High-order Analytical Solutions of Hill’s Equations

The purpose of this paper is the semi-analytical computation of the bounded orbits of Hill’s equations, describing the relative motion of two particles in their Keplerian motion around a central body. We have considered the case in which one of the particles moves along a circular reference orbit. The solutions obtained are the generalisation of the usual periodic orbits obtained for the linearised equations and also of the third-order solution computed by D.L. Richardson and J.W. Mitchell (2003). With the algorithm presented, those orbits can be computed in a fast and efficient way up to a high-order in the in-plane and out-of-plane amplitudes.     

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.     

The post-Newtonian relative orbit of the two-body system. Gravitational radiation

The average loss of energy over one period of the elliptical motion of the two-body system is given, within the quadrupole approximation, by using the relative motion in the post-Newtonian centre of the mass frame. More explicit formulae are derived for the elliptical orbits and detailed results are presented for the circular orbits assuming small orbital velocities compared to the velocity of light. On the other hand, using the defined Lagrangian we give the integrals of motion.     

Low-eccentricity motion of asteroids near the 2/1 Jovian resonance

(903) Nealley moves on an orbit of low eccentricity with a mean motion that is slightly larger than the 2/1 value of resonance. This orbit and some related fictious orbits are studied by numerical integrations of the four-body problem Sun-Jupiter-Saturn-asteroid over an interval of 110000 yr. The author's experience on related cases of resonance allows a study of the variation of suitably defined orbital parameters. The long-term evolution of the orbits is compared with earlier predictions. Some of the librating orbits are temporarily captured in a secondary resonance that refers to three-dimensional motion and is demonstrated by a special example.     

On the Relationship Between Fast Lyapunov Indicator and Periodic Orbits for Continuous Flows

It is already known (Froeschlé et al., 1997a) that the fast Lyapunov indicator (hereafter FLI), i.e. the computation on a relatively short time of a quantity related to the largest Lyapunov indicator, allows us to discriminate between ordered and weak chaotic motion. Using the FLI many results have been obtained on the standard map taken as a model problem. On this model we are not only able to discriminate between a short time weak chaotic motion and an ordered one, but also among regular motion between non resonant and resonant orbits. Moreover, periodic orbits are characterised by constant FLI values which appear to be related to the order of periodic orbits (Lega and Froeschlé, 2001). In the present paper we extend all these results to the case of continuous dynamical systems (the Hénon and Heiles system and the restricted three-body problem). Especially for the periodic orbits we need to introduce a new value: the orthogonal FLI in order to fully recover the results obtained for mappings.     

The kinematic properties of stars in an oscillating Kuzmin disk

We study the kinematic properties of stars under the combined potential of a Kuzmin disk with a simple radial oscillation and a logarithmic halo. The results are: 1) There exist stable, ordered and near-circular orbits. 2) The effect of the oscillating disk is greater on orbits with smaller angular momenta and on that departly greatly from the near-circular orbits. 3) Most of the motion in the disk is ordered motion. 4) Orbits that depart greatly from the near-circular orbits generally have chaotic motion and may eventually escape. But the actual fraction escaped in one Hubble time is small. 5) Disk oscillation may be one of the mechanisms for the formation and long-term maintenance of some star clusters; the larger the amplitude, the greater may be the number of clusters; for a given disk galaxy, there may be more clusters with small than with large angular momenta.     

The structure of motion in a 4-component galaxy mass model

We use a composite galaxy model consisting of a disk-halo, bulge, nucleus and dark-halo components in order to investigate the motion of stars in ther-z plane. It is observed that high angular momentum stars move in regular orbits. The majority of orbits are box orbits. There are also banana-like orbits. For a given value of energy, only a fraction of the low angular momentum stars — those going near the nucleus — show chaotic motion while the rest move in regular orbits. Again one observes the above two kinds of orbits. In addition to the above one can also see orbits with the characteristics of the 2/3 and 3/4 resonance. It is also shown that, in the absence of the bulge component, the area of chaotic motion in the surface of section increases, significantly. This suggests that a larger number of low angular momentum stars are in chaotic orbits in galaxies with massive nuclei and no bulge components.     

Finding how many isolating integrals of motion an orbit obeys

The correlation dimension, that is the dimension obtained by computing the correlation function of pairs of points of a trajectory in phase space, is a numerical technique introduced in the field of non-linear dynamics in order to compute the dimension of the manifold in which an orbit moves, without the need of knowing the actual equations of motion that give rise to the trajectory. This technique has been proposed in the past as a method to measure the dimension of stellar orbits in astronomical potentials, that is the number of isolating integrals of motion the orbits obey. Although the algorithm can in principle yield that number, some care has to be taken in order to obtain good results. We studied the relevant parameters of the technique, found their optimal values, and tested the validity of the method on a number of potentials previously studied in the literature, using the Smaller Alignment Index (SALI), Lyapunov exponents and spectral dynamics as gauges.     

