ.
Observations that meet these criteria are kept for further analysis, while those that do not statisticallyrequire a disc are discarded. Running this model fitting procedure on 283 total observations of GX 339–4 indifferent stages of transition between the low/hard and high/soft states resulted in 177 observations wherea disc component was deemed statistically required from spectral fitting techniques (see Table 2.1).
We caution that the absence of a quasi-blackbody component in a given RXTE spectrum does not
imply the absence or truncation of the physical accretion disc. The low energy cutoff of the RXTE PCA isrestricted to ∼ 3 keV; thus, only the Wien tail of the MCD blackbody spectrum will be in the observableband. Although we may not detect a disc, especially in the HIS and LHS, this is not enough to claim theabsence of a disc component, which may be dominant at energies below the sensitivity of RXTE (e.g., Dunnet al., 2008) and has been observed with XMM-Newton in the LHS of GX 339–4 (e.g., Miller et al., 2006a;
Reis et al., 2008).
Figure 2.2 shows the data-to-model ratios with the disc component removed from the best-fit model
for representative observations of each state transition. Disc fractions are defined by,DF ≡0.1−10 keV
Fdisc,
0.1−100 keV
Ftotal(2.7)

where Fdisc and Ftotal are the unabsorbed fluxes of the disc component and overall model, respectively,
taken over the energy ranges indicated by the superscripts. Fluxes were obtained with the cflux commandin XSPEC. Disc components are robustly detected by the pipeline described above, even for moderately low
disc fractions, which is essential for tracking the evolution of the disc throughout each transition in thesubsequent analysis.

2.5.2

Sensitivity of Spectral Fits to the Continuum Model
The motivation for employing a simple power-law model with limited free parameters is to fit the

portion of the spectrum that is thought to arise from Comptonisation of thermal disc photons by a low41

Fdisc / Ftotal
0.05

0.1

0.2

0.5

5

2
1

2
1

5

2
1

2
1

5

2
1

2
1

5

2
1

2
1

5

2
1

2
1

5

2
1

2
1
4
5
Energy (keV)

96409-01-04-02
96409-01-06-00
96409-01-07-03

10

5

3

95409-01-12-04
95409-01-14-06
95409-01-18-04

10596409-01-04-02
96409-01-06-00
96409-01-07-03

92704-03-08-00
92704-03-11-01
92704-04-02-01

10595409-01-12-04
95409-01-14-06
95409-01-18-04

92428-01-03-00
92428-01-04-02
92085-01-01-00

10592704-03-08-00
92704-03-11-01
92704-04-02-01

90418-01-01-01
60705-01-70-00
90704-02-01-01

10

592428-01-03-00
92428-01-04-02
92085-01-01-00

R02

10

5

90418-01-01-01
60705-01-70-00
90704-02-01-01

70110-01-87-00
60705-01-59-00
80116-02-02-02G

10

D03

Data / Model

10

2
1

R04

Data / Model

10

2
1

R07

Data / Model

10

5

D07

Data / Model

10570110-01-87-00
60705-01-59-00
80116-02-02-02G

40031-03-02-02
70110-01-10-00
40031-03-03-00

10

R10

Data / Model

1040031-03-02-02
70110-01-10-00
40031-03-03-00

1

comptt10

D11

Data / Model

10

Data / Modelpow / bknpow

345
Energy (keV)

10

Figure 2.2: Ratios of the RXTE data to the best-fit models with the diskbb component removed to highlightthe need for a quasi-blackbody feature to model the soft excess. For each transition, three data/model ratios
are shown that representatively span the range in measured disc fractions. Glyph colours denote the discfraction for each fit. For comparison, the best-fit models where Comptonisation is modelled by the powerlaw
or bknpower (left column) and comptt (right column) XSPEC models are shown. For each row, the observation
IDs are given in the upper right corner of the plots and the transition ID on the rightmost ordinate.

42density, energetic electron corona of unknown geometry, which produces a power-law spectrum to first order.
The phenomenological power-law model extends to arbitrarily low energies, which does not mimic a physicalComptonisation process where a sharp cutoff is expected at an energy comparable to the characteristicenergy of the input seed photon distribution, namely, the quasi-blackbody disc spectrum. Therefore, theextension of the power-law component toward arbitrarily low energies results in ‘flux stealing’ from the disccomponent, making a disc more difficult to detect than had we adopted a more physical Comptonisation
model.
To investigate the sensitivity of fitting the disc component on the choice of Comptonisation model, we
explore two different models to characterise the continuum. The Comptonisation model comptt (Titarchuk,
1994; Hua & Titarchuk, 1995) provides an analytic description of the spectrum produced from a soft photon
distribution propagating through a hot plasma cloud. We re-ran the disc detection pipeline with comptt
in place of the power-law component, allowing only the plasma temperature, Tp , plasma optical depth, τp ,and normalisation, Kcomptt , to be free parameters. The remaining comptt parameters that we chose werea redshift z = 0, a seed photon temperature, T0 = Tcol,max (i.e., 0.488 times the diskbb temperature), anda spherical plasma cloud geometry. The additional free parameter introduced by comptt is justified by thephysical, as opposed to phenomenological, nature of the model. For cases where the best-fit converges to theenforced plasma temperature limit, Tp = 100 keV, the data are re-fitted with Tp fixed at 100 keV. When thedisc component is removed to check for the requirement of a disc, T0 becomes a free parameter, as there is
no longer an obvious, physically motivated choice for the seed photon distribution injected into the coronain the absence of a disc. A total of 166/283 observations required a disc component when comptt was used,compared to the 177/283 using the standard phenomenological powerlaw and bknpower models.
An alternative empirical substitute for a power-law model is simpl (Steiner et al., 2009), which
truncates the power-law spectrum at an energy comparable to the characteristic input seed photon energy;
thus, avoiding the unphysical low-energy extension of the powerlaw and bknpower models. Since simpl isa convolution model, we cannot apply the disc detection pipeline of §2.5.1 to determine whether or not adisc component is required. Instead, the previously determined 177 best-fit models are re-fitted with simpl
as the component responsible for modelling Comptonisation, allowing for both up- and down-scattering of

(Data / Model)comptt - (Data / Model)pow / bkn

430.8
92085-01-01-00
92428-01-04-02
92428-01-03-00

0.6

0.4

0.2

0.0

-0.2
3

4

5
6
Energy (keV)

7

8

9

10

Figure 2.3: Dependence of the strength and spectral shape of the disc component on the choice of Comptonisation model for transition R07. The residuals plotted are computed by subtracting the data-to-model
ratio for the pow/bknpow best-fit model from the best-fit comptt data-to-model ratio. The data-to-model
ratios used are those where the disc component was removed, as done in Figure 2.2. Residuals falling on thedashed line indicate no difference in the measured disc component between the two Comptonisation models,
while a positive excess implies flux stealing from the disc component by the pow/bkn model relative to thecomptt model. From top to bottom, the residuals were computed from observation IDs 92428-01-03-00 (redglyphs), 92428-01-04-02 (blue glyphs), and 92085-01-01-00 (green glyphs), which represent the HSS, IS, and
LHS, respectively. The flux stealing effect is most pronounced in the HSS and IS.

4480
pow / bknpow

Number of Observations

simpl
comptt

60

40

20

0
0.0

0.2

0.4
0.6
0.8
Disc Fraction (Fdisc / Ftotal)

1.0

Figure 2.4: Histogram of disc fractions for fits performed with Comptonisation models pow/bknpow (greyfilled), simpl (red forward slashes), and comptt (blue backward slashes). The excess of low disc fractions forthe pow/bknpow model is a consequence of the power-law extending to arbitrarily low energies, effectively
stealing flux from the disc component.

45
the input seed spectrum. Again, as was done for the disc detection pipelines using powerlaw/bknpower andcomptt as the continuum models, the observation is discarded if the best-fit converges on a parameter limit.
This returned 150/177 observations where the best-fit does not encounter a parameter limit. While simpl
cannot capture the high energy break commonly observed in BHB spectra, often resulting in the dismissalof the HEXTE data in order to obtain acceptable fits, this excision of data is justified because a detailed
model of the high-energy spectrum is ancillary to a reliable model of the disc spectrum.
The choice of Comptonisation model influences the measured disc properties. As discussed above,
the three Comptonisation models we explore in this work are: (1) standard power-law or broken powerlaw (hereafter, pow/bknpow), (2) power-law with a low-energy cut-off (hereafter, simpl), and (3) physicalComptonisation model (hereafter, comptt). For representative observations over all transitions fit with the
pow/bknpow and comptt models, Figure 2.2 shows data-to-model ratios with the disc component removed
from the best-fit model, which is a probe of the spectral shape and strength of the disc component. The discis strong in the HSS and gradually weakens, but remains detectable, in the LHS down to DF ' 0.05.
Figure 2.3 shows the residuals for the data-to-model ratios of the comptt fits compared to those forthe best-fit pow/bknpow models for a representative transition. The flux stealing phenomenon inherent to
the low-energy extension of the pow/bknpow model is apparent from the positive excess in the residuals
and affects the measured disc spectrum, particularly for high disc fractions that are characteristic of the
HSS. A histogram of measured disc fractions for all observations deemed to require a disc and for eachchoice of Comptonisation model is shown in Figure 2.4. Generally, replacing the power-law model with more
physically motivated Comptonisation models produces higher disc fractions and yields disc model spectrathat are unplagued by flux stealing. The measured disc fraction for a given observation generally follows
DFsimpl > DFcomptt > DFpow/bknpow .
Given this study on the sensitivity of the spectral fits to the choice of continuum model, we must
decide how to proceed with the subsequent analysis in order to best represent the true character of the discevolution during a state transition. In the interest of comparison and because discrepancies in the measured
disc properties arise between the three different, but seemingly adequate, Comptonisation models, we do
not favour one model over another. Instead, we elect to perform the subsequent analysis in three branches,

46
identical in all respects except for the Comptonisation model (i.e., pow/bknpow, simpl, comptt) being usedin the spectral fits.

2.5.3

Disc Evolution Results from Spectral Fitting
A parameter degeneracy exists between the inner disc radius and the colour correction factor within

the normalisation of the standard diskbb model. Therefore, decomposing the contributions to the discevolution due to changes in the vertical disc structure, parametrized into fcol , and a migrating inner discregion is nontrivial. Combining the models diskpn and ezdiskbb, while appealing to relative measurementsof the disc model normalisation evolution, we examine the degree of change in fcol and Rin required to
explain the disc spectral evolution in state transitions of GX 339–4.

2.5.3.1

Relative Measurements

Obtaining reliable absolute measurements of quantities locked up in the disc normalisation is limitedby constraints on the observables, namely, the mass, distance, and disc inclination, which all appear in thenormalisation. However, relative changes in fcol and/or Rin are accessible from the evolution of the discnormalisation because the intrinsic parameters do not change between successive observations.
Imposing a small systematic error on RXTE PCA data is common practice, resulting in a substantial
improvement in the quality of spectral fits. Applying model fits to data where no systematic error is added
will result in poor fits but with best-fit parameter errors dramatically reduced. While the best-fit parametervalues may be unreliable, they should be affected in a systematic sense and systematic errors associatedwith instrument calibration will be removed from the parameter errors. Provided that relative, rather thanabsolute, measurements are of interest, the evolution of a parameter may be tracked from fits to spectra
without systematic error incorporated into the PCA data.
To test the validity of this proposition, we re-fitted all spectra deemed to require a disc component
with the best-fit models determined from the aforementioned fitting pipeline, but with no systematic error
added to the PCA data. Figure 2.5 compares the distribution of the diskbb normalisation relative error,
defined as σKbb /Kbb , for fits to all GX 339–4 spectra when no systematic error and the standard 0.6%

4740
No systematic error

Number of Observations

0.6% systematic error
30

20

10

0
0.0

0.1

0.2
0.3
0.4
Relative Error (σKbb / Kbb)

0.5

Figure 2.5: Histogram of relative errors on the best-fit diskbb normalisation, σKbb /Kbb , for all RXTE
observations requiring a disc component. Models fit with a systematic error of 0.6% added to all PCA
energy channels and no systematic error added to the PCA data are shown as red forward slashes and grey
bars, respectively. The relative error on the disc normalisation is reduced when systematic error is not added
to the data.

48
2
R02

Kbb, relm = 0.0016
1

0
52380
2

52390

52400

52410

D03

Kbb, relm = -0.0043
1

0
52700

52720

52740

52760

2
R04

Kbb, relm = 0.0019
1

53210

53220

53230

53240

R07

Kbb, rel

0
53190
53200
2
m = 0.0021
1

0
54120

54130

54140

54150

2
D07

Kbb, relm = -0.0035
1

0
54220
2

54230

54240

54250

R10

Kbb, relm = -0.0007
1

55300

55310

55320

55330

D11

Kbb, rel

0
55280
55290
2
m = -0.0016
1

0
55570

55580

55590
MJD

55600

Figure 2.6: Relative comparison of diskbb normalisations, Kbb,rel (see text for definition), for each transition.
The slope, m, from a linear fit to the data is displayed in the upper left corner for each plot. A zero-slope
means that relative changes in Kbb with time are unaffected by neglecting to add systematic errors to the
PCA data, indicating that the systematics remain constant from observation to observation for the timeintervals considered. The flatness of Kbb,rel justifies the use of relative measurements for studying the discevolution throughout a state transition. From top to bottom, the panels show the time evolution of Kbb,rel
for transitions R02, D03, R04, R07, D07, R10, and D11.

49
systematic error were added to the PCA data (Jahoda et al., 2006). As expected, not including systematic
errors results in improved relative errors on the diskbb normalisation. Adopting an analysis technique basedon omitting systematic errors from the PCA data requires that the disc normalisation values are only affected
in a systematic way as a result. For each transition, Figure 2.6 shows Kbb,rel = Kbb,nosys /Kbb , which isdefined as the best-fit diskbb normalisation for fits with no systematic error added, Kbb,nosys , relative tothe normalisation for fits with 0.6% systematic error added, Kbb . The constancy of this ratio throughout
the spectral evolution of all transitions suggests that excluding systematic error for the PCA data to study
the evolution of the disc normalisation in a relative sense is justified in practice.
Occasionally, an observation that was adequately fitted when a 0.6% systematic error was added to the
PCA data will converge on parameter limits or grossly unphysical values when the data without systematic
error are fitted. In these situations, the data without systematic error added are fitted by hand in orderto achieve a fit with more realistic best-fit parameters. All errors presented in this work are 1σ and areappropriately propagated when making relative measurements.6

2.5.3.2

Disc Spectral Evolution: Variable fcol

A variable colour correction factor has been considered from both a numerical (e.g., Shimura &Takahara, 1995; Merloni et al., 2000; Davis et al., 2005) and observational (e.g., Dunn et al., 2011; Reynolds& Miller, 2013) approach, with 1.4 < fcol < 3 generally believed to be a conservatively broad allowable
range, implying at most a factor of ∼ 2 change in fcol . However, this range in fcol does not consider regimes
extending to the low disc luminosities associated with the HIS and LHS nor the effects of disc inhomogeneities
and strong magnetisation. Spectra that are less disc-dominated are harder; thus, requiring an increased fcol .
The possibility of extending the upper fcol bound for discs in the HIS and LHS (i.e., non-disc-dominatedregimes where Ldisc . 0.01 LEdd ) and whether a colour-corrected blackbody is an appropriate description of
the disc spectrum in this regime remain open questions.
Using the diskpn model with the inner disc radius fixed ensures that the evolution of the disc normal6 Two identical data sets, one with a small systematic error added (X) and one without (Y ), are nearly perfectly correlated(i.e., the correlation coefficient, ρX,Y = cov(X, Y )/(σX σY ) ' 1). Since both X and Y only differ in the uncertainties associated
2 . The resulting correlation coefficient becomes,
with the data, their covariance is given by, cov(X, Y ) = cov(X, X) = σX
ρX,Y = σX /σY ' 1, assuming that the systematic errors added to Y are small, which is true for our purposes.

50

pow / bknpow

comptt

simpl

fcol

R02

10
1
0.1

fcol

D03

10
1
0.1

fcol

R04

10
1
0.1

fcol

R07

10
1
0.1

fcol

D07

10
1
0.1

fcol

R10

10
1
0.1

fcol

D11

10
1
0.1
0

20 40
t - t0 (Days)

0

20 40
t - t0 (Days)

0

20 40
t - t0 (Days)

Figure 2.7: Evolution of the colour correction factor over all state transitions derived from best-fit diskpn
normalisations where the inner disc radius was held at 6 Rg . The left, middle, and right panels correspondto fits using the pow/bknpow, comptt, and simpl continuum models, respectively. The abscissa shows time
elapsed since the start time, t0 , of the transition (see Table 2.1). Error bars account for the maximum
conceivable ranges of the mass, distance, and inclination of GX 339–4 (see text). The dashed lines enclose
1.4 < fcol < 3, which is the adopted allowable range in fcol . The transition ID corresponding to each row
is indicated on the rightmost ordinate. Within the large uncertainties, a non-truncated disc with a variablefcol can adequately explain the observed disc spectral evolution.

51
isation is due entirely to a variable colour correction factor, which can be computed directly from Equation
2.6, as demonstrated by Reynolds & Miller (2013). We elect to explore the degree of change in fcol alonethat is required to explain the disc spectral evolution for a fixed inner disc radius of 6 Rg . This choice for
Rin is motivated by considering a disc extending down to the ISCO for a Schwarzschild black hole and is theminimum value allowed by diskpn.
To measure the evolution of absolute fcol values required to explain the disc evolution in each transition
for a fixed inner disc radius, we re-fit all of the spectra with 0.6% systematic error added to the PCA
data using the best-fit models, but replacing the diskbb component with diskpn and freezing Rin at 6
Rg . Figure 2.7 shows the evolution in fcol that is necessary to account for each transition, with the error
bars incorporating the best-fit Kpn error and the range in intrinsic parameters of GX 339–4. The values
adopted for the GX 339–4 observables, along with their (lower, upper) limits, are M = 7 (5.8, 12) M ,D = 8 (6, 10) kpc, and i = 29◦ (10◦ , 60◦ ) (see §2.3.1 and references therein). Figure 2.7 shows that, within
the uncertainties, the disc spectral evolution can potentially be explained by a variable 1.4 . fcol . 3 for an
inner disc radius that remains at 6 Rg throughout each state transition.
Studying the evolution of absolute fcol values for a fixed Rin is limited by the large uncertainties in themass, distance, and inclination of GX 339–4. Given that an inner disc that remains at 6 Rg combined witha modestly variable fcol may be a viable model for accretion disc evolution in the state transitions studied
here, we use the relative measurement technique introduced in §2.5.3.1 to better understand the degree ofchange in fcol that is required to explain the observations without appealing to disc truncation.
The PCA data are replaced with a duplicate copy where no systematic errors have been added. Next,
the diskpn fits where Rin = 6 Rg are re-fitted and a best-fit is found for each observation. This is donefor each of the three Comptonisation models pow/bknpow, simpl, and comptt. For each observation in agiven transition, fcol is computed from Kpn using Equation 2.6, without specifying the intrinsic parameters(i.e., M , D, i). The resulting fcol values are normalised to the fcol from a representative HSS observation,fcol,HSS , and further normalised to the adopted fcol = 1.4 value for the HSS (Davis et al., 2005), yielding

52
the relative colour correction factor,fcol,rel ≡ 1.4

fcol
fcol,HSS


.

(2.8)

Supposing that the inner disc remains fixed at 6 Rg throughout each state transition, Figures 2.8-2.10
show the relative change in fcol alone required to explain the observed disc spectral evolution. Notably,
the pow/bknpow model requires more drastic change in fcol compared to comptt and simpl. Under thepresumption that fcol,rel = 3 is a conservative upper bound, the notion of a variable fcol alone is perhaps
possible for transitions R02, R07, R10, and D11; however, an evolving fcol is an inadequate descriptionof the evolving disc spectrum for transitions D03, R04, and D07, regardless of the Comptonisation model
employed. For the later set of transitions, one must invoke disc truncation or an alternative disc model tosatisfactorily explain the observed disc evolution. Reynolds & Miller (2013) speculate that if the colourcorrected blackbody is a viable model of the LHS, then fcol may be as large as ∼ 5 based on spectral fits to
a comprehensive set of BHB spectra. Inspecting Figures 2.9-2.10, where the comptt and simpl models wereadopted, all seven transitions essentially conform to this extended fcol range; although, making definitive
claims regarding the physically plausible fcol range and applicability of the MCD model in the LHS is beyond
our scope.

2.5.4

A Hot Inner Disc Can Masquerade as an Increased fcol
Observational evidence suggests that GX 339–4 harbors a near-maximally spinning black hole (see§2.3.1). Consequently, the inner disc extends deep into the black hole potential and relativistic effects become
non-negligible, rendering Newtonian disc models inappropriate. The disc models used in this work to derivefcol measurements impose an effective lower limit on the inner disc radius of 6 Rg , whereas the ISCO of a
Kerr black hole with a∗ = 0.998 lies at 1.24 Rg , corresponding to a substantially hotter inner disc. Consider
attempting to use a Newtonian prescription to model an accretion disc around a Kerr black hole, where
Rin = RISCO . The Newtonian disc model is restricted by Rin ≥ 6 Rg and will attempt to fit the true hotter
disc, located at RISCO < 6 Rg , by increasing the colour correction factor as follows. The colour temperature
determined by the model depends on both fcol and Teff according to Tcol = fcol Teff . The maximum effective
temperature of the Kerr disc exceeds that of the Newtonian model with Rin = 6Rg ; therefore, the only way

53

R02

fcol, rel

10

1
52380

52390

52400

52410

D03

fcol, rel

10

1
52700

52720

52740

52760

R04

fcol, rel

10

1
53190

53200

53210

53220

53230

53240

R07

fcol, rel

10

1
54120

54130

54140

54150

D07

fcol, rel

10

1
54220

54230

54240

54250

R10

fcol, rel

10

1
55280

55290

55300

55310

55320

55330

D11

fcol, rel

10

1
55570

55580

55590
MJD

55600

Figure 2.8: Relative measurements of the evolution of fcol for each transition derived from fits using
pow/bknpow to model Comptonisation. The two lower dashed lines enclose 1.4 < fcol < 3, while the upperdashed line marks fcol = 5. The fcol,rel measurements for each transition were made relative to the filled
glyphs and normalised to fcol = 1.4, which is appropriate for the HSS. The disc evolution can be explained
for transitions R02, R07, R10, and D11 by a non-truncated disc with 1.4 ≤ fcol,rel ≤ 5.

54

R02

fcol, rel

10

1
52380

52390

52400

52410

D03

fcol, rel

10

1
52700

52720

52740

52760

R04

fcol, rel

10

1
53190

53200

53210

53220

53230

53240

R07

fcol, rel

10

1
54120

54130

54140

54150

D07

fcol, rel

10

1
54220

54230

54240

54250

R10

fcol, rel

10

1
55280

55290

55300

55310

55320

55330

D11

fcol, rel

10

1
55570

55580

55590
MJD

55600

Figure 2.9: Same as Figure 2.8, but with the Comptonisation model comptt replacing pow/bknpow. The
required evolution in fcol,rel is systematically reduced compared to results from the pow/bknpow continuum
model.

55

R02

fcol, rel

10

1
52380

52390

52400

52410

D03

fcol, rel

10

1
52700

52720

52740

52760

R04

fcol, rel

10

1
53190

53200

53210

53220

53230

53240

R07

fcol, rel

10

1
54120

54130

54140

54150

D07

fcol, rel

10

1
54220

54230

54240

54250

R10

fcol, rel

10

1
55280

55290

55300

55310

55320

55330

D11

fcol, rel

10

1
55570

55580

55590
MJD

55600

Figure 2.10: Same as Figure 2.8, but with the Comptonisation model simpl replacing pow/bknpow. The
required evolution in fcol,rel is systematically reduced compared to results from the pow/bknpow continuum
model.

56
for the model to achieve the colour temperature demanded by the data is to increase fcol to an erroneously
high value.
Using the XSPEC command fakeit none, we generate a suite of simulated disc spectra with the kerrbb
model of a spinning black hole with an RXTE response matrix and integration time of 2000 seconds. The
cases considered are a Schwarzschild black hole (a∗ = 0, η = 0.057) and maximal Kerr black hole (a∗ = 0.998,
η = 0.30) (Thorne, 1974), each having mass 7 M

located at a distance of 8 kpc with a disc inclinationof 30◦ . The inner disc torque is set to zero and the effects of both self-irradiation and limb-darkening on
the disc spectrum are switched on. We pair mass accretion rates with colour correction factors that arereasonably representative of the spectral state corresponding to each Ṁ that we select. For the a∗ = 0
black hole, we simulate disc spectra for Ṁ = 0.05 ṀEdd with fcol = 1.4, attempting to mimic a HSS/SIS
accretion disc; and Ṁ = 0.005 ṀEdd with fcol = 3, which is more representative of a HIS/LHS observation.
Similarly, for the a∗ = 0.998 black hole, we choose Ṁ = 0.01 ṀEdd with fcol = 1.4 and Ṁ = 0.001 ṀEdd
with fcol = 3. Extending Ṁ beyond these ranges for a fixed fcol resulted in either simulated data withunconstrained uncertainties or unacceptable fits (i.e., χ2ν  1) using the procedure described below.
Each spectrum is fit over the energy range 2.8-10 keV using the diskpn model with Rin fixed at
6 Rg . The best-fit colour correction factor is computed from the normalisation using Equation 2.6. Table 2.2
presents the results of this exercise. In general, the psuedo-Newtonian model, diskpn, reliably recovers thetrue fcol value for a non-spinning black hole, but requires a factor of ∼ 1.9 increase in fcol for a maximallyspinning black hole. This demonstrates that the inability of Newtonian disc models to properly model Rin
and Teff for Kerr black holes can result in unphysically large fcol measurements to account for the hotter
inner disc.
One may wonder if relative measurements of fcol derived from Newtonian disc models, as done in thiswork, are affected by the inability of these models to properly represent the hotter discs around spinning black
holes. A colour correction factor evolving in a HSS → LHS transition from fcol = 1.4 → 3 corresponds to afactor of 2.1 increase in fcol . Inspection of Table 2.2 indicates that the diskpn model requires a factor of ∼ 2.1
increase in fcol for an order of magnitude decrease in Ṁ for both the Schwarzschild and maximal Kerr black
holes. This result is fully consistent with the ‘true’ fcol evolution and shows that relative measurements of fcol

57
a∗
0
0
0.998
0.998

Ṁ
(ṀEdd )
0.05
0.005
0.01
0.001

Tcol,max(keV)
0.416(2)0.493(5)0.455(5)0.56(1)Kpn

fcol,true

fcol

χ2ν

0.177(8)0.0098(8)0.015(2)0.0006(1)

1.4
3.0
1.4
3.0

1.39(2)2.87(6)
2.58(9)5.8(2)

1.35
1.26
0.87
0.71

Table 2.2: Results from fitting simulated spectra generated from the kerrbb model of a Kerr black hole withthe pseudo-Newtonian model, diskpn, where the inner disc radius was set to 6 Rg . The first two columns are
the black hole spin and mass accretion rate in Eddington units used to generate the simulated disc spectra.
Proceeding from left to right, the columns are the best-fit maximum disc colour temperature, best-fit discnormalisation, colour correction factor used to generate the disc spectrum, colour correction factor recoveredfrom the best-fit normalisation, and reduced χ2 . All models had 15 degrees of freedom. Uncertainties on
the last significant digit are given in parentheses and correspond to the 90% level.

made with Newtonian disc models are reliable, even when the black hole under consideration is maximallyspinning.

2.5.5

Disc Temperature and Luminosity
The luminosity of a geometrically thin, optically thick disc is theoretically expected to obey a relation4of the form Ldisc = σATeff, where A is the radiating surface area. The disc luminosity depends on both
2
A, which scales as Rin, and effective temperature, which depends on fcol . The purpose of this section is to
2 4
explore the Ldisc − Rin
Teff relation, particularly extended to low luminosities (i.e., the HIS and LHS), andthe effect of introducing the possibility of a variable colour correction factor.
For the ezdiskbb disc model of a Newtonian disc with a zero-torque inner boundary condition,
Zimmerman et al. (2005) finds the disc luminosity is given by,
Ldisc = 73.9σ

Tcol,max
fcol

4

2
Rin
.

(2.9)

Observationally, we measure an unabsorbed disc flux,

Fdisc =

Ldisc
.
4πD2

(2.10)

In practice, Fdisc is obtained by operating the XSPEC cflux command on the disc component with the normalisation frozen to its best-fit value. Unabsorbed disc fluxes and unabsorbed total model fluxes, Ftotal , arecalculated over 0.1-10 keV and 0.1-100 keV, respectively. Figure 2.11 shows the relation between unabsorbed

58

Fdisc / Ftotal
0.05

0.2

0.5

1

10-8

pow / bknpow

Fdisc, unabs (erg s-1 cm-2)

10

0.1

-7

10-9
10-10

10-8
simpl

Fdisc, unabs (erg s-1 cm-2)

10-11

10-9
10-10

10-8
10-9

R02
D03
R04
R07
D07
R10
D11

10-10
10-11
0.1

compttFdisc, unabs (erg s-1 cm-2)

10-11

0.2

0.5
kTeff, max (keV)

1

2

Figure 2.11: Unabsorbed disc flux (0.1-10 keV) vs. maximum disc effective temperature (Teff,max =Tcol,max /fcol ) for all transitions, where Tcol,max is measured using ezdiskbb to model the disc component
and a constant fcol = 1.4 is assumed. Comptonisation is modelled using the XSPEC models pow/bknpow(top panel), simpl (center panel), and comptt (bottom panel). From top to bottom, the horizontal dashedlines indicate Ldisc /LEdd = 10−1 , 10−2 , 10−3 , where Ldisc was converted to Fdisc using Equation 2.10 with
D = 8 kpc. The Eddington luminosity is given by, LEdd = 4πGM mp c/σT , where mp is the proton mass,
σT is the Thomson cross section, and we adopt M = 7 M . Uncertainties in M and D are not incorporated
because the shape of the Fdisc − Teff,max relation is insensitive to these intrinsic parameters. The diagonal
4dashed lines represent the expected Fdisc ∝ Teff,max
relation for an optically thick, geometrically thin, zerotorque disc (Equation 2.9) assuming the aforementioned intrinsic parameters and constant Rin = 1 Rg (blackline), 6 Rg (red line), 20 Rg (blue line), and 100 Rg (green line). Glyph colours denote the disc fraction ofeach observation. The key in the bottom right identifies the glyph associated with each transition ID. When4
a constant fcol is assumed, departure from the Ldisc − Teffis evident for low-luminosity discs characteristicof the HIS and LHS.

59
disc flux and maximum effective temperature for each of the three Comptonisation models pow/bknpow,
comptt, and simpl. Tcol,max is measured from fits to all observations with the disc component modelledby ezdiskbb and Teff,max = Tcol,max /fcol , assuming a constant fcol = 1.4. The disc fractions, defined by
Equation 2.7, are given by the glyph colours. We note that this work includes observations with very low
disc fractions compared to previous investigations of the luminosity-temperature relation for BHBs, which
tend to be restricted to the HSS where DF & 0.8 (e.g., Gierliński & Done, 2004).
If one assumes a constant disc geometry, the disc emitting area is unchanging and we expect to recover4
the scaling, Fdisc ∝ Teff
, denoted by the diagonal lines in Figure 2.11 for various choices of Rin and fcol .

While the theoretical relation is achieved for disc luminosities, Ldisc & 0.05LEdd , clear departure is evident
below this threshold where the BHB is categorized in the HIS or LHS.
The diskbb and ezdiskbb models are purely Newtonian and correction factors accounting for various
general relativistic and viewing angle effects (Cunningham, 1975; Zhang et al., 1997) have been applied inprevious observational studies of the luminosity-temperature relation for BHBs (e.g., Gierliński & Done,
2004; Dunn et al., 2011). Selecting the proper correction factors requires knowledge of the disc inclinationand spin, for which there is considerable debate for GX 339–4 (see §2.3.1). Fortunately, the correction factors
are independent of flux and temperature; therefore, we can safely neglect incorporating these modifications
without altering the shape of the Fdisc −Teff diagram, irrespective of the accuracy of the Fdisc and Teff values.
The effective temperature plotted in Figure 2.11 is calculated under the assumption of a constantcolour correction factor, which will subsequently be relaxed. Deriving relative measurements for the colour
correction factor according to Equation 2.8 and as described in §2.5.3.2, Figure 2.12 shows the relationshipbetween fcol and the maximum colour temperature, which is a parameter in the disc models. Supposingthat the inner disc remains fixed at Rin = 6 Rg , which is the implicit assumption behind all of our fcol,rel
measurements, the colour correction factor increases with colour temperature approximately linearly. The
majority of observations yield colour corrections that fall within the range 1.4 . fcol,rel . 5, in agreementwith Reynolds & Miller (2013). Notably, the models simpl and comptt, which are rooted in physicalconsiderations of Comptonisation, constrict the range of fcol,rel as compared to the phenomenological andwidely used powerlaw and bknpower models. Particularly for best-fit colour temperatures, Tcol,max & 1 keV,

60

Fdisc / Ftotal
0.05

0.1

0.2

0.5

1

20

5

pow / bknpow

fcol, rel

10

R02
D03
R04
R07
D07
R10
D11

2
1

simpl

fcol, rel

10
5

2
1

compttfcol, rel

10
5

2
1
0.5

1
kTcol, max (keV)

2

Figure 2.12: Relative measurements of the colour correction factor shown against the maximum colour
temperature of the disc from best-fits using the XSPEC Comptonisation models pow/bknpow (top panel),
simpl (center panel), and comptt (bottom panel). Results from all transitions are shown together, with thedisc fraction given by the glyph colour. The key in the upper left portion of the top panel identifies theglyph associated with each transition ID. Tcol,max is obtained from the best-fit ezdiskbb parameter andfcol,rel is computed using the same procedure as in Figure 2.8 (see text). Horizontal dashed lines markfcol,rel = 1.4, 3, 5. For a disc inner radius fixed at 6Rg , the majority of observations yield 1.4 . fcol,rel . 5
and show a roughly linear relation between fcol and Tcol .

61fcol, rel
1

5

10

10-8

pow / bknpow

Fdisc, unabs (erg s-1 cm-2)

10

2

-7

10-9
10-10

10-8
simpl

Fdisc, unabs (erg s-1 cm-2)

10-11

10-9
10-10

10-8
10-9

R02
D03
R04
R07
D07
R10
D11

10-10
10-11
0.1

compttFdisc, unabs (erg s-1 cm-2)

10-11

0.2

0.5
kTeff, max (keV)

1

2

Figure 2.13: Unabsorbed disc flux (0.1-10 keV) vs. maximum disc effective temperature for all transitions,where Tcol,max is measured using ezdiskbb. Relaxing the assumption of a constant fcol , all observations were
re-fitted with ezdiskbb replaced by diskpn with Rin = 6 Rg . Relative values of fcol were computed fromthe best-fit diskpn normalisation, using the same procedure as in Figure 2.8. The abscissa shows Teff,max =Tcol,max /fcol,rel . Glyph colours correspond to the fcol,rel of each observation. The various lines are the same4
as those in Figure 2.11. The Ldisc ∝ Teff
relation is recovered for disc luminosities, Ldisc & 10−3 LEdd , whenfcol is considered to be variable.

62

10
R02
D03
R04
R07
D07
R10
D11

9

Rin (Rg)

8

7

6

5
0

20

40
t - t0 (Days)

60

Figure 2.14: Time evolution, as measured from the start of the transition listed in Table 2.1, of the innerdisc radius computed from the ezdiskbb and diskpn normalisations (Equation 2.11) when Rin was fixed in
the diskpn model at 6 Rg . The dashed line marks Rin = 6 Rg . The key in the upper right corner denotes
the glyphs corresponding to each transition. The constancy of Rin implies that relative measurements madefrom combining results from fits made with ezdiskbb and diskpn are unaffected by the minor differences
between those disc models.

63
the powerlaw and bknpower continuum models may return erroneously low values for the disc normalisation.
We attempt to explore whether the luminosity-temperature relation can be extended to lower luminosities characteristic of the HIS and LHS by allowing for a variable fcol . Figure 2.13 shows the Fdisc − Teff
diagram with the only difference to Figure 2.11 being that the effective temperature is now calculated by,
Teff,max = Tcol,max /fcol,rel . In other words, Figure 2.11 is recomputed using the fcol,rel values of Figure
2.12 in place of the constant fcol = 1.4. Accepting the dynamic range in fcol suggested by Figure 2.12,
4Fdisc ∝ Teffis recovered for disc luminosities, Ldisc & 10−3 LEdd . Furthermore, the relation falls neatly on

the expected track corresponding to the emitting surface area remaining constant with the inner disc locatedat Rin = 6 Rg .
Caution must be exercised when combining different disc models. The relative measurements fcol,rel
used to recompute Teff,max in Figure 2.13 called upon diskpn and ezdiskbb separately to model the disccomponent in order to take advantage of the strengths of each model for our purposes. One may reasonably
ask if these two disc models, which both adopt the zero-torque inner boundary condition, are consistent.
The significant difference between the models is that diskpn approximates the general relativistic effects
on the accretion disc with a pseudo-Newtonian potential, while ezdiskbb is strictly Newtonian; therefore,
discrepancies between the two may materialize in the spectral fits. Combining the normalisations from eachdisc model, given by Equations 2.5 and 2.6, the inner disc radius is obtained,
Rin= 0.0677
Rg



Kez
Kpn

1/2
.

