# Integrated Design of Moon-to-Earth Transfer Trajectory Considering Re-Entry Constraints
## 5 Numerical Simulation and Analysis
### Opportunity and Specific Precise Trajectory Verification  
Firstly, determine the re-entry state in ECEF that meets the re-entry conditions. Select the longitude and latitude of the landing field as \(\lambda_{F}=101.45^{\circ}\) E, \(\varphi_{F}=-41.2^{\circ}\) N, the re-entry trajectory inclination as \(i_{r}=45^{\circ}\), the re-entry velocity as \(v_{re}=10.7\) km/s, and the re-entry voyage as \(S_{r}=6456\) km, that is \(U_{r}=58^{\circ}\). Execute the procedures of module M1 in Figure 2 to obtain the re-entry state of the probe in ECEF as \(\mathbf{r}_{e}=\left[4314.9,4783.6,851.5\right]\) km, \(\mathbf{v}_{e}=\left[-7.033,3.535,7.248\right]\) km/s. Select a re-entry date of 3 October 2030, for optimal Moon-to-Earth transfer trajectory design. Set the transfer duration to \(\Delta T=3.0\) days, and the re-entry date range is \(\left[D_{1},D_{1}+1\right]\) = [2,462,777.5, 2,462,778.5] (3 October 2030 00:00:00-4 October 2030 00:00:00). The procedures of module M2 in Figure 2 are implemented to obtain the re-entry time corresponding to the optimal trajectory as JD = 2,462,778.43474 (3 October 2030 22:26:01). In this case, the re-entry velocity with the transfer duration set to the preset value is adjusted to \(\mathbf{v}_{re}=10.6541\) km/s, and the re-entry state of the probe in ECI is \(\mathbf{r}_{i}=\left[5136.5,\,3888.1,\,851.5\right]\) km, \(\mathbf{v}_{i}=\left[-6.501,5.147,7.217\right]\) km/s. The optimal transfer trajectory is shown in Figure 4, and the corresponding radius of the perilune is 2768.5 km. Analyze the opportunity of the Moon-to-Earth transfer in 2030. Set the re-entry angle as \(-6^{\circ}\), and calculate the daily optimal transfer trajectory in 2030, and the corresponding perilune altitude change is shown in Figure 5.  
Set the altitude deviation limit of the perilune as \(\chi\) = 50,000 km. The red dotted line in Figure 5 corresponds to \(\chi\); that is, the date when the perilune altitude of the optimal transfer trajectory is lower than this deviation limit is included in the Moon-to-Earth transfer opportunity. It can be seen that the transfer opportunity is about 6-8 days per month, and the opportunity width is about 74 days in 2030. Similarly, select a re-entry date of 3 October 2030, for precise Moon-to-Earth transfer trajectory design. Choose the inclination and altitude constraints of the perilune as \(i^{\prime}_{m}=85^{\circ}\), \(r^{\prime}_{pm}=200\) km, respectively, and the radius of the SOI of the Moon is selected as \(r_{\text{SOI}}\) = 66,200 km. Adopt the procedures of module **M4** in Figure 3 to correct the semi-major axis, argument of ascending node, and argument of perihelion corresponding to this state. It is obtained that the corrected state at the boundary of the SOI of the Moon is \(r_{\text{SOI}}\) = [\(-186\),\(328.5\), \(-244\),\(850.2\), \(-115\),\(443.3\)] km, \(r_{\text{SOI}}\) = [\(0.5633,0.5225,0.1425\)] km/s. Table 1 shows the orbital elements at the perilune of the Moon-to-Earth transfer trajectory in the MCI. If the probe departs from a 200 km altitude circular orbit coplanar with the Moon-to-Earth transfer trajectory, the applied departure impulse is \(\Delta v_{M}=\left[-217.8,687.1,496.2\right]\) m/s. Finally, corrections are applied to the second segment of the trajectory, so that the three segments are precisely connected. Perform high-precision inverse integration on the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Orbital Elements** & \(\mathbf{a}/\text{km}\) & \(\mathbf{e}\) & \(\mathbf{i}l^{\circ}\) & \(\mathbf{\Omega}l^{\circ}\) & \(\mathbf{\omega}l^{\circ}\) & \(\mathbf{f}f^{\circ}\) \\ \hline Value & \(-4808.3\) & 1.4030 & 85.0 & 291.0 & 55.3 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Orbital elements at perilune of precise Moon-to-Earth transfer trajectory. Figure 4: The optimal transfer trajectory of 3 October 2030. The trajectories near the Earth and the Moon are also shown in magnified form. Figure 5: Transfer opportunity screened by the altitude deviation limit of perilune.