Some special restricted four-body problems—II. From Caledonia to Copenhagen

Special analytical solutions are determined for restricted, coplanar, four-body equal mass problems, including the Caledonian problem, where the masses M_i = M for i = 1,2,3,4. Most of these solutions are shown to reduce to the Lagrange solutions of the Copenhagen problem of three bodies by reducing two of the masses (m_i = m for i = 1,2) in the four-body equal mass problem to zero while maintaining their equality of mass. In so doing, families of special solutions to the four-body problem are shown to exist for any value of the mass ratio μ = m/M.     

Regularization of Motion Equations with L-Transformation and Numerical Integration of the Regular Equations

The sets of L-matrices of the second, fourth and eighth orders are constructed axiomatically. The defining relations are taken from the regularization of motion equations for Keplerian problem. In particular, the Levi-Civita matrix and KS-matrix are L-matrices of second and fourth order, respectively. A theorem on the ranks of L-transformations of different orders is proved. The notion of L-similarity transformation is introduced, certain sets of L-matrices are constructed, and their classification is given. An application of fourth order L-matrices for N-body problem regularization is given. A method of correction for regular coordinates in the Runge–Kutta–Fehlberg integration method for regular motion equations of a perturbed two-body problem is suggested. Comparison is given for the results of numerical integration in the problem of defining the orbit of a satellite, with and without the above correction method. The comparison is carried out with respect to the number of calls to the subroutine evaluating the perturbational accelerations vector. The results of integration using the correction turn out to be in a favorable position.     

Stability of the photogravitational restricted three-body problem with variable masses

This paper investigates the stability of equilibrium points in the restricted three-body problem, in which the masses of the luminous primaries vary isotropically in accordance with the unified Meshcherskii law, and their motion takes place within the framework of the Gylden–Meshcherskii problem. For the autonomized system, it is found that collinear and coplanar points are unstable, while the triangular points are conditionally stable. It is also observed that, in the triangular case, the presence of a constant κ, of a particular integral of the Gylden–Meshcherskii problem, makes the destabilizing tendency of the radiation pressures strong. The stability of equilibrium points varying with time is tested using the Lyapunov Characteristic Numbers (LCN). It is seen that the range of stability or instability depends on the parameter κ. The motion around the equilibrium points L _i (i=1,2,…,7) for the restricted three-body problem with variable masses is in general unstable.     

Magnetic Monopoles in the Early Universe: Pair Production

Magnetic monopoles and antimonopoles with masses M=10¹⁶ Gev and charges q=68.5e in the early universe are considered. Pair production may occur as a result of their Coulomb interaction. Some conditions for formation of such pairs are discussed. In particular, numerical simulations of three particle collisions are carried out. Probabilities for pair production are found in terms of the N-body problem.     

Numerical integration of periodic orbits in the main problem of artificial satellite theory

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ ₂ problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ ₂ andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ ₂, as large as |J ₂|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.     

Two families of simply symmetric orbits in E3

Numerical processes are applied to the classical formulation of the restricted problem of three bodies in order to compute families of periodic orbits that exist inE ³ and are symmetric with respect to a plane that contains the two primaries and their axis of angular velocity. Two such families are found and their stability numbers are computed. In general, the discovery of such families proves to be extremely difficult.     

Numerical exploration of the photogravitational restricted five-body problem

We study numerically the restricted five-body problem when some or all the primary bodies are sources of radiation. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points are given. We found that the number of the collinear equilibrium points of the problem depends on the mass parameter β and the radiation factors q _i , i=0,…,3. The stability of the equilibrium points are also studied. Critical masses associated with the number of the equilibrium points and their stability are given. The network of the families of simple symmetric periodic orbits, vertical critical periodic solutions and the corresponding bifurcation three-dimensional families when the mass parameter β and the radiation factors q _i vary are illustrated. Series, with respect to the mass (and to the radiation) parameter, of critical periodic orbits are calculated.     

