Thesis

Effects of Neutrino Transport on the R-process

Nucleosynthesis in the Supernova Explosion for a

Helium Star of 3.3M⊙

Motoaki Saruwatari

Abstract

Path of stellar evolution depends on the mass in the main sequence stage at birth. Proto-

star, which has been made from the contraction of interstellar gas, continues the contrac-

tion until H burning begins. In this phase, H burning produces He and then He core star

is build up. He stars whose masses exceed 0.46M⊙ cause He burning and forms CO core.

Massive stars (M > 10M⊙) evolve toward the formation of Fe core through red giant

phase. Although light stars (M < 8M⊙) which do not evolve to form the Fe core, massive

star (M > 8M⊙) result in core collapse supernovae and leave black hole or neutron star.

In this phase of supernova explosion, nucleosynthesis called r-process occurs.

Core collapse supernovae are very important astronomical events, while detailed mech-

anism has not been clarified in spite of many simulations have been done. In the present

study, we have first reviewed basic physical and astrophysical processes and afterwards

constructed explosion models with an emphasis on stellar rotation, magnetic fields and

neutrino effect. In particular, we demonstrate that neutrino effects are important for core

collapse supernovae. It has been confirmed that the neutrino emission energy of supernova

is 100 times the explosion energy of supernovae.

We investigate the r-process nucleosynthesis during the magnetohydrodynamical (MHD)

explosion of supernova in a helium star of 3.3M⊙ with the effects of neutrinos included.

Contrary to the case of the spherical explosion, jet-like explosion due to the combined ef-

fects of differential rotation and magnetic field results in the ejection of the lower electron

fraction matter significantly from inside the layers. We find that the ejected materials of

low electron fraction responsible for the r-process comes out from just outside the neutrino

sphere deep inside the Fe-core. This leads to the production of the second to third peak

in the solar r-process pattern of elements. We suggest that there are variations in the

heavy element nucleosynthesis according to the initial conditions of rotational and mag-

netic fields. In particular, the third peak of the distribution is significantly overproduced,

which would indicate a possible new r-process site of MHD jets in supernovae. Therefore,

we can infer that supernova explosions associated to the jet-like explosion such as γ-ray

bursts would be responsible in producing the third peak of the r-process pattern.

In addition to core collapse supernova, we investigate type Ia supernova. Carbon-

oxygen white dwarfs accreting helium in a binary system occur type Ia supernova. We

investigate the effects of a new triple-α reaction rate on the ignition of carbon-oxygen

white dwarfs accreting helium in a binary system. The ignition points determine the

properties of a thermonuclear explosion of a Type Ia supernova. We examine the cases of

different accretion rates of helium and different initial masses of the white dwarf, which

was studied in detail by Nomoto (Astrophys.J. 253 (1982). 798). We find that for all

cases from slow to intermediate accretion rates, nuclear burnings are ignited at the helium

layers. As a consequence, carbon deflagration would be triggered for lower accretion rate

than dM/dt ≃ 4 × 10−8M⊙ yr−1 which has been believed to be the lower limit of the

accretion rate for the deflagration supernova.

Furthermore, off-center helium detonation should result for intermediate and slow

accretion rates, and the region of carbon deflagration for slow accretion rates disappears.

Finally, we should emphasize the importance to investigate the supernova explosion

keeping pace with both the advanced numerical simulations, deep understanding of basic

physics, and the updated observation.

ii

Contents

1 Introduction 1

1.1 Stellar evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Nucleosynthesis in the universe . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Supernova explosion observation and theory 7

2.1 Classification of supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Thermonuclear supernova . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Core-collapse supernova . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Thermonuclear explosion model Ia supernovae and a new triple alpha

reaction 12

3.1 Ignition curves and helium flash on the accreting white dwarfs . . . . . . . 17

3.2 Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Core collapse supernovae 21

4.1 Observations of non-spherical supernova explosion and neutron star . . . . 22

4.2 supernova neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Role of neutrino effects for hydrodynamics . . . . . . . . . . . . . . 26

4.2.2 Neutrino transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

i

4.2.3 Modified leakage scheme . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Accuracy of neutrino transport scheme . . . . . . . . . . . . . . . . . . . . 32

4.3.1 Isotropic diffusion source approximation results . . . . . . . . . . . 32

4.3.2 Comparison between leakage scheme and Boltzmann equation solver 33

4.4 Standing Accretion Shock Instability (SASI) . . . . . . . . . . . . . . . . . 39

5 Nucleosynthesis 40

5.1 Origin of elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 R-process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 basic physics of nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Sites of the r-process 45

6.1 Neutron star merger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.2 Neutrino driven wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.3 Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7 Adiabatic hydrodynamic simulations and r−process nucleosynthesis 49

7.1 Review of simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.2 MHD simulation without neutrino effects . . . . . . . . . . . . . . . . . . . 51

7.2.1 Initial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.2.2 Explosion models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.3 Networks for r -process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.4 Simulations of the r-process nucleosynthesis . . . . . . . . . . . . . . . . . 57

8 Simulations with neutrino effects 67

8.1 MHD equations and physical quantities . . . . . . . . . . . . . . . . . . . . 67

8.2 Initial models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.3 Explosion models and distribution of electron fraction . . . . . . . . . . . . 77

8.4 Nuclear reaction network for the r process nucleosynthesis . . . . . . . . . 80

ii

9 Discussions 82

10 Summary 84

11 Future Work 86

iii

1 Introduction

Supernova event has occupied the established status in astronomy and astrophysics. This

is because the phenomenon presents enormous subjects to be studied by almost every

kinds of scientific fields. In the followings, we introduced briefly some crucial subjects

closely related to our research.

1.1 Stellar evolution

Stellar objects are formed from the contraction of interstellar gas which contains hydro-

gen. Protostars, which contraction of interstellar gas form, continue the contraction and

it raises the inner temperature. When the inner temperature becomes high enough to

begin hydrogen burning, the contraction ceases. The stars in this stage are called the

main sequence. In this stage, hydrogen burning produces He. The resulting He core

temperature is not high enough to start He burning at first but He core contraction raises

the temperature. This He burning makes the thermal pressure and the contraction cease.

In this stage, H burning occurs at the bottom of H layer. H burning and He burning

occur meanwhile and H shell burning make outer layer expand. This stage is called red

giant. For massive stars above 10M⊙, nuclear burnings continue until Fe core is built,

which is the most stable nuclei, in the center of a star. Light star (M < 8M⊙) result

in nuclear burning before the synthesis of Fe and the star finally evolves to white dwarf

having ONeMg or CO core. Massive stars above 10M⊙ result in supernova explosions and

1

subsequently neutron star or black hole are left.

1.2 Supernovae

Supernovae are important astronomical events because they are critically related to neu-

trino, gravitational wave, nucleosynthesis and so on. Supernovae are the most bright and

energetic event in the universe as the astronomical events. We observe several hundreds

of supernovae in a year. SN1987A (Fig. 1.1) is one of the most famous and remarkable

contributer to the development in modern astronomy. 1987A is the first supernova ob-

served in visible radiation in large mageranic cloud. 1987A gave us the opportunity for

observation of near supernova. 1987A also gave us neutrino information. The neutrinos

from 1987A was observed at KAMIOKANDE, IBM, Bakson. It is the first observation of

neutrinos from supernova and the first step of neutrino astronomy. Although they sud-

denly begin to shine brightly, they are not new stars in their origins. They are explosive

events of final fate for massive and low mass stars at the end of their life. There are

some types in supernovae, and the origin is related to both the evolution of massive stars

and the evolution of low mass stars, which leads to the formation of white dwarf. There

are two types of supernova scenario. One is core collapse supernovae and the other is

thermo nuclear reaction supernovae. Massive star having more than 8M⊙ collapse for self

gravity at end of the stellar evolution. This collapse cause shock and supernovae occur.

After supernova occurs, black hole or neutron star remains. Fig. 1.2 shows Which object

remain after supernova. The object depends on metalicity and mass of the star [1].

The other supernovae scenario relate to the binary of red giant and Carbon-oxygen

white dwarf. In binary system, He accrete from red giant star to white dwarf surface.

If accretion rate is high, He burning occur and synthesize Carbon. When the mass of

carbon-oxygen white dwarf core exceed Chandrasekhar mass (∼ 1.4M⊙), white dwarf

collapse and supernovae occur.

2

Fig. 1.1: Supernova 1987A. Left panel shows the large magelanic cloud, where shown as

the pictures of the figures before and after the appearance of supernova. Right panel in the

recent photograph by HST, where a bright around the supernova has been seen. The sign is

flared up by the supernova shock wave. http://www.aao.gov.au/images/captions/aat050.html,

http://hubblesite.org/gallery/album/pr2004009a

3

Fig. 1.2: A schematic scenario for the fate of massive stars from 8M⊙ to 300 M⊙ on the plane

of metallicity vs, initial mass for the early main sequence stage [1]. BH indicates the black hole

which would be left after the core collapse. Neutron star could be left after an explosion, but

if depends on not only the mass of a progenitor but also a mass loss rate. It should be noted

that this classification is based on the calculations of stellar evolution of spherical stars, which

include enormous simplifications of physical and dynamical evolution.

4

1.3 Nucleosynthesis in the universe

Many elements exist in universe but nucleosynthesis processes are some types [2]. Fig. 1.3

shows solar system abundance and nucleosynthesis processes. Origin of the elements goes

back to the epoch of the Big-Bang nucleosynthesis (BBN). In BBN, elements up to Li and

Be can be synthesize mainly. At present, the heavier elements (A > 12) are synthesized

by nuclear fusion in stars. In the stellar core, nuclear burning synthesize mainly elements

lighter than Fe during the stellar evolution. Synthesis of more heavier elements relate

to neutron capture during stellar and high energy explosive astronomical sites such as

supernova explosions.

1.4 Outline of this thesis

We explain two supernovae scenario, core-collapse and thermonuclear explosion, in chapter

2. In chapter 3, we introduce newly triple alpha rate estimated by Ogata et al. 2009 [3]

and its effects for thermonuclear explosion. We focus on core collapse supernovae in

chapter 4. Core collapse supernova is one of r-process sites as we describe in chapter 5

and chapter 6. In chapter 7, we review adiabatic simulations so far. We investigate MHD

simulations with neutrino effects and r-process nucleosynthesis in chapter 8.

5

Fig. 1.3: Solar system abundances compiled and classified by Anders/Grevesse [2]. The hori-

zontal axis indicates the atomic mass numver and the vertical the mass fraction. U means the

nuclear process in the early universe during a few minutes. Other nuclear processes have been

believed to be responsible for the eruption from stars, which indicate the supernova explosion

and mass loss from advanced stage of stellar evolution.

6

2 Supernova explosion observation

and theory

Supernova is a energetic and important event for astronomical research. We explain

about supernova classification at first and secondary supernova scenarios of type Ia and

core collapse supernova.

2.1 Classification of supernovae

Supernova is a explosive phenomena seen in the end of massive stellar evolution. The

bright of supernovae is correspond to a galaxy and explosion energy is estimated 1051erg.

Several hundreds supernovae are observed in a year. 1987A is one of the most famous

supernovae. The types of supernovae is classified by spectra and light curve [4]. Fig.

2.1 shows observational classification. The first classification is performed by existence of

hydrogen line in their spectra: Type I supernovae do not show H line, while those with

the obvious presence of H lines are called type II. Moreover there are three subgroups in

type I supernovae by the presence of Si and He. Only type Ia has Si lines and type Ib

shows presence of He and type Ic does not have He in others. Type II supernovae are also

divided to some subgroups by their shape of light curve. Type IIP has plateau after light

curve peak. SN1987A belongs to this type supernovae. Type IIL has liner decreasing of

light curve. Type IIb is called chameleon star. Although this type supernovae show the

7

Fig. 2.1: Observational classifications of light curves [4]. Thermonuclear explosion model of

accreting white dwarf in called type Ia supernova. Others are all come from the collapse driven

supernova.

property of type II at first, the peculiarity change as type Ib when time passes. These

type II supernovae depend on the mass of hydrogen layer. Hydrogen in outer layer is

ejected as stellar wind in stellar evolution. If a star which has hydrogen layer explode,

the type of supernovae is Ib. On the other hand, if a star which lost hydrogen layer by

stellar wind occur supernovae, the type is Ic. Type IIb is special. Binary systems relate

to this type. It is considered that a star which lost hydrogen layer by companion star

causes this type supernovae. While supernova classified mainly four types observationally

as above, there are two type of supernovae scenario. These are core collapse type and

thermonuclear type by explosion mechanism.

