Stability of the periodic solutions of the restricted three-body problem representing analytic continuations of Keplerian rectilinear periodic motions

The periodic solutions of the restricted three-body problem representing analytic continuations of Keplerian rectilinear periodic motions are well known (Kurcheeva, 1973). Here the stability of these solutions are examined by applying Poncaré's characteristic equation for periodic solutions. It is found that the isoperiodic solutions are stable and all other solutions are unstable.     

On stationary axially symmetric solutions of a scalar-tensor theory of gravitation

In this paper we obtain the general axially symmetric space-times in scalar-tensor theory of gravitation proposed by Sáez and Ballester. The only possible solutions of these space-times in this theory are dust solutions. One of the obtained solutions here has a similar considerations to Van Stockum dust solution. A detailed study of geometric and kinematical properties of the obtained solutions are carried out. Furthermore, closed timelike curves are present in these solutions.     

Neutrino transport with isoenergetic and neutrino-electron scattering

Solution to steady-state, bigroup neutrino transport with isoenergetic and neutrino-electron scattering have been obtained for idealized conditions representative of the outer core or envelope of a star whose inner core is undergoing gravitational collapse. The solutions are presented in a form suitable for comparison with other numerical schemes. The incident neutrino distribution was chosen to be nonequilibrium, and the solutions clearly show the tendency of the neutrinos to equilibrate via neutrino-electron scattering as they propagate through the material. Care is taken to compare solutions obtained by two completely different techniques to ensure that the solutions presented here are accurate and error free. The solutions are compared with those obtained by a simple diffusion scheme, and by a flux limited diffusion scheme. The solutions given by the latter agree well with the solutions derived here, except when the neutrino angular distribution becomes oblate.     

Global Solution of Dynamical Systems a Report to Z. Kopal, on what Followed Since

Two basic problems of dynamics, one of which was tackled in the extensive work of Z. Kopal (see e.g. Kopal, 1978, Dynamics of Close Binary Systems, D. Reidel Publication, Dordrecht, Holland.), are presented with their approximate general solutions. The ‘penetration’ into the space of solution of these non-integrable autonomous and conservative systems is achieved by application of ‘The Last Geometric Theorem of Poincaré’ (Birkhoff, 1913, Am. Math. Soc. (rev. edn. 1966)) and the calculation of sub-sets of ‘solutions précieuses’ that are covering densely the spaces of all solutions (non-periodic and periodic) of these problems. The treated problems are: 1. The two-dimensional Duffing problem, 2. The restricted problem around the Roche limit. The approximate general solutions are developed by applying known techniques by means of which all solutions re-entering after one, two, three, etc, revolutions are, first, located and then calculated with precision. The properties of these general solutions, such as the morphology of their constituent periodic solutions and their stability for both problems are discussed. Calculations of Poincaré sections verify the presence of chaos, but this does not bear on the computability of the general solutions of the problems treated. The procedure applied seems efficient and sufficient for developing approximate general solutions of conservative and autonomous dynamical systems that fulfil the PoincaréBirkhoff theorems. The same procedure does not apply to the sub-set of unbounded solutions of these problems.     

引力常数随时间变化对双星轨道演变的长期效应(椭圆轨道情形)

给出了以偏近点角为自变量的变引力常数的摄动方程组的解.解包括轨道半长轴的长期和周期变化项,其他轨道根数在一阶解中无长期项,只有周期项.近星点经度和平经度在二阶解中显示长期项变化.给出了由于引力常数变化对双星轨道演变情况的数值估计,对结果做了讨论并给出结论.     

Conditionally periodic solutions near the libration points of the restricted circular three-body problem

Families of conditionally periodic solutions have been found by a slightly modified Lyapunov method of determining periodic solutions near the libration points of the restricted three-body problem. When the frequencies of free oscillations are commensurable, the solutions found are transformed into planar or spatial periodic solutions. The results are confirmed by numerically integrating the starting nonlinear differential equations of motion.     

