Poisson Series Processor Research Articles

An improved C++ Poisson series processor with its applications

One of major problems in celestial mechanics is the management of the long developments in Fourier or Poisson series used to describe the perturbed motion in the planetary system. In this work we will develop a software package suitable for managing these objects. This package includes the ordinary arithmetic operations with Poisson series—such as sum or product—as well as the most commonly used functions sin , cos , or exp , among others. Derivation or integration procedures with respect to time of these objects have been implemented and inversion or serialization procedures are also attainable to obtain such series. All the programming has been accomplished overloading the arithmetic and functional operators natural to C++ with the intention of allowing the programmer to work in a more friendly way and treating the series as if they were mere numbers. In this work we extend the processor in order to obtain the solution to perturbed problems which solution is in the form of a Poisson series. These algorithms have been written in C++.

The Implementation of Hori–Deprit Method to the Construction Averaged Planetary Motion Theory by Means of Computer Algebra System Piranha

This article is related to the problem of the construction of planetary motion theory. We have expanded the Hamiltonian of the four-planetary problem into the Poisson series in osculating elements of the second Poincare system. The series expansion is constructed up to the third degree of the small parameter. The averaging procedure of the Hamiltonian is performed by the Hori–Deprit method. It allows to eliminate short-periodic perturbations and sufficiently increase time step of the integration of the equations of motion. This method is based on Lie transformation theory. The equations of motion in averaged elements are constructed as the Poisson brackets of the averaged Hamiltonian and corresponding orbital element. The transformation between averaged and osculating elements is given by the change-variable functions, which are obtained in the second approximation of the Hori–Deprit method. We used computer algebra system Piranha for the implementation of the Hori–Deprit method. Piranha is an echeloned Poisson series processor authored by F. Biscani. The properties of the obtained series are discussed. The numerical integration of the equations of motion is performed by Everhart method for the Solar system’s giant planets.

Generation of a secular system in the analytical theory for the motion of the moon

In order to generate an analytical theory of the motion of the Moon by considering planetary perturbations, a procedure of general planetary theory (GPT) is used. In this case, the Moon is considered as an addition planet to the eight principal planets. Therefore, according to the GPT procedure, the theory of the Moon’s orbital motion can be presented in the form of series with respect to the evolution of eccentric and oblique variables with quasi-periodic coefficients, which are the functions of mean longitudes for principal planets and the Moon. The relationship between evolution variables and the time is determined by a trigonometric solution for the independent secular system that describes the secular motion of a perigee and the Moon node by considering secular planetary inequalities. Principal planetary coordinates required for generating the theory of the motion of the Moon includes only Keplerian terms, the intermediate orbit, and the linear theory with respect to eccentricities and inclinations in the first order relative to the masses. All analytical calculations are performed by means of the specialized echeloned Poisson Series Processor EPSP.

A note on the use of the generalized Sundman transformations as temporal variables in celestial mechanics

The main aim of this paper is to build up a set of semi-analytical integrators based on the use of a class of anomalies defined by means of a generalized Sundman transformation. The integrators are based on the developments of the mean anomaly and the vector radius according to the new anomaly. To manipulate these developments, a Poisson series processor called poison.h has been used. This processor has been written as a C++class and it contains a set of methods to manage the most common arithmetic and functional operations with these objects.

The Conservation of Area Integrals in Averaging Transformations

It is shown for the two-planetary version of the weakly perturbed two-body problem that, in a system defined by a finite part of a Poisson expansion of the averaged Hamiltonian, only one of the three components of the area vector is conserved, corresponding to the longitudes measuring plane. The variability of the other two components is demonstrated in two ways. The first is based on calculating the Poisson bracket of the averaged Hamiltonian and the components of the area vector written in closed form. In the second, an echeloned Poisson series processor (EPSP) is used when calculating the Poisson bracket. The averaged Hamiltonian is taken with accuracy to second order in the small parameter of the problem, and the components of the area vector are expanded in a Poisson series.

Computational tools to construct semi-analytical planetary theories

The main aim of this paper is the construction of a set of computational tools in order to improve the management of large expansions used in celestial mechanics. This package involves an iterative algorithm to develop the inverse of the distance according to an appropriate set of temporal variables, a new Poisson series processor, and an iterative integration method to obtain the solution of Lagrange planetary equations. The method has been applied to compute a set of examples using several kinds of anomalies in order to test the performance of the algorithms.

Expansion of the Hamiltonian of the Two-Planetary Problem into the Poisson Series in All Elements: Application of the Poisson Series Processor

This paper is the third in a series of articles devoted to one of the basic problems of celestial mechanics: the study of the evolution of solar-type planetary systems. In the previous papers a brief review of the history and current state of the problem was given; the plan of the study was outlined; the Jacobi coordinates and the related osculating elements were introduced; the form of the Poisson expansion of the Hamiltonian in all elements was given; and the expansion coefficients for the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem were obtained (though with impure accuracy) by a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. In the present paper the expansion of the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem into the Poisson series in all elements is constructed with the help of the PSP Poisson series processor, which is capable of required accuracy.

