Planar Symmetric Periodic Orbits in Four Dipole Problem

We have extend Stormer’s problem considering four magnetic dipoles in motion trying to justify the phenomena of extreme “orderlines” such as the ones observed in the rings of Saturn; the aim is to account the strength of the Lorentz forces estimating that the Lorentz field, co-acting with the gravity field of the planet, will limit the motion of all charged particles and small size grains with surface charges inside a layer of about 200 m thickness as that which is observed in the rings of Saturn. For this purpose our interest feast in the motion of charged particles with neglected mass where only electromagnetic forces accounted in comparison to the weakness of the Newtonian fields. This study is particularly difficult because in the regions we investigate these motions there is enormous three dimensional instability. Following the Poincare’s hypothesis that periodic solutions are ‘dense’ in the set of all solutions in Hamiltonian systems we try to calculate many families of periodic solutions and to study their stability. In this work we prove that in this environment charged particles can trace planar symmetric periodic orbits. We discuss these orbits in details and we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. We apply numerical procedures to find families of these orbits and to study their stability. Moreover we give the bifurcations of these families with families of planar asymmetric periodic orbits and families of three dimensional symmetric periodic orbits.     

A Note on Second Species Solutions Generated from p–q Resonant Orbits

The first image under the flow of the restricted three-body problem of the p—q resonant strips — that appear in the study of the p—q resonant orbits — do not have, in general, intersection with the strip. In this paper we show some particular situations in which the above intersections exist for some very simple p—q resonant orbits which, at the same time, are periodic second species solutions.This revised version was published online in October 2005 with corrections to the Cover Date.     

Secular Effects on Orbits of Binary Stars Induced by Temporal Variation of Gravitational Constant (Case for Elliptical Orbit)

In this paper we investigate the influence of a varying gravitation constant on the orbits of celestial bodies. Regarding the eccentric anomaly as an independent variable, we find the solutions to the perturbed equations of motion. In the first order solutions, we find the secular and periodic variations in semi-major axis. For the other orbital elements only periodic variations exhibit. However in the second order solutions, the longitude of periastron and the mean longitude have secular terms. Applying the calculations to six selected binaries, we give the numerical estimations of the variations of orbits. These results are then carefully compared and discussed.     

Analytical solutions around planar equilibria of three dipole problem

Proceeding with our investigation into the motion of a particle influenced by the electromagnetic field of three celestial bodies of a magnetic-dipole nature we give here for the first time the analytical expressions of periodic solutions around a planar equilibrium point. These relations are expansions of the planar equations of motion in series of second order power of a parameter in the vicinity of equilibria. The above analytical expressions of periodic solutions give the first members of the family of periodic orbits which emanate from a stable equilibrium point. The whole family can then be calculated using a predictor-corrector algorithm.     

Structures in the phase space of a four dimensional symplectic map

We numerically investigate the projections of non periodic orbits in a 4-dimensional (4-D) symplectic map composed of two coupled 2-dimensional (2-D) maps. We describe in detail the structures that are produced in different planes of projection and we find how the morphology of the 4-D orbits is influenced by the features of the 2-D maps as the coupling parameter increases. We give an empirical law that describes this influence.     

Balanced earth satellite orbits

An analytic model for third-body perturbations and for the second zonal harmonic of the central body's gravitational field is presented. A simplified version of this model applied to the Earth-Moon-Sun system indicates the existence of high-altitude and highly-inclined orbits with their apsides in the equator plane, for which the apsidal as well as the nodal motion ceases. For special positions of the node, secular changes of eccentricity and inclination disappear too (balanced orbits). For an ascending node at vernal equinox, the inclination of balanced orbits is 94.56°, for a node at autumnal equinox 85.44°, independent of the eccentricity of the orbit. For a node perpendicular to the equinox, there exist circular balanced orbits at 90° inclination. By slightly adjusting the initial inclination as suggested by the simplified model, orbits can be found — calculated by the full model or by different methods — that show only minor variations in eccentricity, inclination, argument of perigee, and longitude of the ascending node for 10⁵ revolutions and more. Orbits near the unstable equilibria at 94.56° and 85.44° inclination show very long periodic librations and oscillations between retrogade and prograde motion.Retired from IBM Vienna Software Development Laboratory.     

Equatorial orbits in superposed dipole and uniform magnetic fields

In the present work we study the equatorial motions of charged par ticles that are performed within a field consisting of the superposition of a dipole field—that could represent the magnetic field of a planet — and of a uniform magnetic field normal to the dipole's equator. We use a non-dimensional coordinate system in which the velocity of the charged particle is unit. The model depends on two parameters: the constant of the generalized momentum and the parameter of the interplanetary magnetic field. It is proved that the motion is always bounded. The regions of the motion and the corresponding orbits are studied with respect to the constant of the generalized momentum. We also, investigate numerically conditional periodic and asymptotic orbits.     

Three dimensional periodic orbits in three dipole problem

In the three dipole problem where enormous electromagnetic forces obstruct the three dimensional movement of the charged particle we determined for the first time families of three dimensional asymmetric periodic orbits. We study how these families appear, branching from the planar motion and we develop the procedures we have followed to determine them numerically. Also we give their characteristics and the conical projections and plottings of some orbits.     

