A new solution approach via analytical approximation of the elliptic Kepler equation

A new analytical approach for constructing approximate solutions to the elliptic Kepler equation is proposed. We first establish a high-accuracy initial approximation using the piecewise Padé approximation, subsequently we apply the Schröder method to further improve the accuracy of the initial approximation. In general, one Schröder iteration is sufficient to obtain a highly accurate approximate solution. This is a direct method that requires only solving a cubic equation and evaluating two trigonometric functions. The approximate, analytical solutions are compared with solutions by other numerical procedures to prove the accuracy and effectiveness of the proposed approach.