Black hole accretion disks [Elektronische Ressource] : sources of viscosity and signatures of super-Eddington accretion / by Dominikus Heinzeller

Dissertationsubmitted to theCombined Faculties for the Natural Sciences and for Mathematicsof the Ruperto-Carola University of Heidelberg, Germanyfor the degree ofDoctor of Natural SciencePut forward byDipl.-Phys. Dominikus Heinzellerborn in Weilheim i. OB (Germany)thOral examination: July16 2008Black hole accretion disksSources of viscosity and signaturesof super-Eddington accretionReferees: Prof. Dr. Wolfgang J. DuschlProf. Dr. Shin MineshigeZusammenfassungAkkretionsscheiben um Schwarze LöcherQuellen der Viskosität und Spuren von über-Eddington AkkretionWir untersuchen die Rolle der Konvektion in Akkretionsscheiben um Schwarze Löcher, ins-besondere den Einfluss auf den Energietransport und die Auswirkung konvektiver Turbulenz aufdie Viskosität in der Scheibe. Wir zeigen, dass Konvektion den Energietransport durch Strahlungim Falle einer masselosen Scheibe effizient unterstützt, während es im umgekehrten Fall einerselbstgravitierenden Scheibe zu negativen Rückkopplungseffekten kommt. Obwohl konvektiveTurbulenz einen signifikanten Beitrag zur gesamten Viskosität leistet, kann sie nicht alleine alsErklärung dafür dienen.Im zweiten Teil untersuchen wir die spektrale Energieverteilung von über-Eddington akkretie-renden Schwarzen Löchern, basierend auf 2D strahlungs-hydrodynamischen Simulationsdaten.Wir berechnen die Kontinuumemission und die Emission und Absorption der Eisen-K-Linienmittels einer Ray-tracing Methode.
Publié le : mardi 1 janvier 2008
Lecture(s) : 23
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Source : ARCHIV.UB.UNI-HEIDELBERG.DE/VOLLTEXTSERVER/VOLLTEXTE/2008/8575/PDF/PHD_THESIS.PDF
Nombre de pages : 143
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Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Ruperto-Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Science
Put forward by
Dipl.-Phys. Dominikus Heinzeller
born in Weilheim i. OB (Germany)
thOral examination: July16 2008Black hole accretion disks
Sources of viscosity and signatures
of super-Eddington accretion
Referees: Prof. Dr. Wolfgang J. Duschl
Prof. Dr. Shin MineshigeZusammenfassung
Akkretionsscheiben um Schwarze Löcher
Quellen der Viskosität und Spuren von über-Eddington Akkretion
Wir untersuchen die Rolle der Konvektion in Akkretionsscheiben um Schwarze Löcher, ins-
besondere den Einfluss auf den Energietransport und die Auswirkung konvektiver Turbulenz auf
die Viskosität in der Scheibe. Wir zeigen, dass Konvektion den Energietransport durch Strahlung
im Falle einer masselosen Scheibe effizient unterstützt, während es im umgekehrten Fall einer
selbstgravitierenden Scheibe zu negativen Rückkopplungseffekten kommt. Obwohl konvektive
Turbulenz einen signifikanten Beitrag zur gesamten Viskosität leistet, kann sie nicht alleine als
Erklärung dafür dienen.
Im zweiten Teil untersuchen wir die spektrale Energieverteilung von über-Eddington akkretie-
renden Schwarzen Löchern, basierend auf 2D strahlungs-hydrodynamischen Simulationsdaten.
Wir berechnen die Kontinuumemission und die Emission und Absorption der Eisen-K-Linien
mittels einer Ray-tracing Methode. Wir zeigen, dass relativistische Beaming-Effekte für frontal
betrachtete Scheiben zu über-Eddington Leuchtkräften führen. Die Eisen-Linien erweisen sich
als guter Indikator für den Akkretionsprozess in den inneren Scheibenregionen: Es zeigt sich
eine enge Korrelation zwischen dem Verhältnis der K -Linien zu den K Linen und der Zentral-β α
masse, sowie zwischen der Linienbreite und dem Beobachtungswinkel.
Abstract
Black hole accretion disks
Sources of viscosity and signatures of super-Eddington accretion
We study the role of convection in black hole accretion flows. We investigate the influence of
convection on the energy transport as well as the effect of convective turbulence on the disk’s
viscosity. The results reveal that convection supports the radiative energy transport efficiently
in massless disks, while it can turn into a negative feedback if self-gravity becomes important.
Convective turbulence adds significantly to the total viscosity, but cannot account for it on its
own.
