Abstract
This thesis presents the results of a consistent meaneld study for
the equation of state (EOS) of the stable baryonic matter containing
npe particles in the core of cold neutron star (NS) and hot protoneutron
star (PNS). Within the nonrelativistic HartreeFock formalism, diﬀerent
choices of the inmedium, densitydependent nucleonnucleon (NN) inter
action have been used. Although the considered density dependent NN
interactions have been well tested in numerous nuclear structure and/or
reaction studies, they give rather diﬀerent behaviors of the nuclear sym
metry energy at high baryonic densities which were discussed in the lit
erature as the stiﬀ and soft scenarios for the EOS of asymmetric NM. A
strong impact of the nuclear symmetry energy to the meaneld prediction
of the cooling scenario for NS and thermodynamic properties of the PNS
matter has been found in our study. In addition to the nuclear symmetry
energy, the nucleon eﬀective mass in the highdensity medium was found
also to aﬀect the thermal properties of hot stable baryonic matter of
PNS signicantly.
Given the EOS of the crust of NS and PNS from the compressible
liquid drop model and relativistic meaneld approach, respectively, the
diﬀerent EOS's of the core of NS and PNS were used as input for the
TolmanOppenheimerVolkov equations to obtain the structure of NS and
PNS in the hydrostatic equilibrium, in terms of the gravitational mass,
radius, central baryonic density, pressure and temperature. For the PNS
matter, both the neutrinofree and neutrinotrapped baryonic matters in
equilibrium were investigated at diﬀerent temperatures and entropy per
baryon S=A = 1; 2 and 4. The obtained results show consistently the
strong impact of the nuclear symmetry energy and nucleon eﬀective mass
on the thermal properties and composition of hot PNS matter. Maximal
iigravitation masses obtained with diﬀerent EOS's for the neutrinofree 
stable PNS at S=A = 4 were used to assess the time of the collapse of a
very massive PNS to black hole, based on the results of the hydrodynamic
simulation of a failed supernova of the 40 M⊙ protoneutron progenitor.
The eﬀective, density dependent CDM3Yn interaction has been shown to
be quite reliable in the meaneld description of the EOS of both the cold
and hot asymmetric NM.
iiiAcknowledgements
First and foremost, I gratefully express my best thanks to my super
visor, Prof. Dao Tien Khoa for his longtime tutorial supervision of my
research study at the Institute for Nuclear Science and Technology (INST)
in Hanoi, ever since I graduated from Hanoi University of Pedagogy. Prof.
Khoa has really inspired me to pursuit research in nuclear physics by his
deep knowledge in teaching and coaching his students and young collabo
rators, and his strict demand on every detail of the research work. I would
also like to thank Dr. Jer^ome Margueron from IPN Lyon for his collabo
ration work in the topic of my PhD Thesis and support of my short visit
to IPN Lyon as well as my attendance at some international meetings in
Europe. I have gained good skills of the nuclear physics research during
my short visits to IPN Orsay and IPN Lyon, and I am deeply grateful to
Prof. Nguyen Van Giai from IPN Orsay for his help and encouragement.
I would like to thank my fellow PhD student, Ms. Doan Thi Loan,
who gave very important contribution to our common research project on
the meaneld description of the equation of state of nuclear matter. We
have accomplished together many interesting tasks and share a lot of joint
memories during the years working at INST as PhD students. I wish to
express my thanks also to my colleagues in the nuclear physics center at
INST, in particular, Dr Do Cong Cuong and Mr. Nguyen Hoang Phuc
for their useful discussions and kind friendship that made the working
atmosphere in our group very pleasant and lively. The helpful discussions
on diﬀerent physics problems with Dr. Bui Minh Loc, a frequent visitor at
INST from University of Pedagogy of Ho Chi Minh City, are also thankfully
acknowledged
The present research work has been supported, in part, by National
Foundation for Science and Technology Development (NAFOSTED) of
ivVietnam, Groupe de Physique Theorique of IPN Orsay at Universite Paris
Sud XI Orsay and IPN Lyon, the Palse program of Lyon University, the
LIA collaboration in nuclear physics research between MOST of Vietnam
and CNRS and CEA of France. I am also grateful to INST and Nuclear
Training Center of VINATOM for hosting my research stay at INST within
the PhD program of VINATOM.
