We present the results of the first study of global oscillations of relativistic stars with both elastic crusts and interpenetrating superfluid components. For simplicity, we focus on the axial quasi-normal modes. Our results demonstrate that the torsional crust modes are essentially unaffected by the coupling to the gravitational field. This is as expected since these oscillations are known to be weak gravitational-wave sources. In contrast, the presence of a loosely coupled superfluid neutron component in the crust can have a significant effect on the oscillation spectrum. We show that the entrainment between the superfluid and the crust nuclei is a key parameter in the problem. Our analysis highlights the need for a more detailed understanding of the coupled crust-superfluid at the microphysical level. Our numerical results have, even though we have not considered magnetised stars, some relevance for efforts to carry out seismology based on quasi-periodic oscillations observed in the tails of magnetar flares. In particular, we argue that the sensitive dependence on the entrainment may have to be accounted for in attempts to match theoretical models to observational data.
The inner crust of mature neutron stars, where an elastic lattice of neutron-rich nuclei coexists with a neutron superfluid, impacts on a range of astrophysical phenomena. The presence of the superfluid is key to our understanding of pulsar glitches, and is expected to affect the thermal conductivity and hence the evolution of the surface temperature. The coupling between crust and superfluid must also be accounted for in studies of neutron star dynamics, discussions of global oscillations and associated instabilities. In this paper we develop Lagrangian perturbation theory for this problem, p…
We study the effects of an elastic crust on the oscillation spectrum of superfluid neutron stars. Within the two fluid formalism, we consider Newtonian stellar models that include the relevant constituents of a mature neutron stars. The core is formed by a mixture of superfluid neutrons and a conglomerate of charged particles, while the inner crust is described by a lattice of nuclei permeated by superfluid neutrons. We linearise the Poisson and the conservation equations of nonrotating superfluid stars and study the effects of elasticity, entrainment and composition stratification on the shea…
Pinning of superfluid vortices to the nuclear lattice of the inner crust of a neutron star supports a velocity difference between the superfluid and the solid as the star spins down. Under the Magnus force that arises on the vortex lattice, vortices undergo {\em vortex creep} through thermal activation or quantum tunneling. We examine the hydrodynamic stability of this situation. Vortex creep introduces two low-frequency modes, one of which is unstable above a critical wavenumber for any non-zero flow velocity of the superfluid with respect to the solid. For typical pinning parameters of the i…
A simple description of superfluid hydrodynamics in the inner crust of a neutron star is given. Particular attention is paid to the effect of the lattice of nuclei on the properties of the superfluid neutrons, and the effects of entrainment, the fact that some fraction of the neutrons are locked to the motion of the protons in nuclei.
Pulsar glitches are traditionally viewed as a manifestation of vortex dynamics associated with a neutron superfluid reservoir confined to the inner crust of the star. In this Letter we show that the non-dissipative entrainment coupling between the neutron superfluid and the nuclear lattice leads to a less mobile crust superfluid, effectively reducing the moment of inertia associated with the angular momentum reservoir. Combining the latest observational data for prolific glitching pulsars with theoretical results for the crust entrainment we find that the required superfluid reservoir exceeds …
The motion of superfluid vortices in a neutron star crust is at the heart of most theories of pulsar glitches. Pinning of vortices to ions can decouple the superfluid from the crust and create a reservoir of angular momentum. Sudden large scale unpinning can lead to an observable glitch. In this paper we investigate the scattering of a free vortex off a pinning potential and calculate its mean free path, in order to assess whether unpinned vortices can skip multiple pinning sites and come close enough to their neighbours to trigger avalanches, or whether they simply hop from one pinning site t…
We formulate a low energy effective theory describing phases of matter that are both solid and superfluid. These systems simultaneously break translational symmetry and the phase symmetry associated with particle number. The symmetries restrict the combinations of terms that can appear in the effective action and the lowest order terms featuring equal number of derivatives and Goldstone fields are completely specified by the thermodynamic free energy, or equivalently by the long-wavelength limit of static correlation functions in the ground state. We show that the underlying interaction betwee…
The violent giant flares of magnetars excite QPOs which persist for hundreds of seconds, as seen in the X-ray tail following the initial burst. Recent studies, based on single-fluid barotropic magnetar models, have suggested that the lower-frequency QPOs correspond to magneto-elastic oscillations of the star. The higher frequencies, however, in particular the strong 625 Hz peak, have proved harder to explain, except as high mode multipoles. In this work we study the time evolutions of non-axisymmetric oscillations of two-fluid Newtonian magnetars with no crust. We consider models with superflu…
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 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…
The nature of the interaction between superfluid vortices and the neutron star crust, conjectured by Anderson and Itoh in 1975 to be at the heart vortex creep and the cause of glitches, has been a long-standing question in astrophysics. Using a qualitatively new approach, we follow the dynamics as superfluid vortices move in response to the presence of "nuclei" (nuclear defects in the crust). The resulting motion is perpendicular to the force, similar to the motion of a spinning top when pushed. We show that nuclei repel vortices in the neutron star crust, and characterize the force as a funct…
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…
Pulsar glitches provide a unique way to study neutron star microphysics because short post-glitch dynamics are directly linked to strong frictional processes on small scales. To illustrate this connection between macroscopic observables and microphysics, we review calculations of vortex interactions focusing on Kelvin wave excitations and determine the corresponding mutual friction strength for realistic microscopic parameters in the inner crust. These density-dependent crustal coupling profiles are combined with a simplified treatment of the core coupling and implemented in a three-component …
