Charles University, Prague

New results on the Yarkovsky effect
(Authors: D. Vokrouhlicky and M. Broz)

We present recent results of the long-term dynamics of meteoroids and small asteroids with the Yarkovsky perturbation. In particular, we deal with transport of meteoroids from the main asteroid belt and a possible role of the Yarkovsky effect on dispersion of asteroid families. Attention is also paid to the YORP effect, notably the long-term influence of the Yarkovsky torque on the rotation state. Preliminary results, showing how the YORP evolution of the spin axis may affect the orbital Yarkovsky perturbation, are discussed.

Cornell University (USA)

This talk will describe the Yarkovsky force and some of its ramifications for the orbital histories of meteoroids, asteroids and artificial satellites. Whenever a rotating body's thermal radiation is emitted asymmetrically, it carries energy and momentum away from the body. This re-emission causes the orbit to evolve, most importantly changing the orbital size, thereby allowing resonance zones - where paths can be significantly modified - to be crossed. The recoil force depends on the body's orbit (semi-major axis a, eccentricity e), its spin rate and obliquity, and its physical properties (size, density, thermal diffusivity and surface character). The Yarkovsky acceleration is most important for bodies 1-100m in radius, and can change sign depending on obliquity. Work by P. Farinella, D. Rubincam, D. Vokrouhlicky and others since the late 1990s has suggested that the Yarkovsky effect may be influential in delivering particular meteorites to Earth, in allowing relatively large asteroids to become Earth-crossers and in dispersing asteroid families. The Yarkovsky effect is also believed to account for measured drifts seen in the orbits of the geodynamics satellites LAGEOS 1 and 2, and Etalon 1 and 2. Asteroid rotation may also be noticeably affected by radiation recoil forces.

Cornell University

Asteroid families are generally believed to result from the break-up of a parent body. One problem with this scenario is that the expulsion velocities inferred for the fragments from their observed dispersions in their current orbits seem to be consistently higher than those predicted by collisions in laboratory experiments.

In this work we investigate the degree to which fragments, subsequent to the collision, could be further scattered by encounters with the three most massive asteroids in the Main Belt (Ceres, Pallas, and Vesta) as well as by close encounters with asteroids of diameter bigger than 50 km. We first tried to evaluate the semimajor axis mobility by extrapolating the results of a numerical simulation obtained by Morbidelli and Nesvorny (2000); for 300 asteroids in the main belt and analogous results from a Monte Carlo simulation. We then performed a n-body simulations of the members of a few asteroid families, among which Adeona. Results indicate that such gravitational scattering is effective in dispersing some asteroid families. Thus the inferred original ejection velocities can be significantly reduced once encounters with large asteroids are incorporated in the model.

Politecnico di Torino

Resonant flyby missions to the near-Earth asteroids
(Authors: L. Casalino, E. Perozzi)

Flyby missions to the Near-Earth Asteroids (NEAs) are useful when dealing with high-inclination objects and/or when a limited energy budget is allowed to the mission, thus preventing a more complex rendez-vous mission scenario. When this is the case, a resonant flyby strategy can be searched for, allowing the spacecraft to repeatedly encounter the asteroid in order to increase the scientific return of the mission by performing multiple observations spaced in time. The velocity increment provided by a single impulse on leaving the Earth might in fact be sufficient to insert the spacecraft on a transfer orbit having the same period of revolution of the target. In order to explore this possibility, a target selection strategy and an indirect optimization procedure to minimize the propellant requirements are used to find out favourable mission opportunities. Additional constraints, such as on the relative flyby velocity, can also be imposed to the problem.

Stefano Casotto

Università di Padova

Recent Earth-based and in situ imaging has revealed that the shape of asteroids is far from spherical. This implies that modelling the force field of such bodies near their surfaces cannot be accomplished by the usual expansions in spherical harmonics, which are known to diverge inside the Brillouin sphere. Alternative approaches have been suggested, like the polyhedron method and expansions based on different coordinate systems. This poster will address the expansion of the external potential of an irregularly shaped, homogenous body in spheroidal coordinates, which in general provide a better approximation of its physical surface. In particular, computational problems connected with the Associated Legendre Functions of the second kind and the expansion of the inverse distance in spheroidal harmonics will be addressed. Spheroidal harmonic models for some of the asteroids of known shape will be presented.

Andrea Chessa

Università di Roma "Tor Vergata"

Stability of periodic orbits in the asteroidal belt
(Authors: A. Celletti, A. Chessa, J. Hadjidemetriou, G. Valsecchi)

Periodic orbits in the framework of the planar, circular, restricted three-body problem are analyzed. We investigate the planar linear stability of many resonant motions of objects moving between the orbits of Mars and Jupiter. Results about different behaviour of the resonance stability are presented.

Luigi Chierchia

Università di Roma Tre

Proper degeneracies in the Hamiltonina study of Celestial Mechanics are frequent. Model problems of systems with two degrees of freedom, properly degenerate, and with "intermediate" system depending on the angle conjugate to the non-degenerate action, are discussed from the point of view of stability of the action variables. A connection with the D'Alembert problem (a planet modelled by a nearly flat ellipsoid, revolving on a Keplerian nearly-circular orbit, and subject to the gravitational attraction of a fixed star occupying one of the foci of the ellipse) is discussed.

Piero Cipriani

Istituto Nazionale di Ottica Applicata, Firenze

A class of effective indicators of chaos is presented and the relationships between the qualitative features of their evolution and the diffusive phenomena within sticky regions are discussed.

Guido Colasurdo

Politecnico di Torino

Electric Propulsion and Earth Gravity Assist
(Authors: G. Colasurdo and L. Casalino)

Interplanetary vehicles usually exploit gravity assists to increase the payload. An Earth Gravity Assist (EGA) is scarcely useful when chemical propulsion is employed, as a deep-space Delta-Velocity (V) manoeuvre, which reduces the heliocentric energy of an outbound spacecraft, is necessary to obtain an adequate relative velocity at the Earth reencounter. In particular, (V-EGA trajectories are only effective for missions that use chemical engines and are aimed at Jupiter and beyond. This scenario is changed by the development of electric thrusters that reduce the propellant consumption of the deep-space manoeuvre. This lecture presents an elementary explanation of the reasons that make EGA attractive when electric propulsion is used to reach near Earth bodies. An indirect procedure to optimise this kind of mission is also presented and the corresponding payload improvement is highlighted. Numerical examples concern missions to near-Earth asteroids and planets (namely, Venus and Mars). A complex trajectory to Mercury after a double Venus flyby is also presented.

Astronomy Center of the Academy of Athens

Galactic systems produced by the collapse of initial cosmological perturbations are represented by self-consistent N-body distributions in phase-space. Such systems contain ordered and chaotic domains. Of special interest is the resonance structure of the phase-space distributions.

