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In neutron stars the nuclear asymmetric matter is expected to undergo phase transitions to a superfluid state. According to simple estimates, neutron matter in the inner crust and just below should be in the s-wave superfluid phase, corresponding to the neutron-neutron 1S0 channel. At higher density in the core also the proton component should be superfluid, while in the inner core the neutron matter can be in the 3P2 superfluid phase. Superluidity is believed to be at the basis of the glitches phenomenon and to play a decisive influence on many processes like transport, neutrino emission and cooling, and so on. One of the peculiarity of the superfluid phase is the presence of characteristic collective excitation, the so called ’phonons’, that correspond to smooth modulations of the order parameter and display a linear spectrum at low enough momentum. This paper is a brief review of the different phonons that can appear in Neutron Star superfuid matter and their role in several dynamical processes. Particular emphasis is put on the spectral functions of the different components, that is neutron, protons and electrons, which reveal their mutual influence. The open problems are discussed and indications on the work that remain to be done are given.

Several phenomena occurring in neutron stars are affected by the elementary excitations that characterize the stellar matter. In particular, low-energy excitations can play a major role in the emission and propagation of neutrinos, neutron star cooling and transport processes. In this paper, we consider the elementary modes in the star region where both proton and neutron components are superfluid. We study the overall spectral functions of protons, neutrons and electrons on the basis of the Coulomb and nuclear interactions. This study is performed in the framework of the Random Phase Approximation, generalized to superfluid systems. The formalism we use ensures that the Generalized Ward's Identities are satisfied. We focus on the coupling between neutrons and protons. On one hand this coupling results in collective modes that involve simultaneously neutrons and protons, on the other hand it produces a damping of the excitations. Both effects are especially visible in the spectral functions of the different components of the matter. At high density while the neutrons and protons tend to develop independent excitations, as indicated by the spectral functions, the neutron-proton coupling still produces a strong damping of the modes., 9 pages 6 figures, accepted in Physical Review C. arXiv admin note: substantial text overlap with arXiv:1708.03794

We calculate modified Urca neutrino emission rates in the dense nuclear matter in neutron star cores. We find that these rates are strongly enhanced in the beta-stable matter in regions of the core close to the direct Urca process threshold. This enhancement can be tracked to the use of the in-medium nucleon spectrum in the virtual nucleon propagator. We describe the in-medium nucleon scattering in the non-relativistic Bruckner-Hartree-Fock framework taking into account two-body as well as the effective three-body forces, although the proposed enhancement does not rely on a particular way of the nucleon interaction treatment. Finally we suggest a simple approximate expression for the emissivity of the n-branch of the modified Urca process that can be used in the neutron stars cooling simulations with any nucleon equation of state of dense matter., 8 pages, 3 figures; accepted for publication in PLB. In v.2 misprint in eq.(9) corrected and discussion of cooling curves expanded

Thermal conductivity and shear viscosity of npe$\mu$ matter in non-superfluid neutron star cores are considered in the framework of Brueckner-Hartree-Fock many-body theory. We extend our previous work (Shternin, Baldo and Haensel, 2013) by analysing different nucleon-nucleon potentials and different three-body forces. We find that the use of different potentials leads up to one order of magnitude variations in the values of the nucleon contribution to transport coefficients. The nucleon contribution dominates the thermal conductivity, but for all considered models the shear viscosity is dominated by leptons., Comment: 4 pages, 2 figures; in v.2 misprints in eqs. (5)-(9) corrected

Within a microscopic approach the structure of Neutron Stars is usually studied by modelling the homogeneous nuclear matter of the core by a suitable Equation of State, based on a many-body theory, and the crust by a functional based on a more phenomenological approach. We present the first calculation of Neutron Star overall structure by adopting for the core an Equation of State derived from the Brueckner-Hartree-Fock theory and for the crust, including the pasta phase, an Energy Density Functional based on the same Equation of State, and which is able to describe accurately the binding energy of nuclei throughout the mass table. Comparison with other approaches is discussed. The relevance of the crust Equation of state for the Neutron Star radius is particularly emphasised., Comment: 14 pages, 6 figures, accepted version

