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Studying explosive phenomena in astrophysics by the
example of gamma-ray bursts and supernovae
Anastasia Filina
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Anastasia Filina. Studying explosive phenomena in astrophysics by the example of gamma-ray
bursts and supernovae. Other. Université Nice Sophia Antipolis, 2015. English. <NNT :
2015NICE4038>. <tel-01243560>
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UNIVERSITE NICE SOPHIA ANTIPOLIS - UFR Sciences
Ecole Doctorale en Sciences Fondamentales et Appliquees
THESE
pour obtenir le titre de
Docteur en Sciences
de l’UNIVERSITE Nice Sophia Antipolis
Discipline : (ou spécialité) Asrophysique Relativiste
présentée et soutenue par
AUTEUR Anastasia FILINA
Studying explosive phenomena in astrophysics by the
example of gamma-ray bursts and supernovae
Thèse dirigée par Pascal CHARDONNET
soutenue le 01/07/2015
Jury:
Chardonnet Pascal
LAPTH, Annecy-le-Vieux, France
Directeur
Chechetkin Valery
KIAM, Moscow, Russia
Co-directeur
Chetverushkin Boris
KIAM, Moscow, Russia
Membre de jury
Della Valle Massimo
OACN-INAF, Naples, Italy
Rapporteur
Narozhny Nikolay
MEPhI, Moscow, Russia
Membre de jury
Pozanenko Alexei
IKI, Moscow, Russia
Membre de jury
Titarchuk Lev
Vakili Farrokh
GMU - SPACS, Virginia, USA
OCA, Nice, France
Rapporteur
Membre de jury
i
Contents
Contents
ii
List of Figures
iv
List of Tables
vii
Abstract
viii
1
2
3
Introduction
1.1 Supernovae and gamma-ray bursts
1.1.1 Observations . . . . . . .
1.1.2 Physical processes . . . .
1.1.3 Nucleosynthesis . . . . .
1.2 Thesis outline . . . . . . . . . . .
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1
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Explosive phenomena in stellar physics
2.1 Different modes of combustion . . . .
2.2 The role of instabilities . . . . . . . .
2.2.1 Rayleigh-Taylor instability . .
2.2.2 Landau-Darrieus instability .
2.2.3 Thermal-pulsational instability
2.2.4 Pair instability . . . . . . . .
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Aspherical nucleosynthesis in a core-collapse supernova with 25 M⊙ standard progenitor
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Piecewise Parabolic Method on a Local stencil for hydrodynamics . . .
3.2.2 Conjugate gradients for self-gravity . . . . . . . . . . . . . . . . . . .
3.2.3 Tracer particles method for nucleosynthesis . . . . . . . . . . . . . . .
3.3 Explosive nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Network of nuclear reactions . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Nuclear abundances . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Solving reaction network for nucleosynthesis . . . . . . . . . . . . . .
3.4 SN model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
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iii
Contents
4
5
6
7
Multidimensional simulations of pair-instability supernovae
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Pair-instability supernovae . . . . . . . . . . . . . . . . .
4.3 Numerical approach . . . . . . . . . . . . . . . . . . . . .
4.3.1 Modelling in 1D . . . . . . . . . . . . . . . . . .
4.3.2 Numerical explosion in multi-D . . . . . . . . . .
4.4 Discussions and conclusions . . . . . . . . . . . . . . . .
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On GRB Spectra
5.1 Introduction: snapshot of GRB spectrum . . . . . .
5.2 Black-body and Thermal Bremsstrahlung emission
5.2.1 Our Model . . . . . . . . . . . . . . . . .
5.3 Analysis of gamma-ray bursts . . . . . . . . . . .
5.3.1 Analysis of GRB 090618 . . . . . . . . . .
5.4 Analysis of some other GRBs . . . . . . . . . . . .
5.5 Discussions and conclusions . . . . . . . . . . . .
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Cosmology with GRBs
6.1 General introduction to cosmology
6.2 Cosmology with GRBs . . . . . .
6.3 Number of GRBs per redshift . . .
6.3.1 Luminosity function . . .
6.3.2 Rate of GRB . . . . . . .
6.4 Conclusions . . . . . . . . . . . .
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General conclusions
89
Analysis of some GRBs
91
Abbreviations
100
Physical Constants
101
Symbols
102
Bibliography
103
Acknowledgements
115
List of Figures
1.1
1.2
1.3
1.4
2.1
2.2
2.3
2.4
2.5
2.6
Supernova spectral types. Credit:Daniel Kasen . . . . . . . . . . . . . . . . .
A mosaic of HST images of the hosts of forty-two bursts (left) and the supernovae hosts (right) (Fruchter et al. 2006) . . . . . . . . . . . . . . . . . . . .
Illustration of the light curves of a variety of supernovae on the left (Smith et al.
2007) and comparison it with GRBs on the right (Bloom et al. 2009). . . . . .
Sample of light curves of bright BATSE bursts, showing high diversity. . . . .
Curves of constant shockwave velocity, called Rayleigh lines. . . . . . . . . . .
The shock adiabatic or the Hugoniot adiabatic and Rayleigh line (dashed) . . . .
The detonation adiabatic (continuous line) and the ordinary shock adiabatic
(dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The relation between the different modes of combustion . . . . . . . . . . . . .
Example of Rayleigh-Taylor instability simulations by the PPML code Popov
(2012). The density map is shown . . . . . . . . . . . . . . . . . . . . . . . .
Example of Landau-Darrieus instability simulations for the C/O flame (Bell et
al. 2004). Inflow boundary conditions inject fuel into the bottom of the domain.
The y-velocity of the material with respect to the planar flame is shown in color
for different moments of time. . . . . . . . . . . . . . . . . . . . . . . . . . .
A simplified network of nuclear reactions from 12C to 56 Ni. . . . . . . . . . . .
Comparison of our nucleosynthesis computing code (on the top) and nucleosynthesis computing code from Timmes et al. (2000) (on the bottom). As an example we take evolution of the mass fractions under adiabatic expansion. The
initial conditions are T = 3 × 109 K, ρ = 1 × 109 g cm−3 and an initially half
12C - half 16 O composition for the 13 isotope α-chain network.
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3.3 Presupernova configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Density distribution for t = 60.5 s after inducing the explosion. The coordinates
are shown in the units of solar radius. Color represents the logarithm of density
in the units of ρc = 4.5 × 105 g/cm3 . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Temperature distribution for the same moment and in the same coordinate units
as on fig. 3.4. Color represents the logarithm of temperature in the units of 109 K.
3.6 Tracer locations and the density map for t = 60.5 s, reconstructed from the
recorded tracers data. Color represents the temperature in the units of 109 K. . .
3.7 Distribution of 56 Ni and 52 Fe mass fractions in velocity map for t = 60.5 s. . .
3.8 Distribution of 48Cr and 44 T i mass fractions in velocity map for t = 60.5 s. . . .
3.9 Distribution of 40Ca and 36 Ar mass fractions in velocity map for t = 60.5 s. . .
3.10 Distribution of 32 S and 28 S i mass fractions in velocity map for t = 60.5 s. . . .
3.11 Distribution of 24 Mg and 20 Ne mass fractions in velocity map for t = 60.5 s. . .
3.12 Distribution of 16 O and 12C mass fractions in velocity map for t = 60.5 s. . . .
3.1
3.2
iv
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22
25
26
32
35
36
39
39
40
41
42
42
43
43
44
List of Figures
3.13 Distribution of 4 He mass fraction for t = 60.5 s. The numbers show the values
in some regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.14 The schematic plot of main regions in the ejecta after explosion . . . . . . . . .
3.15 Comparison between the yields in our SN model (red line) and in a SN model
of Maeda et al. (2002) (blue line). The total mass of the produced nuclei in the
units of M⊙ in logarithmic scale is shown. . . . . . . . . . . . . . . . . . . . .
4.1
4.2
4.3
4.4
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
6.1
6.2
6.3
6.4
6.5
6.6
Fate of a star depending on its mass, Mc , and binding energy, Ebind . Explosion
is marked by diamonds and collapse is marked by circles. . . . . . . . . . . . .
Nuclear energy release as a function of maximum temperature (diamonds). The
slope E ∝ T 2 is shown. For comparison data from Arnett (1996) (stars) and
Ober et al. (1983) (triangles) are shown. . . . . . . . . . . . . . . . . . . . . .
SN model with central ignition for t = 28 sec. Logarithm of density (a) is
shown in units of ρc = 2.65 × 105 g/cm3 . Temperature (b) is shown in units of
T c = 2.36 × 109 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SN model with multicore ignition for t = 28 sec. Logarithm of density (a) is
shown in units of ρc = 2.65 × 105 g/cm3 . Temperature (b) is shown in units of
T c = 2.36 × 109 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time-integrated spectra GRB 090618 . . . . . . . . . . . . . . . . . . . . . .
Time-integrated spectra for the different time intervals: from 0 to 50 s, from
50 to 59 s, from 59 to 69 s after the trigger time of GRB 090618. Blue line
shows the fit with the Band function (Eq 5.1), orange line shows the blackbody
+ bremsstrahlung (Eq. 5.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time-integrated spectra for the different time intervals: from 69 to 78 s, from
79 to 105 s after the trigger time of GRB 090618. Blue line shows the fit
with the Band function (Eq 5.1), orange line shows the fit with blackbody +
bremsstrahlung (Eq. 5.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The temperature evolution with time T(t) of GRB 090618 . . . . . . . . . . . .
Example of light curve of GRB 090618 (Izzo et al. 2012) . . . . . . . . . . . .
Hot spots randomly appear on the surface of exploding star. Each spot produces spike of emission. Signals from these spots arrive at different times, so an
observer sees the superposition of the spikes. . . . . . . . . . . . . . . . . . .
2D simulation of an explosion of a 100 solar mass pair instability supernovae explosion in the multicore explosion scenario using PPML methods. This picture
shows the fragmentation of the core and hot spot regions of very high temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cartoon representation of a possible emission mechanism with Black-Body +
Bremmstrahlung emission . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Amati relation with redshift distribution . . . . . . . . . . . . . . . . . . . . .
Isotropic equivalent radiated energy from redshift Eiso (z) . . . . . . . . . . . .
Comparison of luminosity distance computed from Amati relation” and from
cosmological redshift by eq.6.42 . . . . . . . . . . . . . . . . . . . . . . . . .
”Volumetric factor” dV/dz
1+z (z) . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integral of Luminosity Function from Campisi et al. (2010) and Wanderman &
Piran (2010) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rate GRB from Hopkins & Beacom (2006) and rate PISN from Hummel et al.
(2012) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
44
45
47
54
55
56
57
64
65
66
67
67
70
71
72
82
82
83
85
86
87
List of Figures
vi
6.7
Number of observed GRBs with redshift in (z,z+dz) over a redshift interval . .
87
1
2
3
The temperature evolution with time T(t) of GRB 101023 . . . . . . . . . . . .
Light curve of GRB 101023 (Penacchioni et al. 2012) . . . . . . . . . . . . . .
Spectral analysis of GRB 101023 for different intervals of time: [0-45 s] and
[45-89 s]. Blue line shows the fit with the Band function (Penacchioni et al.
2012), orange line shows the fit with blackbody + bremsstrahlung . . . . . . .
Time-integrated spectra of GRB 970111 for the time interval from 21 to 24 s after the trigger time. Blue line shows the fit with the Band function (Eq 5.1)(Frontera et al. 2012), orange line shows the fit with blackbody + bremsstrahlung
(Eq. 5.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The temperature evolution with time T(t) of GRB 970111 . . . . . . . . . . . .
Light curve of GRB 970111 detected with BATSE (Frontera et al. 2012) . . . .
GRB 970111. Time-integrated spectra for different intervals of time: 4-7 s, 713 s, 13-16 s, 16-21 s, 21-24 s, 24-28s, 28-32s, 32-45 s. Blue line shows the
fit with the Band function (Eq 5.1), orange line shows the fit with blackbody +
bremsstrahlung (Eq. 5.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time-integrated spectra of GRB090926A for the time interval from 9.8 to 10.5
s after the trigger time. Blue line shows the fit with the Band function (Eq 5.1)
(Tierney et al. 2013), orange line shows the fit with blackbody + bremsstrahlung
(Eq. 5.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The temperature evolution with time T(t) of GRB 090926A . . . . . . . . . . .
Light curve of GRB 090926A (Tierney et al. 2013) . . . . . . . . . . . . . . .
GRB 090926A. Time-integrated spectra for different intervals of time: form 0.0
to 3.3 s, 3.3 to 9.8 s, 9.8 to 10.5 s and 10.5 to 21.6 s. Blue line shows the fit
with the Band function (Eq 5.1), orange line shows the fit with blackbody +
bremsstrahlung (Eq. 5.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The temperature evolution with time T(t) of GRB 990510 . . . . . . . . . . . .
Light curve of GRB 990510 (Frontera et al. 2012) . . . . . . . . . . . . . . . .
Time-integrated spectra of GRB990510 for the time intervals. Blue line shows
the fit with the Band function (Eq 5.1), orange line shows the fit with blackbody
+ bremsstrahlung (Eq. 5.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The temperature evolution with time T(t) of GRB 980329 . . . . . . . . . . . .
Light curve of GRB 980329 (Frontera et al. 2012) . . . . . . . . . . . . . . . .
Time-integrated spectra of GRB980329 for the time intervals. Blue line shows
the fit with the Band function (Eq 5.1), orange line shows the fit with blackbody
+ bremsstrahlung (Eq. 5.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
91
4
5
6
7
8
9
10
11
12
13
14
15
16
17
92
93
93
93
94
95
95
95
96
96
96
97
98
98
99
List of Tables
3.1
Detailed nucleosynthesis yields. . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.1
Presupernova models and parameters of explosion . . . . . . . . . . . . . . . .
53
5.1
Spectral analysis of GRB 090618 . . . . . . . . . . . . . . . . . . . . . . . . .
68
vii
UNIVERSITE NICE-SOPHIA ANTIPOLIS - UFR Sciences
Abstract
Ecole Doctorale en Sciences Fondamentales et Appliquees
Doctor of Relativistic Astrophysics
Studying explosive phenomena in astrophysics by the example of gamma-ray bursts and
supernovae
by Anastasia Filina
The formation of the first stars hundreds of millions of years after the Big-Bang marks the end
of the so-called Dark Ages. Currently, we have no direct observations on how the primordial
stars (Population III stars) formed, but according to modern theory of stellar evolution these
stars should be very massive (about 100 M⊙ ). Population III stars also have a potential to
produce probably most energetic flashes in the Universe – gamma-ray bursts (GRBs). GRBs
may provide one of the most promising methods to probe directly final stage of life of primordial
stars. Today’s telescopes cannot look far enough into the cosmic past to observe the formation of
the first stars, but the new generation of telescopes will test theoretical ideas about the formation
of the first stars.
Thanks to many years of observations and number of successful space missions we have good
GRB’s data – statistics of occurrence, spectrum, lightcurves. But there are still a lot of questions
in the theory of GRBs. We know that a significant fraction of GRBs, the so called long-duration
GRBs, are related to the death of stars and that they are connected with supernovae. So gammaray bursts are one of the classes of explosive processes in stellar physics that should have a lot of
common with supernovae explosions. In that case GRBs should follow the same physical laws
of explosion as supernovae. This work tries to approach the problem of GRBs as a problem of
stellar explosion.
Necessary instruments of studying stellar explosion were developed as a part of doctoral research: code for solving systems of nuclear reaction equations was incorporated into hydrodynamical code. These tools were applied for supernovae simulations in order to find possible
connection with GRBs. Basing on analysis of supernovae simulations spectral analysis of GRBs
was performed.
Chapter 1
Introduction
The characteristic time scales of the life of stars, from millions to billions years, don’t allow us
to trace the entire life cycle of any concrete star. But huge number of observed stars gives us
an opportunity to observe them at different stages of their existence - from initial formation by
condensation of molecular clouds and up to their death, which for some stars is marked by the
brightest flashes - supernovae explosions. Supernovae explosions are one of the most powerful
events in the universe. The fact that the supernovae explosions occur with a certain regularity
and that regularities were found between different events suggests that this phenomenon is a
natural termination of stellar evolution.
Another example of prompt energetic process in the universe are gamma-ray bursts, which were
discovered few decades after introduction of the concept of supernovae. Apparently these powerful flashes of gamma-ray emission are also associated with explosions of stars. However,
there are significant differences between gamma-ray bursts and usual supernovae. Most of the
radiation is emitted in gamma rays, and the total energy radiated may be one or two orders of
magnitude larger than in usual supernovae. (Assuming that the radiation is isotropic).
These phenomena are two examples of explosive processes in astrophysics, which play a significant role in the history and evolution of the universe.
1.1 Supernovae and gamma-ray bursts
The transient appearance of a ”new star” (nova) in place of sky where none had been observed
previously, was known by astronomers for a long time. But dramatic shift in our understanding
the scale of these phenomena occurred in the beginning of the XX-th century. It was realized that
some of these stars are located in other galaxies, therefore their luminosity should be few order
of magnitudes higher than in usual novae. In the early 1930s Fritz Zwicky, who coined the term
1
Chapter 1. Introduction
2
”super-nova”, realized the potential importance of this phenomenon and began it’s systematic
search and exploration.
Supernovae represent the endpoint of stellar evolution, following an explosion of a star. Supernovae have typical kinetic energy releases of 1051 erg, and total energy releases of up to 1053
erg, roughly equivalent to the Sun’s energy output summed over 1 trillion of years.
Besides accompanying the death of a star, supernovae are also the sites for neutron star and
black hole formation and are a major source of cosmic rays. In addition, they are extremely
effective mechanisms for dispersing the heavy elements produced by the star during its lifetime
into the interstellar gas, thereby enriching the raw material from which new generations of stars
are born. Supernovae are responsible for chemical enrichment of Inter-Stellar Medium (ISM)
and production of heavy elements. The chemical composition of the Universe after the Big
Bang was primarily hydrogen and helium. So the first stars of the Universe (Population III stars,
PopIII) also had very simple chemical composition. During their evolution these stars burned
hydrogen and helium producing heavier elements. If at final stage of evolution star explodes
it expels to ISM chemical elements, that were produced. So next generations of stars would
have more complicated chemical composition with mixtures of metals. Particularly, presence of
heavy chemical elements on the Earth indicates that the Sun was formed from remnants of older
stars, exploded as supernova.
Presence of metals strongly affects the process of formation of new stars. Absorption lines in
atoms of metals increase opacity of the matter, also increasing the role of radiation pressure. As
a result it changes characteristic masses of stars forming by condensation of initial molecular
cloud, thus the next populations of stars have different distribution by their masses (so called
initial mass function, IMF).
Another important role of supernovae is that their remnants are considered as sites of acceleration of cosmic rays. According to the supernova-remnant cosmic-ray hypothesis, protons are
accelerated by the shocks created in a supernova and then further accelerated by the magnetic
fields until they gain enough energy to escape and become a cosmic ray. These energetic protons sometimes collide with other protons and can produce a neutral pion, which decays into two
gamma-ray photons. Recent observations with Fermi satellite detected the characteristic piondecay signature in two supernova remnants within our galaxy — IC 443 and W44 (Ackermann
et al. 2013).
Supernovae are tremendous sources of gravitational waves and neutrinos which makes them
great laboratories for fundamental physics. The galactic or near extragalactic supernova explosion can produce gravitational waves signals that could be detected by current or future experiments. The neutrino emission gives us important information about the physical processes in
Chapter 1. Introduction
3
stellar interiors during explosion, which was proved by the observation of neutrinos from SN
1987A (Bionta et al. 1987; Hirata et al. 1987).
All these facts make supernovae very important field of research which can help us to understand huge variety of physical problems: from evolution of the Universe to details of physical
processes in extreme conditions of stellar explosion. But in some cases the death of a star can be
even more violent and impressive than a supernova. In certain circumstances stars can produce
Gamma-Ray bursts. Gamma-ray bursts (GRBs) are very bright and intense flashes of emission
of gamma rays. They have durations measured in seconds and their isotropic energy can reach
values up to 1054 ergs, making GRBs the most luminous known events. Such huge energy release gives a chance to observe GRBs at cosmological distances. With years of observations,
maximum detected redshift for GRBs increased very fast and from the first detected redshift,
equals 0.695 up to now, maximum detected redshift is z = 9.4 for GBR090429B (Cucchiara
et al. 2011). It is the most distant object ever observed. And for sure, future instruments will
observe more GRBs with even higher redshift. It gives an opportunity to study formation of
the first stars (population III), galaxy formation and resulting reionization of the universe at
z ∼ 6 − 20.
With new satellites we will be able to detect more and more GRBs with known redshift. The
distribution of GRB by redshift could give us an information about the origin of GRB. We can
test theoretical models by predicting Rate of GRB per redshift and comparing with observations.
1.1.1
Observations
First techniques for discovering and studying extragalactic supernovae were developed by Zwicky
and Minkowski in 1930’s. The first few supernovae, discovered by accident, contained examples corresponding to the two types of event. The spectra of one type were not immediately
interpretable even though they were similar from event to event. They showed broad features
which could not be readily associated with any known element. The light curve (luminosity
as a function of time) had a predictable regularity. It rose to peak brightness in the period of
about two weeks and then after declining for two more weeks, it faded in an almost precise exponential law with a half-life of about 50 days (Barbon et al. 1973). Other events were dimmer
at maximum by a factor of a few in luminosity and had light curves which rose and faded in
a manner that varied from supernova to supernova. The most important aspect of these events,
which differentiated them as separate class, was that the spectrum contained the common optical
lines of hydrogen. These two different kinds of supernovae were differentiated by Minkowski
(1941) as Type I and Type II, respectively. The majority of extragalactic supernovae have fallen
into one of these two broad categories - Type I (SN I) for those that displayed no evidence of
hydrogen and Type II (SN II) for those that showed spectral evidence of hydrogen in the ejecta.
Chapter 1. Introduction
4
Further study suggested that Type II events might themselves be differentiated by the shape of
their light curves (Barbon et al. 1979). Some had light curves for which the luminosity declined
roughly in an exponential law (Doggett & Branch (1985) and Young & Branch (1989)), although
with a different time constant than SN I. These came to be known as linear Type II supernovae
(SN II-L). The other class, comprising about two-thirds of a rather small sample of SN II with
no allowance made for selection effects in discovery (Barbon et al. 1979), tend to remain at peak
light for one or two months before declining. These plateau type supernovae (SN II-P) tend to
be less bright than SN I and the brightest SN II-L by a factor of three or four. The presence of
intermediate or very narrow width H emission lines in the spectra came to be known as Type IIn
supernovae (SN IIn), where ”n” denotes narrow.
Figure 1.1 shows the supernova classification scheme, based on the spectrum near maximum
light with the basic differentiating property being whether or not the spectrum shows evidence
of hydrogen. In Type I, If there are lines of Silicon, then this event is called Type Ia. The
events that fail to show strong Si feature near maximum light can be further differentiated by the
presence or absence of strong lines of He I (Wheeler et al. 1987). The events that show no Si
near maximum light, but show He I, are identified as Tybe Ib There are other events that fail to
show either H or Si near maximum light, and very little evidence of He. Wheeler et al. (1986)
proposed the category SN Ic for these events.
A Type IIb supernova has a weak hydrogen line in its initial spectrum, which is why it is classified as a Type II. However, later on the H emission becomes undetectable, and there is also
a second peak in the light curve that has a spectrum which more closely resembles a Type Ib
supernova. As the ejecta of a Type IIb expands, the hydrogen layer quickly becomes more transparent and reveals the deeper layers.
Gamma-ray bursts were discovered by chance. The first GRB was detected on July 2 , 1967, by
the military Vela satellites. These satellites were launched to monitor nuclear explosions on the
Earth and probably on the Moon. Because of secret nature of the mission, the information about
intense flashes of gamma emission remained unpublished until 1973 (Klebesadel et al. 1973).
