Oscillations of rapidly rotating superfluid stars

Using time evolutions of the relevant linearized equations, we study non-axisymmetric oscillations of rapidly rotating and superfluid neutron stars. We consider perturbations of Newtonian axisymmetric background configurations and account for the presence of superfluid components via the standard two-fluid model. Within the Cowling approximation, we are able to carry out evolutions for uniformly rotating stars up to the mass-shedding limit. This leads to the first detailed analysis of superfluid neutron star oscillations in the fast rotation regime, where the star is significantly deformed by the centrifugal force. For simplicity, we focus on background models where the two fluids (superfluid neutrons and protons) corotate, are in β-equilibrium and co-exist throughout the volume of the star. We construct sequences of rotating stars for two analytical model equations of state. These models represent relatively simple generalizations of single fluid, polytropic stars. We study the effects of entrainment, rotation and symmetry energy on non-radial oscillations of these models. Our results show that entrainment and symmetry energy can have a significant effect on the rotational splitting of non-axisymmetric modes. In particular, the symmetry energy modifies the inertial mode frequencies considerably in the regime of fast rotation.     

Crustal oscillations of slowly rotating relativistic stars

We study low-amplitude crustal oscillations of slowly rotating relativistic stars consisting of a central fluid core and an outer thin solid crust. We estimate the effect of rotation on the torsional toroidal modes and on the interfacial and shear spheroidal modes. The results compared against the Newtonian ones for wide range of neutron star models and equations of state.     

Secular instability of g-modes in rotating neutron stars

Gravitational radiation tends to drive gravity modes in rotating neutron stars to become unstable. For an inviscid star, the instability sets in when the rotation frequency is about 0.7 times the corresponding mode frequency of the non-rotating star. Neutron stars with spin frequencies ≳100 Hz are susceptible to this instability, with a growth time of the order of years. However, it is likely that viscous dissipation suppresses the instability except for a narrow range of temperatures around 10⁹ K. We also show that the viscosity-driven instability of g-modes is absent.     

Surface trapping and leakage of low-frequency g modes in rotating early-type stars – II. Global analysis

A global analysis of the surface trapping of low-frequency non-radial g modes in rotating early-type stars is undertaken within the Cowling, adiabatic and traditional approximations. The dimensionless pulsation equations governing these modes are reviewed, and the boundary conditions necessary for solution of the equations are considered; in particular, an outer mechanical boundary condition, which does not enforce complete wave trapping at the stellar surface, is derived and discussed in detail. The pulsation equations are solved for a 7-M_⊙ model star over a range of rotation rates, using a numerical approach.
The results of the calculations confirm the findings of the preceding paper in the series: modes with eigenfrequencies below a cut-off cannot be fully trapped within the star, and exhibit leakage in the form of outwardly propagating waves at the surface. The damping rates resulting from leakage are calculated for such 'virtual' modes, and found to be appreciably larger than typical growth rates associated with opacity-driven pulsation. Furthermore, it is demonstrated that the surface perturbations generated by virtual modes are significantly changed from those caused by fully trapped modes; the latter result suggests differences in the line-profile variations exhibited by these two types of mode.
The findings are discussed in the context of the 53 Per, SPB and pulsating Be classes of variable star. Whilst wave leakage will probably not occur for overstable g modes in the 53 Per and slowly rotating SPB stars, the adoption of the new outer mechanical boundary condition may still affect the pulsational stability of these systems. Wave leakage for overstable modes remains a possibility in Be stars and the more rapidly rotating SPB stars.     

