A new solution approach via analytical approximation of the elliptic Kepler equation. (January 2023)

• Record Type: Journal Article
• Authors: Wu, Baisheng; Zhou, Yixin; Lim, C.W.; Zhong, Huixiang
• Is Part Of: Acta astronautica. Volume 202(2023)
• Page Start: 303
• Page End: 310
• Publication Date: 2023-01
• Subjects: elliptic Kepler equation -- Padé approximation -- Schröder method; Astronautics -- Periodicals; Outer space -- Exploration -- Periodicals; Astronautics; Periodicals; 629.405
• DOI: 10.1016/j.actaastro.2022.10.049
• Languages: English
• ISSNs: 0094-5765

Abstract

Abstract: A new analytical approach for constructing approximate solutions to the elliptic Kepler equation is proposed. We first establish a high-accuracy initial approximation using the piecewise Padé approximation, subsequently we apply the Schröder method to further improve the accuracy of the initial approximation. In general, one Schröder iteration is sufficient to obtain a highly accurate approximate solution. This is a direct method that requires only solving a cubic equation and evaluating two trigonometric functions. The approximate, analytical solutions are compared with solutions by other numerical procedures to prove the accuracy and effectiveness of the proposed approach. Highlights: A new approach is proposed to construct accurate approximate solutions to the elliptic Kepler equation. The Padé approximant is first used to construct an initial approximate solution. Highly accurate approximate solutions can be established by applying just one Schröder iteration. These approximations are valid for all eccentricity and eccentric anomaly.