8

2.2 Thermonuclear supernova

Thermonuclear supernovae occur in binary system of red giant and white dwarf. Red

giant matter out of Roche-Robe, which is the threshold of gravitational force, accrete on

surface of a white dwarf. The matter is composed of hydrogen and helium. Hydrogen

burning occurs easily and synthesize Helium. Therefore, there is helium layer on a white

dwarf. If the helium burning occurs persistently, carbon is synthesized and the mass of CO

core increases. When the mass of CO core exceed Chandrasekhar mass, CO core collapse

and/or carbon ignite. Electron degeneracy pressure sustain CO core. Degeneracy pressure

do not depend on temperature. In spite of increasing temperature by carbon burning, core

pressure do not increase much. Therefore decreasing temperature by expanding core do

not occur. Increasing temperature promote carbon burning reaction in CO core through

convection and this explosive reaction makes white dwarf explode. This is mechanism

for type Ia supernovae. This scenario depend on accretion rate and mass of CO core. If

accretion rate is low, carbon is not produced and temperature at the bottom of helium

layer does not increase. When temperature at the bottom of helium layer increases enough,

helium flash, explosive helium burning, occurs. Helium flash blasts helium layer and a

white dwarf remains. This phenomena have been observed as type .Ia supernova. This

scenario depends on accretion rate and mass of CO core.

1)dM/dt > dMdet/dt

Carbon is synthesized at high temperature He layer and CO core mass increases. When

the mass exceed Chandrasekhar mass, the core collapse and carbon defragration occurs.

2)dMdet/dt > dM/dt > 10−9M⊙/yr

He flash occurs at the bottom of He layer, whose temperature is low, and blasts He layer.

3)dM/dt < 10−9M⊙/yr

The scenario depends on the mass of CO core

We can investigate which scenario a white dwarf pass by description of ignition curve.

9

2.3 Core-collapse supernova

Core collapse supernova has simple scenario. A massive star (M > 8M⊙) collapses at the

end of stellar evolution for the self gravity. Neutrinos emit from inner stellar region at

collapsing stage. When central density increases up to about nuclei density (∼ 1014g/cc),

equation of state becomes stiff drastically and shock formation occurs by bounce. The

shock propagates in outer layer and supernova occurs (Fig. 2.3). Core collapse supernovae

relate to many physical field, for example nucleosynthesis, gravitational wave, neutrino,

and so on. In spite of many simulations, supernova mechanism has not been clarified yet.

Many simulations have been done with considering some effects, for example magnetic

field, neutrino, multi-dimension effect, general relativity and equation of state. These

effects are important for nucleosynthesis, gravitational wave and other researches. Espe-

cially, neutrino effects is critical for supernovae simulations as we write later.

10

Fig. 2.2: Schematic illustration of a progenitor of type Ia supernova: explosion of carbon and

oxygen white dwarf. The explosion is triggered through the accretion of hydrogen and/or helium

from a companion star by the increase of the mass toward the Chandrasekhar mass.

Fig. 2.3: Scenario for the core collapse supernova after the final stage of massive star evolution:

explosion is triggered by the gravitational collapse of a Fe core of presupernova star which is

bounced by the repulsive nuclear potential toward the formation of the shock wave. The last

panel shows an example of supernova remnant of Cas A [5], which has been believed to be the

explosion of a massive star as large las 25M⊙, where shape of an aspherical explosion can be

seen and in particular rather remarkable jet (upper left side in the panel) has been observed.

11

3 Thermonuclear explosion model

Ia supernovae and a new triple alpha

reaction

The triple-α (3α) reaction plays an important role for the helium burning stage on the

stellar evolution of low, intermediate and high mass stars [8, 9, 10], and accreting white

dwarfs [11, 12], where the 3α reaction has been written as a series of the following two

reactions:

α + α −→ 8Be, α + 8Be −→ 12C + γ.

Recently, the reaction rate has been calculated [3], which is very large compared with the

old rate [12, 13, 14] used so far. Fig. 3.1 shows the rate of OKK and NACRE. Although

Fynbo et al. 2005 reported disagreement of triple alpha rate with NACRE (Fig. 3.2), the

disagreement is much less than OKK rate. It should be examined how the new rate affects

astrophysical phenomena, because terrestrial experiments for the 3α reaction are very

difficult such as the study on the QCD phase transition at high densities. We investigate

the effects of a newly calculated 3α reaction rate (OKK rate [3]) on the helium flashes,

which takes place at the center or inside layers of the accreting envelope of the compact

stars. The ignition curve is a fundamental criterion when the nuclear burning begins to

occur and becomes the main energy source to change the stellar structure, where the fates

12

10-80

10-70

10-60

10-50

10-40

10-30

10-20

10-10

100

107

108

109

Rate

log T (K)

NACREOKK

Fig. 3.1: Comparison of two 3α reaction rates between OKK [3] and NACRE [6] (ρNA < σv >

against the temperature). OKK rate is adjusted to accord with that of NACRE at T ∼ 109K.

Significant difference exists up to T ∼ 2 − 3 × 108K. It should be noted that at 107K OKK

is larger than NACRE by a factor of 1026. It should be noted that OKK has been calculated

under the condition of terrestrial condition of vacuum. Therefore, neither the ambient electrons

of screening effect nor other helium particles in the dense matter are not considered.

13

Fig. 3.2: Comparison of the ratios between the 3α rates of NACRE [6] and Fynbo [7] for the

relevant temperature range (T9 = T/109K). The rate by Fynbo et al. [7] is increased at the

low-temperature region of T < 3 × 107K and decreased significantly at the higher temperature

region of T > 2×109K. Dark region comes from the error bars attached to NACRE compilation

14

of the massive stars and/or accreting white dwarfs are determined from the strength of

specified nuclear burnings [11, 15]. While nuclear burning depends on the temperature

severely, the density becomes very important at high density of ρ ≥ 106 g cm−3 and low

temperature of T ≤ 108 K, because the screening effects begin to enhance the reaction

rates.

It has already been shown that the new rate affects significantly the evolutionary

tracks of low mass stars [14]. The evolutions of 1 and 1.5 M⊙ stars have been followed

from the zero-age main sequence through the core He flash/burning. The HR diagram

with use of the OKK rate disagrees considerably with the observations of low mass stars.

It is found that the new rate results in the shortening or disappearence of the red giant

phase, because helium ignites at a much lower temperature and density compared to the

case with the old NACRE rate [13]. Consequently, the OKK rate could be not compatible

with observations. If the new rate is right, we must invoke some new physical processes

such as rotational mixing or other unknown physical effects.

On the other hand, abundances such as helium in globular clusters are open to dispute

[16], which may change the scenario of the stellar evolution of low mass stars. We can see

the effects of the OKK rate on the stellar evolution with use of the ignition properties.

The helium core flash is triggered if the nuclear generation rates εn significantly overcome

the neutrino loss rates εν . Figure. 3.3 shows the ignition curves of the 3α reaction. The

solid line corresponds to the OKK rate and the dashed line is the old rate (labeled with

Nomoto, which is equivalent to NACRE). Also shown are the evolutionary tracks of the

central temperature Tc against the central density ρc for stars of 1 to 40 M⊙ from the

main sequence stages [14, 17, 18]. The evolutions are calculated beyond the core helium

burning with the old 3α reaction rate. We can understand clearly that the helium ignition

occurs at the points of considerably low temperature and density compared to the old

cases [8, 19].

15

7

7.2

7.4

7.6

7.8

8

8.2

8.4

0 1 2 3 4 5 6 7

log T (K)

log ρ (g/cc)

εn=εν: OKK

εn=εν: Nomoto

1M⊙

5M⊙

10M⊙

40M⊙

(Dotter et al. 2009)

Fig. 3.3: Two ignition curves of εn = εν are obtained from the previous rate (dashed line) and

OKK rate (solid line). Evolutionary tracks of (ρc, Tc) with the previous 3α reaction rate are

shown for the stars of 1-40 M⊙. For the star of 1M⊙, the track is taken from Ref. [14]. For

other stars, the evolutions are calculated from the zero-age main-sequence stage [17, 18].

16

3.1 Ignition curves and helium flash on the accreting

white dwarfs

Accreting white dwarfs are considered to be the origin of the Type Ia supernova explosions

[19]. While the white dwarfs are composed mainly of carbon and oxygen (CO), accreting

materials are usually hydrogen and/or helium. Since the hydrogen is converted to helium

through steady hydrogen burning, helium is gradually accumulated on the white dwarf

and the deep layers become hot and dense. Helium flash triggered in the regions composed

of degenerate electrons could develop to the dynamical stage, depending on the accretion

rate dM/dt [19]. The properties of ignition depend on the initial mass of the white dwarf

MC+O for slow accretion rates. If the accretion is rather rapid, dM/dt ≥ 4×10−8M⊙ yr−1,

the carbon deflagration supernova is considered to be triggered at the center [20]. Figure.

3.4 shows the ignition curves concerning helium flash with the old (indicated by Nomoto,

which is equivalent to NACRE) and new 3α (OKK) reaction rates adopted, which gives

(ρc, Tc) for the beginning of helium flashes. Nomoto [11] found that ignition occurs under

the condition of τn = 106 yr, where the time scale of temperature increase by a nuclear

reaction is defined to be τn = Cp T/εn (Cp is the specific heat at the constant pressure

and εn is the nuclear generation rate of the triple-α reaction.). Nomoto [11] computed the

evolution of carbon-oxygen white dwarfs by the Henyey-type implicit-explicit method [8]

from the hydrostatic evolutionary stage of the accretion of helium to the hydrodynamical

stage of thermonuclear explosions of the white dwarfs. Evolutionary tracks of (ρc, Tc)

for cases A-F are taken from the figures in Nomoto’s paper [11] which can be used until

helium flash begins: cases A–F: MC+O (M⊙), dM/dt (M⊙ yr−1); case A: 1.08, 3 × 10−8,

case B: 1.08, 3 × 10−9; case C: 1.28 , 7 × 10−10; case D: 1.35 , 7 × 10−10; case E: 1.13,

4×10−10; case F: 1.28, 4×10−10. Since the convection begins to prevail at the beginning of

helium burning, estimation of the ignition point includes some uncertainties. Therefore,

we draw two ignition curves of τn = 105 yr and 107 yr for OKK in Fig. 3.4 using the same

17

method as described in Ref. [9]. We can find that the helium ignitions occur at a low

density of almost two orders of magnitude if the OKK rate is adopted. Contrary to the

results in Ref. [11], nuclear flashes are triggered for all cases of A-F in the helium layers

accumulated on the CO white dwarfs.

3.2 Concluding remark

The ignition densities that determine the triggering mechanism of Type Ia supernovae will

be changed drastically if we adopt the new 3α reaction rate. We note that the rate revised

by Fynbo et al. [7] does not change our conclusion qualitatively, because the rate is not

different compared with that in the case of NACRE within a factor of 2 for the relevant

range of temperature. Although Nomoto [11] has shown, by simple extrapolation to the

low-temperature side, that the specific nonresonant 3α reaction is crucial in determining

a helium ignition density for accretion that is as slow as dM/dt ≤ 10−9M⊙ yr−1, the

microscopic calculation for the three body problem is found to be much more important

in evaluating the 3α reaction rate. The classification on the basis of the accretion rate on

the dM/dt−MC+O plane proposed by Nomoto [11] and Nomoto et al.[12] will be changed

significantly. It was found that when the density in the burning shell becomes higher

than 2× 106 g cm−3, nuclear flash grows into a detonation or deflagration. We emphasize

that the accretion rates that induce the carbon deflagration supernova become much

lower compared with the standard rate of dM/dt ≃ 4 × 10−8M⊙ yr−1 [8, 11, 12, 19, 20],

because the ignition points for cases C and D shift toward the density of 106g cm−3,

as seen in Fig. 3.4. Contrary to the previous calculations, off-center helium detonation

dominates the mechanism behind Type Ia supernovae for the low helium accretion rate

of dM/dt ≤ 7 × 10−10M⊙ yr−1 without carbon ignition at the center.

