The development of cost effective spacecraft missions places ever-increasing demands on the mission designer to explore and develop new techniques for interplanetary transfer. A key enabling technology to such designs is that of advanced propulsion systems. This generally implies a focus on low thrust systems for the current and next generation of missions. Low thrust systems include a variety of electric propulsion technologies, powered either by solar or radioactive sources, and the next generation of missions are likely to include solar sailing. The application of low thrust gives an opportunity to incorporate some more unusual aspects of astrodynamics to maximise transfer efficiency. The optimisation of such a mission relies on minimisation of the total propulsion unit mass, ie system mass plus fuel mass. This also includes the selection of the best propulsion technology type for a given mission.
A good example arises from the utilisation of the Sun’s gravitational potential to assist in planetary escape and capture. The fundamentals of this effect have been researched previously in the fields of comet dynamics and studies on Weak Stability Boundaries. Its effectiveness can be maximised when considered in conjunction with missions using low thrust interplanetary manoeuvres. Such mission designs may not only result in low propulsion mass but also a robust, failure tolerant, planetary capture scenario A further highly effective application of low thrust systems lies in their combination with multiple planetary gravity assist manoeuvres. An example uses single or multiple Earth gravity assists and low thrust arcs to reach high excess hyperbolic speeds, enabling transfers to Jupiter or Saturn. The manoeuvres take place at modest distances from the Sun and so relatively low mass, solar electric propulsion systems can be used, resulting in an efficient transfer option. The paper will present examples of such optimised propulsion/mission designs.

Z. Knezevich
Belgrade  Astronomical Observatory, Yugoslavia

Chaotic diffusion in the Veritas asteroid family  region

Veritas   asteroid family region is dynamically one of the most   complex in the asteroid belt. Many studies of chaotic motion in the  resonances that cross the family have been published in recent years, with  an aim to understand the dynamical mechanisms at work and establish the age  of the family. Starting from the work by Milani   and Farinella (1994), who introduced the so-called  'chaotic chronology' method and gave first estimate of an upper limit to the age of the Veritas asteroid family, through several later upgrades (Kne \v zevi \'c and Pavlovi \'c 2002) of the same approach, to the more recent attempts using different approaches (  Kne\v zevi\'c et al. 2002,   Nesvorn\'y et al. 2003), all the results indicate an unexpectedly  young age for this family.  Here  we show results of a new study which includes a total of 180 members of the  family. We computed the time series of proper elements for all these bodies  for up to 100~Myr, and we derived the corresponding  Lyapunov exponents and diffusion coefficients.  We confirmed the previously suggested splitting of the family in 5 dynamically  distinct groups, with particular attention payed   to the differences of the two main chaotic strips in terms of the long term diffusion and overlap of the resonant harmonics. Finally, we present two independent estimates of the age of the family, one based on the chaotic  diffusion and involving members of the family located in the resonant strips,  and the other based on remarkable clustering of the nodal longitudes at an  epoch in the past found for a group of family members with regular motion.  The two estimates are in fair agreement with each other, and with the 8.3~Myr  estimate for the age of Veritas family by  Nesvorny et al. (2003).
 

T. Kovács
Eötvös Loránd University Dept. of Astronomy, Hungary

In celestial mechanics the fundamental task is to predict the long time evolution of planetary orbits. The secular evolution of the orbits was investigated first by Lagrange and Laplace in the framwork of the first order secular theory, the so called Laplace-Lagrange theory. In this study the Laplace-coefficients are calculated via different methods, and the corresponding solutions are compared to each other. They are also checked against to a numerical integration of the complete Solar System. These investigations help to choose the most accurate method. The long term evolution of the orbital elements depends on the Laplace-coefficients, the more accurate therefore the Laplace-coefficients are, the more reliable the corresponding solution is.
 

E. Kuznetsov
Astronomical Observatory of the Urals State University

Dynamical evolution of a weakly perturbed Two-Planetary System on a cosmogonic time-scale (K.V.Kholshevnikov, E.D.Kuznetsov)

We consider orbital evolution of planetary systems similar to our Solar one. In the present work we use Jacobian coordinates, construct the Hamiltonian expansions in the Poisson series in all elements for the planetary three-body problem. Values of planetary masses and mean semi-major axes of their orbits may be arbitrary. Further we construct the averaged Hamiltonian by the Hori - Deprit method with accuracy up to second order with respect to the small parameter, the generating function of the Lie transform, change of variables formulae, and right-hand sides of averaged equations. Resonant semi-major axes and estimations of the resonant zones width are obtained for the small parameter values varying from 0.001 to 0.1. The averaged equations are integrated numerically. Orbital evolution of the Sun-Jupiter-Saturn system and several other weakly perturbed two-planetary systems are investigated at the time-scale 10 Gyr. This work was partly supported by the Leading Scientific School, Grant NSh-1078.2003.02.

 

E. Kuznetsov
Astronomical Observatory of the Urals State University

Stochastic dynamics of geosynchronous satellites (E.D.Kuznetsov, G.T.Kaiser) (Poster)

The orbital evolution of geosynchronous satellites moving near separatrices separating of the rotation and libration
motions regions and the libration motions with respect to the one and two stable points was investigated.
We consider the phase plane \"longitude of subsatellite point - semi-major axis\". The separatrices can migrate due to the perturbations. Sizes of the migrate regions are minimal near libration points. Decrease of distance between
separatrices and increase of the separatrix migrate region due to growth of an inclination lead to overlap the
separatrix location regions. The estimations of the stochasticity zones width with respect to the initial semi-major axes values are obtained. The zones width grows at increase of the satellite\\\'s area to mass ratio and decreases with growth of the orbital inclination. The stochasticity zones corresponding various separatrices are separated from each other.
This work was partly supported by the RFBR, Grant 03-02-16313.
 

