Magnetic field of a neutron star with a superconducting quark core in the CFL-phase

The Ginzburg-Landau equations are derived for the magnetic and gluomagnetic gauge fields in the color superconducting core of a neutron star containing a CFL-condensate of diquarks. The interaction of the diquark CFL-condensate with the magnetic and gluomagnetic gauge fields is taken into account. The behavior of the magnetic field in a neutron star is studied by solving the Ginzburg-Landau equations taking correct account of the boundary conditions, including the gluon confinement conditions. The magnetic field distribution in the quark and hadronic phases of a neutron star is found. It is shown that a magnetic field generated in the hadronic phase by the entrainment effect penetrates into the quark core in the form of quark vortex filaments because of the presence of screening Meissner currents. __________ Translated from Astrofizika, Vol. 50, No. 1, pp. 87–98 (February 2007).     

Meissner Effect for “Color” Superconducting Quark Matter

The behavior of the magnetic field inside the superconducting quark matter core of a neutron star is investigated in the framework of the Ginzburg-Landau theory. We take into account the simultaneous coupling of the diquark condensate field to the usual magnetic and to the gluomagnetic gauge fields. We solve the problem for three different physical situations: a semi-infinite region with a planar boundary, a spherical region, and a cylindrical region. We show that Meissner currents near the quark core boundary effectively screen the external static magnetic field.     

Magnetic Field of a Neutron Star With Color Superconducting Quark Matter Core

The behaviour of the magnetic field of a neutron star with a superconducting quark matter core is investigated in the framework of the Ginzburg-Landau theory. We take into account the simultaneous coupling of the diquark condensate field to the usual magnetic and to the gluomagnetic gauge fields. We solve the Ginzburg-Landau equations by properly taking into account the boundary conditions, in particular, the gluon confinement condition. We found the distribution of the magnetic field in both the quark and hadronic phases of the neutron star and show that the magnetic field penetrates into the quark core in the form of quark vortices due to the presence of Meissner currents.     

Vortex structure of neutron stars with triplet neutron superfluidity

The vortex structure of the “npe” phase of neutron stars with a ³P₂ superfluid neutron condensate of Cooper pairs is discussed. It is shown that, as the star rotates, superfluid neutron vortex filaments described by a unitary ordering parameter develop in the “npe” phase. The entrainment of superconducting protons by the rotating superfluid neutrons is examined. The entrainment effect leads to the appearance of clusters of proton vortices around each neutron vortex and generates a magnetic field on the order of 10¹² G. ³P₂ neutron vortex filaments combine with quark semi-superfluid vortex filaments at the boundary of the “npe” and “CFL” phases. At the boundary of the “Aen” and “npe” phases, they combine with ¹S₀ neutron vortex filaments. In this way, a unified vortex structure is formed. The existence of this structure and its collective elastic oscillations explain the observed oscillations in the angular rotation velocity of pulsars.     

Vortex lines in neutron star superfluids and decay of pulsar magnetic fields

We adopt that in the interior of neutron stars both the proton and neutron superfluids are in the vortex state. Thus, in the superconducting core the magnetic field is expected to be organized in the form of quantized fluxoids. It is shown that fluxoids are buoyant. This gives rise to a rapid (5×10⁴ yr) expulsion of the magnetic field out of the superconducting core to the subcrustal region, and a subsequent decay within the outer crust. The effect considered may be the physical reason why the characteristic decay-time of pulsar magnetic fields (10⁶ yr) corresponds to the ohmic dissipation time within the neutron star crust. The intersection of two types of vortex lines with each other and its possible consequence for pulsars is briefly discussed.     

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. Translated from Astrofizika, Vol. 40, No. 4, op. 497–506, October–December, 1997.     