We study coupled axial and polar axisymmetric oscillations of a neutron star endowed with a strong magnetic field, having both poloidal and toroidal components. The toroidal component of the magnetic field is driving the coupling between the polar and axial oscillations. The star is composed of a fluid core as well as a solid crust. Using a two dimensional general relativistic simulation and a magnetic field B = 10^16 G, we study the change in the polar and axial spectrum caused by the coupling. We find that the axial spectrum suffers a dramatic change in its nature, losing its continuum chara…
Detailed structures of the Vela pulsar (PSR B0833-45, with a period of $89.33$ milliseconds) are predicted by adopting a recently-constructed unified treatment of all parts of neutron stars: the outer crust, the inner crust and the core based on modern microscopic Brueckner-Hartree-Fock calculations. To take the pulsar mass ranging from $1.0M_{\odot}$ to $2.0M_{\odot}$, we calculate the central density, the core/crust radii, the core/crust mass, the core/crustal thickness, the moment of inertia, and the crustal moment of inertia. Among them, the crustal moment of inertia could be effectively c…
We have found an attractor for an axially symmetric magnetic field evolving under the Hall effect and subdominant ohmic dissipation, resolving the question of the long term fate of the magnetic field in neutron star crusts. The electron fluid is in isorotation, analogous to Ferraro's law, with its angular velocity being approximately proportional to the poloidal magnetic flux, $Ω\propto Ψ$. This equilibrium is the long term configuration of a magnetic field evolving because of the Hall effect and ohmic dissipation. For an initial dipole dominated field the attractor consists mainly of a dipole…
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…
The superfluid core of a neutron star is usually assumed to corotate with the crust over timescales longer than minutes. I show that the interaction between the neutron superfluid and the type II superconductor of the outer core increases the coupling time to weeks or longer. I suggest that observed post-glitch response over timescales of weeks to years represents recovery of the outer core. Spin glitches could originate in either the inner crust or the outer core.
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 …
The "glitch crisis" of Vela-like pulsars has been a great debate recently. It might challenge the standard two-component glitch model, because large fractions of superfluid neutrons are thought to be entrained in the lattices of the crust part, then there is not enough superfluid neutrons to trigger the large glitches in Vela-like pulsars. But the amount of entrainment which could effectively constrain the fractional moment of inertia of a pulsar, is very uncertain. In order to examine the importance of this parameter on the inner structures of neutron stars, we relax the "glitch crisis" argum…
The crust of a neutron star is thought to be comprised of a lattice of nuclei immersed in a sea of free electrons and neutrons. As the neutrons are superfluid their angular momentum is carried by an array of quantized vortices. These vortices can pin to the nuclear lattice and prevent the neutron superfluid from spinning down, allowing it to store angular momentum which can then be released catastrophically, giving rise to a pulsar glitch. A crucial ingredient for this model is the maximum pinning force that the lattice can exert on the vortices, as this allows us to estimate the angular momen…
We present recent work on using astronomical observations of neutron stars to reveal unique insights into nuclear matter that cannot be obtained from laboratories on Earth. First, we discuss our measurement of the rapid cooling of the youngest neutron star in the Galaxy; this provides the first direct evidence for superfluidity and superconductivity in the supra-nuclear core of neutron stars. We show that observations of thermonuclear X-ray bursts on neutron stars can be used to constrain properties of neutron superfluidity and neutrino emission. We describe the implications of rapid neutron s…
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 …
In this article we present an analytical two-component model for pulsar rotational dynamics. Under the assumption of axial symmetry, implemented by a paraxial array of straight vortices that thread the entire neutron superfluid, we are able to project exactly the 3D hydrodynamical problem to a 1D cylindrical one. In the presence of density dependent entrainment the superfluid rotation is non-columnar: we circumvent this by using an auxiliary dynamical variable directly related to the areal density of vortices. The main result is a system of differential equations that take consistently into ac…
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…
The monitoring of quiescent emission from neutron star transients with accretion outbursts long enough to significantly heat the neutron star crust has opened a new vista onto the physics of dense matter. In this paper we construct models of the thermal relaxation of the neutron star crust following the end of a protracted accretion outburst. We confirm the finding of Shternin et al., that the thermal conductivity of the neutron star crust is high, consistent with a low impurity parameter. We describe the basic physics that sets the broken power-law form of the cooling lightcurve. The initial …
We analyse how the structure of the inner curst is influenced by the pairing correlations. The inner-crust matter, formed by nuclear clusters immersed in a superfluid neutron gas and ultra-relativistic electrons, is treated in the Wigner-Seitz approximation. The properties of the Wigner-Seitz cells, i.e., their neutron to proton ratio and their radius at a given baryonic density, are obtained from the energy minimization at beta equilibrium. To obtain the binding energy of baryonic matter we perform Skyrme-HFB calculations with zero-range density-dependent pairing forces of various intensities…
The linear response of a neutron spin-triplet superfluid onto external weak axial-vector field is studied for the case of $^{3}P_{2}$ pairing with a projection of the total angular momentum $m_{j}=0$. The problem is considered in the BCS approximation discarding Fermi-liquid effects. The anomalous axial-vector vertices of neutron quasiparticles possess singularities at some frequencies which specify existence of undamped spin-density waves in the Cooper condensate. The spin waves are of a low excitation energy and are kinematically able to decay into neutrino pairs through neutral weak current…
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 …
Neutron stars serve as excellent next-generation thermal detectors of dark matter, heated by the scattering and annihilation of dark matter accelerated to relativistic speeds in their deep gravitational wells. However, the dynamics of neutron star cores are uncertain, making it difficult at present to unequivocally compute dark matter scattering in this region. On the other hand, the physics of an outer layer of the neutron star, the crust, is more robustly understood. We show that dark matter scattering solely with the low-density crust still kinetically heats neutron stars to infrared temper…
