Determination of the electromagnetic field produced by a magnetic oblique-rotator

The structure of the corotating region, which forms an inner portion of a stellar magnetosphere, is reconsidered in a quasi-neutral case by taking into account the inertial effects of electrons as well as that of ions up to the first order in their mass ratio (δ=m_?/m₊). It is emphasized first that the magnetosphere is not globally equipotential even in the frame rotating with a central star (i.e. ?#0, where ? is the ‘non-Backus’ potential) due at least to the inertial effects of plasma particles. However, it is shown that the condition ?=0 is asymptotically recovered in the corotating region owing to the presence of the drift current which can be taken into account only when δ is not entirely neglected. This fact suggests that the deviation of the plasma motion in the outer magnetosphere from the corotation can be attributed to the non-zero ?. A globally self-consistent solution is obtained under this condition (?=0). In contrast with the solutions in the ‘force-free’ and the ‘mass-less-electron’ approximations, this solution has a disk structure in the corotation zone in which the plasma and the current density are concentrated to a thin disk near the magnetic equator. Owing to this sheet current in the disk the lines of force of the stellar magnetic field are modified to form a very elongated shape (the magnetodisk) if the plasma β-value is fairly large. Such a disk structure seems to be a common feature in the high β inner magnetospheres of various types of stars.     

Force-Free Rotator

A simple mechanichal problem of the force-free motion of a relativistic bead inside a rotating pipe is examined. A relevant change of shape of the pipe is considered. The force-free motion of the bead along the rotating pipe is examined as the motion in a differentially rotating medium. The differential rotation proves to be resembling a well known case of the Couette flow without outer cylinder. The determination of vectoral fields (magnetic field) in the rotating charged mediums (plasma) is problematic for the laboratory observer. Therefore the same problem is examined in the framework of general relativity, that is the physical quantities are considered in the inertial frame and non-inertial frame with no rotational in homogeneity. The problematic character of the determination of the physical quantity in the differentially rotating madium relative to the inertial observer is explained. It is shown that at certain large distances from the rotation center the pipe does not rotate. However, at rather small distances its shape takes on the appearance of an expanded spiral-like configuration. A possible relevance of the obtained results to the motion of a relativistic plasma flows in pulsar magnetosphere is pointed out. The areas of Crab pulsar's dipole radiation are estimated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Determination of the electromagnetic field produced by a magnetic oblique-rotator

The inertial effect on the structure of the magnetosphere of a rotating star is investigated, in the corotation approximation for a surrounding quasi-neutral plasma. The equation of motion reduces to a usual static balance equation between the electromagnetic and the centrifugal forces, in the rotating frame. However the MHD condition, which can be regarded as a special form of the generalized Ohm's law, is modified by the inclusion of inertial effect, with a violation of the frozen-in condition in case of a general (i.e., not restricted to corotation) plasma motion. The inertial effect on the electromagnetic field is summarized in a partial scalar potential named the non-Backus potential, which is proportional to the centrifugal potential in the corotation approximation.An approximate solution of this corotation problem is given, in which another characteristic radiusr _M appears besides the light radiusr _L . This radius defines a distance beyond which the inertial effect becomes dominant over the electromagnetic one, and is useful in estimating the magnitude of the terminal velocity of a centrifugal wind. A few examples of the modification of dipole magnetic field due to the inertial effect are visualized. In an oblique-rotation case, it can be seen that such a warp of the neutral sheet (the surface ofB _r =0) is reproduced as observed in the Jovian magnetosphere.     

The role of plasma interchange motion for the formation of a plasmapause

Mcllwain's electric and magnetic field distributions (E3H and M2) have been used to calculate the drift path of plasma density irregularities taking into account plasma interchange motion driven by the gravitational and inertial forces acting on the whole mass of the plasma elements.It has been shown that there is a region in the magnetosphere which is unstable with respect to the interchange motion of the cold plasma element. Any plasma hole in the background density drifts ultimately toward an asymptotic trajectory. Along this trajectory the inward gravitational force is balanced by the outward inertial force averaged over one revolution around the Earth. This asymptotic trajectory, along which all plasma holes ultimately accumulate, is identified with the equatorial plasmapause. The maximum velocity for the interchange motion is proportional to the excess (or defect) of density in the plasma element, and inversely proportional to the integrated Pedersen conductivity. Plasma detachment is shown to occur preferentially in the post-midnight sector.     

