An improved algorithm to develop semi-analytical planetary theories using Sundman generalized variables

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++.

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An improved C++ Poisson series processor with its applications

Ortí

Gómez

Rochera

2021

Comp and Math Methods

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CMMSE: Study of a new symmetric anomaly in the elliptic, hyperbolic, and parabolic Keplerian motion

Ortí

Gómez

Rochera

2022

Math Methods in App Sciences

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In the present work, we define a new anomaly, Ψ, termed semifocal anomaly.It is determined by the mean between the true anomaly, 𝑓 , and the antifocal anomaly, 𝑓 ′ ; Fukushima defined 𝑓 ′ as the angle between the periapsis and the secondary around the empty focus. In this first part of the paper, we take an approach to the study of the semifocal anomaly in the hyperbolic motion and in the limit case corresponding to the parabolic movement. From here, we find a relation between the semifocal anomaly and the true anomaly that holds independently of the movement type. We focus on the study of the two-body problem when this new anomaly is used as the temporal variable. In the second part, we show the use of this anomaly-combined with numerical integration methods-to improve integration errors in one revolution. Finally, we analyze the errors committed in the integration process-depending on several values of the eccentricity-for the elliptic, parabolic, and hyperbolic cases in the apsidal region.

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Recent trends on Computational and Mathematical Methods in Science and Engineering (CMMSE)

Vigo‐Aguiar

Cordero

Ramos

2015

Journal of Computational and Applied Mathematics

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An improved algorithm to develop semi-analytical planetary theories using Sundman generalized variables

Cited by 5 publications

References 15 publications

An improved C++ Poisson series processor with its applications

An improved C++ Poisson series processor with its applications

CMMSE: Study of a new symmetric anomaly in the elliptic, hyperbolic, and parabolic Keplerian motion

Recent trends on Computational and Mathematical Methods in Science and Engineering (CMMSE)

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