Keywords: nucleosynthesis yields

viii 6.7 Number of observed GRBs with redshift in (z,z+dz) over a redshift interval . . 1 The temperature evolution with time T(t) of GRB 101023 . . . . . . . . . . . . 2 Light curve of GRB 101023 (Penacchioni et al. 2012) . . . . . . . . . . . . . . 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 . . . . . . . 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). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1 Supernova spectral types. Credit:Daniel Kasen . . . . . . . . . . . . . . . . . 1.2 A mosaic of HST images of the hosts of forty-two bursts (left) and the supernovae hosts (right) (Fruchter et al. 2006) . . . . . . . . . . . . . . . . . . . . 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). . . . . . 1.4 Sample of light curves of bright BATSE bursts, showing high diversity. . . . . 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 × 10 9 K, ρ = 1 × 10 9 g cm -3 and an initially half 12 C -half 16 O composition for the 13 isotope α-chain network. . . . . . . . . Color represents the logarithm of temperature in the units of 10 9 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 10 9 K. . . 3.7 Distribution of 56 Ni and 52 Fe mass fractions in velocity map for t = 60.5 s. . . 3.8 Distribution of 48 Cr and 44 T i mass fractions in velocity map for t = 60.5 s. . . . 3.9 Distribution of 40 Ca 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 12 C mass fractions in velocity map for t = 60.5 s. . . .

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.

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 "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 10 51 erg, and total energy releases of up to 10 53 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 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 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 10 54 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.

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.

Further study suggested that Type II events might themselves be differentiated by the shape of their light curves [START_REF] Barbon | [END_REF]). 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 [START_REF] Barbon | [END_REF], 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. 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 lowenergy 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 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 10 49 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 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 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. 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. 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.

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.

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 44 Ca (made in supernovae as radioactive 44 T i) only in regions of high temperature and large helium mass fraction. This may also explain the production of a few other rare isotopes like 43 Ca 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).

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 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 [START_REF] Landau | Course of theoretical physics[END_REF] , [START_REF] Zeldovich | Theory of Detonation[END_REF] and also [START_REF] Fickett | Detonation: Theory and Experiment[END_REF]. 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 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.

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:

ρ 1 v 1 = ρ 2 v 2 ≡ j,
(2.1)

p 1 + ρ 1 v 1 2 = p 2 + ρ 2 v 2 2 ,
(2.2)

w 1 + v 1 2 2 = w 2 + v 2 2 2 ; (2.3)
where ρ 1 and v 1 is the density and velocity of the undisturbed material (into which the shock wave moves) and ρ 2 and v 2 is the density and velocity of the final state, which remains behind the shock; p 1 and p 2 are the pressures at the initial and final states respectively; w = ǫ + pVenthalpy, 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 v 2 = ρ 1 v 1 ρ 2 ; The result defines a line in the p-V plane called the "Rayleigh line" and expressed by

p 2 -p 1 = ρ 1 v 1 2 -ρ 2 v 1 2 ρ 1 ρ 2 2 = = (ρ 1 v 1 ) 2 1 ρ 1 - 1 ρ 2 ⇒ (ρ 1 v 1 ) 2 = p 2 -p 1 1 ρ 1 -1 ρ 2
R = j 2 - p 2 -p 1 V 1 -V 2 = 0.
A Rayleigh line passes through the point (p 1 , V 1 ) and has slopej 2 . Examples of Rayleigh lines are shown in figure 2.1.The limiting cases are the horizontal, j 2 = 0, and the vertical, j 2 = ∞.

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 V 1 = 1/ρ 1 , V 2 = 1/ρ 2 , we obtain from v 1 = jV 1 , v 2 = jV 2 (2.4)
and, substituting in eq. 2.2,

p 1 + j 2 V 1 = p 2 + j 2 V 2 , (2.5) or j 2 = p 2 -p 1 V 1 -V 2 . (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 j 2 is positive, we see that either

p 2 > p 1 , V 2 < V 1 , or p 2 < p 1 , V 2 > V 1 .
As was shown for example in [START_REF] Landau | Course of theoretical physics[END_REF], the condition that entropy must increase leads to the conclusion that only the former case (p 2 > p 1 ) can actually occur.

We may note the following useful formula for the velocity difference v 1v 2 . Substituting eq.

in

v 1 -v 2 = j(V 1 -V 2 ), we obtain v 1 -v 2 = (p 2 -p 1 )(V 1 -V 2 ).
(2.7)

Next, we can write eq. 2.3 in the form

w 1 + 1 2 j 2 V 1 2 = w 2 + 1 2 j 2 V 2 2 (2.8)
and, substituting j 2 from eq. 2.6, obtain

w 1 -w 2 + 1 2 (V 1 + V 2 )(p 2 -p 1 ) = 0.
(2.9)

If we replace the heat function w by ǫ + pV, we can write this relation as

ǫ 1 -ǫ 2 + 1 2 (V 1 -V 2 )(p 2 + p 1 ) = 0.
(2.10)

These relations defines the relation between the thermodynamic quantities on the two sides of the surface of discontinuity.

For given p 1 , V 1 , equation eq. 2.9 or eq. 2.10 gives the relation between possible values of p 2 and V 2 . 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 (p 1 , V 1 ) (for p 1 = p 2 , V 1 = V 2 we have also ǫ 1 = ǫ 2 , so that eq. 2.10 is satisfied identically). It should be noted that the shock adiabatic cannot intersect the vertical line

V = V 1 except at (p 1 , V 1 )
. The existence of another intersection would mean that two different pressures satisfying eq. 2.10 correspond to the same volume. For V 1 = V 2 , however, we have from eq. 2.10 also ǫ 1 = ǫ 2 , and when the volumes and energies are the same the pressures must be the same. Thus the line V = V 1 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 = p 1 only at the point (p 1 , V 1 ).

