The relativistic equations of motion for a satellite in orbit about a finite-size,rotating Earth

The relativistic equations of motion are derived for N self-gravitating, rotating finite bodies. These equations are then applied to the near-Earth satellite orbit determination problem. The apparent change of the shape of the Earth from the Earth centered frame to the Solar System barycentric frame changes the value of the Newtonian potential term in the metric. This in turn leads to a simplification of the equations of motion in the barycentric frame.     

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.     

Lines formed in rotating and expanding atmospheres

Spectral lines formed in a rotating and expanding atmosphere have been computed in the frame of the observer at infinity. Two kinds of velocity laws are employed: (i) a uniform radial velocity of the gas and (ii) velocity increasing with radius (i.e. velocity gradients). The atmosphere has been assumed to be rotating with constant velocity. We have considered maximum radial and rotational velocities to be 10 and 3 mean thermal units respectively in an atmosphere whose geometrical thickness is 10 times the stellar radius. The total radial optical depth at line centre is taken to be about 100. In all cases, Doppler profile and a source function which is varying as 1/r ² have been used. Generally, the lines are broadened when rotation is introduced. However, when radial motion is also present, broadening becomes asymmetric and the red emission and blue absorption are enhanced.     

Effective potential energy for relativistic particles in the field of inclined rotating magnetized sphere

The dynamics of a charged relativistic particle in electromagnetic field of a rotating magnetized celestial body with the magnetic axis inclined to the axis of rotation is studied. The covariant Lagrangian function in the rotating reference frame is found. Effective potential energy is defined on the base of the first integral of motion. The structure of the equipotential surfaces for a relativistic charged particle is studied and depicted for different values of the dipole moment. It is shown that there are trapping regions for the particles of definite energies.     

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.     

Differential rotation in F stars

Differential rotation can be detected in single line profiles of stars rotating more rapidly than about v sin i = 10km s-1 with the Fourier transform technique. This allows to search for differential rotation in large samples to look for correlations between differential rotation and other stellar parameters. I analyze the fraction of differentially rotating stars as a function of color, rotation, and activity in a large sample of F-type stars. Color and rotation exhibit a correlation with differential rotation in the sense that more stars are rotating differentially in the cooler, less rapidly rotating stars. Effects of rotation and color, however, cannot be disentangled in the underlying sample. No trend with activity is found. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hydrodynamic equations of differentially rotating stellar systems and the velocity ellipsoid

Anisotropic hydrodynamic equations for differentially rotating collisionless stellar systems are derived. These equations can describe the evolution of the systems in a time span longer than their rotation periods.As a by-product of derivation of hydrodynamic equations, the well-known relation that the ratio of the principal axes of the velocity ellipse in a differentially rotating stellar disk is [B/(B-A)]^1/2 is re-found if the system is in a purely circular rotation, whereA andB are the Oort's constants. In addition, we find a systematic mean motion superposed on a purely circular differential rotation makes the directions of axes of the velocity ellipse deviate from the radial and the transverse direction. The observed deviation of directions of axes in our neighbourhood in the Galaxy can be explained if in the mean motion superposed on a purely circular differential rotatin the gas of stars near us is compressed in the radial direction or rarefied in the transverse directions, with irregularities of the order of 5 km/sec in amplitude of velocity and 1 kpc in size. These magnitudes of irregularities agree with those actually observed or with those anticipated from other theoretical considerations.     

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.     

Production of pulses and interpulses in pulsars

Pulsars accelerate the charged particles moving along their magnetic field lines due to their rapidly spinning motion. Particles gain maximum energy from pulsars within the light cylinder when they are moving along the field lines perpendicular to the rotation velocity. In pulsars with non-aligned rotation and magnetic axes, the production of two intense and sharp pulses (main pulse and interpulse) separated by 180° longitude occur at the two regions near the light cylinder where the rotation velocity is perpendicular to the magnetic field. Since the radiating particles move radially along the relativistically compressed magnetic field lines, the observer in the stationary frame receives beamed and transversely compressed radiation pulse. Near the light cylinder position angle varies smoothly during pulsar rotation in a way as Radhakrishnan and Cook (1969) expect its variation near the magnetic pole, as the field lines experience relativistic compression in the direction of rotation. The motion of two charge species along the field lines produce orthogonal modes at each pulse longitude.     

Equilibrium structure of stars obeying a generalized differential rotation law

In the present paper we have considered the problem of determining the equilibrium structure of differentially rotating stars in which the angular velocity of rotation varies both along the axis of rotation and in directions perpendicular to it. For this purpose, a generalized law of differential rotation of the type ² =b ₀+b ₁ s ²+b ₂ s ⁴+b ₃ z ²+b ₄ z ⁴+b ₅ z ² s ² (here is a nondimensional measure of the angular velocity of a fluid element distants from the axis of rotation andz from the plane through the centre of the star perpendicular to the axis of rotation, andb's are suitably chosen parameters) has been used. Whereas Kippenhahn and Thomas averaging approach has been used to incorporate the rotational effects in the stellar structure equations, Kopal's results on Roche equipotentials have been used to obtain the explicit form of the stellar structure equations, which incorporate the rotational effects up to second order of smallness in the distortion parameters. The method has been used to compute the equilibrium structure of certain differentially rotating polytropes. Certain differentially rotating polytropes. Certain differentially rotating models of the Sun have also been computed by using this approach.     

