Classical Physics

    

Simplified Closed-Form Solution of Kepler's Equation with MacLaurin Expansion

Authors: Victor Christianto, Florentin Smarandache

Kepler's equation, a cornerstone of celestial mechanics, relates the mean anomaly (mean angular position) of a celestial body in an elliptical orbit to its eccentric anomaly. While elegant in its formulation (x = y - e sin y), it lacks a general closed-form solution in terms of elementary functions. This article explores the significance of Kepler's equation, briefly traces its historical context, and presents a simplified approach to obtaining a closed-form solution using Mathematica. By expressing the sine function within Kepler's equation as a MacLaurin series, we derive an iterative algorithm that converges rapidly to the eccentric anomaly, providing a practical and computationally efficient method for solving this fundamental equation in celestial mechanics. (submitted to a journal for review.)

Comments: 8 Pages.

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[v1] 2025-01-29 22:10:35

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