Therefore, for a given initial state (i.e. for given p 1 and V 1 ) the shock wave is defined by a single parameter. For example if the pressure behind the shock, p 2 , is known then the density (specific volume V 2 ) can be defined from the shock adiabatic, and using relations 2.6 and 2.4 the mass flax j and velocities v 1 and v 2 are defined.

Let us mention another convenient gafical interpretation of the relation 2.6. Value of j 2 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 (p 1 , V 1 ). For weak shock we can substitute the relation 2.6 by:

j 2 = - ∂p ∂V S .
(2.11)

Then the velocities v 1 and v 2 are equal:

v 1 = v 2 = jV = -V 2 ∂p ∂V S = ∂p ∂ρ S , (2.12)
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:

v 1 > c 1 .
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 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:

w 1 -w 2 + 1 2 (V 1 + V 2 )(p 2 -p 1 ) = 0 (2.13)
is valid for a detonation wave. The curve of p 2 as a function of V 2 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 (p 1 , V 1 ). This is due to the fact that the shock adiabatic deals with w 1 and w 2 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 (p 1 , V 1 ). 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:

j 2 = (p 2 -p 1 )/(V 1 -V 2 ), (2.14)
so that graphicallyj 2 is the slope of the chord from the point (p 1 , V 1 ) to any point (p 2 , V 2 ) on the detonation adiabatic (for instance, the chord ac in figure 2.3). It is seen from the diagram that j 2 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 j min (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 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 = p 1 + j 2 (V 1 -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( j 2 )/d p 2 = 0, i.e. where the shock adiabatic touches the line from (p 1 , V 1 ), the velocity v 2 is equal to the corresponding velocity of sound c 2 and v 2 < c 2 above that point [START_REF] Landau | Course of theoretical physics[END_REF]. 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( j 2 )/d p 2 = 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 (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 v 2 = c 2 holds for a detonation corresponding to the point O (called the Jouguet point).

v 2 ≤ c 2 ,
The velocity of the detonation wave relative to gas 1 is always supersonic (even for the point O):

v 1 > c 1 .
(2.17) This is seen directly from figure 2.3. The velocity of sound c 1 is given graphically by the slope of the tangent to the shock adiabatic for gas 1 (dashed curve) at the point a. The velocity v 1 , 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 v 1 > c 1 . 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 v 1 with which the detonation wave moves relative to the unburnt gas at rest is the velocity of propagation of the detonation. Since

v 1 /V 1 = v 2 /V 2 ≡ j, and V 1 > V 2 , it follows v 1 > v 2 . The difference v 1 -v 2 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 v 1 = jV 1 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 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.

The velocity v 1 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

v 1 ∼ δ/τ ∼ 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 τ fr 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

v 1 c ∼ χ τc 2 ∼ τ fr 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 τ fr ≪ τ, and v 1 ≪ 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:

ρ 1 v 1 = ρ 1 v 2 , or v 1 /v 2 = V 1 /V 2 , (2.21)
where V 1 and V 2 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:

p 1 = p 2 , w 1 = w 2 . (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. First of all, the section AA ′ has no physical significance. For this section

p 2 > p 1 , V 2 > V 1 , and so the mass flux j = (p 2 -p 1 )/(V 1 -V 2 ) is imaginary.
At the points of contact O and O ′ , the derivative d( j 2 )/d p 2 is zero; At such points we have

v 2 /v 1 = 1 and d(v 2 /v 1 )/d p 2 < 0.
Hence it follows that above the points of contact v 2 /c 2 < 1, and below them v 2 /c 2 > 1. The relation between v 1 and c 1 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 v 1 > c 1 , v 2 < c 2 ; on AO v 1 > c 1 , v 2 > c 2 ; on A ′ O ′ v 1 < c 1 , v 2 < c 2 ; below O ′ v 1 < c 1 , v 2 > c 2 . (2.23) At O and O ′ , v 2 = c 2 .
As we approach A, the flux j , and the velocities v 1 , v 2 , tend to infinity.

As we approach A ′ , j and the velocities v 1 , v 2 tend to zero.

The part of the line below O ′ where v 1 < c 1 , v 2 > c 2 is absolutely unstable with respect to infinitesimal displacements in the direction perpendicular to its plane [START_REF] Landau | Course of theoretical physics[END_REF], 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 v 1 and v 2 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

p 2 > p 1 , V 2 < V 1 , v 2 < v 1 .
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 v 1v 2 . In the region below A, the inequalities are reversed:

p 2 < p 1 , V 2 > V 1 , v 2 > v 1
, and the combustion products are less dense than the original gas.

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.

Rayleigh-Taylor instability

The Rayleigh-Taylor instability, or RT instability (Rayleigh in 1900 andTaylor 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 -ρ 1 g∆z, while for the fluid on the underside of the layer the potential is +ρ 2 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

(ρ 1 + ρ 2 )∆z 2 ω 2 ∼ (ρ 1 -ρ 2 )g∆z,
so that we get, for the frequency, 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 Rayleigh-Taylor 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.

ω 2 ∼ ρ 1 -ρ 2 ρ 1 + ρ 2 gk,
In the explosion of a supernova, the envelope is low in density and quite hot.

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 Landau-Darrieus (LD) instability. It was predicted independently by [START_REF] Darrieus | unpublished work presented at La Technique Moderne Della Valle[END_REF]) and [START_REF] Landau | Dynamics of Curved Fronts[END_REF]. This instability wrinkles the flame surface accelerating the combustion and leads to formation of cusps [START_REF] Bell | [END_REF]). 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).

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 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.

Pair instability

When central temperature in stellar interiors reaches sufficiently high values (order of a few times 10 9 K) a significant part of photons of the Plankian spectrum have energies higher than 2m e c 2 (m e 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 [START_REF] Weiss | Cox and Giuli's Principles of Stellar Structure[END_REF].

