Review of the present problems of relativistic celestial mechanics. Advantage is taken of the method suggested earlier by the author and based on using quasi-Galilean coordinates with arbitrary coordinate functions or parameters. As compared with the previous papers the new elements are post-post-Newtonian approximation for the circular motion in the Schwarzschild problem and reduction of the artificial satellite problem including the main solar perturbations to the Schwarzschild problem. Some current questions of time scales definitions, reference frames and reduction of observations are briefly discussed.

We have derived in an explicit form the equations of motion for two spherically-symmetric non rotating bodies in the slow motion approximation. The equations include relativistic corrections of order (v/c)2, (v/c)4 and (v/c)5 to the newtonian equations of motion. It is shown that the equations depend on the only parameter characterizing each body, namely on its relativistic mass, regardless of its internal structure and degree of compactness. This means that the equations can also be applied to bodies with a strong internal gravity, such as neutron stars and black holes. It is shown that in the (v/c)2 and (v/c)4 approximations the equations can be derived from a Lagrangian. The Lagrangian is given in an exact form. The integration of the equations of motion is performed by the method of osculating elements. The formulae for secular change of the semi-major axis and eccentricity coincide precisely with the standard ones whose derivation is based on a calculation of the energy flux in the outgoing gravitational waves.

The definition of the angular momentum of a finite body is given in the post-Newtonian framework. The non-rotating and the rigidly rotating proper reference frame(PRF)s attached to the body are introduced as the basic coordinate systems. The rigid body in the post-Newtonian framework is defined as the body resting in a rigidly rotating PRF of the body. The feasibility of this rigidity is assured by assuming suitable functional forms of the density and the stress tensor of the body. The evaluation of the time variation of the angular momentum in the above two coordinate systems leads to the post-Newtonian Euler's equation of motion of a rigid body. The distinctive feature of this equation is that both the moment of inertia and the torque are functions of the angular velocity and the angular acceleration. The obtained equation is solved for a homogeneous spheroid suffering no torque. The post-Newtonian correction to the Newtonian free precession is a linear combination of the second, fourth and sixth harmonics of the precessional frequency. The relative magnitude of the correction is so small as of order of 10−23 in the case of the Earth.

Solutions of the geodesic equations for bound test particle motion in a Schwarzschild field are expressed using Jacobian elliptic functions. Keplerian orbital elements are identified and related to a set of canonical constants. Relativistic Lagrange planetary perturbation equations are derived.

For a simplified 3-body (Earth, Moon, Sun) problem it is shown how the usual Einstein-Infeld-Hoffmann equations for the lunar motion reduce to the Jacobi-equations after the transformation to the proper reference frame. The dominant relativistic contributions to the lunar laser ranging observables are then obtained in a Hill-Brown calculation. It is argued that in the proper reference frame all post-Newtonian variational terms are proportional to m = n′/(n-n′) [n(n′) = mean motion of Moon (Sun)].

A new software to handle the large expressions involved in high accuracy analytical theories has been developped by H. Claes, M. Moons, J.M. Zune and J. Henrard. The main ideas on which this software is based are described here.

The theory of the motion of Mercury, Venus, Earth and Mars is constructed by numerical integration. The theory takes into account relativistic corrections in the frame of Schwarzschild's space-time metrics. The constants of the theory are determined by discussion of the Soviet and American radar observations of Mercury, Venus and Mars, position astrometric observations of these planets (and the Sun) and observations of the Soviet artificial satellites of Venus. Apart from the planet elements the value AU and corrections to the adopted radii of Mercury, Venus and Mars are determined. Statistics of residuals is given and accuracy of the theory is estimated. The theory presented is an ephemeris base for deep space experiments.

Up to now we have been dealing with the construction of entirely analytical planetary theories such as VS0P82 (Bretagnon, 1982) and T0P82 (Simon, 1983). These theories take into account the whole of the newtonian perturbations of nine point masses: the Sun, the Earth-Moon barycenter, the planets Mercury, Venus, Mars, Jupiter, Saturn, Uranus and Neptune. They also take into account perturbations due to some minor planets, to the action of the Moon and the relativistic effects. The perturbations of these last three types are in a very simple way under analytical form but they considerably increase the computations when introduced in the numerical integration programs.

