The rectilinear restricted problem of three bodies

In this paper we discuss some aspects of the isosceles case of the rectilinear restricted problem of three bodies, where two primaries of equal mass move on rectilinear ellipses, and the particle is confined to the symmetry axis of the system. In particular, the behaviour near a collision of the primaries and also near a collision of all three bodies is investigated. It is shown that this latter singularity is a triple collision in the sense of Siegel's theory. Furthermore, asymptotic expansions for the particle's motion during a parabolic and a hyperbolic escape are derived.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Symmetric rectilinear periodic orbits of three bodies

In this paper the existence of families of symmetric periodic orbits in the rectilinear three body problem with the middle mass much larger than the masses on the outside is rigorously established. A number of these families are continued numerically and their stability properties as orbits of the planar general problem of three bodies are studied.     

Numerical study of the rectilinear isosceles restricted problem

We study the orbit of a particle in the plane of symmetry of two equal mass primaries in rectilinear keplerian motion. Using the surfaces of section we look for periodic orbits, examine their stability and search for quasi-periodic orbits and regions of escape. For large values of the angular momentumC, we verify the validity of the approximation of two fixed centers. However, we also find irregular families of orbits and resonance zones.For small values ofC, the approximation is no longer valid, but we find invariant curves whose interpretation might be interesting.     

The problem of three bodies

The gravitational problem of three bodies is presented in the general case, without restrictions on the distances and masses of the participating bodies. Recent advances are discussed and the consequences of the Laplacean instability in stellar dynamics are described.     

On the generalized restricted problem of three bodies

The particular case of the complete generalized three-body problem (Duboshin, 1969, 1970) where one of the body-points does not exert influence on the other two is analysed. These active material points act on the passive point and also on each other with forces (attraction or repulsion), proportional to the product of masses of both points and a certain function of the time, their mutual distances and their first and second derivatives. Furthermore it is not supposed that generally the third axiom of mechanics (action=reaction) takes place.Here under these more general assumptions the equations of motion of the active masses and the passive point, as well as the diverse transformations of these equations are analogous of the same transformations which are made in the classical case of the restricted three-body problem.Then we determine conditions for some particular solutions which exist, when the three points form the equilateral triangle (Lagrangian solutions) or remain always on a straight line (Eulerian solutions).Finally, assuming that some particular solutions of the above kind exist, the character of solutions near this particular one is envisaged. For this purpose the general variational equations are composed and some conclusions on the Liapunov stability in the simplest cases are made.     

Mass effects in the problem of three bodies

This is a study of the dynamical behavior of three point masses moving under their mutual gravitational attraction in a plane. The initial positions and velocities are identical for all cases studied and only the masses of the participating bodies change in the series of numerical experiments. In this way the effect of the coupling terms in the differential equations of motion are investigated. The motion in all 125 cases begins with an interplay between the three bodies, followed by temporary ejections or by an eventual escape. The total mass of the system is kept constant while the massratios change from 1 to 5. The initial velocities being zero, the total energy is negative in all cases.Approximately 74% of the cases disintegrated (i.e. two bodies formed a binary and the third body escaped) in less than 140 time units, 47% in less than 50 time units and 10% ended in escape in less than 10 time units. Considering three stars with total mass 12M , initially placed at 3, 4 and 5 parsec distances (or three galaxies with mass 2.4×10¹² M , initially placed 30, 40 and 50 kpc apart), the unit of time (approximately the crossing time) becomes 1.5×10⁷ y (3.2×10⁷ y). The average time of disintegration was found to be of the order of 10⁹ y. The average semi-major axis of the binaries left behind after disintegration was 0.7 parsec and the average value of the eccentricity was 0.76. The effect of the masses on the escapes was established and it was found that the bodynot with the smallest mass escaped in 13% of the disintegrated cases. The cases which did not disintegrate in 150 time units were analyzed in detail and the time of their eventual escape was estimated.The numerical results are tabulated regarding escape time, ejection period, total energy, escape energy, terminal velocity, semi-major axis, and eccentricity.The evolution of triple systems is followed from interplays through ejections to escapes and the orbital parameters for the separation of these classes are estimated.     

Numerical exploration of commensurable periodic solutions of the restricted problem of three bodies and their stability

The long period problem provides the initial conditions for numerical computation of close periodic solutions separated in three categories. For each type of commensurability a number of periodic solutions are computed and their stability is studied by computing the characteristic exponents of the matrizant. The Runge-Kutta method for the solution of differential equations of motion was used in all cases. The results obtained are presented for a four cases of commensurability.     

Hill stability of the general problem of three bodies

We discuss Hill stability in the general three-body problem. The Hill curves in the general problem are the same as in the planar problem. We show that the bifurcation points correspond to the five equilibrium solutions, and derive the criterion for stability in the general case. Application of this criterion to 19 natural satellites of the Solar system leads to the result that, apart from Neptune 1, all the other 18 satellites are unstable in the sense of Hill. The dominant factor in producing this result is the finite eccentricity of the planetary orbits around the Sun.     

The invariance of the restricted problem of three bodies

The restricted problem of three bodies in sidereal coordinates is investigated using Lie theoretical methods of extended groups. The application of the theory results in a single scalar generator with an associated invariant-the integral of the motion attributed to Jacobi.     

Bifurcation points in the planar problem of three bodies

An essential parameter in the planar problem of three bodies is the product of the square of the angular momentum and of the total energy (c ² H). The role of this parameter, which may be called abifurcation parameter, in establishing regions of possible motions has been shown by Marchal and Saari (1975) and Zare (1976a). There exist critical values of this parameter below which exchange between bodies cannot occur. These critical values may be calledbifurcation points.This paper gives an analytical criterion to obtain these bifurcation points for any given masses of the participating bodies.     

