Solution of the hyperbolic kepler equation by adomian’s asymptotic decomposition method

The hyperbolic Kepler equation is of practical interest in astronomy. It is often used to describe the eccentric anomaly of a comet of extrasolar origin in its hyperbolic trajectory past the Sun. Efficient determination of the radial distance and/or the Cartesian coordinates of the comet requires accurate calculation of the eccentric anomaly, hence the need for a convenient, robust method to solve Kepler’s equation of hyperbolic type. In this paper, the Adomian’s asymptotic decomposition method is proposed to solve this equation. Our calculations have demonstrated a rapid rate of convergence of the sequence of the obtained approximate solutions, which are displayed in several graphs. Also, we have shown in this paper that only a few terms of the Adomian decomposition series are sufficient to achieve extremely accurate numerical results even for much higher values than those in the literature for the mean anomaly and the eccentricity of the orbit. The main characteristic of the obtained approximate solutions is that they are all odd functions in the mean anomaly, which we have illustrated through graphs. In addition, it is found that the absolute remainder error using only three components of Adomian’s solution decreases across a specified domain and approaches zero as the eccentric anomaly tends to infinity. Moreover, the absolute remainder error decreases by increasing the number of components of the Adomian decomposition series. Finally, the current analysis may be the first to make an effective application of the Adomian’s asymptotic decomposition method in astronomical physics.