Modification of the jeans instability criterion for fractal-structure astrophysical objects in the framework of nonextensive statistics

Unlike classical studies in which the gravitational instability criterion for astrophysical disks is derived in the framework of traditional kinetics or hydrodynamics, we propose to consider the totality of fluffy dust clusters of various astrophysical objects, in particular, protoplanetary subdisks, as a special type of continuous medium, i.e., fractal medium for which there are points and areas not filled with its components. Within the deformed Tsallis statistics formalism, which is intended to describe the behavior of anomalous systems with strong gravitational interaction and fractal nature of phase space, we derive, on the basis of the modified kinetic equation (with the collision integral in the Bhatnagar-Gross-Krook form), the generalized hydrodynamic Euler equations for a medium with the fractal mass dimension. Considering the linearization of the q-hydrodynamics equations, we investigate the instability of an infinitely homogeneous medium to obtain a simplified version of the modified gravitational instability criterion for an astrophysical disk with fractal structure.     

Effect of Coriolis force on the equilibrium structures of rotating stars and stars in binary systems

Kopal (Adv. Astron. Astrophys. 9:1–65, 1972) introduced the concept of Roche equipotentials to incorporate the effects of rotation and tidal distortions on the equilibrium structure and periods of small oscillations of rotating stars and stars in binary systems. However his expression for the Roche equipotential accounts for only the effects of centrifugal and gravitational forces and does not take into account the effect of Coriolis force. In this paper we have suitably modified Kopal’s expression for Roche equipotentials to incorporate into it the effect of Coriolis force as well. The modified expression for the Roche equipotential has then been used to compute the equilibrium structures and shapes of polytropic models of rotating stars and stars in binary systems.     

Dynamical evolution of two associated galactic bars

We study the dynamical interactions of mass systems in equilibrium under their own gravity that mutually exert and ex‐perience gravitational forces. The method we employ is to model the dynamical evolution of two isolated bars, hosted within the same galactic system, under their mutual gravitational interaction. In this study, we present an analytical treatment of the secular evolution of two bars that oscillate with respect to one another. Two cases of interaction, with and without geometrical deformation, are discussed. In the latter case, the bars are described as modified Jacobi ellipsoids. These triaxial systems are formed by a rotating fluid mass in gravitational equilibrium with its own rotational velocity and the gravitational field of the other bar. The governing equation for the variation of their relative angular separation is then numerically integrated, which also provides the time evolution of the geometrical parameters of the bodies. The case of rigid, non‐deformable, bars produces in some cases an oscillatory motion in the bodies similar to that of a harmonic oscillator. For the other case, a deformable rotating body that can be represented by a modified Jacobi ellipsoid under the influence of an exterior massive body will change its rotational velocity to escape from the attracting body, just as if the gravitational torque exerted by the exterior body were of opposite sign. Instead, the exchange of angular momentum will cause the Jacobian body to modify its geometry by enlarging its long axis, located in the plane of rotation, thus decreasing its axial ratios. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibrational stability of the rotating Roche model

The aim of the first part of this investigation will be to establish the explicit form of the linearized systems of differential equations governing arbitrary oscillations (of amplitudes small enough for their squares and higher powers to be negligible) of the rotating Roche model in Clairaut's coordinates (in which their radial component is identified with the total potential). By solving these equations in a closed form we shall prove that this model is incapable of performing such oscillations (for any type of symmetry) about equipotential surfaces representing the figures of equilibrium, as soon as the centrifugal force will cause their equilibrium form to depart from a sphere.In the second part of this paper we shall set up the closed forms of the Laplace equation in Clairaut (non-orthogonal) as well as Roche (orthogonal) coordinates associated with the rotating Roche model; and by a construction of their solution establish successively the explicit forms of the respective harmonic functions associated with such figures (as a generalization of Legendre functions which are similarly associated with a sphere.     

The effect of solar pressure on the motion and stability of interconnected satellites in orbit

The paper deals with the effect of solar pressure on the motion and stability of two satellites connected by an inextensible string in a central gravitational field of force. A system of nonlinear, nonhomogeneous, and non-autonomous equations under the rotating frame of reference in Nechvíle's coordinate system have been obtained. The general solution of the above system of equations is beyond our reach. The particular solutions have been obtained.The particular solution in which the system lies, wholly along the radius vector joining the attracting centre and the centre of mass of the system under the central attracting force along was found to be stable (Singh, 1973). Naturally we got interested in examining the effect of solar radiation pressure on the stability of this particular solution.     

