With the first detections of binary neutron star mergers by gravitational-wave detectors, it proves timely to consider how the internal structure of neutron stars affects the way in which they can be asymmetrically deformed. Such deformations may leave measurable imprints on gravitational-wave signals and can be sourced through tidal interactions or the formation of mountains. We detail the formalism that describes fully-relativistic neutron star models with elastic crusts undergoing static perturbations. This formalism primes the problem for studies into a variety of mechanisms that can deform a neutron star. We present results for a barotropic equation of state and a realistic model for the elastic crust, which enables us to compute relevant quantities such as the tidal deformability parameter. We find that the inclusion of an elastic crust provides a very small correction to the tidal deformability. The results allow us to demonstrate when and where the crust starts to fail during a binary inspiral and we find that the majority of the crust will remain intact up until merger.
Gravitational wave astronomy is expected to provide independent constraints on neutron star properties, such as their dense matter equation of state. This is possible with the measurements of binary components' tidal deformability, which alter the point-particle gravitational waveforms of the late inspiral phase of neutron-star binaries. Although current gravitational wave detectors are not sensitive enough for a precise determination of the individual tidal deformations of the components, a large number of combined observations with future detectors will decrease uncertainties in this quantit…
This study investigates elastic deformation driven by the Hall drift in a magnetized neutron-star crust. Although the dynamic equilibrium initially holds without elastic displacement, the magnetic-field evolution changes the Lorentz force over a secular timescale, which inevitably causes the elastic deformation to settle in a new force balance. Accordingly, elastic energy is accumulated, and the crust is eventually fractured beyond a particular threshold. We assume that the magnetic field is axially symmetric, and we explicitly calculate the breakup time, maximum elastic energy stored in the c…
In the late inspiral phase, gravitational waves from binary neutron star mergers carry the imprint of the equation of state due to the tidally deformed structure of the components. If the stars contain solid crusts, then their shear modulus can affect the deformability of the star and, thereby, modify the emitted signal. Here, we investigate the effect of realistic equations of state (EOSs) of the crustal matter, with a realistic model for the shear modulus of the stellar crust in a fully general relativistic framework. This allows us to systematically study the deviations that are expected fr…
With the recent measurement of GW170817 providing constraints on the tidal deformability of a neutron star, it is very important to understand what features of the equation of state have the biggest effect on it. We therefore study the contribution of the crust to the tidal deformability and the moment of inertia of a neutron star for a variety of well-known equations of state. It is found that the contributions to these quantities from the low density crust are typically quite small and as a result the determination of the tidal deformability provides an important constraint on the equation o…
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 …
Role of the crust on the tidal deformability of a neutron star within a unified treatment of dense matter
Institute of Astronomy and Astrophysics, Université Libre de Bruxelles（布鲁塞尔自由大学天文学与天体物理研究所）；LUTH, Observatoire de Paris, PSL Research University, CNRS, Université Paris Diderot, Sorbonne Paris Cité（巴黎天文台宇宙论与天体物理学实验室，巴黎文理研究大学，法国国家科学研究中心，巴黎第七大学，索邦巴黎西岱大学）
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 $\sim \mathcal{O}(0.1)$ 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, wh…
Nonradial oscillations of stratified neutron stars with solid crusts: Mode characterization and tidal resonances in coalescing binaries
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Yangzhou University（扬州大学）；Kyoto University（京都大学）
We examine the equilibrium of a magnetized neutron-star-crust. We calculate axially symmetric models in which an elastic force balances solenoidal motion driven by a Lorentz force. A large variety of equilibrium models are allowed by incorporating the elastic shear deformation; in addition, toroidal-magnetic-field dominated models are available. These results remarkably differ from those in barotropic fluid stars. We demonstrate some models wherein the magnetic energy exceeds the elastic energy. The excess comes from the fact that a large amount of magnetic energy is associated with the irrota…
