General relativistic electromagnetic fields of a slowly rotating magnetized neutron star – II. Solution of the induction equations

We have solved numerically the general relativistic induction equations in the interior background space–time of a slowly rotating magnetized neutron star. The analytic form of these equations was discussed recently (Paper I), where corrections due to both the space–time curvature and the dragging of reference frames were shown to be present. Through a number of calculations we have investigated the evolution of the magnetic field with different rates of stellar rotation, different inclination angles between the magnetic moment and the rotation axis, as well as different values of the electrical conductivity. All of these calculations have been performed for a constant-temperature relativistic polytropic star and make use of a consistent solution of the initial-value problem which avoids the use of artificial analytic functions. Our results show that there exist general relativistic effects introduced by the rotation of the space–time which tend to decrease the decay rate of the magnetic field. The rotation-induced corrections are however generally hidden by the high electrical conductivity of the neutron star matter, and when realistic values for the electrical conductivity are considered, these corrections become negligible even for the fastest known pulsar.     

Relativistic models of magnetars: the twisted torus magnetic field configuration

We find general relativistic solutions of equilibrium magnetic field configurations in magnetars, extending previous results of Colaiuda et al. Our method is based on the solution of the relativistic Grad–Shafranov equation, to which Maxwell's equations can be reduced. We obtain equilibrium solutions with the toroidal magnetic field component confined into a finite region inside the star, and the poloidal component extending to the exterior. These so-called twisted torus configurations have been found to be the final outcome of dynamical simulations in the framework of Newtonian gravity, and appear to be more stable than other configurations. The solutions include higher-order multipoles, which are coupled to the dominant dipolar field. We use arguments of minimal energy to constrain the ratio of the toroidal to the poloidal field.     

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.     

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.     

Ohm’s law for plasma in general relativity and Cowling’s theorem

The general-relativistic Ohm’s law for a two-component plasma which includes the gravitomagnetic force terms even in the case of quasi-neutrality has been derived. The equations that describe the electromagnetic processes in a plasma surrounding a neutron star are obtained by using the general relativistic form of Maxwell equations in a geometry of slow rotating gravitational object. In addition to the general-relativistic effect first discussed by Khanna and Camenzind (Astron. Astrophys. 307:665, 1996) we predict a mechanism of the generation of azimuthal current under the general relativistic effect of dragging of inertial frames on radial current in a plasma around neutron star. The azimuthal current being proportional to the angular velocity ω of the dragging of inertial frames can give valuable contribution on the evolution of the stellar magnetic field if ω exceeds 2.7×10¹⁷(n/σ) s⁻¹ (n is the number density of the charged particles, σ is the conductivity of plasma). Thus in general relativity a rotating neutron star, embedded in plasma, can in principle generate axial-symmetric magnetic fields even in axisymmetry. However, classical Cowling’s antidynamo theorem, according to which a stationary axial-symmetric magnetic field can not be sustained against ohmic diffusion, has to be hold in the general-relativistic case for the typical plasma being responsible for the rotating neutron star.     

The effects of mass on the radiation of a relativistically rotating neutron star

We investigate the effect of mass on the radiation of a relativistically rotating neutron star. The method of Haxton and Ruffini is used to find the radiation flux from a relativistically rotating neutron star. By extending the idea of a point charge orbiting a black hole, a pulsar is modeled by simulating a relativistically rotating magnetic dipole embedded within a neutron star. The resulting equations retain the mass of the neutron star, thereby introducing effects of general relativity on the radiation from the dipole. We present exact solutions to the modeling equation as well as plots of energy spectra at different rotational velocities and inclination angles. We also present plots of total energy versus mass and two tables containing a comparison of energy ratios. These demonstrate that, for realistic neutron star masses, the high speed enhancement of the radiation is always more than compensated by the frame dragging effect, leading to a net reduction of radiation from the star. It is found that the inclusion of mass not only reduced the special relativistic enhancement, but negates it entirely as the mass of the neutron star approaches the mass limit.     

ROSAT observations of V471 Tauri, showing that stellar activity is determined by rotation, not age

I present pointed ROSAT PSPC observations of the pre-cataclysmic binary V471 Tauri. The hard X-ray emission (>0.4 keV) is not eclipsed by the K star, demonstrating conclusively that this component cannot be emitted by the white dwarf. Instead I show that its spectrum and luminosity are consistent with coronal emission from the tidally spun-up K star. The star is more active than other K stars in the Hyades, but equally active as K stars in the Pleiades with the same rotation periods, demonstrating that rotation — and not age — is the key parameter in determining the level of stellar activity.   The soft X-ray emission (<0.4 keV) is emitted predominately by the white dwarf and is modulated on its spin period. I find that the pulse profile is stable on time-scales of hours and years, supporting the idea that it is caused by the opacity of accreted material. The profile itself shows that the magnetic field configuration of the white dwarf is dipolar and that the magnetic axis passes through the centre of the star.   There is an absorption feature in the light curve of the white dwarf, which occurs at a time when our line of sight passes within a stellar radius of the K star. The column density and duration of this feature imply a volume and mass for the absorber that are similar to those of coronal mass ejections of the Sun.   Finally I suggest that the spin–orbit beat period detected in the optical by Clemens et al. may be the result of the interaction of the K-star wind with the magnetic field of the white dwarf.     

