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<p>Kepler’s equation is fundamental to celestial mechanics, establishing the link between time and the geometric position of a body in an elliptical orbit. This paper introduces a practical and efficient analytical-numerical method for solving this transcendental equation, offering a robust alternative to classical iterative schemes and infinite series expansions. By employing a tangent half-angle transformation, we map the problem into a domain where the relationship between anomalies is effectively modeled by a polynomial scaling of the transformed variable. The core of the method is a cubic polynomial function, dependent on eccentricity, whose coefficients are determined via a global data-driven optimization rather than local Taylor series. The resulting closed-form formula achieves a mean absolute error on the order of <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$10^{-15}$</EquationSource> </InlineEquation> radians, limited only by machine precision and demonstrates a computational speedup of two orders of magnitude compared to standard Newton-Raphson solvers and Bessel series methods (Philcox). This work provides a constant-time, high-precision solution well-suited for large-scale N-body simulations and real-time orbit propagation.</p>

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RETRACTED ARTICLE: An optimized analytical-numerical method for Kepler’s equation with near-machine precision and high computational speedup

  • 期刊论文

  • 发表日期：

    2025年12月22日

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  • Charbel Mamlankou,
  • Bernard Sognombo

摘要

Kepler’s equation is fundamental to celestial mechanics, establishing the link between time and the geometric position of a body in an elliptical orbit. This paper introduces a practical and efficient analytical-numerical method for solving this transcendental equation, offering a robust alternative to classical iterative schemes and infinite series expansions. By employing a tangent half-angle transformation, we map the problem into a domain where the relationship between anomalies is effectively modeled by a polynomial scaling of the transformed variable. The core of the method is a cubic polynomial function, dependent on eccentricity, whose coefficients are determined via a global data-driven optimization rather than local Taylor series. The resulting closed-form formula achieves a mean absolute error on the order of 10 − 15 $10^{-15}$ radians, limited only by machine precision and demonstrates a computational speedup of two orders of magnitude compared to standard Newton-Raphson solvers and Bessel series methods (Philcox). This work provides a constant-time, high-precision solution well-suited for large-scale N-body simulations and real-time orbit propagation.