L.   Iess
Universita' di Roma  " La Sapienza",  Italy
 

The propagation of photons and the motion of Mercury in the gravity field of the Sun are still some of the best available experimental tools for testing general relativity. The spacecraft Cassini has recently measured the spatial metric of the solar system to 500 parts per million in its cruise phase to Saturn. A more complete and accurate set of measurements are planned with BepiColombo, the ESA mission to Mercury. Both experiments are enabled by state-of-the-art microwave tracking systems based upon multi-frequency links, which provide an unprecedented accuracy in range and range rate measurements and therefore a better orbit determination. We review the experimental setup and the results (either achieved or expected) of these experiments, and discuss the future prospects of solar system tests of gravitational theories.

J. Laskar
Observatoire de Paris, France

Chaotic diffusion in the Solar System
 

A. Lemaitre
FUNDP, Namur, Belgium

The 3:2 spin-orbit resonant motion of Mercury

Our purpose is to build a model of rotation for a rigid Mercury,  involving the planetary  perturbations and the non spherical shape of  the planet. The purpose is double: the study of the direct  influence of such a   model on the motion of a satellite (for space missions like BepiColombo) and the building of a reference for the measurements of   the non rigidity (existence and size of a core) of Mercury. Our approach is purely analytical, based on Hamiltonian formalism;  we  start   with a first order basic averaged resonant potential (including $J_2$ and $C_{22}$, and the first powers of the eccentricity and the inclination of Mercury). With this kernel model, we calculate  and identify the equilibria (4); we select the present one, and introduce local canonical variables, describing  the  motion around this 3:2 resonance.  We perform a canonical untangling transformation, so to generate three sets of action-angle  variables,  and identify the three  basic frequencies associated with this motion. We reintroduce the neglected terms (higher powers of the   eccentricity   and variables),  the short periodic terms (lost in the averaging process) and the planetary perturbations (Venus and Jupiter),  and we calcule (through a Lie triangle of order 2 or 3) the three main corrected frequencies and together, thanks to the Lie generator, the  three Complete Euler angles  describing the rotation.  At any point of the development, we use the software SONYR (written by Rambaux and Bois) so to compare and check our calculations. We try  to solve the problem  of the choice of initial conditions for the rotation by the introduction of dissipations and frictions,  which eliminate (in principle)  the so-called free librations.
 
 

University of Namur, FUNDP, Belgium

Exoplanetary systems are quite different from our own planetary system and classical Laplace-Lagrange linear perturbation theory is very limited. We analyze the secular interactions of two coplanar non-resonant planets with a high order expansion of the perturbative potential in powers of eccentricities . We show that this approach models correctly most of exosystems discovered so far. Particular attention is given to apsidal configuration, libration or circulation, oscillation amplitude of the angular difference of the apsidal lines...

Jet Propulsion Laboratory, USA

The SmallTug mission is a technology demonstration for the NASA Space Exploration Program to show the usefulness of low energy trajectories for cargo transport between the Earth and the Moon. These trajectories are generated by the invariant manifolds of unstable quasiperiodic orbit in the Earth-Moon and Sun-Earth systems as coupled 3 body systems. By trading time for less fuel, such trajectories are ideal for the transport of cargo. The trajectory concept is based on previous work by the author on a Lunar Gateway Station concept. For the SmallTug, the trajectory starts with a Geostationary Transfer Orbit and uses low thrust transfer via invariant manifolds to place the microspacecraft in orbit about Lunar L1. After orbiting LL1 for several orbits, the spacecraft will be brought back to Earth orbit to demonstrate the feasibility of this concept. Invariant manifolds continue to play a key role even in the low thrust trajectory domain.

 

U. Locatelli  
Universita' di Roma Tor Vergata, Italy
 

In a recent work by Gabern, Jorba and myself ([GJL], in press on Nonlinearity), we studied the stability of the Trojan asteroids in the framework of the planar circular restricted three-body problem. We succeeded in constructing the invariant tori well approximating the orbits in 23 over 30 considered cases, so that we largely improved the previously existing results. The construction of the invariant tori is based on a careful and quite technical reformulation of the
Kolmogorov\'s normalization algorithm. It is well known that the motions very close to the Lagrangian points are characterized by two frequencies and one of them is much slower than the other one. In order to improve and simplify our approach, we first average the Hamiltonian over the faster frequency. This allow us to produce an integrable approximation of the system, which can be used as a new starting point for the construction of the invariant tori.
The results of this new approach will be compared with those in [GJL].
 