Observatoire de Nice

We investigate the break-down threshold of librational invariant curves.As a model problem, we consider a variant of a mapping introduced by M. Henon, which well describes the dynamics of librational motions surrounding a stable invariant point. We verify in concrete examples the applicability of Greene's method, by computing the instability transition values of a sequence of periodic orbits approaching an invariant curve with fixed noble frequency. However this method requires the knowledge of the location of the periodic orbits within a very good approximation. This task appears to be difficult to realize for a libration regime, due to the different topology of the phase space. To compute the break-down threshold, we tried an alternative method very easy to implement, based on the computation of the Fast Lyapunov Indicators and frequency analysis. Such technique does not require the knowledge of the periodic orbits, but again, it appears very difficult to have a precision better than Greene's method for the computation of the critical parameter.

University of Namur

Mercury: a first model of rotation
(Authors: S. D'Hoedt, A. Lemaitre)

Thanks to the previous works of the Dynamic Systems group (FUNDP) about libration models (e.g. the Moon) and spin-orbit resonances, we are building a new preliminary model for Mercury free rotation. Different perturbations will be added, as well as non rigidity effects in collaboration with the Observatory of Brussels (Belgium).

University of Vienna

We present results of a systematic numerical study concerning the stability of the inner Solar System depending on the masses involved. It seems that up to a factor 100 in the masses of the inner planets the systems is still stable over time scales of millions of years. Additionally we checked the orbits of the Trojans of the inner planets with respect to their stability behaviour depending on the masses of the planets.

Università di Roma Tre

Domains of analyticity for area-preserving maps (in particular the Standard Map) for different rotation numbers. Shape, simmetries, asymptotics when the rotation number is: a) real but closer and closer to a rational or b) rational as best approximant of a real number.

Sylvio Ferraz Mello

Universidade de Sao Paulo

The theory developped by Ferraz-Mello (CMDA vol 66, 1997) is used to compute the solutions of Garfinkel's Ideal Resonance Problem up to the terms of order {\cal O}(\varepsilon\alpha, \sqrt{\vaepsilon}\alpha^4, \alpha^7) in angle, and {\cal O}(\varepsilon^{3/2}\alpha, \varepsilon\alpha^4, \sqrt{\varepsilon}\alpha^7) in the action ($\alpha$ is tha amplitude of libration, in angle).

Observatoire de Nice

The computation on a relatively short time of a quantity, related to the largest Lyapunov Characteristic Exponent, called Fast Lyapunov Indicator, allows to discriminate between ordered and weak chaotic motion and also, under certain conditions, between resonant and non resonant regular orbits. The aim of this poster is to study the behavior of the Fast Lyapunov Indicator for a continuous hamiltonian system: a partucular case of the double star restricted tree bodys problem. Result obtained in the framework of symplectic mapping are recovered for the tangential motion in the orthonormal maniflod to the motion.

Observatoire de la Cote d'Azur, Nice, France

A sample of about one thousand possible orbits of the Tunguska cosmic body (TCB) have been calculated using selected values in azimut, elevation and geocentric velocities. We then analyzed the dynamical properties of the orbits and estimated the probabilities of possible origins of the TCB. From our results it seems that the TCB is more likely of asteroidal origin.

Observatoire de la Cote d'Azur, Nice, France

In the last ten years a large effort has been devoted to the development of fast indicators able to distinguish between chaotic and regular motion, and also, in the case of regular motion, between resonant and non resonant orbits. Although the computation of the Lyapunov Characteristic Indicators represents, both from the theoretical and from the numerical point of view, the best suited method for distinguishing between ordered and chaotic motion, it does not allow to separate resonant orbits from the non resonant ones. This last task is very important, for example, when studying the global stability of a system. The computation, on a relatively short time, of a quantity related to the largest Lyapunov Characteristic Exponent, called the Fast Lyapunov Indicator (FLI), allows to discriminate between ordered and chaotic motion, even weak chaotic motion.

Universitat de Barcelona

We focus on the dynamics of a small particle near the Lagrangian points of the Sun-Jupiter system. To try to account for the effect of Saturn, we develop a specific model based on the computation of a true solution of the planar three-body problem for Sun, Jupiter and Saturn, close to the real motion of these three bodies. Then, we can write the equations of motion of a fourth infinitesimal particle moving under the attraction of these three masses. Using suitable coordinates, the problem is written as a time-dependent perturbation of the well-known spatial Restricted Three-Body Problem. This model is compared with other models to test its accuracy. Finally, we study the dynamics of this model near the triangular points. The tools are based on computing, up to high order, suitable normal forms and first integrals.

Mathematisches Institut, Berlin

Three laws of Kepler type for the restricted 3-body problem, the first law is a geodesic equation; physical understanding and mathematical order of the phenomena of commensurability with methods of Moon theory; elimination of the "problem of small divisors" with the geodesic equation.

Università di Roma "La Sapienza"

I present an algorithmic procedure for further reduction of normal forms; this is a direct extension of the Poincaré-Birkhoff one, along the lines of the Lie algebraic approach of Broer, and requires to solve only linear equations. It can thus be easily implemented on a computer, giving completely explicit formulas.

Università di Milano Bicocca

It is well known that KAM theorem cannot apply to planetary systems because of two difficulties: 1) the degeneracy of the actions in the integrable (Keplerian) part of the Hamiltonian; 2) a too strict hypothesis on the smallness of the perturbation. The first problem can be overcome, by isolating from the rest a main part constituted by the Keplerian part and some terms depending on the secular variables. In this new framework the size of the perturbation is ruled by two small parameters: the mass ratio and the square of the eccentricity. We adapt the original Kolmogorov's normalization algorithm to this case in order to construct the invariant tori corresponding to quasi-periodic strongly non-resonant orbits. By our method, we calculated on a computer a large number of coefficients of the series giving the shape of an invariant torus approximating the orbit of the Sun-Jupiter-Saturn system. The comparison between our semi-analytical method and a numerical integration over a large time scale shows a good agreement. At last, we will briefly discuss the possible strategy that can be used to convert our constructive algorithm in a computer-assisted proof that could ensure the existence of the invariant tori in the real planetary systems.

Universitat de Barcelona

Using numerical procedures we explore the central manifolds of the collinear equilibrium points L_1, L_2 and L_3 of the RTBP problem. We show some of the most relevant families of periodic orbits and 2D tori that organize the phase space around the three libration points.

