The rectilinear three-body problem using symbol sequence I. Role of triple collision

The dynamical structure of phase space of gravitational Newtonian three bodies which lie on a line (rectilinear three-body system) is studied. We take an initial value plane and classify the points on the plane according to the fate of the orbits starting from the points, using symbol sequences. The structure appearing on the initial value plane with this classification was well studied for the equal-mass case (Tanikawa and Mikkola 2000, Chaos 10, 649–657). In this paper, we follow and clarify the changes of this structure with the mass ratio of three particles.     

Strong triple interactions in the general three-body problem

Strong three-body interactions play a decisive role in the course of the dynamical evolution of triple systems having positive as well as negative total energies. These interactions may produce qualitative changes in the relative motions of the components. In triple systems with positive or zero total energies the processes of formation, disruption or exchange of the components of binaries take place as the result of close approaches of the three single bodies or as the result of the passages of single bodies past wide or hard binaries. In the triple systems with negative energies, the strong triple interactions may result in an escape from the system as well as a formation of a hard final binary. This paper summarizes the general results of the studies of the strong three-body interactions in the triple systems with positive and negative energies. These studies were conducted at the Leningrad University Observatory by computer simulations during 1968–1989.     

Close triple approaches and escape in the three-body problem

We have studied a total of 5000 close triple approaches resulting in escape, for equal-mass systems with zero initial velocities. Escape is shown to take place in the majority of the cases after a fly-by close triple approach when the escaper passes near the centre of mass along an almost straight-line orbit. A number of configurational and kinematical parameters are introduced in order to characterize the triple approach. The distributions of these parameters are investigated. A comparison with 831 examples in the vicinity of the so-called Pythagorean problem is carried out. We find that the general features of close triple approaches which result in escape are the same for both types of systems.     

Ejection and collision orbits of the spatial restricted three-body problem

We begin by describing the global flow of the spatial two body rotating problem, =0. The remainder of the work is devoted to study the ejection and collision orbits when >-0. We make use of the blow up techniques to show that for any fixed value of the Jacobian constant the set of these orbits is diffeomorphic to S²×R. Also we find some particular collision-ejection orbits.     

Families of periodic collision orbits in the general three-body problem

In the general three-body problem, in a rotating frame of reference, a symmetric periodic solution with a binary collision is determined by the abscissa of one body and the energy of the system. For different values of the masses of the three bodies, the symmetric periodic collision orbits form a two-parametric family. In the case of equal masses of the two bodies and small mass of the third body, we found several symmetric periodic collision orbits similar to the corresponding orbits in the restricted three-body problem. Starting with one symmetric periodic collision orbit we obtained two families of such orbits. Also starting with one collision orbit in the Sun-Jupiter-Saturn system we obtained, for a constant value of the mass ratio of two bodies, a family of symmetric periodic collision orbits.     

The three-body problem

Book review

The three-body problemby Christian Marchal, Elsevier, Amsterdam, 1990, (ISBN 0444874402)     

The stellar three-body problem

Two applications of von Zeipel's method to the stellar three-body problem eliminate the short period terms and establish two new integrals of the motion beyond the classical integrals. The remaining time averaged problem with only the second order Hamiltonian has one additional integral and can be solved. The motion with the third order averaged Hamiltonian included is more complex, in that there may be additional resonances, and the additional integral does not exist in all cases.     

Binary and triple collisions causing instability in the free-fall three-body problem

Dominant factors for escape after the first triple-encounter are searched for in the three-body problem with zero initial velocities and equal masses. By a global numerical survey on the whole initial-value space, it is found that not only a triple-collision orbit but also a particular family of binary-collision orbits exist in the set of escape orbits. This observation is justified from various viewpoints. Binary-collision orbits experiencing close triple-encounter turn out to be close to isosceles orbits after the encounter and hence lead to escape. Except for a few cases, binary-collision orbits of near-isosceles slingshot also escape. This revised version was published online in July 2006 with corrections to the Cover Date.     