Istituto di Fisica dello Spazio INterplanetario IFSI/INAF, Roma, Italy

The talk is focused on the estimate of the impact of the non-gravitational perturbations on the orbit of the Mercury Planetary Orbiter (MPO), one of the two spacecrafts that will be placed in orbit around the innermost planet of the solar system by the BepiColombo space mission. In particular, the advantages of an on-board accelerometer are outlined with respect to the modelling of the non-gravitational perturbations in the strong radiation environment of Mercury. The readings from the accelerometer guarantees a very significant reduction of the non-gravitational accelerations impact on the space mission accuracy, especially of the dominant direct solar radiation pressure in the very complex radiation environment of Mercury. Practically, we are able to remove from the list of unknowns the non-gravitational accelerations in such a way to transform, aposteriori, the MPO in a drag–free like satellite. The Italian Spring Accelerometer (ISA) has been considered and then selected by the European Space Agency to fly on-board the MPO. ISA is a three–axis instrument with an intrinsic noise level of 10^-9m/s²/Hz^1/2 in the frequency band of 3×10^-5–10^-1 Hz. This noise level matches very well the science requirements of the BepiColombo mission to Mercury with regard to the MPO orbit determination. Through a numerical simulation and analysis we have estimated, over a time span of several years, the behaviours of the disturbing accelerations on the MPO spacecraft produced by the incoming visible solar radiation pressure and by the indirect effects produced by Mercury's albedo. The variations on the orbital parameters of the spacecraft and in their rates have been also estimated over the analysed period. Finally, the impact of the non-gravitational accelerations over the typical arc length that will be used in the MPO orbit analysis is compared with the accelerometer accuracy in order to estimate the advantages of the on-board ISA accelerometer with respect to the best modelling of the subtle non-conservative effects here analysed.

Istituto di Fisica dello Spazio INterplanetario IFSI/INAF, Roma, Italy

The subject of the poster is the method applied since 1996 for the analysis of the orbital residuals of the two LAGEOS satellites in order to derive the frame–dragging effect of their orbit produced by the gravitomagnetic field of the Earth, i.e., the Lense–Thirring effect. The method is based on the difference between the orbital elements of consecutive arcs. It is proved that this ‘’difference method‘’ is excellent for the determination of the secular effects — as in the case of the relativistic precession induced by the Earth’s gravitomagnetic field — but also very useful for the determination and study of the unmodelled (or poorly modelled) long–term periodic effects which influence the satellites orbital elements. A few examples of secular and periodic effects determination will be given using the previous multi–satellite gravity field solution EGM96, as well as the most recent gravity field solutions from the CHAMP and GRACE missions.

Z. Makó
Sapientia University, Romania
 

Several authors studied the capture of small bodies by major planets, introducing different concepts of capture, like weak capture , temporary capture, longest capture, resonant capture, etc.  In all these studies the time is used as measure of the capture. In this paper we try to study the phenomenon of capture using the variation of the angle Dj of
the small body around the capturing planet. We defined the capture effect of the planet to the captured body, as the variation of the angle Dj during the capture, as long as the Kepler-energy of the small body relative to the central
planet is negative. The beginning moment the capture is in the moment when the Kepler-energy of the captured body, relative to the capturing body, becomes negative. The end of capture is in the moment when the Kepler-energy becomes positive. The variation of capture effect around to Jupiter is studied using sections. We are determined several zones, in which, for a given velocity, the capture is occur. These regions we named capture domain.  In this paper we show that the capture domain is chaotic, since this domain has a hyperbolic structure.

V. Martinot
Alcatel Space, France

There have been several projects of visiting the Moons of Mars in the overall goal to better know Mars, some of them even envisaging to establish a first manned base on them. After presenting an overview of the contexts in which these projects were proposed, this article defines possible mission scenarios for a mission inspecting the Moon(s) of Mars, from the Earth departure to a close formation flying with the Moons.

 

Centro de Ciencias Fisicas, U.N.A.M., Cuernavaca, Mexico

 

From Rotating billiards to narrow planetary rings (Poster)
 

The 3:7 inequality between J-3 Ganymede and J-4 Callisto, known as De Haerdtl\'s inequality, induces long period terms in the ephemerides of the Galilean satellites,  but might have also been a source of stochatic behaviour in a recent past. This study describes the stochastic layers crossed by the system because of tidal effects and evaluates their influence on its actual state.

 

K. Meyer

University of Cincinnati, USA

 

Variational Equations for Elliptic Relative Equilibrium

A planar central configuration of the $N$-body problem gives rise to a solution where each particle moves on a specific Keplerian orbit while the totality of the particles move on a homothety motion. If the Keplerian orbit is elliptic then the solution is an equilibrium in pulsating coordinates so we call this solution {\\it elliptic relative equilibrium}. We study the variational equations of these solutions.
 

F. Muñoz-Almaraz
Universitat de Barcelona, Spain

Families of symmetric periodic orbits in the three body problem and the figure eight

Chenciner and Montgomery have proved the existence of a solution (called figure eight) in the three body problem which is symmetric with several time-reversal symmetries. We state a theoretical result for the persistence of symmetric
solutions with respect two time--reversal symmetries and we set a boundary value problem whose \"regular\" continuation is equivalent to the continuation of the symmetric solutions. This method is applied starting from the figure
eight when one of the mass is varied. Several families of symmetric periodic orbits are got and the bifurcations undergo along this family allow us to explain different behauvior depending on the chosen symmetry. Subharmonic bifurcations are also considered. (Work joint with E. Freire, J. Galán and A. Vanderbauwhede)


F. Namouni
Observatoire de Nice, France

On the origin of the eccentricities of extrasolar planets

We present a new a theory that unifies the origin of the   large  eccentricities of extrasolar planets and the small eccentricities  in the  solar system, explains  the preference for apsidal alignment and   anti-alignment in non-resonant multiplanet   systems, and provides clues for the origin of the similarities in the eccentricity distribution of  extrasolar planets and that of  spectroscopic binary stars. We show that if a physical process is weakly dependent on the local  dynamics of the   companion, and  imparts a small relative acceleration to the star-companion system, the eccentricity of the companion's  orbit is   excited to large values. Natural candidates for this process are stellar jets and star-disk winds. In addition to exciting eccentricities, the acceleration gives rise to an escape-driven outward migration  in the  outer parts of the star-companion  system that may have important  consequences for the dynamics of the minor body populations in the solar system.
 