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.     

On Pulsar Electrodynamics in Rotating Frames

In this paper we have considered a rotating, perfectly conducting sphere and have calculated the electric and magnetic field distributions measured by the rotating observer using the anholonomic approach. The calculations have been done for the following two cases: (1) rotating charged spherical shell and (2) a uniformely magnetized sphere. We have shown that in the limiting situation (a/c)² 1 and 1, the magnetic field distribution is the same for both observers, inertial and noninertial. The expressions obtained for the electric field components in the rotating frame have been compared with the corresponding expressions in the inertial frame, where the observer is at rest. Some of the results are in agreement with Post's approach to noninertial electrodynamics.     

Kinetic Alfven wave driven by density and magnetic field inhomogeneities in plasmas of finite β

On the basis of an analysis of the instability of drift caused by density and magnetic field inhomogeneities in plasmas with finite β, the effect of the instability on the excitation of kinetic Alfven wave (KAW) is probed. In the kinetic theory, which correctly treats the effect of the finite Larmor radius and the wave-particle resonant interaction, the motion of the ions is described with the Vlasov equation and the motion of electrons, with the kinetic drift equation. Comparing the effects by inhomogeneities in the density and in the magnetic field in plasmas with finite β, we found that the drift instability is more easily excited by the former, and in the instability so excited, the energy transfer is more intense. This energy transfer provides the physical basis for the excitation of KAW. As shown by numerical solutions, KAWs can be widely excited and produced in the magnetosphere, especially in the cusp of the magnetosphere, in the magnetopause and in the boundary layers of plasma sheets, where inhomogeneities are obvious. The results of the present work further illustrate that the KAW plays an important role in the energy transfer in magnetospheric regions.     

A numerical method to determine the electromagnetic field of the pulsar magnetosphere with inclined magnetic moment

A numerical method to determine the electromagnetic field of a steadily rotating magnetosphere with an inclined magnetic moment under a given boundary condition on an arbitrary shaped boundary surface is presented. The region may include the light cylinder. The present method, together with a companion method giving particle motion and creation, makes an iterative scheme to obtain a global model of the pulsar magnetosphere. A key problem for explaining the particle acceleration in pulsars is to solve field-aligned electric field in an accelerating region bounded by an ideal-MHD region. The present method is fit to connect a solution for the non-ideal-MHD region with another solution for the ideal-MHD region on a boundary surface whose location should also be solved (i.e., a floating boundary). The integration scheme is based on the boundary element method and it has great advantage as compared with other methods like the finite difference method and the Fourier transformation method.     

The effect of an electric field induced by a time-dependent ring current on the particle drift motion

The effect of an electric field induced by a rapidly decaying ring current on the motion of charged particles in the magnetosphere has been investigated using Euler potentials. For a model consisting of the earth dipole and the symmetric ring current, the electric field satisfies the condition E . B = 0.

Under this circumstance, the E × B drift of the particle can be identified as the motion of the magnetic field lines and vice versa. The time dependent electric field induced can be evaluated in a Spherical polar coordinate system by the formula

where and β are Euler potentials.

A model calculation on the particle drift velocity v_D = E × B/B² shows that the radial component of the drift velocity is in good agreement with those deduced from whistler duct studies.     

Energization of charged particles in planetary magnetospheres

A model is presented to describe the energization of charged particles in planetary magnetospheres. The model is based on the stochastic acceleration produced by a random electric field that is induced by the magnetic field fluctuations measured within the magnetospheres. The stochastic behavior of the electric field is simulated through a Monte Carlo method. We solve the equation of motion for a single charged particle—which comprises the stochastic acceleration due to the stochastic electric field, the Lorentz acceleration (containing the local magnetic field and the corotational electric field) and the gravitational planetary acceleration of the particle—under several initial conditions. The initial conditions include the ion species and the velocity distribution of the particles which depends on the sources they come from (solar wind, ionospheres, rings and satellites). We applied this model to Saturn’s inner magnetosphere using a sample of particles (H⁺, H₂O⁺, N⁺, O⁺ and OH⁺) initially located on Saturn’s north pole, above the C-Ring, on the south pole of Enceladus, in the north pole of Dione and above the E-Ring. The results show that the particles tend to increase the value of their energy with time reaching several eV in a few seconds and the large energization is observed far from the planet. We can distinguish three main energization regions within Saturn’s inner magnetosphere: minimum (Saturn’s ionosphere), intermediate (Dione) and high-energy (Enceladus and the E-ring). The resulting energy spectrum follows a power-law distribution (>1 keV), a logistic, an exponential decay or an asymmetric sigmoidal (<1 keV).     

