General Three-body Problem Research Articles

Symmetric doubly asymptotic orbits at collinear equilibrium points in the general three-body problem

This paper concerns the numerical determination of doubly asymptotic orbits at the collinear equilibrium points of the general three-body problem. To compute a doubly asymptotic orbit we use a linear analysis to determine the relevant outgoing eigenvector and a combined iterative method based on the characteristic bisection and Newton's method to solve the resulting system of equations for the mass parameters of the problem. Many such orbits are presented.

An exactly conservative integrator for then-body problem

The two-dimensional n-body problem of classical mechanics is a non-integrable Hamiltonian system for n > 2. Traditional numerical integration algorithms, which are polynomials in the time step, typically lead to systematic drifts in the computed value of the total energy and angular momentum. Even symplectic integration schemes exactly conserve only an approximate Hamiltonian. We present an algorithm that conserves the true Hamiltonian and the total angular momentum to machine precision. It is derived by applying conventional discretizations in a new space obtained by transformation of the dependent variables. We develop the method first for the restricted circular three-body problem, then for the general two-dimensional three-body problem, and finally for the planar n-body problem. Jacobi coordinates are used to reduce the two-dimensional n-body problem to an (n-1)-body problem that incorporates the constant linear momentum and center of mass constraints. For a four-body choreography, we find that a larger time step can be used with our conservative algorithm than with symplectic and conventional integrators.

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 μ =, 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 > 100d, or 1000τ (here d is the mean size, and T 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 E (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.

Stability and bifurcations of the figure-8 solution of the three-body problem.

The stability properties of a recently discovered solution of the general three-body problem with equal masses and the shape of a figure 8 are analyzed as the masses are varied. It is shown by numerical continuation and the evaluation of the characteristic multipliers that the solution is stable only in a narrow mass interval. Other less symmetrical and unstable solutions with equal masses in the same homotopy class as the figure-8 orbit have been found. The branching behavior is also analyzed.

Properties of New Coordinates for the General Three-Body Problem

This is an analysis of the features of the new coordinate system given by the principal axes of inertia, as determined by Euler angles, and twodistances related to the inertia principal moments and an auxiliar angleas coordinates, for studying the general three-body problem, interactingthrough gravitational forces.

Modeling the 5 : 2 Mean-Motion Resonance in the Jupiter–Saturn Planetary System

The motion of the Jupiter–Saturn planetary system near the 5 : 2 mean-motion resonance is modeled analytically in the frame of the planar general three-body problem. The topology of the phase space in the resonance and near-resonance domains is investigated in detail by means of surfaces of section and spectral map techniques. Various regimes of motion of the Jupiter–Saturn system are observed, and their domains are represented on the parameter plane defined by the total energy of the system and the parameter of proximity to the 5 : 2 resonance. The regions of transition between domains of different regimes of motion are characterized by chaotic motion. We show that there is a significant chaotic zone associated with the 5 : 2 mean-motion resonance, but large-scale excursions of the planet eccentricities are not possible in the planar three-body model.

Approximate Analytic Solutions of the Averaged Three-Body Problem at First-Order Resonance with Large Oblateness of the Central Planet

For the general spatial planetary three-body problem at first-order mean motion resonance under the large oblateness of the central planet, the analytic solutions of the averaged motion are obtained with the help of the Weierstrass functions accurate to the third-degree terms in the satellites' eccentricities and inclinations. The behavior of solutions is investigated on the phase plane.

An approach to Mel’nikov theory in celestial mechanics

Using a completely analytic procedure—based on a suitable extension of a classical method—we discuss an approach to the Poincaré–Mel’nikov theory, which can be conveniently applied also to the case of nonhyperbolic critical points, and even if the critical point is located at the infinity. In this paper, we concentrate our attention on the latter case, and precisely on problems described by Kepler-type potentials in one or two degrees of freedom, in the presence of general time-dependent perturbations. We show that the appearance of chaos (possibly including Arnol’d diffusion) can be proved quite easily and in a direct way, without resorting to singular coordinate transformations, such as the McGehee or blowing-up transformations. Natural examples are provided by the classical Gyldén problem, originally proposed in celestial mechanics, but also of interest in different fields, and by the general three-body problem in classical mechanics.

Rigid-Rotator and Fixed-Shape Solutions to the Coulomb Three-Body Problem

Solutions to the general classical Coulomb three-body problem in the form of rigid-rotator and fixed-shape configurations are studied. In the collinear case, some necessary and/or sufficient conditions for the existence of the so-called charge-symmetrical, (−)(+)(−), and charge-asymmetrical, (−)(−)(+), configurations are stated. These conditions involve relations between the geometrical and dynamical parameters of the system under study. The impossibility of the existence of a planar Coulombic rigid rotator is demonstrated. In the two-dimensional case, fixed-shape solutions are studied analytically, and it is shown that, in the three-dimensional case, only fixed-shape solutions involving a triple collision and a static case are possible. Finally, some numerical experimentation, mostly based upon theoretical predictions of the work, is performed, and new bound (although unstable) rotating-oscillating orbits for systems such as the positronium negative ion and helium are found.