Many theories tried to explain these bursts, most of which said that GRBs originate from Galactic neutron stars. Some progress was made until the 1991 launch of the Compton Gamma Ray
Observatory (CGBO) and its Burst and Transient Source Explorer (BATSE) instrument, an extremely sensitive gamma-ray detector. This instrument provided crucial data that showed that
the distribution of GRBs is an isotropic - not biased towards any particular direction in space,
such as toward the galactic plane or the galactic centre. The absence of any such pattern in the
case of GRBs provided strong evidence that gamma-ray bursts have a cosmological origin.
Chapter 1. Introduction
5
Figure 1.1: Supernova spectral types. Credit:Daniel Kasen
The most important event occurred in 1997, when the BeppoSAX satellite succeeded in obtaining high-resolution X-ray images of the predicted fading afterglow of GRB 970228 (Costa et al.
1997), (Piro et al. 1999). This detection, followed by a number of others at an approximate rate
of 10 per year, led to positions accurate to about an arc minute, which allowed the detection and
follow-up of the afterglows at optical and longer wavelengths (e.g., van Paradijs et al. (1997)).
This opened the way for the measurement of redshift distances, the identification of candidate
host galaxies, and the confirmation that GRBs were at cosmological distances (Metzger et al.
1997), (Gehrels et al. 2009). These studies were continued and advanced with launch of further
space missions.
BeppoSAX functioned until 2002 and CGRO (with BATSE) was deorbited in 2000.
One of the important space missions to date, Swift, was launched in 2004 and it is still operational. Swift is equipped with a very sensitive gamma ray detector as well as on-board X-ray and
optical telescopes, which can be rapidly and automatically slewed to observe afterglow emission
following a burst.
The Fermi Gamma-ray Space Telescope was launched in 2008 and its detectors cover energy
range over several decades of keV (∼ 8 keV to ∼ 300 GeV) and it is ideal to study the low-
energy regime of GRBs. Fermi consists of two instruments, the Large Area Telescope (LAT)
operating between ∼ 20 MeV to ∼ 300 GeV (Atwood et al. 2009) and the Gamma-Ray Burst
Chapter 1. Introduction
6
Figure 1.2: A mosaic of HST images of the hosts of forty-two bursts (left) and the supernovae
hosts (right) (Fruchter et al. 2006)
Monitor (GBM) operating between 8 keV - 40 MeV (Meegan et al. 2009). GBM consists of two
types of detectors - two Bismuth Germanate (BGO) scintillating crystals operating between 0.15
- 40 MeV and twelve Sodium Iodide (NaI) scintillating crystals operating between 8 -1000 keV.
The NaI detectors are located in clusters of three around the edges of the satellite and the BGOs
are located on opposing sides of the satellite aligned perpendicular to the LAT boresight. As the
detectors have no active shield and are uncollimated, GBM observes the entire unocculted sky
(Meegan et al. 2009) , (Tierney et al. 2013).
What do we know about GRBs? Space observatories allow us to follow properties of host
galaxies and environment. GRBs appear in young systems with evidence of recent star formation. GRBs are observed less in spiral galaxies and more often in galaxies with low mass and
low metallicity (Perley et al. 2013). GRBs prefer low metallicity and avoid high mass galaxies
(Jimenez & Piran 2013).
There are several cases of association of GRBs with supernovae (Campana et al. 2006; Della
Valle et al. 2004; Galama et al. 1998; Hjorth et al. 2003) In all of these cases the associtated
supernova belongs to type Ibc. This provided the proof that the origin of at least some long
GRBs is in the core collapse of massive stars. Guetta & Della Valle (2007) devided GRBs in two
classes (low-luminosity bursts with luminosity about 1049 ergs/s, and high-luminosity bursts)
and found that their ratios to Type Ibc supernovae (SNe Ibc) are ∼ 1% − 9% and 0.4% − 3%,
respectively.
(Fruchter et al. 2006) compared properties of environment of usual supernovae and supernovae
associated with GRBs. In the images (1.2) the host galaxies for GRBs and supernovae, made by
Chapter 1. Introduction
7
Figure 1.3: Illustration of the light curves of a variety of supernovae on the left (Smith et al.
2007) and comparison it with GRBs on the right (Bloom et al. 2009).
Hubble Space Telescope are shown. Their study showed that the GRBs host galaxies are faint
and irregular.
1.1.2
Physical processes
SNe Ia represent quite uniform class of thermonuclear explosion triggered in low-mass white
dwarfs (with masses close to the Chandrasekhar limit MCh = 1.44 M⊙ ). In such stars nuclear
reactions has stopped after formation of carbon-oxygen core. But if such star increases its
mass (for example by accretion from a companion star) and overcomes critical mass MCh it
losses stability and starts to contract. Increase of the temperature causes explosive thermonuclear
burning of the carbon–oxygen mixture which disrupts the star (Ivanova et al. 1974) . Because
of similar mass and structure of white dwarfs these supernovae have a single pattern of the light
curve. This fact allows to use them as standard candles for cosmology.
Supernovae include various classes of events. Such variety of the light curves explained by the
fact that the explosion mechanism responsible for the different physical processes for different
types of stars. Examples of light curves of different supernovae are presented in figure 1.3.
For more massive stars, the outcomes can be roughly related to the three parameters - mass,
metallicity and rotation rate of a star. In the simplest case of no rotation and no mass loss, one can
delineate five outcomes and assign approximate mass ranges (in some cases very approximate
mass ranges) for each (Woosley & Heger 2015).
From 8 to 30 M⊙ on the main sequence (presupernova helium core masses up to 12 M⊙ ), stars
mostly produce iron cores that collapse to neutron stars leading to explosions that make most of
today’s observable supernovae and heavy elements. Within this range there are probably islands
of stars that either do not explode or explode incompletely and make black holes, especially for
Chapter 1. Introduction
8
helium cores from 7 to 10 M⊙ .
From 30 to 80 M⊙ (helium core mass 10 to 35 M⊙ ), black hole formation is quite likely. Except
for their winds, stars in this mass range may be nucleosynthetically barren. Again though there
will be exceptions, especially when the effects of rotation during core collapse are included.
80 to (very approximately) 150 M⊙ (helium cores 35 to 63 M⊙ ), pulsational-pair instability
supernovae. Violent nuclear-powered pulsations eject the star’s envelope and, in some cases,
part of the helium core, but no heavy elements are ejected and a massive black hole of about 40
M⊙ is left behind.
150 - 260 M⊙ (again very approximate for the main sequence mass range, but helium core 63 to
133 M⊙ ), pair instability supernovae of increasing violence and heavy element synthesis. No
gravitationally bound remnant is left behind.
Over 260 M⊙ (133 M⊙ of helium), with few exceptions, a black hole consumes the whole star.
Rotation generally shifts the main sequence mass ranges (but not the helium core masses) downwards for each outcome. Mass loss complicates the relation between initial main sequence mass
and final helium core mass.
In the most extreme cases, gamma-ray bursts may be produced. The physical origin of Gamma
ray-bursts is still unsolved. It is necessary to take into account many different facts: energetics,
variability of light curve, spectrum. In particular, one of the most important question is nonthermal spectrum of GRBs - any realistic physical model should propose a way to produce huge
amount of energy emitted in gamma-rays without making the source optically thick.
Temporal profiles of the prompt emission of GRBs can have very complex behaviour, no two
GRBs look the same. Some examples of light curves of GRBs are presented in the figure 1.4. It is
seen that there is no any common template, which makes it very difficult to build a classification
of GRBs basing on the morphology of light curves.
Contrary to light curves of the prompt emission, spectra of GRBs are quite similar. Spectra of
most GRBs can be fitted successfully by the empirical function (Band et al. 1993). This function
consists of two smoothly joined power laws. This function describes successfully spectra of the
most GRBs, but it doesnt have physical origin of the emission.
Thermal bremsstrahlung as a possible process responsible for spectra of GRBs was proposed
soon after their discovery (Anzer & Boerner 1976; Fenimore et al. 1982). It was successfully
used to describe many GRBs. This model gives the correct slope of the low-energy part of the
spectrum. But discovery of the cosmological nature of GRBs leaded to the fact that this model
was abandoned. Thermal bremsstrahlung is a low effective process and to explain the observed
intensity it is require to have significant density and volume.
Chapter 1. Introduction
9
Figure 1.4: Sample of light curves of bright BATSE bursts, showing high diversity.
In the eighties, the community of GRB was convinced that the engine was related to neutron
stars, ”it is now widely believed that the bursts come from strongly magnetized neutron stars”
(Lamb 1984). A detailed physical model was elaborated. In view of this plausible model of
neutron star as the standard explanation of GRBs, thermal Bremsstrahlung emission was rejected
and this fact was also explained in many articles (Harding 1991). Now, we know that GRBs have
an other origin and in this case there is a possibility to use a thermal bremsstrahlung for new
models of GRBs.
1.1.3
Nucleosynthesis
The problem of stellar explosion requires consideration of very complex combination of physical
processes and factors: hydrodynamics, nuclear reactions, equaton of state, neutrino cooling, etc.
All these processes affect each other, making the picture of explosion highly complicated. The
same time the details of ongoing physical processes are hidden from observer in the depth of
stellar interiors. What observer can see is the final outcome of an explosion when ejecta expands
and become transparent.
Possible key to understand details of explosions in that case is the nucleosynthesis. Since nuclear
reactions are extremely sensitive to the temperature, the final yields and distribution of isotopes
in expanding ejecta can give as details about dynamics of explosion at very early stages. The
hydrodynamics of explosion is imprinted in the nucleosynthesis.
Chapter 1. Introduction
10
A massive star spends about 90% of its life burning hydrogen and most of the rest burning
helium. Typically these are the only phases of the star that can be studied by astronomers.
These relatively quiescent phases, when convection and radiation transport dominate over neutrino emission, also determine what follows during the advanced burning stages and explosion
(Woosley et al. 2002).
The preexplosive life of a massive star is governed by simple principles. Pressure - a combination of radiation, ideal gas, and, later on, partially degenerate electrons - holds the star up against
the force of gravity, but because it radiates, the star evolves. When the interior is sufficiently
hot, nuclear reactions provide the energy lost as radiation and neutrinos, but only by altering the
composition so that the structure of the star changes with time.
The conditions for explosive nucleosynthesis in massive stars are characterized primarily by the
peak temperature achieved in the matter as the shock passes and the time for which that temperature persists. A typical time for the density to e-fold is the hydrodynamic time. The necessary
condition for explosive modification of the preexplosive composition is that the burning lifetime
at the shock temperature be less than the hydrodynamical time scale. The products of explosive
nucleosynthesis are more sensitive to the peak temperature than to the initial composition. Material heated to 5 billion K will become iron whether it started as silicon or carbon. If explosive
processing is negligible the initial composition is ejected without appreciable modification. This
is the case for most elements lighter than silicon.
Type Ia supernovae are responsible for making part of the iron group (including about one-half
of 56 Fe; Thielemann et al. (1986) ; Timmes et al. (1995)). Rare varieties of type Ia supernovae
may be necessary for the production of a few isotopes not adequately made elsewhere. These
include neutron-rich isotopes of Ca, Ti, Cr, and Fe made in accreting white dwarfs that ignite
carbon deflagration at densities so high that they almost collapse to neutron stars (Iwamoto et
al. 1999; Woosley 1997). Another rare variety of type Ia supernovae are the helium detonations
(Woosley & Weaver 1995). These give temperatures of billions of K in helium-rich zones and
may be necessary in order to understand the relatively large solar abundance of
in supernovae as radioactive
44 T i)
44Ca
(made
only in regions of high temperature and large helium mass
fraction. This may also explain the production of a few other rare isotopes like 43Ca and 47 T i.
Classical novae seem necessary to explain the origin of 15 N (in the beta-limited CNO cycle) and
17 O
(José & Hernanz 1998).
1.2 Thesis outline
Because we know that supernovae and gamma-ray bursts are linked, we propose to study them
as a continuous class of phenomena, in which the process of nuclear explosion and combustion
Chapter 1. Introduction
11
plays a key role. In chapter two, we present the basic concepts of the physics of explosions in
the stars: the propagation of shock waves and hydrodynamic instabilities.
We proposed the model that some of long gamma-ray bursts can be produced by pair-instability
supernova. These supernovae can play a significant role at high-redshift Universe. This theory
requires consideration of many factors.
1. Since we know that gamma-ray bursts are associated with supernova explosions - it is important to understand the conditions under which the star explodes as a usual supernova, and when
as a gamma-ray burst. This requires highly sophisticated numerical simulations of explosions
of stars. For the beginning, we started to develop our hydrodynamics numerical code and the
code of explosive burning. As one of the first tasks was chosen supernova with 25 M⊙ standart
progenitor. This task allowed to test hydrodynamics and nucleosynthesis and compare the results with other works. We got a good agreement with the observational data, as well as with the
other results. However, due to rather complex hydrodynamical code we used, we also obtained
some interesting details of the structure of an exploding star.
2. After we have used this code for the simulation of pair-instability supernova explosion. In the
one-dimensional case, besides the fact that we have reproduced the results of the others works,
we also investigated the effect of mass and initial configuration of the stars on the process and
the final result of the explosion. We got a quantitative assessment that allowed to confirm that
the main parameters, such as energy, the characteristic time and temperature, pair-instability
supernova satisfy to gamma-ray burst. We also have done a preliminary calculation of explosions
of pair-instability supernova in 2D and study the effect of possible irregularity in the star, formed
as a result of convection. Basing on our results we came to the conclusion that the temperature
should be an important parameter that should characterize gamma-ray bursts.
3. Bearing in mind the assumption of the importance of the temperature, we decided to check it
on the observational data of gamma-ray bursts, and to check the spectra of gamma ray bursts and
possible mechanisms of their formation. We have proposed a model that radiation is associated
with the processes of radiation in the expanding shell of the star. We hypothesized that the
two processes are responsible for the formation of the spectrum, bremsstrahlung and blackbody
emission.
4. Since we consider PISN as possible sources of gamma-ray bursts, we can estimate the theoretical frequency of gamma-ray bursts basing on the predictions of the theory of stellar evolution
for very massive stars, and compare with observations. This allowed us on the one hand to make
sure that gamma-ray burst is effectively connected with star formation rate, but on the other hand
to forecast increase of the number of observed gamma-ray bursts with higher redshift, as soon
as the sensitivity of our tools will allow to look so far.
Chapter 2
Explosive phenomena in stellar physics
This section will present the framework, and describe the physics of stellar explosions. It closely
follows the discussion in Landau & Lifshitz (1959) , Zeldovich (1960) and also Fickett & Davis
(2000). There are several kind of supernova explosions of different types of the progenitor stars
with absolutely different physical processes triggering the explosion. But one of the key physical
problems of many types of stellar explosion is the question of flame propagation. For example
the combustion mechanism in thermonuclear explosion (related to type Ia supernovae), or shock
wave propagation (which can induce thermonuclear reactions) in the envelope of a core-collapse
supernova.
There exist two regimes of flame propagation: 1) detonation, in which thermonuclear burning
propagates supersonically with the shock wave front where the temperature jumps up drastically
leading to fast burning; 2) deflagration, when the flame mainly propagates due to dissipative
effects: thermoconductivity or diffusion, or convection, and the flame propagation velocity is
small compared to the speed of sound.
The problem of the regime of flame propagation is crucial for the type of explosions called the
thermonuclear explosions that occur in type Ia supernovae. The progenitors of these supernovae
are white dwarfs, stars in which the matter (mostly carbon and oxygen) is in degenerate state.
The question of explosion in degenerate matter first raised by Arnett (1969). This work studied
the ignition and the formation of detonation wave in the context of degenerate carbon/oxygen cores of intermediate-mass stars. After Ivanova et al. (1974) obtained a sub-sonic flame
propagation (deflagration) in spontaneous regime with pulsations and a subsequent transition to
detonation.
The regime of the flame propagation in type Ia supernovae is still not fully unknown, the observations imply some limitations on it. If the burning of the whole star proceeds in the detonation
12
Chapter 2. Explosive phenomena in stellar physics
13
regime, it burns up to Fe-peak elements. However, it contradicts observations: in a real supernova about half of the star should consist of the intermediate elements. Pure deflagration regime
does not succeed too: the star expands with velocity, which is faster than the flame velocity
so the temperature drastically drops down and all the burning terminates. The only feasible
success- ful regime is the mixed one when the flame starts with deflagration in high density matter, where the expansion coefficient is small. Then it somehow accelerates at the radius, which
is usually characterized by some critical density, and passes to detonation.
2.1 Different modes of combustion
So there are two regimes of flame propagation that have different physical processes underlying:
deflagration front is a subsonic burning front that propagates by a process of heat conduction
or convection; in contrast, a detonation is a shock-induced burning which propagates supersonically. First to define the framework of description of these regimes let us consider the simplest
physical model: the propagation of a shock wave (which assumed to be a jump discontinuity of
physical variables) in one-dimensional flow in a steady state. We can take the coordinate system
where the discontinuity surface is at rest. Then the conservation laws in that case require:
ρ1 v1 = ρ2 v2 ≡ j,
(2.1)
p1 + ρ1 v1 2 = p2 + ρ2 v2 2 ,
(2.2)
w1 +
v1 2
v2 2
= w2 +
;
2
2
(2.3)
where ρ1 and v1 is the density and velocity of the undisturbed material (into which the shock
wave moves) and ρ2 and v2 is the density and velocity of the final state, which remains behind
the shock; p1 and p2 are the pressures at the initial and final states respectively; w = ǫ + pV enthalpy, ǫ - specific internal energy, V = 1/ρ - specific volume. And j denotes the mass flux
density at the surface of discontinuity.
From these equations we have v2 =
ρ1 v 1
ρ2 ;
2
p2 − p1 = ρ1 v1 − ρ2 v1
= (ρ1 v1 )
2
"
2
1
1
−
ρ1 ρ2
⇒ (ρ1 v1 )2 =
ρ1
ρ2
#
p2 − p1
1
1
ρ1 − ρ 2
!2
=
Chapter 2. Explosive phenomena in stellar physics
14
Figure 2.1: Curves of constant shockwave velocity, called Rayleigh lines.
The result defines a line in the p-V plane called the ”Rayleigh line” and expressed by
R = j2 −
p2 − p1
= 0.
V1 − V2
A Rayleigh line passes through the point (p1 , V1 ) and has slope − j2 . Examples of Rayleigh lines
are shown in figure 2.1.The limiting cases are the horizontal, j2 = 0, and the vertical, j2 = ∞.
The vertical Rayleigh line corresponds to an infinite velocity of propagation of a shock wave.
Let’s derive a series of relations which follow from the above conditions. Using the specific
volumes V1 = 1/ρ1 , V2 = 1/ρ2 , we obtain from
v1 = jV1 , v2 = jV2
(2.4)
p1 + j2 V1 = p2 + j2 V2 ,
(2.5)
and, substituting in eq. 2.2,
or
j2 =
p2 − p1
.
V1 − V2
(2.6)
This formula, together with eq.2.4, relates the rate of propagation of a shock wave to the pressures and densities of the gas on the two sides of the surface.
Since j2 is positive, we see that either
p2 > p1 , V2 < V1 , or
Chapter 2. Explosive phenomena in stellar physics
15
p2 < p1 , V2 > V1 .
As was shown for example in Landau & Lifshitz (1959), the condition that entropy must increase
leads to the conclusion that only the former case (p2 > p1 ) can actually occur.
We may note the following useful formula for the velocity difference v1 − v2 . Substituting eq.
2.6 in v1 − v2 = j(V1 − V2 ), we obtain
v1 − v2 =
p
(p2 − p1 )(V1 − V2 ).
(2.7)
Next, we can write eq. 2.3 in the form
w1 +
1
1 2 2
j V1 = w2 + j2 V2 2
2
2
(2.8)
and, substituting j2 from eq. 2.6, obtain
1
w1 − w2 + (V1 + V2 )(p2 − p1 ) = 0.
2
(2.9)
If we replace the heat function w by ǫ + pV, we can write this relation as
1
ǫ1 − ǫ2 + (V1 − V2 )(p2 + p1 ) = 0.
2
(2.10)
These relations defines the relation between the thermodynamic quantities on the two sides of
the surface of discontinuity.
For given p1 , V1 , equation eq. 2.9 or eq. 2.10 gives the relation between possible values of p2
and V2 . This relation is called the shock adiabatic or the Hugoniot adiabatic proposed by W. J.
M. Rankine in 1870 and H. Hugoniot in 1889. It can represented graphically in the p-V plane
(figure 2.2) by a curve passing through the given point (p1 , V1 ) (for p1 = p2 , V1 = V2 we have
also ǫ1 = ǫ2 , so that eq. 2.10 is satisfied identically). It should be noted that the shock adiabatic
cannot intersect the vertical line V = V1 except at (p1 , V1 ). The existence of another intersection
would mean that two different pressures satisfying eq. 2.10 correspond to the same volume. For
V1 = V2 , however, we have from eq. 2.10 also ǫ1 = ǫ2 , and when the volumes and energies
are the same the pressures must be the same. Thus the line V = V1 divides the shock adiabatic
into two parts, each of which lies entirely on one side of the line. Similarly, the shock adiabatic
meets the horizontal line p = p1 only at the point (p1 , V1 ).
Therefore, for a given initial state (i.e. for given p1 and V1 ) the shock wave is defined by a single
parameter. For example if the pressure behind the shock, p2 , is known then the density (specific
Chapter 2. Explosive phenomena in stellar physics
16
Figure 2.2: The shock adiabatic or the Hugoniot adiabatic and Rayleigh line (dashed) .
volume V2 ) can be defined from the shock adiabatic, and using relations 2.6 and 2.4 the mass
flax j and velocities v1 and v2 are defined.
Let us mention another convenient gafical interpretation of the relation 2.6. Value of j2 is the
slope of corresponding Rayleigh line, thus the velocity of a shock wave is determined at each
point of the shock adiabatic by the slope of the chord joining that point to the point (p1 , V1 ). For
weak shock we can substitute the relation 2.6 by:
∂p
j =−
∂V
2
!
.
(2.11)
S
Then the velocities v1 and v2 are equal:
v1 = v2 = jV =
s
−V 2
∂p
∂V
!
S
=
s
∂p
∂ρ
!
,
(2.12)
S
which is velocity of sound c. Thus for velocity of propagation of weak shock waves is in the first
approximation is velocity of sound. And for strong shocks the velocity of shock wave relative
to undisturbed gas is greater than its velocity of sound: v1 > c1 .
The shock wave heats the gas when it passes through - the temperature of gas behind the shock
wave is higher than in front of it. If the shock wave is strong enough, the jump in temperature
can cause thermonuclear reactions to begin. So the passage of the shock wave will cause the
Chapter 2. Explosive phenomena in stellar physics
17
ignition of reactions and combustion will be propagated with velocity of shock. That regime of
combustion propagation is called detonataion.
When the shock wave passes some point of gas the reaction begins at that point, and continues
until all the material there is burnt, i.e. for a time τ which characterises the kinetics of the
reaction concerned. The width of the layer in which combustion is occurring is equal to the
speed of propagation of the shock multiplied by the time τ. It is important to underline that the
width does not depend on the characteristic dimensions of any bodies that are present. When the
characteristic dimensions of the problem are sufficiently large, we can regard the shock wave
and the combustion zone following it as a single surface of discontinuity which separates the
burnt and unburnt gases. We call such a surface a detonation wave.
A detonation wave must satisfy continuity conditions - the flux densities of mass, energy and
momentum must be continuous. All relations derived previously for a shock wave hold for a
detonation wave too, because they were followed from these continuity conditions solely. Thus
the equation:
1
w1 − w2 + (V1 + V2 )(p2 − p1 ) = 0
2
(2.13)
is valid for a detonation wave. The curve of p2 as a function of V2 given by this equation is called
the detonation adiabatic. Unlike the shock adiabatic considered earlier, this curve does not pass
through the given initial point (p1 , V1 ). This is due to the fact that the shock adiabatic deals with
w1 and w2 functions that actually are the same function w(p, V). Whereas this does not hold for a
detonation adiabatic, where gases 1 and 2 have different composition owing to reactions occured.
In figure 2.3 the continuous line shows the detonation adiabatic. The ordinary shock adiabatic for