Euler potential method in three-dimensional stellar wind problems

A theoretical scheme is developed to deal with the problems of stellar winds in three-dimensional situations, and relativistic fluid equations are integrated formally under isentropic and quasi-stationary conditions, in a flat space-time.The relativistic Euler equation for a one-component plasma is expressed in the same form as the ideal-MHD condition for the effective electromagnetic field which combines the inertial and pressure terms with the true electromagnetic field. This equation and that of mass continuity are integrated formally by introducing Euler-type potentials for the effective magnetic field and for the mass flux in the rotating frame, respectively. Functional form of one of these Euler potentials, which represents the total energy per unit charge in the rotating frame, is specified as an integral of motion. For an electron-proton plasma, the integrals for both components are combined to yield the energy integral of the plasma as a whole and the integrated Ohm's law, in the limit of vanishing mass ratio of an electron to a proton.Maxwell's equations are divided in two parts: i.e., the co-rotational and non-corotational parts. It is shown that the electromagnetic potentials for these parts are derived from a scalar super-potential and a vector super-potential, respectively.     

Force-free pulsar magnetospheres

The equations to the steady, axisymmetric, charged pulsar magnetosphere given recently by Michel (1973a, b) and Scharlemann and Wagoner (1972, preprint) are generalized to non-axisymmetric systems that are steady in the frame rotating with the pulsar. It is shown that in all nonsingular models with cylindrical (but not axial) symmetry, and with a non-zero magnetic field component parallel to the axis, the magnetic field lines emanating from the pulsar are all trapped within the light-cylinder, so that there is no net electromagnetic energy flow across the light cylinder.     

General relativistic rotational properties of stellar models

In this paper we give general relativistic expressions for the angular momentum and rotational kinetic energy of slowly rotating stars. These expressions contain contributions from the presure, gravitational red shift, and Doppler shift, and the motion of inertial frames. These contributions are not negligible, e.g., there are stable neutron star models for which the angular velocity of inertial frames at the center is about 70% the angular velocity of the star. These expressions are useful in the study of pulsars if pulsars are rotating neutron stars.     

Dynamics of electrons and heavy ions in Mercury's magnetosphere

Several basic magnetospheric processes at Mercury have been investigated with simple models. These include the adiabatic acceleration and convection of equatorially mirroring charged particles, the current sheet acceleration effect, and the acceleration of Na⁺ and other exospheric ions by the magnetospheric electric and magnetic fields near the planetary surface. The current steady-state treatment of the magnetospheric drift and convection processes suggests that the region of the inner magnetosphere as explored by the Mariner 10 spacecraft during its encounter with Mercury should be largely devoid of energetic (>100 keV) electrons in equatorial mirroring motion. As for ion motion, the large gyroradii of the heavy ions permit surface reimpact as well as loss via intercepting the magnetopause. Because of the kinetic energy gained in the gyromotion, the first effect could lead to sputtering processes and hence generation of secondary ions and neutrals. The second effect could account for the loss of about 50% of Mercury's exospheric ions.     