(2.11)

Since Rin is a separate parameter in diskpn instead of being incorporated into the normalisation, we expect
to exactly recover this value for Rin from Equation 2.11 if diskpn and ezdiskbb are identical. Figure 2.14
shows Rin computed from Equation 2.11 for fits to all observations where Rin = 6 Rg in the diskpn fits.
Values of Rin falling on 6 Rg would indicate that the inner radius enforced in the diskpn fits was exactly
recovered in the independent ezdiskbb fits, while deviations from 6 Rg arise from differences in the ezdiskbb
and diskpn models. Although the inner disc radius value of 6 Rg set in diskpn is not exactly recovered, Rinremains constant throughout each transition at ∼ 7.5 Rg , merely a systematic offset that does not affect theinterpretation of Figure 2.13.

64
2.5.6

Low Frequency Quasi-Periodic Oscillations
LFQPOs are observed in the temporal power spectra of BHBs and remain a long-standing puzzle ofactive observational and theoretical interest. Explanations for the origin of LFQPOs appeal to an exhaustivearray of production models including trapped waves (Kato, 1990), discoseismic modes (Nowak & Wagoner,
1991, 1992, 1993; Reynolds & Miller, 2009), Lense-Thirring precession (Ipser, 1996; Stella et al., 1999;
Ingram et al., 2009), accretion-ejection instability (Tagger & Pellat, 1999; Varnière et al., 2002; Varnière &Tagger, 2002), and a truncated disc LHS geometry (Giannios & Spruit, 2004). Recent simulations of global,
magnetised accretion discs suggest that LFQPOs have an origin in the corona rather than the disc directly(O’Neill et al., 2011). While the origin of LFQPOs is not firmly established, the details of the disc geometry
may provide clues to their nature.
X-ray spectra alone cannot distinguish between a truncated disc and changing disc structure/coronal
activity as the underlying accretion geometry. Given a plausible production mechanism, LFQPOs havethe potential to be diagnostics of the inner accretion flow. In the LFQPO excitation model envisioned by
Giannios & Spruit (2004), quasi-spherical oscillations of the RIAF interior to a truncated disc in the LHS
produce modes with frequencies near the Keplerian frequency of the inner disc,
RKep
=
Rg



GM
2πνQPOc3

−2/3,(2.12)

where νQPO is the frequency of the LFQPO. In this framework, Giannios & Spruit (2004) require that the
RIAF, and hence the recessed inner disc, extend out to Rin ∼ 100 Rg .
LFQPOs observed in GX 339–4 and their frequency migration over the course of the state transitions
studied here has received considerable attention in the literature. The frequency of LFQPOs is observed todecrease (increase) over a decay (rise) transition with a typical range 0.1 Hz . νQPO . 10 Hz, implying that
accretion rate and QPO frequency are intricately linked. In an attempt to test the truncated disc LFQPO
production model, which predicts that RKep ' Rin & 100 Rg in the LHS, we collect measured νQPO values
for all of the GX 339–4 state transitions considered (Belloni et al., 2005; Shaposhnikov & Titarchuk, 2009;
Stiele et al., 2011; Motta et al., 2011; Nandi et al., 2012). Given the uncertainty in black hole mass for GX
339–4, we seek relative measurements of RKep normalized to the ISCO for a non-spinning black hole in the

100
R02

RKep, rel (Rg)

65

10
52390

52400

52410

100
D03

RKep, rel (Rg)

1
52380

10
1
52740

52760

100
10
1
53190

53200

53210

53220

53230

53240

100
R07

RKep, rel (Rg)

52720

R04

RKep, rel (Rg)

52700

10
1
54150

10
54230

54240

54250

100
R10

RKep, rel (Rg)

54140

100

1
54220

10
1
55280

55290

55300

55310

55320

55330

100
D11

RKep, rel (Rg)

54130

D07

RKep, rel (Rg)

54120

10
1
55570

55580

55590
MJD

55600

Figure 2.15: Relative measurements of the evolution of RKep,rel for each transition (see text for definition)
derived from LFQPO frequency evolution. The dashed lines mark Rin = 6, 20, 100 Rg . The RKep,rel measurements for each transition were made relative to the filled glyphs and normalized to Rin = 6 Rg , which isappropriate for a Schwarzschild black hole in the HSS.

66
HSS,

RKep,rel ≡ 6Rg

RKep
RKep,HSS,(2.13)

where RKep,HSS is the Keplerian radius computed for a given transition from the νQPO of the observation
nearest to the HSS in time using Equation 2.12, without specifying M . Supposing that RKep is associatedwith Rin , Figure 2.15 shows the evolution in disc inner radius implied by the observed migrating LFQPOs
for each transition. With the exception of transition D11, the Keplerian frequencies implied by the LFQPOs
reach out to ∼ 50 − 100 Rg in the LHS, which is broadly consistent with the degree of recession requiredby the RIAF/truncated disc LFQPO excitation model. If LFQPOs are associated with the inner disc in asimilar manner to that described by Giannios & Spruit (2004), then a truncated disc geometry may be aplausible description for state transitions. However, the spectral fits in this work suggest that Rin does nottruncate to such extreme disc locations, which would demand reconciling.

2.6

Discussion
We have shown that a moderately increasing fcol as the disc fraction decreases can adequately andconsistently describe the spectral evolution of the accretion disc in GX 339–4 over multiple state transitions.
The purpose of this work is not to debunk a truncated disc geometry as a viable model for the LHS of
BHBs. We recognize that the observations in this work interpreted with simplistic disc models are unable todistinguish between changes in Rin and/or fcol . Rather, we demonstrate that reasonable variations in fcol
provide an alternative explanation to disc truncation for accretion disc evolution. Motivated by this result
and supposing that fcol is variable, we consider the impacts that this would have on the interpretation of
BHB observations, ultraluminous X-ray sources, and models of the accretion flow across spectral states. We
then discuss the physically expected range of attainable values for fcol and conclude the discussion with acautionary commentary on the widely used disc models.

2.6.1

Implications of a Variable Colour Correction
Imposing an unchanging inner accretion disc radius located at the ISCO of a Schwarzschild black hole,

we measured the relative changes in fcol necessary to adequately fit RXTE spectra of GX 339–4 over seven

67
state transitions. Arbitrarily selecting a normalisation of fcol = 1.4 for the HSS and denoting measurementsof fcol relative to this value by fcol,rel (see Equation 2.8), we tracked the evolution of fcol,rel for each transition(see Figures 2.8-2.10). Transitions R02, R07, R10, and D11 require colour corrections conforming to therange 1.4 . fcol,rel . 3, while transition R04 demands the range be extended to 1.4 . fcol,rel . 5. These
ranges in fcol evolution, which can account for the majority of the transitions, correspond to a factor of∼ 2.0 − 3.5 increase in fcol for a HSS → LHS transition. Transitions D03 and D07 require more extreme
evolution in fcol,rel , with values as high as fcol,rel ∼ 10. Our main finding is to demonstrate that an evolving
fcol provides an alternative and adequate description of the changing disc spectrum during a state transition,
which is perhaps a more attractive scenario to imagine than the sudden evacuation or replenishment of theinner accretion flow.

2.6.1.1

4
Recovering L − Teffat Low Luminosities

4
A blackbody source of constant emitting area is expected to obey the L−Teff
scaling, which relates thesource luminosity to its effective temperature. Accretion flows corresponding to the HSS of BHBs, where thethermal accretion disc dominates the total flux and the inner disc location does not deviate from the ISCO,
nicely follow this expected relation (e.g., Gierliński & Done, 2004); however, deviations arise in the HIS andLHS (e.g., Dunn et al., 2011). This work encompasses the disc luminosity range, 10−3 . Ldisc /LEdd . 0.5,
4finding that a fcol that evolves with spectral state can reproduce the theoretical Ldisc −Teffacross all spectral

states. Studying the disc-dominated states (i.e., DF & 0.8) of many BHBs over a typical luminosity range,
40.01 . Ldisc /LEdd . 0.5, Gierliński & Done (2004) found that the Ldisc − Teff
relation holds for a constantfcol . We observe similar and consistent behaviour for GX 339–4 in the HSS. As the disc fraction drops and4
the source enters the HIS or LHS, significant departures from Ldisc − Teffare apparent in Figure 2.11, inagreement with Dunn et al. (2011). These ‘spurs’ on the luminosity-temperature diagram can be interpreted
as either a decrease in the inner disc radius, and hence the emitting area, or the result of neglecting to accountfor a decrease in Teff = Tcol /fcol caused by an increased fcol in the LHS. Of course, some combination of these
two effects is possible as well. Dunn et al. (2011) measured the incremental degree of change in fcol from a4canonical HSS value that was required to return the spurs to the expected Ldisc ∝ Teff
law for a constant

68
emitting area (see their Figure A.1). For GX 339–4, they found that fairly modest changes in fcol , consistent
with the relative changes we determined, could restore the theoretical relation; however, they argue that the4most severe departures from Ldisc − Teff
, which occur for the lowest disc fractions, are not likely the resultof a variable fcol . Here, we suggest that the physically realizable range of fcol is perhaps not as narrow ascommonly assumed (see §2.6.2) and that a thin disc geometry with a fixed inner radius and evolving fcol isa viable phenomenological model for BHB state transitions.
We expand on the work of Dunn et al. (2011) by breaking the observations of a BHB into individualtransitions, exploiting relative measurements, and exploring empirical and physical Comptonisation models.
Comparing Figures 2.8-2.10, we find that employing physically motivated Comptonisation models that avoid
the ‘flux stealing’ phenomenon from the disc component require less severe fcol evolution for any given
4transition. Inspection of the panels of Figure 2.13 shows that the Ldisc − Teff
relation is recovered whena changing fcol is incorporated, regardless of the choice of Comptonisation model. Relative measurementsof fcol broken down by transition in Figures 2.8-2.10 hint at an underlying difference in the details of theaccretion flow depending on whether the source is in the rise or decay stage of outburst. Qualitatively, thedecay transitions demand more extreme and perhaps physically unattainable fcol evolution, while the risetransitions fit more comfortably into the idea of a moderately variable fcol . Given that decoupling the true
behaviour of the inner disc from that of the colour correction factor is impossible with simplistic disc models
and RXTE data, we leave the possibility of different accretion scenarios between rise and decay transitions
as pure speculation.

2.6.1.2

The Ultraluminous State

Adopting a variable colour correction factor parameterises the various physical processes in BHB
accretion into a single factor. This work suggests that moderate changes in fcol may conceivably accountfor the wide range of spectral states observed when a BHB undergoes a state transition. Recently, a newspectral class, dubbed the ultraluminous state (ULS), has been postulated for the class of ultraluminous
X-ray sources (ULXs) (Gladstone et al., 2009). Here, we explore whether the ULS, which has never been
observed in a confirmed BHB, is consistent with the interpretation of spectral states arising from physical69
processes that are observationally manifested as an evolving colour correction factor.
ULXs are X-ray sources that exceed the Eddington luminosity of stellar mass black holes (LX &1039 erg s−1 ), but are not spatially coincident with galactic nuclei; therefore, ULXs are distinct from AGNs.
Two possible explanations for ULXs are: (1) intermediate mass black holes (IMBHs; M ∼ 102 − 104 M )accreting at sub-Eddington rates, but the production rate of binaries composed of an IMBH and donor
star is likely too low to account for the number of observed ULXs (Madhusudhan et al., 2006); and/or(2) stellar mass black holes accreting at super-Eddington rates, which requires a mechanism for achieving
super-Eddington accretion, such as beaming/viewing effects of outflows (e.g., King et al., 2001; Begelman
et al., 2006) or disc inhomogeneities (Begelman, 2001, 2002, 2006).
The spectral characteristics of ULXs were studied by Gladstone et al. (2009), leading them to proposethat ULXs are super-Eddington accreting stellar mass black holes with ubiquitous spectral features. These
commonalities suggest that ULXs are indeed a spectral class of BHBs, dubbed the ultraluminous state(ULS); however, this accretion state does not appear to arise in the well-studied Galactic BHBs. The ULS
is characterised by both a broadened, hardened thermal disc component and a break in the power-law
component at lower energies (& 3 keV) than typically observed in less luminous BHB states. The Gladstone
et al. (2009) ULX sample was adequately fitted with a physically motivated disc plus Comptonising coronamodel, where the best-fit models called for optically thick coronae (τp ∼ 5 − 30) and cool disc colour
temperatures (Tcol,max ∼ 0.2 keV). Physically, the broadened/hardened disc and low-energy spectral break
may be attributed to a complicated inner geometry where an optically thick corona obscures and alters theenergetics of the inner disc. Normally, disc temperatures this low would imply a black hole of mass ∼ 103 M(e.g., Kaaret et al., 2003); however, the measured disc temperature may be lowered if a significant fraction of
the accretion energy is dissipated in the corona (e.g., Svensson & Zdziarski, 1994). Based on the requirementof an optically thick corona, as opposed to the τp ∼ 1 coronae typically observed in BHBs, sub-Eddingtion
accretion onto an IMBH was ruled out.
Figure 2.16 shows the coronal optical depth versus the maximum disc colour temperature for GX 339–4
determined from the best-fits to the disc plus Comptonising corona XSPEC model, phabs×(comptt+diskpn),
with Rin = 6 Rg . Replacing the disc component with either diskbb or ezdiskbb produces essentially70

Fdisc / Ftotal
0.05
100

0.1

0.2

0.5

1

τp

10

1
R02
D03
R04
R07
D07
R10
D11

0.1

0.01
0.1

0.2

0.5
kTcol, max (keV)

1

2

Figure 2.16: Coronal optical depth vs. maximum disc colour temperature for all transitions, where τp andTcol,max are measured using comptt and diskpn with Rin = 6 Rg , respectively. The disc fraction is given by
the glyph colour. The results from a sample of 12 ULXs (Table 8 of Gladstone et al. (2009)) are overlaidin black crosshairs. The HIS/LHS GX 339–4 RXTE data analysed in this work permit moderately opticallythick coronae in the same range as found for ULXs.

71
indistinguishable results. For comparison, the results from Table 8 of Gladstone et al. (2009) are shown fora sample of 12 ULXs determined from fits to high-quality XMM-Newton data, where diskpn and comptt
were also adopted as the disc and Comptonising models, respectively.7 When GX 339–4 is in the HIS/LHS(i.e., low disc fractions), the typical best-fit coronal optical depth falls in the range, τp ∼ 2 − 10. The coronal
optical depth of the ULX sample is described by τp ∼ 5 − 15 (τp ∼ 10 − 100) for discs with temperatures
Tcol,max . 0.5 keV (Tcol,max & 0.5 keV). Figure 2.16 challenges the claim that spectral fits to sub-Eddington
BHBs do not permit moderately optically thick coronae.
Although the data are consistent with an optically thick corona model, here we point out a strong
parameter degeneracy between the colour correction factor and the coronal optical depth. Of the threemodels we considered to represent Comptonisation, all of which provided equally good fits to the data,
only comptt incorporates the coronal temperature and optical depth as physical parameters. Inspection of
Figure 2.12 reveals that the pow/bkn and simpl models require higher colour correction factors than comptt.
Particularly for the HIS/LHS observations of GX 339–4, introducing a physical corona drives fcol to lower
values with preference for an optically thick corona (see Figure 2.12). Therefore, the data cannot distinguishbetween an increased colour correction factor and an optically thick corona.
In the IMBH model, ULXs are massive scaled up versions of BHBs in the LHS (i.e., IMBHs accretingat sub-Eddington rates). The IMBH model for ULXs was discarded by Gladstone et al. (2009) because themeasured coronal optical depths were larger than commonly observed in BHBs. In the case of GX 339–4 inthe HIS/LHS, using the same disc plus Comptonisation model, we see that the best-fit coronal optical depths
lie precisely in the same moderately optically thick range as the ULX sample of Gladstone et al. (2009).
Therefore, the data do permit optically thick coronae in the HIS/LHS of GX 339–4, but we stress that themeasurement of coronal optical depth from basic disc + corona modelling cannot be made definitive dueto the degeneracy between τp and fcol . This possibility alleviates the hesitation for supposing that ULXs
are analogous to scaled up BHBs on the grounds of coronal optical depth. Allowing for a variable colour
correction factor, the broadened/hardened ULX disc spectra and low measured ULX disc temperaturesnaturally fit into the IMBH description; therefore, introducing arguments for reducing the disc temperature
7 Gladstone et al. (2009) do not specify the R
in value used in the diskpn model. Our results suggest that the coronalproperties and disc temperature are largely unchanged in the spectral fits, regardless of what one chooses for Rin .

72
by dissipating accretion energy into the corona become unnecessary. An important discrepancy in coronalproperties between the HIS/LHS GX 339–4 data and the ULX sample is the temperature of the corona.
While the data permit τp ∼ 5 − 10 for both the HIS/LHS and ULXs (see Figure 2.16), the associated coronaltemperature for ULXs (Tp ∼ 1 − 3 keV) is an order of magnitude below that of GX 339–4 in the HIS/LHS(Tp ∼ 10 − 30 keV) (see Figure 2.17). We stress that the purpose of this section is to highlight an interesting
possibility that the ULS may not be an intrinsic feature of BHBs; however, this is highly speculative andfurther investigations are required to shed more light on the nature of ULXs.

2.6.1.3

The Role of fcol in State Transitions

A triggering mechanism for state transitions is not yet well understood. Schnittman et al. (2013)
demonstrated that all of the spectral states of BHBs could be recovered by varying only the mass accretionrate, Ṁ , (i.e., effectively changing the location of the disc photosphere, or the disc/corona boundary) in
the magnetohydrodynamic general relativistic global disc simulation of Noble et al. (2010). The disc wasassumed to produce a diluted blackbody spectrum at the photosphere with a constant fcol = 1.8, which
could be further altered (e.g., broadened/hardened) by bremsstrahlung, synchrotron, and inverse Compton
processes in the corona. The emergent spectrum was computed with a post-processing radiative transfer
technique for choices of Ṁ representative of the observed BHB states. In this simulation, the inner accretion
disc remained extended down toward the ISCO. Only details of the coronal physics for a given location of
the disc photosphere (i.e., for a given value of Ṁ ) provided alterations to a diluted blackbody spectrum andall BHB spectral states were reproduced within this framework. This result emphasises the importance ofcoronal physics in modifying the underlying disc spectrum, suggesting that perhaps fcol is a variable functionof Ṁ and can be dependent on processes external to the disc.
A changing mass accretion rate plays a key role in state transitions, but is likely not the whole picture,nor is it clear what mechanism drives changes in Ṁ . For an inner disc remaining at the ISCO during a statetransition, a large change in Ṁ is implied. In the HSS, the inner portions of the disc are expected to
be radiation pressure-dominated, where instabilities may lead to an inhomogeneous disc structure and a
low fcol ∼ 1 (Begelman, 2006). As Ṁ decreases, the ratio of radiation-to-gas pressures drops and below

73

Fdisc / Ftotal
0.05
100

0.1

0.2

0.5

1

τp

10

1

0.1

0.01
0.1

R02
D03
R04
R07
D07
R10
D11
1

10

100

kTp (keV)
Figure 2.17: Coronal optical depth vs. coronal temperature, both measured using comptt. The disc model
was diskpn with Rin = 6 Rg . The disc fraction is given by the glyph colour. The results from a sample of
12 ULXs (Table 8 of Gladstone et al. (2009)) are overlaid in black crosshairs.

74
some Ldisc /LEdd the density inhomogeneities would disappear and fcol would increase. If the disc is strongly
magnetised, values of fcol & 5 are conceivable (Begelman & Pringle, 2007). For a disc that instead transitions
from a standard thin disc to a truncated disc with an interior RIAF, a relatively constant Ṁ is implied at thisstage with a decreased radiative efficiency. Both scenarios of an inner disc remaining extended to the ISCO
and a truncated disc + RIAF predict different Ṁ behaviour, which is difficult to distinguish observationally.
The phenomenological picture of a variable colour correction factor being capable of describing thedisc spectral evolution observed in BHB outbursts advocates for the development and consideration ofmodels that appeal to an evolving vertical disc structure and/or intricate disc-corona-jet connection. Recent
observational work supports the idea of a coupled disc + corona playing a critical role in the LHS. Adopting
sophisticated, self-consistent modelling techniques, Reis et al. (2013) studied the BHB XTE J1650-500 during
its 2001/2002 outburst and found definite correlations between the reflected X-ray flux and power-law flux
associated with the corona. These results fit into the interpretation of a light bending + collapsing coronamodel for state transitions (e.g., Miniutti & Fabian, 2004) and do not lend support to the truncated disc
picture. An organized campaign of contemporaneous and frequent X-ray/radio monitoring of Cygnus X-1 inthe LHS reveal that the reflected X-ray flux and the radio flux are positively correlated (Miller et al., 2012),
which is consistent with the coronal ejection model of Beloborodov (1999). As the promising prospect of disc
+ corona coupling in BHB outbursts receives more attention, we hope that the relative measurements of fcolpresented here serve as a useful reference for physical models that seek to generate predictive diagnostics ofspectral hardening in accretion systems.

2.6.2

Physical Range of the Colour Correction Factor
The colour correction factor phenomenologically parameterises our ignorance of spectral alterations

arising from the highly unknown and complicated combinations of disc vertical structure, accretion power
dissipation properties, magnetisation, and inhomogeneities. Lumping the effects of these physical processes
into the frequency-independent fcol parameter is attractive from the standpoint of limiting the parametersin MCD blackbody models; however, whether this treatment is an appropriate description of BHB discs inall spectral stages remains disputed. Here, we discuss the results from theoretical endeavors to constrain the

75physically realisable range of the colour correction factor.

2.6.2.1

Radiative Transfer in One-Dimensional Disc Atmospheres

Investigations into the physically reasonable fcol range and the regimes for which the colour-correctedblackbody model is justified have focused on one-dimensional atmospheric disc models. These atmospheres
are assumed to have no azimuthal structure and are constructed at many disc radii from self-similar standard
thin disc equations. In brief, frequency-dependent opacities are computed at each radial position, allowing
the modification of the locally emitted blackbody spectrum to be determined for multiple concentric annuli.
The emergent spectrum, which is modified by the vertical structure, is computed for each annulus. The
integrated disc spectrum is obtained by summing over all annuli and is fitted with the colour-correctedblackbody model, yielding a best-fit fcol .
Following the technique of self-consistently computing the radiative transfer and vertical disc structure(Ross et al., 1992), Shimura & Takahara (1995) considered the effects of electron scattering and free-freeemission/absorption on the emergent disc spectrum. Regardless of the black hole mass and radial positionin the disc, the local spectra (i.e., the emergent spectrum from an annulus) were adequately fit with fcol '1.8 − 2.0 and fcol ' 1.7 for Ṁ ' ṀEdd and Ṁ ' 0.1 ṀEdd , respectively, suggesting a relatively constant fcol
over a large range of mass accretion rates expected in the HSS. For Ṁ . 0.01 ṀEdd , which is characteristicof the LHS, the local spectrum was not well-described by the colour-corrected blackbody model due toineffective (i.e., unsaturated) Comptonisation within the disc.
Merloni et al. (2000) extended the study of Shimura & Takahara (1995) by considering the effectsof Doppler blurring, gravitational redshift, and allowing the dissipated accretion energy to be partitioned
between the corona and disc (Svensson & Zdziarski, 1994). Assuming all of the accretion energy is dissipatedwithin the disc, Merloni et al. (2000) found fcol ' 1.8 for Ṁ = 0.1 − 0.3 ṀEdd , fully consistent with Shimura& Takahara (1995). Exploring the lower accretion rate, Ṁ = 0.05 ṀEdd and allowing a significant fractionof the energy to be dissipated in the corona, Merloni et al. (2000) instead found a variable fcol ranging
from ∼ 1.9 − 2.7. Therefore, if coronal activity is important, fcol cannot be considered constant, with thesuggested physical range being 1.7 . fcol . 3.

76
Davis et al. (2005) presents the most sophisticated study to date of the effects of disc vertical structure on the emergent spectrum, incorporating bound-free opacities and a fully relativistic radiative transfer
treatment. For disc luminosities, Ldisc = 0.01 − 0.3 LEdd , which spans the luminosity regimes of the LHS
and HSS, Davis et al. (2005) found a weakly evolving fcol ' 1.4 − 1.6, concluding that fcol remains fairly
constant over broad ranges in mass accretion rate. The results of Davis et al. (2005) are at odds with thoseof Merloni et al. (2000); however, a direct comparison is complicated by the different methodologies used ineach work.

2.6.2.2

Magnetisation

Although the details of magnetic fields (e.g., topology, strength, variability) in accreting black holesystems are poorly understood, magnetisation is central to many aspects of accretion physics. Perhaps
most notable is the role of the magnetorotational instability (MRI) (Balbus & Hawley, 1991), which is
widely accepted as the mechanism for angular momentum transport; hence, allowing accretion to persist.
The turbulence generated within the disc by the MRI amplifies the toroidal component of magnetic field,
perhaps reaching a state where magnetic pressure dominates over the combined gas and radiation pressures.
Begelman & Pringle (2007) present analytic arguments for the maximal toroidal magnetic field strengthachievable by the MRI and compute the corresponding disc structure, finding that magnetically dominateddiscs are thicker and have harder spectra compared to the standard Shakura-Sunyaev thin disc. Under the
conjecture that the toroidal field attains the limiting value set by the MRI, Begelman & Pringle (2007) derivea lower limit for the colour correction factor,
fcol ≥ 6.9

ρ∗
ρ

−2/9
α

5/27



M
M

−1/108

Ṁ
ṀEdd!5/108 rRg

25/108,

where ρ is the mid-plane density, ρ∗ is the density corresponding to the thermalization layer (i.e., effective
photosphere) of the disc, and ρ∗ /ρ < 1. Shearing box simulations place the disc photosphere a few scaleheights above the mid-plane, corresponding to density contrasts, ρ∗ /ρ ' 10−2 − 10−4 (Hirose et al., 2006,2009). Therefore, if the disc around a stellar mass black hole is magnetically dominated, the colour correction
factor can far surpass the values predicted by thin disc treatments that neglect the effects of a magnetic field77
on the disc structure.
Conceivably, the details of vertical dissipation and coronal properties could alter the emergent disc
spectrum considerably. Beloborodov (1999) envisaged a picture where buoyant magnetic loops subject toParker instability rise into the corona and the magnetic energy available within the loops is subsequently
dissipated by reconnection. Recently, Uzdensky (2013) analytically computed the vertical structure of agas pressure-dominated disc threaded by a poloidal magnetic field where turbulent (i.e., dissipative) heating
arising from the MRI is balanced by radiative cooling. Based on this disc model, Uzdensky (2013) argue
that the minimum fraction of available accretion energy that ends up being released in the corona is, fmin ∼−1/5

β0,max ∼ 0.4, where β0,max ∼ 100 is a reasonable estimate of the gas-to-magnetic pressure ratio at the disc
mid-plane. Under conditions where a substantial fraction of accretion power is dissipated in the corona,
Merloni et al. (2000) demonstrate that substantial spectral hardening is expected.
Numerical simulations show that magnetically supported, gas pressure-dominated discs have harderemergent spectra than their unmagnetised counterparts (Blaes et al., 2006). Blaes et al. (2006) calculated
local accretion disc spectra from stratified shearing box simulations and found the disc spectrum hardened
due to magnetic pressure support dominating in the upper atmosphere. This magnetic pressure acts tovertically extend the disc atmosphere and decrease the density at the effective photosphere, leading tosaturated Comptonisation and spectral hardening. In deriving the local disc spectrum, Blaes et al. (2006)
considered two vertical dissipation profiles: (1) a constant dissipation per unit mass and (2) a broken powerlaw fit to the numerical dissipation profile. Both treatments produced identical emergent spectra, implyingthat the local emergent disc spectrum is insensitive to the vertical dissipation profile and that magnetic
pressure support is solely responsible for the observed spectral hardening. Motivated by the result thatmagnetic pressure support drives the spectral hardening and assuming the dissipation profile has no radial
dependence, Blaes et al. (2006) constructed global disc spectra and found only modest hardening, with fcol
increasing from 1.48 to 1.74 upon introducing magnetic pressure support. Blaes et al. (2006) caution that thenumerical dissipation occurs at the grid scale and may be sensitive to grid resolution. Recent work indicatesthat numerical dissipation in ideal magnetohydrodynamic grid-based simulations does behave like physicaldissipation, but acts to effectively diminish the resolution (Salvesen et al., 2014). Adopting dissipation

78
profiles motivated by numerical simulations is an important step toward merging accretion disc theory withsimulations.

2.6.2.3

Inhomogeneous Discs

Strong disc inhomogeneities are expected to arise in moderate-to-strongly magnetised, radiationpressure-dominated regions of accretion discs (Begelman, 2002, 2006) due to photon bubble instability(Arons, 1992; Gammie, 1998; Begelman, 2001). In this picture, strong density contrasts arise on scales
smaller than the disc scale height, resulting in a porous disc where radiation preferentially ‘leaks’ throughthe underdense regions. Due to the typical densities in photon bubble-dominated discs being higher than instandard thin discs of the same luminosity, thermalization of radiation is more effective in inhomogeneousdiscs. Consequently, radiation that is thermalized near the disc mid-plane and escapes through optically thinchutes can result in a colour correction factor near unity for inhomogeneous discs (Begelman, 2006). Dexter& Quataert (2012) computed inhomogeneous disc spectra based on a toy model (Dexter & Agol, 2011) andfound that the disc spectrum broadens and hardens as the amplitude of the local temperature fluctuations
increases. Inhomogeneous discs can substantially alter the standard disc spectrum and further investigationis required to characterise their properties.

2.6.3

Limitations of Simplistic Accretion Disc Models
While significant strides have been made in exploring the imprints of the disc structure on the emer-

gent spectrum in the HSS, the applicability of the colour-corrected blackbody model to discs in the LHS
remains elusive. From a theoretical perspective, thin discs with low mass accretion rates will be fully gaspressure-dominated with absorption overtaking electron scattering as the dominant opacity source, nullifying
the approximation of the emergent spectrum as a colour-corrected blackbody (e.g., Ebisuzaki et al., 1984).
Nevertheless, standard MCD models consistently produce exceptional fits to LHS accretion disc spectra,
giving credence to the notion that the MCD model successfully describes real astrophysical discs in the
LHS. Analogously, standard thin discs are expected to suffer from thermal (Shakura & Sunyaev, 1976) andviscous (Lightman & Eardley, 1974) instabilities, yet the MCD model provides an acceptable description of

79
the observed disc spectrum, which should not be the case if the disc were plagued by catastrophic instabilities. While it may turn out that fcol has a very narrow allowable range and/or it is an inappropriate
parameterisation to use for low disc luminosities, there is currently no compelling evidence that this should
be so based on fits to observed disc spectra alone.
Direct measurements of fcol should not be made with Newtonian disc models when the black hole is
known to possess a nonzero spin and an inner disc extending to the ISCO. Newtonian disc models have thestrength of being able to describe a standard black hole accretion disc spectrum with very few parameters;
however, the important physical effects of light bending, Doppler blurring, and gravitational redshift inherent
to Kerr black holes are neglected. The pseudo-Newtonian disc model, diskpn, provides accurate fits to fully
relativistic disc spectra from a non-spinning black hole, but requires uncomfortably large values of fcol when
fitting a disc spectrum from an extreme Kerr black hole (see §2.5.4). When attempting to fit the hotter
disc spectra intrinsic to spinning black holes, Newtonian disc models are forced to artificially increase thecolour correction factor to achieve the colour temperature demanded by the data. Fortunately, relative
measurements of fcol do not appear to be appreciably affected in the way that absolute measurements are,at least for the low disc inclination considered in this work. This implies that the relative measurementsof fcol evolution during state transitions of GX 339–4 are sound, regardless of the intrinsic spin possessedby GX 339–4. The possibility that the inner disc recedes radially outward or progresses inward between
successive observations cannot be ruled out and would manifest itself in the relative measurements of fcol ,rendering them moot.
The main outcome of this work is that an evolving colour correction factor in a standard thin disc with
inner radius fixed at the ISCO is a viable, albeit not the only, explanation for the changing disc spectrumof the BHB GX 339–4 over many state transitions. Given the uncertainty in accretion flow geometry as
BHBs transition between spectral states, we emphasise that inner disc radii cannot be reliably computedfrom spectral fits made with simplified disc models. Instead, self-consistent, physically motivated spectral
modelling applied to high-quality CCD X-ray spectra is necessary to distinguish between opposing accretion
scenarios. Future insights into the complicated disc-corona-jet connections in BHBs will likely progressby concentrated observing campaigns and the development of physical models that can provide testable

80
diagnostics.

2.7

Summary & Conclusions
The purpose of this work is to show that physically reasonable changes in the colour correction factor

provide a valid, alternative explanation to disc truncation for explaining the accretion disc spectral evolution
observed in black hole binary state transitions. To this end, we systematically performed spectral fits to
RXTE observations of the low-mass X-ray binary GX 339–4 over seven state transitions and collected theobservations that statistically required a disc component. The disc detection pipeline measured discs withluminosities as low as Ldisc ' 10−3 LEdd and disc fractions down to DF ' 0.05. In a suite of spectral fits
to the observations deemed to require a disc, we imposed an unchanging inner disc radius located at the
ISCO for a Schwarzschild black hole in order to ensure that a changing colour correction factor, rather thana radially receding/advancing inner disc, was responsible for any spectral evolution of the disc component.
We give our conclusions as a bulleted list below.
• For nearly all state transitions of GX 339–4 investigated here, relative changes in fcol in the range 1.4 .
fcol,rel . 5, which corresponds to a factor of ∼ 3.5 change, can fully account for disc evolution in a statetransition without invoking a truncated disc geometry.
• Accounting for an fcol evolution in the range quoted above over the course of state transitions in GX
4
339–4, the Fdisc ∝ Teff
relation is recovered for a constant emitting area (i.e., non-truncated disc) down to

the low luminosities characteristic of the HIS and LHS.
• The validity of the results mentioned above are insensitive to the Comptonisation model adopted forfitting the high-energy continuum. Adopting a continuum model that avoids the low-energy flux stealing
character of a simple power-law reduces the degree of fcol variability required to explain the observed discevolution in state transitions.
• The practice of measuring the inner disc radius from the best-fit disc normalisation is unreliable whenboth Rin and fcol are worked into the disc normalisation. Instead, we suggest that disc models wherethese two parameters are decoupled (e.g., diskpn) be used to break the degeneracy.

81• We stress that this work does not demonstrate that the inner accretion disc remains at the radius of
marginal stability during a state transition. Instead, we argue that variations of fcol within a theoreticallyreasonable range can account for most, if not all, of the disc spectral evolution observed over the courseof a state transition.
• Relative measurements, unlike absolute measurements, of fcol are not appreciably affected by our ignoranceof black hole spin.
• When a physically motivated Comptonisation model is used in conjunction with a disc component to fit
the RXTE data of GX 339–4 in the HIS and LHS, we observe a parameter degeneracy between the coronal
optical depth and the colour correction factor. This calls into question the association of ULXs withsuper-Eddington stellar mass black hole binaries based on measurements of coronal optical depth without
allowing for a variable fcol .
Simple disc models of the MCD family are attractive and useful from the standpoint that they consistently achieve superb fits with limited free parameters to accretion disc spectra, where the intrinsic systemparameters are often poorly constrained. This work illustrates that allowing for the possibility of reasonablevariations in the colour correction factor provides an adequate description of disc spectra over a wide range inluminosity states. We advocate that fcol may be dynamic and should not be dismissed as a mere constant instudies of accretion disc spectra spanning a broad range in luminosity, as is often done in practice. Numerous
factors may contribute to altering the emergent disc spectrum including the vertical disc structure, detailsof accretion energy dissipation, coronal dynamics, magnetisation, and disc inhomogeneities. Given that theavailable X-ray data cannot isolate these effects, common practice is to adopt a blanket term, the colour
correction factor, which is supposed to account for the overall deviations from a multi-colour blackbody disc
spectrum. We encourage an increased awareness for the implications of presuming a constant fcol . Important physics are wrapped up into fcol that should be considered before making definitive claims based on
the assumption of an invariant fcol .
The standard theoretical emergent disc spectrum hinges on the assumptions that the accretion energy
is dissipated within the disc (i.e., where the majority of the mass resides) and that the density structure

82
is smooth and homogeneous. The effects of magnetisation, which surely play a crucial role in accretion
physics and disc structure, are also ignored in the disc models applied to observations. Radiation pressuredominated, magnetised discs, as expected in the HSS, are subject to clumping instabilities, resulting in a
porous disc structure and lowered colour correction factor (Begelman, 2006). Considerations of the corona
as an energy dissipation reservoir (Svensson & Zdziarski, 1994; Merloni et al., 2000; Merloni & Fabian,2002) and of strongly magnetised discs (Begelman & Pringle, 2007) indicate that fcol may extend wellbeyond the canonical value of ∼ 1.7 and into a realm consistent with the values required here and insimilar observational studies (Dunn et al., 2011; Reynolds & Miller, 2013). With the advent of sophisticatedcodes capable of treating magnetohydrodynamics with radiative transport and general relativity, state-ofthe-art numerical simulations provide a promising avenue for studying accretion disc structure with theaforementioned assumptions relaxed. Striving toward the development of sophisticated models that capture
the relevant physics of realistic accreting black hole systems, provide predictive diagnostics, and are accessible
to observers is a crucial step toward disentangling the details of the accretion flow and distinguishing between
different accretion scenarios.

