The inner crust of neutron stars hosts a rich variety of nuclear phenomena and provides a unique environment for exploring microscopic nuclear properties relevant to diverse astrophysical observations. Particularly magnetars, which possess extremely strong magnetic-fields, have attracted increasing attention in connection with nuclear spin dynamics and unconventional pairing correlations. This work is dedicated to develop a comprehensive theoretical framework to describe the structures and properties of two-dimensional (rod-phase) matter in the neutron star inner crust, incorporating band-structure effects, neutron spin-triplet pairing, and strong magnetic-fields on an equal footing. The main results of this study can be summarized as follows. In the first place, the magnetic-fields of the order of $10^{16}\,$G are found to substantially enhance the neutron effective mass by a factor of approximately $1.5$, indicating a significant modification of entrainment properties in strongly magnetized crustal matter. In the second place, while the overall behavior of pairing phase transitions is qualitatively similar to that observed in one-dimensional systems studied previously, the present two-dimensional calculations reveal a nontrivial role of the spin-orbit interaction in inducing spin-polarization under magnetic fields. In the third place, concerning spin-triplet superfluidity, the rank-0 component is shown to emerge as a consequence of magnetic-field-induced spin-polarization, irrespective of the presence of spin-triplet pairing interactions, whereas the rank-2 component appears only when the corresponding interaction channel is included.
In spite of the absence of viscous drag, the neutron superfluid permeating the inner crust of a neutron star cannot flow freely, and is entrained by the nuclear lattice similarly to laboratory superfluid atomic gases in optical lattices. The role of entrainment on the neutron superfluid dynamics is reviewed. For this purpose, a minimal hydrodynamical model of superfluidity in neutron-star crusts is presented. This model relies on a fully four-dimensionally covariant action principle. The equivalence of this formulation with the more traditional approach is demonstrated. In addition, the differ…
We calculate the superfluid density of nucleons in disordered pasta phases in the inner crust of neutron stars using an effective medium approach which parallels that previously used for calculating the electrical conductivity of terrestrial matter. We allow for the effect of entrainment, the fact that the current density of one species of nucleon depends on the gradient of the phase not only of the same species but also of the other species. The superfluid density tensors for the perfectly ordered pasta phases are very anisotropic, and we derive expression for the effective superfluid densiti…
I construct a model of the inner crust of neutron stars using interactions from chiral effective field theory (EFT) in order to calculate its equation of state (EOS), shear properties, and the spectrum of crustal shear modes. I systematically study uncertainties associated with the nuclear physics input, the crust composition, and neutron entrainment, and estimate their impact on crustal shear properties and the shear-mode spectrum. I find that the uncertainties originate mainly in two sources: The neutron-matter EOS and neutron entrainment. I compare the spectrum of crustal shear modes to obs…
Effect of BCS pairing on entrainment in neutron superfluid current in neutron star crust
在中子滴阈以上的中子星壳中传导中子的相对电流密度$n^i$可以预期与相应的粒子动量共矢量$p_i$通过$n^i=K^{ij}p_j$形式的线性关系以物理上良好定义的迁移率张量$K^{ij}$相关。这个结果可以描述为“夹带”，其效果--只要地壳晶格是各向同性的--将简单地将普通中子质量m改变为有效质量$m_\star$，使得根据无约束中子的相关数密度n，我们将得到$K^{ij}=n/m_\starγ^{ij}$。在前面的分析的基础上，一个独立的粒子处理使用布洛赫边界条件，以获得能量分布$E_k$和相关的群速度$v_k^i=\partial E_k/\partial\hbar k_i$作为波数$k_i$的函数，它被证明，迁移率张量将由$K^{ij}\propto\int d^3 k v_k^i v_k^jδ\{E_k-μ\}$，其中$μ$是费米能量。利用Bogoliubov的方法，本文证明了BCS对的能隙Δ_F和相应的准粒子能量函数\EE_k = \sqrt{(E_k-μ)^2+Δ_F^2} $ will just be to replace the Dirac distributional integrand by a smoother distribution : $ K^{ij} \propto \int d^3 k v_k^i v_k^j Δ_F^2 / \EE^3_k $. …
The neutron star crustal EoS and transition point properties are computed within a unified meta-modeling approach. A Bayesian approach is employed including two types of filters: bulk nuclear properties are controlled from low density effective field theory (EFT) predictions as well as the present knowledge from nuclear experiments, while the surface energy is adjusted on experimental nuclear masses. Considering these constraints, a quantitative prediction of crustal properties can be reached with controlled confidence intervals and increased precision with respect to previous calculations: {$…
In order to assess the ability of purely crust-driven glitch models to match the observed glitch activity in the Vela pulsar, we conduct a systematic analysis of the dependence of the fractional moment of inertia of the inner crustal neutrons on the stiffness of the nuclear symmetry energy at saturation density $L$. We take into account both crustal entrainment and the fact that only a fraction $Y_{\rm g}$ of the core neutrons may couple to the crust on the glitch-rise timescale. We use a set of consistently-generated crust and core compositions and equations-of-state which are fit to results …