Filippo Graziani

Università di Roma "La Sapienza"

Space activities are facing a constant trend towards smaller satellites. This fact is basically the answer provided by technology possibilities to the toll that space activities have to pay, which is the high cost of the injection of each kilogram in orbit. As far as technology is able to exploit the same features with a reduced mass, this trend will continue. Indeed, current developments in miniaturised technologies, as the micro-electromechanical systems, clearly guarantee the pursuit of this trend in the upcoming years. The companion aspect of this diffusion of smaller platforms is represented by the distributed architecture of several current mission. This feature is intrinsically connected to microsatellites, as only reduced volume, mass and dimensions allow for allocating several payload in the same launcher, and on the other hand, several small satellites seem to be an obvious solution to divide too large payloads. From this point of view also the current trend initiated by big commercial telecommunication constellation and continued by formation flight mission concept foresees a growth in the close future.
Microsatellite development allows for several considerations. First of all smaller satellites require smaller plants and test facilities, as well as reduced work load and personnel. As a second point to be outlined, the time to delivery can be limited. Therefore, technology used can be the state of the art, opposite to longer, 5 to 10 years development programs typical of space agencies. Stressing this aspect, as from this paper's title, a reason for microsatellite programs can be found in testing new technologies in a convenient time and at a lower cost.
Both these points introduce into play the role of small research entities as research centers and universities. Dealing with these aspects, this paper aims to present the 'first hand' experience of the UNISAT microsatellite, designed, manufactured and recently launched by the GAUSS team at the Università di Roma "La Sapienza". All aspects of the program, since its inception to the in-orbit operation, will be briefly described as a start to present the lessons learned. The global frame of UNISAT program and the future developments and steps of the project will be discussed.
The launch phase analysis will be especially outlined in order to gain a general lesson from the UNISAT program. In fact, microsatellites are usually a part of a multi-payload mission, and therefore all operations have to be carried on following a stricter timetable, as no delay will be possible. At the integration time, usually in a crowded payload bay onboard the launcher, a common understanding and co-operation atmosphere is mandatory (just to name a problem, think about the removal of solar panel protection screens).
Latter aspects enhance another feature of microsatellite programs, which is their high educational value. Again, UNISAT experience, with its success and its drawbacks, holds as a case-test to show the involvement of engineering students, at a level which can not be allowed for in a large space program. Furthermore, the possibility to combine experience of western engineering practice with eastern skills (UNISAT was launched onboard of the russian ñ ukrainian Dnepr rocket) has been relevant to students and researchers, who followed tests in Dnepropetrovsk and launch in Kazachstan.

University of Arizona

Celestial mechanics is responsible for Jupiter's satellite Europa being a possible site for life in the solar system. The Laplace orbital resonance drives a substantial eccentricity. The mutually dependent relationship between orbital and rotational evolution and tidal processes in turn controls Europa's heating and stress. Heat is likely adequate to maintain a liquid water ocean, and keep surface ice thin. Tidal stress can explain crack patterns (global and cycloidal), as well as shear displacement features. The characteristic ridge sets that cover tectonic terrain are likely built by tidal pumping of oceanic fluid and slush through cracks to the surface on a daily basis. Nearly half the surface is chaotic terrain, with morphology and other characteristics indicative of melt-through from below. Formation of both chaotic and tectonic terrains has continually resurfaced the satellite, while connecting the ocean to the surface and providing a variety of evolving environmental niches. As a result of tides, liquid water would daily bathe crustal cracks and surfaces with heat, transporting and mixing substances vertically. Thus a variety of habitable environments likely exist in the crust. Moreover, exposure of the ocean to the surface in the ways described here satisfies a necessary condition for life in the ocean as well, by providing access to oxidants which are available near the surface. These processes were recent, and thus most likely continue today. Longer term changes in environmental conditions in the crust, such as deactivation of individual cracks after thousands of years (due to non-synchronous rotation) and later crustal thawing (releasing any trapped organisms), provided drivers for adaptation, as well as opportunity for evolution.

Ben-Gurion University of the Negev

A gas-kinetic theory approach is used by exploring the combined system of the Boltzmann and the Poisson equations to determine the stability and collective oscillations of mutually gravitating particles of Saturn's rings. The effects of physical collisions between particles are taken into account by using in the Boltzmann kinetic equation a Krook model integral. The present study is aimed above all at explaining the origin and their evolution of the fine-scale 100 m structures in the main Saturn's rings. It is suggested that forthcoming in 2004 Cassini spacecraft high-resolution images may reveal this kind of fine structure in the Saturnian ring system. The results of this theory are tested by N-body computer simulations.

Università di Pisa

The classical averaging principle, used to compute the secular evolution of Main Belt Asteroids, cannot be applied to know the evolution of Near Earth Asteroids (NEA's) because the first order polar singularity appearing in the perturbing function makes the "averaged Hamilton's equations" used in this theory meaningless. We have introduced a generalization of the averaging principle to planet crossing orbits: it allows to predict the long term evolution of NEA's by showing that we can define piecewise-differentiable solutions of the new "averaged equations". This theory has been used to develop an algorithm that actually computes the secular evolution of all the known NEA's; the output of this "averaged integration" has been processed in order to obtain proper elements and proper frequencies for NEA's; they are quasi-integrals of the motion and can be used to understand some qualitative properties of these objects.

John Hadjidemetriou

University of Thessaloniki

The resonances play an important role in the evolution of a dynamical system, because they determine the topology of the phase space. In particular, the resonances in a planetary system correspond to periodic orbits of the two planets, in a rotating frame, where the ratio n/n' of the frequencies of the planets is a rational number (mean motion resonance). We study the case where only one planet, or both planets, have a finite mass.

In the former case, the model is the restricted three body problem, and the major planet could be Jupiter, for the study of asteroid motion, or Neptune for the study of the motion of Kuiper belt objects. We present first the complete structure of resonant and non resonant periodic orbits, in a rotating frame, for a fixed circular orbit of the major planet, both for inner and outer orbits, in the plane. The possibility of continuation of the above planar families of periodic orbits of the circular model to three dimensional motion and for elliptic orbit of the major planet is studied and is related to the appearance of strong chaotic motion, or more confined chaotic motion (stable chaos), for a particular resonance. In addition, the averaged model corresponding to a resonance is studied and its relation to the original, non averaged, model is discussed.

In the case where both planets have finite masses, we study the periodic relative configurations and their stability, and we discuss the dependence of the stability on the mean motion resonance of the two planets. The dependence of the stability on the ratio of the masses of the two planets, and of their eccentricities, for a particular resonance, is also studied. Based on these results, some remarks are made on the stability of exosolar planetary systems.

