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.     

A New Formulation for General Relativistic Force-Free Electrodynamics and Its Applications

We formulate the general relativistic force-free electrodynamics in a new 3 1 language. In this formulation,when we have properly defined electric and magnetic fields,the covariant Maxwell equations could be cast in the traditional form with new vacuum con-stitutive constraint equations. The fundamental equation governing a stationary,axisymmet-ric force-free black hole magnetosphere is derived using this formulation which recasts the Grad-Shafranov equation in a simpler way. Compared to the classic 3 1 system of Thorne and MacDonald,the new system of 3 1 equations is more suitable for numerical use for it keeps the hyperbolic structure of the electrodynamics and avoids the singularity at the event horizon. This formulation could be readily extended to non-relativistic limit and find applica-tions in flat spacetime. We investigate its application to disk wind,black hole magnetosphere and solar physics in both flat and curved spacetime.     

Simulations of the axisymmetric magnetospheres of neutron stars

In this paper we present the results of time-dependent simulations of the dipolar axisymmetric magnetospheres of neutron stars carried out within the frameworks of both relativistic magnetohydrodynamics (MHD) and resistive force-free electrodynamics. The results of force-free simulations reveal the inability of our numerical method to accommodate the equatorial current sheets of pulsar magnetospheres, and raise a question mark about the robustness of this approach. On the other hand, the MHD approach allows us to make significant progress. We start with a non-rotating magnetically dominated dipolar magnetosphere and follow its evolution as the stellar rotation is switched on. We find that the time-dependent solution gradually approaches a steady state that is very close to the stationary solution of the pulsar equation found in 1999 by Contopoulos, Kazanas & Fendt. This result suggests that other stationary solutions that have the Y-point located well inside the light cylinder are unstable. The role of particle inertia and pressure on the structure and dynamics of MHD magnetospheres is studied in detail, as well as the potential implications of dissipative processes in the equatorial current sheet. We argue that pulsars may have differentially rotating magnetospheres which develop noticeable structural oscillations, and that this may help to explain the nature of the subpulse phenomena.     

The force-free dipole magnetosphere in nonlinear electrodynamics

Quantum electrodynamics(QED) effects may be included in physical processes of magnetar and pulsar magnetospheres with strong magnetic fields. Involving the quantum corrections, Maxwell electrodynamics is modified to nonlinear electrodynamics. In this work, we study the force-free magnetosphere in nonlinear electrodynamics in a general framework. The pulsar equation describing a steady and axisymmetric magnetosphere is derived, which now admits solutions with corrections. We derive the first-order nonlinear corrections to the near-zone dipole magnetosphere in some popular nonlinear effective theories.The field lines of the corrected dipole tend to converge on the rotational axis so that the fields in the polar region are stronger compared to the pure dipole case.     

类Crab脉冲星的三维磁层外隙模型

利用三维脉冲星磁层模型研究了磁层外隙的几何结构,首先用自治模型确定“外隙”的垂直尺度,在该模型中外隙尺度受回流的外隙流（带有隙加速的带电粒子发射的曲率光子）加热极帽而产生的热光子的碰撞而成对生成所限,外隙的横向尺度也受本地对生成所限,在脉冲星的磁层中,原则上有两个拓扑分离的外隙,允许同时进入和流出粒子,不过,流入粒子流产生的辐射形态受隙了本地对生成和恒星附近的磁对生成的严格制约,根据外隙及其本地结构的三维模型计算了类Crab脉冲星的辐射和相位分解谱。     

On the electric field screening by electron–positron pairs in a pulsar magnetosphere

We present a steady one-dimensional model for a pulsar polar cap accelerator, where the field-aligned electric field and flow are solved self-consistently with a given current density. It is assumed that no particles return to the star. It is known that the space-charge-limited flow is accelerated to energies high enough to create electron–positron pairs if the assumed current density is high enough. We find that when pairs are created in such a space-charge-limited flow, the accelerating electric field is screened out within a short distance after pair creation, if the pair particle flux is larger than a critical value. We also find that a space charge density wave is excited in the screening region.
We find that a pair flux larger than the critical value M _c=10³–10⁵ must be reached in a layer with thickness equal to the braking distance for the decelerating component. Therefore, the required multiplicity – the number of pairs created by one primary particle – is too large to be realized in the actual pulsar magnetosphere. We suggest that in order to obtain a localized potential drop along the polar cap magnetic flux, one needs to take into account additional effects such as wave–particle interaction or quasi-periodic pair creation.     

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.     

Numerical construction of magnetosphere with relativistic two-fluid plasma flows

We present a numerical model in which a cold pair plasma is ejected with relativistic speed through a polar cap region and flows almost radially outside the light cylinder. Stationary axisymmetric structures of electromagnetic fields and plasma flows are self-consistently calculated. In our model, motions of positively and negatively charged particles are assumed to be determined by electromagnetic forces and inertial terms, without pair creation and annihilation or radiation loss. The global electromagnetic fields are calculated by the Maxwell's equations for the plasma density and velocity, without using ideal magnetohydrodynamic condition. Numerical result demonstrates the acceleration and deceleration of plasma due to parallel component of the electric fields. Numerical model is successfully constructed for weak magnetic fields or highly relativistic fluid velocity, i.e. kinetic energy dominated outflow. It is found that appropriate choices of boundary conditions and plasma injection model at the polar cap should be explored in order to extend present method to more realistic pulsar magnetosphere, in which the Poynting flux is dominated.     

