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Turbulent thermonuclear combustion in degenerate stars [Elektronische Ressource] / Wolfram Schmidt

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Published 01 January 2004
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Max-Planck-Institut fur¨ Astrophysik
Turbulent Thermonuclear
Combustion in Degenerate Stars
Wolfram Schmidt
Vollstandiger¨ Abdruck der von der Fakultat¨ fur¨ Physik der Technischen Uni-
versitat¨ Munchen¨ zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr. L. Oberauer
Prufer¨ der Dissertation: 1. Hon.-Prof. Dr. W. Hillebrandt
2. Univ.-Prof. Dr. M. Lindner
Die Dissertation wurde am 19. 1. 2004 bei der Technischen Universitat¨ Munchen¨
eingereicht und durch die Fakultat¨ fur¨ Physik am 20. 2. 2004 angenommen.ii
’Forgive my ignorance of stellar physics, but I’ve been
studying, so let me see if I get this right.’
’When that big, whirling cloud of dross and corpses
finally collapses, it’s going to dump a tenth of a solar
mass onto the hot, dense surface of that white dwarf. A
dwarf that’s already near its Chandrasekhar limit. Much
of the new material will compress to incredible density
and undergo superfast nuclear fusion, triggering–’
’What Earthlings used to call a “type one” superno-
va,’ the Niss Machine cut in, unable to resist an inbuilt
yen to interrupt.
’Normally, this happens when a large amount of mat-
ter is tugged off a giant star, falling rapidly onto a neigh-
boring white dwarf. In this case, however, the sudden
catalyzing agent will be the flesh of once living beings!
Their body substance will help light a pyre that should
briefly outshine this entire galaxy, and be visible to the
boundaries of the the universe.’
D. Brin, Heaven’s ReachMotivation and Objectives
This Thesis is mainly concerned with turbulence and its significance for thermonu-
clear burning processes in degenerate stars. Contrary to main sequence stars such
as the Sun, in which a feed-back mechanism between reaction rates, expansion and
cooling moderates thermonuclear burning, the physical conditions in a white dwarf
of nearly critical mass entail a thermonuclear runaway, once density and temperature
rise above a certain threshold. The critical mass of a white dwarf is known as Chan-
drasekhar mass. In the course of the run-away, carbon and oxygen is rapidly burned
to heavier elements, in particular, nickel, and the star explodes due to the enormous
energy release. According to our present understanding, burning in these thermonu-
clear supernovae progesses as deflagration, which means that ignition is caused by heat
conduction rather than shock compression. Since the intrinsic propagation speed of a
deflagration front is much less than the speed of sound, there must be something acting
to accelerate the burning process. The agent is thought to be turbulence, which folds
and wrinkles flames and thereby increases the rate of burning. A major difficulty of
describing turbulent burning in numerical simulations stems from the fact that it is im-
possible to resolve the whole range of dynamical scales, even with the most powerful
of currently available computers. This restriction leads to the concept of a large-eddy
simulation, in which only the largest scales are numerically resolved. However, since
the burning process is susceptible to turbulent velocity fluctuations on scales smaller
than the numerical resolution, a model which accounts for effects on these scales is
indispensable. The investigation of several options for such a subgrid scale model is
the research subject of this Thesis. As testing and comparing different subgrid scale
models systematically in full supernova simulations would be quite hard, a simplified
scenario was chosen, where turbulence is artifically produced by a stochastic force
field in a cubic domain. In the beginning, pure hydrodynamical turbulence was in-
vestigated, and then thermonuclear burning was added. The research goal has been
to some extent a phenomenological understanding of turbulence and turbulent burn-
ing, but eventually it aims at a subgrid scale model, which makes physically sound
predictions and is applicable to simulations of thermonuclear supernovae.Acknowledgements
First of all, I thank my advisors W. Hillebrandt and J. C. Niemeyer for their support.
