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.     

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.     

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.     

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.     

Periodic elliptic motions in a planar restricted (N+1)-body problem

LetN2 mass points (primaries) move on a collinear solution of relative equilibrium of theN-body problem; i.e. suitably fixed on a uniformly rotating straight line. Consider the motion of a massless particle in the gravitational field of these primaries with arbitrarily given masses. An existence proof for periodic solutions (i.e. closed trajectories in a rotating coordinate system) will be given, in which the particle performs nearly keplerian elliptic motions about (and close to) any one of the primaries.     

Stability VS inclination of motions inE 3

Five families of three-dimensional doubly symmetric motions are computed after establishing their existence by means of a grid-search technique. It is confirmed that within the same family orbits of lower inclination with respect to the plane of motion of the primaries are stable while the critical inclination at which instability occurs varies between families. The maximum inclination at which stable motions of the type presented here were found is about 52°.     

Periodic motions in the spatial Chermnykh restricted three-body problem

Three-dimensional motions in the Chermnykh restricted three-body problem are studied. Specifically, families of three-dimensional periodic orbits are determined through bifurcations of the family of straight-line periodic oscillations of the problem which exists for equal masses of the primaries. These rectilinear oscillations are perpendicular to the plane of the primaries and give rise to an infinite number of families consisting entirely of periodic orbits which belong to the three-dimensional space except their respective one-dimensional bifurcations as well as their planar terminations. Many of the computed branch families are continued in all mass range that they exist.     

On the non-linear stability of motions around L5 in the elliptic restricted problem of the three bodies

The non-linear stability of motions around L₅ in the elliptic restricted problem of the three bodies is investigated numerically with emphasis on the effect of the orbital eccentricity of the primaries on the shape of the established stability regions. It is shown that with increasing eccentricity, the width of these regions is decreasing.     

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

The existence of new equilibrium points is established in the restricted three-body problem with equal prolate primaries. These are located on the Z-axis above and below the inner Eulerian equilibrium point L ₁ and give rise to a new type of straight-line periodic oscillations, different from the well known Sitnikov motions. Using the stability properties of these oscillations, bifurcation points are found at which new types of families of 3D periodic orbits branch out of the Z-axis consisting of orbits located entirely above or below the orbital plane of the primaries. Several of the bifurcating families are continued numerically and typical member orbits are illustrated.     

Spatial restricted rhomboidal five-body problem and horizontal stability of its periodic solutions

The spatial restricted rhomboidal five-body problem, or shortly, SRRFBP, is a five body problem in which four positive masses, called the primaries, move two by two in coplanar circular motions with the center of mass fixed at the origin such that their configuration is always a rhombus, the fifth mass being negligible and not influencing the motion of the four primaries. The Hamiltonian function that governs the motion of the fifth mass is derived and has three degrees of freedom depending periodically on time. Using a synodical system of coordinates, we fix the primaries in order to eliminate the time dependence. With the help of the Hamiltonian structure, we characterize the regions of possible motion. The vertical $z$ axis is invariant and we study what we call the rhomboidal Sitnikov problem. Unlike the classical Sitnikov problem, no chaos exists and the behavior of the fifth mass is quite predictable, periodic solutions of arbitrary long periods are shown to exist and we study numerically their linear horizontal stability.     

Elliptic Sitnikov five-body problem under radiation pressure

We consider the square configuration of photo-gravitational elliptic restricted five-body problem and study the Sitnikov motions. The four radiating primaries are of equal mass placed at the vertices of square and the fifth body having negligible mass performs oscillations along a straight line perpendicular to the orbital plane of the primaries. The motion of the fifth body is called vertical periodic motion and the main aim of this paper is to study the effect of radiation pressure on these periodic motions in the linear approximation. Moreover, the effects of radiation pressure on the motion of fifth body have been examined with the help of Poincare surfaces of section. By escalating the radiation pressure, surrounding periodic tubes and islands disappear and chaotic motion occurs near the hyperbolic points. Further, by escalating the radiation pressure, the main stochastic region joins the escaping one.     

Parametric evolution of periodic orbits in the restricted four-body problem with radiation pressure

The backbone of the analysis in most dynamical systems is the study of periodic motions, since they greatly assist us to understand the structure of all possible motions. In this paper, we deal with the photogravitational version of the rectilinear restricted four-body problem and we investigate the dynamical behaviour of a small particle that is subjected to both the gravitational attraction and the radiation pressure of three bodies much bigger than the particle, the primaries. These bodies are always in syzygy and two of them have equal masses and are located at equal distances from the third primary. We study the effect of radiation on the distribution of the periodic orbits, their stability, as well as the evolution of the families and their main features.     

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.     