Orbital Decay of a Binary System Consisting of Spin-Down Neutron Stars

{W}e consider the gravitational radiation from two time variable mass stars, orbiting around each other under the influence of gravity. The total rates of the variation of the energy, angular momentum, semimajor axis, eccentricity and orbital period are obtained. The results could be important for the understanding of general relativistic effects in the case of the variation of the gravitational mass due to spinning down of the compact stars, which sensitively depends on the equations of state. The cases of the binary systems PSR 1913+16 and PSR 1534+12 are analyzed in detail, and, for different equations of state of nuclear matter, the corrections to the orbital decay due to gravitational radiation and to the spinning down of the pulsars are calculated. The results show that a future significant improvement in the observational techniques could lead to the observation of the specific general relativistic effect of mass variation of pulsars due to spinning down, via the study of orbital decay, even in slowly rotating binary systems.     

Oscillations of rapidly rotating stratified neutron stars

We use time evolutions of the linear perturbation equations to study the oscillations of rapidly rotating neutrons stars. Our models account for the buoyancy due to composition gradients and we study, for the first time, the nature of the resultant g modes in a fast spinning star. We provide detailed comparisons of non-stratified and stratified models. This leads to an improved understanding of the relationship between the inertial modes of a non-stratified star and the g modes of a stratified system. In particular, we demonstrate that each g mode becomes rotation dominated, i.e. approaches a particular inertial mode, as the rotation rate of the star is increased. We also discuss issues relating to the gravitational wave driven instability of the various classes of oscillation modes.     

Oscillations of rapidly rotating superfluid stars

Using time evolutions of the relevant linearized equations, we study non-axisymmetric oscillations of rapidly rotating and superfluid neutron stars. We consider perturbations of Newtonian axisymmetric background configurations and account for the presence of superfluid components via the standard two-fluid model. Within the Cowling approximation, we are able to carry out evolutions for uniformly rotating stars up to the mass-shedding limit. This leads to the first detailed analysis of superfluid neutron star oscillations in the fast rotation regime, where the star is significantly deformed by the centrifugal force. For simplicity, we focus on background models where the two fluids (superfluid neutrons and protons) corotate, are in β-equilibrium and co-exist throughout the volume of the star. We construct sequences of rotating stars for two analytical model equations of state. These models represent relatively simple generalizations of single fluid, polytropic stars. We study the effects of entrainment, rotation and symmetry energy on non-radial oscillations of these models. Our results show that entrainment and symmetry energy can have a significant effect on the rotational splitting of non-axisymmetric modes. In particular, the symmetry energy modifies the inertial mode frequencies considerably in the regime of fast rotation.     

Gravitational radiation from differentially rotating and oscillating white dwarfs

We examine the possible emission of gravitational waves from white dwarfs undergoing self-similar oscillations driven by the energy released during relaxation of their differential rotation. Two distributions of the initial angular momentum are considered. It is assumed that 1% of the energy dissipated by a rotating white dwarf is converted into the energy of self-similar oscillations and, therefore, into gravitational radiation. The relative amplitude of the gravitational radiation from an isolated white dwarf at a distance of 50 pc is found to be less than 10⁻²⁷. The emission from the galactic population of white dwarfs may create a background which overlaps the random cosmological background of gravitational radiation for the improved decihertz detectors currently being proposed. __________ Translated from Astrofizika, Vol. 49, No. 2, pp. 231–242 (May 2006).     

Effect of Coriolis force on the shapes of rotating stars and stars in binary systems

In this paper an attempt has been made to determine the effect of Coriolis force on the shapes of Roche equipotential surfaces of rotating stars and stars in binary systems. Equations of Roche equipotential surfaces have been obtained for rotating and binary stars which take into account the effects of Coriolis force besides the centrifugal and gravitational forces. Shapes of Roche equipotentials and values of Roche limits are obtained for different values of angular velocity of rotation for rotating stars and for different values of mass ratios for the binary stars. The obtained results have been compared with the corresponding results in which the effect of Coriolis force has not been considered.     

On the gravitational instability of a medium in non-uniform rotation and magnetic field

The effect of a non-uniform magnetic field on the gravitational instability for a non-uniformly rotating, infinitely extending axisymmetric cylinder in a homogeneous medium has been studied. The Bel and Schatzman criterion of gravitational instability for a non-uniformly rotating medium is modified under the effect of a non-uniform/uniform magnetic field acting along the tangential and axial directions. As a consequence the stabilizing and destabilizing effect of the non-uniform magnetic field is obtained, a new criterion for the magneto-gravitational instability is deduced in terms of Alfven’s wave velocity; and it is also found that the Jeans criterion determines the gravitational instability in the absence of rotation and when the non-uniform/uniform magnetic field acts along the axis of the cylinder.     