University of Namur

The Trojan web

We plan to describe and comment the results of a numerical exploration of the numerous natural families of periodic orbits associated with the equilateral equilibria of the restricted problem. Many (if not all) of these families are organized in a very structured network or cobweb. This structure evolves, again in a very organized way, when the mass-ratio varies. Such an organization is not due to chance and indeed most of its features can be explained either with full mathematical rigor or at least as likely possibilities. They are mainly due to the local behaviour of parametrized two degrees of freedom Hamiltonian systems in the vicinity of an equilibrium. But some of the features are of a more global character. Some can be explained by continuity from the local behaviour; others by another factor of organization for small values of the mass ratio: the approximation by the integrable two-body problem in rotating coordinates.

Università di Roma "La Sapienza"

The demand for high capacity radio link in solar system exploration has motivated the development of new technologies based upon the use of higher and higher frequencies in deep space telecommunications. Current missions make use of X-band carriers (7.2-8.4 GHz), but it is likely that toward the end of the decade Ka-band technology (32-34 GHz) will be widely employed. The novel instrumentation under development will benefit also the scientific use of spacecraft tracking and the accuracy of deep space navigation. Indeed, at such high frequencies, interplanetary plasma noise (at present the main noise source in range and range-rate measurements) will be strongly reduced, thus allowing for unprecedented accuracies of the radio-metric data. Significant improvements in the knowledge of the gravitational fields and deep internal structure of solar system bodies is to be expected, as well as more accurate tests of the theories of gravity. Cassini, the most complex planetary spacecraft ever launched, is currently beyond the orbit of Jupiter and en route to Saturn, its final destination, which will be reached in July 2004. The probe is equipped with an outstanding radio science instrumentation, capable of supporting simultaneous links in X- and Ka-band for a complete removal of the plasma noise. The performances of the space instrumentation are matched by a sophisticated ground antenna, especially designed for high frequency stability. The radio system for the mission has been designed for a two-way range-rate measurement precise to 1.5 micron/s over time scales between 1000 and 10000 s. This unprecedented accuracy will be exploited in a number of radio science experiments both in the cruise phase and the Saturn tour. Starting from November 2001, the spacecraft will be extensively tracked for two radio science experiments aiming to measure the relativistic deflection from the sun and to detect low frequency gravitational waves. This data set is expected to provide the cleanest radio-metric measurements ever obtained from a space probe and will be used also to test and refine current models employed in the navigation of the probe. During the tour, spacecraft tracking will allow the determination of the gravity fields of the bodies of the Saturn system, the Love number k2 of Titan and possibly, in combination with SAR imaging, also Titan's moments of inertia. The latter experiment is of compelling interest, as the determination of the rotational state of a planet or satellite from orbit brings crucial information about its internal structure. This method will be tested for the first time with the ESA technology demonstration mission SMART-1 to the moon, to be launched in 2003. The probe will host onboard a X/Ka-band transponder and other instrumentation in view of future ESA deep space mission, such as the cornerstone BepiColombo.

Belgrade Astronomical Observatory

We have applied a spectral formulation of the Nekhoroshev theorem (Guzzo and Benettin 2000) to real asteroids to reveal whether their dynamics satisfy the assumptions of Nekhoroshev theorem. We integrated several bodies in different regions of the asteroid main belt and with different chaotic signature for the time spans up to 100 Myr, and we applied a suitably adapted FFT to the time series of non-singular elements, averaged over the fast variables. We have identified a number of cases with clearcut "band structure" of the spectrum, indicative of the system in Nekhoroshev regime, but we also found cases where spectra were without "band structure"', as well as intermediate cases. We show and discuss the results and their implications for the long term stability of asteroid motion.

University of Thessaloniki, Department of Physics

We study three-dimensional periodic orbits, using the model of the circular restricted three-body problem with the Sun and Neptune as primaries. Families of three-dimensional periodic orbits are numerically computed at the exterior resonances 2/3 and 3/4. The three-dimensional orbits are found by continuation, to the third dimension, of the vertical critical orbits of the corresponding planar problem. The stability of these orbits is studied.

University of Helsinki

Integrator Package for Statistical Orbital Ranging
(Co-authors: J. Virtanen, M. Kaasalainen and K. Muinonen)

A new integrator package for statistical orbital ranging is introduced. The emphasis is on the computational efficiency. Optimization of the N-body integration routine has received special attention. Pre-calculated positions of the perturbing planets are exploited in the N-body integration. Also, modular structure of the implementation allows convenient switching between several numerical integration methods. The applicability of a few fundamental methods is studied as well as their effects to the total computing time in statistical orbital ranging.

Observatoire de Lille

It is often admit that rotational motion of the natural satellites doesn't affect their orbital motion. However, for computation of accurate ephemerids, it is convenient to see to what extend this hypothesis is true, especially in the case of spin-orbit resonances. We treated this question with the help of numerical simulation. Looking at the Galilean system and the system of the main satellites of Saturn, we founded small differences which could be not negligeable especially because of the resonances in these system.

Institut d'Astrophysique Spatiale, Orsay

A strategy combining ion propulsion and gravity assists has been selected by ESA for two solar system exploration missions, as it makes possible missions which were beyond the practical limit with chemical propulsion. It is also considered by NASA for the next Discovery selection.
Ion (or plasma) thrusters provide very high ejection velocities (15 to 40 km/s), so that a much larger on-board delta V can be accommodated with the same mass budget. However, the thrust level is very low, so that the limiting parameter is the thruster lifetime (typically 10,000 hours). Planetary gravity assists drastically lower the delta V requirements. However, such manoeuvers require proper phasing, and the most effective strategies require multiple encounters with planets (in particular with delta-V gravity assists), so that cruise times become very long (7 to 8 years for Cassini, Rosetta, or a chemical mission to Mercury). The combination of low thrust and gravity assists is extremely powerful as the low thrust arcs can be used to correct phasing mismatches and implement delta-V gravity assists with a small impact on the mass budget, while the gravity assists lower the delta V requirements so that the thruster lifetime is not exceeded. Examples will be given for ESA missions with Smart-1, Bepi Colombo, the cornerstone mission to Mercury and Solar Orbiter, a mission orbiting close to the Sun at medium to high inclinations.

Jacques Laskar

Astronomie et Systèmes Dynamiques, BdL, Paris

I will review recent advances in symplectic integrators and their application in planetary dynamics.

University of Namur

Several papers published in the nineties by Beaugé and Ferraz-Mello consider captures in external resonances and asymmetric equilibria, in the frame of the planar averaged three-body problem, for drags like Stockes drag or Poynting-Robertson drag. They find a "universal" eccentric orbit to which the others are converging, for some particular cases of initial conditions and parameters. We study the existence and the stabilty of this universal orbit in the inclined case, using classical lineraization processes and the extended Schubart-like integrator developped by M. Moons, extended to the second order and to the dissipative cases. We give the conditions on the initial conditions (especially the inclination) to keep a capture, for some ranges of perturbing masses and dissipation coefficients. We compare those semi-analytical results with the ones obtained in a "one degree of freedom toy model" developped for a rough automatic calculation of probability of capture.

