A note on relative motion in the general three-body problem

It is shown that the equations of the general three-body problem take on a very symmetric form when one considers only their relative positions, rather than position vectors relative to some given coordinate system. From these equations one quickly surmises some well known classical properties of the three-body problem such as the first integrals and the equilateral triangle solutions. Some new Lagrangians with relative coordinates are also obtained. Numerical integration of the new equations of motion is about 10 percent faster than with barycentric or heliocentric coordinates.     

Bounded motion for a modified three-body problem

The energy and the angular momentum integral of motion for the planar three point problem cannot assure bounded motion. In this paper it is shown that if one of the points is replaced by a homogeneous sphere then bounded motion can be found. The zero velocity surfaces for this modified problem are found and their evolution is described.     

On relative periodic solutions of the planar general three-body problem

We describe two relatively simple reductions to order 6 for the planar general three-body problem. We also show that this reduction leads to the distinction between two types of periodic solutions: absolute or relative periodic solutions. An algorithm for obtaining relative periodic solutions using heliocentric coordinates is then described. It is concluded from the periodicity conditions that relative periodic solutions must form families with a single parameter. Finally, two such families have been obtained numerically and are described in some detail.The present research was carried out partially at the University of California and partially at the Jet Propulsion Laboratory under contract NAS7-100 with NASA.     

On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited

In the framework of the planar and circular restricted three-body problem, we consider an asteroid that orbits the Sun in quasi-satellite motion with a planet. A quasi-satellite trajectory is a heliocentric orbit in co-orbital resonance with the planet, characterized by a nonzero eccentricity and a resonant angle that librates around zero. Likewise, in the rotating frame with the planet, it describes the same trajectory as the one of a retrograde satellite even though the planet acts as a perturbator. In the last few years, the discoveries of asteroids in this type of motion made the term “quasi-satellite” more and more present in the literature. However, some authors rather use the term “retrograde satellite” when referring to this kind of motion in the studies of the restricted problem in the rotating frame. In this paper, we intend to clarify the terminology to use, in order to bridge the gap between the perturbative co-orbital point of view and the more general approach in the rotating frame. Through a numerical exploration of the co-orbital phase space, we describe the quasi-satellite domain and highlight that it is not reachable by low eccentricities by averaging process. We will show that the quasi-satellite domain is effectively included in the domain of the retrograde satellites and neatly defined in terms of frequencies. Eventually, we highlight a remarkable high eccentric quasi-satellite orbit corresponding to a frozen ellipse in the heliocentric frame. We extend this result to the eccentric case (planet on an eccentric motion) and show that two families of frozen ellipses originate from this remarkable orbit.     

Central configurations and hyperbolic-elliptic motion in the three-body problem

A refined classification of motion for the planar three-body problem with zero total energy is presented. In addition, the structure and size of the sets of initial conditions are obtained. Limited results for the spatial problem are also given.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.Supported in part by the Natural Science Fund and the University Research Council of Vanderbilt University.     

On the elliptic restricted three-body problem

The main goal of this paper is to show that the elliptic restricted three-body problem has ejection-collision orbits when the mass parameter µ is small enough. We make use of the blow up techniques. Moreover, we describe the global flow of the elliptic problem when µ = 0 taking into account the singularities due to collision and to infinity.     

On the double averaged three-body problem

The Hamiltonian of three point masses is averaged over fast variablel and ll (mean anomalies) The problem is non-planar and it is assumed that two of the bodies form a close pair (stellar three-body problem). Only terms up to the order of (a/á)⁴ are taken into account in the Hamiltonian, wherea andá are the corresponding semi-major axes. Employing the method of elimination of the nodes, the problem may be reduced to one degree of freedom. Assuming in addition that the angular momentum of the close binary is much smaller than the angular momentum of the motion of the binary around a third body, we were able to solve the equation for the eccentricity changes in terms of the Jacobian elliptic functions.     

Integrals of motion in the elliptic three-dimensional restricted three-body problem

Three integrals of motion have been found in the three-dimensional elliptic restricted three-body problem for small eccentricitye of the relative orbit of the primaries and small distancer and eccentricitye of the orbit of the third body around a primary. The integrals are given in the form of formal series in the mass-ratio , the eccentricitiese, e and the coordinates and velocities. These integrals depend periodically on the time.     

On the motion around the centre of mass of a viscously elastic sphere in the restricted elliptic three-body problem

In the restricted three-body problem we consider the motion of a viscously elastic sphere (planet) with its centre of mass moving in a conditionally-periodic orbit. The approximate equations describing the rotational motion of the sphere in terms of the Andoyer variables are obtained by the method of the separation of motion and averaging; the evolution of the motion is also analysed.     

