4. The series expansions of direct transformations between three anomalies in satellite geodesy

The determination of satellite orbit is one of the fundamental problems in satellite geodesy. A graphical representation of the Keplerian orbit is given in Figure 2, see [20].

Eccentric, mean and true anomalies are used to describe the movement of satellites. Their transformations are often to be dealt with in satellite ephemeris computation and orbit determination of the spacecraft. In Figure 2, E is the Eccentric anomaly, υis the true anomaly. In order to realize the direct transformations between these anomalies, the series expansions of their transformations are derived using the power series method with the help of computer algebra system Mathematica. Their coefficients are expressed in a power series of the orbital eccentricity e and extended up to eighth-order terms of the orbital eccentricity.
