Global stability and the restricted 3-body problem

Global stability regions are found for classi orbits of the circular restricted 3-body problem for primary masses equal and Jacobi constantK>15.5. As this constant decreases, the stability, region shrinks extremely rapidly.     

A new kind of 3-body problem

A new kind of restricted 3-body problem is considered. One body,m ₁, is a rigid spherical shell filled with an homogeneous incompressible fluid of density ₁. The second one,m ₂, is a mass point outside the shell andm ₃ a small solid sphere of density ₃ supposed movinginside the shell and subjected to the attraction ofm ₂ and the buoyancy force due to the fluid ₁. There exists a solution withm ₃ at the center of the shell whilem ₂ describes a Keplerian orbit around it. The linear stability of this configuration is studied assuming the mass ofm ₃ to beinfinitesimal. Explicitly two cases are considered. In the first case, the orbit ofm ₂ aroundm ₁ is circular. In the second case, this orbit is elliptic but the shell is empty (i.e. no fluid inside it) or the densities ₁ and ₃ are equal. In each case, the domain of stability is investigated for the whole range of the parameters characterizing the problem.     

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.     

Effect of electron temperature on small-amplitude electron acoustic solitary waves in non-planar geometry

Effects of electron temperature on the propagation of electron acoustic solitary waves in plasma with stationary ions, cold and superthermal hot electrons is investigated in non-planar geometry employing reductive perturbation method. Modified Korteweg–de Vries equation is derived in the small amplitude approximation limit. The analytical and numerical calculations of the KdV equation reveal that the phase velocity of the electron acoustic waves increases as one goes from planar to non planar geometry. It is shown that the electron temperature ratio changes the width and amplitude of the solitary waves and when electron temperature is not taken into account,our results completely agree with the results of Javidan & Pakzad (2012). It is found that at small values of \(\tau \), solitary wave structures behave differently in cylindrical (\(\text {m} = 1\)), spherical (\(\text {m} = 2\)) and planar geometry (\(\text {m} = 0\)) but looks similar at large values of \(\tau \). These results may be useful to understand the solitary wave characteristics in laboratory and space environments where the plasma have multiple temperature electrons.     

Stability of motion in the Sitnikov 3-body problem

We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m ₁ = m ₂ = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 ≤ m ₃ < 10⁻³, placed initially on the z-axis. We begin by finding for the restricted problem (with m ₃ = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of “islands” of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m ₃ increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m ₃ ≈ 10⁻⁶, the “islands” of bounded motion about the z-axis stability intervals are larger than the ones for m ₃ = 0. Furthermore, as m ₃ increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away “disperse” at larger m ₃ values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries, as observed in the m ₃ = 0 case.     

The restricted 3-body problem with radiation pressure

The restricted 3-body problem is generalised to include the effects of an inverse square distance radiation pressure force on the infinitesimal mass due to the large masses, which are both arbitrarily luminous. A complete solution of the problems of existence and linear stability of the equilibrium points is given for all values of radiation pressures of both liminous bodies, and all values of mass ratios. It is shown that the inner Lagrange point, L₁, can be stable, but only when both large masses are luminous. Four equilibrium points, L₆, L₇, L₈, and L₉ can exist out of the orbital plane when the radiation pressure of the smaller mass is very high. Although L₈ and L₉ are always linearly unstable, L₆ and L₇ are stable for a small range of radiation pressures provided that both large masses are luminous.     

On the planar syzygy solutions of the 3-body problem

Relations between the rectilinear, collinear and syzygy solutions of the N-body problem are first pointed out. It is shown that, along a solution, the set of the non-collinear syzygy configuration instants is formed by isolated points. Then we restrict the study to the planar 3-body problem and prove that for Dirichlet-stable solutions, a non-syzygy solution cannot be as close as possible to a syzygy one. It is also true that, in the case of a syzygy solution, the orbit of one particle crosses the line of the other two and can not be tangent to this line in the transition point. Finally we prove that the set of initial conditions leading to non-collinear syzygy solutions is non-empty and open.     

Periodic orbits of the planar collision restricted 3-body problem

Using the continuation method we prove that the circular and the elliptic symmetric periodic orbits of the planar rotating Kepler problem can be continued into periodic orbits of the planar collision restricted 3-body problem. Additionally, we also continue to this restricted problem the so called “comet orbits”. An erratum to this article can be found at     

Stability of periodic resonance-orbits in the elliptic restricted 3-body problem

Results of families of periodic orbits in the elliptic restricted problem are shown. They are calculated for the mass ratios =0.5 and =0.1 for the primary bodies and for different values of the eccentricity of the orbit of the primaries which is the second parameter. The case =0.5 is also a good model for planetary orbits in binaries. Finally we show detailed stability diagrams and give results according to the stability classification of Contopoulos.     

