Core-crust Transition Density Research Articles

Impact of the hot inner crust on compact stars at finite temperature

We conducted a study on the thermal properties of stellar matter with the nuclear energy density functional BCPM. This functional is based on microscopic Brueckner-Hartree-Fock calculations and has demonstrated success in describing cold neutron stars. To enhance its applicability in astrophysics, we extended the BCPM equation of state to finite temperature for beta -stable neutrino-free matter, taking into consideration the hot inner crust. Such an equation of state holds significant importance for hot compact objects, particularly those resulting from a binary neutron star merger event. Our exploration has shown that with increasing temperature, there is a fast decrease in the crust-core transition density, suggesting that for hot stars it is not realistic to assume a fixed value of this density. The microscopic calculations also reveal that the presence of nuclear clusters persists up to $T=7.21$\,MeV, identified as the limiting temperature of the crust. Above this threshold, the manifestation of clusters is not anticipated. Below this temperature, clusters within the inner crust are surrounded by uniform matter with varying densities, allowing for the distinction between the upper and lower transition density branches. Moreover, we computed mass--radius relations of neutron stars, assuming an isothermal profile for beta -stable neutron star matter at various temperature values. Our findings highlight the significant influence of the hot inner crust on the mass--radius relationship, leading to the formation of larger and more inflated neutron stars. Consequently, under our prescription, the final outcome is a unified equation of state at finite temperature.

Mass–radius relation of neutron stars and massive pulsars with realistic equation of state

We use up-to-date constraints on the mass and radius of 15 neutron star objects and pulsars, from electromagnetic and gravitational wave observables and different theoretical schemes, to extend the nuclear equation of state (EOS) based on realistic Michigan-three-Yukawa (M3Y) nucleon–nucleon interaction, which truly accounts for the low-density EOS of nuclear matter (NM), to describe dense NM. The considered EOSs are employed to map the mass–radius profiles using the Tolman–Oppenheimer–Volkoff equations of hydrostatic equilibrium. We found that the EOSs from CDM3Y-230 to CDM3Y-270, with saturation incompressibility K 0 = 230–270 MeV, successfully reproduce most of the recent constraints on the NS masses and radii. Based on both M3Y-Paris and M3Y-Reid NN interactions, these EOSs indicate radius of 11.67 ± 0.34 km for the NS of 1.4 M ʘ mass, and the expected maximum NS mass (M max) to be 1.93 ± 0.21 M ʘ. The upper limits of constraints indicated stiffer EOSs of K 0 = 300–330 MeV, which have estimated 1arger radii of 12.29 ± 0.14 km for NS (1.4 M ʘ) and heavier M max of 2.31 ± 0.14 M ʘ. Increasing the stiffness of the employed EOS is found to increase the indicated maximum mass of NS, its radius and maximum compactness, the core–crust transition density, the speed of sound in its interior, and slightly the transition proton-fraction, but to decrease the abundance of the proton, muon, and electron over npeμ core matter of NS, as well as the estimated central density.

Neutron star core-crust transition and the crustal moment of inertia in the nonlinear relativistic Hartree approximation* *Supported in part by the National Natural Science Foundation of China (11775049, 12375112)

We investigate the effects of the σ meson mass ( ), symmetry energy, and slope of the symmetry energy on the neutron star core-crust transition density and the crustal moment of inertia ( ) in the nonlinear relativistic Hartree approach (RHA), which includes vacuum polarization. Although the core-crust transition density ( ), pressure ( ), and neutron star radius (R), which are all dependent on the symmetry energy, contribute to determining , we find that changing only the slope of symmetry energy within a reasonable range is not sufficient to reach % to achieve the large glitches of the Vela pulsar. However, since all three factors ( , , and R) increase with the increase in through scalar vacuum polarization, adjusting can easily achieve %.

A Statistical Approach to Neutron Stars' Crust-Core Transition Density and Pressure.

In this paper, a regression model between neutron star crust-core pressure and the symmetry energy characteristics was estimated using the Akaike information criterion and the adjusted coefficient of determination Radj2. The most probable value of the transition density, which should characterize the crust-core environment of the sought physical neutron star model, was determined based on the obtained regression function. An anti-correlation was found between this transition density and the main characteristic of the symmetry energy, i.e., its slope L.

