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In orbital mechanics, the elliptical Kepler equation is a basic nonlinear equation which determines the eccentric anomaly of a planet orbiting the Sun. In this paper, Kepler’s equation has been solved by means of Maclaurin expansion without any need to decompose the involved nonlinearity as in Adomian’s method and the differential transformation method. The obtained approximate solutions are compared with a famous solution to this equation in terms of Bessel function solution. The results showed that the Maclaurin series agrees with the solution in literature in a sub domain of one revolution of the Earth around the Sun. In such a sub domain, the absolute error decreases with increasing the number of terms in the truncated series solution. In order to enhance the numerical results in the whole domain of a complete revolution, the Padé approximation for the Maclaurin series solution are established and then compared with those derived from the Bessel function solution. The comparisons for the eccentric anomaly and the approximate radial distances of the Earth from the Sun reveal that the combined MaclaurinPadé approach is an effective tool to analyze the current problem. In addition, the domain of effectiveness exceeds the domain of a complete revolution. Moreover, the minimal distance (perihelion) and maximal distance (aphelion) approach 147 million kilometers and 152.505 million kilometers, respectively, and these results coincide with known results in astrophysics.

Padé Approximation to the Solution of the Elliptical Kepler Equation