Tiến Sĩ Nghiên cứu phương trình trạng thái của chất hạt nhân cân bằng beta trong sao neutron và sao proto-ne

Abstract
This thesis presents the results of a consistent mean-eld 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 proto-neutron
star (PNS). Within the non-relativistic Hartree-Fock formalism, different
choices of the in-medium, density-dependent nucleon-nucleon (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 mean-eld 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 high-density medium was found
also to affect the thermal properties of hot -stable baryonic matter of
PNS signicantly.
Given the EOS of the crust of NS and PNS from the compressible
liquid drop model and relativistic mean-eld approach, respectively, the
different EOS's of the core of NS and PNS were used as input for the
Tolman-Oppenheimer-Volkov 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 neutrino-free and neutrino-trapped 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 neutrino-free -
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 mean-eld 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 mean-eld 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 Hartree-Fock
BHF Bruckner Hartree-Fock
D Direct
EX Exchange
NS Neutron star
PNS Proto-neutron star
n neutron
p proton
NN nucleon-nucleon
IS Iso-scalar
IV Iso-vector
viContents
Abstract . ii
Acknowledgements . iv
Abbreviations vi
List of tables xi
List of gures xix
1 Introduction 1
2 Hartree-Fock formalism for the mean-eld study of NM 9
2.1 Effective density-dependent NN interaction . 13
2.1.1 CDM3Yn effective interaction 14
2.1.2 M3Y-Pn interactions . 18
2.1.3 Gogny interaction . 20
2.1.4 Skyrme interaction 22
2.2 Explicit Hartree-Fock expressions 23
2.2.1 The nite range interactions . 23
2.2.2 Zero-range 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 Mass-radius relation . 52
3.3.2 Total baryon mass 57
3.3.3 Surface red-shift 59
3.3.4 Binding energy 60
3.3.5 Causality condition 60
4 Hartree-Fock study of hot nuclear matter 63
4.1 Explicit HF expressions 66
4.1.1 The nite range interactions . 66
4.1.2 Zero-range 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 in-medium nucleon effective mass 101
5.3 Proto-neutron 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 M3Y-Pn (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 M3Y-Paris, M3Y-P5, and M3Y-P7
interactions [15, 16] 18
2.4 Parameters of the density-dependent 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 Conguration 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 conguration 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, M3Y-P7
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-
star-facts.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 M3Y-Paris 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 neutron-proton 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 ab-initio 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 density-dependent 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 ab-initio 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 mean-eld potentials given by the M3Y-P5 and
D1N interactions 44
3.2 The same as Fig. 3.1 but using the mean-eld 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 mean-eld 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 stiff-type 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 X-ray burster 4U 1608-52 . 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 J0737-3059 by
Podsiadlowski et al. [81] 58
xiv3.9 The adiabatic sound velocity versus baryon density obtained
with the EOS's given by the stiff-type (upper panel) and
soft-type (lower panel) in-medium 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 M3Y-P7 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
M3Y-P5 (right panel) and M3Y-P7 (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
M3Y-P5 and M3Y-P7 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 prole of the neutron- (upper panel) and proton ef-
fective mass (lower panel) at different neutron-proton 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 M3Y-P7 and M3Y-P5 interactions [15, 16], and the SLy4
version [19] of Skyrme interaction . 81
4.10 Density prole 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 prole 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 M3Y-P7 and M3Y-P5 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 Neutron-proton 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 M3Y-P7 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 prole 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 M3Y-P7 [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 prole 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 conguration . 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 M3Y-P7 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 mean-eld 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 neutron-rich nuclei being at the fore-
front of modern nuclear physics, the determination of the equation of state
(EOS) of asymmetric neutron-rich 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 innite uniform
matter composed of strongly interacting baryons and (almost free) leptons
at different mass densities and neutron-proton asymmetries, it is in fact a
real physical condition existing in neutron stars which can be observed
from Earth through their radiation of X-rays 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 neutron-rich 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 (innite 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 many-body 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 proto-neutron star
(PNS). Proto-neutron stars are compact and very hot and neutrino-rich
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 mass-radius (M=R) relation of NS or PNS can be determined
from the solution of the Tolman-Oppenheimer-Volkov (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 10-20 s PNS is cooling down mainly through electron neutrino
emission, and after about 40-50 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 low-density, 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/neutron-star-facts.html)
liquid drop model and relativistic mean-eld approach, respectively, dif-
ferent EOS's of the uniform core of NS and PNS predicted in the present
mean-eld 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 mean-eld potential that can be obtained from a consistent mean-
eld study, like the relativistic mean-eld (RMF) approach [5] or nonrela-
tivistic Hartree-Fock (HF) formalism [6] based on the realistic choice of the
nucleon-nucleon (NN) interaction in the high-density nuclear medium. To
deduce an in-medium 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 many-body
theories. Therefore, most of the nuclear reaction and structure studies still
use different kinds of the effective (density-dependent) NN interaction as
in-medium interaction between nucleons. Microscopic many-body studies
have shown consistently the strong effect by the Pauli blocking as well
as higher-order 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 mean-eld approaches. For example, the
density dependent CDM3Yn versions [8, 9] of the M3Y interaction, which
was originally constructed to reproduce the G-matrix 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 mean-eld 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 mean-eld
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 so-called symmetry energy. A foremost requisite
to the realistic in-medium 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 M3Y-Paris 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 M3Y-Paris 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
in-medium 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 proton-to-neutron 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 high-density NM is the nucleon effective mass in the medium,
which is determined by the density- and momentum- dependence of the
single-nucleon or nucleon mean-eld potential [9, 23]. Microscopic studies
of the single-nucleon potential in the high-density nuclear medium have
shown the important link of the nucleon effective mass to different nuclear
phenomena such as the dynamics of heavy-ion collisions at intermediate en-
ergies, the damping of low-lying 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 proles 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 mean-eld 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 neutrino-free -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.