The thermal evolution of Neutron Stars is affected by the elementary excitations that characterize the stellar matter. In particular, the low-energy excitations, with a spectrum linear in momentum, can play a major role in the emission and propagation of neutrinos. In this paper, we focus on the elementary modes in the region of proton superfluidity, where the neutron component is expected to have a very small or zero pairing gap. We study the overall spectral functions of protons, neutrons and electrons on the basis of the Coulomb and nuclear interactions. This study is performed in the framework of the Random Phase Approximation, generalized in order to describe the response of a superfluid system. The formalism we use ensures that the Generalized Ward's Identities are satisfied. Despite their relative small fraction, the protons turn out to modify the neutron spectral function as a consequence of the nuclear neutron-proton interaction. This effect is particularly evident at the lower density, just below the crust for a density close to the saturation value, while at increasing density the neutrons and the protons are mainly decoupled. The proton spectral function is characterized by a pseudo-Goldstone mode below $ 2\Delta $, twice the pairing gap, and a pair-breaking mode above $ 2\Delta $. The latter merges in the sound mode of the normal phase at higher momenta. The neutron spectral function develops a collective sound mode only at the higher density. The electrons have a strong screening effect on the proton-proton interaction at the lower momenta, and decouple from the protons at higher momenta., Comment: 6 figures, to appear in Phys. Rev. C

The self-consistent semimicroscopic fully quantum approach developed recently for describing the structure of the inner crust of a neutron star within the Wigner-Seitz method is used to perform a systematic calculation of the properties of the system under study. Only the lowest layers of the crust in the vicinity of the point where a phase transition to a uniform state occurs are excluded from our consideration. Use is made of a realistic microscopic model that treats pairing in neutron matter with allowance for corrections of many-body theory to the Bardeen-Cooper-Schrieffer approximation.

The nuclear symmetry energy characterizes the variation of the binding energy as the neutron to proton ratio of a nuclear system is varied. This is one of the most important features of nuclear physics in general, since it is just related to the two component nature of the nuclear systems. As such it is one of the most relevant physical parameters that affect the physics of many phenomena and nuclear processes. This review paper presents a survey of the role and relevance of the nuclear symmetry energy in different fields of research and of the accuracy of its determination from the phenomenology and from the microscopic many-body theory. In recent years, a great interest was devoted not only to the Nuclear Matter symmetry energy at saturation density but also to its whole density dependence, which is an essential ingredient for our understanding of many phenomena. We analyze the nuclear symmetry energy in different realms of nuclear physics and astrophysics. In particular we consider the nuclear symmetry energy in relation to nuclear structure, astrophysics of Neutron Stars and supernovae, and heavy ion collision experiments, trying to elucidate the connections of these different fields on the basis of the symmetry energy peculiarities. The interplay between experimental and observational data and theoretical developments is stressed. The expected future developments and improvements are schematically addressed, together with most demanded experimental and theoretical advances for the next few years., Comment: 71 pages, 33 figures, accepted for publication by Progress in Particle and Nuclear Physics. arXiv admin note: text overlap with arXiv:1310.2896, arXiv:1205.2585, arXiv:nucl-ex/0610013, arXiv:1407.7625, arXiv:nucl-ex/0310024, arXiv:1101.3648, arXiv:1406.4546, arXiv:1309.5153, arXiv:1401.2074, arXiv:1210.3402, arXiv:1506.04687, arXiv:astro-ph/0407529 by other authors

We derive the equation of state (EOS) of nuclear matter from are alistic constituent quark model for the nucleon-nucleon interaction. We use the Brueckner-Bethe-Goldstone approach with the inclusion of the three hole-line contribution. We find that the resulting EOS reproduces correctly the saturation point, moreover the symmetry energy at low density, its slope, and the incompressibility turn out to be compatible with phenomenology. We calculate the mass-radius relation for neutron stars, and find maximum values close to two solar masses, in agreement with recent observational data.

A self-consistent quantum approach to describe the inner crust structure of neutron stars is developed within the Wigner-Seitz (WS) approximation, based on the generalized energy functional method involving explicitly neutron and proton pairing correlations. The energy functional is constructed by matching the realistic phenomenological nuclear functional by Fayans et al. for describing the pseudonucleus in the center of the WS cell with the one calculated microscopically for neutron matter within the Brueckner approach with the Argonne v18 force. The microscopic description of the neutron superfluidity is based on the BCS approach with the same v18 force. Many-body theory corrections to the BCS method (the correlation and self-energy ones) are included in the calculation scheme in an approximate way. A wide region of average densities was investigated corresponding to the Fermi momentum values kF = 0.6–1.2 fm−1

A critical point determining the interphace between the outer and inner parts of the crust of a neutron star has been found using a previously developed self-consistent semi-microscopic method. The critical density ρd thus obtained is only slightly lower than the commonly accepted value. At the same time, the equilibrium structure of the crust near the critical point differs significantly from that obtained in known calculations.

