Nonequilibrium conditions imposed by neutrino cooling through the liquid-solid transition lead to disorder in the solid crust of neutron stars. Disorder reduces the superfluid fraction, $ρ_s/ρ$, at densities above that of neutron drip, $ρ_d \approx 4\times 10^{11}\,g/cm^3$. For an amorphous solid crust the suppression of $ρ_s$ is small, except in the highest density regions of the crust. In contrast to the strong reduction in neutron conduction predicted for coherent Bragg scattering in a crystalline crust, the disordered solid crust supports sufficient neutron superfluid density to account for pulsar glitches.
The neutron superfluidity in the inner crust of a neutron star has been traditionally studied considering either homogeneous neutron matter or only a small number of nucleons confined inside the spherical Wigner-Seitz cell. Drawing analogies with the recently discovered multi-band superconductors, we have solved the anisotropic multi-band BCS gap equations with Bloch boundary conditions, thus providing a unified description taking consistently into account both the free neutrons and the nuclear clusters. Calculations have been carried out using the effective interaction underlying our recent H…
Despite the absence of viscous drag, the neutron superfluid permeating the inner crust of a neutron star can still be strongly coupled to nuclei due to non-dissipative entrainment effects. Neutron superfluidity and entrainment have been systematically studied in all regions of the inner crust of a cold non-accreting neutron star in the framework of the band theory of solids. It is shown that in the intermediate layers of the inner crust a large fraction of "free" neutrons are actually entrained by the crust. The results suggest that a revision of the interpretation of many observable astrophys…
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…
Nuclear pasta, with non-spherical shapes, is expected near the base of the crust in neutron stars. Large scale molecular dynamics simulations of pasta show long lived topological defects that could increase electron scattering and reduce both the thermal and electrical conductivities. We model a possible low conductivity pasta layer by increasing an impurity parameter Q_{imp}. Predictions of light curves for the low mass X-ray binary MXB 1659-29, assuming a large Q_{imp}, find continued late time cooling that is consistent with Chandra observations. The electrical and thermal conductivities ar…
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 …
Theoretical Constraints on Neutron Star Superfluidity from Her X-1 Precession
School of Physics and Astronomy, Tel Aviv University（特拉维夫大学物理与天文学学院）；Sternberg Astronomical Institute, Lomonosov Moscow State University（罗蒙诺索夫莫斯科大学斯登伯格天文研究所）
Her X-1的观测揭示了其35天超轨道周期中通量、偏振度和偏振角的相关性，表明超流体内壳的进动可能由近自由进动驱动，这对超流体动力学提出了新的挑战。
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…
Calculations of the effects of band structure on the neutron superfluid density in the crust of neutron stars made under the assumption that the effects of pairing are small [N. Chamel, Phys. Rev. C 85, 035801 (2012)] lead to moments of inertia of superfluid neutrons so small that the crust alone is insufficient to account for the magnitude of neutron star glitches. Inspired by earlier work on ultracold atomic gases in an optical lattice, we investigate fermions with attractive interactions in a periodic lattice in the mean-field approximation. The effects of band structure are suppressed when…
We study thermal relaxation in a neutron star after internal heating events (outbursts) in the crust. We consider thin and thick spherically symmetric heaters, superfluid and non-superfluid crusts, stars with open and forbidden direct Urca processes in their cores. In particular, we analyze long-term thermal relaxation after deep crustal heating produced by nuclear transformations in fully or partly accreted crusts of transiently accreting neutron stars. This long-term relaxation has a typical relaxation time and an overall finite duration time for the crust to thermally equilibrate with the c…
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…
Superfluid extension of the self-consistent time-dependent band theory for neutron star matter: Anti-entrainment versus superfluid effects in the slab phase
A number of properties of dense matter can be understood semiquantitatively in terms of simple physical arguments. We begin with the outer parts of neutron stars, and consider the density at which pressure ionization occurs, the density at which electrons become relativistic, the density at which neutrons drip out of nuclei, and the size of the equilibrium nucleus in dense matter. Subsequently, we treat the so-called "pasta" phases expected to occur at densities just below the density at which the transition from the crust to the liquid core of a neutron star occurs. We then consider aspects o…
