Out-of-plane librations of spinning tethered satellite systems

We analyze the out-of-plane librations of a tethered satellite system that is nominally rotating in the orbit plane. To isolate the librational dynamics, the system is modeled as two point masses connected by a rigid rod with the system mass center constrained to an unperturbed circular orbit. For small out-of-plane librations, the in-plane motion is unaffected by the out-of-plane librations and a solution for the in-plane motion is determined in terms of Jacobi elliptic functions. This solution is used in the linearized equation for the out-of-plane librations, resulting in a Hill’s equation. Floquet theory is used to analyze the Hill’s equation, and we show that the out-of-plane librations are unstable for certain ranges of in-plane spin rate. For relatively high in-plane spin rates, the out-of-plane librations are stable, and the Hill’s equation can be approximated by a Mathieu’s equation. Approximate solutions to the Mathieu’s equation are determined, and we analyze the dominant characteristics of the out-of-plane librations for high in-plane spin rates. The results obtained from the analysis of the linearized equations of motion are compared to numerical simulations of the nonlinear equations of motion, as well as numerical simulations of a more realistic system model that accounts for tether flexibility. The instabilities discovered from the linear analysis are present in both the nonlinear system and the more realistic system model. The approximate solutions for the out-of-plane librations compare well to the nonlinear system for relatively high in-plane rotation rates, and also capture the significant qualitative behavior of the flexible system.     

Initial Data Set for Cosmology

The method of conformal metric is used to find an analytic set of consistent initial data for the linearized Einstein field equations. We discuss the relation between independent degrees of freedom and perturbations. This revised version was published online in July 2006 with corrections to the Cover Date.     

An exact spherically symmetric space–time in Einstein–Cartan theory

By adopting the Newman–Penrose–Jogia–Griffith formalism, the field equations in Einstein–Cartan theory for matter with spin creating torsion in space–time are solved in a spherically symmetric space–time by assuming only one non-vanishing component of spin. The exact solution might be the prototype for more realistic models.     

The use of integrals in numerical integrations of theN-body problem

The numerical integration of systems of differential equations that possess integrals is often approached by using the integrals to reduce the number of degrees of freedom or by using the integrals as a partial check on the resulting solution, retaining the original number of degrees of freedom.Another use of the integrals is presented here. If the integrals have not been used to reduce the system, the solution of a numerical integration may be constrained to remain on the integral surfaces by a method that applies corrections to the solution at each integration step. The corrections are determined by using linearized forms of the integrals in a least-squares procedure.The results of an application of the method to numerical integrations of a gravitational system of 25-bodies are given. It is shown that by using the method to satisfy exactly the integrals of energy, angular momentum, and center of mass, a solution is obtained that is more accurate while using less time of calculation than if the integrals are not satisfied exactly. The relative accuracy is ascertained by forward and backward integrations of both the corrected and uncorrected solutions and by comparison with more accurate integrations using reduced step-sizes.     

Unrestricted second-order tensor virial equations for linear oscillations of magnetic configurations

This paper deals with the second-order tensor virial equations for the linear oscillations of a gaseous mass in the presence of a magnetic field. It is shown that the commonly used linearized versions of the tensor virial equations are restricted integral equations that incorporate the linearized equation of motion but not the boundary condition. These restricted equations only allow trial functions that fulfil the boundary condition and are of limited practical value.The unrestricted variational principle for the linear oscillations of a magnetic configuration is used to derive a more general formulation of the second-order tensor virial equations so that the linear trial function _i =X _ij x _j can be used to study the oscillations of a configuration with a magnetic field that extends in the exterior vacuum. The unrestricted virial equations have been applied to Ferraro's model and approximate results for the eigenfrequencies and eigenfunctions have been obtained for nine oscillation modes.     

Plasma wave propagation in the neighborhood of a magnetic neutral point

A linearized magnetohydrodynamic formalism is used to examine the propagation in two dimensions of transverse waves in a plasma in which is embedded a curl-free magnetic field. Only waves of frequency less than the ion cyclotron frequency are considered. The behavior of a wave packet near the magnetic neutral point is deduced from the general solution to the problem, which is found in terms of Bessel functions whose order is frequency dependent. It is shown that a disturbance propagates through the medium with a group velocity that decreases from the speed of light at large distances from the neutral point to zero at the neutral point, and that the amplitudes of the velocities associated with the disturbance diverge there. It is suggested that the stagnation of waves near the origin and the deformation of the magnetic field resulting from the large plasma-flow velocities may provide a clue to the formation of the magnetic neutral sheet required by several flare theories and a theory of the acceleration of cosmic rays.     

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.     

Nonlinear hydrodynamical models of stellar convective zones

Large-eddy simulation (LES) of turbulent convection is discussed in various versions (mixing-length theory, modal theory and spectral theory) in respect to the application to stellar convective zones. For the model construction, the non-local mixing-length formalism is suitable. However, for the determination of basic flow patterns and of mixing-length, the quasi-linear and nonlinear modal theories are useful. The eddy diffusivities are essential in these theories, and the nonlinear treatment of convection consistent with turbulent diffusivities (of effective Reynolds number of about 20 and Prandtl number of 0.4) offers a simple method of constructing stellar models without the use of the mixing-length.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

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.     

Weak field limit and tests of a relativistic theory of gravity

Static spherically-symmetric solutions to the linearized field equations of a generalized scalartensor theory of gravity are derived. The gravitational potential outside such a source is determined; it is found that this potential can have an intermediate range variation. The question of whether such an intermediate range variation would manifest itself in two experiments is addressed.     