We summarize the observational consequences of Type Ia supernovae related to the

new rate.

1. The carbon deflagration model successfully explains the observations of Type Ia

18

7

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8

4 5 6 7 8 9

log T (K)

log ρ (g/cc)

τ=107yr: τ=105yr:OKK OKK

τ=106yr:NomotoA

B

CD

E

F

Fig. 3.4: Ignition curves defined by τn = Cp T/εn yr for helium flashes on the accreting white

dwarfs obtained using the two 3α reaction rates. Ignition curves were obtained from the previous

rate (dashed line, τn = 106 yr: Nomoto) and the OKK rate (solid lines, τn = 105, 107 yrs),

respectively. Evolutionary tracks of (ρc, Tc) are taken from Ref. [11].

19

supernovae. The empirical evidence [20] that carbon deflagration occurs much more fre-

quently compared with the simple detonation is consistent with the new triple-α rate,

because accretion rates responsible for Type Ia supernovae may become lower.

2. For the nucleosynthesis of 56Fe, 56Ni, and Ca-O, the amounts produced may not

be so different compared with W7. However, to elucidate the effects of the new rate on

nucleosynthesis in supernovae, we must calculate several models of lower accretion rates

of dM/dt ≤ 4 × 10−8M⊙ yr−1.

3. There are some observations that the light curves of Type Ia supernovae cannot be

explained by the standard model of W7 [19, 20]. Thus many models different from W7

have been proposed; they are related to the outer layer of exploding white dwarfs. For

example, a late detonation model produces much 56Ni in the outer layer. Since results

obtained using the new rate affects the deflagration model through the accretion rates

and the depths in the accretion layers, models such as late detonation models would be

modified.

The new rate also affects the helium ignition in accreting neutron stars. In particular,

for lower accretion rates, helium burns at lower densities and temperatures, which could

change the epoch of the formation of a helium detonation wave and the modeling of Type

I X-ray bursts [12]. In particular, the mechanism behind superbursts should be studied

again using the new rate, because the amount of 12C plays a crucial role in inducing the

superbursts [22].

20

4 Core collapse supernovae

The energy source of the explosion comes from the gravitational binding energy of the

forming neutron star. Observed kinetic energy of canonical supernova explosions is the

order of 1051erg. This energy is 1% of the total binding energy. The overwhelming fraction

of the binding energy is carried away by neutrinos. Therefore the effect of neutrino is very

important to calculate explosion models.

Two core collapse supernova scenarios have been considered so far. One is called

prompt explosion as described above. The other scenario is called delayed explosion.

Delayed explosion is caused by neutrino effects. High energy neutrinos emit from inner

core after bounce. Although the neutrinos effect as cooling at inner region, neutrino

capture reaction at outer region can give stellar matter energy. Neutrino emission energy

is estimated 1053erg. If only 1% neutrino energy move to stellar matter at outer region,

explosion succeed (Fig. 4.1). Although simulations of Bethe & Wilson [23] succeeded

explosion, there are some parameters of neutrinos (for example, neutrino luminosity and

neutrino mean energy). We need more detailed simulations.

Therefore, not only macrophysics (hydrodynamics, neutrino transport, general rela-

tivity) but also microphysics (neutrino process, neutrino cross section, equation of state)

is important.

Figure. 4.2 shows the results of one-dimensional numerical simulation by Liebendoerfer

[24]. The simulation includes general relativity and neutrino effects with Boltzmann

neutrino transport scheme. Bold line shows the position of propagating shock with general

21

relativity and thin line shows the one with Newtonian. The result shows that explosion

does not succeed for one-dimension.

However, the multi dimensional effects such as convection, rotation and magnetic fields

are expected as keys to understand explosion mechanism.

4.1 Observations of non-spherical supernova explo-

sion and neutron star

There are many observations of supernovae and supernova remnants. Many observational

results show non spherical structure. In particularly, some observations show jet-like

structure (Fig. 4.4). SN1987A also shows jet like structure (Fig. 4.3).

There is a neutron star or black hole in supernovae remnant. Recently, neutron stars

which have strong magnetic field have been observed. These neutron stars are called

magnetar. The origin of magnetic field in magnetar is considered magnetic compression at

the precollapse phase. Magnetic field of magnetar is estimated 1014−15G from synchrotoron

radiation and increasing rotational period of magnetar [25].

Magnetic field effects hydrodynamics as magnetic pressure and causes jet-like explo-

sion. Magnetic field is freeze out and moves with matter under ideal magnetic hydro-

dynamical situation. If there is magnetic field and differential rotation, compression of

magnetic field increase pressure and a jet-like explosion occurs (Fig. 4.5).

Therefore, magnetic field and rotation is important for supernovae simulations.

22

Fig. 4.1: Shock revival due to the delayed heating by neutrinos [23]. The lower dashed curve

is the position of the neutrino sphere, the upper one is the shock.

Fig. 4.2: Shock propagation and damped oscillation after t = 0.1s [24].

23

Fig. 4.3: Growth of supernova jet for the central object in the photograph. The ring-like

structure around the central object can be seen to be flared up by the jet like shock wave. The

bright regions are gradually developed to cover the whole ring. It should be noted that the

central region of supernova shows clearly aspherical shape, which suggests a jet like explosion.

http://hubblesite.org/gallery/album/pr2007010b/

24

Fig. 4.4: Observational evidence of jet propagation from a compact γ-ray source (right figure).

http://www.a.phys.nagoya-u.ac.jp/nuso/2009052.html

Fig. 4.5: Gradually enhanced magnetic pressure to the rotational axis. Magnetic field is shown

schematically, in which initial magnetic field is gotten wound around the axis. A fluid particle

is sticked to the field due to the assumption of the ideal MHD approximation.

25

4.2 supernova neutrinos

We show the importance of neutrino, magnetic field and multi dimensional effects. In this

section, We introduce neutrino effects and neutrino transport in detail.

4.2.1 Role of neutrino effects for hydrodynamics

Bethe & Wilson advocated delayed explosion in 1985 [23]. The basic idea is that neutrinos

emitting from inner region effect cooling at the region but they effect heating at the

outer region through neutrino capture. If stalled shock can stay in heating region, high

energy neutrinos help shock propagate (Fig. 4.1). As described above, one-dimensional

simulations with detailed neutrino treatment do not succeed in explosion. We consider

that the key of explosion is multi dimensional effects [26].

Neutrino effects are very sensitive for supernova explosion simulations. Janka &

Mueller reported the sensitivity of explosion energy for neutrino luminosity [27]. The

result (Fig. 4.6) shows 5% increasing neutrino luminosity artificially in one dimensional

simulation and 15% increasing neutrino luminosity artificially in two dimensional simula-

tion make explosion energy raise up to typical supernova explosion energy (∼ 1051erg).

According to this result, We must treat neutrino as detail as possible. Neutrinos emit

by not only electron capture but also other processes. Neutrino emitting process and

neutrino capture processes are described below:

p + e− −→ νe + n

n + e+ −→ νe + p

e+ + e− −→ νx + νx,

γ −→ νx + νx,

x = e, τ, µ.

Electron and positron capture process, pair creation and plasmon decay effect as cool-

ing. Contrarily, neutrino absorption effect as heating. The rates of these processes depend

26

Fig. 4.6: Sensitivity of the shock propagation [27]. 2D/2.00 means the two dimensional

hydrodynamical calculation and a factor 1052erg multiplied by the neutrino luminosity.

on density and temperature [28]. Fig. 4.7 shows primary neutrino process under the in-

terstellar condition.

Figure. 4.8 shows neutrino situation in Fe core. Neutrino situation reach beta equilib-

rium at near the center of the star because neutrino can not escape easily there in spite

of high neutrino generation rate. The region above equilibrium region is cooling region.

Neutrino generation rates is high and neutrino can escape from this region. Therefore,

neutrino condition can not come to beta equilibrium. There is heating region on cooling

region. In this region, neutrino generation rate is very low because density and temper-

ature is low. Although neutrino generation rate is low, many high energy neutrinos go

through this region and neutrino captures occur.

27

Fig. 4.7: Primary neutrino process against the density [28]. Most dominant neutrino process

for a given density and temperature. TF is the electron Fermi temperature.

4.2.2 Neutrino transport

We showed the importance of neutrino treatment above section. We need to treat neu-

trino transport as accurate as possible. Some schemes for neutrino transport treatment

have been developed so far. The most accurate treatment for neutrino is Boltzmann equa-

tion solver. Although Boltzmann solver is accurate, multi dimensional simulation with

neutrino effect is difficult. For two dimensional simulation, Boltzmann equation is seven

dimensional equation. That is too heavy. Therefore, Other schemes have been developed.

28

Fig. 4.8: Schematic explanation of the region inside the Fe core from neutrino trapped to free

streaming one

We introduce some schemes below.

1) Isotropic diffusion source approximation (IDSA) [29]

Boltzmann equation is solved approximately in IDSA. The basic idea of this scheme is

that accurate energy distribution of neutrino can be described by trapped neutrinos and

streaming neutrinos. Boltzmann equation with relativity is below:

df

cdt+ µ

∂f

∂r+ [µ(

lnρ

cdt+

3v

cr+

1

r)](1 − µ2)

∂f

∂µ+ [µ2(

dlnρ

cdt+

3v

cr) − c

cr]E

∂f

∂E

= j − (j + χ)f − Σ +E2

c(hc)3[

∫Rf ′dµ′ − f

∫Rdµ′]

The most difficult part is integral part in right. This integral part means cross section

effects. Accurate Boltzmann solver calculate the integral part with iteration. The integral

part for perfect trapped neutrino can be calculated easily because scattering neutrino

direction is isotropic.

df t

cdt+

1

3

dlnρ

cdtE

∂f t

∂E= j − (j + χ)f t − Σ, (4.1)

29

where f t, j, (j + χ)f tandΣ means trapped particle distribution, the emitting rate, the

rate of absorption and the rate of converting from trapped particle to streaming particle

respectively.

The integral part for free streaming neutrino is zero. For this situation, Boltzmann

equation can be solved easily in comoving flame with fluid.

df s

cdt+ µ

∂f s

∂r+

1

r(1 − µ2) = (j + χ)f s + Σ (4.2)

where f s means streaming particle distribution. We introduce the results of IDSA simu-

lation to compare with our simulation in later section.

2) Multi-group flux limited diffusion (MGFLD) [30]

This scheme is called MGFLD. Basic idea of MGFLD is transport equation. Transport

equation is below:

1

c

∂I

∂t+ n · I + σI = S (4.3)

where I means intensity of neutrino and S means rate of neutrino generation/capture.

σ means cross section for neutrinos. The cross section depends on neutrino energy. There-

fore, transport equation is deferent from each neutrino energy level. Solving each transport

equation with different energy level improve neutrino treatment. MGFLD need boundary

condition for solving transport.

3) Leakage scheme

Our simulations have been done with this scheme. This scheme includes radiative

transfer idea. There are a border to separate trapped neutrino region from free streaming

neutrino region. The border is called neutrino sphere. Neutrino sphere is defined below:

∫ ∞

Rν

dr

λtot

=2

3. (4.4)

30

Although neutrinos stream freely near or out of neutrino sphere, neutrinos are trapped

deep inner neutrino sphere. We can estimate the time scale of neutrino for escaping from

neutrino sphere. if neutrino scattering occurs by dense matter, the time scale is below:

τdiff =3∆R2

π2cλtot

(4.5)

if neutrinos stream freely, the time scale is estimated:

τfree =∆R

c(4.6)

Effective time scale is larger than another one. Therefore, the time scale of neutrino

for escaping from neutrino sphere is estimated as follow:

τesc = max(∆R

c,

3∆R2

π2cλtot

). (4.7)

We assume that neutrino time dependent in neutrino sphere is described as follow:

dYν

dt= − Yν

τesc

. (4.8)

There is important assumption for leakage scheme. For leakage scheme, neutrino

distribution is Fermi-Dirac distribution everywhere. Leakage scheme is easy to be used

for simple idea and apply multi-dimensional simulation. One of our simulation purpose

is clarifying multi dimensional effects. Therefore, We adopt leakage scheme.

