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.     

Heat blanketing envelopes and thermal radiation of strongly magnetized neutron stars

Strong (B?10⁹ G) and superstrong (B?10¹⁴ G) magnetic fields profoundly affect many thermodynamic and kinetic characteristics of dense plasmas in neutron star envelopes. In particular, they produce strongly anisotropic thermal conductivity in the neutron star crust and modify the equation of state and radiative opacities in the atmosphere, which are major ingredients of the cooling theory and spectral atmosphere models. As a result, both the radiation spectrum and the thermal luminosity of a neutron star can be affected by the magnetic field. We briefly review these effects and demonstrate the influence of magnetic field strength on the thermal structure of an isolated neutron star, putting emphasis on the differences brought about by the superstrong fields and high temperatures of magnetars. For the latter objects, it is important to take proper account of a combined effect of the magnetic field on thermal conduction and neutrino emission at densities ρ?10¹⁰ g?cm^?3. We show that the neutrino emission puts a B-dependent upper limit on the effective surface temperature of a cooling neutron star.     

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.     

Magnetar corona

Persistent high-energy emission of magnetars is produced by a plasma corona around the neutron star, with total energy output of ～10³⁶ erg/s. The corona forms as a result of sporadic starquakes that twist the external magnetic field of the star and induce electric currents in the closed magnetosphere. Once twisted, the magnetosphere cannot untwist immediately because of its self-induction. The self-induction electric field lifts particles from the stellar surface, accelerates them, and initiates avalanches of pair creation in the magnetosphere. The created plasma corona maintains the electric current demanded by curl B and regulates the self-induction e.m.f. by screening. This corona persists in dynamic equilibrium: it is continually lost to the stellar surface on the light-crossing time ～10^?4 s and replenished with new particles. In essence, the twisted magnetosphere acts as an accelerator that converts the toroidal field energy to particle kinetic energy. The voltage along the magnetic field lines is maintained near threshold for ignition of pair production, in the regime of self-organized criticality. The voltage is found to be about ～1 GeV which is in agreement with the observed dissipation rate ～10³⁶ erg/s. The coronal particles impact the solid crust, knock out protons, and regulate the column density of the hydrostatic atmosphere of the star. The transition layer between the atmosphere and the corona is the likely source of the observed 100 keV emission. The corona also emits curvature radiation up to 10¹⁴ Hz and can supply the observed IR-optical luminosity.     

Hydrogen molecular ion in the magnetic field of a neutron star

A two-dimensional potential energy surface of an H ₂ ⁺ molecular ion is calculated for the case of the strong magnetic field of the neutron starB=10¹¹–10¹³ G. It is shown that the dependence of the potential energy from the angle between the magnetic field direction and the internuclear axis becomes very sharp as the magnetic field increases. The obtained potential energy surfaces can be used for studying the vibrational-rotational structure of the H ₂ ⁺ spectrum in a strong magnetic field and the development of the observational methods for the determination of the magnetic field of a neutron star.     

On the differential rotation of the polar caps of neutron stars

The model of a magnetized rotating neutron star with an electric current in the region of its fluid polar magnetic caps is considered. The presence of an electric current leads to differential rotation of the magnetic caps. The rotation structure is determined by the electric current density distribution over the surface. In the simplest axisymmetric configuration, the current flows in one direction near the polar cap center and in the opposite direction in the outer ring (the total current is zero for the neutron star charge conservation). In this case, two rings with opposite directions of rotation appear on the neutron star surface, with the inner ring always lagging behind the star’s main rotation. The differential rotation velocity is directly proportional to the electric current density gradient along the polar cap radius. At a width of the region of change in the electric current from 1 to 10² cm and a period ～1 s and a magnetic field B ～ 10¹² G typical of radio pulsars, the linear differential rotation velocity is ～10^?2–10^?4 cm s^?1 (corresponding to a revolution time of ～0.1–10 yr).     

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.     

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.     

Natural MHD oscillations of a neutron star

Natural, low-frequency, hydromagnetic oscillations of an isolated, nonrotating neutron star, which are localized in the peripheral crust, the structure of which is determined by the electron-nuclear plasma (the Ae phase), are studied. The plasma medium of the outer crust is treated as a homogeneous, infinitely conducting, incompressible continuum, the motions of which are determined by the equations of magnetohydrodynamics. In the approximation of a constant magnetic field inside the crust (the magnetic field outside the star is assumed to have a dipole structure), the spectrum of normal poloidal and toroidal hydromagnetic oscillations, due to presumed residual fluctuations of flow and their associated fluctuations in magnetic field strength, is calculated. Numerical estimates given for the periods of MHD oscillations fall in the range of periods of radio pulsar emission, indicating a close connection between the residual hydromagnetic oscillations and the electromagnetic activity of neutron stars. Translated from Astrofizika, Vol. 40, No. 1, pp. 77–86, January–March, 1997.     

