Consideration is given to the solution by numerical integration of systems of differential equations that are derived from a Hamiltonian function in the extended phase space plus additional forces not included in the Hamiltonian (that is, nearly-Hamiltonian systems). An extended phase space Hamiltonian which vanishes initially will vanish on any solution of the system differential equations. Furthermore, it vanishes in spite of the additional forces, and defines a surface in the extended phase space upon which the solution is constrained.

The exterior gravitational field depending on the Earth’s non-sphericity is usually determined from the analysis of satellite data or by the solution of the exterior boundary value problem. In the latter case some integral equations are solved which correlate the exterior potential with the known vector of gravity and the shape of the Earth’s surface (molodensky problem). In order to carry out the integration the small parameter method is applied. As a result, all the quantities which involve the equations should be expanded in powers of a certain small parameter, among these being the heights of the Earth’s surface points as well as the inclination α of the Earth’s physical surface. Since the angle α can be significant, especially in mountains, and in fact does not depend on any small parameter then the solution of integral equations is possible only for the Earth’s surface which is smoothed enough.

In a development of the reciprocal distance (1/Δ) between two planets, their orbital inclinations, in a sense, improve the convergence, though the development is carried out in powers of the inclinations. This is most clearly shown in the Neptune-Pluto case: if their inclinations, 2° and 17°, are assumed to be 0°, the two orbits cross each other. The development in the actual case ought to be easier than in the assumed case with vanishing inclinations.

In this paper the motion of an orbiter of a satellite of one of the major planets is considered. The orbiter undergoes various perturbing effects. It is shown that the semi-analytical stroboscopic method is well suited to take into account all perturbations.

A brief discussion is given of the basic features of de Sitter’s theory. The main advantage of his theory is that it contains no small divisors, thanks to the use of elliptic rather than circular intermediate orbits in the first approximation. A 50-year extension of the satellite observations available to de Sitter makes it desirable to rederive the elements of his intermediate orbits, whose perijoves have a common retrograde motion. Furthermore, the theory suffers from a convergence problem, which can be avoided by reformulating the theory in terms of canonical variables, a task that is begun here. We adopt a formulation in Poincaré’s canonical relative coordinates rather than, as customary, in ordinary relative coordinates or in the Jacobian canonical coordinates. By means of the generalized Newcomb operators devised by Izsak, the disturbing function is expanded in a form that is very convenient for use with the modified Delaunay variables, G, L – G, H – G, l + ω + Ω, l, and ω and their associated Poincaré variables.

The final results for the theory enabling one to calculate the positions of the Galilean satellites and their partial derivatives are presented, following the techniques outlined in earlier papers. Extensive use of algebraic manipulation software on a digital computer is employed to generate the final expressions. The new theory is, in effect, a revitalization of Sampson’s theory in which we (a) remove algebraic and mathematical errors existing in Sampson’s work, (b) introduce some neglected effects due to solar interactions and the 3-7 commensurability, (c) allow for non-zero amplitude and phase of the free libration, (d) express the final results as analytic functions of variations in 49 arbitrary constants of integration and physical parameters, (e) construct the theory in a manner which readily allows for future revision, and (f) provide analytic expressions for the partial derivatives with respect to the 49 parameters.

The theory of the motion of the Galilean satellites of Jupiter is developed up to the second-order terms. The disturbing forces are those due to mutual attractions, to the non-symmetrical internal mass distribution of Jupiter and to the attraction from the Sun. The mean equator of Jupiter is taken as the reference plane and its motion is considered. The integration of the equations is performed. The geometric equations are solved for the case in which the amplitude of libration is zero. The perturbation method is shortly commented on the grounds of some recent advances in non-linear mechanics.

In a previous paper (Ferraz-Mello, 1974) one perturbation theory has been constructed with special regard to the problem of the motion of the Galilean satellites of Jupiter. In this problem, the motions are nearly circular and coplanar; on the other hand the quasi-resonances lead to strong perturbations. The main characteristic of the theory is that it allows the main frequencies to be kept fixed from the earlier stages, and so, to have a purely trigonometric solution.

