The differences between the Newtonian theory and the general relativity appear both in the “relativistic effects,” such as the advance of perihelion, and in the deflection of light beams that modifies the astronomic observations and the radar reflection measurements.

Over the years there has been a continuing trend toward unification of both the annually printed volumes of ephemerides and the fundamental bases for the ephemerides. Thus, from many completely independent national and private publications, which were based on a multitude of theories, there has developed a continually improving agreement on accurate astronomical constants, planetary, lunar and satellite theories and cooperative methods of printing the annual ephemerides in different languages. This trend is continuing, currently, with the adoption of a new system of astronomical constants in 1976, consideration of revisions of nutation, the planned revision of the A.E. (American Ephemeris and Astronomical Ephemeris) for 1981, and the expected introduction in 1984 of new fundamental planetary and lunar ephemerides based on the new constants and on the FK5. Currently the differences between the printed ephemerides and observations sytematically exceed 2″ for Mars at some times and 6″. for Neptune at all times. It is anticipated that a new set of fundamental theories will be introduced which will be based on a consistent set of astronomical constants and in agreement with the available observational data to the printed accuracy.

The IAU (1976) System of Astronomical Constants and a new set of fundamental theories will expectedly be introduced into the international and national ephemerides for the volumes of 1984 onwards. In order to avoid any confusion in the future, it is necessary to manifest the character of the data published in the current volumes of the Astronomical Ephemeris = American Ephemeris, both abbreviated as A.E. With this end in view, computer programs for the calculations of the ephemerides of the Sun and inner planets based on the Newcomb's (1895, 1898) Tables have been prepared at Tokyo Astronomical Observatory (TAO) and Hydrographic Department of Japan (JHD) independently of each other using different computers and hence different types of FORTRAN. JHD has further prepared the programs for the Moon's ephemerides based on the Brown-Eckert theory and has reproduced the Eckert, Brouwer and Clemence's numerical integrations of the outer planets. Fundamental ephemerides thus calculated are compared with those data tabulated in the A.E. for the year of 1975, as an example, in the present paper.

The present basis for the ephemeris of Mars in the National Ephemerides is the theory of S. Newcomb (1898) as amended by the corrections of Ross (1917). These amendments by Ross, however, are empirical in nature and therefore the present ephemeris of Mars does not have a strictly gravitational basis. In order to provide a gravitationally consistent basis for the ephemeris of Mars, Clemence (1949, 1961) constructed a new general perturbation theory based on the final elements of Mars as derived by Newcomb for the epoch 1850. To test the adequacy and accuracy of this new theory, Clemence compared it against 87 observations from 1802-1839 and 1931-1950. This provided provisional values of the constants (without secular variation) for his new theory. These provisional elements and Clemence's theory were used to produce a heliocentric ephemeris of Mars for the period 1800-2000 (Duncombe and Clemence 1960, Duncombe 1964).

Newcomb's (1898) Tables of Mars with Ross's corrections have been used to compute its ephemeris in national ephemerides. Clemence (1949 and 1961) constructed a more precise theory of Mars and determined preliminary mean elements using a limited number of observations. Laubscher (1971) compared Clemence's theory with about 11000 observations by meridian circles from 1851 through 1969 and about 800 observations by radars and derived an improved set of mean elements. Duncombe and Kubo (1977) compared the ephemeris of Mars, in which Clemence's theory and Laubscher's elements were adopted, with the observations made by the meridian circle at Washington in 1972 and found large systematic residuals, which were larger than residuals by Newcomb's theory. Therefore, Duncombe and Kubo suspected that Laubscher's treatment of data was erroneous. On the other hand we had undertaken a reconstruction of Clemence's theory. In the meantime Kubo and Seidelman independently have found Laubscher's error in the determination of the inclination.

Within the work being carried out at Heidelberg on the establishment of the new fundamental reference coordinate system, the FK5, the determination of the location of celestial equator and the equinox form an important part. The plane of the celestial equator defined by the axis of rotation of the Earth and the plane of the ecliptic defined by the motion of the Earth about the Sun are both in motion due to various causes. The intersection of the equator and the ecliptic, the dynamical equinox, is therefore in motion. Great efforts have been made in the past to determine the location and motion of the dynamical equinox by means of observations of Sun, Moon and planets in such a manner that the dynamical equinox can serve as the origin of the right ascension system of a fundamental catalogue. The results have not been satisfactory, and we have some important evidence that the catalogue equinox of the FK4 is not identical with the “dynamical equinox”. Moreover, is has turned out that the difference α(DYN) - α(FK4) = E(T) depends on the epoch of observation T. Duncombe et al. (1974) have drawn attention to the possible confusion between the catalogue equinox and dynamical equinox; they mention the difference between two Earth longitude systems, one established by the SAO using star positions on the FK4 and the other one established by the JPL using planetary positions measured from the dynamical equinox. This is undoubtedly one legitimate explanation of the difference, even if other sources of errors may also have contributed.

