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.     

Resonance and Capture of Jupiter Comets

A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L ₁and L ₂, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L ₁and L ₂is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L ₁and the other around L ₂) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold tubes associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter's orbit.     

The flattening of three-dimensional periodic orbits in the restricted problem

A number of partly known families of symmetric three-dimensional periodic orbits of the restricted three-body (=0.4) problem are numerically continued in both ends until they terminate with orbits in the plane of motion of the primaries. The families of plane symmetric periodic orbits from which they bifurcate are identified and many orbit illustrations are given.     

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.     

Mass ratios of three bodies for stable low-inclination three-dimensional periodic motion

The intervals of possible stability, on the -axis, of the basic families of three-dimensional periodie motions of the restricted three-body problem (determined in an earlier paper) are extended into regions of the -m ₃ parameter space of the general three-body problem. Sample three-dimensional periodic motions corresponding to these regions are computed and tested for stability. Six regions, corresponding to the vertical-critical orbitsl1v, m1v,m2v, andilv, survive this preliminary stability test-therefore, emerging as the mass parameters regions allowing the simplest types of stable low inclination three-dimensional motion of three massive bodies.     

The family I3υ of the three-dimensional general three-body problem (Parts I and II)

We present the biparametric family I_3υ of symmetric periodic orbits of the three-dimensional general three-body problem, found by numerical continuation of the vertical critical orbit I_3υ of the circular restricted three-body problem. The periodic orbits refer to a suitably chosen rotating frame of reference.     

Cylindrical isomorphic mapping applied to invariant manifold dynamics for Earth–Moon Missions

Several families of periodic orbits exist in the context of the circular restricted three-body problem. This work studies orbital motion of a spacecraft among these periodic orbits in the Earth–Moon system, using the planar circular restricted three-body problem model. A new cylindrical representation of the spacecraft phase space (i.e., position and velocity) is described, and allows representing periodic orbits and the related invariant manifolds. In the proximity of the libration points, the manifolds form a four-fold surface, if the cylindrical coordinates are employed. Orbits departing from the Earth and transiting toward the Moon correspond to the trajectories located inside this four-fold surface. The isomorphic mapping under consideration is also useful for describing the topology of the invariant manifolds, which exhibit a complex geometrical stretch-and-folding behavior as the associated trajectories reach increasing distances from the libration orbit. Moreover, the cylindrical representation reveals extremely useful for detecting periodic orbits around the primaries and the libration points, as well as the possible existence of heteroclinic connections. These are asymptotic trajectories that are ideally traveled at zero-propellant cost. This circumstance implies the possibility of performing concretely a variety of complex Earth–Moon missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining a suitable, convenient end-of-life strategy for spacecraft placed in any of the unstable orbits. The final disposal orbit is an externally confined trajectory, never approaching the Earth or the Moon, and can be entered by means of a single velocity impulse (of modest magnitude) along the right unstable manifold that emanates from the Lyapunov orbit at \(L_2\) .     

Families of periodic orbits in the general three-body problem for the Sun-Jupiter-Saturn mass-ratio and their stability

Several families of planar planetary-type periodic orbits in the general three-body problem, in a rotating frame of reference, for the Sun-Jupiter-Saturn mass-ratio are found and their stability is studied. It is found that the configuration in which the orbit of the smaller planet is inside the orbit of the larger planet is, in general, more stable.We also develop a method to study the stability of a planar periodic motion with respect to vertical perturbations. Planetary periodic orbits with the orbits of the two planets not close to each other are found to be vertically stable. There are several periodic orbits that are stable in the plane but vertically unstable and vice versa. It is also shown that a vertical critical orbit in the plane can generate a monoparametric family of three-dimensional periodic orbits.     

Numerical results to the Sitnikov-problem

We present numerical results of the so-called Sitnikov-problem, a special case of the three-dimensional elliptic restricted three-body problem. Here the two primaries have equal masses and the third body moves perpendicular to the plane of the primaries' orbit through their barycenter. The circular problem is integrable through elliptic integrals; the elliptic case offers a surprisingly great variety of motions which are until now not very well known. Very interesting work was done by J. Moser in connection with the original Sitnikov-paper itself, but the results are only valid for special types of orbits. As the perturbation approach needs to have small parameters in the system we took in our experiments as initial conditions for the work moderate eccentricities for the primaries' orbit (0.33e _primaries 0.66) and also a range of initial conditions for the distance of the 3 ^rd body (= the planet) from very close to the primaries orbital plane of motion up to distance 2 times the semi-major axes of their orbit. To visualize the complexity of motions we present some special orbits and show also the development of Poincaré surfaces of section with the eccentricity as a parameter. Finally a table shows the structure of phase space for these moderately chosen eccentricities.     

Asymptotic solutions in the many-body problem

A three-body problem is considered in which two masses, forming a close binary, orbit a comparatively distant mass. An asymptotic solution of this problem is presented, where the small parameter is related to the distance separating the binary and the remaining mass. Accepting certain model constraints, this solution is accurate within a constant errorO(¹¹) and uniformly valid for time intervalsO(^–3). Two specific examples are chosen to verify the literal solution: one relating to the Sun-Earth-Moon configuration of the solar system, the other to an idealized stellar system where the three masses are in the ratio 20:1:1. In both cases close agreement is found when the analytical solution is compared with an equivalent numerically-generated orbit.     