On oscillatory motion in the problem of three bodies

In the case of oscillatory motion in the problem of three bodies it is shown that ast the mutual distances between particles cannot separate faster thanCt ^2/3 whereC is some positive constant. As bounding functions of time exist for the other classifications of motion in the three body problem, it follows in general that the mutual distances between particles is 0(t) ast.     

Triple close approaches in the problem of three bodies

The gravitational problem of three bodies is treated in the case when the masses of the participating bodies are of the same order of magnitude and their distances are arbitrary. Estimates for the minimum perimeter of the triangle formed by the bodies and for the rate of the expansion of the system are obtained from Sundman's modified general inequality when the total energy of the system is negative. These estimates are used to propose and to describe an escape mechanism based on genuine three-body dynamics and to offer a method to control the accuracy of numerical integrations of the problem of three bodies. The requirements for these two applications are contradictory since an escape is the consequence of a close triple approach which phenomenon is detrimental to the accuracy of the computations. Consequently, the numerical study of escape from a triple system must treat triple close approaches with high reliability.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Almost-square orbits in the restricted problem of three bodies

In the restricted problem of three bodies, it has been discovered that, in a nonrotating frame, almost perfectly square orbits can result. Numerical investigations of these orbits have been made, and a brief explanation of their behaviour is given.     

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.     

A new regularization of the planar problem of three bodies

A new method of simultaneously regularizing the three types of binary collisions in the planar problem of three bodies is developed: The coordinates are transformed by means of certain fourth degree polynomials, and a new independent variable is introduced, too. The proposed transformation is in each binary collision locally equivalent to Levi-Civita's transformation, whereas the singularity corresponding to a triple collision is mapped into infinity. The transformed Hamiltonian is a polynomial of degree 12 in the regularized variables.Presented before the Division of Dynamical Astronomy at the 133rd meeting of the American Astronomical Society, Tampa, Florida, December 6–9, 1970.Department of Aerospace Engineering and Engineering Mechanics.     

Kolmogorov and Nekhoroshev theory for the problem of three bodies

We investigate the long time stability in Nekhoroshev’s sense for the Sun– Jupiter–Saturn problem in the framework of the problem of three bodies. Using computer algebra in order to perform huge perturbation expansions we show that the stability for a time comparable with the age of the universe is actually reached, but with some strong truncations on the perturbation expansion of the Hamiltonian at some stage. An improvement of such results is currently under investigation.     

Integration theory for the restricted problem of three axisymmetric bodies

In this paper the circular planar restricted problem of three axisymmetric ellipsoids S _i (i = 1, 2, 3), such that their equatorial planes coincide with the orbital plane of the three centres of masses, be considered. The equations of motion of infinitesimal body S ₃ be obtained in the polar coordinates. Using iteration approach we have given an approximation for another integral, which independent of the Jacobian integral, in the case of P-type orbits (near circular orbits surrounding both primaries).     

On lagrange solutions in the problem of three rigid bodies

The present paper is a direct continuation of the paper (Duboshin, 1973) in which was proved the existence of one kind of Lagrange (triangle) and Euler (rectilinear) solutions of the general problem of the motion of three finite rigid bodies assuming different laws of interaction between the elementary particles of the rigid bodies. In particular, Duboshin found that the general problem of three rigid bodies permits such solutions in which the centres of mass of the bodies always form an equilateral triangle or always remain on one straight line, and each body possesses an axial symmetry and a symmetry with respect to the plane of the centres of mass and rotates uniformly around its axis orthogonal to this plane. The conditions for the existence of such solutions have also been found. The results in Duboshin's paper have greatly interested the author of the present paper. In another paper (Kondurar and Shinkarik, 1972) considering a more special problem, when two of the three bodies are spheres, either homogeneous or possessing a spherically symmetric distribution of the densities or of the material points, and the third is an axially symmetrical body possessing equatorial symmetry, the present author obtained analogous solutions of the ‘float’ type describing the motion of the indicated dynamico-symmetrical body in assuming its passive gravitation. In the present paper new Lagrange solutions of the considered general problems of three rigid bodies of ‘level’ type are found when the axes of geometrical and mechanical symmetry of all three bodies always lie in the triangle plane, and the bodies themselves rotate inertially around the symmetry axis, independently of the parameters of the orbital motion of the centres of mass as in the ‘float’ case. The study of particular solutions of the general problem of the translatory-rotary motion of three rigid bodies, which are a generalization of Lagrange solutions, is in the author's opinion, a novelty of some interest for both theoretical and practical divisions of celestial mechanics. For example, in recent times the problem of the libration points of the Earth-Moon system has acquired new interest and value. A possible application which should be mentioned is that to the orbits of artificial satellites near the triangular libration points to serve as observation stations with the aim of specifying the physical parameters in the Earth-Moon system (e.g., the relation of the Earth's mass to the Moon's mass for investigating the orientation of the satellite, solar radiation, etc.).     

Surfaces of zero velocity in the restricted problem of three bodies

Recent uses of computer graphics allow the representation of the three-dimensional surfaces of zero velocity, also known as Hill's or the Jacobian surfaces. The purpose of this paper is to show the actual surfaces rather than their projections which are available in the standard literature. The analytical properties of the surfaces are also available; therefore, this paper offers the pertinent references rather than the derivations.     

Properties of the moment of inertia in the problem of three bodies

Sundman's and Birkhoff's results are combined with a recently developed inequality and new qualitative results are given for the problem of three bodies.