The three-dimensional equilibria of magnetic-binary systems with oblate primaries

In this paper the three-dimensional equilibria in a Magnetic-Binary system with oblate primaries are studied. It is also examined how the primaries oblateness affect the equilibria configuration of the spherical case.     

A two-parameter survey of periodic orbits in the restricted problem of three bodies

Within the context of the restricted problem of three bodies, we wish to show the effects, caused by varying the mass ratio of the primaries and the eccentricity of their orbits, upon periodic orbits of the infinitesimal mass that are numerical continuations of circular orbits in the ordinary problem of two bodies. A recursive-power-series technique is used to integrate numerically the equations of motion as well as the first variational equations to generate a two-parameter family of periodic orbits and to identify the linear stability characteristics thereof. Seven such families (comprised of a total of more than 2000 orbits) with equally spaced mass ratios from 0.0 to 1.0 and eccentricities of the orbits of the primaries in a range 0.0 to 0.6 are investigated. Stable orbits are associated with large distances of the infinitesimal mass from the perturbing primary, with nearly circular motion of the primaries, and, to a slightly lesser extent, with small mass ratios of the primaries.Conversely, unstable orbits for the infinitesimal mass are associated with small distances from the perturbing primary, with highly elliptic orbits of the primaries, and with large mass ratios.     

Generalizations of the Jacobian integral

The restricted problem of three bodies is generalized to the restricted problem of 2+n bodies. Instead of one body of small mass and two primaries, the system is modified so that there are several gravitationally interacting bodies with small masses. Their motions are influenced by the primaries but they do not influence the motions of the primaries. Several variations of the classical problem are discussed. The separate Jacobian integrals of the minor bodies are lost but a conservative (time-independent) Hamiltonian of the system is obtained. For the case of two minor bodies, the five Lagrangian points of the classical problem are generalized and fourteen equilibrium solutions are established. The four linearly stable equilibrium solutions which are the generalizations of the triangular Lagrangian points are once again stable but only for considerably smaller values of the mass parameter of the primaries than in the classical problem.     

Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory

This paper deals with the stability analysis of the triangular equilibrium points for the generalized problem of the photogravitational restricted three body where both the primaries are radiating. The problem is generalized in the sense that the eccentricity of the orbits and the oblateness due to both the primaries and infinitesimal are considered. The stability in the case of linear resonance are analyzed based on the Floquet’s theory for finding the characteristic exponent for a system containing periodic coefficients. It was found that the critical value of μ for the stability boundary for parametric excitation is dependent on the oblateness of the primaries as well as infinitesimal.     

Stability and bifurcations of Sitnikov motions

A family of straight line periodic motions, known as the Sitnikov motions and existing in the case of equal primaries of the three body problem, is studied with respect to stability and bifurcations. Continuation of the bifurcations into the case of unequal primaries is also discussed and some of the bifurcating families of three-dimensional periodic motions are computed.     

The libration points of axisymmetric satellite in the gravitation field of two triaxial rigid bodies

In this paper we consider the restricted problem of three rigid bodies (an axisymmetric satellite in the gravitation field of two triaxial primaries). The collinear and triangular equilibrium solutions are obtained. The effect of the primaries on the location of the libration points of a spherical satellite has been studied numerically.     

Parametric dependence of the stationary solutions in the restricted 2 + 2 body problem

The restricted gravitational 2 + 2 body problem, is a particular case of the N body problem and it may be used to approximate the dynamical behaviour of binary asteroids or dual sattelites moving in the gravitational field of two primaries Pi, i = 1,2. By considering oblate primaries, five parameters are needed to describe the model, namely the reduced mass μ of the primary P2, the reduced masses μ1 and μ2 of the minor bodies and the oblatenesses Ii, i = 1,2 of the primaries. This work deals with the effect of those parameters on the location of the stationary solutions. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the restricted three-body problem with generalized forces

By introducing general functions which depend on distance, a general scheme which determines the equilibrium solutions for the generalized restricted three-body problem is given. Applications to problems such as primaries considered as rigid bodies, influence of the radiation pressure of the primaries, and a combination of radiation pressure and rigid body are presented.     

On the circular Sitnikov problem: the alternation of stability and instability in the family of vertical motions

This paper is devoted to the special case of the restricted circular three-body problem, when the two primaries are of equal mass, while the third body of negligible mass performs oscillations along a straight line perpendicular to the plane of the primaries (so called periodic vertical motions). The main goal of the paper is to study the stability of these periodic motions in the linear approximation. A special attention is given to the alternation of stability and instability within the family of periodic vertical motions, whenever their amplitude is varied in a continuous monotone manner.     

Equilibrium points and stability under effect of radiation and perturbing forces in the restricted problem of three oblate bodies

This paper presents a generalized problem of the restricted three body studied in Abdul Raheem and Singh with the inclusion that the third body is an oblate spheroidal test particle of infinitesimally mass. The positions and stability of the equilibrium point of this problem is studied for a model in which the primaries is the binary system Struve 2398 (Gliese 725) in the constellation Draco; which consist of a pair of radiating oblate stars. It is seen that additional equilibrium points exist on the line collinear with the primaries, for some combined parameters of the problem. Hence, there can be up to five collinear equilibrium points. Two triangular points exist and depends on the oblateness of the participating bodies, radiation pressure of the primaries and a small perturbation in the centrifugal force. The stability analysis ensures that, the collinear equilibrium points are unstable in the linear sense while the triangular points are stable under certain conditions. Illustrative numerical exploration is given to indicate significant improvement of the problem in Abdul Raheem and Singh.     

