MINISTRY OF SCIENCE MIMISTRY OF EDUCATION AND TECHNOLOGY AND TRAINING VIETNAM ATOMIC ENERGY INSTITUTE PhD THESIS presented by Ngo Hai Tan EQUATION OF STATE OF THE BETA-STABLE NUCLEAR MATTER IN NEUTRON STARS AND PROTO-NEUTRON STARS Supervisor: Prof Dao Tien Khoa Abstract This thesis presents the results of a consistent mean-field study for the equation of state (EOS) of the β-stable baryonic matter containing npeµν 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) interaction 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 symmetry energy at high baryonic densities which were discussed in the literature as the stiff and soft scenarios for the EOS of asymmetric NM A strong impact of the nuclear symmetry energy to the mean-field 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 β-stable baryonic matter of PNS significantly Given the EOS of the crust of NS and PNS from the compressible liquid drop model and relativistic mean-field 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 β-equilibrium were investigated at different temperatures and entropy per baryon S/A = 1, and 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 ii gravitation masses obtained with different EOS’s for the neutrino-free βstable PNS at S/A = 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-field description of the EOS of both the cold and hot asymmetric NM iii Acknowledgements First and foremost, I gratefully express my best thanks to my supervisor, 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 collaborators, and his strict demand on every detail of the research work I would also like to thank Dr J´erˆome Margueron from IPN Lyon for his collaboration 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-field 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 iv Vietnam, Groupe de Physique Theorique of IPN Orsay at Universite ParisSud 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 v Abbreviations NM Nuclear matter ANM Asymmetric Nuclear matter EOS Equation of state HF Hartree-Fock BHF Br¨ uckner 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 vi Contents Abstract ii Acknowledgements iv Abbreviations vi List of tables xi List of figures xix Introduction Hartree-Fock formalism for the mean-field study of NM 2.1 2.2 2.3 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 Explicit Hartree-Fock expressions 23 2.2.1 The finite range interactions 23 2.2.2 Zero-range Skyrme interaction 26 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 vii HF study of the β-stable NS matter 40 3.1 β equilibrium constraint 41 3.2 EOS of the β-stable npeµ matter 43 3.2.1 Composition of the npeµ matter 43 3.2.2 The cooling of neutron star 47 3.2.3 Pressure of the β-stable npeµ matter 49 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 3.3 Hartree-Fock study of hot nuclear matter 4.1 4.2 63 Explicit HF expressions 66 4.1.1 The finite range interactions 66 4.1.2 Zero-range Skyrme interaction 69 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- 4.2.4 namical properties of NM 79 Entropy 83 HF study of the β-stable PNS matter 89 5.1 β 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 Proto-neutron star in the hydrodynamical equilibrium 103 5.3 viii Conclusion 113 References 118 List of author’s publications in the present research topic ix 129 List of Tables 2.1 Parameters of the central term V (C) (r12 ) in the original M3Y Paris and M3Y-Pn (n=3,4,5) interactions [15] 2.2 Parameters of the density dependence (2.20) of CDM3Yn interaction [8, 9] 2.3 15 17 Ranges and strengths of Yukawa functions used in the radial dependence of the M3Y-Paris, M3Y-P5, and M3Y-P7 interactions [15, 16] 2.4 Parameters