Towards gravitational wave asteroseismology

We present new results for pulsating neutron stars. We have calculated the eigenfrequencies of the modes that one would expect to be the most important gravitational wave sources: the fundamental fluid f mode, the first pressure p mode and the first gravitational wave w mode, for twelve realistic equations of state. From these numerical data we have inferred a set of 'empirical relations' between the mode frequencies and the parameters of the star (the radius R and the mass M ). Some of these relations prove to be surprisingly robust, and we show how they can be used to extract the details of the star from observed modes. The results indicate that, should the various pulsation modes be detected by the new generation of gravitational wave detectors that come online in a few years, the mass and the radius of neutron stars can be deduced with errors no larger than a few per cent.     

r modes and mutual friction in rapidly rotating superfluid neutron stars

We develop a new perturbative framework for studying the r modes of rotating superfluid neutron stars. Our analysis accounts for the centrifugal deformation of the star, and considers the two-fluid dynamics at linear order in the perturbed velocities. Our main focus is on a simple model system where the total density profile is that of an   n = 1  polytrope. We derive a partially analytic solution for the superfluid analogue of the classical r mode. This solution is used to analyse the relevance of the vortex-mediated mutual friction damping, confirming that this dissipation mechanism is unlikely to suppress the gravitational-wave-driven instability in rapidly spinning superfluid neutron stars. Our calculation of the superfluid r modes is significantly simpler than previous approaches, because it decouples the r mode from all other inertial modes of the system. This leads to the results being clearer, but it also means that we cannot comment on the relevance of potential avoided crossings (and associated 'resonances') that may occur for particular parameter values. Our analysis of the mutual friction damping differs from previous studies in two important ways. First, we incorporate realistic pairing gaps which means that the regions of superfluidity in the star's core vary with temperature. Secondly, we allow the mutual friction parameters to take the whole range of permissible values rather than focusing on a particular mechanism. Thus, we consider not only the weak drag regime, but also the strong drag regime where the fluid dynamics are significantly different.     

The magnetosphere of oscillating neutron stars in general relativity

Just as a rotating magnetized neutron star has material pulled away from its surface to populate a magnetosphere, a similar process can occur as a result of neutron-star pulsations rather than rotation. This is of interest in connection with the overall study of neutron star oscillation modes but with a particular focus on the situation for magnetars. Following a previous Newtonian analysis of the production of a force-free magnetosphere in this way Timokhin et al., we present here a corresponding general-relativistic analysis. We give a derivation of the general relativistic Maxwell equations for small-amplitude arbitrary oscillations of a non-rotating neutron star with a generic magnetic field and show that these can be solved analytically under the assumption of low current density in the magnetosphere. We apply our formalism to toroidal oscillations of a neutron star with a dipole magnetic field and find that the low current density approximation is valid for at least half of the oscillation modes, similarly to the Newtonian case. Using an improved formula for the determination of the last closed field line, we calculate the energy losses resulting from toroidal stellar oscillations for all of the modes for which the size of the polar cap is small. We find that general relativistic effects lead to shrinking of the size of the polar cap and an increase in the energy density of the outflowing plasma. These effects act in opposite directions but the net result is that the energy loss from the neutron star is significantly smaller than suggested by the Newtonian treatment.     

Can a slowly rotating neutron star be a radio pulsar?

It is shown that the radius of curvature of magnetic field lines in the polar region of a rotating magnetized neutron star can be significantly less than the usual radius of curvature of the dipole magnetic field. The magnetic field in the polar cap is distorted by toroidal electric currents flowing in the neutron star crust. These currents close up the magnetospheric currents driven by the electron–positron plasma generation process in the pulsar magnetosphere. Owing to the decrease in the radius of curvature, electron–positron plasma generation becomes possible even for slowly rotating neutron stars, with   PB ^−2/3₁₂ < 10 s  , where P is the period of star rotation and   B ₁₂= B /10¹² G  is the magnitude of the magnetic field on the star surface.     