The first detection of a binary neutron star merger has opened the brand new era of multimessenger astronomy. This historic detection has been instrumental in providing constraints on the tidal polarizability of neutron stars. In turn, the tidal polarizability has been used to impose limits on stellar radii and ultimately on the equation of state (EOS). The tidal polarizability is also sensitive to the second tidal Love number k2. It is the main purpose of this work to perform a detailed study of k2 which, for a given compactness parameter, encodes the entire sensitivity of the tidal polarizab…
We investigate new solutions for magnetized neutron stars with a barotropic core in magnetohydrodynamic (MHD) equilibrium and a magneto-elastic crust, which was neglected by previous studies concerning stars in MHD equilibrium. The Lorentz force of the barotropic star is purely irrotational and the structures of magnetic fields are constrained. By contrast, a solenoidal component of the Lorentz force exists in the elastic crust and the structures of the magnetic fields are less restricted. We find that the minor solenoidal component in the elastic crust is important for sustaining the strong m…
As evidenced by the coincident detections of GW170817 and GRB 170817A, short gamma-ray bursts are likely associated with neutron star-neutron star merger events. Although rare, some bursts display episodes of early emission, with precursor flares being observed up to $\sim 10$ seconds prior to the main burst. As the stars inspiral due to gravitational wave emission, the exertion of mutual tidal forces leads to the excitation of stellar oscillation modes, which may come into resonance with the orbital motion. Mode amplitudes increase substantially during a period of resonance as tidal energy is…
We consider the dynamical tidal response of a neutron star in an inspiralling binary, focussing on the impact of the star's elastic crust. Within the context of Newtonian gravity, we add the elastic aspects to the theoretical formulation of the problem and quantify the dynamical excitation of different classes of oscillation modes. The results demonstrate the expectation that the fundamental mode dominates the tidal response and show how the usual tidal deformability (and the Love number) emerge in the static limit. In addition, we consider to what extent the different modes may be excited to …
We analyze how the crust equation of state affects several neutron star properties and how it impacts on possible constraints inferred from astrophysical observations. Using three distinct crusts, we generate three sets of model-independent equations of state describing stellar matter from a Taylor expansion around saturation density. The equations of state are thermodynamically consistent, causal, and compatible with astrophysical observations. The relations between the tidal deformability $Λ$ and compactness $C$, Love number $k_2$ and radius of neutron star with mass $M$ are studied, and the…
We investigate the magneto--elastic equilibrium of a neutron star crust and magnetic energy stored by the elastic force. The solenoidal motion driven by the Lorentz force can be controlled by the magnetic elastic force, so that conditions for the magnetic field strength and geometry are less restrictive. For equilibrium models, the minor solenoidal part of the magnetic force is balanced by a weak elastic force because the irrotational part is balanced by the dominant gravity and pressure forces. Therefore, a strong magnetic field may be confined in the interior, regardless of poloidal or toroi…
In the late inspiral phase of a double neutron star (NS) or NS-black hole system in which one NS is a magnetar, the tidal force on the magnetar arisen from its companion will increase dramatically as the binary approaches. The tidal-induced deformation may surpass the maximum that the magnetar's crust can sustain just seconds or subseconds before the coalescence. A catastrophic global crust destruction may thus occur, and the magnetic energy stored in the magnetar's interior will have the opportunity to be released, which would be observed as a superflare with energy 100s of times larger than …
Background: The nuclear symmetry energy $E_{sym}(ρ)$ encodes information about the energy necessary to make nuclear systems more neutron-rich. While its slope parameter L at the saturation density $ρ_0$ of nuclear matter has been relatively well constrained by recent astrophysical observations and terrestrial nuclear experiments, its curvature $K_{\rm{sym}}$ characterizing the $E_{sym}(ρ)$ around $2ρ_0$ remains largely unconstrained. Over 520 calculations for $E_{sym}(ρ)$ using various nuclear theories and interactions in the literature have predicted several significantly different $K_{\rm{sy…
In this paper, we present some relativistic mean-field inner crust equations of state that have recently been uploaded in the CompOSE online repository. These equations of state fulfill experimental and microscopic constraints, and are also able to reproduce two solar-mass stars. We integrate the TOV equations to obtain the mass-radius relation, and we also calculate the tidal deformability, compactness, and effective tidal deformability to compare with the latest astrophysical data from NICER and LIGO and Virgo.