The rectilinear three-body problem

The rectilinear equal-mass and unequal-mass three-body problems are considered. The first part of the paper is a review that covers the following items: regularization of the equations of motion, integrable cases, triple collisions and their vicinities, escapes, periodic orbits and their stability, chaos and regularity of motions. The second part contains the results of our numerical simulations in this problem. A classification of orbits in correspondence with the following evolution scenarios is suggested: ejections, escapes, conditional escapes (long ejections), periodic orbits, quasi-stable long-lived systems in the vicinity of stable periodic orbits, and triple collisions. Homothetic solutions ending by triple collisions and their dependence on initial parameters are found. We study how the ejection length changes in response to the variation of the triple approach parameters. Regions of initial conditions are outlined in which escapes occur after a definite number of triple approaches or a definite time. In the vicinity of a stable Schubart periodic orbit, we reveal a region of initial parameters that corresponds to trajectories with finite motions. The regular and chaotic structure of the manifold of orbits is mostly defined by this periodic orbit. We have studied the phase space structure via Poincaré sections. Using these sections and symbolic dynamics, we study the fine structure of the region of initial conditions, in particular the chaotic scattering region.     

Motion near total collapse in the planar isosceles three-body problem

We describe in detail the qualitative behavior of solutions of the planar isosceles problem which come close to triple collision. Using a method of McGehee, one may describe a neighborhood of triple collision for any mass ratios. For sufficiently small mass ratios, we exhibit infinitely many distinct periodic and collision/ejection solutions of this problem.Partially supported by N.S.F. Grant MCS 81-01855.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.     

The photogravitational restricted three-body problem

The photogravitational restricted three-body problem is reviewed and the case of the out-of-plane equilibrium points is analysed. It is found that, when the motion of an infinitesimal body is determined only by the gravitational forces and effects of the radiation pressure, there are no out-of-plane stable equilibrium points.     

A search for collision orbits in the free-fall three-body problem I. Numerical procedure

A numerical procedure is devised to find binary collision orbits in the free-fall three-body problem. Applying this procedure, families of binary collision orbits are found and a sequence of triple collision orbits are positioned. A property of sets of binary collision orbits which is convenient to search triple collision orbits is found. Important numerical results are formulated and summarized in the final section.     

Conditionally periodic solutions near the libration points of the restricted circular three-body problem

Families of conditionally periodic solutions have been found by a slightly modified Lyapunov method of determining periodic solutions near the libration points of the restricted three-body problem. When the frequencies of free oscillations are commensurable, the solutions found are transformed into planar or spatial periodic solutions. The results are confirmed by numerically integrating the starting nonlinear differential equations of motion.     

Isoenergetic families of quasi-periodic solutions near the equilateral solution of the three-body problem

The tensor appearing in the equation of geodesic deviation is computed for the equilateral solution of the general three-body problem. The eigenvalues and eigenvectors ofC _k ⁱ are determined. It is found that at least one of the eigenvalues is negative, irrespective of the masses of the bodies. This implies that the equilateral solution is not stable. The eigenvectors with positive eigenvalues generate isoenergetic 1-parameter families of quasi-periodic solutions in the neighborhood of the equilateral solution. The relation between the 1-parameter families constructed here and those known from the literature is discussed.     

Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem

By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j-k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ-C characteristic curves of these families extend to the left of the μ-C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* (μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun-Jupiter and Earth-Moon systems. We found new μ-C and x-C characteristic curves. Along each μ-C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x-C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ-C or the x-C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L₄ and L₅ are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ-C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x-C curve, that spiral into the points L₄ and L₅, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem.     

The elliptical three-body problem. Triangular solution

The restricted three-body problem with eccentric orbit is reviewed and the positions of the triangular Lagrangian points (L₄, L₅) are determined. It is put in evidence the fact the fact L₄ and L₅ are situated at the corners of an isoscales triangle: AB = BC = 1 − e²/)1 + e cos ν )4/3 and AC = 1 − e²/)1 + e cos ν )     

The singly averaged elliptical restricted three-body problem

The plane, singly averaged, elliptical restricted three body problem is considered in the article. The first three terms are taken in the perturbing function. The equations of motion in terms of the canonical elements of Delaunay are obtained. And the change of the elements of motion of the satellite due to the perturbing function is calculated. An application is given in the case of a satellite in the earth-moon system.