Exchange orbits: an interesting case of co-orbital motion

In this investigation we treat a special configuration of two celestial bodies in 1:1 mean motion resonance namely the so-called exchange orbits. There exist—at least—theoretically—two different types: the exchange-a orbits and the exchange-e orbits. The first one is the following: two celestial bodies are in orbit around a central body with almost the same semi-major axes on circular orbits. Because of the relatively small differences in semi-major axes they meet from time to time and exchange their semi-major axes. The inner one then moves outside the other planet and vice versa. The second configuration one is the following: two planets are moving on nearly the same orbit with respect to the semi-major axes, one on a circular orbit and the other one on an eccentric one. During their dynamical evolution they change the characteristics of the orbit, the circular one becomes an elliptic one whereas the elliptic one changes its shape to a circle. This ‘game’ repeats periodically. In this new study we extend the numerical computations for both of these exchange orbits to the three dimensional case and in another extension treat also the problem when these orbits are perturbed from a fourth body. Our results in form of graphs show quite well that for a large variety of initial conditions both configurations are stable and stay in these exchange orbits.     

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.     

On the relation between resonance and instability in planetary systems

The factors which affect the linear stability of a periodic planetary orbit in the plane are studied. It is proved that planetary systems with two planets describing nearly circular orbits in the same direction are linearly stable and no perturbation exists which destroys the stability, unless a resonance of the form 1/3, 3/5, 5/7, ... among the orbits of the planets occurs. This latter resonant case is always unstable. Retrograde motion is always linearly stable. Planetary systems with three or more planets in nearly circular orbits in the same direction are proved to be unstable, in the sense that a Hamiltonian perturbation always exists which destroys the stability. The generation of instability in the case of three or more planets is not only due to the existence of resonances, as in the case of two planets, but also to the nonexistence of integrals of motion, apart from the energy and angular momentum integrals. It is also proved that planetary systems with nearly elliptic orbits of the planets are unstable.     

Slow modes in stellar systems with nearly harmonic potentials: II. Gravitational loss-cone instability

Using a consistent perturbation theory for collisionless disk-like and spherical star clusters, we construct a theory of slow modes for systems having an extended central region with a nearly harmonic potential due to the presence of a fairly homogeneous (on the scales of the stellar system) heavy, dynamically passive halo. In such systems, the stellar orbits are slowly precessing, centrally symmetric ellipses (2: 1 orbits). We consider star clusters with monoenergetic distribution functions that monotonically increase with angular momentum in the entire range of angular momenta (from purely radial orbits to circular ones) or have a growing region only at low angular momenta. In these cases, there are orbits with a retrograde precession, i.e., in a direction opposite to the orbital rotation of the star. The presence of a gravitational loss-cone instability, which is also observed in systems of 1: 1 orbits in near-Keplerian potentials, is associated with such orbits. In contrast to 1: 1 systems, the loss-cone instability takes place even for distribution functions monotonically increasing with angular momentum, including those for systems with circular orbits. The regions of phase space with retrograde orbits do not disappear when the distribution function is smeared in energy. We investigate the influence of a weak inhomogeneity of a heavy halo with a density that decreases with distance from the center.     

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.     

On the effect of the eccentricity of a planetary orbit on the stability of satellite orbits

The effect of the eccentricity of a planet’s orbit on the stability of the orbits of its satellites is studied. The model used is the elliptic Hill case of the planar restricted three-body problem. The linear stability of all the known families of periodic orbits of the problem is computed. No stable orbits are found, the majority of them possessing one or two pairs of real eigenvalues of the monodromy matrix, while a part of a family with complex instability is found. Two families of periodic orbits, bifurcating from the Lagrangian points L₁, L₂ of the corresponding circular case are found analytically. These orbits are very unstable and the determination of their stability coefficients is not accurate, so we compute the largest Liapunov exponent in their vicinity. In all cases these exponents are positive, indicating the existence of chaotic motions     

Periodic Orbits of a Collinear Restricted Three-Body Problem

In this paper we study symmetric periodic orbits of a collinear restricted three-body problem, when the middle mass is the largest one. These symmetric periodic orbits are obtained from analytic continuation of symmetric periodic orbits of two collinear two-body problems.     