the unburnt gas mixture is drawn (dashed) through the point (p1 , V1 ). The detonation adiabatic
always lies above the shock adiabatic, because a high temperature is reached in combustion, and
the gas pressure is therefore greater than it would be in the unburnt gas for the same specific
volume. For the mass flux density:
j2 = (p2 − p1 )/(V1 − V2 ),
(2.14)
so that graphically − j2 is the slope of the chord from the point (p1 , V1 ) to any point (p2 , V2 ) on
the detonation adiabatic (for instance, the chord ac in figure 2.3). It is seen from the diagram
that j2 cannot be less than the slope of the tangent aO. The flux j is just the mass of gas which
is ignited per unit time per unit area of the surface of the detonation wave; we see that, in a
detonation, this quantity cannot be less than a certain limiting value jmin (which depends on the
initial state of the unburnt gas).
Formula 2.14 is a consequence only of the conditions of continuity of the fluxes of mass and
momentum. It holds (for a given initial state of the gas) not only for the final state of the
combustion products, but also for all intermediate states, in which only part of the reaction
Chapter 2. Explosive phenomena in stellar physics
18
Figure 2.3: The detonation adiabatic (continuous line) and the ordinary shock adiabatic (dashed
line).
energy has been evolved. In other words, the pressure p and specific volume V of the gas in any
state obey the linear relation
p = p1 + j2 (V1 − V),
(2.15)
which is shown graphically by the chord ad. This result is of importance in the theory of
detonation. Let us now use a procedure developed by Ya. B. Zel’dovich in 1940 to investigate
the variation of the state of the gas through the layer of finite width which a detonation wave
actually is. The forward front of the detonation wave is a true shock wave in the unburnt gas
1. In it, the gas is compressed and heated to a state represented by the point d (Fig. 2.3) on
the shock adiabatic of gas 1. The chemical reaction begins in the compressed gas, and as the
reaction proceeds the state of the gas is represented by a point which moves down the chord da;
heat is evolved, the gas expands, and its pressure decreases. This continues until combustion
is complete and the whole heat of the reaction has been evolved. The corresponding point is c,
which lies on the detonation adiabatic representing the final state of the combustion products.
The lower point b at which the chord ad intersects the detonation adiabatic cannot be reached
for a gas in which combustion is caused by compression and heating in a shock wave. Thus we
conclude that the detonation is represented, not by the whole of the detonation adiabatic, but
only by the upper part, lying above the point O where this adiabatic touches the straight line aO
drawn from the initial point a.
At the point where d( j2 )/d p2 = 0, i.e. where the shock adiabatic touches the line from (p1 , V1 ),
the velocity v2 is equal to the corresponding velocity of sound c2 and v2 < c2 above that point
Chapter 2. Explosive phenomena in stellar physics
19
Landau & Lifshitz (1959). These results have been obtained only from the conservation laws
for the surface of discontinuity, and are entirely applicable to the detonation wave also. On
the ordinary shock adiabatic for a single gas there are no points with d( j2 )/d p2 = 0. On the
detonation adiabatic, there is such a point, namely the point O. Since detonation corresponds to
the upper part only of the adiabatic, above the point O, we conclude that
v2 ≤ c2 ,
(2.16)
i.e. a detonation wave moves relative to the gas just behind it with a velocity equal to or less than
that of sound; the equality v2 = c2 holds for a detonation corresponding to the point O (called
the Jouguet point).
The velocity of the detonation wave relative to gas 1 is always supersonic (even for the point O):
v1 > c1 .
(2.17)
This is seen directly from figure 2.3. The velocity of sound c1 is given graphically by the slope
of the tangent to the shock adiabatic for gas 1 (dashed curve) at the point a. The velocity v1 , on
the other hand, is given by the slope of the chord ac. Since all the chords concerned are steeper
than the tangent, we always have v1 > c1 . Moving with supersonic velocity, the detonation
wave, like a shock wave, does not affect the state of the gas in front of it. The velocity v1 with
which the detonation wave moves relative to the unburnt gas at rest is the velocity of propagation
of the detonation. Since v1 /V1 = v2 /V2 ≡ j, and V1 > V2 , it follows v1 > v2 . The difference
v1 − v2 is evidently the velocity of the combustion products relative to the unburnt gas. This
difference is positive, i.e. the combustion products move in the direction of propagation of the
detonation wave.
If the detonation is caused by a shock wave which is produced by some external source and is
then incident on the gas, any point on the upper part of the detonation adiabatic may correspond
to the detonation. It is of particular interest to consider a detonation which is due to the combustion process itself. In a number of important cases, such a detonation must correspond to the
Jouguet point, so that the velocity of the detonation wave relative to the combustion products
just behind it is exactly equal to the velocity of sound, while the velocity v1 = jV1 relative to
the unburnt gas has its least possible value. This result was put forward as a hypothesis by D. L.
Chapman in 1899 and E. Jouguet in 1905, but its complete theoretical justification is due to Ya.
B. Zeldovich in 1940.
The deflagration regime takes place in cases where thermonuclear (or chemical) reaction is
strongly exothermic. The speed of reaction strongly depends on the temperature. When the
Chapter 2. Explosive phenomena in stellar physics
20
temperature rises sufficiently, the reaction proceeds rapidly. In the case of endothermic reactions, a continuous supply of heat from an external source is necessary for the reaction to be
maintained; if the temperature is merely raised at the beginning of the reaction, only a small
amount of matter reacts, and thereby reduces the gas temperature to a point where the reaction
ceases. The situation is quite different for a strongly exothermic reaction, where a considerable
quantity of heat is evolved. Here it is sufficient to raise the temperature at a single point; the
reaction which begins at that point evolves heat and so raises the temperature of the surrounding gas, and the reaction, once having begun, will extend to the whole gas. This is called slow
combustion or deflagration.
The combustion of a gas mixture is necessarily accompanied by motion of the gas. In general,
the nature of the combustion process has to be determined by a solution of simultaneous equations which include both those of chemical kinetics for the reaction and those of gas dynamics
for the mixture concerned.
The situation is much simplified in the very important case (the one usually encountered) where
the characteristic dimension l of the problem is large. In such cases, the problems of gas dynamics and chemical kinetics can be considered separately. The region of burnt gas (i.e. the region
where the reaction is over and the gas is a mixture of combustion products) is separated from the
gas where combustion has not yet begun by a transition layer, where the reaction is in progress
(the combustion zone or flame); in the course of time, this layer moves forward, with a velocity
which may be called the velocity of propagation of combustion in the gas. The magnitude of
this velocity depends on the amount of heat transfer from the combustion zone to the cold gas
mixture. The main mechanism of heat transfer is ordinary conduction. The theory of this means
of propagation of combustion was first developed by V. A. Mikhelson in 1890.
We denote by δ the order of magnitude of the width of the combustion zone. It is determined
by the mean distance over which heat evolved in the reaction is propagated during the time τ
for which the reaction lasts (at the point concerned). The time τ is characteristic of the reaction,
and depends only on the thermodynamic state of the gas undergoing combustion (and not on the
parameter l). If χ is the thermometric conductivity of the gas, we have
δ∼
√
χτ.
(2.18)
Let us now make more precise the assumption above: we assume that the characteristic dimension is large compared with the width of the combustion zone (l ≫ δ). When this condition is
true, we can consider the problem of gas dynamics separately from the problem of combustion.
In determining the gas flow, we can neglect the width of the combustion zone, regarding it as
a surface which separates the combustion products from the unburnt gas. On this surface (the
flame front) the state of the gas changes discontinuously, i.e. it is a surface of discontinuity.
Chapter 2. Explosive phenomena in stellar physics
21
The velocity v1 of this discontinuity relative to the gas itself (in a direction normal to the front)
is called the normal velocity of the flame. In a time τ, the combustion is propagated through a
distance of the order of δ, and so the flame velocity is
v1 ∼ δ/τ ∼
p
χ/τ.
(2.19)
The ordinary thermometric conductivity of the gas is of the order of the mean free path of the
particles multiplied by their thermal velocity or, what is the same thing, the mean free time τfr
multiplied by the square of this velocity. Since the thermal velocity of the particles is of the
same order as the velocity of sound, we have
v1
∼
c
r
χ
∼
τc2
r
τfr
.
τ
(2.20)
Not every collision between particles results in a reaction between them; on the contrary, only a
very small fraction of colliding particles react. This means that τfr ≪ τ, and v1 ≪ c. Thus the
flame velocity is, in this case, small compared with the velocity of sound.
On the surface of discontinuity which replaces the combustion zone, the fluxes of mass, momentum and energy must be continuous, as at any discontinuity. The first of these conditions
determines the ratio of the components, normal to the surface, of the gas velocities relative to
the discontinuity: ρ1 v1 = ρ1 v2 , or
v1 /v2 = V1 /V2 ,
(2.21)
where V1 and V2 are the specific volumes of the unburnt gas and the combustion products.
According to the general results for arbitrary discontinuities, the tangential velocity component
must be continuous if the normal component is discontinuous. The streamlines are ”refracted”
at the discontinuity. On account of the smallness of the normal velocity of the flame relative
to that of sound, the condition of continuity of the momentum flux reduces to the continuity of
pressure, and that for the energy flux reduces to the continuity of the heat function:
p1 = p2 , w1 = w2 .
(2.22)
In using these conditions, it must be remembered that the gases on the two sides of the discontinuity under consideration are chemically different, and so the thermodynamic quantities are not
the same functions of one another.
The difference between detonation and deflagration regimes can be demonstrated grafically on
p-V plane.
Chapter 2. Explosive phenomena in stellar physics
22
Figure 2.4: The relation between the different modes of combustion
As was demonstrated above the detonation corresponds to points on the upper part of the detonation adiabatic for the combustion process concerned. Since the equation of this adiabatic is a
consequence only of the conservation laws for mass, momentum and energy (applied to the initial and final states of the burning gas), the points representing the state of the reaction products
must lie on the same curve for any other mode of combustion in which the combustion zone
can be regarded as a surface of discontinuity of some kind. Let us now ascertain the physical
significance of the remainder of the curve.
Let us draw through the point (p1 , V1 ) (point 1 in figure 2.4) vertical and horizontal lines 1A and
1A′ , and the two tangents 1O and 1O′ to the adiabatic. The points A, A′ , O, O′ where these
lines intersect or touch the curve divide the adiabatic into five parts. The part lying above O
corresponds to detonation, as we have said. Let us now consider the other parts of the curve.
First of all, the section AA′ has no physical significance. For this section p2 > p1 , V2 > V1 , and
p
so the mass flux j = (p2 − p1 )/(V1 − V2 ) is imaginary.
At the points of contact O and O′ , the derivative d( j2 )/d p2 is zero; At such points we have
v2 /v1 = 1 and d(v2 /v1 )/d p2 < 0. Hence it follows that above the points of contact v2 /c2 < 1,
and below them v2 /c2 > 1. The relation between v1 and c1 is always found by considering the
slopes of the corresponding chords and tangents, for the part above O. The result is that the
following inequalities hold on the various sections of the adiabatic:
above O
v1 > c1 ,
v2 < c2 ;
Chapter 2. Explosive phenomena in stellar physics
23
on AO
v1 > c1 ,
v2 > c2 ;
on A′ O′
v1 < c1 ,
v2 < c2 ;
below O′
v1 < c1 ,
v2 > c2 .
(2.23)
At O and O′ , v2 = c2 . As we approach A, the flux j , and the velocities v1 , v2 , tend to infinity.
As we approach A′ , j and the velocities v1 , v2 tend to zero.
The part of the line below O′ where v1 < c1 , v2 > c2 is absolutely unstable with respect to
infinitesimal displacements in the direction perpendicular to its plane Landau & Lifshitz (1959),
which means that this part of the curve does not correspond to any mode of combustion that can
be realised in practice.
The section A′ O′ of the adiabatic, on which both velocities v1 and v2 are subsonic, corresponds
to the ordinary slow combustion, i.e. deflagration. An increase in the rate of propagation of
combustion, i.e. in j, corresponds to a movement from A′ (where j = 0) towards O′ .
The AO section of the curve corresponds to non-detonational combustion which propagates
supersonically. This situation can take place when the transfer of heat is very efficient (for
example, transfer by radiation). The value of j then may in principle exceed that corresponding
to the point O′ .
Ordinary slow combustion may spontaneously change into detonation. This transition occurs
owing to an acceleration of the flame, accompanied by an increase in the intensity of the shock
wave preceding it, until the shock becomes strong enough to ignite the gas passing through
it. The mechanism of this spontaneous acceleration of the flame is not clear; it is possible that
turbulence of the flame caused by the walls of the pipe is important. It is also possible that steady
propagation of a flame is unstable when its front is curved by the friction of the gas against the
walls of the pipe.
In conclusion, we may call attention to the following general differences (besides those contained in the inequalities 2.23) between the modes of combustion corresponding to the upper
and lower parts of the adiabatic. Above A we have p2 > p1 , V2 < V1 , v2 < v1 . That is, the reaction products have a pressure and density greater than that of the original gas, and move behind
the combustion front with velocity v1 − v2 . In the region below A, the inequalities are reversed:
p2 < p1 , V2 > V1 , v2 > v1 , and the combustion products are less dense than the original gas.
Chapter 2. Explosive phenomena in stellar physics
24
2.2 The role of instabilities
The hydrodynamic instabilities can strongly affect the regime of combustion. The development
of instability of the flame front in deflagration regime can naturally accelerate the fuel consumption. The hydrodynamic instability leads to distortion and corrugation of the surface of the flame
front increasing its area comparing to the smooth front consequently to an acceleration of the
flame propagation. In experiments it was shown that in some cases such instabilities can lead to
a transition from the regime of slow flame propagation to the regime of detonation. (Gostintsev
et al. 1988) Meantime it is possible that the change in the normal velocity when the front is
deformed can be a stabilizing factor: where the front is concave, v increases (since the heat
transfer into the unburnt mixture in the concavity is improved), while where it is convex v is
reduced.
The factor of instabilities leads to additional difficulties in description and modelling of supernovae. There are a number of hydrodinamic instabilities of importance in stellar explosions. Let
us describe major of them.
2.2.1
Rayleigh-Taylor instability
The Rayleigh-Taylor instability, or RT instability (Rayleigh in 1900 and Taylor in 1950), is a
fingering instability of an interface between two fluids of different densities which occurs when
the lighter fluid is pushing the heavier fluid. Examples include supernova explosions in which
expanding core gas is accelerated into denser shell gas.
Since the flame propagates in gravitational field, and the burned ashes have lower density than
the unburned fuel, the Rayleigh-Taylor (RT) instability is often considered to be the dominant
instability governing the corrugation of the front. The RT instability creates turbulent cascade
providing an acceleration of the flame front.
Let’s consider the boundary conditions. We have set up an interface - ξ, across which pressure
equilibrium. Therefore, δp|ξ = 0. Then the fluid being displaced by ∆z downward across ξ gives
a potential energy term −ρ1 g∆z, while for the fluid on the underside of the layer the potential is
+ρ2 g∆z. It is assumed that a small cough sets up the ripple, so that the energy input is minimal.
The conservation condition for the energy gives
(ρ1 + ρ2 )∆z2 ω2 ∼ (ρ1 − ρ2 )g∆z,
so that we get, for the frequency,
!
ρ1 − ρ2
gk,
ω ∼
ρ1 + ρ2
2
Chapter 2. Explosive phenomena in stellar physics
25
Figure 2.5: Example of Rayleigh-Taylor instability simulations by the PPML code Popov
(2012). The density map is shown
where k ∼ 1/∆z is the wavenumber (spatial frequency of initial perturbation). We find that any
fluid which is set up with a density inversion will inevitably wind up undergoing the RayleighTaylor instability. This is a very simplistic derivation of the condition, but the essentials are
preserved. The growth time shows two roots, but the negative one dies away in time and never
concerns us.
In the explosion of a supernova, the envelope is low in density and quite hot. The blast wave
should become Rayleigh-Taylor unstable, mixing material from the envelope into the blast and
causing knots to appear early in the evolution of the remnant. Detailed hydrodynamic calculations have confirmed the existence of this effect, and it is now taken as one of the mechanisms
operating early in the evolution of the expanding shell. A similar mechanism is acting in the
formation of the interface between H II regions and molecular clouds, and between blast waves
and clouds at the boundary of the cloud.
The Richtmyer-Meshkov instability is the impulsive-acceleration limit of the Rayleigh-Taylor
instability when a shock wave passes through the interface separating two fluids with different
density. Richtmyer provided a theoretical prediction in 1960, and Meshkov provided experimental verification in 1969. In studies of deflagration to detonation transition processes it was
showed that the Richtmyer-Meshkov instability induced flame acceleration can result in detonation.
Chapter 2. Explosive phenomena in stellar physics
26
Figure 2.6: Example of Landau-Darrieus instability simulations for the C/O flame (Bell et al.
2004). Inflow boundary conditions inject fuel into the bottom of the domain. The y-velocity of
the material with respect to the planar flame is shown in color for different moments of time.
2.2.2
Landau-Darrieus instability
Due to thermal expansion across the burning front planar flames are unstable with respect to the
large scale bending. This universal intrinsic hydrodynamic flame instability is called the LandauDarrieus (LD) instability. It was predicted independently by (Darrieus 1938) and (Landau
1944). This instability wrinkles the flame surface accelerating the combustion and leads to
formation of cusps (Bell et al. 2004). The LD instability is driven by the thermal expansion
across the flame and affects a planar flame front even in the absence of gravity.
For the wavelengths much longer than the flame thickness the development of the LD instability
does not depend on complex processes which take place in the burning zone, it is defined only by
the density difference between burned and unburned material. Development of the LD instability
takes place only when the density of burned material is lower than that of unburned material,
which usually is the case in combustion process due to thermal expansion. The LD instability
plays an important role in many thermonuclear burning in supernovae (Blinnikov et al. 1995;
Niemeyer & Hillebrandt 1995). The detailed consideration the of nonlinear stage of the LD
instability and the calculation of the fractal dimension of the flame front for this case is given by
Blinnikov and Sasorov (Blinnikov & Sasorov 1996), Joulin (Joulin 1994).
2.2.3
Thermal-pulsational instability
Both Rayleigh-Taylor and Landau-Darrieus instabilities develop on scales which are much larger
than the flame thickness, thus they can be successfully studied in the approximation of the
discontinuous front. However this approximation is not valid for another instability caused by
a strong temperature dependence of the nuclear reaction rates. This type of instability was first
Chapter 2. Explosive phenomena in stellar physics
27
described by Zeldovich in his studies of the powder combustion. The local perturbation of
temperatures lead to vast difference in reaction rates. These fluctuations can not be stabilized be
the thermal conductivity because of its large timescale. This phenomenon leads to a pulsations
in the flame front propagation effectively changing mean velocity. Such instability can develop
even for one- dimensional perturbations when the plane front preserves its shape.
2.2.4
Pair instability
When central temperature in stellar interiors reaches sufficiently high values (order of a few
times 109 K) a significant part of photons of the Plankian spectrum have energies higher than
2me c2 (me is the electron mass), which is the minimum energy required for the creation of
the electron-positron pairs (e± ). Electrons and positrons annihilate but since the cross-sections
of pair-creation and annihilation are finite, the number of the pairs reaches some dynamical
equilibrium. Thus the kinetic energy of photons transforms to the rest mass energy of the pairs,
altering the equation of state. The influence of e± -pairs on the pressure and the internal energy
should be taken into account in computations of the stellar interiors. The detailed description
of the effects of the electron-positron pairs on the equation of state can be found for example in
Weiss et al. (2004).
Chapter 3
Aspherical nucleosynthesis in a
core-collapse supernova with 25 M⊙
standard progenitor
Explosive nucleosynthesis plays a key role in connection between the hydrodynamical models
of supernovae and observations of individual events.The observable light curves and spectra can
give us a clue about the mechanism of the explosion, including the energy budge and types of
physical processes which take place in the stellar matter.
3.1 Introduction
We investigate the death of a Population I 25 M⊙ star as a supernova (SN). According to standard
evolution theory after leaving main sequence this star passing the stages of a blue supergiant,
a yellow supergiant, a possible cepheid stage and achieves a final red supergiant stage. Due to
high metallicity the stellar wind is strong and the star is losing about a half of the mass during
evolution. By the moment of explosion the central core has a complicated structure with layers
of different nuclei from iron group isotopes to oxygen and has the mass ∼ 8 M⊙ (Woosley et al.
2002). When the star exhausts thermonuclear fuel in the iron core, it starts to collapse until the
moment when it stops by neutronization. The shock from bouncing matter ignites surrounding
matter and the star explodes. Depending on its preceding evolution and presence of hydrogen
envelope the star can explode either as Type Ibc if there is no hydrogen or as Type II supernova
if the H-envelope remained.
28
Chapter 3. Aspherical Nucleosynthesis
29
Normally a presupernova rotates and this could result in nonequilibrium neutronization process
during collapse. This could lead to strongly non-spherical explosion. The entropy excess appears in the central part and leads to development of large-scale convection (Chechetkin et al.
1997). Rotation of the star deforms the shape of this high-entropy area extending it along the
axis of rotation in the direction of the highest pressure gradient – the direction of Archimedes’
force. This results in formation of large-scale convective bubbles, starting to float up along
the axis (Ustyugov et al. 1999). Magnetic fields also favour axisymmetry and lead to higher
collimation of initial motion along the axis due to the additional magnetic pressure from the
sides. According to the magnetorotational mechanism, the magnetic field transforms the energy
of rotation into the energy of the SN shock wave. For a dipole-like initial magnetic field the
supernova explosion develops mainly along the rotational axis forming a mildly collimated jet
(Moiseenko et al. 2010).
The jet-like SNe are observed in a number of cases, e. g. SN 2008D (Soderberg et al. 2008). Its
asymmetry was proved by light polarization analysis, which was possible to do with high accuracy thanks to the neighbor SN 2007uy used as a calibrator (Gorosabel et al. 2010). Also early
observations of
56 Ni
spectral lines in SNe could be explained by non-sphericity of explosion.
Another example is a survey of ejecta in the Cassiopeia A supernova remnant which reveals
strong asymmetry (Hwang et al. 2012). The systematic study of the ejecta asphericity in type
Ib/c supernovae was performed by Taubenberger et al. (2009). The results of their analysis suggest that probably at least half of all SNe Ib/c are aspherical. Recently early 56Ni decay gamma
rays from type Ia SN 2014J suggest an unusual explosion with the break of spherical symmetry
Diehl et al. (2014).
We present our results on nucleosynthesis in aspherical SN explosion with a standard progenitor, studied by 2D hydrodynamic simulations and post-processing nucleosynthesis calculations
(Popov et al. 2014). The explosive burning was computed in the progenitor with the chemical
composition obtained in realistic evolutionary computations (Woosley et al. 2002).
3.2 Methods
3.2.1
Piecewise Parabolic Method on a Local stencil for hydrodynamics
Hydrodynamic simulations were performed with the own numerical code of Mikhail Popov
based on the Piecewise Parabolic Method on a Local stencil (PPML) (Popov & Ustyugov 2007;
Popov 2012). The key PPML-procedure written on FORTRAN could be found in Ustyugov
