Doubly-symmetric motions in the elliptic problem

Poincaré's continuation method is applied to the elliptic restricted problem for the computation of four families of doubly symmetric three-dimensional periodic orbits emanating from similar orbits corresponding to zero eccentricity. The results are given in four tables and the orbits are characterized in regard to their stability.     

Critical generating orbits for second species periodic solutions of the restricted problem

The second species periodic solutions of the restricted three body problem are investigated in the limiting case of μ=0. These orbits, called consecutive collision orbits by Hénon and generating orbits by Perko, form an infinite number of continuous one-parameter families and are the true limit, for μ→0, of second species periodic solutions for μ>0. By combining a periodicity condition with an analytic relation, for criticality, isolated members of several families are obtained which possess the unique property that the stability indexk jumps from ±∞ to ?∞ at that particular orbit. These orbits are of great interest since, for small μ>0, ‘neighboring’ orbits will then have a finite (but small) region of stability.     

Secondary resonances and the boundary of effective stability of Trojan motions

One of the most interesting features in the libration domain of co-orbital motions is the existence of secondary resonances. For some combinations of physical parameters, these resonances occupy a large fraction of the domain of stability and rule the dynamics within the stable tadpole region. In this work, we present an application of a recently introduced ‘basic Hamiltonian model’ \(H_\mathrm{b}\) for Trojan dynamics (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez et al. in Celest Mech Dyn Astron 126:519, 2016): we show that the inner border of the secondary resonance of lowermost order, as defined by \(H_\mathrm{b}\), provides a good estimation of the region in phase space for which the orbits remain regular regardless of the orbital parameters of the system. The computation of this boundary is straightforward by combining a resonant normal form calculation in conjunction with an ‘asymmetric expansion’ of the Hamiltonian around the libration points, which speeds up convergence. Applications to the determination of the effective stability domain for exoplanetary Trojans (planet-sized objects or asteroids) which may accompany giant exoplanets are discussed.     

A numerical study of local stellar motions

The orbits of over 10000 stars are integrated in a steady-state model of the Galaxy for a time 6.0×10⁸ yr. Initially, the stars are placed randomly inside spheres of 500 pc and 50 pc radius and are given random velocities, such that the sample has a Maxwellian or a spheroidal velocity distribution. The spheres are placed at the Sun's distance from the galactic centre (10 kpc) and are given a circular velocity of 250 km s^?1. The mean velocities and dispersions of stars within 1 kpc of an ‘observer’ moving at the circular velocity are calculated as functions of time. The quantities show a strong time-dependence with oscillations of period 10⁸ yr. The oscillations are independent of the mass model and occur also in an inverse square force field. A vertex deviation of the velocity ellipsoid, an asymmetric drift and aK-effect occur as natural consequences of the oscillations. Attempts to apply the Oort method for density determinations in the galactic plane are also influenced by the oscillations. Spiral density waves appear to have a small effect on the motions of the test stars.     

Symmetric motions in the Equatorial Magnetic-Binary Problem

Classical numerical techniques are applied to find families of symmetric periodic orbits in the Equatorial Magnetic-Binary Problem. Stability for each orbit is also studied by means of variational methods. Finally an example in a concrete system is given to verify the procedure proposed.     

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

The existence of new equilibrium points is established in the restricted three-body problem with equal prolate primaries. These are located on the Z-axis above and below the inner Eulerian equilibrium point L ₁ and give rise to a new type of straight-line periodic oscillations, different from the well known Sitnikov motions. Using the stability properties of these oscillations, bifurcation points are found at which new types of families of 3D periodic orbits branch out of the Z-axis consisting of orbits located entirely above or below the orbital plane of the primaries. Several of the bifurcating families are continued numerically and typical member orbits are illustrated.     

