Superfluid Core Research Articles

A Comparison of Measured Crab and Vela Glitch Healing Parameters with Predictions of Neutron Star Models

There are currently two well-accepted models that explain how pulsars exhibit glitches, sudden changes in their regular rotational spin-down. According to the starquake model, the glitch healing parameter, Q, which is measurable in some cases from pulsar timing, should be equal to the ratio of the moment of inertia of the superfluid core of a neutron star (NS) to its total moment of inertia. Measured values of the healing parameter from pulsar glitches can therefore be used in combination with realistic NS structure models as one test of the feasibility of the starquake model as a glitch mechanism. We have constructed NS models using seven representative equations of state of superdense matter to test whether starquakes can account for glitches observed in the Crab and Vela pulsars, for which the most extensive and accurate glitch data are available. We also present a compilation of all measured values of Q for Crab and Vela glitches to date which have been separately published in the literature. We have computed the fractional core moment of inertia for stellar models covering a range of NS masses and find that for stable NSs in the realistic mass range 1.4 +/- 0.2 solar masses, the fraction is greater than 0.55 in all cases. This range is not consistent with the observational restriction Q 0.7) are consistent with the starquake model predictions and support previous conclusions that starquakes can be the cause of Crab glitches.

Radiation of Neutron Stars Produced by Superfluid Core

We find that neutron star interior is transparent for collisionless electron sound, the same way as it is transparent for neutrinos. In the presence of magnetic field the electron sound is coupled with electromagnetic radiation and form the fast magnetosonic wave. We find that electron sound is generated by superfluid vortices in the stellar core. Thermally excited helical vortex waves produce fast magnetosonic waves in the stellar crust which propagate toward the surface and transform into outgoing electromagnetic radiation. The vortex radiation has the spectral index -0.45 and can explain nonthermal radiation of middle-aged pulsars observed in the infrared, optical and hard X-ray bands. The radiation is produced in the stellar interior which allows direct determination of the core temperature. Comparing the theory with available spectra observations we find that the core temperature of the Vela pulsar is T=8*10^8K, while the core temperature of PSR B0656+14 and Geminga exceeds 2*10^8K. This is the first measurement of the temperature of a neutron star core. The temperature estimate rules out equation of states incorporating Bose condensations of pions or kaons and quark matter in these objects. Based on the temperature estimate and cooling models we determine the critical temperature of triplet neutron superfluidity in the Vela core Tc=(7.5\pm 1.5)*10^9K which agrees well with recent data on behavior of nucleon interactions at high energies. Another finding is that in the middle aged neutron stars the vortex radiation, rather then thermal conductivity, is the main mechanism of heat transfer from the stellar core to the surface. Electron sound opens a perspective of direct spectroscopic study of superdense matter in the neutron star interiors.

R‐Modes of Neutron Stars with Superfluid Cores

We investigate the modal properties of the r-modes of rotating neutron stars with the core filled with neutron and proton superfluids, taking account of entrainment effects between the superfluids. The stability of the r-modes against gravitational radiation reaction is also examined considering viscous dissipation due to shear and a damping mechanism called between the superfluids in the core. We find that the r-modes in the superfluid core are split into ordinary r-modes and superfluid r-modes, which we call, respectively, ro- and rs-modes. The two superfluids in the core flow together for the ro-modes, while they countermove for the rs-modes. For the ro-modes, the coefficient κ0 ≡ limΩ→0 ω/Ω is equal to 2m/[l'(l' + 1)], almost independent of the parameter η that parameterizes the entrainment effects between the superfluids, where Ω is the angular frequency of rotation, ω is the oscillation frequency observed in the corotating frame of the star, and l' and m are the indices of the spherical harmonic function representing the angular dependence of the r-modes. For the rs-modes, on the other hand, κ0 is equal to 2m/[l'(l' + 1)] at η = 0 (no entrainment), and it almost linearly increases as η is increased from η = 0. The ro-modes, for which w' ≡ v − v ∝ Ω3, correspond to the r-modes discussed by L. Lindblom & G. Mendell, where v and v are the Eulerian velocity perturbations of the neutron and proton superfluids, respectively. The mutual friction in the superfluid core is found ineffective to stabilize the r-mode instability caused by the ro-mode except in a few narrow regions of η. The r-mode instability caused by the rs-modes, on the other hand, is extremely weak and easily damped by dissipative processes in the star.