Acknowledgments
The authors thank the anonymous referee for her/his constructive comments and suggestions, which
improved this paper. GS thanks Jordan Mirocha for thoughtful discussions and the National Science Foundation for support through the Graduate Research Fellowship Program. RCR thanks the Michigan Societyof Fellows and NASA for support through the Einstein Fellowship Program, grant number PF1-120087. This
work used the JANUS supercomputer, which is supported by the National Science Foundation (award number
CNS-0821794) and the University of Colorado Boulder. The JANUS supercomputer is a joint effort of the
University of Colorado Boulder, the University of Colorado Denver, and the National Center for AtmosphericResearch.

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52400.842

52398.668

52394.445

52391.469

52391.402

52391.336

52390.910

52389.071

52388.065

52387.646

52387.490

52386.934

52386.026

52385.469

52384.979

52384.147

52383.103

52382.114

MJD

pow×(bb+ga)

pow×bbpow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×bbpow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

pow×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

pow×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

Model

Best–fit

2.28(4)3.20(6)2.5(1)

3.0(3)

2.43(2)2.36(2)2.46(7)2.63(9)2.42(6)2.53(2)2.58(1)2.43(1)2.42(1)2.38(1)2.386(7)2.41(1)2.07(1)

1.836(6)1.785(8)1.68(1)1.623(9)1.610(7)1.616(7)1.608(8)1.59(1)1.53(3)1.51(1)1.593(6)1.60(1)1.540(8)1.565(7)1.58(2)

1.517(9)1.526(8)1.52(1)1.47(2)

Γ1

-

-

-

-

-

-

-

-

-

-

-

-

-

28(7)

30(2)

32(3)

28(3)

-

30(1)

33(1)

30.6(9)32.7(7)32.5(8)33.9(7)

32(1)

35(3)

32.1(9)33.1(3)34(2)32.0(7)-33(1)33.6(6)34.0(9)

32(1)

33(1)

(keV)

Eb

-

-

-

-

-

-

-

-

-

-

-

-

-

2.8(3)

2.79(5)2.85(8)2.6(1)-

2.56(4)2.61(6)2.37(3)2.44(3)2.41(3)

2.46(3)2.51(8)2.4(1)

2.39(4)2.44(1)2.43(8)2.37(3)-2.40(6)2.44(3)2.42(4)2.34(6)2.43(6)

Γ2

0.21(2)1.5(2)

0.08(2)0.3(3)

3.3(2)

1.49(8)1.6(3)

2.6(6)

1.5(2)

3.0(2)

4.3(1)

3.8(1)

5.2(2)

5.1(2)

5.1(1)

5.4(1)

3.5(1)

2.50(4)2.23(5)1.81(5)1.60(4)

1.52(3)

1.55(3)1.50(3)1.45(6)1.4(1)

1.33(5)1.44(2)

1.42(6)

1.43(3)1.51(3)1.34(6)1.31(3)1.36(3)1.38(4)1.13(5)Kpow

0.82(4)-0.76(5)0.4(2)

1.08(5)1.10(5)0.89(9)-1.04(9)0.98(4)0.99(3)

1.12(4)1.08(4)1.05(4)1.12(3)

1.11(3)1.10(4)1.01(4)1.05(5)1.10(6)1.05(6)

1.04(6)0.99(6)1.05(6)

1.11(8)1.2(1)

1.26(6)1.05(5)1.04(9)1.30(5)

1.23(5)0.9(1)

1.29(5)1.31(5)1.32(6)

1.26(7)(eV)

σ0.0033(2)-0.0014(1)0.0009(3)0.028(2)0.015(1)0.016(2)-0.017(2)0.024(1)0.027(1)0.030(2)0.036(2)0.042(2)0.041(2)0.040(2)0.043(2)0.036(2)0.036(3)0.037(3)0.036(3)0.033(3)0.031(2)0.032(3)0.038(4)0.043(8)0.045(4)0.033(2)0.029(3)0.045(3)0.042(3)0.024(4)0.044(3)0.046(3)0.044(4)0.040(4)Kline

0.640(6)0.50(1)0.567(3)0.557(8)0.874(6)0.885(4)0.803(5)0.885(6)0.864(5)0.867(5)0.866(5)0.867(6)0.890(8)0.871(8)0.877(8)0.867(8)0.96(3)1.15(7)

1.27(7)1.28(9)1.20(8)1.25(8)1.20(8)1.24(9)1.2(1)

1.2(2)

1.37(8)1.18(7)1.3(1)

1.31(7)1.30(7)1.3(1)

1.28(7)1.21(6)1.35(9)1.38(8)

(keV)

kTin

1429(87)6782(1224)

4401(144)

5467(443)

2017(73)2669(72)3967(154)

2491(77)2957(103)2591(78)2507(84)2147(80)1342(70)1411(76)1431(73)1507(81)243(36)42(13)

31(8)

29(9)

38(12)30(9)35(10)

29(10)

40(15)

46(29)

33(8)38(10)

25(10)

36(8)

32(8)

24(10)

41(10)

54(12)

30(9)

31(8)

Kdisc

1.3(1)

2.9(6)

1.2(3)--

0.87(3)-

1.3(2)

0.95(8)0.99(4)1.02(2)0.99(2)0.99(2)0.97(6)

0.972(7)0.99(1)0.96(2)-0.925(9)0.93(1)0.901(8)0.901(6)0.901(6)0.914(6)0.88(2)0.79(2)0.739(8)0.908(4)0.91(2)0.775(6)-0.91(1)0.765(5)0.749(6)0.719(8)0.76(1)

C

72

74

72

43

43

72

43

74

72

73

72

73

73

70

70

70

70

43

70

70

70

70

71

70

70

70

70

70

70

70

43

70

70

70

70

70

ν

1.55

1.72

1.63

1.05

1.67

1.13

1.22

1.37

0.66

0.83

1.48

1.03

1.11

1.11

1.78

1.76

1.16

1.80

1.05

0.88

1.19

1.07

1.31

0.98

1.07

0.79

1.16

1.90

0.72

1.26

1.62

0.84

1.30

0.96

1.08

0.91

χ2
ν

-8.30(1)-8.07(3)-8.019(6)-7.96(1)-7.601(6)-7.457(4)-7.457(7)-7.488(6)-7.457(6)-7.507(4)-7.523(5)-7.589(5)-7.747(8)-7.763(9)-7.745(7)-7.742(8)-8.35(2)-8.80(3)-8.76(3)-8.79(3)-8.79(3)-8.82(2)-8.82(2)-8.83(2)-8.74(4)-8.65(7)-8.60(2)-8.81(2)-8.84(5)-8.65(2)-8.71(2)-8.78(4)-8.63(2)-8.60(2)-8.67(2)71.7(8)21(3)92.31(9)66(1)

44.4(3)71.1(2)69.4(5)53.6(5)70.0(3)50.4(5)39.9(4)41.5(1)26.53(6)26.4(3)27.8(1)26.3(2)11.5(5)4.3(3)

5.3(3)

5.0(3)

5.0(3)

4.8(2)

4.7(2)

4.7(2)

5.7(5)

6.3(8)

6.9(4)

4.9(2)

4.7(4)

6.2(2)

4.7(2)

5.4(4)

6.6(3)

6.9(2)

5.8(3)

7.2(4)(%)-8.62(3)DF

log(Fdisc )erg/s/cm2

83

90418-01-01-02

60705-01-65-00

90418-01-01-03

60705-01-65-01

R04

R04

R04

90418-01-01-01

R04

R04

90418-01-01-04

90418-01-01-00

R04

R04

70110-01-04-10

60705-01-63-00

D03

D03

60705-01-61-00

60705-01-62-00

D03

80116-02-02-02G

D03

D03

70128-02-03-01

70110-01-02-10

70128-02-03-00

D03

D03

80116-02-02-01

D03

D03

80116-02-01-02

80116-02-02-00

D03

D03

70110-01-98-00

80116-02-01-01G

D03

80116-02-01-00

D03

D03

70110-01-96-00

70110-01-97-00

60705-01-59-00

D03

D03

70110-01-95-00

D03

D03

70110-01-93-00

70110-01-94-00

50117-01-04-02

D03

D03

50117-01-04-01

D03

D03

70110-01-91-00

60705-01-57-00

D03

D03

70109-02-01-01

60705-01-56-00

D03

D03

70110-01-88-00

70110-01-89-00

70110-01-87-00

D03

D03

ID

ID

D03

Observation

Trans.

53201.330

53199.146

53198.295

53198.162

53197.113

53196.212

53196.146

52756.060

52751.569

52750.384

52746.983

52743.305

52743.163

52742.699

52742.346

52742.051

52741.749

52739.606

52737.502

52737.253

52735.660

52734.228

52732.119

52731.583

52727.257

52724.232

52719.234

52718.180

52718.102

52716.732

52713.486

52710.747

52710.012

52707.932

52704.021

52700.602

MJD

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

pow×(bb+ga)

bkn×(bb+ga)

pow×bbpow×(bb+ga)

pow×bbpow×bbpow×bbbkn×(bb+ga)

pow×(bb+ga)

pow×bbbkn×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

bkn×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

Model

Best–fit

1.52(3)

1.48(1)1.48(2)

1.54(1)1.49(2)

1.38(7)

1.52(4)1.51(5)

1.53(9)1.37(6)

1.42(8)1.60(2)

1.59(9)1.57(3)1.60(2)

1.51(2)1.60(2)

1.57(3)

1.63(2)

1.66(4)1.65(1)1.70(5)1.72(5)1.82(2)2.07(5)2.14(5)2.27(7)2.5(1)

2.23(5)2.8(1)

2.43(9)2.48(6)2.29(6)2.7(1)

2.10(6)2.32(5)

Γ1

39(5)

36(3)

32(7)

-

44(7)

-

-

-

-

-

17(2)

-

-

52(17)

-

-

34(4)

35(9)

-

-

-

-

34(9)

-

-

-

-

-

-

-

-

-

-

-

-

-

(keV)

Eb

2.2(2)

1.92(9)1.8(1)-

2.1(2)

-

-

-

-

-

1.8(1)--

2.2(8)--

2.1(2)

2.0(3)----

2.6(7)

-

-

-

-

-

-

-

-

-

-

-

-

-

Γ2

0.22(2)0.192(6)0.187(8)0.209(7)0.182(9)0.12(3)0.19(2)0.022(3)0.029(7)0.026(4)0.041(8)0.075(4)0.07(2)0.069(6)0.078(3)0.064(4)0.082(5)0.079(6)0.100(5)0.11(1)0.109(4)0.13(2)0.14(2)0.19(1)0.24(3)0.23(3)0.22(4)0.4(1)

0.17(2)0.4(1)

0.20(5)0.43(7)0.21(3)0.5(1)

0.14(2)0.19(3)

Kpow

1.0(2)

0.8(1)

0.9(1)

0.8(1)

1.0(1)-

0.0(2)---

1.0(2)

0.8(1)-

1.2(1)

0.7(1)

1.0(1)

0.8(1)

1.0(1)

0.9(1)

1.0(2)

0.86(9)1.0(2)

1.0(1)

1.22(5)0.96(7)0.99(5)0.78(7)0.90(8)0.93(4)0.88(8)0.75(7)0.91(5)0.81(5)0.4(1)

0.96(6)0.72(6)(eV)

σ0.004(1)0.0034(5)0.0039(7)0.0033(4)0.0045(6)-0.0017(3)---0.0008(3)0.0009(2)-0.0015(2)0.0010(1)0.0014(2)0.0013(2)0.0021(3)0.0020(2)0.0024(6)0.0019(2)0.0026(7)0.0034(6)0.0058(3)0.0049(5)0.0053(4)0.0035(4)0.0035(5)0.0035(2)0.0021(3)0.0021(3)0.0038(4)0.0032(3)0.0020(4)0.0040(3)0.0024(2)Kline

1.1(1)

1.21(9)1.1(1)

1.02(6)1.00(5)1.90(8)1.3(2)

2.7(7)

1.8(2)

2.3(1)

1.6(2)

0.92(8)1.8(2)

0.95(6)0.96(6)

1.06(9)0.93(7)0.96(5)0.97(5)0.75(8)1.06(7)1.06(9)0.91(5)0.89(2)0.71(2)0.71(1)0.62(1)0.58(2)0.647(7)0.57(1)0.572(9)0.64(1)0.601(8)0.58(2)

0.662(9)0.624(8)

(keV)

kTin

15(8)

9(3)

11(5)

16(5)

20(5)

1.9(3)

4(3)0.02(3)0.2(1)

0.14(4)0.6(4)

7(4)

0.5(3)

8(3)

7(2)

6(2)

9(4)

12(4)

11(3)

42(34)

8(3)

11(5)

32(10)

44(6)276(42)408(49)1270(119)2212(377)

1096(75)2581(260)

2295(217)

878(97)1492(126)

1712(334)

799(72)1230(100)

Kdisc

0.94(3)0.91(1)0.90(3)0.91(2)0.90(2)-0.91(5)0.79(8)--0.94(7)0.94(3)-0.92(2)--0.96(3)0.97(4)0.91(3)0.86(6)0.92(3)0.93(7)1.09(8)0.68(3)1.2(1)

0.9(1)

1.1(2)-

1.0(1)

2.2(4)

1.4(3)

1.3(1)--

0.7(2)

1.1(2)

C

70

70

70

72

70

45

72

73

45

44

69

72

45

70

43

43

70

70

72

72

72

72

70

71

72

72

72

42

71

71

72

71

43

43

72

72

ν

0.65

0.55

0.71

1.53

1.51

1.21

1.03

1.19

0.73

1.05

0.96

0.87

1.09

0.99

0.91

0.66

0.89

1.09

0.84

0.88

1.18

0.85

0.72

1.43

1.25

1.19

1.58

1.53

1.72

1.02

1.36

1.73

1.97

1.30

1.28

1.12

χ2
ν

-9.36(6)-9.40(2)-9.43(4)-9.45(3)-9.36(3)-9.29(8)-9.62(3)-10.6(2)-10.4(3)-10.1(1)-10.1(2)-9.96(6)-8.0(2)-10.04(6)-9.92(4)-9.80(4)-9.84(5)-9.65(4)-9.71(4)-9.54(8)-9.67(3)-9.51(7)-9.33(5)-9.22(2)-8.83(2)-8.67(2)-8.40(1)-8.27(2)-8.39(1)-8.23(1)-8.29(1)-8.50(1)-8.39(1)-8.38(2)-8.49(1)6.8(8)

6.5(2)

6.1(5)

5.7(4)

7.4(5)9(1)

4.2(3)

3(1)

5(2)

6(1)

6(2)

5.8(8)6(2)4.9(5)

6.1(6)

7.5(7)

7.6(7)

10.6(8)8.2(7)

11(2)

8.6(6)

12(2)

18(1)

17.4(8)

37(1)

48(1)

66.0(6)57.7(6)71(1)57.6(6)73.1(5)42.7(2)67.4(8)43.1(4)70(1)68.3(9)(%)-8.40(1)DF

log(Fdisc )erg/s/cm2

84

92428-01-04-00

92428-01-04-02

92428-01-04-03

R07

R07

92035-01-03-03

R07

R07

92035-01-03-01

92035-01-03-02

R07

R07

92035-01-02-06

92035-01-03-00

R07

R07

92035-01-02-08

92035-01-02-07

R07

R07

92035-01-02-02

92035-01-02-04

R07

R07

92035-01-02-00

92035-01-02-01

R07

R07

92035-01-01-02

92035-01-01-04

R07

R07

92035-01-01-03

92428-01-03-00

R07

R07

92428-01-02-00

R07

92035-01-01-01

92052-07-06-01

R07

92035-01-01-00

60705-01-71-00

R04

R07

90704-02-01-03

R04

R07

90704-01-03-00

90704-02-01-01

R04

R04

90110-02-01-03

60705-01-70-01

R04

R04

60705-01-69-01

60705-01-70-00

R04

90704-01-01-00

R04

R04

60705-01-68-01

60705-01-69-00

R04

R04

60705-01-67-01

60705-01-68-00

60705-01-66-00

R04

R04

ID

ID

R04

Observation

Trans.

54144.883

54144.098

54143.890

54143.039

54142.056

54141.075

54140.225

54139.963

54138.846

54137.857

54137.018

54135.054

54133.953

54133.035

54132.109

54131.127

54130.140

54129.476

54128.976

54127.115

54122.304

54118.975

53238.058

53237.682

53237.547

53235.445

53234.587

53233.000

53230.964

53228.997

53226.452

53225.412

53222.252

53218.113

53215.299

53204.415

MJD

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

pow×(bb+ga)

Model

Best–fit

2.36(2)2.32(1)2.33(1)2.26(1)2.088(7)1.979(8)1.871(8)1.827(7)1.702(6)1.68(1)1.640(7)1.612(8)1.594(8)1.600(7)1.577(8)1.572(8)1.531(8)1.558(9)1.556(8)1.53(2)

1.49(1)1.47(1)2.30(6)2.31(5)2.21(3)2.33(2)2.42(4)2.37(2)2.12(3)

1.78(2)1.64(1)1.63(1)1.58(1)1.54(1)1.55(1)1.57(2)

Γ1

22(8)

33(3)

34(2)

26(5)

31(1)

31(1)

30.0(9)30.8(7)30.4(8)34(1)32.6(7)32.9(8)33.3(7)33.7(7)32.3(7)33.3(9)

32(1)

37(2)

34.2(9)37(2)35.5(9)

37(2)

-

-

-

-

-

-

-

28(3)

34(2)

33(3)

38(2)

32(5)

39(6)

-

(keV)

Eb

2.8(2)

3.1(2)

2.9(1)

2.7(1)

2.82(6)2.74(5)2.59(4)2.64(4)2.49(4)2.56(7)2.46(3)2.41(3)2.42(3)2.41(3)2.35(3)2.35(3)2.26(4)2.33(7)2.27(4)2.41(9)2.21(3)2.12(6)

-

-

-

-

-

-

-

2.2(1)

2.22(7)2.07(9)2.4(1)

1.9(1)

1.9(1)-

Γ2

5.5(3)

5.5(2)

5.4(1)

5.1(1)

3.62(7)3.14(7)2.59(6)2.39(5)1.82(3)

1.76(6)1.60(3)1.36(3)1.29(3)1.30(3)

1.19(3)

1.14(2)0.96(2)0.99(2)0.97(2)0.84(4)0.64(2)0.52(2)0.39(6)0.43(6)0.30(2)0.63(4)0.81(9)1.10(6)0.93(8)0.62(4)0.50(1)0.48(2)0.41(1)0.32(1)0.31(1)0.28(1)Kpow

1.10(5)1.01(5)1.02(4)1.00(4)0.98(3)0.97(4)0.99(4)1.06(4)0.96(4)0.96(7)1.03(5)0.94(7)0.98(6)0.90(6)0.94(7)1.03(6)0.97(5)1.02(6)0.95(6)0.7(1)

0.8(1)

0.9(2)

0.91(5)0.87(5)1.01(4)1.00(3)0.88(4)0.87(4)1.03(6)

1.18(7)1.08(5)1.05(6)1.03(7)0.99(9)0.8(1)

0.9(1)(eV)σ

0.047(3)0.044(3)0.044(2)0.046(2)0.046(2)0.039(2)0.038(2)0.040(2)0.039(2)0.033(3)0.036(2)0.027(3)0.029(3)0.027(2)0.026(2)0.027(2)0.029(2)0.024(2)0.023(2)0.015(3)0.013(2)0.011(3)0.0076(5)0.0079(5)0.0072(3)0.0110(4)0.0111(6)0.0123(6)0.018(1)0.016(1)0.0128(8)0.012(1)0.010(1)0.0072(8)0.005(1)0.0047(8)Kline

0.86(1)0.87(1)0.85(1)0.84(2)0.81(2)0.97(4)1.02(5)1.07(5)0.85(3)

1.11(9)1.05(6)

1.12(8)1.06(7)1.03(7)1.08(7)1.03(6)0.87(3)1.04(6)0.99(6)1.2(1)

1.18(9)1.2(1)

0.683(4)0.676(4)0.699(4)0.752(4)0.726(4)0.770(6)0.78(1)0.96(4)1.07(4)1.09(5)

1.12(7)1.04(6)1.1(1)

0.91(9)

(keV)

kTin

1645(129)1201(116)

1391(119)1114(119)942(129)

211(49)130(36)92(24)

325(80)69(29)

84(25)

60(21)

80(27)

87(30)

71(24)

87(27)

249(56)70(21)

93(29)

36(18)

33(12)

28(12)

3078(108)3229(126)

2629(93)

1863(71)2243(81)1509(73)694(62)89(17)

39(6)

33(7)

28(8)

25(7)

13(7)

28(16)

Kdisc

1.04(5)0.99(2)0.97(1)0.98(3)0.944(9)0.937(8)0.935(8)0.912(7)0.935(7)0.92(1)0.905(6)0.900(6)0.916(7)0.924(6)0.937(7)0.923(7)0.921(9)0.92(1)0.936(7)0.93(1)0.918(7)0.93(1)-0.94(9)1.06(8)1.04(5)1.1(1)

1.19(6)0.98(4)0.96(3)0.90(1)0.91(2)0.93(2)0.98(2)0.93(2)0.92(3)

C

70

70

70

70

70

70

70

70

70

70

70

70

70

70

70

70

70

70

70

70

70

70

43

72

72

72

72

72

72

70

70

70

70

70

70

72

ν

1.72

1.54

1.71

1.82

1.75

1.38

1.49

1.28

1.68

1.48

1.19

1.41

1.41

1.55

1.43

1.89

1.45

1.00

1.20

0.82

1.69

1.32

1.82

1.49

1.67

1.81

1.76

1.27

1.70

1.18

1.44

1.11

1.07

0.91

0.89

1.05

χ2
ν

-9.32(4)-7.73(1)-7.84(1)-7.81(1)-7.92(2)-8.07(2)-8.41(3)-8.52(3)-8.59(2)-8.44(3)-8.65(3)-8.66(3)-8.70(3)-8.66(3)-8.68(3)-8.69(3)-8.68(3)-8.52(3)-8.76(3)-8.72(3)-8.79(3)-8.87(2)-8.91(3)-7.848(6)-7.847(7)-7.878(6)-7.900(6)-7.880(6)-7.948(8)-8.27(1)-8.79(3)-8.96(2)-9.01(2)-9.02(2)-9.20(3)27.0(3)22.2(4)23.5(4)20.3(5)19.2(8)15.4(8)11.6(7)

9.7(5)

11.0(8)8.2(6)

6.8(4)

6.8(4)

7.6(5)

7.2(5)

7.4(5)

7.6(5)

11.4(7)

6.9(4)

7.7(6)

7.3(4)

7.2(3)

7.3(4)

79.26(2)77.82(5)82.0(4)67.7(3)62.66(8)52.0(2)36.6(8)15.0(8)10.2(4)9.1(4)

9.6(4)

7.1(4)

5.7(5)

4.2(7)(%)-9.50(9)DF

log(Fdisc )erg/s/cm285

92704-04-02-00

92704-04-02-02

92704-03-15-00

D07

D07

92704-03-14-03

D07

D07

92704-03-14-01

92704-03-14-02

D07

D07

92704-03-13-02

92704-03-14-00

D07

92704-03-13-01

D07

D07

92704-04-01-06

92704-03-13-00

D07

92704-03-12-01

D07

D07

92704-04-01-04

92704-04-01-05

D07

D07

92704-03-12-00

92704-04-01-03

D07

D07

92704-04-01-01

92704-04-01-02

92704-04-01-00

D07

D07

92704-03-11-01

D07

D07

92704-03-10-11

92704-03-11-00

D07

92704-03-10-00

D07

D07

92704-03-09-00

D07

92704-03-07-01

D07

92704-03-08-00

92704-03-07-00

D07

92704-03-08-01

92704-03-06-03

D07

D07

92704-03-06-02

D07

D07

92085-01-01-04

92085-01-01-05

R07

92085-01-01-00

R07

R07

92035-01-04-00

92035-01-04-01

92428-01-04-04

R07

R07

ID

ID

R07

Observation

Trans.

54246.873

54245.892

54245.794

54244.979

54243.864

54242.884

54241.570

54240.308

54239.743

54238.750

54237.704

54237.501

54237.433

54237.365

54237.298

54236.597

54236.523

54236.455

54236.383

54235.797

54234.858

54232.682

54231.616

54228.399

54227.624

54226.10654225.253

54224.124

54223.293

54223.227

54152.855

54151.816

54150.761

54148.153

54147.031

54145.974

MJD

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×bbpow×(bb+ga)

pow×bbpow×(bb+ga)

bkn×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×bbpow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

bkn×(bb+ga)

bkn×(bb+ga)

Model

Best–fit

1.52(4)1.51(3)

1.48(2)

1.52(4)1.48(5)

1.52(2)

1.49(5)

1.63(3)

1.57(3)

1.66(4)1.80(2)

1.78(3)1.74(3)

1.78(3)

1.78(4)

1.83(5)1.80(3)1.80(3)1.90(3)

1.85(3)

1.98(3)2.16(4)2.22(6)2.24(6)2.08(7)2.33(4)2.34(7)3.06(7)2.40(9)2.34(7)2.37(2)2.38(2)2.36(2)2.47(3)2.51(1)2.41(1)

Γ1

-

-

-

-

-

-

-

-

39(5)

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

29(3)

41(6)

(keV)

Eb

-

-

-

-

-

-

-

-

2.8(6)

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

2.9(1)

3.3(5)

Γ2

0.074(8)0.078(6)0.070(3)0.083(4)0.08(1)

0.094(5)0.09(1)0.14(1)0.12(1)0.15(2)0.20(1)0.18(1)0.16(1)0.18(2)0.18(2)0.21(3)0.20(2)0.19(1)0.25(2)0.20(2)0.26(2)0.22(3)0.24(4)0.22(4)0.13(3)0.34(4)0.28(5)

1.6(3)

0.35(9)0.30(5)1.04(7)

1.86(9)1.29(6)2.1(1)

5.4(2)

5.8(2)

Kpow

0.0(4)

0.8(3)

0.8(1)

0.0(2)-

0.7(1)-

0.6(1)

0.8(2)

1.1(1)

0.87(9)0.93(7)1.00(7)0.96(7)1.0(1)

0.9(2)

0.86(8)0.90(6)0.76(7)0.96(7)0.82(6)0.93(4)0.83(5)0.76(6)0.86(6)0.82(4)0.75(7)-0.73(8)0.71(6)0.91(4)1.03(4)0.94(4)0.84(5)1.03(4)1.01(5)(eV)

σ0.0005(1)0.0012(4)0.0015(2)0.0008(1)-0.0018(3)-0.0024(4)0.0025(6)0.0039(7)0.0039(4)0.0050(5)0.0053(5)0.0054(5)0.0052(7)0.0039(9)0.0046(5)0.0055(4)0.0049(5)0.0058(5)0.0056(4)0.0060(3)0.0052(4)0.0038(3)0.0042(4)0.0054(3)0.0038(4)-0.0044(6)0.0048(4)0.016(1)0.024(1)0.018(1)0.019(1)0.036(2)0.039(2)Kline

1.1(2)

1.0(1)

0.84(4)1.2(1)

1.53(8)1.05(8)

1.58(7)0.95(8)1.10(9)0.95(6)0.96(4)0.89(3)0.94(3)0.90(4)0.89(4)0.97(7)0.96(4)0.82(2)0.84(4)0.80(2)0.77(2)0.672(7)0.665(8)0.637(8)0.633(7)0.668(6)0.633(9)0.53(2)0.62(1)0.642(7)0.823(4)0.838(4)0.842(4)0.826(5)0.876(7)0.86(1)

(keV)

kTin

3(3)

11(7)

25(7)

5(2)

2.0(3)10(4)2.4(4)

21(11)

13(5)

31(12)

36(8)62(13)

51(10)

62(14)

64(15)

36(13)

43(10)

113(18)81(21)

135(23)214(32)1153(72)1256(97)1575(132)

1843(137)

1431(81)

1802(157)

4845(1518)

2273(248)

1939(136)

4454(137)

3806(123)

3936(115)

4196(154)

2628(123)

1770(125)

Kdisc

-

0.89(4)---0.82(4)0.91(5)0.88(4)0.93(4)0.95(5)-0.89(4)---0.92(7)0.89(5)0.94(5)0.94(5)0.91(6)0.90(7)1.1(1)

1.0(1)

1.0(1)

1.1(2)

1.0(1)-

1.9(4)

1.4(3)

1.3(2)

1.01(4)0.97(3)0.99(3)

1.09(4)1.00(2)0.94(2)

C

43

72

43

43

45

72

74

72

70

72

43

72

43

43

43

72

72

72

72

72

72

72

72

72

72

72

43

74

72

72

72

72

72

72

70

70

ν

0.86

0.95

1.09

0.89

1.07

0.83

1.13

0.79

0.98

1.19

1.65

1.67

1.21

1.14

0.81

1.41

1.12

1.30

1.12

1.67

1.41

1.94

1.60

1.92

1.85

1.82

1.64

1.74

1.14

1.27

1.33

1.74

1.49

1.90

1.48

1.51

χ2
ν

-7.484(7)-9.90(7)-9.80(5)-9.57(4)-9.72(5)-9.64(6)-9.58(4)-9.51(6)-9.44(5)-9.38(4)-9.42(5)-9.18(3)-9.08(3)-9.07(3)-9.07(3)-9.06(3)-9.28(4)-9.11(3)-8.97(2)-9.06(4)-8.92(3)-8.80(2)-8.30(1)-8.29(1)-8.26(1)-8.21(1)-8.220(9)-8.22(1)-8.10(5)-8.14(2)-8.16(1)-7.362(5)-7.400(5)-7.377(4)-7.381(6)5.5(7)

6.2(6)

10.6(9)7.2(8)

8(1)

8.5(7)

10(1)

10(1)

13.4(9)

11(1)

18(1)

22(1)

23(1)

22(2)

23(2)

14(1)

21(1)

27(1)

22(2)

30(2)

35(2)

69.3(9)

70(1)

73(1)

82(1)

65.4(5)69.4(7)22(2)68.3(6)71.2(5)81.5(2)69.2(3)77.3(2)66.5(5)38.0(3)27.3(4)(%)-7.68(1)DF

log(Fdisc )erg/s/cm2

86

96409-01-06-02

96409-01-07-00

96409-01-07-03

D11

D11

96409-01-06-01

D11

D11

96409-01-05-03

96409-01-06-00

D11

D11

96409-01-05-04

96409-01-05-05

96409-01-04-08

D11

D11

96409-01-04-07

D11

D11

96409-01-02-02

96409-01-04-02

D11

D11

95409-01-14-06

95409-01-18-04

R10

R10

95409-01-14-00

95409-01-13-06

R1095409-01-14-02

95409-01-13-01

R10

R10

95409-01-13-05

R10

R10

95409-01-13-03

95409-01-13-00

R10

R10

95409-01-12-00

92704-04-03-00

D07

95409-01-12-04

92704-03-24-00

D07

R10

92704-03-22-00

D07

R10

92704-04-02-03

92704-04-02-04

D07

D07

92704-03-19-00

92704-04-02-01

D07

D07

92704-03-17-00

92704-03-18-00

92704-03-16-00

D07

D07

ID

ID

D07

Observation

Trans.

55604.912

55603.992

55601.891

55598.677

55597.264

55594.902

55592.739

55590.441

55588.554

55587.503

55586.502

55573.469

55327.044

55299.787

55297.883

55295.008

55294.129

55293.095

55292.796

55289.628

55288.380

55286.738

55281.592

54256.612

54256.302

54254.925

54251.848

54251.780

54251.708

54250.731

54249.556

54248.836

54247.923

MJD

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×bbpow×bbpow×bbpow×bbpow×(bb+ga)

pow×(bb+ga)

pow×(bb+ga)

pow×bbbkn×(bb+ga)

bkn×bbModel

Best–fit

1.63(4)

1.67(6)

1.66(4)1.90(5)

1.94(6)

2.11(6)2.1(1)

2.4(1)

2.3(2)

2.2(1)

2.3(1)

2.34(8)2.43(6)2.07(1)

1.852(8)1.66(1)1.624(9)1.625(9)1.61(1)

1.587(9)1.575(9)1.59(1)1.53(1)1.48(5)1.51(8)

1.57(6)1.50(5)1.50(3)

1.47(3)1.45(4)1.48(5)

1.46(4)

1.49(6)

Γ1

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

29(12)

29(11)

(keV)

Eb

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

1.8(3)

1.8(3)

Γ2

0.088(9)0.10(2)0.09(1)0.14(2)0.17(3)0.20(3)0.15(5)0.3(1)

0.18(8)0.2(1)

0.31(8)0.35(8)2.0(3)

3.01(8)2.03(4)1.32(4)1.24(3)

1.25(3)

1.18(4)1.02(2)0.97(2)1.00(3)0.75(2)0.031(4)0.036(7)0.043(7)0.044(6)0.044(4)0.042(3)0.044(5)0.053(7)0.055(6)0.06(1)

Kpow

0.7(2)

0.9(2)

1.05(9)0.79(7)1.0(1)

0.75(7)0.94(9)0.6(1)

0.8(1)

0.8(2)

0.8(1)

0.76(6)0.96(7)0.95(4)0.96(4)0.82(6)0.97(5)0.80(6)0.93(7)0.95(5)0.96(5)0.80(6)0.87(7)

-

-

-

-

0.8(2)

1.1(2)

0.2(2)-

0.9(2)-(eV)σ0.0014(4)0.0020(6)0.0041(5)0.0041(4)0.0045(6)0.0040(4)0.0046(6)0.0031(6)0.0031(6)0.0038(9)0.0044(6)0.0057(6)0.028(2)0.039(2)0.037(2)0.026(2)0.032(2)0.025(2)0.031(2)0.026(2)0.026(2)0.021(2)0.019(1)

-

-

-

-

0.0006(2)0.0009(2)0.0004(1)-0.0011(3)

-

Kline

0.76(7)0.8(1)

0.83(3)0.74(2)0.76(3)0.66(2)0.63(2)0.58(3)0.57(2)0.60(2)0.62(2)0.61(1)0.768(8)0.77(2)0.84(3)0.80(4)0.85(3)0.88(4)0.82(4)0.81(3)0.80(3)0.78(4)0.85(4)1.7(1)

1.4(2)

1.2(3)

1.6(1)

1.1(1)

0.96(9)1.2(1)

1.5(1)

1.0(1)

1.4(2)

(keV)

kTin

45(31)

28(23)

63(14)

198(34)163(40)571(97)1024(196)

1787(610)

2307(590)

1640(422)

1765(285)

2662(271)

3461(198)

1045(153)

391(73)352(106)

264(63)212(63)346(100)

340(78)367(82)349(104)

204(50)0.38(8)0.6(4)1(2)

0.5(2)

3(2)

6(3)

2(2)

1.1(4)

8(6)1.4(5)

Kdisc

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.89(6)--0.99(7)-0.87(4)0.72(5)0.91(6)0.91(8)1.01(9)

C

43

43

43

43

43

43

43

43

43

43

43

43

43

43

43

43

43

43

43

43

43

43

43

74

45

45

74

43

72

72

74

70

72

ν

0.93

0.88

0.72

1.24

1.40

1.86

1.93

1.29

1.35

1.08

1.86

1.96

1.61

1.85

1.89

1.54

1.33

1.80

1.49

1.87

1.98

1.85

1.57

1.06

1.05

1.19

1.18

0.69

0.82

0.83

1.23

0.71

1.18

χ2
ν

-9.50(7)-9.52(9)-9.19(3)-8.89(2)-8.94(3)-8.63(2)-8.47(2)-8.38(4)-8.28(4)-8.34(3)-8.27(2)-8.10(2)-7.59(1)-8.11(2)-8.39(2)-8.51(4)-8.54(3)-8.57(4)-8.49(4)-8.51(3)-8.51(3)-8.56(4)-8.64(3)-10.2(1)-10.3(2)-10.3(2)-10.2(1)-10.09(7)-9.99(7)-9.99(6)-8.0(1)-9.81(7)

14(2)

13(3)

25(2)

41(2)

35(2)

53(2)

68(3)

58(3)

75(4)

68(3)

65(1)

71(1)

57.3(2)19.7(9)12.5(7)10.2(9)9.4(7)

8.8(7)

10.6(9)10.9(8)11.2(8)10.1(9)9.8(7)

6(1)

5(2)

4(1)

5(1)

5.5(8)

6.6(9)

6.0(7)

5(1)

8(1)

6(1)

(%)

-9.9(1)DFlog(Fdisc )erg/s/cm2

87

ID

ID

MJD
Model

Best–fit

Γ1
(keV)

Eb

Γ2

Kpow
(eV)

σ

Kline
(keV)

kTin

Kdisc

C

ν

χ2
ν
DF
(%)

log(Fdisc )erg/s/cm2bApplies for fits using the HEXTE data. The full version of this table for all fits used in this work is available as an electronic supplement.

flux over the 0.1-10 keV band, and disc fraction. Dashes indicate that a particular parameter was not applicable in the fit. a Applies for fits using the bknpower model;

normalisation, disc colour temperature as returned by diskbb, disc normalisation, constant applied to the HEXTE data b , number of degrees of freedom, reduced χ2 , disc

PCA observation, the best-fit XSPEC model, power-law index, break energya , high-energy power-law indexa , power-law normalisation, Gaussian line width, Gaussian line

0.6% systematic error is added to all PCA energy channels. From left to right, the columns give the state transition ID, observation ID, MJD for the midpoint of thephabs with NH = 5.7 × 1021 cm−2 (Miller et al. 2009b) and an energy-independent constant is applied to the HEXTE data to account for offset relative to the PCA. A

disc and continuum were diskbb and powerlaw (or bknpower), respectively. The Fe Kα line at 6.4 keV is fit with the gauss model. The entire model is absorbed using

Table 2.3: Best-fit parameters and 1σ standard uncertainty in parentheses for all observations where a disc was deemed required. The spectral models employed for theObservation

Trans.