Statistical analysis of the effect of the symmetry energy on the crust-core transition density and pressure in neutron stars

Statistical analysis of the effect of the symmetry energy on the crust-core transition density and pressure in neutron stars

Strong magnetic fields and pasta phases reexamined

In this work, we compute the structure and composition of the inner crust of a neutron star in the presence of a strong magnetic field, such as it can be found in magnetars. To determine the geometry and characteristics of the crust inhomogeneities, we consider the compressible liquid drop model, where surface and Coulomb terms are included in the variational equations, and we compare our results with previous calculations based on more approximate treatments. For the equation of state (EoS), we consider two non-linear relativistic mean-field models with different slopes of the symmetry energy, and we show that the extension of the inhomogeneous region inside the star core due to the magnetic field strongly depends on the behavior of the symmetry energy in the crustal EoS. Finally, we argue that the extended spinodal instability observed in previous calculations can be related to the presence of small amplitude density fluctuations in the magnetar outer core, rather than to a thicker solid crust. The compressible liquid drop model formalism, while in overall agreement with the previous calculations, leads to a systematic suppression of the metastable solutions, thus allowing a more precise estimation of the crust-core transition density and pressure, and therefore a better estimation of the crustal radius.

Strong correlation of the neutron star core-crust transition density with the σ-meson mass via vacuum polarization

We study the neutron star core-crust transition density ρt with the inclusion of the vacuum polarization in the dielectric function in the nonlinear relativistic Hartree approach (RHAn). It is found that the strong correlation between the ρt and the scalar meson mass mσ strikingly overwhelms the uncertainty of the nuclear equation of state in the RHAn models, in contrast to the usual awareness that ρt is predominantly sensitive to the isovector nuclear potential and symmetry energy. The accurate extraction of ρt through the future gravitational wave measurements can thus provide a strong constraint on the longstanding uncertainty of mσ, which is of significance to better infer the vacuum property. As an astrophysical implication, it suggests that the correlation between ρt and mσ is very favorable to reconcile the difficulty in reproducing the large crustal moment of inertia for the pulsar glitches with the well constrained symmetry energy.

Unified neutron star EOSs and neutron star structures in RMF models

In the framework of the Thomas-Fermi approximation, we systematically study the EOSs and microscopic structures of neutron star matter in a vast density range with n b ≈ 10−10-2 fm−3, where various covariant density functionals are adopted, i.e., those with nonlinear self couplings (NL3, PK1, TM1, GM1, MTVTC) and density-dependent couplings (DD-LZ1, DDME-X, PKDD, DD-ME2, DD2, TW99). It is found that the EOSs generally coincide with each other at n b ≲ 10−4 fm−3 and 0.1 fm−3 ≲ n b ≲ 0.3 fm−3, while in other density regions they are sensitive to the effective interactions between nucleons. By adopting functionals with a larger slope of symmetry energy L, the curvature parameter K sym and neutron drip density generally increases, while the droplet size, proton number of nucleus, core-crust transition density, and onset density of non-spherical nuclei, decrease. All functionals predict neutron stars with maximum masses exceeding the two-solar-mass limit, while those of DD2, DD-LZ1, DD-ME2, and DDME-X predict optimum neutron star radii according to the observational constraints. Nevertheless, the corresponding skewness coefficients J are much larger than expected, while only the functionals MTVTC and TW99 meet the start-of-art constraints on J. More accurate measurements on the radius of PSR J0740 + 6620 and the maximum mass of neutron stars are thus essential to identify the functional that satisfies all constraints from nuclear physics and astrophysical observations. Approximate linear correlations between neutron stars’ radii at M = 1.4M ⊙ and 2M ⊙, the slope L and curvature parameter K sym of symmetry energy are observed as well, which are mainly attributed to the curvature-slope correlations in the functionals adopted here. The results presented here are applicable for investigations of the structures and evolutions of compact stars in a unified manner.