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.

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.

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 [START_REF] Moiseenko | AIP Conf. Proc[END_REF]).

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). 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 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).

Methods

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.

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.

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 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. 2002Maeda et al. , 2010;;Nagataki 1999;Nagataki et al. 2006;Ohkubo et al. 2006;[START_REF] Patat | [END_REF]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 240 2 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.

Explosive nucleosynthesis

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 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)).

→ 23 Na + p , 12 C + 16 O → 24 Mg + α , → 27 Al + p , 16 O + 16 O → 28 S i + α ,

Nuclear abundances

We describe the nuclear population in each cell of a simulation in terms of the nuclear abundance of each species

Y = n/ρN A ,
where n is the number density of the species, ρ is the total mass density, and N A is Avagadro's number. For a nucleus with atomic mass number A, AY ≡ X is commonly referred to as the mass fraction of this nucleus and the sum of all mass fractions ( A i Y i ) is by construction equal to 1.

The formula A i Y i = 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 in an astrophysical model. The equation of charge conservation becomes

Z i Y i = Y e ,
where Y e = n e /ρN A 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 dY i /d t with dependence only on nuclear reaction rates and local thermodynamic conditions.

dY i d t = j N i j λ j Y j + j,k N i j,k ρN A σv j,k Y j Y k + j,k,l N i j,k,l ρ 2 N A 2 σv j,k,l Y j Y k Y l (3.1)
where the decay constants (λ) and velocity-integrated reaction cross sections σv contain the essential nuclear data. Rate of the forward reaction R j,k = N A σv j,k : R j,k = exp a 0 + a 1 + a 2 + a 3 T 1/3 9 + a 4 T 9 + a 5 T 5/3 9 + a 6 log T 9 ; rate of the reverse reaction of photodissociation λ: λ = exp a 0 + T 9 + T 1/3 9 + a 3 T 1/3 9 + a 4 T 9 + a 5 T 5/3 9 + a 6 log T 9 ;

where T 9 = T 10 9 K and a i -the adjustable parameters.

The N s provide for proper accounting of numbers of nuclei with

N i j = N i , N i j,k = N i / n m m=1 |N j m |!, and N i j,k,l = N i / n m m=1 |N j m |!
where the N ′ i s are positive (or negative) numbers that specify how many particles of species i are created (or destroyed) in a reaction, while the denominators, including 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.

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

τ k ( j) = 1 σv j,k Y k or τ j ( j) = 1 λ j (3.2)
for a decay. Systems whose solutions depend on a wide range of timescales are termed stiff.

Gear [START_REF] Gear | Numerical Initial Value Problems in Ordinary Differential Equations[END_REF]) 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 [START_REF] Lambert | Computational Techniques for Ordinary Differential Equations[END_REF] is that a system of equations is stiff if the ratio of the real part of the largest eigenvalue λ max of the Jacobian matrix of the system of equations is much larger than the real part of the smallest eigenvalue λ min . Because of the wide range in timescales between strong, electromagnetic and weak reactions, λ max /λ min > 10 15 is not uncommon in astrophysics, making thermonuclear reaction networks extraordinarily stiff.

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 [START_REF] Hairer | Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems[END_REF].

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 × 10 9 K, ρ = 1 × 10 9 g cm -3

and an initially composition as half 12 C -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.

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 ρ c ∼ 4.5 × 10 5 g/cm 3 and T c ∼ 1 × 10 9 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 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/cm 3 . The total initial mass inside the computational 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 ⊙ , M iron = 1.8M ⊙ ):

X( 12 C) = 0.03, X( 54 Fe) -the rest.
For the oxygen layer (0.0208R ⊙ < r < 0.09R ⊙ , M O = 5.53M ⊙ ):

X( 20 Ne) = 0.2, X( 24 Mg) = 0.04, X( 12 C) = 0.015, X( 54 Fe) = 3 × 10 -4 , X( 16 O) -the rest.
For the helium layer (0.09R ⊙ < r < 0.98234R ⊙ , M He = 1.33M ⊙ ):

X( 12 C) = 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 ⊙ , M He+H ∼ 0.002M ⊙ ):

X( 1 H) = 0.5, X( 14 N) = 7.5 × 10 -3 , X( 16 O) = 5.1 × 10 -3 , 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 × 10 51 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: v z = αz in the vertical direction and v r = 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.

Results

The simulation of supernova hydrodynamics with the initial parameters described above is presented on fig. 3 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 × 10 10 K. Behind the shock the flow is moving more uniformly with lower temperature T ∼ (0.4 ÷ 1) × 10 9 K. The most cold part is an inner structure (the region showed on fig. 3.5 by blue) with T ∼ (0.35 ÷ 1) × 10 8 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. 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/cm 3 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 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 ⊙ .

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 × 10 52 ergs (twice more than in our case) 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 [START_REF] Hamuy | [END_REF]. SNe associated with GRBs. The estimated progenitor masses for them are 45 ± 5, 32.5 ± 7.5 and 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 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 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.

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 [START_REF] Fowler | [END_REF]. 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 × 10 9 K intensive creation of electron-positron pairs occurs. It increases the energy losses by neutrino through a pair annihilation process, e + + e -→ ν e + ν e , 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 into explosion by nuclear burning in the center. Nuclear energy release was ∼ 6 × 10 51 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 [START_REF] Arnett | Explosive Nucleosynthesis[END_REF][START_REF] Arnett | Late Stages of Stellar Evolution[END_REF]. The cores exploded and the lighter one left a 2.2 M ⊙ silicon remnant. [START_REF] Wheeler | [END_REF] found that the cores with masses 10 3 -10 4 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.

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 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.

