Abstract
The Kepler’s equation of elliptic orbits is one of the most significant fundamental physical equations in Satellite Geodesy. This paper demonstrates symbolic iteration method based on computer algebra analysis (SICAA) to solve the Kepler’s equation. The paper presents general symbolic formulas to compute the eccentric anomaly (E) without complex numerical iterative computation at run-time. This approach couples the Taylor series expansion with higher-order trigonometric function reductions during the symbolic iterative progress. Meanwhile, the relationship between our method and the traditional infinite series expansion solution is analyzed in this paper, obtaining a new truncation method of the series expansion solution for the Kepler’s equation. We performed substantial tests on a modest laptop computer. Solutions for 1,002,001 pairs of (e, M) has been conducted. Compared with numerical iterative methods, 99.93% of all absolute errors δE of eccentric anomaly (E) obtained by our method is lower than machine precision epsilon over the entire interval. The results show that the accuracy is almost one order of magnitude higher than that of those methods (double precision). Besides, the simple codes make our method well-suited for a wide range of algebraic programming languages and computer hardware (GPU and so on).
Highlights
The Kepler’s equation of elliptic orbits is one of the most significant fundamental physical equations in Satellite Geodesy
As all the satellite orbits are nearly-circular, this paper focuses on the situation of nearly-circular motions, providing a symbolic iteration method based on computer algebra analysis for Kepler’s equation
In order to improve the accuracy of the eccentric anomaly (E) in the field of satellite geodesy, this paper proposes a solution for Kepler’s equation for the nearly-circular orbits, which improves the accuracy to the order of 1 0−17
Summary
The Kepler’s equation of elliptic orbits is one of the most significant fundamental physical equations in Satellite Geodesy. The paper presents general symbolic formulas to compute the eccentric anomaly (E) without complex numerical iterative computation at run-time. This approach couples the Taylor series expansion with higher-order trigonometric function reductions during the symbolic iterative progress. N ijenhuis[1] further presented a non-iterative method that represented a gain obtained through a starter algorithm, which required two trigonometric evaluations. These methods are classic and efficient at that time, their performances are limited by the historical scientific conditions with a relatively poor accuracy
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