We systematically examine torsional shear oscillations of neutron star crusts by newly taking into account the possible presence of the phase of cylindrical nuclei. In this study, we neglect an effect of magnetic fields, under which the shear oscillations can be damped by the magnetic interaction. First, by identifying the low frequency quasi-periodic oscillations (QPOs) observed in the soft-gamma repeaters (SGRs) as the fundamental torsional oscillations, we constrain the slope parameter of the nuclear symmetry energy, $L$, for reasonable values of the star's mass $M$ and radius $R$. Meanwhile, we find that the 1st overtone of torsional oscillations obtained for given $M$ and $R$ can be expressed well as a function of a new parameter $ς\equiv (K_0^4 L^5)^{1/9}$, where $K_0$ is the incompressibility of symmetric nuclear matter. Assuming that the lowest of the QPO frequencies above 500 Hz observed in SGR 1806-20 comes from the 1st overtone, we can constrain the value of $ς$. Then, for each neutron star model, such a value of $L$ as can be obtained from the observed low frequency QPOs translates to the optimal value of $K_0$ via the above constraint on $ς$. Finally, its consistency with allowed values of $K_0$ from empirical giant monopole resonances leads to neutron star models with relatively low mass and large radius, which are qualitatively similar to the prediction in earlier investigations. This result suggests that $L\simeq 58-73$ MeV, even when uncertainties in the neutron superfluid density inside the phase of cylindrical nuclei are allowed for.
Crustal torsional oscillations depend on not only crust properties but also the stellar mass and radius. Thus, one could extract stellar information by identifying the observed frequencies of stellar oscillations with the crustal torsional oscillations. Owing to the confinement of torsional oscillations inside the crust region of neutron stars, we successfully derive an empirical formula for the fundamental crustal torsional oscillations as a function of the stellar mass, radius, the so-called slope parameter of the nuclear symmetry energy, and the angular index of oscillations, with which one…
Quasi-periodic oscillations (QPOs) discovered in soft-gamma repeaters (SGRs) are expected to help us to study the properties of matter in neutron stars. In earlier investigations, we identified the QPOs of frequencies below $\sim100$ Hz observed in giant flares of SGR 1806$-$20 and SGR 1900+14 as the crustal torsional oscillations. For this purpose, we calculated the frequencies of the fundamental torsional oscillations with various angular indices $\ell$, by changing the stellar mass and radius. In this work, we try to explain the additional QPO frequencies recently reported by Huppenkothen e…
The shear modulus of neutron star matter is one of the important properties for determining torsional oscillations in neutron stars. We take into account the effects of finite sizes of spherical nuclei on the shear modulus and examine the frequencies of crustal torsional oscillations. The shear modulus decreases owing to the finite-size effect, which in turn decreases the frequencies of torsional oscillations. In particular, the finite-size effect becomes more crucial for oscillations with a larger azimuthal quantum number and for neutron star models with a weaker density dependence of nuclear…
We systematically examine the crustal torsional oscillations as varying the stellar mass and radius, where we take into account the effect of electron screening due to the inhomogeneity of electron distribution. In the examinations, we adopt two different equations of state (EOSs) for the inner crust region of neutron stars. As a result, we find that the frequencies depend obviously on the EOS, even if the neutron star models are almost independent of the EOS. That is, one could solve the degeneracy of the stellar models with different EOS via the observations of shear oscillations frequencies…
We consider how superfluidity of dripped neutrons in the crust of a neutron star affects the frequencies of the crust's fundamental torsional oscillations. A nonnegligible superfluid part of dripped neutrons, which do not comove with nuclei, act to reduce the enthalpy density and thus enhance the oscillation frequencies. By assuming that the quasi-periodic oscillations observed in giant flares of soft gamma repeaters arise from the fundamental torsional oscillations and that the mass and radius of the neutron star is in the range of 1.4 < M/M_SUN < 1.8 and 10 km < R < 14 km, we constrain the d…