Tame and chaotic behavior in the planar isosceles 3-body problem

We show that every planar isosceles solution of the three-body problem encounters a collision of the symmetric particles, either forwards or backwards in time. Regularizing analytically this collision, the solution has at least a syzygy configuration and/or leads to a total collapse. Some further simple results support the intuitive image on the tame local behavior of the motion as long as it does not lead to a triple collision. As a main result we prove that total collapse singularities, can be regularized in aC ¹-fashion with respect to time, for all values of the masses. Using symbolic dynamics, the chaotic character of theC ¹-regularized solutions is pointed out.     

On a restricted (2n+3)-body problem

We consider the case of (2n+1) bodies (n0) each of mass m, which are placed on a circle with radius r, such that they form a regular polygon: an equilateral (2n+1)-angle. In the centre of the circle a body of mass b times m is placed, where b is chosen large enough to ensure stability of the system; only gravitational interaction is considered. Each of the bodies rotates uniformly around the centre with angular velocity . In addition to the (2n+2) bodies, considered to be point masses, we have another point mass with negligible mass compared to the former ones; we are then interested in the motion of the small body in the gravitational field of force generated by the large ones, moving themselves in an equilibrium configuration reacting to each other's fields of force but not to the (2n+3)-d body.     

An alternative deduction of the hill-type surfaces of the spatial 3-body problem

In the present paper, inequalities stronger than Sundman's and the best possible zero velocity surfaces of the spatial 3-body problem first obtained by Saari (1987) are deduced using a modified version of the transformation developed by Zare (1976). The notion of inertia ellipsoid is used to show the equivalence of the present authors' result to that of Saari's.     

Constraints on bounded motion and mutual escape for the full 3-body problem

When gravitational aggregates are spun to fission they can undergo complex dynamical evolution, including escape and reconfiguration. Previous work has shown that a simple analysis of the full 2-body problem provides physically relevant insights for whether a fissioned system can lead to escape of the components and the creation of asteroid pairs. In this paper we extend the analysis to the full 3-body problem, utilizing recent advances in the understanding of fission mechanics of these systems. Specifically, we find that the full 3-body problem can eject a body with as much as 0.31 of the total system mass, significantly larger than the 0.17 mass limit previously calculated for the full 2-body problem. This paper derives rigorous limits on a fissioned 3-body system with regards to whether fissioned system components can physically escape from each other and what other stable relative equilibria they could settle in. We explore this question with a narrow focus on the Spherical Full Three Body Problem studied in detail earlier.     

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°.     

Numerical continuation of periodic orbits from the restricted to the general 3-dimensional 3-body problem

In this paper several monoparametric families of periodic orbits of the 3-dimensional general 3-body problem are presented. These families are found by numerical continuation with respect to the small massm ₃, of some periodic orbits which belong to a family of 3-dimensional periodic orbits of the restricted elliptic problem.     

The influence of a resisting medium on 3-dimensional periodic orbits of the restricted 3-body problem

Summary The evolution of the extreme values of the functionsx(t),y(t) andz(t) which are the coordinates of the third bodyP in the barycentric rotating frame of reference, when friction is present, is discussed. These values, which are constant for periodic orbits, change due to the presence of the resisting medium. It is shown that either the orbit tends to become circular and coplanar with the two primaries or to collide with one of the primaries.     

The dynamical decay of unstable 4-body systems

Unstable 4-body systems with negative energy can ultimately decay to (1) a binary plus two single stars, (2) two separate binaries, or (3) a stable triple plus a single star. One hundred random 2-dimensional and one hundred random 3-dimensional 4-body systems have been numerically integrated to determine the statistics of the end products. Of the final stable triples and binaries, 19% were triples, which agrees well with observational estimates of the ratio of triples to binaries. The results were essentially the same for 2- and 3- dimensional systems.     

New proper motions for some AGK3 stars

In a program conducted to isolate AGK3 stars with large proper motions, it has been found that more than one hundred stars seem to be affected by large errors in their published proper motions. For some of those objects south of +25 degrees, new proper motions are being obtained using, as first-epoch positions, the published material of the Astrographic Catalogue. Second-epoch positions are derived from new plates taken with the Yale Southern Observatory double astrograph.