Evolution of a ‘Class Two’ family of periodic orbits in the general planar problem of three bodies

The elliptic restricted problem of three bodies with unit eccentricity of the primaries is used to generate a family of periodic orbits in the general problem of three bodies. The parameter of the family is the mass of one of the participating bodies. This varies from zero to a termination value. The mass ratio of the primaries of the unperturbed problem (three to five) is maintained throughout the generation of the family. In this way an asymmetry is introduced generalizing the Copenhagen elliptic problem as the generating model. All members of the family experience a close approach and a collision between the primaries during half of the period of the orbit, therefore, the family is classified as Class Two.     

Parametric evolution of periodic orbits in the restricted four-body problem with radiation pressure

The backbone of the analysis in most dynamical systems is the study of periodic motions, since they greatly assist us to understand the structure of all possible motions. In this paper, we deal with the photogravitational version of the rectilinear restricted four-body problem and we investigate the dynamical behaviour of a small particle that is subjected to both the gravitational attraction and the radiation pressure of three bodies much bigger than the particle, the primaries. These bodies are always in syzygy and two of them have equal masses and are located at equal distances from the third primary. We study the effect of radiation on the distribution of the periodic orbits, their stability, as well as the evolution of the families and their main features.     

Numerical explorations of the rectilinear problem of three bodies

We present some results of a numerical exploration of the rectilinear problem of three bodies, with the two outer masses equal. The equations of motion are first given in relative coordinates and in regularized variables, removing both binary collision singularities in a single coordinate transformation. Among our most important results are seven periodic solutions and three symmetric triple collision solutions. Two of these periodic solutions have been continued into families, the outer massm ₃ being the family parameter. One of these families exists for all masses while the second family is a branch of the first at a second-kind critical orbit. This last family ends in a triple collision orbit.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

The rectilinear three-body problem as a basis for studying highly eccentric systems

The rectilinear elliptic restricted three-body problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity \(e'=1\), but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (AIAA J 7:1003–1009, 1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter \(\mu =0.5\) (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke’s computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to \(\mu \) and \(e'<1\). Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449.     

The libration points of axisymmetric satellite in the gravitation field of two triaxial rigid bodies

In this paper we consider the restricted problem of three rigid bodies (an axisymmetric satellite in the gravitation field of two triaxial primaries). The collinear and triangular equilibrium solutions are obtained. The effect of the primaries on the location of the libration points of a spherical satellite has been studied numerically.     

A note on the elliptic restricted three-body problem

This work considers periodic solutions, arc-solutions (solutions with consecutive collisions) and double collision orbits of the plane elliptic restricted problem of three bodies for =0 when the eccentricity of the primaries,e _p , varies from 0 to 1. Characteristic curves of these three kinds of solutions are given.     

Families of first kind periodic solutions of the spatial and planar elliptic restricted three-body problem with collision

We consider a restricted three-body problem where the primaries are moving in an elliptic collision orbit and the infinitesimal mass moves in a three dimensional space. This paper is devoted to prove analytically the existence of several families of symmetric periodic solutions as continuation of Keplerian circular orbits. In our approach the perturbing parameter is related with the energy of the primaries.     

Families of periodic orbits in the restricted four-body problem

In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented. Three bodies of masses m ₁, m ₂ and m ₃ (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal and finally when we have three unequal masses. Series, with respect to the mass m ₃, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of the mass parameters are also calculated.     

Étude du choc de deux petites masses en présence d'un troisième corps de masse prépondérante

The analytical study of the evolution in the rectilinear problem of three bodies, leads us to consider the collision between two bodies,M ₂ andM ₃, in the presence of the third body,M ₁. This problem, which seems to be difficult to approach in the general case, can be partly solved if the masses ofM ₂ andM ₃ are equal and can be neglected in regard toM ₁. In this particular case of the general problem, the mechanical study of a collision betweenM ₂ andM ₃, leads to two distinct types of collisions: ‘instantaneous collisions’, and ‘collisions with repetition’, according to the value of a parameter which depends on the position and the speed of the binaryM ₂ M ₃, relative toM ₁, in the collision. In the first type, the collision exchanges the speeds ofM ₂ andM ₃, while in the second type, there is a series of collisions succeeding each other.     

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.     

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.     

Analysis of some degenerate quadruple collisions

We consider the trapezoidal problem of four bodies. This is a special problem where only three degrees of freedom are involved. The blow up method of McGehee can be used to deal with the quadruple collision. Two degenerate cases are studied in this paper: the rectangular and the collinear problems. They have only two degrees of freedom and the analysis of total collapse can be done in a way similar to the one used for the collinear and isosceles problems of three bodies. We fully analyze the flow on the total collision manifold, reducing the problem of finding heteroclinic connections to the study of a single ordinary differential equation. For the collinear case, from which arises a one parameter family of equations, the analysis for extreme values of the parameter is done and numerical computations fill up the gap for the intermediate values. Dynamical consequences for possible motions near total collision as well as for regularization are obtained.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.Dedicated to Prof. Szebehely on the occasion of his sixtieth birthday.     

Two‐Body Problems with Drag or Thrust: Qualitative Results

We consider two‐body problems in which the drag is proportional to the velocity divided by the square of the distance and whose radial and tangential components have distinct coefficients. For all parameters, we study the flow of the system obtained by suitable coordinate and time transformations and draw conclusions about the qualitative behavior of solutions. In each case, we examine the existence of collision–ejection, collision–escape, capture–collision, capture–escape, and oscillatory rectilinear orbits, study the motion near collision, and show that if periodic orbits exist they must be limit cycles. This revised version was published online in July 2006 with corrections to the Cover Date.     