A semi-microscopic self-consistent quantum approach developed recently to describe the inner crust structure of neutron stars within the Wigner-Seitz (WS) method with the explicit inclusion of neutron and proton pairing correlations is further developed. In this approach, the generalized energy functional is used which contains the anomalous term describing the pairing. It is constructed by matching the realistic phenomenological functional by Fayans et al. for describing the nuclear-type cluster in the center of the WS cell with the one calculated microscopically for neutron matter. Previously the anomalous part of the latter was calculated within the BCS approximation. In this work corrections to the BCS theory which are known from the many-body theory of pairing in neutron matter are included into the energy functional in an approximate way. These modifications have a sizable influence on the equilibrium configuration of the inner crust, i.e. on the proton charge Z and the radius R_c of the WS cell. The effects are quite significant in the region where the neutron pairing gap is larger., Comment: 24 pages, 14 figures; LaTeX, submitted to EPJA

The structure of neutron stars is determined by the nuclear Equation of State well above the nuclear saturation density. Since only limited hints from heavy ion reactions at intermediate energy can be used to constraint the Equation of State, theoretical predictions are essential. Different microscopic many-body methods for symmetric matter as well as for pure neutron matter are compared to the aim of assessing the theoretical uncertainties. The possible transition to quark matter in the interior of neutron stars is then briefly discussed and it is argued that recent observational data, if confirmed, would point towards a stiffening of the quark Equation of State with respect to the asymptotic freedom regime.

We study the hadron-quark phase transition in the interior of neutron stars (NS). For the hadronic sector, we use a microscopic equation of state (EOS) involving nucleons and hyperons derived within the Brueckner-Bethe-Goldstone many-body theory, with realistic two-body and three-body forces. For the description of quark matter, we employ both the MIT bag model with a density dependent bag constant, and the color dielectric model. We calculate the structure of NS interiors with the EOS comprising both phases, and we find that the NS maximum masses are never larger than 1.7 solar masses, no matter the model chosen for describing the pure quark phase., 11 pages, 5 figures, submitted to Phys. Rev. D

We study the hadron-quark phase transition in the interior of neutron stars (NS's). We calculate the equation of state (EOS) of hadronic matter using the Brueckner-Bethe-Goldstone formalism with realistic two-body and three-body forces, as well as a relativistic mean field model. For quark matter we employ the MIT bag model constraining the bag constant by using the indications coming from the recent experimental results obtained at the CERN SPS on the formation of a quark-gluon plasma. We find necessary to introduce a density dependent bag parameter, and the corresponding consistent thermodynamical formalism. We calculate the structure of NS interiors with the EOS comprising both phases, and we find that the NS maximum masses fall in a relatively narrow interval, $1.4 M_\odot \leq M_{\rm max} \leq 1.7 M_\odot$. The precise value of the maximum mass turns out to be only weakly correlated with the value of the energy density at the assumed transition point in nearly symmetric nuclear matter., Comment: 25 pages, Revtex4, 16 figures included as postscript

We review our knowledge on the properties of the nuclear medium that have been studied, along many years, on the basis of many-body theory, laboratory experiments and astrophysical observations. First we consider the realm of phenomenological laboratory data and astrophysical observations, and the hints they can give on the characteristics that the nuclear medium should possess. The analysis is based on phenomenological models, that however have a strong basis on physical intuition and an impressive success. More microscopic models are also considered, and it is shown that they are able to give invaluable information on the nuclear medium, in particular on its Equation of State. The interplay between laboratory experiments and astrophysical observations are particularly stressed, and it is shown how their complementarity enriches enormously our insights into the structure of the nuclear medium. We then introduce the nucleon-nucleon interaction and the microscopic many-body theory of nuclear matter, with a critical discussion about the different approaches and their results. The Landau Fermi Liquid theory is introduced and briefly discussed. As illustrative example, we discuss neutron matter at very low density, and it is shown how it can be treated within the many-body theory. A section is dedicated to the pairing problem. The connection with nuclear structure is then discussed, on the basis of the Energy Density Functional method. The possibility to link the physics of exotic nuclei and the astrophysics of neutron stars is particularly stressed. Finally we discuss the thermal properties of the nuclear medium, in particular the liquid-gas phase transition and its connection with the phenomenology on heavy ion reactions and the cooling evolution of neutron stars. The presentation has been taken for non-specialists and possibly for non-nuclear physicists., Comment: 90 pages, 29 figures, revised version