88

Chapter 3

A Physical Model for State Transitions in Black Hole X–ray Binaries

3.1

PrefaceThis paper was published in Monthly Notices of the Royal Astronomical Society, Volume 437, Issue4, pp. 3994-3999 on February 01, 2014. The author list is: Nixon, C. & Salvesen, G., with both of us
contributing equally. Specifically, I am responsible for most of the connections to black hole X-ray binary
phenomenology, while Chris is responsible for the details of disc tearing. Here, we propose a toy model forblack hole state transitions based on the phenomenon of accretion disc tearing.

Abstract
We present an accretion cycle which can explain state transitions and other observed phenomena inblack hole X–ray binaries. This model is based on the process of disc tearing, where individual rings ofgas break off the disc and precess effectively independently. This occurs when the Lense–Thirring effect isstronger than the local disc viscosity. We discuss implications of this model for quasi–periodic oscillations
and the disc–jet–corona coupling. We also speculate on applying this model to active galactic nuclei andother accreting systems.

3.2

Introduction
Binary systems occur frequently in astrophysics, from supermassive black hole binaries in galaxy

mergers to protostellar binaries in star forming regions. When one component of a stellar binary undergoes

90
a supernova explosion, the result may be an X–ray binary, where the companion star feeds the newly formed
neutron star or black hole. These systems show evidence for accretion proceeding through a gas disc (e.g.
Pringle & Rees, 1972; Pringle, 1981).
X–ray binaries display a wide variety of spectral and timing properties, with many observable features such as jets, disc winds, coronae, quasi–periodic oscillations and strong disc emission and reflection.
Black hole state transitions exhibit all of these complex and interacting features, making a theoreticalunderstanding challenging to construct. However, a powerful phenomenological description emerges frommodel–independent spectral and timing observations (for a recent review, see Belloni, 2010). Magnetohydrodynamical (MHD) effects are often supposed to be the underlying physics responsible for this behaviour.
However, the aim of this paper is to propose a physically motivated and self–consistent mechanism, using
mainly hydrodynamical effects, to explain black hole state transitions.
Accretion discs are often thought to be warped, with conclusive evidence from maser discs in active
galactic nuclei (AGN; Greenhill et al., 2003) and suggestive observations in X–ray binaries (e.g. Katz, 1973;
Wijers & Pringle, 1999; Ogilvie & Dubus, 2001; Miller et al., 2006c), protostellar discs (e.g. Hughes et al.,2009), and perhaps ultraluminous X–ray sources (e.g. Pasham & Strohmayer, 2013). In X–ray binaries theblack hole spin can be strongly misaligned to the binary orbit by a supernova kick during formation (e.g.
Jonker & Nelemans, 2004). This misalignment can persist for the entire lifetime of the system as the blackhole angular momentum is much larger than the total angular momentum transferred through the disc (e.g.
Maccarone, 2002; Nixon & King, 2013). For discs with angular momentum misaligned to the spin of theblack hole, the differential precession induced by the Lense & Thirring (1918) effect is communicated throughthe disc by a viscosity (Bardeen & Petterson, 1975). This (effective) viscosity most likely arises from MHD
turbulence induced by the magnetorotational instability (Balbus & Hawley, 1991). There is observational
evidence for Lense–Thirring precession in X–ray binaries (e.g. Miller & Homan, 2005; Schnittman et al.,2006). Assuming the disc can efficiently communicate the precession, this results in an aligned inner discjoined to a misaligned outer disc by a smooth warp (see e.g. Pringle, 1992; Lodato & Pringle, 2006).
Most investigations into warped discs use simplistic forms of the disc viscosity (e.g. Pringle, 1992) as
this allows for simulation of thin discs with α > H/R (Papaloizou & Pringle, 1983). Some simulations of

91
tilted discs explicitly include MHD effects to generate a turbulent viscosity, but these are currently restricted
to thick discs (Fragile et al., 2007, 2009; McKinney et al., 2013; Sorathia et al., 2013a). To make progresson thin discs, Nixon & King (2012) simulated the disc evolution using the method of Pringle (1992) withthe constrained isotropic viscosities derived from the fluid equations for a warped disc by Ogilvie (1999).
Their simulations suggest that often the disc cannot communicate the precession as efficiently as required tomaintain a smooth warp – instead the disc breaks. Follow–up 3D hydrodynamical simulations found that thedisc is indeed unable to communicate the precession, and therefore breaks into distinct planes which precesseffectively independently – this is disc tearing (Nixon et al., 2012a). These simulations also show significant
dynamical evolution of the disc, predominantly infall of gas between interacting torn rings. In this paper,
we explore an accretion picture for black hole state transitions which includes the disc tearing process.

3.3

Accretion picture
We describe an accretion cycle for black hole binary state transitions incorporating disc tearing (Nixon

et al., 2012a). We begin with plausible, generic initial conditions summarised as follows.
1) A black hole with non–negligible spin. A modest value of the black hole spin, a > 0.01, is more thansufficient to drive disc tearing (eq. 8 of Nixon et al., 2012a). Current estimates of stellar mass black holespins suggest moderate–to–high values with 0.1 . a . 1 (e.g. Miller et al., 2009b; McClintock et al., 2014;
Reynolds, 2014).
2) A geometrically thin, optically thick disc of gas at some large radius from the black hole. This is consistent
with the accretion event being triggered by a disc instability, such as the thermal–viscous instability driven
by hydrogen ionization (Meyer & Meyer-Hofmeister, 1982; Smak, 1982; Lasota, 2001).
3) A misalignment between the angular momenta of the accretion disc and the black hole. Again only amodest disc–spin misalignment angle, θ > a few degrees, is required (Nixon et al., 2012a). Large disc–spin misalignments may be common (e.g. Maccarone, 2002; Martin et al., 2010). There is currently little
constraint on the disc–spin misalignment. Population synthesis studies (e.g. Fragos et al., 2010) can generate
the likelihood of disc–spin misalignments, but much of the input physics of binary formation and black holekicks is not understood. Therefore it is likely that the results of Fragos et al. (2010), that most binaries

92

(5)
103

(4)

PCU2 Intensity (counts s-1)

(3)
(6)

(2)

(7)

102

(8)
(1)
0.1

Hardness Ratio

1

Figure 3.1: The left panel shows the different stages of the tearing accretion cycle. In each part the black
circle represents the black hole, with the arrow showing the direction of its spin. Solid black lines show the(sometimes warped) Shakura–Sunyaev disc, which in (4) & (5) has torn, with the thickness approximately
denoting its surface density. “X”s mark regions of hot, low–density gas. The quasi–spherical blue component
is the corona. Possible jets are shown by lines leaving the central black hole, green solid lines suggest a steadyjet and red dashed lines suggest a transient jet. The figure shows the progression of the system, with each
black hole state labelled. The cycle starts with a disc approaching the black hole (1) → (2). The initial discwarps (3) and the first ring is torn off (4). The main tearing phase is (5): see Section 3.3.3. Once the innerdisc aligns, the accretion is more stable (6). As the surface density drops, the hot inner disc is revived (7),and the system returns to its original quiescent state (8) & (1). We note that a precessing, warped disc isa 3D structure (see e.g. Fig. 3.2), so the 2D representation shown here is a simple conceptualisation. The
right panel shows where these eight stages lie on a representative hardness–intensity diagram (data from the
2002/2003 GX 339–4 outburst, see e.g. Salvesen et al., 2013).

93
have θ . 10◦ but up to a third may have θ & 10◦ , represents a lower–bound on the possible disc–spin
misalignments. Also, analysis of individual sources (e.g. Steiner & McClintock, 2012) can reveal some
aspects of the disc–spin misalignment by comparing the binary inclination with the jet inclination. However,this only tests for misalignment along the line–of–sight, offering no constraint perpendicular to this plane.
Therefore, one cannot constrain the true disc–spin misalignment, as noted by Steiner & McClintock (2012).

3.3.1

Disc evolution
Fig. 3.1 shows the disc evolution starting from the initial conditions described above. The eight stages

in the accretion cycle are:1) An outer Shakura–Sunyaev disc moves inwards towards the black hole, but is still at large radii. Interior
to this disc, there is hot, low–density gas present from the last accretion event. This gas may provide thenecessary conditions to produce the steady jet sometimes observed in the quiescent state.
2) The outer disc approaches the black hole, with increasing surface density, decreasing the size of the hot,
low–density region.
3) The outer disc reaches a radius where the Lense–Thirring precession begins to affect the disc, driving awarp on the inner edge. This occurs at a radius where the inflow time (R2 /ν) becomes comparable to theLense–Thirring precession time (1/Ωp ), which is at
Rtilt ∼ 22/3

 a 2/3  R 4/3
αHRg ≈ 1 × 103 Rg(3.1)

where Rg = GM/c2 is the gravitational radius of a black hole of mass M and we have assumed a Shakura& Sunyaev (1973) viscosity ν = αH 2 Ω where α is the dimensionless viscosity parameter, H is the discsemi–thickness at a radius R and Ω is the disc angular velocity. We adopt typical parameters throughout:a = 0.5 (e.g. Miller et al., 2009b), α = 0.1 (e.g. King et al., 2007) and H/R = 0.02 (e.g. Shakura & Sunyaev,
1973). Equation 1 is derived through analogous reasoning to Natarajan & Pringle (1998) who compared theprecession time to the vertical viscous time to find the location of a warp in a steady state disc. Here weuse the usual planar disc viscosity, appropriate for a disc approaching the black hole from large radius.
4) The first ring of gas is torn from the disc, precessing effectively independently. Any interaction of this94
ring with the outer disc causes shocks which produce hot gas and dynamical infall after cooling. Any gasthat cannot cool quickly enough fuels the corona. There may or may not be an inner disc during this phase.
Tearing of disc rings occurs inside (Nixon et al., 2012a)
Rbreak ∼4aR|sin θ|3
αH

2/3
Rg ≈ 50 Rg .

(3.2)

This radius depends (weakly) on the disc–spin inclination angle. For angles . a few × H/R the disc isunlikely to tear (see e.g. the discussion of Nixon et al., 2013). For a modest inclination angle of 30◦ thetearing radius is Rbreak ≈ 30Rg .
5) The disc reaches a quasi–steady tearing phase, where successive rings are torn off and interact (see Fig. 3.2).
This produces both an aligned inner disc from gas that cools, and a strong corona from gas that remains
hot (for more detail see Section 3.3.3). The main physics in this interaction is the shocks between adjacentrings. The inner disc, which extends down to the innermost stable circular orbit (ISCO), has a variable massaccretion rate and probably drives short–lived powerful jets while the accretion rate is high. The tearing
phase continues as long as misaligned gas is supplied to the tearing region, which persists until the discaligns from the inner regions outwards. Therefore, the tearing process proceeds for a time . talign (Rtilt ).
The alignment timescale is given by King et al. (2013b), which (with M = 10 M ) gives
1
talign (Rtilt ) ∼
4αa2



H
R

2 

Rtilt
Rg

9/2

GM
≈ 35 days.
c3

(3.3)

We choose the radius for this timescale as Rtilt , rather than Rbreak , as the viscous communication of angularmomentum causing alignment is significantly reduced in a strong warp (Ogilvie, 1999; Nixon & King, 2012;
Lodato & Gerosa, 2013). To prevent tearing entirely the disc must be unable to feed the tearing region
with misaligned angular momentum. This estimate is subject to some uncertainty, and requires further
investigation (see e.g. the discussion by King et al., 2013b). As the disc is not in a steady state, it isplausible that alignment out to ∼ Rtilt is required to halt tearing, but it is also possible that disc tearing
can be halted sooner by alignment at a smaller radius, closer to Rbreak . However, the two main uncertainties
in the alignment timescale are both likely to increase talign by reducing the vertical viscosity responsible
for transferring the misaligned angular momentum. First, this viscosity is expected to weaken as the warp
amplitude increases (Lodato & Gerosa, 2013). Second, MHD effects are expected to weaken this viscosity

95
further (see e.g. Pringle, 1992; Nixon et al., 2012a; Sorathia et al., 2013b, and Section 3.4).
6) The inner disc is now closely aligned with the black hole spin beyond the break radius, which inhibits
tearing. A strong Shakura–Sunyaev disc component extends down to the ISCO. The outer disc remains
misaligned and joined to the inner disc by a warped region. The innermost disc is not subject to the strong
depositions of gas present in (5). Therefore the jet probably switches off in this state (cf. King et al., 2004).
7) As the accretion event runs out of mass supplied from large radius, the disc surface density drops.
Eventually, this allows inefficient cooling to revive the inner hot disc.
8) Finally, the accretion event is over and the system returns to the initial state.

3.3.2

State transitions
We now describe the behaviour of observed systems by stepping through the state transition phe-nomenology, while matching the observations (see e.g. Belloni, 2010, and references therein) to each of theeight stages of the disc tearing accretion picture in Fig. 3.1.
Quiescent state (QS): In quiescence there is no Shakura–Sunyaev disc inside ∼ 102 Rg , but a hot gas component may be present interior to this (e.g. Esin et al., 2001; Tomsick et al., 2009), possibly driving a steady
low–luminosity jet. The accretion cycle starts (1) and ends (8) here.
Low/hard state (LHS): As the system comes out of quiescence, a Shakura–Sunyaev disc appears (e.g. Miller
et al., 2006a; Reis et al., 2010). A persistent steady jet is seen (e.g. Fender, 2001), suggesting that some inner
hot gas is still present (2) & (3). By the end of the LHS, type–C low–frequency quasi–periodic oscillations(LF QPOs), discussed below, emerge.
Hard–intermediate state (HIMS): This state is not marked by any sudden spectral or timing changes andis consistent with a continuation of the LHS. The steady jet turns off and the type–C LF QPOs evolve tohigher frequencies in the LHS → HIMS transition. An inner disc extending to the ISCO is often observed
contemporaneously with the LHS → HIMS transition and a corona emerges. In our accretion picture, this iswhen the first ring is torn from the disc (4). Shocks between this ring and the outer disc create the corona(see Section 3.3.3) and possibly a disc that spreads down to the ISCO. This gas sweeps away the hot flow
still present in (3), turning off the steady jet.

96

Figure 3.2: Column density projection showing the disc structure for a tearing disc simulation with a disc–spin misalignment of 45◦ . The two plots show the same disc from different angles. The left panel has theblack hole spin pointing up the page, while the right panel has the spin pointing out of the page. The
simulation, to be presented in a future paper, follows the method of Nixon et al. (2012a), but uses a smalleraccretion radius allowing the innermost disc to be resolved. The innermost disc is closely aligned to theblack hole spin because the tilted components of the angular momentum vectors cancel when the precessing
rings interact, leading to dynamical infall (see eqns. 3 & 4 of Nixon et al., 2013).

97
Soft–intermediate state (SIMS): The transition from the HIMS to SIMS is marked by both the appearanceof type–B LF QPOs, and relativistic transient (i.e. ballistic) jets. Much weaker type–A LF QPOs are sometimes seen, but type–C LF QPOs no longer persist. The compact corona becomes more prominent and astrong disc extends down to the ISCO, where it remains throughout HIMS ↔ SIMS transitions (Reis et al.,2013). Section 3.3.3 elaborates on the details of this disc–jet–corona connection. This transition between
the HIMS and SIMS is the main tearing phase (5), described in detail in Section 3.3.3. The tearing rings
quasi–periodically fuel the corona and the inner disc. A powerful jet can be launched when the accretion
rate through the inner disc is high.
High/soft state (HSS): This state is characterised by a spectrally dominant Shakura–Sunyaev disc component
present down to the ISCO. This is well–matched by (6) where a stable, long–lived accretion disc extends
down to the ISCO, aligned to the black hole spin. In this phase, the disc is thin, so jets are probably strongly
suppressed (e.g. Tananbaum et al., 1972; Fender et al., 1999a). However, disc winds are common (e.g. Miller
et al., 2008; King et al., 2013a). Tentative evidence for a warped disc in a microquasar was found by Miller
et al. (2006c) who observed an obscured disc wind consistent with modulation by a disc warp. A standard
4
relation is robustly detected across a broad range of mass accretion
thin disc closely following the Ldisc − Teffrates in the HSS, with departures from this relation arising in the intermediate states and LHS (e.g. Dunnet al., 2011). However, if a variable spectral hardening factor is permitted, this scaling relation may extend
into the intermediate states and LHS (Salvesen et al., 2013; Reynolds & Miller, 2013).
Decay HSS → HIMS → LHS/QS: In the outburst decay, the system returns to the HIMS when the aligned
inner disc is accreted and inefficient cooling allows the hot flow to again pervade the inner regions (7). The
SIMS is not re-entered in outburst decay, which is consistent with our accretion picture of the inner discaligning to the black hole spin (6), halting further disc tearing events that are responsible for the SIMS. The
system finally evolves from the HIMS (7) to the LHS (8), accompanied by type-C LF QPOs of decreasing
frequency, completing the state transition accretion cycle.
Quasi–periodic oscillations (QPOs): There are three main types of LF QPO observed (Casella et al., 2005).
Type–C LF QPOs are predominantly observed in both the rise and decay of the system (e.g. Belloni et al.,2005), whereas types A & B are mostly observed in the SIMS and mark the transitions between the HIMS

98
and SIMS (Casella et al., 2004). In our accretion picture, the initial warping and precession of the inwardly
propagating disc (3) may manifest itself as the evolving type-C LF QPO, which is consistent with previous
suggestions (e.g. Ipser, 1996; Stella et al., 1999; Ingram et al., 2009). Type–C LF QPOs are also observed,
decreasing in frequency, as the system decays back to the LHS (e.g. Kalemci et al., 2004).
Motta et al. (2011) suggest that type–C and type–A LF QPOs might be the result of the sameunderlying physics, at different evolutionary stages, with type–B being physically distinct. In the context
of this disc tearing model, this result fits naturally with type–A being precessing torn rings or a precessing
oblate corona. This also suggests a plausible reason why type–A are weak and sometimes not seen, as theflux from precessing rings is dominated by the stronger components in the system at that time (e.g. innerdisc, corona and jet).

3.3.3

Disc–jet–corona coupling
In the transitions between the HIMS and SIMS, corresponding to (5) in Fig. 3.1 and the torn discstructure in Fig. 3.2, observations confirm the presence of an inner disc extending down to the ISCO (e.g.
Dunn et al., 2011; Reis et al., 2013), powerful transient jets (e.g. Fender et al., 2004) and a strong dynamic
corona (e.g. Reis et al., 2013). The production and interaction between the disc, jet and corona naturally
arise from our tearing disc picture as follows. Multiple rings break from the disc in the quasi–steady disctearing phase (see Fig. 3.2). As these rings precess they shock, cancelling some of their angular momentum
but also heating strongly. The cancellation of angular momentum is dependent on the orientation of therings, with larger disc–spin misalignment angles resulting in more efficient cancellation. Some of the gas
cools efficiently to form an inner disc which accretes on to the black hole, while the rest of the gas cannot
cool fast enough and so fuels the corona (see right panel of Fig. 3.3). While the accretion rate through theinner disc is high, and the disc therefore thick (cf. King et al., 2004), a powerful jet is launched. Recurring
accretion events from interacting rings re–supply the inner disc, driving the observed transient jet. The
observed decrease in hardness from the HIMS to the SIMS is therefore given by the cooling gas forming an
inner disc, whose spectrum peaks in the soft X–ray band.
If all of the shocked gas from tearing rings is unable to cool, which seems unlikely but possible

99

Figure 3.3: Column density projection showing the disc structure from a misaligned rings simulation in
Nixon et al. (2012b). The left panel uses an isothermal equation of state where gas heating is assumed to
be lost as radiation. In contrast, the right panel retains the shock heating in the gas, causing it to expand
vigorously. In reality the equation of state lies somewhere between these two extremes. Most likely somegas is able to cool and subsequently fall to form an inner disc, while the gas that is unable to cool forms alow–density, quasi–spherical structure – the corona.

100
dependent on the local conditions (density, etc.), the resultant distribution of gas is quasi–spherical (H/R ∼1) and robbed of a significant part of its angular momentum (implying an effective α ∼ 1). These are
precisely the conditions required to generate radiatively inefficient accretion flows (Narayan & Yi, 1994; Esin
et al., 1997; Blandford & Begelman, 1999) or magnetically arrested discs (Bisnovatyi-Kogan & Ruzmaikin,
1974; Narayan et al., 2003; Tchekhovskoy et al., 2011).

3.4

Discussion & conclusions
We have presented a new accretion picture for black hole X–ray binaries. The only assumptions

needed in this accretion picture are the three generic initial conditions: (1) A black hole with non–negligible
spin. (2) A geometrically thin, optically thick accretion disc far from the black hole. (3) A misalignment
between the disc and black hole angular momenta. The new ingredient in the accretion cycle is the processof disc tearing (Nixon et al., 2012a, 2013), which is present in the hard–intermediate and soft–intermediate
states. Tearing discs produce a variety of behaviour capable of explaining state transitions and many of theother observed black hole X–ray binary phenomena, including QPOs and the disc–jet–corona connection.
The phenomenological description of accretion in black hole X–ray binaries (e.g. Belloni, 2010) is
powerful in explaining the general observed properties. However, these systems are complex and show
different behaviour, not just from source to source, but from outburst to outburst in the same source. Ourmodel may well prove robust in this regard as the equations for the various radii and timescales in Section 3.3
are not weakly (nor too strongly) dependent on the disc parameters. Therefore modest changes in the discconditions can lead to a rich variety of behaviour. For example, the initial ring torn from the disc (4) may
occur while the disc has a low surface–density, so cooling is probably inefficient. In this case, tearing maylead to heating of the disc, which temporarily stabilizes it against tearing while the surface density remains
low. These details, and other important physical effects (e.g. radiation warping; Pringle, 1996), determine
the exact evolution of any given system.
In addition to QPOs discussed in Section 3.3.2, black hole binaries show other timing properties. For
example, a positive correlation is observed between rms variability and X–ray flux (Uttley & McHardy, 2001).
The trend of increasing varibility with increasing flux is generally consistent with the accretion picture in

101
Fig 3.1, but requires further investigation. Strong X–ray flux above 2 keV is expected when a strong coronalcomponent is present, which coincides with the most variable phases of accretion.
The viscous evolution of the disc is probably driven by turbulence induced by the magnetorotational
instability (MRI; Balbus & Hawley, 1991). We do not anticipate that including MHD will restrict thetearing behaviour, but this is simply unknown. Progress is being made in this area: investigations using
MHD grid codes are starting to look at disc warps (Fragile et al., 2007; Sorathia et al., 2013b), and smoothed
particle hydrodynamics simulations are beginning to reliably include MHD (e.g. Price et al., 2012). Some
investigations into the effective viscosities in a warped disc have already been performed. For example,
Torkelsson et al. (2000) measured the coefficients of the viscosity tensor from MHD turbulence and concluded
that they are consistent with an isotropic viscosity, as used in the disc tearing simulations by Nixon et al.
(2012a) (cf. Ogilvie, 1999; Lodato & Price, 2010). Also, Ogilvie (2003) developed an analytical model for the
dynamical evolution of magnetorotational turbulent stresses in good agreement with Torkelsson et al. (2000).
These investigations, numerical and analytical, allow for the effective viscosity from MHD turbulence to beanisotropic, but conclude the result is (near) isotropic. Recently King et al. (2013b) placed observational
constraints on this viscosity, finding that the precession in Her X-1 is consistent with an isotropic viscosity.
Sorathia et al. (2013a) have since claimed the opposite result, that the effective vertical viscosity ismuch stronger than predicted by an isotropic model. They simulate a misaligned disc subject to Lense–
Thirring precession, including MHD to self-consistently generate a turbulent viscosity. However, due totheir choice of parameters (thick, low–viscosity disc and a precession which is faster than dynamical at the
inner edge of the grid) it is unclear how relevant this work is to the discs discussed here (cf. Papaloizou &Pringle, 1983). They report that the dominant mechanism transporting angular momentum is bending waves
induced by radial pressure gradients, as shown for α < H/R by Papaloizou & Pringle (1983); Papaloizou &Lin (1995) and discussed by Lodato & Pringle (2007). Therefore it appears that this work is consistent withthe literature. The main difference is the measurement of a significantly anisotropic effective viscosity, withthe vertical viscosity much stronger than isotropy implies. This result appears at odds with previous work
on MHD turbulence (Torkelsson et al., 2000; Ogilvie, 2003), but as remarked by Sorathia et al. (2013a),hydrodynamic effects dominate MHD effects and therefore this difference is probably not due to MHD.

102
Indeed for the parameters used in their simulation, the disc may be vulnerable to hydrodynamic instabilities(see e.g. the parametric instability; Gammie et al., 2000). Such extra dissipation can explain the discrepancy,
but without further simulations we can only speculate. However, for typical black hole disc parameters such
instabilities are not thought to be important, but certainly worthy of investigation (e.g. Ogilvie & Latter,2013). For now it appears prudent to continue with the assumption of a (near) isotropic viscosity supported
by numerical, analytical and observational results (Torkelsson et al., 2000; Ogilvie, 2003; King et al., 2013b).
The accretion picture proposed here is not particular to black hole X–ray binaries. We therefore
briefly discuss some other systems in the context of this accretion picture:
Persistent sources: It is plausible that tearing is entirely suppressed for some system parameters. For
example, the mass transfer rate in the X–ray binary SS433 is probably highly super–Eddington, so the discremains thick all the way down to the ISCO (Begelman et al., 2006) and is therefore unable to experiencetearing. Other sources which show persistent or “failed” state transition behaviour may have negligible blackhole spin or disc–spin misalignment, thus inhibiting tearing.
Other stellar binaries: There appears no reason to assume that accretion far from the central object occurs
differently for different central objects (e.g. Lasota, 2001). Therefore, any model should reduce to thestandard accretion model (e.g. Frank et al., 2002) for white dwarf and magnetic neutron star accretion. For
neutron stars with weak magnetic fields, a similar cycle as proposed above may arise, as the Lense–Thirring
precession is also expected to act in these systems.
Ultraluminous X–ray sources (ULXs): These may be extreme stellar mass black hole binaries with thelargest disc–spin misalignments, so the strong shocks induced by disc tearing produce both powerful X–ray
emission and significantly enhance the accretion rate. If ULXs contain intermediate mass black holes they
may correspond to (5) in Fig. 3.1, evolving on much longer timescales.
Active galactic nuclei: The timescales predicted for state changes in AGN are substantially longer thanfor X–ray binaries. Therefore, state transitions can only be understood by considering AGN populations.
However, it is possible that the different stages of the proposed accretion cycle correspond to different types
of AGN. For example, the type I versus type II AGN dichotomy may be states (5) & (6) in Fig. 3.1 viewed
from the required angle to reveal or obscure the central accretion (cf. Nayakshin, 2005).

103
There are significant observational data with which we can test this model; high–quality spectral
and timing information is available for numerous systems. Substantial progress can also be made on thetheoretical side. For example, simulations of interacting rings of gas, with a realistic equation of state, arerequired to confirm our reasoning in Section 3.3.3. Also, the shocks between rings and the geometry ofthe disc may provide a physical motivation for disc reflection modelling. We are actively exploring bothobservational and theoretical considerations of the accretion picture presented in this paper to put thisphysically motivated black hole state transition model to the test.

Acknowledgments
We thank Andrew King, Jon Miller & Jim Pringle for valuable discussions. We also thank Phil
Armitage and Mitch Begelman for thoughtful comments on the manuscript. We thank the referee for athorough examination of our paper, which stimulated us to make it (we believe) considerably clearer. CN
thanks NASA for support through the Einstein Fellowship Program, grant PF2-130098. GS thanks the
National Science Foundation for support through the Graduate Research Fellowship Program. Some of the
figures used splash (Price, 2007) for the visualization.

Chapter 4Black Hole Spin Measurements are Sensitive to Accretion Disc Spectral
Hardening

4.1

PrefaceThis paper is submitted to Monthly Notices of the Royal Astronomical Society. The author is:

Salvesen, G., & Begelman, M. C. The two well-established techniques used to measure black hole spinin X-ray binaries often yield conflicting results, which is troublesome if either method is to be deemedrobust. Dissecting one of these methods, we show that incorporating uncertainties associated with the discatmospheric structure dominates the error budget, particularly for low and moderate black hole spins.

abstract
Black hole spin has profound astrophysical importance, influencing the cosmological evolution ofgalaxies and offering an energy reservoir for powering relativistic jets. The two well-established techniques
for measuring black hole spin often yield conflicting results, which must be resolved before either method
may be deemed robust. In practice, the continuum fitting technique does not marginalize over the spectral
hardening factor, fcol , when measuring black hole spin. Considering representative model X-ray binaries, wedemonstrate that realistic uncertainties in fcol dominate the error budget on the black hole spin. Moderateto-low black hole spins cannot be reliably constrained when adopting erroneous fcol values and/or modest
uncertainties on fcol ; however, high spins (a & 0.9) are relatively insensitive to ignorance of fcol . To
demonstrate this, we incorporate fcol uncertainties into the original black hole spin probability distributions

105
for X-ray binaries with spin measurements vetted by the continuum fitting community. Finally, we suggestthat by adopting black hole spin measurements from the iron line technique, absolute measurements of fcol
can be made with the continuum fitting technique. This could potentially offer a novel way to observationallyprobe accretion disc atmospheric physics.

4.2

Introduction
The only intrinsic properties of an astrophysical black hole are its mass and angular momentum, or

spin (e.g., Misner et al., 1973). While elegant, this “no-hair” theorem reveals a fundamental limitation to
the available observational avenues with which to study black holes.
Black hole mass undoubtedly has important astrophysical implications across the mass spectrum,from understanding how supermassive black holes co-evolve with their host galaxies (Magorrian et al., 1998)
to the details of the supernovae explosions that produce stellar mass black holes (Bailyn et al., 1998).
Fortunately, black hole mass is observationally accessible with a variety of independent techniques. The
masses of supermassive black holes can be measured with reverberation mapping (Blandford & McKee, 1982;
Peterson et al., 2004), maser discs (e.g., Miyoshi et al., 1995), stellar dynamics (Kormendy & Richstone,
1995; Ghez et al., 2005), gas dynamics (e.g., Barth et al., 2001), and empirical methods such as the M − σrelation (Gebhardt et al., 2000; Ferrarese & Merritt, 2000). The masses of stellar mass black holes can beobtained from optical light curve modeling of X-ray binaries (e.g., Orosz & Bailyn, 1997) and microlensing(Agol et al., 2002).
Black hole spin, while currently less-constrained than black hole mass, also has high astrophysicalrelevance. For instance, the cosmological evolution of the coupled growth between black holes and their
host galaxies is imprinted on the spin distribution of supermassive black holes (Volonteri et al., 2013).
Relativistic jets, which inject energy into their surroundings and are observed in many black hole systems,
may be powered by tapping into the black hole spin energy reservoir (Penrose, 1969; Blandford & Znajek,
1977). Black holes in X-ray binaries cannot change the magnitude of their natal spin appreciably over thesystem lifetime (King & Kolb, 1999); therefore, these systems offer a laboratory for studying the core-collapse
supernovae that produce stellar mass black holes. Accretion discs are also subject to interesting behaviour

106
in the presence of a spinning black hole. When the black hole spin vector is misaligned to the accretion discrotational axis, the disc experiences a differential precession (Lense & Thirring, 1918). If strong enough, thisprecession can cause the disc to break into distinct interacting rings of gas (Nixon et al., 2012a), which may
be the underlying mechanism responsible for black hole state transitions (Nixon & Salvesen, 2014). Robust
measurements of black hole spin are necessary to realize these far-reaching astrophysical implications.
Observational X-ray astronomy recently commenced an era of measuring black hole spin, with two
sophisticated techniques at the forefront of this effort. One method is based on modeling the fluorescentiron emission line profile (Fabian et al., 1989; Laor, 1991), which is a feature in the disc reflection spectrum(Lightman & White, 1988). The other method appeals to modeling the accretion disc continuum spectrum(Zhang et al., 1997). In practice, only the iron line technique can be applied to supermassive black holes.
However, independent verification of the robustness of the iron line technique is required if the measured
supermassive black hole spins are to be used to constrain cosmic evolution scenarios and other black holespin-related phenomena. Both spin measurement methods are valid for stellar mass black holes in X-ray
binaries; thus, providing two complementary and independent ways to measure black hole spin in thesesystems.
Miller et al. (2009b) were the first to employ both the iron line and continuum fitting techniques
simultaneously to measure black hole spin for a sample of X-ray binaries by linking the spin parameter
between the disc reflection and disc continuum spectral components. While broad agreement was foundwith previous spin measurements from the iron line technique alone, at least two systems (4U 1543–475
and GRO J1655–40) showed strong discrepancies with previous spin measurements from the continuum
fitting technique. Steiner et al. (2011) measured the spin of the black hole X-ray binary XTE J1550–564
independently with the iron line and continuum fitting techniques, finding weak agreement between the two
methods as a consequence of large uncertainties.
Currently, both black hole spin measurement methods are mature and well-established, each having
been applied to a modest population of black hole X-ray binaries (Reynolds, 2014; McClintock et al., 2014).
For the six black hole X-ray binaries with spins measured by both the iron line and continuum fitting
techniques, there is either disagreement or broad consistency between the derived spins (see Figure 4.1 and107
Table 4.1). Significant discrepancies in the black hole spin obtained with two different methods for the same
source are troublesome. This warrants an investigation into the validity of the assumptions behind both ofthese spin measurement techniques, but in this paper we focus on the continuum fitting method.
The continuum fitting technique relies on reliably modeling the observed accretion disc spectrum.
Standard models for the emitted disc spectrum neglect the effects of the vertical (i.e., perpendicular to thedisc mid-plane) gas structure (e.g., Shakura & Sunyaev, 1973; Novikov & Thorne, 1973). However, as this“seed” disc spectrum propagates through the disc atmosphere, the emergent spectrum will be hardened
due to the collective effects of electron scattering and absorptive/emissive opacities. The degree of disc
spectral hardening is parametrized by the phenomenological spectral hardening factor (also called the colour
correction factor), which is defined as the ratio of the colour-to-effective disc temperatures, fcol = Tcol /Teff(e.g., Shimura & Takahara, 1995). In practice (e.g., McClintock et al., 2006), the observed X-ray disccontinuum is fit using a hybrid model consisting of a general relativistic multi-colour blackbody disc arounda black hole (Li et al., 2005) and a tabulated grid of spectral energy distributions computed from discatmospheric modeling, which determines fcol (Davis & Hubeny, 2006). In these disc atmospheric models,
fcol is effectively confined to the narrow range 1.5 . fcol . 1.7 (Davis et al., 2005).
However, these determinations of fcol do not incorporate potentially important physics that may
drastically alter the degree of spectral hardening. In particular, vertical magnetic pressure support in the discis neglected, which alters the properties of the effective photosphere and hence the emergent disc spectrum.
Blaes et al. (2006) showed that even relatively weak magnetic pressure support enhances fcol from 1.48 to
1.74, which if neglected would result in an erroneous spin measurement of a ' 0.5 for a non-spinning blackhole (i.e., a = 0). Magnetic pressure-dominated discs would likely produce far more substantial disc spectral
hardening (e.g., Begelman & Pringle, 2007). Indeed, the recent constraint on the strength of the magnetic
field driving the accretion disc wind in the black hole X-ray binary GRS 1915+105 implies that the disc isstrongly magnetized (Miller et al., 2016).
This paper is organized as follows. We outline the methodology for the basic continuum fitting
technique, which implicitly assumes that the degree of accretion disc spectral hardening is modest andconfined to a narrow range (§4.3). We then relax this assumption and show that the measured black hole

Continuum Fitting Black Hole Spin, aCF

108

1.0

LMC GRSX−1 1915+105

0.8CygnusX−1
4U 1543−47
GRO J1655−40

0.6

0.4

XTE J1550−564
0.2

0.0
0.0

Iron Line Black Hole Spin, aFeKα
0.2

0.4

0.6

0.8

1.0

Figure 4.1: Black hole spin measurements from the iron line technique (aFeKα ) and the continuum fitting
technique (aCF ) plotted against one another for the sources in Table 4.1. Spin measurements for 4U 1543–47,
GRO J1655–40, and XTE J1550–564 are in disagreement at the 1σ level, as these points do not lie on thedashed line that indicates agreement between the two spin measurement techniques.

109
spin is sensitive to the adopted level of spectral hardening and its uncertainty (§4.4.2). We demonstrate thatexisting continuum fitting black hole spin measurements can become unconstrained when fcol uncertainty is
included (§4.4.3). We discuss our results (§5.6), followed by a summary and conclusions (§5.7).

4.3

Disc Continuum Fitting Methodology
Here, we outline the greatly simplified version of the continuum fitting technique for measuring blackhole spin that we adopt in our subsequent analysis. The spectrum of a geometrically thin, optically thickaccretion disc (Shakura & Sunyaev, 1973) around a black hole is described by the two-parameter multicolour disc blackbody model (e.g., Mitsuda et al., 1984; Makishima et al., 1986; Gierliński et al., 1999; Franket al., 2002; Zimmerman et al., 2005). One parameter is the maximum colour temperature in the disc,max
Tcol= Tcol [(49/36)Rin ] = 66 /77

1/4

T∗ , where,Tcol (R) = T∗RRin

−3/4 "


1−β

R
Rin

−1/2 #1/4
.