vAbbreviations
NM Nuclear matter
ANM Asymmetric Nuclear matter
EOS Equation of state
HF HartreeFock
BHF Bruckner HartreeFock
D Direct
EX Exchange
NS Neutron star
PNS Protoneutron star
n neutron
p proton
NN nucleonnucleon
IS Isoscalar
IV Isovector
viContents
Abstract . ii
Acknowledgements . iv
Abbreviations vi
List of tables xi
List of gures xix
1 Introduction 1
2 HartreeFock formalism for the meaneld study of NM 9
2.1 Eﬀective densitydependent NN interaction . 13
2.1.1 CDM3Yn eﬀective interaction 14
2.1.2 M3YPn interactions . 18
2.1.3 Gogny interaction . 20
2.1.4 Skyrme interaction 22
2.2 Explicit HartreeFock expressions 23
2.2.1 The nite range interactions . 23
2.2.2 Zerorange Skyrme interaction 26
2.3 HF results for the cold asymmetric nuclear matter . 27
2.3.1 Saturation properties . 27
2.3.2 Total energy of cold NM . 31
2.3.3 Nuclear matter pressure . 33
2.3.4 Symmetry energy . 35
vii3 HF study of the stable NS matter 40
3.1 equilibrium constraint . 41
3.2 EOS of the stable npe matter 43
3.2.1 Composition of the npe matter . 43
3.2.2 The cooling of neutron star . 47
3.2.3 Pressure of the stable npe matter 49
3.3 Cold neutron star in hydrodynamical equilibrium . 51
3.3.1 Massradius relation . 52
3.3.2 Total baryon mass 57
3.3.3 Surface redshift 59
3.3.4 Binding energy 60
3.3.5 Causality condition 60
4 HartreeFock study of hot nuclear matter 63
4.1 Explicit HF expressions 66
4.1.1 The nite range interactions . 66
4.1.2 Zerorange Skyrme interaction 69
4.2 HF results for the EOS of hot ANM . 70
4.2.1 Helmholtz free energy 70
4.2.2 Free symmetry energy 75
4.2.3 Impact of nucleon eﬀective mass on the thermaldy
namical properties of NM 79
4.2.4 Entropy 83
5 HF study of the stable PNS matter 89
5.1 equilibrium constraint . 90
5.2 EOS of PNS matter 93
5.2.1 Impact of the free symmetry energy . 93
5.2.2 Impact of the inmedium nucleon eﬀective mass 101
5.3 Protoneutron star in the hydrodynamical equilibrium . 103
viiiConclusion 113
References 118
List of author's publications in the present research topic 129
ixList of Tables
2.1 Parameters of the central term V (C) (r 12 ) in the original M3Y
Paris and M3YPn (n=3,4,5) interactions [15] 15
2.2 Parameters of the density dependence (2.20) of CDM3Yn
interaction [8, 9] 17
2.3 Ranges and strengths of Yukawa functions used in the ra
dial dependence of the M3YParis, M3YP5, and M3YP7
interactions [15, 16] 18
2.4 Parameters of the densitydependent term v (DD) (n b ; r 12 )[15,
16] . 19
2.5 Ranges and strengths of Gaussian functions used in the ra
dial dependence of the D1S and D1N interactions [10, 11]. 21
2.6 HF results for the NM saturation properties using the con
sidered eﬀective NN interactions. The nucleon eﬀective mass
m
=m is evaluated at = 0 and E 0 = E(n 0 ; = 0)=A. K sym
is the curvature parameter of the symmetry energy (2.6),
and K τ is the symmetry term of the nuclear incompressibil
ity (2.57) determined at the saturation density n δ of asym
metric NM . 29
x3.1 Conguration of static neutron star given by diﬀerent NS
equations of state: maximum gravitational mass M G , radius
R G , and moment of inertia I G ; maximum central densities
n c ; c and pressure P c ; total baryon number A; surface red
shift z surf ; binding energy E bind 56
5.1 Properties of the free and trapped, stable PNS at en
tropy per baryon S=A = 0; 1; 2 and 4, given by the solu
tions of the TOV equations using the EOS's based on the
CDM3Y3, CDM3Y6 interactions [9] and their soft CDM3Y3s,
CDM3Y6s versions [6]. M max and R max are the maximum
gravitational mass and radius; n c ; c ; P c , and T c are the
baryon number density, mass density, total pressure, and
temperature in the center of PNS. T s is temperature of the
outer core of PNS, at baryon density s 0:63 10 15 g/cm 3 .