Pulsational properties of V2109 Cyg

In this study V2109 Cyg (a pulsating δ Scuti star) has been modelled. In treating the oscillation equations perturbation in gravitational potential energy has been taken into account. Both radial and nonradial oscillations are treated with adiabatic approximation. The so called radial fundamental frequency (5.3745 c/d) and the nonradial frequency (5.8332 c/d) were obtained within a satisfactory precision. It was found that the Cowling approximation introduced more error as one went from low overtones to high overtones in radial oscillations. A similar trend was observed in nonradial case with low values of l. By keeping the effective temperatures almost the same as with V2109 Cyg two more models with different masses have also been calculated to see the effect of inclusion of perturbation in gravitational potential energy on oscillation frequencies in different masses. Conclusion arrived is that one must be careful to employ the Cowling approximation especially for high nonradial oscillation frequencies. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Forced oscillations in a hydrodynamical accretion disk and QPOs

This is the second of a series of papers aimed to look for an explanation on the generation of high frequency quasi-periodic oscillations (QPOs) in accretion disks around neutron star, black hole, and white dwarf binaries. The model is inspired by the general idea of a resonance mechanism in the accretion disk oscillations as was already pointed out by Abramowicz and Klu’zniak (2001). In a first paper (P'etri, 2005a, paper I), we showed that a rotating misaligned magnetic field of a neutron star gives rise to some resonances close to the inner edge of the accretion disk. In this second paper, we suggest that this process does also exist for an asymmetry in the gravitational potential of the compact object. We prove that the same physics applies, at least in the linear stage of the response to the disturbance in the system. This kind of asymmetry is well suited for neutron stars or white dwarfs possessing an inhomogeneous interior allowing for a deviation from a perfectly spherically symmetric gravitational field. After a discussion on the magnitude of this deformation applied to neutron stars, we show by a linear analysis that the disk initially in a cylindrically symmetric stationary state is subject to {three kinds of resonances: a corotation resonance, a Lindblad resonance due to a driven force and a parametric resonance}. In a second part, we focus on the linear response of a thin accretion disk in the 2D limit. {Waves are launched at the aforementioned resonance positions and propagate in some permitted regions inside the disk, according to the dispersion relation obtained by a WKB analysis}. In a last part, these results are confirmed and extended via non linear hydrodynamical numerical simulations performed with a pseudo-spectral code solving Euler's equations in a 2D cylindrical coordinate frame. {We found that for a weak potential perturbation, the Lindblad resonance is the only effective mechanism producing a significant density fluctuation}. In a last step, we replaced the Newtonian potential by the so called logarithmically modified pseudo-Newtonian potential in order to take into account some general-relativistic effects like the innermost stable circular orbit (ISCO). The latter potential is better suited to describe the close vicinity of a neutron star or a black hole. However, from a qualitative point of view, the resonance conditions remain the same. The highest kHz QPOs are then interpreted as the orbital frequency of the disk at locations where the response to the resonances are maximal. It is also found that strong gravity is not required to excite the resonances.     

Torsional oscillations of magnetized relativistic stars

Strong magnetic fields in relativistic stars can be a cause of crust fracturing, resulting in the excitation of global torsional oscillations. Such oscillations could become observable in gravitational waves or in high-energy radiation, thus becoming a tool for probing the equation of state of relativistic stars. As the eigenfrequency of torsional oscillation modes is affected by the presence of a strong magnetic field, we study torsional modes in magnetized relativistic stars. We derive the linearized perturbation equations that govern torsional oscillations coupled to the oscillations of a magnetic field, when variations in the metric are neglected (Cowling approximation). The oscillations are described by a single two-dimensional wave equation, which can be solved as a boundary-value problem to obtain eigenfrequencies. We find that, in the non-magnetized case, typical oscillation periods of the fundamental     torsional modes can be nearly a factor of 2 larger for relativistic stars than previously computed in the Newtonian limit. For magnetized stars, we show that the influence of the magnetic field is highly dependent on the assumed magnetic field configuration, and simple estimates obtained previously in the literature cannot be used for identifying normal modes observationally.     

Gravitational Field of Fractal Distribution of Particles

In this paper we consider the gravitational field of fractal distribution of particles. To describe fractal distribution, we use the fractional integrals. The fractional integrals are considered as approximations of integrals on fractals. Using the fractional generalization of the Gauss’s law, we consider the simple examples of the fields of homogeneous fractal distribution. The examples of gravitational moments for fractal distribution are considered.     

