An application of the scaled Runge-Kutta algorithms to some problems of celestial mechanics

New Runge-Kutta algorithms are applied for determining the solution of a system of ordinary differential equations at any point within a given integrating step. In this paper we propose the application of these new algorithms in order to determine, with the smallest possible cost, the exact point of intersection of a symmetric orbit, with the axis or plane of symmetry, which appear in various problems of celestial mechanics.     

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.     

Applications of computers to celestial mechanics

The development of digital computers induced major new developments in Celestial Mechanics. At present, one can hardly mention a project in Celestial Mechanics that does not use computers as the principal tool. One can distinguish many different manners of using computers in Celestial Mechanics. Among them, the following are presented and typical examples are given: algebraic manipulations for literal and semi-numerical theories, numerical integration of the equations of motion, determination of physical parameters, numerical checks of analytical results, studies of families of solutions, search for new conjectures, scanning the phase space. In all these approaches, and in others omitted here, major scientific achievements were obtained in the last 20 years and new problems can now be envisaged that were unaccessible even a few years ago.     

A FORTRAN-based Poisson series processor and its applications in celestial mechanics

The present article describes the design and applications of our Poisson series subroutine package developed and maintained since 1968. The programs are written in standard FORTRAN-77 and are almost independent of the wordlength of the particular computer. The system has no restriction on the number of polynomial and angular variables and the storage allocations are completely automatic, invisible to the user.The nucleus of the system consists of about 20 basic traffic subroutines that handle the terms of the different series. Besides these subroutines, we have a number of I/O routines as well as arithmetic subroutines and a large number of Celestial Mechanics applications such as the classical expansions of the Kepler Problem and several expansions of Disturbing Functions. A preprocessor has also been built, allowing the user to write code in a high-level language, such as Jefferys' Trigman, and then translate it in our call-statements.The system was developed on several different computers: first on the Univac 1108 at the Jet Propulsion Laboratory in 1968, then the IBM 360-91 at UCLA, Los Angeles, California and finally the CDC 6600 and CYBER 170/750 at the University of Texas, Austin, Texas. The latest version is entirely geared towards the IBM-PC and compatibles.     

On one special case of parametric resonance in problems of celestial mechanics

We consider a periodic (in time) linear Hamiltonian system that depends on a small parameter. At a zero value of this parameter, the matrix of the system is constant, has two identical pairs of purely imaginary roots, and is not reducible to diagonal form. Therefore, the unperturbed system is unstable. We propose an algorithm for determining the boundaries of the instability regions for the system at nonzero values of the small parameter. This algorithm was used to analyze the stability of triangular libration points in the elliptical restricted three-body problem and in the stability problem in one special case of stationary rotation of a satellite relative to the center of mass.     

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.     

The Lyapunov characteristic exponents-applications to celestial mechanics

After a presentation of Lyapunov characteristic exponents (LCE) we recall their basic properties and numerical methods of computation. We review some numerical computations which are concerned with LCEs and mainly related to celestial mechanics problems.     

Intrinsic nonlinearity and method of disturbed observations in inverse problems of celestial mechanics

Orbit determination from a small sample of observations over a very short observed orbital arc is a strongly nonlinear inverse problem. In such problems an evaluation of orbital uncertainty due to random observation errors is greatly complicated, since linear estimations conventionally used are no longer acceptable for describing the uncertainty even as a rough approximation. Nevertheless, if an inverse problem is weakly intrinsically nonlinear, then one can resort to the so-called method of disturbed observations (aka observational Monte Carlo). Previously, we showed that the weaker the intrinsic nonlinearity, the more efficient the method, i.e. the more accurate it enables one to simulate stochastically the orbital uncertainty, while it is strictly exact only when the problem is intrinsically linear. However, as we ascertained experimentally, its efficiency was found to be higher than that of other stochastic methods widely applied in practice. In the present paper we investigate the intrinsic nonlinearity in complicated inverse problems of Celestial Mechanics when orbits are determined from little informative samples of observations, which typically occurs for recently discovered asteroids. To inquire into the question, we introduce an index of intrinsic nonlinearity. In asteroid problems it evinces that the intrinsic nonlinearity can be strong enough to affect appreciably probabilistic estimates, especially at the very short observed orbital arcs that the asteroids travel on for about a hundredth of their orbital periods and less. As it is known from regression analysis, the source of intrinsic nonlinearity is the nonflatness of the estimation subspace specified by a dynamical model in the observation space. Our numerical results indicate that when determining asteroid orbits it is actually very slight. However, in the parametric space the effect of intrinsic nonlinearity is exaggerated mainly by the ill-conditioning of the inverse problem. Even so, as for the method of disturbed observations, we conclude that it practically should be still entirely acceptable to adequately describe the orbital uncertainty since, from a geometrical point of view, the efficiency of the method directly depends only on the nonflatness of the estimation subspace and it gets higher as the nonflatness decreases.     

Perturbation equations of celestial mechanics

Perturbation equations of celestial mechanics in terms of orbital elements are completely derived in application to the motion of interplanetary dust particle in the gravational field of the Sun and under the action of disturbing forces. Consideration of change of mass of interplanetary dust particle is the most important feature of this derivation. The results obtained are completely general in the case of constant masses.     

TheN-body problem of celestial mechanics

A selective survey of then-body problem of celestial mechanics is given where the emphasis is on the asymptotic behavior of all solutions ast, the possible configurations the particles can assume in phase space and in physical space, and collision and non-collision singularities.Supported in part by NSF Grant MPS 71-03407 A03.     

Transformations to extend the range of application of power series solutions of differential equations of motion

Let the solution of a differential equation, expanded in powers of the independent variablet, have radius of convergenceT. let , wheret=t(), be a new independent variable, and let the corresponding power series in have radius of convergenceS. Thent(S) will not in general be equal toT. Ift(S)>T, then the series in powers of may have advantages over those in powers oft. Mathematical consequences of this distinction have been appreciated since the time of Poincaré. In this note the practical applications of some transformations are investigated.     

Miscellanea from the history of celestial mechanics

The KS-transformation introduced by P. Kustaanheimo and E. Stiefel into celestial mechanics is derived straight from the Kepler Formulas. There follow the treatment of the inverse Newton problem comprising the derivation of the differential equations of mechanics by J. Hermann and L. Euler and also remarks concerning the fundamental papers by Euler about the planet problem and then-body problem. The conclusion is a simple example given by A. Voss and H. Liebmann, for the differential equations of mechanics with non-holonomic condition, which is of pseudoplanetary quality.This paper originated from several stays at the Eidgenössische Technische Hochschule, Zürich; Seminar of Professor E. Stiefel in 1973/74.     

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.     

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.     

Generalized perturbation equations of celestial mechanics

Generalized perturbation equations of celestial mechanics in terms of orbital elements are derived. The most general case is considered: Keplerian motion of two bodies caused by gravitational forces between them is disturbed by disturbing acceleration acting on each of the bodies separately and by changes of masses of these bodies. It is also pointed out why derivation presented in Klaka (1992a) is completely physically correct only for constant masses.