Dynamical behaviour of planetesimals temporarily captured by a planet from heliocentric orbits: basic formulation and the case of low random velocity

Planetesimals encountering with a planet cannot be captured permanently unless energy dissipation is taken into account, but some of them can be temporarily captured in the vicinity of the planet for an extended period of time. Such a process would be important for the origin and dynamical evolution of irregular satellites, short-period comets, and Kuiper-belt binaries. In this paper, we describe the basic formulation for the study of temporary capture of planetesimals from heliocentric orbits using three-body orbital integration, such as the definition of the duration and rate of temporary capture, and present results in the case of low random velocity of planetesimals. In the case of planetesimals initially on circular orbits, we find that planetesimals undergo a close encounter with the planet before they become temporarily captured. When planetesimals are scattered by the planet into the vicinity of one of periodic orbits around the planet, the duration of temporary capture tends to be extended. Typically, these capture orbits are in the retrograde direction around the planet. We evaluate the rate of temporary capture of planetesimals, and find that the ratio of this rate to their collision rate on to the planet increases with increasing semimajor axis of the planet. Similar results are obtained for planetesimals with non-zero but small random velocities, as long as Kepler shear dominates the relative velocity between the planet and planetesimals. For larger initial random velocities of planetesimals, temporary capture in both prograde and retrograde directions with much longer duration becomes possible.     

Chaos in Relativity and Cosmology

Chaos appears in various problems of Relativity and Cosmology. Here we discuss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes. (a) The Mixmaster equations have a general solution (i.e. a solution depending on 6 arbitrary constants) of Painlevé type, but there is a second general solution which is not Painlevé. Thus the system does not pass the Painlevé test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (α + β + γ) has only one maximum for τ = τ_m and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus the Lyapunov Characteristic Number (LCN) is zero. The "finite time LCN" is positive for finite τ and tends to zero when τ → ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics of a chaotic scattering problem. (b) In the case of two fixed black holes M₁ and M₂ the orbits of photons are separated into three types: orbits falling into M₁ (type I), or M₂ (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are orbits of the third type. This is a typical chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcation ratio is the same as in classical conservative systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Some manifolds of periodic orbits in the restricted three-body problem

In the present paper we give some numerical results about natural families of periodic orbits, which emanate from limiting orbits around the equilateral equilibrium points of the Restricted Three-Body Problem, when the mass ratio is greater than Routh's critical one.     

Analytical and numerical manifolds in a symplectic 4-D map

We study analytically the orbits along the asymptotic manifolds from a complex unstable periodic orbit in a symplectic 4-D Froeschlé map. The orbits are given as convergent series. We compare the analytic results by truncating the series at various orders with the corresponding numerical results and we find agreement along a more extended length, as the order of truncation increases. The agreement is improved when the parameters approach those of the stability domain. Along the manifolds no terms with small divisors appear in the series. The same result is found if we use a parametrization method along the asymptotic curves. In the case of orbits starting close to the manifolds small divisors appear, but the orbits remain close to the manifolds for an extended period of time. If the parameters of the map are close to the stable domain the orbits recede and approach the origin several times and remain confined in a certain volume around the origin for a long time before escaping to large distances. For special sets of parameters we see resonance phenomena and the orbits take particular forms near every resonance.     

Dynamical friction of bodies orbiting in a gaseous sphere

The dynamical friction experienced by a body moving in a gaseous medium is different from the friction in the case of a collisionless stellar system. Here we consider the orbital evolution of a gravitational perturber inside a gaseous sphere using three-dimensional simulations, ignoring however self-gravity. The results are analysed in terms of a 'local' formula with the associated Coulomb logarithm taken as a free parameter. For forced circular orbits, the asymptotic value of the component of the drag force in the direction of the velocity is a slowly varying function of the Mach number in the range 1.0–1.6. The dynamical friction time-scale for free decay orbits is typically only half as long as in the case of a collisionless background, which is in agreement with E. C. Ostriker's recent analytic result. The orbital decay rate is rather insensitive to the past history of the perturber. It is shown that, similarly to the case of stellar systems, orbits are not subject to any significant circularization. However, the dynamical friction time-scales are found to increase with increasing orbital eccentricity for the Plummer model, whilst no strong dependence on the initial eccentricity is found for the isothermal sphere.     

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.     