Crust–core coupling and r-mode damping in neutron stars: a toy model

r-modes in neutron stars with crusts are damped by viscous friction at the crust–core boundary. The magnitude of this damping, evaluated by Bildsten & Ushomirsky (BU) under the assumption of a perfectly rigid crust, sets the maximum spin frequency for neutron stars spun up by accretion in low-mass X-ray binaries (LMXBs). In this paper we explore the mechanical coupling between the core r-modes and the elastic crust, using a toy model of a constant-density neutron star having a crust with a constant shear modulus. We find that, at spin frequencies in excess of ≈50 Hz, the r-modes strongly penetrate the crust. This reduces the relative motion (slippage) between the crust and the core compared with the rigid-crust limit. We therefore revise down, by as much as a factor of 10²–10³ , the damping rate computed by BU, significantly reducing the maximal possible spin frequency of neutron stars with solid crusts. The dependence of the crust–core slippage on the spin frequency is complicated, and is very sensitive to the physical thickness of the crust. If the crust is sufficiently thick, the curve of the critical spin frequency for the onset of the r-mode instability becomes multivalued for some temperatures; this is related to avoided crossings between the r-mode and higher-order torsional modes in the crust. The critical frequencies are comparable to the observed spins of neutron stars in LMXBs and millisecond pulsars.     

Seismic signatures of strange stars with crust

We study acoustic oscillations (eigenfrequencies, velocity distributions, damping times) of normal crusts of strange stars. These oscillations are very specific because of huge density jump at the interface between the normal crust and the strange matter core. The oscillation problem is shown to be self-similar. For a low (but non-zero) multipolarity l , the fundamental mode (without radial nodes) has a frequency of ∼300 Hz and mostly horizontal oscillation velocity; other pressure modes have frequencies ≳20 kHz and almost radial oscillation velocities. The latter modes are similar to radial oscillations (having approximately the same frequencies and radial velocity profiles). The oscillation spectrum of strange stars with crust differs from the spectrum of neutron stars. If detected, acoustic oscillations would allow one to discriminate between strange stars with crust and neutron stars and constrain the mass and radius of the star.     

Minimal models of cooling neutron stars with accreted envelopes

We study the 'minimal' cooling scenario of superfluid neutron stars with nucleon cores, where the direct Urca process is forbidden and enhanced cooling is produced by neutrino emission due to the Cooper pairing of neutrons. Extending our recent previous work, we include the effects of surface accreted envelopes of light elements. We employ the phenomenological density-dependent critical temperatures   T _cp(ρ)  and   T _cnt(ρ)  of singlet-state proton and triplet-state neutron pairing in a stellar core, as well as the critical temperature   T _cns(ρ)  of singlet-state neutron pairing in a stellar crust. We show that the presence of accreted envelopes simplifies the interpretation of observations of thermal radiation from isolated neutron stars in the scenario of our recent previous work and widens the class of models for nucleon superfluidity in neutron star interiors consistent with the observations.     

Differential rotation of relativistic superfluid in neutron stars

It is shown how to set up a mathematically elegant and fully relativistic superfluid model that can provide a realistic approximation (neglecting small anisotropies due to crust solidity, magnetic fields, etc., but allowing for the regions with vortex pinning) of the global structure of a rotating neutron star, in terms of just two independently moving constituents. One of these represents the differentially rotating neutron superfluid, while the other part represents the combination of all the other ingredients, including the degenerate electrons, the superfluid protons in the core, and the ions in the crust, the electromagnetic interactions of which will tend to keep them locked together in a state of approximately rigid rotation. Order of magnitude estimates are provided for relevant parameters such as the resistive drag coefficient.     