Results at S=A = 0 represent the stable conguration of cold
(free) NS [6] . 105
5.2 The same as Table 5.1 but obtained with the EOS's based
on the SLy4 version [19] of Skyrme interaction, M3YP7
interaction parametrized by Nakada [16], and D1N version
[18] of Gogny interaction 107
xiList of Figures
1.1 Structure of neutron star. (http://www.buzzle.com/articles/neutron
starfacts.html) 4
1.2 Density dependence of the energy (per baryon) of NM en
ergy given by the HF calculation using the CDM3Y3 and
CDM3Y6 versions [8] of the M3YParis interaction, which
are associated with the nuclear incompressibility K = 218
and 252 MeV, respectively . 6
2.1 Total NM energy per particle E=A at diﬀerent neutron
proton asymmetries given by the HF calculations using
the CDM3Y3 (lower panel) and CDM3Y6 (upper panel) in
teractions. The solid circles are the saturation densities of
the NM at the diﬀerent neutronproton asymmetries 30
2.2 Total NM energy per particle E=A for symmetric NM (upper
panel) and pure neutron matter (lower panel) given by the
HF calculation using diﬀerent interactions. The circles and
crosses are results of the abinitio calculation by Akmal,
Pandharipande and Ravenhall (APR) [52] and microscopic
Monter Carlo (MMC) calculation by Gandol et al. [53],
respectively . 32
xii2.3 Pressure of symmetric NM (upper panel) and pure neutron
matter (lower panel) calculated in the HF approximation
using the eﬀective NN interactions given in Table 2.6. The
shaded areas are the empirical constraints deduced from the
HI
ow data [59] 34
2.4 HF results for the NM symmetry energies E sym (n b ) given
by the densitydependent NN interactions under study. The
shaded (magenta) region marks the empirical boundaries de
duced from the analysis of the isospin diﬀusion data and
double ratio of neutron and proton spectra data of HI col
lisions [60, 61]. The square and triangle are the constraints
deduced from the consistent structure studies of the GDR
[62] and neutron skin [50], respectively. The circles and
crosses are results of the abinitio calculation by Akmal,
Pandharipande and Ravenhall (APR) [52] and microscopic
Monter Carlo (MMC) calculation by Gandol et al. [53],
respectively . 36
3.1 The fractions x j = n j =n b of constituent particles of the NS
matter obtained from the solutions of Eqs. (3.4) and (3.6)
using the meaneld potentials given by the M3YP5 and
D1N interactions 44
3.2 The same as Fig. 3.1 but using the meaneld potentials
given by the M3YP7 and Sly4 interactions. The circles are
n j values calculated at the maximum central densities n c
given by the solution of the TOV equations . 45
3.3 The same as Fig. 3.2 but using the meaneld potentials
given by the CDM3Y6 and its soft version interactions . 46
xiii3.4 The proton fraction x p of the stable NS matter obtained
from the solutions of Eqs. (3.4) and (3.6) using the mean
eld potentials given by the stiﬀtype CDM3Yn and Sly4
interactions. The circles are n p values calculated at the max
imum central densities n c given by the TOV equations. The
thin lines are the corresponding DU thresholds (3.11) . 47
3.5 The pressure inside the NS matter obtained with the in
medium NN interactions that give stiﬀ (upper panel) and
soft (lower panel) behavior of E sym (n b ), in comparison with
the empirical data points deduced from the astronomical ob
servation of neutron stars [83]. The shaded band shows the
uncertainties associated with the data determination. The
circles are P values calculated at the corresponding maxi
mum central densities given by the TOV equations . 50
3.6 The NS gravitational mass versus its radius in comparison
with the empirical data (shaded contours) deduced by
Ozel
et al [83] from recent astronomical observations of neutron
stars. The circles are values calculated at the maximum cen
tral densities. The thick solid (red) line is the limit allowed
by General Relativity [79] . 53
3.7 The same as Fig. 3.6, but in comparison with the empirical
data (shaded contours) deduced by Steiner et al. [80] from
the observation of the Xray burster 4U 160852 . 54
3.8 The gravitational mass M given by diﬀerent EOS's of the NS
matter plotted versus the corresponding total baryon mass