Evolution of Pulsars with Energy Release in the Superfluid Core of a Neutron Star

The effect of a neutron-proton vortex system on the rotation dynamics of neutron stars is examined. The dynamics of the motion of a two component superfluid system in the core of a neutron star yields an equation for the evolution of the pulsar's rotation period. The spin down of the star owing to energy release at the core boundary, which is associated with a contraction of the length of the neutron vortex as it moves radially and magnetic energy of the vortical cluster is released, is taken into account. Evolutionary curves are constructed for pulsars with different magnetic fields and stellar radii. For certain values of the coefficient of friction between the superfluid and normal components in the core of the neutron star, at the end of its evolution a radio pulsar may become an anomalous x-ray pulsar or a source of soft gamma radiation with a period on the order of 10 seconds.

Cooling of neutron stars: Two types of triplet neutron superfluidity

We consider the cooling of neutron stars with superfluid cores composed of neutrons, protons, and electrons (for singlet proton pairing and triplet neutron pairing). The emphasis is on triplet neutron pairing with the component of the total moment of neutron pairs along the quantization axis |m J | = 2. This case stands out in that it leads to power-law rather than exponential suppression of the main neutrino processes by neutron superfluidity. For the chosen critical neutron temperatures Tcn, the cooling with |m J | = 2 proceeds either almost in the same way as the commonly considered cooling with m J =0 or appreciably faster. The cooling with variable (over the core) critical temperatures Tcn(ρ) and Tcp(ρ) can generally be described by the cooling with some effective constant temperatures Tcn and Tcp. The hypothesis of strong neutron superfluidity with |m J | = 2 is in conflict with the observational data on the thermal radiation from isolated neutron stars; the hypothesis of weak neutron superfluidity of any type is consistent with the observations.

Non–linear magnetic field decay in neutron stars

There exists both theoretical and observational evidence that the magnetic field decay in neutron stars may proceed in a pronounced non–linear way during a certain episode of the neutron star's life. In the presence of a strong magnetic field the Hall–drift dominates the field evolution in the crust and/or the superfluid core of neutron stars. Analysing observations of P and for sufficiently old isolated pulsars we gain strong hints for a significantly non–linear magnetic field decay. Under certain conditions with respect to the geometry and strength of a large–scale magnetic background field an instability is shown to occur which rapidly raises small–scale magnetic field modes. Their growth rates increase with the background field strength and may reach ~ 104 times the ohmic decay rate. Consequences for the rotational and thermal evolution as well as for the cracking of the crust of neutron stars are discussed.

Neutrino emission from superfluid neutron-star cores: Various types of neutron pairing

We calculate and provide analytic fits of the factors which describe the reduction of the neutrino emissivity of modified Urca and nucleon-nucleon bremsstrahlung processes by superfluidity of neutrons and protons in neutron-star cores. We consider $^1$S$_0$ pairing of protons and either $^1$S$_0$ or $^3$P$_2$ pairing of neutrons. We analyze two types of $^3$P$_2$ pairing: the familiar pairing with zero projection of the total angular momentum of neutron pairs onto quantization axis, $m_J=0$; and the pairing with $|m_J|=2$ which leads to the gap with nodes at the neutron Fermi surface. Combining the new data with those available in the literature we fully describe neutrino emission by nucleons from neutron star cores to be used in simulations of cooling of superfluid neutron stars.

Hall-drift induced magnetic field instability in neutron stars.

In the presence of a strong magnetic field and under conditions as realized in the crust and the superfluid core of neutron stars, the Hall drift dominates the field evolution. We show by a linear analysis that, for a sufficiently strong large-scale background field depending at least quadratically on position in a plane conducting slab, an instability occurs which rapidly generates small-scale fields. Their growth rates depend on the choice of the boundary conditions, increase with the background field strength, and may reach 10(3) times the Ohmic decay rate. The effect of that instability on the rotational and thermal evolution of neutron stars is discussed.