Simulations of glitches in isolated pulsars

Many radio pulsars exhibit glitches wherein the star's spin rate increases fractionally by ∼10⁻¹⁰–10⁻⁶. Glitches are ascribed to variable coupling between the neutron star crust and its superfluid interior. With the aim of distinguishing among different theoretical explanations for the glitch phenomenon, we study the response of a neutron star to two types of perturbations to the vortex array that exists in the superfluid interior: (1) thermal motion of vortices pinned to inner crust nuclei, initiated by sudden heating of the crust, (e.g., a starquake), and (2) mechanical motion of vortices (e.g., from crust cracking by superfluid stresses). Both mechanisms produce acceptable fits to glitch observations in four pulsars, with the exception of the 1989 glitch in the Crab pulsar, which is best fitted by the thermal excitation model. The two models make different predictions for the generation of internal heat and subsequent enhancement of surface emission. The mechanical glitch model predicts a negligible temperature increase. For a pure and highly conductive crust, the thermal glitch model predicts a surface temperature increase of as much as ∼2 per cent, occurring several weeks after the glitch. If the thermal conductivity of the crust is lowered by a high concentration of impurities, however, the surface temperature increases by ∼10 per cent about a decade after a thermal glitch. A thermal glitch in an impure crust is consistent with the surface emission limits following the 2000 January glitch in the Vela pulsar. Future surface emission measurements coordinated with radio observations will constrain glitch mechanisms and the conductivity of the crust.     

Two-fluid models of superfluid neutron star cores

Both relativistic and non-relativistic two-fluid models of neutron star cores are constructed, using the constrained variational formalism developed by Brandon Carter and co-workers. We consider a mixture of superfluid neutrons and superconducting protons at zero temperature, taking into account mutual entrainment effects. Leptons, which affect the interior composition of the neutron star and contribute to the pressure, are also included. We provide the analytic expression of the Lagrangian density of the system, the so-called master function, from which the dynamical equations can be obtained. All the microscopic parameters of the models are calculated consistently using the non-relativistic nuclear energy density functional theory. For comparison, we have also considered relativistic mean field models. The correspondence between relativistic and non-relativistic hydrodynamical models is discussed in the framework of the recently developed 4D covariant formalism of Newtonian multifluid hydrodynamics. We have shown that entrainment effects can be interpreted in terms of dynamical effective masses that are larger in the relativistic case than in the Newtonian case. With the nuclear models considered in this work, we have found that the neutron relativistic effective mass is even greater than the bare neutron mass in the liquid core of neutron stars.     

Relativistic models of magnetars: structure and deformations

We find numerical solutions of the coupled system of Einstein–Maxwell equations with a linear approach, in which the magnetic field acts as a perturbation of a spherical neutron star. In our study, magnetic fields having both poloidal and toroidal components are considered, and higher order multipoles are also included. We evaluate the deformations induced by different field configurations, paying special attention to those for which the star has a prolate shape. We also explore the dependence of the stellar deformation on the particular choice of the equation of state and on the mass of the star. Our results show that, for neutron stars with mass   M = 1.4 M_⊙  and surface magnetic fields of the order of 10¹⁵ G, a quadrupole ellipticity of the order of 10⁻⁶ to 10⁻⁵ should be expected. Low-mass neutron stars are in principle subject to larger deformations (quadrupole ellipticities up to 10⁻³ in the most extreme case). The effect of quadrupolar magnetic fields is comparable to that of dipolar components. A magnetic field permeating the whole star is normally needed to obtain negative quadrupole ellipticities, while fields confined to the crust typically produce positive quadrupole ellipticities.     

On the solution space of differentially rotating neutron stars in general relativity

A highly accurate, multidomain spectral code is used in order to construct sequences of general relativistic, differentially rotating neutron stars in axisymmetry and stationarity. For bodies with a spheroidal topology and a homogeneous or an   N = 1  polytropic equation of state, we investigate the solution space corresponding to broad ranges of degree of differential rotation and stellar densities. In particular, starting from static and spherical configurations, we analyse the changes of the corresponding surface shapes as the rate of rotation is increased. For a sufficiently weak degree of differential rotation, the sequences terminate at a mass-shedding limit, while for moderate and strong rates of differential rotation they exhibit a continuous parametric transition to a regime of toroidal fluid bodies. In this article, we concentrate on the appearance of this transition, analyse in detail its occurrence and show its relevance for the calculation of astrophysical sequences. Moreover, we find that the solution space contains various types of spheroidal configurations, which were not considered in previous work, mainly due to numerical limitations.     