Three-body Coulomb problem in the dipole approximation: and states

In the asymmetric doubly excited states of the two-electron atom one electron is excited much more than the other. The characteristic separation of the strongly excited electron from the atomic nucleus significantly exceeds that for the weakly excited electron . The electron - electron interaction is approximated by the first non-trivial (dipole) term of its expansion over . The same approach is easily extended to the general Coulomb three-body problem with the particles of arbitrary mass and charge. The dynamics of three-body states with and symmetry is analysed in quantum, semiclassical and classical approximations in comparison with the S and states considered previously. Mathematically, the problem is reduced to two coupled three-term recursion relations which were not studied previously. The exact and approximate (semiclassical) degeneracies of the levels with different values of the total orbital momentum L are revealed, and the energies of the `planetary atom' states are estimated.

Wide Binaries from Few-Body Interactions

The orbital properties of wide binaries are modified by stellar encounters during their lifetimes as field stars. The few body process is particularly efficient right after the star formation in star clusters. In this article we review the orbital parameters which we expect from the few-body process. First, the solutions of the General Three-Body Problem are described, and then a detailed discussion of the theory relevant to binaries follows. Among other things, Opik's law of the distribution of semi- major axes of the binaries is derived. In addition, we summarize recent results from the few-body problem of extended bodies which may be relevant to the early dynamical evolution of protostellar clouds.

VERTICAL-CRITICAL PERIODIC ORBITS IN THE GENERAL THREE-BODY PROBLEM

We present special generating plane orbits, the vertical-critical orbits, of the coplanar general three-body problem. These are determined numerically for various values of m3, for the entire range of the mass ratio of the two primaries. The vertical-critical orbits are necessary in order to specify the vertically stable segments of the families of plane periodic orbits, and they are also the starting points of the families of the simplest possible three-dimensional periodic orbits, namely the simple and double periodic. The initial conditions of the vertical-critical periodic orbits of the basic families l, m, i, h, b and c and their stability parameters are determined.

A study of the plane unrestricted three-body problem

The general (unrestricted) three-body problem is investigated in the case when the force of mutual attraction between the bodies is proportional to the nth power of their distance, where n is an arbitrary real number. A new description is given of the plane problem, based on the introduction of the following Lagrange variables: r—the square root of half the polar moment of inertia, ψ—the angle between the two sides of the triangle, and γ—the natural logarithm of the quotient of those two sides. The first variable characterizes the size of the triangle, and the other two, its configuration. Routh's equations are derived, in which the variable r is ‘almost separated’ from γ and ψ; the system of equations is reversible. In the special case of the restricted problem, i.e. when the mass of one of the bodies tends to zero, the variables are completely separable, so that the problem describes only the change in the configuration of the triangle. It is shown that the qualitative results, known for Newtonian interaction ( n = −2), are valid throughout the range −3 < n < −1. In particular, for these values of n ‘elementary’ methods of analysis are used to solve the problems of Hill stability for a pair of bodies, the existence of final motions relating to hyperbolic-elliptic motions is established for n = −2, and a local analysis is carried out of the neighbourhoods of the classical liberation points. Local analysis showed that in the neighbourhood of collinear points two families of Lyapunov periodic motions exist, into which the family of two-dimensional “whiskered” tori degenerates. In the linear approximation, the problems of the stability of triangular points in the restricted and unrestricted formulations are equivalent to one another. Hence the triangular elliptical solutions of the unrestricted problem are stable throughout the domain constructed by Danby for the restricted problem. Allowance for the small non-zero mass of one of the bodies may make the other two bodies leave the unperturbed circular orbit; there is no such effect in the restricted problem.

A fast method for quantitatively measuring stability in a three-body dynamical system

The three body problem can be used to investigate many types of dynamical systems when they possess an attraction for each other. The circular restricted three-body problem is a special case of the general three-body problem. The circular restricted three-body problem has no closed form solution and must be numerically integrated to obtain a trajectory of the motion of the third body. A quantitative stability investigation usually requires a tremendous amount of computation due to numerically integrating each individual trajectory. By reducting the amount of numerical integration time, more time can be directed toward numerically investigating the stability relationship of the trajectory sets for the various mass ratios. In this paper a closed form measure of stability for a bounded region in the three-body system is found and used to perform a global (examination of all mass ratio combinations of the three bodies) stability investigation. The developed stability criteria is a function of the initial conditions of the third body and the strength of the potential field. In this study the potential field influencing the bodies is considered to be a gravitational field and it is shown that there are certain mass ratio combinations of the three bodies that result in an optimally stable system.

