Classification of orbits in the plane isosceles three-body problem

The general plane isosceles three-body problem is considered for different ratios of the central body mass to the masses of other bodies. The central body goes through the middle of the segment connecting the other bodies along the perpendicular to this segment. The initial conditions are chosen by two parameters: the virial ratio k and the parameter     , where r˙ is the relative velocity of the 'outer' bodies, and R˙ is the velocity of the 'central' body with respect to the mass centre of the 'outer' bodies. The equations of motion are numerically integrated until one of three times: the time of escape of the central body, its time of ejection with   R >100 d   , or 1000 τ (here d is the mean size, and τ is the mean crossing time of the triple system). The regions corresponding to escapes of the central body after different numbers of triple approaches are found at the plane of parameters   k ∈(0,1)  and   μ ∈(-1,1)  . The regions of stable motions are revealed. The zones of regular and stochastic orbits are outlined. The fraction of stochastic trajectories increases with the central mass. The fraction of stable orbits is highest for equal masses of the bodies.     

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.     

Dynamics of rotating triple systems

We investigate the dynamical evolution of 100 000 rotating triple systems with equal-mass components. The system rotation is specified by the parameter ω=?c²E, where c and E are the angular momentum and total energy of the triple system, respectively. We consider ω=0.1,1, 2, 4, 6 and study 20 000 triple systems with randomly specified coordinates and velocities of the bodies for each ω. We consider two methods for specifying initial conditions: with and without a hierarchical structure at the beginning of the evolution. The evolution of each system is traced until the escape of one of the bodies or until the critical time equal to 1000 mean system crossing times. For each set of initial conditions, we computed parameters of the final motions: orbital parameters for the final binary and the escaping body. We analyze variations in the statistical characteristics of the distributions of these parameters with ω. The mean disruption time of triple systems and the fraction of the systems that have not been disrupted in 1000 mean crossing times increase with ω. The final binaries become, on average, wider at larger angular momenta. The distribution of their eccentricities does not depend on ω and generally agrees with the theoretical law f(e)=2e. The velocities of the escaping bodies, on average, decrease with increasing angular momentum of the triple system. The fraction of the angles between the escaping-body velocity vector and the triple-system angular momentum close to 90° increases with ω. Escapes in the directions opposite to rotation and prograde motions dominate at small and large angular momenta, respectively. For slowly rotating systems, the angular momentum during their disruption is, on average, evenly divided between the escaping body and the final binary, whereas in rapidly rotating systems, about 80% of the angular momentum is carried away by the escaping component. We compare our numerical simulations with the statistical theory of triple-system disruption.     

Periodic orbits in the general three-body problem and the relationship between them

We study the regions of finite motions in the vicinity of three simple stable periodic orbits in the general problem of three equal-mass bodies with a zero angular momentum. Their distinctive feature is that one of the moving bodies periodically passes through the center of mass of the triple system. We consider the dynamical evolution of plane nonrotating triple systems for which the initial conditions are specified in such a way that one of the bodies is located at the center of mass of the triple system. The initial conditions can then be specified by three parameters: the virial coefficient k and the two angles, φ₁ and φ₂, that characterize the orientation of the velocity vectors for the bodies. We scanned the region of variation in these parameters k∈(0, 1); φ₁, φ₂∈(0, π) at steps of δk=0.01; δφ₁=δφ₂=1° and identified the regions of finite motions surrounding the periodic orbits. These regions are isolated from one another in the space of parameters (k, φ₁, φ₂). There are bridges that correspond to unstable orbits with long lifetimes between the regions. During the evolution of these metastable systems, the phase trajectory can “stick” to the vicinity of one of the periodic orbits or move from one vicinity to another. The evolution of metastable systems ends with their breakup.     

Particle motions in Maxwell’s ring dynamical systems

The focus of this contribution is an effort to review and report the main results obtained so far, concerning the periodic motions of a small body in the combined gravitational field created by a regular ν-gon arrangement of ν big bodies with equal masses, where ν > 7, and another central primary with different mass. Various types of planar periodic motions are presented and networks of characteristic curves of families are depicted, in order to show their distribution in the space of the initial conditions, as well as the evolution of their members that are also examined under the variation of the parameters of the system. Furthermore, the regions of the allowed three-dimensional motions, as well as their variation, are illustrated by means of the zero-velocity surfaces. All this new material is added to the already existing data, and completes thus the profile of the dynamical behavior of the system.     

Comparative Study of the 2:3 and 3:4 Resonant Motion with Neptune: An Application of Symplectic Mappings and Low Frequency Analysis

The 2:3 and 3:4 exterior mean motion resonances with Neptune are studied by applying symplectic mapping models. The mappings represent efficiently Poincaré maps for the 3D elliptic restricted three body problem in the neighbourhood of the particular resonances. A large number of trajectories is studied showing the coexistence of regular and chaotic orbits. Generally, chaotic motion depletes the small bodies of the effective resonant region in both the 2:3 and 3:4 resonances. Applying a low frequency spectral analysis of trajectories, we determined the phase space regions that correspond to either regular or chaotic motion. It is found that the phase space of the 3:4 resonant motion is more chaotic than the 2:3 one.     