4.2.3 Modified leakage scheme

For our simulation, we modify leakage scheme in two points. First point is about out of

neutrino sphere. For original leakage scheme, neutrinos which is created out of neutrino

31

sphere emit with infinity velocity. In other words, all neutrinos which is created in a

time step do not stay there in next time step. This is inconvenient for neutrino transport

because neutrinos emit with almost light speed out of neutrino sphere. Therefore, we

solve free streaming transport equation there.

∂nν

∂t+ ∇ · (nνc) = S, (4.9)

where nν , S mean neutrino number density and neutrino capturing rate.

Second point is about estimated time scale inside of neutrino sphere. Diffusion time

scale and free streaming time scale of neutrinos is the time scale of moving neutrino

between ∆R. Inside of neutrino sphere, neutrinos move isotropically as trapped neutrino

assumption of IDSA. Therefore, net escaping neutrino is factor of 1/2. In addition to it,

average radial velocity is c/√

3 if neutrinos are isotropic. Therefore, we multiply a factor

of 2√

3 by the time scales.

4.3 Accuracy of neutrino transport scheme

4.3.1 Isotropic diffusion source approximation results

Although Boltzmann equation solver is accurate, multi dimensional simulation is difficult.

Other neutrino transport schemes is necessary for multi dimensional simulations. For the

simulations, accuracy of neutrino transport scheme is important to discuss. We show

IDSA results [29] for comparing with Boltzmann, Fermi-Dirac approximation.

Figure. 4.9 shows neutrino and anti-neutrino fraction of function as radius. Blue line,

red line, and dash line means Boltzmann equation solver, Fermi-Dirac approximation and

IDSA result respectively. According to the results, IDSA is almost accurate. In addition

to IDSA result, the point that Fermi-Dirac approximation is accurate is important. The

point means that neutrino scattering/absorption processes out of neutrino sphere do not

32

break Fermi-Dirac distribution because these processes are too rare. From the point of

view, we can assume Fermi-Dirac distribution as leakage scheme.

4.3.2 Comparison between leakage scheme and Boltzmann equa-

tion solver

In this section, we compare with leakage scheme and Boltzmann equation solver (Liebendoerfer

et al. 2001) for discussing accuracy of leakage scheme. The simulation with Boltzmann

equation solver were carried out with a new general relativistic neutrino radiation hy-

drodynamics code, AGILE-BOLTZMANN, based on a conservative formulation of GR

radiation hydrodynamics in spherical symmetry and comoving coordinates. The initial

model is 3.3M⊙ He core model [10]. Equation of state is Lattimer-Swesty [31]. Out

simulation with leakage scheme were carried out with hydrodynamics code, ZEUS-2D,

newtonian magneto-hydrodynamics code. For our case, Equation of state is, different

from Liebendoerfer et al. 2009, Shen EoS. Therefore, we can not perfectly compare with

the results of Boltzmann and leakage. Hydrodynamics, neutrino generation and neutrino

transport will be described later in detail.

Figure. 4.10 shows Ye distribution of Boltzmann and leakage. For Boltzmann result,

Ye distribution is very steep for radius because increasing temperature by shock promotes

electron capture behind the shock. Ye distribution is almost corresponding to each results

qualitatively.

Figure. 4.11 shows time dependent of neutrino luminosity for Boltzmann and leakage

results. Each results show steep peak after bounce. This is called neutrino burst or

neutronization burst. Neutrino burst is caused disintegrated nuclei by the shock. The

shock disintegrate nuclei and there are many free neutrons and protons behind the shock.

Electron capture rate for free protons is very high. When shock propagates to neutrino

sphere, rapid electron capture occurs and neutrinos escape from neutrino sphere rapidly.

This is neutrino burst mechanism. Neutrino burst does not effect hydrodynamics because

33

neutrino burst continues for very short time and integral neutrino energy is not so large.

Although the peak of neutrino burst is different each other, each results show corre-

sponding results. Neutrino luminosity in later phase is composed of photo neutrino and

plasmon decay because neutrinos do not emit around neutrino sphere for low Ye.

Although our leakage scheme is Newtonian formation, we introduce general relativistic

effect for neutrino heating/cooling finally. Figure. 4.12 shows net heating/cooling rate

with general relativity and Newtonian. General relativity makes gravity strong. This

makes temperature increase. More energetic neutrinos emit from inner region because

neutrino generation rate depends on temperature. This means increasing cooling rate in

inner region. Energetic neutrinos also cause increasing heating rate for large neutrino cross

section. If neutrino luminosity and mean energy of neutrino is same, general relativistic

effect is minus for neutrino heating. Red shift, general relativistic effect, decreases neutrino

energy at outer region and neutrino cross section [32].

34

Fig. 4.9: Comparison of the mean energy (lower panel) and neutrino fraction (upper panel)

at a typical postbounce time of 150ms for different ν-transport schemes [29]. The thick solid

line shows the neutrino abundance with Boltzmann transport. The IDSA leads to a trapped

neutrino abundance (dashed line) and a streaming neutrino abundance (dash-dotted line). They

sum up to the total neutrino abundance (thin solid line), which is comparable to the Boltzmann

result.

35

0

0.1

0.2

0.3

0.4

0.5

0.6

105 106 107 108

Yν,

Ye

radius (cm)

initialYe,12Ye,13Ye,14Yν,13Yν,14

Fig. 4.10: Electron fraction (Ye) and neutrino fraction (Yν against the radius (upper figure)

and Ye using the Boltzmann equation solver [24].)

36

Fig. 4.11: Comparison of neutrino luminosity. Upper figures are obtained by the ’exact’

calculation. Lower are by our leakage scheme.

37

Fig. 4.12: Effects of general relativity on the heating/cooling rate against the radius [24]. The

dashed lines mark the separate heating and cooling rates whose superposition (solid) leads to

the net rate.

38

Fig. 4.13: Revival/growth of the shock wave propagation due to the accretion shock instability.

Left panel indicate the delayed explosion of 3.3 M⊙ helium star due to a neutrino heating. Right

panel illustrates the convection due to hydrodynamical instability caused by late time neutrino

heating.

4.4 Standing Accretion Shock Instability (SASI)

Hydrodynamical instability may be a key for supernovae. Recently, standing accretion

shock Instability (SASI) [33] is focused on. Strong convection occurs behind shock by

SASI. If the convection makes shock stay in heating region, neutrino heating help shock

propagation (Fig. 4.13). The possibility of SASI explosion was reported [34]. The simula-

tion was carried with Newtonian hydrodynamics code (ZEUS-2D) and IDSA. Figure. 4.13

shows shock propagation. Upper solid red line means max radius of the shock and lower

solid red line means average radius of the shock. Max radius of the shock propagate to

outside of Fe core. Although they suggest explosion is succeeded in, mean shock radius is

a few hundreds km at the end of simulation. The result is considered too early to suggest

that explosion occurs.

39

5 Nucleosynthesis

In this section, we explain the origin of elements and r-process.

5.1 Origin of elements

Some processes categorize the origin of elements as we described above. We can recognize

solar system abundance as cosmic abundances. Neutron capture reactions are the most

natural process to synthesize heavy nuclei which are mass number A is larger than that

of Fe group isotopes. Heavy elements can be divided to three different categories. s- and

r-processes are induced neutron capture and p-process is mainly induced by photodisin-

tegrations.

5.2 R-process

R-process contains rapid neutron capture and beta decay process. When temperature is

high in stellar matter, matter is under NSE. Rapid neutron capture occurs after decreasing

temperature. If neutron capture is rapider than beta decay, elements near neutron drip

line is synthesized and mass number raise. After decreasing temperature enough, beta

decay time scale is rapider than neutron capture. Beta decay process increases Z number

of elements (Fig. 5.1).

It has been considered that the origin of heavy neutron-rich elements like uranium is

40

Fig. 5.1: Nuclear chart on which primary nuclei with magic numbers and basic related to the

r-process nuclear processes are attached.

http://www.rarf.riken.go.jp/pub/newcontents/contents/facility/RIBF.html

mainly due to the r-process nucleosynthesis that occurs during the supernova explosions

and/or neutron star mergers. The main issue concerning the r-process research is to

reproduce the three peaks (A ∼ 80, 130, and 195) in the abundance pattern for the

r-elements in the solar system. Among models of the r-process, it has been believed

that supernovae are the most plausible astrophysical site. The explosion is triggered

by the gravitational collapse of massive stars of M > 10M⊙. Since a proto-neutron

star is formed after the explosion, neutron-rich elements seem to be easily ejected by

41

Fig. 5.2: ’R-nuclei’ are compared with calculational results for three typical tracer particles,

which have been expelled by the shock wave propagation (model 5 in Chap. 8) [35].

the supernova shock. Unfortunately all realistic numerical simulations concerning the

collapse-driven supernovae have failed to explode the outer layer outside the Fe-core.

Therefore, plausible site/mechanism of the r-process has not yet been clarified. On the

other hand, explosive nucleosynthesis that produces most elements up to Fe-group nuclei

has been calculated under the assumption that the explosion is triggered from outside the

Fe-core whose location is defined as the mass cut; the calculated abundances from C to

Ge are consistent with the supernova observations and the chemical evolution of galaxies.

However, this situation can not be applied to the r-process due to the large electron

fraction Ye (¿ 0.4) and the low entropy distribution above the mass cut. A specific model

of neutrino-wind with the very high entropy per baryon has been suggested to reproduce

42

up to the third peak in the abundance pattern of the r-process. It should be noted that

this model includes artificial parameters such as mass loss rates and initial conditions

of hydrodynamical calculations. On the other hand, detailed r-process calculations that

include the fission have not been fully performed. If the r-process occurs along the paths

of the neutron-drip line, the fission process should become important.

5.3 basic physics of nucleosynthesis

We give a brief review of thermonuclear reaction rates for applying to astrophysical nu-

cleosynthesis, for more detail, Thielemann et al. [44] and there are many books on astro-

nuclearphysics. There are three main types of reactions in thermonuclear reactions: those

are (1) Decays, photodisintegrations, electron and positron captures and neutrino induced

reactions, (2) Two-body reactions and (3)Three-body reactions. In the second reaction

group, reaction can be expressed following:

a + x −→ b + Y (5.1)

in which a particle a strikes an nucleus X and they are transformed into a new nucleus

Y and a particle b. Here, we assume that a and X are gases with a uniform density, the

reaction rate, r, is given by:

r = (Nav)(σ(v)NX) (5.2)

where na, nX and v are the number of particles a and X, and relative velocity, respec-

tively. In the astrophysical plasma, which is in thermodynamic equilibrium, the relative

velocities follow Maxwell-Boltzmann distribution of speed. Thus

ra,X = NaNX⟨σv⟩ = NaNX(8/pi)2

µ1/2(KBT )3/2

∫ ∞

0

Eσ(E)exp(−E

kBT)dE (5.3)

43

where T, KB and E are temperature of plasma, energy of plasma and Bolztmann

constant, respectively. 3.2.2 Nuclear reaction networks The abundance evolution equation

of element i

dYi

dt=

∑j

N ijλjYj +

∑j

N ij,kρNA⟨σv⟩j,kλjYjYk +

∑j

N ij,k,lρNA⟨σv⟩λjYjYkYl (5.4)

where in the abundance in unit. The numerical factors are determined in the following

way:

N ij = Ni, N i

j,k =Ni

Nj!Nk!, N i

j,k,l =Ni

Nj!Nk!Nl!(5.5)

where the numbers, Nm, are positive or negative and specify the number of isotopes

m created (positive) or destroyed in the considered reaction.

44

6 Sites of the r-process

We explain plausible sites of r-process in this section. It is one of the main goals of nu-

clear astrophysics to understand in which events the elements that constitute our physical

universe are synthesized. Nuclei beyond the iron group have to be formed via successive

captures of neutron capture because of high coulomb barriers for charged particle reac-

tions. The actual astrophysical environment for the r-process, however, is still unknown.

6.1 Neutron star merger

One of r-process sites is neutron star merger. Gravitational wave emits from binary

system of rotating two neutron stars. The timescale of gravitational wave is longer than

one of the revolution. The system changes quasi-statically. Gravitational wave extracts

rotation energy from the binary system and two neutron stars get close with rotating (Fig.

