Families of Periodic Orbits Emanating From Homoclinic Orbits in the Restricted Problem of Three Bodies

We describe and comment the results of a numerical exploration on the evolution of the families of periodic orbits associated with homoclinic orbits emanating from the equilateral equilibria of the restricted three body problem for values of the mass ratio larger than μ ₁. This exploration is, in some sense, a continuation of the work reported in Henrard [Celes. Mech. Dyn. Astr. 2002, 83, 291]. Indeed it shows how, for values of μ. larger than μ ₁, the Trojan web described there is transformed into families of periodic orbits associated with homoclinic orbits. Also we describe how families of periodic orbits associated with homoclinic orbits can attach (or detach) themselves to (or from) the best known families of symmetric periodic orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

Periodic Orbits in Rotating Second Degree and Order Gravity Fields

Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these periodic orbits, we remove the two unity eigenvalues from the state transition matrix to find a robust, non-singular linear map to solve for the periodic orbits. The algorithm converges well, especially for stable periodic orbits. Using the searching procedure, which is relatively automatic, we find five basic families of periodic orbits in the rotating second degree and order gravity field for planar motion, and discuss their existence and stability at different central body rotation rates.     

Periodic Orbits in Rotating Second Degree and Order Gravity Fields

Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these periodic orbits, we remove the two unity eigenvalues from the state transition matrix to find a robust, non-singular linear map to solve for the periodic orbits. The algorithm converges well, especially for stable periodic orbits. Using the searching procedure, which is relatively automatic, we find five basic families of periodic orbits in the rotating second degree and order gravity field for planar motion, and discuss their existence and stability at different central body rotation rates.     

Displaced non-Keplerian orbits using impulsive thrust

This paper investigates new families of displaced, highly non-Keplerian orbits in the two-body problem and artificial equilibria in the circular restricted three-body problem. The families of orbits presented extend prior work by using periodic impulses to generate displaced orbits rather than continuous thrust. The new displaced orbits comprise a sequence of individual Keplerian arcs whose intersection is continuous in position, with discontinuities in velocity removed using impulses. For frequent impulses the new families of orbits approximate continuous thrust non-Keplerian orbits found in previous studies. To generate approximations to artificial equilibria in the circular restricted three-body problem, periodic impulses are used to generate a sequence of connected three-body arcs which begin and terminate at a fixed position in the rotating frame of reference. Again, these families of orbits reduce to the families of artificial equilibria found using continuous thrust.     

2/1 resonant periodic orbits in three dimensional planetary systems

We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to 50^?–60^?, which may be related with the existence of real planetary systems.     

Energy and stability in the Full Two Body Problem

The conditions for relative equilibria and their stability in the Full Two Body Problem are derived for an ellipsoid–sphere system. Under constant angular momentum it is found that at most two solutions exist for the long-axis solutions with the closer solution being unstable while the other one is stable. As the non-equilibrium problem is more common in nature, we look at periodic orbits in the F2BP close to the relative equilibrium conditions. Families of periodic orbits can be computed where the minimum energy state of one family is the relative equilibrium state. We give results on the relative equilibria, periodic orbits and dynamics that may allow transition from the unstable configuration to a stable one via energy dissipation.      

Periodic orbits of the elliptic restricted problem for the Sun-Jupiter-Saturn system

A systematic approach to generate periodic orbits in the elliptic restricted problem of three bodies in introduced. The approach is based on (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to periodic orbits in the elliptic restricted problem. Two families of periodic orbits of the elliptic restricted problem are found by this approach. The mass ratio of the primaries of these orbits is equal to that of the Sun-Jupiter system. The sidereal mean motions between the infinitesimal body and the smaller primary are in a 2:5 resonance, so as to approximate the Sun-Jupiter-Saturn system. The linear stability of these periodic orbits are studied as functions of the eccentricities of the primaries and of the infinitesimal body. The results show that both stable and unstable periodic orbits exist in the elliptic restricted problem that are close to the actual Sun-Jupiter-Saturn system. However, the periodic orbit closest to the actual Sun-Jupiter-Saturn system is (linearly) stable.     

Second Species Periodic Orbits of the Elliptic 3 Body Problem

We consider the plane restricted elliptic 3 body problem with small mass ratio and small eccentricity and prove the existence of many periodic orbits shadowing chains of collision orbits of the Kepler problem. Such periodic orbits were first studied by Poincaré for the non-restricted 3 body problem. Poincaré called them second species solutions.     

on the relationship between fast lyapunov indicator and periodic orbits for symplectic mappings

The computation on a relatively short time of a quantity, related to the largest Lyapunov Characteristic Exponent, called Fast Lyapunov Indicator allows to discriminate between ordered and weak chaotic motion and also, under certain conditions, between resonant and non resonant regular orbits. The aim of this paper is to study numerically the relationship between the Fast Lyapunov Indicator values and the order of periodic orbits. Using the two-dimensional standard map as a model problem we have found that the Fast Lyapunov Indicator increases as the logarithm of the order of periodic orbits up to a given order. For higher order the Fast Lyapunov Indicator grows linearly with the order of the periodic orbits. We provide a simple model to explain the relationship that we have found between the values of the Fast Lyapunov Indicator, the order of the periodic orbits and also the minimum number of iterations needed to obtain the Fast Lyapunov Indicator values.     