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…
Estimation of neutron star mass and radius of FRB 20240114A by identification of crustal oscillations
Kochi University（高知大学）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所交叉理论数学科学中心（iTHEMS））；University of Tübingen（蒂宾根大学）；University of Maryland, College Park（马里兰大学帕克分校）；NASA Goddard Space Flight Center（美国国家航空航天局戈达德太空飞行中心）
A quasi-statistical equilibrium model is constructed to simulate the multicomponent composition of the crust of an accreting neutron star. The ashes of rp-process nucleosynthesis are driven by accretion through a series of electron captures, neutron emissions, and pycnonuclear fusions up to densities near the transition between the neutron star crust and core. A liquid droplet model which includes nuclear shell effects is used to provide nuclear masses far from stability. Reaction pathways are determined consistently with the nuclear mass model. The nuclear symmetry energy is an important unce…
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: {$…
The deep crustal heating, associated with exothermal nuclear reactions, is believed to be a key parameter for describing the thermal evolution of accreting neutron stars. In this paper, we present the first thermodynamically consistent calculations of the crustal heating for realistic compositions of thermonuclear ashes. In contrast to previous studies based on the traditional approach, we account for neutron hydrostatic/diffusion (nHD) equilibrium condition imposed by superfluidity of neutrons in a major part of the inner crust and rapid diffusion in the remaining part of the inner crust. We …
We study breaking stress of deformed Coulomb crystals in a neutron star crust, taking into account electron plasma screening of ion-ion interaction; calculated breaking stress is fitted as a function of electron screening parameter. We apply the results for analyzing torsional oscillation modes in the crust of a non-magnetic star. We present exact analytic expression for the fundamental frequencies of such oscillations and show that the frequencies of all torsional oscillations are insensitive to the presence of the outer neutron star crust. The results can be useful in theoretical modeling of…
The spin-down of a neutron star, e.g. due to magneto-dipole losses, results in compression of the stellar matter and induces nuclear reactions at phase transitions between different nuclear species in the crust. We show that this mechanism is effective in heating recycled pulsars, in which the previous accretion process has already been compressing the crust, so it is not in nuclear equilibrium. We calculate the corresponding emissivity and confront it with available observations, showing that it might account for the likely thermal ultraviolet emission of PSR J0437-4715.
Observations of soft X-ray transients in quiescence suggest the existence of heat sources in the crust of accreting neutron stars. Heat is thought to be released by electroweak and nuclear processes triggered by the burying of ashes of X-ray bursts. The heating is studied using a fully quantum approach taking consistently into account nuclear shell effects. We have followed the evolution of ashes made of $^{56}$Fe employing the nuclear energy-density functional theory. Both the outer and inner crusts are described using the same functional, thus ensuring a unified and thermodynamically consist…
With various sets of the parameters that characterize the equation of state (EOS) of nuclear matter, we systematically examine the thickness of a neutron star crust and of the pasta phases contained therein. Then, with respect to the thickness of the phase of spherical nuclei, the thickness of the cylindrical phase, and the crust thickness, we successfully derive fitting formulas that express the ratio of each thickness to the star's radius as a function of the star's compactness, the incompressibility of symmetric nuclear matter, and the density dependence of the symmetry energy. In particula…
We study the properties of pasta structures and their influence on the neutron star observables employing the effective relativistic mean-field (E-RMF) model. The compressible liquid drop model is used to incorporate the finite size effects, considering the possibility of nonspherical structures in the inner crust. The unified equation of states are constructed for several E-RMF parameters to study various properties such as pasta mass and thickness in the neutron star's crust. The majority of the pasta properties are sensitive to the symmetry energy in the subsaturation density region. Using …
