Magnetic fields in axisymmetric neutron stars

We derive general equations for axisymmetric Newtonian magnetohydrodynamics and use these as the basis of a code for calculating equilibrium configurations of rotating magnetized neutron stars in a stationary state. We investigate the field configurations that result from our formalism, which include purely poloidal, purely toroidal and mixed fields. For the mixed-field formalism, the toroidal component appears to be bounded at less than 7 per cent. We calculate distortions induced both by magnetic fields and by rotation. From our non-linear work, we are able to look at the realm of validity of perturbative work: we find for our results that perturbative-regime formulae for magnetic distortions agree to within 10 per cent of the non-linear results if the ellipticity is less than 0.15 or the average field strength is less than 10¹⁷ G. We also consider how magnetized equilibrium structures vary for different polytropic indices.     

Equilibrium configurations of strongly magnetized neutron stars with realistic equations of state

We investigate equilibrium sequences of magnetized rotating stars with four kinds of realistic equations of state (EOSs) of SLy, FPS, Shen and LS, employing the Tomimura–Eriguchi scheme to construct the equilibrium configurations. We study the basic physical properties of the sequences in the framework of Newtonian gravity. In addition, we take a new step by taking into account a general relativistic effect to the magnetized rotating configurations. With these computations, we find that the properties of the Newtonian magnetized stars, e.g. structure of magnetic field, highly depends on the EOSs. The toroidal magnetic fields concentrate rather near the surface for Shen and LS EOSs than those for SLy and FPS EOSs. The poloidal fields are also affected by the toroidal configurations. Paying attention to the stiffness of the EOSs, we analyse this tendency in detail. In the general relativistic stars, we find that the difference due to the EOSs becomes small because all the employed EOSs become sufficiently stiff for the large maximum density, typically greater than  10¹⁵ g cm⁻³  . The maximum baryon mass of the magnetized stars with axis ratio   q ∼ 0.7  increases about up to 20 per cent for that of spherical stars. We furthermore compute equilibrium sequences at finite temperature, which should serve as an initial condition for the hydrodynamic study of newly born magnetars. Our results suggest that we may obtain information about the EOSs from the observation of the masses of magnetars.     

Axisymmetric magnetic fields in stars: relative strengths of poloidal and toroidal components

In this third paper in a series on stable magnetic equilibria in stars, I look at the stability of axisymmetric field configurations and, in particular, the relative strengths of the toroidal and poloidal components. Both toroidal and poloidal fields are unstable on their own, and stability is achieved by adding the two together in some ratio. I use Tayler's stability conditions for toroidal fields and other analytic tools to predict the range of stable ratios and then check these predictions by running numerical simulations. If the energy in the poloidal component as a fraction of the total magnetic energy is written as E_p / E , it is found that the stability condition is a ( E / U ) < E_p / E ≲ 0.8 where E /U is the ratio of magnetic to gravitational energy in the star and a is some dimensionless factor whose value is of order 10 in a main-sequence star and of order 10³ in a neutron star. In other words, whilst the poloidal component cannot be significantly stronger than the toroidal, the toroidal field can be very much stronger than the poloidal–given that in realistic stars we expect E / U < 10⁻⁶. The implications of this result are discussed in various contexts such as the emission of gravitational waves by neutron stars, free precession and a 'hidden' energy source for magnetars.     

A new numerical scheme for structures of rotating magnetic stars

We have developed a new numerical scheme for obtaining structures of rapidly rotating stars with strong magnetic fields. In our scheme, both poloidal and toroidal magnetic fields can be treated for stars with compressibility and infinite conductivity. By introducing the vector potential and its integral representation, we can treat the boundary condition for the magnetic fields across the surface properly. We show structures and distributions of magnetic fields as well as the distributions of the currents of rotating magnetic polytropic stars with polytropic index   N = 1.5  . The shapes of magnetic stars are oblate as long as the magnetic vector potential decreases as 1/ r when   r →∞  . For extremely strong magnetic fields, equilibrium configurations can be of toroidal shapes.     

Gravitational collapse of polytropic, magnetized, filamentary clouds

For the case in which the gas of a magnetized filamentary cloud obeys a polytropic equation of state, gravitational collapse of the cloud is studied using a simplified model. We concentrate on the radial distribution and restrict ourselves to a purely toroidal magnetic field. If the axial motions and poloidal magnetic fields are sufficiently weak, we could reasonably expect our solutions to be a good approximation. We show that while the filament experiences gravitational condensation and the density at the centre increases, the toroidal flux-to-mass ratio remains constant. A series of spatial profiles of density, velocity and magnetic field for several values of the toroidal flux-to-mass ratio and the polytropic index, is obtained numerically and discussed.     