Search for and study of nearly periodic orbits in the plane problem of three equal-mass bodies

We analyze nearly periodic solutions in the plane problem of three equal-mass bodies by numerically simulating the dynamics of triple systems. We identify families of orbits in which all three points are on one straight line (syzygy) at the initial time. In this case, at fixed total energy of a triple system, the set of initial conditions is a bounded region in four-dimensional parameter space. We scan this region and identify sets of trajectories in which the coordinates and velocities of all bodies are close to their initial values at certain times (which are approximately multiples of the period). We classify the nearly periodic orbits by the structure of trajectory loops over one period. We have found the families of orbits generated by von Schubart’s stable periodic orbit revealed in the rectilinear three-body problem. We have also found families of hierarchical, nearly periodic trajectories with prograde and retrograde motions. In the orbits with prograde motions, the trajectory loops of two close bodies form looplike structures. The trajectories with retrograde motions are characterized by leafed structures. Orbits with central and axial symmetries are identified among the families found.     

Analysis of the neighborhood of the 2: 1 resonance in the equal-mass three-body problem

We consider the trajectories in the neighborhood of a 2: 1 resonance (in periods of osculating motions of the outer and inner binaries) in the plane equal-mass three-body problem. We identified the zones of motions that are stable on limited time intervals. All of them correspond to the retrograde motions of the outer and inner subsystems. The prograde motions are unstable: the triple system breaks up into a final binary and an escaping component. In the barycentric nonrotating coordinate system, the trajectories occasionally form symmetric structures composed of several leaves. These structures persist for a long time, and, subsequently, the trajectories of the bodies fill compact regions in coordinate space.     

A symplectic mapping model for the 3/4 exterior

We present a symplectic mapping model to study the evolution of a small body at the 3/4 exterior resonance with Neptune, for planar and for three dimensional motion. The mapping is based on the averaged Hamiltonian close to this resonance and is constructed in such a way that the topology of its phase space is similar to that of the Poincaré map of the elliptic restricted three-body problem. Using this model we study the evolution of a small object near the 3/4 resonance. Both chaotic and regular motions are found, and it is shown that the initial phase of the object plays an important role on the appearance of chaos. In the planar case, objects that are phase-protected from close encounters with Neptune have regular orbits even at eccentricities up to 0.44. On the other hand objects that are not phase protected show chaotic behaviour even at low eccentricities. The introduction of the inclination to our model affects the stable areas around the 3/4 mean motion resonance, which now become thinner and thinner and finally at is=10° the whole resonant region becomes chaotic. This may justify the absence of a large population of objects at this resonance.     

Regions of a satellite��s motion in a Maxwell��s ring system of?N?bodies

The study of a dynamical system comprises a variety of processes, each one of which requires careful analysis. A fundamental preliminary step is to detect and limit the regions where solutions may exist. In the case of the ring problem of (N+1)-bodies or, otherwise, the regular polygon problem of (N+1) bodies, the existence of a Jacobian-type integral of motion constitutes the key for the investigation of the areas where the motions of the small particle are realized. Based on the aforementioned integral, we present an extended study of the parametric evolution of the regions where 3-D particle motions may exist.     

The region of stable motions around a periodic eight-like orbit in the general three-body problem

We investigate the neighborhood of the periodic eight-like orbit found by Moore (1993) and Chenciner and Montgomery (2000). One-, two-, and three-dimensional scans in body coordinates, velocities, and masses were constructed. We found the regions of initial conditions in which the maximum mutual separation did not exceed 5 distance units during 2000 time units (about 300 periods of the initial solution). Larger deviations from the periodic solution lead to distant body ejections and escapes. The identified regions of finite motions are complex in structure. In some sections, these are simple-connected manifolds, while in other sections, stability zones alternate with escape zones. We estimated the fractal dimensions of the stability regions in three-dimensional scans: it typically ranges from 2 to 3. In some cases, we found transitions between motions along the figure of eight in its neighborhood and motions in the vicinity of a periodic Broucke orbit in the isosceles three-body problem.     

Multiple Periodic Orbits in the Ring Problem: Families of Triple Periodic Orbits

The ring problem deals with the motion of a small body which is subjected to the combined gravitational attraction of N massive bodies arranged in an annular configuration. In this paper we study the distribution of the triple periodic orbits in the phase space of the initial conditions and we discuss their evolution and their principal features. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamics of close binary systems with mass exchange

The effects of non-isotropic ejection of mass from either component of a binary system on the orbital elements are studied, for the case of a small initial eccentricity of the relative orbit, when all the ejected mass falls on the other component. The problem is transformed to an equivalent two-body problem with isotropic variation of mass, plus a perturbing force which is a function of the intial conditions of ejection of the particles and their final, positions and velocities when they fall on the surface of the other star. The variation of the orbital elements are derived. It is shown that, to first-order terms in the eccentricity, the secular change of the semimajor axis is equal to the one corresponding to the case of zero initial eccentricity. On the contrary, the secular change of the eccentricity is smaller and it depends on the variations of mass ejection due to the finite eccentricity.     

