Hierarchy of periodic solutions families of spatial Hill’s problem

Many modern space projects require the knowledge of orbits with certain properties. Most of these projects assume the motion of a space vehicle in the neighborhood of a celestial body, which in turn moves in the field of the Sun or another massive celestial body. A good approximation of this situation is Hill’s problem. This paper is devoted to the investigation of the families of spatial periodic solutions to the three-dimensional Hill’s problem. This problem is nonintegrable; therefore, periodic solutions are studied numerically. The Poincare theory of periodic solutions of the second kind is applied; either planar or vertical impact orbits are used as generating solutions.     

Radiating Collapse Solutions

Exact gravitational solutions with radial pressure and heat flow are obtained by integrating the field equations. Junction conditions which match the collapse solutions to the exterior Vaidya metric show that, at the boundary, the pressure is proportional to the magnitude of the heat flow vector. This condition allows us to determine the time-dependent functions of the interior solutions.     

A new shock locus for similarity solutions in one-dimensional unsteady gas dynamics

This paper deals with shock conditions for the progressing wave (or similarity) solutions of one-dimensional, unsteady gas dynamics. These solutions have hitherto been used to deal with the flow behind shocks moving into stationary atmospheres. By generalising the shock conditions to the case of moving atmospheres, it is shown that the progressing wave solutions can be used to describe a certain class of flows, and a new shock locus can be constructed in the phase plane of the solutions. It is hoped that such solutions will be of use in describing the unsteady flow behind shocks propagating into the ambient solar wind.     

Periodic orbits near infinity in the restrictedN-body problem

This paper shows that there exist two families of periodic solutions of the restrictedN-body problem which are close to large circular orbits of the Kepler problem. These solutions are shown to be of general elliptic type and hence are stable. If the restricted problem admits a symmetry, then there are symmetric periodic solutions which are close to large elliptic orbits of the Kepler problem.     

Periodic solutions in Hill's relativistic problem

In this article the existence of periodic solutions in Hill's relativistic problem is demonstrated using Poincaré's small parameter method. This method guarantees the convergence of the series representing the periodic solutions.     

Second species periodic solutions with an 0(μ) near-moon passage

This paper uses the results of second-order asymptotic matching in the restricted three body problem to establish the existence and first-order asymptotic approximation of various families of second species periodic solutions with one near-moon passage during a half-period. In this way, the existence and asymptotic approximation of second species solutions with any number of near-moon passages during a half-period can be established based on higher order asymptotic matching. Second species solutions with near-moon passages have not been studied numerically due to the difficult nature of this problem.This research was supported in part by the National Science Foundation under Grant GP42739 and in part by Northern Arizona University under a university research grant.     

Optimized magnetic reconnection solutions in three dimensions

It has recently been shown that there is a well defined upper limit to the rate of magnetic merging for two-dimensional flux pile-up solutions. This rate, derived by equalizing the dynamic and magnetic pressures in the reconnection region and saturating the magnetic field in the current layer, leads to a significant enhancement of the classical Sweet–Parker merging limit. In this study we explore optimal merging rates in the case of three-dimensional fan and spine reconnection solutions. The ideas of optimization and saturation are first illustrated using an exact fan solution. We go on to show that while spine solutions seem ineffective as flare release mechanisms, optimized fan solutions have energy release characteristics typical of modest events.     

Approximate General Solution Of The Two-Dimensional Duffing Dynamical System

This paper presents the approximate general solution of the triple well, double oscillator non-linear dynamical system. This system is non-integrable and the approximate general solution is calculated by application of the Last Geometric Theorem of Poincaré (Birkhoff, 1913, 1925). The original problem, known as the Duffing one, is a 1 degree of freedom system that, besides the conservative force component, includes dumping and external forcing terms (see details in the web site: http://www.uncwil.edu/people/hermanr/chaos/ted/chaos.html). The problem considered here is a 2 degree of freedom, autonomous and conservative one, without dumping, and of axisymmetric potential. The space of permissible motions is scanned for identification of all solutions re-entering after from one to nine oscillations and the precise families of periodic solutions are computed, including their stability parameter, covering all cases with periods T corresponding to 4osc/T. Seven sub-domains of the space of solutions were investigated in detail by zooming, an operation that proved the possibility to advance the accuracy of the approximate general solution to the level permitted by the integration routine. The approximation of the general solution, although impressive, provides clear evidence of the complexity of the problem and the need to proceed to larger period families. Nevertheless, it allows prediction of the areas where chaos and order regions in the Poincaré surfaces of section are to be expected. Examples of such surfaces of sections, as well as of types of closed solutions, are given. Two peculiar points of the space of solutions were identified as crossing, or source points from which infinite families of periodic solutions emanate. The morphology and stability of solutions of the problem are studied and discussed.     