Exact Brans-Dicke bianchi type-I vacuum solutions with a cosmological constant

Exact cosmological solutions of the vacuum Brans-Dicke field equations are derived. The solutions are the anisotropic generalizations of the isotropic de Sitter solutions of the general theory of relativity.     

Approximate solutions for isothermal flows behind strong spherical shocks with variable energy

Self-similar unsteady flows with zero temperature gradient behind strong spherical shocks propagating in non-uniform perfect gas at rest are investigated. The total energy of the flow is assumed to be varying with the shock radius obeying a power law. Approximate solutions in a closed analytical form are obtained using the integral method. Also these solutions are shown to be useful to calculate easily and quickly the shock temperature, X-ray surface brightness and luminosity which are important in astrophysical problems. It is found that these approximate solutions are in close agreement with numerical solutions.     

Black hole stability in multidimensional gravity theory

Exact static, spherically symmetric solutions to the Einstein-Maxwell-scalar equations, with a dilatonic-type scalar-vector coupling, in D-dimensional gravity with a chain of n Ricci-flat internal spaces are considered. Their properties and special cases are discussed. A family of multidimensional dilatonic black-hole solutions is singled out, depending on two integration constants (related to black hole mass and charge) and three free parameters of the theory (the coordinate sphere, internal space dimensions, and the coupling constant). The behaviour of the solutions under small perturbations preserving spherical symmetry, is studied. It is shown that the black-hole solutions without a dilaton field are stable, while other solutions, possessing naked singularities, are catastrophically unstable.     

Gravitational collapse of gas clouds with spherical,cylindrical or plane symmetry in the linear wave flow approximation

The equations of gas dynamics are solved, quasi-analytically by applying McVittie's method for spherical, cylindrical and plane configurations. The hypothesis of linear wave flow is applied and it is assumed that the final state of collapsing clouds is a hydrostatic equilibrium state, determined by complete polytropes. Complete analytical solutions are found when the generalized (to the three symmetries) Emden equation admits of analytical solutions. Otherwise the solutions are left in terms of the numerical solutions of the Emden equation. Numerical solutions to the Emden equation in the plane case are found and tabulated. A strong dependence of amplification, of density, pressure and temperature of the gas, on the symmetry is found. In addition, it is conclude that the flow remains subsonic, during the collapse, except toward the boundaries of the collapsing clouds.     

A new class of relativistic stellar models

Einstein field equations for a static and spherically symmetric perfect fluid are considered. A formulation given by Patiño and Rago is used to obtain a class of nine solutions, two of them are Tolman solutions I, IV and the remaining seven are new. The solutions are the correct ones corresponding to expressions derived by Patiño and Rago which have been shown by Knutsen to be incorrect. Similar to Tolman solution IV each of the new solutions satisfies energy conditions inside a sphere in some range of two independent parameters. Besides, each solution could be matched to the exterior Schwarzschild solution at a boundary where the pressure vanishes and thus the solutions constitute a class of new physically reasonable stellar models.     

Three-dimensional bifurcations of periodic solutions around the triangular equilibrium points of the restricted three-body problem

Vertically critical, planar periodic solutions around the triangular equilibrium points of the Restricted Three-Body Problem are found to exist for values of the mass parameter in the interval [0.03, 0.5]. Four series of such solutions are computed. The families of three-dimensional periodic solutions that branch off these critical orbits are computed for µ = 0.3 and are continued till their end. All orbits of these families are unstable.     

Radiative source-function predictions for finite and semi-infinite non-conservative atmospheres

The probabilistic method of Sobolev and Case's method of normal mode expansion are combined to predict source-function distributions for radiative transfer in non-conservative, planeparallel atmospheres. The solutions obtained for semi-infinite atmospheres are exact and can be expressed in terms of functions and parameters associated with the non-conservative Milne problem. The predictions for finite atmospheres are approximate and are constructed from the semi-infinite solutions. Tabular values of the requisite functions and parameters are provided to facilitate rapid numerical evaluation of the solutions. Although the finite solutions corresponds to the zeroth-order (optically thick) approximation by Case's method, an assessment of the accuracy indicates that the results are useful for optical thicknesses as small as one or even less. The close connection between the results obtained and the method of point-direction gain of Van de Hulst is discussed.     