Invariant sets and polhodes in the rigid body problem

In this work we will describe the sets in the rigid body phase space where the energy and angular momentum are constant, and it will turn out that in nontrivial cases they will simply take the form of cartesian products of the polhodes byS ¹. These sets are important for the global study of said geodesic mechanical system for being invariant under Euler's equations (energy and momentum are constant along their solutions).To motivate from something more familiar in celestial mechanics, we will begin to relate the problem to Smale's study of the planarn-body problem (Smale, 1970) and Easton's study of the planar 3-body problem (Easton, 1971), exemplifying in particular with the central force problem.In the last Sections 4 and 5, we extent our methods to give results for generalized solids on Lie groups, mentioning the further extensions to transitive mechanical systems.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.This work was partially supported by the Consejo Nacional de Ciencia y Tecnología (México) under grant PNCB-049.     

Determination of Libration Characteristics of Asteroids near Mean Motion Commensurabilities 1/1, 4/3, 3/2, 2/1, 7/3, 5/2, and 3/1

The possibility of using a generalized perfect resonance for the study of libration motions of asteroids near the (p+ q)/p-type commensurabilities of the mean motions of asteroids and Jupiter is considered. Based on the equations of the planar circular restricted three-body problem, the libration-motion equations are derived and their solutions for the intermediate Hamiltonian, as well as a solution taking into account perturbations of the order O(m ^3/2), are determined.     

Effects of resonances on the stability of retrograde satellites

The stability evolution of family f of the planar circular restricted three-body problem in the Earth–Moon case is explored numerically using the Poincaré surface of section. It is shown that third order resonances are the main cause of the reduction of the stability region of retrograde satellites. Several branches of family f are also computed and these are seen by the configuration of their family characteristics to roughly determine the stability region. Previous results on smaller mass ratios of primaries are thus extended to the Earth–Moon system.     

Computation of elliptic Hansen coefficients and their derivatives

The problem of computation of elliptic Hansen coefficients and their derivatives is considered for constructing a motion theory of an artificial Earth satellite with large eccentricity. An algorithm for analytical and numerical computation of these coefficients and their derivatives is described. The recurrence relations for derivatives of the first and second order and initial values for recurrences are obtained. As an example, numerical values of some elliptic Hansen coefficients are given for the orbit with eccentricityk=0.74.     

Non-schubart periodic orbits in the rectilinear three-body problem

In the present paper, in the rectilinear three-body problem, we qualitatively follow the positions of non-Schubart periodic orbits as the mass parameter changes. This is done by constructing their characteristic curves. In order to construct characteristic curves, we assume a set of properties on the shape of areas corresponding to symbol sequences. These properties are assured by our preceding numerical calculations. The main result is that characteristic curves always start at triple collision and end at triple collision. This may give us some insight into the nature of periodic orbits in the N-body problem.     

THE HILL PROBLEM WITH OBLATE SECONDARY: NUMERICAL EXPLORATION

We introduce a three-dimensional version of Hill’s problem with oblate secondary, determine its equilibrium points and their stability and explore numerically its network of families of simple periodic orbits in the plane, paying special attention to the evolution of this network for increasing oblateness of the secondary. We obtain some interesting results that differentiate this from the classical problem. Among these is the eventual disappearance of the basic family g′ of the classical Hill problem and the existence of out-of-plane equilibrium points and a family of simple-periodic plane orbits non-symmetric with respect to the x-axis.     

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.     

Particle Dynamics in a Maxwell’s Ring-Type Configuration with a Radiating Central Primary

Maxwell’s ring-type configuration (i.e. an N-body model where the ν = Ν − 1 bodies have equal masses and are located at the vertices of a regular ν-gon while the N-th body with a different mass is located at the center of mass of the system) has attracted special attention during the last 15 years and many aspects of it have been studied by considering Newtonian and post-Newtonian potentials (Mioc and Stavinschi 1998, 1999), homographic solutions (Arribas et al. 2007) and relative equilibrium solutions (Elmabsout 1996), etc. An equally interesting problem, known as the ring problem of (N + 1) bodies, deals with the dynamics of a small body in the combined force field produced by such a configuration. This is the problem we are dealing with in the present paper and our aim is to investigate the variations in the dynamics of the small body in the case that the central primary is also a radiating source and therefore acts on the particle with both gravitation and radiation. Based on the general outlines of Radzievskii’s model, we study the permitted and the existing trapping regions of the particle, its equilibrium locations and their parametric variations as well as the existence of focal points in the zero-velocity diagrams. The distribution of the characteristic curves of families of planar symmetric periodic orbits and their stability for various values of the radiation coefficient of the central body is additionally investigated.     