We study the structure of newly born neutron stars (protoneutron stars) within the finite temperature Brueckner-Bethe-Goldstone theoretical approach including also hyperons. We find that for purely nucleonic stars both finite temperature and neutrino trapping reduce the value of the maximum mass. For hyperonic stars the effect is reversed, because neutrino trapping shifts the appearance of hyperons to larger baryon density and stiffens considerably the equation of state., 11 pages, 7 figures, submitted to Astronomy & Astrophysics

Within the Wigner-Seitz approximation, a self-consistent fully quantum-mechanical calculation of the structure of the inner crust of a neutron star is performed over a wide range of densities with allowance for superfluidity effects. Within the approach used, the Wigner-Seitz cell consists of a nuclear-like cluster surrounded by a nearly uniform neutron gas. An effective energy functional is constructed by matching, at the cluster surface, the realistic phenomenological nuclear functional for the cluster due to S. A. Fayans and his coauthors and the energy functional calculated microscopically for neutron matter. The microscopic component of the functional is calculated within the Brueckner method by using the v18 Argonne interaction.

A self-consistent quantum approach to describe the inner crust structure of neutron stars is developed, within the Wigner–Seitz approximation. It is based on the generalized energy functional method involving explicitly neutron and proton pairing correlations. The energy functional is constructed by matching the realistic phenomenological functional by Fayans et al. for describing the nuclear-type cluster in the center of the Wigner–Seitz cell with the one calculated microscopically for neutron matter. It is shown that the neutron and proton superfluidity influences significantly the ground state structure of the inner crust.

The different superfluid states that can occur inside neutron stars are reviewed, both in the crust region and in the core. The role of correlations in determining the values of the pairing gaps is emphasized. Comparison is made on one hand with pairing in nuclear structure calculations in ordinary (terrestrial) nuclei, and on the other hand with hints from phenomenological analysis of observational data.

A self-consistent, completely quantum calculation of the structure of the inner crust of neutron stars is carried out in the Wigner-Seitz approximation with a realistic phenomenological nuclear energy functional, where pair correlations of neutrons and protons are included in the explicit form. It has been shown that the superfluidity of neutrons and protons affects the structure of the ground state of the crust.

The density of nuclear matter in the interior of neutron stars can reach values, for the largest masses, which can be compatible with the onset of hadron deconfinement. For the study of this possibility the only viable method at present is the comparison between the available nucleon and quark Equations of State (EoS) at increasing baryon density. It is then possible to trace the transition to the deconfined phase or the appearence of a mixed phase. We present recent results on the structure of neutron stars based on this procedure. For the nucleon matter, the microscopic many-body theory of the Nuclear Equation of State is discussed in the framework of the Bethe-Brueckner-Goldstone method. The expansion is extended up to the three hole-line diagrams contribution. For the quark matter, different models are used to generate the quark EoS. Despite the maximum mass of neutron stars turns out to be only marginally sensitive to the considered quark EoS, it is found that the structure of neutron stars can drastically depend on the adopted model.

The microscopic many-body theory of the Nuclear Equation of State is discussed in the framework of the Bethe–Brueckner–Goldstone method. The expansion is extended up to the three hole-line diagrams contribution. The Brueckner equation for the two-body G-matrix and the Bethe–Fadeev equation for the three-body scattering matrix are solved both for the gap and continuous choices of the single particle potential. For symmetric and pure neutron matter strong evidence of convergence in the expansion is found. Once three-body forces are introduced, the phenomenological saturation point is reproduced. In order to study neutron stars static properties, the theory is extended to include strangeness, and the possible quark-gluon plasma component is described in the simplified MIT bag model. The results for the mass and radius of neutron stars are briefly discussed.

