A note on the solution of the variational equations of a class of dynamical systems

Some properties are derived for the solutions of the variational equations of a class of dynamical systems. It is shown that in rather general conditions the matrix of the linearized Lagrangian equations of motion have an important property for which the word skew-symplectic has been introduced. It is also shown that the fundamental matrix of solutions is symplectic, the word symplectic being used here in a more general sense than in the classical literature. Two consequences of the symplectic property are that the fundamental matrix is easily invertible and that the eigenvalues appear in reciprocal pairs. The effect of coordinate transformations is also analyzed; in particular the change from Lagrangian to canonical systems.     

A linear solution of the equations of motion of an earth-orbiting satellite based on a Lie-series

Using Hill's variables, an analytical solution of a canonical system of six differential equations describing the motion of a satellite in the gravitational field of the earth is derived. The gravity field, expanded into spherical harmonics, has to be expressed as a function of the Hill variables. The intermediary is chosen to include the main secular terms. The first order solution retains the highly practical formal structure of Kaula's linear solution, but is valid for circular orbits and provides of course a spectral decomposition of radius vector and radial velocity. The resulting eccentricity functions are much simpler than the Hansen functions, since a series evaluation of the Kepler equation is avoided. The present solution may be extended to higher order solutions by Hori's perturbation method.     

Analytic solution of transfer problem linearized with respect to a small parameter in the problem of an almost gray atmosphere

Cosmic Research NPO, Azerbaidzhan Academy of Sciences. Translated from Astrofizika, Vol. 28, No. 2, pp. 455–262, March–April, 1988.     

The Roche Coordinates in three dimensions and their application to problems of double-star astronomy

In a preceding paper (Kopal, 1969; in what follows referred to as Paper I) we introduced a new system of curvilinear coordinates-hereafter referred to as Roche Coordinates — in which spheres of constant radius in spherical polars have been replaced by surfaces of constant potential of a rotating gravitational dipole; while the angular coordinates are orthogonal to the equipotentials. In Paper I we established an explicit form of such a transformation, and related the Roche coordinates with polar coordinates (with which they coalesce in the immediate neighbourhood of each one of the two finite mass-points) in the plane case. The aim of the present investigation will be to generalize the definition of the Roche coordinates to three dimensions.The opening Section 1 of this paper will contain a general outline of the proposed three-dimensional transformation; and in Section 2 details of this transformation will be explicitly worked out correctly to quantities of first order in superficial distortion — an approximation which should prove adequate in regions surrounding the two finite masses; while in Section 3 we shall evaluate (to this degree of accuracy) the metric coefficients of the respective transformation, and its direction cosines, in both polar and curvilinear coordinates. Section 4 will then contain a formulation of the fundamental equations of hydrodynamics in terms of the three-dimensional Roche coordinates; and their advantages for a treatment of certain classes of dynamical problems encountered in doublestar astronomy will be illustrated in the concluding Section 5 by an investigation of the vibrational stability of the Roche model. We shall show that this model is capable of performing free radial oscillations which remain barotropic only if its equilibrium form is spherical (i.e., in the absence of any external mass in the neighbourhood); but not if it is distorted to any extent by rotation or tides.     

Einstein-Maxwell field equations in isotropic coordinates: an application to neutron star and quark star

The photogravitational restricted three bodies within the framework of the post-Newtonian approximation is carried out. The mass of the primaries are assumed changed under the effect of continuous radiation process and oblateness effects of the two primaries. New perturbed locations of the triangular points are computed. In order to introduce a semi-analytical view, A Mathematica program is constructed so as to draw the locations of triangular points versus the whole range of the mass ratio μ taking into account the photo-gravitational effects, the relativistic corrections and/or oblateness effects. All the obtained figures are analyzed.     

