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.     

On the rotation of co-orbital bodies in eccentric orbits

We investigate the resonant rotation of co-orbital bodies in eccentric and planar orbits. We develop a simple analytical model to study the impact of the eccentricity and orbital perturbations on the spin dynamics. This model is relevant in the entire domain of horseshoe and tadpole orbit, for moderate eccentricities. We show that there are three different families of spin–orbit resonances, one depending on the eccentricity, one depending on the orbital libration frequency, and another depending on the pericenter’s dynamics. We can estimate the width and the location of the different resonant islands in the phase space, predicting which are the more likely to capture the spin of the rotating body. In some regions of the phase space the resonant islands may overlap, giving rise to chaotic rotation.     

The dynamics of two massive planets on inclined orbits

The significant orbital eccentricities of most giant extrasolar planets may have their origin in the gravitational dynamics of initially unstable multiple planet systems. In this work, we explore the dynamics of two close planets on inclined orbits through both analytical techniques and extensive numerical scattering experiments. We derive a criterion for two equal mass planets on circular inclined orbits to achieve Hill stability, and conclude that significant radial migration and eccentricity pumping of both planets occurs predominantly by 2:1 and 5:3 mean motion resonant interactions. Using Laplace-Lagrange secular theory, we obtain analytical secular solutions for the orbital inclinations and longitudes of ascending nodes, and use those solutions to distinguish between the secular and resonant dynamics which arise in numerical simulations. We also illustrate how encounter maps, typically used to trace the motion of massless particles, may be modified to reproduce the gross instability seen by the numerical integrations. Such a correlation suggests promising future use of such maps to model the dynamics of more coplanar massive planet systems.     

Long term evolution of quasi-circular Trojan orbits

Trojan asteroids undergo very large perturbations because of their resonance with Jupiter. Fortunately the secular evolution of quasi circular orbits remains simple—if we neglect the small short period perturbations. That study is done in the approximation of the three dimensional circular restricted three-body problem, with a small mass ratio μ—that is about 0.001 in the Sun Jupiter case. The Trojan asteroids can be defined as celestial bodies that have a “mean longitude”, M + ω + Ω, always different from that of Jupiter. In the vicinity of any circular Trojan orbit exists a set of “quasi-circular orbits” with the following properties: (A) Orbits of that set remain in that set with an eccentricity that remains of the order of the mass ratio μ. (B) The relative variations of the semi-major axis and the inclination remain of the order of ${\sqrt{\mu}}$ . (C) There exist corresponding “quasi integrals” the main terms of which have long-term relative variations of the order of μ only. For instance the product c(1 – cos i) where c is the modulus of the angular momentum and i the inclination. (D) The large perturbations affect essentially the difference “mean longitude of the Trojan asteroid minus mean longitude of Jupiter”. That difference can have very large perturbations that are characteristics of the “horseshoes orbit”. For small inclinations it is well known that this difference has two stable points near ±60° (Lagange equilibrium points L₄ and L₅) and an unstable point at 180° (L₃). The stable longitude differences are function of the inclination and reach 180° for an inclination of 145°41′. Beyond that inclination only one equilibrium remains: a stable difference at 180°.     

On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited

In the framework of the planar and circular restricted three-body problem, we consider an asteroid that orbits the Sun in quasi-satellite motion with a planet. A quasi-satellite trajectory is a heliocentric orbit in co-orbital resonance with the planet, characterized by a nonzero eccentricity and a resonant angle that librates around zero. Likewise, in the rotating frame with the planet, it describes the same trajectory as the one of a retrograde satellite even though the planet acts as a perturbator. In the last few years, the discoveries of asteroids in this type of motion made the term “quasi-satellite” more and more present in the literature. However, some authors rather use the term “retrograde satellite” when referring to this kind of motion in the studies of the restricted problem in the rotating frame. In this paper, we intend to clarify the terminology to use, in order to bridge the gap between the perturbative co-orbital point of view and the more general approach in the rotating frame. Through a numerical exploration of the co-orbital phase space, we describe the quasi-satellite domain and highlight that it is not reachable by low eccentricities by averaging process. We will show that the quasi-satellite domain is effectively included in the domain of the retrograde satellites and neatly defined in terms of frequencies. Eventually, we highlight a remarkable high eccentric quasi-satellite orbit corresponding to a frozen ellipse in the heliocentric frame. We extend this result to the eccentric case (planet on an eccentric motion) and show that two families of frozen ellipses originate from this remarkable orbit.     