Innovative methods of correlation and orbit determination for space debris

We propose two algorithms to provide a full preliminary orbit of an Earth-orbiting object with a number of observations lower than the classical methods, such as those by Laplace and Gauss. The first one is the Virtual debris algorithm, based upon the admissible region, that is the set of the unknown quantities corresponding to possible orbits for a given observation for objects in Earth orbit (as opposed to both interplanetary orbits and ballistic ones). A similar method has already been successfully used in recent years for the asteroidal case. The second algorithm uses the integrals of the geocentric 2-body motion, which must have the same values at the times of the different observations for a common orbit to exist. We also discuss how to account for the perturbations of the 2-body motion, e.g., the J ₂ effect.     

Complex statistics in Hamiltonian barred galaxy models

We use probability density functions (pdfs) of sums of orbit coordinates, over time intervals of the order of one Hubble time, to distinguish weakly from strongly chaotic orbits in a barred galaxy model. We find that, in the weakly chaotic case, quasi-stationary states arise, whose pdfs are well approximated by q-Gaussian functions (with 1 <?q < 3), while strong chaos is identified by pdfs which quickly tend to Gaussians (q =?1). Typical examples of weakly chaotic orbits are those that ??stick?? to islands of ordered motion. Their presence in rotating galaxy models has been investigated thoroughly in recent years due to their ability to support galaxy structures for relatively long time scales. In this paper, we demonstrate, on specific orbits of 2 and 3 degree of freedom barred galaxy models, that the proposed statistical approach can distinguish weakly from strongly chaotic motion accurately and efficiently, especially in cases where Lyapunov exponents and other local dynamic indicators appear to be inconclusive.     

Constructive-analytical solution of the problem of the secular evolution of polar satellite orbits

The well-known twice-averaged Hill problem is considered by taking into account the oblateness of the central body. This problem has several integrable cases that have been studied qualitatively by many scientists, beginning with M.L. Lidov and Y. Kozai. However, no rigorous analytical solution can be obtained in these cases due to the complexity of the integrals. This paper is devoted to studying the case where the equatorial plane of the central body coincides with the plane of its orbital motion relative to the perturbing body, while the satellite itself moves in a polar orbit. A more detailed qualitative study is performed, and an approximate constructive-analytical solution of the evolution system in the form of explicit time dependences of the eccentricity and pericenter argument of the satellite orbit is proposed. The methodical accuracy for the polar orbits of lunar satellites has been estimated by comparison with the numerical solution of the system.     

Straight-line oscillations generating three-dimensional motions in the photogravitational restricted three-body problem

The Sitnikov configuration is a special case of the restricted three-body problem where the two primaries are of equal masses and the third body of a negligible mass moves along a straight line perpendicular to the orbital plane of the primaries and passes through their center of mass. It may serve as a toy model in dynamical astronomy, and can be used to study the three-dimensional orbits in more applicable cases of the classical three-body problem. The present paper concerns the straight-line oscillations of the Sitnikov family of the photogravitational circular restricted three-body problem as well as the associated families of three-dimensional periodic orbits. From the stability analysis of the Sitnikov family and by using appropriate correctors we have computed accurately 49 critical orbits at which families of 3D periodic orbits of the same period bifurcate. All these families have been computed in both cases of equal and non-equal primaries, and consist entirely of unstable orbits. They all terminate with coplanar periodic orbits. We have also found 35 critical orbits at which period doubling bifurcations occur. Several families of 3D periodic orbits bifurcating at these critical Sitnikov orbits have also been given. These families contain stable parts and close upon themselves containing no coplanar orbits.     

The theory of asynchronous relative motion II: universal and regular solutions

Two fully regular and universal solutions to the problem of spacecraft relative motion are derived from the Sperling–Burdet (SB) and the Kustaanheimo–Stiefel (KS) regularizations. There are no singularities in the resulting solutions, and their form is not affected by the type of reference orbit (circular, elliptic, parabolic, or hyperbolic). In addition, the solutions to the problem are given in compact tensorial expressions and directly referred to the initial state vector of the leader spacecraft. The SB and KS formulations introduce a fictitious time by means of the Sundman transformation. Because of using an alternative independent variable, the solutions are built based on the theory of asynchronous relative motion. This technique simplifies the required derivations. Closed-form expressions of the partial derivatives of orbital motion with respect to the initial state are provided explicitly. Numerical experiments show that the performance of a given representation of the dynamics depends strongly on the time transformation, whereas it is virtually independent from the choice of variables to parameterize orbital motion. In the circular and elliptic cases, the linear solutions coincide exactly with the results obtained with the Clohessy–Wiltshire and Yamanaka–Ankersen state-transition matrices. Examples of relative orbits about parabolic and hyperbolic reference orbits are also presented. Finally, the theory of asynchronous relative motion provides a simple mechanism to introduce nonlinearities in the solution, improving its accuracy.     

