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12 pages, 1592 KiB

Open AccessArticle

Revisiting the Dynamics of Two-Body Problem in the Framework of the Continued Fraction Potential

by Sergey Ershkov, Ghada F. Mohamdien, M. Javed Idrisi and Elbaz I. Abouelmagd

Mathematics 2024, 12(4), 590; https://doi.org/10.3390/math12040590 - 16 Feb 2024

Cited by 12 | Viewed by 2199

In this analytical study, a novel solving method for determining the precise coordinates of a mass point in orbit around a significantly more massive primary body, operating within the confines of the restricted two-body problem (R2BP), has been introduced. Such an approach entails the utilization of a continued fraction potential diverging from the conventional potential function used in Kepler’s formulation of the R2BP. Furthermore, a system of equations of motion has been successfully explored to identify an analytical means of representing the solution in polar coordinates. An analytical approach for obtaining the function t = t(r), incorporating an elliptic integral, is developed. Additionally, by establishing the inverse function r = r(t), further solutions can be extrapolated through quasi-periodic cycles. Consequently, the previously elusive restricted two-body problem (R2BP) with a continued fraction potential stands fully and analytically solved. Full article

(This article belongs to the Section E4: Mathematical Physics)

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29 pages, 10962 KiB

Open AccessArticle

Design of a Low-Energy Earth-Moon Flight Trajectory Using a Planar Auxiliary Problem

by Ilya Nikolichev and Vladimir Sesyukalov

Appl. Sci. 2023, 13(3), 1967; https://doi.org/10.3390/app13031967 - 2 Feb 2023

Cited by 1 | Viewed by 1878

Abstract

The paper presents a sufficiently simple technique for designing a low-energy flight trajectory of a spacecraft (SC) from the Earth to the Moon with insertion into a low circular orbit of the latter. The proposed technique is based on the solution and subsequent analysis of a special model problem, which is a variant of the restricted circular four-body problem (RC4BP) Earth-Sun-SC-Moon; for which it is assumed that the planes of the orbits of all considered bodies coincides. The planar motion of the center of mass of the SC relative to the Earth is considered as perturbed (Sun, Moon). To describe it, equations in osculating elements are used, obtained by using the method of variation of constants based on the analytical solution of the planar circular restricted problem of two bodies (RC2BP)—Earth-SC, for which the rotation of the main axes of the coordinate system (the main plane) is synchronized with the motion of the Sun. The trajectory problem of designing a SC flight from a low circular near-Earth orbit to a low circular selenocentric one (“full” motion model—a restricted four-body problem (R4BP), an ephemeris model) is considered as an optimization one in the impulse formulation. The solution of the main problem is carried out in few (three) stages, on each the appropriate solution of the current variant of the auxiliary problem is determined, which is subsequently used as the basis of the initial approximation to the main one. Full article

(This article belongs to the Special Issue Advanced Schemes for Lunar Transfer, Descent and Landing)

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