of the density-dependent term v (DD) (nb , r12 )[15, 16] 2.5 19 Ranges and strengths of Gaussian functions used in the radial dependence of the D1S and D1N interactions [10, 11] 2.6 18 21 HF results for the NM saturation properties using the considered effective NN interactions The nucleon effective mass m∗ /m is evaluated at δ = and E0 = E(n0 , δ = 0)/A Ksym is the curvature parameter of the symmetry energy (2.6), and Kτ is the symmetry term of the nuclear incompressibility (2.57) determined at the saturation density nδ of asymmetric NM x 29 Chapter HF study of the β-stable PNS matter 115 density, the smaller the nucleon effective mass the larger the temperature of NM The density dependent NN interactions were further used to generate the EOS of baryon matter in the uniform PNS core, which was smoothly connected to the EOS of the inhomogeneous PNS crust by Shen et al [5] for the mean-field study of β-stable PNS matter at finite temperature Our study considered two different scenarios: the ν-trapped matter with the electron lepton fraction Ye ≈ 0.4 and the total entropy per baryon S/A = 1, and 4, which mimics the initial stage of PNS; the ν-free matter at S/A = 1, and 4, which is close to the late stage of PNS when most neutrinos have escaped The impact of nuclear symmetry energy was found significant in the ν-free PNS matter where the dynamic neutron-proton asymmetry, baryon and lepton compositions obtained with the asy-stiff and asy-soft interactions are quite different The high-density behaviors of the density profiles of temperature and entropy of the ν-free, β-stable PNS matter are strongly affected not only by the symmetry energy but also by the nucleon effective mass Although the impact of the symmetry energy was found less significant in the presence of trapped neutrinos, we found that the symmetry energy still affects strongly the neutrino fraction in the ν-trapped PNS matter, with xν high baryon densities, and xν 0.1 given by the asy-stiff interactions at 0.1 given by the asy-soft interactions Using the inputs of the TOV equation based on different EOS’s, we obtained the β-stable hydrostatic configuration of PNS at the total entropy per baryon S/A = 1, and in both the ν-free and ν-trapped scenarios In the absence of trapped neutrinos, different slopes of the symmetry energy at high baryon densities were shown to give a difference of 0.3 to 0.5 M⊙ in the maximum gravitational mass Mmax predicted with the CDM3Yn interactions For the ν-trapped, β-stable PNS, the effect of the symmetry energy is diminished already at the entropy S/A = and 2, and the stable 116 Chapter HF study of the β-stable PNS matter values of the maximum gravitational mass and radius of the PNS are rather close, independent of the behavior of the symmetry energy However, the impact of the nucleon effective mass remains drastic in both ν-trapped and ν-free cases, with temperature of the PNS matter being inversely proportional to m∗ In particular, the difference in temperature in both the outer core and center of PNS given by different density dependent NN interactions at S/A = is mainly due to the difference in m∗ because the effect of the symmetry energy is diminished at this high entropy A special attention was given to the configuration of the ν-free PNS at entropy S/A = 4, which was shown by Hempel et al [76] to occur at the onset of the collapse of a massive (40 M⊙ ) protoneutron progenitor to black hole We found that at very high temperatures of PNS matter at S/A = 4, the impact of the symmetry energy becomes weaker and the similar β-stable configurations of PNS were obtained with both the asy-stiff and asy-soft