Virial oscillations of celestial bodies-I. The effect of electrostatic interactions

The Hamiltonian form of Jacobi's virial equation, which permits obtaining solution of the equation while considering both gravitational and Coulomb interactions, is given for the system of the material points constituting a celestial body.On the basis of the numerical solutions, in the framework of the plasma model of a celestial body, it is shown that for the Coulomb interactions of charged particles the product of the form-factors and , entering expressions for the potential energy and the moment of inertia, remains constant.Without any model restrictions this conclusion is confirmed in case of the asymptotic time limit of simultaneous collision of all the charged particles of the system.A relationship between the potential energy of a spherically symmetrical celestial body and its mass through a phenomenological parameter, which is the sound velocity, is found from the consideration of the hydrostatic equilibrium condition of the body, taking the Coulomb interactions into account.     

Moment of inertia of a neutron star. I. Relativistic equation for the accumulated moment of inertia

A relativistic, first-order differential equation is derived for the accumulated moment of inertia of a spherically symmetric celestial body. An approximate equation is proposed to describe the contribution of relativistic effects to the moment of inertia of a superdense star. For configurations of an incompressible fluid, this approximation describes the results of the numerical calculations of Chandrasekhar and Miller to within 5% in the entire range of central pressures from 0 to ∞. Translated from Astrofizika, Vol. 40, No. 1, pp. 87–96, January–March, 1997.     

Four Dipole Problem Equilibrium Configurations

In this article we study for first time the motion of charged particleswith neglected mass under the influence of Lorentz and Coulomb forces offour moving celestial bodies. For this problem we give the equations ofmotion, variational equations and energy integral. Also we give theequilibrium configurations demonstrating an extensive numericalinvestigation of the equilibrium points with comments about theirappearance.     

Analysis of the potential field and equilibrium points of irregular-shaped minor celestial bodies

The equilibrium points of the gravitational potential field of minor celestial bodies, including asteroids, comets, and irregular satellites of planets, are studied. In order to understand better the orbital dynamics of massless particles moving near celestial minor bodies and their internal structure, both internal and external equilibrium points of the potential field of the body are analyzed. In this paper, the location and stability of the equilibrium points of 23 minor celestial bodies are presented. In addition, the contour plots of the gravitational effective potential of these minor bodies are used to point out the differences between them. Furthermore, stability and topological classifications of equilibrium points are discussed, which clearly illustrate the topological structure near the equilibrium points and help to have an insight into the orbital dynamics around the irregular-shaped minor celestial bodies. The results obtained here show that there is at least one equilibrium point in the potential field of a minor celestial body, and the number of equilibrium points could be one, five, seven, and nine, which are all odd integers. It is found that for some irregular-shaped celestial bodies, there are more than four equilibrium points outside the bodies while for some others there are no external equilibrium points. If a celestial body has one equilibrium point inside the body, this one is more likely linearly stable.     

On the fluid and viscoelastic deformations of the planets

The shape of the Earth and of planets depends on the exciting forces and on the rheology of the medium. From the equilibrium equation, we present the main modelisations of the viscous and inviscid fluids and we essentially describe the characteristics of linear viscoelastic deformations, for the Maxwell viscoelastic model of rheology. We use the elastic, viscoelastic and fluid Love numbers in order to investigate the associated relaxation modes. For these various kinds of rheology of the planets interior, we compute the geoid and the topography induced by an internal mass distribution. Finally, we show the importance of this viscoelastic deformation calculations in the study of the celestial body rotations.     

Empirical evidence for tidal evolution in transiting planetary systems

Most transiting planets orbit very close to their parent star, causing strong tidal forces between the two bodies. Tidal interaction can modify the dynamics of the system through orbital alignment, circularization, synchronization and orbital decay by exchange of angular moment. Evidence for tidal circularization in close-in giant planet is well known. Here, we review the evidence for excess rotation of the parent stars due to the pull of tidal forces towards spin-orbit synchronization. We find suggestive empirical evidence for such a process in the present sample of transiting planetary systems. The corresponding angular momentum exchange would imply that some planets have spiralled towards their star by substantial amounts since the dissipation of the protoplanetary disc. We suggest that this could quantitatively account for the observed mass–period relation of close-in gas giants. We discuss how this scenario can be further tested and point out some consequences for theoretical studies of tidal interactions and for the detection and confirmation of transiting planets from radial velocity and photometric surveys.     

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.     

Compact models with regular charge distributions

We model a compact relativistic body with anisotropic pressures in the presence of an electric field. The equation of state is barotropic, with a linear relationship between the radial pressure and the energy density. Simple exact models of the Einstein–Maxwell equations are generated. A graphical analysis indicates that the matter and electromagnetic variables are well behaved. In particular, the proper charge density is regular for certain parameter values at the stellar center unlike earlier anisotropic models in the presence of charge. We show that the electric field affects the mass of stellar objects and the observed mass for a particular binary pulsar is regained. Our models contain previous results of anisotropic charged matter with a linear equation of state for special parameter values.     