Pulsars are highly magnetized rotating neutron stars with a very stable rotation speed. Irrespective of their stable rotation rate, many pulsars have been observed with the sudden jump in the rotation rate, which is known as pulsar glitch. The glitch phenomena are considered to be an exhibit of superfluidity of neutron matter inside the neutron star's crustal region. The magnitude of such rapid change in rotation rate relative to their stable rotation frequency can quantify the moment of inertia of the crustal region to the total moment of inertia of the star called as the fractional moment of…
The mass, radius and crustal fraction of moment of inertia in neutron stars are calculated using $β$-equilibrated nuclear matter obtained from Skyrme effective interaction. The transition density, pressure and proton fraction at the inner edge separating the liquid core from the solid crust of the neutron stars are determined from the thermodynamic stability conditions using the KDE0v1 set. The neutron star masses obtained by solving the Tolman-Oppenheimer-Volkoff equations using neutron star matter obtained from this set is able to describe highly massive compact stars $\sim$2$M_{\odot}$. The…
Background: In the solid crust of neutron stars, a variety of crystalline structure may exist. Recently the band theory of solids has been applied to the inner crust of neutron stars and significance of the entrainment between dripped neutrons and the solid crust was advocated. Since it influences interpretations of various phenomena of neutron stars, it has been desired to develop deeper understanding of the microphysics behind. Purpose: The purpose of the present article is to propose a fully self-consistent microscopic framework for describing time-dependent dynamics of neutron star matter,…
If the observed quasi-periodic oscillations in magnetar flares are partially confined to the crust, then the oscillation frequencies are unique probes of the nuclear physics of the neutron star crust. We study crustal oscillations in magnetars including corrections for a finite Alfven velocity. Our crust model uses a new nuclear mass formula that predicts nuclear masses with an accuracy very close to that of the finite range droplet model. This mass model for equilibrium nuclei also includes shell corrections and an updated neutron-drip line. We perturb our crust model to predict axial crust m…
We study the long-wavelength excitations of the inner crust of neutron stars, considering three phases: cubic crystal at low densities, rods and plates near the core-crust transition. To describe the phonons, we write an effective Lagrangian density in terms of the coarse-grained phase of the neutron superfluid gap and of the average displacement field of the clusters. The kinetic energy, including the entrainment of the neutron gas by the clusters, is obtained within a superfluid hydrodynamics approach. The potential energy is determined from a model where clusters and neutron gas are conside…
The inner crust of neutron stars is supposed to be inhomogeneous and composed of dense structures (clusters) that are immersed in a dilute gas of unbound neutrons. Here we consider spherical clusters forming a BCC crystal and cylindrical rods arranged in a hexagonal lattice. We study the relative motion of these dense structures and the neutron gas using superfluid hydrodynamics. Within this approach, which relies on the assumption that Cooper pairs are small compared to the crystalline structures, we find that the entrainment of neutrons by the clusters is very weak since neutrons of the gas …
Superfluid extension of the self-consistent time-dependent band theory for neutron star matter: Anti-entrainment versus superfluid effects in the slab phase
The pasta phases predicted to occur near the inner boundary of the crust of a neutron star resemble liquid crystals, a smectic A in the case of sheet-like nuclei (lasagna) and the columnar phase in the case of rod-like nuclei (spaghetti). An important difference compared with usual liquid crystals is that the nucleons are superfluid. We develop the hydrodynamic equations for this system and use them to study collective oscillations. Nucleon superfluidity leads to important qualitative differences in the spectra of these oscillations and also increases their frequencies compared with ordinary l…
Nuclear matter properties based on a relativistic approach suitable for the description of multi-component systems are calculated. We use a set of nuclear relativistic mean-field models that satisfy acceptable nuclear matter properties and neutron star observations. The effects of the density dependence of the symmetry energy and of the Landau quantization due to the presence of a strong external magnetic field are discussed. Properties such as the proton fraction, the Landau mass, Landau parameters and entrainment matrix, the adiabatic index and speed of sound are calculated for cold $β$-equi…
