Abstract
This paper proposes a highly efficient method for solving the hyperbolic Kepler equation (HKE). The hyperbolic eccentric anomaly interval is divided into two parts: a finite interval and an infinite interval. For the finite interval, a piecewise Padé approximation is first used to establish an initial approximate solution of the HKE. For the infinite interval, an analytical initial approximate solution of the HKE is constructed. These initial approximations are highly accurate and can be further improved to higher accuracy with only one step of Schröder iteration. The proposed method only requires the evaluation of no more than three transcendental functions.
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