We derive a new equation of state (EoS) for neutron stars (NS) from the outer crust to the core based on modern microscopic Brueckner-Hartree-Fock (BHF) calculations using the Argonne $v_{18}$ potential plus three-body forces computed with the Urbana model. To deal with the inhomogeneous structures of matter in the NS crust, we use the recent Barcelona-Catania-Paris-Madrid (BCPM) nuclear energy density functional that is directly based on the same microscopic BHF calculations, and which is able to reproduce the ground-state properties of nuclei along the periodic table. The EoS of the outer crust requires the masses of neutron-rich nuclei, which are obtained through Hartree-Fock-Bogoliubov calculations with the BCPM functional when they are unknown experimentally. To compute the inner crust, Thomas-Fermi calculations in Wigner-Seitz cells are performed with the same functional. Existence of nuclear pasta is predicted in a range of average baryon densities between $\simeq$0.067 fm$^{-3}$ and $\simeq$0.0825 fm$^{-3}$, where the transition to the core takes place. The NS core is computed from the nuclear EoS of the BHF calculation assuming non-exotic constituents (core of $npe\mu$ matter). In each region of the star, we discuss the comparison of the new EoS with previous EoSes for the complete NS structure, in particular, with the Lattimer-Swesty EoS and with the Shen et al. EoS widely used in astrophysical calculations. The new microscopically derived EoS fulfills at the same time a NS maximum mass of 2~$M_\odot$ with a radius of 10 km, and a 1.5~$M_\odot$ NS with a radius of 11.7 km., Comment: 23 pages, 17 figures, revised version accepted for publication in Astronomy & Astrophysics

Starting from a realistic constituent quark model for the nucleon-nucleon interaction, we derive the equation of state (EOS) of nuclear matter within the Bethe-Brueckner-Goldstone approach up to three-hole-line level, without need to introduce three-nucleon forces. To estimate the uncertainty of the calculations both the gap and the continuous choices for the single-particle potential are considered and compared. The resultant EOS is compatible with the phenomenological analysis on the saturation point, the incompressibility, the symmetry energy at low density and its slope at saturation, together with the high-density pressure extracted from flow data on heavy ion collisions. Although the symmetry energy is appreciably larger in the gap choice in the high-density region, the maximum neutron star masses derived from the continuous-choice EOS and the gap-choice EOS are similar and close to two solar masses, which is again compatible with recent observational data. Comparison with other microscopic EOS is presented and discussed., 15 pages, 11 figures, 8 tables

We calculate the effective masses of neutrons and protons in dense nuclear matter within the microscopic Brueckner-Hartree-Fock many-body theory and study the impact on the neutrino emissivity processes of neutron stars. We compare results based on different nucleon-nucleon potentials and nuclear three-body forces. Useful parametrizations of the numerical results are given. We find substantial in-medium suppression of the emissivities, strongly dependent on the interactions., Comment: 5 pages, 3 figures, accepted for publication as Brief Report in Phys. Rev. C

The Equation of State of pure neutron matter at zero temperature is calculated up to the three hole-line level of approximation in the Bethe-Brueckner-Goldstone expansion. The calculations are performed for the Av14 and the Av18 nucleon-nucleon potentials, up to the Fermi momentum kF=2.8 fm−1. The three hole-line contribution turns out to be smaller than in symmetric nuclear matter, and indeed a minor correction if the so-called “continuous choice” for the single particle potential is adopted. These findings are in line with the corresponding values of the wound parameter. The results are expected to be relevant for the theory of neutron stars structure.

We determine fully self-consistent single-particle potentials and chemical potentials of nucleons and hyperons in asymmetric nuclear matter, using an extended Brueckner-Hartree-Fock formalism. We carefully analyze the onset of ${\ensuremath{\Sigma}}^{\ensuremath{-}}$ and $\ensuremath{\Lambda}$ formation in $\ensuremath{\beta}$ stable and charge neutral matter. The role played by the three-body nucleon interaction is widely discussed. The results indicate that formation of hyperons sets in at about (2--3) times normal nuclear matter density, for all the different nucleonic equations of state that are considered.

We present results for the ${}^{3}{P}_{2}{\ensuremath{-}}^{3}{F}_{2}$ pairing gap in neutron matter with several realistic nucleon-nucleon potentials, in particular with recent phase-shift-equivalent potentials. We find that their predictions for the gap cannot be trusted at densities above $\ensuremath{\rho}\ensuremath{\approx}1.7{\ensuremath{\rho}}_{0},$ where ${\ensuremath{\rho}}_{0}$ is the saturation density for symmetric nuclear matter. In order to make predictions above that density, potential models which fit the nucleon-nucleon phase shifts up to about 1 GeV are required.

