Proof of a conjecture of E. Strömgren

The conjecture of Strömgren according to which, in the restricted problem, a class of doubly asymptotic orbits are limit members of families of periodic orbits is examined in the more general framework of analytic Hamiltonian system with two degrees of freedom. Sufficient conditions for the conjecture to become a theorem are established. Theses conditions amount to a transversality condition for the doubly asymptotic orbits and are likely to be verified in the cases considered in the literature of numerical explorations of the restricted problem.     

Asymptotic Orbits at the Triangular Equilibria in the Photogravitational Restricted Three-Body Problem

We study numerically the asymptotic homoclinic and heteroclinic orbits associated with the triangular equilibrium points L ₄ and L ₅, in the gravitational and the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these critical points, are also presented. Hundreds of asymptotic orbits for equal mass of the primaries and for various values of the radiation pressure are computed and the most interesting of them are illustrated. In the Copenhagen case, which the problem is symmetric with respect to the x- and y-axis, we found and present non-symmetric heteroclinic asymptotic orbits. So pairs of heteroclinic connections (from L ₄ to L ₅ and vice versa) form non-symmetric heteroclinic cycles. The termination orbits (a combination of two asymptotic orbits) of all the simple families of symmetric periodic orbits, in the Copenhagen case, are illustrated.     

Asymptotic theory of the transfer of polarized radiation with resonance scattering in the doppler core of the line

In the first of the series of papers by Ivanov et al. it was shown that the model problem of the transfer of polarized radiation as a result of resonance scattering from two-level atoms in a homogeneous plane atmosphere in the absence of LTE comes down, in the approximation of complete frequency redistribution, to the solution of an integral matrix equation of the Wiener-Hopf type for a (2 × 2) matrix source function S(τ). In the second paper in this series, devoted to the vector Milne problem, complete asymptotic expansions of the matrix I(z) [which is essentially a Laplace transform of the matrix S(τ)] for the case of a Doppler profile of the coefficient of absorption, and the coefficients of asymptotic expansions of S(τ) (τ » 1) are expressed in terms of coefficients of the expansions of I(z). We show that asymptotic expansions of S(τ) can be found directly from an integral matrix equation of the Wiener-Hopf type for S(τ). We give new recursive equations for the coefficients of these expansions, as well as a new derivation of asymptotic expansions of the matrix I, including its second column, which was considered only briefly by Ivanov et al.     

Asymptotic expansions for acoustical modes of the homogeneous compressible model

The behavior ofp-modes of high degree and high order in the homogeneous compressilbe model is examined. The second-order differential equation of Pekeris is used to construct asymptotic expansions near the centre and near the surface, which are singular points, and near the turning point of that equation. An equation for the frequencies is obtained by requiring the continuity of the asymptotic solutions and of their first derivatives. Numerical applications are considered.     

Asymptotic expansions in the perturbed two-body problem with application to systems with variable mass

The main theorems of the theory of averaging are formulated for slowly varying standard systems and we show that it is possible to extend the class of perturbation problems where averaging might be used. The application of the averaging method to the perturbed two-body problem is possible but involves many technical difficulties which in the case of the two-body problem with variable mass are avoided by deriving new and more suitable equations for these perturbation problems. Application of the averaging method to these perturbation problems yields asymptotic approximations which are valid on a long time-scale. It is shown by comparison with results obtained earlier that in the case of the two-body problem with slow decrease of mass the averaging method cannot be applied if the initial conditions are nearly parabolic. In studying the two-body problem with quick decrease of mass it is shown that the new formulation of the perturbation problem can be used to obtain matched asymptotic approximations.     

Astrophysical spectroscopy and neutron reactions: Integral transforms and Voigt functions

The Voigt functionsK(x, y) andL(x, y) which play an essential role in astrophysical spectroscopy and neutron physics are investigated and generalized from the viewpoint of integral operators. Unified representations and series expansions involving classical functions of mathematical physics and multivariable hypergeometric functions are established. From the delicate asymptotic analysis of Laplace and Hankel integral transforms we extract complete and rigorous asymptotic expansions of the generalized Voigt functions for large values of the variablesx andy which are of great value in the theory of spectral line profiles.     