By means of two dimensional, general-relativistic, magneto-hydrodynamical simulations we investigate the oscillations of magnetized neutron star models (magnetars) including the description of an extended solid crust. The aim of this study is to understand the origin of the QPOs observed in the giant flares of SGRs. We confirm the existence of three different regimes: (a) a weak magnetic field regime B<5 x 10^13 G, where crustal shear modes dominate the evolution; (b) a regime of intermediate magnetic fields 5 x 10^13 G<B< 10^15 G, where Alfvén QPOs are mainly confined to the core of the neutr…
We study the tidal response of a superfluid neutron star in a binary system, focussing on Newtonian models with superfluid neutrons present throughout the star's core and the inner crust. Within the two-fluid formalism, we consider the main aspects that arise from the presence of different regions inside the star, with particular focus on the various interfaces. Having established the relevant theory, we determine the tidal excitation of the most relevant oscillation modes during binary inspiral. Our results suggest that superfluid physics has a negligible impact on the static tidal deformatio…
Corrections to elastic moduli, including the effective shear modulus, of a solid neutron star crust due to electron screening are calculated. At any given mass density, the crust is modelled as a body-centred cubic Coulomb crystal of fully ionized atomic nuclei of a single type with a polarizable charge-compensating electron background. Motion of the nuclei is neglected. The electron polarization is described by a simple Thomas-Fermi model of exponential electron screening. The results of numerical calculations are fitted by convenient analytic formulae. They should be used for precise neutron…
We study the time-evolution of axisymmetric oscillations of superfluid magnetars with a poloidal magnetic field and an elastic crust, working in Newtonian gravity. Extending earlier models, we study the effects of composition gradients and entrainment on the magneto-elastic wave spectrum and on the potential identification of the observed quasi periodic oscillations (QPOs). We use two-fluid polytropic equations of state to construct our stellar models, which mimic realistic composition gradient configurations. The basic features of the axial axisymmetric spectrum of normal fluid stars are repr…
We consider the dynamics of a mechanical failure induced by a shear stress in a strongly magnetized neutron-star crust. We show that even if the elastic properties of the crust allow the creation of a shear crack, the strongly sheared magnetic field around the crack leads to a back-reaction from the Lorentz force which does not allow large relative displacement of the crack surfaces. Instead, the global evolution of the crack proceeds on a slow resistive time scale, and is unable to release any substantial mechanical energy. Our calculations demostrate that for {\it some} magnetic-field config…
In some short gamma-ray bursts, precursor flares occurring $\sim$ seconds prior to the main episode have been observed. These flares may then be associated with the last few cycles of the inspiral when the orbital frequency is a few hundred Hz. During these final cycles, tidal forces can resonantly excite quasi-normal modes in the inspiralling stars, leading to a rapid increase in their amplitude. It has been shown that these modes can exert sufficiently strong strains onto the neutron star crust to instigate yieldings. Due to the typical frequencies of $g$-modes being $\sim 100\text{ Hz}$, th…
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）
This paper considers the yielding response of a neutron star crust to smooth, unbalanced Maxwell stresses imposed at the core-crust boundary, and the coupling of the dynamic crust to the external magnetic field. Stress buildup and yielding in a magnetar crust is a global phenomenon: an elastic distortion radiating from one plastically deforming zone is shown to dramatically increase the creep rate in distant zones. Runaway creep to dynamical rates is shown to be possible, being enhanced by in situ heating and suppressed by thermal conduction and shearing of an embedded magnetic field. A global…
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, a…
Crustquake events may be connected with both rapid spin-up `glitches' within the regular slowdown of neutron stars, and high-energy magnetar flares. We argue that magnetic field decay builds up stresses in a neutron star's crust, as the elastic shear force resists the Lorentz force's desire to rearrange the global magnetic-field equilibrium. We derive a criterion for crust-breaking induced by a changing magnetic-field configuration, and use this to investigate strain patterns in a neutron star's crust for a variety of different magnetic-field models. Universally, we find that the crust is most…