Two-fluid models of superfluid neutron star cores

Both relativistic and non-relativistic two-fluid models of neutron star cores are constructed, using the constrained variational formalism developed by Brandon Carter and co-workers. We consider a mixture of superfluid neutrons and superconducting protons at zero temperature, taking into account mutual entrainment effects. Leptons, which affect the interior composition of the neutron star and contribute to the pressure, are also included. We provide the analytic expression of the Lagrangian density of the system, the so-called master function, from which the dynamical equations can be obtained. All the microscopic parameters of the models are calculated consistently using the non-relativistic nuclear energy density functional theory. For comparison, we have also considered relativistic mean field models. The correspondence between relativistic and non-relativistic hydrodynamical models is discussed in the framework of the recently developed 4D covariant formalism of Newtonian multifluid hydrodynamics. We have shown that entrainment effects can be interpreted in terms of dynamical effective masses that are larger in the relativistic case than in the Newtonian case. With the nuclear models considered in this work, we have found that the neutron relativistic effective mass is even greater than the bare neutron mass in the liquid core of neutron stars.     

Cosmology without Big Bang within the framework of the Projective Unified Field Theory and the influence on the motion of cosmic objects

The Projective Unified Field Theory having been developed by the author since 1957 is applied to a closed homogeneous isotropic cosmological model. By postulating a (new) conservation law for the scalaric substrate mass (derived from the balance equation of matter and leading to a basically new view on the concept of mass in context with the Mach principle) we obtained a coupled system of differential equations for the world radius and the scalaric world function, which could be treated numerically by Maple. Detailed calculations show that the big bang singularity of the Einstein theory can be avoided. Since the mass density and the temperature of the background radiation exhibit maxima at the same time after the cosmological “big start”, this fact could be of interest for cosmogonic activities. Within the framework of this theory there are hints for an antigravitational world era with repulsive forces after the big start. Furthermore, the balance equations for the energy and the angular momentum of a body as well as the equation of motion of a body (with a series of consequences) are derived. In this context we found a formula for the time dependence of the “effective Newtonian gravitational constant”. Further results refer to certain aspects for understanding the observed rotational curves of cosmic objects within galaxies as well as to the conservation of the number of photons and baryons and their mutual ratio etc.     

Some nonlinear effects in the relativistic two-body problem

The test-particle motion in the centrally symmetric gravitational field can be described by the equation in the form appropriate for a nonlinear oscillator — the nonlinear terms being due to the nonrelativistic effects. This enables us to apply to this equation the well-known asymptotic methods of the theory of nonlinear oscillations. Typical nonlinear oscillation phenomena arising from the action of external forces are shown to take place. The form of equations and the main results remain valid in the problem of two bodies of comparable mass in the post-Newtonian approximation.     

Self-modulation of pulsar radiation in strongly magnetized electron-positron plasmas

A nonlinear Schrödinger equation is obtained for linearly polarized electromagnetic waves propagating across the ambient magnetic field in an electron-positron plasma. The nonlinearities arising from wave intensity induced particle mass modulation, as well as harmonic generation are incorporated. Modulational instability and localization of pulsar radiation are investigated.     

Self-consistent solution for gaseous shock and stellar density wave

In this paper, the self-consistent density wave theory containing both a gaseous shock and a linear stellar density wave is studied, and a quasi-stable, tightly-wound, two-arm solution is obtained. The solution is convergent if the incomplete, linearized hydrodynamic equations are used, and the solution then gives the same dispersion relation as the local, asympotic solution, but the density and field profiles will be non-sinusoidal. The stellar wave will be unstable if the complete, linearized hydrodynamic equations are used.     

Black hole spin dependence of general relativistic multi-transonic accretion close to the horizon

We introduce a novel formalism to investigate the role of the spin angular momentum of astrophysical black holes in influencing the behavior of low angular momentum general relativistic accretion. We propose a metric independent analysis of axisymmetric general relativistic flow, and consequently formulate the space and time dependent equations describing the general relativistic hydrodynamic accretion flow in the Kerr metric. The associated stationary critical solutions for such flow equations are provided and the stability of the stationary transonic configuration is examined using an elegant linear perturbation technique. We examine the properties of infalling material for both prograde and retrograde accretion as a function of the Kerr parameter at extremely close proximity to the event horizon. Our formalism can be used to identify a new spectral signature of black hole spin, and has the potential of performing the black hole shadow imaging corresponding to the low angular momentum accretion flow.     

Angular momentum and gyroscope in flat space-time theory of gravitation

The study of a previously proposed theory of gravitation in flat space-time (Petry, 1981a) is continued. A conservation law for the angular momentum is derived. Additional to the usual form, there must be added a term coming from the spin of the gravitational field. The equations of motion and of spin angular momentum for a spinning test particle in a gravitational field are given. An approximation of the equations of the spin angular momentum in the rest frame of the test particle is studied. For a gyroscope in an orbit of a rotating massive body (e.g., the Earth) the precession of the spin axis agrees with the result of Einstein's general theory of relativity.     

A linear scattering problem in magnetohydrodynamics: Transmission resonances in a magnetic slab

The reduced linearized equations of ideal magnetohydrodynamics which are highly nonlinear in the eigenvalue parameter, are linearized about a prescribed value of that parameter, enabling the equation to be expressed as a Schrödinger equation with piecewise uniform coefficients. Reflection and transmission coefficients are obtained using standard techniques, and in addition to the possibility of total transmission of an incident wave occurring (together with complex-valued resonance energies), the magnetic field introduces other total transmission energy levels which have no counterpart in the absence of a magnetic field.