et al. (2009). PPML is an improvement over the popular Piecewise Parabolic Method (PPM)
suggested by Colella et al. (1984) for compressible flows with strong shocks. PPM was implemented in many modern hydrodynamic codes and is widely used in computational practice
Chapter 3. Aspherical Nucleosynthesis
30
now. The principal difference between PPM and PPML is the way how the interface values of
the variables are computed between the adjacent cells. Instead of 4-point interpolation procedure in the standard PPM we have suggested to evolve them with time using the conservation
property of Riemann invariants. The PPML has demonstrated high accuracy on both smooth
and discontinued solutions. It could be more suitable for problems where the requirement of
low dissipation is crucial. For example, it was successfully implemented for direct turbulence
simulation (Kritsuk et al. 2009).
For SN simulation we have used a cylindrical coordinate system where all the variables depend
on the vertical coordinate z and the distance r to the z-axis, i.e. we assume the rotational symmetry. The sequential description of the PPML-algorithm in cylindrical geometry is presented
in Popov (2012). Reflecting boundary conditions have been imposed along the cylindrical axis
and the equatorial plane, allowing free outflow across the outer boundaries. In our case we have
performed simulations in one quadrant with a 3200 × 3200 grid resolution.
The application of hydrodynamic code to the considered supernova model showed artificially
accelerated velocities of the matter near the cylindrical axis. This is a well known pathological
behavior ”carbuncle” which could appear in a region where the shock is almost but not exactly
parallel to the grid edges (e. g. Quirk (1994)). The artificial dissipation was inserted according
to Loh & Jorgenson (2009) for cure.
3.2.2
Conjugate gradients for self-gravity
The Poisson equation to recompute gravitational field was solved with standard method of conjugate gradients. We did not use any precondition matrix for the reason of saving memory. The
routine started every 4th time step with previously known gravity distribution as an initial approximation, which gave about ∼ 400 ÷ 500 iterations at the beginning of the explosion and
∼ 10 ÷ 40 at the end with an accuracy threshold 1 × 10−12 . The gravity was computed with
1600 × 1600 grid resolution which twice more rough than the hydrodynamic grid. To obtain
the components of free-fall acceleration at necessary points a bicubic interpolation procedure,
which returns all the derivatives, was applied to gravity potential.
3.2.3
Tracer particles method for nucleosynthesis
The simplest approach to the computation of explosive nucleosynthesis is to avoid coupling between hydrodynamics and nuclear reactions and to use the tracer particles method (TPM) which
is the post-processing calculations of the nuclear yields. For the first time it was applied to compute the nucleosynthesis in an axisymmetric explosion in core collapse SN with 6 M⊙ helium
core with comparison to the observations of SN 1987A and solar system abundances (Nagataki
Chapter 3. Aspherical Nucleosynthesis
31
et al. 1997). It has become popular nowadays because of the simplicity and the possibility to
follow very wide nuclear reactions network (Chamulak et al. 2012; Maeda et al. 2002, 2010;
Nagataki 1999; Nagataki et al. 2006; Ohkubo et al. 2006; Patat et al. 2001; Travaglio et al.
2004a,b).
The tracers (or markers) are the points on the grid map – virtual particles which are passively
advected by the flow without any coupling with it via gravity or inertia. Every tracer behaves
like a massless particle. All of the tracers are advected with the flow individually with some
velocity, to obtain a new location of the tracer at the next moment t + τ we have used a secondorder Runge-Kutta time integration procedure. With the tracers we can follow the fluid elements
and record the current density and temperature. These recorded values were used in the set of
equations for nuclear abundances.
We have distributed 2402 tracers over the grid in a random way at the initial moment. According
the divergence study of TPM, a number of 128 tracers per axis gives an accuracy better than
2% for nuclei with mass fraction & 10−5 for the grid resolution 512 × 512 in one hemisphere
(Seitenzahl et al. 2010).
In common approach the mass which corresponds to each tracer is just the total mass inside the
computational domain divided by the number of tracers. Thus all the tracers represent the same
mass fraction. This mass fraction is a contribution of each tracer to the total mass of produced
nuclei. We suppose this approach is not precise because the tracers are distributed in regions with
different density, which could change by several orders, and can not represent the same mass.
In our computations we modified standard TPM and used tracers just to reconstruct a spatial
distribution of each nuclide (its mass fraction) at the final moment of hydrodynamic simulation
using the triangulation technique. The mass of the nuclei produced in each computational cell
was computed by multiplication of the cell’s volume by the density and by the mass fraction of
a particular nuclide in the center of the cell. We are going to discuss the difference between two
approaches for TPM in a future paper.
3.3 Explosive nucleosynthesis
3.3.1
Network of nuclear reactions
We have implemented a network of nuclear reactions from 4 He up to 56 Ni (fig. 3.1). These are
basic nuclei connected by all possible two-particle reactions with protons and α-particles in the
input and output channels (the α network). Also we have included heavy-ion reactions
12
C + 12C → 20 Ne + α ,
Chapter 3. Aspherical Nucleosynthesis
32
→ 23 Na + p ,
12
C + 16 O → 24 Mg + α ,
→ 27 Al + p ,
16
O + 16 O → 28 S i + α ,
→ 31 P + p
and the reverse reactions due to photodisintegration. The heavy-ion reactions are shown on
fig. 3.1 by the arcs. The triple-α reaction for carbon production was also implemented
3 4 He → 12C + γ + 7.281 MeV .
In total we have introduced 30 nuclei, which seems to be a good quantity required for explosive
nucleosynthesis calculations with a good accuracy (Timmes et al. 2000) . We did not include
heavier than
56 Ni
nuclei since beyond the iron pike the reverse reactions are much faster than
the direct ones. According to the results in other researches, production of other Ni isotopes is
much less than of 56 Ni (see for example Maeda et al. (2002)).
Figure 3.1: A simplified network of nuclear reactions from 12C to 56 Ni.
Chapter 3. Aspherical Nucleosynthesis
3.3.2
33
Nuclear abundances
We describe the nuclear population in each cell of a simulation in terms of the nuclear abundance
of each species
Y = n/ρNA ,
where n is the number density of the species, ρ is the total mass density, and NA is Avagadro’s
number. For a nucleus with atomic mass number A, AY ≡ X is commonly referred to as the mass
P
fraction of this nucleus and the sum of all mass fractions ( Ai Yi ) is by construction equal to 1.
P
The formula Ai Yi = 1 is in reality an expression of nucleon number conservation rather than
mass conservation, since the nuclear binding energy can approach 1% of the nuclear mass. In
fact the conversion of mass into energy is one of the primary reasons to include nuclear reactions
P
in an astrophysical model. The equation of charge conservation becomes Zi Yi = Ye , where
Ye = ne /ρNA is the electron abundance, which provides a measure of the neutron-richness of the
matter.
In general case the evolution of the nuclear abundances in a thermonuclear reaction network
takes the form of a set of ordinary differential equations for the time derivatives dYi /d t with
dependence only on nuclear reaction rates and local thermodynamic conditions.
X
X
dYi X i
N ij,k ρNA hσvi j,k Y j Yk +
=
N jλ jY j +
N ij,k,l ρ2 NA 2 hσvi j,k,l Y j Yk Yl
dt
j
j,k
j,k,l
(3.1)
where the decay constants (λ) and velocity-integrated reaction cross sections hσvi contain the
essential nuclear data. Rate of the forward reaction R j,k = NA hσvi j,k :
R j,k = exp a0 + a1 + a2 + a3 T 91/3 + a4 T 9 + a5 T 95/3 + a6 log T 9 ;
rate of the reverse reaction of photodissociation λ:
λ = exp a0 + T 9 + T 91/3 + a3 T 91/3 + a4 T 9 + a5 T 95/3 + a6 log T 9 ;
where T 9 =
T
109 K
and ai – the adjustable parameters.
Qm
|N jm |!,
The N s provide for proper accounting of numbers of nuclei with N ij = Ni , N ij,k = Ni / nm=1
Q
′
n
m
|N jm |! where the Ni s are positive (or negative) numbers that specify how
and N ij,k,l = Ni / m=1
many particles of species i are created (or destroyed) in a reaction, while the denominators, in-
cluding factorials, run over the nm different species destroyed in the reaction and avoid double
counting of the number of reactions when identical particles react with each other.
Chapter 3. Aspherical Nucleosynthesis
34
The network nuclear reaction initial value problem can be solved using any of the many methods
(for exampleArnett (1996) or recent review Travaglio & Hix (2013)). But the physical nature
of the problem, reflected in the decay constants and velocity-integrated reaction cross sections,
greatly restricts the optimal choice. The large number of reactions display a wide range of
reaction timescales, with the mean lifetime of particle j against destruction by reaction with
particle k given by
τk ( j) =
1
1
or τ j ( j) =
hσvi j,k Yk
λj
(3.2)
for a decay. Systems whose solutions depend on a wide range of timescales are termed stiff.
Gear (Gear 1971) demonstrated that even a single equation can be stiff if it has both rapidly and
slowly varying components. Practically, stiffness occurs when the limitation of the timestep size
is due to numerical stability rather than accuracy. A more rigorous definition (Lambert 1980)
is that a system of equations is stiff if the ratio of the real part of the largest eigenvalue λmax of
the Jacobian matrix of the system of equations is much larger than the real part of the smallest
eigenvalue λmin . Because of the wide range in timescales between strong, electromagnetic and
weak reactions, λmax /λmin > 1015 is not uncommon in astrophysics, making thermonuclear
reaction networks extraordinarily stiff.
3.3.3
Solving reaction network for nucleosynthesis
To implement the reaction network we used our code written in FORTRAN using the RADAU5
solver for stiff problems, based on 5th-order Runge-Kutta method (Hairer et al. 1991).
All the necessary reactions rates, based on experimental information, can be found in Caughlan
et al. (1988) and Rauscher et al. (2000).
Our code for computing nucleosynthesis has been tested on different tasks for constant values of
density and temperature. Let’s take as an example following test. The reaction network suite is a
13 isotope α-chain plus heavy-ion reaction network, which is suitable for most multidimensional
simulations of stellar phenomena where having a reasonably accurate energy generation rate is
of primary concern. The initial conditions were taken as T = 3 × 109 K, ρ = 1 × 109 g cm−3
and an initially composition as half
12C
- half
16 O.
In figure 3.2 we can see that our reaction
network reproduces the reaction network from Timmes et al. (2000) with a high accuracy.
Chapter 3. Aspherical Nucleosynthesis
35
* hydrostatic: btemp= 3.000E+09 bden= 1.000E+09
1
10
0
10
ε
o16
c12
ε
ε
ε
o16
c12
o16
Mass Fraction
s32
ar36
s32
ε
mg24
-4
si28
si28
si28
s32
s32
s32
ar36
ar36
ca40
ar36
ca40
mg24
s32
-3
ar36
ar36
he4
10
si28
o16
ε
ne20
-2
he4
10
mg24
ne20
si28
mg24
c12
si28
-1
10
10
o16
ε
ne20
o16
mg24
ca40
εne20
ni56
ca40
ε
c12
si28
fe52
10
-5
ne20
ca40
ca40
ε
mg24
o16
ti44
ti44
cr48
ne20
fe52
he4
-6
ti44
10
-7
10
s32
ti44
10
-8
c12
he4
ti44
10
-13
10
-9
cr48
ti44
o16
ε
he4
si28
mg24
mg24
-5
10
cr48
c12
-1
10
Time (s)
ni56
c12
10
3
Figure 3.2: Comparison of our nucleosynthesis computing code (on the top) and nucleosynthesis computing code from Timmes et al. (2000) (on the bottom). As an example we
take evolution of the mass fractions under adiabatic expansion. The initial conditions are
T = 3 × 109 K, ρ = 1 × 109 g cm−3 and an initially half 12C - half 16 O composition for
the 13 isotope α-chain network.
Chapter 3. Aspherical Nucleosynthesis
36
Iron-core
R=0.0208 R , M=1.8 M
O-layer
R=0.09 R , M=5.53 M
R=1 R
He-layer
, M=1.33 M
Figure 3.3: Presupernova configuration.
3.4 SN model
For our SN model we have used a polytropic model of a star with index γ = 4/3 which corresponds to Eddington standard model with purely radiative heat transport. The pressure in this
case is determined by radiation. This assumption is valid only in the central hot core of a star.
Initial configuration was calculated from the condition of hydrostatic equilibrium of non-rotating
ideal gas sphere with a simple equation of state p = ρRT/µ, where the mean molecular weight
was assumed as µ = 0.7. We did not recompute gravity field assuming the explosion does not
violate its initial spherically symmetric distribution.
To construct initial equilibrium profiles of density and temperature we took central values ρc ∼
4.5 × 105 g/cm3 and T c ∼ 1 × 109 K what corresponded to the realistic presupernova star model
(Woosley et al. 2002). This model was computed by KEPLER code, which is a 1D implicit
hydrodynamics package adopted for stellar evolution simulations (Weaver et al. 1978). The
input parameters for the KEPLER code were the solar abundances.
In our presupernova model we have not set the exact stellar structure according to fig. 9 from
Woosley et al. (2002), but very close to it. Upon the equilibrium profile of density three main
regions have been set – an iron core, a layer of oxygen and a layer of helium (fig. 3.3). The
size of each region have been chosen to give the correct mass of each type of the matter. Since
we approximate the star by politropic equation of state, the density profile in our equilibrium
configuration does not perfectly match the correct one obtained in realistic evolutionary computations. To obtain the correct mass of the iron core we assume its radius as 0.0208R⊙ and not
0.014R⊙ as in Woosley et al. (2002). We assume this difference is negligible for our simulation
Chapter 3. Aspherical Nucleosynthesis
37
because the total size of computational domain is 3.2 × 3.2 R⊙ and the most important here is
the correct gravity force.
For computational reasons beyond the helium layer we have joined the rarefied helium-hydrogen
envelope with uniform density ρ ∼ 2.2 × 10−10 g/cm3 . The total initial mass inside the compu-
tational domain is ∼ 8.66 M⊙ . In the chemical composition the iron group isotopes have been
substituted with only one 54 Fe. This approximation does not make a significant impact on the
results. The initial mass fractions have been set according to Woosley et al. (2002) as follows.
For the iron core (r < 0.0208R⊙ , Miron = 1.8M⊙ ):
X(12C) = 0.03,
X(54 Fe) − the rest.
For the oxygen layer (0.0208R⊙ < r < 0.09R⊙ , MO = 5.53M⊙ ):
X(20 Ne) = 0.2,
X(24 Mg) = 0.04,
X(12C) = 0.015,
X(54 Fe) = 3 × 10−4 ,
X(16 O) − the rest.
For the helium layer (0.09R⊙ < r < 0.98234R⊙ , MHe = 1.33M⊙ ):
X(12C) = 1.05 × 10−2 ,
X(20 Ne) = 1.1 × 10−2 ,
X(54 Fe) = 3 × 10−4 ,
X(4 He) − the rest.
For the helium-hydrogen envelope (r > 0.98234R⊙ , MHe+H ∼ 0.002M⊙ ):
X(1 H) = 0.5,
X(14 N) = 7.5 × 10−3 ,
X(16 O) = 5.1 × 10−3 ,
Chapter 3. Aspherical Nucleosynthesis
38
X(20 Ne) = 1.8 × 10−3 ,
X(54 Fe) = 3 × 10−4 ,
X(4 He) − the rest.
A common way in simulations to initiate explosion is to deposit energy to the central region of
the star (Woosley & Weaver 1995). As a parameter the size of the region, where the energy is
deposited, should be defined. In our case the energy E ∼ 5 × 1051 ergs (which is typical for SNe)
is deposited in the layer of oxygen, surrounding the iron core, starting from radius 0.0208R⊙ to
0.026R⊙ . The mass of this layer is 1M⊙ and this is a free parameter of the model. The iron
core is excluded from hydrodynamic simulation since we assume it collapsed and acts only as a
gravity source.
The energy is divided between thermal and kinetic parts equally. In order to take into account
rotation of the star the kinetic energy is distributed in an axisymmetric way by imposing different
initial velocities in different directions: vz = αz in the vertical direction and vr = βr on the
equatorial plane. The ratio of the coefficients α : β = 8 : 1. This is a similar approach as it was
used in Maeda et al. (2002).
The tracers have been distributed randomly at initial moment but non-uniformly because of
significant difference in density for the different regions. In the exploding layer 7200 tracers
have been inserted, in the rest part of the oxygen layer – 14400, in the helium layer – 21600, in
the helium-hydrogen envelope – 14400.
A grid of 3200 × 3200 cells allows to resolve the boundary between the iron core and oxygen
layer and to separate accurately the collapsed core from the ejecta.
3.5 Results
The simulation of supernova hydrodynamics with the initial parameters described above is presented on fig. 3.4-3.5 where the density and the temperature are shown respectively for the
moment t = 60.5 s. The coordinates are in the units of solar radius. Color represents the logarithm of density and temperature in the units of central values. The line shows the initial position
of the He -layer boundary.
The obtained structure of the hydrodynamic flow contains the contact discontinuity between the
undisturbed envelope and expanding hot matter. The contact discontinuity is followed by the
shock. The narrow region between the shock and the contact discontinuity has the highest temperature T ∼ 1.5 × 1010 K. Behind the shock the flow is moving more uniformly with lower
temperature T ∼ (0.4 ÷ 1) × 109 K. The most cold part is an inner structure (the region showed
Chapter 3. Aspherical Nucleosynthesis
39
Log
! , 4.5
10 5 g/cm3
+
z, R
3
-4.17
initial He-layer
boundary
2.5
-4.62
-5.07
-5.53
2
-5.98
-6.43
-6.88
1.5
-7.34
-7.79
1
-8.24
-8.69
0.5
-9.15
-9.60
0
0.5
1
1.5
2
2.5 r, R
3
Figure 3.4: Density distribution for t = 60.5 s after inducing the explosion. The coordinates
are shown in the units of solar radius. Color represents the logarithm of density in the units of
ρc = 4.5 × 105 g/cm3 .
z, R
3
Log T, 10 9 K
1.50
hydrodynamic
waves
2.5
1.25
1.00
0.75
2
0.50
0.25
0.00
1.5
-0.25
-0.50
1
-0.75
-1.00
0.5
-1.25
-1.50
0
0.5
1
1.5
2
2.5 r, R
3
Figure 3.5: Temperature distribution for the same moment and in the same coordinate units as
on fig. 3.4. Color represents the logarithm of temperature in the units of 109 K.
Chapter 3. Aspherical Nucleosynthesis
3
40
z, R
T,10 9 K
28.87
26.47
2.5
24.06
4
21.66
2
19.25
16.85
14.44
1.5
12.04
9.63
3
1
7.23
4.82
2
0.5
2.42
0.01
1
0
0
0.5
1
1.5
2
2.5
r, R
3
Figure 3.6: Tracer locations and the density map for t = 60.5 s, reconstructed from the recorded
tracers data. Color represents the temperature in the units of 109 K.
on fig. 3.5 by blue) with T ∼ (0.35 ÷ 1) × 108 K. This large-scale hydrodynamic picture is in
agreement with self-similar analytical solutions of the problem of uniformly expanding gas in
a stationary medium if the profiles of density in both mediums are changing as a power law
(Chevalier 1982). In limiting case when the explosion could be considered as point-like, the
gravitational energy is much less than the expanding kinetic energy and the pressure of undisturbed medium is much less than the pressure inside, the solution is tending to Sedov-Taylor
self-similar solution obtained analytically for the first time in Sedov (1946). The numerical
solution of Sedov problem by PPML code is presented in Popov (2012) .
The hydrodynamic waves marked on fig. 3.5 appeared as a result of influence the reflecting
boundary conditions since they force strict axial symmetry and symmetry between the north
and the south hemisphere. The large-scale ’carbuncle’ developed along the equatorial plane is a
result of interacting between a shock and a reflecting boundary. At early stage of simulation this
structure is very similar to the one which obtained in Double Mach Reflection hydrodynamic
test (Woodward et al. 1984).
The smoothing of the boundary of the ’cold’ bubble (the region showed on fig. 3.5 by blue) near
the cylindrical axis is the result of artificial dissipation, inserted to suppress ’carbuncle’ effect
according to Loh & Jorgenson (2009). Otherwise we would obtain a large-scaled patalogy with
very high unphysical velocities along the axis because the simulation time is rather long.
The tracers distribution for t = 60.5 s is shown on fig. 3.6. Color represents the density map, but
here it has been reconstructed from the recorded tracers data. Due to initial non-uniformity of
Chapter 3. Aspherical Nucleosynthesis
z, R
X( 56 Ni)
2.0
41
z, R
0.98
X( 52 Fe),10 -2
5
3.75
0.73 2.0
2.5
0.49
1.25
0.25
1.5
1.5
0
0
1.0
1.0
0.5
0.5
r, R
0.0
0.0
0.5
1.0
1.5
0.0
0.0
r, R
0.5
1.0
1.5
Figure 3.7: Distribution of 56 Ni and 52 Fe mass fractions in velocity map for t = 60.5 s.
tracers’ distribution we can observe the lines, formed by tracers, which separate the matter originated from different regions of the initial configuration. The line marked by 1 is formed by the
tracers which were initially the closest to the boundary of the iron core. They were shifted by the
flow leaving the space with decreasing density behind. The density in the close neighbourhood
of the iron core reaches the values ρ ∼ 10−10 g/cm3 which from numerical point of view could
be considered as vacuum. The region between the lines 1 and 2 is a matter originated from the
exploding inner oxygen-layer (the energy deposit region). The region between the lines 2 and 3
is the rest part of the oxygen core. The region between the lines 3 and 4 is the expanding matter
from former He-layer. As can be seen, the tracers are being captured by the propagating outer
shock and are piled up within a narrow hottest region between the shock (line 4) and the contact
discontinuity.
Following the fluid elements we have computed the composition change within every tracer with
time. Spatial distribution of the nuclei of main diagonal of α-chain (fig. 3.1), for the moment
t = 60.5 s is shown on figs. 3.7-3.12. The mass fraction of every nuclide is shown by color.
Superimposing figs. 3.7-3.12 and the regions on fig. 3.6 we can see that the main nucleosynthesis
takes place within the matter of former oxygen core in the initial layer-like structure of the
presupernova. The structure of spatial distribution of the nuclei is complicated and contains
many fragments but one clearly can distinguish two main regions: one contains the matter from
the exploded layer, another – from the rest part of the oxygen core. In the both regions there
Chapter 3. Aspherical Nucleosynthesis
z, R
X( 48 Cr),10 -2
2.0
42
z, R
3
X( 44 Ti),10 -2
2
2.25 2.0
1.5
1.5
1
0.75
1.5
0
0.5
1.5
1.0
1.0
0.5
0.5
r, R
0.0
0.0
0.5
1.0
1.5
0
0.0
0.0
r, R
0.5
1.0
1.5
Figure 3.8: Distribution of 48Cr and 44 T i mass fractions in velocity map for t = 60.5 s.
z, R
X( 40 Ca)
2.0
1.5
z, R
0.2
0.09
0.1
0.06
0.05
0.03
0
1.5
1.0
0.5
0.5
r, R
0.5
1.0
0.12
0.15 2.0
1.0
0.0
0.0
X( 36Ar)
1.5
0.0
0.0
0
r, R
0.5
1.0
1.5
Figure 3.9: Distribution of 40Ca and 36 Ar mass fractions in velocity map for t = 60.5 s.
Chapter 3. Aspherical Nucleosynthesis
z, R
X( 32 S)
2.0
43
z, R
0.3
X( 28 Si)
0.32
0.225 2.0
0.24
0.15
0.16
0.075
1.5
0.08
1.5
0
0
1.0
1.0
0.5
0.5
r, R
0.0
0.0
0.5
1.0
1.5
0.0
0.0
r, R
0.5
1.0
1.5
Figure 3.10: Distribution of 32 S and 28 S i mass fractions in velocity map for t = 60.5 s.
z, R
X( 24 Mg)
2.0
z, R
0.15
X( 20 Ne)
0.2
0.113 2.0
0.15
0.075
0.1
0.05
0.038
1.5
1.5
0
0
1.0
1.0
0.5
0.5
0.0
0.0
r, R
0.5
1.0
1.5
0.0
0.0
r, R
0.5
1.0
1.5
Figure 3.11: Distribution of 24 Mg and 20 Ne mass fractions in velocity map for t = 60.5 s.
Chapter 3. Aspherical Nucleosynthesis
z, R
44
X( 16 O)
z, R
0.86
X( 12 C), 10 -2
3.5
2.0
1.5
1.56
0.65
1.17
0.43 3.0
0.78
0.22
0.39
2.5
0
0
2.0
1.0
1.5
1.0
0.5
0.5
0.0
0.0
r, R
0.5
1.0
0.0
0.0
1.5
r, R
0.5
1.0
1.5
2.0
2.5
Figure 3.12: Distribution of 16 O and 12C mass fractions in velocity map for t = 60.5 s.
3
z, R
1
X( 4He)
0.75
2.5
0.9984
0.4853
0.5
0.25
0.9973
2
0
0.9825
1.5
0.925
1
0.5
no 4He
0
0.5
1
1.5
2
2.5 r, R
3
Figure 3.13: Distribution of 4 He mass fraction for t = 60.5 s. The numbers show the values in
some regions.
Chapter 3. Aspherical Nucleosynthesis
3
45
z, R
H+He atmosphere
2.5
enriched by 12 C
helium cloud
of low density
trace of
2
56
1.5
52
Ni
Fe
48
intermediate
mass nuclei
Cr
4 He
56
1
44
Ni
Ti
40
Ca
0.5
light nuclei
0
0.5
1
1.5
2
2.5 r, R
3
Figure 3.14: The schematic plot of main regions in the ejecta after explosion
are two large fragments of
56 Ni
with the mass fraction up to 0.98 (fig. 3.7, left). The upper
fragment, which is placed near the axis and propagates in the polar direction, was produced in
the region of the oxygen core out of the energy deposit layer. The second large fragment of 56 Ni
was produced from the matter of the exploded layer. Inside this bag of
56 Ni
there is a bubble
which contains intermediate and light nuclei from 40Ca to 16 O (figs. 3.9-3.10), fig. 3.11 (left),
fig. 3.12 (left).
The distributions of light nuclei are presented on figs. 3.11-3.12. The significant part of 24 Mg
and
16 O
originates from the initial composition of oxygen core and is carried out by the flow
along the equatorial plane.
20 Ne
was consumed by nuclear reactions everywhere except the
region shown on figs. 3.11 (right), its residue is moving together with the other light nuclei.
The distribution of mass fraction of 4 He is presented on fig. 3.13. The figures show the values in
the corresponded regions and agree with initial presupernova composition. The mass fraction of
4 He
is gradually reducing in the direction from the outer shock to the nuclear production region,
substituting by 52 Fe, 48Cr, 44 T i and 40Ca. The He-cloud is enriched by 12C (fig. 3.12, right) but
this carbon is only ∼ 0.00022 ÷ 0.012 by mass fraction.
An interesting result is the region marked by asterisk on fig. 3.13. This almost pure 4 He was
produced from the matter of the energy deposit region in the part where the highest kinetic
energy was applied. This impact resulted in dissociation of all types of nuclei. The total mass
of 4 He produced in this region is ∼ 0.22M⊙ .
Chapter 3. Aspherical Nucleosynthesis
46
Table 3.1: Detailed nucleosynthesis yields.
Species
12C
13 N ∗
16 O
20 Ne
23 Na
24 Mg
26 Mg
27 Al
28 S i
30 S i
31 P
32 S
34 S
35Cl
36 Ar
Yield (M⊙ )
6.45 × 10−2
4.27 × 10−5
3.31
0.710
1.28 × 10−6
0.245
7.14 × 10−6
4.90 × 10−5
0.172
1.23 × 10−5
2.02 × 10−5
0.118
2.76 × 10−5
9.03 × 10−5
3.71 × 10−2
Species
38 Ar
39 K
40Ca
42Ca
43 S c∗
44 T i∗
46 T i
47 V ∗
48Cr ∗
50Cr
51 Mn∗
52 Fe∗
54 Fe
55Co∗
56 Ni∗
Yield (M⊙ )
1.87 × 10−5
4.05 × 10−4
6.23 × 10−2
5.91 × 10−5
5.75 × 10−5
1.73 × 10−3
4.86 × 10−5
5.25 × 10−4
4.86 × 10−3
2.60 × 10−5
4.43 × 10−4
1.19 × 10−2
1.60 × 10−3
1.95 × 10−3
0.521