PICsar: A 2.5D axisymmetric,relativistic, electromagnetic,Particle in Cell code with a radiation absorbing boundary

We present PICsar – a new Particle in Cell code geared towards efficiently simulating the magnetosphere of the aligned rotator. PICsar is a special relativistic, electromagnetic, charge conservative code that can be used to simulate arbitrary electromagnetics problems in axisymmetry. It features stretchable body-fitted coordinates that follow the surface of a sphere, simplifying the application of boundary conditions in the case of the aligned rotator; a radiation absorbing outer boundary, which allows a steady state to be set up dynamically and maintained indefinitely from transient initial conditions; and algorithms for injection of charged particles into the simulation domain. The code is parallelized using MPI and scales well to a large number of processors. We discuss the numerical methods used in PICsar and present tests of the code. In particular, we show that PICsar can accurately and efficiently simulate the magnetosphere of the aligned monopole rotator in the force-free limit. We present simulations of the aligned dipole rotator in a forthcoming paper.     

The force-free magnetosphere around an oblique rotator

The force-free magnetosphere around an obliquely rotating pulsar is studied. The basic equations reduce to two equations for two Euler potentials. One of the Euler potentials is regarded as a generalization of the stream function of the poloidal magnetic field lines in an axisymmetric rotator. Two divergence-free vectors become tangential to the surface on which this Euler potential is constant.     

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.     

Auroral-polar currents during periods of moderate magnetospheric activity

Data from a meridian line of three component magnetometers were used to investigate the character of the equivalent overhead current flow at high latitudes during periods of moderately strong magnetospheric activity. The polar cap equivalent current flow is inexplicable in terms of our knowledge of the polar cap electric field configuration and probably represents the combined effect of several real current systems seated in the auroral oval and the polar cap. An important contributing factor is the current system associated with the interaction of the magnetosphere with the azimuthal component of the interplanetary magnetic field. The region of the Harang discontinuity is identifiable through the intrusion of polar cap equivalent current flow into the latitudinal regime normally occupied by the eastward and westward electrojets. The Harang discontinuity exhibits marked changes in scale size in association with substorm activity.     

Numerical simulations of aligned neutron star magnetospheres

We present detailed numerical simulations of the magnetosphere of an isolated neutron star in which the spin and magnetic dipole axes of the star are aligned. We demonstrate that stable charge distributions are always found, rather than particle outflows. A stable magnetosphere consists of a dome above the polar cap containing plasma of one charge and an equatorial belt containing plasma of the other sign: E · B =0 inside both of these. These are separated by a vacuum gap in which E · B ≠0 ( ρ =0 instead). We show that the charge distribution used in the 'standard' Goldreich–Julian pulsar model is inherently unstable: it collapses to a stable configuration that is very similar to the others illustrated here. An instructive video of this collapse is available at http://spacsun.rice.edu/~ian/. For typical pulsars, the stable solution has no particles near to the light cylinder, and if there were any there then their loss from the system would not lead to a replacement from the star (in contradiction to the explicit assumption used in the Goldreich–Julian model). We discuss the generic effects of pair creation, in particular as an additional source of ionization in the vacuum gap. The overall effect is simply to reduce the value of E · B in the vacuum gap so that the pair-production rate drops towards zero. A dome, disc and gap geometry is still the resulting solution. In conclusion, we confirm previous studies that the aligned rotator cannot make an active pulsar.     

The origin of Ganymede's polar caps

Since their discovery in Voyager images, the origin of the bright polar caps of Ganymede has intrigued investigators. Some models attributed the polar cap formation to thermal migration of water vapor to higher latitudes, while other models implicated plasma bombardment in brightening ice. Only with the arrival of Galileo at Jupiter was it apparent that Ganymede possesses a strong internal magnetic field, which blocks most of the plasma from bombarding the satellite's equatorial region while funneling plasma onto the polar regions. This discovery provides a plausible explanation for the polar caps as related to differences in plasma-induced brightening in the polar and the equatorial regions. In this context, we analyze global color and high resolution images of Ganymede obtained by Galileo, finding a very close correspondence between the observed polar cap boundary and the open/closed field lines boundary obtained from new modeling of the magnetic field environment. This establishes a clear link between plasma bombardment and polar cap brightening. High resolution images show that bright polar terrain is segregated into bright and dark patches, suggesting sputter-induced redistribution and subsequent cold trapping of water molecules. Minor differences between the location of the open/closed field lines boundary and the observed polar cap boundary may be due to interaction of Ganymede with Jupiter's magnetosphere, and our neglect of higher-order terms in modeling Ganymede's internal field. We postulate that leading-trailing brightness differences in Ganymede's low-latitude surface are due to enhanced plasma flux onto the leading hemisphere, rather than darkening of the trailing hemisphere. In contrast to Ganymede, the entire surface of Europa is bombarded by jovian plasma, suggesting that sputter-induced redistribution of water molecules is a viable means of brightening that satellite's surface.     