Scattering of an Electromagnetic Wave on a Dielectric Layer Bounded on Two Sides by Two Different Homogeneous, Semi-Infinite Media

The problem of the passage of a plane electromagnetic wave through an arbitrary, inhomogeneous dielectric layer bounded on two sides by two different homogeneous, semi-infinite media is considered. Algebraic relations are obtained between the amplitudes of transmission and reflection (the scattering amplitudes) for the problem under consideration and the wave scattering amplitudes when the layer is bounded on both sides by a vacuum. It is shown that for s and p polarized fields the scattering problem (a boundary-value problem) can be formulated as a Cauchy problem directly for the s and p wave equations. It is also shown that the problem of finding the field inside the layer also reduces to a Cauchy problem in the general case.     

Motions in the field of two rotating magnetic dipoles. II. Stability of the equilibrium points

The stability of the equilibrium points found to exist (cf. Goudaset al., 1985, referred to henceforth as Paper I) in the problem of two parallel, or antiparallel, magnetic dipoles that rotate about the centre of mass of their carrier stars, is studied by computing the characteristic roots of their variational equations. The characteristic equation, a biquadratic, solved for many combinations of and showed that all equilibrium points of this problem are unstable.     

The selection of filters for reduction of optical contamination in astronomical CCD X-ray images

The European Photon Imaging Camera(EPIC), is the X-ray imaging and medium spectroscopy instrument for theESA X-ray Multi Mirror telescope(XMM) mission. TheCCD detectors to be used in the three focal plane cameras will provide images in the energy band from 0.1 to 10 keV. However, spectral studies may be compromised by low energy, optical photon contamination. In order to reduce this effect, a number of filters will be incorporated onto a rotating mechanism in the camera head. The filters will be chosen to provide a significant reduction in the optical contamination from a source whilst minimising the attenuation of the X-ray flux. Four commercial filters are described here and their effects on calculated typical source fluxes evaluated. In addition, two alternative filter designs are described and their effects on a simulated source spectra are debated. In both cases, particular attention is given to the problem of maintaining high sensitivity at soft X-ray energies (less than 2 keV).     

On the triangular libration points in photogravitational restricted three-body problem with variable mass

This paper investigates the triangular libration points in the photogravitational restricted three-body problem of variable mass, in which both the attracting bodies are radiating as well and the infinitesimal body vary its mass with time according to Jeans’ law. Firstly, applying the space-time transformation of Meshcherskii in the special case when q=1/2, k=0, n=1, the differential equations of motion of the problem are given. Secondly, in analogy to corresponding problem with constant mass, the positions of analogous triangular libration points are obtained, and the fact that these triangular libration points cease to be classical ones when α≠0, but turn to classical L ₄ and L ₅ naturally when α=0 is pointed out. Lastly, introducing the space-time inverse transformation of Meshcherskii, the linear stability of triangular libration points is tested when α>0. It is seen that the motion around the triangular libration points become unstable in general when the problem with constant mass evolves into the problem with decreasing mass.     

A study of collision orbits with the Moon by the method of surface of section

Non-periodic orbits of a natural satellite of the Moon are studied, for the case of the circular three-body problem with the method of surface of section. According to this method, each orbit is represented by a point, in the plane x0\.x, which corresponds to y = 0 and \.y > 0 and a fixed energy. Conclusions are deduced from the shape of this curve for probable collisions of the satellite on the lunar surface. This method of surface of section can be used for the study of orbits which collide with the Moon's surface after a large number of revolutions around the Moon and their study would be difficult to explore with other methods.