interactions The Mmax and Rmax values were found strongly increased at the entropy per baryon S/A = 4, with the difference ∆Mmax ≈ 0.3 − 0.5 M⊙ and ∆Rmax ≈ − km compared to the results obtained at S/A = Thus, the hot PNS at S/A = is substantially expanded in size, with the decreased central pressure and density In the outer core of hot PNS being at entropy S/A = 4, only temperature given by the asy-stiff CDM3Yn interaction is comparable to that predicted by the hydrodynamic simulation The maximum gravitational masses Mmax obtained for the β-stable and ν-free PNS at S/A = using different EOS’s of hot NM were used to estimate the time tBH of collapse of the 40 M⊙ progenitor to black hole, based on a correlation found between tBH and Mmax from the hydrodynamic simulation [76, 4] From a more general viewpoint, the present mean-field study illustrates the large impact of the nuclear symmetry energy and nucleon effective mass in dense and hot baryon matter This effect becomes more complicated Chapter HF study of the β-stable PNS matter 117 after the inclusion of trapped neutrinos which weakens the correlation of the thermodynamic properties of PNS with the symmetry energy The densities and temperatures reached in the core of PNS at entropy of S/A ≈ or higher might imply a phase transition to new degrees of freedom as discussed, e.g., in Refs [95] Thus, PNS is the most extreme compact object which requires the most advanced knowledge in the nuclear and QCD physics It is, however, very short-lived because it lasts only a minute or so before being collapsed to black hole or cooled down to neutron star Thus, the observation of the next supernova explosion in our galaxy will certainly provide the new and fascinating information about the hot PNS matter that is extremely difficult to obtain in the terrestrial nuclear physics laboratories The present mean-field results were obtained in the absence of an external electromagnetic field This is, however, only an ideal condition In reality, numerous magnetars (neutron stars with an extremely powerful magnetic field that emit high-energy electromagnetic radiation in the form of X-rays and gamma rays) have been detected, and the astrophysics studies of NS or PNS being in a strong magnetic field are now of high interest Therefore, an extended topics of the present mean-field study is the inclusion of a strong magnetic field into the hot β-stable matter inside the core of NS or PNS We also expect to learn more about properties of the spin-polarized NM as well as effects of a strong magnetic field to the configuration of a compact massive star Bibliography [1] J.M Lattimer and M Prakash,“The Physics of Neutron Stars”, Science 304, 536 (2004); J.M Lattimer and M Prakash, “Neutron star observations: Prognosis for equation of state constraints”, Phys Rep 442, 109 (2007) [2] J.M Lattimer, “The Nuclear Equation of State and Neutron Star Masses”, Annual Review of Nuclear and Particle Science 62, 485 (2012) [3] M Prakash, I Bombarci, M Prakash, P.J Ellis, J.M Lattimer, and R Knorren, “Composition and Structure of Protoneutron Stars”, Phys Rep 280, (1997) [4] A W Steiner, M Hempel, and T Fischer, “Core-Collapse Supernova Equations of State Based on Neutron Star Observations”, Astrophys J 774, 17 (2013) [5] H Shen, H Toki, K Oyamatsu, and K Sumiyoshi, “Relativistic equation of state of nuclear matter for supernova and neutron star”, Prog Theor Phys 100, 1013 (1998); http://user.numazuct.ac.jp/ sumi/eos/index.html [6] D.T Loan, N.H Tan, D.T Khoa, and J Margueron, “ Equation of state of neutron star matter, and the nuclear symmetry energy”, Phys Rev C 83, 065809 (2011) 118 Bibliography 