An analytic investigation into the behavior of Kepler’s equation1

Sie sind hier: Homepage > Suche

An analytic investigation into the behavior of Kepler’s equation1

Turner, James D.

Abstract Kepler’s Equation is solved over the entire range of elliptic and parabolic motion. Global solution accuracy is maintained by defining four M-e plane solution sub-domains. Three computational methods are concatenated in each sub-domain to solve the problem. Perturbation solutions obtain the basic solutions in each sub-domain. Accelerated series convergence is achieved by transforming the perturbation solutions with Padé approximations. A variable-order Newton extrapolation method is used to further transform the Padé results to yield thirteen digits of precision over the entire range of elliptic and parabolic motion. Only four trigonometric evaluations are required for half of the M-e plane. The remaining half of the M-e plane, including the parabolic special case, only requires a cube root and four trigonometric evaluations. Along the M∼0 and e∼0.7 boundary, a closed-form solution is obtained for a pair of complex-valued roots that possess inflection points that limit the radius of convergence of series approximations for Kepler’s equation. The proposed algorithm effectively provides a closed-form solution for Kepler’s equation, in the sense that the solution accuracy exceeds any projected operational mission need.