M b . The shaded rectangle is the empirical value inferred
from observations of the double pulsar PSR J07373059 by
Podsiadlowski et al. [81] 58
xiv3.9 The adiabatic sound velocity versus baryon density obtained
with the EOS's given by the stiﬀtype (upper panel) and
softtype (lower panel) inmedium NN interactions. The
thick solid (red) lines are the subluminal limit (v s 6 c),
and the vertical arrows indicate the baryon densities above
which the NS matter predicted by the M3YP7 interaction
becomes superluminal (see details in the text) 61
4.1 Free energy per particle F=A of symmetric nuclear matter
(SNM) and pure neutron matter (PNM) at diﬀerent temper
atures given by the HF calculation using the CDM3Y3 (right
panel) and CDM3Y6 (left panel) interactions [9] (lines), in
comparison with the BHF results (symbols) by Burgio and
Schulze [27] . 71
4.2 The same as Fig. 4.1 but for the HF results obtained with the
M3YP5 (right panel) and M3YP7 (left panel) interactions
parametrized by Nakada [15, 16] . 72
4.3 The same as Fig. 4.1 but for the HF results obtained with
the D1N version [18] of Gogny interaction (left panel) and
SLy4 version [19] of Skyrme interaction (right panel) 73
4.4 Pressure of ANM (upper panel) and PNM (lower panel) at
T=0,20,40 MeV compare to the analysis of the collective
data measured in relativistic HI collision [59] 74
4.5 Free symmetry energy per particle F sym =A of pure neutron
matter ( = 1) at diﬀerent temperatures given by the HF
calculations (4.29) using the CDM3Y3 and CDM3Y6 inter
actions [9] and their soft versions CDM3Y3s and CDM3Y6s
[6] (lines), in comparison with the BHF results (symbols) by
Burgio and Schulze [27] 75
xv4.6 The same as Fig. 4.4 but for the HF results given by the
M3YP5 and M3YP7 interactions parametrized by Nakada
[15, 16], and the D1N version [18] of Gogny interaction and
SLy4 version [19] of Skyrme interaction . 76
4.7 Free symmetry energy (4.29) (lower panels) and internal
symmetry energy (4.31) (upper panels) at diﬀerent temper
atures, given by the HF calculations using the CDM3Y6
interaction [9]. (F sym =A)= 2 curves must be very close if the
quadratic approximation (4.30) is valid 77
4.8 Density prole of the neutron (upper panel) and proton ef
fective mass (lower panel) at diﬀerent neutronproton asym
metries given by the HF calculation using the CDM3Y6
and CDM3Y3 interactions [9], and the D1N version of Gogny
interaction [18], in comparison with the BHF results (sym
bols) by Baldo et al. [71] . 80
4.9 The same as Fig. 4.8 but for the HF results obtained with
the M3YP7 and M3YP5 interactions [15, 16], and the SLy4
version [19] of Skyrme interaction . 81
4.10 Density prole of temperature in the isentropic and symmet
ric NM given by the HF calculation using diﬀerent density
dependent NN interactions, in comparison with that given
by the approximation (4.34) for the fully degenerate Fermi
(DF) system at T ≪ T F 82
4.11 Density prole of entropy per particle S=A of symmetric
nuclear matter (SNM) and pure neutron matter (PNM) at
diﬀerent temperatures, deduced from the HF results (lines)
obtained with the CDM3Y6 and CDM3Y3 interactions [9],
in comparison with the BHF results by Burgio and Schulze
(symbols) [27] 84
xvi4.12 The same as Fig. 4.6 but for the HF results obtained with
the M3YP7 and M3YP5 interactions [15, 16] 85
4.13 The same as Fig. 4.6 but for the HF results obtained with
the D1N version [18] of Gogny interaction and SLy4 version
[19] of Skyrme interaction . 86
4.14 Symmetry part of the entropy per particle (4.33) of pure
neutron matter at diﬀerent temperatures given by the CDM3Y3
and CDM3Y6 interactions [9] and their soft CDM3Y3s and
CDM3Y6s versions [6]. S sym =A is scaled by the correspond
ing temperature to have the curves well distinguishable at
diﬀerent T . 87
5.1 Total pressure (5.10) of the isentropic free (left panel) and
trapped (right panel) stable PNS matter at diﬀerent
baryon densities n b and entropy per baryon S=A = 1; 2 and
4. The EOS of the PNS crust is given by the RMF calcu
lation by Shen et al. [5], and the EOS of the uniform PNS
core is given by the HF calculation using the CDM3Y6 in
teraction [9] (upper panel) and its soft CDM3Y6s version
[6] (lower panel). The transition region matching the PNS