I am indepted to Wolfgang, who offered me the opportunity of doing a doctoral the-
sis in the supernova team of the Max-Plack-Institute for Astrophysics. Jens initiated
my research with a paper on dynamical subgrid scale modelling, and the idea of us-
ing stochastic stirring in a cubic box is also his. I thank my colleague F. Ropk¨ e for
many discussions and, particularly, M. Reinecke for answering a plethora of questions
regarding the code and running simulations on various platforms. Without Martin’s
help I would hardly have managed to get it going! Valuable last minute input came
from F. Kupka, who has already played an influencial role in the course of my diploma
thesis. The research presented in this Thesis would not have been possible at all with-
out high-end computing resources. The required computing power was provided by
the Hitachi SR-8000 supercomputer of the Leibniz Computing Centre in Munich and
the IBM p690 of the Computing Centre of the Max-Planck-Society in
There are always people behind the scenes, whose support is nonetheless indis-
pensable. Most of all, I am grateful for the continual encouragement and financial aid
of my parents. Without naming them individually, I finally thank many friends of mine
for their share in recreational activities, which rank highly among the preconditions for
creative work of any kind.Contents
Motivation and Objectives iii
Acknowledgements v
1 White Dwarfs and Thermonuclear Supernovae 1
1.1 Physical Foundations .... ...... ..... ...... ..... 2
1.1.1 Degeneracy ..... 2
1.1.2 Turbulence ...... ..... ...... ..... 5
1.1.3 Combustion ..... 10
1.2 Models and Observations . . ...... ..... ...... ..... 16
1.2.1 Progenitor Scenarios 16
1.2.2 Ignition and Explosion Models . ..... ...... ..... 19
1.2.3 Lightcurves and Spectra .... 21
2 Forced Isotropic Turbulence 25
2.1 Dynamical Equations and Spectral Representation ...... ..... 25
2.1.1 The Piece-Wise Parabolic Method .... 26
2.1.2 Fourier Transforms of Periodic Dynamical Variables ..... 28
2.2 Stochastic Forcing ..... ...... ..... ...... 29
2.2.1 The Ornstein-Uhlenbeck Process ..... 29
2.2.2 Spectral Representation of the Force Field...... 30
2.2.3 The Physical Stochastic Force Field . . . ..... 33
2.3 Physical Scales . . ..... ...... ..... ...... 34
2.3.1 Mass Density .... ..... 35
2.3.2 Fermi Energy and the Speed of Sound . . ...... 35
2.3.3 Characteristic Velocity and Integral Length Scale . . ..... 37
2.4 Global Statistics and Flow Structure . . ..... ...... 38
2.4.1 Dimensionless Physical Quantities .... ..... 38
2.4.2 Solenoidal Forcing at Low Density . . . ...... 38
2.4.3 F at High . . . ..... 42
2.4.4 Partially Dilatational Forcing . . ..... ...... 45
2.5 Spectral Analysis . ..... ...... ..... 49
2.5.1 The Energy Spectrum Function ..... ...... 49
2.5.2 Numerical Dissipation ..... ..... 52
2.5.3 Turbulence Energy Spectra . . . ..... ...... 53
2.5.4 Dissipation Length Scales . . . ..... 56viii CONTENTS
3 Subgrid Scale Models 61
3.1 The Filtering Approach . . . ...... ..... ...... ..... 62
3.1.1 Turbulence Energy . 63
3.1.2 A Non-Linear Algebraic Model ..... ...... ..... 64
3.1.3 Decomposition of the Kinetic Energy Conservation Law . . . 66
3.1.4 Closures for the Turbulence Energy Equation .... ..... 68
3.1.5 Dynamical Procedures ..... ..... ...... 70
3.2 The Self-Similarity of Turbulence . . . ..... 73
3.2.1 Hierarchical Filtering ...... ..... ...... 73
3.2.2 Production ..... ..... 76
3.2.3 Dissipation ...... ..... ...... 80
3.2.4 Diffusion . ..... ..... 80
4Deflagration in the Cube 83
4.1 Burning and the Problem of Flame Tracking . . . ...... ..... 84
4.1.1 The Level Set Method ..... ..... 84
4.1.2 The Turbulent Flame Speed . . ...... ..... 87
4.2 Transient Laminar Burning in a Developing Flow 89
4.2.1 Critical Parameters . ...... ..... ...... ..... 89
4.2.2 Evolution of the Burning Process .... 91
4.3 Turbulent Burning . ..... ...... ..... ...... ..... 94
4.3.1 The Algebraic vs. the Dynamical Model . 94
4.3.2 Statistical vs. Localised Closures..... ...... ..... 98
4.3.3 The Similarity Closure for Production . . 101
4.3.4 The Semi-Localised Model . . . ..... ...... ..... 104
4.3.5 Active Subgrid Scale Modelling 107
4.3.6 The Evolution of Turbulent Burning . . . ...... ..... 112
5Resum´ e´ 117
5.1 Concerning Turbulence . . . ...... ..... ...... ..... 117
5.1.1 Localisation of the Subgrid Scale Closures ..... 118
5.1.2 Flame Physics and the Level Set Method ...... ..... 120
5.2 Towards New Supernova Explosion Models . . . 121
Bibliography 123
A Numerical Techniques 131
A.1 Implementation of Discrete Filters . . . ..... ...... ..... 131
A.1.1 Spatial Filtering . . . ...... 132
A.1.2 Temporal . ..... ...... ..... 134
A.2 Implementation of Subgrid Scale Diffusion . . . 134Chapter 1
White Dwarfs and
Thermonuclear Supernovae
Der wichtigste Stern istα CMa, Sirius, der Hundsstern;
¨bei den Agyptern stand der Große Hund fur¨ Anubis, den
schakalkop¨ figen Wachter¨ der Totenstadt. Sirius ist mit
m−1 .4 der hellste Stern am Himmel, ein A1-Stern in we-
niger als 9 Lichtjahren Entfernung; er hat einen kleinen
Belgeiter, der ihn in einem Abstand von 2,9 Milliarden
Kilometer in rund 50 Jahren einmal umrundet: Sirius B
mmit einer Helligkeit von 9 .1 ist der erste Weiße Zwerg,
den man beobachtet hat.
