An introduction to the theory of internal structure of supermassive compact celestial bodies

Different in principle from the contemporary standard black hole accretion models, a new approach to the understanding of the internal structure of highly compact stationary supermassive celestial bodies has been worked out. The equations of equilibrium configurations of baryonic protomatter (ECBP) have been discussed. In a particular case of ideal degenerated neutron gas in absence of a process of inner distortion of the space and time, it has been shown that the theory suggested by Ter-Kazarian (1989c) leads to the same results as those obtained by Oppenheimer and Volkoff (1939) based on Einstein's theory. The numerical integration of equations of ECBP in the most simple case of equilibrium single-component configurations of degenerated ideal gas of neutrons in a presence of one-dimensional space-like inner distortion of space-time continuum is carried out. It has been shown that the stable stationary supermassive cores are formed in the central parts of the considered configurations. As the models of active galactic nuclei (AGNs) one has considered only the configurations that consisted of these cores surrounded by accretion disks. The fundamental difference from the standard black-hole accretion models is the fact that the central cores are in a stable equilibrium state with certain radial distributions of density and pressure and with a number of integral characteristics. The significant effect of metric singularity cut-off has been established, due to the action of which a singularity of metric ceased to be significant. The numerous integrations have also revealed the other fact of great importance, the presence within the outlined theory of a rigorous restriction on the upper limit of possible values of total masses of considered equilibrium configurations, which is to beM3.5×10⁸ M . In the last section one has proceeded to the direct modelling of concrete AGNs (for 61 sources), the whole point of which comes to the solving of the inverse problem. The results of all calculations that have been carried out in the present work are summarized in Tables I–VII and represented by means of numerous figures. Finally, one should emphasize the important fact of the existence of BL Lac objects OJ 287, 3C 66A, and B2 1308+32, the observed time-scale for flux variations of which are inconsistent with contemporary black hole aceretion models. The case is quite different within the scope of the suggested theory. It seems that a decisive significance for these objects has the action of metric singularity cut-off effect. Due to this their observed sizes are less than the sizes of corresponding spheres of the event horizon. This may serve as a further indication that the suggested theory is preferable to the standard models.     

Stability of equilibrium self-gravitating models and the possibility of black hole formation in the jordan-brans-dicke theory

We show that, starting from the qualitative analysis of the solutions of the vacuum equations, one can conclude that, in contrast to the general theory of relativity, in the Jordan-Brans-Dicke theory gravitational collapse does not lead to the formation of black holes, but that no reliable determination of the existence or nonexistence of black holes in the Jordan-Brans-Dicke theory will be possible until the time-dependent equations have been solved numerically. We propose a method of solving the problem of stability of equilibrium self-gravitating configurations with respect to radial pulsations and state a criterion for stability.Translated fromAstrofizika, Vol. 38, No. 3, 1995.     

Vibrational stability of double-star Roche model

The aim of the present paper will be to extend the methods of our previous investigations (Kopal, 1980, 1987) by employing the Clairaut coordinates (in which the radial component is identified with the total potential) to analyze the nature of small oscillations about the equilibrium form of Roche double-star model (identical, in fact, with zero-velocity surfaces of the restricted problem of three bodies).Linearized equations of this problem have been set up in Clairaut coordinates, and solved in a closed form. This solution turns out to be closely analogous to that obtained already for the rotating single-star Roche model, and discloses that (like in the preceding case) the terms secular in time appear already in the linear approximation. However, whether or not a retention of nonlinear terms in the equations of motion can regain secular stability of the respective configurations remains yet to be clarified by future investigations.     

Modified equations in the theory of induced gravity. Solution to the cosmological constant problem

This research is an extension of the author’s works, in which conformally invariant generalization of string theory was suggested to higher-dimensional objects. Special cases of the proposed theory are Einstein’s theory of gravity and string theory. This work is devoted to the formation of self-consistent equations of the theory of induced gravity in the presence of matter in the form of a perfect fluid that interacts with scalar fields. The study is done to solve these equations for the case of the cosmological model. In this model time-evolving gravitational and cosmological “constants” take place which are determined by the square of scalar fields. The values of which can be matched with the observational data. The equations that describe the theory have solutions that can both match with the solutions of the standard theory of gravity as well as it can differ from it. This is due to the fact that the fundamental “constants” of the theory, such as gravitational and cosmological, can evolve over time and also depend of the coordinates. Thus, in a rather general case the theory describes the two systems (stages): Einstein and “evolving”. This process is similar to the phenomenon of phase transition, where the different phases (Einstein gravity system, but with different constants) transit into each other.     