On some magnetohydrostatic similarity solutions

It is shown that some recently constructed similarity solutions in force-free configurations belong to a wider class of magnetohydrostatic equilibria, which allows more flexibility in the selection of the parameters of problems. For limiting configurations, solutions are constructed analytically, by means of matched asymptotic expansions, to a second-order approximation.     

Asymptotic orbits in the (<Emphasis Type="Italic">N</Emphasis>+1)-body ring problem

In this paper we study the asymptotic solutions of the (N+1)-body ring planar problem, N of which are finite and ν=N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m ₀=β m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov periodic orbits, which emanate from the collinear equilibrium points L ₁ and L ₂. We construct numerically, from the intersection points of the appropriate Poincaré cuts, homoclinic symmetric asymptotic orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points. All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented.     

Mass spectrum of the system of inhomogeneous intergalactic clouds

The possible influence of the internal structure of inter galactic clouds on their asymptotic mass spectrum n(m, t→ ∞) is investigated by numerically solving the Smoluchovski equation. Allowance for internal structure (inhomogeneity, fractality) leads to a relationship between cloud mass and radius of the type m oc R^k; values of the parameter k from the range [1.5, 3] were used in the calculations. Values of the slope q(k) of the asymptotic mass spectrum are obtained for different sections of the spectrum and different initial conditions n(m, t = 0). It is shown that the slope of the asymptotic spectrum depends strongly on the parameter k, which characterizes the degree of inhomogeneity of the mass distribution in the clouds. A self-consistent estimate of the fractal dimension of intergalactic clouds is obtained, improving the value found earlier. Translated from Astrofizika, Vol. 40, No. 4, pp. 535–544, October-December, 1997.     

First-order resonances in three-degrees-of-freedom systems

Hamiltonian systems with three degrees of freedom which have a resonance-ratio 1∶2∶ω with ω=1, 2, 3, or 4 are studied here. The periodic orbits are determined together with their stability characteristics. Furthermore we shall study the fundamental question of asymptotic integrability of these systems.     

Friedmann's cosmology in Lyra's manifold

A unified treatment of Friedmann-Lemaître-Robertson-Walker models is given within the context of Lyra's manifold, which is a modification of Riemannian geometry, and comparisons are made with corresponding models in the latter geometry. The asymptotic behaviour of the models for small and largeR is discussed.     

Line Formation in a Purely Scattering, Optically Thick Atmosphere

A model problem in the theory of line formation in an optically thick, purely scattering, stellar atmosphere is considered. The integral equation of radiation transfer at line frequencies is solved numerically for a two-level atom in the approximation of complete frequency redistribution in scattering. The numerical results are compared with those calculated from equations of the asymptotic theory. On the basis of the asymptotic theory, the positions of intensity maxima in a line are found for different absorption profiles.     

Escapes and Recurrence in a Simple Hamiltonian System

Many physical systems can be modeled as scattering problems. For example, the motions of stars escaping from a galaxy can be described using a potential with two or more escape routes. Each escape route is crossed by an unstable Lyapunov orbit. The region between the two Lyapunov orbits is where the particle interacts with the system. We study a simple dynamical system with escapes using a suitably selected surface of section. The surface of section is partitioned in different escape regions which are defined by the intersections of the asymptotic manifolds of the Lyapunov orbits with the surface of section. The asymptotic curves of the other unstable periodic orbits form spirals around various escape regions. These manifolds, together with the manifolds of the Lyapunov orbits, govern the transport between different parts of the phase space. We study in detail the form of the asymptotic manifolds of a central unstable periodic orbit, the form of the escape regions and the infinite spirals of the asymptotic manifolds around the escape regions. We compute the escape rate for different values of the energy. In particular, we give the percentage of orbits that escape after a finite number of iterations. In a system with escapes one cannot define a Poincaré recurrence time, because the available phase space is infinite. However, for certain domains inside the lobes of the asymptotic manifolds there is a finite minimum recurrence time. We find the minimum recurrence time as a function of the energy.This revised version was published online in October 2005 with corrections to the Cover Date.     