 

B. Noyelles

IMCCE - Observatoire de Paris

 

Stochastic behaviour around the 3:7 inequality between Ganymede and Callisto
 

The 3:7 inequality between J-3 Ganymede and J-4 Callisto, known as De Haerdtl\'s inequality, induces long period terms in the ephemerides of the Galilean satellites,  but might have also been a source of stochatic behaviour in a recent past. This study describes the stochastic layers crossed by the system because of tidal effects and evaluates their influence on its actual state.
 

 

 

M. Olle

Universitat Politecnica de Catalunya, Spain

Horseshoe motion in the restricted three body problem

In the planar RTBP, the invariant manifolds of the collinear equilibrium point $L_3$, for different values of the mass parameter $\\mu$, give rise to horseshoe motion of different kinds: periodic orbits and homoclinic orbits. We give a mechanism, for $\\mu =0$, that explains the families of horseshoe periodic orbits that exist for $\\mu >0$ and small. Horseshoe periodic motion is also analyzed for any value of $\\mu \\in (0,1/2]$. Finally, we consider the spatial RTBP where bifurcating families of 3D horseshoe periodic orbits and horseshoe invariant tori also exist.

 

E.Olmedo
Universitat de Barcelona, Spain

Computing quasi-periodic motions for a particle in the Earth-Moon system: 2,3,4D-tori and their corresponding invariant manifolds (joint work with Angel Jorba)
 

We are interested in the motion of a particle, such as a spacecraft or an asteroid, in the Earth-Moon System. The simplest model for this problem is the well known Restricted Three Body Problem (RTBP). Here we consider more realistic models that take into account the main perturbations coming from the Sun and the non-circular motion of Earth and Moon. These models are written as quasi-periodic time-dependent perturbations of the RTBP. In our case, depending on the number of effects considered, the perturbation can have 2, 3 or 4 basic frequencies. In the RTBP, there are five equilibrium that here will become tori of dimension 2, 3 or 4, depending on the number of basic frequencies of the perturbation. Our goal is to compute the tori substituting the collinear equilibrium $L_{1,2}$, as well as their invariant unstable and stable manifolds. We will also talk about other quasi-periodic solutions appearing from the centre directions of these tori. In that case the dimension of the tori increase in one or two units.
Due to the strong instability of these points, we use a parallel shooting technique that has the collateral effect of increasing the dimension of the phase space up to 24. We represent these tori and the Floquet transformations by (truncated) Fourier series. Due to the huge amount of computer resources needed, we have modified a previous software of the authors --that was already parallelised and adapted or a Beowulf cluster-- to adapt it to the particularities of this problem. In the talk we will discuss some details of this computation and the results obtained.

 

 

 

J. F. Palacian and P. Yanguas

 

Universidad publica de Navarra, Spain

 

The spatial elliptic restricted three-body problem: periodic orbits

Using a combination of analytical and numerical techniques we calculate new families of periodic orbits of the spatial elliptic restricted three-body problem. The initial conditions to numerically compute the orbits are given in an analytical way. After doubly averaging the Hamiltonian over the time and the mean anomaly we calculate its relative equilibria and study their linear and Lyapunov stability. This is used, not only to reconstruct invariant manifolds in the original system, but also to determine six families of periodic orbits with large semimajor axes having any eccentricity. Four families lie on a plane perpendicular to the plane of the primaries, two of them being stable and the other two unstable. The other two families are stable families of closed orbits coplanar with the primaries.
 

E. Perozzi 

Studi e Progetti Innovativi, Telespazio Spa, Roma, Italy

Planets around a 
pulsar: the co-orbital hypothesis (Poster)

Authors: 

E. Perozzi1, S. DAll'Osso2, G.L. Israe2, L. Stella2, A. Possenti3
1Telespazio, Roma, Italy, 2Osservatorio Astronomico di Roma, 3Osservatorio
Astronomico di Cagliari

In order to explain the occurrence of peculiar glitches in anomalous X-ray pulsars, a system of two large eccentricity co-orbital massive planets has
been proposed. The resulting dynamics falls in the domain of the so-called general Three-Body Problem, where no simplifying assumptions can be made and
the extension of what is already known for the restricted case depends on the specific case. Numerical experiments have been therefore carried out for
the co-orbital planetary hypothesis, suggesting that there is sufficient dynamical ground for such a system to survive over its typical lifetime,
which is of the order of 104 - 105 years.

G. Pinzari 

Universita' di Roma Tre, Italy

Four classical methods for determining planetary elliptic elements (Authors: A. Celletti, G. Pinzari)

The discovery of the asteroid Ceres by G. Piazzi in 1801 motivated the development of a mathematical technique proposed by C.F.Gauss, which allows to recover the orbit of a celestial body starting from a minimum of three
observations. Here we compare the method proposed by Gauss with the techniques (based on three observations) developed by P.S. Laplace and by O.F. Mossotti. We also consider another method developed by O.F.Mossotti, based on four observations. We provide a theoretical and numerical comparison among the different procedures. As an application, we consider the computation of the orbit of the asteroid Juno.
 