Solar perturbations in Saturnian satellite motions are computed with accuracy of 10-5 to try to analyze observed data of orbital elements for Titan. Perturbations due to Iapetus in Titan’s orbit are also developed by taking into account the motion of the orbital plane of Iapetus. Then the mass of Iapetus is determined by the motion of the orbital plane of Titan. Also oblateness parameters of Saturn and the mass of Rhea are determined by seven secular motions for six satellites. It is also found in the analysis that G. Struve’s values for the semi-major axes adopted in almanacs differ from the computed values by using the new data.

A programme of photographic observations of the satellites of Saturn, and the method of reduction and analysis are described. The observations have been used to update existing theories of Tethys, Dione, Rhea and Titan. A new theory of the motion of lapetus has been fitted to the observations.

A program is underway at the McDonald Observatory to extend the series of photographic observations of the satellites of the outer planets (Abbot, Mulholland and Shelus, A.J. 80, 1975), and concurrent theoretical studies have led to a new orbital theory for the resonant pair of satellites, Enceladus and Dione (Jefferys and Ries, A.J. 80, 1975). The construction of the new theory, using the computer software system TRIGMAN, has provided Fortran subroutines for the computation of the planetocentric coordinates of the two satellites, as well as partial derivatives for the orbit elements and certain other physical parameters of the orbit problem, including some of the harmonics of the gravitational field of Saturn. The available photographic observations for these two objects are currently being discussed with the new theory, and improved values of the orbital parameters are expected in the near future.

A method for generating long periodic variations in satellite orbital elements when perturbed by discrete gravity anomalies is presented. The method consists of developing a disturbing potential as a function of orbital and gravity anomaly parameters, and generating partial derivatives of the potential with respect to the orbital elements. The partials are averaged over the period of the satellite to eliminate the short periodic variations. The averaged partials are substituted into the variation of parameter equations to give the mean orbital rates. Classically orbital elements are used in generating gravity field and thus the method is dynamic in nature. The problem is extremely cumbersome and complex when multi-state parameters have to be estimated from a considerably large data set. However, when mean orbital rates are used, the problem reduces to a simple linear static case, where only the gravity parameters have to be estimated, and it is a simple matrix inversion problem. Thus the method developed here was utilized in reducing Appolo 15 and 16 subsatellite radio tracking data to produce a lunar gravity field represented by point masses.

A third-order solution is developed for the motions of artificial satellites moving in the gravitational field of the Earth, whose potential includes the second-, third-, and fourth-order zonal harmonics. Third-order periodic perturbations with fourth-order secular perturbations are derived by Hori’s perturbations method. All quantities are expanded into power series of the eccentricity, but the solution is obtained so as to be closed with respect to the inclination. A comparison with the results of numerical integration of the equations of motion indicates that the solution can predict the position of a close-earth satellite with a small eccentricity with an accuracy of better than 1 cm over 1 month.

We consider the determination of the potential from the orbit of a satellite. The potential obtained is not unique because the satellite might describe the same orbit with different forces. The classical problem of determination of orbits has a unique solution, for given initial conditions. The problem is reduced to the planar case with a conservative force. The mathematical results depend on the arbitrary functions of the system of coordinates used.

A new approach to the study of the Solar System and planetary systems in general is proposed, through the use of periodic planetary-type orbits of the general N-body problem. In such an orbit, one body (called Sun) has a large mass and the rest N-l bodies (called planets) have small but not negligible masses and it can be proved that monoparametric families of periodic orbits of the N-body problem exist in a rotating frame of reference, all being of the planetary type.