We compare a set of equations of state derived within microscopic many-body approaches, and study their predictions as far as phenomenological data on nuclei from heavy ion collisions, and astrophysical observations on neutron stars are concerned. We find that all the data, taken together, put strong constraints not easy to be fulfilled accurately. However, no major discrepancies are found among the selected equations of state and with respect to the data. The results provide an estimate of the uncertainty on the theoretical prediction at a microscopic level of the nuclear equation of state., 10 pages, 7 figures

The thermomagnetic evolution of the young neutron star in Cassiopea A is studied by considering fast neutrino emission processes. In particular, we consider neutron star models obtained from the equation of state computed in the framework of the Brueckner-Bethe-Goldstone many-body theory and variational methods, and models obtained with the Akmal-Pandharipande-Ravenhall equation of state. It is shown that it is possible to explain a fast cooling regime as the one observed in the neutron star in Cassiopea A if the Joule heating produced by dissipation of the small-scale magnetic field in the crust is taken into account. We thus argue that it is difficult to put severe constraints on the superfluid gap if the Joule heating is considered., Comment: 4 pages, 2 figures, to appear on A&A Letters

We study the hadron-quark phase transition at finite temperature in the interior of protoneutron stars, combining the Dyson-Schwinger model for quark matter with the Brueckner-Hartree-Fock approach for hadronic matter. We discuss the dependence of the results on different nuclear three-body forces and on details of the quark model. %Compared to the bag model, the transition from the hadron phase %(without hyperons) to the quark phase appears at large density. A maximum mass exceeding two solar masses can be obtained with a strong three-body force and suitable parameter values in the Dyson-Schwinger model. With a hybrid configuration, the maximum mass of protoneutron stars is larger then that of cold neutron stars, such that a delayed collapse might be possible in principle., Comment: 10 pages, 5 figures

We study the hadron-quark phase transition in the interior of neutron stars. For the hadronic sector, we use a microscopic equation of state involving nucleons and hyperons derived within the Brueckner-Hartree-Fock many-body theory with realistic two-body and three-body forces. For the description of quark matter, we employ the Dyson-Schwinger approach and compare with the MIT bag model. We calculate the structure of neutron star interiors comprising both phases and find that with the Dyson-Schwinger model, the hadron-quark phase transition takes place only when hyperons are excluded, and that a two-solar-mass hybrid star is possible only if the nucleonic equation of state is stiff enough., Comment: 10 pages, 8 figures

The thermal evolution of neuron stars depends on the elementary excitations affecting the stellar matter. In particular, the low-energy excitations, whose energy is proportional to the transfered momentum, can play a major role in the emission and propagation of neutrinos. In this paper, we focus on the density modes associated with the proton component in the homogeneous matter of the outer core of neutron stars (at density between one and three times the nuclear saturation density, where the baryonic constituants are expected to be neutrons and protons). In this region, it is predicted that the protons are superconductor. We study the respective roles of the proton pairing and Coulomb interaction in determining the properties of the modes associated with the proton component. This study is performed in the framework of the Random Phase Approximation, generalized in order to describe the response of a superfluid system.The formalism we use ensures that the Generalized Ward's Identities are satisfied. An important conclusion of this work is the presence of a pseudo-Goldstone mode associated with the proton superconductor in neutron-star matter. Indeed, the Goldstone mode, which characterizes a pure superfluid, is suppressed in usual superconductors due to the long-range Coulomb interaction, which only allows a plasmon mode. However, for the proton component of stellar matter, the Coulomb field is screened by the electrons and a pseudo-Goldstone mode occurs, with a velocity increased by the Coulomb interaction., Comment: Submitted for publication

The neutron-proton superfluidity for nuclear matter in the 3 S 1  3 D 1 channel is studied within the BCS formalism. In agreement with previous calculations based on the T -matrix approach, a large energy gap is found if one adopts the bare nucleon-nucleon interaction in the gap equation. The dependence of the results on the single-particle spectrum and on the nucleon-nucleon interaction is analyzed and discussed. The relevance of the results for the structure of neutron stars, nuclear matter saturation and superfluidity in finite nuclei is briefly discussed.

The 3 P 2 neutron superfluidity and the 1 S 0 proton superfluidity in neutron star matter are investigated by solving the gap equation exactly for a realistic nucleon-nucleon potential, namely the Argonne v 14 potential. For the 3 P 2 case, our results point to a nearly isotropic pairing function, in close analogy with the rigorous result obtained by Balian and Werthamer for S = 1 pairing without J -dependence. The proton abundance and the 1 S 0 proton gap are calculated and their relationship with the symmetry energy is discussed.