E. Pitjeva
Russia

The asteroid impact on planets and mass estimations of asteroids and the main asteroid belt from ranging
 

The significant impact of main belt asteroids on the inner planets and Kuiper belt objects on the position of the
Sun was shown by numerical integration and should be taken into account when high-accuracy planetary ephemerides are constructed. From an analysis of motions of the major planets some physical parameters of the asteroids were obtained by processing of precise measurements of ranging (1961-2003). The masses of the 6 asteroids were determined individually, masses of the most relevant 296 asteroids were derived from their latest published diameters
making use of the corresponding densities. The total contribution of all remaining small asteroids was modeled as an acceleration caused by a solid ring. As a sequence the total mass of the main asteroid belt was obtained: M=(15+-1)*10E-10 mass of Sun. An derived expression for estimating the total number of minor planets in any unit interval of absolute magnitude H was compared with the observed distributions of the 233001 asteroids.

 

M. Post
University of Paderborn, Germany

On Target for Venus - Set Oriented Numerical Methods for the Construction of Energy-Efficient Low-Thrust Trajectories

Recently, new techniques for the design of energy efficient trajectories for space missions have been proposed that are based on the circular restricted three body problem as the underlying mathematical model. These techniques exploit the structure of certain invariant sets and associated invariant manifolds in phase space in order to systematically construct energy efficient flight paths. We extend this model in order to account for a control force continuously applied on the spacecraft as realized by low thrust propulsion systems. We show how the techniques for the trajectory design can be suitably augmented. As an example we compute approximations to trajectories for a mission to Venus.
 

G. Pucacco
Università di Roma "Tor Vergata", Italy

Stability of periodic orbits in galactic potentials

A survey of the stability properties of periodic orbits in non integrable analytic potentials suitable to describe elliptical galaxies is performed through a normal form analysis. Normal modes stability is determined from the nature of relative equilibria of the reduced systems. The results are compared with the analysis performed with other classical approaches.
 

N. Rambaux
University of Namur and Royal Observatory of Belgium

An accurate theory of the rotation of Mercury

We have used the SONYR model of Bois and Rambaux (acronym of Spin-Orbit N-bodY Relativisitic model) of the solar System in order to reach a very accurate rotational motion of Mercury. The approach of the BJV model (Bois, Journet and Vockrouhlicky) as well as its SONYR extension derive from the DSX formulation of the post-Newtonian theory of motion for a system of N arbitrary extended, weakly self-graviting, rotating and deformable bodies in mutual interactions. As a consequence, the SONYR model gives an accurate simultaneous integration of the spin-orbit motion of Mercury. Moreover, the model includes the Mercury^Òs core-rotation couplings by the way of different internal structure models. The obtained rotation of Mercury is in agreement with the analytical theory of D^ÒHoedt and Lemaître. The SONYR theory of the Mercury^Òs rotation gives a complete analysis of the Mercury^Òs librations as well as the origin of the oscillation in its obliquity.

 

 

P. Robutel
IMCCE, Observatoire de Paris, France
 

Diffusion in Jupiter's Trojan swarm

In a previous paper, we have shown the existence of a resonant web in the space phase of Jupiter^Òs Trojans.
After a brief description of this resonant structure, we will show how it generates diffusion in the frequency space.
Then, we will try to connect the diffusion to the long-term erosion of the Trojan's swarm and its long-term stability.
 

A. Rossi
CNUCE-CNR, Pisa, Italy

Collision risk against space debris in low and medium Earth orbits

The space debris polluting the circumterrestrial space are now widely recognized as a jeopardizing factor for every space mission.  Whereas the issue was initially raised and studied for the Low Earth Orbits (LEO), nowadays there is growing concern also for the Medium Earth Orbits (MEO), where the important navigation constellations (e.g., the GPS) are located, and for the unique zone of the Geosynchronous Orbits (GEO). The dynamics of the different populations in these orbital regimes will be described, also in view of the most recent observational data.   The impact risk on the sensible spacecraft orbiting LEO and MEO will be outlined, along with the methods required to study the particular problems posed by the different orbital dynamics. In particular, for the LEO region, the impact risk is analyzed in the framework of the analytical theory of close encounters between a small body on a generic keplerian orbit and a target in a circular orbit. We study how the size of the collision region in orbital elements space varies as a function of the orbital elements of the projectiles and of the interval of time between the epoch under consideration and the time of impact.   For the MEO region, the dynamics of the navigation constellations, close the 2:1 resonance with the Earth rotation, will be discussed. In particular, the long term stability of these orbits and the collision risk, posing new interesting problems related to the space debris management in the MEO regime, will be outlined.

L. M. Saha
Zakir Husain College, Dehli University, India

On the effect of Poynting-Robertson drag on nonlinear stability of motion binary star system

Authors: L.M.Saha, M.K.Das, Pankaj Narang and Manabu Yuasa

Nonlinear stability of motion in the presence of Poynting-Robertson drag in a binary star system has been investigated within the framework of restricted three body problem. For different radiation parameters of binaries and varying initial conditions the transition from regular to chaotic motion of trajectories has been investigated using nonlinear forcasting method as used in nonlinear dynamics. Results are shown to compare well with those obtained using Poincare' surface of sections.
 

 

Z. Sándor

Eötvös University, Budapest, Hungary

Determination of orbital parameters of exoplanets from transit fotometry by using dynamical constraints
 

Space missions in the near future (for example COROT and KEPLER) devoted to discover Earth-like exoplanets will use the transit photometry as detection method. The transit of an unseen planet before the disc of the hosting star results in the periodic dimming of the star\'s light intensity. From the period, the duration, and the shape of the light curve of the transit the orbital parameters of the exoplanet can be estimated only within some error
limits. In this paper we investigate the possibility of refining the orbital parameters of an unseen exoplanet discovered by transit photometry when other massive planets are present in the system. The presence of a giant exoplanet results in
appearing dynamically unstable regions in the phase space. Thich means a dynamical constraint in the sense that only those orbital parameters will be selected, which originate ordered orbits. By using chaos-detection methods (RLI, FLI, SALI), orbital solutions, which result in chaotic orbits can be avoided.
 