Flux-Vortex Pinning and Neutron Star Evolution

G. Srinivasan et al. (1990) proposed a simple and elegant explanation for the reduction of the neutron star magnetic dipole moment during binary evolution leading to low mass X-ray binaries and eventually to millisecond pulsars: Quantized vortex lines in the neutron star core superfluid will pin against the quantized flux lines of the proton superconductor. As the neutron star spins down in the wind accretion phase of binary evolution, outward motion of vortex lines will reduce the dipole magnetic moment in proportion to the rotation rate. The presence of a toroidal array of flux lines makes this mechanism inevitable and independent of the angle between the rotation and magnetic axes. The incompressibility of the flux-line array (Abrikosov lattice) determines the epoch when the mechanism will be effective throughout the neutron star. Flux vortex pinning will not be effective during the initial young radio pulsar phase. It will, however, be effective and reduce the dipole moment in proportion with the rotation rate during the epoch of spindown by wind accretion as proposed by Srinivasan et al. The mechanism operates also in the presence of vortex creep.     

On the torque exerted by a warped,magnetically threaded accretion disk

Most astrophysical accretion disks are likely to be warped.In X-ray binaries,the spin evolution of an accreting neutron star is critically dependent on the interaction between the neutron star magnetic field and the accretion disk.There have been extensive investigations on the accretion torque exerted by a coplanar disk that is magnetically threaded by the magnetic field lines from the neutron stars,but relevant works on warped/tilted accretion disks are still lacking.In this paper we develop a simplified twocomponent model,in which the disk is comprised of an inner coplanar part and an outer,tilted part.Based on standard assumption on the formation and evolution of the toroidal magnetic field component,we derive the dimensionless torque and show that a warped/titled disk is more likely to spin up the neutron star compared with a coplanar disk.We also discuss the possible influence of various initial parameters on the torque.     

The upper limit of magnetic field strength in dense stellar hadronic matter

It is shown that in strongly magnetized neutron stars, there exist upper limits of magnetic field strength, beyond which the self energies for both neutron and proton components of neutron star matter become complex in nature. As a consequence they decay within the strong interaction time scale. However, in the ultra-strong magnetic field case, when the zeroth Landau level is only occupied by protons, the system again becomes stable against strong decay.      

A hydrodynamic model for asymmetric explosions of rapidly rotating collapsing supernovae with a toroidal atmosphere

We numerically solved the two-dimensional axisymmetric hydrodynamic problem of the explosion of a low-mass neutron star in a circular orbit. In the initial conditions, we assumed a nonuniform density distribution in the space surrounding the collapsed iron core in the form of a stationary toroidal atmosphere that was previously predicted analytically and computed numerically. The configuration of the exploded neutron star itself was modeled by a torus with a circular cross section whose central line almost coincided with its circular orbit. Using an equation of state for the stellar matter and the toroidal atmosphere in which the nuclear statistical equilibrium conditions were satisfied, we performed a series of numerical calculations that showed the propagation of a strong divergent shock wave with a total energy of ～0.2×10⁵¹ erg at initial explosion energy release of ～1.0×10⁵¹ erg. In our calculations, we rigorously took into account the gravitational interaction, including the attraction from a higher-mass (1.9M_⊙) neutron star located at the coordinate origin, in accordance with the rotational explosion mechanism for collapsing supernovae. We compared in detail our results with previous similar results of asymmetric supernova explosion simulations and concluded that we found a lower limit for the total explosion energy.     

Magnetic field evolution in neutron stars

Neutron stars contain persistent, ordered magnetic fields that are the strongest known in the Universe. However, their magnetic fluxes are similar to those in magnetic A and B stars and white dwarfs, suggesting that flux conservation during gravitational collapse may play an important role in establishing the field, although it might also be modified substantially by early convection, differential rotation, and magnetic instabilities. The equilibrium field configuration, established within hours (at most) of the formation of the star, is likely to be roughly axisymmetric, involving both poloidal and toroidal components. The stable stratification of the neutron star matter (due to its radial composition gradient) probably plays a crucial role in holding this magnetic structure inside the star. The field can evolve on long time scales by processes that overcome the stable stratification, such as weak interactions changing the relative abundances and ambipolar diffusion of charged particles with respect to neutrons. These processes become more effective for stronger magnetic fields, thus naturally explaining the magnetic energy dissipation expected in magnetars, at the same time as the longer-lived, weaker fields in classical and millisecond pulsars. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A novel look at the pulsar force-free magnetosphere

