The long period behavior of the orbits of the Galilean satellites of Jupiter

Some of the results of an investigation into the long period behavior of the orbits of the Galilean satellites of Jupiter are presented. Special purpose computer programs were used to perform all the algebraic manipulations and series expansions that are necessary to describe the mutual interactions among the satellites.The disturbing function was expanded as a Poisson series in the modified Keplerian elements referred to a Jovicentric coordinate system. The differential equations for the modified Keplerian elements were then formed, and all short period perturbations were removed using Kamel's perturbation method. Approximate analytical solutions for these differential equations are derived, and the general form of the solutions are given.     

Invariant Tori in the Secular Motions of the Three-body Planetary Systems

We consider the problem of the applicability of KAM theorem to a realistic problem of three bodies. In the framework of the averaged dynamics over the fast angles for the Sun–Jupiter–Saturn system we can prove the perpetual stability of the orbit. The proof is based on semi-numerical algorithms requiring both explicit algebraic manipulations of series and analytical estimates. The proof is made rigorous by using interval arithmetics in order to control the numerical errors.     

A general precompiler for algebraic manipulation

A generalized precompiler for systems performing algebraic manipulation of Poisson series has been written. It accepts a trigonometric superset of FORTRAN IV similar to Jefferys' TRIGRUN language (Jefferys, 1972) and generates a valid FORTRAN IV program which drives an abstract formula manipulation machine. This machine is designed to be generally compatible with any manipulation system, and has been implemented with two such systems. The precompiler is written in standard FORTRAN IV and was designed to allow simple conversion for use on most computers.     

Un formulaire pour le calcul des perturbations d'ordres élevés dans les problèmes planétaires

The authors present formulas in compact form for constructing high order planetary perturbations with respect to the disturbing masses. They have been built by an iterative process and give the variations of osculating elements. Singularities due to vanishing eccentricities and inclinations are not present in the differential equations. All elementary operations are manipulations of Fourier series with numerical coefficients, and great care has been taken to economize algebraic operations. Results are presented in three forms:

1. vectorial form, with real components which may be useful in numerical integrations;
2. complex form, to put in evidence the symmetries of the system of variables;
3. scalar form, which is the most elaborate. This last form has been used for constructing the first order perturbations for any pair of planets. Two illustrations are given (Jupiter and Saturn, Venus and Earth). Further remarks are made about the practical manipulation of Fourier series, resolution of Kepler's equation in complex form and construction by iteration of the inverse of the distance between two bodies.

     

A semi-analytic algorithm for constructing lower dimensional elliptic tori in planetary systems

We adapt the Kolmogorov’s normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure can also provide some analytic expansions of the motions on elliptic tori. By extensively using algebraic manipulations on a computer, we explicitly apply our method to a planar four-body model not too different with respect to the real Sun–Jupiter–Saturn–Uranus system. The frequency analysis method allows us to check that our location of the initial conditions on an invariant elliptic torus is really accurate.     

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.     

Computer-developed construction of analytic expressions for the coordinates and partial derivatives of Jupiter's Galilean satellites

In his effort to develop series expressions for the coordinates of the Galilean satellites accurate to one are second (Jovicentric), R. A. Sampson was forceda priori to adopt certain numerical values for several constants imbedded in his theory. His final numerical values for the series expressions are not amenable to adjustment of the constants of integration nor of physical constants which affect the motion of the satellites. A method which utilizes computer-based algebraic manipulation software has been developed to reconstruct Sampson's theory, to remove existing errors, to introduce neglected effects and to provide analytical expressions for the coordinates as well as for the partial derivatives with respect to orbital parameters, Jupiter and satellite masses, Jupiter's oblateness (J ₂,J ₄) and Jupiter's pole and period of rotation. The computer-based manipulations enable one to perform, for example, the approximately 10⁸ multiplications required in calculating some perturbations (and their partial derivatives) of Satellite II by Satellite III with ease, and provide algebraic expressions which can readily be adjusted to generate theories corresponding to revised constants of integration and physical parameters.     

脉动变星的非线性分析

本文从支配恒星内部物理过程的整套基本方程组出发，应用阵发混沌机制，研究其非线性特性，经过理论分析和数学演算，最后得出结论，即恒星结构方程本身，在一定参数范围内，能够自发出现阵发混沌脉动，这与已观测到的一些脉动变星的不规则光变性质，定性符合得很好。     

Contribution to the solution of the three-body problem in power series form

After reviewing the existing procedures for solving the three-body problem by convergent power series, the author develops two algebraic methods in terms of the independent variable which is either the time t or Levi-Civita's regularizing variable u. These power series solve in Weierstrass' and Painlevé's sense the problem formulated in its greatest generality, since no restrictions at all are made on the order of magnitude of masses and none of the three bodies is restricted to moving along a prescribed conic section. Besides, the reference system used is a tridimensional Cartesian one. In the t-domain, the expressions for the high-order derivatives of the coordinates are computed using repeatedly Leibnitz's rule for derivatives of products of functions. In the u-domain, an extremely simple successive approximation procedure is established by means of a single recursion formula which requires elementary operations to be performed on polynomials of increasing degrees.     

Branching solutions and Lie series

Branching solutions of algebraic equations are treated using Lie series. A new method is proposed to derive Puiseux expansions. Newton's diagram is considered in the context of Lie series. An application of finding equilibrium points of a Hamiltonian system near resonances is also demonstrated.     