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 M c , we built several configurations by choosing different values of central density, ρ c . This allowed us to consider models with different values of binding energy, E bind , for each M c (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. T max the maximum temperature at the center at the moment of reversal of collapse, E nuc 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, M c , there is some critical value E * bind of initial binding energy that determines the fate of the core: explosion (PISN) if E bind > E * bind or core-collapse 1 (CC) for E bind < E * bind . Two regions could be seen clearly in the M c -E bind diagram (Fig. 4.1). This behavior could be explained by the fact that models with lower E bind (higher absolute value of E bind ) 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. E * bind tends to zero with a growth of M c . 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, E nuc , 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.

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. 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/cm 3 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 = Kρ γ , where K = 8.225 × 10 15 , 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 × 10 52 ergs was uniformly deposited within the radius r e = 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 × 10 52 ergs. This nonuniformity could present some inhomogeneities in the core that occur prior to explosion. 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 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.

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

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;[START_REF] Paczyński | Gamma-Ray Bursts, 4th Hunstville Symposium[END_REF]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 L iso , 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) :

N E (E) =        CE α e -E/E 0 , (α -β)E 0 ≥ E, C ′ E β , (α -β)E 0 < E, (5.1)
where N E (E) is the photon number spectrum, E is the photon energy in keV, E 0 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 N E (E) a continuous and continuously differentiable function:

C ′ = C ((α -β)E 0 ) α-β e -(α-β) .
Spectral hardness is usually measured by the peak energy E peak of the EF E spectrum (F E = EN E ). This energy is related to E 0 as

E peak = (2 + α)E 0 and (α -β)E 0 = E break .
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 E break represents the maximum of emission for GRB spectrum, i.e. where the maximum emission is concentrated. In some sense, we expect that E break 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 [START_REF] Larsson | [END_REF]. 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.

Black-body and Thermal Bremsstrahlung emission

It is generally accepted by scientists that the GRB spectrum is a Band spectrum represented by the function N B defined in paragraph 1. Following the analysis of Ryde and other, we will also follow the idea of a thermal emission defined by:

N BB = A BB E 2 e E/kT -1 (5.2)
where A BB 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 ∼ 10 9 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 n e n i Z 2 g (5.3) in erg • s -1 • cm -3 .
T is the electron temperature, n e and n i 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 × 10 9 , therefore it is completely ionized and we have n i = n e = 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 ≃ 10 25 cm -3 . Therefore, we deduce the corresponding value for

ǫ f f ≃ 7.4 × 10 24 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 L Br . We have: L Br = ǫ f f × V and E Br = 3 2 kT e × n e × V. The cooling time, which measure how fast the process of thermal bremsstrahlung emission is efficient, is simply given by: τ Br = E Br L Br . Assuming T ≃ 2 × 10 9 , we obtain: τ Br ≃ 5.2 × 10 18 n in seconds. Even with a small density of ∼ 10 15 , 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 × 10 9 and n ≃ 10 25 cm -3 .

A luminosity of ∼ 10 49 erg • s -1 , corresponding to an individual spike, could be obtained if the total emitted volume is the order of 10 25 cm 3 . This corresponds to a spherical blob of radius ∼ 1000 km.

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):

I Br (E) = n e n i 32 3 π 3 1/2 e 4 m 2 e c 4
Ry kT

1/2
e -E/kT ḡ, (5.4)

where I Br is the power emitted per unit volume per unit of time per unit energy, n e and n i 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 I E in physical units:

I Br (E) = 2 × 10 -20 n e n i √ T e -E/kT ḡ erg s • cm 3 • keV . (5.5)
Then the photon number spectrum measured by the observer is:

N Br (E) = I Br (E)V 4π E D 2 , (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 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 :

N Br (E) = A Br 1 E e -E/kT , (5.7)
where A Br is a normalization factor that can be computed from Eqs. (5.5, 5.6) 1 . Thermal bremsstrahlung naturally explains N E ∝ 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):

N BB (E) = A BB E 2 e E/kT -1 , (5.8)
where A BB 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) = N BB (E) + N Br (E) = A BB E 2 e E/kT -1 + A Br 1 E e -E/kT .
(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.

Analysis of gamma-ray bursts

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 of the time-integrated E 2 • N E 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.

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 E iso = 10 53 erg 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.

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 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 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 ∼ 10 8 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 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 ∼ 10 53 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. In [START_REF] Baranov | Proceedings of the 48th Rencontres de Moriond[END_REF] 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 E iso 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. 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 star. Here we have used a simple function in time

I i = I 0,i e -(t-t 0,i) 2
for the luminosity function of each blob. Here I i is the intensity, I 0,i is normalization factor, which is assumed to be proportional to the size of a spot i; t 0,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 the burst could be explained by the sum (superposition) of the individual light curve I = i I i .

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 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 [START_REF] Baranov | Proceedings of the 48th Rencontres de Moriond[END_REF] 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 [START_REF] Weinberg | Cosmology[END_REF].

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 t 1 with wavelength λ 1 will be observed at t 0 with wavelength

λ 0 = λ 1 a(t 0 ) a(t 1 ) . (6.1)
Since a(t 0 ) > a(t 1 ), the wavelength of the light increases, λ 0 > λ 1 .

