In this paper we show that the Conditional Entropy of nearby orbits may be a useful tool to explore the phase space associated to a given Hamiltonian. The arc length parameter along the orbits, instead of the time, is used as a random variable to compute the entropy. In the first part of this work we summarise the main analytical results to support this tool while, in the second part, we present numerical evidence that this technique is able to localise (stable) periodic and quasiperiodic orbits, ‘aperiodic’ orbits (chaotic motion) and unstable periodic orbits (the ‘source’ of chaotic motion). Besides, we show that this technique provides a measure of chaos which is similar to that given by the largest Lyapunov Characteristic Number. It is important to remark that this method is very simple to compute and does not require long time integrations, just realistic physical times.

Two simple and efficient numerical methods to explore the phase space structure are presented, based on the properties of the “dynamical spectra”. 1) We calculate a “spectral distance” D of the dynamical spectra for two different initial deviation vectors. D → 0 in the case of chaotic orbits, while D → const ≠ 0 in the case of ordered orbits. This method is by orders of magnitude faster than the method of the Lyapunov Characteristic Number (LCN). 2) We define a sensitive indicator called ROTOR (ROtational TOri Recongnizer) for 2D maps. The ROTOR remains zero in time on a rotational torus, while it tends to infinity at a rate ∝ N = number of iterations, in any case other than a rotational torus. We use this method to locate the last KAM torus of an island of stability, as well as the most important cantori causing stickiness near it.

The resonant structure near a noble cantorus is found. Islands of stability are located near the gaps of the cantorus. The crossing of the gaps of the cantorus by the asymptotic curves of unstable periodic orbits is shown numerically (non-schematically). We discuss how these structures influence stickiness.

This paper reviews three recent works on the numerical methods to integrate ordinary differential equations (ODE), which are specially designed for parallel, vector, and/or multi-processor-unit (PU) computers. The first is the Picard-Chebyshev method (Fukushima, 1997a). It obtains a global solution of ODE in the form of Chebyshev polynomial of large (> 1000) degree by applying the Picard iteration repeatedly. The iteration converges for smooth problems and/or perturbed dynamics. The method runs around 100-1000 times faster in the vector mode than in the scalar mode of a certain computer with vector processors (Fukushima, 1997b). The second is a parallelization of a symplectic integrator (Saha et al., 1997). It regards the implicit midpoint rules covering thousands of timesteps as large-scale nonlinear equations and solves them by the fixed-point iteration. The method is applicable to Hamiltonian systems and is expected to lead an acceleration factor of around 50 in parallel computers with more than 1000 PUs. The last is a parallelization of the extrapolation method (Ito and Fukushima, 1997). It performs trial integrations in parallel. Also the trial integrations are further accelerated by balancing computational load among PUs by the technique of folding. The method is all-purpose and achieves an acceleration factor of around 3.5 by using several PUs. Finally, we give a perspective on the parallelization of some implicit integrators which require multiple corrections in solving implicit formulas like the implicit Hermitian integrators (Makino and Aarseth, 1992), (Hut et al., 1995) or the implicit symmetric multistep methods (Fukushima, 1998), (Fukushima, 1999).

The word “average” and its variations became popular in the sixties and implicitly carried the idea that “averaging” methods lead to “average” Hamiltonians. However, given the Hamiltonian H = H0(J) + ϵR(θ,J),(ϵ ≪ 1), the problem of transforming it into a new Hamiltonian H* (J*) (dependent only on the new actions J*), through a canonical transformation given by zero-average trigonometrical series has no general solution at orders higher than the first.

Due to the existence of invariant tori, chaotic sea and hyperbolic structures in higher dimensional phase space of a volume-preserving map, the diffusion route of chaotic orbits will be complicated. The velocity of diffusion will be very slow if the orbits are near an invariant torus. In order to realize this complicated diffusion phenomenon, in this paper we study the diffusion characters in the different regions, i.e., chaotic, hyperbolic and invariant tori’s regions. We find that for the three different regions, the diffusion velocities are different. The diffusion velocity in the vicinity of an invariant torus is the slowest one.

The planar rotational motion of a non-symmetric satellite in an elliptic orbit is considered. A two-dimensional map is constructed, describing the motion in a vicinity of the separatrix of the synchronous spin-orbit resonance. This map is a generalization of Chirikov’s separatrix map, in the sense that the asymmetry of perturbation is taken into account. Phase portraits of the generalized map perfectly reproduce well-known examples of surfaces of section (first computed by Wisdom et al. (Wisdom et al., 1984), Wisdom (Wisdom, 1987)) of the phase space of spin-orbit coupling for non-symmetric natural satellites. Moreover, it provides a straightforward analytical description of the phase space: analysis of properties of the map allows one to precalculate, by means of compact analytical relations, the locations of resonances and chaos borders, the emergence of marginal resonances, and even to describe bifurcations of the synchronous resonance’s center, though far from the separatrix.