Baker and Hardy, Der Kosmos-Sternfuhr¨ er
The mountain smoked beneath the moon;
The dwarves, they heard the tramp of doom.
J. R. R. Tolkien, The Hobbit
In 1862, A. G. Clarke found the dim companion of the bright star Sirius A, after F.
W. Bessel had deduced from parallax measurements that Sirius, in fact, is a binary star
system. It came as a great surprise to astronomers when in 1915 W. Adams discovered
that Sirius B is a blue-white star, much too hot for its luminosity L≈ 0.03L . Indeed,
the startling consequence inferred from the Stefan-Boltzmann law was that Sirius B
must be extraordinarily compact, a result first dismissed by astronomers as being ab-
4surd. Later the temperature of Sirius B was determined to be 2.7· 10 K, implying a
6 −3radius of a mere 5500 km and a density of about 3· 10 gcm . The mystery was lifted
by R. H. Fowler in 1926. He applied the newly discovered exclusion principle of Pauli
and was able to show that a star like Sirius B must be entirely supported by electron
degeneracy pressure against gravity. Today a great variety of these white dwarfs are
known, with masses typically in the range from 0.4to0.8 M and an average density
5 −31of 4.7· 10 gcm .
1Sirius B has a mass of about 1 M which is larger than the mass of most known white dwarfs. More
on the phenomenology of white dwarfs can be found in Liebert [1980] or Carroll and Ostlie [1996,
Section 15.2].2 Physical Foundations
The formation of a white dwarf is the final stage in the evolution of a star less mas-
sive than about 8M , which does not cease in a core collapse supernova. Isolated white
dwarfs are cooling over many billions of years. Eventually, they will fade and become
dark. White dwarfs which are members of close binary star systems, on the other hand,
may accrete mass from their companions and thereby go through violent evolutionary
changes. Such events are observable as novae. They belong to the class of cataclysmic
variables, which are characterised by short outbursts of radiation following long qui-
escent intervals. These outbursts are caused by the explosive thermonuclear burning
of a hydrogen surface layer, which is accumulated through accretion from the com-
panion star [cf. Carroll and Ostlie, 1996, Section 17.4]. In the course of nova, most of
the accreted mass is ejected into space. However, under certain circumstances a white
dwarf can gradually accrete material without entering a nova phase. In this case, the
mass will steadily increase and finally approach the Chandrasekhar limit, which is the
largest mass that can be supported by the degeneracy pressure of electrons. Close to
the Chandrasekhar mass, the density and temperature in the core of the white dwarf
reach a critical threshold. At this point, the rate of thermonuclear reactions increases
rapidly and a catastrophic runaway sets in, which incinerates and disrupts the whole
51star within a few seconds. The released energy is of the order 10 erg and gives rise
to one of the most luminous events in the universe, a type Ia supernova. Although
alternative progenitor systems have been suggested, the best match between observa-
tional properties and predictions from numerical simulations is found for the scenario
outlined above, which is known as the single degenerate (SD) scenario. The physics
of thermonuclear supernova explosions is introduced in the first part of this Chapter.
In particular, it is shown that turbulence plays a crucial role in the modelling of ther-
monuclear burning. Furthermore, theory and observations of type Ia supernovae are
discussed in a broader astrophysical context in the second part.
1.1 Physical Foundations
Our current theoretical understanding of thermonuclear supernovae in the SD scenario
rests on three fundamental pillars. Firstly, the physics of degenerate matter, secondly,
hydrodynamics including turbulence and, finally, thermonuclear combustion physics.
The underlying principles and some of the most important facts are outlined in the
1.1.1 Degeneracy
Degeneracy pressure is a non-thermal property of a fermion gas. In the limit of zero
temperature, each fermion occupies the state of lowest energy available without violat-
ing Pauli’s exclusion principle. Let us consider a gas consisting of electrons and ignore
electrostatic interactions. According to Heisenberg’s uncertainty relation, the minimal
3phase space volume occupied by any electron is (2π)/g, whereg is the multiplicity
given by the number of possible spin orientations. For electrons, which are spin-1/2
particles,g= 2. In the ground state, all electrons occupy states within a sphere of
radius p in momentum space. p is called the Fermi momentum. It is related to theF F