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.     

Families of periodic planetary-type orbits in the three-body problem and their stability

Several families of the planar general three-body problem for fixed values of the three masses are found, in a rotating frame of reference, where the mass of two of the bodies is small compared to the mass of the third body. These families were obtained by the continuation of a degenerate family of periodic orbits of three bodies where two of the bodies have zero masses and describe circular orbits around a third body with finite mass, in the same direction.The above families represent planetary systems with the body with the large mass representing the Sun and the two small bodies representing two planets or comets. One section of a family is shown to represent the Jupiter family of comets and also a model for the Sun-Jupiter-Saturn system is found.The stability analysis revealed that stability exists for small masses and small eccentricities of the two planets. Planetary systems with relatively large masses and eccentricities are proved to be unstable. In particular, the Jupiter family of comets, for small masses of the two small bodies, and the Sun-Jupiter-Saturn system are proved to be stable. Also, it was shown that resonances are not necessarily associated with instabilities.     

The 1/1 resonance in extrasolar systems

We present families of symmetric and asymmetric periodic orbits at the 1/1 resonance, for a planetary system consisting of a star and two small bodies, in comparison to the star, moving in the same plane under their mutual gravitational attraction. The stable 1/1 resonant periodic orbits belong to a family which has a planetary branch, with the two planets moving in nearly Keplerian orbits with non zero eccentricities and a satellite branch, where the gravitational interaction between the two planets dominates the attraction from the star and the two planets form a close binary which revolves around the star. The stability regions around periodic orbits along the family are studied. Next, we study the dynamical evolution in time of a planetary system with two planets which is initially trapped in a stable 1/1 resonant periodic motion, when a drag force is included in the system. We prove that if we start with a 1/1 resonant planetary system with large eccentricities, the system migrates, due to the drag force, along the family of periodic orbits and is finally trapped in a satellite orbit. This, in principle, provides a mechanism for the generation of a satellite system: we start with a planetary system and the final stage is a system where the two small bodies form a close binary whose center of mass revolves around the star.     

Equilibrium points and Lyapunov families in the circular restricted three-body problem with an oblate primary and a synchronous rotating dipole secondary: Application to Luhman-16 binary system

We numerically study a version of the synchronous circular restricted three-body problem, where an infinitesimal mass body is moving under the Newtonian gravitational forces of two massive bodies. The primary body is an oblate spheroid while the secondary is an elongated asteroid of a combination of two equal masses forming a rotating dipole which is synchronous to the rotation of the primaries of the classic circular restricted three-body problem. In this paper, we systematically examine the existence, positions, and linear stability of the equilibrium points for various combinations of the model's parameters. We observe that the perturbing forces have significant effects on the positions and stability of the equilibrium points as well as the regions where the motion of the particle is allowed. The allowed regions of motion as determined by the zero-velocity surface and the corresponding isoenergetic curves as well as the positions of the equilibrium points are given. Finally, we numerically study the binary system Luhman-16 by computing the positions of the equilibria and their stability as well as the allowed regions of motion of the particle. The corresponding families of periodic orbits emanating from the collinear equilibrium points are computed along with their stability properties.     

The equilibria and periodic orbits around a dumbbell-shaped body

This paper investigates the equilibria, their stability, and the periodic orbits in the vicinity of a rotating dumbbell-shaped body. First, the geometrical model of dumbbell-shaped body is established. The gravitational potential fields are obtained by the polyhedral method for several dumbbell-shaped bodies with various length–diameter ratios. Subsequently, the equilibrium points of these dumbbell-shaped bodies are computed and their stabilities are analyzed. Periodic orbits around equilibrium points are determined by the differential correction method. Finally, in order to understand further motion characteristic of dumbbell-shaped body, the effect of the rotating angular velocity of the dumbbell-shaped bodies is investigated. This study extends the research work of the orbital dynamics from simple shaped bodies to complex shaped bodies and the results can be applied to the dynamics of orbits around some asteroids.     

Periodic attitudes and bifurcations of a rigid spacecraft in the second degree and order gravity field of a uniformly rotating asteroid

In this work, periodic attitudes and bifurcations of periodic families are investigated for a rigid spacecraft moving on a stationary orbit around a uniformly rotating asteroid. Under the second degree and order gravity field of an asteroid, the dynamical model of attitude motion is formulated by truncating the integrals of inertia of the spacecraft at the second order. In this dynamical system, the equilibrium attitude has zero Euler angles. The linearised equations of attitude motion are utilised to study the stability of equilibrium attitude. It is found that there are three fundamental types of periodic attitude motions around a stable equilibrium attitude point. We explicitly present the linear solutions around a stable equilibrium attitude, which can be used to provide the initial guesses for computing the true periodic attitudes in the complete model. By means of a numerical approach, three fundamental families of periodic attitudes are studied, and their characteristic curves, distribution of eigenvalues, stability curves and stability distributions are determined. Interestingly, along the characteristic curves of the fundamental families, some critical points are found to exist, and these points correspond to tangent and period-doubling bifurcations. By means of a numerical approach, the bifurcated families of periodic attitudes are identified. The natural and bifurcated families constitute networks of periodic attitude families.     