(4.1)

Here, R is the radial location in the disc and β describes the boundary condition on the inner disc edge, Rin ,with β = 1 commonly adopted and corresponding to zero torque. The characteristic temperature crudely
describing the inner disc colour temperature is,T∗ = fcol

3GM Ṁ
38πσSB Rin

!1/4
,

(4.2)

where fcol is the spectral hardening factor (also called the colour correction factor), M is the black hole mass,Ṁ is the mass accretion rate, G is the gravitational constant, and σSB is the Stefan-Boltzmann constant.
The colour temperature is related to the effective temperature of the disc by Teff = Tcol /fcol . The second
parameter of the multi-colour disc blackbody model is the flux normalization,
Kflux =

1
4
fcol



Rin
D

2
cos (θdisc ) ,(4.3)

where D is the distance to the black hole and the inner disc inclination is θdisc = 0◦ and 90◦ for face-on andedge-on discs, respectively.
Once the disc spectrum is adequately fitted, independent knowledge of D, θdisc , and fcol give Rinfrom Equation 4.3. Assuming that Rin coincides with the innermost stable circular orbit (ISCO), RISCO ,110
a measurement of M then gives rISCO = RISCO /Rg , where Rg = GM/c2 is the gravitational radius, withc being the speed of light. This dimensionless location of the ISCO, rISCO , can then be unambiguouslyinterpreted as a measurement of the black hole spin parameter, a = cJ/(GM 2 ), where J is the black holeangular momentum (Bardeen et al., 1972).

4.4

Analysis and Results
In §4.4.1, we show that modest uncertainty in fcol can easily dominate the uncertainty in the blackhole spin parameter measured using the continuum fitting technique. Adopting a model black hole X-ray
binary with known intrinsic parameters in §4.4.2, we show how uncertainties in fcol affect the black hole spinmeasured from continuum fitting. Finally, in §4.4.3 we fold uncertainties in fcol into vetted measurementsof black hole spin from continuum fitting.

4.4.1

Continuum Fitting Technique: Individual Parameter Uncertainties
Measuring Rin from Equation 4.3 is the essence of the continuum fitting technique. Consequently, themain challenge in measuring black hole spin from disc continuum fitting is the need for tight constraints on
Kflux , M , D, θdisc , and fcol . Uncertainties associated with these parameters all contribute to the uncertainty
in the measured black hole spin.
Here, we isolate the contribution to the measured black hole spin uncertainty due to each of these
individual parameters with the following straightforward exercise. We construct a representative black hole00
X-ray binary with true parameter values: a0 = 0, M 0 = 7.5 M , D0 = 8.5 kpc, θdisc= 45◦ , and fcol
= 1.6. Aprime on a parameter denotes a true value as opposed to an unprimed parameter, which is to be interpreted
as a “measured” value with an associated uncertainty.
The simplified continuum fitting method will now be applied to make a mock measurement of theblack hole spin probability density function (PDF), f (a), for this model X-ray binary, where all but one
parameter are taken to be measured perfectly (i.e., with zero uncertainty). For this single parameter ofinterest x — with Kflux , M , D, θdisc , or fcol being acceptable choices — we suppose that x is measured with

111
PDF,
f (x) = N (x, µx = x0 , σx = ξx0 ) .

(4.4)

Here, N is the normal distribution,

N (x, µx , σx ) =

σx

1
√

"

− (x − µx )
exp
2σx2
2π

2

#
,

(4.5)

where µx is the mean of the distribution and σx is the standard deviation. We choose to center our mockmeasurement of f (x) on its true value (i.e., µx = x0 ) with a 10% measurement uncertainty relative to thetrue value (i.e., σx = ξx0 with ξ = 0.1). This level of measurement uncertainty for a given isolated parameterx is typical when applying the continuum fitting method to a black hole X-ray binary. All other parameters
needed to measure the inner disc radius, rin , are taken to be their true values. Therefore, when considering
only the uncertainties on the parameter of interest, x, the PDF for the inner disc radius is obtained bychange of variables from x to rin ,fx (rin ) = f (x) |Jx |,(4.6)

where Jx = ∂x/∂rin is the Jacobian determinant for the change of variables and Equation A.1 relates x andrin . We calculate fx (rin ) for x = (Kflux , M, D, θdisc , fcol ), with each expression given in Appendix B. The
black hole spin parameter PDF for the isolated parameter of interest, fx (a), then follows from assumingthat rISCO = rin and performing the change of variables from rISCO to a (see §A.0.3).
Figure 4.2 shows the individual parameter contributions to the uncertainty in the black hole spin PDF
as measured by the continuum fitting technique. For each parameter of interest with an adopted 10% relative
uncertainty, the measured black hole spin, ax , and its ±1σ uncertainties are listed in the upper left cornerof Figure 4.2. Uncertainty in fcol leads to the broadest black hole spin parameter PDF. Uncertainties in
M and D contribute modestly to black hole spin measurement uncertainty, while uncertainties in Kflux andθdisc also contribute to a lesser, but still significant, extent. Therefore, this demonstrated sensitivity of blackhole spin measurement error to uncertainty in fcol justifies a targeted investigation into the assumptions
regarding fcol made by the disc continuum fitting technique.

112

6

9

8

Inner Disc Radius, rin
7

6

5

4

3

21

aθ =0.00−+00..0608
aK =0.00−+00..0909
aD =0.00−+00..1819
aM =0.00−+00..1620
af =0.01−+00..3439

Black Hole Spin PDF, f(a)

disc

5

4

flux

col

3

2

1

0
−1.0

Black Hole Spin, a

−0.5

0.0

0.5

1.0

Figure 4.2: Black hole spin parameter PDFs, f (a), resulting from isolating individual parameter uncertainties
when measuring spin with the continuum fitting technique. The model black hole X-ray binary has true00parameters a0 = 0, M 0 = 7.5 M , D0 = 8.5 kpc, θdisc= 45◦ , and fcol
= 1.6. When generating a given PDF,
the 1σ uncertainty on the parameter of interest is taken to be 10% of its true value. The largest contribution
to uncertainties in the measured black hole spin comes from uncertainties in fcol .

113
4.4.2

Effects of Spectral Hardening on Black Hole Spin Measurements: Model X-ray
Binaries
We now consider the full grid of true black hole spins, −1 ≤ a0 ≤ 1, for our model black hole X-ray

0
0
binary with true parameter values: M 0 = 7.5 M , D0 = 8.5 kpc, θdisc= 45◦ , and fcol
= 1.6.

4.4.2.1

Marginalizing Over fcol Uncertainties

Our objective here is to determine how uncertainties in fcol contribute to the error budget in the blackhole spin as measured by the continuum fitting technique. Our approach is to obtain mock measurementsof the black hole spin PDF, f (a), given mock measurements of the PDFs f (Kflux ), f (M ), f (D), f (θdisc ),and f (fcol ). The procedure for obtaining this mock spin measurement is as follows.
0(seeStarting with a known true spin parameter, a0 , we compute the true location of the ISCO, rISCO
0
Equation A.2). Given our model black hole X-ray binary with true intrinsic parameters M 0 , D0 , θdisc
, and
0
0
fcol
, we obtain the true flux normalization, Kflux(see Equation A.1). We are now in a position to construct

the following mock observable PDFs required by the simplified continuum fitting technique outlined in §4.3
(see Equation 4.5),
0
f (Kflux ) = N (Kflux , µKflux = Kflux, σKflux = ξKflux0 )(4.7)

f (M ) = N (M, µM = M 0 , σM = ξM 0 )(4.8)

f (D) = N (D, µD = D0 , σD = ξD0 )

(4.9)

0
0
f (θdisc ) = N (θdisc , µθdisc = θdisc, σθdisc = ξθdisc)(4.10)00
f (fcol ) = N (fcol , µfcol = fcol
, σfcol = ξfcol),(4.11)

where we choose ξ = 0.1 for all mock observables.
We transform f (Kflux ) to f (rin ) using change of variables (see §A.0.2),
Z ∞

Z π/2

Z ∞ Z ∞f (rin ) =dM dD dθdisc dfcol
fcol =1

θdisc =0

D=0

M =0

× f (M ) f (D) f (θdisc ) f (fcol ) f (Kflux ) |JKflux →rin | ,(4.12)

± 1σ Uncertainty in Measured BH Spin

114

1.0

µfcol = fcol′
σfcol =0.1µfcol

0.5

0.0

dfcol

fcol

−0.5

θdisc D M

dM dD dθdiscfcol θdisc D M

dM dD dθdisc dfcol

−1.0
−1.0

True Black Hole Spin, a′
−0.5

0.0

0.5

1.0

Figure 4.3: The ±1σ uncertainties on the black hole spin measured from the continuum fitting method,plotted against the true black hole spin, a0 , for a representative model black hole X-ray binary. The adopted
0
fcol PDF has mean µfcol = fcoland standard deviation σfcol = 0.1µfcol . Solid black lines and error bands show
the ±1σ uncertainties on a from marginalizing over all parameters M , D, θdisc , and fcol . Dashed blue lines
result from marginalizing over only M , D, θdisc . Dotted red lines result from marginalizing over only fcol .
Folding fcol uncertainties into the black hole spin error budget leads to significantly larger spin measurement
uncertainties, especially for low-to-moderate a0 values. Green hatched regions show the uncertainty levels at
which a black hole spin measurement becomes a lower or upper limit.

115
where JKflux →rin is the Jacobian determinant for the change of variables from Kflux to rin (see Equation
A.14). The integration of Equation 4.12 is done numerically and verified with Monte Carlo techniques.
Assuming that f (rin ) = f (rISCO ), the output black hole spin as measured by the continuum fitting
technique is then obtained by transforming f (rISCO ) to f (a) using change of variables (see §A.0.3). This
procedure is repeated across the full range of true black hole spins, −1 ≤ a0 ≤ 1.
Figure 4.3 plots the ±1σ uncertainties on the black hole spin that are measured from the above
procedure against the true spin value. In this case, the measurement technique recovers the true black holespin by design because the PDFs for M , D, θdisc , and fcol are all centered on their true values. The ±1σ
error band enclosed by the solid black lines comes from marginalizing over M , D, θdisc , and fcol , as done in
Equation 4.12. In practice, the continuum fitting technique does not marginalize over fcol ,0
Z fcol+ Z π/2

Z ∞ Z ∞f (rin ) =dfcol dM dD dθdisc
0 −
fcol

θdisc =0

D=0

M =0

0
× δ (fcol − fcol
) f (M ) f (D) f (θdisc )× f (Kflux ) |JKflux →rin | ,(4.13)

which is shown by dashed blue lines. For the sake of comparison, dotted red lines show the results frommarginalizing over only fcol ,Z ∞0
Z M 0 + Z D0 + Z θdisc+

dfcol dM dD dθdiscf (rin ) =fcol =1

M 0 −

D 0 −

0
θdisc
−

0
× f (fcol ) δ (M − M 0 ) δ (D − D0 ) δ (θdisc − θdisc)× f (Kflux ) |JKflux →rin | .

(4.14)In Equations 4.13 and 4.14,  > 0 such that the Dirac delta function, δ (x), is integrated over its peak.
Figure 4.3 shows that if fcol is known accurately and precisely, then the continuum fitting techniqueused in practice can provide fairly tight constraints on the black hole spin (dashed blue lines). However,the uncertainties on the measured black hole spin increase dramatically when fcol uncertainties are also
marginalized over (solid black lines). Indeed, marginalizing over only fcol (dotted red lines) leads to greater
uncertainty in the measured black hole spin than from marginalizing over M , D, and θdisc (dashed blue lines),
supposing that all of these parameters have a 10% relative uncertainty. If we instead adopt a 7% relative

116

1.0

µfcol =(1−0.1)fcol′
σfcol =0.1µfcol

0.5

0.0

dfcol

fcol

−0.5

θdisc D M

Measured Black Hole Spin, aMeasured Black Hole Spin, a

1.0

θdisc D M
0.5

dM dD dθdiscfcol θdisc D M

dM dD dθdisc dfcol

0.0

−0.5

dM dD dθdiscfcol θdisc D M

dM dD dθdisc dfcol

−1.0
−1.0

dfcol

fcol

µfcol =(1+0.1)fcol
σfcol =0.1µfcol′−1.0

True Black Hole Spin, a
−0.5

0.0

0.5

′

1.0

−1.0

True Black Hole Spin, a
−0.5

0.0

0.5

1.0

′

Figure 4.4: Continuum fitting measured black hole spin, a, plotted against the true black hole spin for arepresentative model black hole X-ray binary, a0 . Left: The adopted fcol PDF has mean µfcol = 1.44 andstandard deviation σfcol = 0.1µfcol . Right: The adopted fcol PDF has mean µfcol = 1.76 and standard
deviation σfcol = 0.1µfcol . The line styles and error bands have the same interpretation as in Figure 4.3.
The dash-dotted black line marks agreement between the true and measured black hole spins. These figures
show that adopting even a mildly erroneous fcol leads to black hole spin measurements that are inconsistent
with the true values.

117
uncertainty for fcol , the error contribution from fcol alone (dotted red lines) can be made to match that fromM , D, and θdisc (dashed blue lines). Therefore, we suggest as a general guideline that if fcol is known towithin ∼ 5%, then fcol will probably not dominate the error budget on the measured black hole spin.
Figure 4.3 also shows that high prograde black hole spins are relatively insensitive to uncertainties
in fcol . For a0 = 0.9, the standard continuum fitting practice finds a = 0.90+0.05−0.05 while also marginalizing
over fcol yields the similar constraint a = 0.90+0.09−0.08 . However, low-to-moderate prograde spins cannot betightly constrained given modest uncertainties in fcol . Taking a0 = 0.5 (a0 = 0) as examples, neglecting fcol
+0.30
uncertainties results in a = 0.50+0.17−0.16 (a = 0.00−0.27 ), while incorporating all parameter uncertainties gives+0.47
a = 0.50+0.27−0.26 (a = 0.00−0.43 ). Most affected by uncertainties in fcol are retrograde spinning black holes, as

all but the most extreme retrograde spins could be distinguished from a prograde spin at the 1σ confidence
level (i.e., σa+ & a).

4.4.2.2

Adopting Erroneous fcol Values

0
We now aim to determine how adopting an erroneous fcol value (i.e., µfcol 6= fcol
in Equation 4.11)

affects the black hole spin parameter, a, as measured by the continuum fitting method.
To quantify the impact of choosing an erroneous fcol value on black hole spin measurements, weconsider two different spectral hardening factor PDFs: one slightly underestimating and one slightly overes0
timating the true spectral hardening factor fcol
= 1.6, respectively,−−f − (fcol ) = N fcol , µfcol = fcol
, σfcol = ξfcol



(4.15)

++f + (fcol ) = N fcol , µfcol = fcol
, σfcol = ξfcol.

(4.16)−+
00Here, fcol
= (1 − 0.1) fcol
= 1.44, fcol
= (1 + 0.1) fcol
= 1.76, and ξ = 0.1 to represent a 1σ measurement

uncertainty at the 10% level. We then repeat the procedure of §4.4.2.1, but replace f (fcol ) in Equations 4.12−0
and 4.14 with either f − (fcol ) or f + (fcol ). In Equation 4.13 we replace δ (fcol − fcol
) with either δ fcol − fcol+or δ fcol − fcol
and adjust the fcol integration limits accordingly. The Jacobians in these equations also−+
replace fcol with the appropriate fcol
or fcol.

Figure 4.4 shows the black hole spin that is measured (compared to its true value) when the adopted

118
mean of the fcol PDF undershoots (µfcol = 1.44; left panel) or overshoots (µfcol = 1.76; right panel) the true
0value fcol
= 1.6 by modest amounts. Figure 4.4 demonstrates that adopting even these mildly incorrect

values for fcol leads to black hole spin measurements that are generally inconsistent with the true spin values
at the 1σ level.
We now consider how adopting the correct M , D, and θdisc , but an erroneous value of the spectral
hardening factor over the range 1.0 ≤ fcol ≤ 3.0, affects the measured black hole spin value. Using Equation
4.14, we obtain a mock measurement of the black hole spin parameter, a, for each assumed fcol . Figure 4.5
shows the grid of assumed spectral hardening factors, fcol , and true black hole spins, a0 , where the contour
lines show the measured black hole spin parameter, a, and the colours show the difference between the trueand measured black hole spins, ∆a = a0 − a. Blue (red) regions are where the black hole spin is measured tobe less (greater) than the true value; therefore, selecting a value for fcol that is greater (less) than the true
value causes the measured black hole spin to be too small (large). Erroneous black hole spin measurements
due to choosing an incorrect fcol are most severe in the darker regions.
0
The inset of Figure 4.5 shows that choosing fcol values that deviate from fcol
at the 10% level (i.e.,0= 1.6), can lead to substantially incorrect spin measurements (e.g.,fcol = 1.76 or fcol = 1.44, when fcol|∆a| ' 0.4 for a0 = 0). The lines plotted in the inset merely show the measured black hole spins from Figure
4.4 subtracted from the true spins. With decreasing true black hole spin and for a given erroneous fcol“measurement,” the measured spin deviates further from its true value. This means that low-to-moderate
black hole spin measurements (e.g., a < 0.9) from continuum fitting suffer greatly (i.e., |∆a > 0.1|) fromadopting fcol values that are incorrect by an amount & 10% of the true value. High spin measurements (e.g.,a > 0.9) are less affected from adopting mildly erroneous fcol values.

4.4.3

Effects of Spectral Hardening on Black Hole Spin Measurements: Observed X-ray
Binaries

119∆a =True Black Hole Spin (a′ ) − Measured Black Hole Spin (a)
−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

3.0

2.5

∆a = a′ −aSpectral Hardening Factor, fcol

0.6

ol = 1.
7

fc

0.4
0.2

fcol

0.0

−0.2

f col

−0.4

= fcol′ =1.6

6

= 1.44

−0.6
−1.0

−0.5

0.0

a′

0.5

2.0

a

.0
1
−
=
a
1.5

1.0

=

.5
0
−

.5
0
=
a

0
a =0
.

.9
0
=
a
1.0
−1.0

−0.5

0.0

0.5

True Black Hole Spin, a′

1.0

Figure 4.5: Consider a representative black hole X-ray binary with true black hole spin, a0 , (x-axis) and0
true spectral hardening factor, fcol
= 1.6. Adopting some measured fcol value (y-axis) to be used by thecontinuum fitting technique, black contour lines show the resulting measured black hole spin, a, and thecoloured contours show ∆a = a0 − a. White regions are where no solution for a exists under the rin = rISCO
assumption (i.e., rin < 1 or rin > 9). The discrepancy between a and a0 worsens (dark regions) with either0decreasing a0 or increasing |fcol− fcol |. The inset shows ∆a plotted against a0 for fcol = 1.76 (solid line)
and fcol = 1.44 (dashed line), emphasizing that even mildly incorrect values of fcol cause non-negligible
discrepancies between a and a0 , particularly for low-to-moderate a0 values.

0.9985+0.0005−0.0148 (7)+0.05
0.92−0.07 (5)ad
0.97+0.014
(3)
−0.02
+0.02
0.97−0.25 (27)0.3 ± 0.1 (13)Cygnus X–1

4U 1543–47

27.1 ± 0.8 (18)66 ± 2c (4)θbinary[◦ ]74.6 ± 1.0 (16)69.2 ± 0.7 (20)0.84 ± 0.05 (11) 1.765 ± 0.015 (11) 15.65 ± 1.45 (16)0.25+0.13−0.16 (28)0.20+0.34−0.33 (26)0.12 ± 0.19 (6)

−
−
−
−

M33 X–7

LMC X–3

H1743–322

A0620–00

6.61 ± 0.25 (2)51.0 ± 0.9 (2)75 ± 3 (26)74.7 ± 3.8 (19)1.06 ± 0.12 (2)8.5 ± 0.8 (26)48.10 ± 2.22 (17)840 ± 20 (16)4.38+0.58−0.41 (19)3.2 ± 0.2 (10)7.5 ± 1.0f (24)

Table 4.1: Black hole X-ray binaries with spin measurements vetted by the continuum fitting community (McClintock et al., 2014). Six of these sources
also have black hole spins vetted by the iron line community (Reynolds, 2014). From left to right the columns are the source name, iron line spin,
continuum fitting spin, spectral hardening factor, black hole mass, binary orbital inclination, and distance to the source. Errors are at the 1σ confidence
level unless indicated otherwise. Instances where the confidence interval was unclear are assumed to be 1σ. The values listed for fcol , M , θbinary , and D
are those from the original cited works used in determining aCF . Each measurement has its reference number in parentheses according to the following
list: (1) Blum et al. (2009); (2) Cantrell et al. (2010); (3) Fabian et al. (2012); (4) Fender et al. (1999b); (5) Gou et al. (2009); (6) Gou et al. (2010);(7) Gou et al. (2011); (8) Greene et al. (2001); (9) Harlaftis & Greiner (2004); (10) Hjellming & Rupen (1995); (11) Liu et al. (2008); (12) McClintock
et al. (2006); (13) Miller et al. (2009b); (14) Orosz et al. (1998); (15) Orosz (2003); (16) Orosz et al. (2007); (17) Orosz et al. (2009); (18) Orosz et al.
(2011a); (19) Orosz et al. (2011b); (20) Orosz et al. (2014); (21) Özel et al. (2010); (22) Reid et al. (2011); (23) Reis et al. (2009); (24) Shafee et al.
(2006); (25) Steiner et al. (2011); (26) Steiner et al. (2012a); (27) Steiner et al. (2012b); (28) Steiner et al. (2014). a The wider error range suggested
in Table 1 of McClintock et al. (2014) is adopted. b The fiducial fcol = 1.6 is adopted because a value is not given explicitly in the cited paper. c The
binary orbital plane is not measured directly, but instead assumed to be perpendicular to the measured inclination of the jet axis. d Errors are at the
90% confidence level. e fcol is estimated from the figures in the cited paper. f This distance appears to be unpublished, with Orosz et al. (1998) quoting
D = 9.1 ± 1.1 kpc. g See also Miller et al. (2009b), who find a = 0.76 ± 0.01. h In the absence of a dynamical mass measurement, the mass distributionof Özel et al. (2010) was assumed.

1.6 (−)− (21)bh

7.0 ± 0.6 (20)9.10 ± 0.61 (19)

1.6 (−)b

1.6 (−)b

1.6 (−)XTE J1550–564

70.2 ± 1.2 (8)bg
0.55+0.15−0.22 (25)

6.30 ± 0.27 (8)1.65e (24)

GRO J1655–40 0.92 ± 0.02d (23) 0.70 ± 0.10a (24)9.4 ± 1.0 (14, 15) 20.7 ± 1.5 (14, 15)

1.5e (24)0.34+0.20−0.28 (25)1.86+0.12−0.11 (22)11.2 ± 0.8 (4)

D
[kpc]

1.562 ± 0.008 (5) 10.91 ± 1.41 (17) 36.38 ± 1.92 (17) 48.10 ± 2.22 (17)14.8 ± 1.0 (18)14.0 ± 4.4 (9)M[M ]0.80 ± 0.10 (24)

LMC X–1

1.6b (−)> 0.95a (12)0.98 ± 0.01 (1)GRS 1915+105
1.610 (7)

fcol

aCF

aFeKα

Source120

121
At the time of writing, spin measurements that pass stringent quality criteria using the iron line
technique exist for at least 20 supermassive black holes in active galactic nuclei and 14 stellar mass black
holes in X-ray binaries (Reynolds, 2014). Similarly, the continuum fitting technique has yielded at least
10 stellar mass black hole spins that also underwent a rigorous vetting process (McClintock et al., 2014).
Of the stellar mass black hole spins measured independently by both the iron line and continuum fitting
communities, there are six overlapping X-ray binary sources. These are listed in Table 4.1.
Figure 4.1 plots the continuum fitting spin, aCF , against the iron line spin, aFeKα , for these six
sources. Three sources are in agreement with near-maximal spin (Cygnus X–1, GRS 1915+105, and LMC
X–1), one source is in very weak agreement due to large uncertainties (XTE J1550–564), and two sources
are in disagreement at the 1σ level (4U 1543–47 and GRO J1655–40). By adopting the slightly different
values fcol = 1.96, 1.46, and 1.32 than those listed in Table 4.1 for 4U 1543–47, XTE J1550–564, and
GRO J1655–40, respectively, the continuum fitting black hole spins can be made to match those from theiron line method. This illustrates that physically plausible deviations from the fcol values selected by thecontinuum fitting methodology can completely account for discrepancies between the two leading black holespin measurement techniques.
Figures 4.6 and 4.7 show the normalized black hole spin PDFs measured by the continuum fitting
technique, f (aCF ). We extracted published black hole spin PDFs with the DataThief program (B. Tummers
2006; http://datathief.org/) for the X-ray binaries: LMC X–1 (Gou et al., 2009), XTE J1550–564 (Steiner
et al., 2011), M33 X–7 (Liu et al., 2008), LMC X–3 (Steiner et al., 2014), H1743–322, (Steiner et al., 2012a),and A0620–00 (Gou et al., 2010). We assumed normal distributions for 4U 1543–47 and GRO J1655–40
based on the aCF values and uncertainties listed in Table 4.1, as the actual PDFs were not published.
As show in §4.4.2.2, adopting even a slightly incorrect fcol can cause the continuum fitting technique tofail. Alleviating this problem would require either direct fcol measurements or a firmer understanding of thedominant physical processes in accretion disc atmospheres (see §4.5.2). Therefore, we cannot easily explore
how an erroneous fcol choice alters the measured black hole spin PDFs for real X-ray binaries. Instead, we
suppose that the fcol values for each source listed in Table 4.1 are correct — we stress that this may not
be true — in order to investigate how including uncertainties in fcol modifies the measured black hole spin

Black Hole Spin, a
LMC X−1

1

1.0

0.5

0.9

0

-0.5

-1

.05
aCF =0.92−+00.07.03
a =0.97−+0
0.10

0.8
0.7

0.6
0.5

0.4
0.3
0.2
0.1

0.0
0

1

2

Inner Disc Radius, rin
3

4

5

6

7

8

9

10

11 12

Black Hole Spin, a
Inner Disc Radius PDF, f(rin)Inner Disc Radius PDF, f(rin)

122

1

1.0

0.5

0.5

0

-0.5

-1

GRO J1655−40

0.9

.10
aCF =0.70−+00.10.18a =0.72−+0
0.19

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0

1

2

3

4

5

6

7

8

9

Inner Disc Radius, rin

10

11 12

Inner Disc Radius PDF, f(rin)Inner Disc Radius PDF, f(rin)

1.0

-0.5

-1

4U 1543−47

0.9

.10
aCF =0.80−+00.10.16a =0.83−+0
0.18

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0

1

2

Black Hole Spin, a

1

0

3

5

6

7

8

9

0

-0.5

-1

Inner Disc Radius, rinBlack Hole Spin, a

1

0.50

4

0.5

10

11 12

XTE J1550−564

0.45

.20
aCF =0.34−+00.28.47a =0.38−+0
0.45

0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0

1

2

3

4

5

6

7

8

9

Inner Disc Radius, rin

10

11 12

Figure 4.6: Inner disc radius PDFs measured with the continuum fitting technique (solid lines) for X-ray
binaries LMC X–1 (top left; µfcol = 1.562), 4U 1543–47 (top right; µfcol = 1.5), GRO J1655–40 (bottom left;µfcol = 1.65), and XTE J1550–564 (bottom right; µfcol = 1.6). Dashed blue lines show the original black holespin PDFs obtained from the continuum fitting technique. Solid black lines show the black hole spin PDFs
when fcol is also marginalized over assuming a representative dispersion of σfcol = 0.1µfcol . Shaded areas
enclose the ±1σ confidence regions.

Black Hole Spin, a
M33 X−7

1

0.5

1.4

0

-0.5

-1

.05
aCF =0.84−+00.05.12
a =0.84−+0
0.14

1.2
1.0

Inner Disc Radius PDF, f(rin)Inner Disc Radius PDF, f(rin)123Black Hole Spin, a
LMC X−3

1

1.0

0.5

0.9

-0.5

-1

=0.25−+00 1613
40
a =0.23−+0
0 37

0.8

.
.
.
.

aCF

0.7

0.6

0.8

0.5

0.6

0.4
0.3

0.4
0.2

0.0

0.2
0.1

0.0

0

1

2

Inner Disc Radius, rin
3

4

5

6

7

8

9

10

11 12

0

1

2

1

0.50

0.5

0

-0.5

H1743−322

0.45

.34
aCF =0.20−+00.33.53
a =0.39−+0
0.56

0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0

1

2

3

4

5

6

7

8

9

Inner Disc Radius, rin

Inner Disc Radius, rin
3

4

5

6

7

8

9

10

11 12

Black Hole Spin, a

-1

10

11 12

Inner Disc Radius PDF, f(rin)Black Hole Spin, a

Inner Disc Radius PDF, f(rin)

0

1

1.0

0.5

0

-0.5

-1

A0620−00

0.9.19
aCF =0.12−+00.19.47a =0.15−+0
0.43

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0

1

2

3

4

5

6

7

8

9

Inner Disc Radius, rin

10

11 12

Figure 4.7: Inner disc radius PDFs measured with the continuum fitting technique (solid lines) for X-ray
binaries M33 X–7 (top left; µfcol = 1.765), LMC X–3 (top right; µfcol = 1.6), H1743–322 (bottom left;µfcol = 1.6), and A0620–00 (bottom right; µfcol = 1.6). Dashed blue lines show the original black hole spin
PDFs obtained from the continuum fitting technique. Solid black lines show the black hole spin PDFs whenfcol is also marginalized over assuming a representative dispersion of σfcol = 0.1µfcol . Shaded areas enclosethe ±1σ confidence regions.

124
PDF. In practice, when measuring the black hole spin PDF, the continuum fitting technique either does
not marginalize over the spectral hardening factor or its uncertainty is taken to be incredibly small (i.e.,
σfcol . 0.01). Our aim here is to incorporate reasonable uncertainties on fcol (e.g., at the 10% level) into
the actual measured black hole spin PDFs with the following procedure.
Given a measured black hole spin PDF, f (aCF ), we convert this to a PDF for the ISCO, f (rISCO,CF ),by change of variables from aCF to rISCO,CF (see §A.0.4). Adopting the assumption that the ISCO coincides
with the location of the inner disc edge, we obtain the measured PDF for the inner disc radius, f (rin,CF ) =f (rISCO,CF ). Next, we obtain f (Kflux ) by change of variables from rin,CF to Kflux and marginalizing overM , D, and θdisc (see §A.0.1),
Z fcol,CF +∆ Z π/2

Z ∞ Z ∞f (Kflux ) =dfcol dM dD dθdiscfcol,CF −∆θdisc =0

D=0

M =0

× δ (fcol − fcol,CF ) f (M ) f (D) f (θdisc )× f (rin,CF ) |Jrin →Kflux | ,(4.17)

where we adopt normal distributions for M , D, and θdisc , and a Dirac delta function for fcol based on thevalues and uncertainties used by the continuum fitting technique in Table 4.1 (denoted by subscript “CF”),
f (M ) = N (M, µM = MCF , σM = σMCF )(4.18)

f (D) = N (D, µD = DCF , σD = σDCF )(4.19)f (θdisc ) = N θdisc , µθdisc = θdisc,CF , σθdisc = σθdisc,CF .

(4.20)

We integrate Equation 4.17 numerically and confirm the result with Monte Carlo integration. Importantly,
in obtaining f (Kflux ) we do not marginalize over fcol because the original f (aCF ) for a given source did not
marginalize over fcol .
With the “observed” flux normalization PDF in hand from Equation 4.17, we can now perform achange of variables from Kflux to rin (see §A.0.2), marginalizing over all continuum fitting parameters — M ,D, θdisc , and fcol (see Equation 4.12) — to arrive at f (rin ). In doing so, we choose a normal distribution
for fcol centered on the continuum fitting value listed in Table 4.1,
f (fcol ) = N (fcol , µfcol = fcol,CF , σfcol = ξfcol,CF ) ,(4.21)

125
with ξ = 0.1 to assume a 10% relative uncertainty on fcol,CF . Under the usual assumption that rISCO = rin ,we have f (rISCO ) = f (rin ). Finally, we arrive at the desired PDF for the black hole spin parameter, f (a),by change of variables from rISCO to a (see §A.0.3).
For black hole X-ray binaries with aCF < 0.95 in Table 4.1, Figures 4.6 and 4.7 show how incorporating
uncertainties in fcol at the 10% level modify the black hole spin PDFs. We found that our methodology
has difficulty with particularly steep PDFs associated with high-spin measurements, such as Cygnus X–1(Gou et al., 2011) and GRS 1915+105. However, we saw in §4.4.2.1 that spins measured to be near-maximal
will not be strongly affected by including fcol uncertainties. Figures 4.6 and 4.7 show that black hole spin
becomes poorly constrained for X-ray binaries with a . 0.5 measured from continuum fitting when a 10%
relative uncertainty in fcol is included in the error budget.

4.5

Discussion

4.5.1

Continuum Fitting Technique in Practice
The description of the continuum fitting procedure for measuring black hole spin outlined in §4.3 is a

simplified version of the more sophisticated modern technique used in practice.
Great care is taken to select observations where the black hole X-ray binary of interest is in thehigh/soft state where the disc is the dominant spectral component (e.g., McClintock et al., 2006); thus,
minimizing the number of model components and free parameters required to fit the spectrum. Observations
4of black hole X-ray binaries in this disc-dominated state support the Ldisc − Teff
scaling relation expected fora multi-colour disc blackbody model for disc luminosities 0.01 . Ldisc /LEdd . 0.5 (Gierliński & Done, 2004);
thus, justifying the basic continuum fitting methodology. Furthermore, only the observations corresponding
to luminosity states Ldisc /LEdd . 0.3 are used because the assumption of a geometrically thin disc breaks
down for disc luminosities in excess of this limit (e.g., Shafee et al., 2006; McClintock et al., 2006).
A fully relativistic version of the multi-colour disc blackbody model is adopted in practice (e.g.,
Novikov & Thorne, 1973; Gierliński et al., 2001), which can also handle effects such as limb-darkening, selfirradiation, and a non-zero inner disc torque (Li et al., 2005). This disc continuum model chooses mass

126
max
accretion rate, Ṁ , and black hole spin, a, as free parameters, rather than Tcol
and Kflux as presented in§4.3. Given independent measurements of M , D, and θdisc , the final parameter that must be constrained isfcol , which is determined from disc atmospheric modeling as follows.
The seed disc spectrum is altered by various physical processes that conspire to produce a hardened
emergent spectrum, which is parametrized by fcol (e.g., Shimura & Takahara, 1995). Davis et al. (2005)
calculated non-local thermodynamic equilibrium relativistic accretion disc models that self-consistently treat
the radiative transfer and vertical disc temperature + ionization structure, accounting for Comptonizationand free-free/bound-free opacities. For restricted values of the disc luminosity and effective α-viscosity
parameter, Davis & Hubeny (2006) ran a suite of the Davis et al. (2005) models over a grid of relevant
parameter space. Each realization produced an emergent disc spectrum with an associated fcol value. These“look-up tables” were implemented into XSPEC (Arnaud, 1996) and coupled to a relativistic multi-colour
disc blackbody model (McClintock et al., 2006). Therefore, when using the current disc continuum fitting
approach to constrain black hole spin, the chosen value for fcol is based on the detailed disc atmosphericmodeling of Davis et al. (2005). However, the uncertainty range in fcol is inherently restricted by the physics
included in this disc atmospheric model. As a result, uncertainties in fcol are either ignored or so small thatfcol is effectively not marginalized over in the error analysis. We suggest in §4.5.2 that additional physics
could significantly expand the allowable range of fcol values.
Another assumption of the continuum fitting technique that deserves mentioning here is that the
inclination of the inner disc regions, θdisc , is not measured directly. Rather, θdisc is regularly assumed to
be aligned with the binary orbital inclination, θbinary , which is measured from the periodic eclipsing by theaccretion disc of the companion star’s optical light curve (e.g., Orosz & Bailyn, 1997). However, θdisc andθbinary are not necessarily aligned (e.g., Maccarone, 2002; Martin et al., 2010).