An accretion outburst onto a neutron star transient heats the neutron star's crust out of thermal equilibrium with the core. After the outburst the crust thermally relaxes toward equilibrium with the neutron star core and the surface thermal emission powers the quiescent X-ray light curve. Crust cooling models predict that thermal equilibrium of the crust will be established $\approx 1000 \, \mathrm{d}$ into quiescence. Recent observations of the cooling neutron star transient MXB 1659-29, however, suggest that the crust did not reach thermal equilibrium with the core on the predicted timescal…
The thermomagnetic evolution of the young neutron star in Cassiopea A is studied by considering fast neutrino emission processes. In particular, we consider neutron star models obtained from the equation of state computed in the framework of the Brueckner-Bethe-Goldstone many-body theory and variational methods, and models obtained with the Akmal-Pandharipande-Ravenhall equation of state. It is shown that it is possible to explain a fast cooling regime as the one observed in the neutron star in Cassiopea A if the Joule heating produced by dissipation of the small-scale magnetic field in the cr…
We investigate the cooling of neutron stars with relativistic and non-relativistic models of dense nuclear matter. We focus on the effects of uncertainties originated from the nuclear models, the composition of elements in the envelope region, and the formation of superfluidity in the core and the crust of neutron stars. Discovery of $2 M_\odot$ neutron stars PSR J1614-2230 and PSR J0343+0432 has triggered the revival of stiff nuclear equation of state at high densities. In the mean time, observation of a neutron star in Cassiopeia A for more than 10 years has provided us with very accurate da…
Nuclear matter and finite nuclei exhibit the property of superfluidity by forming Cooper pairs. We review the microscopic theories and methods that are being employed to understand the basic properties of superfluid nuclear systems, with emphasis on the spatially extended matter encountered in neutron stars, supernova envelopes, and nuclear collisions. Our survey of quantum many-body methods includes techniques that employ Green functions, correlated basis functions, and Monte Carlo sampling of quantum states. With respect to empirical realizations of nucleonic and hadronic superfluids, this r…
We examine to what extent the inferred surface temperature of magnetars in quiescence can constrain the presence of a superfluid in the neutron star core and the role of magnetic field decay in the core. By performing detailed simulations of neutron star cooling, we show that extremely strong heating from field decay in the core cannot produce the high observed surface temperatures nor delay the onset of neutron superfluidity in the core. We verify the results of Kaminker et al., namely that the high magnetar surface temperatures require heating in the neutron star crust, and crust heating is …
I summarize some constraints on the physics of neutron stars arising from X-ray observations of the surfaces of neutron stars, focusing on using models of low-magnetic-field neutron star atmospheres to interpret their X-ray spectra. I discuss observations of spectral lines, pulsation profiles, X-ray bursts, radius measurements of transiently accreting neutron stars in quiescence, crust and core cooling measurements of transiently accreting neutron stars, and cooling of young neutron stars. These observations have constrained the neutron star mass and radius (and thus the internal composition, …
Pulsars are known for their superb timing precision, although glitches can interrupt the regular timing behavior when the stars are young. These glitches are thought to be caused by interactions between normal and superfluid matter in the crust of the star. However, glitching pulsars such as Vela have been shown to require a superfluid reservoir that greatly exceeds that available in the crust. We examine a model in which glitches tap the superfluid in the core. We test a variety of theoretical superfluid models against the most recent glitch data and find that only one model can successfully …
We have studied systematically microscopic properties of a quantum vortex in neutron matter at finite temperatures and densities corresponding to different layers of the inner crust of a neutron star. To this end and in preparation of future simulations of the vortex dynamics, we have carried out fully self-consistent 3D Hartree-Fock-Bogoliubov calculations, using one of the latest nuclear energy-density functionals from the Brussels-Montreal family, which has been developed specifically for applications to neutron superfluidity in neutron-star crusts. By analyzing the flow around the vortex, …
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.