Adiabatic pulsations and convective instability of uniformly rotating gaseous masses

Third-order virial equations are used to investigate the oscillations and the stability of the sequence of uniformly rotating compressible Maclaurin spheroids, referred to in an inertial frame. It is seen that in the case of the oscillations belonging to the third harmonics, the frequency spectrum of the Maclaurin sequence referred to in an inertial frame is distinct from the spectrum of the Maclaurin sequence considered stationary in a rotating frame of reference. Considering the Maclaurin sequence in an inertial frame, the neutral point and the point of onset of dynamical instability (corresponding to the third harmonic deformations) are isolated. They occur for the values of the eccentricitye=0.73113 and 0.96696, respectively. The neutral point is the analogue of the first point of bifurcation along the Dedekind sequence of ellipsoids and is distinct from the neutral point (e=0.89926) along the Maclaurin sequence considered stationary in a rotating frame; this latter point is the analogue of the first point of bifurcation along the Jacobian sequence. Both the Maclaurin sequences in an inertial frame and in a rotating frame become, however, dynamically unstable for the same eccentricitye=0.96696.     

Stellar models under homoaxial rotation: Eulerian equations and simulation of non-uniformly rotating stars

The Eulerian equations are set up for a model subject to homoaxial rotation and suitable for simulation of a non-uniformly rotating star. These equations are formulated in a non-inertial frame of reference, rotating uniformly (i.e., rigidly) with respect to the inertial common frame.     

The frequencies of high frequency quasi-periodic oscillations in two different frames in black hole LMXBs

We discuss the modes of the Alfve'n waves in the accretion disk with a toroidal magnetic field in black hole low mass X-ray binaries in a rotating frame.By solving the perturbed general relativistic magnetohydrodynamic equations in the rotating frame,we find two stable modes of the Alfve'n wave which are the same as those in the fiducial observer frame.This gives a feasible way to transform between the two different frames,which validates the possible Alfve'n wave modes in the accreting celestial bodies wit...     

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.     

Newtonian analogue of force and motion of a free particle in the gravitational field of theC-metric

In the first part of the paper the Newtonian analogue of force for theC-metric has been investigated. To the first-order of approximation in the absence of acceleration of the particle generating theC-metric, one component of the force vector corresponds to the Newtonian analogue of force. In general there are relativistic correction terms due to acceleration term in theC-metric. In the second part of the paper the motion of a freely falling body has been investigated. It is found that plane orbits are not possible. Also the radial fall is not possible and in the equation of the orbit there are terms having no classical analogue. They can be interpreted as the effect of the dragging of the inertial frame produced by the rectilinear acceleration.     

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.     

Instabilities in rotating relativistic stars driven by viscosity

We investigate the instability driven by viscosity in rotating relativistic stars by means of an iterative approach. We focus on polytropic rotating equilibrium stars and impose an m=2 perturbation in the lapse. We vary both the stiffness of the equation of state and the compactness of the star to study these factors on the critical value T/W for the instability. For a rigidly rotating star, the criterion T/W, where T is the rotational kinetic energy and W the gravitational binding energy, mainly depends on the compactness of the star and takes values around 0.13–0.16, which slightly differ from that of Newtonian incompressible stars (∼0.14). For differentially rotating stars, the critical value of T/W is found to span the range 0.17–0.25. The value is significantly larger than in the rigidly rotating case with the same compactness of the star. Finally we discuss the possibility of detecting gravitational waves from viscosity-driven instabilities using ground-based interferometers.      

Equations of general relativistic radiation hydrodynamics in Kerr space–time

Equations of fully general relativistic radiation hydrodynamics in Kerr space–time are derived. While the interactions between matter and radiation are introduced in the comoving frame, the derivatives used when describing the global evolutions of both the matter and the radiation are given in the Boyer–Lindquist frame (BLF) which is a frame fixed to the coordinate describing the central black hole. Around a rotating black hole, both the matter and the radiation are influenced by the frame-dragging effects due to the black hole's rotation. As a fixed frame, we use the locally non-rotating reference frame (LNRF) which is one of the orthonormal frame. While the special relativistic effects such as beaming effects are introduced by the Lorentz transformation connecting the comoving frame and the LNRF, the general relativistic effects such as frame dragging and gravitational redshift are introduced by the tetrads connecting the LNRF and the BLF.     

Equilibrium structure of differentially rotating polytropic models of the stars

In this paper we propose a method for computing the equilibrium structure of differentially rotating polytropic models of the stars. A general law of differential rotation of the type ²=b ₀+b ₁ s ²+b ₂ s ⁴, which can account for a reasonably large variety of possible differential rotations in the stars has been used. The distortional effects have been incorporated in the structure equations up to second order of smallness in distortion parametersb ₀,b ₁, andb ₂ using Kippenhahn and Thomas' averaging approach in conjunction with Kopal's results on Roche equipotentials in manner similar to the one earlier used by Mohan and Saxena for computing the equilibrium structure of polytropes having solid body rotation. Numerical results have been obtained for various types of differentially rotating polytropic models of stars of polytropic indices 1.5, 3, and 4. Certain differentially rotating models of the Sun which are possible with such a type of law of differential rotation, have also been computed.