6.1). After neutron stars merger, black hole and neutron rich disk remains. R-process

occurs in this scenario and r-elements are ejected as disk. Such neutron star mergers

occur 10−5/yr/galaxy estimated recently [36]. According to newtonian hydrodynamics

simulations for neutron star merger, the ejected mass is about 10−4M⊙. The synthesized

r-elements depend on some parameters, rotation and Ye. Figure. 6.2 shows synthesized

elements for other Ye input. If Ye is 0.1, heavy elements (A > 130) of solar system

abundance is reproduced. The ejected mass and occurring rate, however, is too low to

effect and reproduce solar system abundance.

45

6.2 Neutrino driven wind

Next r-process site is neutrino driven wind. High energy neutrinos emit from protoneu-

tron star. The neutrinos heat matter on surface of neutron star and the matter is ejected

as neutron star wind. Neutrino driven wind was considered low Ye because the the mat-

ter on surface of proto-neutron star is low Ye. The r−process of neutrino driven wind

depends on entropy, dynamical timescale and Ye. Entropy decides distribution of nuclei

in equilibrium and the ratio of free nuclei. If entropy is high, there are more free nuclei.

Therefore, synthesis of heavier nuclei need high entropy. Dynamical timescale, timescale

for decreasing temperature, can be regarded as timescale of synthesizing seed nuclei. If

the timescale is short, more neutrons remain.

According to recent simulation [38], much neutrino captures occur and neutrino driven

wind reaches proton rich condition (Fig. 6.3). It is suspected for a site of r-process.

6.3 Supernovae

Supernovae is a plausible site of r-process. Figure. 6.3 shows r-process scenario in super-

novae. The falling matter by collapse decrease Ye through electron capture process. The

temperature of matter which is heated by shock reaches T> 9 × 109K and the matter

condition reaches NSE. Timescale of synthesis is shorter than dynamical one under NSE

condition. Therefore, temperature, density, Ye decide composition in NSE.When temper-

ature drops, the condition leave from NSE and rapid neutron capture occurs. heavier

nuclei synthesis need low Ye condition as we write above. Although the condition reaches

low Ye through electron capture process, neutron capture process increases Ye. The bal-

ance decides nuclei mass number. Our simulations have been done so far. Nishimura [39]

reported the possibility of heavy element synthesize as we write later. However, the sim-

ulation was carried out without neutrino effects. In present study, we do it with neutrino

effect using leakage scheme.

46

Fig. 6.1: Hydrodynamical simulation of neutron star merger [37]. R-process could be resulted

at the outer wing bound to the gravitational potential.

Fig. 6.2: Produced abundances are compared to the solar values for artificially fixed values of

Ye [37].

47

Fig. 6.3: Changes of Ye against the radius from the center are shown for neutrino winds expelled

by neutrino heating of 1D calculations [38].

Fig. 6.4: R-process scenario of supernovae.

48

7 Adiabatic hydrodynamic

simulations and r−process

nucleosynthesis

The origin of heavy neutron-rich elements such is mainly due to the r-process nucleosyn-

thesis that occurs during the supernova explosions [40, 41]. The study of the r-process

has been developed considerably keeping pace with the terrestrial experiments of nu-

clear physics far from the stability line of nuclides. In particular, among the three peaks

(80Se,130 Te and 195Pt) in the abundance pattern for the r-elements in the solar system,

the transition from the second to third peak elements has been advocated by nuclear

physicists. [40] Although supernovae are the most reliable astrophysical sites of the r-

process [42], the explosion mechanism is still unclear. It has been believed that the

explosion is triggered by the gravitational collapse of massive stars of M > 10 M⊙ [10]. It

has been the great hope that neutron-rich elements could be ejected during the shock wave

propagation. As far as the one-dimensional calculations, almost all realistic numerical sim-

ulations concerning the collapse-driven supernovae have failed to explode the outer layer

above the Fe-core due to drooping of the energetic shock wave propagation [43]. There-

fore, a plausible site/mechanism of the r-process has not yet been clarified. Explosive

nucleosynthesis that produces most elements up to Fe-group nuclei has been calculated

under the plausible assumption that the explosion is triggered from outside the Fe-core

49

whose location is defined to be mass cut [44, 45]. The abundance pattern from C to Ge are

rather consistent with the supernova observations, the chemical evolution of galaxies, and

the solar system abundances [46, 47]. Although this artificial explosion has succeeded to

select parameters to explain the observed light curves, the method can not be applied to

the r-process due to the large electron fraction Ye (> 0.4) and the low entropy distribution

above the mass cut.

During the development of many hydrodynamical simulations, explosive nucleosyn-

thesis of the jet-like explosion has been investigated with use of the two-dimensional

hydrodynamical code [48], and they found the strong α-rich freeze out region after the

shock wave passage. Two-dimensional (2D) magnetohydrodynamical (MHD) calculations

have been performed under the various initial parameters concerning rotation and mag-

netic field [49, 50]. The ZEUS-2D code [51] has been modified to include a tabulated

equation of state [52], electron captures and neutrino (ν-) transport [49]. (W ). For the

cylindrical profiles of the rotation and magnetic field, it is found that the shape of the

shock wave becomes prolate compared to the case without magnetic fields though de-

tailed studies on ν-transport must be developed. Furthermore, the confined magnetic

fields behind the shock front push the shock wave strongly. It remains unclear whether

some significant differences are found in the hydrodynamical features if the initial models

of stellar evolution include magnetorotational effects or not [53, 54].

In this adiabatic study, we perform the calculations of the MHD explosion of the He-

star of 3.3 M⊙ until the final simulation time tf ≃ 600 ms. For the MHD calculations, two

models are adopted for the initial configuration of rotation and magnetic fields. Contrary

to the previous investigation of the r -process under adiabatic MHD explosion [39], we

include the effects of neutrinos using a leakage scheme [55, 56]. Moreover we investigate

the possibility of the r-process in the MHD jets with use of our large nuclear reaction

network. We find the region that produces the r-process elements under the particular

distribution of low Ye.

50

7.1 Review of simulations

At first, we review our simulation so far [39] to compare with this study. The simulation

was carried out with Newtonian magneto-hydrodynamics code (ZEUS-2D). We accepted

four models for this simulations.

7.2 MHD simulation without neutrino effects

7.2.1 Initial Models

The presupernova model has been calculated from the evolution of He-core of 3.3 M⊙ that

corresponds to 13 M⊙ in the main sequence stage [10]. The mass of the Fe-core is 1.18

M⊙ that is the smallest Fe-core in massive stars obtained from the stellar evolutionally

calculation with the limitation of the spherical symmetry. The edge of the Fe-core that

has steep density gradient is at R = 8.50 × 107 cm from the center. The mass of the Si-

rich layer is 0.33 M⊙ and the layer extends to 5.47× 108 cm above the Fe-core. Since the

central density exceeds 1010 g cm−3 (ρ = 2.79 × 1010 g cm−3) and temperature T9 = 9.04

in units of 109 K, the Fe-core just begins to collapse.

Initial models for the collapse calculations (precollapse models) are constructed by

using the density and temperature distributions of the original Fe+Si core. We adopt

cylindrical properties of the angular velocity Ω and the toroidal component of the magnetic

field Bϕ as follows [49]:

Ω(X,Z) = Ω0 ×X2

0

X2 + X20

· Z40

Z4 + Z40

, Bϕ(X,Z) = B0 ×X2

0

X2 + X20

· Z40

Z4 + Z40

(7.1)

where X and Z are the distances from the rotational axis and the equatorial plane with

X0 and Z0 being model parameters. Both Ω0 and B0 are the initial values at X = 0

and Z = 0. Initial parameters of four precollapse models are given in Table 7.1. The

spherically symmetric case is denoted by model 1. In model 2, the profiles of rotation and

51

magnetic field in the Fe-core are taken to be nearly uniform. We present model 4 as the

case having a differentially rapid rotating core and strong magnetic fields. An intermediate

example, model 3 between model 2 and model 4 is prepared for reference. Since the value

of T/|W | is higher compared to that used in [50] by a few percents, we regard the present

case of T/|W | = 0.5 % as rather rapid rotating stars with the moderate magnetic field. In

all computations, spherical coordinates (r, θ) are adopted. The computational region is

set to be 0 ≤ r ≤ 4000 km and 0 ≤ θ ≤ π/2, where the included mass in the precollapse

models amounts to 1.42 M⊙. The first quadrant of the meridian section is covered with

400(r) × 30(θ) mesh points. To get information of mass elements, five thousand tracer

particles are placed within the region of 0.449 ≤ Ye ≤ 0.49 between 0.8 M⊙ (r = 410 km)

and 1.3 M⊙ (r = 2200 km).

Table 7.1: Initial parameters of precollapse models.

Model T/|W | (%) Em/|W | (%) X∗0 Z∗

0 Ω0 (s−1) B0 (G)

model 1 0 0 0 0 0 0

model 2 0.5 0.1 1 1 5.2 5.4 ×1012

model 3 0.5 0.1 0.5 1 7.9 1.0 ×1013

model 4 0.5 0.1 0.1 1 42.9 5.2 ×1013

MHD equations and physical quantities

Basic MHD equations are enumerated as follows:

Dρ

Dt+ ρ∇ · −→v = 0, (7.2)

ρDv

Dt= −∇P − ρ∇Φ +

1

4π(∇×

−→B ) ×

−→B , (7.3)

∂−→B

∂t= ∇× (−→v ×

−→B ), (7.4)

52

∇2Φ = 4πGρ, (7.5)

ρD(e/ρ)

Dt= −P∇ · −→v , (7.6)

where D/Dt is a Lagrange derivative. ρ is the density, v the velocity, P the pressure, B

the magnetic field, e the internal energy density respectively. The gravitational potential

Φ is solved from the poisson equation.

7.2.2 Explosion models

We perform the calculations of the collapse, bounce, and the propagation of the shock wave

with use of ZEUS-2D in which the realistic equation of state [52] has been implemented by

[49]. We do not include the neutrino transport, since our aim is to clarify the differences

in the nucleosynthesis between spherical and MHD jet explosion. It is noted that the

contribution of the nuclear energy generation is usually negligible compared to the shock

energy. In Table 8.2, our results of MHD calculations are summarized. Eexp is the

explosion energy when the shock reaches the edge of the Fe-core [57]. In model 3, the

explosion is failed due to the specific combination of rotation and magnetic field between

the values of model 2 and model 4; although Eexp still exists, the radial distance from the

center in the generated shock front at the bounce shrinks gradually after a few oscillations

of the front. Therefore, it does not always depend on T/|W | and/or Em/|W | whether the

explosion succeeds or not. In Figs. 7.1 and 7.2 trajectories of tracer particles are shown

for some specified values of Ye. While the jet-like explosion occurs along the equator in

model 2 (Fig. 7.1), collimated jet is emerged from the rotational axis in model 4 (Fig.

7.2). Figure 7.6 shows the density, temperature, and entropy per baryon in kB of the

tracer particles in Fig. 7.2. We find after the jet-like explosion of model 4 that it remains

1.24 M⊙ proto-neutron star inside the radius 300 km accompanying successive accretion

onto the star with dM/dt = 0.43 M⊙ s−1 at t = 0.35 s.

53

0.5

1

0.5 1 1.5

Z/1

08 [cm

]

X/108 [cm]

Ye=Ye= 0.2280.1830.2290.1770.1780.177

Fe-core

Fig. 7.1: Trajectories of each tracer particle from the initial stage (+) to the final stage (•)

of 565 ms during the simulation (model 2). The edge of the Fe-core in the precollapse model is

shown by the thick-dotted line. The values of Ye correspond to those in the last stage of NSE

for each tracer particle [39].

54

0.5

1

1.5

0.5 1

Z/1

08 [cm

]

X/108 [cm]

Ye=0.4490.1830.1580.1760.1820.1700.179

Fe-core

Fig. 7.2: Same as Fig. 7.1 but for the final stage (•) of 507 ms (model 4) [39].

55

Table 7.2: Calculated quantities that are crucial in the r-process.