Numerical determination of three-dimensional periodic orbits generated from vertical self-resonant satellite orbits

The mechanism by which ‘vertical’ branches consisting of symmetric, three-dimensional periodic orbits bifurcate from families of plane orbits at ‘veertical self-resonant’ orbits is discussed, with emphasis on the relationship between symmetry properties and multiplicity, and methods for the numerical determination of such branches are described. As examples, eight new families of all symmetry classes which branch vertically from the familyf of retrograde satellite orbits in the Sun-Jupiter case of the restricted problem (μ=0.000 95), are given in their entirety; these branches are found, as expected, to occur in pairs, each pair arising from the same self-resonant orbit, and their symmetry properties following the predicted pattern. The stability and other properties of the branch orbits are discussed.     

Geostationary secular dynamics revisited: application to high area-to-mass ratio objects

The long-term dynamics of the geostationary Earth orbits (GEO) is revisited through the application of canonical perturbation theory. We consider a Hamiltonian model accounting for all major perturbations: geopotential at order and degree two, lunisolar perturbations with a realistic model for the Sun and Moon orbits, and solar radiation pressure. The long-term dynamics of the GEO region has been studied both numerically and analytically, in view of the relevance of such studies to the issue of space debris or to the disposal of GEO satellites. Past studies focused on the orbital evolution of objects around a nominal solution, hereafter called the forced equilibrium solution, which shows a particularly strong dependence on the area-to-mass ratio. Here, we (i) give theoretical estimates for the long-term behavior of such orbits, and (ii) we examine the nature of the forced equilibrium itself. In the lowest approximation, the forced equilibrium implies motion with a constant non-zero average ‘forced eccentricity’, as well as a constant non-zero average inclination, otherwise known in satellite dynamics as the inclination of the invariant ‘Laplace plane’. Using a higher order normal form, we demonstrate that this equilibrium actually represents not a point in phase space, but a trajectory taking place on a lower-dimensional torus. We give analytical expressions for this special trajectory, and we compare our results to those found by numerical orbit propagation. We finally discuss the use of proper elements, i.e., approximate integrals of motion for the GEO orbits.     

Computing with certainty individual members of families of periodic orbits of a given period

The accurate computation of families of periodic orbits is very important in the analysis of various celestial mechanics systems. The main difficulty for the computation of a family of periodic orbits of a given period is the determination within a given region of an individual member of this family which corresponds to a periodic orbit. To compute with certainty accurate individual members of a specific family we apply an efficient method using the Poincaré map on a surface of section of the considered problem. This method converges rapidly, within relatively large regions of the initial conditions. It is also independent of the local dynamics near periodic orbits which is especially useful in the case of conservative dynamical systems that possess many periodic orbits, often of the same period, close to each other in phase space. The only computable information required by this method is the signs of various function evaluations carried out during the integration of the equations of motion. This method can be applied to any system of celestial mechanics. In this contribution we apply it to the photogravitational problem.     

Periodic orbits and bifurcations in the Sitnikov four-body problem when all primaries are oblate

We study the motions of an infinitesimal mass in the Sitnikov four-body problem in which three equal oblate spheroids (called primaries) symmetrical in all respect, are placed at the vertices of an equilateral triangle. These primaries are moving in circular orbits around their common center of mass. The fourth infinitesimal mass is moving along a line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries. A relation between the oblateness-parameter ‘A’ and the increased sides ‘ε’ of the equilateral triangle during the motion is established. We confine our attention to one particular value of oblateness-parameter A=0.003. Only one stability region and 12 critical periodic orbits are found from which new three-dimensional families of symmetric periodic orbits bifurcate. 3-D families of symmetric periodic orbits, bifurcating from the 12 corresponding critical periodic orbits are determined. For A=0.005, observation shows that the stability region is wider than for A=0.003.     

Stability of interplay motions

A family of rectilinear periodic solutions of the three-body problem, in which the central body collides alternately with each of the two other bodies, is investigated numerically for all values of the three masses. It is found that for every mass combination there exists just one solution of this kind. The linear stability of the orbits with respect to arbitrary three-dimensional perturbations is also investigated. Domains of stability and instability are displayed in a triangular mass diagram. Their boundaries form one-parameter families of critical orbits, which are tabulated. Limiting cases where one or two masses vanish are studied in detail. The domains of stability cover nearly one half of the total area in the mass diagram: this reinforces the conclusion that real triple stars might have motions of a kind entirely different from the usual hierarchical arrangement.     