A semi-analytical method to study perturbed rotational motion

A semi-analytical method is presented to study the system of differential equations governing the rotational motion of an artificial satellite. Gravity gradient and non gravitational torques are considered. Operations with trigonometric series were performed using an algebraic manipulator. Andoyer's variables are used to describe the rotational motion. The osculating elements are transformed analytically into a mean set of elements. As the differential equations in the mean elements are free of fast frequency terms, their numerical integration can be performed using a large step size.     

On the numerical transformation of variables in perturbation theory

Once the generating function of a Lie-type transformation is known, canonical variables can be transformed numerically by application of a Runge-Kutta type integration method or any other appropriate numerical integration algorithm. The proposed approach avails itself of the fact, that the transformation is defined by a system of differential equations with a small parameter as the independent variable. The integration of such systems arising in the perturbation theories of Hori and Deprit is discussed. The method allows to compute numerical values of periodic perturbations without deriving explicitly the perturbation series. This saving of an algebraic work is achieved at the expense of multiple evaluations of the generator's derivatives.     

Orbital maneuver optimization using time-explicit power series

Orbital maneuver transfer time optimization is traditionally accomplished using direct numerical sampling to find the mission design with the lowest delta-v requirements. The availability of explicit time series solutions to the Lambert orbit determination problem allows for the total delta-v of a series of orbital maneuvers to be expressed as an algebraic function of only the individual transfer times. The delta-v function is then minimized for a series of maneuvers by finding the optimal transfer times for each orbital arc. Results are shown for the classical example of the Hohmann transfer, a noncoplanar transfer as well as an interplanetary fly-by mission to the asteroids Pallas and Juno.     

Derivatives of the Gravity Potential with Respect to Rectangular Coordinates

Starting from a complex operator of derivation, we give expressions for derivatives of arbitrary order of the gravity potential with respect to rectangular coordinates. These expressions have a form similar to the original potential expanded in spherical harmonics and are free of singularity at the poles. Computing sets of numerical coefficients once for all, we can compute the derivatives with a very limited work: the same functions are used to compute all derivatives by means of a unique parametrized formula. This is very comfortable for further algebraic manipulations. Numerical tests prove the accuracy and the efficiency of the algorithm derived from our formula to compute the gravity acceleration vector and the gravity gradient tensor.     

User's guide to REDUCE subroutines for algebraic computations in general relativity

We describe the use of a package of subroutines for general-relativistic algebraic computations written in the LISP-based algebraic programming system REDUCE developed by Anthony Hearn. The first group of routines calculates concomitants of the metric tensor such as the Riemann tensor, Ricci tensor, Ricci scalar, Einstein tensor and Weyl tensor from given covariant or covariant and contravariant metric tensor components. One of these procedures includes the evaluation of the Maxwell equations and invariants as well as the Maxwell energy momentum tensor. A second group of routines takes the components of a null tetrad as input and evaluates the null frame projections of the Riemann tensor, the Einstein tensor as well as – if required – the 14 local invariants of the Riemann tensor. It also includes a determination of the Petrov type. Perturbation calculations may be performed and run effectively. The output can be presented in a flexible format chosen according to the needs of the user. A number of further special-purpose programs are available on request. The article should enable a reader familar with General Relativity but unfamilar with formula manipulating system, to employ the package.     

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.     

Nonlinear periodic and algebraic ion-acoustic solitary waves in a relativistic plasma

We derive a mixed modified Korteweg-de Vries (MK-dV) equation from a semi-relativistic ion acoustic wave with hot ions by the fluid approximation. The positive cubic nonlinearity of the mixed MK-dV equation give rise to the periodic progressive waves and the algebraic solitary waves. The periodic wave bears a series of solitary pulses, and the algebraic solitary wave reduces the rarefactive solitary wave in the limit of the particular boundary condition. These nonlinear wave modes explain, respectively, the periodic pulse of the potential and the rarefactive solitary wave of the fine structure observed in space.     

A New Symbolic Processor for the Earth Rotation Theory

In this paper we present a new symbolic processor specially suited for the Earth rotation theory. This processor works with a more general kind of Poisson series called Kinoshita series, which has resulted to be very useful in the Earth rotation theory. Its structure is adapted for dealing with the more general analytical expressions that appear in the Earth rotation theory. This new algebraic processor has been successfully used for computing different contributions to the nutation series of the rigid Earth.This revised version was published online in October 2005 with corrections to the Cover Date.     

Quasi-periodic orbits about the translunar libration point

Analytical solutions for quasi-periodic orbits about the translunar libration point are obtained by using the method of Lindstedt-Poincaré and computerized algebraic manipulations. The solutions include the effects of nonlinearities, lunar orbital eccentricity, and the Sun's gravitational field. For a small-amplitude orbit, the orbital path as viewed from the Earth traces out a Lissajous figure. This is due to a small difference in the fundamental frequencies of the in-plane and out-of-plane oscillations. However, when the amplitude of the in-plane oscillation is greater than 32 379 km, there is a corresponding value of the out-of-plane amplitude that will produce a path where the fundamental frequencies are equal. This synchronized trajectory describes a halo orbit of the Moon.     

An efficient method to calculate Ambarzumian-Chandrasekhar's and hopf's functions

An efficient method is proposed to calculate scalar Ambarzumian-Chandrasekhar's and Hopf's functions. This method is based on the approximation of Sobolev's resolvent function using exponent series, the coefficients of which are readily found from approximate characteristic equation and from a system of linear algebraic equations.The approximate expressions for the above functions are given. For checking purposes the calculations were carried out in single, double, and quadruple precision. For isotropic, Rocard, and Rayleigh scattering we present a sample of results in 14 significant figures.The Hopf function for isotropic and Rayleigh scattering is presented in 18 significant figures and the well-known Hopf constantq() is found in 59 significant figures.