We calculate crustal properties of neutron stars, namely, mass ($M_{\rm crust}$), radius ($R_{\rm crust}$) and fraction of moment of inertia ($ΔI/I$) from parametrizations of hadronic relativistic mean-field (RMF) model consistent with symmetric and asymmetric nuclear matter constraints, as well as some stellar boundaries. We verify which one are also in agreement with restrictions of $ΔI/I \geqslant 1.4\%$ and $ΔI/I \geqslant 7\%$ related to the glitching mechanism observed in pulsars, such as the Vela one. The latter constraint explains the glitches phenomenon when entrainment effects are ta…
We perform a systematic assessment of models for the equation of state (EOS) of dense matter in the context of recent neutron star mass and radius measurements to obtain a broad picture of the structure of neutron stars. We demonstrate that currently available neutron star mass and radius measurements provide strong constraints on moments of inertia, tidal deformabilities, and crust thicknesses. A measurement of the moment of inertia of PSR J0737-3039A with 10% error, without any other information from observations, will constrain the EOS over a range of densities to within 50%$-$60%. We find …
We demonstrate that observations of glitches in the Vela pulsar can be used to investigate the strength of the crust-core coupling in a neutron star, and suggest that recovery from the glitch is dominated by torque exerted by the re-coupling of superfluid components of the core that were decoupled from the crust during the glitch. Assuming that the recoupling is mediated by mutual friction between the superfluid neutrons and the charged components of the core, we use the observed magnitudes and timescales of the shortest timescale components of the recoveries from two recent glitches in the Ve…
A mixture of superconducting and superfluid nuclear liquids of protons coupled to the ultrarelativistic electron gas, and neutrons is considered. In the magnetohydrodynamic (MHD) approximation, the energy-momentum (stress) tensor is derived, and the entrainment contribution is found in the explicit form. It is shown that this contribution generates a force density, when a superfluid velocity lag and the magnetic field are simultaneously present. This force may be important in the nuclear "pasta" phase in neutron stars, if the proton and neutron Cooper pairing in the pasta phase is taken into a…
We consider torsional oscillations that are trapped in a layer of spherical-hole (bubble) nuclear structure, which is expected to occur in the deepest region of the inner crust of a neutron star. Because this layer intervenes between the phase of slab nuclei and the outer core of uniform nuclear matter, torsional oscillations in the bubble phase can be excited separately from usual crustal torsional oscillations. We find from eigenmode analyses for various models of the equation of state of uniform nuclear matter that the fundamental frequencies of such oscillations are almost independent of t…
We investigate neutron star moments of inertia from Bayesian posterior probability distributions of the nuclear equation of state that incorporate information from microscopic many-body theory and empirical data of finite nuclei. We focus on PSR J0737-3039A and predict that for this 1.338 M_sun neutron star the moment of inertia lies in the range $1.04 \times 10^{45}$ g cm$^{2} < I < 1.51 \times 10^{45}$ g cm$^{2}$ at the 95% credibility level, while the most probable value for the moment of inertia is $\tilde I = 1.36 \times 10^{45}$ g cm$^{2}$. Assuming a measurement of the PSR J0737-3039A m…
Nonuniform structure of low-density nuclear matter, known as nuclear pasta, is expected to appear not only in the inner crust of neutron stars but also in core-collapse supernova explosions and neutron-star mergers. We perform fully three-dimensional calculations of inhomogeneous nuclear matter and neutron-star matter in the low-density region using the Thomas-Fermi approximation. The nuclear interaction is described in the relativistic mean-field approach with the point-coupling interaction, where the meson exchange in each channel is replaced by the contact interaction between nucleons. We i…
Nuclear density functional theory (DFT) is the tool of choice in describing properties of complex nuclei and intricate phases of bulk nucleonic matter. It is a microscopic approach based on an energy density functional representing the nuclear interaction. An attractive feature of nuclear DFT is that it can be applied to both finite nuclei and pasta phases appearing in the inner crust of neutron stars. While nuclear pasta clusters in a neutron star can be easily characterized through their density distributions, the level of clustering of nucleons in a nucleus can often be difficult to assess.…
In the framework of the relativistic mean field model with Thomas-Fermi approximation, we study the structures of low density nuclear matter in a three-dimensional geometry with reflection symmetry. The numerical accuracy and efficiency are improved by expanding the mean fields according to fast cosine transformation and considering only one octant of the unit cell. The effect of finite cell size is treated carefully by searching for the optimum cell size. Typical pasta structures (droplet, rod, slab, tube, and bubble) arranged in various crystalline configurations are obtained for both fixed …