C. Skokos 

Research Center for Astronomy and Applied Mathematics, Athens, Greece

The importance of periodic orbits in three-dimensional galactic bars and modern numerical techniques for tracing them 

Finding the periodic orbits is of great importance for the understanding of the dynamical behavior of galactic models. In our contribution we briefly report
some resent results obtained from the orbital study of analytic three-dimensional (3D) models representing barred galaxies, emphasizing on the
connection of periodic orbits to observed morphologies in real galaxies. In 3D models, the planar x1 family of periodic orbits has in general large unstable
parts and, thus, its orbits are not sufficient in building the bar. However, other families of periodic orbits that bifurcate from x1 have large stable
parts that support the bar. These families built the so-called 'x1-tree'. Specific families of the x1-tree are associated with certain morphological
features like peanut edge-on profiles and face-on boxy isophotes. Other families, not belonging to the x1-tree, influenced by the 4:1, 6:1 and 8:1
resonances are related to the appearance of various types of inner rings. 

We also propose and apply a new numerical technique for locating periodic orbits, based on the Particle Swarm Optimization (PSO) method. PSO belongs to
the category of Swarm Intelligence methods, which are closely related to the methods of Evolutionary Computation. We develop an appropriate scheme that
transforms the problem of finding periodic orbits into the problem of detecting global minimizers of a function, which is defined on the Poincare Surface of
Section (PSS) of a Hamiltonian system. By combining the PSO method with deflection techniques, we succeeded in tracing systematically several periodic
orbits in a 3D model of a Ferrers bar, which are reported for the first time. The method succeeded in tracing the initial conditions of periodic orbits in
cases where Newton iterative techniques had difficulties. In particular, we found families of 2D and 3D periodic orbits associated with the inner 8:1 to
12:1 resonances, between the radial 4:1 and corotation resonances. The main advantages of the proposed algorithm are its simplicity, as well as its ability
to locate many periodic orbits per run at a given Jacobian constant. The method is particular useful for tracing orbits in the corotation region in order to
construct self-consistent Schwarzschild-type models of disk galaxies.

V. Sidorenko
Keldysh Institute of Applied Mathematics, Russia

Long-term evolution of the asteroid orbits at the 3:1 mean motion resonance with Jupiter (planar problem)

We consider the 3:1 mean-motion resonance of the planar elliptic restricted three-body problem (Sun-Jupiter-asteroid.
Using the numeric averaging both over the orbital motion and over resonant angle librations/oscillations, we obtained the evolutionary equations, which describe the long-term behaviour of the asteroid\'s argument of pericentre and eccentricity (without any restriction on its value). Then a detailed classification of the possible evolution paths was developed. It significantly generalized the similar results on secular effects in the discussed problem, recently obtained within the scope of the well known Wisdom model (Neishtadt, Sidorenko, 2004). A special attention was given to the very-high-eccentricity asteroidal motion. Being limited to relatively small values of the eccentricity, the Wisdom model did not allow for studying possible transitions from moderate values of the eccentricity (e~0.2-0.3) to the values 0.9-0.95 (Ferraz-Mello, Klafke, 1991). We have demonstrated that,  under certain conditions, the existing region of adiabatic chaos can be a place where such transitions are permissible.
 

V. Sidorenko
Keldysh Institute of Applied Mathematics, Russia

The Comet Nucleus Rotation: The Possible Influence of The Reactive Torques (Poster)

Nucleus rotation affects many processes studied in cometary physics at a fundamental level. Additionally, hypotheses on likely nucleus rotation states are needed to constrain the mathematical models being developed to simulate and analyse the navigation problems that arise in spacecraft missions to comets. Hence, it is important to understand the long-term dynamics of comet nucleus rotation. Reactive torques due to anisotropic sublimation of cometary ice will result in slow variations of a nucleus' rotation parameters. We study the spin evolution of comet nuclei using the averaging method. The general equations of nucleus attitude motion are derived and separately averaged over the fast rotational dynamics terms and the comet orbit. Analysis of the averaged equations allows us to extract the relevant physical parameters that control the evolution of a comet's rotation state. The rotation states of some nuclei, recently observed by spacecrafts (Borelly, Wild 2, Tempel 1) are discussed as an example.
 

C. Simo'
Universitat de Barcelona, Spain

Transitions chains close to tori in the 3D restricted  three-body problem : theory, examples and applications

The spatial restricted three-body problem is considered for small values  of the mass parameter and for energy close to the one of the collinear equilibrium  point between the primaries. That point is unstable and it has a 4D   center manifold. For fixed values of the energy most of the points  on the center manifold are on KAM   tori. The homoclinic and   heteroclinic connections between these tori   are studied. For simplicity the study is restricted to primary intersections,  that is, the ones which occur after one revolution around the main primary.  On the mass parameter-energy plane the values for which such connections exist are characterized. A theorem about the existence of transition chains is proved. Explicit examples of transitions chains are constructed. Applications  to spacecraft missions and to astronomy will be examined.
This is a joint work with Regina Mart\'{\i   }nez and Anna Sam\`a.