Magnetic deformation of the white dwarf surface structure

The influence of strong, large‐scale magnetic fields on the structure and temperature distribution in white dwarf atmospheres is investigated. Magnetic fields may provide an additional component of pressure support, thus possibly inflating the atmosphere compared to the non‐magnetic case. Since the magnetic forces are not isotropic, atmospheric properties may significantly deviate from spherical symmetry. In this paper the magnetohydrostatic equilibrium is calculated numerically in the radial direction for either for small deviations from different assumptions for the poloidal current distribution. We generally find indication that the scale height of the magnetic white dwarf atmosphere enlarges with magnetic field strength and/or poloidal current strength. This is in qualitative agreement with recent spectropolarimetric observations of Grw+10°8247. Quantitatively, we .nd for e.g. a mean surface poloidal field strength of 100 MG and a toroidal field strength of 2‐10 MG an increase of scale height by a factor of 10. This is indicating that already a small deviation from the initial force‐free dipolar magnetic field may lead to observable effects. We further propose the method of finite elements for the solution of the two‐dimensional magnetohydrostatic equilibrium including radiation transport in the diffusive approximation. We present and discuss preliminary solutions, again indicating on an expansion of the magnetized atmosphere.     

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.     

Constraints on the magnetic field geometry of magnetars

We study the effect of the magnetic field geometry on the oscillation spectra of strongly magnetized stars. We construct a configuration of magnetic field where a toroidal component is added to the standard poloidal one. We consider a star with a type I superconductor core so that both components of the magnetic field are expelled from the core and confined in the crust. Our results show that the toroidal contribution does not influence significantly the torsional oscillations of the crust. On the contrary, the confinement of the magnetic field in the crust drastically affects the torsional oscillation spectrum. A comparison with estimations for the magnetic field strength, from observations, excludes the possibility that magnetars will have a magnetic field solely confined in the crust, that is, our results suggest that the magnetic field in whatever geometry has to permeate the whole star.     

Pseudo-Newtonian models for the equilibrium structures of rotating relativistic stars

We obtain equilibrium solutions for rotating compact stars, including special relativistic effects. The gravity is assumed to be Newtonian, but we use the active mass density, which takes into account all energies such as the motion of the fluid, internal energy and pressure energy in addition to the rest-mass energy, in computing the gravitational potential using Poisson's equation. Such a treatment could be applicable to neutron stars with relativistic motions or a relativistic equation of state. We applied Hachisu's self-consistent field (SCF) method to find spheroidal as well as toroidal sequences of equilibrium solutions. Our solutions show better agreement with general relativistic solutions than the Newtonian relativistic hydrodynamic approach, which does not take into account the active mass. Physical quantities such as the peak density and equatorial radii in our solutions agree with the general relativistic ones to within 5 per cent. Therefore our approach can be used as a simple alternative to the fully relativistic one when a large number of model calculations is necessary, as it requires much fewer computational resources.     

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.     

Toroidal versus poloidal magnetic fields in Sun-like stars: a rotation threshold

From a set of stellar spectropolarimetric observations, we report the detection of surface magnetic fields in a sample of four solar-type stars, namely HD 73350, HD 76151, HD 146233 (18 Sco) and HD 190771. Assuming that the observed variability of polarimetric signal is controlled by stellar rotation, we establish the rotation periods of our targets, with values ranging from 8.8 d (for HD 190771) to 22.7 d (for HD 146233). Apart from rotation, fundamental parameters of the selected objects are very close to the Sun's, making this sample a practical basis to investigate the specific impact of rotation on magnetic properties of Sun-like stars.
We reconstruct the large-scale magnetic geometry of the targets as a low-order  (ℓ < 10)  spherical harmonic expansion of the surface magnetic field. From the set of magnetic maps, we draw two main conclusions. (i) The magnetic energy of the large-scale field increases with rotation rate. The increase in chromospheric emission with the mean magnetic field is flatter than observed in the Sun. Since the chromospheric flux is also sensitive to magnetic elements smaller than those contributing to the polarimetric signal, this observation suggests that a larger fraction of the surface magnetic energy is stored in large scales as rotation increases. (ii) Whereas the magnetic field is mostly poloidal for low rotation rates, more rapid rotators host a large-scale toroidal component in their surface field. From our observations, we infer that a rotation period lower than ≈12 d is necessary for the toroidal magnetic energy to dominate over the poloidal component.     

On the quasi-periodic oscillations in magnetars

We study torsional Alfvén oscillations of magnetars, that is neutron stars with a strong magnetic field. We consider the poloidal and toroidal components of the magnetic field and a wide range of equilibrium stellar models. We use a new coordinate system  ( X , Y )  , where     and     and a ₁ is the radial component of the magnetic field. In this coordinate system, the one+two-dimensional evolution equation describing the quasi-periodic oscillations (QPOs), see Sotani et al., is reduced to a one+one-dimensional equation where the perturbations propagate only along the y -axis. We solve the one+one-dimensional equation for different boundary conditions and the open magnetic field lines, that is magnetic field lines that reach the surface and there match up with the exterior dipole magnetic field as well as closed magnetic lines, i.e. magnetic lines that never reach the stellar surface. For the open field lines, we find two families of QPO frequencies: a family of 'lower' QPO frequencies which is located near the x -axis and a family of 'upper' frequencies located near the y -axis. According to Levin, the fundamental frequencies of these two families can be interpreted as the turning point of the continuous spectrum. We find that the upper frequencies are multiples of the lower ones by a constant equalling  2 n + 1  . For the closed lines, the corresponding factor is   n + 1  . By using these relations, we can explain both the lower and the higher observed frequencies in SGR 1806−20 and SGR 1900+14.     