Triple approaches in the plane isosceles equal-mass three-body problem

We analyze flyby-type triple approaches in the plane isosceles equal-mass three-body problem and in its vicinity. At the initial time, the central body lies on a straight line between the other two bodies. Triple approaches are described by two parameters: virial coefficient k and parameter $\mu = \dot r/\sqrt {\dot r^2 + \dot R^2 }$ , where $\dot r$ is the relative velocity of the extreme bodies and $\dot R$ is the velocity of the central body relative to the center of mass of the extreme bodies. The evolution of the triple system is traceable until the first turn or escape of the central body. The ejection length increases with closeness of the triple approach (parameter k). The longest ejections and escapes occur when the extreme bodies move apart with a low velocity at the time of triple approach. We determined the domain of escapes; it corresponds to close triple approaches (k>0.8) and to μ in the range ?0.2<μ<0.7. For small deviations from the isosceles problem, the evolution does not differ qualitatively from the isosceles case. The domain of escapes decreases with increasing deviations. In general, the ejection length increases for wide approaches and decreases for close approaches.     

The regular motion of an axisymmetrical satellite in the field of a triaxial body

This paper investigates the regular motions of an axisymmetrical satellite in the field of Newton's attraction of a triaxial body. Both the orbital and the self rotational motions of the two bodies are taken into consideration. The exact solutions are discussed using Poincaré's method of small parameter. In the decomposition of the force function all the harmonic terms up to the third order are taken into account.The results show the existence of eight solutions. The stability of the new group of solutions is discussed using two methods to get the necessary and sufficient conditions required for the stability of these motions.     

Triple collisions in the one-dimensional three-body problem

We consider the motions of particles in the one-dimensional Newtonian three-body problem as a function of initial values. Using a mapping of orbits to symbol sequences we locate the initial values leading to triple collisions. These turn out to form curves which give clear structure to the region in which the motions depend sensitively on initial conditions. In addition to finding the triple collision orbits we also locate orbits which end up to a triple collision in both directions of time, that is, orbits which are finite both in space and time. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Main features of dynamical escape from three-dimensional triple systems

The dynamical evolution of triple systems with equal and unequal-mass components and different initial velocities is studied. It is shown that, in general, the statistical results for the planar and three-dimensional triple systems do not differ significantly. Most (about 85%) of the systems disrupt; the escape of one component occurs after a triple approach of the components. In a system with unequal masses, the escaping body usually has the smallest mass. A small fraction (about 15%) of stable or long-lived systems is formed if the angular momentum is non-zero. Averages, distributions and coefficients of correlations of evolutionary characteristics are presented: the life-time, angular momentum, numbers of wide and close triple approaches of bodies, relative energy of escapers, minimum perimeter during the last triple approach resulting in escape, elements of orbits of the final binary and escaper.     

Effects of stellar wind,dynamical friction,and star mergers on the dynamical evolution of multiple stars

The dynamical evolution of small stellar groups composed of N=6 components was numerically simulated within the framework of a gravitational N-body problem. The effects of stellar mass loss in the form of stellar wind, dynamical friction against the interstellar medium, and star mergers on the dynamical evolution of the groups were investigated. A comparison with a purely gravitational N-body problem was made. The state distributions at the time of 300 initial system crossing times were analyzed. The parameters of the forming binary and stable triple systems as well as the escaping single and binary stars were studied. The star-merger and dynamical-friction effects are more pronounced in close systems, while the stellar wind effects are more pronounced in wide systems. Star-mergers and stellar wind slow down the dynamical evolution. These factors cause the mean and median semimajor axes of the final binaries as well as the semimajor axes of the internal and external binaries in stable triple systems to increase. Star mergers and dynamical friction in close systems decrease the fraction of binary systems with highly eccentric orbits and the mean component mass ratios for the final binaries and the internal and external binaries in stable triple systems. Star mergers and dynamical friction in close systems increase the fraction of stable triple systems with prograde motions. Dynamical friction in close systems can both increase and decrease the mean velocities of the escaping single stars, depending on the density of the interstellar medium and the mean velocity of the stars in the system.     

A numerical investigation of the one-dimensional newtonian three-body problem

Numerical orbit integrations have been conducted to characterize the types of trajectories in the one-dimensional Newtonian three-body problem with equal masses and negative energy. Essentially three different types of motions were found to exist. They may be classified according to the duration of the bound three-body state. There are zero-lifetime predictable trajectories, finite lifetime apparently chaotic orbits, and infinite lifetime quasi-periodic motions. The quasi-periodic orbits are confined to the neighbourhood of Schubart's stable periodic orbit. For all other trajectories the final state is of the type binary + single particle in both directions of time. The boundaries of the different orbit-type regions seem to be sharp. We present statistical results for the binding energies and for the duration of the bound three-body state. Properties of individual orbits are also summarized in the form of various graphical maps in a two-dimensional grid of parameters defining the orbit. Supported by the Academy of Finland.