The Ephemeris Development Program has been in existence for nearly 20 years at JPL, providing high precision present-day knowledge of the positions of the moon and major planets. The resultant ephemerides are used extensively in the navigation of spacecraft and in the reduction of astrometric observations. They also provide a key element in the testing of various theories of gravitation and a means for the determination of various relevant astronomical constants. The ephemerides and the process of creating them are both shown to be viable tools for the measurement of various gravitational effects which govern the motions of the objects in the solar system.

This paper gives an outline of the least-squares adjustment of the ephemerides to the observations, the present physical (dynamical) model, the present observations to which the ephemerides are fit, the expected accuracies of various ephemeris elements, recent and future observations and features of the solar system which are poorly determined (and thereby place limits upon the accuracies). Recent comparisons with similar work at the Center for Astrophysics (formerly at MIT) are serving as valuable independent checks on formulations and procedures used at each institution; they also lend insight toward what are the realistic accuracies being attained. The export procedure, by which an outside user may obtain and use the JPL ephemerides, is described.

The relativistic and lunar perturbations must be included in a realistic theory of the secular evolution of the planetary elements. In our general theory, we include the first order of these perturbations. Comparison with more elaborated studies shows that it is sufficient with respect to the accuracy of our theory.

The theory of the motion of Halley comet has been built using ground optical positional observations of 1682, 1759, 1835, 1910 and 1982–84. The orbital elements of the comet and two coefficients characterizing non-gravitational perturbations were determined precisely as a result of the statistical treatment of the observations. The estimations of the accuracy of the theory and the data describing its agreement with the observations are presented. The theory has been applied to ballistic-navigational calculations for the Soviet Vega project.

This paper is a summary of the referenced complete paper which includes figures and references.

Long term numerical integrations of planetary orbits designed to study the stability of the Solar System over timescales comparable to its age have become very promising thanks to the availability of very powerful computers and to a substantial improvement in our methods of investigating the stability of hierarchical dynamical systems. The stability of such numerical integrations relies on the ability to control all possible sources of error. Among the errors caused by the inadequacy of the physical model are those due to the fact that Newton's theory of gravitation is used instead of general relativity. We show that the secular advance of perihelia predicted by general relativity can be simulated exactly by a 1/r2 perturbing potential with almost negligible additional cost in computer time.

Predictions of tracks of occultations of stars by minor planets give a strong test of the accuracy of observations, star places, and dynamical theories. A critical discussion of the 1983 September 11 occultation of 14 Piscium by (51) Nemausa shows that the systematic regional errors of fundamental stars introduce errors in the long-term predictions of occultations that are larger than relativistic effects.

Recent measurements in the infra-red of the energy emitted by Io suggest that if the observed emission is due to tidal dissipation in Io, then one should observe a very large secular acceleration of Io in its orbit. The purpose of this paper is to investigate the observational evidence which might be examined to verify that hypothesis. We have assembled the largest collection of eclipse observations extant and investigate the possibility of observing secular drifts in the period of Io. Analysis of eclipse observations requires the knowledge of good tables of ΔT and we explore the implications of the standard lunar acceleration of Spencer-Jones ṅMoon = −22.44 arcsec/cy2 versus that of Morrison and Ward ṅMoon = −26.0 arcsec/cy2. Preliminary results suggest that large tidal accelerations of Jupiter's satellites do not exist at the suggested magnitude and hence the heat flux measurements probably represent an episodic occurrence.

The two manners to improve the knowledge of the motions of the galilean satellites are the elaboration of a new theory and the use of high precision observations. In this paper, we present new results on these subjects.

We have applied the Sagnier's method in order to get a new theory of the motions and we describe here the second approximation which leads to a third order solution. In this solution appear coupled terms in longitude of the perijoves and longitudes of the nodes. The de Haerdtl's inequalities, which come from the 3-7 commensurability between satellites 3 and 4 are included in this solution. These developments, first computed by de Heardtl, have been improved by Lieske (1973). In our solution we introduce the use of the variables in inclination, and some new terms appear.

In order to get the best precision for the final ephemerides, the accuracy of the observations have to be improved: best results may be obtained with the use of the observations of mutual events. Simulations have shown the interest to take into account the variations of the albedo as a function of geographic longitudes and latitudes on the satellites themselves. So, the theoretical flux of light is closer to the observed one and the accuracy is improved. In 1985, mutual events occur and we have organized a campaign in order to obtain results of high precision.