The elastic crust of a neutron star fractures repeatedly as it spins down electromagnetically. An idealised, macroscopic model of inhomogeneous crustal failure is presented based on a cellular automaton with nearest-neighbour tectonic interactions involving strain redistribution and thermal dissipation. Predictions are made of the size and waiting-time distributions of failure events, as well as the rate of failure as the star spins down. The last failure event typically occurs when the star spins down to approximately 1% of its birth frequency with implications for rotational glitch activity.…
In this study, we examine the magnetic field evolution occurring in a neutron star crust. Beyond the elastic limit, the lattice ions are assumed to act as a plastic flow. The Ohmic dissipation, Hall drift, and bulk fluid velocity driven by the Lorentz force are considered in our numerical simulation. A magnetically induced quadrupole deformation is observed in the crust during the evolution. Generally, the ellipticity decreases as the magnetic energy decreases. In a toroidal-field-dominated model, the sign of the ellipticity changes. Namely, the initial prolate shape tends to become oblate. Th…
The configuration of the magnetic field in the interior of a neutron star is mostly unknown from observations. Theoretical models of the interior magnetic field geometry tend to be oversimplified to avoid mathematical complexity and tend to be based on axisymmetric barotropic fluid systems. These static magnetic equilibrium configurations have been shown to be unstable on a short time scale against an infinitesimal perturbation. Given this instability, it is relevant to consider how more realistic neutron star physics affects the outcome. In particular, it makes sense to ask if elasticity, whi…
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…
We study the effect of superfluidity on the tidal response of a neutron star in a general relativistic framework. In this work, we take a dual-layer approach where the superfluid matter is confined in the core of the star. Then, the superfluid core is encapsulated with an envelope of ordinary matter fluid which acts effectively as the low-density crustal region of the star. In the core, the matter content is described by a two-fluid model where only the neutrons are taken as superfluid and the other fluid consists of protons and electrons making it charge neutral. We calculate the values of va…
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…
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, (…
Modelling mountains on accreting magnetized neutron stars
Università di Pisa（比萨大学）；Scuola Normale Superiore（比萨高等师范学院）；Università di Milano（米兰大学）；INFN, Sezione di Milano（意大利国家核物理研究所，米兰分部）；INFN, Sezione di Pisa（意大利国家核物理研究所，比萨分部）；INFN, Sezione di Ferrara（意大利国家核物理研究所，费拉拉分部）；Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences（波兰科学院尼古拉·哥白尼天文学中心）
We present an ensemble of unified neutron star crust and core equations of state, constructed using an extended Skyrme energy density functional through the crust and outer core, and appended by two piecewise polytropes at higher densities. The equations of state are parameterized by the first three coefficients in the density expansion of the symmetry energy $J,L$ and $K_{\rm sym}$, the moment of inertia of a 1.338 M$_{\odot}$ star $I_{1.338}$ and the maximum neutron star mass $M_{\rm max}$. We construct an ensemble with uniform priors on all five parameters, and then apply data filters to th…
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 neutron star maximum mass and the radius are investigated within the framework of the relativistic mean-field (RMF) model. The variation in the radius at the canonical mass, $R_{1.4}$, using different inner crust equation of state (EoS) with different symmetry energy slope parameter is studied. It is found that although the NS maximum mass and the corresponding radius do not vary much with different inner crust EoSs, the radius and the tidal deformability at 1.4$M_{\odot}$ vary with the different choice of crust EoS and variation of about 1-2 km is seen in the radius at the canonical mass.…
We study the effects of nuclear symmetry energy on the mass-radius relation and tidal deformability of neutron stars, considering the self-consistency of the equation of state (EOS). We first construct a set of unified EOSs based on relativistic mean-field models with different density dependence of the symmetry energy. For the description of pasta phases appearing in the inner crust of neutron stars, we perform a self-consistent Thomas-Fermi calculation using the same nuclear interaction as that for the uniform matter in the core. To examine possible effects from the self-consistency of the E…