The rectilinear three-body problem using symbol sequence I. Role of triple collision

The dynamical structure of phase space of gravitational Newtonian three bodies which lie on a line (rectilinear three-body system) is studied. We take an initial value plane and classify the points on the plane according to the fate of the orbits starting from the points, using symbol sequences. The structure appearing on the initial value plane with this classification was well studied for the equal-mass case (Tanikawa and Mikkola 2000, Chaos 10, 649–657). In this paper, we follow and clarify the changes of this structure with the mass ratio of three particles.     

The equilibrium solutions of restricted problem of three axisymmetric rigid bodies

In this paper we consider the restricted problem of three axisymmetric rigid bodies under the central forces. The collinear and triangular equilibrium solutions are obtained. Finally a numerical study of the influence of the non-sphericity and the rotation of the primaries in the location of the libration points is made.     

Einstein-Maxwell field equations in isotropic coordinates: an application to neutron star and quark star

The photogravitational restricted three bodies within the framework of the post-Newtonian approximation is carried out. The mass of the primaries are assumed changed under the effect of continuous radiation process and oblateness effects of the two primaries. New perturbed locations of the triangular points are computed. In order to introduce a semi-analytical view, A Mathematica program is constructed so as to draw the locations of triangular points versus the whole range of the mass ratio μ taking into account the photo-gravitational effects, the relativistic corrections and/or oblateness effects. All the obtained figures are analyzed.     

THE PHASE SPACE STRUCTURE OF THE EXTENDED SITNIKOV-PROBLEM

In this article we treat the 'Extended Sitnikov Problem' where three bodies of equal masses stay always in the Sitnikov configuration. One of the bodies is confined to a motion perpendicular to the instantaneous plane of motion of the two other bodies (called the primaries), which are always equally far away from the barycenter of the system (and from the third body). In contrary to the Sitnikov Problem with one mass less body the primaries are not moving on Keplerian orbits. After a qualitative analysis of possible motions in the 'Extended Sitnikov Problem' we explore the structure of phase space with the aid of properly chosen surfaces of section. It turns out that for very small energies H the motion is possible only in small region of phase space and only thin layers of chaos appear in this region of mostly regular motion. We have chosen the plane ( ) as surface of section, where r is the distance between the primaries; we plot the respective points when the three bodies are 'aligned'. The fixed point which corresponds to the 1 : 2 resonant orbit between the primaries' period and the period of motion of the third mass is in the middle of the region of motion. For low energies this fixed point is stable, then for an increased value of the energy splits into an unstable and two stable fixed points. The unstable fixed point splits again for larger energies into a stable and two unstable ones. For energies close toH = 0 the stable center splits one more time into an unstable and two stable ones. With increasing energy more and more of the phase space is filled with chaotic orbits with very long intermediate time intervals in between two crossings of the surface of section. We also checked the rotation numbers for some specific orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

Periodic orbits in the restricted four-body problem with two equal masses

The restricted (equilateral) four-body problem consists of three bodies of masses m ₁, m ₂ and m ₃ (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them.     

Low-thrust propulsion in a coplanar circular restricted four body problem

This paper formulates a circular restricted four body problem (CRFBP), where the three primaries are set in the stable Lagrangian equilateral triangle configuration and the fourth body is massless. The analysis of this autonomous coplanar CRFBP is undertaken, which identifies eight natural equilibria; four of which are close to the smaller body, two stable and two unstable, when considering the primaries to be the Sun and two smaller bodies of the Solar System. Following this, the model incorporates ‘near term’ low-thrust propulsion capabilities to generate surfaces of artificial equilibrium points close to the smaller primary, both in and out of the plane containing the celestial bodies. A stability analysis of these points is carried out and a stable subset of them is identified. Throughout the analysis the Sun-Jupiter-asteroid-spacecraft system is used, for conceivable masses of a hypothetical asteroid set at the libration point L ₄. It is shown that eight bounded orbits exist, which can be maintained with a constant thrust less than 1.5 × 10⁻⁴ N for a 1000 kg spacecraft. This illustrates that, by exploiting low-thrust technologies, it would be possible to maintain an observation point more than 66% closer to the asteroid than that of a stable natural equilibrium point. The analysis then focusses on a major Jupiter Trojan: the (624) Hektor asteroid. The thrust required to enable close asteroid observation is determined in the simplified CRFBP model. Finally, a numerical simulation of the real Sun-Jupiter-(624) Hektor-spacecraft is undertaken, which tests the validity of the stability analysis of the simplified model.     

An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem

The model of extended Sitnikov Problem contains two equally heavy bodies of mass m moving on two symmetrical orbits w.r.t the centre of gravity. A third body of equal mass m moves along a line z perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the primaries plane the third body describes an oscillatory motion around it. The motion of the three bodies is described by a coupled system of second order differential equations for the radial distance of the primaries r and the third mass oscillation z. This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non integrable and therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence of the function iteration method is given and the iterations are carried out to order 1 in r and to order 2 in z. For small values of the initial oscillation amplitude of the third mass we obtain results in very good agreement to numerically obtained solutions.