Alfvén polar oscillations of relativistic stars

We study polar Alfvén oscillations of relativistic stars endowed with a strong global poloidal dipole magnetic field. Here, we focus only on the axisymmetric oscillations which are studied by numerically evolving the two-dimensional perturbation equations. Our study shows that the spectrum of the polar Alfvén oscillations is discrete in contrast to the spectrum of axial Alfvén oscillations which is continuous. We also show that the typical fluid modes, such as the f and p modes, are not significantly affected by the presence of the strong magnetic field.     

A radiation-driven disc wind model for massive young stellar objects

The generation of magnetic fields by a battery, operating in an ion–electron plasma around a Kerr black hole, is studied in the 3+1 split of the Kerr metric. It is found that the gravitomagnetic contributions to the electron partial pressure are able to drive currents. The strength of the equilibrium magnetic field should be higher than for the classical Biermann battery, which is found to operate in this relativistic context as well, since the gravitomagnetic driving terms can less easily be quenched than the classical ones. In axisymmetry the battery can induce only toroidal magnetic fields. Once a toroidal magnetic field is present, however, the coupling of gravitomagnetic and electromagnetic fields generates a poloidal magnetic field even in axisymmetry. A rotating black hole, embedded in plasma, will therefore always generate toroidal and poloidal magnetic fields.     

Magnetic fields of Ap stars as a result of the Tayler instability

Ap star magnetism is often attributed to fossil magnetic fields which have not changed much since the pre‐main‐sequence epoch of the stars. Stable magnetic field configurations are known which could persist probably for the entire mainsequence life of the star, but they may not show the complexity and diversity exhibited by the Ap stars observed. We suggest that the Ap star magnetism is not a result of stable configurations, but is the result of an instability based on strong toroidal magnetic fields buried in the stars. The highly nonaxisymmetric remainders of the instability are reminiscent of the diversity of fields seen on Ap stars. The strengths of these remnant magnetic fields are actually between a few per cent up to considerable fractions of the internal toroidal field; this means field strengths of the order of kGauss being compatible with what is observed. The magnetic fields emerge at the surface rather quickly; rough estimates deliver time‐scales of the order of a few years. Since rotation stabilizes the instability, normal A stars may still host considerable, invisible toroidal magnetic fields (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relativistic expansion of magnetic loops at the self-similar stage

We obtained self-similar solutions of relativistically expanding magnetic loops taking into account the azimuthal magnetic fields. We neglect stellar rotation and assume axisymmetry and a purely radial flow. As the magnetic loops expand, the initial dipole magnetic field is stretched into the radial direction. When the expansion speed approaches the light speed, the displacement current reduces the toroidal current and modifies the distribution of the plasma lifted up from the central star. Since these self-similar solutions describe the free expansion of the magnetic loops, i.e.  d v /d t = 0  , the equations of motion are similar to those of the static relativistic magnetohydrodynamics. This allows us to estimate the total energy stored in the magnetic loops by applying the virial theorem. This energy is comparable to that of the giant flares observed in magnetars.     

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.     

Physical characteristics of critically distorted or almost spherically shaped rotating magnetic polytropic models

We use the so-called complex plane iterative technique (CIT) to the computation of polytropic stars distorted by rotation (either rigid or differential) and magnetic fields (both toroidal and poloidal). We give emphasis on computing(i) critically rotating configurations, and (ii) configurations that – dueto the counterbalancing of the effects of rotation and poloidal magnetic field with the effects of toroidal magnetic field – obtain an almost spherical shape. This revised version was published online in July 2006 with corrections to the Cover Date.     

Magnetic Neutron Stars. I. Static Field Configurations

The paper is the first in a series dealing with the structure of magnetic and rotating neutron stars including general relativistic effects. The geometry of fossile magnetic fields frozen in the highly conductive neutron star matter in a non-rotating (or weakly rotating) star is studied. § 2 treats the general poloidal field in a vacuum outside the star. The geometry of magnetic fields within the star — whose matter is governed by a barytropic equation of state — is restricted by the condition that the magnetic force density should be curl-free to maintain equilibrium (§ 3). Numerical results are obtained for a poloidal and a toroidal dipole field (§ 4).     

Axisymmetric stars with both poloidal and toroidal magnetic fields

Chandrasekhar and Prendergast have established a result which has been assumed to imply that axisymmetric stars with an internal toroidal magnetic field should have zero external poloidal field. By considering mildly singular functions, the range of solutions is increased, and models can then be constructed which have toroidal and poloidal fields in the interior and a non-zero, external, poloidal field. Both the magnetic field and its associated current are continuous everywhere.