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 τ 1 , 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 τ 0 = τ 1 +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, ∆t 1 = a(τ 1 )∆τ (6.2) and ∆t 0 = a(τ 0 )∆τ. (6.3)

If ∆t is the period of the light wave, the light is emitted with wavelength λ 1 = ∆t 1 (in units where c = 1), but is observed with wavelength λ 0 = ∆t 0 , so that

λ 0 λ 1 = a(τ 0 ) a(τ 1 ) . (6.4)
If a(t) is increasing, then this is a redshift, a decrease in frequency by a factor a(t 1 ) a(t 0 ) , equivalent to an increase in wavelength by a factor conventionally called 1 + z:

1 + z = a(t 0 ) a(t 1 ) . (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(t 0 )[1 + (t 1 -t 0 )H 0 + • • • ] (6.6)
where H 0 is a coefficient known as the Hubble constant:

H 0 ≡ ȧ(t 0 ) a(t 0 ) . (6.7)
Eq.6.5 then gives the fractional increase in wavelength as

z = H 0 (t 0 -t 1 ) + • • • . (6.8)
For close objects, t 0t 1 is simply the proper distance d (in units with c = 1). We therefore find that the redshift increases linearly with distance z ≈ H 0 d. (6.9)

The slope in a redshift-distance diagram therefore measures the current expansion rate of the universe, H 0 . These measurements used to come with very large uncertainties. Since H 0 normalizes everything else (see below), it became conventional to define 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 4πd 2 l, so l = L 4πd 2 .

H 0 ≡ 100 h kms -1 M pc -1 , ( 6
(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 6.13) where dχ = dr/ √ 1kr 2 , and

ds 2 = dt 2 -a 2 (t)[dχ 2 + S 2 k (χ)dΩ 2 ], (
S k (χ) =                R 0 sinh(χ/R 0 ), k = -1, hyperbolic χ, k = 0, Euclidean R 0 sin(χ/R 0 ), k = +1, spherical (6.14)
The distance multiplying the solid angle element dΩ 2 is the metric distance,

d m = 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 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.

χ(z) = t 0 t 1 dt a(t) = z 0 dz H(z) , (
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 4πχ 2 .
(6.17)

In an FRW spacetime, this result is modified for three reasons: 1. At the time t 0 that the light reaches the Earth, the proper area of a sphere drawn around the supernova and passing through the Earth is 4πd 2 m The fraction of the light received in a telescope of aperture A is thereforeA/4πd 2 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 E 0 of the photons when they are received is less than the energy E 1 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 4πd m 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

d L = d m (1 + z). (6.19)
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 t 1 . A naive astronomer could decide to measure the distance d A to the object by measuring its angular size δθ and using the Euclidean formula for its distance, .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 = D δθ . ( 6
D = a(t 1 )S k (χ)δθ = d m 1 + z δθ. (6.21)
Hence, we get

d A = d m (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

d A = d L (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 a

2 = 8πG 3 ρ - k a 2 , (6.24) ä a = - 4πG 3 (ρ + 3P). (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 ρ r for the contribution from radiation (with ρ γ for photons and ρ ν for neutrinos), ρ m for the contribution by matter (with ρ c for cold dark matter and ρ b for baryons) and ρ Λ for the vacuum energy contribution. The first Friedmann equation is often written in terms of the Hubble parameter, H ≡ ȧ/a,

H 2 = 8πG 3 ρ - k a 2 . (6.26)
Let us use subscripts '0' to denote quantities evaluated today, at t = t 0 . A flat universe (k = 0) corresponds to the following critical density today

ρ crit,0 = 3H 0 2 8πG = 1.9×10 -29 h 2 grams cm -3 = 2.8×10 11 h 2 M ⊙ M pc -3 = 1.1×10 -5 h 2 photons cm -3 . (6.27)
We use the critical density to define dimensionless density parameters

Ω I,0 ≡ ρ I,0 ρ crit,0 .
(6.28)

The Friedmann equation can then be written as

H 2 (a) = H 0 2 Ω r,0 a 0 a 4 + Ω m,0 a 0 a 3 + Ω k,0 a 0 a 2 + Ω Λ,0 , (6.29)
where we have defined a "curvature" density parameter, Ω k,0 ≡ -k/(a 0 H 0 ) 2 . It should be noted that in the literature, the subscript '0' is normally dropped, so that e.g. Ω m 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, a 0 ≡ 1. Eq. 6.29 then becomes The equation of state of dark energy seems to be that of a cosmological constant,w Λ ≈ -1. The matter splits into 5% ordinary matter (baryons, 'b') and 27% (cold) dark matter (CDM, 'c'):

H 2 H 0 2 = Ω r a -4 + Ω m a -3 + Ω k a -2 + Ω Λ . ( 6
Ω b = 0, 05, Ω c = 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

Ω Λ + Ω m + Ω r + Ω k = 1, Ω k ≡ - k a 2 0 H 2 0 (6.33)
and after we have

d t = dx H 0 x Ω Λ + Ω k x -2 + Ω m x -3 + Ω r x -4 = = -dz H 0 (1 + z) Ω Λ + Ω k (1 + z) 2 + Ω m (1 + z) 3 + Ω r (1 + z) 4 (6.34)
where x ≡ a/a 0 = 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 t

(z) = 1 H 0 1/(1+z) 0 dx x Ω Λ + Ω k x -2 + Ω m x -3 + Ω r x -4 . (6.35)
In particular, by setting z = 0, we find the present age of the universe: .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

t 0 = 1 H 0 1 0 dx x Ω Λ + Ω k x -2 + Ω m x -3 + Ω r x -4 . ( 6
r(z) = S t 0 t(z) dt a(t) = S        1 a 0 H 0 1 1/(1+z) dx x 2 Ω Λ + Ω k x -2 + Ω m x -3 + Ω r x -4        , where S [χ] =                sinh(χ), k = -1, χ, k = 0, sin(χ), k = +1
This can be written more conveniently by using Eq. 6.33 to express a 0 H 0 in terms of Ω k . We then have a single formula (6.37) which can be used for any curvature. (Eq. 6.36 has a smooth limit for Ω k → 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

a 0 r(z) = 1 H 0 Ω k 1/2        Ω k 1/2 1 1/(1+z) dx x 2 Ω Λ + Ω k x -2 + Ω m x -3 + Ω r x -4        ,
d L (z) = a 0 r(z)(1 + z) = 1 + z H 0 Ω k 1/2        Ω k 1/2 1 1/(1+z) dx x 2 Ω Λ + Ω k x -2 + Ω m x -3 + Ω r x -4        . (6.38)
For k = 0 we have Ω k = 0 and Eq. 6.38 becomes

d L (z) = a 0 r 1 (1 + z) = 1 + z H 0 1 1 1+z dx x 2 • (Ω Λ + Ω m • x -3 + Ω r • x -4 ) 1/2 .
(6.39)