Besides their intrinsic nuclear-structure value, nuclear mass models are essential for astrophysical applications, such as r-process nucleosynthesis and neutron-star structure. To overcome the intrinsic limitations of existing "state-of-the-art" mass models, we propose a refinement based on a Bayesian Neural Network (BNN) formalism. A novel BNN approach is implemented with the goal of optimizing mass residuals between theory and experiment. A significant improvement (of about 40%) in the mass predictions of existing models is obtained after BNN refinement. Moreover, these improved results are …
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…
The accretion of matter onto the surface of a neutron star in a low-mass X-ray binary triggers X-ray bursts, whose ashes are buried and further processed thus altering the composition and the properties of the stellar crust. In this second paper of a series, the impact of accretion on the equation of state and on the global properties of neutron stars is studied in the framework of the nuclear energy-density functional theory. Considering ashes made of $^{56}$Fe, we calculated the equations of state using the same Brussels-Montreal nuclear energy-density functionals BSk19, BSk20, and BSk21, as…
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 systematically examine the fundamental frequencies of shear torsional oscillations in neutron star crusts in a manner that is dependent on the parameter $L$ characterizing the poorly known density dependence of the symmetry energy. The identification of the lowest quasiperiodic oscillation (QPO) among the observed QPOs from giant flares in soft-gamma repeaters as the $/ell=2$ fundamental torsional oscillations enables us to constrain the parameter $L$ as $L/ge 47.4$ MeV, which is the most conservative restriction on $L$ obtained in the present work that assumes that the mass and radius of t…
We use the effective relativistic mean-field (E-RMF) model to study the crustal properties of the neutron star. The unified equations of state (EoS) are constructed using recently developed E-RMF parameter sets, such as FSUGarnet, IOPB-I, and G3. The outer crust composition is determined using the atomic mass evaluation 2020 data [Chinese Physics C {\bf 45}, 030002 (2021)] along with the available Hartree-Fock-Bogoliubov mass models [Phys. Rev. C {\bf 88}, 024308 (2013)] for neutron-rich nuclei. The structure of the inner crust is estimated by performing the compressible liquid drop model calc…
In the crust of a neutron star, global torsional oscillations could occur in two elastic layers. The outer and inner layers are composed of spherical and cylindrical nuclei and of cylindrical holes (tubes) and spherical holes (bubbles), respectively, while between these two layers, a phase of slab-like (lasagna) nuclei with vanishingly small elasticity is sandwiched. In this work, we update systematic calculations of the eigenfrequencies of the fundamental oscillations in the inner layer by newly allowing for the presence of tubes. We find that the frequencies still depend strongly on the slop…
The quasiperiodic oscillations found in the three giant flares of soft gamma-ray repeaters observed to date have been interpreted as crustal oscillations caused by a starquake following a dramatic rearrangement of the stellar magnetic field. Motivated by these observations, we study the influence of the magnetic field geometry in the frequencies of the torsional oscillations of magnetized neutron stars. We use realistic tabulated equations of state for the core and crust of the stars and model their magnetic field as a dipole plus a toroidal component, using the relativistic Grad-Shafranov equ…
In a realistic model of magneto-elastic oscillations in magnetars, we find that crustal shear oscillations, often invoked as an explanation of quasi-periodic oscillations (QPOs) seen after giant flares in soft gamma-ray repeaters (SGRs), are damped by resonant absorption on timescales of at most 0.2s, for a lower limit on the dipole magnetic field strength of 5 x 10^13 G. At higher magnetic field strengths (typical in magnetars) the damping timescale is even shorter, as anticipated by earlier toy-models. We have investigated a range of equations of state and masses and if magnetars are dominat…