119 [7] N.H Tan, D.T Loan, D.T Khoa and Jerome Margueron, “Meanfield study of hot β-stable protoneutron star matter: Impact of the symmetry energy and nucleon effective mass”, Phys Rev C 93, 035806 (2016) [8] D.T Khoa, G.R Satchler, and W von Oertzen, “Nuclear incompressibility and density dependent NN interactions in the folding model for nucleus-nucleus potentials”, Phys Rev C 56,954 (1997) [9] D.T Loan, B.M Loc, and D.T Khoa, “Extended Hartree-Fock study of the single-particle potential: The nuclear symmetry energy, nucleon effective mass, and folding model of the nucleon optical potential”, Phys Rev C 92,034304 (2015) [10] G Bertsch, J Borysowicz, H McManus, and W.G Love, “Interactions for inelastic scattering derived from realistic potentials”, Nucl Phys A 284, 399 (1977) [11] N Anantaraman, H Toki, and G.F Bertsch, “An effective interaction for inelastic-scattering derived from the Paris potential”, Nucl Phys A 398, 269 (1983) [12] D.T Khoa, W von Oertzen, and A.A Ogloblin, “Study of the equation of state for asymmetric nuclear matter and interaction potential between neutron-rich nuclei using density dependent M3Y interaction”, Nucl Phys A 602,98 (1996) [13] D.T Khoa, H.S Than, and D.C Cuong, “Folding model study of the isobaric analog excitation: Isovector density dependence, Lane potential, and nuclear symmetry energy”, Phys Rev C 76,014603 (2007) 120 Bibliography [14] D.T Khoa, W von Oertzen, H.G Bohlen, and S Ohkubo, “Nuclear rainbow scattering and nucleus–nucleus potential”, J Phys G34,R111 (2007) [15] H Nakada, “Mean-field approach to nuclear structure with semirealistic nucleon-nucleon interactions”, Phys Rev C 78,054301 (2008) [16] H Nakada, “Semi-realistic nucleon-nucleon interactions with improved neutron-matter properties”, Phys Rev C 87,014336 (2013) [17] J.F Berger, M Girod, and D Gogny, “Time-dependent quantum collective dynamics applied to nuclear fission”, Comp Phys Comm 63,365 (1991) [18] F Chappert, M Girod, and S Hilaire, “Towards a new Gogny force parameterization: Impact of the neutron matter equation of state”, Phys Lett B 668,420 (2008) [19] E Chabanat, P Bonche, P Haensel, J Meyer, and R Schaeffer, “A Skyrme parametrization from subnuclear to neutron star densities Part II Nuclei far from stabilities”, Nucl Phys A 635,231 (1998) [20] E Chabanat, P Bonche, P Haensel, J Meyer, and R Schaeffer, “A Skyrme parametrization from subnuclear to neutron star densities”, Nucl Phys A 627,710 (1997) [21] B.A Li, L.W Chen, and C.M Ko, “Recent progress and new challenges in isospin physics with heavy-ion reactions”, Phys Rep 464,113 (2008) [22] H.S Than, D.T Khoa, and N.V Giai, “Neutron star cooling: A challenge to the nuclear mean field”, Phys Rev C 80, 064312 (2009) Bibliography 121 [23] B.A Li and L.W Chen, Modern Physics Letters A 30, 1530010 (2015) [24] K.A Brueckner, S.A Coon, J Dabrowski, “Nuclear symmetry energy”, Phys Rev 168, 1184 (1967) [25] G.E Brown, “ Landau, Brueckner-Bethe, and Migdal theories of Fermi systems”, Rev Mod Phys 43, (1971) [26] N.M Hugenholtz and L Van Hove, “A theorem on the single particle energy in a Fermi gas with interaction”, Physica 24, 363 (1958) [27] G.F Burgio and H.J Schulze, “The maximum and minimum mass of protoneutron stars in the Brueckner theory”, Astronomy & Astrophys 518, A17 (2010) [28] M Baldo and C Maieron, “Equation of State of Nuclear Matter at high baryon density”, J Phys G 34, R243 (2007) [29] H.A Bethe, “Supernova mechanisms”, Rev Mod Phys 62,801 (1990) [30] D T Khoa, E Khan, G Col`o, and N V Giai, “Folding model analysis of elastic and inelastic proton scattering on sulfur isotopes”, Nucl Phys A 706,61 (2002) [31] D.T Khoa and H.S