crust with the uniform core is shown as the dotted lines 94
5.2 Neutronproton asymmetry of the free (left panels) and
trapped (right panels) stable PNS matter at diﬀerent
baryon number densities n b and entropy per baryon S=A =
1; 2 and 4. The EOS of the homogeneous PNS core is given
by the HF calculation using the CDM3Y6 interaction [9] and
its soft CDM3Y6s version [6] . 95
xvii5.3 Entropy per baryon (upper panel) and temperature (lower
panel) as function of baryon number density n b of the 
stable PNS matter given by the CDM3Y6 interaction [13, 9]
(thick lines) and its soft CDM3Y6s version [6] (thin lines)
in the free (left panel) and trapped (right panel) cases. 96
5.4 Particle fractions as function of baryon number density n b in
the free and stable PNS matter at entropy per baryon
S=A = 1; 2 and 4, given by the CDM3Y6 interaction [9]
(upper panel) and its soft CDM3Y6s version [6] (lower panel). 97
5.5 The same as Fig. 5.4 but for the trapped, stable matter
of the PNS . 98
5.6 The same as Fig. 5.4, but given by the SLy4 version [19] of
Skyrme interaction (upper panel) and M3YP7 interaction
parametrized by Nakada [16] (lower panel) 99
5.7 The same as Fig. 5.6 but for the trapped, stable PNS
matter 100
5.8 Density prole of neutron and proton eﬀective mass in the 
free and stable PNS matter at entropy per baryon S=A =
1; 2 and 4, given by the HF calculation using the CDM3Y6
[9] and M3YP7 [16] interactions (left panel), the D1N ver
sion of Gogny interaction [18] (right panel) and SLy4 version
[19] of Skyrme interaction (right panel) 102
5.9 Density prole of temperature in the free and stable
PNS matter at entropy per baryon S=A = 1; 2 and 4, de
duced from the HF results obtained with the same density
dependent NN interactions as those considered in Fig. 5.8. 103
xviii5.10 Gravitational mass (in unit of solar mass M⊙) of the 
stable, free (left panel) and trapped (right panel) PNS
at entropy S=A = 1; 2 and 4 as function of the radius (in
km), based on the EOS of the homogeneous PNS core given
by the CDM3Y6 interaction [9] (upper panel) and its soft
CDM3Y6s version [6] (lower panel). The circle at the end
of each curve indicates the last stable conguration . 104
5.11 The same as Fig. 5.10 but given by the CDM3Y3 interaction
[9] (upper panel) and its soft CDM3Y3s version [6] (lower
panel) 106
5.12 The same as Fig. 5.10 but given by the SLy4 version of
Skyrme interaction [19] (upper panel) and M3YP7 interac
tion parametrized by Nakada [16] (lower panel) . 108
5.13 Delay time t BH from the onset of the collapse of a 40 M⊙
progenitor until the black hole formation as function of the
enclosed gravitational mass M G (open squares) given by the
hydrodynamic simulation [76, 4], and M max values given by
the solution of the TOV equations using the same EOS for
the free and stable PNS at S=A = 4 (open circles). The
M max values given by the present meaneld calculation of
the free and stable PNS at S=A = 4 using diﬀerent den
sity dependent NN interactions are shown on the correlation
line interpolated from the results of simulation . 111
xixChapter 1
Introduction
With the physics of unstable neutronrich nuclei being at the fore
front of modern nuclear physics, the determination of the equation of state
(EOS) of asymmetric neutronrich nuclear matter (NM) becomes also an
important research goal in many theoretical and experimental studies. Al
though, in general concept, asymmetric NM is an idealized innite uniform
matter composed of strongly interacting baryons and (almost free) leptons
at diﬀerent mass densities and neutronproton asymmetries, it is in fact a
real physical condition existing in neutron stars which can be observed
from Earth through their radiation of Xrays or radio signals. Up to now,
about 2000 neutron stars have been detected (mostly as radio pulsars) in
the Milky Way and Large Magellanic Cloud, with the observed gravitation
mass of the most massive neutron stars reaching around or slightly above
two solar masses (M G 2:01 0:04 M⊙). Above this value stars evolve
into black holes. For diﬀerent theoretical studies, such a large neutron star
mass should be possible with the realistic EOS of neutron star matter.
