On resonance in celestial mechanics

This brief survey of the author's contribution to the theory of resonance in celestial mechanics begins with the genesis of the Small Divisor. The fundamental distinction between theshallow anddeep resonance is illustrated by the 52 Jupiter-Saturn and the 3-2 Neptune-Pluto resonances in the planetary system.The search for aglobal solution through a removal of the small divisor is put into a historical perspective through the work of Laplace, Bohlin, and Poincaré. The author's own contribution to the methodology is the formulation and the solution of the Ideal Resonance Problem. If the resonance issimple, all the singularities in the solution are removed by means of aregularizing function. On the other hand, if the resonance isdouble, the second critical divisor seems irremovable, and a global solution may be precluded.Invited paper, IAU 1979, Commission 7, Montreal, Canada.     

Comments on the application of power series solutions to problems in celestial mechanics

The author relates his experiences in utilizing the power series method to generate trajectories for orbital and sub-orbital vehicles and for then-body problem.     

PARSEC: An interactive Poisson series processor for personal computing systems

PARSEC is a PC-based, interactive algebraic manipulation package designed to manipulate series of the kind frequently found in Celestial Mechanics applications and perturbation procedures. The system is fundamentally an input/command interpreter which allows the user to enter algebraic expressions and procedures and to control the flow of series manipulations interactively. The system is designed to allow easy manipulations of polynomial-trigonometric series within the environment of an electronic scratchpad.     

Kepler discretization in regular celestial mechanics

In this paper, a special extrapolation method for the numerical integration of perturbed Kepler problems (given in KS-formulation) is worked out and analyzed in detail. The underlying so-called Kepler discretization isexact for the pure (elliptic) Kepler motion. A numerically stable realization is presented together with a backward error analysis: this analysis shows that the effect of the arising rounding errors can be regarded as a small perturbation inferior to the physical perturbation. For test purposes, a well-known example describing the motion of an artificial Earth satellite in an equator plane subject to the oblateness perturbation is used to demonstrate the efficiency of the new extrapolation method.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

Variable constant of gravitation and celestial mechanics

In the present paper the celestial mechanics consequences (light deflection, radar ranging of the planet, geodetic precession and secular effects in the orbital elements in the two-body problem) for the class of the theories based on the vacuum Jordan's Lagrangian has been considered. In these theories the gravitational constantG is proportional to ^–, being a scalar field and , some dimensionless parameter and the local law of conservation of the energy-momentum tensor holds. Of all these theories with different the most interesting one is that corresponding to =0. In the postnewtonian approximation this gravitational theory is completely equivalent to the general theory of relativity.     

Astronomical measurements and coordinate conditions in relativistic celestial mechanics

Determination of dynamical effects from the equations of motion and calculation of ephemerides in terms of measurable quantities on the basis of the equations of light should be performed in one and the same coordinate system. The choice of coordinate system is arbitrary. For illustration we consider coplanar circular motions of the Earth and one of the inner planets in the solar gravitational field described by the generalized three-parametric Schwarzschild metric. Specific values of the metric parameters characterize the adopted gravitational theory, as well as a definite coordinate system (for example, isotropic or standard coordinates). Coordinates of the planets and radii of the orbits are coordinate-dependent quantities and cannot be directly reconciled with measurable quantities such as the round-trip transit times of radar signals or the angular distance between the planet and the distant fixed source (quasar). These ephemeris data may be calculated in terms of the initial measured values independently of the employed coordinate system. Relativistic ephemeris corrections should be taken into account both in radar reflection measurements and astrometric observations.     

Boundary manifolds for energy surfaces in celestial mechanics

We complete Mc Gehee's picture of introducing a boundary (total collision) manifold to each energy surface. This is done by constructing the missing components of its boundary as other submanifolds. representing now the asymptotic behavior at infinity.It is necessary to treat each caseh=0,h>0 orh<0 separately. In the first case, we repeat the known result that the behavior at total escape is the same as in total collision. In particular, we explain why the situation is radically different in theh>0 case compared with the zero energy case. In the caseh<0 we have many infinity manifold components. and the general situation is not quite well understood.Finally, our results forh0 are shown to be valid for general homogeneous potentials.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.The research conducted in this paper has been partially supported by CONACYT (México), under grant PCCBNAL 790178.Partially supported by an Ajut a l'Investigacio of the University of Barcelona.     

Poincaré's hydrodynamic analogy in celestial mechanics

The analogy between the differential equations describing dynamical systems of two degrees of freedom on one hand and two and three dimensional flow on the other hand is investigated in some detail.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Symplectic integrators for long-term integrations in celestial mechanics

We discuss the use of symplectic integration algorithms in long-term integrations in the field of celestial mechanics. The methods' advantages and disadvantages (with respect to more common integration methods) are discussed. The numerical performance of the algorithms is evaluated using the 2-body and circular restricted 3-body problems. Symplectic integration methods have the advantages of linear phase error growth in the 2-body problem (unlike most other methods), good conservation of the integrals of the motion, good performance for moderately eccentric orbits, and ease of use. Its disadvantages include a relatively large number of force evaluations and an inability to continuously vary the step size.     