Model tb tf T/|W |f Em/|W |f E∗exp Y ∗

e Mej/M⊙ Mrej/M⊙

model 1 114 424 0 0 2.050 0.272 1.97 × 10−1 5.64 × 10−2

model 2 132 565 6.2 0.042 0.728 0.177 4.71 × 10−2 9.72 × 10−3

model 3 138 492 6.0 0.143 0.559 – – –

model 4 141 507 6.0 0.130 0.306 0.158 1.65 × 10−2 2.00 × 10−3

7.3 Networks for r-process

We have developed the nuclear reaction network that had been constructed for the r -

process. The full network consists of about 4000 nuclear species up to Z=100. We include

two body reactions, i.e., (n, γ), (p, γ), (α, γ), (p, n), (α, p), (α, n), and their inverses. This

network contains specific reactions such as three body reactions, heavy ion reactions

and weak interactions. We construct two kinds of the network A and B that consist

of different nuclear data set. For nuclear masses, the experimental data [58] is used if

available; otherwise, the theoretical data by mass formula FRDM [59] is adopted in the

range Z < 83 and/or ETFSI [60] in 8 < Z < 110. Two types of mass formula are

developed, following.

FRDM

This mass formula constructed by the Nilsson-Struntisky model considering effects of

shell and microscopic part. Calculations are based on the finite-range droplet macroscopic

model and the folded-Yukawa single-particle microscopic model

ETFSI

The ETFSI approach is a semi-classical approximation to the Hartree-Fock method

in which the shell corrections are calculated with the integral version of the Strutinsky

theorem. BCS corrections are added with a delta-pairing force.

Most reaction rates are taken from the compilation REACLIB [61, 62] that in cludes

56

experimental and theoretical data for the reaction rates and partition functions with use

of FRDM(NETWORK A) or ETFSI (NETWORK B). Reaction rates for Z > 83 that are

not available in REACLIB are take from ref.

The same fission data is adopted for both NETWORK A and B.

(a)Spontaneous fission:

Ezperimental half lives and branching ratios of spontaneous fission are taken from

ref. While theoretical formula of half life [63, 64] with empirical fission barrier [65, 66]

is adopted for nuclei whose half lives are not known experimentally for all nuclei of both

N > 155 and A > 240, the life times of the decay are set to be 10−20s.

(b) β delayed fission

Branching ratios of β-delayed fission are taken from Staudt et al.1992. [67].

(c) Fission yields

Empirical formula is adopted about decay products. Since many charged particles

participate in the nucleosynthesis during the explosion, we have included the screening

effects for all relevant reactions [68]. We also use theoretical weak interaction rates that

are the function of the density and temperature [69, 70].

7.4 Simulations of the r-process nucleosynthesis

During the explosion, the temperature exceeds 1010 K around the layers of the Si+Fe

core, where the region of the nuclear statistical equilibrium (NSE) is realized as shown in

Fig. 7.6. Therefore, we follow the change in Ye of the ejected tracer particle due to the

weak interactions of electron/positron captures, and β±-decays until the last stage in the

calculation of NSE. We set this stage to be T9 ∼ 9; afterward the temperature decreases

in time as shown in Fig. 7.6. The change in Ye without the effects of neutrino interactions

is calculated from the relation

dYe

dt=

∑all i

[λ+ − λ−] yi, (7.7)

57

where for the abundance yi, λ+ consists of the β− and positron capture rates, and λ−

consists of the β+ and electron capture rates, respectively. Time evolutions of Ye relevant

to the r-process are shown in Fig. 7.3 (left panels). The trajectories of tracer particles with

0.177 ≤ Ye ≤ 0.44 are depicted in Fig. 7.1 for model 2 and those with 0.158 ≤ Ye ≤ 0.46

in Fig. 7.2 for model 4, respectively; the values of Ye indicate those of the last stage in

the NSE calculation. In model 4, the polar region is ejected having rather high value of

Ye ≃ 0.45. The lowest value of Ye ≃ 0.16 is discovered from around the region inclined at

20 – 30 degrees from the rotational axis. If we include the neutrino captures by nucleons

and nuclei in equation (7.7), the decrease in Ye would be modified. We discuss and

estimate uncertainties concerning the change in Ye later.

Thereafter, using the compositions obtained from the last NSE stage and the profiles

of the density and temperature during the explosion, we perform the r-process nucleosyn-

thesis with the nuclear reaction network described above. We remark that after the last

stage in NSE, the changes in Ye are obtained from the calculations by the full network.

After the time t0 ≈ tf with ρ = ρ0 and T = T0, both the temperature and density are

extrapolated to t = 2.0 s (T9 ∼ 0.1) according as ρ = ρ0e−at and T = T0e

−bt, where a ∼ 3b

[71]. To check the effects of the expansion law on the r-process nucleosynthesis, we adopt

the functional forms of the free expansion: ρ = ρ0 exp(−t/τexp) and T = T0 exp(−t/3τexp).

Here, the expansion time scale is assumed to be τexp = 446/√

ρ0 s [72].

Figures 7.3 (right panels) show the ejected mass in M⊙ against Ye in the range 0.15 ≤

Ye ≤ 0.46. For the spherical explosion, materials with 0.272 ≤ Ye ≤ 0.46 is ejected. The

ejection for Ye < 0.4 occurs from inside the Fe-core in the range of R = 660 − 767 km.

On the other hand, ejection occurs in the direction of the equator with 0.177 < Ye < 0.44

for model 2. As shown in Fig. 7.2, materials with 0.158 < Ye < 0.46 are emerged for

the jet-like explosion along the rotational axis (model 4). For both models 2 and 4, the

ejection for Ye < 0.46 comes from the Si-rich layer. We recognize that as against the

spherical explosion, jet-like explosion of model 4 decreases Ye significantly.

58

We calculate the r-process nucleosynthesis during the MHD explosion using NET-

WORKs A and B. Since Ye decreases to 0.158 at the last stage of NSE, the r-process paths

reach to the neutron drip line. Figure 7.4 shows the comparison of the solar r-process

abundances with obtained abundances. Generally, compared to the spherical explosion

the jet-like explosion results in the increase of the nuclei for A ≥ 120. The reproduction of

the peaks in the distributions of the r-elements depends on the decrease in Ye during the

early phase of the explosion (T9 ≥ 9). In model 1, the produced r-elements are not enough

to explain even the second peak in the r-process pattern. In model 2, although the second

peak is reproduced well, the amount of the produced r-elements are too small to explain

the third peak. For model 4 we succeed in making the global abundance pattern of the

r-elements from the first to the third peak. To check the extrapolation to ρ and T after tf ,

we show the abundance patterns with the thick dotted line in Fig. 7.4 that is calculated

under the assumption of the free expansion; the curve of FRDM in Fig. 7.4 is described

by the thin line. Since the case of the free expansion causes the rapid decrease in ρ and T ,

the abundance flow reaches to the neutron rich side. For example, the temperature of the

mass particle # 1 that produces three peaks in the r-element distribution is lower by a

factor of two at t ∼ 1.2 s than that for the original extrapolation. On the other side, The

abundance peaks depend on both the β-decay rates and the location of the neutron drip

line. Though the global distribution of the produced r-elements do not depend on the

expansion law, we need to continue MHD calculations after tf to get quantitative results

under the present parameter ranges.

Moreover, we find that the fission cycling leads to the normal r-process nucleosynthesis

based on (n,γ) ­ (γ,n) equilibrium accompanied with β-decays for the low Ye region

(Ye < 0.2). On the other hand, distribution of final products are found to be much

sensitive to the mass formula as seen in the differences between Fig. 7.4. This is because

the global r-process pattern and the profiles of the peaks in the abundance pattern depend

on the β-decay rates that have been calculated using the mass formulae.

59

In the present investigation, as the first step we ignored the effects of neutrino trans-

port. It is known that neutrinos not only take off the explosion energy but also are

captured by nucleons/nuclei significantly.

In order to estimate the neutrino absorption rates, the time evolution of the neu-

trino luminosities/energies is the indispensable information, which we cannot obtain in

the current purely-adiabatic simulations. Since there are no calculations employing 2D

neutrino transport with MHD, all we can do is to estimate them from the state-of-the-art

1D neutrino transport calculations. We take these values from Liebendoefer et al.2001.

[24] because the same progenitor model was adopted. Using the data obtained in our

numerical simulation of model 4 such as the density, the proton/neutron fractions, and

the radius, we can estimate the neutrino absorption rates according to the prescription

described in [73].

We add following reactions to NSE calculation.

νe + n −→ p + e− (7.8)

νe + p −→ n + e− (7.9)

Therefore, the evolution equation of Ye is modified to following:

λνen = 4.83Lνe,51(ϵνe,MEV + 2∆MEV + 1.2∆2

νe,MEV

ϵνe,MEV

)r−26 s−1 (7.10)

which need neutrino quantities which are luminosity and mean energy.

We adopt the neutrino quantities from Liebendofer et al. 2001, which is similar cal-

culation depend on the same stellar pre-collapse models. In Fig. 7.5 we show the changes

in Ye with and without neutrino reactions, where we give the cases for the total neu-

trino reactions multiplied by a factor of 0, 0.05, 0.1, 0.2, 0.5 and 1. As a result, if the

anisotropic effects of neutrino emissions are not included, the neutrino absorption rates

60

would become much higher compared to the emission rates in the later phases after the

core-bounce. Since Ye does not decrease as before, the r-process does not occur.

However, as easily seen, there exist inconsistencies in the above estimation. As sug-

gested in the previous studies [74, 75], the neutrino emissions/absorptions become highly

anisotropic when the iron core rotates rapidly. Anisotropies in the neutrino fluxes (the

ratio of the neutrino flux along the rotation axis to that in the equatorial plane) have been

remarked to be a significant factor up to 2. Here it is noted that the neutrino capture

rates are proportional to the neutrino luminosities. Note that the particles producing

the 3rd peaks in our simulations stay for a while due to convection near the regions in

the equatorial plane. When we boldly extrapolate the fact about the asymmetry of the

neutrino radiation to our results, the neutrino luminosities near in the equatorial plane

would be reduced, which may work for reducing the neutrino capture rates in the region.

Since we cannot estimate the effects of anisotropies on the neutrino reactions, we suggest

only the effects with the limitation of the spherical model as seen in Fig. 7.5; if the total

neutrino reactions are reduced by 80 %, r-process may proceed.

Furthermore, if we select another initial models having parameters different from those

given in Table 7.1, values of Ye lower than those obtained in §7.4 would be obtained.

Fission should be included in the r-process calculation for the situation of very low Ye

such as a neutron star merger with 0.05 < Ye < 0.15 [76, 77]. To draw a robust conclusion

to these issues, multi-dimensional radiation MHD simulations are indispensable, which

we are currently preparing for.

As shown in Table 7.2 ejected mass of the r-elements amounts to Mrej = 2× 10−3 M⊙

in case of model 4 that is one tenth of the total ejected mass. Since the ejected mass of

oxygen MO is 0.15 M⊙ for the spherical explosion of the 3.3 M⊙ helium core [10], the ratio

Mrej/MO ∼ 10−2 is large by a factor of 100 compared to the corresponding solar ratio of

1.6×10−4. It has been pointed out for the explosion of massive stars the underproduction

of the p-nuclides with respect to oxygen, when normalized to the solar values [78]. However

61

it is found that as far as the case in the 3.3 M⊙ helium core, produced p-nuclides are free

from the problem of the underproduction [78]. Considering uncertainties neglected in the

present simulations and the differences in the nucleosynthesis between the spherical and

jet-like explosion, the problem of the overproduction in the r-elements should be worth

while to examine in detail.

62

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

Ye

time [s]

Ye=0.2910.2840.3020.3560.445

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

Ye

time [s]

Ye=0.2280.1830.2290.1770.1780.177

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

Ye

time [s]

Ye=0.4490.1830.1580.1760.1820.1700.179

10-5

10-4

10-3

10-2

10-1

0 0.1 0.2 0.3 0.4

ejec

ted

mas

s [M

]

Ye

o.o.

10-5

10-4

10-3

10-2

10-1

0 0.1 0.2 0.3 0.4

ejec

ted

mas

s [M

]

Ye

o.o.o.o.

10-5

10-4

10-3

10-2

10-1

0 0.1 0.2 0.3 0.4

ejec

ted

mas

s [M

]

Ye

o.o.o.

Fig. 7.3: Time evolutions of Ye (left panels) and ejected mass vs. Ye (right panels) for each

tracer particle in model 1 (upper), model 2 (middle), and model 4 (lower) [39]. Rather uniform

distribution of Ye has been obtained due to adiabatic assumption.

63

10-10

10-9

10-8

10-7

10-6

10-5

50 100 150 200 250

Abu

ndan

ce

Mass number

Fig. 7.4: Abundances obtained from model 4 with use of NETWORK A; the thick dotted line

is the case of the the free expansion [39]. Extrapolation after t = tf do not affect significantly

the abundance pattern.