Nonlinear analytic solution for perturbed relative motion using differential equinoctial elements

This paper develops a nonlinear analytic solution for satellite relative motion in J₂-perturbed elliptic orbits by using the geometric method that can avoid directly solving the complex differential equations. The differential equinoctial elements (DEEs) are used to remove any singularities for zero-eccentricity or zero-inclination orbits. Based on the relationship between the relative states and the DEEs, state transition tensors (STTs) for transforming the osculating DEEs and propagating the mean DEEs have been derived. The formulation of these STTs has been split into a set of vector and matrix operations, which avoids directly expanding the complex second-order terms, and thus, the obtained STTs could be easy-to-understand and easy-to-code. Numerical results show that the proposed nonlinear solution is valid for zero-eccentricity and zero-inclination reference orbit and is more accurate than the previous linear or nonlinear methods for the long-term prediction of satellite relative motion.     

Superosculating intermediate orbits: Fourth-and fifth-order tangencies to trajectories of perturbed motion

The theory of superosculating intermediate orbits previously suggested by the author is developed. A new class of orbits with a fourth-order tangency to the actual trajectory of a celestial body at the initial time is constructed. Orbits with a fifth-order tangency have been constructed for the first time. The motion in the constructed orbits is represented as a combination of two motions: the motion of a fictitious attracting center with a variable mass and the motion relative to this center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden—Mestschersky problem. The variation in the mass of the fictitious center obeys Mestschersky’s first and combined laws. The new orbits represent more accurately the actual motion in the initial segment of the trajectory than an osculating Keplerian orbit and other existing analogues. Encke’s generalized methods of special perturbations in which the constructed intermediate orbits are used as reference orbits are presented. Numerical simulations using the approximations of the motions of Asteroid Toutatis and Comet P/Honda—Mrkos—Pajdu?áková as examples confirm that the constructed orbits are highly efficient. Their application is particularly beneficial in investigating strongly perturbed motion.     

Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations

Finding relative satellite orbits that guarantee long-term bounded relative motion is important for cluster flight, wherein a group of satellites remain within bounded distances while applying very few formationkeeping maneuvers. However, most existing astrodynamical approaches utilize mean orbital elements for detecting bounded relative orbits, and therefore cannot guarantee long-term boundedness under realistic gravitational models. The main purpose of the present paper is to develop analytical methods for designing long-term bounded relative orbits under high-order gravitational perturbations. The key underlying observation is that in the presence of arbitrarily high-order even zonal harmonics perturbations, the dynamics are superintegrable for equatorial orbits. When only J ₂ is considered, the current paper offers a closed-form solution for the relative motion in the equatorial plane using elliptic integrals. Moreover, necessary and sufficient periodicity conditions for the relative motion are determined. The proposed methodology for the J ₂-perturbed relative motion is then extended to non-equatorial orbits and to the case of any high-order even zonal harmonics (J _2n , n ≥ 1). Numerical simulations show how the suggested methodology can be implemented for designing bounded relative quasiperiodic orbits in the presence of the complete zonal part of the gravitational potential.     

Chaos and elliptical galaxies

Recent results on chaos in triaxial galaxy models are reviewed. Central mass concentrations like those observed in early-type galaxies - either stellar cusps, or massive black holes — render most of the box orbits in a triaxial potential stochastic. Typical Liapunov times are 3–5 crossing times, and ensembles of stochastic orbits undergo mixing on timescales that are roughly an order of magnitude longer. The replacement of the regular orbits by stochastic orbits reduces the freedom to construct self-consistent equilibria, and strong triaxiality can be ruled out for galaxies with sufficiently high central mass concentrations.     

Planetary dynamics in the system α Centauri: The stability diagrams

The stability of the motion of a hypothetical planet in the binary system ?? Cen A?CB has been investigated. The analysis has been performed within the framework of a planar (restricted and full) three-body problem for the case of prograde orbits. Based on a representative set of initial data, we have obtained the Lyapunov spectra of the motion of a triple system with a single planet. Chaotic domains have been identified in the pericenter distance-eccentricity plane of initial conditions for the planet through a statistical analysis of the data obtained. We have studied the correspondence of these chaotic domains to the domains of initial conditions that lead to the planet??s encounter with one of the binary??s stars or to the escape of the planet from the system. We show that the stability criterion based on the maximum Lyapunov exponent gives a more clear-cut boundary of the instability domains than does the encounterescape criterion at the same integration time. The typical Lyapunov time of chaotic motion is ??500 yr for unstable outer orbits and ??60 yr for unstable inner ones. The domain of chaos expands significantly as the initial orbital eccentricity of the planet increases. The chaos-order boundary has a fractal structure due to the presence of orbital resonances.