Effect of perturbed potentials on the stability of libration points in the restricted problem

The location and the stability in the linear sense of the libration points in the restricted problem have been studied when there are perturbations in the potentials between the bodies. It is seen that if the perturbing functions satisfy certain conditions, there are five libration points, two triangular and three collinear. It is further observed that the collinear points are unstable and for the triangular points, the range of stability increases or decreases depending upon whetherP> or <0 wherep depends upon the perturbing functions. The theory is verified in the following four cases:

1. There are no perturbations in the potentials (classical problem).
2. Only the bigger primary is an oblate spheroid whose axis of symmetry is perpendicular to the plane of relative motion (circular) of the primaries.
3. Both the primaries are oblate spheroids whose axes of symmetry are perpendicular to the plane of relative motion (circular) of the primaries.
4. The primaries are spherical in shape and the bigger is a source of radiation.

     

Evolution of a ‘Class Two’ family of periodic orbits in the general planar problem of three bodies

The elliptic restricted problem of three bodies with unit eccentricity of the primaries is used to generate a family of periodic orbits in the general problem of three bodies. The parameter of the family is the mass of one of the participating bodies. This varies from zero to a termination value. The mass ratio of the primaries of the unperturbed problem (three to five) is maintained throughout the generation of the family. In this way an asymmetry is introduced generalizing the Copenhagen elliptic problem as the generating model. All members of the family experience a close approach and a collision between the primaries during half of the period of the orbit, therefore, the family is classified as Class Two.     

Sitnikov five-body problem with combined effects of radiation pressure and oblateness

We study the motion of negligible mass in the frame work of Sitnikov five-body problem where four equal oblate spheroids known as primaries symmetrical in all respect are placed at the vertices of square. These primaries are also considered as source of radiations moving in a circular orbit around their common center of mass. The fifth negligible mass performs oscillations along z-axis which is perpendicular to the orbital plane of motion of the primaries and passes through the center of mass of the primaries. Under the combined effects of radiation pressure and oblateness, we have developed the series solution by the Lindstedt-Poincare technique and established averaged Hamiltonians by applying the Van der Pol transformation and averaging technique of Guckenheimer and Holmes (1983). The orbits such as regular, periodic, quasi-periodic, chaotic, or stochastic have been examined with the help of Poincare surfaces of section.     

Particle motions around,between or outside the two dipoles of a magnetic binary system

In this paper we study the simple symmetric motions of a charged particle around, between or outside the two primaries of a magnetic-binary system. We also attempt to reveal the influence of the oblateness of the primaries on the evolution of the families of these motions and on their orbital characteristics.     

Numerical experiments on planetary orbits in double stars

This is a numerical study of orbits in the elliptic restricted three-body problem concerning the dependence of the critical orbits on the eccentricity of the primaries. They are defined as being the separatrix between stable and unstable single periodic orbits. As our results are adapted to the existence of planetary orbits in double stars we concentrated first on the P-orbits (defined to surround both primaries). Due to the complexity of the elliptic problem there is no analytical approach possible. Using the results of some 300 integrated orbits for 10³ to 3. 10³ periods of the primaries we established lower and upper bounds for the critical orbits for different values of the eccentricity.     

Planar motions of a charged particle in a magnetic-binary system with spherical or oblate primaries

In this paper we present for the first time simple symmetric motions in the planar magnetic-binary problem where both primaries are spherical bodies or oblate spheroids. From the study of this case it follows that there is a dense and complicated distribution of the families of such motions in the phase space. Our results also show that the orbital characteristics of the particle and the configuration of the phase space are appreciably affected by the oblateness of the primaries only if these parameters become sufficiently large.     

Can intergalactic suprathermal grains produce extensive air showers?

The events following the impact of intergalactic suprathermal grains with atmosphere are examined, and some similarity is found between the expected air shower and observations of largest cosmic ray showers. It is concluded that the largest air showers are, in any case, initiated by primaries of intergalactic origin. Whether the primaries are suprathermal dust grains or single nuclei is inconclusive.     

Periodic orbits and bifurcations in the Sitnikov four-body problem when all primaries are oblate

We study the motions of an infinitesimal mass in the Sitnikov four-body problem in which three equal oblate spheroids (called primaries) symmetrical in all respect, are placed at the vertices of an equilateral triangle. These primaries are moving in circular orbits around their common center of mass. The fourth infinitesimal mass is moving along a line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries. A relation between the oblateness-parameter ‘A’ and the increased sides ‘ε’ of the equilateral triangle during the motion is established. We confine our attention to one particular value of oblateness-parameter A=0.003. Only one stability region and 12 critical periodic orbits are found from which new three-dimensional families of symmetric periodic orbits bifurcate. 3-D families of symmetric periodic orbits, bifurcating from the 12 corresponding critical periodic orbits are determined. For A=0.005, observation shows that the stability region is wider than for A=0.003.     

Motions in the field of two rotating magnetic dipoles I: Equilibrium points

The equilibrium points of charged particles moving under the Lorentz forces of two parallel or anti-parallel magnetic dipoles located at the two primaries of the Restricted Three-Body Problem are calculated. The magnetic moments of the dipoles are taken perpendicular to the plane of motion of the primaries. The configuration studied simulates the case of close-binary stars with dominant zero-order harmonics in the magnetic field of each star. Depending on the mass parameter and the magnetic moment ratio of the two dipoles, it is found that there exist from one to nine equilibrium points in the plane of motion of the primaries. On the synodical axis of the primaries, depending on the above two ratios, there exist one, two three, or five equilibrium points, while off this axis the number of equilibrium points is zero, two or four.     

A Search for Pulsating, Mass-Accreting Components in Algol-Type Eclipsing Binaries

We present a status report on the search for pulsations in primary componants of Algols systems (oEA stars). Analysis of 21 systems with A0-F2 spectral type primaries revealed pulsations in two systems suggesting that of the order of ten persent of Algols primaries in this range are actually pulsators.