University of San Juan, Argentina

We carried out a series of numerical simulations of the dynamical evolution of test particles in the region a < 2 AU. We explored two different scenarios: 1) Only Jupiter and Saturn are present and 2) Including the inner planets. We found that only those asteroids from well defined narrow regions in semimajor axis, associated to the nu16 secular resonance and 5:1 mean motion commesurability with Jupiter, can reach Mars and Earth crossing orbits in time scale of 1.0 E+7-8 yr. Secular resonances with the inner planets and mean motion commesurability with both the inner and outer planets play a key role in the primordial sculpting of this region.

Bureau des longitudes, Paris

We present a new reduction for the general n-body problem,associated to the angular momentum integral. The proposed reduction is performed in two steps. A first reduction, called partial is based only on the fixed direction of the angular momentum. The reduction can then be completed using the norm of the angular momentum. In fact, the partial reduction presents many advantages. In particular, in the reduced secular system, we can construct a birkhoff normal form at any order. Moreover, the topology of this problem remains the same as for the non reduced system, contrarily to the Jacobi reduction where a singularity is present for zero inclinations. For three bodies, these reductions can be done in a very simple way in Poincare's rectangular variables. In the general n-body case, the reduction can be performed up to a fixed degree in inclinations and excentricities, using expansions made with computer algebra. As an example, we provide the truncated expressions for the change of variable in the four bodies case, obtained using the computer algebra system TRIP.

ONERA, France

Many demonstrations of existence of interesting n-body solutions use the method of minimization of action. However some of these demonstrations have been considered as insufficient because of the possible interference of singularities. The analysis of this dificulty shows that :
A) A n-body solution minimizing the action between given terminal conditions has no discontinuities : all n bodies have a continuous and bounded motion and thus all eventual singularities are collisions.
B) A beautiful extension of Lambert's theorem shows that no double collision can occur at an intermediate time.
C)The demonstration can be extended to triple and to multiple collisions.
Thus the method of the minimzation of action leads to pure n-body motions without singularities at intermediate times, even if one or several collisions are imposed at initial or final times. That method is suitable only for non-infinitesimal masses, fortunately a similar method, with the same general property, can be extended to n-body problems including infinitesimal masses.

Università di Pisa

Abstract TBD

Moscow Institute for Physics & Technology

Spin rotation of a magnetically stabilized satellite effected by various disturbances
(Co-authors: K. Kurmakaev, M. Ovchinnikov)

Spin motion of a satellite stabilized along the local geomagnetic field is considered. The satellite is provided with a magnetic coil having a permanent dipole moment and, also, a number of hysteresis rods fabricated from soft magnetic material. The elements of satellite structure have various characteristics for reflecting solar radiation. The dipole moment is assumed to be sufficiently large to provide the magnetic orientation of the satellite. We use recent results from our investigation of the periodic motions of a magnetically stabilized satellite. The gravity-gradient and damping torques, and also a torque developed by solar radiation pressure, are considered to provide disturbances of the steady-state motion. Equilibrium positions, librations and uniform spin rotations are investigated. The equations of motion admit periodic solutions. The boundaries of the areas of stability in space of the satellite parameters are mostly determined by the ratios of its moments of inertia. Conditions of stability are also obtained. The microsatellite UNISAT-1 is taken as an example.

Anna M. Nobili
(for the GG collaboration)

Universita' di Pisa, Dipartimento di Matematica

The Equivalence Principle (EP) stated by Galileo, reformulated by Newton and revisited by Einstein to become the founding Principle of General Relativity, has attracted the attention of space agencies around the world. NASA, ASI and CNES (in the US, Italy and France) are considering space missions for EP testing.
All missions will fly experiments to test the most direct consequence of the EP: the Universality of Free Fall, whereby all bodies fall with the same acceleration regardless of their mass and composition. The most accurate EP experiments on the ground have been carried out with test bodies suspended on a torsion balance finding no violation to about 10^-13. Test bodies in low Earth orbit are subject to a driving acceleration much stronger than on torsion balances on the ground, by about 3 orders of magnitude. Moreover, absence of weight is ideal in small force experiments. As a consequence, space missions can potentially improve by several orders of magnitude the current sensitivity in EP tests. The goals are: 10^-15 for the French muSCOPE,10^-17 for the Italian ``GALILEO GALILEI'' (GG), 10^-18 for the American STEP. While the STEP and muSCOPE designs are similar (the main difference being that STEP is cryogenic and muSCOPE is not), GG (which in the current baseline is a room temperature mission) differs from both of them in that the test bodies are weakly coupled cylinders in supercricital rotation. GG is currently undergoing an advanced mission study within the Italian Space Agency (ASI) which includes laboratory tests of a payload prototype. The talk will describe the main features of the GG dynamical system (in a 6-body configuration) and give an update on the mission study, the error budget and the analysis of measurement data taken with the laboratory prototype.

Universitat Politècnica de Catalunya

Estimates and intrincacities of the normalized Hamiltonian near a critical periodic orbit
(Authors: M. Ollé, J. Villanueva, J.R. Pacha)

In this work, we study a three degree of freedom real analytic Hamiltonian system, H, having a one-parametric family of periodic orbits which undergoes a transition stable-complex unstable for some critical value of the parameter: this means that when we cross this critical value, the four non-trivial Floquet multipliers of the corresponding orbit go from the unit circle to the complex plane by colliding pairwise on the unit circle for the critical value. There are several phenomena linked to this kind of instabilization which has been observed through numerical simulations. Among them, we center this contribution to understand the particular unfolding of families of invariant two dimensional tori (the so-called Hamiltonian Hopf bifurcation). Our method of study involves the construction of an "ad hoc" resonant normal form around the critical periodic orbit. So we cast the transformed Hamiltonian into one of the form $H\circ \Psi = N_r + \mathfrak{R}$, where $N_r$ stands for the normal form (up to some order $r$), whereas $\mathfrak{R}$ is the remainder.
First, the appearing of two-parameter family of invariant tori is proved for $N_r$ and second, we study the persistence of these solutions when $\mathfrak{R}$ is added. To apply KAM perturbation techniques we need to be able to consider the remainder as an (small) contribution in front of the normal form, so both, an appropriate selection of the optimal order $r$ of the normal form process and a later bounding of the remainder, with an estimation of the domain of convergence of the transformation; are two necessary steps.

