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.     

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.     

Spatial p-q Resonant Orbits of the RTBP

The purpose of this paper is to extend the study of the so called p-q resonant orbits of the planar restricted three-body problem to the spatial case. The p-q resonant orbits are solutions of the restricted three-body problem which have consecutive close encounters with the smaller primary. If E, M and P denote the larger primary, the smaller one and the infinitesimal body, respectively, then p and q are the number of revolutions that P gives around M and M around E, respectively, between two consecutive close approaches. For fixed values of p and q and suitable initial conditions on a sphere of radius around the smaller primary, we will derive expressions for the final position and velocity on this sphere for the orbits under consideration.     

On quasi-satellite periodic motion in asteroid and planetary dynamics

Applying the method of analytical continuation of periodic orbits, we study quasi-satellite motion in the framework of the three-body problem. In the simplest, yet not trivial model, namely the planar circular restricted problem, it is known that quasi-satellite motion is associated with a family of periodic solutions, called family f, which consists of 1:1 resonant retrograde orbits. In our study, we determine the critical orbits of family f that are continued both in the elliptic and in the spatial models and compute the corresponding families that are generated and consist the backbone of the quasi-satellite regime in the restricted model. Then, we show the continuation of these families in the general three-body problem, we verify and explain previous computations and show the existence of a new family of spatial orbits. The linear stability of periodic orbits is also studied. Stable periodic orbits unravel regimes of regular motion in phase space where 1:1 resonant angles librate. Such regimes, which exist even for high eccentricities and inclinations, may consist dynamical regions where long-lived asteroids or co-orbital exoplanets can be found.     

Inclined asymmetric librations in exterior resonances

Librational motion in Celestial Mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or \(\pi \), the libration is called asymmetric. In the context of the planar circular restricted three-body problem, asymmetric librations have been identified for the exterior mean motion resonances (MMRs) 1:2, 1:3, etc., as well as for co-orbital motion (1:1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the three-dimensional space. We employ the computational approach of Markellos (Mon Not R Astron Soc 184:273–281,  https://doi.org/10.1093/mnras/184.2.273, 1978) and compute families of asymmetric periodic orbits and their stability. Stable asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun–Neptune–TNO system (TNO = trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1:2 resonant, inclined asymmetric librations. For the stable 1:2 TNO librators, we find that their libration seems to be related to the vertically stable planar asymmetric orbits of our model, rather than the three-dimensional ones found in the present study.     

Apsidal motion in the eclipsing binary system OX cas

Multi-colourWBVR photoelectric observations of the eclipsing binary OX Cas were carried out. The photometric elements, absolute parameters and the angular rate of the apsidal motion ( = 9.1 deg yr^–1 were obtained. The apsidal parameterk ₂ derived for this system is by 15–25% smaller than the theoretical parameterk ₂.     

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 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.     

Stability criterion for a light binary attracted by a heavy body

Dynamical behaviour of a small binary with equal components, each of mass m, is considered under attraction of a heavy body of mass M. Differential equations of the general three-body problem are integrated numerically using the code by S. J. Aarseth (Aarseth, Zare 1974) for mass ratios m/M within 10⁻¹¹–10⁻⁴ range. The direct and retrograde orbits of light bodies about each other are considered which lie either in the plane of moving their center of mass or in the plane perpendicular to it. It is shown numerically that the critical separation between the binary components which leads to disruption of binary is proportional to (m/M)^1/3. The criterion can be used for studying (in the first approximation) the motion of double stars and binary asteroids or computing the parameters of magnetic monopol and antimonopol pairs.     

Numerical determination of families of three-dimensional double-symmetric periodic orbits in the restricted three-body problem,II

New families of three-dimensional double-symmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits ( _v = – 1, b _v – 0) of the basic plane familiesi, g ₁, g₂, c andI. Further, the predictor-corrector procedure employed to reveal these families has been described and interesting numerical results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.     

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.     

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.     

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.     

Three-dimensional periodic motion of three finite masses around collinear equilibrium configurations

Three-dimensional periodic motions of three bodies are shown to exist in the infinitesimal neighbourhood of their collinear equilibrium configurations. These configurations and some characteristic quantities of the emanating three-dimensional periodic orbits are given for many values of the two mass parameters, =m ₂/(m ₁+m ₂) andm ₃, of the general three-body problem, under the assumption that the straight line containing the bodies at equilibrium rotates with unit angular velocity. The analysis of the small periodic orbits near the equilibrium configurations is carried out to second-order terms in the small quantities describing the deviation from plane motion but the analytical solution obtained for the horizontal components of the state vector is valid to third-order terms in those quantities. The families of three-dimensional periodic orbits emanating from two of the collinear equilibrium configurations are continued numerically to large orbits. These families are found to terminate at large vertical-critical orbits of the familym of retrograde periodic orbits ofm ₃ around the primariesm ₁ andm ₂. The series of these termination orbits, formed when the value ofm ₃ varies, are also given. The three-dimensional orbits are computed form ₃=0.1.     