Than, “Isospin dependence of He+p optical potential and the symmetry energy”, Phys Rev C 71,044601 (2005) [32] N.D Chien and D.T Khoa, “Neutron transition strengths of 2+ states in the neutron-rich oxygen isotopes determined from inelastic proton scattering”, Phys Rev C 79,034314 (2009) [33] D.T Khoa and W von Oertzen, “A nuclear matter study using the density dependent M3Y interaction”, Phys Lett B 304, (1993) 122 Bibliography [34] D.T Khoa and W von Oertzen, “Refractive alpha-nucleus scattering: a probe for the incompressibility of cold nuclear matter”, Phys Lett B 342,6 (1995) [35] D.T Khoa, B.M Loc, and D.N Thang, “Folding model study of the charge-exchange scattering to the isobaric analog state and implication for the nuclear symmetry energy”, The European Physical Journal A 50 p1–15 (2014) [36] P Haensel, A.Y Potekhin, and D.G Yakovlev, “Neutron Stars 1, Equation of State and Structure”, Astrophysics and Space Science Library, Vol 326, Springer Science+Business Media, LLC (2007) [37] D.G Yakovlev, A.D Kaminker, O.Y Gnedin, P Haensel, “Neutrino Emission from Neutron Stars”, Phys Rep 354, (2001) [38] S L Shapiro and S A Teukolsky, “Black Holes, White Dwarfs, and Neutron Stars”, (Wiley-VCH Verlag, Weinheim (2004) [39] P Ring, P Schuck, “The Nuclear Many-body Problem”, Springer: Springer-Verlag Berlin Heidelberg, (1980) [40] R Reid, “Local phenomenological nucleon-nucleon potentials”, Ann of Phys 50,411 (1968) [41] D.N Basu, P Roy Chowdlhury, C Samanta, “Nuclear equation of state at high baryonic density and compact star constraints”, Nucl Phys A 811, 140 (2008); P Roy Chowdlhury, C Samanta, and D.N Basu, “Isospin dependent properties of asymmetric nuclear matter”, Phys Rev C 80, 011305(R) (2009) [42] J.P Jeukenne, A Lejeune and C Mahaux, “Optical-model potential in finite nuclei from Reid’s hard core interaction”, Phys Rev C 16, 80 (1977) Bibliography 123 [43] A Lejeune, “Low-energy optical model potential in finite nuclei from Reid’s hard core interaction”, Phys Rev C 21, 1107 (1980) [44] D Gogny, Proceeding of the International Conference on Nuclear Selfconsistent Fields, Trieste, 1975 G Ripka and M Porneuf, Eds North Holland, Amsterdam, 1975 [45] J Decharg´e and D Gogny, “Hartree-Fock-Bogolyubov calculations with the D effective interaction on spherical nuclei”, Phys Rev C 21, 1568 (1980) [46] D.G Vautherin, D.M Brink, “Hartree-Fock Calculations with Skyrme’s Interaction I Spherical Nuclei”, Phys Rev C 5, 626 (1972) [47] C.S Wang, K.C Chung and A J Santiago, “Systematics of nuclear central densities”, Phys Rev C 60 034310 (1999) [48] T Li, U Garg, Y Liu, R Marks, B K Nayak, P V Madhusudhana Rao, M Fujiwara, H Hashimoto, K Kawase, K Nakanishi, S Okumura, M Yosoi, M Itoh, M Ichikawa, R Matsuo, T Terazono, M Uchida, T Kawabata, H Akimune, Y Iwao, T Murakami, H Sakaguchi, S Terashima, Y Yasuda, J Zenihiro, and M N Harakeh, “Isotopic dependence of the giant monopole resonance in the even-A Sn112−124 isotopes and the asymmetry term in nuclear incompressibility”, Phys Rev Lett 99, 162503 (2007) [49] B A Brown, “Neutron radii in nuclei and the neutron equation of state”, Phys Rev Lett 85, 5296 (2000) [50] R J Furnstahl, “Neutron radii in mean field models”, Nucl Phys A 706, 85 (2002) 124 Bibliography [51] C J Horowitz and J Piekarewicz, “Neutron star structure and the neutron radius of 208Pb”, Phys Rev Lett 86, 5647 (2001) [52] A Akmal, V R Pandharipande, and D G Ravenhall, “Equation of state of nucleon matter and neutron star structure”, Phys Rev C 58, 1804 (1998) [53] S Gandolfi, A Yu Illarionov, S Fantoni, J C Miller, F Pederiva and K E Schmidt, “Microscopic calculation of the equation of state of