A new solution approach via analytical approximation of the elliptic Kepler equation

Wu, Baisheng; Zhou, Yixin; Lim, C.W.; Zhong, Huanhuan

doi:10.1016/j.actaastro.2022.10.049

Cited by 8 publications

(5 citation statements)

References 23 publications

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“…We can further derive the following formulas by simply deriving the two Formulas (12) and (13) above:…”

Section: The Derivation Of the Expansion Formula For The Relationship...mentioning

confidence: 99%

New Methods of Series Expansions between Three Anomalies

Zhao,

Li,

Bian

et al. 2024

Applied Sciences

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The calculation of satellite orbit involves some very complex formula derivations and expansions, which are very difficult to manually derive and prone to errors. And the efficiency of manual derivation is not high. We can use computer algebra systems to derive complex formulas related to satellite orbits. This can avoid some of the drawbacks of manual derivation and significantly improve computational efficiency and accuracy. In the past, the relationship among three anomalies was generally represented in the form of a trigonometric series with the first eccentricity e as the parameter. In this paper, the trigonometric series with the parameter m=1−1−e2e is used, as determined by the Lagrange conjugate series. We can use the formula of the Lagrange conjugate series to derive the relationship between the true anomaly and elliptic anomaly. And the relationship between the elliptic anomaly and the mean anomaly is derived by using the symbolic iteration method. In this research paper, we calculated the accuracy of the trigonometric series expansion among three types of anomalies at the first eccentricity e equal to values of 0.01, 0.1, and 0.2. The calculation results indicate that the accuracy of the trigonometric series expansion with m as the parameter is better than 10−5. Moreover, in some cases, the trigonometric series expansion among the three anomalies with m as a parameter is simpler in form than the expansion expressed with parameter e. This paper also derived and calculated the symbolic expressions and extreme values of the difference among three anomalies and expressed the extreme values of the difference in the form of a power series of e. It can be seen that the extreme value increases with the increase in eccentricity e. And the absolute values of the extreme value of the difference between the elliptic anomaly and the mean anomaly, the true anomaly and the elliptic anomaly, and the true anomaly and the mean anomaly increase in this order. When the eccentricity is small, the absolute value of the extreme value of the difference between the true anomaly and the mean anomaly is about twice as large as the elliptic anomaly and the mean anomaly and the true anomaly and the mean anomaly.

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“…We can further derive the following formulas by simply deriving the two Formulas (12) and (13) above:…”

Section: The Derivation Of the Expansion Formula For The Relationship...mentioning

confidence: 99%

New Methods of Series Expansions between Three Anomalies

Zhao,

Li,

Bian

et al. 2024

Applied Sciences

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“…( 3), and vice-versa, which has led to a vast literature concerning alternative approaches [3]. A detailed description of the different works presented in the literature on this problem can be found in [3,14]. The remainder of this paper is structured as follows.…”

Section: Of 16mentioning

confidence: 99%

“…Kepler's laws provide an empirical mathematical model to describe the dynamics of the motion of the planets around the Sun (later proven by Newton's analytical results [1,2]. Therefore, they are of fundamental importance in astrophysics, celestial mechanics, and earth science [3]. In this way and under the Copernican hypothesis, where the Earth rotates on itself and around the Sun, phenomena such as the apparent motion of the stars and the flattening of the Earth, among others, could be explained.…”

Section: Introductionmentioning

confidence: 99%

“…Some of the procedures give satisfactory approximations, but they are very complex and often require many terms in order to obtain a solution with a given degree of accuracy and may involve high computational cost [3,7]. These solutions normally consist of calculating infinite series, or high-order iterative approaches [3,[7][8][9][10]12,14] and by integral solutions [13]. Recently, some methods which use artificial intelligence were proposed [11].…”

Section: Introductionmentioning

confidence: 99%

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Quasi-Analytical Solution of Kepler’s Equation as an Explicit Function of Time

Beloiarov,

Cruz-Gómez

et al. 2024

Preprint

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Although the empirical Kepler’s laws can be proven by applying Newton’s laws to the dynamics of two particles attracted due to the gravitational interaction, there is no explicit formula for the motion as a function of time. In this paper, a quasi-analytical solution for this problem is proposed. It approximates the real behavior of celestial bodies with an acceptable degree of accuracy. All calculations involved require a low computational cost. This problem is closely related to Kepler’s equation, since, the solution for the equations of motion as a function of time gives us the solution to Kepler’s equation as well. The results are presented for each planet of the solar system (including Pluto), and the solution is compared against the real orbits.

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“…However, since 1993 (the year of publication of Colwell's book), this list has has grown so rapidly that it is difficult to account for all of the strategies that have been developed. Just to gain an idea, quoting some of the papers that have been published on the subject in the last five years would be sufficient [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

On the Bessel Solution of Kepler’s Equation

Borghi

2024

Mathematics

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Since its introduction in 1650, Kepler’s equation has never ceased to fascinate mathematicians, scientists, and engineers. Over the course of five centuries, a large number of different solution strategies have been devised and implemented. Among them, the one originally proposed by J. L. Lagrange and later by F. W. Bessel still continue to be a source of mathematical treasures. Here, the Bessel solution of the elliptic Kepler equation is explored from a new perspective offered by the theory of the Stieltjes series. In particular, it has been proven that a complex Kapteyn series obtained directly by the Bessel expansion is a Stieltjes series. This mathematical result, to the best of our knowledge, is a new integral representation of the KE solution. Some considerations on possible extensions of our results to more general classes of the Kapteyn series are also presented.

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A new solution approach via analytical approximation of the elliptic Kepler equation

Cited by 8 publications

References 23 publications

New Methods of Series Expansions between Three Anomalies

New Methods of Series Expansions between Three Anomalies

Quasi-Analytical Solution of Kepler’s Equation as an Explicit Function of Time

On the Bessel Solution of Kepler’s Equation

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