4.5.2

Measuring fcol with the Continuum Fitting Technique
Estimating fcol from a physically motivated model when characterizing the disc continuum is com-mendable and is what the continuum fitting technique now does in practice. However, the physics consideredin the disc atmospheric modeling effectively confines fcol to the range 1.5 . fcol . 1.7 (see Table 4.1; Davis

127& Hubeny, 2006). Recent observational investigations suggest that the spectral hardening factor can exceed
fcol ' 2 (e.g., Salvesen et al., 2013; Reynolds & Miller, 2013), even in the disc-dominated high/soft state2
applicable to the continuum fitting technique (Maitra et al., 2014). Given the scaling rin ∝ fcol(see Equation

A.1), realistic fcol values and uncertainties should be fully considered when measuring black hole spin.
There may be additional physics not included in the disc atmospheric modeling adopted by thecontinuum fitting technique that could have a strong influence on the degree of spectral hardening. A
diminishment to fcol ' 1 can arise from disc inhomogeneities, which allow the preferential escape of radiation
through low-density channels. Strong disc inhomogeneity can arise in radiation pressure-dominated discs
susceptible to photon bubble instability (Begelman, 2002, 2006) and/or magnetic pressure-dominated discs(e.g., Salvesen et al., 2016b,a). An enhancement to fcol ' 2 can arise from considering dissipation of accretion
power in the corona (Merloni et al., 2000) and/or significant vertical magnetic pressure support (e.g., Blaes
et al., 2006). Indeed, strongly magnetized accretion discs may achieve fcol ∼ 5 (Begelman & Pringle, 2007)
and we are currently working to determine how the emergent disc spectrum is altered due to substantial
vertical magnetic pressure support (Salvesen et al., in preparation).
Current X-ray data are sufficient to constrain the two free parameters in the multi-colour disc blackbody model (i.e., Ṁ and a); however, the data cannot also constrain fcol as a third free parameter. Placing
firm observational constraints on fcol is challenging, with previous relative (e.g., Salvesen et al., 2013) andabsolute (Maitra et al., 2014) measurements of fcol suggesting that the fiducial 1.5 . fcol . 1.7 range may
be too restrictive.
We point out that the continuum fitting technique is capable of making absolute measurements offcol . The key to making these fcol measurements is to adopt the black hole spin as a known intrinsic (i.e.,unchanging) system parameter, as is done for M , D, and θdisc . With this approach, fcol becomes a freeparameter in the X-ray spectral fitting, rather than being inferred from the model-dependent approach usedin practice. The advantage to this modified version of continuum fitting is its ability to simultaneously
constrain two extrinsic parameters; namely, Ṁ and fcol (instead of a). Figure 4.1 shows there are at least six
viable sources for which the iron line black hole spin could be adopted and the continuum fitting techniqueused to measure fcol . Directly measuring fcol has the exciting potential to probe disc atmospheric physics

128
and variability in an unprecedented way.
To determine the continuum fitting technique’s potential to constrain fcol , we performed the following
0
exercise. For a model black hole X-ray binary with true parameters M 0 = 7.5 M , D0 = 8.5 kpc, θdisc= 45◦ ,
00fcol
= 1.6, and inner disc radius rin
coinciding with the ISCO, we suppose that each of these parameters ismeasured accurately with a 10% relative uncertainty. Using the methodology of §4.4.2, but now with fcol
as the parameter of interest rather than rin , we find that the simplified continuum fitting technique yields
fcol = 1.60 ± 0.09 from marginalizing over only rin and fcol = 1.60 ± 0.15 from marginalizing over M , D,
θdisc , and rin . Whether continuum fitting could achieve these encouraging fcol constraints remains to be
seen. In practice, modeling black hole X-ray binary spectra will always require a hard X-ray component.
The difficultly in measuring where this hard spectral component turns over leads to significant uncertainty inthe flux, which is likely to be a major limitation to using continuum fitting to make absolute measurementsof fcol .

4.6

Summary and Conclusions
We quantified the consequences of neglecting uncertainties in the spectral hardening factor, fcol , onblack hole spin measurements obtained with the continuum fitting technique. Our main conclusions are thefollowing:• When all parameters required by the continuum fitting method adopt the same relative uncertainty,
fcol dominates the error budget on the measured black hole spin (see Figure 4.2).
• Allowing for modest uncertainty in fcol (e.g., at the 10% level) leads to significantly larger black holespin measurement uncertainty (see Figure 4.3).
• For black holes with low-to-moderate intrinsic spin (i.e., a < 0.9), adopting a slightly incorrect fcolvalue (e.g., 10% off from the true value) results in a spin measurement that is inconsistent withthe true spin (see Figures 4.4 and 4.5). Therefore, non-maximal black hole spins derived from thecontinuum fitting technique are only robust if fcol is known with high accuracy.
• In some cases (e.g., XTE J1550–564, LMC X–3, H1743–322, A0620–00), considering a fiducial 10%

129
uncertainty in fcol leads to substantial broadening of the black hole spin PDFs obtained with thecontinuum fitting technique (see Figures 4.6 and 4.7).
• Of the six black hole X-ray binaries with vetted spin measurements from two independent techniques,
the three non-extreme (i.e., a < 0.9) cases are in disagreement at the 1σ level (see Figure 4.1). Thesesources could be brought into agreement with fcol values in the range 1.3 ≤ fcol ≤ 2.0.
• The physically plausible range for fcol may be wider than is typically considered by the continuum
fitting community. We are currently working to determine just how large fcol can become as a resultof strong vertical magnetic pressure support in the accretion disc atmosphere (Salvesen et al., in
preparation).
• Observational studies of disc atmospheric physics could be conducted by making absolute measurements of fcol with the continuum fitting technique. This would require adopting independentmeasurements of black hole spin, as is already done for the other intrinsic system parameters; namely,
black hole mass, source distance, and inner disc inclination.

Acknowledgements
GS thanks Eric Coughlin and Jordan Mirocha for helpful discussions. GS acknowledges support
through the NASA Earth and Space Science Graduate Fellowship program. MCB acknowledges support from
NSF grant AST-1411879. This work used the Janus supercomputer, which is supported by the National
Science Foundation (award number CNS-0821794) and the University of Colorado Boulder. The Janus
supercomputer is a joint effort of the University of Colorado Boulder, the University of Colorado Denver,
and the National Center for Atmospheric Research. Janus is operated by the University of Colorado Boulder.

Chapter 5

Accretion Disc Dynamo Activity in Local Simulations Spanning
Weak-to-Strong Net Vertical Magnetic Flux Regimes

5.1

PrefaceThis paper was published in Monthly Notices of the Royal Astronomical Society, Volume 457, Issue

1, pp. 857-874 on March 21, 2016. The author list is: Salvesen, G., Simon, J. B., Armitage, P. J., &Begelman, M. C. We performed local simulations of accretion discs spanning magnetic pressure-dominatedto gas pressure-dominated regimes. We characterized the relationship between the disc magnetization leveland dynamo activity, showing that the numerical results are well-described by a relatively simple analytic
model.

Abstract
Strongly magnetized accretion discs around black holes have attractive features that may explain
enigmatic aspects of X-ray binary behaviour. The structure and evolution of these discs are governed bya dynamo-like mechanism, which channels part of the accretion power liberated by the magnetorotational
instability (MRI) into an ordered toroidal magnetic field. To study dynamo activity, we performed threedimensional, stratified, isothermal, ideal magnetohydrodynamic shearing box simulations. The strength of
the self-sustained toroidal magnetic field depends on the net vertical magnetic flux, which we vary acrossalmost the entire range over which the MRI is linearly unstable. We quantify disc structure and dynamo
properties as a function of the initial ratio of mid-plane gas pressure to vertical magnetic field pressure,131
β0mid = pgas /pB . For 105 ≥ β0mid ≥ 10 the effective α-viscosity parameter scales as a power-law. Dynamo
activity persists up to and including β0mid = 102 , at which point the entire vertical column of the disc ismagnetic pressure-dominated. Still stronger fields result in a highly inhomogeneous disc structure, with large
density fluctuations. We show that the turbulent steady state β mid in our simulations is well-matched by
the analytic model of Begelman et al. (2015) describing the creation and buoyant escape of toroidal field,
while the vertical structure of the disc can be broadly reproduced using this model. Finally, we discuss theimplications of our results for observed properties of X-ray binaries.

5.2

Introduction
The magnetorotational instability (MRI; Balbus & Hawley, 1991) is a well-understood mechanism forgenerating turbulence and angular momentum transport in accretion discs. In the limit where the disc has
no net magnetic field, the MRI acts as a true dynamo in which the majority of the magnetic energy resides inlarge-scale toroidal magnetic fields. Local magnetohydrodynamic (MHD) simulations with zero net verticalflux show that the rate of angular momentum transport, parametrized by the effective α-viscosity (Shakura& Sunyaev, 1973), is then of the order of α ∼ 0.01 (Davis et al., 2010).
The zero net field limit of the MRI provides an answer to the fundamental question of why discs
accrete, but it may not be the relevant regime for most astrophysical systems. The rate of angular momentum
transport increases with net magnetic flux (Hawley et al., 1995) above a threshold — an initial mid-planeratio of gas to vertical magnetic field pressure β0mid ∼ 105 — that is quite low. At much stronger fieldstrength, β0mid ∼ 102 , this turbulent transport can be substantial with α ∼ 1 (Bai & Stone, 2013). The
importance of magnetic pressure also increases with net flux. In simulations with weak net flux, magnetic
pressure dominates only in a coronal region (Miller & Stone, 2000), whereas fields stronger than β0mid ∼ 103
lead to suprathermal toroidal fields in the disc mid-plane (Bai & Stone, 2013) and qualitative changes to thedisc structure. Finally, disc winds appear to accompany the MRI whenever a dynamically significant net
field is present (Suzuki & Inutsuka, 2009; Fromang et al., 2013; Bai & Stone, 2013).
Except in special circumstances (such as near magnetospheres, or when discs in binaries are threaded
by a secondary star’s magnetic field; Meyer-Hofmeister et al., 1996), determining the net field from first

132
principles is a hard problem. Depending on circumstances, a local net field may be the remnant of that
present when the disc formed (Sikora & Begelman, 2013), the product of dynamo action from zero net field
conditions (Beckwith et al., 2011), or the result of a competition between advection and diffusion processes(Lubow et al., 1994). None of these processes is fully understood, but it is plausible that they could typicallylead to net fields stronger than the very low threshold for changes to α. At a more phenomenologicallevel, disc models that invoke strong fields show promise for modeling observed disc properties that areotherwise inexplicable. The structure of magnetically dominated discs is stable against thermal/viscous
instability (Begelman & Pringle, 2007), and less susceptible to gravitational fragmentation (Pariev et al.,
2003; Begelman & Pringle, 2007; Gaburov et al., 2012). Moreover, since the evolution of the net field occurs
on a time scale that is intermediate between the dynamical and viscous time scales, it has the potential toact as a slowly varying control parameter in models of X-ray binary state transitions (Begelman & Armitage,2014).
Our goal in this study is to quantify the properties of MRI disc turbulence across almost the entire
range of net fluxes that admit linearly unstable MRI modes. Following a long line of prior work, we adopt a
local approximation and consider vertically stratified isothermal discs for which the net flux is a conserved
quantity. We adopt the basic computational approach of Bai & Stone (2013), who performed shearing box
simulations with a net vertical magnetic flux that had a ratio of gas to magnetic pressure, β ≡ pgas /pB ,whose initial value at the disc mid-plane ranged from β0mid = 102 − 104 . We extend their work to stronger
fields (β0mid = 10, previously considered by Lesur et al., 2013), and test the sensitivity of the results tothe size of the simulation domain, which affects the variability properties of the zero net flux MRI (Simon
et al., 2012). More important than these technical differences, we focus our analysis on the properties of the“MRI-dynamo” — the periodic reversals of the large-scale toroidal magnetic field that are characteristic of
the weak-field MRI in both the local (Brandenburg et al., 1995; Shi et al., 2010; Davis et al., 2010; Simon
et al., 2011, 2012) and global limits (O’Neill et al., 2011; Beckwith et al., 2011). If the MRI-dynamo persists
at high magnetizations, the Poynting flux associated with the periodic expulsion of magnetic field from thedisc could play a dominant role in the energetics of the disc atmosphere (Begelman et al., 2015). Whether
this occurs is unclear. Johansen & Levin (2008), using short duration simulations, found apparently stable

133
highly magnetized disc structures, while Bai & Stone (2013) found that the MRI-dynamo petered out once
the net field reached β0mid = 102 . Perhaps as a consequence of our larger domain size, we find instead that
MRI-dynamo activity can be present even when the toroidal magnetic field is suprathermal throughout thedisc. We also find that the strongest net fields result in the formation of a highly inhomogeneous disc, andwe show that the structure of the simulated discs is well approximated by a simple analytic model of buoyant
toroidal field escape.
The outline of the paper is as follows. After describing our simulations (§6.3), we characterize theproperties of MRI turbulence (§5.4) and the MRI-dynamo (§5.5) as a function of net vertical magnetic flux.
We discuss our results in the context of X-ray binary phenomenology and strongly magnetized accretion
discs (§5.6), followed by a summary and conclusions (§5.7).

5.3

Numerical Simulations
We use the Athena code to solve the equations of ideal MHD in the shearing box approximation. We

refer the reader to Gardiner & Stone (2005, 2008) for descriptions of the Athena algorithms and to Stoneet al. (2008) for descriptions of their implementation.

5.3.1

Shearing Box Simulations
The shearing box (Goldreich & Lynden-Bell, 1965) models a relatively small patch of a differentiallyrotating fluid by expanding the equations of motion in a locally co-rotating frame. To apply the shearing box
geometry to an accretion disc (Hawley et al., 1995; Brandenburg et al., 1995), we go from a cylindrical frame(R, φ, z) into a local Cartesian frame (x, y, z) using the coordinate transformations: x = R − R0 , y = R0 φ,z = z, where R0 is the reference radius corresponding to the center of the shearing box and co-rotating withthe disc at angular frequency Ω = Ωk̂. For a shearing box with vertical density stratification, the equationsof compressible, isothermal, ideal MHD in conservative form with unit vectors î, ĵ, k̂ are (Hawley et al., 1995;

134
Stone et al., 1996),∂ρ= − ∇ · (ρv)
∂t


 
∂(ρv)
B2
= − ∇ · ρvv − BB + pgas +I∂t2+ 2qρΩ2 xî − ρΩ2 z k̂ − 2Ω × (ρv)

(5.1)

(5.2)

∂B
= − ∇ · (vB − Bv) .
∂t

(5.3)

Here ρ is the gas density, pgas is the gas pressure, v is the velocity, ρv is the momentum density, and B
is the magnetic field, which in our convention absorbs a factor of µ/√
4π , where µ = 1 is the adopted

magnetic permeability. I is the identity matrix that acts on the total pressure pgas + B 2 /2. We adopt an
isothermal equation of state, pgas = ρc2s , where cs is the sound speed. The shear parameter is defined as,
q = −dln (Ω) /dln (R), and we choose the value q = 3/2 corresponding to a Keplerian accretion disc.
There are various options for numerically integrating Equations 6.1-6.3 within Athena. We use theshearing box implementation of Stone & Gardiner (2010), with an orbital advection scheme to separately
evolve the background shear flow, vsh = −qΩxĵ, and the velocity fluctuations, v0 = v − vsh . Using orbital
advection gives a substantial improvement in computational performance and solution accuracy. We use
the Harten-Lax-van Leer discontinuities (HLLD) Riemann solver (Miyoshi & Kusano, 2005; Mignone, 2007)(see, e.g., Salvesen et al. (2014) for a comparison of solvers in Athena). Radial (x), toroidal (y), and vertical(z) boundary conditions are shearing periodic, strictly periodic, and outflowing, respectively (Hawley et al.,
1995; Simon et al., 2011).
The shearing box has well-known limitations, including the neglect of curvature terms and radial
gradients, and symmetry between positive and negative x that leaves the location of the central object
undefined. Two limitations are of particular concern for our application. First, the shearing box is not
an ideal setup for representing disc winds, which are present in all net flux MRI simulations. Although
the shearing box effective potential is locally the same as for a global (Blandford & Payne, 1982) wind,the derived solutions can have an unphysical geometry and additionally depend in detail on the verticalboundary conditions (Bai & Stone, 2013; Fromang et al., 2013). Winds are not the focus of this paper, butthey are present and we cannot readily quantify the extent to which uncertainties in the strength of disc

135
winds propagate to other aspects of the solution. Second is the issue of domain size. The effective turbulent
α measured in local MRI simulations converges at modest domain size, but other properties of interest (suchas the level of variability) continue to vary even in spatially extended shearing boxes (Simon et al., 2012).
There is no way to internally assess whether an unconverged solution in a small box is better or worse thana converged solution in a box so large that neglect of curvature terms (for some assumed disc thickness) isformally unjustified. Ultimately, global simulations are needed to avoid these limitations that are inherent
to the shearing box.

5.3.2

Initial Setup and Parameters

(Nx , Ny , Nz )[Nzones ](480, 960, 480)
(360, 720, 360)
(360, 720, 360)
(240, 480, 240)
(240, 480, 240)
(180, 360, 180)

Grid Res.
[Nzones /H]
48
36
36
24
24
18

MRI Res.
[λfg /∆xi ]
15
1.0
3.3
7.0
22
52
413
105
104
103
102
101

β0mid
0
4.47e-3
1.41e-2
4.47e-2
1.41e-1
4.47e-1Bz
18.1(8)17.3(6)29(1)6(2)e1
1.2(2)e2
1.1(1)e2

hQmid
x it
47(2)
46(2)
70(3)
1.4(3)e2
3.3(7)e2
2.4(4)e2

hQmid
y it

11.2(5)10.9(3)19.7(8)4(1)e19(1)e11.2(1)e2

hQmid
z it

Initial Field
Configuration
Flux Tube
Bz +Sinusoid
Bz +Sinusoid
Bz +Sinusoid
Bz +Sinusoid
Bz +Sinusoid

Table 5.1: Basic information for the suite of shearing box simulations. From left to right the columns are: simulation identification label, total numberof grid zones in each dimension, grid resolution (applies to all dimensions), number of grid zones spanning the fastest growing MRI wavelength (applies
to all dimensions), initial ratio of gas pressure-to-magnetic pressure at the disc mid-plane, net vertical magnetic flux density (code units), saturated
state time-averaged Qx , Qy , Qz (see Equation 5.9) at the disc mid-plane, and the initial magnetic field configuration. Parentheses indicate the ±1σ
range on the last digit. The domain size for all simulations is (Lx , Ly , Lz ) = (10H0 , 20H0 , 10H0 ) and all simulations begin at time t = 0 and terminate
at time tf = 225 orbits, while the time-averaging is done over the domain [ti , tf ] = [25, 225] orbits.

ZNVF
NVF-β5
NVF-β4
NVF-β3
NVF-β2
NVF-β1

ID

136

137
Our initial setup is based on Stone et al. (1996) and closely follows that of Simon et al. (2012).
We initialize all simulations with a vertically stratified density profile according to isothermal hydrostatic
equilibrium,ρ0 (x, y, z) = ρmid
0 exp



−z 2
2H02


,

(5.4)

where ρmidis the initial (i.e., t = 0) gas density at the disc mid-plane (i.e., z = 0) and H0 is the initial gas
0density scale height of the disc1 ,
H0 =

cs
.
Ω

(5.5)

In code units the initial parameter choices are Ω = 1, ρmid
= 1, and cs = 1, which correspond to H0 = 1 and0initial disc mid-plane gas pressure pmid
gas,0 = 1.
We run shearing box simulations with and without an imposed net vertical magnetic flux. The
magnetization of the gas is parametrized by the ratio of gas pressure-to-magnetic pressure, defined as,
β=

pgas
ρc2
= 2s ,
pB
B /2

(5.6)

with β0 being the initial plasma-β parameter and β0mid being its value at the disc mid-plane. Our net
vertical magnetic flux simulations adopt an initial magnetic field configuration, B0 , consisting of a uniform
vertical field with an additional sinusoidal component (added to improve numerical stability during the initialtransient growth of the MRI),
Bx,0 (x, y, z) = 0
By,0 (x, y, z) = 0



1
2πx
Bz,0 (x, y, z) = B0 1 + sin
.
2
Lx

(5.7)

Because of the spatial variation of Bz,0 in the initial conditions, slightly different definitions of the relationq
between B0 and β0mid are possible. We adopt B0 = 2pgas,0 /β0mid . For these simulations initialized with a
net vertical magnetic flux, β0 refers only to the vertical component.
The only physical parameter we vary in our suite of net vertical magnetic flux simulations is β0mid (seeTable 5.1). We choose initial net vertical magnetic flux — NVF prefix in the simulation naming convention —1 We note that Simon et al. (2012) define H =0theirs.

√2cs /Ω; therefore, one must keep this in mind if comparing our results to138values corresponding to very weak (NVF-β5; β0mid = 105 ), weak (NVF-β4; β0mid = 104 ), moderate (NVF-β3;
β0mid = 103 ), strong (NVF-β2; β0mid = 102 ), and very strong (NVF-β1; β0mid = 101 ) magnetization levels.
For comparison we also run a zero net vertical magnetic flux simulation (ZNVF; see Table 5.1), which
adopts an identical initial magnetic field configuration to that of Simon et al. (2012), based on the twisted
toroidal flux tube setup of Hirose et al. (2006). As was done in Simon et al. (2012), the initial toroidal andpoloidal plasma-β parameters for this simulation are βy,0 = 100 and βp,0 = 1600, respectively, corresponding
to β0mid = 413.
In order to seed the MRI, we populate the grid at time t = 0 with perturbations — drawn randomly
from a uniform distribution with zero mean value — to the gas density and each velocity component. The
maximum density and velocity perturbation amplitudes are |δρ|/ρ = 0.01 and |δvi | = (0.01/5) cs , respectively(e.g., Hawley et al., 1995), where i = (x, y, z) denotes the spatial dimension.
To prevent time-steps from becoming too small as a result of large Alfvén speeds in regions of very
low gas density, we enforce a gas density floor of ρfloor = 10−4 ρmid
0 . For all simulations, the vertical profilesof horizontally-averaged gas density remain well above this lower limit for all heights above/below the disc
mid-plane.
For all simulations, we choose a domain size of (Lx , Ly , Lz ) = (10H0 , 20H0 , 10H0 ). This choice
is motivated by the vertically stratified shearing box simulations of Simon et al. (2012), which provide
compelling evidence for non-local, “mesoscale” structures (i.e., on scales  H) contributing to angularmomentum transport as the domain size increases (see also Guan & Gammie, 2011). It also allows us to
explore the sensitivity of the results to changes in domain size by comparison (where our parameters overlap)
with the results of Bai & Stone (2013), who used very similar computational methods. As we have alreadyremarked, the neglect of curvature terms in large local domains, such as ours, is only formally justified
when modeling very thin discs (with H/R ∼ 0.01). Explicit comparisons of dynamo behavior in mesoscaleand global simulations (Beckwith et al., 2011), however, show that this aspect of the dynamics remains
well-modeled locally even when the formal criterion for validity is only marginally satisfied. Simulations
NVF-β5 and NVF-β4 have a spatial resolution of 36 grid zones/H0 in each dimension, or (Nx , Ny , Nz ) =(360, 720, 360). Simulations NVF-β3 and NVF-β2 have a spatial resolution of 24 grid zones/H0 in each

139
dimension, or (Nx , Ny , Nz ) = (240, 480, 240). Simulation NVF-β1 has a spatial resolution of 18 grid zones/H0
in each dimension, or (Nx , Ny , Nz ) = (180, 360, 180). The zero net magnetic flux simulation ZNVF has aspatial resolution of 48 grid zones/H0 in each dimension, or (Nx , Ny , Nz ) = (480, 960, 480). Table 5.1
summarizes the suite of shearing box simulations considered in this work.
Ideally, we would like to resolve both the most unstable linear MRI modes in the initial conditions(which can be assessed analytically), and the non-linear properties of turbulence in the saturated state (which
can only be assessed empirically). This aspiration is very hard to achieve across the range of net fluxes weconsider, because the most unstable linear modes are of much smaller scale in the weak field simulations2 .
Considering first the linear criterion, the fastest growing wavelength for the MRI for an unstratified isothermal
Keplerian accretion disc is (Hawley et al., 1995),
8π vA,0λfg = √
,
15 Ω

(5.8)

√
where vA,0 = B0 / ρ0 is the initial Alfvén speed. Table 5.1 lists the number of grid zones resolving λfg at thedisc mid-plane for each of our simulations. Clearly our two weakest net field runs (NVF-β5 and NVF-β4)
fail to properly resolve this mode (the run with zero net vertical flux is initialized with a moderately strong
toroidal field, whose instability is easier to resolve). We can also consider the resolution needed to reproduce
the fastest growing modes in stratified discs. Latter et al. (2010) performed one-dimensional numericalsimulations to compute MRI growth rates in this case and suggest that at least 25, 50, 200 grid zones/H are
needed for simulations with β0mid = 102 , 103 , 104 , respectively, in order to resolve the fastest growing linear
MRI modes that develop into channel flows. We meet this requirement for NVF-β2, but not for the weaker
net field cases.
The above discussion indicates that the linear growth of the MRI in the two weakest net field runs
cannot be reliably captured by our simulations. Our analyses exclude this phase. All of the runs, however,are unstable and develop into a turbulent saturated state in which the magnetic field is substantially stronger(and hence easier to resolve). Empirically, the resolution of the turbulent steady state can be parametrized2

In principle, the weak field simulations could be run in smaller boxes, with higher spatial resolution, but this would
introduce different but equally problematic issues of convergence with domain size.

140by the quality factor (Sano et al., 2004),
Qi =λMRI,i,∆xi

where λMRI,i = 2πvA,i /Ω is the characteristic MRI wavelength along spatial dimension i, vA,i =

(5.9)
p

Bi2 /ρ is

the Alfvén speed in direction i, and ∆xi is the size of a grid zone in direction i. Sano et al. (2004) suggest aturbulent steady state time-averaged quality factor hQi it & 6, while Hawley et al. (2011) recommend a more
stringent hQy it & 20 and hQz it & 10. These criteria are good rules of thumb, though clearly the specifichQi it values needed to properly resolve turbulence are code dependent. Table 5.1 lists the horizontally- andtime-averaged quality factors evaluated at the disc mid-plane (i.e., where Qi is minimized). For all of oursimulations, the turbulent steady state is well-resolved by the standards just outlined.
Due to the net vertical magnetic flux we impose, significant mass outflows develop that would deplete
the entire disc on timescales shorter than the 200 orbits that we study in the turbulent steady state. Therefore, after every time step, we maintain a constant mass in the domain by multiplying the gas density inevery grid zone by an appropriate common factor (Bai & Stone, 2013).

5.3.3

Notation Convention
In the analyses described in subsequent sections, angled brackets surrounding a quantity denote an

average. The subscript on the brackets indicates the dimension being averaged over, where hGit , hGixy ,and hGiV indicate a time average, horizontal (i.e., disc-plane) average, and volume average of the quantity
G (x, y, z, t), respectively, given by,1
hGit =
tf − ti

Z tf
Gdt(5.10)ti

Z + 21 Ly Z + 12 Lx
1hGixy =Gdxdy
Lx Ly − 12 Ly − 12 Lx
Z + 21 Lz
1hGiV =
hGixy dz.
Lz − 12 Lz(5.11)(5.12)

Time averages span 200 orbits over the range [ti , tf ] = [25, 225], where we measure time units in orbits at thecenter of the shearing box. Horizontal averages encompass the domain (Lx , Ly ) = (10H0 , 20H0 ). Volume
averages encompass the domain (Lx , Ly , Lz ) = (10H0 , 20H0 , 10H0 ).

141
Nested angled brackets denote averages over multiple dimensions, where hhGiV it means that thequantity G was volume-averaged and then time-averaged. A quantity with the superscript Gmid means that
G was evaluated at the disc mid-plane. A quantity with the subscript G0 means that G was evaluated at
t = 0. Quoted uncertainty ranges or error bands shown in figures denote the ±1σ range corresponding only
to the dimension indicated by the subscript on the outermost angled brackets.
Finally, we introduce the notations of a bar (−) and a hat (∧) over a parameter that is defined as theratio of two quantities, such as β ≡ pgas /pB (Equation 6.6) and α ≡ Txy /pgas (Equation 5.20). Using β as
an example, the bar and hat have the following meanings,
hpgas iV
hpB iV(5.13)

hpgas ixy.
βb ≡hpB ixy(5.14)

β≡5.4

Basic Properties of MRI Turbulence
A standard diagnostic of MRI turbulence is the rate of angular momentum transport. For a Keple-

rian accretion disc, this is parametrized by the non-dimensional effective α-viscosity parameter (Shakura &Sunyaev, 1973),
Txy
pgas

(5.15)

αRey =

Txy,Reypgas(5.16)

αMax =

Txy,Max,
pgas(5.17)

α=where Txy = Txy,Rey + Txy,Max is the xy (i.e., rφ) component of the total stress tensor, with Reynolds stressand Maxwell stress components,
Txy,Rey = ρvx vy0(5.18)

Txy,Max = −Bx By .

(5.19)For each simulation, Table 5.2 provides values for βb and αb, along with their individual components, allevaluated at the disc mid-plane and time-averaged.

142

Figure 5.1: Time evolution of the effective viscosity parameter α = hTxy iV /hpgas iV for each simulation:
ZNVF (purple lines), NVF-β5 (blue lines), NVF-β4 (green lines), NVF-β3 (orange lines), NVF-β2 (redlines), and NVF-β1 (black lines). The vertical dashed line at time t = 25 orbits marks the start of all time
averaging in this work. The effective viscosity parameter is relatively small for the zero net vertical magnetic
flux simulation ZNVF, but increases dramatically with increasing net vertical magnetic flux, even exceeding
unity for simulations NVF-β2 and NVF-β1.

143

Figure 5.2: Time-averaged effective α-viscosity parameter as a function of the initial disc mid-plane ratio ofgas pressure-to-magnetic pressure for simulations with a net vertical magnetic flux. The angular momentum
transport rate hαit is set by β0mid , following a power-law relation with the net vertical magnetic flux. Rather
than normalizing the total stress by the gas pressure, α = hTxy iV /hpgas iV (circles), if we instead normalizeby the magnetic pressure, αmag = hTxy iV /hpB iV (squares), then αmag becomes essentially independent of
β0mid .

144

Figure 5.3: Volume renderings of the ratio of gas pressure-to-magnetic pressure, β ≡ pgas /pB , at time t = 125
orbits. From top to bottom, the left column shows simulations ZNVF, NVF-β5, NVF-β4 and the right column
shows simulations NVF-β3, NVF-β2, NVF-β1. Red, white, and blue colours denote regions with β < 1, β ' 1,and β > 1, respectively, with the colours being logarithmically rendered over the range log10 (β) = [−2, 2].
Larger scale turbulent structures develop with decreasing β0mid . The full simulation domain is magnetic
pressure-dominated for NVF-β2 and NVF-β1.

145

Figure 5.4: Vertical profiles of the time-averaged βb (Equation 5.14) for each simulation (see Figure 5.1 forline colour conventions). Coloured bands show the ±1σ range about the average for the dimension indicatedby the outermost angled bracket (time in this case). The horizontal dashed line marks equipartition between
gas pressure and magnetic pressure. A magnetic pressure-dominated corona forms for all simulations. The
location of the transition from gas to magnetic pressure-domination (i.e., β = 1) moves toward the disc
mid-plane with increasing net vertical magnetic flux, with the entire domain becoming magnetic pressuredominated (i.e., β < 1) for simulations NVF-β2 and NVF-β1.

146

Figure 5.5: Vertical profiles of the horizontally- and time-averaged gas density fluctuations relative to thehorizontally- and time-averaged gas density weighted by mass (see Figure 5.1 for line colour conventions).
For the zero net vertical magnetic flux simulation ZNVF, the normalized gas density r.m.s. fluctuations areat the ' 10% (∼ 20 − 50%) level in the disc (corona) regions. These gas density fluctuations increase withthe amount of net vertical magnetic flux, with large variations about the ' 65% (∼ 75 − 100%) level in thedisc (corona) regions for the most strongly magnetized simulations NVF-β2 and NVF-β1.

147
Figure 5.1 shows the time evolution of α for each simulation. The zero net vertical magnetic flux
simulation ZNVF shows a relatively small hαit = 0.015 ± 0.003 with little temporal variability, consistent
with the results from Davis et al. (2010) and Simon et al. (2012). When a net vertical magnetic flux is
introduced, the rate of angular momentum transport becomes both enhanced and more variable, achieving
hαit = 2.3 ± 0.5 for the simulation with the strongest net vertical magnetic flux NVF-β1.
Treating the initial value of β at the disc mid-plane as a control parameter, Figure 5.2 shows how
α scales with β0mid . We find that the scaling is well-fit by a single power-law, which closely matches the
expected slope of −1/2. Our best fit has the form,
hαit = 1.1 × 101 β0mid

−0.53.

(5.20)

This relation holds over four orders of magnitude in β0mid (two orders of magnitude in Bz ), and covers almost
the entire range of net fluxes for which the flux (a) boosts the efficiency of transport as compared to zero-net
field simulations and (b) allows linearly unstable MRI modes. Alternatively, we can choose to normalize thestress by the magnetic pressure, αmag = hTxy iV /hpB iV . With this definition, the effective viscosity becomes
essentially independent of β0mid ,hαmag it = 0.30 β0mid

−0.040.

(5.21)

These results are consistent with previous works that find power-law and nearly constant scalings with β for
α and αmag (Hawley et al., 1995; Blackman et al., 2008). Finally, we can combine Equations 5.20 and 5.21
to assess how the disc magnetization (which in the saturated state is dominated by the toroidal component)
scales with the imposed net flux. We find,hβit ∼ 0.03 β0mid

1/2

,

(5.22)

which implies that the net flux threshold for the disc to become magnetically dominated (in a volumeaveraged sense) is at approximately β0mid ∼ 103 . As a caveat, we mention that results derived from volume
averages of α and β will be somewhat dependent on the vertical domain size. For instance, results would beweighted more to the corona for a more vertically extended box.
Figure 5.3 shows volume renderings of β at t = 125 orbits for each simulation. The gas is turbulent,with blue and red indicating gas pressure-dominated (β > 1) and magnetic pressure-dominated (β < 1)

148

Figure 5.6: Normalized probability density function of gas density relative to its mean value, evaluated at
z/H0 = 0 (left) and z/H0 = 2.5 (right) for each simulation (see Figure 5.1 for line colour conventions). Simulations ZNVF, NVF-β5, and NVF-β4 have highly homogeneous density structures at the disc mid-plane
region, with modest inhomogeneity developing in the coronal regions. The strongly magnetized simulations NVF-β3 and NVF-β2 are inhomogeneous throughout the domain, while the very strongly magnetized
simulation NVF-β1 is highly inhomogeneous.

149
regions, respectively, while white shows regions near equipartition (β ' 1). As the amount of net verticalmagnetic flux increases, the simulation domain becomes more magnetic pressure-dominated and the turbulence develops larger-scale structure. Figure 6.1 shows the corresponding vertical profile of the horizontallyand time-averaged plasma-β. All of the simulations show the formation of a low-density magnetic pressureb t = 1 transition occurs moves
dominated corona (Miller & Stone, 2000). The vertical location where the hβi
closer to the disc mid-plane with increasing net vertical magnetic flux. The entire domain, including the
mid-plane, becomes magnetic pressure-dominated for simulations NVF-β2 and NVF-β1.
Our two strongest net field runs become fully magnetically dominated, reducing the mid-plane densityand pressure in the saturated state as compared to the initial conditions. As a result, while the average
vertical magnetic flux is a conserved quantity, the mid-plane value of β associated with the vertical field
is not. The reduction in β0mid due to the onset of magnetic pressure support means that it is conceivablethat an initially unstable disc, with a vertical flux close to the linear stability threshold, could evolve into amagnetically dominated state where the vertical field was no longer unstable. We see no evidence for such aneffect, though a simple estimate suggests that our strongest net field simulation does access a regime where thetension of the vertical field is dynamically highly important on all scales. Nominally, magnetic tension from2
& 6c2s /π 2 (Balbus & Hawley, 1998), where vA,za strong vertical magnetic field will suppress MRI when vA,z√is the vertical component of the Alfvén speed, vA = B/ ρ. In the context of our simulations this stability
criterion can be written as βz . 3.3. Simulation NVF-β1, with hβzmid it = 1.8 ± 0.3, is sufficiently magnetized
as to approach this limit, but only in the saturated state whose structure is dramatically different from theequilibrium used to compute the linear stability threshold. To quantify this, we measure the variance of thehorizontally-averaged gas density at every snapshot in time for each simulation, weighted by mass in orderto account for how mass is distributed across regions of a given density,
2
hσm
ixy =



ρ3
hρixy




−
xy

ρ2
hρixy

2
.

(5.23)

xy

Figure 5.5 shows the vertical profile of the horizontally- and time-averaged mass-weighted gas density fluctuations, δρm =p

2 , normalized to the horizontally- and time-averaged mass-weighted gas density vertical
σmprofile. For the zero-net field3 and weak field runs, the r.m.s. density fluctuations near the mid-plane are
3 The boundary features seen in Figures 6.1 and 5.5 for simulation ZNVF are a consequence of a non-negligible fraction of

150
ID
ZNVF
NVF-β5
NVF-β4
NVF-β3
NVF-β2
NVF-β1hβbmid it
68(6)
41(3)
17(1)
2.1(9)
0.4(1)
0.31(8)hβbymid it
81(7)
49(3)
21(2)
3(1)
0.5(2)
0.5(1)

hβbpmid it
1.6(1)e3
9.3(6)e2
2.9(4)e23(1)e17(2)2.8(5)

hb
αmid it
0.0088(7)0.014(1)0.035(3)0.3(1)
1.0(4)
1.1(3)

mid
hb
αRey
it
0.0020(2)0.0034(3)0.008(1)0.06(2)0.16(8)0.12(5)

mid
hb
αMax
it
0.0067(5)0.0111(8)0.027(2)0.22(9)0.9(3)
1.0(3)

Table 5.2: Fundamental properties of the MRI turbulence, all evaluated at the disc mid-plane and timeaveraged. From left to right the columns are: simulation identification label, plasma-βb parameter, toroidalb poloidal component of β,b effective α
component of β,b-viscosity parameter, Reynolds component of αb, andMaxwell component of αb. Parentheses indicate the ±1σ range on the last digit.

at the 10% level, consistent with the usual expectation that compressibility is ignorable for mid-plane MRI
dynamics. (Note that even in these simulations, substantially stronger density fluctuations occur in thecorona.) For NVF-β2 and NVF-β1, conversely, the density fluctuation amplitude is extremely strong, withδρ/ρ ≈ 0.6 − 1.
To further investigate the inhomogeneous nature of strongly magnetized discs, we compute the normalized probability density function (PDF) of the gas density relative to its mean value at a given heightin the disc. Figure 5.6 shows these PDFs at the disc mid-plane (z/H0 = 0; left panel) and in the corona(z/H0 = 2.5; right panel). In other words, Figure 5.6 shows what fraction of volume at a given height isoccupied by gas with density ρ/hρixy . For the zero-net field and weak field runs, the gas density PDFs are
narrowly concentrated about the mean value, which indicates a relatively homogeneous density structure.
However, the moderate and strong field runs have an inhomogeneous density structure, as evidenced by thepeak in the gas density PDF shifting to lower ρ/hρixy values and the distribution broadening. Interestingly, a
two-component structure develops for the strongest net field simulation NVF-β1. As expected for a disc withα ∼ 1, NVF-β1 has become highly inhomogeneous, such that the relevance of the linear stability analysis forthe vertical field is limited. Indeed, we cannot exclude the possibility that clumpy MRI-unstable discs witheven stronger net fields might be possible, at least in situations where the net field evolves slowly towardhigher values.
grid zones reaching the imposed gas density floor.

