三近点角间变换的级数展开式

The Series Expansions of Transformations between Eccentric,Mean and True Anomalies

针对偏近点角、真近点角和平近点角之间数值变换十分困难的问题,为实现近点角间的直接变换,借助计算机代数系统Mathematica,提出了一种幂级数法,推导出它们之间变换的级数展开式,并将式中系数统一表示为轨道偏心率的幂级数形式,且扩展至偏心率的8次方。本文首先阐述方法的原理,最后通过算例分析验证,结果表明对于偏心率小于0.05的卫星轨道,该方法导出的公式计算精度优于10-5″,可供实际使用。

Aiming at computational difficulties between eccentric, mean and true anomalies, in order to realize the direct transformations between them, 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 and extended up to eighth-order terms of the orbital eccentricity. The principle of the method is described firstly, and verified by the numerical examples. The results show that the precision of these expansions is higher than 10-5 when the orbital eccentricity is smaller than 0.05, which could satisfy practical application.