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 E iso :

(E iso /10 52 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 = L 4πd 2 L and fluence S = E iso 4πd 2 L • (1 + z). Observed peak energy E peak,obs = E peak • (1 + z) -1 , isotropic equivalent radiated energy E iso = k • (1 + z) α • (E p,obs ) α .
In this case fluence S is:

S = k • (1 + z) α+1 • (E p,obs ) α 4πd 2 L ,
and we can express the luminosity distance from this equation: and compare with theoretical value (eq. 6.39):

d obs L =       k • (1 + z) α+1 • (E p,obs ) α 4πS       1/2 ,
d L (z) = 1 + z H 0 1 1 1+z dx x 2 • (Ω Λ + Ω m • x -3 + Ω r • x -4 ) 1/2

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. 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 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: For luminosity distance we used the theoretical value:

dN GRB d z = Ω obs 4π • P z • R grb (z) • dt 1 + z • dV dz (
d L (z) = 1 + z H 0 1 1 1+z dx x 2 • (Ω Λ + Ω m • x -3 + Ω r • x -4 ) 1/2
(6.42) Another expression for luminosity function is from Campisi et al. (2010):

Φ(L) = K • ( L L * ) ψ • exp(- L * L )
L-isotropic equivalent intrinsic burst luminosity, ψ -asymptotic slope at the faint end, K =

[L 0 Γ(-ψ -1)] -1 for ψ < -1 -normalization constant, L * = L(z) = L 0 (1 + z) δ -characteristic cutoff luminosity. We take sample of the parameters δ, L 0 , ψ from (Campisi et al. 2010):δ = 2.0, L 0 = 0, 347 • 10 50 erg • s -1 , ψ = -1, 709.

To find a number of observed GRBs (eq.6.41) we should know integral over luminosity function:

log L lim (z) Φ(L)d log L We can see in the figure 6.5, that between these two luminosity functions there are no fundamental difference.

Rate of GRB

Let's consider R GRB (z) -rate of GRB. There are different ways too. Generally it is assumed that the rate of GRB is proportional to star formation rate R S FR (z) :

R GRB (z) = k GRB • R S FR (z). (6.43)

For SFR we used approximation formula by Hopkins & Beacom (2006) :

R S FR (z) = (a 1 +a 2 •z)•h 1+( z a 3
) a 4 , a 1 , a 2 , a 3 , a 4 , h -parameters a 1 = 0, 017, a 2 = 0, 13, a 3 = 3, 3, a 4 = 4, 3, h = 0, 7, coefficient

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 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.

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

Physical Constants

Speed of Light c = 2.997 58 × 10 10 cm s -1

Gravitational Constant G = 6.6742 × -8 cm 3 g -1 s -1

2. 1

 1 Curves of constant shockwave velocity, called Rayleigh lines. . . . . . . . . . . 2.2 The shock adiabatic or the Hugoniot adiabatic and Rayleigh line (dashed) . . . . 2.3 The detonation adiabatic (continuous line) and the ordinary shock adiabatic (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The relation between the different modes of combustion . . . . . . . . . . . . . 2.5 Example of Rayleigh-Taylor instability simulations by the PPML code Popov (2012). The density map is shown . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A simplified network of nuclear reactions from 12 C to 56 Ni. . . . . . . . . . . . 3.2 Comparison of our nucleosynthesis computing code (on the top) and nucleosynthesis computing code from

3. 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 ρ c = 4.5 × 10 5 g/cm 3 . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Temperature distribution for the same moment and in the same coordinate units as on fig. 3.4.

v 3 .

 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 Fate of a star depending on its mass, M c , and binding energy, E bind . Explosion is marked by diamonds and collapse is marked by circles. . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . 4.3 SN model with central ignition for t = 28 sec. Logarithm of density (a) is shown in units of ρ c = 2.65 × 10 5 g/cm 3 . Temperature (b) is shown in units of T c = 2.36 × 10 9 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 SN model with multicore ignition for t = 28 sec. Logarithm of density (a) is shown in units of ρ c = 2.65 × 10 5 g/cm 3 . Temperature (b) is shown in units of T c = 2.36 × 10 9 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Time-integrated spectra GRB 090618 . . . . . . . . . . . . . . . . . . . . . . 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). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The temperature evolution with time T(t) of GRB 090618 . . . . . . . . . . . . 5.5 Example of light curve of GRB 090618 (Izzo et al. 2012) . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Cartoon representation of a possible emission mechanism with Black-Body + Bremmstrahlung emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Amati relation with redshift distribution . . . . . . . . . . . . . . . . . . . . . 6.2 Isotropic equivalent radiated energy from redshift E iso (z) . . . . . . . . . . . . 6.3 Comparison of luminosity distance computed from Amati relation" and from cosmological redshift by eq.6.42 . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 "Volumetric factor" dV/dz 1+z (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Integral of Luminosity Function from Campisi et al. (2010) and Wanderman & Piran (2010) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Rate GRB from Hopkins & Beacom (2006) and rate PISN from Hummel et al. (2012) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1

  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).

Figure 1 . 1 :

 11 Figure 1.1: Supernova spectral types. Credit:Daniel Kasen

Figure 1

 1 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)

Figure 1 . 3 :

 13 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).