The discussed details of nucleosynthesis are summarized on fig. 3.14 where the main zones are
presented on a single plot. The colors distinguish different regions according to the location
in initial presupernova configuration – the matter from the energy deposit region is colored by
red, the rest part of the oxygen core – by green, the helium envelope – by blue. The fragments
of
56 Ni
are pushed away by the explosion in the axial direction with the estimated velocity
vz ∼ (1.5 ÷ 2.0) × 109 cm/s while the light nuclei are carried by the flow along the equatorial
plane with vr ∼ (2 ÷ 3) × 108 cm/s. The light nuclei should form a thorus-like structure around
of bipolar jet iron-group isotopes. This result fully corresponds to the model of SN 2008D
suggested in Tanaka et al. (2009) – a bipolar explosion with a torus-like distribution of oxygen.
The detailed yields obtained by TPM in our SN model are presented in Tab. 3.1. The nuclei,
corresponded to the main diagonal of α-chain, are shown in bold. The radioactive nuclei are
marked by asterisks.
3.6 Conclusions
It is interesting to compare our results and the results of Maeda et al. (2002), where a hypernova
model was investigated in an axisymmetric explosion. By the moment of explosion a progenitor star had the mass ∼ 13.8 M⊙ and contained an uncollapsed layer of carbon-oxygen and a
silicon core. The mass of a corresponding main sequence star is ∼ 40 M⊙ . The burning mainly
in a carbon-oxygen matter was computed by TPM with a wide network of nuclear reactions
including 222 isotopes. The inserted energy of ∼ 1 × 1052 ergs (twice more than in our case)
Chapter 3. Aspherical Nucleosynthesis
2
47
Log(MX, M )
16O
1
20 Ne
0
28 Si
24 Mg
56 Ni
32 S
36 Ar
-1
40 Ca
54 Fe
12C
-2
44 Ti
-3
-4
31 P
-5
39 K
35 Cl
27Al
38 Ar
42 Ca 46
Ti
50 Cr
26 Mg
-6
23 Na
Figure 3.15: Comparison between the yields in our SN model (red line) and in a SN model of
Maeda et al. (2002) (blue line). The total mass of the produced nuclei in the units of M⊙ in
logarithmic scale is shown.
was distributed in the central core (but not in a layer) with the same axial symmetry in kinetic
energy. Fig. 3.15 gives the comparison between the total mass of the produced nuclei in the units
of M⊙ in logarithmic scale obtained in our SN model (red line) and in a SN model of Maeda et
al. (2002) (blue line). In Maeda et al. (2002) the quantity of 56 Ni ∼ 0.4 M⊙ was a criteria to
set a mass cut to reproduce the peak of the light curve of SN 1998bw (Nakamura et al. 2001).
Taking into account the difference of the progenitors, in the energies of the explosions and in
the way of the energy distribution we can say that the total masses of the main nuclei (along the
diagonal of the α-chain) obtained in our SN model are reliable. The masses of the other nuclei
seems to be underestimated because our network of nuclear reactions is not so rich.
The radiation from decay of radioactive 56 Ni contributes to the a SN luminosity and a spectral
peak. The mass of
56 Ni
produced in SNe could be estimated from observations by tail phase
luminosity analysis using a number of methods, see for example Hamuy (2003). The fainter
luminosity corresponds to a smaller amount of
56 Ni.
The summarized data on the known SNe
type II-P (with plateau lightcurves) progenitors, including SN 1987A, shows that the mass of
the produced 56 Ni does not exceed a value of 0.1 M⊙ (Smartt et al. 2009). Still in some events
more
56 Ni
production is expected by theoretical modelizations, for example 0.55 ± 0.05 M⊙
for SN 2003lw (Mazzali et al. 2006), 0.4 ± 0.125 M⊙ for SN 2003dh (Deng et al. 2005),
0.43 ± 0.05 M⊙ for SN 1998bw (Nakamura et al. 2001). All of these are extremely powerful
SNe associated with GRBs. The estimated progenitor masses for them are 45 ± 5, 32.5 ± 7.5 and
40 ± 5 M⊙ respectively.
Chapter 3. Aspherical Nucleosynthesis
48
According to observations it seems that we overestimated the mass of 56 Ni for this type of SN
progenitor. This could be fitted by adjusting of free parameters of the model – the explosion
energy and the size of the layer where it is inserted. Also it is important to obtain asymmetry in
a self-consistent way by considering the rotation. Since the size of the iron core is much smaller
than the computational domain it is necessary to use nestled grids in the neighbourhood of the
core. All this will be a topic of future research.
Chapter 4
Multidimensional simulations of
pair-instability supernovae
4.1 Introduction
The first stars of the Universe, called Population III stars (Pop III), are rapidly becoming an
important subject of investigation from the point of view of theory and observations. The formation of these stars hundreds of millions of years after the Big Bang marks the end of what is
called the “Dark Age”. Today’s telescopes cannot look far enough into the cosmic past, so we
do not have direct observations of how the primordial stars were formed. This new window is
paramount for astrophysics and cosmology. Population III stars are formed with primordial nucleosynthesis elements and are responsible for the formation of the first metals in the Universe.
The change in chemical composition also affects the initial mass function of stars. The energy
scattered in the newborn Universe will drastically influence its history. Certainly, the new generation of instruments will give us an opportunity to test theoretical ideas about the formation of
the first stars.
To study these stars, we need sophisticated numerical simulations (Bromm et al. 2003; Bromm
& Larson 2004). Pop III stars also have the potential to produce gamma-ray bursts (GRBs),
which may provide one of the most promising methods of directly probing the final stages of
Pop III stars (Chardonnet et al. 2010; de Souza et al. 2011).
Among these first–generation stars, an important role was played by massive stars. The Jean’s
mass favors the creation of very massive objects during star formation by condensing a nuclear
cloud. Numerical simulations predict that Pop III stars could have masses as high as a few
49
Chapter 4. Pair-instability supernovae
50
hundred solar masses. As these stars evolve, physical conditions in the center lead to the development of specific type of hydrostatic instability through electron-positron pairs creation (pair
instability).
As shown by many numerical simulations (Barkat et al. 1967; Waldman 2008; Woosley et al.
2002), these very massive stars could end their life, depending on mass, either by producing
pair-instability supernovae (PISNe), leaving no remnant, or by collapse to a black hole. In the
case of PISNe, the energy release is tremendous and could possibly be seen with new telescopes
(James Webb Space Telescope, European Extremely Large Telescope).
We present the results of one-dimensional simulations and analysis of the fate of a star depending
on physical conditions. Many recent articles (Chen et al. 2011; Joggerst et al. 2011) address
the problem of multidimensional simulations of a PISN explosion. Such studies are needed to
compute light curves and time of appearance of lines of chemical elements, so they allow us to
predict the observational signatures of these supernovae. We present 2D simulations of PISN
explosion based on the idea of nonuniform explosion and compare the results with the case of
uniform explosion in stellar core.
4.2 Pair-instability supernovae
One of the first works that has considered electron-positron pairs creation inside stars and
predicted instability caused by this process was the paper of Fowler & Hoyle (1964). The
density–temperature relations for massive stars (30M⊙ and higher) were computed, using a polytropic structure approximation and the assumption that a star evolves through series of quasiequilibrium states. A variable number of electron–positron pairs was taken into account by the
authors. They discovered that when the central temperature of a star reaches values ∼ 2 × 109 K
intensive creation of electron–positron pairs occurs. It increases the energy losses by neutrino
through a pair annihilation process, e+ + e− → νe + νe , and accelerates the contraction, raising
central temperature. The rise of temperature creates new pairs, and at some point, energy supply
from the contraction becomes insufficient to maintain production of additional pairs. As a result implosion of the core develops until the temperature reaches values necessary for explosive
nuclear burning.
Instability caused by pair production could also be described in terms of thermodynamics. Under
the assumption of an adiabatic process, the electron–positron pairs creation reduces adiabatic
index Γ = (∂ ln p/∂ ln ρ)S to values less than 4/3. If a significant part of a star reaches conditions
where Γ ≤ 4/3, the star becomes very “soft” and dynamically unstable.
First dynamical calculations of a star that undergoes pair instability were performed by Barkat
et al. (1967). It was shown in this paper that implosion of 40 M⊙ oxygen star could be reversed
Chapter 4. Pair-instability supernovae
51
into explosion by nuclear burning in the center. Nuclear energy release was ∼ 6 × 1051 ergs,
corresponding to consumption of several solar masses of oxygen. It was enough not only to
reverse the collapse but also to disrupt the star totally without any remnant. Further investigation
of the evolution of very massive stars is found in numerous works (Rakavy & Shaviv 1967;
Rakavy et al. 1967; Rakavy & Shaviv 1968). It was concluded that stars heavier than 25 M⊙
are pair-unstable. The cores lighter than this limit pass pair-instability zone and evolve up to the
Fe-He transition region. Dynamical calculations of unstable oxygen cores for 45–60 M⊙ were
performed by Fraley (1968), who obtained an oscillation for the lightest core and explosion for
the heavier cores from this interval. First detailed evolutional calculations of helium cores were
performed by Arnett (1972). He has demonstrated that the cores consist mainly of oxygen when
reaching the pair-instability zone. Dynamical calculations of helium cores with masses 64 M⊙
and 100 M⊙ were presented by Arnett (1973, 1974). The cores exploded and the lighter one
left a 2.2 M⊙ silicon remnant. Wheeler (1977) found that the cores with masses 103 –104 M⊙
collapse.
These numerical simulations have shown that the fate of a star depends on its mass and its
configuration. Apparently there is a window of masses when a star explodes as pair-instability
supernova (Woosley et al. 2002). If the star is too light it avoids pair instability and could
collapse to a black hole or neutron star owing to dissociation of iron at higher temperatures.
If the star is too heavy, nuclear burning is not enough to reverse the implosion caused by pair
creation. In the middle mass range, there is a possibility of explosion. A detailed scenario
of the dynamical stage depends on the balance between the rate of neutrino losses, speed of
contraction, and rate of nuclear energy release.
4.3 Numerical approach
To investigate the behavior of pair-unstable stars, we performed various hydrodynamical simulations. With the one-dimensional (1D) Lagrangian code, we studied the fate of oxygen cores
depending on mass and initial configuration. Application of 1D Lagrangian schemes (not only
for evolutionary, but also for dynamical calculations) is common in modern research (Kasen
et al. 2011; Waldman 2008; Woosley et al. 2007) since it has some advantages in solving the
system of equations including nuclear burning.
To study the last stage of explosion when shockwave propagates outward, we applied a twodimensional (2D) code. There are a few recent modelizations of PISNe in 2D (Chen et al. 2011;
Joggerst et al. 2011). In both cases a modern astrophysical code, CASTRO, has been used. This
code is based on a high-order radiation-hydrodynamics solver (Almgren et al. 2010), includes
self-gravity, and allows following nucleosynthesis and energy output from nuclear burning during the explosion. But to investigate the influence of hydrodynamical solvers we applied our
Chapter 4. Pair-instability supernovae
52
own numerical code based on the piecewise parabolic method on a local stencil (PPML) (Popov
& Ustyugov 2007; Popov 2012).
Of course a real star is three-dimensional, so detailed and precise description requires consideration of all physical processes inside a star in 3D. Extension of PPML code to full 3D case will
be made in future.
4.3.1
Modelling in 1D
We considered dynamics of massive stellar cores that are unstable due to pair creation. One
of the key questions of supernova modeling is the choice of the initial configuration. A very
common approach is to compute the pre-supernova configuration by evolutionary codes. But
evolutionary codes usually assume spherical symmetry, and any asymmetric process, such as
convection or rotation, can only be taken into account in some effective way with approximation.
Also detailed computation of stellar evolution requires considering many physical processes that
have uncertain descriptions. This lead to precise presupernova configurations possibly being
unknown.
Instead of taking initial conditions from evolutionary calculations, we used a polytropic structure
approximation with index 4/3. For each core having mass Mc , we built several configurations
by choosing different values of central density, ρc . This allowed us to consider models with
different values of binding energy, Ebind , for each Mc (see Table 4.1). Although this approach
simplifies description of the stellar structure, it gives a possibility to explore a wider variety explosion scenarios that depend on initial configuration. Thermodynamical quantities at the center
that we have chosen and values of binding energy are very close to the results of evolutionary
calculations (Waldman 2008).
The question of the presence of massive hydrogen envelope in very massive stars is under debate.
It is not clear that such stars keep this envelope up to the moment of explosion since it is very
weakly bound, for example, it was shown by Woosley et al. (2007) that the envelope could be
expelled in previous stages of evolution. In some works (e.g. Nozawa et al. 2003) the dynamics
of very massive stars was considered by taking the massive envelope into account. In others (e.g.
Kasen et al. 2011), only the cores were considered. In this work we only followed the dynamics
of the cores without an envelope.
For the 1D computations we developed a numerical code based on the standard Lagrangian
approach (e.g. Dunina-Barkovskaya & Imshennik 2003). The equation of state that we used
(Blinnikov et al. 1996) takes the birth of electron–positron pairs into account. Energy release
from nuclear burning and neutrino losses were taken into account. Nuclear burning was followed
by α-chain of reactions up to 56 Ni.
Chapter 4. Pair-instability supernovae
53
Table 4.1: Presupernova models and parameters of explosion
Mc
[M⊙ ]
60
60
Ebind
[1051 erg]
-5.54
-5.48
70
70
70
70
70
-6.10
-5.78
-5.27
-4.87
-3.67
78
78
78
ρc
g/cc]
0.87
1.15
Fate
T max
[109 K]
4.07
4.07
Enuc
[1051 erg]
20.9
21.0
PISN
PISN
1.0
1.3
1.7
2.0
3.0
—
—
4.52
4.16
3.74
—
—
29.4
25.9
18.8
CC
CC
PISN
PISN
PISN
-4.59
-3.73
-2.96
2.0
2.5
3.0
—
4.18
3.84
—
28.8
23.1
CC
PISN
PISN
90
90
90
90
90
90
-5.19
-3.78
-3.01
-2.07
-1.85
-1.43
1.5
2.0
2.3
2.7
2.8
3.0
—
—
—
4.77
4.30
4.11
—
—
—
38.7
34.9
38.7
CC
CC
CC
PISN
PISN
PISN
95
95
95
95
-1.87
-1.11
-0.98
-0.63
1.5
2.6
2.8
3.0
—
4.39
4.31
4.20
—
39.9
37.1
35.0
CC
PISN
PISN
PISN
[105
100
-2.43
2.0
—
—
CC
100
-1.12
2.5
4.56
44.7
PISN
100
-0.69
2.65
4.32
41.5
PISN
Mc is the mass of the oxygen core, Ebind the initial binding energy, ρc the initial central density,
T max the maximum temperature at the center at the moment of reversal of collapse, Enuc the
total nuclear energy release. PISN – the model explodes without any remnant, CC – collapse at
the center of the core.
Summarized results of our computations are presented in Table 4.1. The first important fact
∗
is that for each core mass, Mc , there is some critical value Ebind
of initial binding energy that
∗
determines the fate of the core: explosion (PISN) if Ebind > Ebind
or core-collapse1 (CC) for
∗ . Two regions could be seen clearly in the M –E
Ebind < Ebind
c
bind diagram (Fig. 4.1). This
behavior could be explained by the fact that models with lower Ebind (higher absolute value of
Ebind ) gain higher kinetic energy to the moment of oxygen ignition and proceeds faster to Fe-He
transition zone (photodissociation). Thus outer layers of the core do not have enough time to
bounce and expand. Therefore the pressure on the central part could not be reduced. Photodissociation dramatically drops the pressure in the center, and the core collapses. The critical value
1
Collapse of the central part of the core with possible ejection of outer layers.
Chapter 4. Pair-instability supernovae
54
0
Explosion
-4
E
bind
, 10
51
erg
-2
Collapse
-6
60
70
80
90
100
M , M
c
Figure 4.1: Fate of a star depending on its mass, Mc , and binding energy, Ebind . Explosion is
marked by diamonds and collapse is marked by circles.
∗
Ebind
tends to zero with a growth of Mc . Considering that for a stable nonrotating configuration
the binding energy should be negative, we can propose the mass limit for the explosion of nonrotating oxygen core at a value of about 110 M⊙ . This value is in good agreement with results
of the previous works (Bond et al. 1984; Ober et al. 1983). Rotation could push the mass limit
of explosion to higher values (Glatzel et al. 1985).
An interesting correlation has been found for the models that explode: value of total nuclear
energy release, Enuc , increases with maximum temperature T max reached at the center (Fig. 4.2).
This dependence could be described in a similar way to Ayasli & Joss (1982). For the case of
thermonuclear flash in neutron star they have shown that the mass of consumed nuclear fuel is
proportional to the square of burning temperature.
4.3.2
Numerical explosion in multi-D
To study the role of hydrodynamical instabilities on process of explosion we have performed 2D
computations and compared them with recent works (Chen et al. 2011; Joggerst et al. 2011).
Hydrodynamic simulations were performed with a numerical code based on the PPML (Popov &
Ustyugov 2007; Popov 2012). The key PPML-procedure written on FORTRAN could be found
in Ustyugov et al. (2009). This method is an improvement over the popular piecewise parabolic
method (PPM) suggested by Colella et al. (1984) for compressible flows with strong shocks and
extended by Dai et al. (1994) for magnetohydrodynamics. The PPML method has demonstrated
high accuracy in both smooth and discontinued solutions. It could be more suitable for problems
where the requirement of low dissipation is crucial. In our case low dissipation of PPML is
needed to describe shocks and contact discontinuities with higher accuracy.
Chapter 4. Pair-instability supernovae
55
E
nuc
, ergs
1053
1052
E
2
nuc
~T
max
1051
4
T
max
6
9
, 10
K
Figure 4.2: Nuclear energy release as a function of maximum temperature (diamonds). The
slope E ∝ T 2 is shown. For comparison data from Arnett (1996) (stars) and Ober et al. (1983)
(triangles) are shown.
For current study we chose a simplified physical model of explosion. We neglected the energy
feedback from nuclear reactions and gravity changes in the source of explosion, trying to obtain
the principal possibility of a total disruption of the stellar core to many fragments in the case of
a very massive progenitor.
With our hydrodynamic SN model we investigated a Pop III star with a 100 M⊙ oxygen core
assuming rotational symmetry. The oxygen core has the radius r = 0.29 R⊙ , beyond which
a uniform density of ∼ 1 g/cm3 was set for the uniform atmosphere. We used a polytropic
model of a star with index γ = 4/3. The initial configuration was calculated from the condition
of hydrostatic equilibrium of a nonrotating ideal gas sphere with a polytropic equation of state
(EOS) p = Kργ , where K = 8.225 × 1015 , p, and ρ are expressed in CGS units. Temperature was
computed as p = ρRT/µ, where R is the universal gas constant and µ the mean molecular weight,
which was assumed to be µ = 0.37143. This value of µ gives the ideal gas approximation, which
is very close to the full realistic EOS in the considered range of density and temperature.
We simulated the explosion by depositing the thermal energy in the central region in the two
variants. In the first case the energy E = 5 × 1052 ergs was uniformly deposited within the radius
re = 0.1 R⊙ , which contains 60 M⊙ . In the second variant the energy was inserted by the series
of ten ignition bubbles at t = 0 sec. All of the bubbles had different energy values and sizes
distributed in a stochastic way. The bubbles were randomly put within 0.06 R⊙ of the center.
The total energy deposited was also E = 5 × 1052 ergs. This nonuniformity could present some
inhomogeneities in the core that occur prior to explosion.
Chapter 4. Pair-instability supernovae
56
Z, R
(a)
Z, R
(b)
r, R
r, R
Figure 4.3: SN model with central ignition for t = 28 sec. Logarithm of density (a) is shown
in units of ρc = 2.65 × 105 g/cm3 . Temperature (b) is shown in units of T c = 2.36 × 109 K.
Nuclear burning in the center of a star could cause the development large-scale convection (Arnett & Meakin 2011). If convection occurs prior to the moment of pair instability, the contraction and explosion could be nonsymmetrical. Inhomogeneities in temperature and density could
cause ignition spots to occur in the core.
The results of computing of the uniform explosion are presented in Fig. 4.3. It shows density on a
logarithmic scale and temperature for the moment t = 28 sec. The solid line represents the initial
position of the boundary of the core. The shock, produced by the explosion, is split on two fronts
that are propagating through the rarefied matter and heating it. In the central part of the core,
there is a region with a Rayleigh-Taylor instability. The radius at which this instability occurs
Chapter 4. Pair-instability supernovae
57
Z, R
(a)
Z, R
(b)
r, R
r, R
Figure 4.4: SN model with multicore ignition for t = 28 sec. Logarithm of density (a) is shown
in units of ρc = 2.65 × 105 g/cm3 . Temperature (b) is shown in units of T c = 2.36 × 109 K.
is very close to the value obtained in Chen et al. (2011). The development of the instability
requires more precise computations with a more physical configuration of the envelope, but we
suppose that this instability could result in the formation of large-scale nonuniform structures
on a large time scale.
To cure the pathological “carbuncle” behavior that could appear in a region where the shock is
almost but not exactly parallel to the grid edges, the artificial dissipation was inserted according
to Loh & Jorgenson (2009). This is the reason for the instability development in Fig. 4.3 being
smoothed near the z-axis.
Figure 4.4 presents the density and temperature for multicore ignition. In this scenario the front
Chapter 4. Pair-instability supernovae
58
of the shockwave is not as smooth as in the previous case. Many fragments of hot matter appear
behind the shockwave. This could lead to disrupting of the star in many fragments. As a result
the light curves of such supernova could be more complex than in a spherically symmetric case.
4.4 Discussions and conclusions
We have presented our analysis of a PISN explosion. Results of a 1D simulations are in good
agreement with previous works. We proposed the initial binding energy of a star as the criteria
of its subsequent fate. An interesting correlation between total nuclear energy release and maximum temperature was found. We also performed 2D numerical simulations. We proposed to
explore two possible models of PISN explosion mechanism: uniform and nonuniform explosion.
The spherically symmetric explosion in general reproduces the results of other numerical codes
(Chen et al. 2011; Joggerst et al. 2011); however depending on how nuclear explosion occurs in
massive star, we could imagine different pictures of the stellar explosion. If the stellar core is
nonuniform prior to explosion, nuclear burning could start in some parts of the core. To simulate
this model we proposed a multicore ignition scenario. This could be an “exotic scenario” and
not the standard explored to now. But if the explosion is nonuniform, it could change the light
curve, chemical production, and also the spectrum.
Another key question of PISN explosion phenomena is the role of the envelope. Woosley et
al. (2007) propose the idea that quite small pulsations of a pair-unstable star could eject the
envelope. They applied their results to the case of SN2006gy, which is known to be a PISN.
If we apply an idea of nonuniform explosion to a star without an envelope, it could produce a
light curve with very complex behavior that is different from the plateau type. But this problem
requires additional investigation by radiation-transfer computations.
Chapter 5
On GRB Spectra
5.1 Introduction: snapshot of GRB spectrum
Almost fifty years ago, the Vela satellites discovered the gamma-ray bursts (GRBs), intense
flashes of X and gamma radiations (Strong et al. 1974; Strong & Klebesadel 1974). Since that
time, many scientists from around the world have built new experiments to study this enigmatic
phenomena, developing a multidisciplinary field that explore the origin of GRBs (Gehrels &
Razzaque 2013). In those fifty years, there have been great advances in GRBs. Among this
we can cite: BATSE experiment has shown that practically all GRB spectrum follow a phenomenological law called BAND spectrum, BeppoSAX satellite has demonstrated that GRB
have a cosmological origin, SWIFT and FERMI satellites have defined a more precise picture
of GRB light curve and spectra. Many other experiments in space and on the Earth have also
contributed to clarification the definition of a GRB at least from experimental point of view. On
the other hand, great advances have been made from the theoretical point of view. Many models
have been abandoned and it is well establish that GRBs with long duration are related to the
end of massive stars. We can cite (Bisnovatyi-Kogan et al. 1975; Paczyński 1998; Woosley &
Wallace 1982) and a lot of others. Although it is true that many scientists favoured the idea of
the fireball, there is still no undisputed evidence for the GRB engine. However it is now widely
accepted that the progenitor of GRB is related to the end of massive stars thanks to many optical
observations. From the point of view of numbers any GRBs could be characterized by it total
isotropic luminosity Liso , a peak energy spectrum E peak and a very short duration of the prompt
emission ∼ 100 seconds compare to the relatively long duration afterglow up to years. In 1993,
Band proposed a simple phenomenological law to interpret the GRB spectrum as follows (Band
et al. 1993) :
59
Chapter 5. On GRB Spectra
60