Ettore Perozzi

Telespazio, Roma & DESPA - Observatoire de Paris Meudon

Every name tells a story - whatever sad or happy: no wonder that the question posed by poor Giulietta in trying to escape her fate - what's in a name? that which we call a rose by any other name would smell as sweet - has quickly become a standard quotation from Shakespeare. Going through the list of asteroid names is therefore an intriguing exercise, and while I was doing so some time ago it hit my mind a name with a beautiful story which could deserve a place in the sky. After the necessary official steps were done with the kind support of Ted Bowell (who generously offered an asteroid discovered by himself), the proposal to name CACCIOPPOLI asteroid N° 9934 (provisional designation 1985UC) was sent to the International Astronomical Union by the end of March 2001. Here is the story behind the name.

Marco Pettini

Osservatorio Astrofisico di Arcetri (Firenze)

Dynamics and thermodynamics of gravitational clustering

The thermodynamic behaviour of self-gravitating N-body systems has been worked out by means of a standard method in Molecular Dynamics: the time averages of suitable quantities are numerically computed along the dynamical trajectories to yield thermodynamic observables. The link between dynamics and thermodynamics is made in the microcanonical ensemble of statistical mechanics. Through the computation of basic thermodynamic observables and of the equation of state in the P - V plane, new evidence is given of the existence of a phase transition from a homogeneous phase to a clustered phase. The dynamical-microcanonical averages are compared to their corresponding canonical ensemble averages, obtained through standard Monte Carlo computations. A major disagreement is found, because the canonical ensemble seems to have completely lost any information about the phase transition. It is also shown that the clustering phase transition in self-gravitating systems has a counterpart in a major change in the degree of chaoticity of the dynamics. Finally, we show how to properly tackle chaos in gravitational N-body systems within a Riemannian-geometric framework.

University of Vienna

A detailed stability study about planetary motion in the binary Gliese 86 will be presented. Gliese 86 is one of the four double star systems, where a planet was detected and it is the only system where the distance of two stars is less than 20 AU. Our numerical investigation is concerned with the stability of planets moving around one component - which is called S-type motion

University of Roma "Tor Vergata"

We investigate the dynamics of some simple prototype systems which are integrable at zero energy, outside the zero-energy hypersurface. We find that, in general, integrability is not preserved at arbitrary values of the energy. The separating coordinates at zero energy allow a perturbation treatment of these systems at energies slightly different from zero, by which one can obtain analytical proofs of non-integrability and can explore the qualitative behavior of the non-integrable dynamics.

European Space Agency

The design of spacecraft trajectories is a crucial part of a space mission design. Often the mission goal is tightly related to the spacecraft trajectory. A geostationary orbit is indeed mandatory for a stationary equatorial position. Visiting a solar system planet implies that a proper trajectory is used to bring the spacecraft from Earth to the vicinity of the planet. The first planetary missions were based on conventional trajectories obtained with chemical engine rockets. The manoeuvres could be considered "impulsive" and clear limitations to the possible missions were set by the energy required to reach certain orbits. The gravity assists trajectories opened a new way of wandering through the solar system, by exploiting the gravitational field of some planets. The advent of other propulsion techniques, as electric or ion propulsion and solar sail, opened a new dimension to the planetary trajectory, while at the same time posing new challenges. These "low thrust" propulsion techniques cannot be considered "impulsive" anymore and require for their study mathematical techniques which are substantially different from before. The optimisation of such trajectories is also a new field of flight dynamics, which involves complex treatments especially in multi-revolution cases as in a lunar transfer trajectory. One advantage of these trajectory is that they allow to explore regions of space where different bodies gravitationally compete with each other. We can exploit therefore this gravitational perturbations to save fuel or reduce time of flight. The SMART-1 spacecraft, first European mission to the Moon will test for the first time all these techniques. The paper is a summary report on various activities conducted by the project team in these areas.

ADS/IMC, Bureau des Longitudes

Using Laskar's Frequency Map Analysis, we have performed a complete study of the dynamics of massless particles in the Solar System, from Mercury to the outer parts of the Kuiper belt, for all values of the eccentricities. This provides a complete dynamical map of the Solar System. This general method is also applied to various planetary systems.

FGUP NPC "Nedra" - Russia

The applications of the computer algebra methods for the perturbation function expansion

The classical expansion of the perturbation function in the 3-body problem, derived first time by LeVerrier, obtained with Maple computer algebra system. Then, the performance of the perturbation function in the different co-ordinate frames, include KS-variables, is compared. After that, the some particular cases of the problem setting are considered - the co-orbital motion and close encounters. In case of low perturbation, the method of the intermediate orbit construction is given. The convergence for the obtained perturbation function expansion is estimated. The proposed method to calculate relevant sums can be important for practical application. Some another applications of Maple system are discussed. First of all - for the Hamilton function performance in the different co-ordinate system. The expansions for H are given both for resonance and non-resonance motion. The results may be important for the construction of the different mapping schemas and their accuracy estimations. Author thanks to the Watherloo company for the trial version of Maple system.

The example of a stable kN+1 body central configuration

The k configurations of N points, each having mk mass, and a central mass M, forming a central configuration, is considered. The motion equations for a testing particle are given in different co-ordinates. Different forms of motion equations are compared. The study of this system showed, that it may be stable in linear approximation in case that the outer mass is larger than the inner one: mo>mi. The main kinds of a finite and infinite motion in the system are described. The system of configurations, replaced by angle one from another, is considered. The dependence of the character of motion and stability from this angle is investigated. The numerical method is used to search the collisionless configurations. The stability in the second order by perturbations and some possible applications of results are discussed.

Alessandro Rossi

CNUCE-CNR, Pisa

The growing number of space debris in Earth orbit represents a threat to the current missions, especially in the most crowded zones of the Low Earth Orbit region (approximately up to 2000 km of altitude). The risk of collision between a spacecraft and a piece of debris can be estimated by several means. We developed a method, based on the \"Opik's theory of planetary encounters that allows us to efficiently calculate and visualize the collision risk faced by a given spacecraft in a circular LEO. A description of the method is first given. Then an estimate of the collision risk, and of its evolution for several decades in the future, is given, for some sensitive spacecraft currently in orbit (e.g. the International Space Station, the IRIDIUM and GLOBALSTAR constellation satellites, etc.) under different scenarios of the possible future space traffic.

University of Glasgow and Glasgow Caledonian University

It is shown that the application of Sundman's Inequality to the Caledonian Symmetric Four Body Problem (CSFBP) enables an equation independent of the initial conditions to be found, together with a constant $\C_0$ that is defined solely by the initial conditions. From them the topology of the surfaces of separation defining regions of real motion for any given CSFBP can be determined. The connectivity of the surfaces of separation enables predictions to be made of the possible hierarchical evolution of the system.

