After analysing the recent developments of the theory of the nutation for a simplified rigid Earth model (Kinoshita and Souchay, 1990) with new corrections and new contributions (Williams, 1994; Souchay and Kinoshita, 1996), we will study the effect of these developments on the calculation of the coefficients of nutation for a non rigid Earth model, based on the transfer function given by Wahr (1979). The relative improvements characterized by the residuals with the observations are explained in the following.

We begin with the development of a new analytical theory for the rotational motion of deformable celestial bodies. In this work we present a new unperturbed Hamiltonian which includes the main term affecting the Chandler period.

Once, we numerically integrated the precession and nutation of a spheroidal rigid Earth (Kubo and Fukushima 1987). As a natural extension, we tried to integrate the rotation of a triaxial rigid Earth numerically and faced a problem: a loss of precision in long-term integration. This is due to the smallness of the characteristic period of the problem: 1 day. Of course, one can integrate the rotational motion in higher precision arithmetics with a smaller stepsize. However, the quadruple precision integration is roughly 30 times more time-consuming than the double precision integration. See Table 1. Therefore, it is desirable if there is a formulation 1) reducing the overall integration error, 2) being independent on the choice of the integrator and 3) requiring no extra computations. The key points to achieve this goal will be to find a set of variables which 1) are efficiently convertible to the physical quantities required finally, say, the orientation matrix in the case of the rotational dynamics, and 2) vary with time as smoothly as possible. In this note, we report a discovery of such an example.

We consider a generalized Euler-Poinsot problem for a stationary gyrostat whose first two components of the gyrostatic momentum are null. The problem is formulated in the Serret-Andoyer canonical variables and analytically integrated by means of the Hamilton-Jacobi equation in terms of elliptic functions and integrals. The obtained solutions are just the same as those for rigid bodies if a specific constant is annulled. Finally, two applications are proposed: 1) to obtain the action-angle variables of this problem, and 2) to the problem of the rotation of the Earth, using a triaxial gyrostat as a model.

The general expression of the tidal function between two real bodies of arbitrary mass distribution has been derived. The case has been regarded when the distances between the bodies are comparable to their sizes. In this case, the effect of the body figure is substantial.

The relations obtained are utilized for the tidal influence of Jupiter on the surface of Io. It has been shown that Jupiter's shape (with J2 = 0.015) causes a deformation in the sub-Jupiter point of about 15m. This is approximately 10% of the total tidal deformation of Io caused by Jupiter's gravitation.

This paper reviews the main methods for constructing mapping models for Hamiltonian systems, for the study of motion in the Solar System. The emphasis is given to the relation between the various mapping techniques, the methods to check how close is a mapping model to the original system and the effects of an incomplete model on the evolution of the system.

Que dire de l'algèbre symbolique en mécanique céleste si ce n'est que les astronomes furent les premiers sur le chantier. Rien là qui doive surprendre.

A specialized Poisson Series Processor (PSP) is proposed. It is designed for manipulating long Poisson series. The Keplerian Processor and analytical generator of special celestial mechanics functions based on the PSP are proposed as well.

In this article are discussed classical and modern interpretations of the perturbation theory methods.

We consider a restricted planar circular three-body problem (Sun-Jupiter-asteroid) in a non-resonance case. There are two new algorithms developed for construction of a quasi-periodic solution in a trigonometric form by means of computer algebra. The first corresponds to classical method of simple iterations leading to series in powers of small mass mJ, the second, to iterations with rapid (quadratic) convergence, but having ordinary type and not involving a successive coordinate transformations. All these iterations require a realization of algebraic operations on trigonometric polynomials with the help of computers of high capacity. It would be interesting to compare the solutions obtained with the two algorithms and to estimate the domain of their practical convergence.

In a recent paper, Morbidelli and Giorgilli (1995) proved the superexponential stability of invariant tori. As usual in the theory of dynamical systems, the results are rigorously proved assuming that the perturbation is small enough. The numerical experiments show, however, that invariant tori persist up to much larger perturbation magnitudes. Therefore, it is interesting to check numerically if the superexponential stability and the other properties outlined in Morbidelli and Giorgilli's theorem persist up to the value of the perturbation for which the torus actually breaks up. Moreover, one would like to have a numerical indication about the size of the super-exponentially stable region existing around a torus. Is the superexponential stability just an asymptotic result, or does it concern a macroscopic region of physical interest?

