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Symbolic iteration method based on computer algebra analysis for Kepler's equation.

Ruichen Zhang1, Shaofeng Bian2, Houpu Li2

  • 1Department of Navigation, Naval University of Engineering, 717 Jiefang Avenue, Wuhan, China. zrcrc1009@sina.com.

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Summary
This summary is machine-generated.

This study introduces a new symbolic method to solve Kepler's equation for satellite geodesy, providing highly accurate eccentric anomaly calculations without complex numerical iterations. The approach offers superior precision compared to traditional methods.

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Area of Science:

  • Satellite Geodesy
  • Celestial Mechanics
  • Computational Physics

Background:

  • Kepler's equation is fundamental for describing elliptic orbits in celestial mechanics.
  • Traditional numerical methods for solving Kepler's equation can be computationally intensive and may lack precision.

Purpose of the Study:

  • To develop a novel symbolic iteration method based on computer algebra analysis (SICAA) for solving Kepler's equation.
  • To derive general symbolic formulas for computing eccentric anomaly (E) without runtime numerical computations.
  • To analyze the relationship between the new method and traditional series expansion solutions.

Main Methods:

  • Symbolic iteration method based on computer algebra analysis (SICAA).
  • Coupling Taylor series expansion with higher-order trigonometric function reductions.
  • Development of a new truncation method for series expansion solutions.

Main Results:

  • Achieved highly accurate symbolic formulas for eccentric anomaly (E).
  • 99.93% of computed errors were below machine precision, exceeding traditional double-precision methods.
  • Demonstrated accuracy approximately one order of magnitude higher than existing numerical methods.

Conclusions:

  • The SICAA method provides a precise and efficient alternative for solving Kepler's equation.
  • The symbolic approach simplifies computation and enhances accuracy in satellite geodesy applications.
  • The method's simplicity and efficiency make it adaptable for various programming languages and hardware, including GPUs.