Bifurcation of Relative Equilibria in the Main Problem of Artificial Satellite Theory for a Prolate Body

In this paper, we study circular orbits of the J ₂ problem that are confined to constant-z planes. They correspond to fixed points of the dynamics in a meridian plane. It turns out that, in the case of a prolate body, such orbits can exist that are not equatorial and branch from the equatorial one through a saddle-center bifurcation. A closed-form parametrization of these branching solutions is given and the bifurcation is studied in detail. We show both theoretically and numerically that, close to the bifurcation point, quasi-periodic orbits are created, along with two families of reversible orbits that are homoclinic to each one of them.     

Simple three-dimensional periodic orbits in a galactic-type potential

We study the families of simple periodic orbits in a three-dimensional system that represents the inner parts of a perturbed triaxial galaxy. The perturbations depend on two control parameters. We find the regions where each family is stable, simply unstable, doubly unstable, or complex unstable. the stable and simply unstable families produce other families by bifurcation. Several families reach a maximum (or minimum) perturbation and then are continued by other families. The bifurcations are direct or inverse. The transition from one type of bifurcation to the other is theoretically explained. Another important phenomenon is the splitting of one family into two, or the joining of two families into one. We do not have any complex instability in the limiting cases of two-dimensional motions (when one control parameter is zero).The two main families of periodic orbits are in most cases stable when the energy is smaller than the escape energy. Most high energy orbits are unstable. However, we found stable orbits even for energies about four times larger than the escape energy.     

A Family of Symmetrical Schubart-Like Interplay Orbits and their Stability in the One-Dimensional Four-Body Problem

We locate members of an important category of periodic orbits in the Newtonian four-body problem. These systems perform an interplay motion similar to that of the periodic three-body orbit discovered by Schubart. Such orbits, when stable, have been shown to be a key feature and influence on the dynamics of few-body systems. We consider the restricted case where the masses are collinear and are distributed symmetrically about their centre of mass. A family of orbits is generated from the known (three-dimensionally) unstable equal masses case by varying the mass ratio, whilst maintaining the symmetry. The stability of these orbits to perturbation is studied using linear stability analysis, analytical approximation of limiting cases and nonlinear simulation. We answer the natural question: are there any stable periodic orbits of this kind? Three ranges of the mass ratio are found to have stable orbits and three ranges have unstable orbits for three-dimensional motion. The systems closely resemble their three-body counterparts. Here the family of interplay orbits is simpler requiring just one parameter to characterise the mass ratio. Our results provide a further insight into three-body orbits studied previously.     

On the long-period evolution of the sun-synchronous orbits

The dynamic evolution of sun-synchronous orbits at a time interval of 20 years is considered. The numerical motion simulation has been carried out using the Celestial Mechanics software package developed at the Institute of Astronomy of the University of Bern. The dependence of the dynamic evolution on the initial value of the ascending node longitude is examined for two families of sun-synchronous orbits with altitudes of 751 and 1191 km. Variations of the semimajor axis and orbit inclination are obtained depending on the initial value of the ascending node longitude. Recommendations on the selection of orbits, in which spent sun-synchronous satellites can be moved, are formulated. Minimal changes of elements over a time interval of 20 years have been observed for orbits in which at the initial time the angle between the orbit ascending node and the direction of the Sun measured along the equator have been close to 90° or 270°. In this case, the semimajor axis of the orbit is not experiencing secular perturbations arising from the satellite’s passage through the Earth’s shadow.     

Topology of the Two Fixed Centers Problem

In this paper we give a complete topological characterization of the Two Fixed Centers (TFC) problem flow. This characterization is based on the link formed by some basic periodic orbits. The Restricted Circular Three-Body (RCTB) problem is considered as a perturbation of the TFC in the case of two primaries with equal masses. The basic periodic orbits of the integrable problem can be continued in the non-integrable one.This revised version was published online in October 2005 with corrections to the Cover Date.     

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 convergent solution in kopal''s iterative method of solving eclipsing binary orbits

In /1/, we have discussed the question whether Kopal's iterative method of solving eclipsing binary orbits is always convergent. In this paper, we show, by means of an example, that, even when the solution is convergent, it may not be the true solution but a false one. Through calculations, we empirically identify the circumstances in which Kopal's method will either be divergent or lead to a false solution. We also make an improvement on the optimization procedure in /1/.