Jovian proton Aurora

The planet Jupiter possesses a magnetic field and is surrounded by a magnetosphere. The occurrence of auroral and polar cap phenomena similar to those found on earth is very likely. In this work auroral and polar cap emissions in a model Jovian atmosphere are determined for proton precipitation. The incident protons, which are characterized by representative spectra, are degraded in energy by applying the continuous slowing down approximation. All secondary and higher generation electrons are assumed to be absorbed locally and their contributions to the total emissions are included. Volume emission rates are calculated from the total direct excitation rates with corrections for cascading applied. Results show that most molecular hydrogen and helium emissions for polar cap precipitation are below the ambient dayglow values. Charge capture by precipitating protons is an important source of Lyman α and Balmer α emissions and offers a key to the detection of large fluxes of low energy protons.     

Thermal emission areas of heated neutron star polar caps

Observed hot spots on neutron stars are often associated with polar caps heated by the backflow of energetic electrons or positrons from accelerators on bundles of open magnetic field lines. Three effects are discussed that may be relevant to formation of hot spots and their areas. (1) The area of a polar cap is proportional to the ratio of the star’s surface dipole field to the local field at the polar cap. Because the field is coupled to the evolving spin of the superfluid core of the star, this ratio can depend on the stellar spin and its history. (2) The hot emission area may appear smaller to a distant observer when emitted X-rays propagate through electron-positron plasma created in the magnetosphere. The X-rays then change their energy spectrum because of cyclotron resonant scattering by pairs. (3) Hot spots may form on the star’s surface as a result of crust motions that are driven by the pull of core flux tubes pinned to the crust. Such motions twist the footprints of closed magnetic loops of the magnetosphere and induce an electric current in the loop, which will heat those footprints.     

The structure of polar caps and the equatorial belt of pulsars

The self-consistent balanced pulsar magnetosphere of a magnetic neutron star with aligned magnetic and rotational axes is considered. It is shown that the magnetosphere consists of electron polar caps separated by empty space from a positron equatorial belt. The shape of the cold polar caps at a large distance from the star is calculated. It is shown that the cap shape at a large distance is independent of the magnetospheric structure near the neutron star. The shape of the equatorial belt is calculated. It is shown that a part of the equatorial belt rotates differentially, and its angular velocity is larger than that of the star (superrotation). It is shown that under certain conditions the space charge density of the belt can be very large. In principle, the formation of a surface charge placed in vacuum on a magnetic surface is possible. Magnetospheric vibrations are considered. A connection is established between drifting subpulses and the equatorial belt superrotation and also between drifting subpulses and cap vibrations. The characteristic frequency of vibrations and the angular velocity of superrotation are estimated.     

Plasma magnetosphere of rotating magnetized neutron star in the braneworld

Plasma magnetosphere surrounding rotating magnetized neutron star in the braneworld has been studied. For the simplicity of calculations Goldreich-Julian charge density is analyzed for the aligned neutron star with zero inclination between magnetic field and rotation axis. From the system of Maxwell equations in spacetime of slowly rotating star in braneworld, second-order differential equation for electrostatic potential is derived. Analytical solution of this equation indicates the general relativistic modification of an accelerating electric field and charge density along the open field lines by brane tension. The implication of this effect to the magnetospheric energy loss problem is underlined. It was found that for initially zero potential and field on the surface of a neutron star, the amplitude of the plasma mode created by Goldreich-Julian charge density will increase in the presence of the negative brane charge. Finally we derive the equations of motion of test particles in magnetosphere of slowly rotating star in the braneworld. Then we analyze particle motion in the polar cap and show that brane tension can significantly change conditions for particle acceleration in the polar cap region of the neutron star.     

The global magnetic topology of AB Doradus

We have used Zeeman–Doppler maps of the surface field of the young, rapid rotator AB Dor  ( P _rot=0.514 d)  to extrapolate the coronal field, assuming it to be potential. We find that the topology of the large-scale field is very similar in all three years for which we have images. The corona divides cleanly into regions of open and closed field. The open field originates in two mid-latitude regions of opposite polarity separated by about 180° of longitude. The closed field region forms a torus extending almost over each pole, with an axis that runs through these two longitudes. We have investigated the effect on the global topology of different forms of flux in the unobservable hemisphere and in the dark polar spot where the Zeeman signal is suppressed. The flux distribution in the unobservable hemisphere affects only the low-latitude topology, whereas the imposition of a unidirectional polar field forces the polar cap to be open. This contradicts observations that suggest that the closed field corona extends to high latitudes and leads us to propose that the polar cap may be composed of multipolar regions.