We compare a set of equations of state derived within microscopic many-body approaches, and study their predictions as far as phenomenological data on nuclei from heavy ion collisions, and astrophysical observations on neutron stars are concerned. All the data, taken together, put strong constraints not easy to be fulfilled accurately. However the results provide an estimate of the uncertainty on the theoretical prediction at a microscopic level of the nuclear equation of state.

A semimicroscopic self-consistent quantum approach was developed recently to describe the inner crust structure of neutron stars within the Wigner-Seitz method and with the explicit inclusion of neutron and proton pairing correlations is used for finding the neutron drip point that separates the outer and inner crusts. The equilibrium configurations of the crust are examined in vicinity of the drip point and in the upper part of the inner crust, for the density region corresponding to average Fermi momenta k F = 0.2 ÷ 0.5 fm -1 .

Recently, it has been shown that the standard Nambu-Jona-Lasinio (NJL) model is not able to reproduce the correct QCD behavior of the gap equation at large density, and therefore a different cutoff procedure at large momenta has ben proposed. We found that, even with this density dependent cutoff procedure, the pure quark phase in neutron stars (NS) interiors is unstable, and we argue that this could be related to the lack of confinement in the original NJL model., 2 pages, 1 figure, to be published in the proceedings of the conference EXOCT07, Catania, 11-15 June, 2007

We discuss the high-density nuclear equation of state within the Brueckner-Hartree-Fock approach. Particular attention is paid to the effects of nucleonic three-body forces, the presence of hyperons, and the joining with an eventual quark matter phase. The resulting properties of neutron stars, in particular the mass-radius relation, are determined. It turns out that stars heavier than 1.3 solar masses contain necessarily quark matter., Comment: 21 pages, 12 figures, Talk given at the NATO Advanced Research Workshop on Superdense QCD Matter and Compact Stars, 27 Sep - 4 Oct 2003, Yerevan, Armenia

The influence of the boundary conditions used in the Wigner-Seitz (WS) approximation applied to the neutron star inner crust is examined. The generalized energy functional method which includes the neutron and proton pairing correlations is used. Predictions of two versions of the boundary conditions are compared with each other. The uncertainties in the equilibrium configuration (Z,R_c) of the crust, where Z is the proton charge and R_c, the radius of the WS cell, correspond to variation of Z by 2 - 6 units and of R_c, by 1 - 2 fm. The effect of the boundary conditions is enhanced at increasing density. These uncertainties are smaller than the variation of Z and R_c coming from the pairing effects. In the case of high densities, k_F greater than 1 1/fm, the most important uncertainty occurs in the value of the neutron gap. In the WS approximation, it originates from the shell effect in the neutron single-particle spectrum which is rather pronounced in the case of larger k_F and, correspondingly, small R_c values, but it becomes negligible at lower density near the drip point. An approximate recipe to avoid this uncertainty is suggested., Comment: Extended version of arXiv: nucl-th/0601096

We explore the relevance of color superconductivity inside a possible quark matter core for the bulk properties of neutron stars. For the quark phase we use an Nambu--Jona-Lasinio (NJL) type model, extended to include diquark condensates. For the hadronic phase, a microscopic many-body model is adopted, with and without strangeness content. In our calculations, a sharp boundary is assumed between the hadronic and the quark phases. For NJL model parameters fitted to vacuum properties we find that no star with a pure quark core does exist. Nevertheless the presence of color superconducting phases can lower the neutron star maximum mass substantially. In some cases, the transition to quark matter occurs only if color superconductivity is present. Once the quark phase is introduced, the value of the maximum mass stays in any case below the value of two solar masses., 11 pages, 3 figures, v2: minor corrections in the text, layout of the figures improved, references added, v3: transition densities from hadronic to quark matter added, version accepted for publication in PLB

We study the relevance of the energy correlation produced by the two-body spin-orbit coupling present in realistic nucleon-nucleon potentials. To this purpose, the neutron matter Equation of State (EoS) is calculated with the realistic two-body Argonne $v_8'$ potential. The shift occuring in the EoS when spin-orbit terms are removed is taken as an estimate of the spin-orbit correlation energy. Results obtained within the Bethe-Brueckner-Goldstone expansion, extended up to three hole-line diagrams, are compared with other many-body calculations recently presented in the literature. In particular, excellent agreement is found with the Green's function Monte-Carlo method. This agreement indicates the present theoretical accuracy in the calculation of the neutron matter EoS., Comment: 5 pages, 2 figures, 2 tables; to appear in Phys. Rev. C

