Characteristic exponents atL 4 andL 5 in the elliptic restricted problem of three bodies

We study the linear stability of the triangular points in the elliptic restricted problem by determining the characteristic exponents with a convergent method of iteration which in essence was introduced by Cesari (1940). We obtain the general term of such exponents as a power series in the eccentricity of the primaries, valid for sufficiently small and at all values of except one in the interval of stability of the circular problem.     

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.     

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.     

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 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.     

The restricted problem of three bodies with rigid dumb-bell satellite

This paper is concerned with an extension of the classical restricted problem of three bodies in three dimensions. Usually, the satellite is considered to be a point mass. Here, the satellite is assumed to have a simple structure. The equations of motion are obtained and some of their consequences are discussed.     

Stability of triangular equilibrium points in the elliptic restricted problem of three bodies with radiating and triaxial primaries

This paper studies the stability of infinitesimal motions about the triangular equilibrium points in the elliptic restricted three body problem assuming bigger primary as a source of radiation and the smaller one a triaxial rigid body. The perturbation technique developed by Bennet (Icarus 4:177, 1965b) has been used for determination of characteristic exponents. This technique is based on Floquet’s Theory for determination of characteristic exponents in the system with periodic coefficients. The results of the study are analytical and numerical expressions are simulated for the transition curves bounding the region of stability in the μ–e plane, accurate to O(e ²). The unstable region is found to be divided into three parts. The effect of radiation parameter is significant. For small values of e, the results are in favor with the numerical analysis of Danby (Astron. J. 69:166, 1964), Bennet (Icarus 4:177, 1965b), Alfriend and Rand (AIAA J. 6:1024, 1969). The effect of radiation pressure is significant than the oblateness and triaxiality of the primaries.     

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.     

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).     

Stationary motions in a restricted problem of three rigid bodies

The present paper is a continuation of papers by Shinkaric (1972), Vidyakin (1976), Vidyakin (1977), and Duboshin (1978), in which the existence of particular solutions, analogues to the classic solutions of Lagrange and Euler in the circular restricted problem of three points were proved. These solutions are stationary motions in which the centres of mass of the bodies of the definite structures always form either an equilateral triangle (Lagrangian solutions) or always remain on a straight line (Eulerian solutions) The orientation of the bodies depends on the structure of the bodies. In this paper the usage of the small-parameter method proved that in the general case the centre of mass of an axisymmetric body of infinitesimal mass does not belong to the orbital plane of the attracting bodies and is not situated in the libration points, corresponding to the classical case. Its deviation from them is proportional to the small parameter. The body turns uniformly around the axis of symmetry. In this paper a new type of stationary motion is found, in which the axis of symmetry makes an angle, proportional to the small parameter, with the plane created by the radius-vector and by the normal to the orbital plane of the attracting bodies. The earlier solutions-Shinkaric (1971) and Vidyakin (1976)-are also elaborated, and stability of the stationary motions is discussed.     

Investigation of the hydrodynamic analogy in the restricted problem of three bodies

The hydrodynamic analogy concept is examined from the standpoint of the possibility of physical realization of an analog device. The conditions that must be satisfied, conservation of mass and momentum and uniqueness of physical properties, are discussed in detail and applied to examples, including the Birkhoff formulation. The transformation of the restricted problem into a velocity field does not insure that an analog flow can be constructed, as demonstrated by two cases that correspond to physically impossible flows. Conservative force fields are certainly incapable of producing the desired results. The physical possibility of simulating the Birkhoff velocity field remains uncertain, but a practical consideration in the visualization seemingly eliminates all possibility of this analogy being useful.     

Equations of motion of the restricted problem of three bodies with variable mass

In this paper we have deduced the differential equations of motion of the restricted problem of three bodies with decreasing mass, under the assumption that the mass of the satellite varies with respect to time. We have applied Jeans law and the space time transformation contrast to the transformation of Meshcherskii. The space time transformation is applicable only in the special casen=1,k=0,q=1/2. The equations of motion of our problem differ from the equations of motion of the restricted three body problem with constant mass only by small perturbing forces.     