University of Michigan

The dynamical problem of a binary asteroid system is posed and its stability discussed. In the general binary asteroid problem we consider the motion of two bodies of arbitrary mass distribution as they move relative to each other. Basic results from the classical n-body problem can still apply to such a system and some basic concepts continue to hold. Specifically, the total energy of the system, now including both rotational and translational energy, controls whether the system may ultimately disrupt under its own interactions. Whether or not an excess of positive energy (contained for example in the rotational motion of an asteroid) can be transfered to a positive excess of energy in the translational motion depends on the amount of spin-orbit coupling that can occur between the bodies. Some methods for estimating the amount of energy and angular momentum transfer between rotational and translational motions are introduced. These methods can be used to develop constraints on the long-term stability of a binary asteroid system.

Bruno Sicardy

Observatoire et Universite de Paris

Planetary ring dynamics

Planetary rings are natural laboratories where dynamical processes can be observed, and theories can be tested. Recent observations at high spatial resolution, from spacecraft or from the Earth, have revealed a great wealth of complex dynamical behaviors in rings. We will discuss some of them, especially those connected to ring-satellite interactions. Resonances between point-like satellites and fluid-like rings give a special status to these interactions. In particular, celestial mechanics must in this case be combined with hydrodynamics (or at the least, N-body problem), in order to correctly describe the collective behavior of perturbed particles, due to mutual collisions.
This gives rise to pressure, viscosity and self-gravitation terms which complicate the equations of motion. This yields in countepart a rich context of various processes: spiral density and bending waves, ringlet radial confinement, arc confinement, clearing of gaps, creation of very sharp edges, rigid precession, resonances with the magnetic field, etc...
We will present some of these processes, in view of recent data acquired on rings. We will also discuss the numerous unresolved problems associated with ring dynamics, and show how they can be related to general problems of disks in astrophysics.

Research Center for Astronomy, Academy of Athens

The problem of stability of the Trojan asteroids is investigated in the light of the Nekhoroshev theory of stability over large time intervals. We consider the two-dimensional planar, and the three-dimensional spatial restricted three body problem (Sun Jupiter asteroid) as simple models for describing the motion of an asteroid. Using these models we find regions of effective stability around the Lagrangian point L4 such that if the initial conditions of an orbit are inside these regions the orbit is confined in a slightly larger neighborhood of the equilibrium for a very long time. By combining analytical methods and numerical approximations we are able to prove that stability over the age of the universe is guaranteed in a realistic region, big enough to include some real asteroids. This significantly improves previous works on the subject.

Considering a 3D potential describing a Ferrers bar embedded in an axisymmetric Miyamoto disk, we investigate the families of periodic orbits that could populate the central region of such a system and thus make the bulge. We study the importance of the inclusion of an explicit bulge component in our potential and its influence on the bulge dynamics. We estimate the significance of the various families found and we study the dynamical phenomena that bring these families in the system. Our goal is to make a library of periodic orbits, that could help in building a self-consistent model for the bulge of a barred galaxy.

Besides the computation of the Lyapunov Characteristic Number (LCN), in the last decade a number of new techniques have been developed for distinguishing between ordered and chaotic motion in dynamical systems, e. g. the frequency analysis, the Fast Lyapunov Indicator (FLI) and the study of dynamical spectra of quantities like stretching numbers, helicity angles and twist angles. In the present paper we introduce a new, fast and easy to compute indicator for the same purpose based on the differential stretching along orbits. The efficiency of the new indicator is shown in weakly chaotic multidimensional systems such as a four dimensional (4D) symplectic map composed of two weakly coupled 2D maps and a comparison with the above methods is given.

Slovak Academy of Sciences

Three possible applications of the constructed analytical theory of the "stellar" three-body problem are presented. The "stellar" problem is a particular case of the motion of three points, in which the masses of the components are comparable, and the ratio of the semi-major axis of their orbits is the small parameter. As the intermediate orbit we used the solution of the simplified canonical system of differential equations, in which the Hamiltonian does not contain terms of the third and higher orders. The secular and long-periodic terms to the second order were taken into account in the intermediate orbit. The solution was obtained in terms of hyperelliptic integrals. It allows to investigate the dynamical evolution of stellar systems over long time intervals, to calculate maximum and minimum values of the eccentricities, and to consider the question about the stability.
In the first case the theory was applied to the stellar systems for which all six Keplerian elements and the direction of radial velocities nearby to nodes are known. Such orbits of stars are defined unambiguously. All dynamical characteristics of the systems can be calculated by the formulae of the theory.
In the second case the theory was applied to the systems for which the inclinations in stellar catalogues are given with double signs. The theory allows to get the precise sign of the inclinations, because for one of the signs the system is unstable. To check this result the three real stellar systems were taken from the catalogue.
The third application deals with close binary systems, supposing the existence of a far third component. For such a system all orbital elements and the mass of the third component cannot be obtained from observations. Only the orbital period and the light equation may be known for the third component. The theory allows to get orbital elements of the supposed component so that the close binary system should stay stable. For the illustration the system $AS$ Camelopardalis was considered.

Observatoire de Paris

Inter-comparisons between numerical integrations and VSOP analytical ephemerides for the solar system
(Co-authors: D. Gauchez, M. Fouchart, J. Souchay)

The purpose of this paper is to compare numerical integration with analytical solution of the n body problem. The first part of the article is dedicated to the comparison of the precision evolution with respect to the cpu time of several numerical integrator applied to the 2 body problem with incresing eccentricite. The integrator used here are: Runge-Kutta-Nystrˆm, RADAU, Burlirsh-Stoer and a method based on Lie Series. In the second part the comparison between numerical integration and analytical solution is done. The problem studied is the 8 body problem and the analytical solution is given by VS0P 88. We compare the evolutions of semi-major axis and eccentricities as given by the methods.

Università di Roma "La Sapienza"

Low DV orbit insertion in interplanetary missions
(Authors: P.Teofilatto and C. Circi)

A key issue in interplanetary missions is the attempt to reduce as much as possible the on board propellant, which has a direct impact on the payload weight. Basically this means to minimize the variation of velocity DV needed for the spacecraft orbit insertion. Then it is of interest to look for arrival conditions close to (temporary) ballistic capture of the spacecraft by the target planet. For the case of lunar transfer orbits, such a ballistic capture can be achieved by the close passage of the spacecraft through the unstable Lagrangian points in the Earth-Sun and Earth-Moon systems. The same kind of low orbit insertion technique can be pursued in the case of Jovian satelliets, and more generally in all the cases where a four body effect can be relevant. In interplanetary missions four body effects are rather weak, in general, and the restricted three body problem can approximate the spacecraft motion rather well. In this framework the conditions for temporary capture are found for orbit insertion to Mars and Mercury. It is found that it is possible to take advantage of the eccentricity of these planets in order to reduce the spacecraft DV at arrival. A simple analytical formula it is found which determines the value of the local eccentricity to allow capture for Mars and Mercury transfer orbits as function of the spacecraft pericenter distance and planet anomaly. This formula generalizes to satellites of any eccentricity and to any value of the planet anomaly a formula given by Hoppenheimer. Several numerical tests have been performed and the accuracy of the formula is shown. The possible payload mass increment for these low DV Mars/Mercury orbit insertions are evaluated.