64

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Ye

time [s]

No Neutrino1

0.50.20.1

0.05

Fig. 7.5: Time evolutions of Ye due to the inclusion of the neutrino absorption reactions(7.10)

multiplied by a factor of 0, 0.05, 0.1, 0.2, 0.5, and 1. The effects of ν increase the value of Ye

tremendously if we apply results of spherical explosion model, where explosion has been failed

[39].

65

106

107

108

109

1010

1011

0 0.1 0.2 0.3 0.4 0.5

dens

ity [g

/cm

3 ]

time [s]

109

1010

1011

0 0.1 0.2 0.3 0.4 0.5

tem

pera

ture

[K]

time [s]

100

101

102

0 0.1 0.2 0.3 0.4 0.5

entr

opy

[kB/b

aryo

n]

time [s]

Fig. 7.6: Time evolution of the density, temperature, and entropy per baryon in kB. Each

curve corresponds to that in Fig. 7.2 (model 4).

66

8 Simulations with neutrino effects

There are some problem for nucleosynthesis simulation in Nishimura et al.2006 as follows:

1) It is adiabatic simulation without neutrino effect.

2) Network include neutrino effect qualitatively.

3) Neutrino luminosity is the result 1D simulation [24].

Therefore, we calculate MHD hydrodynamics simulations with neutrino effect using

leakage scheme.

8.1 MHD equations and physical quantities

Basic MHD equations are enumerated as follows:

Dρ

Dt+ ρ∇ · −→v = 0, (8.1)

ρDv

Dt= −∇P − ρ∇Φ +

1

4π(∇×

−→B ) ×

−→B , (8.2)

∂−→B

∂t= ∇× (−→v ×

−→B ),∇2Φ = 4πGρ, (8.3)

ρD(e/ρ)

Dt= −P∇ · −→v + Q+ − Q−, (8.4)

where D/Dt is a Lagrange derivative. ρ is the density, v the velocity, P the pressure,

B the magnetic field, e the internal energy density, Q+ and Q− are the neutrino heating

67

and cooling rates, respectively. The gravitational potential Φ is solved from the poisson

equation.

Neutrino luminosity Lν can be estimated from the average ν-energy ϵν,esc (see Eq. (8.17)

later) that escapes freely:

Lν =

∫V

ϵν,escnν

τesc

dV, (8.5)

where nν is the ν-number density and τesc is the escape time for a neutrino to reach the

ν-sphere Rν that is obtained from the leakage scheme [74]. Rν is defined by using the

total neutrino’s mean free path λtot as follows:∫ ∞

Rν

dr

λtot

=2

3. (8.6)

Contrary to the original leakage scheme, we do not adopt free stream approximation for

neutrinos outside Rν . We include the effects of neutrinos, because for each time step of

∆t, neutrino runs by c∆t, which is typically 104cm during the simulation of explosion.

Q+ and Q− are evaluated from [23, 79]

Q+ = σabnnLν

4πr2, Q− = ϵν,esc

nν

τesc

, (8.7)

where nn is the number density of free neutrons and σab is the ν-absorption cross section

by free neutrons that is the most important heating source.

The change in Ye that is governed by ν-interactions with protons and/or nuclei is given

by

dYe

dt= −ΓpYp − ΓNYN . (8.8)

The electron capture rate by a proton (p + e− −→ n + νe) with Qp = 1.3 MeV is

Γp =1

2π3~G2

F C2V (1 + 3a2)

(h3c3)2Ip, (8.9)

Ip =

∫ µe

Q

dEeE2eE

lνfe(1 − fν). (8.10)

Here l = 2, Q = Qp + µν , and fα is Fermi-Dirac distribution of particles α (= e, ν)

fα =1

1 + e(ϵα−µα)/Tα, (8.11)

68

where ϵα and µα are the single particle energy and the chemical potential, respectively, in

units of the Boltzmann constant. We note that outside the equilibrium region of neutrinos

and baryons, significant thermal deviation comes out between them, i.e., Tν = Tm. GF is

the Fermi coupling constant, CV = 0.97 the pseudo vector coupling constant and CA the

axial vector coupling constant. We set the ratio |CA/CV | = 1.27. The capture rate by a

nucleus of the atomic number Z with the Q-value QN is

ΓN =12

7

1

2π3h

G2F C2

A(Z − 20)

(h3c3)2IN , (8.12)

(8.13)

where IN is obtained from Eq. (8.10) with the values of l = 2 and Q = Qn + µν . Note

that this capture process is inhibited above the neutron number N = 40 due to the effects

of shell-blocking. [80, 81]

Energies of emitted neutrinos by the individual electron captures are [82, 83]

Eν,p =Jp

Ip

, Eν,N =JN

IN

, (8.14)

where Jp and JN are got from Ip and In with l = 3, respectively. The average energy of

neutrinos emitted by electron captures are written as

ϵν =Eν,pYp + Eν,N YN

Yp + YN

, (8.15)

where Yp, YN and Ye are the number fractions of protons, nuclei and electrons, relative

to baryons. This average energy is added to obtain the ν-energy density at the next time

step of our simulation.

Inside the ν-sphere, Tν and µν are evaluated in terms of nν and the ν-energy density

eν

nν =T 3

ν

(~c)3

∫4πϵ2

νfνdϵν , eν =T 4

ν

(~c)3

∫4πϵ3

νfνdϵν , (8.16)

where the Fermi-Dirac distribution is assumed for neutrinos. Outside the ν-sphere, neu-

trino radiation can be approximated as the black body with µν = 0. The average ν-energy

69

is written as

ϵν,esc =F3

F2

Tν,sp, (8.17)

where Tν,sp is the ν-temperature at the ν-sphere and the Fermi integrals, F2 and F3, are

obtained from

Fi =

∫ ∞

0

xi

1 + exdx, (8.18)

where index i takes 2 and 3.

If Tν ≥ Tm inside the ν-sphere, the baryon energy density em increases due to the

energy flow from neutrinos

dem

dt=

c

λν

eν , (8.19)

where λν is the mean free path of neutrinos [84]. In this region, we replace Tν,sp by Tm to

get ϵν,esc in (8.17) to avoid numerical problems, which would be underestimate of ν-energy.

The similar procedure can be applied to anti neutrinos of the reaction, n+e+ −→ p+νe,

with the substitutions

µe −→ −µe, µν −→ µν , Tν −→ Tν . (8.20)

We include the effects of other neutrino processes. They are expressed as,

e+ + e− −→ νx + νx,

γ −→ νx + νx,

where three flabors of neutrinos, x = e, τ , and µ. These processes are important for the

late stage of the explosion. The most neutrino luminosity for this stage is governed by

the processes. The reaction rates of these processes are obtained by Ref. [85]. Electron

neutrinos emitted contribute both the neutrino cooling and heating. On the other hand,

µ and τ neutrinos do only neutrino cooling. The emission rate of νe and νe by electron-

positron pair annihilation is given by

Ree(νe, νe) =(C1 + C2)νe,νe

36

σ0c

(mec2)2ϵe−ϵe+ · (1 − fνe(ϵ))ee · (1 − fνe(ϵ))ee (8.21)

70

where σ0 = 1.76 × 10−44cm2, and ϵe−/ϵe+ indicate electron/positron energy density. The

weak interaction constants are (C1 + C2) = (CV − CA)2 + (CV + CA)2 and (1 − fνe(ϵ))ee

means blocking factor in the neutrino phase space. This factor is approximately taken

into account by

(1 − fνi(ϵ))ee = (1 + exp(−(

1

2

F4(ηe)

F3(ηe)+

1

2

F4(−ηe)

F3(−ηe)− ηνi

))−1. (8.22)

Fn(η) means Fermi integral (n = 3 and 4),

Fn(η) =T n+1

(~c)n+1

∫ ∞

η

xn

1 − ex−ηdx (8.23)

where η = µe/T .

For the production of (νµ, νµ), and (ντ ,ντ ), the corresponding rate is

Ree(νx) =(C1 + C2)νx,νx

9

σ0c

(mec2)2ϵe−ϵe+ · (1 − fνx(ϵ))

2ee (8.24)

where (C1 + C2)νx,νx = (CV − CA)2 + (CV + CA − 2)2.

The rate of creation of νe or νe by the decay of transversal plasmons can be written with

sufficient accuracy as

Rγ(νe, νe) =π3

3α∗C2V

σ0c2

(mec2)2

T 8

(hc)6γ6exp(−γ)(1 + γ)(1 − fνe(ϵ))γ(1 − fνe(ϵ))γ (8.25)

and the corresponding rate for producing νx is

Rγ(νx, νx) =4π3

3α∗ (CV − 1)2 σ0c2

(mec2)2

T 8

(hc)6γ6exp(−γ)(1 + γ)(1 − fνx)γ. (8.26)

Here, α∗ is the fine-structure constant, α∗ = 1/137.036, γ = 5.5565 × 10−2√

13(π2 + 3η2

e)

and (1 − fνx)γ is blocking factor:

(1 − fνx)γ = 1 + exp[−(1 +1

2

γ2

1 + γ− ηνi

)]−1

(8.27)

Outside Rν , we include the effects of ν-radiation terms in Eq.(8.8),

dYe

dt= −ΓpYp − ΓNYN +

c

λν

− c

λν

Yn, (8.28)

where λν is the mean free path of electron neutrino and λν is that of anti-neutrino.

71

8.2 Initial models

Table 8.1: Initial parameters of precollapse models.

Model T/|W | (%) Em/|W | (%) X∗0 Z∗

0 Ω0 (s−1) B0 (G)

model 1 0 0 0 0 0 0

model 2 0.5 0.1 1 1 5.2 5.4 ×1012

model 3 0.5 0.1 0.5 1 7.9 1.0 ×1013

model 4 0.5 0.1 0.1 1 42.9 5.2 ×1013

model 5 1.5 0.1 0.1 1 72.9 5.2 ×1013

In all computations, spherical coordinates (r, θ) are adopted. The computational re-

gion is set to be 0 ≤ r ≤ 4000 km and 0 ≤ θ ≤ π/2, where the included mass in the

pre-collapse models amounts to 1.42 M⊙. The first quadrant of the meridian section is

covered with 300 (r)×30 (θ) mesh points (Fe-core plus some amounts of s: layer). To get

information of mass elements, ten thousand tracer particles are placed within the region

of 0.449 ≤ Ye ≤ 0.49 between 0.8 M⊙ (r = 410 km) and 1.3 M⊙ (r = 2200 km).

Table 8.2: Calculated quantities that are crucial in the r-process.

Model tb tf T/|W |f Em/|W |f E∗exp Mej/M⊙ Mrej/M⊙

model 1 111 283 0 0 0.023 - -

model 2 125 311 6.91 0.053 0.127 - -

model 3 129 329 8.74 0.116 0.164 - -

model 4 133 433 8.80 0.142 1.13 0.111 -

model 5 180 624 15.3 0.339 0.484 0.022 5.90 × 10−3

72

2⋅108

1⋅108

2⋅108 1⋅108

Z [c

m]

X [cm] 2⋅108

1⋅108

2⋅108 1⋅108

Z [c

m]

X [cm]Fig. 8.1: Snapshots of tracer particles at t = 100 ms (top) and t = 200 ms (bottom) after the

bounce in model 4.

73

2⋅108

1⋅108

2⋅108 1⋅108

Z [c

m]

X [cm] 2⋅108

1⋅108

2⋅108 1⋅108

Z [c

m]

X [cm]Fig. 8.2: Snapshots of tracer particles at t = 100 ms (top) and t = 200 ms (bottom) after the

bounce in model 5.

74

Fig. 8.3: Contour of Ye over the range 0.1 – 0.5 at final stage of calculation in model 4 (left)

and model 5 (right).

75

0.1

0.2

0.3

0.4

0.5

0.2 0.4 0.6

Ye

time (s)

0.1

0.2

0.3

0.4

0.5

0.2 0.4 0.6

Ye

time(s)

Fig. 8.4: Time evolution of Ye in model 4 (upper) and model 5 (lower). For the model 4, the

explosion resembles to spherical one and all Ye has increased to Ye > 0.4 such as deduced from

neutrino capture process.