In the terrestrial laboratories, the interior of a heavy neutronrich nucleus
like lead or uranium can be considered as a small fragment of asymmetric
NM, and some basic properties of asymmetric NM were deduced from the
structure studies of heavy nuclei with neutron excess. Very important are
12 Chapter 1. Introduction
the saturation properties of NM, in particular, the internal energy pressure
of the symmetric NM (innite nuclear matter with the same neutron and
proton densities) around the saturation baryon number density n 0 0:17
fm
3 . In terms of thermodynamics, EOS often means the dependence of
the pressure P on the mass density and temperature T of NM, while in
the manybody studies of NM it is often discussed as the dependence of
the internal NM energy on the baryon number density and temperature.
From the nuclear astrophysics viewpoint, a realistic EOS of neutron
star matter is a vital input for the astrophysical studies for the structure
and formation of cold neutron star (NS) as well as hot protoneutron star
(PNS). Protoneutron stars are compact and very hot and neutrinorich
stellar objects which have the shortest stellar life time in the Universe (it
is around one minute between the birth of PNS following the gravitational
collapse of a massive progenitor and the appearance of a black hole or
a neutron star). Nevertheless, many complex physics phenomena occur
during these seconds, with PNS contracting, cooling down and eventually
losing all its neutrino content. Very important for the hydrodynamical
modeling of a compact PNS or NS are its gravitational mass and radius in
the hydrostatic equilibrium. With a given EOS of the stable neutron rich
matter, the massradius (M=R) relation of NS or PNS can be determined
from the solution of the TolmanOppenheimerVolkov (TOV) equations
[1], which were derived from the Einstein theory of the general relativity
assuming the spherical symmetry of the stellar object
dP
dr
= G
m
r 2
(
1 +
P
c 2
) (
1 +
4Pr 3
mc 2
) (
1
2Gm
rc 2
) 1
;
dm
dr
= 4r
2
; (1.1)
where G is the universal gravitational constant, P and are the pressure
and mass energy density of NS or PNS, r is the radial coordinate in the
Schwarzschild metric, and m is the gravitational mass enclosed within the3
sphere of radius r. As discussed below in the present thesis, the TOV equa
tions (1.1) are solved numerically using the realistic P() relation given by
a chosen EOS of the matter inside NS or PNS. As such, EOS means not
only P() but also the composition of the stellar matter. Nuclear astro
physics is an interdisciplinary eld that is actively developed and carried
out at diﬀerent nuclear physics centers in recent years [2]. The nuclear
astrophysical modeling of the stellar object is based in many cases on the
astrophysical observations and extrapolations of what we consider reliable
physical knowledge of the EOS of NM tested in terrestrial laboratories.
In fact, the combined use of the astrophysical and nuclear physics data in
the astrophysical studies also oﬀers a unique opportunity to test diﬀerent
theoretical nuclear models.
In the hot environment of PNS, the entropy per baryon S=A is believed
to be of the order of 1 or 2 Boltzmann constant k B [3] (in the present work
we assume k B = 1). However, the recent astrophysical studies have sug
gested that S=A might well exceed 4 in the hot core of PNS during a failed
supernova [4], when a very massive progenitor collapses directly to black
hole. Such an environment is so extreme that neutrinos are rst trapped
within the PNS matter in the beginning of the core collapse. Then, on a
time scale of 1020 s PNS is cooling down mainly through electron neutrino
emission, and after about 4050 s the PNS matter becomes transparent to
neutrinos. For a newborn NS, the cooling via neutrino emission can take
place for 100 to 10 5 years before the
cooling period begins [1]. Thus,
the knowledge about the EOS of the hot, asymmetric NM is vital for the
astrophysics studies of both NS and PNS. From the surface of NS and
PNS inwards, the baryon matter rst forms a lowdensity, inhomogeneous
crust and, with the increasing baryon density, the matter gradually forms
a uniform core (see Fig. 1.1).
Given the EOS's of the crust of NS and PNS given by the compressible4 Chapter 1. Introduction
Figure 1.1: Structure of neutron star.