151
ID
ZNVF
NVF-β5
NVF-β4
NVF-β3
NVF-β2
NVF-β1

Pdyn[orbits]
12(3)
10(2)
8(2)
15(4)
8(5)e1−ξhhηiV it

mid
hαB
it

mid
hβB15
it

0.5(1)
0.6(1)
0.7(2)
0.4(1)
0.08(5)−0.029(4)0.043(3)0.068(6)0.12(2)0.21(3)0.30(5)0.016(2)0.029(3)0.09(1)0.3(1)
0.3(2)
0.4+0.2−0.4

7(2)e14(1)e11.7(4)e1
2(1)
0.7+1
−0.7
0.6+1
−0.6

Table 5.3: Fundamental properties of the MRI-dynamo. From left to right the columns are: simulation
identification label, period of MRI-dynamo cycles (Pdyn ), ξ parameter (see Equation 5.24), volume- andtime-averaged efficiency parameter for the rise of toroidal magnetic flux (η), disc mid-plane evaluated andtime-averaged parameter for the toroidal magnetic flux production (αB ), and disc mid-plane evaluated andmidtime-averaged plasma-β predicted from Equation 5.31 (βB15). All of these parameters are defined in the
text. Parentheses indicate the ±1σ range on the last digit.

5.5

Properties of the MRI-Dynamo
Having established the basic properties of MRI turbulence in our simulations, this section quantifies

the properties of the MRI-dynamo.

5.5.1

Toroidal Magnetic Field Reversals
For each simulation, Figure 5.7 shows the space-time diagram of the horizontally-averaged toroidal

magnetic field, hBy ixy . These so-called “butterfly diagrams” exhibit quasi-periodic reversals of hBy ixy , which
is a hallmark of accretion disc MRI-dynamo activity (e.g., Brandenburg et al., 1995). The amount of net
vertical magnetic flux increases from top to bottom in Figure 5.7 and black contours show the transitionalvertical location where βb = 1. Reversals in the large-scale toroidal magnetic field showcase regular periodicity
for the zero net vertical magnetic flux simulation and simulations with β0mid & 104 . This is a well-known
result for stratified shearing boxes (e.g., Simon et al., 2012) and global disc simulations (e.g., O’Neill et al.,2011) with zero net vertical magnetic flux. However, previous shearing box simulations with a net verticalmagnetic flux and β0mid & 104 (Fromang et al., 2013; Bai & Stone, 2013) show highly irregular dynamo
patterns.
In the moderately magnetized regime with β0mid = 103 , Figure 5.7 shows that the toroidal magnetic
field reversals become less frequent and more sporadic, with temporary episodes where the entire disc domain
becomes magnetic pressure-dominated. This behaviour was also observed in the β0mid = 103 shearing box

152

Figure 5.7: Time evolution of the horizontally-averaged toroidal magnetic field, hBy ixy , with height from thedisc mid-plane, z/H0 . From top to bottom are space-time diagrams for simulations ZNVF, NVF-β5, NVFβ4, NVF-β3, NVF-β2, NVF-β1. MRI-dynamo behaviour occurs in all but the most strongly magnetized
simulation NVF-β1. The period of the MRI-dynamo cycle generally becomes longer and reversals become
more sporadic with increasing net vertical magnetic flux. Black lines show the βb = 1 contour. The entire
disc becomes magnetic pressure-dominated for brief episodes in simulation NVF-β3 and for all times insimulations NVF-β2 and NVF-β1.

153

Figure 5.8: Butterfly diagrams for simulations NVF-β4 (left) and NVF-β2 (right), zoomed in over 10 orbits
to highlight the hBy ixy structure during a reversal. Black lines show the βb = 1 contour. The current sheet
sweeping through the vertical domain during a reversal in hBy ixy propagates faster in strongly magnetizeddiscs (right) than in weakly magnetized discs (left).

Figure 5.9: Magnetic field structure during an MRI-dynamo reversal in the large-scale toroidal magnetic field(see left panel of Figure 5.8) for the weak net vertical magnetic flux simulation NVF-β4. The top panels show
colour renderings of the toroidal magnetic field and poloidal magnetic field components for a slice throughthe xz-plane at y = 0. The bottom panels show colour renderings of the vertical magnetic field and horizontal
magnetic field components for a slice through the xy-plane at z = 0. Panels from left to right show times
t = 113, 114, 115, 116 orbits. By is fairly turbulent, but relatively well-organized, near the equator and ismore organized above/below the disc mid-plane.

154

Figure 5.10: Magnetic field structure during an MRI-dynamo reversal in the large-scale toroidal magnetic
field (see right panel of Figure 5.8) for the strong net vertical magnetic flux simulation NVF-β2. The top
panels show colour renderings of the toroidal magnetic field and poloidal magnetic field components for aslice through the xz-plane at y = 0. The bottom panels show colour renderings of the vertical magnetic field
and horizontal magnetic field components for a slice through the xy-plane at z = 0. Panels from left to rightshow times t = 114, 115, 116, 117 orbits. Organized By structures dominate throughout the domain and arehighly organized on the equator.

155

Figure 5.11: Time evolution of the volume-averaged toroidal magnetic field (left panels) and its power
spectrum (right panels). From top to bottom are simulations ZNVF, NVF-β5, NVF-β4, NVF-β3, NVFβ2, NVF-β1. For simulations with a zero or weak (β0mid . 104 ) net vertical magnetic flux, a coherent peak
appears in the power spectrum resulting from a regular MRI-dynamo period of Pdyn ' 10 orbits. This feature
is substantially broadened for the moderately magnetized simulation NVF-β3 due to the MRI-dynamo flips
becoming more sporadic and having a longer period. The strongly magnetized simulation NVF-β2 shows
no discernible power spectral peak. The very strongly magnetized simulation NVF-β1 does not exhibit
MRI-dynamo cycles and has a flat power spectrum with a high-frequency cutoff at ' 1 orbit−1 .

156
simulations of Bai & Stone (2013). In the β0mid = 102 strong net vertical magnetic flux simulation, the entire
disc domain achieves βb < 1 for all times and the toroidal field reversals become still less frequent and more
sporadic. However, they still occur. To our knowledge this is the first time that the MRI-dynamo butterfly
pattern has been observed to persist in a magnetic pressure-dominated disc. The toroidal magnetic field
flip-flops disappear for the very strongly magnetized β0mid = 101 simulation, though it is possible that thesimulation was not evolved to an adequately long time and/or a larger domain size is necessary to observe
this behaviour.
Figure 5.7 also shows that the MRI-dynamo generates an ordered toroidal magnetic field near the disc
mid-plane, which buoyantly rises and reaches a maximal amplification near the height where βb ' 1. The
colour bars in Figure 5.7 show that the peak strength that the toroidal magnetic field reaches is similar forthe zero net vertical magnetic flux simulation ZNVF and the very weak net vertical magnetic flux simulation
NVF-β5. This peak toroidal field strength roughly doubles with each order of magnitude decrease in β0mid
until the β0mid = 101 simulation, where the toroidal field strength appears to saturate to the same level asin the β0mid = 102 simulation. This is suggestive of a threshold value to the toroidal magnetic field that the
MRI-dynamo can produce from an imposed net vertical magnetic flux.
The period of MRI-dynamo cycles can be measured as Pdyn = ∆t/Ncycles . Over the time interval∆t = tf − ti = 200 orbits, we measure the number of MRI-dynamo cycles, Ncycles = Nflips /2, by counting
the number of hBy iV reversals separately in the top (z > 0) and bottom (z < 0) simulation domains and


top
bot
averaging them together to get Nflips = Nflips+ Nflips/2. We measure Pdyn and estimate its uncertainty
as Pdyn /pNcycles . The MRI-dynamo period (in units of orbits) can also be parametrized as,
Pdyn =

2π[orbits] ,ξΩ(5.24)

and we list the results for Pdyn and ξ in Table 5.3. The ξ parameter will be important in §5.5.2 and §5.5.3.
Figure 5.8 shows zoomed-in subsets of Figure 5.7 for simulations NVF-β4 (left) and NVF-β2 (right)
to highlight the space-time structure of toroidal magnetic field reversals for weak and strong net verticalmagnetic flux cases, respectively. Notably, the rise speed of the current sheets launched during each reversalof the toroidal magnetic field increases with net vertical magnetic flux.

157
Figures 5.9 and 5.10 show the detailed magnetic field structure for simulations NVF-β4 and NVF-β2,
respectively, over the course of the hBy ixy reversals highlighted in Figure 5.8. Comparing Figures 5.9 and5.10 demonstrates that the MRI-dynamo behaves differently in the weak and strong net vertical flux cases.
For the weakly magnetized NVF-β4, the azimuthal slice (xz-plane) of Figure 5.9 shows small-scale,turbulent magnetic field structure concentrated to the disc mid-plane regions, with By becoming moreorganized at heights where β . 1. While there is a mildly dominant By polarity in the mid-plane regions at
any given time, a blend of both positive and negative By structures is always present. The equatorial slice(xy-plane) shows that even at the equator, By is relatively well-organized.
For the strongly magnetized NVF-β2, Figure 5.10 shows that By dominates at all heights and is highlyorganized at the disc mid-plane. In the leftmost panel, a precursor of the MRI-dynamo flip is present in
the form of a few ribbon-like positive (blue) By structures. While the entire domain is dominated by either
positive or negative By at a given time preceding or following an MRI-dynamo reversal, wispy By structuresof the non-dominant polarity always persist in the vicinity of the disc mid-plane. Indeed, the rightmost
panel shows that a negative (red) By ribbon-like precursor is now present near the mid-plane following an
MRI-dynamo flip. For both NVF-β4 and NVF-β2, the MRI-dynamo launches current sheets into the corona— although they do not propagate perfectly synchronously above/below the disc mid-plane — and thelarge-scale toroidal magnetic field in the coronal region reverses sign.
Several authors have attempted to understand the origin of the MRI-dynamo field reversals, andquantitative properties such as the period, using αΩ dynamo theory (see e.g. Gressel & Pessah, 2015, andreferences therein). In a different approach, Herault et al. (2011) computed exactly time-periodic dynamo
solutions to an incompressible model at modest Reynolds and magnetic Reynolds numbers. They argued
that the MRI-dynamo results from interactions between the dominant toroidal field and non-axisymmetricperturbations. The relevance of these simplified models to our simulations — which as we have noted arehighly compressible when the net flux is strong — is unclear, but it is possible that sufficient small-scale
turbulence is a prerequisite for MRI-dynamo reversals. We observe a clear decrease in small-scale turbulenceas the net field increases (see Figures 5.9 and 5.10).
The left panel of Figure 5.11 shows the time-evolution of the volume-averaged toroidal magnetic field,

158
hBy iV , for each simulation. The MRI-dynamo cycles are apparent from the sinusoidal behaviour of hBy iV .
A dynamo flip in one disc hemisphere is nearly always mirrored by a similar response in the other dischemisphere; however, this response is not necessarily instantaneous. For instance, consider the time ranget ' 150 − 200 in simulation NVF-β4. Figure 5.7 shows a well-defined butterfly pattern for hBy ixy , but
Figure 5.11 shows that the top (z > 0) and bottom (z < 0) domains are anti-synchronized, as evidenced by
hBy iV ' 0 in this time range.
The right panel of Figure 5.11 shows the temporal power spectrum generated from hBy iV for eachsimulation. These are not significantly altered by the occasional offset nature of hBy iV in the top andbottom disc hemispheres just described. The weakly magnetized simulations with a well-defined MRIdynamo pattern (ZNVF, NVF-β5, NVF-β4) display a coherent peak in the power spectrum, while thisfeature broadens/disappears for the strongly magnetized simulations because the MRI-dynamo cycles areeither sporadic (NVF-β3, NVF-β2) or non-existent (NVF-β1).

5.5.2

Production and Escape of Toroidal Magnetic Field
The MRI-dynamo channels accretion power into ordered toroidal magnetic fields near the disc mid-plane, which buoyantly rise through the disc atmosphere and escape. In the model for the MRI-dynamo
proposed by Begelman et al. (2015), accretion energy is being liberated locally by the MRI and some fractionof this is channeled into the toroidal magnetic field. This toroidal field rises due to the Parker instability
and carries a vertical flux of magnetic energy (i.e., Poynting flux). However, this buoyant rise of toroidal
field is hampered by field buckling and reconnection, causing energy to be lost to the gas by the magnetic
field as it rises. The details of this complicated energy exchange process are simplified by assuming that allof the energy lost by the toroidal magnetic field as it rises goes into locally heating the gas. In this modelfor toroidal magnetic flux production and escape, the governing energy balance equation is,d2
p B vB= ΩvB [αB (pgas + pB ) − 2ξpB ] ,dz(5.25)

where αB measures the portion of accretion energy liberated locally by the MRI that goes into the toroidal
magnetic field and we recall the MRI-dynamo period parameter ξ from Equation 5.24. The vertical profile

159

Figure 5.12: Vertical profiles of the horizontally- and time-averaged rise speed (in code units) of the toroidal
magnetic flux (see Figure 5.1 for line colour conventions). For all simulations, vB ' 0 within the gas-pressureb t ' 1). Outside of this vertical location, vB > 0
dominated regions (i.e., interior to the z-locations where hβi
and increases with disc altitude. As the net vertical magnetic flux increases, vB increases and even reaches
supersonic speeds (i.e., vB > cs = 1) for simulations NVF-β2 and NVF-β1.

160
for the rise speed of the toroidal magnetic flux can be expressed as,vB (z) = ηΩz,

(5.26)

where η = 1 would correspond to the field rising at the free-fall speed. Values of η ≤ 1 phenomenologicallyaccount for effects that slow down the rise speed of the toroidal magnetic field, such as tangling of magnetic
field lines and magnetic reconnection.
Here, we use our simulations to measure quantities that characterize toroidal magnetic flux production(αB ) and escape (vB , η) within the context of this model. We measure the horizontally-averaged rise
speed of the toroidal magnetic field, hvB ixy , as follows. The flux of magnetic energy — or Poynting flux,
FB = B × (v × B) — has vertical component,
FB,z = Bx (vz Bx − vx Bz ) − By (vy Bz − vz By ) .

(5.27)

If we assume that the magnetic field is purely toroidal, B ' By , which is justified by our simulations, thenthe z-component of the Poynting flux becomes,
FB,z = vB By2 ,(5.28)

where vB = vz is a measure of the rise speed of the toroidal magnetic field defined by Equation 5.26. To
measure vB from our simulations, we equate Equations 5.27 and 5.28, solve for vB , and horizontally-average
the result to obtain,
hvB ixy =hBx (vz Bx − vx Bz ) − By (vy Bz − vz By )ixy
By2 xy.

(5.29)

Figure 5.12 shows the horizontally- and time-averaged vB for each simulation. The toroidal magnetic field rise
speed is essentially zero in the gas pressure-dominated regions where the disc exhibits small-scale turbulence
and the large-scale toroidal magnetic field is not strongly ordered. For vertical locations exterior to theb t ' 1 transition (see Figure 6.1), where the toroidal magnetic field dominates the vertical pressure support
hβi
in the disc, vB is increasing with disc altitude. As the amount of net vertical magnetic flux increases, vB at
any given height above/below the disc mid-plane also increases, even becoming supersonic for simulations
NVF-β2 and NVF-β1. Recall that cs = 1 throughout the domain for our isothermal simulations.

161
We measure the vertical profile of the toroidal magnetic flux escape speed parameter η (z) by insertinghvB ixy from Equation 5.29 into Equation 5.26. We average over z to obtain hηiV , ignoring numerical artifacts
concentrated only very close to the disc mid-plane and the vertical boundaries. Table 5.3 lists hhηiV it foreach simulation. As the amount of net vertical magnetic flux increases, the toroidal magnetic field is able
to rise more efficiently (i.e., η increases), with the speed reaching 20-30% of the free-fall speed for the twomagnetically dominated runs. Presumably, this is a consequence of the toroidal magnetic field becoming
more ordered with increasing net vertical magnetic flux (see Figures 5.9 and 5.10), i.e., there is less tangling
and reconnection of magnetic field lines.
We measure the vertical profile of the toroidal magnetic flux production parameter αB by rearranging
Equation 5.25 to obtain,

αB (z) =1
hpgas ixy + hpB ixy



1
Ω




dhpB ixydhvB ixyhvB ixy+ 2hpB ixy+ 2ξhpB ixy .
dz
dz(5.30)In the vertical locations near the disc mid-plane αB has an essentially flat profile, which is not shown here
for the sake of brevity. We adopted a constant value for ξ; therefore our measured values for αB accountfor energy dissipation during MRI-dynamo flips in a volume-averaged sense and not as a function of disc
altitude. Given that αB parametrizes the amount of accretion power liberated by the MRI that goes into
the toroidal magnetic field and that we measure αB to have a flat vertical profile in the MRI-active disc midmid
it forplane regions (i.e., |z/H0 | < 1), we choose to evaluate αB at the disc mid-plane. Table 5.3 lists hαBeach simulation. As the amount of the net vertical magnetic flux increases, the MRI channels a fractionallylarger amount of the liberated accretion energy into toroidal magnetic fields.
Notably, as shown by inspecting Equation 5.30, the values reported for αB are heavily influenced by thefrequency of MRI-dynamo cycles, as parametrized by ξ. A current sheet sweeps vertically through the domain
with each MRI-dynamo reversal in the large-scale toroidal magnetic field. These strong current sheets arelikely sites of magnetic reconnection and could potentially deposit substantial amounts of accretion power inthe coronal regions, presumably through heating by magnetic reconnection. For the zero and weak/moderate(β0mid & 103 ) net vertical magnetic flux simulations, this heating caused by MRI-dynamo cycles dominates
the αB values we measure. For the NVF-β2 simulation, MRI-dynamo cycles are infrequent and contribute

162
in an ancillary way to αB .
By considering the interplay of toroidal magnetic flux production and its buoyant escape, along withthe effects of heating by current sheets launched during each MRI-dynamo cycle, Begelman et al. (2015)
show that the plasma-β at the disc mid-plane can be determined via a regularity condition,
mid
βB15
=

2η
+ ν − 1,αB

(5.31)

where ν ≡ 2ξ/αB is the reconnection efficiency parameter. For each simulation, Table 5.3 gives the values of
mid
hβB15
it calculated from the values of ξ, η, and αB also listed in Table 5.3. Comparing these model-dependent

values for the mid-plane plasma-β to those that we measure directly in our simulations (see hβbmid it in Table5.2), we find extremely good agreement. The mid-plane magnetization can thus be understood in termsof a very simple model that balances toroidal field production against vertical escape at a rate tied to thefree-fall speed, though simulations are needed (especially for weak net fields) to measure the efficiency ofthese processes as a function of the net field strength.

5.5.3

Vertical Disc Structure
We can also compare our simulation results to the analytic predictions for the vertical disc structurein the context of the MRI-dynamo model discussed in §5.5.2. Figure 5.13 shows the vertical profiles of
the horizontally- and time-averaged gas pressure (left panels) and magnetic pressure (right panels), wheredisc altitude is in units of the gas density scale height H that develops in the turbulent steady state ofeach simulation and not its initial value H0 . Given the isothermal equation of state that we adopt, the gaspressure profiles are equivalent to the gas density profiles.
Using Equation 5.31 and combining equations for hydrostatic equilibrium and the Poynting flux,
Begelman et al. (2015) derive coupled ordinary differential equations that govern the vertical structure,dp̃B + β mid p̃gas = −β mid ρ̃
dy
y

dp̃B
β mid
=
(p̃gas − p̃B ) ,dy
1 + β mid − ν

mid22
where ρ̃ = ρ/ρmid , p̃gas = pgas /pmid
gas , p̃B = pB /pB , and y = z /(2H ).

(5.32)(5.33)

163

Figure 5.13: Dotted lines show the vertical profiles of the horizontally- and time-averaged gas pressure (leftpanel) and magnetic pressure (right panel), each normalized to their value at the disc mid-plane (see Figure
5.1 for line colour conventions). Also plotted are the analytic model predictions (Equations 5.32 and 5.33;
mid
mid
Begelman et al., 2015): solid lines adopt β mid = hβB15
it and ν = 2ξ/hαB
it from Table 5.3; dashed lines
mid
mid
b
adopt β= hβ it from Table 5.2 and ν = 0. From top to bottom are simulations ZNVF, NVF-β5,
NVF-β4, NVF-β3, NVF-β2, NVF-β1. We note that z is plotted in units of the gas density scale height, H,
that develops for each individual simulation and not its initial value, H0 .

164
Combining Equations 5.32 and 5.33 with an isothermal equation of state, Figure 5.13 shows thevertical profiles of gas pressure (left panels) and magnetic pressure (right panels) predicted by the Begelman
et al. (2015) model and compared to the simulation results. The solid lines show the results from choosing
mid
mid
β mid = hβB15
it and ν = 2ξ/hαB
it (see Table 5.3). The dashed lines show the results from choosing

β mid = hβbmid it (see Table 5.2) and ν = 0. The main effect of including the magnetic reconnection efficiency
parameter ν is to steepen the magnetic pressure profile. The model that does not include ν (dashed lines)
roughly matches the magnetic pressure profiles for the zero and weak/moderate (β0mid & 103 ) net verticalmagnetic flux simulations. The model that does include ν (solid lines) does a better job of capturing
the steepening of the magnetic pressure profiles observed in the magnetic pressure-dominated simulations(β0mid . 102 ).

5.6

Discussion

5.6.1

MRI-Dynamo Cycles
We find that the cyclic MRI-dynamo reversals in the large-scale toroidal magnetic field have a regular' 10 orbit periodicity for simulations with zero and weak (β0mid & 103 ) net vertical magnetic flux. For
β0mid . 103 , these reversals become more scattered and increase in period with increasing net verticalmagnetic flux, until they are no longer observed for the most strongly magnetized simulation with β0mid = 101 .
Previous shearing box simulations with a zero net vertical magnetic flux show the same MRI-dynamo
cycle regularity and periodicity that we observe, provided that the domain size is sufficiently large (e.g.,
Simon et al., 2012). The strong (β0mid . 103 ) net vertical magnetic flux simulations of Bai & Stone (2013)
also display MRI-dynamo cycle characteristics that are consistent with our simulations. However, somedifferences from our results arise for weaker net fluxes, with previous works finding the butterfly pattern tobe highly irregular for β0mid & 104 (e.g., Fromang et al., 2013; Bai & Stone, 2013). Given that Bai & Stone(2013) used very similar methods to ours, the most likely origin for the differences we observe is differences inthe domain size. The shearing box simulations of Bai & Stone (2013) used a substantially smaller horizontal
domain (Lx , Ly ) = (4H0 , 8H0 ), which has been shown in zero net vertical magnetic flux simulations to affect165
the dynamo properties (Simon et al., 2012).
The smooth transition that we observe in the cyclic MRI-dynamo behaviour across the zero-to-weak
net vertical magnetic flux regimes suggests that we have obtained numerically converged solutions, that arephysically reasonable in the local limit. However, the requirement for such large horizontal domains meansthat we are pushing the limits of the shearing box’s validity by neglecting curvature terms. We cannot
rule out the possibility that the MRI-dynamo is intrinsically periodic in a shearing box and even the strong
field models may become periodic for larger domain sizes. Similarly, the absence of MRI-dynamo cycles weobserve for β0mid = 101 may be a consequence of an insufficiently small domain size and/or short integration
time. Ultimately, global disc simulations are needed to determine if the sporadic nature of MRI-dynamo
cycles that we see in the strong (β0mid . 103 ) net vertical magnetic flux regime is an intrinsic property of
MRI-dynamo behaviour.

5.6.2

Quasi-Periodic Oscillations
Black hole X-ray binaries showcase complicated evolutionary cycles, called state transitions (e.g.,

Remillard & McClintock, 2006; Belloni, 2010), where the source transitions between “low/hard” and “high/soft”
states. The low/hard state is characterized by a low luminosity (. 10−3 LEdd ) and a non-thermal hard X-ray
spectrum, while the high/soft state is characterized by a high luminosity (& 0.1LEdd ) and a quasi-thermal
soft X-ray spectrum. The soft quasi-thermal X-rays are attributed to a geometrically thin, optically thickaccretion disc (Shakura & Sunyaev, 1973), while the non-thermal hard X-rays are thought to arise fromenergetic electrons in a surrounding “corona” that inverse Compton scatter the seed disc photons (Haardt& Maraschi, 1991). In a complete state transition hysteresis cycle (∼ 100 days), the source rises out of
the low/hard state along a high-luminosity track into the high/soft state, but then decays on a different
low-luminosity track back to the low/hard state, passing through hybrid “intermediate” states along theway. The outburst decay track is typically ∼ 10 − 100 times fainter than the outburst rise track.
During a low/hard ↔ high/soft state transition, low-frequency quasi-periodic oscillations (QPOs)
are observed (van der Klis, 1989; Casella et al., 2005), which are coherent features in the temporal power
spectrum of the X-ray light curve. The QPO centroid frequency evolves from νQ,0 ' 0.01 → 10 Hz during

166
a low/hard → high/soft state transition (e.g., Belloni et al., 2005; Belloni, 2010). These QPOs ultimately
disappear in the high/soft state. The origin of QPOs is still unknown, but if understood, QPOs would be a
powerful diagnostic for measuring disc evolution (e.g., Miller & Miller, 2015).
The MRI-dynamo cycles of flip-flopping toroidal magnetic field shown in Figure 5.7 are promising candidates for QPOs (e.g., O’Neill et al., 2011; Begelman et al., 2015). Figure 5.11 shows that a coherent QPOlike feature appears for the weakly magnetized discs (β0mid & 104 ) with peak frequency νpeak ∼ 0.1 orbits−1 .
A strong peak is not seen for the strongly magnetized discs (β0mid . 103 ) due to the irregularity of the MRIdynamo cycles, but this result may be a consequence of domain size restrictions as previously mentioned.
We consider the increase in MRI-dynamo period with net flux to be a robust result; therefore, the frequency
of a resulting power spectral peak would increase with decreasing magnetization. However, we emphasizethat Figure 5.11 cannot be directly compared to an observed power spectrum because the toroidal magnetic
field was not mapped to an emission mechanism. If, as suggested by Begelman & Armitage (2014), the discbecomes decreasingly magnetized during the low/hard → high/soft state transition, MRI-dynamo cycles
could potentially produce a QPO-like feature that fits the observed evolutionary behaviour of νQ . Global
simulations are needed to address these speculations (e.g., O’Neill et al., 2011) because QPO frequency andstrength would likely be set far from the black hole, and the modulations would have to take into accountradial propagation of the MRI-dynamo cycles.

5.6.3

Spectral Hardening
Understanding the consequences of vertical magnetic pressure support for the observed accretion disc

spectrum is necessary for assessing the robustness of any observational results based on disc continuum
modeling. A disc with magnetic pressure support has a lower gas density at the effective photosphere, which
enhances electron scattering and leads to a harder spectrum compared to a disc without magnetic pressure
support (Davis et al., 2005; Blaes et al., 2006).
The degree of spectral hardening is parametrized by the colour correction factor fcol (Shimura &Takahara, 1995). Discs without a vertical magnetic pressure contribution tend to have fcol values confinedto the narrow range fcol ' 1.5 − 1.7 (e.g., Davis et al., 2005). However, observations of disc evolution

167
in X-ray binary state transitions (Salvesen et al., 2013; Reynolds & Miller, 2013) and sophisticated jointspectral fitting of the disc in a steady high/soft state (Maitra et al., 2014) demonstrate that the data permit
fcol & 2. Post-processing of a weakly magnetized shearing box simulation with zero net vertical magnetic
flux showed that magnetic pressure support in the disc surface layers leads to enhanced spectral hardening,with fcol = 1.74 (1.48) when magnetic pressure support was included (neglected) in the vertical hydrostaticbalance (Blaes et al., 2006). Analytic models of very strongly magnetized discs predict substantial spectral
hardening of fcol ∼ 5 (Begelman & Pringle, 2007). The two main contributing effects to spectral hardening
are the ratio of electron scattering to absorption being higher at the disc photosphere and the profile of MRI
dissipation being concentrated higher in the disc (i.e., there is a lot of dissipation just below the photosphere).
However, the quantitative details of how significant magnetic pressure support influences the observed disc
spectrum are not yet understood.
Measurements of black hole spin with the disc continuum fitting technique (Zhang et al., 1997; McClintock et al., 2006) ignore vertical magnetic pressure support in the disc when modeling the disc spectrum.
This may lead to erroneously overestimated spins and explain the inconsistencies between the two leading
black hole spin measurement techniques. Given a non-spinning black hole (a = 0) with typical X-ray binary
system parameters, one would obtain the incorrect spin parameter a ' 0.5 by ignoring fairly weak verticalmagnetic pressure support (Blaes et al., 2006). We are currently quantifying the impact of spectral hardening
on black hole spin measurements (Salvesen et al., in preparation).

5.7

Summary and Conclusions
We have studied the structure and dynamo-related variability of accretion discs, using isothermal local

simulations that include vertical stratification. The simulations span almost the full range of net verticalfluxes that lead to MRI-unstable initial conditions (β0mid = 101 − 105 ), and were run in large domainsthat capture mesoscale structures that are important for disc variability. Our two strongest net field runs,
with β0mid = 10 and 100, result in the formation of magnetically dominated discs whose structure differs
qualitatively from that of gas pressure-dominated solutions. Our main conclusions are:168
• The α-viscosity parameter, α = hTxy iV /hpgas iV , follows a power-law spanning two orders of magnitude in net vertical magnetic flux (four orders of magnitude in β0mid ). Alternatively, normalizing thestress to the magnetic pressure yields a constant viscosity parameter αmag = hTxy iV /hpB iV that is
independent of the initial mid-plane β (Hawley et al., 1995; Blackman et al., 2008).
• Gas density fluctuations (relative to the time-averaged gas density vertical profile) increase with net
vertical magnetic flux, reaching the ∼ 60 − 100% level for our magnetic-pressure dominated discsimulations NVF-β2 and NVF-β1. These highly magnetized but still MRI-unstable discs are highly
inhomogeneous, clumpy structures.
• MRI-dynamo cycles are highly regular for simulations with either zero net vertical magnetic flux
or weak (β0mid & 104 ) net vertical magnetic flux. The MRI-dynamo period increases and becomes
more sporadic for moderate-to-strong (β0mid . 103 ) net vertical magnetic flux simulations. A large
horizontal domain appears to be necessary to capture this behaviour, in net field as well as zero net
field simulations (Simon et al., 2012).
• Entry into the fully magnetic pressure-dominated regime is not a sufficient condition for the complete suppression of cyclic MRI-dynamo reversals. We observe multiple reversals in the magnetic
pressure-dominated NVF-β2 simulation, and cannot exclude the possibility that very long time scale
reversals occur for even more strongly magnetized discs. With increasing magnetization, even longer
integration times and/or larger domain sizes than we considered may be necessary to properly study
these MRI-dynamo cycles.
• The magnetization at the disc mid-plane is very well-described by the regularity condition (seeEquation 5.31) given by Begelman et al. (2015), which is derived from a model based on the balance
between the MRI-dynamo production of toroidal field and its buoyant escape.
• The Begelman et al. (2015) model provides a reasonable first-order description of the vertical discstructure seen in the simulation results. Incorporating a magnetic reconnection efficiency parameterν into the model helps to match the steep magnetic pressure profiles from the magnetic pressure-169
dominated disc simulations (β0mid . 102 ).
Our results support aspects of a currently speculative scenario in which the net poloidal flux is the
key parameter controlling the structure and evolution of accretion discs in black hole X-ray binaries. Net
fluxes that remain in the MRI-unstable regime can still be strong enough to stimulate qualitative changes
to the disc structure, and lead to the formation of magnetically supported and highly inhomogeneous discs.
Analytic models of such discs suggest that many of the observationally interesting consequences occur as aconsequence of the work done by the toroidal field as it escapes, and future simulations will need to relax
the isothermal assumption adopted here to study such effects.

Acknowledgments
We thank the anonymous referee for her/his constructive comments and suggestions, which improved
this paper. GS acknowledges support through the NASA Earth and Space Science Graduate Fellowship
program. J.B.S.’s support was provided in part under contract with the California Institute of Technology(Caltech) and the Jet Propulsion Laboratory (JPL) funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. PJA acknowledges support from NASA under
Astrophysics Theory Program awards NNX11AE12G and NNX14AB42G, and from the NSF under award
AST-1313021. MCB acknowledges support from NSF grant AST-1411879. This work used the Janus supercomputer, which is supported by the National Science Foundation (award number CNS-0821794) andthe University of Colorado Boulder. The Janus supercomputer is a joint effort of the University of Colorado Boulder, the University of Colorado Denver, and the National Center for Atmospheric Research. This
work used the yt project (Turk et al., 2011), an open source data analysis and visualization toolkit for
astrophysical simulations.

Chapter 6

Strongly Magnetized Accretion Discs Require Poloidal Flux

6.1

PrefaceThis paper was published in Monthly Notices of the Royal Astronomical Society, Volume 460, Issue4, pp. 3488-3493 on August 21, 2016. The author list is: Salvesen, G., Armitage, P. J., Simon, J. B., &Begelman, M. C. We find that local simulations of accretion discs with realistic vertical boundary conditions
can only develop into a magnetic pressure-dominated state if a sufficiently strong background poloidal net
flux is present.

Abstract
Motivated by indirect observational evidence for strongly magnetized accretion discs around black
holes, and the novel theoretical properties of such solutions, we investigate how a strong magnetization
state can develop and persist. To this end, we perform local simulations of accretion discs with an initiallypurely toroidal magnetic field of equipartition strength. We demonstrate that discs with zero net verticalmagnetic flux and realistic boundary conditions cannot sustain a strong toroidal field. However, a magnetic
pressure-dominated disc can form from an initial configuration with a sufficient amount of net vertical flux
and realistic boundary conditions. Our results suggest that poloidal flux is a necessary prerequisite for thesustainability of strongly magnetized accretion discs.

171

6.2

Introduction
Magnetic fields are fundamental to the physics of accretion discs. The magnetorotational instability(MRI) causes an accretion disc to enter a self-sustaining turbulent steady state, wherein an effective viscosity
is generated that drives accretion (Balbus & Hawley, 1991, 1998). The disc magnetization is parametrized
by the ratio of gas-to-magnetic pressure, β ≡ pgas /pB . While some numerical simulations examine MRI
turbulence in strongly magnetized discs (β . 1; e.g., Johansen & Levin, 2008; Bai & Stone, 2013; Salvesen
et al., 2016b), the majority focus on the weakly magnetized regime.
Several lines of evidence suggest that real accretion discs around black holes in X-ray binaries andgalactic nuclei may be strongly magnetized. Observations of powerful disc winds (e.g., Miller et al., 2006b)
and relativistic jets (e.g., Fender et al., 2004) in black hole X-ray binary systems suggest the presence of asignificant poloidal flux, which catalyzes a much stronger toroidal field in MRI-active discs. On larger scales,
the gas supplied to the disc by the donor star is expected to be strongly magnetized (β . 1; Johansen &Levin, 2008). In the Galactic Center, dust polarization measurements in the inner ∼ 102 pc reveal both atoroidal magnetic field near the Galactic plane, and a large-scale poloidal field at higher Galactic latitudes(Nishiyama et al., 2010).
At a more circumstantial level, strongly magnetized discs provide a promising theoretical framework
for addressing longstanding problems that plague their weakly magnetized counterparts. In particular,
magnetically dominated discs are not prone to thermal instability in radiation pressure dominated regions(Begelman & Pringle, 2007), a result which appears consistent with the low levels of variability seen inluminous X-ray binary states. They are also less susceptible to gravitational fragmentation that would
inhibit continued accretion onto the black hole in active galactic nuclei (Pariev et al., 2003; Begelman &Pringle, 2007; Gaburov et al., 2012). Furthermore, the effective α-viscosity (Shakura & Sunyaev, 1973), whichparametrizes the rate of angular momentum transport, is observationally constrained to be α ∼ 0.1 − 0.4 in
dwarf novae systems (King et al., 2007). Local disc simulations with weak magnetic pressure support andzero net vertical flux find α ∼ 0.01 (Stone et al., 1996; Davis et al., 2010; Simon et al., 2011) . However,
local disc simulations with net vertical flux allow the disc to develop significant magnetic pressure support

172
and boost α to values ∼ 0.1 − 1 (e.g., Hawley et al., 1995; Salvesen et al., 2016b). Finally, recent theoreticalmodels for X-ray binary state transitions appeal to strongly magnetized accretion discs (Begelman et al.,2015) and the evolution of a net poloidal flux (Begelman & Armitage, 2014).
In this paper, we address the question of whether there are multiple routes by which an accretion disccan sustain a strongly magnetized state. One well-studied route requires a net poloidal flux (typically in thewindow 103 > βp > 10) that is strong enough to affect the MRI dynamics but weak enough to admit MRIdriven turbulence. Simulations show that this leads to a supra-thermal mid-plane field that is replenishedby dynamo action as it buoyantly escapes (Bai & Stone, 2013; Salvesen et al., 2016b). An alternate route
could involve processes other than the MRI. In particular, Johansen & Levin (2008) argued that a dynamo
mediated by the Parker instability could maintain a strongly magnetized state from an initial configurationof a purely toroidal field in equipartition with the gas. This suggestion is particularly interesting as — ifcorrect — it would imply that discs with zero net poloidal flux could exist in two distinct states depending
upon the history of the system.
Here we consider local disc simulations that have zero net vertical magnetic flux and are initializedwith an equipartition toroidal field. We adopt two sets of vertical boundary conditions, one which traps
toroidal magnetic flux and one which allows it to escape freely. We show that the disc cannot maintain its
strongly magnetized initial state for the more physically realistic outflow boundary conditions, and hence
conclude that strongly magnetized astrophysical accretion discs require a background poloidal magnetic flux.