Low Density Neutron Star Matter with Quantum Molecular Dynamics: The Role of Isovector Interactions

The effect of isospin-dependent nuclear forces on the inner crust of neutron stars is modeled within the framework of Quantum Molecular Dynamics (QMD). To successfully control the density dependence of the symmetry energy of neutron-star matter below nuclear saturation density, a mixed vector-isovector potential is introduced. This approach is inspired by the baryon density and isospin density-dependent repulsive Skyrme force of asymmetric nuclear matter. In isospin-asymmetric nuclear matter, the system shows nucleation, as nucleons are arranged into shapes resembling nuclear pasta. The dependence of clusterization in the system on the isospin properties is also explored by calculating two-point correlation functions. We show that, as compared to previous results that did not involve such mixed interaction terms, the energy symmetry slope L is successfully controlled by varying the corresponding coupling strength. Nevertheless, the effect of changing the slope of the nuclear symmetry energy L on the crust-core transition density does not seem significant. To the knowledge of the authors, this is the first implementation of such a coupling in a QMD model for isospin asymmetric matter, which is relevant to the inner crust of neutron and proto-neutron stars.

Theoretical Study of the Role of Symmetry Energy as Well as Its Density Slope and Curvature on Neutron Star Core Crust Transition Density Using Finite Range Effective Interaction

Theoretical Study of the Role of Symmetry Energy as Well as Its Density Slope and Curvature on Neutron Star Core Crust Transition Density Using Finite Range Effective Interaction

Crust-core transition of a neutron star: effect of the temperature under strong magnetic fields

The effect of temperature on the crust-core transition of a magnetar is studied. The thermodynamical spinodals are used to calculate the transition region within a relativistic mean-field approach for the equation of state. Magnetic fields with intensities $5\times 10^{16}$ G and $5\times 10 ^{17}$ G are considered. It is shown that the effect on the extension of the crust-core transition is washed away for temperatures above $10^{9}$ K for magnetic field intensities $ \lesssim 5\times 10^{16}$ G but may still persist if a magnetic field as high as $5\times 10 ^{17}$G is considered. For temperatures below that value, the effect of the magnetic field on crust-core transition is noticeable and grows as the temperature decreases and, in particular, it is interesting to identify the existence of disconnected non-homogeneous matter above the $B=0$ crust core transition density. Models with different symmetry energy slopes at saturation show quite different behaviors. In particular, a model with a large slope, as suggested by the recent results of PREX-2, predicts the existence of up to four disconnected regions of non-homogeneous matter above the zero magnetic field crust-core transition density.

Crust-core transition and dynamical instabilities in neutron stars for model with point-coupling interactions * *Supported by Changsha Municipal Natural Science Foundation (kq2007004) and the construct program of the key discipline in Hunan province

We explore the effects of the density dependence of symmetry energy on the crust-core phase transition and dynamical instabilities in cold and warm neutron stars in the relativistic mean field (RMF) theory with point-coupling interactions using the Vlasov approach. The role of neutrino trapping is also considered. The crust-core transition density and pressure, distillation effect, and cluster size and growth rates are discussed. The present work shows that the slope of symmetry energy at saturation, temperature, and neutrino trapping have non-negligible effects.

Constraints on the nuclear symmetry energy from asymmetric-matter calculations with chiral NN and 3N interactions

The nuclear symmetry energy is a key quantity in nuclear (astro)physics. It describes the isospin dependence of the nuclear equation of state (EOS), which is commonly assumed to be almost quadratic. In this work, we confront this standard quadratic expansion of the EOS with explicit asymmetric nuclear-matter calculations based on a set of commonly used Hamiltonians including two- and three-nucleon forces derived from chiral effective field theory. We study, in particular, the importance of non-quadratic contributions to the symmetry energy, including the non-analytic logarithmic term introduced by Kaiser [Phys.~Rev.~C \textbf{91}, 065201 (2015)]. Our results suggest that the quartic contribution to the symmetry energy can be robustly determined from the various Hamiltonians employed, and we obtain 1.00(8) MeV (or 0.55(8) MeV for the potential part) at saturation density, while the logarithmic contribution to the symmetry energy is relatively small and model-dependent. We finally employ the meta-model approach to study the impact of the higher-order contributions on the neutron-star crust-core transition density, and find a small 5\% correction.