The stationary axisymmetric force-free magnetosphere of a pulsar is considered. We present an exact dipolar solution of the pulsar equation, construct the magnetospheric model on its basis and examine its observational support. The new model has toroidal rather than common cylindrical geometry, in line with that of the plasma outflow observed directly as the pulsar wind nebula at much larger spatial scale. In its new configuration, the axisymmetric magnetosphere consumes the neutron star rotational energy much more efficiently, implying re-estimation of the stellar magnetic field, \(B_{\mathrm{new}}^{0}=3.3\times10^{-4}B/P\), where \(P\) is the pulsar period. Then the 7-order scatter of the magnetic field derived from the rotational characteristics of the pulsars observed appears consistent with the \(\cot\chi\)-law, where \(\chi\) is a random quantity uniformly distributed in the interval \([0,\pi/2]\). Our result is suggestive of a unique actual magnetic field strength of the neutron stars along with a random angle between the magnetic and rotational axes and gives insight into the neutron star unification on the geometrical basis.     

Cooling of neutron stars with strong toroidal magnetic fields

We present models of temperature distribution in the crust of a neutron star in the presence of a strong toroidal component superposed to the poloidal component of the magnetic field. The presence of such a toroidal field hinders heat flow toward the surface in a large part of the crust. As a result, the neutron star surface presents two warm regions surrounded by extended cold regions and has a thermal luminosity much lower than in the case the magnetic field is purely poloidal. We apply these models to calculate the thermal evolution of such neutron stars and show that the lowered photon luminosity naturally extends their life-time as detectable thermal X-ray sources. Work partially supported by UNAM-DGAPA grant #IN119306.     

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.     

Tayler instability with Hall effect in young neutron stars

Collapse calculations indicate that the hot young neutron stars rotate differentially so that strong toroidal magnetic field components should exist in the outer shell where also the Hall effect appears to be important when the Hall parameter = ω_Bτ exceeds unity. The amplitudes of the induced toroidal magnetic fields are limited by the current‐induced Tayler instability. An important characteristics of the Hall effect is its distinct dependence on the sign of the magnetic field. We find for fast rotation that positive (negative) Hall parameters essentially reduce (increase) the stability domain. It is thus concluded that the toroidal field belts in young neutron stars induced by their differential rotation should have different amplitudes in both hemispheres which later are frozen in. Due to the effect of magnetic suppression of the heat conductivity also the brightness of the two hemispheres should be different. As a possible example for our scenario the isolated neutron star RBS 1223 is considered which has been found to exhibit different X‐ray brightness at both hemispheres (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Towards self-consistent models of isolated neutron stars

We propose a self–consistent model to explain all observational properties reported so far on the isolated neutron star (INS) RX J0720-3125 with the aim of giving a step forward towards our understanding of INSs. For a given magnetic field structure, which is mostly confined to the crust and outer layers, we obtain theoretical models and spectra which account for the broadband spectral energy distribution (including the apparent optical excess), the X-ray pulsations, and for the spectral feature seen in the soft X-ray spectrum of RX J0720-3125 around 0.3 keV. By fitting our models to existing archival X-ray data from 6 different XMM–Newton observations and available optical data, we show that the observed properties are fully consistent with a normal neutron star, with a proper radius of about 12 km, a temperature at the magnetic pole of about 100 eV, and a magnetic field strength of 2–3×10¹³ G. Moreover, we are able to reproduce the observed long–term spectral evolution in terms of free precession which induces changes in the orientation angles of about 40 degrees with a periodicity of 7 years. In addition to the evidence of internal toroidal components, we also find strong evidence of non–dipolar magnetic fields, since all spectral properties are better reproduced with models with strong quadrupolar components.      

Thermal Evolution of Neutron Stars

We present results from simulations of protoneutron star thermal evolution using neutrino opacities that are consistently calculated with the equation of state. When hyperons are allowed to appear in the system, we obtain metastable configurations that after the deleptonization stage become unstable. Concerning the evolution of old neutron stars, we present the results of our investigation on the effect of the Joule heating due to magnetic field dissipation. We conclude that this mechanism can be efficient in maintaining the surface temperature of the star above 3 × 10⁴ - 10⁵ K during a very long time (≥ 100 Myr), comparable with the decay time of the magnetic field. This revised version was published online in July 2006 with corrections to the Cover Date.