 

 

J. Soler
Universitat de Girona, Spain

Periodic Solutions in the Elliptic Restricted Three-Body Problem
 

A discrete family of symmetric periodic solutions in the restricted elliptic three-body problem is shown to exist in the case of arbitrary masses of the primaries and arbitrary eccentricity of their orbit. The infinitesimal body moves on a nearly circular orbit of very large radius with a period multiple of that of the primaries. Averaging and and an implicit function theorem of Arenstorf are the main tools in the proof of the result.

 

B. Steves
Glasgow Caledonian University,  UK

The Caledonian Symmetrical N-Body problem for planetary systems and stellar clusters

Insight into the stability of symmetrical stellar clusters with planetary systems may be obtained by generalising the investigations of Steves and Roy on symmetrical four body problems to symmetrical N-Body problems. Analytical stability criterion valid for all time have been derived for a special symmetric configuration of the general four-body problem, called the Caledonian Symmetric Four-Body Problem (CSFBP). This problem exhibits many of the salient characteristics of the general four body problem, yet the symmetry greatly reduces the number of degrees of freedom. The CSFBP analytical stability criterion is now generalised to n-body symmetrical problems which include problems with a mass located at the centre of mass of the system. This enables the stability of subsets such as symmetrical even number of bodies stellar clusters and symmetrical planetary systems with a large central mass/star to be investigated.

 

A. Süli
Eötvös Loránd University Dept. of Astronomy, Hungary

A detailed comparison of the different chaos detection methods

Preliminary results are presented on a systematic and detailed comparison of the different Chaos Detection Methods (CDM). The method of LCE, FLI, SALI, MEGNO, RLI and FSN are shortly introduced and applied both to mappings and to continuous Hamiltonian systems. The results of the different methods are compared and a detailed discussion is presented.

F. Szenkovits
Babeº-Bolyai University, Cluj-Napoca, Romania

Hill stability in the elliptic restricted three-body problem

We use a three dimensional generalization of Szebehely^Òs invariant relation obtained in the planar case of the elliptic restricted three-body problem to establish more accurate criteria of the Hill stability. By using these criteria, the stability of the natural satellites of our planetary system and of possible extra solar satellites is studied.

P.   Teofilatto (in collaboration with Christian Circi)
Universita' di Roma  "La Sapienza", Italy
 

Weak Stability Boundary trajectories for the deployment of lunar spacecraft constellations

Several ideas about the possible use of the Moon have been recently proposed, for instance the development of habited lunar bases to study  the long term effects of space environment on human health, and the use of the Moon as a convenient launch base for interplanetary exploration. If the lunar colonization will have such a relevance on the future space missions, it will be of crucial importance to find out economical roots to the Moon, such as the low energy orbits based on the Weak Stability Boundary (WSB) effect. In the present paper the physical and mathematical meaning of the WSB effect are revised and the convenience of WSB trajectories to display a constellation of lunar satellites is proved.

 

S. Terracini
Universita'  di Milano Bicocca, Italy

Collisions and symmetries in the periodic N-body problem

This talks concerns trajectories of the newtonian N-body poblem that are minimals for the action in the space of equivariant loops with respect to a given group. A main question is whether minimizing paths admit collisions.
 

G. Tommei
Università di Pisa, Italy

Canonical elements for Opik theory

Opik theory (Opik 1976) deals with planetary close encounters of  a small body moving on a planetocentric hyperbola. The usual elements used to study the dynamics of the encounter (Valsecchi et al. 2003) are not canonical. We search for canonical ones: starting from hyperbolic Delaunay elements we derive a set of canonical elements for hyperbolic collision orbits (eccentricity e -> 1+, semi-major axis a fixed) and then we introduce the unperturbed velocity of the small body U and the space covered along the asymptote. An interesting result would be to get a canonical set containing the coordinates on the Target Plane, useful for a complete analysis of the future encounters: we discuss this possibility.
 

F. Topputo

Politecnico di Milano, Italy

Sixth-Order Linear Multi-Point Method: an Astrodynamical Application (F. Topputo, R. Armellin, L. P. Quartapelle)
 

TPBVP (Two-Point Boundary Value Problem) arises when, solving a set of ODEs, conditions on the state variables are prescribed on both sides of the integration domain. Solving a TPBVP is a usual task in astrodynamics when a
nominal trajectory is required with specified properties at its ends. A typical example occurs in preliminary mission analysis, in the frame of two-body problem, when looking for an arc linking two fixed points in a given time: this is the Lambert\'s problem. Such a problem can be solved by using efficient algorithms since an analytic solution is available in the case of a two-body model. In the present paper a sixth-order linear multipoint method (LMPM) has been derived to solve TPBVPs arising in more difficult models, as the three-body  context, where the chaotic regime makes the TPBVP solution a more challenging task. The sixth-order accuracy is achieved by approximating the dynamical system over a uniform grid by means of general four-point computational molecules and two special five-point molecules defined at the two ends of the integration interval. The sixth-order LMPM attains accuracy beyond the first Dahlquist stability barrier since the integration scheme guarantees an order of accuracy of six while it is expected to be four from the Dahlquist^Òs theorem. The sixth-order Linear Multi-Point Method has been proven to be effective both for the computation of halo orbits and for the solution of Lambert like problem in the restricted three-body model.

 

S. Valk

University of Namur, FUNDP, Belgium

Is the theory of mean orbital motion convenient to compute orbits of earth space debris? (Poster)
 

Most of studies of orbits of clouds of space debris are based on statistical methods, to deduce from the trajectories the main characteristics of the motion, in particular over large time scales. The paper we propose here should be the first step to compute explicitly the long time evolution of orbits of a set of numerous space debris (10,000 typically). We propose, hence, to study if the theory of mean orbital motion, implemented in the CODIOR software, is adapted, or not, to such a goal. In particular, we will analyse, for LEO & MEO orbits, the impact of each term appearing in the
equations of motion, deduced from analytical transformation. As a result, we will quantify and test the cpu load to compute simultaneously a wide range of trajectories.