The Structure of Periodic Solutions in the Restricted Problem of the Three Bodies II

In this work we consider four families of plane periodic orbits direct around the Sun which approach Jupiter but they are sufficiently far from it so as to be considered as predominantly two body orbits of the Sun-asteroid system. We study their horizontal and vertical stabilities and we give the exact orbits of bifurcations of these families with three-dimensional families of the same multiplicity or twice the multiplicity of the above families of plane symmetric periodic orbits. Moreover, we give the first segments of the three dimensional families of symmetric periodic orbits which emanate from these plane bifurcations and we study their stability relating it with the stability of the plane bifurcations.     

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 symplectic property of matrizants

The matrizants of simply and doubly symmetric orbits of the three-dimensional restricted problem of three bodies because of their symplectic character, can be transformed by means of multiplication with constant matrices into symmetric ones. As a result the bilinear relations between their elements, existing on account of the symplectic property, take linear and very simple forms. This fact is very useful in computer applications where these relations are used as criteria of accuracy.     

Particle motion in the vicinity of the protosun with variable luminosity

Variations of luminosity of the protosun during its Hayashi stage produced variations of its repulsive radiative action on small particles in its vicinity — or, in other words, variations of the effective mass of the protosun. Changes of the effective mass produced changes of size of orbits for particles circling around the protosun. When the luminosity increased, the effect of variable luminosity (EVL) diminished or overbalanced the Poynting-Robertson effect (PRE), hindering the small particles in their drift toward the protosun, and reinforced PRE when the luminosity decreased. An analysis of quasicircular motion of small uncharged particles moving in transparent circumsolar space under both effects — EVL and PRE — is given.     

Discrete Virial Theorem

We reexamine the classical virial theorem for bounded orbits of arbitrary autonomous Hamiltonian systems possessing both regular and chaotic orbits. New and useful forms of the virial theorem are obtained for natural Hamiltonian flows of arbitrary dimension. A discrete virial theorem is derived for invariant circles and periodic orbits of natural symplectic maps. A weak and a strong form of the virial theorem are proven for both flows and maps. While the Birkhoff Ergodic Theorem guarantees the existence of the relevant time averages for both regular and chaotic orbits, the convergence is very rapid for the former and extremely slow for the latter. This circumstance leads to a simple and efficient measure of chaoticity. The results are applied to several problems of current physical interest, including the Hénon–Heiles system, weak chaos in the standard map, and a 4D Froeschlé map.     

The evolution of periodic orbits close to homoclinic points

For conservative dynamical systems having two degrees of freedom Birkhoff has established the existence of two classes of periodic orbits. The first consists of stable-unstable pairs close to periodic orbits of the stable type, and the second of orbits having fixed points (in a suitable surface of section) close to homoclinic points. In this paper orbits of the latter type are listed, and their evolution followed as a function of the energy. For the energy at which they were first computed, all were unstable; but they evolved, with diminishing energy, into one orbit of the stable type which appears to be a member of the first class of orbits mentioned above.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

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.     

Hamiltonian Stability of Spin–Orbit Resonances in Celestial Mechanics

The behaviour of ‘resonances’ in the spin-orbit coupling in celestial mechanics is investigated in a conservative setting. We consider a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. The continuous system is reduced to a mapping by integrating the equations of motion through a symplectic algorithm. We study numerically the stability of periodic orbits associated to the above mapping by looking at the eigenvalues of the matrix of the linearized map over the full cycle of the periodic orbit. In particular, the value of the trace of the matrix is related to the stability character of the periodic orbit. We denote by ε_* (p/q) the value of the perturbing parameter at which a given elliptic periodic orbit with frequency p/q becomes unstable. A plot of the critical function ε_* (p/q) versus the frequency at different orbital eccentricities shows significant peaks at the synchronous resonance (for low eccentricities) and at the synchronous and 3:2 resonances (at higher eccentricities) in good agreement with astronomical observations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Existence of periodic orbits of the second kind in the elliptic restricted problem of three bodies

Hale's method is used to show the existence of symmetric periodic orbits of the second kind for the particular case of the elliptic restricted problem of three bodies. In this treatment we also obtain a new proof of the existence of periodic orbits of the first and second kinds in the circular restricted problem.     

On the families of periodic orbits which bifurcate from the circular Sitnikov motions

In this paper we deal with the circular Sitnikov problem as a subsystem of the three-dimensional circular restricted three-body problem. It has a first analytical part where by using elliptic functions we give the analytical expressions for the solutions of the circular Sitnikov problem and for the period function of its family of periodic orbits. We also analyze the qualitative and quantitative behavior of the period function. In the second numerical part, we study the linear stability of the family of periodic orbits of the Sitnikov problem, and of the families of periodic orbits of the three-dimensional circular restricted three-body problem which bifurcate from them; and we follow these bifurcated families until they end in families of periodic orbits of the planar circular restricted three-body problem. We compare our results with the previous ones of other authors on this problem. Finally, the characteristic curves of some bifurcated families obtained for the mass parameter close to 1/2 are also described.     

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.     

Application of Gauss's method in the formation of periodic solutions in the relativistic restricted problem of three bodies

The paper discusses the existence of periodic and quasi-periodic solutions in the space relativistic problem of three bodies with the help of Poincaré's small parameter method starting from non-Keplerian generating solutions, i.e., using Gauss's method. The main peculiarity of these periodic orbits is the fact that they close, in general, after many revolutions. It is worth noticing that these periodic orbits give a new class of periodic solutions of the classical circular problem of three bodies, if relativistic effects are neglected.