Two cases are studied in detail, N=3 and N=4. In N=3, apart from a general discussion, we present a detailed analysis of the Sun-Jupiter-Saturn system and a study is made on which configurations with the masses of these two planets, or a multiple of them, are stable or unstable. Also, part of a family is shown to represent the Jupiter family of comets. It was found that commensurabilities are not in general associated with instabilities. For N=4 we present three families of periodic orbits. The motion corresponding to a branch of one of the above families has many similarities with the actual motion of the three inner satellites of Jupiter.

It is shown that there exist many commensurable cases in the obtained periodic orbits and that the resonant orbits increase as the number of bodies increases. Based on these results, an attempt is made to explain the existence of commensurabilities in the Solar System.

Finally, it is mentioned that a periodic motion of the planetary type can be used as a reference orbit for accurate computations for the actual motions of the planets or satellites of the Solar System. In this way the small divisor difficulties existing in the classical approach will not appear.

Based upon the authors’ result (Astron. Astrophys. 43, 331-3, 1975) that oblateness of the more massive primary decreases the value of the critical mass, this note is concerned primarily with a few observations on Szebehely’s result (Astron. J. 72, 7-9, 1967) that the Coriolis force is a stabilizing force. It has been deduced that a small change ε in the Coriolis force from unity results in a perturbation of the critical mass by a factor of -8ε/81.

In the frame of the gravitational restricted three-body problem, we study by numerical simulation the retrograde satellites -and their stability-at large distance. In the circular plane case, the stable satellites mostly surround (in the phase-space X°-V°) the characteristic of a family of single-periodic orbits where this family is stable, and librates (in the physical space) around a curve which corresponds to the nearest (in the phase-space X°-V°) periodic orbit. An analytical analysis of this libration is made for Hill’s case. The beginning of the study of the three—dimensional orbits is presented.

In the restricted problem of three point masses, the positions of the equilibrium points are well known and are tabulated. When the satellite is a rigid body, these values no longer correspond to the equilibrium points. This paper seeks to determine the magnitudes of the discrepancies.

A great deal of human and computer effort has been directed in recent decades to the determination of the periodic orbits of the restricted three-body problem and the study of their properties for well known reasons of significance and feasibility.

Previous work on the plane circular restricted problem of three bodies (Message 1953, 1959, 1970, and Fragakis 1973) has shown the existence, in association with each of the commensurabilities 2:1 and 3:1 of the orbital periods, of a pair of families of asymmetric periodic solutions, branching from the stable series of symmetric periodic solutions of Poincaré’s second sort associated with that commensurability. (Each solution of either family is the mirror image, in the line of the two finite bodies, of a member of the other family of solutions associated with the commensurability.) The stability is transferred at the bifurcation to the two series of asymmetric orbits, each of which is therefore stable. Recent numerical integrations carried out by one of us (P.J.M.) have found such asymmetric periodic orbits associated also with the 4:1 commensurability, and quantities describing orbits of one of the two series are given in Table 1, showing the run of such orbits up to a second bifurcation with the same series of symmetric periodic orbits from which it sprang. Quantities describing some members of this series of symmetric orbits are given in Table 2. It is seen that stability is transferred back to the symmetric series at the second bifurcation. (The unit of distance is the distance between the two finite bodies, the unit of speed is the speed, of their relative motion, and the initial conditions given (x°, ẋ°, ẏ°) are for a crossing of the line of the two finite bodies, this line being taken as axis of “x” in a rotating Cartesian frame in the usual way. The mean values of the major semi-axis and eccentricity are denoted by ā and ē, respectively, C is Jacobi’s constant, and ȳ2 is the mean value of the critical argument ȳ2 = 4λ – λ′ – 3ω. The mass ratio used is 0.000954927, T is the period of the solution in units of the period of the motion of the two finite bodies, and 2π c/T is the non-zero characteristic exponent.)

Les équations du mouvement du troisième corps, dans les variables sans dimensions définies par Szebehely (1967, page 591) ont été données dans un article précédent (Sergysels-Lamy, 1975).