Periodic solutions about the collinear lagrangian solution in the general problem of three bodies

The article describes the solutions near Lagrange's circular collinear configuration in the planar problem of three bodies with three finite masses. The article begins with a detailed review of the properties of Lagrange's collinear solution. Lagrange's quintic equation is derived and several expressions are given for the angular velocity of the rotating frame.The equations of motion are then linearized near the circular collinear solution, and the characteristic equation is also derived in detail. The different types of roots and their corresponding solutions are discussed. The special case of two equal outer masses receives special attention, as well as the special case of two small outer masses.Finally, the fundamental family of periodic solutions is extended by numerical integration all the wap up to and past a binary collision orbit. The stability and the bifurcations of this family are briefly enumerated.     

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.     

The 1/1 resonance in extrasolar planetary systems

We study orbits of planetary systems with two planets, for planar motion, at the 1/1 resonance. This means that the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit, at a certain epoch, take different values. We consider the general case of different planetary masses and, as a special case, we consider equal planetary masses. We start with the exact resonance, which we define as the 1/1 resonant periodic motion, in a rotating frame, and study the topology of the phase space and the long term evolution of the system in the vicinity of the exact resonance, by rotating the orbit of the outer planet, which implies that the resonance and the eccentricities are not affected, but the symmetry is destroyed. There exist, for each mass ratio of the planets, two families of symmetric periodic orbits, which differ in phase only. One is stable and the other is unstable. In the stable family the planetary orbits are in antialignment and in the unstable family the planetary orbits are in alignment. Along the stable resonant family there is a smooth transition from planetary orbits of the two planets, revolving around the Sun in eccentric orbits, to a close binary of the two planets, whose center of mass revolves around the Sun. Along the unstable family we start with a collinear Euler–Moulton central configuration solution and end to a planetary system where one planet has a circular orbit and the other a Keplerian rectilinear orbit, with unit eccentricity. It is conjectured that due to a migration process it could be possible to start with a 1/1 resonant periodic orbit of the planetary type and end up to a satellite-type orbit, or vice versa, moving along the stable family of periodic orbits.     

On the circular Sitnikov problem: the alternation of stability and instability in the family of vertical motions

This paper is devoted to the special case of the restricted circular three-body problem, when the two primaries are of equal mass, while the third body of negligible mass performs oscillations along a straight line perpendicular to the plane of the primaries (so called periodic vertical motions). The main goal of the paper is to study the stability of these periodic motions in the linear approximation. A special attention is given to the alternation of stability and instability within the family of periodic vertical motions, whenever their amplitude is varied in a continuous monotone manner.     

Spin axis evolution of two interacting bodies

We consider the solid-solid interactions in the two body problem. The relative equilibria have been previously studied analytically and general motions were numerically analyzed using some expansion of the gravitational potential up to the second order, but only when there are no direct interactions between the orientation of the bodies. Here we expand the potential up to the fourth order and we show that the secular problem obtained after averaging over fast angles, as for the precession model of Boué and Laskar [Boué, G., Laskar, J., 2006. Icarus 185, 312-330], is integrable, but not trivially. We describe the general features of the motions and we provide explicit analytical approximations for the solutions. We demonstrate that the general solution of the secular system can be decomposed as a uniform precession around the total angular momentum and a periodic symmetric orbit in the precessing frame. More generally, we show that for a general n-body system of rigid bodies in gravitational interaction, the regular quasiperiodic solutions can be decomposed into a uniform precession around the total angular momentum, and a quasiperiodic motion with one frequency less in the precessing frame.     

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.     

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.     

Dynamics Of 2/1 Resonant Extrasolar Systems Application to HD82943 and GLIESE876

A complete study is made of the resonant motion of two planets revolving around a star, in the model of the general planar three body problem. The resonant motion corresponds to periodic motion of the two planets, in a rotating frame, and the position and stability properties of the periodic orbits determine the topology of the phase space and consequently play an important role in the evolution of the system. Several families of symmetric periodic orbits are computed numerically, for the 2/1 resonance, and for the masses of some observed extrasolar planetary systems. In this way we obtain a global view of all the possible stable configurations of a system of two planets. These define the regions of the phase space where a resonant extrasolar system could be trapped, if it had followed in the past a migration process.The factors that affect the stability of a resonant system are studied. For the same resonance and the same planetary masses, a large value of the eccentricities may stabilize the system, even in the case where the two planetary orbits intersect. The phase of the two planets (position at perihelion or aphelion when the star and the two planets are aligned) plays an important role, and the change of the phase, other things being the same, may destabilize the system. Also, the ratio of the planetary masses, for the same total mass of the two planets, plays an important role and the system, at some resonances and some phases, is destabilized when this ratio changes.The above results are applied to the observed extrasolar planetary systems HD 82943, Gliese 876 and also to some preliminary results of HD 160691. It is shown that the observed configurations are close to stable periodic motion.     

