A new analytical solution of the hyperbolic Kepler equation using the Adomian decomposition method

In this paper, the Adomian decomposition method (ADM) is proposed to solve the hyperbolic Kepler equation which is often used to describe the eccentric anomaly of a comet of extrasolar origin in its hyperbolic trajectory past the Sun. A convenient method is therefore needed to solve this equation to accurately determine the radial distance and/or the Cartesian coordinates of the comet. It has been shown that Adomian's series using a few terms are sufficient to achieve extremely accurate numerical results even for much higher values of eccentricity than those in the literature. Besides, an exceptionally rapid rate of convergence of the sequence of the obtained approximate solutions has been demonstrated. Such approximate solutions possess the odd property in the mean anomaly which are illustrated through several plots. Moreover, the absolute remainder error, using only three components of Adomian's solution decreases across a specified domain, approaches zero as the eccentric anomaly tends to infinity. Also, the absolute remainder error decreases by increasing the number of components of the Adomian decomposition series. In view of the obtained results, the present method may be the most effective approach to treat the hyperbolic Kepler equation.