Existence of libration points in the restricted problem of three bodies with radiation pressure

In this paper the photogravitational forces restricted of three bodies be considered. We have assumed the infinitesimal mass of the shape of an axisymmetric body and one the finite masses be spherical luminous body while the other be an axisymmetric body non-luminous body. It is seen that there is a possibility of nine libration points for small values of oblatenesses.     

The structure of periodic solutions in the restricted problem of three bodies

We consider the basic families of plane-symmetric simply-periodic orbits in the Sun-Jupiter case of the plane restricted three-body problem and we study their horizontal and vertical stabilities. We give the critical orbits of these families, corresponding to the vertical stability parameter = 1 and in future communications we shall give the three-dimensional families which emanate from these plane bifurcations.     

The onset of chaotic motion in the restricted problem of three bodies

A full characterization of a nonintegrable dynamical system requires an investigation into the chaotic properties of that system. One such system, the restricted problem of three bodies, has been studied for over two centuries, yet few studies have examined the chaotic nature of some ot its trajectories. This paper examines and classifies the onset of chaotic motion in the restricted three-body problem through the use of Poincaré surfaces of section, Liapunov characteristic numbers, power spectral density analysis and a newly developed technique called numerical irreversibility. The chaotic motion is found to be intermittent and becomes first evident when the Jacobian constant is slightly higher thanC ₂.     

The formal integrals of the restricted planar problem of the three bodies

The first integrals of motion of the restricted planar circular problem of three bodies are constructed as the formal power series in r^1/2, r being the distance of a moving particle from the primary. It is shown that the coefficients of these series are trigonometric polynomials of an angular variable. Some particular solutions have been found in a closed form. The proposed method for constructing the formal integrals can be generalized to a spatial problem of three bodies.     

Motion near the triangular points in the elliptic restricted problem of three bodies

In this paper the first variational equations of motion about the triangular points in the elliptic restricted problem are investigated by the perturbation theories of Hori and Deprit, which are based on Lie transforms, and by taking the mean equations used by Grebenikov as our upperturbed Hamiltonian system instead of the first variational equations in the circular restricted problem. We are able to remove the explicit dependence of transformed Hamiltonian on the true anomaly by a canonical transformation. The general solution of the equations of motion which are derived from the transformed Hamiltonian including all the constant terms of any order in eccentricity and up to the periodic terms of second order in eccentricity of the primaries is given.     

The equilibrium solutions of the restricted problem of three rigid bodies with variable mass

In this paper we consider the circular planar restricted problem of three rigid bodiesS _i(i=1, 2, 3), two of them are axisymmetric ellipsoids and a third bodyS ₃ is a spherical satellite with decreasing mass, under the gravitational forces. The effect of small perturbations in the Coriolis force and the centrifugal forces on the location of equilibrium points has been studied. It is found only in the case when the primaries have equal differences between their respective principal moments of inertial the pointsL ₄ andL ₅ form nearly equilateral tringles with the primaries. The equilibrium pointsL ₁,L ₂,L ₃ remain collinear an ies on the line joining the primaries.     

Existence of particular solutions in the generalised restricted problem of three bodies with variable masses

The restricted problem of three bodies with variable masses is considered. It is assumed that the infinitesimal body is axisymmetric with constant mass and the finite bodies are spherical with variable masses such that the ratio of their masses remains constant. The motion of the finite bodies are determined by the Gyldén-Meshcherskii problem. It is seen that the collinear, triangular, and coplanar solutions not exist, but these solutions exist when the infinitesimal body be a spherical.     

Some families of periodic oscillations in the restricted problem with small mass-ratios of three bodies

This paper reports on the numerical determination of families of periodic oscillations in the case =0.000 95 of the restricted problem. The families emanating out of the collinear Lagrangian pointsL ₁,L ₂,L ₃ are examined as well as some asymmetric periodic oscillations related to them. An effort is made to complete the global picture of simple-periodic symmetric oscillations in the present case of the problem (the S-J case). This is done by examining the orbits with initial conditions such that the infinitesimal body starts from a position on the ₁-axis (₀₂ = 0) with a negative initial velocity perpendicular to this axis . In a previous article this investigation has been carried out for negative values of ₀₁, where the position of the small primary defines ₁=0. Now we proceed to consider orbits with ₀₁>0. The phase portrait of asymmetric periodic orbits is also examined.