We introduce a new kind of canonical variables that prove very useful when the order of a Hamiltonian system can be reduced by one, as in the case of isoenergetic reduction, and of what we call homogeneous reduction. The Kepler Problem, Geometrical Optics and McGehee Blow-up are discussed as examples. Finally we carry out the isoenergetic reduction of the general N-Body Problem using the new variables, and briefly discuss its application to the problem of collision.

Many of the important applications of the circular planar restricted problem of three bodies involve motion in the vicinity of the unit circle, (as defined in canonical units). It is then of interest to develop simplified models which are valid in this region. These models preserve the gross characteristics of the original system but they possess simpler equations of motion.

We will also show that several simplified models can be seen as a perturbation of a very well known simple linear system: the Clohessy-Wiltshire equations used by NASA in all their rendezvous operations. These are actually very close to the well-known Hill problem. We will thus consider the Restricted problem as a perturbed Hill or Clohessy-Wiltshire problem. We also introduce the Clohessy-Wiltshire Lagrangian in polar coordinates.

The isosceles three-body problem with Sitnikov-type symmetry has been reduced to a two-dimensional area-preserving Poincaré map depending on two parameters: the mass ratio, and the total angular momentum. The entire parameter space is explored, contrasting new results with ones obtained previously in the planar (zero angular momentum) case. The region of allowable motion is divided into subregions according to a symbolic dynamics representation. This enables a geometric description of the system based on the intersection of the images of the subregions with the preimages. The paper also describes the regions of allowable motion and bounded motion, and discusses the stability of the dominant periodic orbit.

The problem of two gyrostats in a central force field is considered. We prove that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system posseses symmetries. Using them we perform the reduction of the number of degrees of freedom. We show that at every stage of the reduction process, equations of motion are Hamiltonian and give explicit forms corresponding to non-canonical Poissson brackets. Finally, we study the case where one of the gyrostats has null gyrostatic momentum and we study the zero and the second order approximation, showing that all equilibria are unstable in the zero order approximation.

The past decade brought direct evidence of the previously surmised exoplanetary systems. A variety of planetary system types exist those around pulsars, around both young and old main-sequence stars (as evidenced by planetesimal disks of the Beta Pictoris-type), and the mature giant exoplanets found in radial velocity surveys. The surprising diversity of the exoplanetary systems is addressed by several theories of their origin.

Analytical complements have been brought to Moons’ lunar libration theory concerning tidal effects, direct perturbations due to the Earth’s figure, and indirect non periodic perturbations. Comparisons to JPL numerical integrations DE245 and DE403 have been performed and the residuals treated by frequency analysis, allowing the determination of fitted free libration parameters and numerical complements.

Range of values of the Sun’s mass quadrupole moment of coefficient J2 arising both from experimental and theoretical determinations enlarge across literature on two orders of magnitude, from around 10−7 until to 10−5. The accurate knowledge of the Moon’s physical librations, for which the Lunar Laser Ranging data reach an outstanding precision level, prove to be appropriate to reduce the interval of J2 values by giving an upper bound of J2. A solar quadrupole moment as high as 1.1 10−5 given either from the upper bounds of the error bars of the observations, or from the Roche’s theory, is not compatible with the knowledge of the lunar librations accurately modeled and observed with the LLR experiment The suitable values of J2 have to be smaller than 3.0 10−6.

As a consequence, this upper bound of 3.0 10−6 is accepted to study the impact of the Sun’s quadrupole moment of mass on the dynamics of the Earth-Moon system. Such an effect (with J2 = 5.5 ± 1.3 × 10−6) has been already tested in 1983 by Campbell & Moffat using analytical approximate equations, and thus for the orbits of Mercury, Venus, the Earth and Icarus. The approximate equations are no longer sufficient compared with present observational data and exact equations are required. As if to compute the effect on the lunar librations, we have used our BJV relativistic model of solar system integration including the spin-orbit coupled motion of the Moon. The model is solved by numerical integration. The BJV model stems from general relativity by using the DSX formalism for purposes of celestial mechanics when it is about to deal with a system of n extended, weakly self-gravitating, rotating and deformable bodies in mutual interactions.

The resulting effects on the orbital elements of the Earth have been computed and plotted over 160 and 1600 years. The impact of the quadrupole moment of the Sun on the Earth’s orbital motion is mainly characterized by variations of , and Ė. As a consequence, the Sun’s quadrupole moment of mass could play a sensible role over long time periods of integration of solar system models.

The effect of the C22 gravity field term on a particle is evaluated analytically over one orbit to find the change in orbit energy and angular momentum as an explicit function of the orbital inclination, argument of pericenter, longitude of the ascending node, orbit parameter and eccentricity. Changes in orbit energy and angular momentum are shown to be proportional to a family of integrals which can be parameterized in terms of eccentricity and non-dimensional pericenter radius.

This paper begins with a brief review of a form of the Lie series transformation, and then reports some new results in the study, using Lie series methods, of the orbit of Saturn’s satellite Hyperion. In particular, improved expressions are given for the long-period perturbations of the orbital elements which describe the motion in the orbit plane, and also first results for expressions for the short-period perturbations in the apse longitude, derived from the Lie series generating function.