On the stability of compact supermassive objects

Proceeded from the gravitation equations proposed by one of authors it was argued in a previous paper that there can exist supermassive compact configurations of degenerated Fermi‐gas without events horizon. In the present paper we consider the stability of these objects by a method like the one used in the theory of stellar structure. It is shown that the configurations are stable.     

Central configurations of the five-body problem with equal masses

In this paper we present a complete classification of the isolated central configurations of the five-body problem with equal masses. This is accomplished by using the polyhedral homotopy method to approximate all the isolated solutions of the Albouy-Chenciner equations. The existence of exact solutions, in a neighborhood of the approximated ones, is then verified using the Krawczyk method. Although the Albouy-Chenciner equations for the five-body problem are huge, it is possible to solve them in a reasonable amount of time.     

Tidal evolution of close-in exoplanets in co-orbital configurations

In this paper, we study the behavior of a pair of co-orbital planets, both orbiting a central star on the same plane and undergoing tidal interactions. Our goal is to investigate final orbital configurations of the planets, initially involved in the 1/1 mean-motion resonance (MMR), after long-lasting tidal evolution. The study is done in the form of purely numerical simulations of the exact equations of motions accounting for gravitational and tidal forces. The results obtained show that, at least for equal mass planets, the combined effects of the resonant and tidal interactions provoke the orbital instability of the system, often resulting in collision between the planets. We first discuss the case of two hot-super-Earth planets, whose orbital dynamics can be easily understood in the frame of our semi-analytical model of the 1/1 MMR. Systems consisting of two hot-Saturn planets are also briefly discussed.     

Partial Reduction in the N-Body Planetary Problem using the Angular Momentum Integral

We present a new set of variables for the reduction of the planetary n-body problem, associated to the angular momentum integral, which can be of any use for perturbation theory. The construction of these variables is performed in two steps. A first reduction, called partial is based only on the fixed direction of the angular momentum. The reduction can then be completed using the norm of the angular momentum. In fact, the partial reduction presents many advantages. In particular, we keep some symmetries in the equations of motion (d'Alembert relations). Moreover, in the reduced secular system, we can construct a Birkhoff normal form at any order. Finally, the topology of this problem remains the same as for the non-reduced system, contrarily to Jacobi's reduction where a singularity is present for zero inclinations. For three bodies, these reductions can be done in a very simple way in Poincaré's rectangular variables. In the general n-body case, the reduction can be performed up to a fixed degree in eccentricities and inclinations, using computer algebra expansions. As an example, we provide the truncated expressions for the change of variable in the 4-body case, obtained using the computer algebra system TRIP.     

Non-isothermal magnetohydrostatic equilibrium structure in the solar atmosphere

The non-isothermal magnetohydorstatic equilibrium is studied in this paper, on the basis of two-dimensional solutions of an isothermal case^[9]. We present two models of temperature distribution and derive the two relevant partial differential equations of non-isothermal magnetohydrostatic equilibrium. Using the solution of the isothermal case as an approximation of the 1st degree and choosing appropriate boundary conditions, we obtain several solutions of the equations with an iterative method numerically, and obtain the equilibrium configurations and the distributions of temperature and magnetic energy. From these results, we find that the equilibrium configurations will change obviously as the temperature changes from the isothermal case slightly. Change of the temperature may play an important role in producing relatively violent solar activities through instability.     

First order planetary perturbations with elliptic functions

The differential equations of planetary theory are solved analytically to first order for the two-dimensional case, using only Jacobian elliptic functions and the elliptic integrals of the first and second kind. This choice of functions leads to several new features potentially of importance for planetary theory. The first of these is that the solutions do not require the expansion of the reciprocal of the distance between two planets, even for those variables which depend on two angular arguments. A second result is that the solution is free from small divisors with the exception of two special resonances. In fact, not only are the solutions for resonant orbits free from small divisors, the perturbations for all variables are expressible in closed form. A subset of the resonant orbits maintains this form and in addition has the remarkable feature that the first order perturbations are purely periodic; they contain no secular terms. A solution for the 13 resonance case is given as an example.     