6.3

Numerical Simulations
Following Salvesen et al. (2016b), we use the Athena code (Gardiner & Stone, 2005, 2008; Stone et al.,2008) to solve the equations of compressible, isothermal, ideal magnetohydrodynamics in the “shearing box”

173
approximation (Hawley et al., 1995; Stone & Gardiner, 2010) including vertical density stratification,∂ρ= − ∇ · (ρv)
∂t


 
B2
∂(ρv)
= − ∇ · ρvv − BB + pgas +I∂t2+ 2qρΩ2 xî − ρΩ2 z k̂ − 2Ω × (ρv)∂B= − ∇ · (vB − Bv) .
∂t

(6.1)

(6.2)
(6.3)

In order, Equations 6.1 - 6.3 describe the conservation of mass, the conservation of momentum, and magnetic
induction, with the various parameters having their usual meanings: ρ is the gas density, pgas is the gaspressure, v is the gas velocity, and B is the magnetic field. Our magnetic field definition subsumes a√factor of µ/ 4π, with µ = 1 being the magnetic permeability. We choose an equation of state, pgas = ρc2s ,corresponding to an isothermal gas with sound speed cs . I is the identity matrix and (î, ĵ, k̂) are the Cartesian(x, y, z) unit vectors. Specific to the shearing box geometry, Ω = Ωk̂ is the angular frequency corresponding
to co-rotation with the disc. The simulation domain is centered on the arbitrary reference radial location
R = R0 and q = −dln (Ω) /dln (R) = 3/2 is the shear parameter corresponding to Keplerian rotation.
The question we are interested in answering is whether a background poloidal magnetic flux is necessary to sustain a strongly magnetized accretion disc. The net vertical magnetic flux simulations of Salvesen
et al. (2016b) with β0mid . 103 and outflow boundary conditions developed into a magnetic pressuredominated state. The zero net vertical flux simulations of Johansen & Levin (2008) were initialized withan equipartition toroidal field and remained strongly magnetized, but adopted boundary conditions that
confined the magnetic flux within the domain. The simulations presented in this Letter (see Table 6.1) arevery similar to those of Salvesen et al. (2016b), but have the following two differences: (1) initial conditions
with a purely toroidal magnetic field configuration (§6.3.1) and (2) boundary conditions that either do or do
not permit magnetic flux to escape (§6.3.2).

174
6.3.1

Initial Conditions
As did Johansen & Levin (2008), we initialize zero net vertical magnetic flux simulations with avertical gas density profile in hydrostatic equilibrium and a purely toroidal magnetic field configuration,
ρ (x, y, z) = ρmid
0 exp

−z 2
2Hβ2

!

"
By,0 (x, y, z) =with ρmid= 1. The gas scale height is Hβ =0q

2β0−1 c2s ρmid
0 exp

(6.4)
−z 2
2Hβ2

!#1/2
,

(6.5)

1 + β0−1 cs /Ω. Because we choose cs = 1 and Ω = 1, the scaleheight resulting from thermal pressure support alone opposing vertical gravity is H0 = cs /Ω = 1, which weadopt as our unit of length. The domain size for all simulations is (Lx , Ly , Lz ) = (10H0 , 20H0 , 10H0 ). Discmagnetization is parametrized by the ratio of gas-to-magnetic pressure,
β≡

pgas
ρc2
= 2s .
pB
B /2

(6.6)

We choose an equipartition initial magnetic field β0 = 1, which makes the vertical domain size 7.1Hβ whenthe magnetic field contribution to hydrostatic equilibrium is included. Random perturbations to the gas
density and velocity are introduced at t = 0 in order to initiate the MRI (Hawley et al., 1995). We impose
a density floor ρfloor = 10−4 ρmid
0 .

Grid Res. β0mid[Nzones /H0 ]
36
1
24
1
24
102
0
0
0.141

Bz
Outflowing
Periodic
Outflowing

Vertical BCs

tf
hQmid
y it
[orbits]
125
44(2)
125
35(2)
225
3.3(7)e2
10.3(5)8.4(4)
9(1)e1

hQmid
z it
0.0038(5)0.008(1)0.10(6)mid
it
hTxy,Rey

0.013(1)0.022(2)0.5(1)

midit
hTxy,Max

0.013(1)0.019(2)1.0(4)

hb
αmid it

45(4)
32(3)
0.4(1)

hβbmid it

Table 6.1: Summary of simulations. From left to right the columns are: simulation identification label, grid resolution (applies to all dimensions), initialplasma-β at the disc mid-plane, net vertical magnetic flux density (code units), vertical boundary conditions, and simulation termination time. All
subsequent quantities, which are defined in the text, are evaluated at the disc mid-plane and time-averaged from ti = 25 to tf . Parentheses indicate
the ±1σ range on the last digit from the time averaging. ∗ Reference net vertical magnetic flux simulation from Salvesen et al. (2016b).

ZNVF-OZNVF-P
NVF-β2∗

ID

175

176
6.3.2

Boundary Conditions
We adopt the usual shearing box boundary conditions of shearing periodic in x (radial) and strictly

periodic in y (toroidal) (Hawley et al., 1995; Simon et al., 2011). Table 6.1 summarizes our simulations,which fundamentally differ only in their choice of z (vertical) boundary conditions:• ZNVF-O adopts the modified outflowing vertical boundary conditions of Simon et al. (2011) andSimon et al. (2013), in which the gas density, pressure, and in-plane components of the magnetic field
are exponentially extrapolated from the physical domain into the ghost zones. This method prevents
the unphysical accumulation of magnetic flux at the vertical boundaries, thus allowing magnetic flux
to escape the simulation domain.
• ZNVF-P adopts periodic vertical boundary conditions, which do not allow magnetic flux to escapethe simulation domain.
Neither condition is fully realistic, but outflowing boundary conditions are closer to the physicalsituation. Any significant differences between the zero net vertical flux simulations ZNVF-O and ZNVF-P
can be ascribed to their differences in vertical boundary conditions. We performed resolution and verticaldomain size convergence studies that corroborated the results from simulation ZNVF-P. We contrast theseruns to NVF-β2 from Salvesen et al. (2016b), which combines outflow boundary conditions with a net
poloidal field β0mid = 102 .

6.4

Analysis and Results
We are interested in the turbulent steady states arising from zero net vertical flux shearing box

simulations that differ in their choice of vertical boundary conditions. Empirically, the saturated state isconsidered well-resolved for quality factors Qi & 6 (Sano et al., 2004), given by1 ,
Qi =

λMRI
,
∆xi

(6.7)

where, for spatial dimension i, the characteristic MRI wavelength is λMRI,i = 2πvA,i /Ω and the Alfvén speed
is vA,i =pBi2 /ρ. Table 6.1 shows that all of our simulations are well-resolved according to the Qy and Qz

1 Hawley et al. (2011) suggest Q

y & 20 and Qz & 10.

177
benchmarks.
Table 6.1 also lists diagnostics of MRI turbulence, each evaluated at the disc mid-plane and timeaveraged over the saturated state (i.e., t ≥ 25 orbits).

The rate of angular momentum transport isparametrized by the effective α-viscosity (Shakura & Sunyaev, 1973),

αb=hTxy,Rey ixy + hTxy,Max ixy,
hpgas ixy(6.8)

where the notation h·ixy indicates that the quantity is horizontally-averaged. The Reynolds and Maxwell
contributions to the xy component of the total stress tensor are Txy,Rey = ρvx vy0 and Txy,Max = −Bx By ,where vy0 are the remaining fluctuations when the background shear is subtracted from vy . The plasma-β
parameter in Table 6.1 is defined as,
hpgas ixyβb =.
hpB ixy(6.9)

Figure 6.1 shows the vertical profiles of βb for each simulation.
To our surprise, the mid-plane diagnostics of MRI turbulence are only weakly dependent on the verticalboundary conditions for the zero net flux simulations. In the saturated turbulent steady state of runs ZNVFmid
mid
O and ZNVF-P, we find Txy,Max' 4Txy,Rey,αbmid ∼ 0.01, and βbmid  1, consistent with previous shearing

box simulations with zero net vertical magnetic flux (e.g., Stone et al., 1996; Davis et al., 2010; Simon et al.,2011). Conversely, the strong net vertical magnetic flux simulation NVF-β2 shows an enhanced αbmid ' 1.0
and a magnetic pressure-dominated disc with βbmid ' 0.4 (Salvesen et al., 2016b).
Larger differences between the zero net flux simulations are apparent from the vertical profiles ofmagnetization shown in Figure 6.1. With outflowing boundary conditions, we find that the initially strong(equipartition level) toroidal field buoyantly escapes in the transient stage. The disc evolves into a generally
weakly magnetized state, with βb ∼ 1 being reached only in a coronal region at |z| > 3.5H0 . With periodic
boundary conditions, on the other hand, the final magnetized state is determined by the redistribution of
the initial equipartition toroidal field. After a sufficiently long interval, this leads to a weakly magnetizeddisc core within about one scale height of the disc mid-plane, with extremely strong magnetization at higher
altitudes. We are only able to sustain a strongly magnetized disc mid-plane with the aid of poloidal field.
Simulation NVF-β2, which also adopts outflowing boundary conditions, evolves from a net vertical flux

178

Figure 6.1: Vertical profiles of the time-averaged plasma-β for zero net vertical flux simulations with outflowing (solid black line) and periodic (dashed blue line) vertical boundary conditions. The strongly magnetized
net vertical flux simulation with outflowing boundary conditions (dotted red line) from Salvesen et al. (2016b)
is shown for reference. Coloured bands show the respective ±1σ range in βb from the time averaging. The
horizontal dashed line marks equipartition βb = 1.

179

Figure 6.2: Left panels: Space-time diagrams of the horizontally-averaged toroidal magnetic field, hBy ixy .
Right panels: Space-time diagrams of the horizontally-averaged radial magnetic field, hBx ixy . Shown are
simulations ZNVF-O (top panels), ZNVF-P (middle panels), and NVF-β2 (botom panels). Black lines markthe β = 1 contour, exterior (interior) to which β < 1 (β > 1). The magnetic field reversals characteristic of
the MRI-dynamo are only observed in the simulations with outflowing boundary conditions (i.e., ZNVF-O
and NVF-β2). This suggests that simulations with vertical boundary conditions that confine magnetic flux
within the domain cannot capture important accretion disc physics.

180
configuration with β0mid = 102 into a strongly magnetized state with βb . 1. In this run the MRI-dynamo
constantly replenishes the escaping toroidal field from the imposed vertical field.
Figure 6.2 (left panels) shows the space-time diagrams of the toroidal magnetic field for each simulation.
The simulations with outflowing vertical boundary conditions (ZNVF-O and NVF-β2) exhibit the toroidal
field reversals characteristic of MRI-dynamo activity, albeit with differing periods. Simulation ZNVF-P,
which confines magnetic flux within the domain, does not show this behaviour of periodically launchingcurrent sheets across the full vertical domain. This illustrates that other important aspects of MRI-dynamo
activity in accretion discs, beyond the vertical profile of magnetization, are sensitive to the vertical boundary
conditions and cannot be captured by simulations that do not permit magnetic flux to buoyantly escape thedomain.
Figure 6.2 (right panels) also shows space-time diagrams for the radial component of the magnetic
field. Similarly to the toroidal field, the radial field experiences dynamo cycles of escape and replenishment
for simulations ZNVF-O and NVF-β2. In simulation ZNVF-P, the radial field develops long-lived banded
structures at the β ' 1 interfaces (|z/H0 | ' 1.5). The simulations of Johansen & Levin (2008) developed
similar peaks in the radial field, but at larger scale heights and also at the disc mid-plane, which we do not
observe in simulation ZNVF-P.
Figure 6.3 shows the magnetic field structure at the snapshot in time t = 100 orbits. In all cases, thetoroidal magnetic field is the dominant component. Small-scale turbulent magnetic structures exist in thevicinity of the disc mid-plane for the weakly magnetized simulations (ZNVF-O and ZNVF-P). Comparedto ZNVF-O, simulation ZNVF-P shows an abrupt transition from the highly-turbulent disc core regions
into the magnetically dominated disc atmosphere. The ordered toroidal magnetic field structure in the disc
mid-plane is similar for both ZNVF-O and ZNVF-P. As discussed in Salvesen et al. (2016b), the strongly
magnetized disc simulation NVF-β2 develops ribbon-like field structures near the disc mid-plane and thefield is remarkably well-ordered at the mid-plane.
Our results do not support suggestions that there are non-MRI dynamo routes to sustaining strong
magnetization in the absence of poloidal field. For zero net vertical flux simulations with vertical boundary
conditions that enforce the confinement of toroidal magnetic flux, we do find a very strongly magnetized

181

Figure 6.3: Magnetic field structure for simulations ZNVF-O (left panels), ZNVF-P (center panels), and
NVF-β2 (right panels) at the snapshot in time t = 100 orbits. The top panels show streamlines for thepoloidal magnetic field components and colours show the toroidal magnetic field strength for a slice throughthe xz-plane at y = 0. The bottom panels show streamlines for the horizontal magnetic field components andcolours show the vertical magnetic field strength for a slice through the xy-plane at z = 0. In all cases, thetoroidal magnetic field dominates and is well-organized at the disc mid-plane. Notably, the toroidal magnetic
field has large-scale coherence and is most highly organized for the strongly magnetized disc simulation NVFβ2.

182
corona (βb < 0.1 for |z| > 2H0 ). However, even in this case the disc evolves toward a weakly magnetizedcore (βbmid ' 32). For more physically meaningful boundary conditions that allow mass and magnetic flux
to buoyantly escape, the initially strong toroidal flux is quickly expelled and the disc settles into a weakly
magnetized state (βbmid ' 45) that appears identical to similar simulations initialized with much weakertoroidal fields. We find no local limit in which a strongly magnetized disc (i.e., βbmid . 1) can be sustained
given an initial configuration with zero net vertical flux and outflow boundary conditions.

6.5

Discussion and Conclusions
In this paper we have argued that a strong poloidal magnetic flux threading the accretion disc isnecessary in order to generate and sustain a magnetically dominated state. This conclusion differs fromthat of Johansen & Levin (2008), who observed a strong magnetization (β mid ' 2) in zero net vertical flux
simulations that was sustained for at least a moderate duration (20-30 orbits). It appears likely that thedifference between our results is a consequence of the different vertical boundary conditions. Although wenever find a strongly magnetized disc mid-plane, we do find strong magnetization at quite small heights above
the mid-plane if we adopt periodic boundary conditions that — like those used by Johansen & Levin (2008)— confine the toroidal flux within the simulation domain2 . In the more physically-motivated situation
where magnetic flux can escape, the initially strong flux is buoyantly expelled and the disc settles into aweakly magnetized state characteristic of local disc simulations with zero/weak net vertical flux. As we havenoted previously (Salvesen et al., 2016b), the properties of the MRI dynamo do vary when the disc becomesstrongly magnetized, with the characteristic reversals of toroidal field becoming increasingly infrequent andpossibly entirely absent. Given this, it is possible that the Parker instability (Parker, 1966; Shu, 1974) plays
a role in the dynamo process in highly magnetized discs, as envisaged by Tout & Pringle (1992) and Johansen& Levin (2008). We suggest, however, that a net poloidal flux is a pre-requisite for any dynamo that leads
to strongly magnetized accretion discs.
Even without a zero net vertical flux route to strong magnetization, there are plausible pathways toforming strongly magnetized discs via poloidal fields. The requisite field threading the disc may be generated
2 For technical reasons, we have not been able to run a case with identical boundary conditions to those used by Johansen& Levin (2008), and hence some differences are expected between our run ZNVF-P and theirs.

183
in situ and/or externally. In the in situ scenario, random small-scale field generated by dynamo activity
within local patches of the disc could combine coherently to form a large-scale poloidal magnetic field (Tout& Pringle, 1996; King et al., 2004). It is known that this process generates poloidal fields of sufficient strength
to locally modify the MRI (Sorathia et al., 2010; Beckwith et al., 2011), but whether the ultimate outcomeof such an inverse cascade is sufficient to form a strongly magnetized disc is unclear. In the external field
scenario, a pervading background magnetic field could provide the poloidal flux. This “fossil field” could
be an interstellar magnetic field or a galactic field in the context of accretion discs in X-ray binaries andaround supermassive black holes, respectively. Regardless of the origin of the net poloidal flux, the efficiencyof radial transport of flux through the accretion disc depends upon the effective magnetic diffusivity andthe geometric thickness of the disc (Lubow et al., 1994; Guilet & Ogilvie, 2012). Given favorable conditions
for accumulation of poloidal magnetic flux, a strongly magnetized disc would necessarily follow. In Xray binaries, where there is strong evidence that the disc alternates between geometrically thick and thinconfigurations near the black hole, changes in the transport efficiency could lead to cyclic accumulation andloss of flux (Begelman & Armitage, 2014).
Direct measurement of disc magnetic fields remains difficult. Numerous accretion disc phenomena,
however, appear to be sensitive to the degree of magnetization, and further study of these may allow for
useful constraints. Examples include the variability properties of radiation pressure dominated discs, which
are predicted to be viscously/thermally stable when magnetic pressure-dominated (Begelman & Pringle,
2007), unlike standard disc models without magnetic pressure support (Lightman & Eardley, 1974; Shakura& Sunyaev, 1976), and the size of discs in active galactic nuclei, which are predicted to be more stableagainst fragmentation into stars when strongly magnetized (Pariev et al., 2003; Begelman & Pringle, 2007;
Gaburov et al., 2012) than otherwise (Shlosman & Begelman, 1987). From a more directly observational
perspective, vertical magnetic pressure support acts to harden the emergent accretion disc spectrum (e.g.,Blaes et al., 2006). For extremely magnetized discs, Begelman & Pringle (2007) predict dramatic spectral
hardening with a colour correction factor of fcol ∼ 5. We have previously argued that the evolution of thedisc spectral component during black hole X-ray binary state transitions can be explained by a variable fcol ,arising from changes in the disc vertical structure (Salvesen et al., 2013), and we are currently working to184quantify the degeneracies between fcol and black hole spin determined via the continuum fitting technique(Salvesen et al., in preparation).
For a geometrically thin disc supporting an MRI-dynamo, buoyant fields escape vertically on a timescale (comparable to the free-fall time scale; Begelman et al., 2015; Salvesen et al., 2016b) that is shortcompared to the time scale for viscous radial advection of Bφ . The local approximation, used by bothJohansen & Levin (2008) and ourselves, should then be appropriate. For somewhat thicker discs, longduration global accretion disc simulations with varying levels of background poloidal magnetic flux areneeded. Promising initial studies of strongly magnetized discs in the global limit have already been completed(Machida et al., 2000; Gaburov et al., 2012; Sa̧dowski, 2016). Currently, however, the presence of a net
poloidal magnetic flux is the only demonstrated way to generate and maintain an accretion disc with adynamically important magnetic field (i.e., β mid ∼ 1). Future work on disc accretion physics in this strongly
magnetized regime may help decipher the enigmatic phenomena associated with accretion discs in X-ray
binaries and galactic nuclei.

Acknowledgments
We thank the referee, Anders Johansen, for his careful reading and constructive comments. GS
acknowledges support through the NASA Earth and Space Science Graduate Fellowship program. PJA acknowledges support from NASA under Astrophysics Theory Program awards NNX11AE12G and NNX14AB42G,
and from the NSF under award AST-1313021. J.B.S.’s support was provided in part under contract with
the California Institute of Technology (Caltech) and the Jet Propulsion Laboratory (JPL) funded by NASAthrough the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. MCB acknowledges support from NSF grant AST-1411879. This work used the Janus supercomputer, which is supportedby the National Science Foundation (award number CNS-0821794) and the University of Colorado Boulder.
The Janus supercomputer is a joint effort of the University of Colorado Boulder, the University of Colorado
Denver, and the National Center for Atmospheric Research.

Chapter 7

Conclusions

Over 40 years ago, Shakura & Sunyaev (1973) revolutionized our understanding of astrophysicalaccretion discs around compact objects. The X-ray spectrum emitted by an accretion disc in a black hole Xray binary system was subsequently definitively observed (Mitsuda et al., 1984; Makishima et al., 1986); thus,
validating the basic multi-colour disc blackbody model. Some time later, Balbus & Hawley (1991) identified
the magnetorotational instability (MRI) as the physical mechanism that allows accretion to proceed, which
solidified magnetic fields as playing a fundamental role in disc accretion.
With the launch of X-ray observatories with unprecedented timing (e.g., RXTE) and spectral (e.g.,
XMM-Newton, Chandra) capabilities, a rich phenomenology of black hole X-ray binaries emerged (Remillard & McClintock, 2006). Any explanations for the enigmatic behavior of black hole X-ray binaries (e.g.,spectral state transitions) would have to consider the time-dependent interplay between the accretion disc,wind, corona, and jet. Empirically, this behavior could be explained by a truncated disc model (e.g., Esin
et al., 1997). However, this picture has not yet been unambiguously confirmed observationally nor has aphysical mechanism been identified for a migrating inner disc edge. Chapter 2 was inspired by the largely
unconstrained geometry of the inner disc regions. Fitting X-ray data of the transient black hole X-ray binary
GX 339–4, we showed that a radially fixed (i.e., non-truncated) disc with a variable vertical structure —perhaps due to an evolving magnetic state of the disc — is a viable explanation for the observed disc spectral
evolution. In Chapter 3, we introduced a toy model that appeals to disc tearing — a process that awaits
observational (in)validation — as a potential mechanism for black hole state transitions. While this modelis in its infancy, it is one of few attempts to understand the complete black hole state transition accretion

186
cycle.
Computational advances and the application of the “shearing box” approximation ushered in a new
era for accretion disc physics (Hawley et al., 1995). Numerical simulations of accretion discs in this local
limit opened the door to studying the MRI and its non-linear evolution into turbulence. A major finding
from these simulations is that the MRI turbulent stresses — arising from correlated radial and azimuthal
fluctuations in the magnetic field and fluid velocity — responsible for transporting angular momentum anddriving accretion appear to be inherently local (but see Simon et al., 2012). This is precisely the basis forthe α prescription of Shakura & Sunyaev (1973), which therefore might be an adequate description of MRI
turbulence. Shearing box simulations with vertical density stratification and zero net vertical magnetic flux
show that α converges to ∼ 0.01 (Davis et al., 2010), while a net vertical magnetic flux couples the field
to the fluid and enhances turbulent transport of angular momentum (Hawley et al., 1995). In Chapter 5,
we performed shearing box simulations that included a background net vertical magnetic flux of varying
strength that spanned nearly the full range over which the MRI is linearly unstable. We found that the
MRI-driven turbulent stress parametrized by α is tightly correlated with the net vertical flux, achieving
α ∼ 0.01 − 1. Accretion discs in dwarf novae (a subclass of cataclysmic variables) have α ∼ 0.1 − 0.4 (Kinget al., 2004), while α has yet to be measured in a black hole accretion disc.
Dynamo activity materializes in simulations that incorporate stratification of the disc atmosphere
due to the vertical gravity of the black hole (Brandenburg et al., 1995). The linearly unstable vertical field
generates radial field, which is stretched into a dominant azimuthal field by the differential rotation. This
growing azimuthal field buoyantly rises, producing vertical field and completing the cycle. What’s more,
the MRI-dynamo continually produces quasi-periodic alternating signs of azimuthal magnetic field thatbuoyantly escapes. In the presence of a net vertical magnetic flux, the dynamo can constantly feed off of thisbackground field to generate an even stronger azimuthal field. In Chapter 5, we characterized the propertiesof the MRI-dynamo for a very wide range in net vertical magnetic flux and found that the quasi-periodic
flip-flops in the azimuthal field can persist even when the disc mid-plane is magnetic pressure-dominated.
We also found that these simulations support the simple model for dynamo activity given by Begelman et al.
(2015).

187
Inspired by observations of accretion disc winds in black hole X-ray binaries that imply strong magnetic
fields (e.g., Miller et al., 2006b, 2016), we investigated how such a strongly magnetized accretion disc mightform. In Chapter 5, we showed that given a sufficiently strong background poloidal flux, a strong azimuthal
field will develop in the disc mid-plane regions and be sustained by dynamo activity. In Chapter 6, weconsidered the alternative initial condition of a zero net vertical magnetic flux and an initial azimuthal
magnetic field in equipartition with the gas pressure. This initially strong azimuthal field could not besustained by dynamo activity; therefore, we conclude that strongly magnetized discs require poloidal flux.
While understanding the phenomena and physics of black hole accretion discs is interesting in its own
right, these machines can also be used as tools for measuring black hole spin as follows. The location of
the innermost stable circular orbit (ISCO) is uniquely set by the black hole spin. Because regions interior
to the ISCO are thought to be devoid of emitting gas, a measurement of the ISCO interpreted as the inneredge of the accretion disc acts as a measurement of the black hole spin. In Chapter 4 we showed that theblack hole spin measured by two well-established techniques are often at odds with each other for the same
X-ray binary. Dissecting the continuum fitting method for measuring black hole spin, we demonstrated
that uncertainties related to the vertical structure of the disc dominate the error budget, yet these are notproperly taken into consideration. We emphasize that the observed accretion disc spectrum is sensitive tothe contribution from magnetic pressure support to the vertical hydrostatic equilibrium (Blaes et al., 2006),
particularly if the disc is strongly magnetized (Begelman & Pringle, 2007). Therefore, the magnetic state ofthe disc should be considered before these spin measurements may be deemed robust.
The compelling evidence for and important implications of strongly magnetized accretion discs justifies
their further study, which is what I plan to do in the near future. However, whether or not black hole accretion
discs turn out to be strongly magnetized remains to be seen.

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Appendix A

Probability Density Functions and Change of Variables

We wish to obtain the probability density function (PDF) for the black hole spin parameter fromthe simplified version of the continuum fitting technique described in §4.3. This requires developing thenecessary mathematics for transforming between various PDFs by change of variables as follows.
Using change of variables, we develop the equations to transform between the flux normalization PDF,
f (Kflux ), and the inner disc radius PDF, f (rin ), in §A.0.1 and §A.0.2. We can relate the flux normalization,
Kflux , to the non-dimensional inner disc radius, rin = Rin /Rg , by rewriting Equation 4.3 as,
Kflux =

1
4
fcol



GM/c2
D

2

2
cos (θdisc ) rin
.

(A.1)

Similarly, we will also derive the transformation between the innermost stable circular orbit PDF,
f (rISCO ), and the black hole spin parameter PDF, f (a), in §A.0.3 and §A.0.4. The equation relating theinnermost stable circular orbit, rISCO , to the black hole spin parameter, a, is (Bardeen et al., 1972),

p



3 + Z2 − (3 − Z1 ) (3 + Z1 + 2Z2 ) : 0 ≤ a ≤ +1
rISCO =
(A.2)




p


3 + Z2 + (3 − Z1 ) (3 + Z1 + 2Z2 ) : −1 ≤ a < 0,where,i1/3 h1/3
1/3
Z1 ≡ 1 + 1 − a 2(1 + a) + (1 − a)q
Z2 ≡ 3a2 + Z12 .

(A.3)
(A.4)

200
A.0.1

Flux Normalization PDF, f (Kflux )Given the set of independent1

random variables M , D, θdisc , fcol , and rin with corresponding

PDFs f (M ), f (D), f (θdisc ), f (fcol ), and f (rin ), we wish to find the PDF f (Kflux ) for random variable Kflux , where Kflux = Kflux (M, D, θdisc , fcol , rin ) according to Equation A.1. The multivariate PDF
f (M, D, θdisc , fcol , rin ) transforms to f (M, D, θdisc , fcol , Kflux ) by change of variables,f (M, D, θdisc , fcol , Kflux ) = f (M, D, θdisc , fcol , rin ) |Jrin →Kflux | ,(A.5)

where, since the random variables are independent,f (M, D, θdisc , fcol , rin ) = f (M ) f (D) f (θdisc ) f (fcol ) f (rin ) ,(A.6)

and Jrin →Kflux is the Jacobian determinant for the change of variables given by,
Jrin →Kflux = det∂ (M, D, θdisc , fcol , rin )∂ (M, D, θdisc , fcol , Kflux )∂rin∂Kflux


D
1
1 2
−1/2
p
Kflux .
= fcol2
GM/c2
cos (θdisc )
=

(A.7)

To find the flux normalization PDF, f (Kflux ), we marginalize f (M, D, θdisc , fcol , Kflux ) over M , D, θdisc ,and fcol to get,
ZZZZ
f (x) f (rin ) |Jrin →Kflux | dx,

f (Kflux ) =

(A.8)

Rx

where,
Z ∞

ZZZZ

Z π/2

Z ∞ Z ∞

≡
Rx

(A.9)
fcol =1

θdisc =0

D=0

M =0

f (x) ≡ f (M ) f (D) f (θdisc ) f (fcol )dx ≡ dM dD dθdisc dfcol .

(A.10)(A.11)

Equation A.8 allows us to transform from f (rin ) to f (Kflux ).
1 Here and throughout, we assume that the random variables undergoing a transformation by change of variables are
independent.

201
A.0.2

Inner Disc Radius PDF, f (rin )We will also have cause to run the PDF transformation described in §A.0.1 in reverse, as follows.
Given the set of independent random variables M , D, θdisc , fcol , and Kflux with corresponding PDFs f (M ),f (D), f (θdisc ), f (fcol ), and f (Kflux ), we wish to find the PDF f (rin ) for random variable rin , whererin = rin (M, D, θdisc , fcol , Kflux ) according to Equation A.1. The multivariate PDF f (M, D, θdisc , fcol , Kflux )transforms to f (M, D, θdisc , fcol , rin ) by change of variables,f (M, D, θdisc , fcol , rin ) = f (M, D, θdisc , fcol , Kflux ) |JKflux →rin | ,(A.12)

where, since the random variables are independent,f (M, D, θdisc , fcol , Kflux ) = f (M ) f (D) f (θdisc ) f (fcol ) f (Kflux )(A.13)and JKflux →rin is the Jacobian determinant for the change of variables given by,
JKflux →rin = det∂ (M, D, θdisc , fcol , Kflux )∂ (M, D, θdisc , fcol , rin )∂Kflux
∂rin

2
2
GM/c2
= 4
cos (θdisc ) rin .
fcol
D

=

(A.14)

To find the inner disc radius PDF, f (rin ), we marginalize f (M, D, θdisc , fcol , rin ) over M , D, θdisc , and fcol
to get,ZZZZ
f (x) f (Kflux ) |JKflux →rin | dx.

f (rin ) =(A.15)

Rx

Equation A.15 allows us to transform from f (Kflux ) to f (rin ).

A.0.3

Black Hole Spin Parameter PDF, f (a)While the measurable quantity is the inner disc radius PDF, we are interested in converting this toa black hole spin parameter PDF under the assumption that rin = rISCO . Given the PDF f (rISCO ) for
random variable rISCO , we can transform this to the PDF f (a) for random variable a according to Equation
A.2 by change of variables,f (a) = f (rISCO ) |JrISCO →a | ,(A.16)

202
where JrISCO →a is the Jacobian determinant for the change of variables given by,
JrISCO →a ==drISCOdadZ1
2(3 − Z1 ) dZ
 dZ2
da − (Z1 + Z2 ) da− p
: 0 ≤ a ≤ +1
 da
(3 − Z1 ) (3 + Z1 + 2Z2 )(A.17)




dZ1

2

(3 − Z1 ) dZ
dZ2
da − (Z1 + Z2 ) da
+ p
: −1 ≤ a < 0,da
(3 − Z1 ) (3 + Z1 + 2Z2 )where,
i
1/3
dZ1
1h
−2/3
−2/3
(1 + a)− (1 − a)=
1 − a2
da
3
i
−2/3
2 h
1/3
1/3
1 − a2− a (1 + a) + (1 − a)31
3a + Z1 dZ
dZ2
da
= p
.
da
3a2 + Z12

(A.18)(A.19)

Equation A.16 allows us to transform from f (rISCO ) to f (a).

A.0.4

Innermost Stable Circular Orbit PDF, f (rISCO )The reverse of the PDF transformation process in §A.0.3 is to compute the PDF f (rISCO ) for randomvariable rISCO given the PDF f (a) for random variable a. However, solving Equation A.2 for a is analyticallyintractable, which prohibits computing the Jacobian determinant for the change of variables, Ja→rISCO =da/drISCO . Instead, we compute the unnormalized innermost stable circular orbit PDF, f ∗ (rISCO ), by
iteratively solving Equation A.2 for the black hole spin parameter, a, across all values of rISCO . We thenrenormalize f ∗ (rISCO ) to obtain the desired innermost stable circular orbit PDF, f (rISCO ). This process
allows us to transform from f (a) to f (rISCO ).

Appendix B

Effects of Individual Parameter Uncertainties on Black Hole Spin
Measurements

For the basic continuum fitting technique of §4.3, Equation A.1 relates the “ancillary observables,”Kflux , M , D, θdisc , and fcol to the “primary observable,” rin , which is needed to obtain a black hole spinmeasurement. Consider a black hole X-ray binary for which all but one ancillary observable are measured
exactly. This one ancillary observable of interest, x, is measured with probability density function (PDF),
f (x) = N (x, µx = x0 , σx = 0.1x0 ) ,(B.1)

where N (x, µx , σx ) is the normal distribution (Equation 4.5) and a prime represents the true parametervalue. The PDFs for all other ancillary observables, y, are Dirac delta functions centered on the true
parameter values,
f (y) = δ (y − y0 ) .

(B.2)

We now seek to determine the contribution to the uncertainty in the primary observable rin (andultimately the black hole spin parameter a) by each individual ancillary observable. The PDF for the
ancillary observable of interest, f (x), is related to the PDF for the inner disc radius fx (rin ) by change ofvariables from x to rin ,fx (rin ) = f (x) |Jx |,(B.3)

where the Jacobian determinant for the change of variables is Jx = ∂x/∂rin and Equation A.1 relates x to204
rin . The Jacobian determinants for each individual ancillary parameter of interest are,
JKflux =0∂Kflux2Kflux
rin
=
02
∂rin
rin

(B.4)

JM =

∂M
0 −2
= −M 0 rin
rin
∂rin

(B.5)

JD =

∂D
D0
= 0
∂rin
rin

(B.6)

Jθdisc =∂θdisc∂rin"

002= 2cos (θdisc) rin

Jfcol =

1 − cos

20(θdisc) 0 4 #−1/2
rin
−3
rin
rin

f0
∂fcol
−1/2
= pcol0 rin .
∂rin
2 rin

(B.7)
(B.8)

Pairing each f (x) (Equation B.1) with the corresponding Jx , Equation B.3 gives,
fKflux (rin ) =0
2Kflux√ rin
02
rin σKflux 2π


2 
 2
rin
0
− µKflux 0 − Kflux rin

× exp 

2
2σK


flux


(B.9)

2 

0
0 rin
−
µ
−
M
M
rin


−2
√ rin
fM (rin ) =
exp 

2
2σM
σM 2π

(B.10)


2 
0 rin
−
µ
−
D
0
D
rin
D


√ exp fD (rin ) = 0
22σDrin σD 2π(B.11)

0
M 0 rin0" 0 4 #−1/2
002
r2cos (θdisc
) rin
−3
0
√
fθdisc (rin ) =1 − cos2 (θdisc
) in
rin
r
σθdisc 2π
in
 

2 
 0 2 rin−10− µθdisc cos (θdisc ) rin − cos× exp 2σθ2disc


f0
−1/2
ffcol (rin ) = p 0 col √ rin
2 rin σfcol 2π
 
2 
q
rin
0
−
f
−
µ
0
fcol


col
rin


× exp .
2
2σfcol

(B.12)(B.13)

Finally, we obtain fx (a) from fx (rin ) from the methodology of §A.0.3 and plot the results in Figure

205
4.2.