Generation of the magnetic fields of pulsars

The influence of the effect of entrainment of superconducting protons by superfluid neutrons on the distribution of neutron vortices in a rotating neutron star is investigated. It is shown that the proton vortex clusters generated by entrainment currents create the magnetic structure of a neutron vortex. The average magnetic field induction in a neutron vortex is calculated. The presence of the magnetic field of a neutron vortex considerably alters the radius of the vortex zone. The width of the vortex-free zone at the surface of the neutron star’s core increases, reaching macroscopic values on the order of several meters. This result considerably changes earlier concepts of the distribution of neutron vortices in a neutron star. Translated from Astrofizika, Vol. 43, No. 3, pp. 377-386, July–September, 2000.     

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.     

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) > H_cl (H_cl is the first critical field). Translated from Astrofizika, Vol. 42, No. 2, pp. 225–234, April–June, 1999     

Vortex lattice oscillations in rotating neutron stars with quark “CFL” cores

Collective elastic oscillations of a lattice of nonabelian quark semisuperfluid vortex filaments in the superfluid core of a rotating neutron star are examined. It is shown that in the incompressible fluid approximation, transverse long wavelength oscillations (Tkachenko oscillations) owing to shear deformation of the vortex lattice propagate in a plane perpendicular to the axis of rotation. The periods of these oscillations are consistent with rotational variations on the order of 100-1000 days observed in the pulsars PSR B0531+21 and PSR B1828-11. Translated from Astrofizika, Vol. 52, No. 1, pp. 165–169 (February 2009).     

Relaxation of quantum vortices in type II superconductors and in neutron stars

The nonstationary dynamics of vortices in conventional type II superconductors and in neutron stars is examined in the Newtonian approximation. A relaxation equation is obtained for vortices approaching an equilibrium distribution after a change in an external magnetic field. The relaxation times are estimated for vortices in low-temperature superconductors and for proton vortices in the superconducting core of a neutron star. It is shown that the proton vortex system created by entrainment currents is rigidly coupled to the neutron vortices. Translated from Astrofizika, Vol. 52, No. 2, pp. 291–300 (May 2009).     

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.     

Electric and magnetic fields inside the superconducting core of neutron stars

Feynman's approach has been used to derive the equation of dynamics for type II superconductors from the Schr?dinger equation. A closed set of equations for the study of vortex dynamics has been obtained. These equations have been used for calculating electric and magnetic fields inside the core of neutron stars. In particular, the contribution of vortices to the generation of electric and magnetic fields inside the core of the star is explicitly displayed.     

Vortex–interface interactions and generation of glitches in pulsars

We show that the crust–core interface in neutron stars acts as a potential barrier to the peripheral neutron vortices approaching the interface in the model in which these are coupled to the proton vortex clusters. This elementary barrier arises because of the interaction of vortex magnetic flux with the Meissner currents set up by the crustal magnetic field at the interface. The dominant part of the force is derived from the cluster–interface interaction. As a result of the stopping of the continuous neutron vortex current through the interface, angular momentum is stored in the superfluid layers in the vicinity of the crust–core interface during the interglitch period. Discontinuous annihilation of proton vortices at the boundary restores the neutron vortex current and spins up the observable crust on short time-scales, leading to a glitch in the spin characteristics of a pulsar.     

Magnetohydrodynamics in superconducting-superfluid neutron stars

Magnetohydrodynamic (MHD) equations are presented for the mixture of superfluid neutrons, superconducting protons and normal electrons believed to exist in the outer cores of neutron stars. The dissipative effects of electron viscosity and mutual friction resulting from electron-vortex scattering are also included. It is shown that Alfvén waves are replaced by cyclotron-vortex waves that have not been previously derived from MHD theory. The cyclotron-vortex waves are analogous to Alfvén waves with the tension arising from the magnetic energy density replaced by the vortex energy density. The equations are then put into a simplified form useful for studying the effect of the interior magnetic field on the dynamics. Of particular interest is the crust–core coupling time, which can be inferred from pulsar glitch observations. The hypothesis that cyclotron-vortex waves play a significant role in the core spin-up during a glitch is used to place limits on the interior magnetic field. The results are compared with those of other studies.     

