On the integration of Cid’s radial intermediary

Abad, Alberto; Calvo, M.; Elipe, A.

doi:10.1016/j.actaastro.2020.11.025

Cited by 6 publications

(2 citation statements)

References 22 publications

(29 reference statements)

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“…There are several procedures for solving Equation (19). Most of them consist in converting the equations of motion into the ones of a harmonic oscillator by means of a of time and length transformation and then introduce a generalized Kepler equation (Abad et al 2001), or use the Krylov-Bogoliubov averaging method (Krylov & Bogoliubov 1950) that provides error bounds for the solution in time intervals of size the inverse of the small parameter, which is quite convenient for studies requiring long time intervals and its complexity is similar to a pure Kepler's problem (Abad et al 2021). Other alternatives use a linearization function to reach the harmonic oscillator problem (Belen'kii 1981; Cid et al 1986;Ferrándiz 1986) and then obtain the relation between the solution of the harmonic oscillator and the original problem by means of elliptic functions (Ferrándiz 1986).…”

Section: Analytical Solutionmentioning

confidence: 99%

“…Since the Hamiltonian in Equation (2) depends on only one variable, the radial distance r, the problem is integrable and one could think that there is no need to dedicate more time to such "a simple" problem. Albeit the fact of its integrability, the list of works dedicated to the problem is long, facing several aspects of motion; in particular, the time evolution of orbital elements (Lara & Gurfil 2012;Belen'kii 1981;Cid et al 1986;Ferrándiz 1986;Deprit & Ferrer 1987;Abad et al 2020Abad et al , 2021.…”

Section: Introductionmentioning

confidence: 99%

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Close Binary Stars Modeled by Two Prolate Ellipsoids in Synchronous Rotation

Elipe,

da Costa,

Piccotti

et al. 2023

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The presence of tidal deformations in close binary stars has already been confirmed by astronomical observations. The present paper aims to simply address an astronomy problem, studying the relative movement of close binaries disturbed by their mutual deformation through some basic concepts and tools of celestial mechanics. For this purpose, the tidal effect is modeled by considering that each star is an elongated revolution ellipsoid in such a way that axes of revolution are coincident, and their largest axes point toward each other along the motion. The potential for mutual attraction is then obtained, resulting in a perturbed Keplerian system with perturbation proportional to the inverse of the cubic distance between the stars, thus being a one-degree-of-freedom problem and, therefore, integrable. The effective potential, the integrals of energy and angular momentum, and the Laplace vector are used to obtain qualitative information about the dynamics before integrating it. The motion describes a rosette-like orbit with periodic osculating elements, or a circle when the energy is a local minimum. Finally, an analytical solution is presented in terms of elliptic functions by using a regularizing and linearizing function.

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Section: Analytical Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Close Binary Stars Modeled by Two Prolate Ellipsoids in Synchronous Rotation

Elipe,

da Costa,

Piccotti

et al. 2023

Self Cite

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Wrong hypotheses in the generalized RTBP

Elipe

2024

Astrophys Space Sci

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Generalized restricted three body problems consist of adding some extra hypotheses to the Restricted three body problem (RTBP) in order to have a new problem, not very different of the original RTBP. However, not any additional hypothesis is allowed; it must satisfy the laws of Physics. Among the several generalizations found in literature, we prove that at least there are two hypotheses that cannot be used, namely: 1) Perturbation in Coriolis and/or centrifugal forces, and 2) primaries are spheroids moving on elliptical orbits.

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A set of orbital elements to fully represent the zonal harmonics around an oblate celestial body

Arnas

Linares

2021

Monthly Notices of the Royal Astronomical Society

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This work introduces a new set of orbital elements to fully represent the zonal harmonics problem around an oblate celestial body. This new set of orbital elements allows to obtain a complete linear system for the unperturbed problem and, in addition, a complete polynomial system when considering the perturbation produced by the zonal harmonics from the gravitational force of an oblate celestial body. These orbital elements present no singularities and are able to represent any kind of orbit, including elliptic, parabolic and hyperbolic orbits. In addition, an application to this formulation of the Poincaré-Lindstedt perturbation method is included to obtain an approximate first order solution of the problem for the case of the J2 perturbation.

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On the integration of Cid’s radial intermediary

Cited by 6 publications

References 22 publications

Close Binary Stars Modeled by Two Prolate Ellipsoids in Synchronous Rotation

Close Binary Stars Modeled by Two Prolate Ellipsoids in Synchronous Rotation

Wrong hypotheses in the generalized RTBP

A set of orbital elements to fully represent the zonal harmonics around an oblate celestial body

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