76

1.0⋅10-5

1.0⋅10-4

1.0⋅10-3

1.0⋅10-2

1.0⋅10-1

0.05 0.1 0.2 0.3 0.4 0.5

Eje

cted

mas

s (s

olar

mas

s un

it)

Ye

model 4model 5

Fig. 8.5: Ejected mass as a function of Ye. The thick lines indicate the results of model 4. Two

separate distributions of Ye are due to different neutrino irradiation just outside the neutrino

sphere.

8.3 Explosion models and distribution of electron frac-

tion

We perform the calculations of the collapse, bounce and propagation of the shock wave

with use of ZEUS-2D where the realistic equation of state [52] has been implemented

[49]. It is noted that the contribution of the nuclear energy generation is usually negli-

gible compared to the shock energy. Our results of MHD calculations are summarized

in Table 8.2, where Eexp is the explosion energy when the shock reaches the edge of the

Fe-core [57] and Mej is the mass summed over the ejected tracer particles. it does not

always depend on T/|W | and/or Em/|W | whether the explosion succeeds or not. While

77

Fig. 8.6: Density contour over 108 − 1014g cm−3 at t = 200 ms after the bounce in model 4

(upper) and model 5 (lower).

78

10-12

10-10

10-8

10-6

10-4

10-2

100

50 100 150 200 250

mas

s fr

actio

n

mass number

solar:model5:

10-12

10-10

10-8

10-6

10-4

10-2

100

50 100 150 200 250

mas

s fr

actio

n

mass number

solar:model5:

Fig. 8.7: Nucleosynthesis network calculation results using ETFSI (upper) and FRDM (lower).

The third peak is reproduced for both mass formula. However, the case of FRDM is too steep

to compare with the solar pattern, that is, the shell effects may be exaggerated. The last peak

of A > 200 can not be reproduced in both cases.

79

the jet-like explosion occurs along the equator (up to 40 from the equator) in model 4, a

collimated jet is emerged from the rotational axis in model 5 (Fig. 8.2). A proto-neutron

star remains after the jet-like explosion. During the explosion, temperature exceeds 1010

K around the original layers of the Si+Fe core, where the nuclear statistical equilibrium

is realized.

In model 4, the equatorial region is ejected as show in Fig. 8.2 having rather high

value of Ye ≃ 0.50 (Figs. 8.3). In model 5, materials are ejected with the jets along the

polar regions where total angle is collimated around 20 from the axix (see Fig. 8.2) The

corresponding evolutions of Ye relevant to the r-process are shown in Fig. 8.4. The lowest

value of Ye ≃ 0.20 is found around the polar region as seen in Fig. 8.3.

Figure. 8.5 shows the ejected mass against Ye in the range 0.05 ≤ Ye ≤ 0.50. In model

4 the ejection for Ye > 0.40 comes from the Si-rich layer along the equatorial region, which

is attributed to the centrifugal force relative to magnetic one. We recognize that as against

the spherical explosion, Ye decreases significantly for model 5, due to the collimated jet

along the rotational axis.

8.4 Nuclear reaction network for the r process nucle-

osynthesis

We calculate the r−process nucleosynthesis for the explosion model 5. We have devel-

oped the nuclear reaction network that had been constructed for the r−process based on

Ref. [84]. The network has been extended toward the neutron-rich side till the neutron-

drip line. The full network consists of about 4000 nuclear species up to Z = 100. We

include two body reactions, i.e.,(n,γ), (p,γ), (α, γ), (p,n), (α,p), (α,n), and their inverses.

We construct two kinds of the network, called FRDM and ETFSI. The mass formula of

FRDM is constructed by the Nilsson-Struntinsky model considering effects of shell and

microscopic part. ETFSI approach is a semi-classical approximation to the Hartree-Fock

80

method in which the shell corrections are calculated with the integral version of the Struti-

nsky theorem. Most reaction rates are taken from experimental ones if available which

are supplemented by theoretical data with inverse reaction rates and partition functions

with use of FRDM or ETFSI.

Before the nucleosynthesis calculation, we have assumed abundances to be in nuclear

statistical equilibrium state (NSE) as has been done (Ref. [84, 86]) The NSE code is used

just after the temperature drops 1010 K until around 9×109 K. Then, the nuclear reaction

network of the r-process has been operated till the temperature decreases to 2−3×109 K

(t ∼ 600 ms) using the data of the MHD calculations. After that, network calculations

are performed until T ∼ 107 K (t ∼ 10 s) with the method in Ref. [84].

81

9 Discussions

We can indicate the possibility of the r-process nucleosynthesis during MHD explosion in

a massive star of 13 M⊙, where we have examined two pre-collapse models. They have

been constructed changing the plausible distributions of rotation and magnetic fields

parametrically [39].

We include the effect of neutrinos by using leakage scheme. This scheme treats neu-

trino effect approximately and assume Fermi-Dirac distribution for neutrinos. Other

schemes are developed for solving neutrino transport in resent research. For example,

IDSA (Isotropic Diffusion Source Approximation) solves Boltzmann equation approxi-

mately and the result obtained by using this scheme is consistent with one dimension

simulations, where the Fermi-Dirac distribution has been found to be good approxima-

tion. We will use it in future work.

In case of the rapid rotation and strong magnetic field (model 4), barely jet like

explosion is obtained in the direction of the equatorial region (left panel in Fig. 8.6). It is,

however, impossible to reproduce the r-elements even up to the second peak of the solar r-

process abundance pattern, because Ye of the ejected materials distributes in the range of

high values of Ye ≥ 0.4. In model 5, where we have adopted a special initial configuration

of concentrated magnetic field with strong differential rotation, jet-like explosion emerges

in the direction of the rotational axis (right panel in Fig. 8.6). The difference is that

model 5 has larger value of the angular velocity compared to model 4 by a factor of 1.7.

We compared the produced heavy elements with the solar r−elements in Fig. 8.7, where

82

results for two different mass formula are shown. Model 5 may present an appropriate

site for reproducing only the third peak with the first and second peaks underproduced.

Contrary to the negative conclusion against the possible r−process in the previous study

[39], we succeed in reproducing the elemental distributions of the r−elements as far as

the third peak of the solar pattern is concerned. At the same time, we have showed the

possibility for the lower Ye materials to be ejected significantly if the neutrino transport

works appropriately.

Observations of γ-ray bursts associated with supernovae are rare at the present obser-

vations. Therefore, the new site of the r-process to produce significantly only the third

peak of the solar abundance patter should be also rare, which is not to conflict with the

chemical evolution of galaxies. The nuclear process responsible after the third peak would

have some relations to our MHD jet model. Since γ-ray bursts should have continued from

the formation of the first star, a new model beyond our jet model would give a clue of

nuclear cosmo-chronology represented by such as a nucleus of 232Th which half life is as

long as the age of the universe [86].

We could conclude that supernova explosions of massive stars associated with the r-

process could come true if a progenitor has special distributions of rotational/magnetic

fields inside the stellar core if our simple neutrino transport scheme like leakage one can

be applied.

83

10 Summary

In spite of many detail simulations, supernova explosion has not succeeded in 1D simula-

tions. One of the key effect of explosion is considered to be nonspherical effects (rotation,

magnetic field), and neutrino effects. On the other hand, supernova simulations are impor-

tant not only for hydrodynamic study but also nucleosynthesis. Therefore, we calculate

MHD simulation considering neutrino effects, rotation and magnetic field. In addition

to hydrodynamic simulations, r−process nucleosynthesis simulations also is calculated in

this study.

Although it is better to use Boltzmann equation solver for including neutrino trans-

port, Boltzmann equation solver is difficult to use for multi-dimentional simulation. There-

fore, we have used leakage scheme for neutrino transport. Leakage scheme is an ap-

proximate approach for neutrino transport and we add modification about the physical

process just outside neutrino sphere and timescale of neutrino drift instead of original

leakage scheme. At first step, we have compared our 1D simulation results with those

of Boltzmann equation solver for checking accuracy of our leakage scheme. We can not

compare two simulations perfectly because different EoS’s was used for the simulations.

Nonetheless, we have obtained plausible results using a simple leakage scheme.

We have calculated MHD simulations by varying two parameters (the distribution of

rotation and magnetic field). The previous adiabatic simulations without neutrino effects

have succeeded in explosion [39], However, only two models with neutrino effects whose

distribution of magnetic field and rotation concentrated in the central region succeed

84

explosion. Generally, for strong and concentrated rotation model with some magnetic

field, collimated jet like explosion occurs. For weak and concentrated rotation model, large

region around from rotational axis to equational axisesis blasted. This is for magnetic

field effect. For this study, ejected region is different from Nishimura et al.2006. For our

model 4, deep region is ejected in comparison to Nishimura’s model 4. It means that the

composition depends on rotation parameter. The model 5 may present an appropriate

site for reproducing only the third peak with the first and second peaks underproduced.

Contrary to the negative conclusion against the possibility of the r−process in the previous

study [39], we have succeeded in demonstrating the possible r−process.

85

11 Future Work

We have demonstrated in this thesis that the supernova mechanism and nucleosynthesis

remain some crucial problems.This is the reason that whole consistency for the origin of

elements could have not been obtained.We should include new effects in future as follows:

1) Detailed neutrino transport scheme

In the present study, we use a simple leakage scheme for description of neutrino trans-

port. Leakage scheme can describe neutrino transport in very easy way compared to

the Boltzmann equation solver. Neutrino effect, however, is very sensitive as we show in

Fig. 4.6. We need to adopt more detailed neutrino transport scheme. We plan to include

the scheme of IDSA. IDSA is rather easy to apply to multi dimensional simulations. For

difficulty of multi dimensional simulation with Boltzmann equation solver, IDSA simula-

tion will become important role in the near future.

2) Neutrino oscillation

We do not consider neutrino oscillation because transport with neutrino oscillation is

difficult to handle. Since, mean energies of τ , µ neutrinos are higher than that of electron

neutrino, the effect causes significant effects for neutrino heating. Neutrino oscillation

could affect effect neutrino heating rate. We are planning to include the effects in the

transport equation with a simple scheme.

86

3) SASI

SASI has been focused on hydrodynamics simulations. SASI is considered to help the

shock stay increase in heating regions. Although SASI is in a good sense for explosion

mechanism, it may play minus role for heavy nuclei synthesis. It has been reported that a

strong convection occurs behind shock by SASI but the convection effect tend to average

the Ye distribution.

4) Distribution of magnetic field

In our study, we consider only toroidal magnetic field. If we input polar magnetic field as

an initial parameter, a weak rotation model may lead to succeed in explosion and draw up

low Ye matter from deep inside the region. We need to simulate with many parameters.

5) Simulations of other massive star model

We have adopted 3.3M⊙ He core model, because it has been rather easy to succeed in

explosion. We need to simulate for more heavier stellar models. Neutrino luminosity may

rise itself for more massive models. Increasing neutrino luminosity could cause neutrino

capture process furthermore and heavier nuclei may not be synthesized.

6) Magnetar

In our simulation, proto neutron star has very strong magnetic field. This may suggest

the formation of magnetar. Our simulation may show the relation between magnetar and

heavy elements synthesis and this should be checked also from the point of observations

of magnetars.

87

ACKNOWLEDGEMENTS

This is my opportunity to thank all the people who have helped me during the course

of this work. First of all, I wish to warmly thank my supervisor, Professor Masa-aki

Hashimoto giving me opportunity to begin my research theme, his continuous scientific

advices and environment of my undergraduate to graduate course education. I could not

write this thesis without his advices and encouragement. I also want to show my gratitude

to Professor Kenzo Arai, Professor Yoshifumi Shimizu and Professor Shin-ichiro Fujimoto.

I had many fruitful discussions and suggestions. I thank all of the collaborators, Professor

Sho-ichi Yamada, Dr. Kei Kotake. I thank colleague of all the members of our research

group, especially I am grateful to Professor. Yamaoka and Professor. Machida giving me

advises. I am also grateful to Dr. Riou Nakamura, Nobutoshi Yasutake, Tsuneo Noda

and Shin-jiro Kouzuma, Masaomi Ono, Masafumi Shimada, Shohei Mimata, E. P. B.

A. Thushari, Yusuke. Sakane, Hideyuki Tsujimoto, Yasuhide Matsuo, Yukihiro Kikuchi,

Yusuke. Hayashi, Jun. Shimanoe, Tadahumi. Sato.

88

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