Families of periodic collision orbits in the general three-body problem

In the general three-body problem, in a rotating frame of reference, a symmetric periodic solution with a binary collision is determined by the abscissa of one body and the energy of the system. For different values of the masses of the three bodies, the symmetric periodic collision orbits form a two-parametric family. In the case of equal masses of the two bodies and small mass of the third body, we found several symmetric periodic collision orbits similar to the corresponding orbits in the restricted three-body problem. Starting with one symmetric periodic collision orbit we obtained two families of such orbits. Also starting with one collision orbit in the Sun-Jupiter-Saturn system we obtained, for a constant value of the mass ratio of two bodies, a family of symmetric periodic collision orbits.     

Periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body

We have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body. We have determined the periodic orbits for different values of μ, h and A (h is energy constant, μ is mass ratio of the two primaries and A is an oblateness factor). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by Karimov and Sokolsky (Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of oblateness by taking some fixed values of μ, A and h. As starters for our method, we use some known periodic orbits in the classical restricted three body problem.     

Families of periodic orbits in the planar three-body problem

A periodic orbit of the restricted circular three-body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massm ₃ of the third body. This family is continued numerically up to a maximum value of the mass of the originally small body, which corresponds to a mass ratiom ₁:m ₂:m ₃?5:5:3. From that point on the family continues for decreasing massesm ₃ until this mass becomes again equal to zero. It turns out that this final orbit of the family is a periodic orbit of the elliptic restricted three body problem. These results indicate clearly that families of periodic motions of the three-body problem exist for fixed values of the three masses, since this continuation can be applied to all members of a family of periodic orbits of the restricted three-body problem. It is also indicated that the periodic orbits of the circular restricted problem can be linked with the periodic orbits of the elliptic three-body problem through periodic orbits of the general three-body problem.     

From the circular to the spatial elliptic restricted three-body problem

We deal with the study of the spatial restricted three-body problem in the case where the small particle is far from the primaries, that is, the so-called comet case. We consider the circular problem, apply double averaging and compute the relative equilibria of the reduced system. It appears that, in the circular problem, we find not only part of the equilibria existing in the elliptic case, but also new ones. These critical points are in correspondence with periodic and quasiperiodic orbits and invariant tori of the non-averaged Hamiltonian. We explain carefully the transition between the circular and the elliptic problems. Moreover, from the relative equilibria of elliptic type, we obtain invariant 3-tori of the original system.     

Resonant periodic orbits in the exoplanetary systems

The planetary dynamics of 4/3, 3/2, 5/2, 3/1 and 4/1 mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to 60^° of mutual planetary inclination, but in most families, the stability does not exceed 20^°–30^°, depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the 4/3 and 5/2 resonance, respectively.     

Families of periodic orbits in the three-body problem

We show by a general argument that periodic solutions of the planar problem of three bodies (with given masses) form one-parameter families. This result is confirmed by numerical investigations: two orbits found earlier by Standish and Szebehely are shown to belong to continuous one-parameter families of periodic orbits. In general these orbits have a non-zero angular momentum, and the configuration after one period is rotated with respect to the initial configuration. Similar general arguments whow that in the three-dimensional problem, periodic orbits form also one-parameter families; in the one-dimensional problem, periodic orbits are isolated.     

New periodic solutions of the three dimensional restricted problem

A global review of the symmetric solutions of the restricted problem made in the Introduction opens a window on new symmetric periodic orbits of the two body problem in rotating axes which could be ‘trivially’ continuable to symmetric periodic orbits of the three dimensional restricted problem for small values of μ (see Figure 3). The proof of this possibility of continuation is given in Sections 1, 2, 3 using regularizing variables.     

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.     

Periodic orbits and bifurcations in the Sitnikov four-body problem

We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż₀ varies within a finite interval (while z ₀ tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity.     

Symmetric and asymmetric 3:1 resonant periodic orbits with an application to the 55Cnc extra-solar system

We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit.     

Properties of linearized mappings associated with periodic orbits in the three-body problem

In this paper three results on the linearized mapping associated with the plane three body problem near a periodic orbit are established. It is first shown that linear stability of such an orbit is independent of initial position on the orbit and of coordinate system. Second, the relation of Hénon connecting the rates of change of rotation angle and period on an isoenergetic family of periodic orbits is proved, together with a similar relation for families of orbits closing exactly in a rotating coordinate system. Finally, a condition for a critical orbit is given which is applicable to any family of periodic orbits.