The dynamo theory of plasma turbulence wave in rotating celestial bodies

The various modes of plasma turbulence waves (including MHD waves) are easily excited under cosmic circumstances. In this paper, if we consider that the celestial bodies rotate, there is a source term generated for the magnetic induced equation by the excited plasma turbulence waves. If we expand the turbulent field in the Fourier series and include rotation velocity, the dynamo equation for turbulent waves is obtained. We have also obtained the solutions of various wave forms corresponding to different rotation velocities and then we significantly discuss the magnetic fields in the Sun, planets, and other celestial bodies.     

Virial oscillations of celestial bodies

A general approach to the solution of the perturbed oscillation problem for celestial bodies is considered. The solution sought describes unperturbed virial oscillations (zero approximation) affected by external perturbing effects. In the general case, these perturbations can be expressed by an arbitrary given function of time, Jacobi's function and its first derivative. Standard methods and modes of perturbation theory are used for solution of the problem.It is shown that while studying the evolution of a celestial body as a dissipative system in the framework of perturbed virial oscillations, the analytical expression for perturbing function can be derived, assuming the celestial body to be an oscillating electrical dipole emitting electromagnetic energy.The general covariant form of Jacobi's equation is derived and its spur is examined. It is shown that the scalar form of Jacobi's equation appears to be more universal than Newton's laws of motion from which it is derived.     

Virial oscillations of celestial bodies: V. The structure of the potential and kinetic energies of a celestial body as a record of its creation history

The physical meaning of the terms of the potential and kinetic energy expressions, expanded by means of the density variation function for a nonuniform self-gravitating sphere, is discussed. The terms of the expansions represent the energy and the moment of inertia of the uniform sphere, the energy and the moment of inertia of the nonuniformities interacting with the uniform sphere, and the energy of the nonuniformities interacting with each other. It follows from the physical meaning of the above components of the energy structure, and also from the observational fact of the expansion of the Universe that the phase transition, notably, fusion of particles and nuclei and condensation of liquid and solid phases of the expanded matter accompanied by release of energy, must be the physical cause of initial thermal and gravitational instability of the matter. The released kinetic energy being constrained by the general motion of the expansion, develops regional and local turbulent (cyclonic) motion of the matter, which should be the second physical effect responsible for the creation of celestial bodies and their rotation.     

Ocean Worlds in the Outer Regions of the Solar System (Review)

There is an astonishing variety of celestial bodies in the outer regions of the Solar System: Europa, with its bizarre surface features, Enceladus, small but geologically active, Titan, the only moon with a significant atmosphere, Pluto, with its nitrogen glaciers, and many others. Over the past 25 years, measurements from spacecraft have shown that many of these celestial bodies are ocean worlds with large volumes of liquid water trapped under icy surfaces. This new group of celestial bodies, ocean worlds, is important for research for several reasons, but the most convincing and at the same time the simplest reason is that they can be potential habitats. Life, as we know it, requires liquid water in addition to energy, nutrients, and a sustainable environment. All these requirements can be met for some of these celestial bodies. The moons of the giant planets on which the presence of the subsurface ocean is established (Europa, Ganymede, Titan, and Enceladus) and their astrobiological potential are discussed.     

Virial oscillations of celestial bodies III. The solution of the evolutionary problem in non-newtonian time scale

Generalized Jacobi's equation is derived by introducing the friction force into the equations of motion of mass points constituting the system.The exact solution of the equation of virial oscillations of celestial bodies written for non-conservative systems is obtained using non-linear time scale in the course of the change of variables for a particular friction force law.The nature of the undamped virial oscillations of celestial bodies is though to be related to the system unstability near the state determined by the virial theorem. Thus, the friction force changes its sign near the unstable equilibrium state and due to dissipation of energy during evolution of the system the undamped virial oscillations can be described as self-exited oscillations.     

A phenomenological model for the precession of planets and bending of light

We presented a phenomenological mode that attributes the precession of perihelion of planets to the relativistic correction. This modifies Newton’s equation by adding an inversely cube term with distance. The total energy of the new system is found to be the same as the Newtonian one. Moreover, we have deduced the deflection of light formula from Rutherford scattering. The relativistic term can be accounted for quantum correction of the gravitational potential and/or self energy of objects.     

Spectrophotometric variability of the magnetic CP star 56 Arietis in spectral region from 1950 to 3200 Å

We presented a phenomenological mode that attributes the precession of perihelion of planets to the relativistic correction. This modifies Newton’s equation by adding an inversely cube term with distance. The total energy of the new system is found to be the same as the Newtonian one. Moreover, we have deduced the deflection of light formula from Rutherford scattering. The relativistic term can be accounted for quantum correction of the gravitational potential and/or self energy of objects.     