Communications between the Poisson series processor PSPC and general scientific software

Special efficient Poisson series processors (PSP) have been created. Poison series appear frequently in problems of non-linear dynamics and celestial mechanics and their size makes their manipulation by means of general computer algebra systems (CAS) inefficient, but the characteristics of the problems suggest the use of general CAS with other general scientific software like compilers, libraries of compiled programs or word processors. To combine these specialized and general tools we need to communicate between them. In this paper we describe the tools to share information between PSPC, our own PSP, and other general software.

A New Echeloned Poisson Series Processor (EPSP)

A specialized Echeloned Poisson Series Processor (EPSP) is proposed. It is a typical software for the implementation of analytical algorithms of Celestial Mechanics. EPSP is designed for manipulating long polynomial-trigonometric series with literal divisors. The coefficients of these echeloned series are the rational or floating-point numbers. The Keplerian processor and analytical generator of special celestial mechanics functions based on the EPSP are also developed.

PSP: A new Poisson series processor

A specialized Poisson Series Processor (PSP) is proposed. It is designed for manipulating long Poisson series. The Keplerian Processor and analytical generator of special celestial mechanics functions based on the PSP are proposed as well.

Automated Computation of General Perturbations for the Flora Asteroidal Group

AbstractThe general perturbations of the asteroids of the Flora group (975″ < n < 1150″, n - the mean daily motion) due to Jupiter have been automatically computed on an IBM 4341 computer with the Hansen method by manipulating Broucke’s Symbolic Poisson Series Processor, and the results have been compared with that computed previously with the Bohlin group method.

Automatic computer evaluation of general perturbation for Flora asteroids. I. The Hansen method

General perturbations of the asteroids of Flora group due to Jupiter have been automatically computed on IBM 4341 Computer according to Hansen's method using Broucke's Symbolic Poisson Series Processor. Previous results using Bohlin's method are compared. We conclude: 1. The computerized code is accurate and efficient. 2. In the 1000′ < n< 1100′ case, Strömberg's table is suitable for moderate eccentricities and inclinations, with maximum error less than 20'. 3. In the n∼1047′ case, close to the 2 7 commensurability, the long period term (2–7) should be included, but de la Villemarque's formula for this term is insufficiently precise. 4. In the 975′ < n< 1000′ case, Strömberg's table is unsuitable, especially when both e and i are large, contrary to de la Villemarque's view. 5. In the n∼997′ case, near to the 3 10 commensurability, the long period (3–10) term should be included.

Processing Poisson series in parallel

A massively parallel processor proves to be a powerful tool for manipulating large Poisson series. Assuming the series to be sparse, as in most problems of non-linear dynamics, they map into the computer one term per processor. The parallel processor then executes the basic operations in the free algebra of Poisson series over the field of real or rational numbers. We describe a prototype Poisson series processor currently running on the Connection Machine. Timing the calculation of various expansions in the two-body problem demonstrates massive parallelism's potential for tackling the enormous symbolic calculations customary in non-linear dynamics.

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.

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

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

Expansion of the Third-Body Disturbing Function

HE expansion of the disturbing function still remains one of the most important problems in celestial mechanics, both in planetary applications and artificial satellite problems. The expansion of the gravitational disturbing function in zonal and tesseral harmonics based on the Legendre polynomials is now relatively easy to perform with the help of modern digital computers. On the other hand, the disturbing function for the thirdbody effects (solar and lunar perturbations on artificial satellites) is more difficult to expand, even with the availability of fast digital computers. The classically known expansions come in two categories: with the Laplace coefficients l and with the Legendre polynomials. 27 In the present Note we develop a new form of the thirdbody disturbing function which is in the category of the Legendre expansions. The purpose of this work is to find an expansion which is especially efficient and easy to carry out to a high order with the use of a package of Poisson series programs for algebraic operations on a computer. The reader should be aware of the fact that the most efficient expansion for hand calculations is usually not the most efficient expansion with a computerized algebraic processor. Simplicity of programming is another factor that needs to be considered on a computer. In general, the computer will prefer simple recurrence relations. The Poisson series processor will usually have a certain number of standard operations available: repeated partial differentiation, substitution of a series for a polynomial variable, the bionomial theorem, and the Bessel series of the Kepler problem are all standard one-line operation with our processor. It is clear that many of the classical celestial mechanics problems must be completely reformulated in the light of these modern tools. The expansion of the disturbing function, which is the object of the present text, is a literal expansion of the following general form: with the remarkable property that it is separated in factors each depending only on one body. In other words, Xn depends only on the elements (e, /, M, co, ft) while X^3) depends only on the elements (e3,i3,M3,u3,Q3) of the third (perturbing) body. These factors are therefore Poisson series in five variables only. They are relatively short series, in comparison with the resulting series R which is an expansion

A more general system for Poisson Series manipulation

We describe the design of a working Poisson Series Processor that is more general than others in use today. We try to show that the price of generality is worth paying in active research areas in celestial mechanics.