Interaction of energetic particles with HM-waves in the magnetosphere

Energetic particle response in electromagnetic fields of ULF HM-waves in the magnetosphere is reviewed. Pc4–5 geomagnetic pulsations observed at the synchronous altitude are classified into three types, in respect to their major magnetic field polarization in different directions, local time dependence, and different characteristics of accompanied flux modulations of energetic particles, i.e., two nearly transverse waves with the azimuthal and the radial polarization, and the compressional stormtime pulsations. Firstly, we formulate the drift kinetic theory of particle flux modulations under the constraint of the magnetic moment conservation. A generalized energy integral of the particle motion interacting with a ULF-wave with the three-dimensional structure propagating to the azimuthal direction is obtained in the L-shell coordinate of a mirror magnetic field. Its linearized form is reduced to the same form as the previously derived energy change, including the bounce-drift resonant interaction. It is shown that the perturbed guiding center distribution function of energetic particles consists of four contributions, the adiabatic mirror effect corresponding to pitch-angle change, the kinetic effects due to energy change and the accompanying L-shell displacement, and the bounceaveraged drift phase bunching. Secondly, the basic HM-wave modes constitutingcoupling ULF oscillations in non-uniform plasmas are discussed in different models of approach for different plasma states. The diamagnetic drift Alfvén wave and the compressional drift wave with a larger azimuthal mode number in a high-beta plasma are candidates for the stormtimes pulsations. The former is intrinsically a guided localized mode, while the latter is a non-localized mode. By making use of the above preparation, we apply the developed drift kinetic theory to interpret the phase relationships between the ion flux modulation and the geomagnetic pulsation in some selected examples of observations, demonstrating a fair agreement in theoretical results with the observations.     

The effect of the atmosphere and ionosphere on long period magnetospheric micropulsations

A study is made of the screening effect of the atmosphere and ionosphere which lets only part of a magnetospheric micropulsation signal reach the ground. We obtain analytical expressions relating the magnetic field in the magnetosphere to that on the ground, which are confirmed by computing a full solution. Only the part of the signal with no vertical current will be seen on the ground; this agrees with previous work. Using this result, we find that the reflection condition of a transverse hydromagnetic wave incident on the top of the ionosphere indicates a ‘fixed-end’ boundary condition for the field line. To a first approximation, micropulsation polarisations observed on the ground must be rotated through a right angle to obtain those in the magnetosphere. The strength of the electric field predicted in the ionosphere agrees with radio aurorae observations.     

Stochastic Acceleration of Energetic Particles in the Magnetosphere of Saturn

Voyager's plasma probe observations suggest that there are at least three fundamentally different plasma regimes in Saturn: the hot outer magnetosphere, the extended plasma sheet, and the inner plasma torus. At the outer regions of the inner torus some ions have been accelerated to reach energies of the order of 43 keV. We develop a model that calculates the acceleration of charged particles in the Saturn's magnetosphere. We propose that the stochastic electric field associated to the observed magnetic field fluctuations is responsible of such acceleration. A random electric field is derived from the fluctuating magnetic field – via a Monte Carlo simulation – which then is applied to the momentum equation of charged particles seeded in the magnetosphere. Taking different initial conditions, like the source of charged particles and the distribution function of their velocities, we find that particles injected with very low energies ranging from 0.129 eV to 5.659 keV can be strongly accelerated to reach much higher energies ranging from 22.220 eV to 9.711 keV as a result of 125,000 hitting events (the latter are used in the numerical code to produce the particle acceleration over a predetermined distance).     

The example of effective plasma acceleration in a magnetosphere

The problem of effective transform of Poynting flux energy into the kinetic energy of relativistic plasma outflow in a magnetosphere is considered. In this article we present an example of such acceleration. In order to perform it, we use the approach of ideal axisymmetric magnetohydrodynamics (MHD). For highly magnetized plasma outflow we show that a linear growth of Lorentz factor with a cylindrical distance from the rotational axis is a general result for any field configuration in the sub-magnetosonic flow. In the far region the full magnetohydrodynamics problem for one-dimensional flow is considered. It turns out that the effective plasma outflow acceleration is possible in the paraboloidal magnetic field. It is shown that such an acceleration is due to the drift of charged particles in the crossed electric and magnetic field. The clear explanation of the absence of acceleration in the monopole magnetic field if given.      

The motion of charged particles in a strong electromagnetic field and curvature radiation

The features of the relativistic charge particle motion and emission due to the radiative slamping in the strong electromagnetic fields are investigated. It is shown that the radiative force responsible for curvature radiation is associated with the particle drift in an inhomogeneous magnetic field. The adiabatic trajectory is obtained for the relativistic particle, moving in a strong static electron-magnetic field, particle energy being determined by the balance of the work of the electric field and the energy losses through curvature radiation.