We study long-term thermal evolution of neutron stars in soft X-ray transients (SXTs), taking the deep crustal heating into account consistently with the changes of the composition of the crust. We collect observational estimates of average accretion rates and thermal luminosities of such neutron stars and compare the theory with observations. We perform simulations of thermal evolution of accreting neutron stars, considering the gradual replacement of the original nonaccreted crust by the reprocessed accreted matter, the neutrino and photon energy losses, and the deep crustal heating due to n…
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…
We model the quiescent luminosity of accreting neutron stars with several equation of states (EOSs), including the effect of pion condensation and superfluidity. As a consequence of comparison with the observations, we show that the results with Togashi EoS (the strong direct Urca process is forbidden) and TM1e EoS (mass at direct Urca process is $2.06 M_\odot$) can explain the observations well by considering pion condensation and the effect of superfluidity, while LS220 EoS and TM1 EoS can explain the observations well by considering the baryon direct Urca process and the effect of superflui…
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…
Phase transitions in the inner crust of neutron stars within the superfluid band theory: Competition between $^1\text{S}_0$ pairing and spin polarization under finite temperature and magnetic field
As mature neutron stars are cold (on the relevant temperature scale), one has to carefully consider the state of matter in their interior. The outer kilometer or so is expected to freeze to form an elastic crust of increasingly neutron-rich nuclei, coexisting with a superfluid neutron component, while the star's fluid core contains a mixed superfluid/superconductor. The dynamics of the star depend heavily on the parameters associated with the different phases. The presence of superfluidity brings new degrees of freedom -- in essence we are dealing with a complex multi-fluid system -- and addit…
We study the relative importance of several recent updates of microphysics input to the neutron star cooling theory and the effects brought about by superstrong magnetic fields of magnetars, including the effects of the Landau quantization in their crusts. We use a finite-difference code for simulation of neutron-star thermal evolution on timescales from hours to megayears with an updated microphysics input. The consideration of short timescales ($\lesssim1$ yr) is made possible by a treatment of the heat-blanketing envelope without the quasistationary approximation inherent to its treatment i…
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…
X-ray observations of the neutron star in the Cas A supernova remnant over the past decade suggest the star is undergoing a rapid drop in surface temperature of $\approx$ $2-5.5\%$. One explanation suggests the rapid cooling is triggered by the onset of neutron superfluidity in the core of the star, causing enhanced neutrino emission from neutron Cooper pair breaking and formation (PBF). Using consistent neutron star crust and core equations of state (EOSs) and compositions, we explore the sensitivity of this interpretation to the density dependence of the symmetry energy $L$ of the EOS used, …
The observed rapid cooling of the Cassiopeia A neutron star (Cas A NS) can be interpreted as being triggered by the onset of neutron superfluidity in the core of the star, causing enhanced neutrino emission from neutron Cooper pair breaking and formation (PBF). This provides a unique possibility for probing the neutron condensate in the core. Using consistent neutron star core and crust equation of state and composition, I explore the sensitivity of this interpretation to the phase state of the triplet superfluid condensate. Modeling cooling within an expected range of neutron star masses and …
Neutron star models with maximum mass close to $2 \ M_{\odot}$ reach high central densities, which may activate nucleonic and hyperon direct Urca neutrino emission. To alleviate the tension between fast theoretical cooling rates and thermal luminosity observations of moderately magnetized, isolated thermally-emitting stars (with $L_γ \gtrsim 10^{31}$ erg s$^{-1}$ at $t \gtrsim 10^{5.3}$ yr), some internal heating source is required. The power supplied by the internal heater is estimated for both a phenomenological source in the inner crust and Joule heating due to magnetic field decay, assumin…