Glitches induced by the core superfluid in a neutron star

The long-term evolution of the relative rotation of the core superfluid in a neutron star with respect to the rest of the star, at different radial distances from the rotation axis, is determined through model calculations. The core superfluid rotates at a different rate (faster, in young pulsars), while spinning down at the same steady-state rate as the rest of the star, because of the assumed pinning between the superfluid vortices and the superconductor fluxoids. We find that the magnitude of this rotational lag changes with time and also depends on the distance from the rotation axis; the core superfluid supports an evolving pattern of differential rotation. We argue that the predicted change of the lag might occur as discrete events which could result in a sudden rise of the spin frequency of the crust of a neutron star, as is observed at glitches in radio pulsars. This new possibility for the triggering cause of glitches in radio pulsars is further supported by an estimate of the total predicted excess angular momentum reservoir of the core superfluid. The model seems to offer also resolutions for some other aspects of the observational data on glitches.

Structure of the Magnetic Field of a Neutron Star

The magnetic field distribution in the superfluid, spherical, hadronic core of a rotating neutron star, which consists of vortex and vortex-free zones, is investigated. Due to the effect of “entrainment” of superconducting protons by rotating superfluid neutrons, a nonuniform magnetic field, the average value of which is constant, is formed in the vortex zone of the neutron star, directed parallel to the star's axis of rotation. It is shown that at the stellar surface, near the equatorial plane, there is a vortex-free zone of macroscopic size in which there is no magnetic field. The magnetic field near the boundaries of the vortex-free zone falls off exponentially with depth into the interior of this zone. This result essentially alters earlier concepts about the magnetic field distribution in the superfluid hadronic core of a neutron star. Outside the hadronic core the magnetic field has a dipole character with a magnetic moment on the order of 1030 g×cm3.

On the dynamics of superfluid neutron star cores

We discuss the nature of the various modes of pulsation of superfluid neutron stars using comparatively simple Newtonian models. The matter in these stars is described in terms of a two-fluid model, where one fluid is the neutron superfluid, which is believed to exist in the core and inner crust of mature neutron stars, and the other fluid represents a conglomerate of all other constituents (crust nuclei, protons, electrons, etc.). In our model, we incorporate the non-dissipative interaction due to the so-called entrainment effect, whereby the momentum of one constituent (e.g. the neutrons) carries along part of the mass of the other constituent. We show that there is no independent set of propagating g-modes in a superfluid neutron star core. We also approach the toroidal pulsation modes of a slowly rotating superfluid star. We show that the superfluid equations support a new class of r-modes. Finally, the role of the entrainment effect on the superfluid mode frequencies is shown explicitly via solutions to local dispersion relations.

Probing neutron-star superfluidity with gravitational-wave data.

We discuss the possibility that future gravitational-wave detectors may be able to detect various modes of oscillation of old, cold neutron stars. We argue that such detections would provide unique insights into the superfluid nature of neutron-star cores, and could also lead to a much improved understanding of pulsar glitches. Our estimates are based on a detector configuration with several narrow-band (cryogenic) interferometers operating as a "xylophone" which could lead to high sensitivity at high frequencies. We also draw on recent advances in our understanding of the dynamics of pulsating superfluid neutron-star cores.

On the nature of the residual magnetic fields in millisecond pulsars

We consider the expulsion of proton fluxoids along neutron vortices from the superfluid/superconductive core of neutron star with weak ($B<10^{10}$ G) magnetic field. The velocity of fluxoids is calculated from the balance of buoyancy, drag and crustal forces. We show, that the proton fluxoids can leave the superfluid core sliding {\it along} the neutron vortices on a timescale of about $10^7$ years. An alternative possibility is that fluxoids are aligned with the vortices on the same timescale. As the result, non--aligned surface magnetic fields of millisecond pulsars can be sustained for $\ga 10^9$ years only in case of a comparable dissipation timescale of the currents in the neutron star crust. This defines upper limits of the impurity concentration in the neutron star crust: $Q \la 0.1$ if a stiff equation of state determines the density profile.

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.

R-modes in superfluid neutron stars

The analogs of r-modes in superfluid neutron stars are studied here. These modes, which are governed primarily by the Coriolis force, are identical to their ordinary-fluid counterparts at the lowest order in the small angular-velocity expansion used here. The equations that determine the next order terms are derived and solved numerically for fairly realistic superfluid neutron-star models. The damping of these modes by superfluid ``mutual friction'' (which vanishes at the lowest order in this expansion) is found to have a characteristic time-scale of about 10^4 s for the m=2 r-mode in a ``typical'' superfluid neutron-star model. This time-scale is far too long to allow mutual friction to suppress the recently discovered gravitational radiation driven instability in the r-modes. However, the strength of the mutual friction damping depends very sensitively on the details of the neutron-star core superfluid. A small fraction of the presently acceptable range of superfluid models have characteristic mutual friction damping times that are short enough (i.e. shorter than about 5 s) to suppress the gravitational radiation driven instability completely.

