Two fast and accurate routines for solving the elliptic Kepler equation for all values of the eccentricity and mean anomaly

Context. The repetitive solution of Kepler’s equation (KE) is the slowest step for several highly demanding computational tasks in astrophysics. Moreover, a recent work demonstrated that the current solvers face an accuracy limit that becomes particularly stringent for high eccentricity orbits. Aims. Here we describe two routines, ENRKE and ENP5KE, for solving KE with both high speed and optimal accuracy, circumventing the abovementioned limit by avoiding the use of derivatives for the critical values of the eccentricity e and mean anomaly M, namely e > 0.99 and M close to the periapsis within 0.0045 rad. Methods. The ENRKE routine enhances the Newton-Raphson algorithm with a conditional switch to the bisection algorithm in the critical region, an efficient stopping condition, a rational first guess, and one fourth-order iteration. The ENP5KE routine uses a class of infinite series solutions of KE to build an optimized piecewise quintic polynomial, also enhanced with a conditional switch for close bracketing and bisection in the critical region. High-performance Cython routines are provided that implement these methods, with the option of utilizing parallel execution. Results. These routines outperform other solvers for KE both in accuracy and speed. They solve KE for every e ∈ [0, 1 − ϵ], where ϵ is the machine epsilon, and for every M, at the best accuracy that can be obtained in a given M interval. In particular, since the ENP5KE routine does not involve any transcendental function evaluation in its generation phase, besides a minimum amount in the critical region, it outperforms any other KE solver, including the ENRKE, when the solution E(M) is required for a large number N of values of M. Conclusions. The ENRKE routine can be recommended as a general purpose solver for KE, and the ENP5KE can be the best choice in the large N regime