On theories of gravitation in which the dynamical equations do not follow from the field equations and the Birkhoff Theorem

In nicht-Einsteinschen Gravitationstheorien gilt das Birkhoff-Theorem im allgemeinen nicht mehr. Es wird gezeigt, daß in Theorien, in denen die dynamischen Gleichungen nicht aus den Feldgleichungen folgen, zeitabhängige Vakuumlösungen nötig sind, um nichtstatische kugelsymmetrische inkohärente Materieverteilungen anschließen zu können. Für die Trederschen Tetradentheorien wird gezeigt, daß solche Lösungen existieren und der Anschluß stetig und eindeutig möglich ist In Verallgemeinerung dieser Ergebnisse wird untersucht, in welchen Gravitationstheorien die dynamischen Gleichungen nicht aus den Feldgleichungen folgen. Dies führt auf nicht-Einsteinsche Theorien, wie bimetrische Theorien oder Tredersche Tetradentheorien, die zusätzliche geometrische Größen enthalten, die nicht dynamische Variable der Theorie sind.     

The solution of the Rankine-Hugoniot equations for fast shocks in an anisotropic kappa distributed medium

In this paper, we concentrate on the analysis of the anisotropic Rankine-Hugoniot equations for perpendicular and oblique fast shocks. In particular, as additional information to the anisotropic set of equations, the threshold conditions of the fire-hose and mirror instability are used to bound the range of the pressure anisotropy downstream of the discontinuity. These anisotropic threshold conditions of the plasma instabilities are obtained via a kinetic approach using a generalized Lorentzian distribution function, the so-called kappa distribution function. Depending on up-stream conditions, these instabilities further define stable and unstable regions with regard to the pressure anisotropy downstream of the shock. The calculations are done for different upstream Alfvén Mach numbers. We found that low values of the parameter kappa reduce the pressure anisotropy downstream of the shock.     

On the shape of meteor light curves, and a fully analytic solution to the equations of meteoroid ablation

The shape and characteristics (beginning and end heights, and height of maximum brightness) of meteor light curves are investigated under the constraint that the surface area S that a meteoroid presents to the oncoming air flow varies as a power law in the meteoroid mass m such that   S ∼ m ^α  . We investigate the meteoroid ablation for a range of values of α, and find that the  α= 1  condition allows for a fully analytic solution to the coupled differential equations of meteoroid ablation when the density profile is that of an isothermal atmosphere. The possible geometrical properties of Geminid meteoroids are discussed in terms of the  α= 1  ablation model and it is shown that they are consistent with being derived from an asteroidal, rather than cometary, parent body.     

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.     

A perturbative solution of the transfer equations for the Stokes parameters in a magnetic field

The transfer equations for the Stokes parameters, as derived under the assumption of LTE and in the general case of anomalous Zeeman multiplet, are solved by a perturbative method that reduces the system of four coupled linear differential equations to an iterative series of well known linear differential equations. The perturbation parameter turns out to be of the order of magnitude of the ratio of the Larmor frequency to the line Doppler half-width, so that the method appears to be particularly suitable for small magnetic fields.Some qualitative results on the importance of Faraday rotation and the orders of magnitude of the line profiles of the Stokes parameters are obtained.     

Third-order secular Solution of the variational equations of motion of a satellite in orbit around a non-spherical planet

We constructed an analytical theory of satellite motion up to the third order relative to the oblateness parameter of the Earth (J ₂). Equations of secular variations was developed for the first three orbital elements (a, e, i) of an artificial satellite. The secular variations are solved in a closed form.     

Nonlinear stability of an infinite cylinder in the presence of a uniform axial magnetic field

The effect of a uniform axial magenetic field on the nonlinear instability of a self-gravitating infinite cylinder is examined. Using the method of multiple scales, it is found that while the nonlinear (modulational) instability cannot be completely suppressed, the presence of a magnetic field does increase the range of stable wave numbers. The evolution of the amplitude is governed by a non-linear Schrödinger equation which gives the criterion for modulational instability.Department of Chemical Engineering and Technology.Department of Mathematics.     