Asymptotic solutions in the many-body problem

A small particle moves in the vicinity of two masses, forming a close binary, in orbit about a distant mass. Unique, uniformly valid solutions of this four-body problem are found for motion near both equilateral triangle points of the binary system in terms of a small parameter , where the primaries move in accordance with a uniformly-valid three-body solution. Accuracy is maintained within a constant errorO(⁸), and the solutions are uniformly valid as tends to zero for time intervalsO(^–3). Orbital position errors nearL ₄ andL ₅ of the Earth-Moon system are found to be less than 5% when numerically-generated periodic solutions are used as a standard of comparison.     

The rectilinear three-body problem using symbol sequence II: role of the periodic orbits

We study the change of phase space structure of the rectilinear three-body problem when the mass combination is changed. Generally, periodic orbits bifurcate from the stable Schubart periodic orbit and move radially outward. Among these periodic orbits there are dominant periodic orbits having rotation number (n − 2)/n with n ≥ 3. We find that the number of dominant periodic orbits is two when n is odd and four when n is even. Dominant periodic orbits have large stable regions in and out of the stability region of the Schubart orbit (Schubart region), and so they determine the size of the Schubart region and influence the structure of the Poincaré section out of the Schubart region. Indeed, with the movement of the dominant periodic orbits, part of complicated structure of the Poincaré section follows these orbits. We find stable periodic orbits which do not bifurcate from the Schubart orbit.     

Family <Emphasis Type="Italic">h</Emphasis> of periodic solutions of the restricted problem for big μ

The results of the computation of the family h of symmetric periodic solutions of the circular planar restricted three-body problem for μ = 0.3, 0.4, and 0.5 are presented. This family begins with retrograde circular orbits around a massive body. Associated with each value of μ are the table of critical orbits, the orbit pictures, the graphs of characteristics of the family in four coordinate systems, and the graphs of the period and traces (planar and vertical). Regularities on the family and its evolution as μ increased were observed.     

Family <Emphasis Type="Italic">h</Emphasis> of periodic solutions of the restricted problem for small μ

The results of the calculation of the family h of symmetric periodic solutions of the planar restricted three-body problem for four values of μ = 0, 10^?3, 0.1, and 0.2 are presented. This family begins with retrograde circular orbits around the body of bigger mass. Associated with each value of μ are the table of critical orbits, the orbit pictures, graphs of the characteristics of the family in four coordinate systems, and graphs of the period and of traces (planar and vertical). Regularities on the family and its connection to the generating family are observed.     

Asymmetric Periodic Solutions in the Restricted Problem of Three Bodies

A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincaré surface of sections, that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion.     

Periodic orbits of the general three-body problem for the Sun-Jupiter-Saturn system

Two families of symmetric periodic orbits of the planar, general, three-body problem are presented. The masses of the three bodies include ratios equal to the Sun-Jupiter-Saturn system and the periods of the orbits of Jupiter and Saturn are in a 25 resonance. The (linear) stability of the orbits are studied in relation to eccentricity and mass variations. The generation of the two families of periodic orbits follows a systematic approach and employs (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to the elliptic restricted problem and from the circular and elliptic problems to the general problem through bifurcation phenomena relating the three dynamical systems. The approach also provides insight into the evolutionary process of periodic orbits continued from the restricted problems to the general problem.     

The continuation of periodic orbits from the restricted to the general three-body problem

Celestial Mechanics and Dynamical Astronomy - It is proved that a symmetric periodic orbit of the circular planar restricted three-body problem can be continued analytically, when the mass of the...     

Bifurcations of Plane to 3d Periodic Orbits in the Restricted Three-Body Problem with Oblateness

We consider the bifurcation of 3D periodic orbits from the plane of motion of the primaries in the restricted three-body problem with oblateness. The simplest 3D periodic orbits branch-off at the plane periodic orbits of indifferent vertical stability. We describe briefly suitable numerical techniques and apply them to produce the first few such vertical-critical orbits of the basic families of periodic orbits of the problem, for varying mass parameter and fixed oblateness coefficent A₁ = 0.005, as well as for varying A₁ and fixed = 1/2. The horizontal stability of these orbits is also determined leading to predictions about the stability of the branching 3D orbits.     

Global Bifurcations of Periodic Solutions of the Restricted Three Body Problem

We describe global bifurcations from the libration points of non-stationary periodic solutions of the restricted three body problem. We show that the only admissible continua of non-stationary periodic solutions of the planar restricted three body problem, bifurcating from the libration points, can be the short-period families bifurcating from the Lagrange equilibria L ₄, L ₅. We classify admissible continua and show that there are possible exactly six admissible continua of non-stationary periodic solutions of the planar restricted three body problem. We also characterize admissible continua of non-stationary periodic solutions of the spatial restricted three body problem. Moreover, we combine our results with the Déprit and Henrard conjectures (see [8]), concerning families of periodic solutions of the planar restricted three body problem, and show that they can be formulated in a stronger way. As the main tool we use degree theory for SO(2)-equivariant gradient maps defined by the second author in [25].This revised version was published online in October 2005 with corrections to the Cover Date.     

General three-body problem: Families of three-dimensional periodic orbits (part II)

In this paper we present a two-parametric family of symmetric periodic orbits of the three-dimensional general three-body problem, found numerically by continuation of a vertical critical orbit of the circular restricted three-body problem. The periodic orbits refer to a suitably defined rotating frame of reference.