New self-similar solutions of polytropic gas dynamics

We explore semicomplete self-similar solutions for the polytropic gas dynamics involving self-gravity under spherical symmetry, examine behaviours of the sonic critical curve and present new asymptotic collapse solutions that describe 'quasi-static' asymptotic behaviours at small radii and large times. These new 'quasi-static' solutions with divergent mass density approaching the core can have self-similar oscillations. Earlier known solutions are summarized. Various semicomplete self-similar solutions involving such novel asymptotic solutions are constructed, either with or without a shock. In contexts of stellar core collapse and supernova explosion, a hydrodynamic model of a rebound shock initiated around the stellar degenerate core of a massive progenitor star is presented. With this dynamic model framework, we attempt to relate progenitor stars and the corresponding remnant compact stars: neutron stars, black holes and white dwarfs.     

Chaotic spiral galaxies

We study the role of asymptotic curves in supporting the spiral structure of a N-body model simulating a barred spiral galaxy. Chaotic orbits with initial conditions on the unstable asymptotic manifolds of the main unstable periodic orbits follow the shape of the periodic orbits for an initial interval of time and then they are diffused outwards along the spiral structure of the galaxy. Chaotic orbits having small deviations from the unstable periodic orbits, stay close and along the corresponding unstable asymptotic manifolds, supporting the spiral structure for more than 10 rotations of the bar. Chaotic orbits of different Jacobi constants support different parts of the spiral structure. We also study the diffusion rate of chaotic orbits outwards and find that the orbits that support the outer parts of the galaxy are diffused outwards more slowly than the orbits supporting the inner parts of the spiral structure.     

Non-equidistant spectrum of gravity modes of a solar model with a mixed core

The asymptotic properties of the gravity modes of solar models with a mixed core have been investigated. In this model, the Brunt-Väissälä frequency has a strong enhancement in the region of variable chemical composition at the boundary of the mixed core, giving rise to a non-equidistant spectrum of gravity modes periods. An asymptotic expression for the periods is derived, which relates the main feature of the departure from period equidistance to the stratification of the model. Qualitative agreement with the numerical periods of the model is obtained.     

Homoclinic and heteroclinic orbits in the photogravitational restricted three-body problem

We study numerically the asymptotic homoclinic and heteroclinic orbits around the hyperbolic Lyapunov periodic orbits which emanate from Euler's critical points L ₁ and L ₂, in the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these Lyapunov orbits, are also presented. Poincaré surface of sections of these manifolds on appropriate planes and several homoclinic and heteroclinic orbits for the gravitational case as well as for varying radiation factor q ₁, are displayed. Homoclinic-homoclinic and homoclinic-heteroclinic-homoclinic chains which link the interior with the exterior Hill's regions, are illustrated. We adopt the Sun-Jupiter system and assume that only the larger primary radiates. It is found that for small deviations of its value from the gravitational case (q ₁ = 1), the radiation pressure exerts a significant impact on the Hill's regions and on these asymptotic orbits.     

Stickiness effects in chaos

Stickiness is a temporary confinement of orbits in a particular region of the phase space before they diffuse to a larger region. In a system of 2-degrees of freedom there are two main types of stickiness (a) stickiness around an island of stability, which is surrounded by cantori with small holes, and (b) stickiness close to the unstable asymptotic curves of unstable periodic orbits, that extend to large distances in the chaotic sea. We consider various factors that affect the time scale of stickiness due to cantori. The overall stickiness (stickiness of the second type) is maximum near the unstable asymptotic curves. An important application of stickiness is in the outer spiral arms of strong-barred spiral galaxies. These spiral arms consist mainly of sticky chaotic orbits. Such orbits may escape to large distances, or to infinity, but because of stickiness they support the spiral arms for very long times.