Computation of analytical solutions of the relative motion about a Keplerian elliptic orbit
Computation of analytical solutions of the relative motion about a Keplerian elliptic orbit
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DOI：
10.1016/j.actaastro.2012.07.026
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发表时间：
2012-12
影响因子：
3.5
通讯作者：
Yuan Ren;J. Masdemont;M. Marcote;G. Gómez
中科院分区：
文献类型：
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作者：
Yuan Ren;J. Masdemont;M. Marcote;G. Gómez
本文的目的是获得椭圆 Hill-Clohessy-Wiltshire 非线性方程解的面内和面外振幅的三阶表达式。所得到的三阶解在真实异常方面是明确的。展开系数作为先导轨道偏心率 e 的函数给出（即对于所有 e 值都有效）。对于 e=0，我们恢复 Richardson 和 Mitchell 针对圆形情况给出的解；对于 e≠0，解的线性项恢复了 Lawden 为线性化椭圆 HCW 方程（也称为 Tschauner-Hempel 方程）找到的解。在本文的最后部分，我们解释了如何使用 Lindstedt-Poincaré 程序获得椭圆 HCW 非线性方程（以两个振幅和偏心率的幂）的形式级数解。
The purpose of this paper is to obtain a third-order expression, for the in-plane and out-of-plane amplitudes, of the solutions of the elliptic Hill–Clohessy–Wiltshire non-linear equations. The resulting third-order solution is explicit in terms of true anomaly. The coefficients of the expansions are given as functions of the eccentricity e of the orbit of the leader (i.e., are valid for all values of e). For e=0 we recover the solution given by Richardson and Mitchell for the circular case; for e≠0 the linear terms of the solution recover the solution found by Lawden for the linearised elliptic HCW equations, also known as the Tschauner–Hempel equations. In the last part of the paper we explain how a formal series solution of the elliptic HCW non-linear equations (in powers of the two amplitudes and the eccentricity) can be obtained, using the Lindstedt–Poincaré procedure.