To determine the frequencies of magnetic oscillations in neutron stars with highly tangled magnetic fields, we derive the perturbation equations. We assume that the field strength of the global magnetic structure is so small that such fields are negligible compared with tangled fields, which may still be far from a realistic configuration. Then, we systematically examine the spectra of the magnetic oscillations, as varying the magnetic field strength and stellar mass. The frequencies without crust elasticity are completely proportional to the strength of the magnetic field, whose proportionali…
We present a new mechanism for deep crustal heating in accreting neutron stars. Charged pions ($π^+$) are produced in nuclear collisions on the neutron star surface during active accretion and upon decay they provide a flux of neutrinos into the neutron star crust. For massive and/or compact neutron stars, neutrinos deposit $\approx 1\textrm{--} 2 \, \mathrm{MeV}$ of heat per accreted nucleon into the inner crust. The strength of neutrino heating is comparable to the previously known sources of deep crustal heating, such as from pycnonuclear fusion reactions, and is relevant for studies of coo…
We extend a general-relativistic ideal magneto-hydrodynamical code to include the effects of elasticity. Using this numerical tool we analyse the magneto-elastic oscillations of highly magnetised neutron stars (magnetars). In simulations without magnetic field we are able to recover the purely crustal shear oscillations within an accuracy of about a few per cent. For dipole magnetic fields between 5 x 10^13 and 10^15 G the Alfvén oscillations become modified substantially by the presence of the crust. Those quasi-periodic oscillations (QPOs) split into three families: Lower QPOs near the equat…
Bayesian analysis of the shear modulus in the neutron-star crust
Grand Accélérateur National d’Ions Lourds (GANIL), CEA/DRF - CNRS/IN2P3（大离子重加速器（GANIL），CEA/DRF-CNRS/IN2P3）；Université de Caen Normandie, ENSICAEN, CNRS/IN2P3, LPC Caen UMR6534（诺曼底卡昂大学，ENSICAEN，CNRS/IN2P3，LPC Caen UMR6534）；Institut Universitaire de France (IUF)（法国高等研究院）
Strange stars are one of the hypothetical compact stellar objects that can be formed after a supernova explosion. The existence of these objects relies on the absolute stability of strange {\it collapsed} quark matter with respect to standard nuclear matter. We discuss simple models of strange stars with a bare quark matter surface, thus standard nuclear matter is completely absent. In these models an electric dipole layer a few hundreds Fermi thick should exist close to the star surface. Studying the torsional oscillations of the electrically charged layer we estimate the emitted power, findi…
GRB 211211A was accompanied by a kilonova, implying a merger origin for the event. A precursor flare, modulated by quasi-periodic oscillations at $\approx 22\, \mbox{Hz}$, was also seen $\approx 1\, \mbox{s}$ prior to the main emission. It is suggested here that the precursor resulted from the resonant shattering of one star's crust prior to coalescence. Seismic aftershocks and low-frequency torsional modes may emanate from the rupture site in this case, explaining the underlying oscillations. This interpretation is directly analogous to proposals for giant flares from soft gamma repeaters, wh…
We general relativistically calculate the frequency of fundamental torsional oscillations of neutron star crusts, where we focus on the crystalline properties obtained from macroscopic nuclear models in a way depending on the equation of state of nuclear matter. We find that the calculated frequency is sensitive to the density dependence of the symmetry energy, but almost independent of the incompressibility of symmetric nuclear matter. By identifying the lowest-frequency quasi-periodic oscillation in giant flares observed from soft gamma-ray repeaters as the fundamental torsional mode and all…
In this contribution, we have made use of the contemporary equation of states (EOSs) for the complete neutron star structure, and confronted them with one particular glitch constrain for the crustal moment of inertia (MOI). We find that with these EOSs, the radii of three millisecond pulsars selected by NICER: PSR J0437-4715, PSR J1614-2230, PSR J0751+1807, are all around 12.5 km. Also, a star with $M \lesssim 1.55 M_{\odot}$ would fulfill the MOI calculation for glitch constrain with the latest neutron superfluidity density, and the glitch crisis might not be present.