α −E/E0 , (α − β)E ≥ E,


0
 CE e
NE (E) = 

 C′ Eβ,
(α − β)E0 < E,
(5.1)
where NE (E) is the photon number spectrum, E is the photon energy in keV, E0 is the efolding energy, α and β are the asymptotic power-law indices, C is the amplitude in photons
s−1 cm−2 keV−1 and C ′ is chosen to make NE (E) a continuous and continuously differentiable
function:
C ′ = C ((α − β)E0 )α−β e−(α−β) .
Spectral hardness is usually measured by the peak energy E peak of the EF E spectrum (F E =
ENE ). This energy is related to E0 as E peak = (2 + α)E0 and (α − β)E0 = Ebreak .
This law represents the most successful way of fitting GRB spectrum. In absence of any physical interpretation, we could advance the reasonable assumption that the energy peak Ebreak
represents the maximum of emission for GRB spectrum, i.e. where the maximum emission is
concentrated. In some sense, we expect that Ebreak represents some fundamental property of
GRB physical emission processes.
Fenimore et al. (1982) noticed in 1982, that the Comptonization mechanism for GRB emission was certainly more adapted. Later, the idea of synchrotron emission proposed by (Katz
1994) was accepted by scientists. Granot et al. (1999) have used a variant, the synchrotron
self-absorption, to reproduce the different breaks in frequencies and the necessary presence of
magnetic field GRB engine makes this idea plausible. Many bursts were with synchrotron emission (Crider et al. 1997; Ghirlanda et al. 2003; Preece et al. 1998). But nowadays it is shown,
that optically thin synchrotron emission is incompatible with the hard low energy slopes observed in substantial fraction of GRBs (Beloborodov 2013; Kaneko et al. 2006; Lundman et al.
2013; Preece et al. 1998).
One of the popular models is the theory of photospheric emission (Larsson et al. 2011). This
model takes into account the hard low-energy spectra, photospheric emission (Mészáros & Rees
2000; Mészáros et al. 2002; Pe’er et al. 2006; Rees & Mészáros 2005). Also this model explains
the correlations observed between the peak energy and luminosity (Borgonovo & Ryde 2001;
Golenetskii et al. 2006; Yonetoku et al. 2004). Some of the authors take a combination of
blackbody models and an additional non-thermal component. This model could describe many
bursts very well (Ghirlanda et al. 2003; Ryde 2004; Ryde & Pe’er 2009). We have tried to do an
other attempt based only on physical consideration.
We consider the thermal Bremsstrahlung associated with black-body emission without any initial physical constraints or a priori models. We have done fits of different GRB spectra and
compare also our results with the light curves. In the last part we will draw a possible scenario
in which this emission mechanism could work.
Chapter 5. On GRB Spectra
61
5.2 Black-body and Thermal Bremsstrahlung emission
It is generally accepted by scientists that the GRB spectrum is a Band spectrum represented by
the function NB defined in paragraph 1. Following the analysis of Ryde and other, we will also
follow the idea of a thermal emission defined by:
NBB = ABB
E2
eE/kT − 1
(5.2)
where ABB is a normalization factor, which simply represents the abundance of thermal emission
in the GRB spectrum. kT represents a characteristic energy in the GRB engine, written in
temperature.
We simplify the problem assuming that the BB spectrum is at high temperature and represent the
peak of BAND spectrum: kT ∼ E peak . If this first indication is correct then we can also deduce
that the typical temperature of this thermal plasma is ∼ 109 K. It is also an experimental fact that
most of long duration GRB observed have an energy peak E peak in the range [(100, 300] keV
(Nava et al. 2011).
However planckian spectrum cannot reproduce entire GRB spectrum. We propose that an additional component of GRB spectrum is thermal Bremsstrahlung. This physical process has low
energy asymptotic in the form of
e−E/kT
E
, which is compatible with most GRB spectra.
We can conclude this paragraph by some interesting numbers. The thermal bremsstrahlung
emission from a hot plasma is characterized by temperature T and by the emissivity : :
√
ǫ f f = 1.4 × 10−27 T ne ni Z 2 g
(5.3)
in erg · s−1 · cm−3 . T is the electron temperature, ne and ni are the number densities of electrons
and protons, Z - electric charge of ions, g is the Gaunt factor, a phenomenological way to take
into account for the variety of elements. Typically, our plasma has a temperature ≃ 2 × 109 ,
therefore it is completely ionized and we have ni = ne = n. The dependence in Z 2 could change
the emissivity by 3 orders of magnitude in case of heavy elements, which are expected to be
produced in stellar explosion. If we take the maximum density allowed to have a optically
thin plasma for γ radiation : ∼ 10 g · cm−3 , this gives a maximum value for the density of
electrons and ions: n ≃ 1025 cm−3 . Therefore, we deduce the corresponding value for ǫ f f ≃
7.4 × 1024 erg · s−1 · cm−3 .
Assuming for simplicity a uniform emission inside a volume V, it is easy to compute the total
energy reservoir E Br and the luminosity LBr . We have: LBr = ǫ f f × V and E Br = 32 kT e × ne × V.
The cooling time, which measure how fast the process of thermal bremsstrahlung emission is
5.2 × 1018
Br
. Assuming T ≃ 2 × 109 , we obtain: τBr ≃
efficient, is simply given by: τBr = ELBr
n
Chapter 5. On GRB Spectra
62
in seconds. Even with a small density of ∼ 1015 , the time scale is ∼ 500, i.e. the duration of a
long GRB. The smaller time scale are obtained for higher value of density. It is a well known
fact, that thermal bremsstrahlung is a very efficient cooling process for high temperature thermal
plasma. Therefore, we expect that this mechanism with black body emission are certainly the
main dominant processes. It depends on two main parameters: the temperature of the plasma T
and the electron density n. We can also estimate the volume in order to reproduce a typical GRB
spike emission in the light curve. For this scenario, we assume T ≃ 2 × 109 and n ≃ 1025 cm−3 .
A luminosity of ∼ 1049 erg · s−1 , corresponding to an individual spike, could be obtained if the
total emitted volume is the order of 1025 cm3 . This corresponds to a spherical blob of radius
∼ 1000 km.
5.2.1
Our Model
In order to explain non-thermal spectrum we propose an idea that different physical processes
are responsible for different part of the spectrum. Regions with lower density radiate thermal
bremsstrahlung spectrum owing to interaction of ionized electrons with protons and ions. Thermal bremsstrahlung is responsible for formation of low-energy part of GRB spectrum. Central
part of GRB spectrum in our model is dominated by thermal emission from the regions with
higher density.
The spectrum of the radiated energy can be described as (Greene 1959):
IBr (E) = ne ni
32 π 1/2 e4 Ry
3 3
m2e c4 kT
!1/2
e−E/kT ḡ,
(5.4)
where IBr is the power emitted per unit volume per unit of time per unit energy, ne and ni are the
number densities of electrons and protons, E is the photon energy, T is the electron temperature,
Ry is the Rydberg energy constant, ḡ is the Gaunt factor. For the task of spectrum fitting it is
convenient to express IE in physical units:
erg
ne ni
.
IBr (E) = 2 × 10−20 √ e−E/kT ḡ
s · cm3 · keV
T
(5.5)
Then the photon number spectrum measured by the observer is:
NBr (E) =
IBr (E)V
,
4π E D2
(5.6)
where V is the volume of the emitting region, D is the distance towards the observer.
This simple relation explains why this scenario was rejected for GRB spectra in the case of
neutron star scenario in the 80’s (Harding 1991). Accretion on NS scenario used a volume V,
too small in order to reproduce the observed spectra. Another critics at that time was also the
Chapter 5. On GRB Spectra
63
fact that in this NS model, magnetic fields were high enough that Bremsstrahlung was in direct
competition with synchrotron emission which is more efficient. However, this relation tells us
that in the case of hot plasma (where T ∼ E peak ) this possibility should be analysed completely
and the conclusions of such analysis put in confront to other observational data, like light curves.
It should be only at the end, with a more detail picture, that we should reject or consider such
possibility.
Therefore the photon number density is proportional to E −1 e−E/kT :
NBr (E) = ABr
1 −E/kT
e
,
E
(5.7)
where ABr is a normalization factor that can be computed from Eqs. (5.5, 5.6)1 . Thermal
bremsstrahlung naturally explains NE ∝ E −1 low-energy slope observed in many bursts (Preece
et al. 1998).
The spectrum of thermal emission is described by a Planck function (blackbody emission):
NBB (E) = ABB
E2
eE/kT − 1
,
(5.8)
where ABB is a normalization factor which depends on the area of the emission surface and the
distance to observer. The resulting spectrum is thus the sum of two contributions:
N(E) = NBB (E) + NBr (E) = ABB
E2
eE/kT
−1
+ ABr
1 −E/kT
e
.
E
(5.9)
We want to underline that in our model we use a single value of the temperature for both processes. So as a result this model has three free parameters comparing to the four parameters in
the Band function.
5.3 Analysis of gamma-ray bursts
5.3.1
Analysis of GRB 090618
In order to test our model we checked whether it is possible to reproduce characteristic Band like
spectra by combination of two processes - BB and Br. We underline that we did not performed
a full cycle analysis such as choosing the correct detectors, analysing noise, etc, since the data
analysis is not the primary goal of the work. Instead we have used a simplified approach - we
did our analysis basing on already analysed bursts for which we were able to find the substracted
data (the data for GRB090618 was generously presented by L.Izzo). Figure 5.1 shows the fit
1
In keV–MeV energy range Gaunt factor ḡ is a slowly varying function of the photon energy and can be assumed
to be constant order of 10 (Brussaard & van de Hulst 1962; Anzer & Boerner 1976).
Chapter 5. On GRB Spectra
64
Figure 5.1: Time-integrated spectra for the time interval from 50 to 59 s after the trigger time
of GRB 090618. Blue line shows the fit with the Band function (Eq 5.1), orange line shows the
blackbody + bremsstrahlung (Eq. 5.9).
of the time-integrated E 2 · NE spectra (the energy spectra per logarithm of the energy interval,
measured in photons · keV · cm−2 · s−1 ) for the time interval from 50 to 59 s after the trigger
time with blackbody + bremsstrahlung (Eq. 5.9). We can see that the curve describes the data
very well - resulting function reproduce very close the observation data. But of course more
precise analysis should be done and for more bursts which is the aim for future works. Since
we didn’t consider high-energy part of the spectrum the difference between these two fittings is
significant for the photon energies above 500 keV, but our fitting is still within error boxes of
the data. The value of the temperature that we obtain from the fitting is kT = 117.5 keV which
can be easily obtained in PISN explosion. Also we checked five time intervals for this GRB.
Each of the intervals corresponds to a particular pulse (Rao et al. 2011), (Izzo et al. 2012) . In
the table 5.1 there is the result of our computations. Now if we look at the time dependence
we have also succeeded to obtain a promising result (figure 5.3 ). In our fit we can plot the
temperature evolution with time T(t) (figure 5.4, figure 5.5). The evolution of temperature with
time corresponds to behavior of the lightcurve.
5.4 Analysis of some other GRBs
We also have analysed several other bursts and succeed to reproduce the spectrum. And here are
examples of other bursts that can be successfully reproduced by our model.
We analysed GRB 101023. Redshift of this GRB z = 0.9 and isotropic energy Eiso = 1053 erg
(Penacchioni et al 2012). In the figure 3 in Appendix it is shown the fit for GRB 101023 of the
time-integrated E 2 ·NE spectra (the energy spectra per logarithm of the energy interval, measured
Chapter 5. On GRB Spectra
65
interval [0-50 s]
interval [50-59 s]
interval [59-69 s]
Figure 5.2: Time-integrated spectra for the different time intervals: from 0 to 50 s, from 50 to
59 s, from 59 to 69 s after the trigger time of GRB 090618. Blue line shows the fit with the
Band function (Eq 5.1), orange line shows the blackbody + bremsstrahlung (Eq. 5.9).
Chapter 5. On GRB Spectra
66
interval [69-78 s]
interval [78-105 s]
Figure 5.3: Time-integrated spectra for the different time intervals: from 69 to 78 s, from 79
to 105 s after the trigger time of GRB 090618. Blue line shows the fit with the Band function
(Eq 5.1), orange line shows the fit with blackbody + bremsstrahlung (Eq. 5.9).
in photons · keV · cm−2 · s−1 ) for two time intervals: 0-45 s and 45-89 s after the trigger time with
the Band function (Eq 5.1) and blackbody + bremsstrahlung (Eq. 5.9).
For GRB 970111 ((Frontera et al. 2013) ) also was done time resolved spectral analysis dividing
the count lightcurve and we can see the temperature evolution with time (figure 5, figure 6) in
Appendix.
Chapter 5. On GRB Spectra
67
Figure 5.4: The temperature evolution with time T(t) of GRB 090618
ergs
Flux
cm2 s
8. ! 10"6
6. ! 10"6
4. ! 10"6
2. ! 10"6
50
100
150
Time!s"
Figure 5.5: Example of light curve of GRB 090618 (Izzo et al. 2012)
5.5 Discussions and conclusions
In the first decade of GRB observations, their spectra appeared to be well fitted by thermal
bremsstrahlung with temperature of ∼ 100 keV (Mazets & Golenetskii 1981). However, in the
80’s the most popular model for GRB was the idea that the bursts are coming from strongly
magnetized neutron stars (Harding 1991). Consequently, even if thermal bremsstrahlung provides very good fits of GRB spectra, the condition of this emission in a scenario of accreting
neutron stars became less plausible and has been abandoned. The high magnetized neutron
stars naturally favour synchrotron emission. We consider that the emission mechanism of GRBs
is certainly a key to understand properly the engine of this phenomena. In fact, any emission
mechanism needs special geometry or assumptions and could provide also additional values of
GRB predictions. In this work, we have reconsidered thermal bremsstrahlung and we propose
a plausible interpretation of the GRB spectra compatible with other observations. In this paragraph we will present our results and derive some predictions in order to test this scenario. Our
fundamental assumption is not to restrict the size of the source in order to not eliminate, a priori
any processes. We would like to derive possible geometrical consequences from this simple
assumption.
The first result presented in Figure 5.1, represents a very good fit of GRB090618 where we have
Chapter 5. On GRB Spectra
68
Table 5.1: Spectral analysis of GRB 090618
GRB 090618
Time int. [s]
[0 − 50]
[50 − 59]
[59 − 69]
[69 − 78]
[78 − 105]
161
−1.04
−2.42
124
−1.06
−2.62
Redshift = 0.54
Eiso [erg] = 2.90 × 1053
Band fitting:
E0 [keV]
αBand
βBand
119
−0.74
−2.32
195
−1.13
−2.5
321
−0.99
−2.60
Our model:
ABr
ABb
kT [keV]
RBB [cm]
LBr [erg/s]
LBb [erg/s]
%Br
Ltot [erg/s]
48.33
3.25 × 10−8
163.8
1.70 × 108
1.04 × 1051
2.69 × 1050
75.78
7.72 × 10−8
138.7
2.61 × 108
1.48 × 1051
3.28 × 1050
320.5
11.8 × 10−9
304.8
1.02 × 108
9.63 × 1051
1.17 × 1051
223.2
2.17 × 10−7
145.8
4.39 × 108
4.50 × 1051
1.12 × 1051
135.4
2.64 × 10−7
111.2
4.84 × 108
2.349 × 1051
4.63 × 1050
79, 4%
81.9%
89.1%
80.0%
83.5%
1.31 × 1051
1.81 × 1051
1.08 × 1052
1.63 × 1051
2.81 × 1051
used both black-body and thermal bremsstrahlung. This combination tells us that the environment should be very hot ∼ 100 keV. From this first result we do expect to have a hot plasma
cooling quickly.
We notice also that at low energy ∼ 1 keV the GRB spectrum is essentially due to the bremsstrahlung.
The percentage of BB emission increase gradually with increase of energy (see table 5.1).
It is well known that GRB spectra is characterized by hard to soft emission. In our interpretation
the hard part of the spectrum correspond to the thermal black-body emission while the soft
part is related to the thermal bremsstrahlung. This difference tells us also about the thickness
of the emission region: the thermal emission is coming from optically thick region while the
bremsstrahlung emission comes from optically thin regions. This result should be keep in mind
for a dynamical point of view. To explain hard to soft evolution of the spectra in view of our
result, a possibility could be that dense regions, more thick, emits first a thermal emission; and
with time, due to fast expansion, it become less dense and therefore a thermal bremsstrahlung
emission could be produced. These are possible signatures on the Figure 5.1 and table 5.1.
We have obtained similar results for other bursts. Bremsstrahlung works out very well. In hot
thermal plasma we naturally expect to have photons copiously emitted by black body emission
and by thermal bremsstrahlung depending on thickness. Table 5.1 gives us an indication of the
Chapter 5. On GRB Spectra
69
importance of thermal bremsstrahlung in GRB 090618. As we said, this abundance is certainly
related to the dynamics and geometry. Almost all GRB spectra present a turn-off called E peak
in the BAND terminology, where we have E peak ∼ kT . If we assume this model is correct we
do expect a very hot thermal plasma. In hot thermal plasma where many heavy elements are
present abundantly, we expect in addition to thermal bremsstrahlung to have also lines. These
lines may be difficult to detect now but this could be an additional proof of the bremmsstrahlung
emission.
Our second result involves the light curves. It has been noticed since times that GRB spectra
are complex due to time variability (Minaev et al. 2013). Norris et al. (1986) noticed that each
pulse inside a GRB present also this tendency of hard to soft emission. Therefore, we have
apply our combined fit ( black-body + thermal Bremsstrahlung) with time-sequence analysis of
GRB090618. If we analyse carefully our results in table 5.1, we see clearly that T is not constant
with time and it is not a monotonic variation of time. The variation with time of the temperature
of the hot plasma is represented in Figure 5.4, while the light curve of the associated GRB is in
Figure 5.5. We have found that the temperature correlates with the variation of the light curve.
We have also test these results on other GRBs. Here we present the case of GRB970111 in
Figures 5 and 6. In the table 5.1, we have also computed a radius associated with the black-body
emission. This radius is very small ∼ 108 cm. This radius is an equivalent radius for a BB
emission. The true radius of the star, which corresponds to hydrodynamical process related to
the explosion, is certainly much larger. It means that the BB emission is not uniform on a unique
surface but only bright hot spots located on a sphere with larger radius (see figures 5.6 and 5.8).
Here, we have assumed uniform density of electrons and ions and will take a constant radius for
duration of spectrum. In addition we assume identical temperature for electrons and ions. With
this crude assumption, the density derived are consistent with values of stellar explosion. The
correct picture is certainly more complex with non uniform density and with possible clumps.
Following our time-resolved spectra analysis we expect that the temperature introduced in our
first conclusion is correlated in the same way as the variation of the light curve.
Our third result is a natural consequence of the previous results for time-resolved and timeintegrated spectra. This result is very important since it establishes a physical link between one
parameter of the spectra: the peak energy, and another parameter of the light curve: the maximum flux. We have shown that this link is valid even for a time-resolved spectra. The physical
consequence, independent of any model, is important because this is a bridge between the two
sets of data: light curves and spectra. The simplicity of the spectra through the phenomenological modelling with BAND spectrum is certainly related to a simple emission mechanism. On
the other hand, the complexity of the light curve should also incorporate this simple emission
mechanism. We need to consider the light curve not as a whole complex structure but as complex superposition of simple structures: individual spikes with the same emission mechanism,
which being applied to entire light curve also works. In short, a GRB light curve could be the
Chapter 5. On GRB Spectra
70
Figure 5.6: Hot spots randomly appear on the surface of exploding star. Each spot produces
spike of emission. Signals from these spots arrive at different times, so an observer sees the
superposition of the spikes.
result of many secondary sources. The analogous idea supported by fitting a number of different
pulses was discussed in the paper Minaev et al. (2014).
For us this paradigm is essential because this conclusion is a consequence of fits of data without
any assumption of any engine model. It is generally believed that GRB light curves are more
complex than GRB spectra, at least it appears like this. This result gives a possible connection
between both and invites us to understand a light curve not in totality but in the individual spike.
We believe that we could be able to find a similarity in each spike, despite to global diversity
of all GRB light curves. As example, in our cartoon (figure 5.6), we have used an exponential
behaviour for the time profile. Each spike, as a fingerprint, could keep the memory of the GRB
physical process. Maybe the secret of the GRB engine reside in the study of single spike.
Now, the central question, is to understand how gamma-ray burst scenario could have black
body and thermal bremsstrahlung emission. The paradigm of gamma-ray burst is centralized
on compactness problem and on the necessity of ultra-relativistic motion. This is the first ingredient of almost all GRB models. However, in a seminal paper, Ruderman introducing the
physical problem of gamma-ray bursts did not exclude the possibility that different spikes might
originate in different regions separate by larger distances (Ruderman 1975). It is interesting to
see that Bussard and Lamb noticed also in 1982, that thermal Bremsstrahlung will need large
area (Bussard & Lamb 1982). In the light of these results, we think that it could be interesting to
investigate the possibility of emission in large regions. In 2010, we have proposed a possibility
that gamma-ray bursts paradigm are related to the end of massive stars (Chardonnet et al. 2010).
The explosion of very massive stars, PISNe, gives a natural energy reservoir of ∼ 1053 ergs in a
very short time ∼ 100 seconds. The temperature at the instant of explosion of the stars gives us
a correct value of the mean energy in the spectrum ∼ 100 keV.
Chapter 5. On GRB Spectra
71
!
Figure 5.7: 2D simulation of an explosion of a 100 solar mass pair instability supernovae
explosion in the multicore explosion scenario using PPML methods. This picture shows the
fragmentation of the core and hot spot regions of very high temperature.
In Baranov et al. (2013) we have shown that the so called Amati relation (Amati et al. 2002)
could be explained as a natural consequence of this assumption. If we plot the nuclear energy
versus the temperature of the star at the instant of the explosion the points of different masses are
in perfect coincidence, without fine tuning, of the Amati diagram. In short the Amati relation
tells us the mass of the progenitor in first order. The metallicity is the other key parameter.
In our paper, we have proposed a new paradigm in which the core explodes by fragmentation
(figure 5.7). Many blobs are ejected outside the envelop and therefore could radiate in X and γ
rays. This new idea is supported by 2D hydrodynamical simulations (Popov & Ustyugov 2007;
Popov 2012). Consequently, with expansion, these bubbles become less dense and a thermal
bremsstrahlung could occurs related to thin regions.
The central idea is related to a very hot plasma expelled outside due to the fragmentation of
the core. This plasma is characterized by the temperature T. Now, if we assume that Eiso is
represented by the total nuclear energy available in the explosion of a PISNe and the energy
peak represents a measure of the temperature of the stars, we could propose also an associated
spectrum. The natural mechanism is thermal emission and also bremsstrahlung due to present
of a hot plasma. On the other hand, the explosion of the core by fragmentation made many hot
bubbles which correspond to the different spikes of the light curves. On a cartoon presented in
figure 5.6 we show that due to the propagation time, the superposition of each spikes is able
to reproduce an apparent chaotic light cure. However, each spike has the memory of the initial
plasma of the star. This is why, despite a great distance, each spike has in average the same
value of T. We have found an interesting correlation in this scenario between the temperature of
each spike the variation in time of the light curve.
Chapter 5. On GRB Spectra
72
Br
BB
Figure 5.8: Cartoon representation of a possible emission mechanism with Black-Body +
Bremmstrahlung emission
According to our spectra analysis we expect to have zones in the hot spots depending on density:
the core, a thick region, is associated to BB emission while the output region of these blobs less
dense, corresponding to bremsstrahlung emission as we proposed in figure 5.8. A detailed study
of this mechanism is under investigation with our 3D code.
Our scenario is similar to many secondary sources originated from the initial destruction of the
2
star. Here we have used a simple function in time Ii = I0,i e−(t−t0,i ) for the luminosity function
of each blob. Here Ii is the intensity, I0,i is normalization factor, which is assumed to be proportional to the size of a spot i; t0,i is the time delay of arrival of a signal to observer, which
depends on position of a spot on the surface. This simple function shows that the light curve of
P
the burst could be explained by the sum (superposition) of the individual light curve I = i Ii .
This new interpretation also fits with the idea of the photospheric emission developed in many
articles. The radius of the photosphere introduced in many articles corresponds to the radius of
emission of our blobs seen in figure 5.6. We have analysed also: GRB990510, 980329, 101023
and 090926A (see in Appendix). The conclusions are the same: the variation of T, in our model
of BB+thermal Bremsstrahlung is correlate with the variation of the flux in the light curve.
Therefore, this results is robust and have a natural interpretation in our global scenario of thermal blob ejecta. We have propose the scenario in which the fragmentation of the core creates a
multiplicity of fragment of hot nuclear matter emitting in black body or thermal Bremsstrahlung
depending on the their thickness. Thermal Bremsstrahlung emission for GRBs was proposed by
(Mazets & Golenetskii 1981) and by also other authors. However, in the scenario favour at that
time, the neutron star, it was not possible to justify enough matter to explain the observed luminosity. In our PISNe scenario, the large quantity of matter is enough to explain the luminosity
of GRBs. This possibility, if confirmed proposes also to revisit the paradigm of the explosion of
PISNe. We have start to analyse some different scenarios of explosion with fragmentations. For
us a GRB light curve could be the result of many secondary sources.
Now, the central question of the validity of such scenario is more related to understanding of the
Chapter 5. On GRB Spectra
73
PISNe explosion and the role of the envelop and more generally to the link between GRB and
SN. If this scenario is correct, it means that the link of Gamma-Ray Bursts and SN should be
now considered with this new possibility. Recently, Yusof et al. have computed the evolution
and fate of massive stars (Yusof et al. 2013). They revised the death of massive stars. Especially,
for the range of metallicity used in their paper, all PINSe produce a SN type Ib or a type Ic but
not a type II. This is certainly another encouraging element since it is also the same type of SN
associated with GRBs. In a recent paper, Chakraborti et al. (2014) have studied the missing link
between GRBs and SN. With the actual standard scenario of GRB with ultra-relativistic motion,
we see from the figures 3 and 4 in their article that this missing link is far from being found.
SN have completely different characteristic from GRBs. In standard models GRB needs ultra
relativistic motion γ ∼ 100 − 1000 and needs very little mass: ∼ 10−6 solar mass. In contrast SN
does not have ultra relativistic motion and needs more mass ejecta. In a recent article (Baranov
et al. 2013) we have analysed the fragmentation of the core in the explosions of PISNe. Further
works is on progress using a new version of 3D hydrodynamical code.
Chapter 6
Cosmology with GRBs
6.1 General introduction to cosmology
All information about the universe comes to us from the observation of a shift in the frequencies of spectral lines from distant galaxies as compared with their values observed in terrestrial
laboratories.
The light emitted by a distant galaxy can be viewed either quantum mechanically as freelypropagating photons, or classically as propagating electromagnetic waves. To interpret the observations correctly, we need to take into account that the wavelength of the light gets stretched
(or, equivalently, the photons lose energy) by the expansion of the universe (Weinberg 2008).
Redshifting of photons. In the quantum mechanical description, the wavelength of light λ is
inversely proportional to the photon momentum, λ = h/p. Where h is Hubble constant in units
of 100 km sec−1 M pc−1 and p is a pressure. Since according to the momentum of a photon
evolves as a(t)−1 , the wavelength scales as a(t). Light emitted at time t1 with wavelength λ1 will
be observed at t0 with wavelength
λ0 = λ1
a(t0 )
.
a(t1 )
(6.1)
Since a(t0 ) > a(t1 ), the wavelength of the light increases, λ0 > λ1 .
Redshifting of classical waves. We can derive the same result by treating light as classical electromagnetic waves. Consider a galaxy at a fixed comoving distance d. At a time τ1 , the galaxy
emits a signal of short conformal duration ∆τ. According to ∆τ = ∆χ (the radial coordinate), the
light arrives at our telescopes at time τ0 = τ1 +d. The conformal duration of the signal measured
by the detector is the same as at the source, but the physical time intervals are different at the
points of emission and detection,
∆t1 = a(τ1 )∆τ
74
(6.2)
Chapter 6. Cosmology with GRBs
75
and
∆t0 = a(τ0 )∆τ.
(6.3)
If ∆t is the period of the light wave, the light is emitted with wavelength λ1 = ∆t1 (in units where
c = 1), but is observed with wavelength λ0 = ∆t0 , so that
λ0 a(τ0 )
=
.
λ1 a(τ1 )
If a(t) is increasing, then this is a redshift, a decrease in frequency by a factor
(6.4)
a(t1 )
a(t0 ) ,
equivalent to
an increase in wavelength by a factor conventionally called 1 + z:
1+z=
a(t0 )
.
a(t1 )
(6.5)
Alternatively, if a(t) is decreasing then we have a blueshift, a decrease in wavelength given by
the factor Eq. 6.5 with z negative. These results are frequently interpreted in terms of the familiar
Doppler effect; For an increasing or decreasing a(t), the proper distance to any comoving source
of light like a typical galaxy increases or decreases with time, so that such sources are receding
from us or approaching us, which naturally produces a redshift or blueshift. For this reason,
galaxies with redshift (or blueshift) z are often said to have a cosmological radial velocity cz.
(The meaning of relative velocity is clear only for z ≪ 1, so the existence of distant sources
with z > 1 does not imply any violation of special relativity.) However, the interpretation of the
cosmological redshift as a Doppler shift can only take us so far. In particular, the increase of
wavelength from emission to absorption of light does not depend on the rate of change of a(t) at
the times of emission or absorption, but on the increase of a(t) in the whole period from emission
to absorption. We can also understand the frequency shift 6.4 by reference to the quantum theory
of light: The momentum of a photon of frequency ν is hν/c (where h is Planck’s constant), and
the momentum varies as 1/a(t).
For nearby sources, we may expand a(t) in a power series, so
a(t) ≈ a(t0 )[1 + (t1 − t0 )H0 + · · · ]
(6.6)
where H0 is a coefficient known as the Hubble constant:
H0 ≡
ȧ(t0 )
.
a(t0 )
Eq.6.5 then gives the fractional increase in wavelength as
(6.7)
Chapter 6. Cosmology with GRBs
76
z = H0 (t0 − t1 ) + · · · .
(6.8)
For close objects, t0 − t1 is simply the proper distance d (in units with c = 1). We therefore find
that the redshift increases linearly with distance
z ≈ H0 d.
(6.9)
The slope in a redshift-distance diagram therefore measures the current expansion rate of the
universe, H0 . These measurements used to come with very large uncertainties. Since H0 normalizes everything else (see below), it became conventional to define
H0 ≡ 100 h kms−1 M pc−1 ,
(6.10)
where the parameter h is used to keep track of how uncertainties in H0 propagate into other
cosmological parameters. Today, measurements of H0 have become much more precise
h ≈ 0.7 ± 0.01.
(6.11)
Another important moment is how astronomical distances are measured.
The most common method of determining distances in cosmology is based on the measurement
of the apparent luminosity of objects of known (or supposedly known) absolute luminosity. The
absolute luminosity L is the energy emitted per second, and the apparent luminosity l is the
energy received per second per square centimetre of receiving area. If the energy is emitted
isotropically, then we can find the relation between the absolute and apparent luminosity in
Euclidean geometry by imagining the luminous object to be surrounded with a sphere whose
radius is equal to the distance d between the object and the earth. The total energy per second
passing through the sphere is 4πd2 l, so
l=
L
.
4πd2
(6.12)
Metric distance. We first define a distance that isn’t really observable, but that will be useful in
defining observable distances. Consider the Friedmann-Robertson-Walker (FRW) metric in the
form
ds2 = dt2 − a2 (t)[dχ2 + S 2 k (χ)dΩ2 ],
(6.13)
Chapter 6. Cosmology with GRBs
77
√
where dχ = dr/ 1 − kr2 , and



