New Methods of Series Expansions between Three Anomalies

Abstract

The calculation of satellite orbit involves some very complex formula derivations and expansions, which are very difficult to manually derive and prone to errors. And the efficiency of manual derivation is not high. We can use computer algebra systems to derive complex formulas related to satellite orbits. This can avoid some of the drawbacks of manual derivation and significantly improve computational efficiency and accuracy. In the past, the relationship among three anomalies was generally represented in the form of a trigonometric series with the first eccentricity e as the parameter. In this paper, the trigonometric series with the parameter m=1−1−e2e is used, as determined by the Lagrange conjugate series. We can use the formula of the Lagrange conjugate series to derive the relationship between the true anomaly and elliptic anomaly. And the relationship between the elliptic anomaly and the mean anomaly is derived by using the symbolic iteration method. In this research paper, we calculated the accuracy of the trigonometric series expansion among three types of anomalies at the first eccentricity e equal to values of 0.01, 0.1, and 0.2. The calculation results indicate that the accuracy of the trigonometric series expansion with m as the parameter is better than 10−5. Moreover, in some cases, the trigonometric series expansion among the three anomalies with m as a parameter is simpler in form than the expansion expressed with parameter e. This paper also derived and calculated the symbolic expressions and extreme values of the difference among three anomalies and expressed the extreme values of the difference in the form of a power series of e. It can be seen that the extreme value increases with the increase in eccentricity e. And the absolute values of the extreme value of the difference between the elliptic anomaly and the mean anomaly, the true anomaly and the elliptic anomaly, and the true anomaly and the mean anomaly increase in this order. When the eccentricity is small, the absolute value of the extreme value of the difference between the true anomaly and the mean anomaly is about twice as large as the elliptic anomaly and the mean anomaly and the true anomaly and the mean anomaly.