Impact of strong magnetic fields on the inner crust of neutron stars

We study the impact of strong magnetic fields on the pasta phases that are expected to exist in the inner crust of neutron stars. We employ the relativistic mean field model to describe the nucleon interaction and use the self-consistent Thomas-Fermi approximation to calculate the nonuniform matter in neutron star crust. The properties of pasta phases and crust-core transition are examined. It is found that as the magnetic field strength $B$ is less than ${10}^{17}$ G, the effects of magnetic field are not evident comparing with the results without magnetic field. As $B$ is stronger than ${10}^{18}$ G, the onset densities of pasta phases and crust-core transition density decrease significantly, and the density distributions of nucleons and electrons are also changed obviously.

Probing Crust Meltdown in Inspiraling Binary Neutron Stars.

Thanks to recent measurements of tidal deformability and radius, the nuclear equation of state and structure of neutron stars are now better understood. Here, we show that through resonant tidal excitations in a binary inspiral, the neutron crust generically undergoes elastic-to-plastic transition, which leads to crust heating and eventually meltdown. This process could induce ∼O(0.1) phase shift in the gravitational waveform. Detecting the timing and induced phase shift of this crust meltdown will shed light on the crust structure, such as the core-crust transition density, which previous measurements are insensitive to. A direct search using GW170817 data has not found this signal, possibly due to limited signal-to-noise ratio. We predict that such a signal may be observable with Advanced LIGO Plus and more likely with third-generation gravitational-wave detectors such as the Einstein Telescope and Cosmic Explorer.

Curvature-slope correlation of nuclear symmetry energy and its imprints on the crust-core transition, radius, and tidal deformability of canonical neutron stars

Background: The nuclear symmetry energy $E_{sym}(\rho)$ encodes information about the energy necessary to make nuclear systems more neutron-rich. While its slope parameter L at the saturation density $\rho_0$ of nuclear matter has been relatively well constrained by recent astrophysical observations and terrestrial nuclear experiments, its curvature $K_{\rm{sym}}$ characterizing the $E_{sym}(\rho)$ around $2\rho_0$ remains largely unconstrained. Over 520 calculations for $E_{sym}(\rho)$ using various nuclear theories and interactions in the literature have predicted several significantly different $K_{\rm{sym}}-L$ correlations. Purpose: If a unique $K_{\rm{sym}}-L$ correlation of $E_{sym}(\rho)$ can be firmly established, it will enable us to progressively better constrain the high-density behavior of $E_{sym}(\rho)$ using the available constraints on its slope parameter L. We investigate if and by how much the different $K_{\rm{sym}}-L$ correlations may affect neutron star observables. Method: A meta-model of nuclear Equation of States (EOSs) with three representative $K_{\rm{sym}}-L$ correlation functions is used to generate multiple EOSs for neutron stars. We then examine effects of the $K_{\rm{sym}}-L$ correlation on the crust-core transition density and pressure as well as the radius and tidal deformation of canonical neutron stars. Results:The $K_{\rm{sym}}-L$ correlation affects significantly both the crust-core transition density and pressure. It also has strong imprints on the radius and tidal deformability of canonical neutron stars especially at small L values. The available data from LIGO/VIRGO and NICER set some useful limits for the slope L but can not distinguish the three representative $K_{\rm{sym}}-L$ correlations considered.

Neutron star inner crust: Effects of rotation and magnetic fields

We study the role of the pasta phases on the properties of rotating and magnetized neutron stars. In order to investigate such systems, we make use of two different relativistic mean-field unified inner-crust--core equations of state, with a different density dependence of the symmetry energy, and an inner crust computed within a Thomas-Fermi calculation. Special attention is given to the crust-core transition density, and the pasta phases effects on the global properties of stars. The effects of strong magnetic fields and fast rotation are computed by solving the Einstein-Maxwell equations self-consistently, taking into account anisotropies induced by the centrifugal and the Lorentz force. The location of the magnetic field neutral line and the maximum of the Lorentz force on the equatorial plane are calculated. The conditions under which they fall inside the inner crust region are discussed. We verified that models with a larger symmetry energy slope show more sensitivity to the variation of the magnetic field. One of the maxima of the Lorentz force, as well as the neutral line, and for a certain range of frequencies, fall inside the inner crust region. This may have consequences in the fracture of the crust, and may help explain phenomena associated with star quakes.