R0 sinh(χ/R0 ), k = −1, hyperbolic





S k (χ) = 
χ,
k = 0, Euclidean





 R0 sin(χ/R0 ),
k = +1, spherical
(6.14)
The distance multiplying the solid angle element dΩ2 is the metric distance,
dm = S k (χ).
(6.15)
In a flat universe (k = 0), the metric distance is simply equal to the comoving distance χ. The
comoving distance between us and a galaxy at redshift z can be written as
χ(z) =
Z
t0
t1
dt
=
a(t)
Z
z
0
dz
,
H(z)
(6.16)
where the redshift evolution of the Hubble parameter, H(z), depends on the matter content of the
universe . We emphasize that the comoving distance and the metric distance are not observables.
Luminosity distance. Type IA supernovae (see 1.2.1) are called ’standard candles’ because they
are believed to be objects of known absolute luminosity L (= energy emitted per second). The
observed flux F (= energy per second per receiving area) from a supernova explosion can then
be used to infer its (luminosity) distance. Consider a source at a fixed comoving distance χ. In
a static Euclidean space, the relation between absolute luminosity and observed flux is
F=
L
.
4πχ2
(6.17)
In an FRW spacetime, this result is modified for three reasons: 1. At the time t0 that the
light reaches the Earth, the proper area of a sphere drawn around the supernova and passing
through the Earth is 4πd2 m The fraction of the light received in a telescope of aperture A is
thereforeA/4πd2 m . 2. The rate of arrival of photons is lower than the rate at which they are
emitted by the redshift factor 1/(1 + z). 3. The energy E0 of the photons when they are received
is less than the energy E1 with which they were emitted by the same redshift factor 1/(1 + z).
Hence, the correct formula for the observed flux of a source with luminosity L at coordinate
distance χ and redshift z is
F=
L
4πdm
2 (1
+
z)2
≡
L
4πd L 2
(6.18)
where we have defined the luminosity distance, dL, so that the relation between luminosity, flux
and luminosity distance is the same as in 6.17. Hence, we find
dL = dm (1 + z).
(6.19)
Chapter 6. Cosmology with GRBs
78
Angular diameter distance. Sometimes we can make use of ’standard rulers’, i.e. objects of
known physical size D. (This is the case, for example, for the fluctuations in the CMB.) Let
us assume again that the object is at a comoving distance χ and the photons which we observe
today were emitted at time t1 . A naive astronomer could decide to measure the distance dA to
the object by measuring its angular size δθ and using the Euclidean formula for its distance,
dA =
D
.
δθ
(6.20)
This quantity is called the angular diameter distance. The FRW metric 6.13 implies the following
relation between the physical (transverse) size of the object and its angular size on the sky
D = a(t1 )S k (χ)δθ =
dm
δθ.
1+z
(6.21)
Hence, we get
dA =
dm
.
(1 + z)
(6.22)
The angular diameter distance measures the distance between us and the object when the light
was emitted. We see that angular diameter and luminosity distances aren’t independent, but
related by
dA =
dL
.
(1 + z)2
(6.23)
Therefore if we have measured the luminosity distance at a given redshift (and if we are convinced of the correctness of the Robertson-Walker metric), then we learn nothing additional
about a(t) if we also measure the angular diameter distance at that redshift. Neither galaxies
nor supernovas have well-defined edges, so angular diameter distances are much less useful
in studying the cosmological expansion than are luminosity distances. However, they play an
important role in the theoretical analysis of gravitational lenses and of the fluctuations in the
cosmic microwave radiation background.
Friedmann equations.
Friedmann equations (Friedmann 1924) are the fundamental equations governing the expansion
of the universe:
ȧ 2
a
ä a
=
=−
k
8πG
ρ − 2,
3
a
(6.24)
4πG
(ρ + 3P).
3
(6.25)
Here, G is Newton’s gravitational constant, and ρ and P should be understood as the sum of all
contributions to the energy density and pressure in the universe. We write ρr for the contribution
from radiation (with ργ for photons and ρν for neutrinos), ρm for the contribution by matter (with
ρc for cold dark matter and ρb for baryons) and ρΛ for the vacuum energy contribution. The first
Chapter 6. Cosmology with GRBs
79
Friedmann equation is often written in terms of the Hubble parameter, H ≡ ȧ/a,
H2 =
8πG
k
ρ − 2.
3
a
(6.26)
Let us use subscripts ’0’ to denote quantities evaluated today, at t = t0 . A flat universe (k = 0)
corresponds to the following critical density today
ρcrit,0 =
3H 0 2
= 1.9×10−29 h2 grams cm−3 = 2.8×1011 h2 M⊙ M pc−3 = 1.1×10−5 h2 photons cm−3 .
8πG
(6.27)
We use the critical density to define dimensionless density parameters
ΩI,0 ≡
ρI,0
.
ρcrit,0
(6.28)
The Friedmann equation can then be written as
2
2
"
H (a) = H 0 Ωr,0
a 4
0
a
+ Ωm,0
a 3
0
a
+ Ωk,0
a 2
0
a
#
+ ΩΛ,0 ,
(6.29)
where we have defined a ”curvature” density parameter, Ωk,0 ≡ −k/(a0 H0 )2 . It should be noted
that in the literature, the subscript ’0’ is normally dropped, so that e.g. Ωm usually denotes
the matter density today in terms of the critical density today. From now on we will follow
this convention and drop the ’0’ subscripts on the density parameters. We will also use the
conventional normalization for the scale factor, a0 ≡ 1. Eq. 6.29 then becomes
H2
= Ωr a−4 + Ωm a−3 + Ωk a−2 + ΩΛ .
H02
(6.30)
Observations show that the universe is filled with radiation (’r’), matter (’m’) and dark energy
(’Λ’):
|Ωk | ≤ 0.01, Ωr = 9.4 × 10−5 , Ωm = 0.32, ΩΛ = 0.68.
(6.31)
The equation of state of dark energy seems to be that of a cosmological constant,wΛ ≈ −1. The
matter splits into 5% ordinary matter (baryons, ’b’) and 27% (cold) dark matter (CDM, ’c’):
Ωb = 0, 05, Ωc = 0.27.
(6.32)
We see that even today curvature makes up less than 1% of the cosmic energy budget. At earlier
times, the effects of curvature are then completely negligible (recall that matter and radiation
scale as a−3 and a−4 , respectively, while the curvature contribution only increases as a−2 ).
Using Friedmann equations we have
Chapter 6. Cosmology with GRBs
80
ΩΛ + Ωm + Ωr + Ωk = 1,
Ωk ≡ −
a2
k
2
0H 0
(6.33)
and after we have
dt =
=
dx
=
p
H0 x ΩΛ + Ωk x−2 + Ωm x−3 + Ωr x−4
−dz
p
(6.34)
H0 (1 + z) ΩΛ + Ωk (1 + z)2 + Ωm (1 + z)3 + Ωr (1 + z)4
where x ≡ a/a0 = 1/(1 + z). Therefore, if we define the zero of time as corresponding to an
infinite redshift, then the time at which light was emitted that reaches us with redshift z is given
by
1
t(z) =
H0
Z
1/(1+z)
dx
.
p
x ΩΛ + Ωk x−2 + Ωm x−3 + Ωr x−4
0
(6.35)
In particular, by setting z = 0, we find the present age of the universe:
1
t0 =
H0
Z
1
0
dx
.
p
x ΩΛ + Ωk x−2 + Ωm x−3 + Ωr x−4
(6.36)
In order to calculate luminosity or angular diameter distances, we also need to know the radial
coordinate r(z) of a source that is observed now with redshift z. According to Eq. 6.34, this is
r(z) = S
"Z
t0
t(z)
#
dt
=S
a(t)
where



 1 Z 1
dx
 ,

p
a0 H0 1/(1+z) x2 ΩΛ + Ωk x−2 + Ωm x−3 + Ωr x−4 



sinh(χ), k = −1,





S [χ] = 
χ,
k = 0,





 sin(χ),
k = +1
This can be written more conveniently by using Eq. 6.33 to express a0 H0 in terms of Ωk . We
then have a single formula
1
a0 r(z) =
H0 Ωk 1/2

 1/2 Z
Ωk
1
1/(1+z)
dx
p
x2 ΩΛ + Ωk x−2 + Ωm x−3 + Ωr x−4


 ,
(6.37)
Chapter 6. Cosmology with GRBs
81
which can be used for any curvature. (Eq. 6.36 has a smooth limit for Ωk → 0, which gives the
result for zero curvature. Also, for k < 0, the argument of the hyperbolic sine is imaginary, and
we can use sinh ix = isinx). Using Eq. 6.37 gives the luminosity distance of a source observed
with redshift z as
1+z
dL (z) = a0 r(z)(1 + z) =
H0 Ωk 1/2

 1/2 Z
Ωk
1
dx
1/(1+z)
p
x2 ΩΛ + Ωk x−2 + Ωm x−3 + Ωr x−4


 . (6.38)
For k = 0 we have Ωk = 0 and Eq. 6.38 becomes
1+z
dL (z) = a0 r1 (1 + z) =
H0
Z
1
1
1+z
x2
dx
.
· (ΩΛ + Ωm · x−3 + Ωr · x−4 )1/2
(6.39)
6.2 Cosmology with GRBs
Since GRBs are very bright, it was a hope to use them as the standard candles. The idea was to
use the“Amati relation”.
We can try to use “Amati relation” (Amati et al. 2002) to do cosmology with GRB. “Amati
relation” is the correlation between the spectrum peak energy E peak and the isotropic equivalent
radiated energy Eiso :
(Eiso /1052 ergs) = k · (E peak /keV)α
(6.40)
where k = 80, α = 0, 57 . If we assume that “Amati relation” is valid for each GRB (no
scattering), we can use it for cosmology. Let’s consider flux F =
(1 + z). Observed peak energy E peak,obs = E peak · (1 +
Eiso = k · (1 +
z)α
· (E p,obs
z)−1 ,
In this case fluence S is:
k · (1 + z)α+1 · (E p,obs )α
4πdL2
,
and we can express the luminosity distance from this equation:
dLobs
and fluence S =
Eiso
4πdL2
·
isotropic equivalent radiated energy
)α .
S =
L
4πdL2
1/2

 k · (1 + z)α+1 · (E p,obs )α 
 ,
= 
4πS
Chapter 6. Cosmology with GRBs
82
Figure 6.1: Amati relation with redshift distribution
Figure 6.2: Isotropic equivalent radiated energy from redshift Eiso (z)
and compare with theoretical value (eq. 6.39):
dL (z) =
1+z
H0
Z
1
1
1+z
dx
x2 · (ΩΛ + Ωm · x−3 + Ωr · x−4 )1/2
6.3 Number of GRBs per redshift
Assuming that GRBs can be produced by Pair-Instability Supernovae we tried to compute the
number of GRBs per redshift.
Chapter 6. Cosmology with GRBs
83
Figure 6.3: Comparison of luminosity distance computed from Amati relation” and from cosmological redshift by eq.6.42
In order to compute the number of GRBs per redshift, it is necessary to know the rate of GRB per
unit of volume, RGRB . Number of observed GRBs is proportional to the rate of GRB as function
of redshift, comoving volume and observational time. It is necessary to have a model, that can
define RGRB . One of the standard approaches is to assume that the rate of GRB is proportional
to the star formation rate. Another way how to calculate the theoretical event rate of gammaray bursts from the collapse of massive first-generation (Population III) stars was shown by de
Souza et al. (2011). With a semi-analytical approach the star formation rate for Pop III.1 and
III.2 stars was estimated including all relevant feedback effects: photo-dissociation, reionization,
and metal enrichment.
But we also assumed the that rate of GRB can be proportional to the rate of pair-instability
supernova.
Since GRBs are detected by instruments from the satellites, we also need to take into account
limitations of instruments: such as field of view of the telescopes, limited sensitivity, etc. Because of limited sensitivity only intense bursts can be detected for very high distances. So we had
to consider the distribution of GRB by luminosity (so called luminosity function). Luminosity
functions are generally constructed basing on observational data.
David Wanderman and Tsvi Piran (Wanderman & Piran 2010) assumed that the luminosity
function is redshift independent. This common assumption is reasonable since a priori there is
no competing reason why the luminosity function should depend on the redshift. They have
tested the validity of this assumption and have shown that it is accepted with a high statistical
significance.
Chapter 6. Cosmology with GRBs
84
Cristiano Porciani and Piero Madau (Porciani & Madau 2001) used realistic estimates for the
evolution of the stellar birthrate in galaxies to model the number counts, redshift distribution, and
time-delay factors of GRBs. They presented luminosity function fits to the BATSE log N-log P
relation for different redshift distributions of the bursts. Autors studied the expected cosmological distribution of GRBs in the massive star progenitor scenario and showed that the brightness
distribution of the BATSE bursts can be well reproduced by assuming a proportionality between the GRB rate density and observationally based SFR estimates, once the standard-candle
hypothesis is relaxed (Totani 1997) . Autors’ starting hypothesis was that the rate of GRBs
is proportional to comoving rate densities of star formation and core-collapse (Type II) supernovae. The constant of proportionality is a free parameter of the model. Popular scenarios for
GRBs include also merging neutron stars (Paczynski 1986) or the formation of black holes in
supernova-like events (collapsars; (MacFadyen & Woosley 1999) ). They assumed that GRBs
are produced by stellar systems that evolve rapidly from their formation to the explosion epoch.
The same assumption has Campisi et al. (2010) .
In our computations we have considered different functions.
The number of observed GRBs with redshift in (z,z+dz) over a time interval dt is:
dNGRB Ωobs
dt
dV
=
· Pz · Rgrb (z) ·
·
(z) ·
dz
4π
1 + z dz
Z
Φ(L)d log L
(6.41)
In this equation there are two unknown functions: luminosity function Φ(L) and rate of GRB
RGRB (z). The other parameters are the following: Ωobs - field of view of experiment, Ωobs = 1, 4
for Swift, Pz = 0, 28 - normalization factor, dt - observation time,
dV
dz (z)
-comoving volume
element per unit of redshift.
4πdL2 (z)
dV
,
(z) = c ·
dz
H(z) · (1 + z)2
c - speed of light 299 792 458 m/s, dL - luminosity distance [Mpc], H(z) = H0 · ((ΩΛ + Ωk ·
(1 + z)2 + Ωm · (1 + z)3 + Ωr · (1 + z)4 )1/2 , H0 = 72[km · c−1 · M pc−1 ] - Hubble constant,
ΩΛ = 0, 73, Ωm = 0, 27, Ωr = 0, Ωk = 0 - cosmological parametres. ”Volumetric factor”
dV/dz
1+z (z)
is plotted on Fig. 6.4.
For luminosity distance we used the theoretical value:
dL (z) =
1+z
H0
Z
1
1
1+z
dx
x2 · (ΩΛ + Ωm · x−3 + Ωr · x−4 )1/2
(6.42)
Chapter 6. Cosmology with GRBs
85
Figure 6.4: ”Volumetric factor”
6.3.1
dV/dz
1+z (z)
Luminosity function
Let’s consider Φ(L) - luminosity function. It can be calculated in different ways. For example:
luminosity function from Wanderman & Piran (2010):