Accretion disc–stellar magnetosphere interaction: field line inflation and the effect on the spin-down torque

We calculate the structure of a force-free magnetosphere which is assumed to corotate with a central star and which interacts with an embedded differentially rotating accretion disc. The magnetic and rotation axes are aligned, and the stellar field is assumed to be a dipole. We concentrate on the case when the amount of field line twisting through the disc–magnetosphere interaction is large , and consider different outer boundary conditions. In general the field line twisting produces field line inflation (e.g. Bardou & Heyvaerts), and in some cases with large twisting many field lines can become open. We calculate the spin-down torque acting between the star and the disc, and we find that it decreases significantly for cases with large field line twisting. This suggests that the oscillating torques observed for some accreting neutron stars could be caused by the magnetosphere varying between states with low and high field line inflation. Calculations of the spin evolution of T Tauri stars may also have to be revised in the light of the significant effect that field line twisting has on the magnetic torque resulting from star–disc interactions.     

Unveiling the thermal and magnetic map of neutron star surfaces though their X-ray emission: method and light-curve analysis

Recent Chandra and XMM–Newton observations of a number of X-ray 'dim' pulsating neutron stars have revealed quite unexpected features in the emission from these sources. Their soft thermal spectrum, believed to originate directly from the star surface, shows evidence for a phase-varying absorption line at some hundred eVs. The pulse modulation is relatively large (pulsed fractions in the range ∼12–35 per cent), the pulse shape is often non-sinusoidal, and the hard X-ray colour appears to be anticorrelated in phase with the total emission. Moreover, the prototype of this class, RX J0720.4−3125, has been found to undergo rather sensible changes in both its spectral and timing properties over a time-scale of a few years. All these new findings seem difficult to reconcile with the standard picture of a cooling neutron star endowed with a purely dipolar magnetic field, at least if surface emission is produced in an atmosphere on top of the crust. In this paper we explore how a dipolar+quadrupolar star-centred field influences the properties of the observed light curves. The phase-resolved spectrum has been evaluated accounting for both radiative transfer in a magnetized atmosphere and general relativistic ray-bending. We computed over 78 000 light curves, varying the quadrupolar components and the viewing geometry. A comparison of the data with our model indicates that higher-order multipoles are required to reproduce the observations.     

Topology of magnetars external field – I. Axially symmetric fields

There is an increasing theoretical and observational evidence that the external magnetic field of magnetars may contain a toroidal component, likely of the same order of the poloidal one. Such 'twisted magnetospheres' are threaded by currents flowing along the closed field lines which can efficiently interact with soft thermal photons via resonant cyclotron scatterings (RCS). Actually, RCS spectral models proved quite successful in explaining the persistent ∼1–10 keV emission from the magnetar candidates, the soft γ-ray repeaters (SGRs) and the anomalous X-ray pulsars (AXPs). Moreover, it has been proposed that, in the presence of highly relativistic electrons, the same process can give rise to the observed hard X-ray spectral tails extending up to  ∼200 keV  . Spectral calculations have been restricted up to now to the case of a globally twisted dipolar magnetosphere, although there are indications that the twist may be confined only to a portion of the magnetosphere, and/or that the large-scale field is more complex than a simple dipole. In this paper, we investigate multipolar, force–free magnetospheres of ultramagnetized neutron stars. We first discuss a general method to generate multipolar solutions of the Grad-Schlüter-Shafranov (GSS) equation, and analyse in detail dipolar, quadrupolar and octupolar fields. The spectra and lightcurves for these multipolar, globally twisted fields are then computed using a Monte Carlo code and compared with those of a purely dipolar configuration. Finally, the phase-resolved spectra and energy-dependent lightcurves obtained with a simple model of a locally sheared field are confronted with the International Gamma-Ray Astrophysics Laboratory ( INTEGRAL ) observations of the AXPs 1RXS J1708−4009 and 4U 0142+61. Results support a picture in which the field in these two sources is not globally twisted.     