The integrable cases of the general spatial three-body problem at third-order resonance

For the general three-body problem at third-order resonance an analytical solution is obtained by the use of the Weierstrass functions.

Bifurcations of separatrices in the averaged planar general three-body problem at first-order resonance

It is known that an abridged case of the averaged planar general three-body problem, at first-order resonance, is analytically integrated, using an expansion of the disturbing function linear in the eccentricities. There exist different methods with the help of which the integration can be performed. For the first time Sessin and Ferraz-Mello in the years 1981–88 (Sessin, 1981, 1983; Ferraz-Mello and Sessin, 1984; Ferraz-Mello, 1987, 1988) did an analytic integration for the restricted elliptic three-body problem, in terms of the variablesK andH (K=ΣD j e j cos (ψ1−π j ),H=ΣD j e j sin (ψ1−π j ),D j = const, wheree j and π j are, respectively, the eccentricity and the longitude of the periapsis of thei-th planet, ψ1 is the Delaunay's anomaly), which is inconvenient for the analytical investigation of the evolution of the major semi-axesa j , the eccentricitiese j and the resonance phases ϕ j =ψ1−π j . Later, a different method for the analytical integration of the general three-body problem, in the variablesa j ,e j and ϕ j , was considered by the author (Shinkin, 1993). A disadvantage of both methods is the fact that they use non-canonical changes of variables. But there exists a third very beautiful canonical method of analytical integration of the general planetary problem, which is briefly considered in the present paper and allows us to describe the bifurcations of separatrices (i.e. appearance, disappearance, splitting and confluence of separatrices) separating the domains of libration and circulation of the resonance phase on the phase plane in the averaged planar general three-body problem at first-order resonance. The bifurcation parameter is analytically found and plays an important role in a qualitative description of all kinds of motion in the examined problem.

The integrable cases of the planetary three-body problem at first-order resonance

This paper investigates the stability of the motion in the averaged planar general three-body problem in the case of first-order resonance. The equations of the averaged motion of bodies near the resonance surface is obtained and is analytically integrated by quadratures. The stability of the averaged motion is analytically investigated in relation to the semi-major axes, the eccentricities and the resonance phases. An autonomous second-order equation is obtained for the deviation of semiaxes from the resonance surface. This equation has an energy integral and is analytically integrated by quadratures. The quasi-periodic dependence on time with two-frequency basis of the averaged motion of bodies is found. The basic frequencies are analytically calculated. With the help of the mean functionals calculated along integral curves of the averaged problem the new analytic first integrals are constructed with coefficients periodic in time. The analytic conditions of librations of resonance phases are obtained.

On basic families of three-dimensional periodic orbits of three massive bodies and their stability

The basic families of three-dimensional periodic orbits of the general three-body problem are determined numerically in order to obtain a global view of the simpler patterns of periodic three-body motion in three dimensions. The stability of the orbits is also computed. It is found that most of the orbits are unstable but stability intervals do exist for some of the families.

Numerical Simulations of Planetary Systems of the Jupiter-Saturn Type

We made a numerical study of the General Three-Body Problem in two dimensions, with the intention to obtain some statistical estimates of the outcome of the system after a long time. Two different sets of masses were used. In the first series of experiments we use masses in the ratio of 0.95, 0.04 and 0.01. In the second series, we use masses that are exactly in the Sun-Jupiter-Saturn ratio. To facilitate the discussion, we use the names Sun, Jupiter and Saturn for the three masses, in both cases. In all our experiments, the orbit of Jupiter starts with zero eccentricity and with a unit radius. However, the orbit of Saturn varies in two ways: the initial value of the semi-major axis varies from 1.1 to 3.5 and the eccentricity from 0.0 to 0.75. In total about 4000 cases were run for the two series of masses. All the numerical integrations were done with the method of recurrent power series of order 14, in a heliocentric frame of reference, integrating thus eight simultaneous first-order differential equations. All integrations were performed for a maximum of 12,500 canonical units of time, corresponding to about 2000 revolutions of Jupiter. The cause of termination or type of catastrophe for the system has been determined in all cases. In most cases, this is a close approach of Saturn with Jupiter, followed by ejection of Saturn from the system.

Mass ratios of three bodies for stable low-inclination three-dimensional periodic motion

The intervals of possible stability, on the μ-axis, of the basic families of three-dimensional periodie motions of the restricted three-body problem (determined in an earlier paper) are extended into regions of the μ-m3 parameter space of the general three-body problem. Sample three-dimensional periodic motions corresponding to these regions are computed and tested for stability. Six regions, corresponding to the vertical-critical orbitsl1v, m1v,m2v, andilv, survive this preliminary stability test-therefore, emerging as the mass parameters regions allowing the simplest types of stable low inclination three-dimensional motion of three massive bodies.