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 …
We discuss outer and inner crusts of neutron stars in strong magnetic fields. Here, we demonstrate the effect of Landau quantization of electrons on the ground state properties of matter in outer and inner crusts in magnetars. This effect leads to the enhancement of the electron number density in strong magnetic fields with respect to the zero field case. For the outer crust, we adopt the magnetic Baym-Pethick-Sutherland model and obtain the sequence of nuclei and equation of state (EoS). The properties of nuclei in the inner crust in the presence of strong magnetic fields are investigated usi…
The interior of a neutron star is expected to exhibit different states of matter. In particular, complex non-spherical configurations known as `pasta' phases may exist at the highest densities in the inner crust, potentially having an impact on different neutron-star phenomena. We study the properties of the pasta phase and the uncertainties in the pasta observables which are due to our incomplete knowledge of the nuclear energy functional. To this aim, we employed a compressible liquid-drop model approach with surface parameters optimised either on experimental nuclear masses or theoretical c…
Strong magnetic fields have important effects on the crustal properties of magnetars. Here we study the magneto-elastic oscillations of magnetars taking into consideration the effect of strong magnetic fields on the crustal composition (magnetised crust). We calculate global magneto-elastic (GME) modes as well as modes confined to the crust (CME) only. The composition of the crust directly affects its shear modulus which we calculate using three different nucleon-nucleon interactions: SLy4, SkM and Sk272. The shear modulus of the crust is found to be enhanced in strong magnetic fields $\geq 10…
A number of observed phenomena associated with individual neutron star systems or neutron star populations find explanations in models in which the neutron star crust plays an important role. We review recent work examining the sensitivity to the slope of the symmetry energy $L$ of such models, and constraints extracted on $L$ from confronting them with observations. We focus on six sets of observations and proposed explanations: (i) The cooling rate of the neutron star in Cassiopeia A, confronting cooling models which include enhanced cooling in the nuclear pasta regions of the inner crust, (…
We show that the fundamental seismic shear mode, observed as a quasi-periodic oscillation in giant flares emitted by highly-magnetized neutron stars, is particularly sensitive to the nuclear physics of the crust. The identification of an oscillation at ~ 30 Hz as the fundamental crustal shear mode requires a nuclear symmetry energy that depends very weakly on density near saturation. If the nuclear symmetry energy varies more strongly with density, then lower frequency oscillations, previously identified as torsional Alfven modes of the fluid core, could instead be associated with the crust. I…
Transient neutron star (NS) LMXBs undergo episodes of accretion, alternated with quiescent periods. During an accretion outburst, the NS heats up due to exothermic accretion-induced processes taking place in the crust. Besides the long-known deep crustal heating of nuclear origin, a likely non-nuclear source of heat, dubbed 'shallow heating', is present at lower densities. Most of the accretion-induced heat slowly diffuses into the core on a timescale of years. Over many outburst cycles, a state of equilibrium is reached when the core temperature is high enough that the heating and cooling (ph…
We systematically study the symmetry energy effects of the transition density $n_{\rm t}$ and the transition pressure $P_{\rm t}$ around the crust-core interface of a neutron star in the framework of the dynamical and the thermodynamical method respectively. We employ both the parabolic approximation and the full expansion, for the definition of the symmetry energy. We use various theoretical nuclear models, which are suitable for reproducing the bulk properties of nuclear matter at low densities, close to saturation density as well as the maximum observational neutron star mass. Firstly we de…
We study the torsional magneto-elastic oscillations of relativistic superfluid magnetars and explore the effects of a phase transition in the crust-core interface (nuclear pasta) which results in a weaker elastic response. Exploring various models with different extension of nuclear pasta phases, we find that the differences in the oscillation spectrum present in purely elastic modes (weak magnetic field), are smeared out with increasing strength of the magnetic field. For magnetar conditions, the main characteristic and features of models without nuclear pasta are preserved. We find in genera…
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…
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…
Observed thermal emission from accreting neutron stars (NSs) in a quiescent state is believed to be powered by nonequilibrium nuclear reactions that heat the stellar crust (deep crustal heating paradigm). We derive a simple universal formula for the heating efficiency, assuming that an NS has a fully accreted crust. We further show that, within the recently proposed thermodynamically consistent approach to the accreted crust, the heat release can be parametrized by the only one parameter -- the pressure $P_{\rm oi}$ at the outer-inner crust interface (as we argue, this pressure should not nece…
Even though the crystallize nature of the neutron star crust plays a pivotal role in describing various fascinating astrophysical observations, its microscopic structure is not fully understood in the presence of a colossal magnetic field. In the present work, we study the crustal properties of a neutron star within an effective relativistic mean field framework in the presence of magnetic field strength $\sim 10^{17}$G. We calculate the equilibrium composition of the outer crust by minimizing the Gibbs free energy using the most recent atomic mass evaluations. The magnetic field significantly…
The modeling of many neutron star observables incorporates the microphysics of both the stellar crust and core, which is tied intimately to the properties of the nuclear matter equation of state (EoS). We explore the predictions of such models over the range of experimentally constrained nuclear matter parameters, focusing on the slope of the symmetry energy at nuclear saturation density $L$. We use a consistent model of the composition and EoS of neutron star crust and core matter to model the binding energy of pulsar B of the double pulsar system J0737-3039, the frequencies of torsional osci…
