An efficient code to solve the Kepler equation. Elliptic case
An efficient code to solve the Kepler equation. Elliptic case
复制标题
求解开普勒方程的有效代码。
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
J. Peláez
中科院分区：
文献类型：
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作者：
V. Raposo;J. Peláez
本文提出了一种求解椭圆轨道开普勒方程的新方法。这种新的方法利用了修改后的牛顿？Raphson方法当初始种子接近寻找的解时。为了确定一个好的初始种子，偏心异常域[0，？]的离散化在几个区间，并为这些区间的每一个五次插值多项式的介绍。多项式的六个系数是通过在相应区间的两端要求六个条件来获得的。因此，真实的函数和多项式在区间的两端具有相等的值。两个一阶导数也有类似的关系。在奇异角的开普勒方程，M小于1和1？e接近于零的渐近展开。在大多数情况下，产生的种子导致达到机器误差精度与修改后的牛顿？无迭代或仅一次迭代的Raphson方法。与目前使用的其他方法相比，这种方法提高了计算时间。
A new approach for solving Kepler equation for elliptical orbits is developed in this paper. This new approach takes advantage of the very good behaviour of the modified Newton?Raphson method when the initial seed is close to the looked for solution. To determine a good initial seed the eccentric anomaly domain [0, ?] is discretized in several intervals and for each one of these intervals a fifth degree interpolating polynomial is introduced. The six coefficients of the polynomial are obtained by requiring six conditions at both ends of the corresponding interval. Thus the real function and the polynomial have equal values at both ends of the interval. Similarly relations are imposed for the two first derivatives. In the singular corner of the Kepler equation, M smaller than 1 and 1 ? e close to zero an asymptotic expansion is developed. In most of the cases, the seed generated leads to reach machine error accuracy with the modified Newton?Raphson method with no iterations or just one iteration. This approach improves the computational time compared with other methods currently in use.