Spherical radiation-driven shock waves in a non-uniform atmosphere

Similarity solutions are obtained for spherical radiation-driven shock waves propagating in a non-uniform atmosphere at rest obeying a density power law. Approximate analytical solutions are also obtained and found to be in good agreement with the numerical solutions. The effect of the parameter characterizing the initial density distribution of the gas on solutions of the flow field is studied in detail. It is also shown analytically that the shock wave propagates as an overdriven detonation.     

New exact cosmologies on the brane

We develop a method for constructing exact cosmological solutions in brane world cosmology. New classes of cosmological solutions on Randall–Sandrum brane are obtained. The superpotential and Hubble parameter are represented in quadratures. These solutions have inflationary phases under general assumptions and also describe an exit from the inflationary phase without a fine tuning of the parameters. Another class solutions can describe the current phase of accelerated expansion with or without possible exit from it.     

Asymmetric periodic solutions of the averaged Hill problem with allowance for a planet’s oblateness

We consider asymmetric periodic solutions of the double-averaged Hill problem by taking into account oblateness of the central planet. They are generated by steady-state solutions, which are stable in the linear approximation and correspond to satellite orbits orthogonal to the line of intersection of the planet’s equatorial plane with the orbital plane of a disturbing point. For two model systems [(Sun+Moon)-Earth-satellite] and [Sun-Uranus-satellite], these periodic solutions are numerically continued from a small vicinity of the equilibrium position. The results are illustrated by projecting the solutions onto the (pericenter argument-eccentricity) and (longitude-inclination) planes.     

Analytical solutions for the equations of motion of a space vehicle during the atmospheric re-entry phase on a 2-D trajectory

A practical and important problem encountered during the atmospheric re-entry phase is to determine analytical solutions for the space vehicle dynamical equations of motion. The author proposes new solutions for the equations of trajectory and flight-path angle of the space vehicle during the re-entry phase in Earth’s atmosphere. Explicit analytical solutions for the aerodynamic equations of motion can be effectively applied to investigate and control the rocket flight characteristics. Setting the initial conditions for the speed, re-entering flight-path angle, altitude, atmosphere density, lift and drag coefficients, the nonlinear differential equations of motion are linearized by a proper choice of the re-entry range angles. After integration, the solutions are expressed with the Exponential Integral, and Generalized Exponential Integral functions. Theoretical frameworks for proposed solutions as well as, several numerical examples, are presented.     

A class of new Bianchi I perfect fluid space-times

A procedure to generate new exact solutions to Einstein equations for perfect fluids is applied to LRS Bianchi type I line-element. Starting from some known solutions a class of new perfect fluid solutions of Bianchi type I are presented. The physical and kinematical properties of spatially homogeneous and anisotropic cosmological models are studied.     

Equilibrium solutions of the restricted problem of 2 + 2 axisymmetric rigid bodies

The restricted problem of 2 + 2 homogenous axisymmetric ellipsoids such that their equatorial planes coincide with the orbital plane of the centers of mass is considered. The equilibrium solutions of this problem are shown to exist. Six of these solutions are located about the collinear points of the restricted problem of three axisymmetric ellipsoids. A special case of this problem is studied and sixteen solutions are found in the neighborhood of the triangular Lagrangian points.     

Closed-form expressions for the rate of growth of perturbations in zero-pressure Friedmann-Lemaître universes

The general solutions to the density perturbation equation derived by Bonnor are pressented for the zero-pressure Friedmann-Lemaître universes in terms of elliptic functions. The particular solutions already published to this equation in terms of elementary functions are exhibited as special cases arising from the general solutions. These occur when the elliptic functions simplify to either trigonometric or hyperbolic functions.