Differential changes of time involving two parameters are considered. Universal expressions for dynamical variables of interest in Keplerian motion allow us to reduce the integration of the time transformations to that of integrands depending on an eccentric-like universal anomaly. Elliptic integrals and functions are required to complete the integration.

The problem of relativistic equations of motion for extended celestial bodies in the first post-Newtonian approximation is reviewed. It is argued that the problems dealing with kinematical aspects have been solved in a satisfactory way, but more work has to be done on the dynamical side. Concepts like angular velocity, moments of inertia, Tisserand axes etc. still have to be introduced in a rigorous manner at the 1PN level.

We consider rotational motion of an arbitrarily composed and shaped, deformable weakly self-gravitating body being a member of a system of N arbitrarily composed and shaped, deformable weakly self-gravitating bodies in the post-Newtonian approximation of general relativity. Considering importance of the notion of angular velocity of the body (Earth, pulsar) for adequate modelling of modern astronomical observations, we are aimed at introducing a post-Newtonian-accurate definition of angular velocity. Not attempting to introduce a relativistic notion of rigid body (which is well known to be ill-defined even at the first post-Newtonian approximation) we consider bodies to be deformable and introduce the post-Newtonian generalizations of the Tisserand axes and the principal axes of inertia.

Relativistic spin effects involved in the Earth-Moon dynamics are reviewed. They enclose: (i) the coordinate system effects, and (ii) the relativistic physical librations. The geodetic precession is the only relativistic spin phenomenon which has been firmly detected so far. The best candidates of the effects which might be detected in the forthcoming period are the lunar physical librations and coordinate nutations. As for the latter, however, a fine cancellation between the geodetic and the Lense-Thirring coordinate effects results in decreasing their amplitude just below the possibility of the Lunar Laser Ranging technology.

It is a fact that, none of the diverse coordinate systems unified in the Post-Newtonian formalism used to describe the exterior Schwarzschild field can be regarded as being materialized by a reference frame. Only the polar Gaussian coordinates (ρ, ϑ, ϕ, t), or their naturally associated Fermi coordinates, can be shown to have this property (Synge, 1960).

The concept of the screening of interparticle interactions has its origin in electrolyte and plasma theories. The most known example is the Debye-Hückel screening of the potential of the resting test charge provided with the availability of charges of the opposite sign and the total electroneutrality of the plasma. When this charge is moving, the static Debye screening decreases and an anisotropic dynamic screening develops due to excitation of waves of charge density. As a result, an effective potential of the positive moving ion becomes alternating with a characteristic length of space oscillations of order of the Debye length (Peter, 1990). Such dynamic screening is due to perturbations of charges of both signs by the varying field of moving test charge. This effect does not call for an electroneutrality of the system and is associated with the long-range character of the Coulomb interaction potential only.

There has been, and continues to be, a close interaction between celestial mechanics used for the generation of ephemerides, mathematical techniques, and computer technology. As the computer capabilities of the ephemerides offices and the users of ephemerides have improved, the methods of determining and the accuracies of ephemerides have changed and the medium and representation of the ephemerides provided to the user have evolved.

Ephemerides have been provided in the form of theories, tables, tabulations, polynomials, graphics, and subroutines by means of the printed page, punched cards, magnetic tape, floppy disks, CD/ROMs, and electronic mail. As mathematical techniques and computer technology continue to develop and the requirements for ephemerides evolve, the methods of representing and diffusing ephemerides will continue to improve.

Le Bureau des longitudes est chargé de rédiger la connoissance des temps, qui sera imprimée aux frais de la République, de manière que l'on puisse toujours avoir les éditions de plusieurs années à l'avance; il perfectionnera les tables astronomiques, et les méthodes des longitudes, et s'occupera de la publication des observations astronomiques et météorologiques. (Extrait du Rapport sur l'établissement du Bureau des longitudes par Grégoire, Séance du 7 Messidor, l'an 3 de la République une et indivisible; Suivi du décret de la Convention nationale, et Imprimé par son ordre. Le document, de seize pages se termine par la mention: Adopté; il est dit: A Paris, De l'Imprimerie nationale, Messidor l'An III)

Normally the planetary and satellite ephemerides are provided in tabular form, where the user interpolates between points in order to obtain the ephemerides. There are other methods of providing ephemerides by means of polynomial representations. The user is supplied with the coefficients of a set of polynomials which allow him a fast ephemerides evaluation.