In ELP, the computation of planetary perturbations is about 20 years old. A better knowledge of lunar and planetary parameters, new planetary solutions under construction and progresses in numerical tools, are factors that should contribute to their improvements. The construction of planetary perturbations takes widely its inspiration from Brown’s method. In a first step, we only consider the main problem (Earth, Moon, and Sun with a Keplerian motion). The solution of the main problem is actually of a high precision and is used as a reference (Chapront-Touzé, 1980). This solution is expressed in Fourier series of the 4 Delaunay arguments, with numerical coefficients, and partials with respect to integration constants.

The method based on the variation of arbitrary constants is described in (M.Chapront-Touzé, J.Chapront, 1980). Equations of Moon’s motion are written in a rotating frame where the reference plane is the mean ecliptic. In this frame, the absolutec acceleration is expressed by means of disturbing forces acting on the Moon, by the Sun, the Earth and a planet. It is the gradient of F which can be divided into several components: Fc related to the main problem, FD and FI giving rise to direct and indirect planetary perturbations.

An analytical long-term theory of the motion of the Moon is constructed within the framework of the general planetary theory (Brumberg, 1995). A method, different from the one of (Ivanova, 1997) designated below as (*), for the determination of the perturbations depending on the eccentricities and inclinations of lunar and planetary orbits is used which allows to obtain the solution of the problem in the purely trigonometric form up to any order with respect to the small parameters.

The aim of this paper is to construct the long-term Lunar theory in the form consistent with the general planetary theory (Brumberg, 1995). For this purpose the Moon is considered as an additional planet in the field of eight major planets (Pluto being excluded). In the result the coordinates of the Moon may be represented by means of the power series in the evolutionary eccentric and oblique variables with trigonometric coefficients in mean longitudes of the Moon and the planets. The long-period perturbations are determined by solving a secular system in Laplace-type variables describing the secular motions of the lunar perigee and node and taking into account the secular planetary inequalities.

Following efforts to construct an accurate modelisation of Mars rotation starting from canonical equations in an Hamiltonian theoretical frame (Bouquillon and Souchay, 1996), we use recent results from radio tracking data of the Mars Pathfinder mission (Folkner et al., 1997) to modelize in the best way the motion of precession and nutation of the planet. A complete set of coefficients related to these two motions is presented, including the main effect due to the Sun and also those due to the two satellites Phobos and Deimos as well as to the planets. Morever, Oppolzer terms are calculated and included.

The role of the 200 yr-long period found recently in the mean longitude of Mimas (Vienne and Duriez 1992) is investigated through numerical integrations. It is shown that it has a deciding effect on the descriptions of the resonance motion of the Mimas-Tethys system, as considered up to now. As a result, Mimas’s inclination before capture may have been higher (up to 0.7°) or lower (down to 0.03°) than the value previously considered (0.42°). Also, Tethys’s eccentricity on capture may have been quite higher (≈ 0.008 versus 0). Moreover, the probability of capture is found to be very sensitive to Tethys’s eccentricity, and possibly much higher (up to 1) than the value considered before (0.04).

In order to obtain corrections to orbital elements of minor planets and to determine parameters of rotation of a reference frame, a geometrical model with restrictions in the declination, longitude of the ascending node and argument of the perihelion to minimize a residual function involving all named parameters was presented (Marco et al., 1996, 1997). On the other hand, such rotational model of correction should not modify the semiaxis, eccentricity nor the mean anomaly, so new restrictions on these parameters seem to be necessary. These conditions should reflect the invariability in the size of the orbit and they should be included in the model to complete a consistent set of restrictions.

In this work we attempt to make progress into assessing the importance of secular interactions between planetary satellites. In recent years, discrepancies have been observed in the expected positions of small planetary satellites (Bosh & Rivkin, 1996; Roddier et al., 1998). The existing ephemerides-producing algorithms for these objects assume fixed, elliptical and inclined orbits whose rate of precession is determined by oblateness alone. Even though the masses of these satellites are quite small relative to the planet (∼ 10−9 – 10−10) their small mutual separations and the existence of much larger satellites further out leaves open the possibility that in some cases at least the fixed-orbit assumption is only a crude approximation to reality. Two important dynamical mechanisms through which these orbits may evolve are resonant or secular interactions. In order to explore the possibility of the latter we have set up a simple planar system where an satellite in a circular orbit around a spherical planet is perturbing a massless particle which moves in proximity to various mean motion resonances. We aim to examine the effect of the resonance on the particle’s reference orbit by measuring the secular frequency. The effects of oblateness have not been taken into account as they are adequately modeled by orbit-fitting theories and can thus be readily subtracted.

According to many authors, there is no reason to believe that inner planets never had satellites in the past. Indeed, Alfvén and Arrhenius (1972), Burns (1973), Ward and Reid (1973), etc, explain their disappearance using essentially tidal effects which cause secular variations in the semimajor axes. Here, taking into account only gravitational forces, we show that the extinction or the non existence of Earth’s satellites could also be related to some strong instabilities which increase the eccentricities e to prohibitive values.