Protodiscs around hot magnetic rotator stars

We develop equations and obtain solutions for the structure and evolution of a protodisc region that is initially formed with no radial motion and super-Keplerian rotation speed when wind material from a hot rotating star is channelled towards its equatorial plane by a dipole-type magnetic field. Its temperature is around 10⁷ K because of shock heating and the inflow of wind material causes its equatorial density to increase with time. The centrifugal force and thermal pressure increase relative to the magnetic force and material escapes at its outer edge. The protodisc region of a uniformly rotating star has almost uniform rotation and will shrink radially unless some instability intervenes. In a star with angular velocity increasing along its surface towards the equator, the angular velocity of the protodisc region decreases radially outwards and magnetorotational instability (MRI) can occur within a few hours or days. Viscosity resulting from MRI will readjust the angular velocity distribution of the protodisc material and may assist in the formation of a quasi-steady disc. Thus, the centrifugal breakout found in numerical simulations for uniformly rotating stars does not imply that quasi-steady discs with slow outflow cannot form around magnetic rotator stars with solar-type differential rotation.     

The radio lighthouse CU Virginis: the spin-down of a single main-sequence star

The fast rotating star CU Virginis is a magnetic chemically peculiar star with an oblique dipolar magnetic field. The continuum radio emission has been interpreted as gyrosynchrotron emission arising from a thin magnetospheric layer. Previous radio observations at 1.4 GHz showed that a 100 per cent circular polarized and highly directive emission component overlaps to the continuum emission two times per rotation, when the magnetic axis lies in the plane of the sky. This sort of radio lighthouse has been proposed to be due to cyclotron maser emission generated above the magnetic pole and propagating perpendicularly to the magnetic axis. Observations carried out with the Australia Telescope Compact Array at 1.4 and 2.5 GHz one year after this discovery show that this radio emission is still present, meaning that the phenomenon responsible for this process is steady on a time-scale of years. The emitted radiation spans at least 1 GHz, being observed from 1.4 to 2.5 GHz. On the light of recent results on the physics of the magnetosphere of this star, the possibility of plasma radiation is ruled out. The characteristics of this radio lighthouse provide us a good marker of the rotation period, since the peaks are visible at particular rotational phases. After one year, they show a delay of about 15 min. This is interpreted as a new abrupt spinning down of the star. Among several possibilities, a quick emptying of the equatorial magnetic belt after reaching the maximum density can account for the magnitude of the breaking. The study of the coherent emission in stars like CU Vir, as well as in pre-main-sequence stars, can give important insight into the angular momentum evolution in young stars. This is a promising field of investigation that high-sensitivity radio interferometers such as Square Kilometre Array can exploit.     

Magnetic braking in magnetic binary stars

The role of an external magnetic field in the magnetic braking of a star with a dipolar field is investigated. In a magnetic cataclysmic variable system (i.e. the primary compact star has a strong magnetic field), the field external to the braking star (a late-type main-sequence star with a dynamo-generated field) originates from the compact star. A closed field region — the system dead zone — is formed within the binary system, and it does not take part in magnetic braking. The overall braking rate depends on the extent of this region and of the open flux, and is dependent on centrifugal effects. In the case of two interacting dipoles, the dipole orientations relative to the spin axes and to each other are found to be important, leading to different amounts of open flux and therefore of magnetic braking, owing to different centrifugal effects on closed field regions. However, in circumstances consistent with observations and dynamo theory, the white dwarf's field reduces the magnetic braking of the secondary significantly, a finding qualitatively similar to the results previously obtained for two anti-aligned dipoles perpendicular to the orbital plane. In the cases where the two dipole axes are not perpendicular to the orbital plane, but are inclined in the plane that links them, the 'cut-off' in magnetic braking is less abrupt, and this effect is more obvious as the inclinations increase. Only in the extreme cases when the two dipole axes are aligned in the orbital plane does the braking increase with white dwarf field strength. We conclude that detailed evolutionary modelling of AM Herculis systems needs to take account of the inclination effect.     

Analysis of magnetic field measurements for T Tau

The presence of hot spots on the surface of T Tau attributable to mass accretion from the protoplanetary disk is shown to have virtually no effect on the accuracy of estimating the magnetic field strength for this star. By comparing the magnetic field strengths for T Tau at the photospheric level measured by various methods, we found that if the angle i at which we see T Tau does not exceed 10°, then the magnetic field of the star could be dipolar with the angle between the dipole axis and the rotation axis of the star ?85°. If, however, it later emerges that i > 10°, its magnetic field is essentially nondipolar and/or nonstationary.     

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.     