 

V. Varin

Keldysh Institute of Applied Mathematics, Russia

Computation of families of periodic solutions of the restricted three-body problem

We compute natural two-parametric families of symmetric periodic solutions (SPS) of the planar circular restricted three-body problem for all values of the mass ratio $\mu \in [0,1/2]$. The  computations are organized as follows. Starting with a generating family for $\mu=0$, we compute the family for various fixed values of the mass ratio $\mu \in (0,1/2]$. For each SPS we compute its period and two traces, namely, the plane and the vertical ones. Two characteristics of the family, i.e.\ its intersection
with the symmetry plane, are plotted in the three coordinate systems: one global and two local ones
related to the primaries. We isolate bifurcations of families, which can be of the two types: (I)
an intersection with another family; (II) a self-intersection of the family.
 

 

A. Venturelli

Université d'Avignon, Spain

Parabolic Solutions of the three-body problem asymptotic to a lagrangian configuration

Using variation methods, we show that given a configuration of three body in the plane, there exists a parabolic starting from that given configuration and asymptotic to a lagrangian one. Moreover, this solution has zero angular momentum.
 

A. Vienne

IMCCE - Observatoire de Paris

 

Stochastic behaviour around the 3:7 inequality between Ganymede and Callisto
 

See Benoit Noyelles

 

A. Vigueras
Universidad Politecnica de Cartagena, Spain

Relative equilibria for a gyrostat in the order k approximate dynamics of the three-body problem
 

We consider the non-canonical Hamiltonian dynamics of a gyrostat in the three-body problem. As in Mondejar et al. (2001), we obtain the equations of the reduced dynamics, then we give necessary and sufficient conditions for existence of relative equilibria type Euler, Lagrange and other (that we call \"planar\" rotations) in different approximations of the potential function. When the gyrostat is an axis-symmetric homogeneous ball (oblate or elongated), has a plane of symmetry and the gyrostatic momentum is constant, several families of relative equilibria of the previously mentioned types are obtained. In particular, for the first order approximation dynamics a complete study of the bifurcations of these relative equilibria is made. In this way, we generalise the classical results for the three-body problem and others (in which one of them is a rigid body or gyrostat) due to Fanny and Badoui (1998) and Mondejar et al. (2001).

 

R. Vilhena de Moraes
FEG/UNESP, Brazil

Attitude Equilibrium of a Satellite Subject to Gravity Gradient Torque

The stability analysis of the rotational motion of an artificial satellite under the influence of external torques can be mandatory for the success of some space missions. In this paper the stability analysis is performed using Hamiltonian formalism and a normal form for the Hamiltonian. Using Lie-Hori theory the Hamiltonian is normalized up to order four, in the neighborhoods of the equilibrium points. The equations considered for the rotational motion includes the gravity gradient torque and are described by the Andoyer variables. The equilibrium points and stability regions are established.

 

G. Voyatzis
Dept. of Physics, Univ. of Thessaloniki, 54124, Greece

Symmetric and asymmetric librations of 3:1 resonant planetary systems. An application to the extrasolar system 55Cnc

We study the dynamics of 3:1 resonant motion for planetary systems with two planets based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating
frame) and we computed all the basic families of symmetric and asymmetric periodic orbits. Four symmetric families bifurcate from the family of circular orbits of the two planets and the asymmetric families bifurcate from the symmetric families at the points where the stability index becomes critical. There exist also asymmetric families that are independent of the above mentioned families. Resonant librating motion is found around stable periodic orbits. Therefore, the stable periodic orbits (symmetric or asymmetric) determine the possible stable configurations of the system even if the orbits of the two planets intersect. In the particular study, many periodic orbits are weakly unstable, but the Poincare sections indicate that these periodic orbits are also surrounded by invariant tori of regular motion and chaos is practically absent for low or moderate values of the eccentricity. An application is made for the extrasolar planetary system 55Cnc.
 

J. Waldvogel
ETH Zuerich, Switzerland
 

Quaternions and the perturbed Kepler problem

Quaternions, introduced in 1856 by W. R. Hamilton as a generalization of complex numbers, lead to a remarkably simple representation of the perturbed three-dimensional Kepler problem in regularized variables. The talk gives an overview of this technique, including applications to perturbation theories.
 

P. Waz

Uniwersytet Mikolaja Kopernika, Torun, Poland

 

Dynamics of Planetary and Star Systems Including Perturbing Forces (Poster)

 

In this presentation the influence of the the nonsphericity of the potential on the orbital motion and, as a consequence, on the stability of extrasolar planetary systems is described. A significance of this effects, though in entirely different context (the evolution of the orbit of Phobos, a Moon of Mars) has already been demonstrated [1,2]. The systems of planets of Jupiter type which are close to the star are considered. The construction of an analytical model of the influence of the
perturbing forces on the dynamics of the system is attempted. The derived formulas are used in studies of other systems including multiple systems of stars.

This work has been supported by the Polish KBN.

References
[1] Waz P., Analytical Theory of the Motion of Phobos - Analysis of the Perturbational Function, A&A 348, 300-310, 1999.
[2] Waz P., Analytical Theory of the Motion of Phobos - Comparison with Numerical Integration, A&A 416, 1187-1192, 2004.