We have seen (Stellmacher, 1996) that the long-period terms for Hyperion’s motion are very well represented by a second kind and second genius periodic orbit, after Poincaré’s classification.

We have shown how to construct such an orbit with, as only data, the observed periods, which characterise the resonance of the Titan-Hyperion couple, and the Titan’s motion which is an elliptical one. The physical quantities as the masses and the J2 term of Saturn’s flattening are given. We will present the results that we obtained, and compare them with those that other authors obtained by fitting the series to the observations.

The orbits of planet-crossing asteroids (and comets) can undergo close approaches and collisions with some major planet. This introduces a singularity in the N-body Hamiltonian, and the averaging of the equations of motion, traditionally used to compute secular perturbations, is undefined. We have shown (Gronchi and Milani, 1998) that it is possible to define in a rigorous way some generalised averaged equations of motion, in such a way that the generalised solutions are unique and piecewise smooth, with corners on the node crossing lines.

The model is the averaged equations of motion first introduced by Kozai (1962): the perturbing planets are assumed to move in circular, coplanar orbits, and the equations of motion are averaged over the anomalies of the asteroid and of the planets. In the non-crossing case the averaging is integrable; in the planet-crossing case there is a polar singularity of order two in the equations of motion, and averaging is not possible. To define a generalized solution, we decrease the order of the polar singularity by the method of extraction of the singularities by Kantorovich. The singularity of the perturbing function is approximated by a modified inverse distance, the one between the straight lines tangent to the two orbits at the nodal points. In this approximation the averaged perturbing function has an analytical expression, allowing explicit computation with elliptic integrals and elementary functions.

There are several important reasons to consider relativistic effects in rotational motion of celestial bodies. General Relativity is now recommended by the International Astronomical Union and International Union of Geodesy and Geophysics as a theoretical framework for modeling of high-precision observational data. On the other hand, various geodynamical observations provide data which are widely used for testing General Relativity itself.

In Newtonian mechanics it is well known how to describe rotational motion of an extended body. In General Relativity this is a rather subtle issue. The concept of a precessing extended rigid body in general relativity encounters fundamental difficulties and cannot be introduced even in the first post-Newtonian approximation. From a practical point of view, however, the rotational motion of the Earth even at the Newtonian level is defined operationally through the time-dependence of geocentric quasi-inertial coordinates of observing sites. An analogous operational definition can be applied in general relativity. To this end, we need a set of physically adequate reference systems.

Let us consider the orbital problem in which a particle is subject to the force (per unit mass)

The force consists of the Newtonian two body attraction term and a drag part which is linear in both components of velocity (radial vr and transverse vt). Depending on a physical interpretation of the parameters μ and α, as well as on the particular choice of the dimensionless constant γ, the model (1) can match various two body problems with dissipation. They include the classical Poynting-Robertson effect (Robertson, 1936) with γ = 1, Poynting’s (1903) version with γ = ½ and the two body drag cases recently studied by Mittleman and Jezewski (1982) and by Mavraganis and Michalakis (1994) under the name of Danby’s drag.

The spatial (planar) three-body problem admits the ten (six) integrals of energy, center of mass, linear momentum and angular momentum. Fixing these integrals defines an eight (six) dimensional algebraic set called the integral manifold, 𝔐(c, h) (m(c, h)), which depends on the energy level h and the magnitude c of the angular momentum vector. The seven (five) dimensional reduced integral manifold, 𝔐R(c, h) (mR(c, h)), is the quotient space 𝔐(c, h)/SO2 (m(c, h)/SO2) where the SO2 action is rotation about the angular momentum vector. We want to determine how the geometry or topology of these sets depends on c and h. It turns out that there is one bifurcation parameter, ν = −c2h, and nme (six) special values of this parameter, νi, i = 1, …, 9.

At each of the special values the geometric restrictions imposed by the integrals change, but one of these values, ν5, does not give rise to a change in the topology of the integral manifolds 𝔐(c, h) and 𝔐R(c, h). The other eight special values give rise to nine different topologically distinct cases. We give a complete description of the geometry of these sets along with their homology. These results confirm some conjectures and refutes several others.

Progress has been made in understanding the stability of hierarchical three-body systems where the third body moves on an approximately Keplerian orbit about the centre of mass of the binary, at a distance large compared to the binary separation. Harrington (1968,1969) showed analytically that provided the third body was sufficiently distant from the binary no secular terms appeared in the semi-major axis and the system was stable. Harrington (1972,1975,1977) established numerically a critical minimum separation distance (or period) for a stable system in terms of the masses, unaffected by the relative inclinations of the orbits, except for angles close to 90°. Most subsequent investigations have therefore used planar configurations. Graziani & Black (1981), Black (1982) and Pendleton & Black (1983) again using long-term integration of the orbits obtained a criterion for high and low mass binaries. Donnison & Mikulskis (1992,1994,1995) carried out numerical integrations on prograde, retrogade, planetary and stellar triple systems and found for prograde systems very good quantitative agreement with the c2H method. Eggleton & Kieselva (1995) suggested a critical distance ratio approximation determined by the masses in the system. Systems with eccentric orbits are covered using the period ratio determined by Kepler’s third law.

Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks like

where (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.

The Lagrangian equilateral points L4 and L5 of the restricted circular three-body problem are elliptic for all values of the reduced mass μ below Routh’s critical mass μR ≈ .0385. In the spatial case, because of the possibility of Arnold diffusion, KAM theory does not provide Lyapunov-stability. Nevertheless, one can consider the so-called ‘Nekhoroshev-stability’: denoting by d a convenient distance from the equilibrium point, one asks whether

for any small єe > 0, with positive a and b. Until recently this problem, as more generally the problem of Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, was studied only under some arithmetic conditions on the frequencies, and thus on μ (see e.g .Giorgilli, 1989). Our aim was instead considering all values of μ up to μR. As a matter of fact, Nekhoroshev-stability of elliptic equilibria, without any arithmetic assumption on the frequencies, was proved recently under the hypothesis that the fourth order Birkhoff normal form of the Hamiltonian exists and satisfies a ‘quasi-convexity’ assumption (Fassò et al, 1998; Guzzo et al, 1998; Niedermann, 1998).

Transport in Hamiltonian systems, in the case of strong perturbation, can be modeled as a diffusion process, with the diffusion coefficient being constant and related to the maximal Lyapunov number (Konishi 1989). In this respect the relation found by Lecar et al. (1992) between the escape time of asteroids, TE, and the Lyapunov time, TL, can be easilly recovered (Varvoglis & Anastasiadis 1996). However, for moderate perturbations, chaotic trajectories may have a peculiar evolution, owing to stickiness effects or migration to adjacent stochastic regions. As a result, the function χ(t), which measures the exponential divergence of nearby trajectories, changes behaviour within different time intervals. Therefore, trajectories may be divided into segments, i = 1,..., n, each one being assigned an “Effective” Lyapunov Number (ELN), λi = χ(ti).

Mixed-variable symplectic integrators provide a fast, moderately accurate way to study the long-term evolution of a wide variety of N-body systems (Wisdom & Holman 1991). They are especially suited to planetary and satellite systems, in which a central body contains most of the mass. However, in their original form, they become inaccurate whenever two bodies approach one another closely. Here, I will show how to overcome this difficulty using a hybrid integrator that combines symplectic and conventional algorithms.

A symplectic integrator works by splitting the Hamiltonian, H, for an N-body system, into two or more parts H = H0 + H1 + …, where є i = Hi/H0 ≪ 1 for i = 1, 2 …. An integration step consists of several substeps, each of which advances the system due to the effect of one part of the Hamiltonian only. The error incurred over the whole step is ∼ є τn1, where τ is the timestep, n is the order of the integrator, and є is the largest of єi.

We investigate the dynamics of a dissipative standard mapping defined by the equations

where y ∈ R, x ∈ T and ε is a real parameter, we refer to 0 < α < 1 as the “dissipative parameter” and to ψ as the “dissipative coefficient” (ε = α = 0 provides an integrable mapping). Notice that the dynamics is contractive, since the jacobian of the above mapping equals to 1 − α. In particular, we want to compare (see Celletti et al., 1997) the solutions associated to the conservative map (i.e., α = 0) with that related to (1) (α ≠ 0). For simplicity, we consider the case when α = ε2 and construct explicit approximate solutions to the conservative and dissipative systems, using a suitable parametrization like in (Celletti and Chierchia, 1988).

The following paper is devoted to the theoretical exposition of the obtention of second order perturbations of elliptic elements and is a follow-up of previous papers (Marco et al., 1996; Marco et al., 1997) where the hypothesis was made that the matrix of the partial derivatives of the orbital elements with respect to the initial ones is the identity matrix at the initial instant only. So, we must compute them through the integration of Lagrange planetary equations and their partial derivatives.

Such developments have been applied to the individual corrections of orbits together with the correction of the reference system through the minimization of a quadratic form obtained from the linearized residual. In this state two new targets emerged:

1. 1. To be sure that the most suitable quadratic form was to be considered.

2. 2. To provide a wider vision of the behavior of the different orbital parameters in time.

Both aims may be accomplished through the consideration of the second order partial derivatives of the elliptic orbital elements with respect to the initial ones.

To take advantage of the linear and regular formulation and treatment of Celestial Mechanics problems (Kustaanheimo & Stiefel 1965; Stiefel & Scheifele 1971; Deprit, Elipe & Ferrer 1994), Sharaf & Saad (1997) have given an analytical expansion of the Earth’s gravitational zonal potential in terms of Kustaanheimo-Stiefel (KS) regular elements (Stiefel & Scheifele 1971, §19), with special emphasis on its application to elliptic-type two-body orbits and, consequently, using a generalized (elliptic) eccentric anomaly as the independent variable.

Motivated by these and other considerations based on the definition and use of KS elements, and following a treatment similar to that of Stiefel & Scheifele (1971, §19), we develop element equations corresponding to a DEF-formulation of the satellite problem under the effect of the zonal potential.