Moscow State Academy of Instrument-making and Computer Science

The theory of the motion of a binary planet is proposed.

Moscow State Academy of Instrument-making and Computer Science

In the photogravitational three body problem we study the motion of a particle under action of gravitational attraction by two primaries and at the same time under light pressure by one or both primaries. Dynamical equations for this problem belong to a class of the conservative reversible systems with two degrees of freedom. Equations of motion are reduced to the autonomous reversible system of 3-rd order and to the periodic reversible system of 2-nd order. The method for constructing of all symmetric periodic orbits is proposed. Results are represented for families of orbits adjointing to collinear libration points. For each of collinear libration points the number of such families is defined in dependence on values of parameters for double stellar system which includes two identical primaries. The method mentioned above is used and the numerical calculation of these families is carried out. The stability of orbits is investigated, and the evolution of families is studied.

Kleomenis Tsiganis

Aristotle University of Thessaloniki

Chaos in mean motion resonances results in a slow, diffusive-like, evolution of an asteroid's elements. A numerical calculation of the relevant local transport coefficients is made, for several mean motion resonances with Jupiter, using short-term integrations. Predictions for the escape time of asteroids is then made, through the solution of the associated kinetic equation, and compared with the results of long-term numerical integration.

Institut de Mecanique Celeste et de Calcul des Ephemerides

The faint satellites of Jupiter J14 Thebe, J15 Adrastea and J16 Metis have their orbits inside the one of Io, nearby the planet. Consequently, the astrometric and dynamical studies of these satellites may allow to explore the planetary environment. In order to perform this study we have developed a numerical model of the motions and we began to collect the available astrometric observations. We also started a campaign of new observations. First results have been obtained by comparing this model to observations and we present a discussion of these results. We also provide estimates of the faint non gravitational effects due to the closeness of Jupiter.

University of Texas at Austin

We consider several Hamiltonian systems close to stability. We show that the Arnold diffusion time is bounded by a power of the homoclinic splitting.

University of Thessaloniki

We show that the occurrence of stable chaos in mean motion resonances with Jupiter, where cases of real asteroids on stable-chaotic orbits have been identified, is related to the fact that there do not exist families of periodic orbits in the planar elliptic restricted problem and in the 3-D circular problem corresponding to this resonance. This property may provide a "protection mechanism", leading to semi-confinement of chaotic orbits and extremely slow migration in the space of proper elements, so that diffusion is practically not related to the value of the Lyapunov time of chaotic orbits. We show, also, that stable-chaotic orbits have a characteristic spectrum of autocorrelation times: for the action conjugate to the critical argument, the autocorrelation time is of the order of the Lyapunov time, while for the eccentricity- and inclination-related actions the autocorrelation time may be longer than 1,000 Lyapunov times. This fact reflects the disability of these "flawed" resonances to modify secular motion significantly, at least for times of the order of 200 Myrs.

Observatoire de Paris

In attempt to forecast the Leonid meteor stream, we looked at non gravitationnal forces effects on very narrow particles in the solar system.

IMCCE/Observatoire de Lille

Some recent studies have shown that the eccentricity of Tethys has a deciding influence on the evolution, under tidal effects, of the resonance of the Mimas-Tethys system. Tethys' eccentricity induces secondary resonances in which the system can be temporarily captured. These secondary resonances lead also to a chaotic zone separating libration from circulation. As a consequence, the inclination of Mimas before capture may have been very different than the value previously considered. Another important result is that the probability of capture in the main resonance may have been much greater than the previous admitted value 0.04 (up to 1). Unfortunately, the value of the eccentricity of Tethys is badly known. In the present work, we present a new determination of this eccentricity. This determination uses the newest representation TASS1.7 of the motion of the main satellites of Saturn. We use also an analysis of the mean longitude of Mimas in which the eccentricity of Tethys has a notable influence. The recent reduction of the CCD observations of Mimas allows to make such analysis.

Observatory, University of Helsinki

Optimized statistical ranging of asteroid orbits
(Co-authors: Karri Muinonen, Teemu Laakso, Mikko Kaasalainen, Edward Bowell)

We present an upgraded version of statistical orbital ranging for asteroids. We have optimized the computational algorithm by improving the iteration of the topocentric ranges and the selection of the pair of observations used to compute the orbital elements. Moreover, we will be making use of a new integrator package, which should prove useful for NEOs with short observational arcs. Altogether, this faster version of statistical ranging will enable the use of N-body integrations when they are required. We have also developed methods to study quantitatively the validity of linear approximation in orbit determination problems. With our statistical ranging method we can find the region of qualification for the linear approximation as a function of time elapsed from the observations. In addition, we are examining the invariance of the orbital element probability density in transformations between different orbital element sets.

Institute of Astronomy, Prague

New results on the Yarkovsky effect
(Authors: D. Vokrouhlicky and M. Broz)

We present recent results of the long-term dynamics of meteoroids and small asteroids with the Yarkovsky perturbation. In particular, we deal with transport of meteoroids from the main asteroid belt and a possible role of the Yarkovsky effect on dispersion of asteroid families. Attention is also paid to the YORP effect, notably the long-term influence of the Yarkovsky torque on the rotation state. Preliminary results, showing how the YORP evolution of the spin axis may affect the orbital Yarkovsky perturbation, are discussed.

ETH - Zuerich

The formation of double stars or double asteroids is sometimes attributed to ''capture`` of one body by another under purely Newtonian forces. We investigate this process by means of the planar problem of three bodies with two small masses, and by its limiting case, Hill's lunar problem. It is found that in the absence of dissipative forces the state of capture generally does not last forever. However, temporary capture may last for arbitrarily long times.

National Astronomical Observatory of Japan

It is known that the symplectic mapping obtained as a symplectic integration method is formally an exact time evolution of the modified Hamiltonian which is close to the original Hamiltonian. In the case when the original Hamiltonian has an additional first integral, it is shown that the modified first integral, which is defined to be an integral for the modified Hamiltonian, does not necessarily exist in general. This non-existence of the modified first integral is demonstrated by an example of the 2D harmonic oscillator with an integer frequency ratio.

Cairo University

A sufficiently accurate solution is given to account for tidal effect that weak gravitational waves exert on a bound system of two bodies in elliptical orbits. The dimensionless amplitude of the wave is considered a small parameter of the first order. Functions of the true anomaly are developed in Fourier series in the mean anomaly retaining powers of the eccentricity up to the third. The perturbations are obtained both in Keplerian and in canonical elements.

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