Piecewise rational Padé and Hermite approximations for the elliptic Kepler equation

Accurate and efficient solution of the Elliptic Kepler Equation (EKE) is fundamental in orbital mechanics and spacecraft trajectory analysis. In this work, we present a family of piecewise rational approximations for solving the EKE, F ( E ; e ) ≡ E − e sin E = M , based on Padé and Hermite-type formulations. The proposed approaches replaces the transcendental term sin E with Hermite and Piecewise Padé-Type (PPT) approximants, the later originally introduced by Brezinski, providing higher accuracy than the traditional Piecewise Padé (PP) method of Wu and co-workers without increasing computational cost. With these approximants, the resulting rational form reduces the EKE to a cubic equation that can be solved analytically, making it suitable for onboard implementations or large-scale orbit propagation tasks. Numerical experiments demonstrate that the PPT-based solution significantly improves accuracy for moderate and high eccentricities, including near-parabolic cases. Additionally, optimized parameter selection in general [3/2] rational representations yields further accuracy gains. These results show that the proposed piecewise rational method offers a reliable and computationally efficient alternative for precise orbital position determination across a wide range of eccentricities. • A family of piecewise rational methods is proposed for the Elliptic Kepler Eq. • The approach employs Hermite and Piecewise Padé-Type (PPT) approximants. • It outperforms the traditional Piecewise Padé (PP) method in accuracy. • The rational form reduces the EKE to an analytically solvable cubic equation. • PPT-based models remain precise for high and near-parabolic eccentricities.