Mean orbital elements are obtained from their instantaneous, osculating counterparts by removal of the short periodic perturbations. They can be computed by means of different theories, analytical or numerical, depending on the problem and accuracy required. The most advanced contemporary analytical theory (Knežević 1988) accounts only for the perturbing effects due to Jupiter and Saturn, to the first order in their masses and to degree four in eccentricity and inclination. Nevertheless, the mean elements obtained by means of this theory are of satisfactory accuracy for majority of the asteroids in the main belt (Knežević et al. 1988), for the purpose of producing large catalogues of mean and proper elements, to identify asteroid families, to assess their age, to study the dynamical structure of the asteroid belt and chaotic phenomena of diffusion over very long time spans. In the vicinity of the main mean motion resonances, however, especially 2:1 mean motion resonance with Jupiter, these mean elements are of somewhat degraded accuracy.

Asteroid masses are the largest source of unmodeled forces in current planetary ephemerides research. Williams (1984) showed that the asteroids produce km size perturbations in the position of Mars. However, the masses of only three asteroids are known to better than 10%, and only six other asteroid masses have been determined at all.

Detecting the mass of an asteroid is difficult because the observed quantity is the change in the mean motion of a second, perturbed asteroid. Asteroid masses are small, so the change in the mean motion is typically on the order of 0."015 yr−1. Thus, excellent orbit determinations are needed both before and after the perturbing encounter. This requires high precision observations over as many oppositions as possible.

Hilton (1997) determined the mass of 15 Eunomia to within 25% by detecting perturbations of 1313 Berna. The greatest source of uncertainty in determining the mass of Eunomia was the very poor coverage and accuracy of pre-encounter observations. Hilton (1998) has determined the masses of 1 Ceres, 2 Pallas and 4 Vesta, all based on mutual interactions. The uncertainties in the masses are 1% for Ceres, 3% for Pallas, and 7% for Vesta. The masses of Ceres and Pallas are the best so far, and the mass for Vesta corroborates previous determinations of its mass.

A large fraction of the asteroids observed so far are to be considered lost, that is they cannot be recovered by pointing the telescope at the predicted position only. The catalogues of asteroid orbits are therefore polluted by large numbers of low accuracy orbits, which cannot be easily improved by observation. Two of these inaccurate orbits can belong to the same physical object, and indeed identifications are regularly found when the orbital elements are close; an unknown number of real identifications have not been found in the existing catalogues because the orbital elements as computed are far apart. It is impossible to systematically check for possible identification of all the couples of orbital element sets; an algorithm is required to single out a comparatively small number of proposed identifications, and to find a suitable first guess within the region of convergence of the differential corrections procedure to be used to confirm the identification.

In the linear approximation the target function to be minimized is a quadratic form; thus a fully analytic algorithm can be given to compute both the first guess for an identification and the cost, that is the increase in the target function resulting from joining the two sets of observations. This allows to define a number of distances in the elements space, which are functions, not only of the two orbital element sets, but also of their uncertainties, as measured by the Gaussian covariance matrices. Tests on the values of these distances are used as preliminary filters, to allow more extensive computations to be performed on a fraction of the total number of possible couples.

The interaction between resonances and dissipative effects has long been investigated in relationship with the dynamics of dust particles or the evolution of planetesimals in the primordial solar nebula. More recently, it has been shown that another important application of this kind of dynamics is the delivery of asteroid fragments (precursors of meteorites and NEAs) to Earth-crossing orbits. The relevant dissipative mechanism, the so-called Yarkovsky effect (Öpik 1951; Peterson 1976; Burns et al. 1979) has been known for a long time, but its “seasonal” and “diurnal” variants, as well as its interaction with the collisional process taking place in the asteroid belt, have been studied quantitatively only in the last few years (Rubincam 1995, 1998; Hartmann et al. 1997, 1998; Farinella et al. 1998; Vokrouhlický and Farinella 1998a,b; Vokrouhlický 1998a,b; Farinella and Vokrouhlický 1998a,b).

In the Farinella et al. (1998) paper we have provided a unified discussion of the Yarkovsky effect in both its variants. After computing the rate of the corresponding semimajor axis drift as a function of size and spin rate, and comparing the relevant timescales with those for collisional disruption and spin axis reorientation, we have rediscussed some issues in meteorite science which are put in a new light by the relevance of the Yarkovsky effect. In particular, this mechanism provides a good explanation for the fact that meteorite cosmic ray exposure ages (in particular for irons) are much longer than the dynamical lifetimes of objects delivered to the Earth-crossing region through resonances. Thanks to the Yarkovsky effect, small asteroid fragments in the belt undergo a slow drift in semimajor axis (with a random-walk component related to their rotational state) and therefore have enough mobility to reach the resonances after comparatively long times spent in nonresonant main-belt orbits. Metal-rich fragments have slower Yarkovsky drift rates than stones, but their much longer collisional lifetimes can explain why iron meteorites appear to sample a larger number of asteroid parent bodies compared to ordinary chondrites.