Complete spin and orbital evolution of close-in bodies using a Maxwell viscoelastic rheology

In this paper, we present a formalism designed to model tidal interaction with a viscoelastic body made of Maxwell material. Our approach remains regular for any spin rate and orientation, and for any orbital configuration including high eccentricities and close encounters. The method is to integrate simultaneously the rotation and the position of the planet as well as its deformation. We provide the equations of motion both in the body frame and in the inertial frame. With this study, we generalize preexisting models to the spatial case and to arbitrary multipole orders using a formalism taken from quantum theory. We also provide the vectorial expression of the secular tidal torque expanded in Fourier series. Applying this model to close-in exoplanets, we observe that if the relaxation time is longer than the revolution period, the phase space of the system is characterized by the presence of several spin-orbit resonances, even in the circular case. As the system evolves, the planet spin can visit different spin-orbit configurations. The obliquity is decreasing along most of these resonances, but we observe a case where the planet tilt is instead growing. These conclusions derived from the secular torque are successfully tested with numerical integrations of the instantaneous equations of motion on HD 80606 b. Our formalism is also well adapted to close-in super-Earths in multiplanet systems which are known to have non-zero mutual inclinations.     

Neutron stars in the bimetric scalar—tensor theory of gravitation. II. solutions with a variable scalar field

Solutions of the equations of the bimetric scalar—tensor theory of gravitation with a variable scalar field are found for configurations of superdense matter with different versions of the equation of state. The possible existence of static superdense and supermassive configurations is established for all of the versions of the equation of state of superdense matter used. Translated from Astrofizika, Vol. 41, No. 1, pp. 131–135, January-March, 1998.     

Certain problems of solar and heliospheric physics in the light of novel satellite data

Recent satellite data have given a better insight into the possible nature of extremely strong disturbances on the Sun and in the heliosphere by relating them to processes in the solar interior. The energy, momentum, and mass transfer on various spatiotemporal scales are organized in the Sun into a hierarchy of coupled nonlinear processes. Confirmation has been given to the fact that coronal mass ejections and solar flares are not linked causally but merely reflect the existence of two channels of free-energy dissipation in the solar atmosphere in the form of plasma motion and plasma emission; their relative role can be described by a corresponding nondimensional parameter. Information on the global asymmetry of the solar emission and active processes has been gained. A great diversity in the geometry of eruptive events (not necessarily associated with magnetic reconnection) has been revealed. In our opinion, the basic unresolved problems in the investigation of solar activity dictate the necessity of carrying out more accurate, absolutely calibrated measurements of the whitelight solar emission at appropriately high spatiotemporal resolutions. The development of direct and indirect techniques of measuring the electric fields and currents with the aim of reconstructing the solar and heliospheric current system remains a challenging task.     

Homologous prominence non-radial eruptions: A case study

The present study provides important details on homologous eruptions of a solar prominence that occurred in active region NOAA 10904 on 2006 August 22. We report on the pre-eruptive phase of the homologous feature as well as the kinematics and the morphology of a forth from a series of prominence eruptions that is critical in defining the nature of the previous consecutive eruptions. The evolution of the overlying coronal field during homologous eruptions is discussed and a new observational criterion for homologous eruptions is provided. We find a distinctive sequence of three activation periods each of them containing pre-eruptive precursors such as a brightening and enlarging of the prominence body followed by small surge-like ejections from its southern end observed in the radio 17 GHz. We analyse a fourth eruption that clearly indicates a full reformation of the prominence after the third eruption. The fourth eruption although occurring 11 h later has an identical morphology, the same angle of propagation with respect to the radial direction, as well as similar kinematic evolution as the previous three eruptions. We find an important feature of the homologous eruptive prominence sequence that is the maximum height increase of each consecutive eruption. The present analysis establishes that all four eruptions observed in Hα are of confined type with the third eruption undergoing a thermal disappearance during its eruptive phase. We suggest that the observation of the same direction of the magnetic flux rope (MFR) ejections can be consider as an additional observational criterion for MFR homology. This observational indication for homologous eruptions is important, especially in the case of events of typical or poorly distinguishable morphology of eruptive solar phenomena.     

Expanding,homogeneous, ellipsoidal density perturbations. III. A simple theory for the acquisition of angular momentum by tidal torques with the effects of the spin growth on the expansion included

The effects of the acquisition of angular momentum on the expansion of homogeneous, ellipsoidal density perturbations is investigated by generalizing the theory of previous papers of this series, where spin grows to the first order in overdensity. A small difference is found to be between the two cases, except for the fact that the body under consideration becomes unbound earlier in the current approach. A comparison is also made with the results of a different theory, where spin grows to the second order in overdensity. Quasi-oblate triaxial configurations turn out to gain less angular momentum in respect to both oblate and more elongated configurations with same minor to major axis ratio. In all cases of physical interest, the spin growth from the beginning to the maximum ellipsoidal volume exceeds the spin growth from the maximum ellipsoidal volume to the turnaround of the major axis, by a factor of at least 3. It is also inferred that the spin growth from the turnaround of the major axis on, probably does not exceed about one third the angular momentum previously gained.     