Emergence of magnetic field due to spin-polarized baryon matter in neutron stars

A model of the ferromagnetic origin of magnetic fields of neutron stars is considered. In this model, the magnetic phase transition occurs inside the core of neutron stars soon after formation. However, owing to the high electrical conductivity the core magnetic field is initially fully screened. We study how this magnetic field emerges for an outside observer. After some time, the induced field that screens the ferromagnetic field decays enough to uncover a detectable fraction of the ferromagnetic field. We calculate the time-scale of decay of the screening field and study how it depends on the size of the ferromagnetic core. We find that the same fractional decay of the screening field occurs earlier for larger cores. We conjecture that weak fields of millisecond pulsars, B ∼10⁸–10⁹ G, could be identified with ferromagnetic fields of unshielded fraction ε ∼10⁻⁴–10⁻³ resulting from the decay of screening fields by a factor 1− ε in ∼10⁸ yr since their birth.     

Magnetic torques between accretion discs and stars

I show in this paper that two types of magnetic torques can appear in the interaction between an accretion disc and a magnetic accretor. There is the well-known torque resulting from the difference in angular velocity between the accretion disc and the star, but in addition there is a torque coming from the interaction between the stellar magnetic field and the disc's own magnetic field. The latter form of magnetic torque decreases in strength more slowly with increasing radius, and will therefore dominate at large radii. The direction of the disc field is not determined by the difference in angular velocity between the star and the disc as in the Ghosh &38; Lamb model, but rather is a free parameter. The magnetic torque may therefore either spin up or spin down the star, and the torque changes sign if the magnetic field in the disc reverses. I suggest that this mechanism can explain the torque reversals that have been observed in some disc-fed X-ray pulsars.     

The first magnetic maps of a pre-main-sequence binary star system – HD 155555

We present the first maps of the surface magnetic fields of a pre-main-sequence binary system. Spectropolarimetric observations of the young, 18 Myr, HD 155555 (V824 Ara, G5IV+K0IV) system were obtained at the Anglo-Australian Telescope in 2004 and 2007. Both data sets are analysed using a new binary Zeeman–Doppler imaging (ZDI) code. This allows us to simultaneously model the contribution of each component to the observed circularly polarized spectra. Stellar brightness maps are also produced for HD 155555 and compared to previous Doppler images.
Our radial magnetic maps reveal a complex surface magnetic topology with mixed polarities at all latitudes. We find rings of azimuthal field on both stars, most of which are found to be non-axisymmetric with the stellar rotational axis. We also examine the field strength and the relative fraction of magnetic energy stored in the radial and azimuthal field components at both epochs. A marked weakening of the field strength of the secondary star is observed between the 2004 and 2007 epochs. This is accompanied by an apparent shift in the location of magnetic energy from the azimuthal to radial field. We suggest that this could be indicative of a magnetic activity cycle. We use the radial magnetic maps to extrapolate the coronal field (by assuming a potential field) for each star individually – at present ignoring any possible interaction. The secondary star is found to exhibit an extreme tilt (≈75°) of its large-scale magnetic field to that of its rotation axis for both epochs. The field complexity that is apparent in the surface maps persists out to a significant fraction of the binary separation. Any interaction between the fields of the two stars is therefore likely to be complex also. Modelling this would require a full binary field extrapolation.     

Model pulsar magnetospheres: the perpendicular rotator

The simplest model illustrating the effect of the magnetospheric charge-current field on the structure of a pulsar magnetic field has the region within the light-cylinder filled with the GoldreichJulian charge density which corotates with the neutron star, but has no electric currents along the magnetic field lines. This model has previously been studied for the axisymmetric case, with the rotation and magnetic dipolar axes aligned. The analogous problem is now solved with the two axes mutually perpendicular, so that not only the material current arising from the rotating charges but also the displacement current contributes. Again, the constructed magnetic field B ₀ crosses the light-cylinder normally, and there is no energy flux to infinity. However, in a more realistic model there is a flow of current along B ₀, generating a field B ₁ which has a non-vanishing toroidal component at the light-cylinder, so yielding a finite integrated Poynting flux.     

Electromagnetic fields of slowly rotating compact magnetized stars in braneworld

We study the structure of electromagnetic field of slowly rotating magnetized star in a Randall-Sundrum II type braneworld. The star is modeled as a sphere consisting of perfect highly magnetized fluid with infinite conductivity and frozen-in dipolar magnetic field. Maxwell’s equations for the external magnetic field of the star in the braneworld are analytically solved in approximation of small distance from the surface of the star. We have also found numerical solution for the electric field outside the rotating magnetized neutron star in the braneworld in dependence on brane tension. The influence of brane tension on the electromagnetic energy losses of the rotating magnetized star is underlined. Obtained “brane” corrections are shown to be relevant and have non-negligible values. In comparison with astrophysical observations on pulsars spindown data they may provide an evidence for the brane tension and, thus, serve as a test for the braneworld model of the Universe.