3. *The Arenstorf orbit*

Another interesting orbit describes the stable movement of a spacecraft around Earth and Moon ([22], pg. 129).

$$\begin{aligned} \, \, ^1y^{\prime\prime} &= \, \, ^1y + 2 \cdot ^2y^{\prime} - \zeta^{\prime} \cdot \frac{^1y + \zeta}{P\_1} - \zeta \cdot \frac{^1y - \zeta^{\prime}}{P\_2}, \\\, \, ^2y^{\prime\prime} &= \, \, ^2y + 2 \cdot ^1y^{\prime} - \zeta^{\prime} \cdot \frac{^2y}{P\_1} - \zeta \cdot \frac{^2y}{P\_2}, \end{aligned}$$

with

$$\begin{array}{rcl}P\_1 &=& \sqrt{\left(^1y + \zeta\right)^2 + ^2y^2}^3, P\_2 = \sqrt{\left(^1y - \zeta'\right)^2 + ^2y^2}^3, \\ \zeta &=& 0.012277471, \zeta' = 0.987722529,\end{array}$$

initial values

 $\text{s}^{1}\text{y}(0) = 0.994$ ,  $\text{s}^{1}\text{y}'(0) = 0$ , }\text{s}^{2}\text{y}(0) = 0,  $\text{s}^{2}\text{y}'(0) = -2.00158510637908252$ ,

and with *xA* = 17.0652165601579625589 the solution is periodic.

We also transformed this problem to a system of four first-order equations and solved it to *xA* and 2*xA*. After recording the endpoint errors and the costs we present the efficiency measures ratios of DLMP6(5) vs. NEW6(5) for Arenstorf in Table 7.

**Table 7.** Efficiency measure ratios of DLMP6(5) vs. NEW6(5) for Arenstorf.