In the second part, we study the spectral energy distribution of super-Eddington accretion
flows onto a black hole, based on 2D RHD simulation data. We model the continuum emission as
well as the iron K line emission and absorption features with a ray-tracing radiative transfer code.
We find that mild relativistic beaming effects become important, leading to super-Eddington
luminosities for face-on seen disks. We confirm the diagnostic power of the iron K lines on the
accretion process in the inner disk region, finding a strong correlation between the central black
hole mass and the ratio of the K to the K lines. We also detect a trend of line broadening forα β
edge-on seen disks.Contents
1 A small bubble in the Universe 3
2 The diffusion limit and convective feedback in selfgravitating disks 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Adaptable viscosity parameterβ and numerical techniques . . . . . . . . 12
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
˙2.3.1 Influence ofM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Influence ofχ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.3 Influence ofβ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17max
2.4 Comparison with the classical diffusion limited case . . . . . . . . . . . . . . . . 17
2.5 The influence of turbulence on the energy transport . . . . . . . . . . . . . . . . 20
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 The role of convection in black hole accretion disks 25
3.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Set-up and nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Radial structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Vertical stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.1 Structure equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.3 Opacityκ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.4 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Numerical solution of the vertical stratification . . . . . . . . . . . . . . . . . . 36
3.5.1 Set of discretized equations . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5.2 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.3 Numerical implementation ofκ . . . . . . . . . . . . . . . . . . . . . . 41
3.5.4 Agreement of disk and atmosphere solution . . . . . . . . . . . . . . . . 41
3.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6.1 Parameters, simulation characteristics . . . . . . . . . . . . . . . . . . . 41
3.6.2 Disk properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 SED of super-Eddington flows I – continuum processes 57
4.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.1 RHD simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.2 Equation of radiative transfer . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.3 Frequency-dependent radiation quantities . . . . . . . . . . . . . . . . . 61
1Contents
4.2.4 Flux limited diffusion approximation . . . . . . . . . . . . . . . . . . . 62
4.2.5 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.6 Color-corrected temperatures . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.1 Overall spectral properties . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.2 Angular dependence of the luminosity . . . . . . . . . . . . . . . . . . . 68
4.3.3 Blackbody fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5 SED of super-Eddington flows II – the iron K line complex 75
5.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Frequency-dependent bound-free absorption . . . . . . . . . . . . . . . . . . . . 77
5.3.1 Bound-free absorption coefficients . . . . . . . . . . . . . . . . . . . . . 78
5.3.2 Number densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.3 Cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3.4 Upper limits on the contribution of excitation levels . . . . . . . . . . . . 81
5.4 Line transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4.1 Atomic population calculations and Debye’s theory . . . . . . . . . . . . 83
5.4.2 Partition functions for metals . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4.3 Saha equation for metals . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.4 Line profile functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.5 Standard line transition data . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4.6 Fluorescence lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4.7 Supplement to the numerical calculation . . . . . . . . . . . . . . . . . . 89
5.5 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5.1 Basic assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5.2 Atmosphere model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.5.3 Modification of the radiative transfer equation . . . . . . . . . . . . . . . 97
5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.6.1 Data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.6.2 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Epilog 119
Acknowledgements 121
References 123
Appendix 129
A.1 Physical constants in the cgs system of units . . . . . . . . . . . . . . . . . . . . 129
A.2 The Henyey method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.3 Iron K-shell fluorescence data . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
21 A small bubble in the Universe
When I began my physical studies [in Munich in 1874] and sought advice
from my venerable teacher Philipp von Jolly . . . he portrayed to me physics
as a highly developed, almost fully matured science . . . Possibly in one or
another nook there would perhaps be a dust particle or a small bubble to be
examined and classified, but the system as a whole stood there fairly secured,
and theoretical physics approached visibly that degree of perfection which,
for example, geometry has had already for centuries.
– from a 1924 lecture by Max Planck (Sci. Am, Feb 1996)
On the very day when this introduction was written (April 23, 2008), we were celebrating the
th150 birthday of Max Planck. Fortunately, Planck decided to study physics despite the bleak
future for research that was presented to him. In 1901, he published an article with the title “On
the Law of Distribution of Energy in the Normal Spectrum”, describing the spectral radiance of
electromagnetic radiation at all wavelengths of a blackbody at a given temperature. This was
not only the cornerstone in his career, it also paved the way for modern physics and astronomy.
Only through Planck’s law, astronomers were able to model the emitted spectrum of a star, a
blackbody radiator in zeroth order, without running into the ultraviolett catastrophe. Quantum
mechanics and atomic physics would not exist without this fundamental discovery.