Isoenergetic families of quasi-periodic solutions near the equilateral solution of the three-body problem

The tensor appearing in the equation of geodesic deviation is computed for the equilateral solution of the general three-body problem. The eigenvalues and eigenvectors ofC _k ⁱ are determined. It is found that at least one of the eigenvalues is negative, irrespective of the masses of the bodies. This implies that the equilateral solution is not stable. The eigenvectors with positive eigenvalues generate isoenergetic 1-parameter families of quasi-periodic solutions in the neighborhood of the equilateral solution. The relation between the 1-parameter families constructed here and those known from the literature is discussed.     

Stable Chaos versus Kirkwood Gaps in the Asteroid Belt: A Comparative Study of Mean Motion Resonances

We have shown, in previous publications, that stable chaos is associated with medium/high-order mean motion resonances with Jupiter, for which there exist no resonant periodic orbits in the framework of the elliptic restricted three-body problem. This topological “defect” results in the absence of the most efficient mechanism of eccentricity transport (i.e., large-amplitude modulation on a short time scale) in three-body models. Thus, chaotic diffusion of the orbital elements can be quite slow, while there can also exist a nonnegligible set of chaotic orbits which are semiconfined (stable chaos) by “quasi-barriers” in the phase space. In the present paper we extend our study to all mean motion resonances of order q≤9 in the inner main belt (1.9-3.3 AU) and q≤7 in the outer belt (3.3-3.9 AU). We find that, out of the 34 resonances studied, only 8 possess resonant periodic orbits that are continued from the circular to the elliptic three-body problem (regular families), namely, the 2/1, 3/1, 4/1, and 5/2 in the inner belt and the 7/4, 5/3, 11/7, and 3/2 in the outer belt. Numerical results indicate that the 7/3 resonance also carries periodic orbits but, unlike the aforementioned resonances, 7/3-periodic orbits belong to an irregular family. Note that the five inner-belt resonances that carry periodic orbits correspond to the location of the main Kirkwood gaps, while the three outer-belt resonances correspond to gaps in the distribution of outer-belt asteroids noted by Holman and Murray (1996, Astron. J.112, 1278-1293), except for the 3/2 case where the Hildas reside. Fast, intermittent eccentricity increase is found in resonances possessing periodic orbits. In the remaining resonances the time-averaged elements of chaotic orbits are, in general, quite stable, at least for times t∼250 Myr. This slow diffusion picture does not change qualitatively, even if more perturbing planets are included in the model.     

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 main mean motion commensurabilities in the planar circular and elliptic problem

We study the motion of asteroids in the main mean motion commensurabilities in the frame of the planar restricted three-body problem. No assumption is made about the size of the eccentricity of the asteroid. At small to moderate eccentricity, we recover existing results (shape of the phase space and location of secondary resonances). We also provide global pictures of the dynamics in the region of secondary resonances. At high eccentricity, the phase space portraits of the integrable part of the Hamiltonian show new families of stable orbits for the 3:2 and 2:1 cases and the secular resonances ₅ and ₆ are located.     

Some families of periodic oscillations in the restricted problem with small mass-ratios of three bodies

This paper reports on the numerical determination of families of periodic oscillations in the case =0.000 95 of the restricted problem. The families emanating out of the collinear Lagrangian pointsL ₁,L ₂,L ₃ are examined as well as some asymmetric periodic oscillations related to them. An effort is made to complete the global picture of simple-periodic symmetric oscillations in the present case of the problem (the S-J case). This is done by examining the orbits with initial conditions such that the infinitesimal body starts from a position on the ₁-axis (₀₂ = 0) with a negative initial velocity perpendicular to this axis . In a previous article this investigation has been carried out for negative values of ₀₁, where the position of the small primary defines ₁=0. Now we proceed to consider orbits with ₀₁>0. The phase portrait of asymmetric periodic orbits is also examined.     

Forbidden Zones for Circular Regular Orbits of the Moons in Solar System,R3BP

Previously, we have considered the equations of motion of the three-body problem in a Lagrange form (which means a consideration of relative motions of 3-bodies in regard to each other). Analysing such a system of equations, we considered the case of small-body motion of negligible mass m ₃ around the second of two giant-bodies m ₁, m ₂ (which are rotating around their common centre of masses on Kepler’s trajectories), the mass of which is assumed to be less than the mass of central body. In the current development, we have derived a key parameter η that determines the character of quasi-circular motion of the small third body m ₃ relative to the second body m ₂ (planet). Namely, by making several approximations in the equations of motion of the three-body problem, such the system could be reduced to the key governing Riccati-type ordinary differential equations. Under assumptions of R3BP (restricted three-body problem), we additionally note that Riccati-type ODEs above should have the invariant form if the key governing (dimensionless) parameter η remains in the range 10^?2 Open image in new window 10^?3. Such an amazing fact let us evaluate the forbidden zones for Moon’s orbits in the inner solar system or the zones of distances (between Moon and Planet) for which the motion of small body could be predicted to be unstable according to basic features of the solutions of Riccati-type.