On the use of STF-tensors in celestial mechanics

The purpose of this article is to emphasize the usefulness of STF-tensors in celestial mechanics. Using STF-mass multipole moments and Cartesian coordinates the derivations of equations of motion, the interaction- and tidal-potentials for an isolated system ofN arbitrarily shaped and composed, purely gravitationally interacting bodies are particularly simple. Using simple relations between STF-tensors and spherical harmonics it is shown how all Cartesian formulas can be converted easily into the usual spherical representations. Some computational aspects of STF-tensors and spherical harmonics are discussed. A list of useful formulas for STF-tensors is provided.     

A survey of Poisson series processors

The sheer magnitude of the work involved in the construction of perturbation theories (the astronomical computations) made it inevitable that astronomers would very early become interested in the possibility of constructing them with the aid of computers. Since then general systems for symbolic manipulations have been developed and used widely. Nevertheless there still remain problems for which these general systems are not well adapted and many specialized systems of algebraic manipulation (mostly Poisson series processors) are in use. An attempt is made to review this field by sketching some of the ideas on which these Poisson series processors are built.     

Satellites of minor planets: A new frontier for celestial mechanics

Most minor planets have satellites. This observation is predicted to have a major impact on the theory of minor planets, comets, and meteors.Presented at the Symposium Star Catalogues, Positional Astronomy and Celestial Mechanics, held in honor of Paul Herget at the U.S. Naval Observatory, Washington, November 30, 1978.     

Application of Lie-series to regularized problems in celestial mechanics

In the following paper we tried to apply the Lie-formalism to the regularized restricted three body problem. It will be shown that this algorithm leads to a very simple structure program which is also fast.     

Elliptical orbit with non-central force field in celestial mechanics

Equations describing planar motion along a constant elliptical orbit in non-central force field are derived in a simple way and discussed. The angular moment along the elliptical orbit is not a constant of motion. The problem appears to be related to the problem of planar motion with drag force. Relations with related works are discussed. In presence of drag force, conditions for the force field to be derived from the gradient of a function are discussed, and -elation of this function with the potential function is indicated.     

Normal CO-ordinates and asymptotic expansions in resonance cases in celestial mechanics

In the presence of a single small-integer near commensurability of orbital period, the construction of a complete formal solution of the equations for the mutual perturbations in a planetary or satellite system, entirely in periodic terms, can be carried out after the use of a transformation of the variables which brings the quadratic terms of the Hamiltonian to a suitable normal form. A method for finding such a transformation is described.     

The roche coordinates and their use in hydrodynamics or celestial mechanics

The aim of the present paper will be to introduce a new system of curvilinear coordinateshereafter referred to as Roche coordinates-in which spheres of constant radius are replaced by equipotential surfaces of a rotating gravitational dipole (which consists of two discrete points of finite mass, revolving around their common center of gravity); while the remaining coordinates are orthogonal to the equipotentials. It will be shown that the use of such coordinates offers a new method of approach to the solution of certain problems of particle dynamics (such as, for instance, the construction of certain types of trajectories in the restricted problem of three bodies); as well as of the hydrodynamics of gas streams in close binary systems, in which the equipotential surfaces of their components distorted by axial rotation and mutual tidal interaction constitute essential boundary conditions.Following a general outline of the problem in Section 1, the Roche coordinates associated with the equipotentials of a rotating gravitational dipole will be constructed in the plane case (Section 2), and their geometrical properties discussed. In Section 3, we shall transform the fundamental equations of hydrodynamics to their forms appropriate in the curvilinear Roche coordinates. The metric coefficients of this transformation will be formulated in a closed form in Section 4 in terms of the respective partial derivatives of the potential; while in Section 5 analytic expressions for the Roche coordinates will be given in the orbital plane of the dipole, which are exact as far as the distortion of the equipotential curves from circular form can be described by the second, third and, fourth harmonics.The concluding Section 6 will be devoted to a formulation of the equations of a mass-point in the restricted problem of three bodies in the Roche coordinates. Three special cases will be considered: (a) motion in the neighborhood of the equipotential curves; (b) motion in the direction normal to such curves; and (c) motion in the neighbourhood of the Lagrangian points. It will be shown that motion in one coordinate is possible only in limiting cases which will be enumerated; but twodimensional motions in which one velocity component is very much smaller than the other invite further study.A generalization of the plane Roche coordinates to three dimensions, with application to additional classes of problems, is being postponed for a subsequent paper.