The possible detection of a compact object in the remnant of SN 1987A presents an unprecedented opportunity to follow its early evolution. The suspected detection stems from an excess of infrared emission from a dust blob near the compact object's predicted position. The infrared excess could be due to the decay of isotopes like 44Ti, accretion luminosity from a neutron star or black hole, magnetospheric emission or a wind originating from the spindown of a pulsar, or thermal emission from an embedded, cooling neutron star (NS 1987A). It is shown that the last possibility is the most plausible…
Spinning neutron stars can emit long-lived gravitational waves. There are several mechanisms that can produce such continuous wave emission. These mechanisms relate to the strains in the elastic crust, the star's magnetic field, superfluidity of the neutron fluid, and bulk oscillations of the entire star. In this chapter we describe how the frequency content of the gravitational wave signal, and its relation to any electromagnetically observed spin frequency, can be used to constrain the mechanism producing the gravitational waves. These ideas will be of use in the event of the first detection…
We investigate the neutron stars spin evolution (breaking, inclination angle evolution and radiative precession), taking into account the superfluidity of the neutrons in the star core. The neutron star is treated as a two-component system consisting of a "charged" component (including the crust and the core protons, electrons and normal neutrons) and a core superfluid neutron component. The components are supposed to interact through the mutual friction force. We assume that the "charged" component rotates rigidly. The neutron superfluid velocity field is calculated directly from linearized h…
The dissipation of intense crustal electric currents produces high Joule heating rates in cooling neutron stars. Here it is shown that Joule heating can counterbalance fast cooling, making it difficult to infer the presence of hyperons (which accelerate cooling) from measurements of the observed thermal luminosity $L_γ$. Models with and without hyperon cores match $L_γ$ of young magnetars (with poloidal-dipolar field $B_{\textrm{dip}} \gtrsim 10^{14}$ G at the polar surface and $L_γ \gtrsim 10^{34}$ erg s$^{-1}$ at $t \lesssim 10^5$ yr) as well as mature, moderately magnetized stars (with $B_{…
Thermal states of neutron stars in soft X-ray transients (SXRTs) are thought to be determined by "deep crustal heating" in the accreted matter that drives the quiescent luminosity and cooling via emission of photons and neutrinos from the interior. In this study, we assume a global thermal steady-state of the transient system and calculate the heating curves (quiescent surface luminosity as a function of mean accretion rate) predicted from theoretical models, taking into account variations in the equations of state, superfluidity gaps, thickness of the light element layer and a phenomenologica…
We study the quiescent luminosities of accreting neutron stars by a new mechanism as neutrino heating for additional deep crustal heating, where the neutrino heating is produced by charged pions decay from the nuclear collisions on the surface of neutron star during its active accretion. For low mass neutron star($\lesssim1.4~M_{\odot}$), as the neutrino heating is little($\lesssim1$ MeV per accreted nucleon) or there would be no neutrino heating, the quiescent luminsoity will be not affected or slightly affected. While for massive neutron star ($\gtrsim2~M_{\odot}$), the quiescent luminosity …
In the magneto-centrifugal mechanism for jet formation, accreting neutron stars are assumed to produce relativistic jets only if their surface magnetic field is weak enough ($B \sim 10^8$ G). However, the most common manifestation of neutron stars are pulsars, whose magnetic field distribution peaks at $B \sim 10^{12}$ G. If the neutron star magnetic field has at least this strength at birth, it must decay considerably before jets can be launched in binary systems. We study the magnetic field evolution of a neutron star that accretes matter from the wind of a high-mass stellar companion so tha…
The interiors of mature neutron stars are expected to be superfluid. Superfluidity of matter on the microscopic scale can have a number of large scale, potentially observable consequences, as the superfluid component of the star can now flow relative to the `normal' component that is tracked by electromagnetic emission. The most spectacular of such phenomena are pulsar glitches, sudden spin-ups observed in many pulsars. A background flow of a normal-fluid component with respect to the superfluid is also known to lead to a number of instabilities in laboratory superfluids, possibly leading to t…