Nowadays, more than 100 years later, we know that von Jolly could not have been further off
the mark. Although our understanding of the Universe broadened to an extent almost beyond
belief, it seems that with the answer of one question, at least ten others are rising. And so it
happens that this thesis deals with two out of many open questions in one of von Jolly’s small
bubbles. And even there, we again encounter Planck’s blackbody radiation law and its offsprings
in atomic physics.
In this dissertation, we study the properties of accretion disks around black holes. Having
called them one of von Jolly’s small bubbles, it needs to be put into the right context. Yes, it
is certainly only one corner of physics where questions remain to be answered. At the same
time, accretion disks are ubiquitous! They can be found almost everywhere in the Universe,
from today back in time until the Cosmos was less than one million years old, from sizes of
11 18about one solar radius (10 cm) in low mass X-ray binary systems up to one parsec (10 cm)
in active galactic nuclei, and around a wealth of objects like protostars, white dwarfs, neutron
stars or black holes. Despite this huge variety, the driving physical principle, the accretion of
matter onto a central object through a disk-like structure, remains the same. Understanding the
key process of accretion is therefore one of the big challenges, but also one of the big chances of
stastronomy in the21 century. Here, we investigate two pixels of the overall picture.
31 A small bubble in the Universe
The source of viscosity in astrophysical disks Theoretical modeling of accretion disks
dates back to the year 1948, when Weizsäcker published his article about the rotation of cosmic
gas (Weizsäcker, 1948). A key ingredient to describe the accretion process is the origin of the
viscosity, which causes friction in the disk and an inward motion of the material. In those early
years, it was generally believed that molecular viscosity is responsable for this effect. However,
the first observations of accretion disks in cataclysmic variables, which allowed to deduce the
typical timescales of the accretion process, threw over the theoretical expectations (see, e. g.,
Prendergast & Burbidge (1968); Pringle & Rees (1972)). They revealed, that the numbers mea-
sured in the lab and those needed to account for the observations differed by about ten orders of
magnitude. The following decades saw a plethora of attempts to resolve this puzzling situation,
but none of them succeeded. Only Shakura & Sunyaev (1973) proposed a parameterization, the
α-viscosity, by which all of a sudden most observations could be fitted adequately. Despite this
success, the Shakura-Sunyaev viscosity remains a purely empirical description and no physical
explanation, and it is limited to thin disks with negligible disk masses.
Among the physical theories, the most promising ones are:
• Differential rotation. Accretion disks are mostly showing a nearly Keplerian rotation
profile. An obvious candidate for the turbulence in such a disk is therefore differential ro-
tation. From early laboratory experiments on rotating Couette-Taylor flows (Wendt, 1933;
Taylor, 1936), this possibility was first ruled out. However, in recent re-investigations,
Richard & Zahn (1999) and Richard (2001) concluded that differential rotation can give
rise to turbulence, despite published arguments. At the same time, Duschl et al. (2000)
formulated the β-viscosity description. Although being a parameterization like its an-
cestor, it can actually be related to the process of differential rotation. Contrary to the
α-prescription, the β-viscosity accounts properly for the selfgravity of the disk. At the
same time, it includes the α-viscosity in the case of a shock dissipation limited, non-
selfgravitating disk.
• Convection. The process of accretion is the most efficient way of producing energy, out-
classing nuclear fission by a factor of at least 100. In order to account for the transport
of these huge amounts of energy, convection is considered to support or even dominate
in some cases over radiation. It is therefore natural to regard the turbulence caused by
convective motion as a possible candidate for viscosity. Again, first (semi-)analytical in-
vestigations gave discouraging results, since they led to hugely massive disks which could
not be explained, given theα-viscosity description (Vila, 1981; Duschl, 1989). Ruden et
al. (1988); Ryu & Goodman (1992) studied convective instabilities in thin gaseous disks
and confirmed that angular momentum transport can be supported by convective turbu-
lence. Goldman & Wandel (1995) investigated accretion disks where viscosity is given by
convection solely and where the energy transport is maintained by radiation and convec-
tion. They found the resulting viscosity being too low by a factor of 10 to 100, but could
1not draw a final conclusion, since their disk model was oversimplified.
• Magneto-rotational instability. The magneto-rotational instability (MRI) was first no-
ticed in a non-astrophysical context by Velikhov (1959); Chandrasekhar (1960) when con-
sidering the stability of a Taylor-Couette flow of an ideal hydromagnetic fluid. More than
1The authors applied a one-zone approximation (c. f., Sect.2.1) in their models and estimated the contribution of
convection to the energy transport in an overly simple way.
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