In a non-inertial reference frame, of which the motion is given, the differential equations describing the rotation of a planet around its center of mass can be derived either under the form of Euler type equations, or from certain relations between angular velocity vectors or even angles. Unfortunately these equations are not well adapted to an analytical integration. Nevertheless, with some simplifications, Ward (1973), and Christensen (1977), by considering the motion of the spin axis in a non-inertial reference frame attached to the moving orbital plane, have found very large periodic variations in the obliquity of the Mars planet over a period of the order 1.2 10+5 years, which would be of great importance to the climatic history of the planet.

The development of automatic, computer-controlled, electronic equipment, under the direction of an intelligent user, has significantly affected the output of scientific knowledge from astronomical research. This increased output is as much a result of the easing and/or elimination of heretofore tedious tasks as it is a result of vast increases in data acquisition rates via observing systems which make more efficient use of in-coming electromagnetic radiation. One such automated system (Shelus et al, 1977) has been implemented by the present authors to predict occultations of stars by Solar System objects. In the past these predictions have been a time consuming task which was made even more onerous since only a very few observable phenomena were found. Perhaps of even more importance is that, since candidate objects to be occulted typically were obtained from star catalogs, such a search was incomplete. Note that phenomena involving stars fainter than normal catalog limits are certainly relevant in the cases of Uranus, Neptune, Pluto, minor planets and the natural satellites of the major planets.

Most of the natural satellites of the planets of the solar system may be put into one of three main groups, according as to which of three main influences dominate the perturbation of their motion from Keplerian motion about the primary planet. The first of these is the attraction of the Sun, which governs the perturbations of the Moon's motion about the Earth, and those of the outer satellites of Jupiter (satellites VI to XIII), and Saturn's satellite Phoebe. The second is the departure of the gravitational field of the planet from that of a spherically symmetric body (the “figure terms”), and this governs the perturbations of the two satellites of Mars, Jupiter's satellite Amalthea (V), Neptune's satellite Triton, is probably the most important influence on Uranus' satellites, and is important, though not dominant, for the inner satellites of Saturn. The third influence is the mutual attraction of the satellites themselves. An order of magnitude argument suggests that periodic perturbations from this cause could scarcely be expected to be measureable from Earth, were it not that the frequent appearance of small-integer near-commensurabilities of pairs of orbital periods, and the consequent argumentation of the associated perturbations by a variety of types of resonance effects, in the systems of Jupiter and Saturn, causes mutual perturbations to dominate the orbital theories of three of the four great satellites of Jupiter, and six of the nine satellites of Saturn, and enables the masses of most of the satellites involved to be determined with otherwise unexpected relative precision (in some favourable cases, of the order of one per-cent) from Earth based data. Let us now consider the satellite systems of each of the outer planets in a little more detail.

For the improvement of the orbital elements of Hyperion, a program of photographic observations of Saturnian satellites has been made since the opposition of 1966. In this paper are presented the reduction method and results derived from the observations during the oppositions of 1968 and 1977.

Upcoming missions to the outer planets have made evident the need for better theories of their satellites. We have been involved in an effort to provide better satellite theories through new observations (Abbot, Mulholland and Shelus 1975; Mulholland, Shelus and Abbot 1976; Mulholland and Shelus 1977; and Benedict, Shelus and Mulholland 1978), and new analytical theories (Jefferys and Ries 1975; Jefferys 1976. Hereinafter these are denoted Paper I and Paper II, respectively). In this paper we report on our incorporation of new and old observations into our theories, and on our progress on the theoretical front.

As spacecraft and sophisticated ground-based observations measure physical properties of many planets and satellites, dynamical theory and astrometry remain a principal source of such knowledge of the Uranian system. Study of the motions of Uranus' satellites thus has broad application to planetary studies as well as to celestial mechanics. Moreover, the structure and dynamics of the system provide important cosmogonical constraints; any theory of solar system origin and evolution must account for the formation within it of analogous systems of regular satellites.

The capture of satellites is considered to be a slow process, in which particles approaching the planet lose their energy gradually and fall down into the stable orbits around it. Therefore, in order to investigate the capture problem, it seems to be necessary to clarify the behaviors of orbits near the planet. Recently Hayashi et al. (1977) and Heppenheimer and Porco (1977) found that in the restricted three body problem, particles incident through Lagrange points L1 and L2 revolve always directly around Jupiter as long as the potential windows at L1 and L2 are small. This suggests the impossibility of the capture of retrograde satellites when the potential windows are small.