Central configurations of four bodies with an axis of symmetry

A complete solution is given for a symmetric case of the problem of the planar central configurations of four bodies, when two bodies are on an axis of symmetry, and the other two bodies have equal masses and are situated symmetrically with respect to the axis of symmetry. The positions of the bodies on the axis of symmetry are described by angle coordinates with respect to the outside bodies. The solution is such, that giving the angle coordinates, the masses for which the given configuration is a central configuration, can be computed from simple analytical expressions of the angles. The central configurations can be described as one-parameter families, and these are discussed in detail in one convex and two concave cases. The derived formulae represent exact analytical solutions of the four-body problem.     

Maximum mass of magnetic white dwarfs

We revisit the problem of the maximum masses of magnetized white dwarfs(WDs).The impact of a strong magnetic field on the structure equations is addressed.The pressures become anisotropic due to the presence of the magnetic field and split into parallel and perpendicular components.We first construct stable solutions of the Tolman-Oppenheimer-Volkoff equations for parallel pressures and find that physical solutions vanish for the perpendicular pressure when B(?) 10~(13) G.This fact establishes an upper bound for a magnetic field and the stability of the configurations in the(quasi) spherical approximation.Our findings also indicate that it is not possible to obtain stable magnetized WDs with super-Chandrasekhar masses because the values of the magnetic field needed for them are higher than this bound.To proceed into the anisotropic regime,we can apply results for structure equations appropriate for a cylindrical metric with anisotropic pressures that were derived in our previous work.From the solutions of the structure equations in cylindrical symmetry we have confirmed the same bound for B ~ 10~(13) G,since beyond this value no physical solutions are possible.Our tentative conclusion is that massive WDs with masses well beyond the Chandrasekhar limit do not constitute stable solutions and should not exist.     

Evolution of the photospheric magnetic field and coronal null points before solar flares

Based on a topological model for the magnetic field of a solar active region (AR), we suggest a criterion for the existence of magnetic null points on the separators in the corona. With the problem of predicting solar flares in mind, we have revealed a model parameter whose decrease means that the AR evolves toward a major eruptive flare. We analyze the magnetic field evolution for AR 9077 within two days before the Bastille Day flare on July 14, 2000. The coronal conditions are shown to have become more favorable for magnetic reconnection, which led to a 3B/X5.7 eruptive flare.     

Particle content and degrees of freedom of a gravitational field in 4th order theories of gravity

In gravitational theories of 4-th order, the influence of certain properties of the field equations (tracelessness, conformal invariance, scale in variance respectively their breaking) for the “particle content” (number of degrees of freedom, mass, spin) is investigated. Using the plane-wave ansatz valid in linearized theory it is possible to determine the mass content of the theory, but one cannot get assertions about the number of degrees of freedom and the spin states corresponding to the field quanta. In the linearized theory, this can be done with a spin projection formalism. Using the Cauchy initial value problem and a counting method first developed by EINSTEIN one can get, however, a useful definition of the concept of the degrees of freedom for the full nonlinear theory. This is due to the fact that this method allows to incorporate the concrete structure of the field equations (and thus their nonlinearities). Analysing different general-relativistic field theories via these approaches the influence of the various structures of nonlinearities is discussed. It is, in particular, shown that those results obtained by the spin projection formalism can be reproduced by “nonlinear methods”.     

Formation of emission spectra in movin media

Conclusions It can be seen from the above review that the theory of radiative transfer in moving media in its classical field of applications is a well-developed branch of theoretical astrophysics. Studies i n recent years have clarified important questions such as the asymptotic behavior of the kernel functions and the characteristic lengths of the theory. It has been established that there are two types of radiative coupling, and the influence of nonlocal radiative coupling on the formation of spectral lines and radiation pressure has been investigated. The theory now has at its disposal a large selection of asymptotic, approximate, and numerical methods for solving different applied problems.Despite the competition from numerical methods, the overwhelming majority of calculations of emission spectra in the region of supersonic motions has been made on the basis of the escape-probability method and its generalization to the case of nonlocal radiative coupling. This is explained not only by the simplicity and economy of the method but even more by the fact that the greater accuracy of the calculations that can be achieved by using numerical methods is frequently spurious, since it certainly exceeds the accuracy of the basic assumptions. The real way to increase the reliability in the diagnosis of a radiating gas is, first, to solve simultaneously many-level problems for the group of elements whose lines are observed in the spectrum of the object. Second, where possible, one must solve simultaneously the stationary equations and the heat balance equations.Crimean Astrophysical Observatory. Translated from Astrofizika, Vol. 20, No. 2, pp. 365–417, March–April, 1984.