In neutron star matter, there exist $^{1}S_{0}$ superfluids in lower density in the crust while $^{3}P_{2}$ superfluids are believed to exist at higher density deep inside the core. In the latter, depending on the temperature and magnetic field, either the uniaxial nematic phase, the D$_{2}$-biaxial nematic phase, or the D$_{4}$-biaxial nematic phase appears. In this paper, we discuss a mixture of the $^{1}S_{0}$ and $^{3}P_{2}$ superfluids and find their coexistence. Adopting the loop expansion and the weak-coupling approximation for the interaction between two neutrons, we obtain the Ginzbur…
We investigate superfluid phase transitions of asymmetric nuclear matter at finite temperature ($T$) and density ($ρ$) with a low proton fraction ($Y_{\rm p} \le 0.2$) which is relevant to the inner crust and outer core of neutron stars. A strong-coupling theory developed for two-component atomic Fermi gases is generalized to the four-component case and is applied to the system of spin-$1/2$ neutrons and protons. The empirical phase shifts of neutron-neutron (nn), proton-proton (pp) and neutron-proton (np) interactions up to $k = 2$ ${\rm fm}^{-1}$ are described by multi-rank separable potenti…
Thermal evolution of neutron stars is known to depend on the properties of superdense matter in neutron star cores. We suggest a statistical analysis of isolated cooling middle-aged neutron stars and old transiently accreting quasi-stationary neutron stars warmed up by deep crustal heating in low-mass X-ray binaries. The method is based on simulations of the evolution of stars of different masses and on averaging the results over respective mass distributions. This gives theoretical distributions of isolated neutron stars in the surface temperature--age plane and of accreting stars in the phot…
We study the effect of superfluidity on torsional oscillations of highly magnetised neutron stars (magnetars) with a microphysical equation of state by means of two-dimensional, magnetohydrodynamical- elastic simulations. The superfluid properties of the neutrons in the neutron star core are treated in a parametric way in which we effectively decouple part of the core matter from the oscillations. Our simulations confirm the existence of two groups of oscillations, namely continuum oscillations that are confined to the neutron star core and are of Alfvénic character, and global oscillations wi…
When a neutron star cools to below the critical temperature for the onset of superfluidity, nucleon pair breaking and formation (PBF) processes become the dominant mechanism for neutrino emission, while the modified URCA and the nuclear bremsstrahlung processes are suppressed. The PBF processes in neutron stars have also been used to set upper limits on the properties of axions, which are comparable to those set by supernova SN 1987A. We apply this constraint on Weinberg's Higgs portal model, in which the dark radiation particles (the Goldstone bosons) and the dark matter candidate (a Majorana…
Understanding the average motion of a multitude of superfluid vortices in the interior of a neutron star is a key ingredient for most theories of pulsar glitches. In this paper we propose a kinetic approach to compute the mutual friction force that is responsible for the momentum exchange between the normal and superfluid components in a neutron star, where the mutual friction is extracted from a suitable average over the motion of many vortex lines. As a first step towards a better modelling of the repinning and depinning processes of many vortex lines in a neutron star, we consider here only…
Modelling of crust heating and cooling across multiple accretion outbursts of the low mass X-ray binary MXB 1659-29 indicates that the neutrino luminosity of the neutron star core is consistent with direct Urca reactions occurring in $\sim 1\%$ of the core volume. We investigate this scenario with neutron star models that include a detailed equation of state parametrized by the slope of the nuclear symmetry energy $L$, and a range of neutron and proton superfluid gaps. We find that the predicted neutron star mass depends sensitively on $L$ and the assumed gaps. We discuss which combinations of…
The standard cooling scenario in the presence of nucleon superfluidity fits rather well to the observation of the neutron stars. It implies that the stellar cooling arguments could place a stringent constraint on the properties of novel particles. We study in particular the cooling rate induced by dark gauge bosons for very young neutron stars: remnants of Cassiopeia A and SN1987A. The cooling is dominantly contributed either by the nucleon pair breaking and formation in the core or by the electron bremsstrahlung in the crust, depending on the age of the stars and the form of the couplings. We…