Sensitivity study of high eccentricity orbits for Mars gravity recovery

By linear perturbation theory, a sensitivity study is presented to calculate the contribution of the Mars gravity field to the orbital perturbations in velocity for spacecrafts in both low eccentricity Mars orbits and high eccentricity orbits(HEOs). In order to improve the solution of some low degree/order gravity coefficients, a method of choosing an appropriate semimajor axis is often used to calculate an expected orbital resonance, which will significantly amplify the magnitude of the position and velocity perturbations produced by certain gravity coefficients. We can then assess to what degree/order gravity coefficients can be recovered from the tracking data of the spacecraft. However, this existing method can only be applied to a low eccentricity orbit, and is not valid for an HEO. A new approach to choosing an appropriate semimajor axis is proposed here to analyze an orbital resonance. This approach can be applied to both low eccentricity orbits and HEOs. This small adjustment in the semimajor axis can improve the precision of gravity field coefficients and does not affect other scientific objectives.     

New periodic solutions of the three dimensional restricted problem

A global review of the symmetric solutions of the restricted problem made in the Introduction opens a window on new symmetric periodic orbits of the two body problem in rotating axes which could be ‘trivially’ continuable to symmetric periodic orbits of the three dimensional restricted problem for small values of μ (see Figure 3). The proof of this possibility of continuation is given in Sections 1, 2, 3 using regularizing variables.     

The evolution of periodic orbits close to heteroclinic points

Several families of periodic orbits close to heteroclinic points were computed, and their evolution, as a function of the energy, was followed. All appeared spontaneously at the occasion of the formation of the heteroclinic points, and had no genealogical links with other families. This behavior, which confirms a prediction by Dr. Contopoulos, is in contrast to the behavior of periodic orbits close to homoclinic points, some of which have genealogical links with periodic orbits existing in the ‘quasi-integrable’ phase of the system. The orbits found here appeared in regions where stochasticity was well established; so their appearance does not seem to be connected with the onset of stochasticity.     

Area-preserving mappings and deterministic chaos for nearly parabolic motions

The present work investigates a mechanism of capturing processes in the restricted three-body problem. The work has been done in a set of variables which is close to Delaunay's elements but which allows for the transition from elliptic to hyperbolic orbits. The small denominator difficulty in the perturbation theory is overcome by embedding the small denominator in an analytic function through a suitable analytic continuation. The results indicate that motions in nearly parabolic orbits can become chaotic even though the model is deterministic. The theoretical results are compared with numerical results, showing an agreement of about one percent. Some possible applications to cometary orbits are also given.     

Cometary motions

The history of the study of cometary motions is described, from the pre-Copernican era until the present time, with emphasis on the determination of orbits, the calculation of special perturbations, and the analysis of nongravitational effects. A brief survey is made of the statistics of cometary orbits and of the implications concerning cometary origin and evolution. 1. Pre-Newtonian Ideas It is said... that the highest region of the air follows the celestial motion. This is demonstrated by those stars that suddenly appear —I mean those stars that the Greeks calledcometae orpogoniae. The highest region is considered their place of generation, and just like other stars they also rise and set. We can say that this part of the air is deprived of the terrestrial motion because of its great distance from the Earth.     

Regular motions in double bars – I. Double-frequency orbits and loops

Bars in galaxies are mainly supported by particles trapped around stable periodic orbits. These orbits represent oscillatory motion with only one frequency, which is the bar driving frequency, and miss free oscillations. We show that a similar situation takes place in double bars: particles get trapped around parent orbits, which in this case represent oscillatory motion with two frequencies of driving by the two bars, and which also lack free oscillations. Thus the parent orbits, which constitute the backbone of an oscillating potential of two independently rotating bars, are the double-frequency orbits. These orbits do not close in any reference frame, but they map on to closed curves called loops. Trajectories trapped around the parent double-frequency orbit map on to a set of points confined within a ring surrounding the loop.