Figure 1 . 4 :

 14 Figure 1.4: Sample of light curves of bright BATSE bursts, showing high diversity.

Figure 2

 2 Figure 2.1: Curves of constant shockwave velocity, called Rayleigh lines.

Figure 2

 2 Figure 2.2: The shock adiabatic or the Hugoniot adiabatic and Rayleigh line (dashed) .

Figure 2 . 3 :

 23 Figure 2.3: The detonation adiabatic (continuous line) and the ordinary shock adiabatic (dashed line).

Figure 2 . 4 :

 24 Figure 2.4: The relation between the different modes of combustion

Figure 2 . 5 :

 25 Figure 2.5: Example of Rayleigh-Taylor instability simulations by the PPML code Popov (2012). The density map is shown

  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.

Figure 2

 2 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.

  3.2.1 Piecewise Parabolic Method on a Local stencil for hydrodynamicsHydrodynamic 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 inUstyugov et al. (2009). PPML is an improvement over the popular Piecewise Parabolic Method (PPM)

→

  fig.3.1 by the arcs. The triple-α reaction for carbon production was also implemented

Figure 3

 3 Figure 3.1: A simplified network of nuclear reactions from 12 C to 56 Ni.

Figure 3 . 2 :

 32 Figure 3.2: Comparison of our nucleosynthesis computing code (on the top) and nucleosynthesis computing code fromTimmes 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 × 10 9 K, ρ = 1 × 10 9 g cm -3 and an initially half 12 C -half 16 O composition for the 13 isotope α-chain network.

  .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.

Figure 3 Figure 3

 33 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 ρ c = 4.5 × 10 5 g/cm 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 10 9 K.

Figure 3

 3 Figure 3.7: Distribution of 56 Ni and 52 Fe mass fractions in velocity map for t = 60.5 s.

Figure 3 Figure 3

 33 Figure 3.8: Distribution of 48 Cr and 44 T i mass fractions in velocity map for t = 60.5 s.

Figure 3 Figure 3

 33 Figure 3.10: Distribution of 32 S and 28 S i mass fractions in velocity map for t = 60.5 s.

Figure 3 Figure 3 Figure 3

 333 Figure 3.12: Distribution of 16 O and 12 C mass fractions in velocity map for t = 60.5 s.

  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 of56 Ni are pushed away by the explosion in the axial direction with the estimated velocity v z ∼ (1.5 ÷ 2.0) × 10 9 cm/s while the light nuclei are carried by the flow along the equatorial plane with v r ∼ (2 ÷ 3) × 10 8 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.

Figure 3

 3 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.

  The fainter luminosity corresponds to a smaller amount of56 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 ⊙[START_REF] Smartt | [END_REF]. Still in some events more56 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

4. 1

 1 IntroductionThe 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.

Figure 4

 4 Figure 4.1: Fate of a star depending on its mass, M c , and binding energy, E bind . Explosion is marked by diamonds and collapse is marked by circles.

Figure 4

 4 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.

Figure 4

 4 Figure 4.3: SN model with central ignition for t = 28 sec. Logarithm of density (a) is shown in units of ρ c = 2.65 × 10 5 g/cm 3 . Temperature (b) is shown in units of T c = 2.36 × 10 9 K.

Figure 4

 4 Figure 4.4: SN model with multicore ignition for t = 28 sec. Logarithm of density (a) is shown in units of ρ c = 2.65 × 10 5 g/cm 3 . Temperature (b) is shown in units of T c = 2.36 × 10 9 K.

Figure 5

 5 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).

  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).

FigureFigure 5 . 5 :

 55 Figure 5.4: The temperature evolution with time T(t) of GRB 090618

Figure 5

 5 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.

Figure 5

 5 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.

  Figure 5.8: Cartoon representation of a possible emission mechanism with Black-Body + Bremmstrahlung emission

  .30) Observations show that the universe is filled with radiation ('r'), matter ('m') and dark energy ('Λ'):|Ω k | ≤ 0.01, Ω r = 9.4 × 10 -5 , Ω m = 0.32, Ω Λ = 0.68.(6.31)

Figure 6

 6 Figure 6.1: Amati relation with redshift distribution

Figure 6

 6 Figure 6.3: Comparison of luminosity distance computed from Amati relation" and from cosmological redshift by eq.6.42

  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.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

  z) • Φ(L)d log L (6.41) In this equation there are two unknown functions: luminosity function Φ(L) and rate of GRB R GRB (z). The other parameters are the following: Ω obs -field of view of experiment, Ω obs = 1, 4 for Swift, P z = 0, 28 -normalization factor, dt -observation time, dV dz (z) -comoving volume element per unit of redshift.dV dz(z) = c • 4πd 2 L (z) H(z) • (1 + z) 2 , c -speed of light 299 792 458 m/s, d L -luminosity distance [Mpc], H(z) = H 0 • ((Ω Λ + Ω k • (1 + z) 2 + Ω m • (1 + z) 3 + Ω r • (1 + z) 4 ) 1/2 , H 0 = 72[km • c -1 • M pc -1 ] -Hubble constant, Ω Λ = 0, 73, Ω m = 0, 27, Ω r = 0, Ω k = 0 -cosmological parametres. "Volumetric factor" dV/dz1+z (z) is plotted on Fig.6.4.

Figure 6

 6 Figure 6.4: "Volumetric factor" dV/dz 1+z (z)

Figure 6

 6 Figure 6.5: Integral of Luminosity Function from Campisi et al. (2010) andWanderman & Piran (2010)

k

  GRB = 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:R GRB ∼ R PIS N (6.44)The rate of PISN we took fromHummel et al. (2012) fig. 6.6 red line).