Stability of motion of the restricted circular and charged three-body problem

Stabiliity is applied to characterize type of motion in which the moving body is confined to certain limited regions and in this sense we may say that the motion of the body in question is stable. This method has been used in the past chiefly in connection with the classical restricted problem of three bodies.In this paper we consider a dynamical system defined by the Lagrangian

Contribution to the solution of the three-body problem in power series form

After reviewing the existing procedures for solving the three-body problem by convergent power series, the author develops two algebraic methods in terms of the independent variable which is either the time t or Levi-Civita's regularizing variable u. These power series solve in Weierstrass' and Painlevé's sense the problem formulated in its greatest generality, since no restrictions at all are made on the order of magnitude of masses and none of the three bodies is restricted to moving along a prescribed conic section. Besides, the reference system used is a tridimensional Cartesian one. In the t-domain, the expressions for the high-order derivatives of the coordinates are computed using repeatedly Leibnitz's rule for derivatives of products of functions. In the u-domain, an extremely simple successive approximation procedure is established by means of a single recursion formula which requires elementary operations to be performed on polynomials of increasing degrees.     

Parameter values for stable low-inclination periodic motion in the restricted three-body problem with oblateness

The known intervals of possible stability, on the mgr-axis, of basicfamilies of 3D periodic orbits in the restricted three-body problem areextended into -A₁ regions for oblate larger primary, A ₁ beingthe oblateness coefficient. Eight regions, corresponding to the basicstable bifurcation orbits l1v, l1v, l2v, l3v, m1v, m1v,m2v, i1v are determined and related branching 3D periodic orbits arecomputed systematically and tested for stability. The regions for l1v,m1v and m2v survive the test emerging as the regions allowing thesimplest types of stable low inclination 3D motion. For l1v, l2v,l3v, m1v and m2v oblateness seems to have a stabilising effect,while stability of i1v survives only for a very small range of A ₁values.     

On the restricted three-body problem with generalized forces

By introducing general functions which depend on distance, a general scheme which determines the equilibrium solutions for the generalized restricted three-body problem is given. Applications to problems such as primaries considered as rigid bodies, influence of the radiation pressure of the primaries, and a combination of radiation pressure and rigid body are presented.     

On topological stability in the general three-body problem

The confining curves in the general three-body problem are studied; the role of the integralc ² h (angular momentum squared times energy) as bifurcation parameter is established in a very simple way by using symmetries and changes of scale. It is well known (Birkhoff, 1927) that the bifurcations of the level manifolds of the classical integrals occur at the Euler-Lagrange relative equilibrium configurations. For small values of the mass ratio ε=m ₃/m ₂ both the positions of the collinear equilibrium points and thec ² h integral are expanded in power series of ε. In this way the relationship is found between the confining curves resulting from thec ² h integral in the general problem, and the zero velocity curves given by the Jacobi integral in the corresponding restricted problem. For small values of ε the singular confining curves in the general and in the restricted problem are very similar, but they do not correspond to each other: the offset of the two bifurcation values is, in the usual, system of units of the restricted problem, about one half of the eccentricity squared of the orbits of the two larger bodies. This allows the definition of an approximate stability criterion, that applies to the systems with small ε, and quantifies the qualitatively well known destabilizing effect of the eccentricity of the binary on the third body. Because of this destabilizing effect the third body cannot be bounded by any topological criterion based on the classical integrals unless its mass is larger than a minimum value. As an example, the three-body systems formed by the Sun, Jupiter and one of the small planets Mercury, Mars, Pluto or anyone of the asteroids are found to be ‘unstable’, i.e. there is no way of proving, with the classical integrals, that they cannot cross the orbit of Jupiter. This can be reliably checked with the approximate stability criterion, that given for the most important three-body subsystems of the Solar System essentially the same information on ‘stability’ as the full computation of thec ² h integral and of the bifurcation values.     

Numerical stabilization of the differential equations of Keplerian motion

A stabilization of the classical equations of two-body motion is offered. It is characterized by the use of the regularizing independent variable (eccentric anomaly) and by the addition of a control-term to the differential equations. This method is related to the KS-theory (Stiefel, 1970) which performed for the first time a stabilization of the Kepler motion. But in contrast to the KS-theory our method does not transform the coordinates of the particle. As far as the theory of stability and the numerical experiments are concerned we restrict ourselves to thepure Kepler motion. But, of course, the stabilizing devices will also improve the accuracy of the computation of perturbed orbits. We list, therefore, also the equations of the perturbed motion.     

About regions of convergence of expansions of differential equations of the three-dimensional restricted three-body problem in the vicinity of collinear libration points

The analysis of regions of convergence of expansions of the right-hand sides of the differential equations of motion in the vicinity of the collinear libration points in the circular restricted three-body problem in powers of coordinates of the infinitesimal body due to F. R. Moulton (Moulton, 1920) is shown to be erroneous and his results are corrected. The generalisation of Moulton's results to analogous expansions of the equations in the elliptic problem of three bodies made by R. W. Farquhar (Farquhar, 1968) is shown to be groundless.     

The photogravitational three-body problem: The approximation of the collinear equilibria in a simple compact form

The collinear equilibrium points of the three-body problem for cases where the moving particle is further acted upon by a weak radiation pressure are given approximately in a simple compact form.