Analytical expansions of torque-free motions for short and long axis modes

Torque-free motion of a rigid body is integrable and its solution is expressed in terms of elliptic functions and elliptic integrals. The conventional analytical expression of the solution, however, is complicated and not suitable for hand-calculation. Recently the rotational motions of small celestial bodies in the solar system are frequently investigated by numerically integrating the equations of motion instead of using the analytical solution, since the numerical evaluation of the analytical and exact solution is a little bit difficult. As the observational accuracy of the rotational motions of the small bodies in the solar system is quite low, what we need for the reduction of these observations are rough estimates of the period of Eulerian motion ( or the free precession period) and the amplitudes of the main periodic terms. Here we give simple analytical expansions of torque-free motions for short- and long-axis modes, which are correct up to the second-order of a small parameter. These expressions include only trigonometric functions and are easily evaluated by hand calculation for estimates of the essential quantities from which we can determine a global rotational motion of the torque-free motion. They can also be used as the zero-th order solution in a perturbation method, when the motion is perturbed by external torques.     

A family of periodic solutions of the planar three-body problem,and their stability

We describe a one-parameter family of periodic orbits in the planar problem of three bodies with equal masses. This family begins with Schubart's (1956) rectilinear orbit and ends in retrograde revolution, i.e. a hierarchy of two binaries rotating in opposite directions. The first-order stability of the orbits in the plane is also computed. Orbits of the retrograde revolution type are stable; more unexpectedly, orbits of the interplay type at the other end of the family are also stable. This indicates the possible existence of triple stars with a motion entirely different from the usual hierarchical arrangement.     

Effect of radiation on the stability of a retrograde particle orbit in different stellar systems

We have numerically investigated the stability of retrograde orbits/trajectories around Jupiter and the smaller of the primaries in binary systems RW-Monocerotis (RW-Mon) and Krüger-60 in the presence of radiation. A trajectory is considered as stable if it remains around the smaller mass for at least few hundred binary periods. In case of circular binary orbit, we find that the third order resonance provides the basis for reduction of stability region of retrograde motion of particle in RW-Mon and Sun-Jupiter system both in the presence and absence of radiation. Considering finite ellipticity in Sun-Jupiter system we find that for distant retrograde orbits, radiation from the Sun increases the width of the stable region and covers a significant portion of the region obtained in the absence of solar radiation. Further, due to solar radiation pressure, the stable region in the neighborhood of Jupiter has been found to shift much below the characteristic asymptotic line for the periodic retrograde orbits. In case of Krüger-60 we observe the distant retrograde orbits around the smaller of the primaries get affected considerably with increase in radiation parameter β₁. Further the range of velocities for which stable motion may persist narrows down for distant retrograde orbits in this system.     

Bounded motion in a generalized two-body problem

In this paper we prove the existence of ring-type bounded motion in an isolated system consisting of a massive point particle and a homogeneous cube. We study the case of planar motion where the particle moves in a symmetry plane of the cube and we use a rotating frame of reference with its center at the mass center of the cube and its axes coinciding with the symmetry axes of the cube. We prove that, for negative values of the total energy and properly chosen values of the total angular momentum, the relative distance of the bodies has an upper and a lower bound-i.e., the regions of possible motion lie inside an annulus around the cube (motion inside a ring or an island).     

Periodic solutions of circular-elliptic type in the planarN-body problem

Letn2 mass points with arbitrary masses move circularly on a rotating straight-line central-configuration; i.e. on a particular solution of relative equilibrium of then-body problem. Replacing one of the mass points by a close pair of mass points (with mass conservation) we show that the resultingN-body problem (N=n+1) has solutions, which are periodic in a rotating coordinate system and describe precessing nearlyelliptic motion of the binary and nearlycircular collinear motion of its center of mass and the other bodies; assuming that also the mass ratio of the binary is small.     

Stability and bifurcations of Sitnikov motions

A family of straight line periodic motions, known as the Sitnikov motions and existing in the case of equal primaries of the three body problem, is studied with respect to stability and bifurcations. Continuation of the bifurcations into the case of unequal primaries is also discussed and some of the bifurcating families of three-dimensional periodic motions are computed.