Self-gravitational instability of rotating viscous Hall plasma with arbitrary radiative heat-loss functions and electron inertia

The effects of arbitrary radiative heat-loss functions and Hall current on the self-gravitational instability of a homogeneous, viscous, rotating plasma has been investigated incorporating the effects of finite electrical resistivity, finite electron inertia and thermal conductivity. A general dispersion relation is obtained using the normal mode analysis with the help of relevant linearized perturbation equations of the problem, and a modified Jeans criterion of instability is obtained. The conditions of modified Jeans instabilities and stabilities are discussed in the different cases of our interest. We find that the presence of arbitrary radiative heat-loss functions and thermal conductivity modifies the fundamental Jeans criterion of gravitational instability into a radiative instability criterion. The Hall parameter affects only the longitudinal mode of propagation and it has no effect on the transverse mode of propagation. For longitudinal propagation, it is found that the condition of radiative instability is independent of the magnetic field, Hall parameter, finite electron inertia, finite electrical resistivity, viscosity and rotation; but for the transverse mode of propagation it depends on the finite electrical resistivity, the strength of the magnetic field, and it is independent of rotation, electron inertia and viscosity. From the curves we find that the presence of thermal conductivity, finite electrical resistivity and density-dependent heat-loss function has a destabilizing influence, while viscosity and magnetic field have a stabilizing effect on the growth rate of an instability. The effect of arbitrary heat-loss functions is also studied on the growth rate of a radiative instability.     

On the Evolution of Balloon Satellite Motions in a Plane Restricted Three-Body Problem with Light Pressure

We investigate the evolution of high Earth satellite orbits under gravitational perturbations from the Sun and light pressure forces, without the Earth shadow effect. We present the disturbing function of the problem provided that the satellite is a sphere. The mean value of the disturbing function in the absence of resonances between the mean unperturbed motion of the satellite and the mean motion of the Sun has also been obtained. The semimajor axis of the satellite orbit and the mean value of the disturbing function are shown to be integrals of the averaged osculating equations. TheHill version of the problem, whereby the distance to the satellite is much smaller than the Earth–Sun distance, has been studied in detail: we have constructed the phase portraits of the oscillations at various parameters and described three types of quasiperiodic satellite trajectories—librational and rotational trajectories as well as Earth collision trajectories. Numerical simulations of trajectories have allowed the additional effects caused by light pressure to be described: the displacement of the bounded trajectory of the satellite as a whole relative to the trajectory of the classical three-body problem into a region more distant from the Sun.     

Linear kinetic theory of instabilities of a gravitatingstellar disk

Linear kinetic theory is developed to describe collective oscillations (and their instabilities) propagating in a rapidly rotating disk of stars, representing a highly flattened galaxy. The analysis is carried out for the special case of a self-gravitating, infinitesimally thin, and spatially inhomogeneous system, taking into account the effects both of thermal movements of stars and of gravitational encounters between stars and giant molecular clouds of an interstellar medium. The star–cloud encounters are described with the use of the Landau collision integral. The dynamics of gravity perturbations with rare interparticle encounters is considered. Such a disk is treated by employing the well elaborated mathematical formalisms from plasma perturbation theory using normal-mode analysis. In particular, the method of solving the Boltzmann equation is applied by integration along paths, neglecting the influence of star–cloud encounters on the distribution of stars in the zeroth-order approximation. We are especially interested in important kinetic effects due to wave–star resonances, which we have little knowledge about. The kinetic effects are introduced via a minor drift motion of stars which is computed from the equations of stellar motion in an unperturbed central force field of a galaxy. The dispersion laws for two main branches of disk's oscillations, that is the classical Jeans branch and an additional gradient branch, are deduced. The resonant Landau-type instabilities of hydrodynamically stable Jeans and gradient gravity perturbations is considered to be a long-term generating mechanism for propagating density waves, thereby leading to spiral-like and/or ring-like patterns in the flat galaxies. This revised version was published online in July 2006 with corrections to the Cover Date.     

Slowly rotating superdense configurations in Rosen's bimetric theory of gravitation

In the limits of the Bimetric Theory of Gravitation the equations of axial-symmetric gravitational field, describing the field of rotating configurations, are obtained and examined. The analytical solutions of these equations out of configurations are found. The equations are solved numerically in the first approximation in the angular velocity for the configurations consisting of the real baryon gas (cf. Sahakian, 1972). The moments of inertia of these configurations in BTG are calculated. It is shown that the equations and the appropriate solution, obtained by Falik and Opher (1979), are incorrect.