We study the consequences of the hadron‐quark phase transition for the values of the maximum mass of neutron stars (NS’s). For the hadronic phase, we use an equation of state (EOS) derived within the Brueckner‐Bethe‐Goldstone formalism with realistic two‐body and three‐body forces. For quark matter we employ the MIT bag model with a density dependent bag parameter. We calculate the structure of NS interiors with the EOS comprising both phases, and we find that the NS’s maximum masses fall in a relatively narrow interval, 1.4 M⊙ ⩽ Mmax ⩽ 1.7 M⊙, the precise value being only weakly correlated with the value of the energy density at the assumed transition point in nearly symmetric nuclear matter.

The Bethe-Brueckner-Goldstone many-body theory of the Nuclear Equation of State is reviewed in some details. In the theory, one performs an expansion in terms of the Brueckner two-body scattering matrix and an ordering of the corresponding many-body diagrams according to the number of their hole-lines. Recent results are reported, both for symmetric and for pure neutron matter, based on realistic two-nucleon interactions. It is shown that there is strong evidence of convergence in the expansion. Once three-body forces are introduced, the phenomenological saturation point is reproduced and the theory is applied to the study of neutron star properties. One finds that in the interior of neutron stars the onset of hyperons strongly softens the Nuclear Equation of State. As a consequence, the maximum mass of neutron stars turns out to be at the lower limit of the present phenomenological observation., 30 pages, 21 postscript figures included. Lecture presented at the workshop on "Physics of Neutron Stars Interior" held at the European Center for Theoretical Nuclear Physics, June 2000, Trento, Italy. To be published in Lecture Notes in Physics, eds. D. Blaschke, N. K. Glendenning and A. Sedrakian (Springer, 2001)

We make a first calculation of eigenfrequencies of radial pulsations of neutron stars with quark cores in a general relativistic formalism given by Chandrasekhar. The equations of state (EOS) used to estimate such eigenfrequencies have been derived by taking proper care of the hadron-quark phase transition. The hadronic EOS's have been obtained in the framework of the Brueckner-Hartree-Fock and relativistic mean field theories, whereas the quark EOS has been derived within the MIT bag model. We find that the periods of oscillations of neutron stars with a quark core show a kink, which is associated with the presence of a mixed phase region. Also, oscillation periods show significant differences between ordinary neutron stars and neutron stars with dynamical quark phases., 10 pages, TeX, submitted to Astrophysical Journal Letters

In the framework of the non-relativistic Brueckner-Bethe-Goldstone theory, we derive a microscopic equation of state for asymmetric and $\beta$-stable matter containing $\Sigma^-$ and $\Lambda$ hyperons. We mainly study the effects of three-body forces (TBFs) among nucleons on the hyperon formation and the equation of state (EoS). We find that, when TBFs are included, the stellar core is almost equally populated by nucleons and hyperons. The resulting EoS, which turns out to be extremely soft, has been used in order to calculate the static structure of neutron stars. We obtain a value of the maximum mass of 1.26 solar masses (1 solar mass $M_o \simeq 1.99 \cdot 10^{33} g$). Stellar rotations increase this value by about 12%., Comment: 4 pages, Latex, 2 figures included. To appear in the Proceedings of '' Bologna 2000 - Structure of the Nucleus at the Dawn of the Century'', May 29- June 3, 2000, Bologna, Italy

Massive neutron stars (NS) are expected to possess a quark core. While the hadronic side of the NS equation of state (EOS) can be considered well established, the quark side is quite uncertain. While calculating the EOS of hadronic matter we have used the Brueckner-Bethe-Goldstone formalism with realistic two-body and three-body forces, as well as a relativistic mean field model. For quark matter we employ the MIT bag model constraining the bag constant by exploiting the recent experimental results obtained at CERN on the formation of a quark-gluon plasma. We calculate the structure of NS interiors with the EOS comprising both phases, and we find that the NS maximum masses fall in a relatively narrow interval, $1.45 M_\odot \leq M_{\rm max} \leq 1.65 M_\odot$, near the lower limit of the observational range., Comment: 12 pages, TeX, submitted to Physics Letters B