−αL


(L/L∗ ) , if L < L∗ ,
Φ(L) = 



(L/L∗ )−βL , if L > L∗
L∗ = 1052,5 erg · s−1 αL = 0, 2, βL = 1, 4
Another expression for luminosity function is from Campisi et al. (2010):
Φ(L) = K · (
L ψ
L∗
) · exp(− )
L∗
L
L- isotropic equivalent intrinsic burst luminosity, ψ - asymptotic slope at the faint end, K =
[L0 Γ(−ψ − 1)]−1 for ψ < −1 -normalization constant, L∗ = L(z) = L0 (1 + z)δ - characteristic
cutoff luminosity. We take sample of the parameters δ, L0 , ψ from (Campisi et al. 2010):δ =
2.0, L0 = 0, 347 · 1050 erg · s−1 , ψ = −1, 709.
To find a number of observed GRBs (eq.6.41) we should know integral over luminosity function:
Z
Φ(L)d log L
log Llim (z)
Chapter 6. Cosmology with GRBs
86
Figure 6.5: Integral of Luminosity Function from Campisi et al. (2010) and Wanderman &
Piran (2010)
Llim (z) = 4π · dL 2 (z) · Flim - luminosity threshold of a satellite, Flim = 1, 2 · 10−8 erg · cm−2 · s−1 -
bolometric energy flux limit of the instrument of Swift. (Fig. 6.5)
We can see in the figure 6.5, that between these two luminosity functions there are no fundamental difference.
6.3.2
Rate of GRB
Let’s consider RGRB (z) - rate of GRB. There are different ways too. Generally it is assumed that
the rate of GRB is proportional to star formation rate RS FR (z) :
RGRB (z) = kGRB · RS FR (z).
(6.43)
For SFR we used approximation formula by Hopkins & Beacom (2006) :
RS FR (z) =
(a1 +a2 ·z)·h
,
1+( az )a4
3
a1 , a2 , a3 , a4 , h - parameters a1 = 0, 017, a2 = 0, 13, a3 = 3, 3, a4 = 4, 3, h = 0, 7, coefficient
kGRB = 10−8 [M⊙ −1 ] (Fig. 6.6 light blue line). But according to our modelling of GRB as PISN
it is natural to propose that the rate of GRB equals (or proportional) to the rate of PISN:
RGRB ∼ RPIS N
The rate of PISN we took from Hummel et al. (2012) fig. 6.6 red line).
(6.44)
Chapter 6. Cosmology with GRBs
87
Figure 6.6: Rate GRB from Hopkins & Beacom (2006) and rate PISN from Hummel et al.
(2012)
Figure 6.7: Number of observed GRBs with redshift in (z,z+dz) over a redshift interval
As we can see, if PISN could produce GRB, they will play significant role only at high redshifts
∼> 5 − 10. At this redshift Population III stars were more massive, more frequently producing
PISN.
6.4 Conclusions
Basing on these simple assumptions we were able to compute the number of observed GRBs
with redshift and compare it with observational data (6.7). It was shown that general behaviour is
reproduced, but there are certainly some deviations. While the PISN model predicts an increase
Chapter 6. Cosmology with GRBs
88
of the number of GRBs with higher redshift, since primordial stars have higher masses and PISN
explosions were more frequent.
Chapter 7
General conclusions
In order to develop the model that gamma-ray bursts are related to the deaths of very massive
stars, different physical aspects of this phenomena was studied.
In Chapters 1 and 2, a connection between supernovae and gamma-ray bursts was analysed as
for continuous class of phenomena with the important role of nuclear explosion and combustion.
The general concepts of fluid dynamics of explosion and the role of hydrodynamic instabilities
were presented.
In order to develop and adjust tools for studying stellar explosions, such as supernovae and
gamma ray bursts, the supernova explosion phenomenon was studied from the perspective of
hydrodynamics and explosive burning modelling in Chapter 3. The problem of nucleosynthesis
was studied within an aspherical supernova model. The explosive burning was computed in a
star of 25 M⊙ initial mass on its final stage of evolution. Calculations of detailed yield of chemical elements was performed as a post-processing step using the tracer particles method. The
produced nuclei formed a layer-like structure enclosing large fragments of nickel and iron-group
isotopes that were pushed away from the central region by an explosion along the polar direction. The light nuclei were preferentially moving along the equatorial plane forming a torus-like
structure. Obtained results are in a good agreement with works of other teams. Moreover the
usage of precise hydrodynamical code allowed us to obtain interesting details on fine structure
of exploding matter and distribution of elements in the ejecta. This could help us in future works
to understand better the hydrodynamics of explosion and to propose an explanation of variability
of lightcurves in GRBs.
In Chapter 4 multidimensional simulations of pair-instability supernovae were studied. According to theoretical models, massive stars with masses within the 100 − 250 M⊙ range should
explode as pair-instability supernovae. Since the first stars of the Universe are believed to be
very massive, these supernovae should play a significant role in the early stages of its history.
89
Chapter 7. General conclusions
90
But these stars represent the last unobserved population, owing to detection limits of current
telescopes. In this chapter pair-instability supernovae explosions using various numerical codes
were analysed. Basing on this results it was shown that the temperature is an important parameter that could characterize GRBs.
In Chapter 5 the assumption of the importance of the temperature was checked on the observational data of gamma-ray bursts. Also the spectra of gamma ray bursts were studied. The
model explaining non-thermal spectra of gamma-ray bursts was suggested. It was proposed that
different physical processes are responsible for low- and high-energy parts of the spectrum. Low
energy emission could be explained by a combination of thermal bremsstrahlung process in regions with lower density and blackbody emission from more dense regions. This combination
naturally explains photon number spectrum with power-law index -1 observed in many bursts.
The high-energy part is formed by inverse-Compton scattering of soft photons on electrons in hot
regions. It was shown for the case of some gamma-ray bursts that with this approach and taking
into account only thermal bremsstrahlung and blackbody emission it is possible to reconstruct
most part of the spectrum. This model needs further investigation, including multidimensional
simulations of stellar explosions.
The idea about the sources of gamma-ray bursts was studied in Chapter 6. Basing on the theory
of stellar evolution, the theoretical frequency of gamma-ray bursts was estimated and compared
with observations. It was shown that GRB is effectively connected with SFR, and an increase of
the number of observed gamma-ray bursts with higher redshift was proposed.
Analysis of some GRBs
Figure 1: The temperature evolution with time T(t) of GRB 101023
Figure 2: Light curve of GRB 101023 (Penacchioni et al. 2012)
91
Appendix A. Analysis of some GRBs
Figure 3: Spectral analysis of GRB 101023 for different intervals of time: [0-45 s] and [45-89
s]. Blue line shows the fit with the Band function (Penacchioni et al. 2012), orange line shows
the fit with blackbody + bremsstrahlung
92
Appendix A. Analysis of some GRBs
Figure 4: Time-integrated spectra of GRB 970111 for the time interval from 21 to 24 s after
the trigger time. Blue line shows the fit with the Band function (Eq 5.1)(Frontera et al. 2012),
orange line shows the fit with blackbody + bremsstrahlung (Eq. 5.9).
Figure 5: The temperature evolution with time T(t) of GRB 970111
Figure 6: Light curve of GRB 970111 detected with BATSE (Frontera et al. 2012)
93
Appendix A. Analysis of some GRBs
94
int 3
nuFnu (keV)
10 4
1000
1000
100
100
10
10
1
1
1
10
Energy (keV)
100
1000
1
int 5
nuFnu (keV)
10 4
1000
1000
100
100
10
10
1
1
1
10
Energy (keV)
100
10 4
1000
1000
100
100
10
10
1
1
1
10
Energy (keV)
100
10 4
1000
1000
100
100
10
10
1
1
10
Energy (keV)
100
1000
10
Energy (keV)
100
1000
10
Energy (keV)
100
1000
int 10
nuFnu (keV)
10 4
1
1000
int 8
1
1000
int 9
100
nuFnu (keV)
10 4
nuFnu (keV)
Energy (keV)
int 6
1
1000
int 7
10
nuFnu (keV)
10 4
nuFnu (keV)
int 4
nuFnu (keV)
10 4
1
10
Energy (keV)
100
1000
Figure 7: GRB 970111. Time-integrated spectra for different intervals of time: 4-7 s, 7-13
s, 13-16 s, 16-21 s, 21-24 s, 24-28s, 28-32s, 32-45 s. Blue line shows the fit with the Band
function (Eq 5.1), orange line shows the fit with blackbody + bremsstrahlung (Eq. 5.9).
Appendix A. Analysis of some GRBs
Figure 8: Time-integrated spectra of GRB090926A for the time interval from 9.8 to 10.5 s after
the trigger time. Blue line shows the fit with the Band function (Eq 5.1) (Tierney et al. 2013),
orange line shows the fit with blackbody + bremsstrahlung (Eq. 5.9).
Figure 9: The temperature evolution with time T(t) of GRB 090926A
Figure 10: Light curve of GRB 090926A (Tierney et al. 2013)
95
Appendix A. Analysis of some GRBs
96
nuFnu
4 (keV)
int 1
10
10
5000
5000
1000
1000
500
500
100
100
50
50
10
100
Energy (keV)
1000
10 4
10
int 3
nuFnu
4 (keV)
10
1000
1000
500
500
100
100
50
50
Energy (keV)
1000
10 4
Energy (keV)
1000
10 4
int 4
10
5000
100
100
nuFnu
4 (keV)
5000
10
int 2
nuFnu
4 (keV)
10
100
Energy (keV)
1000
10 4
Figure 11: GRB 090926A. Time-integrated spectra for different intervals of time: form 0.0 to
3.3 s, 3.3 to 9.8 s, 9.8 to 10.5 s and 10.5 to 21.6 s. Blue line shows the fit with the Band function
(Eq 5.1), orange line shows the fit with blackbody + bremsstrahlung (Eq. 5.9).
Figure 12: The temperature evolution with time T(t) of GRB 990510
Figure 13: Light curve of GRB 990510 (Frontera et al. 2012)
Appendix A. Analysis of some GRBs
97
int 1
nuFnu (keV)
1000
500
500
100
100
50
50
10
10
5
5
5
10
50
100
Energy (keV) 500
1000
5
5000
int 6
nuFnu (keV)
1000
500
500
100
100
50
50
10
10
5
5
5
10
50
100
Energy (keV)
500
1000
5000
int 8
5
1000
500
500
100
100
50
50
10
10
5
5
5
10
50
100
Energy (keV)
500
1000
5000
int 10
500
500
100
100
50
50
10
10
5
5
50
100
Energy (keV) 500
1000
5000
Energy (keV)500
1000
5000
int 7
10
5
1000
10
100
50
100
Energy (keV) 500
1000
5000
int 9
10
50
100
Energy (keV) 500
1000
5000
int 11
nuFnu (keV)
1000
5
50
nuFnu (keV)
1000
nuFnu (keV)
10
nuFnu (keV)
1000
nuFnu (keV)
int 2
nuFnu (keV)
1000
10
50
100
Energy (keV)
500
1000
Figure 14: Time-integrated spectra of GRB990510 for the time intervals. Blue line shows the
fit with the Band function (Eq 5.1), orange line shows the fit with blackbody + bremsstrahlung
(Eq. 5.9).
Appendix A. Analysis of some GRBs
Figure 15: The temperature evolution with time T(t) of GRB 980329
Figure 16: Light curve of GRB 980329 (Frontera et al. 2012)
98
Appendix A. Analysis of some GRBs
99
int 2
nuFnu (keV)
1000
500
500
100
100
50
50
10
10
5
5
5
10
50
100
Energy (keV) 500
1000
5
5000
int 4
nuFnu (keV)
1000
500
500
100
100
50
50
10
10
5
5
5
10
50
100
Energy (keV)500
1000
5
5000
int 6
1000
500
500
100
100
50
50
10
10
5
5
10
50
100
Energy (keV)
500
1000
5000
50
100
Energy (keV)
500
1000
5000
int 5
10
50
100
Energy (keV)
500
1000
5000
int 7
nuFnu (keV)
1000
5
10
nuFnu (keV)
1000
nuFnu (keV)
int 3
nuFnu (keV)
1000
5
10
50
100
Energy (keV)
500
1000
5000
Figure 17: Time-integrated spectra of GRB980329 for the time intervals. Blue line shows the
fit with the Band function (Eq 5.1), orange line shows the fit with blackbody + bremsstrahlung
(Eq. 5.9).
Abbreviations
BATSE
Burst and Transient Source Explorer
BB
Black body
BGO
Bismuth Germanate
CBM
Circumburst medium
CC
Core Collapse
CGRO
Compton Gamma Ray Observatory
CMB
Cosmic microwave background
EOS
Equation of State
FRW
Friedmann-Robertson-Walker metric
GBM
Gamma-Ray Burst Monitor
GRB
Gamma-Ray Burst
IMF
Initial Mass Function
ISM
Interstellar Medium
LAT
Large Area Telescope
PISN
Pair-Iinstability Supernova
PPML
Piecewise Parabolic Method on a Local stencil
SD
Spectroscopy detector
SN
Supernova
TPM
Tracer particles method
100
Physical Constants
Speed of Light
c
=
2.997 924 58 × 1010 cm s−1
Gravitational Constant
G
=
6.6742 × 10−8 cm3 g−1 s−1
Plank’s Constant
h
=
6.626 × 10−27 erg s
~=h/2π
=
1.0546 × 10−27 erg s
Boltzmann Constant
k
=
1.3807 × 10−16 erg K−1
Avogadro’s Number
NA
=
6.022 × 1023 mole−1
Gas constant
R
=
8.314 × 107 erg K−1 mole−1
Stefan-Boltzmann Constant
σ
=
5.670 × 10−5 erg K−4 cm−2 s−1
Radiation Density Constant
a=4σ/c
=
7.566 × 10−15 erg K−4 cm−3
Electron’s Rest Mass
me
=
9.109 × 10−28 g
me c2
=
0.511 MeV
Thomson cross section
σT
=
0.66524 × 10−24 cm2
Electron volt
1 eV
=
9.602 × 10−12 erg
1 eV/k
=
11604 K
Rydberg
1 Ry
=
13.606 eV
Hubble constant
H0
=
72 km s−1 Mpc−1
Solar Mass
M⊙
=
1.989 × 1033 g
Solar Radius
R⊙
=
6.955 × 1010 cm
Solar Luminosity
L⊙
=
3.846 × 1033 erg s−1
101
Symbols
dL
luminosity distance
Mpc
E
energy
erg
F
flux
erg cm−2 s−1
λ
rate of spontaneous decay
s−1
L
luminosity
erg s−1
NE
photon number spectrum
photons cm−2 s−1 keV−1
n
number density
cm−3
p
pressure
erg cm−3
ρ
density
g cm−3
S
fluence
erg cm−2
T
temperature
K
t
time
s
v
velocity
cm3 g−1
V
specific volume
cm s−1
X
mass fraction
Y
mole fraction
mole g−1
102
Bibliography
Ackermann, M., Asano, K., Atwood, W. B., et al. 2010, ApJ, 716, 1178
Ackermann, M., Ajello, M., Asano, K., et al. 2011, ApJ, 729, 114
Ackermann, M., Ajello, M., Allafort, A., et al. 2012, Apj, 748, 151
Ackermann, M., Ajello, M., Allafort, A., et al. 2013, Science, 339, 807
Almgren, A. S., et al. 2010, ApJ, 715, 1221
Amati, L., Frontera, F., Tavani, M., et al. 2002, A & A, 390, 81
Anzer, U., & Boerner, G. 1976, Mitteilungen der Astronomischen Gesellschaft Hamburg, 38,
121
Arnett, W. D. 1969, Apss, 5, 180
Arnett, W. D. 1972, ApJ, 176, 699
Arnett, W. D. 1973, Explosive Nucleosynthesis, ed. D. N. Schramm & W. D. Arnett (Univ. of
Texas Press, Austin), 236
Arnett, W. D. 1974, Late Stages of Stellar Evolution, ed. R. J. Tayler & J. E. Hesser, IAUS 66
(D. Reidel Publishing, Dordrecht), 1
Arnett, W. D. 1980, Apj, 237, 541
Arnett, W. D. 1982, Apj, 253, 785
Arnett, W. D. 1996, Supernovae and Nucleosynthesis, Princeton: Princeton University Press,
380
Arnett, W. D., & Meakin, C. 2011, ApJ, 733, 78
103
Bibliography
104
Atwood, W. B., Abdo, A. A., Ackermann, M., et al. 2009, Apj, 697, 1071
Ayasli, S., & Joss, P. S. 1982, ApJ, 256, 637
Band, D., Matteson, J., Ford, L., et al. 1993, ApJ, 413, 281
Baranov, A. A., Chardonnet, P., Chechetkin, V. M., Filina, A. A., & Popov, M. V. 2013, A & A,
558, AA10
Baranov, A. A., Chardonnet, P., Filina, A. A., Chechetkin, V. M., & Popov, M. V. 2013, Proceedings of the 48th Rencontres de Moriond, p.289-292
Barbon, R., Ciatti, F., & Rosino, L. 1979, aap, 72, 287
Barbon, R., Ciatti, F., & Rosino, L. 1973, aap, 29, 57
Barkat, Z., Rakavy, G., & Sack, N. 1967, prl, 18, 379
Baumann Daniel 2009, DAMTP Cosmology lectures
Bell, J. B., Day, M. S., Rendleman, C. A., Woosley, S. E., & Zingale, M. 2004, Apj, 606, 1029
Bellm, E. C. 2010, ApJ, 714, 881
Beloborodov, A. M. 2013, ApJ, 764, 157
Bionta, R. M., Blewitt, G., Bratton, C. B., Casper, D., & Ciocio, A. 1987, Physical Review
Letters, 58, 1494
Bisnovatyi-Kogan, G. S., Imshennik, V. S., Nadyozhin, D. K., & Chechetkin, V. M. 1975, apss,
35, 23
Bissaldi, E., von Kienlin, A., Lichti, G., et al. 2009, Experimental Astronomy, 24, 47
Blinnikov, S. I., Sasorov, P. V., & Woosley, S. E. 1995, Ssr, 74, 299
Blinnikov, S., IV, & Sasorov, P. V. 1996, Pre, 53, 4827
Blinnikov, I. S., Dunina-Barkovskaya, N. V., & Nadyozhin, D. K. 1996, ApJs, 106, 171
Bloom, J. S., Perley, D. A., Li, W., et al. 2009, ApJ, 691, 723
Bond, J. R., Arnett, W. D., & Carr, B. J. 1984, ApJ, 280, 825
Borgonovo, L., & Ryde, F. 2001, ApJ, 548, 770
Bibliography
105
Branch, D. 1986, Apjl, 300, L51
Briggs, M. S., Connaughton, V., Wilson-Hodge, C., et al. 2011, grl, 38, L02808
Bromm, V., Yoshida, N., & Hernquist, L. 2003, ApJ, 596, L135
Bromm, V., & Larson, R. B. 2004, araa, 42, 79
Brussaard, P. J., & van de Hulst, H. C. 1962, Reviews of Modern Physics, 34, 507
Bussard, R. W., & Lamb, F. K. 1982, Gamma Ray Transients and Related Astrophysical Phenomena, 77, 189
Campisi, M. A., Li, L.-X., & Jakobsson, P. 2010, mnras, 407, 1972
Campana, S., Mangano, V., Blustin, A. J., et al. 2006, Nature, 442, 1008
Caughlan, G. R., & Fowler, W. A. 1988, At. Data Nucl. Data Tables, 40, 283
Chakraborti, S., Soderberg, A., Chomiuk, L., et al. 2014, arXiv:1402.6336
Chamulak, D. A., Meakin, C. A., Seitenzahl, I. R., & Truran, J. W. 2012, ApJ, 744, 27
Chardonnet, P., Chechetkin, V., & Titarchuk, L. 2010, apss, 325, 153
Chechetkin, V. M., Ustyugov, S. D., Gorbunov, A. A., & Polezhaev, V. I. 1997, Astron. Let., 23,
30.
Chen, K.-J., Heger, A., & Almgren, A. S. 2011, Comput. Phys. Communicat., 182, 254
Chevalier, R. A. 1982, ApJ, 258, 790
Colella, P., & Woodward, P. R. 1984, J. Comput. Phys., 54, 174
Costa, E., Frontera, F., Heise, J., et al. 1997, Nature, 387, 783
Crider, A., Liang, E. P., Smith, I. A., et al. 1997, ApJl, 479, L39
Cucchiara, A., Levan, A. J., Fox, D. B., et al. 2011, apj, 736, 7
Dai, W., & Woodward, P. R. 1994, J. Comput. Phys., 115, 485
Darrieus, G., 1938, unpublished work presented at La Technique Moderne
Della Valle, M., Malesani, D., Benetti, S., et al. 2004, Gamma-Ray Bursts: 30 Years of Discovery, 727, 403
106
Bibliography
Deng, J., Tominaga, N., Mazzali, P. A., Maeda, K., & Nomoto, K. 2005, ApJ, 624, 898
de Souza, R. S., Yoshida, N., & Ioka, K. 2011, A & A, 533, A32
Diehl, R., Siegert, T., Hillebrandt, W., et al. 2014, Science, 345, 1162
Doggett, J. B., & Branch, D. 1985, aj, 90, 2303
Dunina-Barkovskaya, N. V., & Imshennik, V. S. 2003, Astron. Lett., 29, 10
Elias, J. H., Matthews, K., Neugebauer, G., & Persson, S. E. 1985, Apj, 296, 379
Fenimore, E. E., Klebesadel, R. W., Laros, J. G., Stockdale, R. E., & Kane, S. R. 1982, Nature,
297, 665
Fickett & Davis. 2000, Detonation:
Theory and Experiment. ISBN 0486414566,
9780486414560. Published by Courier Corporation, 2000
Fowler, W. A., & Hoyle, F. 1964, ApJs, 9, 201
Fraley, G. S. 1968, apss, 2, 96
Friedmann, A. 1924, Zeitschrift fur Physik, 21, 326
Frontera, F., Amati, L., Guidorzi, C., Landi, R., & in’t Zand, J. 2012, ApJ, 754, 138
Frontera, F., Amati, L., Farinelli, R., et al. 2013, ApJ, 779, 175
Fruchter, A. S., Levan, A. J., Strolger, L., et al. 2006, 441, 463
Galama, T. J., Vreeswijk, P. M., van Paradijs, J., et al. 1998, Nature, 395, 670
Gear, C. Numerical Initial Value Problems in Ordinary Differential Equations, Englewood Cliffs,
NJ: Prentice-Hall, 1971
Gehrels, N., Kniffen, D. A., & Ormes, J. F. 1992, Current Topics in Astrofundamental Physics,
602
Gehrels, N., Ramirez-Ruiz, E., & Fox, D. B. 2009, araa, 47, 567
Gehrels, N., & Razzaque, S. 2013, Frontiers of Physics, 8, 661
Ghirlanda, G., Celotti, A., & Ghisellini, G. 2003, A & A, 406, 879
Ghirlanda, G., Bosnjak, Z., Ghisellini, G., Tavecchio, F., & Firmani, C. 2007, MNRAS, 379, 73
Bibliography
107
Glatzel, W., El Eid, M. F., & Fricke, K. J. 1985, A & A 149, 413
Glazyrin, S. I., Blinnikov, S. I., & Dolgov, A. 2013, Mnras, 433, 2840
Golenetskii, S., Aptekar, R., Mazets, E., et al. 2006, GRB CoordiNaturees Network, 5722, 1
Gorosabel, J., et al. 2010, A&A, 522, A14
Gostintsev Y. A., Istratov A. G., and Shulenin Y. V. , Combust. Expl. Shock Waves 24, 563
(1988)
Granot, J., Piran, T., & Sari, R. 1999, ApJ, 527, 236
Greene, J. 1959, ApJ, 130, 693
Guetta, D., & Della Valle, M. 2007, APJl, 657, L73
Hairer, E., & Wanner, G. 1991, Solving Ordinary Differential Equations II. Stiff and DifferentialAlgebraic Problems (Springer-Verlag, second revised edition 1996)
Hamuy, M. 2003, ApJ, 582, 905
Harding, A. K. 1991, physrep, 206, 327
Harkness, R. P., Wheeler, J. C., Margon, B., et al. 1987, Apj, 317, 355
Heger, A., & Woosley, S. E. 2002, ApJ, 567, 532
Hirata, K., Kajita, T., Koshiba, M., Nakahata, M., & Oyama, Y. 1987, Physical Review Letters,
58, 1490
Hjorth, J., Sollerman, J., Møller, P., et al. 2003, Nature, 423, 847
Hopkins, A. M., & Beacom, J. F. 2006, apj, 651, 142
Hummel, J. A., Pawlik, A. H., Milosavljević, M., & Bromm, V. 2012, apj, 755, 72
Hwang U., & Laming M. 2012, ApJ, 746, 130
Ivanova, L. N., Imshennik, V. S., & Chechetkin, V. M. 1974, Apss, 31, 497
Iwamoto, K., Brachwitz, F., Nomoto, K., et al. 1999, Apjs, 125, 439
Izzo, L., Ruffini, R., Penacchioni, A. V., et al. 2012, A & A, 543, A10
Jimenez, R., & Piran, T. 2013, ApJ, 773, 126
Bibliography
108
Joggerst, C. C., & Whalen, D. J. 2011, ApJ, 728, 129
José, J., & Hernanz, M. 1998, Apj, 494, 680
Joulin, G. 1994, Pre, 50, 2030
Kaneko, Y., Preece, R. D., Briggs, M. S., et al. 2006, ApJs, 166, 298
Kasen, D., Woosley, S. E., & Heger, A. 2011, ApJ, 734, 102
Katz, J. I. 1994, ApJl, 432, L107
Kistler, M. D., Yüksel, H., Beacom, J. F., & Stanek, K. Z. 2008, apjl, 673, L119
Klebesadel, R. W., Strong, I. B., & Olson, R. A. 1973, Apjl, 182, L85
Kritsuk, A. G., Ustyugov, S. D., Norman, M. L., & Padoan, P. 2009, J. of Phys.: Conf. Ser., 180,
012020
Lamb, D. Q. 1984, Annals of the New York Academy of Sciences, 422, 237
Lambert, J. in: Computational Techniques for Ordinary Differential Equations, edited by I.
Gladwell and D. Sayars, New York: Academic Press, 1980: 19
Landau, L. D., 1944, Acta Physicochimica, USSR, 19, 77, reprinted in P. Pelce, Dynamics of
Curved Fronts, Academic Press, Berkeley, CA, 1988
Landau, L. D., & Lifshitz, E. M. 1959, Course of theoretical physics, Oxford: Pergamon Press,
1959,
Lang, K. R. 1980, Astrophysical Formulae. A Compendium for the Physicist and Astrophysicist.
Springer, Berlin, 1980
Larsson, J., Ryde, F., Lundman, C., et al. 2011, MNRAS, 414, 2642
Lin, L., Göǧüş, E., Baring, M. G., et al. 2012, Apj, 756, 54
Loh, C. Y. & Jorgenson P. C. E. 2009, J. Comput. Phys., 228, 1343
Lundman, C., Pe’er, A., & Ryde, F. 2013, MNRAS, 428, 2430
MacFadyen, A. I., & Woosley, S. E. 1999, apj, 524, 262
Maeda, K., Nakamura, T., Nomoto, K., Mazzali, P. A., Patat, F., & Hachisu, I. 2002, ApJ, 565,
405
Bibliography
109
Maeda, K., Röpke, F. K., Fink, M., Hillebrandt, W., Travaglio, C., & Thielemann, F.-K. 2010,
ApJ, 712, 624
Mazets, E. P., & Golenetskii, S. V. 1981, Astrophysics and Space Physics Reviews, 1, 205
Mazzali, A., et al. 2006, ApJ, 645, 1323
Meegan, C., Lichti, G., Bhat, P. N., et al. 2009, Apj, 702, 791
Mészáros, P., & Rees, M. J. 2000, ApJ, 530, 292
Mészáros, P., Ramirez-Ruiz, E., Rees, M. J., & Zhang, B. 2002, ApJ, 578, 812
Metzger, M. R., Djorgovski, S. G., Kulkarni, S. R., et al. 1997, Nature, 387, 878
Minaev, P., Pozanenko, A., Grebenev, S., & Molkov, S. 2013, arXiv:1304.0348
Minaev, P., Pozanenko, A., Molkov, S. & Grebenev, S. ”Catalog of short gamma-ray transients
detected in the SPI/INTEGRAL experiment” Astronomy Letters, Volume 40, Issue 5, pp.235267, 2014.
Minkowski, R. 1941, pasp, 53, 224
Moiseenko, S. G., Bisnovatyi-Kogan, G. S., & Ardeljan, N. V. 2010, AIP Conf. Proc., 1206, 282
Muccino, M., Ruffini, R., Bianco, C. L., Izzo, L., & Penacchioni, A. V. 2013, ApJ, 763, 125
Nagataki, S., Hashimoto, M., Sato, K., & Yamada, S. 1997, ApJ, 486, 1026
Nagataki, S. 1999, ApJ, 511, 341
Nagataki, S., Mizuta, A., & Sato, K. 2006, ApJ, 647, 1255
Nakamura, T., Mazzali, P. A., Nomoto, K., & Iwamoto, K. 2001, ApJ, 550 991
Nava, L., Ghirlanda, G., Ghisellini, G., & Celotti, A. 2011, A & A, 530, AA21
Niemeyer, J. C., & Hillebrandt, W. 1995, Apj, 452, 779
Norris, J. P., Share, G. H., Messina, D. C., et al. 1986, ApJ, 301, 213
Nozawa, T., Kozasa, T., Umeda, H., Maeda, K., & Nomoto, K. 2003, ApJ, 598, 785
Ober, W. W., El Eid, M. F., & Fricke, K. J. 1983, A & A, 119, 61
Bibliography
110
Ohkubo, T., Umeda, H., Maeda, K., Nomoto, K., Suzuki, T., Tsuruta, S., & Rees, M. J. 2006,
ApJ, 645, 1352
Paczynski, B. 1986, apjl, 308, L43
Paczyński, B. 1998, Gamma-Ray Bursts, 4th Hunstville Symposium, 428, 783
Patat, F., et al. 2001, ApJ, 555, 900
Pe’er, A., Mészáros, P., & Rees, M. J. 2006, ApJ, 642, 995
Penacchioni, A. V., Ruffini, R., Izzo, L., et al. 2012, A & A, 538, A58
Perley, D. A., Levan, A. J., Tanvir, N. R., et al. 2013, ApJ, 778, 128
Piro, L., Costa, E., Feroci, M., et al. 1999, Apjl, 514, L73
Popov, M. V., & Ustyugov, S. D. 2007, Computational Mathematics and Mathematical Physics,
47, 1970
Popov, M. V. 2012, Computational Mathematics and Mathematical Physics, 52, 1186
Popov, M. V., Filina, A. A., Baranov, A. A., Chardonnet, P., & Chechetkin, V. M. 2014, ApJ,
783, 43
Porciani, C., & Madau, P. 2001, apj, 548, 522
Porter, A. C., & Filippenko, A. V. 1987, aj, 93, 1372
Preece, R. D., Briggs, M. S., Pendleton, G. N., et al. 1996, Apj, 473, 310
Preece, R. D., Briggs, M. S., Mallozzi, R. S., et al. 1998, ApJl, 506, L23
Quirk J. 1994, Int. J. Num. Meth. Fluids 18, 555
Rakavy, G., & Shaviv, G. 1967, ApJ, 148, 803
Rakavy, G., Shaviv, G., & Zinamon, Z. 1967, ApJ, 150, 131
Rakavy, G., & Shaviv, G. 1968, ApJs, 1, 429
Rao, A. R., Malkar, J. P., Hingar, M. K., et al. 2011, ApJ, 728, 42
Rauscher, T., & Thielemann, F.-K. 2000, At. Data Nucl. Data Tables 75, 1
Rees, M. J., & Mészáros, P. 2005, ApJ, 628, 847
Bibliography
111
Ruderman, M. 1975, Seventh Texas Symposium on Relativistic Astrophysics, 262, 164
Ryde, F. 2004, ApJ, 614, 827
Ryde, F., & Pe’er, A. 2009, ApJ, 702, 1211
Sedov, L. I. 1946, Prikl. Mat. Mekh, 10, 241
Seitenzahl, I. R., Röpke, F. K., Fink, M., Pakmor, R. 2010, MNRAS, 407, 2297
Shore, Steven N. 1992, An introduction to astrophysical hydrodynamics. ISBN 0126406707,
9780126406702. Published by Academic Press, 1992
Smartt, S. J., Eldridge, J. J., Crockett, R. M., & Maund J. R. 2009, MNRAS, 395, 1409
Smith, N., Li, W., Foley, R. J., et al. 2007, Apj, 666, 1116
Soderberg, A. M, et al. 2008, Nature, 453, 469
Strong, I. B., Klebesadel, R. W., & Olson, R. A. 1974, ApJl, 188, L1
Strong, I. B., & Klebesadel, R. W. 1974, Nature, 251, 396
Taubenberger, S., Valenti, S., Benetti, S., et al. 2009, MNRAS, 397, 677
Tanaka, M., et al. 2009, ApJ, 700, 1680
Tegmark, M., & Zaldarriaga, M. 2009, PRD, 79, 083530
Thielemann, F.-K., Nomoto, K., & Yokoi, K. 1986, Aap, 158, 17
Tierney, D., McBreen, S., Hanlon, L., et al. 2011, American Institute of Physics Conference
Series, 1358, 397
Tierney, D., McBreen, S., Preece, R. D., et al. 2013, Aap, 550, AA102
Timmes, F. X., Woosley, S. E., & Weaver, T. A. 1995, Apjs, 98, 617
Timmes, F. X., Hoffman, R. D., & Woosley, S. E. 2000, Apjs, 129, 377
Titarchuk, L., Farinelli, R., Frontera, F., & Amati, L. 2012, ApJ, 752, 116
Totani, T. 1997, apjl, 486, L71
Travaglio, C., Kifonidis, K., & Müller, E. 2004a, New Astron. Rev., 48, 25
Bibliography
112
Travaglio, C., Hillebrandt, W., Reinecke, M., & Thielemann, F.-K. 2004b, A&A, 425, 1029
Travaglio, C., & Hix, W. R. 2013, Frontiers of Physics, 8, 199
Ustyugov, S. D., Chechetkin, V. M. 1999, Astron. Rep., 43, 718.
Ustyugov, S. D., Popov, M. V., Kritsuk, A. G., & Norman, M. L. 2009, J. Comput. Phys., 228,
7614
van Paradijs, J., Groot, P. J., Galama, T., et al. 1997, Nature, 386, 686
von Kienlin, A., Briggs, M. S., Connoughton, V., et al. 2009, American Institute of Physics
Conference Series, 1133, 446
Waldman, R. 2008, ApJ, 685, 1103
Wanderman, D., & Piran, T. 2010, mnras, 406, 1944
Weaver, T. A., Zimmerman, G. B., & Woosley, S. E. 1978, ApJ, 225, 1021
Weinberg, S. 2008, Cosmology, by Steven Weinberg. ISBN 978-0-19-852682-7. Published by
Oxford University Press, Oxford, UK, 2008.,
Weiss, A., Hillebrandt, W., Thomas, H.-C., & Ritter, H. 2004, Cox and Giuli’s Principles of
Stellar Structure, by A. Weiss, W. Hillebrandt, H-C. Thomas, H. Ritter. Cambridge, UK:
Princeton Publishing Associates Ltd, 2004.,
Wheeler, J. C. 1977, apss, 50, 125
Wheeler, J. C., Harkness, R. P., Barkat, Z., & Swartz, D. 1986, pasp, 98, 1018
Wheeler, J. C., Harkness, R. P., Barker, E. S., Cochran, A. L., & Wills, D. 1987, Apjl, 313, L69
Wilson-Hodge, C. A., Case, G. L., Cherry, M. L., et al. 2012, Apjs, 201, 33
Woodward, P. R., & Colella, P. 1984, J. Comput. Phys., 54, 115
Woosley, S. E., & Wallace, R. K. 1982, ApJ, 258, 716
Woosley, S. E., & Weaver, T. A. 1995, Apjs, 101, 181
Woosley, S. E. 1997, Apj, 476, 801
Woosley, S. E., Heger, A., & Weaver, T. A. 2002, Rev. Mod. Phys., 74, 1015
Bibliography
113
Woosley, S. E., Blinnikov, S., & Heger, A. 2007, Nature, 450, 390
Woosley, S. E., & Heger, A. 2015, Astrophysics and Space Science Library, 412, 199
Yonetoku, D., Murakami, T., Nakamura, T., et al. 2004, ApJ, 609, 935
Young, T. R., & Branch, D. 1989, Apjl, 342, L79
Yusof, N., Hirschi, R., Meynet, G., et al. 2013, MNRAS, 433, 1114
Zeldovich, Ya. B, Kompaneets A. 1960, Theory of Detonation. Published by Academic Press,
1960
Zeldovich, Y. B., Barenblatt, G. I., Librovich, V. B., & Markhviladze, G. M. 1985, The Mathematical Theory of Combustion and Explosions (New York: Consultants Bureau)
Bibliography
114
Acknowledgements
First of all I would like to thank my supervisor Pascal Chardonnet for inviting me to LAPTH
and for giving me an opportunity to start, accomplish and present this work. Thanks for the
ideas, discussions and sharing of experience. I thank Pascal a lot for his comprehension and
help whenever I needed.
I am also very grateful to him for his organisational work as a coordinator of Erasmus Mundus
PhD program. Owing to this, I had a great opportunity to get to know France, its culture and
language. As well as to meet interesting people from all over the world.
I would like to thank my co-supervisor Valery Chechetkin for his ongoing participation in my
work, his knowledge sharing and advice.
My special thanks to Andrey Baranov and Mikhail Popov for our work together, for their help
and useful discussions. They made a considerable contribution to this work.
It is important for me to express my gratitude to all Laboratoire d’Annecy-le-Vieux de Physique
Théorique. Especially I want to thank Fawzi Boudjema for his support and sharing positive
mood.
I am indebted to the referees of this manuscript - Massimo Della Valle, Alexei Pozanenko and
Lev Titarchuk for useful and critical comments, which helped to improve this work.
This thesis work had carried out in the financial support of Erasmus Mundus Joint Doctorate
Program by Grants Number 2012-1740 from the agency EACEA of the European Commission.
Finally, I want to thank my family for their every day distant support and encouragement during
my research work. My special thanks to my husband. I thank him for his endless patience and
his help to find new forces and inspiration.
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