Report on the dynamical evolution of an axially symmetric quasar model

The role of the angular momentum in the regular or chaotic character of motion in an axially symmetric quasar model is examined. It is found that, for a given value of the critical angular momentumL _zc , there are two values of the mass of the nucleusM _n for which transition from regular to chaotic motion occurs. The [L _zc – M _n ] relationship shows a linear dependence for the time independent model and an exponential dependence for the evolving model. Both cases are explained using theoretical arguments together with some numerical evidence. The evolution of the orbits is studied, as mass is transported from the disk to the nucleus. The results are compared with the outcomes derived for galactic models with massive nuclei.     

A note on the solution of the saturation flux limited solar wind equations

The solution curves of the differential equations determining the behavior of the solar wind are calculated for the case where the heat flux has its maximum value 3/2 nkTv _th. All the supersonic solutions are asymptotically adiabatic, T r ^-4/3.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The fifth-order analytical solution of the equations of motion of a satellite in orbit around a non-spherical planet

A high-precise analytical theory of a satellite in orbit around a non-spherical planet has been developed. The Poisson's small parameter method has been used. All secular and short-periodic perturbations proportional up to and including a product of five arbitrary harmonic coefficients of the planetary potential expansion are calculated. Long-periodic perturbations are derived with the accuracy of up to the fourth-order, inclusive. The influence of the high-order perturbations on the motion of ETALON-1 satellite has been investigated. The results of comparison of the numerical and analytical integration of the equations of its motion over a five year interval are as follows:

Capillary action in a crack on the surface of asteroids with an application to 433 Eros

Some asteroids contain water ice, and a space mission landing on an asteroid may take liquid to the surface of the asteroid. Gas pressure is very weak on the surface of asteroids. Here we consider the capillary action in a crack on the surface of irregular asteroids. The crack is modeled as a capillary which has a fixed radius. An asteroid's irregular gravitational potential influences the height of the liquid in the capillary. The height of the liquid in the capillary on the surface of such asteroids is derived from the asteroid's irregular gravitational potential. Capillary mechanisms are expected to produce an inhomogeneaous distribution of emergent liquid on the surface. This result is applied to asteroid 433 Eros, which has an irregular, elongated, and concave shape. Two cases are considered: (1) we calculate the height of the liquid in the capillary when the direction of the capillary is perpendicular to the local surface of the asteroid; (2) we calculate the height of the liquid in the capillary when the direction of the capillary is parallel to the vector from the center of mass to the surface position. The projected height in the capillary on the local surface of the asteroid seems to depend on the assumed direction of the capillary.     

Nonlinear break up of an infinite cylinder in the presence of a uniform axial magnetic field

A third-order theory has been developed to study the instability of an infinite cylinder in the presence of a uniform axial magnetic field. By use of strained coordinates, the growth of the surface waves is calculated for wave numbers less than the critical wave number. It is observed that the perturbed surface of an infinite cylinder assumes a non-sinusoidal shape resulting in secondary waves due to energy transfer between the harmonics of different orders.Department of Chemical Engineering and Technology.Department of Mathematics.     

Surface vacuum energy and stresses on a brane in AdS space with an application to the brane-world model

The vacuum expectations of the surface energy-momentum tensor generated on a brane in AdS space-time by quantum fluctuations of a scalar field with an arbitrary coupling parameter are studied. It is assumed that the field satisfies mixed boundary conditions on the brane. A generalized zeta function method is used as a regularization procedure. Two regions, located to the left (L-region) and right (R-region) of the brane, are considered. It is shown that the surface energies for both these regions contain pole and finite contributions. Analytic expressions are derived for both parts. When calculating the total surface energy including the contributions from the L- and R-regions, the pole terms add out in odd spatial dimensions. The surface energy-momentum tensor induced by vacuum quantum effects corresponds to the generation of a cosmological constant on the brane. These results are applied to the second Randall-Sundrum model.Translated from Astrofizika, Vol. 48, No. 1, pp. 151–164 (February 2005).