(http://www.buzzle.com/articles/neutronstarfacts.html)
liquid drop model and relativistic meaneld approach, respectively, dif
ferent EOS's of the uniform core of NS and PNS predicted in the present
meaneld approach have been used as the input for the TOV equations
(1.1) to obtain the basic stellar characteristics in the hydrostatic equi
librium, like the M=R relation between the gravitational mass and radius,
central matter density and pressure. These results were compared with the
available empirical data to verify the reliability of the considered EOS's of
asymmetric NM in the astrophysical modeling of NS and PNS.
The key quantity to distinguish diﬀerent nuclear EOS's of NM is the5
nuclear meaneld potential that can be obtained from a consistent mean
eld study, like the relativistic meaneld (RMF) approach [5] or nonrela
tivistic HartreeFock (HF) formalism [6] based on the realistic choice of the
nucleonnucleon (NN) interaction in the highdensity nuclear medium. To
deduce an inmedium NN interaction starting from the free NN interaction
to the form amenable for diﬀerent nuclear structure and reaction calcu
lations still remains a challenge for the microscopic nuclear manybody
theories. Therefore, most of the nuclear reaction and structure studies still
use diﬀerent kinds of the eﬀective (densitydependent) NN interaction as
inmedium interaction between nucleons. Microscopic manybody studies
have shown consistently the strong eﬀect by the Pauli blocking as well
as higherorder NN correlations with the increasing baryon density. Such
medium eﬀects are normally considered as the physics origin of the em
pirical density dependence introduced into various versions of the eﬀective
NN interaction used in the HF meaneld approaches. For example, the
density dependent CDM3Yn versions [8, 9] of the M3Y interaction, which
was originally constructed to reproduce the Gmatrix elements of the Reid
[10] and Paris [11] NN potentials in an oscillator basis.
In searching for a realistic choice of the eﬀective NN interaction for the
consistent use in the meaneld studies of NM and nite nuclei as well as
in the nuclear reaction calculations, we have performed in the present work
a systematic HF study of asymmetric NM at both zero and nite tempera
tures using the CDM3Yn interactions, which have been used mainly in the
folding model studies of the nuclear scattering [8, 12, 13, 14], and the M3Y
Pn interactions carefully parametrized by Nakada [15, 16] for use in the
HF studies of nuclear structure. For comparison, the same HF study has
also been done with the D1S and D1N versions of the Gogny interaction
[17, 18] and Sly4 version of the Skyrme interaction [19]. In the meaneld
studies, the EOS of NM is usually associated with density dependence of6 Chapter 1. Introduction
the total energy of NM (per baryon) which is expressed as
E
A
(n b ; ) =
E
A
(n b ; = 0) + E sym (n b )
2
+ O(
4
) + ::: (1.2)
where the baryon number density (n b = n n + n p ) is the sum of the neutron
and proton number densities, = (n n n p )=n b is the asymmetry parame
ter, and E sym (n b ) is the socalled symmetry energy. A foremost requisite
to the realistic inmedium NN interaction is that it should give the proper
description of the saturation properties of symmetric NM, i.e., the binding
energy of symmetric NM of around
E
A
(n 0 ; = 0) 16 MeV at the satu
ration density n 0 , like the HF results obtained with the density dependent
CDM3Y3 and CDM3Y6 versions [8] of the M3YParis interaction shown
in Fig 1.2.
0 1 2 3 4
20
0
20
40
60
80
E/A(MeV)
0
CDM3Y3 (K=218 MeV)
CDM3Y6 (K=252 MeV)
EOS for cold Nuclear Matter
Figure 1.2: Density dependence of the energy (per baryon) of NM energy given by the
HF calculation using the CDM3Y3 and CDM3Y6 versions [8] of the M3YParis inter
action, which are associated with the nuclear incompressibility K = 218 and 252 MeV,
respectively.