Virial oscillations of celestial bodies IV: The Lyapunov stability of motion

It is shown that with a virial approach to the solution of the many-body problem the integral characteristics of a system (Jacobi's function and total energy), being present in Jacobi's equation, are immanent to its own integrals. Estimating the Lyapunov stability of motion of a system they play the role of Lyapunov functions.Studying Lyapunov stability of the virial oscillations of celestial bodies we used the Duboshin criterion applicable when permanent perturbations are present. In the case of conservative systems the potential energy of the system plays the role of such a perturbation. Thus, the nature of the virial oscillations can be understood as an effect of non-linear resonance between the kinetic and the potential energies.It is shown that the stability of virial oscillations of conservative systems relative to variations of the form-factors product is only a necessary condition in the proof of the hypothesis that =const. for celestial bodies. The sufficient condition for the proof of this equality consists of the given direct derivation of the equation of virial oscillations of celestial bodies from Einstein's equation, as well as of the equivalence of Schwarzschild's solution and the solution of Jacobi's equation at .The stability of virial oscillations for dissipative systems is studied. It is shown that the stability is limited by the period of time of its bifurcation.     

Slowly rotating relativistic stars

Equations are given which determine the moment of inertia of a rotating relativistic fluid star to second order in the angular velocity with no other approximation being made. The equations also determine the moment of inertia of matter located between surfaces of constant density in a rotationally distorted star; for example, the moments of inertia of the crust and core of a rotationally distorted neutron star can be calculated in this way. The method is applied ton=3/2 relativistic polytropes and to neutron star models constructed from the Baym-Bethe-Pethick-Sutherland-Pandharipande equation of state. Supported in part by the National Science Foundation. Alfred P. Sloan Research Fellow.     

Relativistic gravitational potential and its relation to mass-energy

From the general theory of relativity a relation is deduced between the mass of a particle and the gravitational field at the position of the particle. For this purpose the fall of a particle of negligible mass in the gravitational field of a massive body is used. After establishing the relativistic potential and its relationship to the rest mass of the particle, we show, assuming conservation of mass-energy, that the difference between two potential-levels depends upon the value of the radial metric coefficient at the position of an observer. Further, it is proved that the relativistic potential is compatible with the general concept of the potential also from the standpoint of kinematics. In the third section it is shown that, although the mass-energy of a body is a function of the distance from it, this does not influence the relativistic potential of the body itself. From this conclusion it follows that the mass-energy of a particle in a gravitational field is anisotropic; isotropic is the mass only. Further, the possibility of an incidental feed-back between two masses is ruled out, and the law of the composition of the relativistic gravitational potentials is deduced. Finally, it is shown, by means of a simple model, that local inhomogeneities in the ideal fluid filling the Universe have negligible influence on the total potential in large regions.     

Moment of inertia of a neutron star. II. Relativistic corrections

In part I we suggested an approximate equation to determine the contribution of relativistic effects to the moment of inertia of a superdense star. In the present paper it is tested on model neutron stars with nine different variants of the equation of state of superdense matter. It is established that the approximation error does not exceed 5% for stable configurations. A more accurate version of the Ravenhall—Pethick equation [D. G. Ravenhall and C. J. Pethick, Astrophys. J., 424, 846 (1994)] for the moment of inertia as a function of the mass and radius of a neutron star is derived. Translated from Astrofizika, Vol. 40, No. 4, pp. 507–516, October–December, 1997.     

The solution for seasonal variations in the Earth's rate of rotation

New physical principles for an explanation of seasonal variations in the Earth's rate of rotation are proposed. It is thought that the variations are caused by a variation of the total energy of the Earth's atmosphere in the course of the planet's revolution about the Sun in elliptic orbit. Jacobi's virial equation for the Earth's atmosphere is derived from the Eulerian equations. The virial theorem is obtained. The existence of the relationship between Jacobi's function and potential energy of the atmosphere is confirmed. In the framework of this relationship, Jacobi's equation is reduced to the equation of unperturbed virial oscillations. The solution of the above-mentioned equation expresses the periodic virial oscillations of Jacobi's function (moment of inertia) of the Earth's atmosphere with time. The solution of the perturbed virial oscillation problem of the atmosphere-solid Earth system is obtained. The perturbation term in Jacobi's virial equation regards, in explicit form, the energy changes occurring in the atmosphere in the course of the planet's revolution about the Sun in elliptic orbit. The annual and semi-annual periodic variations in the Earth's rate of rotation can be considered as an astrometrical result following from the obtained solution. A satisfactory accord of the theoretical results with experimental data is shown.