Constraints on the inner edge of neutron star crusts from relativistic nuclear energy density functionals

The transition density nt and pressure Pt at the inner edge between the liquid core and the solid crust of a neutron star are analyzed using the thermodynami- cal method and the framework of relativistic nuclear energy density functionals. Starting from a functional that has been carefully adjusted to experimental binding energies of finite nuclei, and varying the density dependence of the cor- responding symmetry energy within the limits determined by isovector prop- erties of finite nuclei, we estimate the constraints on the core-crust transition density and pressure of neutron stars: 0.086 fm−3 ≤ nt < 0.090 fm−3 and 0.3 MeV fm−3 < Pt ≤ 0.76 MeV fm−3 [1].

Properties of the nuclear symmetry energy studied by the radioactivity of unstable nuclei

In this paper, the properties of the symmetry energy in asymmetric nuclear matter, including the quadratic and quartic symmetry energies as well as their density slopes at the saturation density, are studied by the radioactivity of unstable nuclei. The symmetry energy was initially introduced as a quantum correction energy in the Weizsacker-Bethe liquid-drop model, which was proposed to describe the binding energies of finite nuclei. In order to systematically study the properties of the symmetry energy, the concept of the nuclear matter was introduced, which consists of neutrons and protons with finite neutron and proton densities but infinite neutron and proton numbers. By comparing the properties of the isospin symmetric nuclear matter, in which the neutron density is equal to the proton density, and those of the isospin asymmetric nuclear matter, in which the two densities are unequal, the symmetry energy is found to approximately represent the difference of the energy per nucleon in them. That is to say, the symmetry energy is an evaluation of the energy cost to convert protons in symmetric nuclear matter to neutrons in asymmetric nuclear matter. Since the symmetry energy characterizes the isospin asymmetric effect in nuclear matter, it plays critical roles in not only nuclear physics but also astrophysics. For example, the symmetry energy is closely related to the nuclear masses, the structures and properties of nuclei far away from stability line and near drip lines, the mechanism of heavy-ion reactions, the structures and component of neutron stars, the masses and radii of neutron stars, the cooling process of neutron stars, and so on. Besides of the quadratic symmetry energy, the quartic symmetry energy can have obvious effects on the properties of neutron stars as well. The quartic symmetry energy can significantly affect both the proton fraction in β-stable neutron stars and the critical density for the direct Urca process, which leads to faster cooling of neutron stars. In addition, the quartic symmetry energy is found to be very important for the location of the inner edge of crusts and the core-crust transition density and pressure in neutron stars. So the quartic symmetry energy can also have an influence on the structure of neutron stars. Because of the very important roles of the symmetry energy playing in not only nuclear physics but also astrophysics, it is of great significance to study the properties of both the quadratic and quartic symmetry energies. Based on the fundamental Hugenholtz-Van Hove (HVH) theorem, the properties of the symmetry energy are found to be closely related to the isoscalar and the isovector parts of the nucleon single-particle potentials, where the latter two parts can be extracted from the nuclear radioactivity. So by linking both the symmetry energy and the nuclear radioactivity to the nucleon single-particle potentials, the symmetry energy can be directly studied by nuclear radioactivity. In this paper, both the heavy-cluster radioactivity and the proton radioactivity are employed to constrain the properties of not only the quadratic symmetry energy but also the quartic symmetry energy. Besides, the effect of both the spectroscopic factor of proton radioactivity and the deformation in the daughter nucleus on the extracted results are also discussed, which are subsequently found to be very limited.

Bayesian modeling of the nuclear equation of state for neutron star tidal deformabilities and GW170817

We present predictions for neutron star tidal deformabilities obtained from a Bayesian analysis of the nuclear equation of state, assuming a minimal model at high-density that neglects the possibility of phase transitions. The Bayesian posterior probability distribution is constructed from priors obtained from microscopic many-body theory based on realistic two- and three-body nuclear forces, while the likelihood functions incorporate empirical information about the equation of state from nuclear experiments. The neutron star crust equation of state is constructed from the liquid drop model, and the core-crust transition density is found by comparing the energy per baryon in inhomogeneous matter and uniform nuclear matter. From the cold $\beta$-equilibrated neutron star equation of state, we then compute neutron star tidal deformabilities as well as the mass-radius relationship. Finally, we investigate correlations between the neutron star tidal deformability and properties of finite nuclei.