Fluctuational mechanism of formation of proton vortices in the “npe” phase of a neutron star

The superfluid core (“npe” phase) of a neutron star, consisting of superfluid neutrons, superconducting protons, and normal electrons, is considered. The Gibbs thermodynamic potential of a superconducting proton vortex in a proton superconductor of the second kind, interacting with the normal core of a neutron vortex of radius r ≪ λ parallel to it (λ is the depth of penetration), is calculated. It is shown that under this assumption, the capture by the core of only one vortex turns out to be energetically favored. The force exerted on the proton vortex by the entrainment current, always directed toward the core, is found. The corresponding force for a proton antivortex is directed outward toward the outer boundary of the neutron vortex. It is shown that the fluctuational formation of a vortex-antivortex pair is possible at a large distance from the core under the action of the entrainment current. Under the action of the entrainment current, the antivortex travels outward, while the vortex remains inside the neutron vortex. It is shown that the formation of new proton vortices is possible only in the region in which the entrainment magnetic field strength is H(ρ) > Hcl (Hcl is the first critical field).

Dynamical Coupling of the Core Superfluid in Neutron Stars

Coupling of the neutron superfluid in the core of a neutron star to the rest of the star is discussed, assuming no pinning for the superfluid vortices. The earlier calculation of the relevant dynamical coupling timescale is shown to be incomplete. The corrected timescale is shorter by an order of magnitude; hence the observationally inferred upper limit on the core-coupling time remains much larger than, and in agreement with, the new predicted value.

Neutron Star Magnetic Field Evolution, Crust Movement, and Glitches

Spinning superfluid neutrons in the core of a neutron star interact strongly with co-existing superconducting protons. One consequence is that the outward(inward) motion of core superfluid neutron vortices during spin-down(up) of a neutron star may alter the core's magnetic field. Such core field changes are expected to result in movements of the stellar crust and changes in the star's surface magnetic field which reflect those in the core below. Observed magnitudes and evolution of the spin-down indices of canonical pulsars are understood as a consequence of such surface field changes. If the growing crustal strains caused by the changing core magnetic field configuration in canonical spinning-down pulsars are relaxed by large scale crust-cracking events, special properties are predicted for the resulting changes in spin-period. These agree with various glitch observations, including glitch activity, permanent shifts in spin-down rates after glitches in young pulsars, the intervals between glitches, families of glitches with different magnitudes in the same pulsar, the sharp drop in glitch intervals and magnitudes as pulsar spin-periods approach 0.7s, and the general absence of glitching beyond this period.

Type-I superconductivity of protons and magnetic fields of pulsars

The magnetic structure of neutron vortices in the superfluid cores of neutron stars is determined assuming that the proton condensate forms a type-I superconductor. It is shown that the entrainment currents induced by the neutron vortex circulation cause the proton superconductor to break into normal and superconducting domains of cylindrical form coaxial with the neutron vortex. The minimum of the energy functional corresponds to a tube radius a(0.1 – 0.5) b, where b is the outer radius of the neutron vortex. The magnetic field within the tube is of the order of 5 × 10 14 G.

Relaxation time of the superfluid core of a neutron star

The superconducting proton condensate in the “npe” phase of a neutron star is considered. It is shown to be a type II superconductor in the outer layer of the “npe” phase and a type I superconductor in the inner layer. Relaxation times are found for elastic scattering of normal relativistic electrons from the magnetic fields of proton vortex clusters in the case of a type II superconductor and elastic scattering from the magnetic field at the center of a neutron vortex in the case of a superconductor of the first kind. The dynamical relaxation times obtained for the angular velocity of the pulsar PSR 0833—45 vary, as a function of the density of the layers taking part in the relaxation process, within a fairly wide range: from several hours to l09 years. This means that the characteristic times of variation of pulsar angular velocity may be observed to lie in the indicated time range.