This research is part of a wider one (Carusi & Pozzi, 1978a, b) concerning a detailed study of the dynamics of close encounters between a giant planet and a minor object. A special result of that investigation was the recognization of some satellite-capture events, already found by Everhart (1973). An important remark about this previous work is that all satellite-captures occurred with low inclination objects which orbits were initially near-tangent to the Jupiter's one. Starting from this consideration, a hundred fictitious orbits have been generated in order to study the phenomenon in greater detail. Their initial distribution is shown in fig. 1. The initial angular parameters i, ω, Ω, were chosen to be equal to those of the most interesting case of the previous research. Eccentricities were selected regularly in the range .01-.5, with a step of .01, giving the same value to an object of the upper band and to the next of the lower. The semimajor axes were chosen at random between limits computed so that the aphelion for the lower band, or the perihelion for the upper band, would lie within a distance of 108km from Jupiter's orbit. As the orbital planes do not coincide with that of Jupiter, the minimum distance point between the two orbits does never coincide with the object perihelion or aphelion, but is always close to them. Fig. 2 shows the final situation of this population: we note that in no case a permanent binding occurred. We can do some remarks on this picture. First of all we note that 56% of objects experienced a temporary binding to the planet. Secondly, 67% of objects, bounded or not, had a final orbit lying, on the a-e diagram, on the opposite band with respect to their initial one. This kind of transition is especially significant if compared with the case of observed comets, because it gives a simple mechanism to transform long-period comets in short-period ones, and to transfer comets from one family to another. Actually, on the basis of the computations of Kazimirchak-Polonskaya (1972), we can say that a similar process has been experienced, for example, by comets Whipple, Oterma, Brooks 2, Lexell, Kearns-Kwee and others. A third remarkis that, between the initial and final situations, small eccentricities are increased, as a consequence of the encounter, whilst the great ones are decreased. This phenomenon leads to a clustering of final orbits in the eccentricity range .1-.2.

Nature has presented us with two systems of planetary rings. They are not at all similar. Saturn has bright, broad rings separated by narrow gaps. The rings of Uranus are dark, narrow and widely spaced. Presumably both sets of rings lie inside the Roche limit and this is why the ring material has not condensed into satellites.

During this symposium on the dynamics of the solar system, we have mainly studied the movements of the bodies of the solar system submitted to gravitational perturbations. The next step is to take into account the physical collisions. Indeed, there can be little doubt that collisions between “macroscopic bodies” are of frequent occurence in the Universe. All kinds of quite different objects undergo such collisions: these may range from large interstellar clouds to small solid bodies in the solar system. Collisions have surely played an important role in the formation of planets and satellites and continue to play a central role in the behaviour of the planetary discs. For example for Saturn's rings, one can see intuitively that until the optical depth drops much below unity, the rings are still evolving. Each orbiting particle can be taken as occupying a kind of torus, and collisions will continue until there is only one particle in each such “orbital tube”; this corresponds to a very small optical depth.

It is shown that the hypothesis that comets originate in the vicinity of each of the giant planets is consistent with the idea that these planets possess rings. It is suggested that the rings of Uranus, the probable gaseous ring around Jupiter and the clouds surrounding Io are rapidly evolving phenomena - the same may also even be true of Saturn's rings - which is to be expected of agglomerations of cometary meteoroids losing their icy constituents by sublimation.

This paper consists of a review about work done on three-dimensional motion at commensurabilities of either the mean motions, or of secular periods, and of a report on the author's recent results on some special cases. Real and fictitious asteroidal orbits and the corresponding long-period effects are the main subject of interest. At first, methods are listed.

The frequency distribution of the asteroidal mean motions shows the well known Kirkwood gaps which are found near the commensurabilities 3/1, 5/2, 7/3, and 2/1. The number of asteroids in the gaps is significantly smaller than those in the corresponding regions around these commensurabilities. In the outer part of the asteroidal belt, the frequency distribution is reversed: The number of asteroids at the commensurabilities 3/2, 4/3, and 1/1 is much larger than in the region around these commensurabilities.

It is well known that the semi-major axes of the asteroids between the orbits of Mars and Jupiter are not uniformly distributed. The depleted regions in the outer part and the Kirkwood gaps in the inner part of the belt represent singularities in the frequency distribution of the asteroidal semi-major axes. The question whether these depleted regions and the Kirkwood gaps are due to gravitational perturbations of Jupiter or due to cosmogonic effects has not yet been solved and the discussion is still continuing.