The construction of averaged planetary motion theory by means of computer algebra system Piranha (original) (raw)
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Mathematics in Computer Science, 2019
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.
The Use of CAS Piranha for the Construction of Motion Equations of the Planetary System Problem
Applications of Computer Algebra, 2017
In this paper, we consider the using of the computer algebra system Piranha as applied to the study of the planetary problem. Piranha is an echeloned Poisson series processor, which is written in C++ language. It is new, specified, highefficient program for analytical transformations of polynomials, Fourier and Poisson series. We used Piranha for the expansion of the Hamiltonian of four-planetary problem into the Poisson series and the construction of motion equations by the Hori-Deprit method. Both of these algorithms are briefly presented in this work. Different properties of the series representation of the Hamiltonian and motion equations are discussed.
Solar System Research, 2004
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.
Solar System Research, 2016
We consider an algorithm to construct averaged motion equations for four-planetary systems by means of the Hori-Deprit method. We obtain the generating function of the transformation, change-variable functions and right-hand sides of the equations of motion in elements of the second Poincaré system. Analytical computations are implemented by means of the Piranha echeloned Poisson processor. The obtained equations are to be used to investigate the orbital evolution of giant planets of the Solar system and various extrasolar planetary systems.
Programming and Computer Software, 2019
The classical two-planet problem of three bodies of variable masses is studied in the general case when the body masses vary anisotropically at different rates. Differential equations of motion in terms of osculating elements of aperiodic motion along quasi-conic sections are derived. An algorithm for computing the perturbation function in the form of power series in small parameters and the derivation of differential equations determining the secular perturbations of the orbital elements are discussed. All symbolic computations are performed using Mathematica.
A Parallel Integration Method for Solar System Dynamics
The Astronomical Journal, 1997
We describe how long-term solar system orbit integration could be implemented on a parallel computer. The interesting feature of our algorithm is that each processor is assigned not to a planet or a pair of planets but to a time-interval. Thus, the 1st week, 2nd week,. . . , 1000th week of an orbit are computed concurrently. The problem of matching the input to the (n + 1)-st processor with the output of the n-th processor can be solved efficiently by an iterative procedure. Our work is related to the so-called waveform relaxation methods in the computational mathematics literature, but is specialized to the Hamiltonian and nearly integrable nature of solar system orbits. Simulations on serial machines suggest that, for the reasonable accuracy requirement of 1 ′′ per century, our preliminary parallel algorithm running on a 1000-processor machine would be about 50 times faster than the fastest available serial algorithm, and we have suggestions for further improvements in speed.
The theory of Enceladus and Dione: An application of computerized algebra in dynamical astronomy
1974
The orbits of the satellites of the outer planets are poorly known, due to lack of attention over the past half century. We have been developing a new theory of Saturn's satellites Enceladus and Dione which is literal (all constants of integration appear explicitly), canonically invariant (the Hori-Lie method is used), and which correctly handles the eccentricity-type resonance between the two satellites. The algebraic manipulations are being performed using the TRIGMAN formula manipulation language, and the programs have been developed so that with minor modifications they can be used on the Mimas-Tethys and Titan-Hyperion systems.
Review of the works on the orbital evolution of solar system major planets
Solar System Research, 2007
The cognition history of the basic laws of motion of Solar system major planets is presented. Before Newton, the description of motion was purely kinematic, without relying on physics in view of its underdevelopment. From the standpoint of the modern mathematical theory of approximation, all of the models from Ptolemy's predecessors to Kepler inclusive differ only in details. The mathematical theory worked on an infinite time scale; the motion was represented by P. Bohl's quasi-periodic functions (a special case of H. Bohr's quasi-periodic functions). After Newton, the mathematical description of motion came to be based on physical principles and took the form of ordinary differential equations. The advent of General Relativity (GR) and other relativistic theories of gravitation in the 20th century changed little the mathematical situation in the field under consideration. Indeed, the GR effects in the Solar system are so small that the post-post-Newtonian approximation is sufficient. Therefore, the mathematical description using ordinary differential equations is retained. Moreover, the Lagrangian and Hamiltonian forms of the equations are retained. From the 18th century until the mid-20th century, all the theories of planetary motion needed for practice were constructed analytically by the small parameter method. In the early 20th century, Lyapunov and Poincaré established the convergence of the corresponding series for a sufficiently small time interval. Subsequently, K. Kholshevnikov estimated this interval to be on the order of several tens of thousands of years, which is in agreement with numerical experiments. The first works describing analytically (in the first approximation) the evolution on cosmogonic time scales appeared in the first half of the 19th century (Laplace, Lagrange, Gauss, Poisson). The averaging method was developed in the early 20th century based on these works. Powerful analytical and numerical methods that have allowed us to make significant progress in describing the orbital evolution of Solar system major planets appeared in the second half of the 20th century. This paper is devoted to their description.
Planetary Three-Body Problem with Poisson Series Processor
m2 = "m0m2 as masses of the Sun, Jupiter and Saturn respectively. Small parameter " can be chosen as " … max(f m1; f m2)=m0 or " … (f m1 + f m2)=m0. For the system the Sun - Jupiter - Saturn we choose the small parameter " equal to 10¡3. In this case the dimensionless masses m1 and m2 are the unity order values (m1 … 1, m2 … 1=3). Let us represent the Hamiltonian as a sum