Region of motion in the Sitnikov four-body problem when the fourth mass is finite

This article deals with the region of motion in the Sitnikov four-body problem where three bodies (called primaries) of equal masses fixed at the vertices of an equilateral triangle. Fourth mass which is finite confined to moves only along a line perpendicular to the instantaneous plane of the motions of the primaries. Contrary to the Sitnikov problem with one massless body the primaries are moving in non-Keplerian orbits about their centre of mass. It is investigated that for very small range of energy h the motion is possible only in small region of phase space. Condition of bounded motions has been derived. We have explored the structure of phase space with the help of properly chosen surfaces of section. Poincarè surfaces of section for the energy range ?0.480≤h≤?0.345 have been computed. We have chosen the plane (q ₁,p ₁) as surface of section, with q ₁ is the distance of a primary from the centre of mass. We plot the respective points when the fourth body crosses the plane q ₂=0. For low energy the central fixed point is stable but for higher value of energy splits in to an unstable and two stable fixed points. The central unstable fixed point once again splits for higher energy into a stable and three unstable fixed points. It is found that at h=?0.345 the whole phase space is filled with chaotic orbits.     

The Sitnikov family and the associated families of 3D periodic orbits in the photogravitational RTBP with oblateness

We consider the photogravitational restricted three-body problem with oblateness and study the Sitnikov motions. The family of straight line oscillations exists only in the case where the primaries are of equal masses as in the classical Sitnikov problem and have the same oblateness coefficients and radiation factors. A perturbation method based on Floquet theory is applied in order to study the stability of the motion and critical orbits are determined numerically at which families of three-dimensional periodic orbits of the same or double period bifurcate. Many of these families are computed.     

Numerical results to the Sitnikov-problem

We present numerical results of the so-called Sitnikov-problem, a special case of the three-dimensional elliptic restricted three-body problem. Here the two primaries have equal masses and the third body moves perpendicular to the plane of the primaries' orbit through their barycenter. The circular problem is integrable through elliptic integrals; the elliptic case offers a surprisingly great variety of motions which are until now not very well known. Very interesting work was done by J. Moser in connection with the original Sitnikov-paper itself, but the results are only valid for special types of orbits. As the perturbation approach needs to have small parameters in the system we took in our experiments as initial conditions for the work moderate eccentricities for the primaries' orbit (0.33e _primaries 0.66) and also a range of initial conditions for the distance of the 3 ^rd body (= the planet) from very close to the primaries orbital plane of motion up to distance 2 times the semi-major axes of their orbit. To visualize the complexity of motions we present some special orbits and show also the development of Poincaré surfaces of section with the eccentricity as a parameter. Finally a table shows the structure of phase space for these moderately chosen eccentricities.     

Secondary resonances of co-orbital motions

The size distribution of the stability region around the Lagrangian point L ₄ is investigated in the elliptic restricted three-body problem as the function of the mass parameter and the orbital eccentricity of the primaries. It is shown that there are minimum zones in the size distribution of the stability regions, and these zones are connected with the secondary resonances between the frequencies of librational motions around L ₄. The results can be applied to hypothetical Trojan planets for predicting values of the mass parameter and the eccentricity for which such objects can be expected or their existence is less probable.     

Periodic motions around a collinear equilibrium solution of the general three-body problem

The collinear equilibrium position of the circular restricted problem with the two primaries at unit distance and the massless body at the pointL ₃ is extended to the planar three-body problem with respect to the massm ₃ of the third body; the mass ratio μ of the two primaries is considered constant and the constant angular velocity of the straight line on which the three masses stay at rest is taken equal to 1. As regards periodic motions ‘around’ the equilibrium pointL ₃, four possible extensions from the restricted to the general problem are presented each of them starting with a simple or a doubly periodic orbit of the family α of the Copenhagen category (μ=0.50). Form ₃=0.10, μ=0.50 (i.e. for fixed masses of all three bodies) the characteristic curve of the extended family α is found. The qualitative differences of the families corresponding tom ₃=0 andm ₃=0.10 are discussed.     

Periodic solutions of the third sort for restricted problem of three bodies and their stability

In this paper periodic solutions of the third sort for restricted problem of three bodies in the three-dimensional space are derived numerically by starting from generating solutions obtained by one of the authors (1969) and by increasing the mass-ratio of the two primaries stepwise from zero to about 1000 for 21, 32 and 61 cases of commensurable mean motions. Periodic solutions both for circular and elliptic orbits of the primaries are obtained.The stability of the periodic solutions for the 21 circular case is discussed and it is found that none of them is linearly stable.     

Application of a Fictitious Attracting Center to the Study of the Motion of Minor Bodies Approaching Major Planets

The motion of minor Solar System bodies having close encounters with major planets is described using the model of motion within the framework of the perturbed restricted three-body problem. The actual motion of a minor body is represented as a combination of two motions, namely, the motion of a fictitious attracting center with a variable mass and the motion with respect to the fictitious center. The position and mass of the fictitious center are chosen so that, when the minor body collides with any of the primaries, the fictitious center carries into the center of inertia of the colliding body and the mass of the fictitious center becomes identical to the mass of this body. The regularizing KS-transformation and Sundman’s time transformation were applied to coordinates and velocities. As a result, a system of differential equations of motion that are quasilinear within the nearest vicinity of each of the primary attracting bodies was obtained. These equations are characterized by a numerical behavior during the encounters of the minor body with the primaries that is essentially better than that of the initial equations of motion. The motion of comets Brooks 2 and Gehrels 3, which have fairly close encounters with Jupiter, is simulated.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 3, 2005, pp. 272–280.Original Russian Text Copyright © 2005 by Shefer.