The r-mode instability windows of hybrid stars

Taking into account the peculiar properties of hybrid stars, stars containing both a core of strange quark matter and the solid crust of a neutron star, and employing a fully self-consistent second-order correction technique, we study the time scale of bulk viscosity dissipation at the low temperature limit (T < 10⁹ K) and with this time scale we calculate the critical spin frequency of the hybrid star. It is found that its minimal value is 704.42 Hz (corresponding to a pulse period of 1.42 ms). When this is compared with the periods of neutron and strange stars, a better interpretation of the observational data is obtained.     

Maximum mass of a hot neutron star with a quark core

We have considered a hot neutron star with a quark core,a mixed phase of quark-hadron matter,and a hadronic matter crust and have determined the equation of state of the hadronic phase and the quark phase.We have then found the equation of state of the mixed phase under the Gibbs conditions.Finally,we have computed the structure of a hot neutron star with a quark core and compared our results with those of the neutron star without a quark core.For the quark matter calculations,we have used the MIT bag model...     

Planar solitonic filaments in solar physics

A new class of analytical solution of the coupled system of Da Rios-diffusion equation in magnetohydrodynamic (MHD) representing solitonic vortex filaments is obtained. One of the solutions is similar to the one found by Rogers and Schief describing a solitary wave propagating along a constant torsion vortex filament. Scalar magnetic diffusion equations are obtained by decomposing the magnetic filament along Frenet frame. The integral invariant of curvature is used to place limits on the diffused 100 eV plasma filament curvature. The resistivity of η=5×10^?5 ohm?cm—close to the stainless steel limit is used to approximate the Frenet curvature. From the scalar diffusion equations the vortex filaments are constrained to move along torsionless (planar) trajectories. Da Rios equations are coupled to diffusion equation to obtain a solitonic vortex diffused filaments. Due to bounds in time and length L we show that the model discussed is particularly useful in solar physics.     

Fluctuational mechanism for the formation of proton vortices in the superfluid core of neutron stars

The Gibbs thermodynamic potential of a proton vortex interacting with the normal core of a neutron vortex of radius r << λ (λ is the penetration depth) that is parallel to it and has an outer boundary of radius b is calculated. It is shown that, under this assumption, the capture of only one vortex by the core is energetically favorable. The force acting on the proton vortex owing to the entrained current is found and it is always directed toward the core. The corresponding force for a proton antivortex is directed toward the outer boundary of the neutron vortex. The Ginzburg-Landau equation is solved for a vortex-antivortex system and its Gibbs function is calculated. It is shown that at large distances from the core, vortex-antivortex pairs can form because of fluctuations. Acted on by the entrainment current, the antivortex moves outward, while the vortex stays inside the neutron vortex. It is shown that the best conditions for fluctuational pair production, followed by separation, exist near the outer boundary. It is shown that new proton vortices can develop only in a region where the entrainment magnetic field strength H (ρ) > H_C1 (H_C1 is the lower critical field). __________ Translated from Astrofizika, Vol. 51, No. 1, pp. 139–149 (February 2008).     

Theory of superdense matter and degenerate stellar configurations

During the last two decades the theory of degenerate stellar configurations has been developed in works by Ambartsumian and Sahakian, as well as in some other papers. This article is further progress in this direction. Systematic investigations of thermodynamic properties of the ground and metastable states of degenerate plasma have been carried out over the total range of pressures. It was found that in the range of densities 3×10¹⁰???3×10¹⁴ g cm^?3 there exists a pionization effect which plays an important role in the thermodynamics of degenerate plasma. The pion condensate present in nuclear matter promotes the existence of metastable nuclear clusters with the nuclear numberA?10⁶. The equation of state of degenerate stellar matter has been notably revised and, accordingly, the neutron star parameters have been calculated anew. The role of the pion condensate in generating strong magnetic fields observed in the pulsars is discussed.