Structural stability of homothetic solutions of the collinearn-body problem

We investigate specific homothetic solutions of then-body problem which both begin and end in a simultaneous collision of all of the particles. Under a suitable change of variables, these solutions become heteroclinic orbits, i.e., they lie in the intersection of the stable and unstable manifolds of distinct equilibrium points. Our main result is that these manifolds intersect transversely along these orbits. This proves that the homothetic solutions are structurally stable.Partially supported by NSF Grant MCS 77-00430.     

Periodic solutions of the restricted problem that are analytic continuations of periodic solutions of Hill's problem for small μ>0

This paper establishes the existence and first order perturbation approximation of an infinite number of one-parameter families of symmetric periodic solutions of the restricted three body problem that are analytic continuations of symmetric periodic solutions of Hill's problem for small values of the mass ratio μ>0.     

The general class of Bianchi cosmological models with viscous fluid and particle creation in Brans-Dicke theory

This paper deals with the general class of Bianchi cosmological models with bulk viscosity and particle creation described by full causal thermodynamics in Brans-Dicke theory. We discuss three types of average scale-factor solutions for the general class of Bianchi cosmological models by using a special law for the deceler- ation parameter which is linear in time with a negative slope. The exact solutions to the corresponding field equations are obtained in quadrature form and solutions to the Einstein field equations are obtained for three different physically viable cosmologies. All the physical parameters are calculated and discussed in each model.     

A family of well behaved charge analogues of Durgapal’s perfect fluid exact solution in general relativity II

This paper presents a family of two-parametric interior solutions of Einstein–Maxwell field equations in general relativity for a static spherically symmetric distribution of a charged perfect fluid with particular form of charge distribution. This class of solutions gives us wide range of parameters, n and K, for which the solutions are well behaved hence, suitable for modeling of compact star (e. g. bare strange quark star). The mass of star is maximized with all degree of suitability by assuming the stellar “surface” density equal to strange (quark) matter density at zero pressure. It is hoped that our investigation may be of some help in connection of some study of stellar structure.     

Self-Similar Solutions for Hypersonic Source With Variable Mass Loss Rate

For the 1D radial hypersonic flow the development of the configuration with two shocks and contact discontinuity is considered. At small and large moments of time solutions in explicit form are found. As follows from these solutions the contact surface accelerates in time. This acceleration makes possible the Rayleigh-Taylor instability to develop. The 2D numerical investigation of the problem has confirmed the instability of the 1D solution. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical boundary conditions for solar magnetohydrodynamic flows using the method of characteristics

We discuss the self-consistent time-dependent numerical boundary conditions on the basis of theory of characteristics for magnetohydrodynamics (MHD) simulations of solar plasma flows. The importance of using self-consistent boundary conditions is demonstrated by using an example of modeling coronal dynamic structures. This example demonstrates that the self-consistent boundary conditions assure the correctness of the numerical solutions. Otherwise, erroneous numerical solutions will appear.     

A note on the elliptic restricted three-body problem

This work considers periodic solutions, arc-solutions (solutions with consecutive collisions) and double collision orbits of the plane elliptic restricted problem of three bodies for =0 when the eccentricity of the primaries,e _p , varies from 0 to 1. Characteristic curves of these three kinds of solutions are given.     

Analytical solutions for J 2-perturbed unbounded equatorial orbits

While solutions for bounded orbits about oblate spheroidal planets have been presented before, similar solutions for unbounded motion are scarce. This paper develops solutions for unbounded motion in the equatorial plane of an oblate spheroidal planet, while taking into account only the J ₂ harmonic in the gravitational potential. Two cases are distinguished: A pseudo-parabolic motion, obtained for zero total specific energy, and a pseudo-hyperbolic motion, characterized by positive total specific energy. The solutions to the equations of motion are expressed using elliptic integrals. The pseudo-parabolic motion unveils a new orbit, termed herein the fish orbit, which has not been observed thus far in the perturbed two-body problem. The pseudo-hyperbolic solutions show that significant differences exist between the Keplerian flyby and the flyby performed under the the J ₂ zonal harmonic. Numerical simulations are used to quantify these differences.     

Approximate Analytic Solutions of the Averaged Three-Body Problem at First-Order Resonance with Large Oblateness of the Central Planet

For the general spatial planetary three-body problem at first-order mean motion resonance under the large oblateness of the central planet, the analytic solutions of the averaged motion are obtained with the help of the Weierstrass functions accurate to the third-degree terms in the satellites' eccentricities and inclinations. The behavior of solutions is investigated on the phase plane.This revised version was published online in October 2005 with corrections to the Cover Date.     

Solitons and other solutions to the quantum Zakharov-Kuznetsov equation

In this paper, we present G′/G-expansion method, exp-function method, modified F-expansion method as well as the traveling wave hypothesis for finding the exact traveling wave solutions of the quantum Zakharov-Kuznetsov equation which arises in quantum magneto-plasmas. By these methods, rich families of exact solutions have been obtained, including soliton solutions. This work continues to reinforce the idea that the proposed methods, with the help of symbolic computation, provide a powerful mathematical tool for solving nonlinear partial differential equations.