nuclear matter and neutron star structure”, Monthly Notices of the Royal Astronomical Society: Letters 404, L35–L39 (2010) [54] Z H Li, U Lombardo, H J Schulze, W Zuo, L W Chen, and H R Ma, “Three-body force rearrangement effect on single particle properties in neutron-rich nuclear matter”, Phys Rev C 74, 047304 (2006) [55] R B Wiringa, V Fiks, and A Fabrocini, “Equation of state for dense nucleon matter”, Phys Rev C 38, 1010 (1988) [56] J.M Lattimer, C.J Pethick, M Prakash, and P Haensel, “Direct URCA Process in Neutron stars”, Phys Rev Lett 66, 2701 (1991) [57] M Onsi and J M Pearson, “Equation of state of stellar nuclear matter and the effective nucleon mass”, Phys Rev C 65, 047302 (2002) [58] P E Hodgson, “Effective mass in nuclei”, Contemp Phys 24, 491 (1983) [59] P.Danielewicz, R Lacey and W.G Lynch, “Determination of the Equation of State of Dense Matter”, Science 298, 1592 (2002) Bibliography 125 [60] A Ono, P Danielewicz, W A Friedman, W G Lynch, and M B Tsang, “Isospin fractionation and isoscaling in dynamical simulations of nuclear collisions”, Phys Rev C 68, 051601(R) (2003) [61] M B Tsang, Y Zhang, P Danielewicz, M Famiano, Z Li, W G Lynch, and A W Steiner, “Constraints on the density dependence of the symmetry energy”, Phys Rev Lett 102, 122701 (2009) [62] L Trippa, G Col`o, and E Vigezzi, “Giant dipole resonance as a quantitative constraint on the symmetry energy”, Phys Rev C 77, 061304(R) (2008) [63] D V Shetty, S J Yennello, and G A Souliotis, “Density Dependence of The Symmetry Energy and The Nuclear Equation of State: A Dynamical and Statistical Model Perspective”, Phys Rev C 76, 024606 (2007) [64] J Piekarewicz and M Centelles, “Incompressibility of neutron-rich matter”, Phys Rev C 79, 054311 (2009) [65] M Hempel, “Nucleon self-energies for supernova equations of state”, Phys Rev C 91, 055807 (2015) [66] G Col`o, U Garg, and H Sagawa, “Symmetry energy from the nuclear collective motion: constraints from dipole, quadrupole, monopole and spin-dipole resonances”, Eur Phys J A 50, 26(2014) [67] B A Li and X Han, “10.1140/epja/i2014-14026-9”, Phys Lett B 727, 276 (2013) [68] C Xu, B A Li, and L W Chen, “Relationship between the symmetry energy and the single-nucleon potential in isospin-asymmetric nucleonic matter”, Eur Phys J A 50, 21 (2014) 126 Bibliography [69] J.M Lattimer, K.A Van Riper, M Prakash, and M Prakash, “Rapid cooling and the structure of neutron star”, Astrophys J 425, 802 (1994) [70] D Page, J.M Lattimer, M Prakash, and A.W Steiner, “Minimal cooling of neutron stars: A new paradigm”, Astrophys J Suppl Series 155, 623 (2004) [71] M Baldo, G F Burgio, H J Schulze, and G Taranto, “Nucleon effective masses within the Brueckner-Hartree-Fock theory: Impact on stellar neutrino emission”, Phys Rev C 89, 048801 (2014) [72] A W Steiner, M Prakash, and J M Lattimer, “Quark-hadron phase transitions in Young and old neutron stars”, Phys Lett B 486, 239 (2000) [73] J Xu, L W Chen, C M Ko, and B A Li, “Transition density and pressure in hot neutron stars”, Phys Rev C 81, 055805 (2010) [74] A Burrows and J M Lattimer, “The birth of neutron stars”, Astrophys J 307, 178 (1986) [75] K Kotake, K Sato, and K Takahashi, “Explosion mechanism, neutrino burst and gravitational wave in core-collapse supernovae”, Rep Prog Phys 69, 971 (2006) [76] M Hempel, T Fischer, J Schaffner-Bielich, and M Liebend¨orfer, “New equation of state in simulation core-collapse supernovae”, Astrophys J 748, 70 (2012) [77] F Douchin and P Haensel, “A unified equation of state of dense matter and neutron star structure”, Astron Astrophys 380, 151 (2001) Bibliography 127 [78] P B Demorest, T Pennucci, S M Ransom, M S E