Analytical investigations of quasi-circular frozen orbits in the Martian gravity field

Frozen orbits are always important foci of orbit design because of their valuable characteristics that their eccentricity and argument of pericentre remain constant on average. This study investigates quasi-circular frozen orbits and examines their basic nature analytically using two different methods. First, an analytical method based on Lagrangian formulations is applied to obtain constraint conditions for Martian frozen orbits. Second, Lie transforms are employed to locate these orbits accurately, and draw the contours of the Hamiltonian to show evolutions of the equilibria. Both methods are verified by numerical integrations in an 80 × 80 Mars gravity field. The simulations demonstrate that these two analytical methods can provide accurate enough results. By comparison, the two methods are found well consistent with each other, and both discover four families of Martian frozen orbits: three families with small eccentricities and one family near the critical inclination. The results also show some valuable conclusions: for the majority of Martian frozen orbits, argument of pericentre is kept at 270° because J ₃ has the same sign as J ₂; while for a minority of ones with low altitude and low inclination, argument of pericentre can be kept at 90° because of the effect of the higher degree odd zonals; for the critical inclination cases, argument of pericentre can also be kept at 90°. It is worthwhile to note that there exist some special frozen orbits with extremely small eccentricity, which could provide much convenience for reconnaissance. Finally, the stability of Martian frozen orbits is estimated based on the trace of the monodromy matrix. The analytical investigations can provide good initial conditions for numerical correction methods in the more complex models.     

Dynamical relaxation and the orbits of low-mass extrasolar planets

We consider the evolution of a system containing a population of massive planets formed rapidly through a fragmentation process occurring on a scale on the order of 100 au and a lower mass planet that assembles in a disc on a much longer time-scale. During the formation phase, the inner planet is kept on a circular orbit owing to tidal interaction with the disc, while the outer planets undergo dynamical relaxation. Interaction with the massive planets left in the system after the inner planet forms may increase the eccentricity of the inner orbit to high values, producing systems similar to those observed.     

Stability of inclined orbits of terrestrial planets in habitable zones

In long-term stability studies of terrestrial planets moving in the habitable zone (HZ) of a sun-like star, we distinguish four different configurations: (i) planets moving in binary star systems, (ii) the inner type (where the gas giant moves outside the HZ), (iii) the outer type (where the gas giant is closer to the star, than the HZ) and (iv) the Trojan type (where the gas giant moves in the HZ). Since earlier calculations indicated, that the stability of the motion in the HZ also depends on the inclination of the terrestrial planet orbits, we present a detailed numerical investigation to show correlations between the eccentricity, the mass and the distance of the giant planet for various inclinations of the terrestrial planets. The orbital stability of the HZ was examined for all four configurations stated above. While we could find hardly any stable orbits for the first three types for inclinations higher than 40^°, the Trojan planets can be stable up to an inclination of 60^°. Additionally, we could also find some stabilizing effects of the inclination for the first three types. As dynamical model we used the elliptic restricted three-body problem, which consists of two massive and one mass-less body. This allows an application to all detected and future extrasolar single planet systems.     

On linking coefficient of two Keplerian orbits

We define a function of the set of pairs of Keplerian ellipses so that the sign of the function will be a topological invariant of their configuration. The sign is negative if and only if the related ellipses are linked. Two modifications of the coefficient which are more reliable in the case of closed to coplanar orbits are proposed. Explicit formulae representing the linking coefficients as functions of orbital elements are deduced. Extension in the case of unbounded orbits is obtained. We suggest different ways to use these coefficients for determining intersections of pairs of osculating Keplerian orbits. If we study dynamical behaviour of geometric configuration of pairs of Keplerian orbits, we can fix the moments of their intersections. These moments correspond exactly to the vanishing of linking coefficients. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical periodic orbits of charged grains around magnetic planets

We apply a numerical searching method to investigate three-dimensional periodic orbits of charged dust particles in planetary magnetospheres. A classic generalized Stormer model of magnetic planets along with the parameters of Saturn is employed. More periodic orbits are found, besides the already known circular periodic orbits in or parallel to the equatorial plane. We divide all these orbits into six categories based on their appearances. By calculating the characteristic multipliers of the orbits, we investigate the stabilities of these periodic orbits.     

On the coplanar eccentric non-restricted co-orbital dynamics

We study the phase space of eccentric coplanar co-orbitals in the non-restricted case. Departing from the quasi-circular case, we describe the evolution of the phase space as the eccentricities increase. We find that over a given value of the eccentricity, around 0.5 for equal mass co-orbitals, important topological changes occur in the phase space. These changes lead to the emergence of new co-orbital configurations and open a continuous path between the previously distinct trojan domains near the \(L_4\) and \(L_5\) eccentric Lagrangian equilibria. These topological changes are shown to be linked with the reconnection of families of quasi-periodic orbits of non-maximal dimension.     

The planetary masses and the orbits of the first four minor planets

Numerical tests are the basis of a study about the effects caused in the orbits of the planets (1)–(4) by possible errors in the system of planetary masses. The masses of five major and three minor planets are considered. Especially, the effects caused by (1) Ceres in the orbit of (2) Pallas since the time of discovery are found to be large enough for a determination of the mass of Ceres. A first result for this mass is (6.7±0.4)×10^–10 solar masses.     

Periodic orbits of galactic motion

Poincaré's surface of section method is used to find and classify the main periodic orbits in a two-dimensional galactic potential first introduced by Hénon and Heiles. The stability of these periodic orbits is studied. Numerical integration with Bulirsch-Stoer method is used.     