Figure 6

 6 Figure 6.6: Rate GRB from Hopkins & Beacom (2006) and rate PISN from Hummel et al. (2012)

Figure 1 :Figure 3 :

 13 Figure 1: The temperature evolution with time T(t) of GRB 101023

Figure 5 :

 5 Figure 5: The temperature evolution with time T(t) of GRB 970111

Figure 9 :Figure 11 :

 911 Figure 9: The temperature evolution with time T(t) of GRB 090926A

Figure 12 :Figure 14 :

 1214 Figure 12: The temperature evolution with time T(t) of GRB 990510

Figure 15 :Figure 17 :

 1517 Figure 15: The temperature evolution with time T(t) of GRB 980329

AvogadroElectron

  Plank's Constant h = 6.626 × 10 -27 erg s=h/2π = 1.0546 × -27 erg s Boltzmann Constant k = 1.3807 × -16 erg K -1 314 × 10 7 erg K -1 mole -1 Stefan-Boltzmann Constant σ = 5.670 × 10 -5 erg K -4 cm -2 s -1Radiation Density Constant a=4σ/c = 7.566 × 10 -15 erg K -4 cm -3

  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 40M ⊙ 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.

Table 3

 3

	.1: Detailed nucleosynthesis yields.
	Species Yield (M ⊙ ) Species Yield (M ⊙ ) 12 C 6.45 × 10 -2 38 Ar 1.87 × 10 -5 13 N * 4.27 × 10 -5 39 K 4.05 × 10 -4 16 O 3.31 40 Ca 6.23 × 10 -2 20 Ne 0.710 42 Ca 5.91 × 10 -5 23 Na 1.28 × 10 -6 43 S c * 5.75 × 10 -5 24 Mg 0.245 44 T i * 1.73 × 10 -3 26 Mg 7.14 × 10 -6 46 T i 4.86 × 10 -5 27 Al 4.90 × 10 -5 47 V * 5.25 × 10 -4 28 S i 0.172 48 Cr * 4.86 × 10 -3 30 S i 1.23 × 10 -5 50 Cr 2.60 × 10 -5 31 P 2.02 × 10 -5 51 Mn * 4.43 × 10 -4 32 S 0.118 52 Fe * 1.19 × 10 -2 34 S 2.76 × 10 -5 54 Fe 1.60 × 10 -3 35 Cl 9.03 × 10 -5 55 Co * 1.95 × 10 -3 36 Ar 3.71 × 10 -2 56 Ni

* 0.521

The discussed details of nucleosynthesis are summarized on fig.

3

.14 where the main zones are

Table 4

 4 is the mass of the oxygen core, E bind the initial binding energy, ρ c the initial central density,

		.1: Presupernova models and parameters of explosion
	M c	E bind	ρ c	T max	E nuc	Fate
	[M ⊙ ] [10 51 erg] [10 5 g/cc] [10 9 K] [10 51 erg] 60 -5.54 0.87 4.07 20.9	PISN
	60	-5.48	1.15	4.07	21.0	PISN
	70	-6.10	1.0	-	-	CC
	70	-5.78	1.3	-	-	CC
	70	-5.27	1.7	4.52	29.4	PISN
	70	-4.87	2.0	4.16	25.9	PISN
	70	-3.67	3.0	3.74	18.8	PISN
	78	-4.59	2.0	-	-	CC
	78	-3.73	2.5	4.18	28.8	PISN
	78	-2.96	3.0	3.84	23.1	PISN
	90	-5.19	1.5	-	-	CC
	90	-3.78	2.0	-	-	CC
	90	-3.01	2.3	-	-	CC
	90	-2.07	2.7	4.77	38.7	PISN
	90	-1.85	2.8	4.30	34.9	PISN
	90	-1.43	3.0	4.11	38.7	PISN
	95	-1.87	1.5	-	-	CC
	95	-1.11	2.6	4.39	39.9	PISN
	95	-0.98	2.8	4.31	37.1	PISN
	95	-0.63	3.0	4.20	35.0	PISN
	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
	M c

Table 5

 5

			.1: Spectral analysis of GRB 090618
			GRB 090618
	Time int. [s]	[0 -50]	[50 -59]	[59 -69]	[69 -78]	[78 -105]
			Redshift = 0.54
			E iso [erg] = 2.90 × 10 53
			Band fitting:
	E 0 [keV]	119	195	321	161	124
	α Band β Band	-0.74 -2.32	-1.13 -2.5	-0.99 -2.60	-1.04 -2.42	-1.06 -2.62
			Our model:
	A Br	48.33	75.78	320.5	223.2	135.4
	A Bb kT [keV]	3.25 × 10 -8 7.72 × 10 -8 11.8 × 10 -9 2.17 × 10 -7 2.64 × 10 -7 163.8 138.7 304.8 145.8 111.2
	R BB [cm] L Br [erg/s] L Bb [erg/s]	1.70 × 10 8 1.04 × 10 51 1.48 × 10 51 9.63 × 10 51 4.50 × 10 51 2.349 × 10 51 2.61 × 10 8 1.02 × 10 8 4.39 × 10 8 4.84 × 10 8 2.69 × 10 50 3.28 × 10 50 1.17 × 10 51 1.12 × 10 51 4.63 × 10 50
	% Br	79, 4%	81.9%	89.1%	80.0%	83.5%
	L tot [erg/s]	1.31 × 10 51 1.81 × 10 51 1.08 × 10 52 1.63 × 10 51	2.81 × 10 51
	used both black-body and thermal bremsstrahlung. This combination tells us that the environ-
	ment should be very hot ∼ 100 keV. From this first result we do expect to have a hot plasma cooling quickly.

  .10) where the parameter h is used to keep track of how uncertainties in H 0 propagate into other cosmological parameters. Today, measurements of H 0 have become much more precise

	h ≈ 0.7 ± 0.01.	(6.11)
	Another important moment is how astronomical distances are measured.

C + 12 C → 20 Ne + α ,

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).

Acknowledgements

List of Figures

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

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).

Abbreviations