Roberts, and J W T Hessels, “A two-solar-mass neutron star measured using Shapiro delay”, Nature (London) 467, 1081 (2010) [79] N K Glendenning, “Compact Stars: Nuclear Physics, Particle Physics and General Relativity”, (Springer, New York, 2000) [80] A W Steiner, J M Lattimer, and E F Brown, “The equation of state from observed masses and radii of neutron star”, Astrophys J 722, 33 (2010) [81] Ph Podsiadlowski, J D M Dewi, P Lesaffre, J C Miller, W G Newton, and J R Stone, “The double pulsar J0737-3039: Testing the neutron star equation of state”, Monthly Notices of the Royal Astronomical Society 361, 1243 (2005) [82] B G Todd-Rutel and J Piekarewicz, “Neutron-rich nuclei and neutron stars: A new accurately calibrated interaction for the study of neutron-rich matter”, Phys Rev Lett 95, 122501 (2005) ¨ [83] F Ozel, G Baym, and T G¨ uver, “Astrophysical Measurement of the Equation of State of Neutron Star Matter”, Phys Rev D 82, 101301(R) (2010) [84] I Bombaci, “Isospin Physics in Heavy Ion Collisions at Intermediate Energies”, edited by B A Li and W U Schr¨oder (Nova Science, New York, 2001), p 35 [85] J Cottam, F Paerls, and M Mendez, “Gravitationally redshifted absorption lines in the X-ray burst spectra of a neutron star”, Nature (London) 420, 51 (2002) [86] C E Rhoades Jr and R Ruffini, “Maximum Mass of a Neutron Star”, Phys Rev Lett 32, 324 (1974) 128 Bibliography [87] H.A Bethe and J Goldstone, “Effect of repulsive core in the theory of complex nuclei”, Proc Roy Soc A 238, 157 (1957) [88] C Samanta, D Bandyopadhyay and J.N De, “Incompressibility of asymmetric nuclear matter”, Phys Lett B 217 (1989) 381 [89] C Constantinou, B Muccioli, M Prakash, and J M Lattimer, “Thermal properties of supernova matter: The bulk homogeneous phase”, Phys Rev C 89, 065802 (2014) [90] C Wellenhofer, J W Holt, and N Kaiser, “Thermodynamics of isospin-asymmetric nuclear matter from chiral effective field theory”, Phys Rev C 92, 015801 (2015) [91] A Fedoseew and H Lenske, “Thermal properties of asymmetric nuclear matter”, Phys Rev C 91, 034307 (2015) [92] R B Wiringa, V G J Stoks, and R Schiavilla, “Accurate nucleonnucleon potential with charge-independence breaking”, Phys Rev C 51, 38 (1995) [93] C Constantinou, B Muccioli, M Prakash, and J M Lattimer, “Thermal properties of hot and dense matter with finite range interactions”, Phys Rev C 92, 025801 (2015) [94] I Vida˜ na, I Bombaci, A Polls, and A Ramos, “Microscopic study of neutrino trapping in hyperon stars”, Astron Astrophys 399, 687 (2003) [95] J A Pons, S Reddy, P J Ellis, M Prakash, and J M Lattimer, “Kaon Condensation in Proto-Neutron Star Matter”, Phys Rev C 62, 035803 (2000) List of publications Doan Thi Loan, Ngo Hai Tan, Dao Tien Khoa and J Margueron, “ Equation of state of neutron star matter, and the nuclear symmetry energy”, Phys Rev C 83, 065809 (2011) Ngo Hai Tan, Doan Thi Loan, Dao T Khoa and Jerome Margueron, “Mean-field study of hot β-stable protoneutron star matter: Impact of the symmetry energy and nucleon effective mass”, Phys Rev C 93, 035806 (2016) 129 ... soft scenarios for the EOS of asymmetric NM A strong impact of the nuclear symmetry energy to the mean-field prediction of the cooling scenario for NS and thermodynamic properties of the PNS matter. .. thesis presents the results of a consistent mean-field study for the equation of state (EOS) of the β -stable baryonic matter containing npeµν particles in the core of cold neutron star (NS) and. .. to the form amenable for different nuclear structure and reaction calculations still remains a challenge for the microscopic nuclear many-body theories Therefore, most of the nuclear reaction and