Magnetic and spin evolution of neutron stars in close binaries

The evolution of neutron stars in close binary systems with a low-mass companion is considered, assuming the magnetic field to be confined within the solid crust. We adopt the standard scenario for the evolution in a close binary system, in which the neutron star passes through four evolutionary phases ('isolated pulsar'–'propeller'– accretion from the wind of a companion – accretion resulting from Roche-lobe overflow). Calculations have been performed for a great variety of parameters characterizing the properties of both the neutron star and the low-mass companion. We find that neutron stars with more or less standard magnetic field and spin period that are processed in low-mass binaries can evolve to low-field rapidly rotating pulsars. Even if the main-sequence life of a companion is as long as 10¹⁰ yr, the neutron star can maintain a relatively strong magnetic field to the end of the accretion phase. The model that is considered can account well for the origin of millisecond pulsars.     

Kelvin–Helmholtz instability and circulation transfer at an isotropic–anisotropic superfluid interface in a neutron star

A recent laboratory experiment suggests that a Kelvin–Helmholtz (KH) instability at the interface between two superfluids – one rotating and anisotropic, the other stationary and isotropic – may trigger sudden spin-up of the stationary superfluid. This result suggests that a KH instability at the crust–core (  ¹ S₀–³ P₂  –superfluid) boundary of a neutron star may provide a trigger mechanism for pulsar glitches. We calculate the dispersion relation of the KH instability involving two different superfluids including the normal fluid components and their effects on stability, particularly entropy transport. We show that an entropy difference between the core and crust superfluids reduces the threshold differential shear velocity and threshold crust–core density ratio. We evaluate the wavelength of maximum growth of the instability for neutron star parameters and find the resultant circulation transfer to be within the range observed in pulsar glitches.     

The evolution of the core and surface magnetic fields in isolated neutron stars

We apply the model of flux expulsion from the superfluid and superconductive core of a neutron star, developed by Konenkov & Geppert, both to neutron star models based on different equations of state and to different initial magnetic field structures. Initially, when the core and the surface magnetic field are of the same order of magnitude, the rate of flux expulsion from the core is almost independent of the equation of state, and the evolution of the surface field decouples from the core field evolution with increasing stiffness. When the surface field is initially much stronger than the core field, the magnetic and rotational evolution resembles that of a neutron star with a purely crustal field configuration; the only difference is the occurrence of a residual field. In the case of an initially submerged field, significant differences from the standard evolution only occur during the early period of the life of a neutron star, until the field has been re-diffused to the surface. The reminder of the episode of submergence is a correlation of the residual field strength with the submergence depth of the initial field. We discuss the effect of the re-diffusion of the magnetic field on the difference between the real and the active age of young pulsars and on their braking indices. Finally, we estimate the shear stresses built up by the moving fluxoids at the crust–core interface and show that these stresses may cause crust cracking, preferentially in neutron stars with a soft equation of state.     

Non-linear hydrodynamical evolution of rotating relativistic stars: numerical methods and code tests

We present numerical hydrodynamical evolutions of rapidly rotating relativistic stars, using an axisymmetric, non-linear relativistic hydrodynamics code. We use four different high-resolution shock-capturing (HRSC) finite-difference schemes (based on approximate Riemann solvers) and compare their accuracy in preserving uniformly rotating stationary initial configurations in long-term evolutions. Among these four schemes, we find that the third-order piecewise parabolic method scheme is superior in maintaining the initial rotation law in long-term evolutions, especially near the surface of the star. It is further shown that HRSC schemes are suitable for the evolution of perturbed neutron stars and for the accurate identification (via Fourier transforms) of normal modes of oscillation. This is demonstrated for radial and quadrupolar pulsations in the non-rotating limit, where we find good agreement with frequencies obtained with a linear perturbation code. The code can be used for studying small-amplitude or non-linear pulsations of differentially rotating neutron stars, while our present results serve as testbed computations for three-dimensional general-relativistic evolution codes.