Computing Radial Anomaly in Kepler’s Equation via Weierstrass Elliptic Function
Abstract
The Weierstrass elliptic function is presented in the light of the astrodynamical equation. We synchronise the Weierstrass elliptic function with the elliptic curve which relates the set of Sato weights with the genus $(n,s)$, where $n<s$ and $n$, $s$ are co-prime, making all equations homogeneous. The duplication formula of the Weierstrass formula, as previously used in Uwamusi [14], is introduced as an indicator of how the function behaves near and far away from the origin of a complex number. The differential equation satisfied by the Weierstrass function is explained, and the invariant discriminant function of the Weierstrass elliptic function is taken as an important tool. A method for speeding up the computation process in the radial anomaly in Kepler’s equation, which provides the time of root passage between a pseudo-time and a stable variable time, is introduced via the Chebyshev–Halley iteration formula of third order in the light of the Weierstrass elliptic function. This thus provides in-depth information regarding the radius of peri-center passage and the real root closest to the exact solution. Also in the paper is a computation of the Schwarzian derivative for the Weierstrass elliptic function. Numerical examples are demonstrated with these methods, and the results obtained are quite impressive.
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References
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