Johann Heinrich Lambert and the determination of orbits for planets and comets

Johann Heinrich Lambert, a Swiss-German scholar, was counted among the most famous men of his time. Being an autodidact and without formal university studies, he won access to the modern sciences, in particular to philosophy, mathematics, astronomy and physics; and indeed he was most successful in all these fields. In the area of mathematics, for example, he made contributions to the theory of irrational numbers (continued fractions), to the problem of parallel lines and non-Euclidian geometry, to trigonometry (group-theoretical formulation of Nepper's rule), the foundations of the perspective, he gave a proof for the irrationality of e and and moreover some remarkable indications for their transcendency, to mention only a few. In the field of astronomy, he made contributions not only to the foundations of photometry but also to the orbital determination of planets and comets, which culminated in Lambert's Theorem. This, as well as his Cosmologische Briefe (with which he made a contribution to the structure of the world parallel to the Kantian view) is the subject matter of the following report. Even in philosophy Lambert kept pace with Kant (Criterium veritatis, 1761 when he said: Wir wissen, dass Gestalt und Grösse vom Orte umabhängige Bestimmungen sind.).

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Zusammenfassung Johann Heinrich Lambert, ein deutsch-schweizer Wissenschaftler, gehörte zu den berühmtesten Männern seiner Zeit. Er hat als Autodidakt ohne Studium an einer Hochschule sich den Zugang zu den modernen Wissenschaften, insbesondere zue Philosophie, zur Mathematik, zu Astronomie und Physik geschaffen und auf diesen Gebieten Fortschrittliches geleistet. Auf dem Gebiete der Mathematik z.B. waren es seine Beiträge zur Theorie der Irrationalzahlen (Kettenbrüche), zum Problem der Parallellinien und der nichteuklidischen Geometrie, zur Trigonometrie (gruppentheoretischer Ansatz für die Neppersche Regel), die Begründung der Perspektive, Beweis der Irrationalität von e und mit Ansätzen zum Nachweis ihrer Transzendenz, um nur einiges herauszuheben. Auf dem Gebiete der Astronomie waren es neben seinen Grundlagen für die Photometrie seine Beiträge zur Bahnbestimmung von Planeten und Kometen, die er mit dem nach ihm benannten Lambertschen Theorem krönte. Darüber und über seine Cosmologischen Briefe, mit denen er einen Beitrag zur Struktur der welt parallel zu der von Kant gegeben hat, wird im folgenden ausführlich gehandelt. Auch in der Philosophie war er im Gleichschritt mit Kant (Criterium veritatis 1761 mit dem Ausspruch: Wir wissen, daß Gestalt und Größe vom Orte unabhängige Bestimmungen sind.).

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Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

Dynamics of planets in retrograde mean motion resonance

In a previous paper (Gayon and Bois 2008a), we have shown the general efficiency of retrograde resonances for stabilizing compact planetary systems. Such retrograde resonances can be found when two-planets of a three-body planetary system are both in mean motion resonance and revolve in opposite directions. For a particular two-planet system, we have also obtained a new orbital fit involving such a counter-revolving configuration and consistent with the observational data. In the present paper, we analytically investigate the three-body problem in this particular case of retrograde resonances. We therefore define a new set of canonical variables allowing to express correctly the resonance angles and obtain the Hamiltonian of a system harboring planets revolving in opposite directions. The acquiring of an analytical “rail” may notably contribute to a deeper understanding of our numerical investigation and provides the major structures related to the stability properties. A comparison between our analytical and numerical results is also carried out.     

Direct perturbations of the planets on the Moon's motion

The aim of the present paper is to present the theoretical background of a method to compute the planetary perturbations on the Moon's motion. We formulate an algorithm based upon the Lie transform method and well-suited to the particular problem at hand.This algorithm is being implemented using Henrard's Semi-Analytical Lunar Ephemeris (SALE) as solution of the Main Problem and Bretagnon's planetary theory. The accuracy of the solution is intended to be about 0".001 for terms of period up to 2000 years.To illustrate the interest of our approach, we comment on some preliminary results obtained about the direct perturbations due to Venus on the Moon's longitude. The final results will be the subject of another paper.     

Tidal interactions and disruptions of giant planets on highly eccentric orbits

We calculate the evolution of planets undergoing a strong tidal encounter using smoothed particle hydrodynamics (SPH), for a range of periastron separations. We find that outside the Roche limit, the evolution of the planet is well-described by the standard model of linear, non-radial, adiabatic oscillations. If the planet passes within the Roche limit at periastron, however, mass can be stripped from it, but in no case do we find enough energy transferred to the planet to lead to complete disruption. In light of the three new extrasolar planets discovered with periods shorter than two days, we argue that the shortest-period cases observed in the period-mass relation may be explained by a model whereby planets undergo strong tidal encounters with stars, after either being scattered by dynamical interactions into highly eccentric orbits, or tidally captured from nearly parabolic orbits. Although this scenario does provide a natural explanation for the edge found for planets at twice the Roche limit, it does not explain how such planets will survive the inevitable expansion that results from energy injection during tidal circularization.