Very vital quantity for the determination of the EOS of asymmetric
NM is the nuclear symmetry energy, especially, the behavior of E sym (n b )7
with the increasing baryon number density. Because the symmetry energy
of NM is determined entirely by the isospin and density dependence of the
inmedium NN interaction, it is directly related to the realistic description
of the structure of nite nuclei with neutron excess. Moreover, the knowl
edge of the symmetry energy of NM is essential for the determination of
the chemical potentials of the constituent baryons and leptons that in turn
determine the particle abundances in the stellar objects prior to or after
the supernova [21]. The recent HF studies of NM [22] have shown that the
density dependence of the symmetry energy of NM given by the eﬀective
NN interactions considered in the present work is associated with two dif
ferent (soft and stiﬀ ) behaviors at high baryon densities. As a result, these
two families predict very diﬀerent behaviors of the protontoneutron ratio
in the equilibrium that can imply two drastically diﬀerent mechanisms
for the neutron star cooling (with or without the direct Urca process). Fur
thermore, the diﬀerence in the NM symmetry energy is showing up also in
the main properties of NS in the hydrostatic equilibrium [6] that are read
ily obtained from the solutions of the TOV equations (1.1). In this thesis,
the strong impact of the nuclear symmetry energy to the basic properties
of the NS matter as well as PNS matter is illustrated in details based on
the results of a consistent HF study of the NS and PNS matter in the
equilibrium.
Another fundamental physics quantity that also impacts the behav
ior of the highdensity NM is the nucleon eﬀective mass in the medium,
which is determined by the density and momentum dependence of the
singlenucleon or nucleon meaneld potential [9, 23]. Microscopic studies
of the singlenucleon potential in the highdensity nuclear medium have
shown the important link of the nucleon eﬀective mass to diﬀerent nuclear
phenomena such as the dynamics of heavyion collisions at intermediate en
ergies, the damping of lowlying nuclear excitations and giant resonances .8 Chapter 1. Introduction
as well as the thermodynamical properties of the collapsing stellar mat
ter [23]. Therefore, an interesting research topic discussed in this thesis is
how the nucleon eﬀective mass given by diﬀerent density dependent NN
interactions aﬀects the basic properties of the asymmetric NM at zero and
nite temperatures. In particular, the density proles of the temperature
and entropy of the hot stable PNS matter. The impact of symmetry
energy as well as impact of nucleon eﬀective mass to the properties of NS
and PNS matter have been discussed in the two recent publishes by author
of this thesis and collaborations [6, 7].
The structure of this thesis is as follows: the next chapter presents the
HF formalism for the meaneld study of the EOS of asymmetric NM. The
EOS of the stable NS matter and its composition at zero temperature
are discussed in Chapter 3, with the emphasis on the impact of the nu
clear symmetry energy. The results of a consistent HF study of the hot
asymmetric NM and stable PNS matter are presented in Chapters 4 and
5, where the strong impact of the nuclear symmetry energy and nucleon
eﬀective mass on the thermal properties and composition of hot PNS mat
ter is investigated. In particular, the maximal gravitation masses obtained
with diﬀerent EOS's for the neutrinofree stable PNS at the entropy per
baryon S=A 4 were used to assess the time of the collapse of a very
massive, hot PNS to black hole, based on the results of the hydrodynamic
simulation of a failed supernova of the 40 M⊙ protoneutron progenitor.
The summary of the present research and the main conclusions are given
in the nal chapter.
Xem Thêm: Nghiên cứu phương trình trạng thái của chất hạt nhân cân bằng beta trong sao neutron và sao protoneutron (Equation of state of the betastable nuclear matter for neutron and protoneutron stars)Nội dung trên chỉ thể hiện một phần hoặc nhiều phần trích dẫn. Để có thể xem đầy đủ, chi tiết và đúng định dạng tài liệu, bạn vui lòng tải tài liệu. Hy vọng tài liệu Nghiên cứu phương trình trạng thái của chất hạt nhân cân bằng beta trong sao neutron và sao protoneutron (Equation of state of the betastable nuclear matter for neutron and protoneutron stars) sẽ giúp ích cho bạn.

Nghiên cứu phương trình trạng thái của chất hạt nhân cân bằng beta trong sao neutron và sao protoneutron (Equation of state of the betastable nuclear matter for neutron and protoneutron stars)
TIẾN SĨ Nghiên cứu phương trình trạng thái của chất hạt nhân cân bằng beta trong sao neutron và sao protoneutron (Equation of state of the betastable nuclear matter for neutron and protoneutron stars)
Nghiên cứu phương trình trạng thái của chất hạt nhân cân bằng beta trong sao neutron và sao protoneutron (Equation of state of the betastable nuclear matter for neutron and protoneutron stars)
Nghiên cứu phương trình trạng thái của chất hạt nhân cân bằng beta trong sao neutron và sao protoneutron (Equation of state of the betastable nuclear matter for neutron and protoneutron stars)
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