# Integrated Design of Moon-to-Earth Transfer Trajectory Considering Re-Entry Constraints
## 5 Numerical Simulation and Analysis
### Design Parameters Analysis
#### 5.2.1 Position of Landing Site  
On the premise that other design parameters remain unchanged, the latitude and longitude of the landing field affect the position of the re-entry point according to the geometric relationship of the spherical triangle. Figure 8 reveals the moment distribution in a day based on the single transfer trajectory example when the longitude of the landing field varies in full phase space, while Figure 9 shows the perilune altitudes of several latitudes and the distribution of opportunities when the latitude of the landing field changes within a certain range in the first three months of 2030.  
It can be seen from Figure 8 that when the longitude of the landing field changes in the whole space, the daily optimal transfer moment changes with the same trend of longitude. This is easy to understand, for the direction of the velocity of the re-entry point in ECEF will change with the longitude of the landing field, while the angle of the Moon's revolution changes very little within a day, so the date distribution of opportunities has little changes. On the same day, the directions of the optimal transfer trajectories corresponding to the landing field with different longitudes in the ECI should basically be the same. That is, when the longitude traverses a circle uniformly, the optimal transfer time in a day also changes uniformly within 24 h.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Source** & \(\mathbf{\gamma}\) & \(\mathbf{\Delta T}\) & **Opportunity** & **Minimum \(\mathbf{\Delta V}\)** \\ \hline Paper & \(-6^{\circ}\) & 3 days & 1 January–4 January, 25 January–31 January & 889.5 m/s \\ Reference [45] & \(-6^{\circ}\) & 3 days & 1 January–3 January, 26 January–31 January & 890.1 m/s \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of the obtained opportunity between this paper and the reference.  
Figure 7: RAAN and \(\Delta V\) of the precise trajectories in the first three months of 2030. **(a)** The RAAN of the perilune in the first three months of 2030; **(b)** The \(\Delta V\) required at the perilune in the first three months of 2030.  
In contrast, the latitude of the landing field has a greater impact on the distribution of opportunities, which is shown in Figure 9. The velocity directions of the re-entry points in ECEF corresponding to different latitudes are different, especially in the z-direction, which leads to the fact that when the latitude changes on a feasible date, the height of the perilune, especially the distance in the z-direction, exceeds the deviation limit. Thus, this opportunity no longer exists. However, correspondingly, when the angle of the Moon can compensate for the z-direction deviation caused by latitude changes on other dates, this date is a new transfer opportunity. Thus, the difference of the latitudes of the landing field leads to the diversification of the transfer opportunity. When the re-entry inclination is set to 45\({}^{\circ}\), the opportunity lengths in the first three months of 2030 are 18 days, 35 days, 18 days, 17 days, and 34 days corresponding to the latitudes of \(0^{\circ},10^{\circ},20^{\circ},30^{\circ},40^{\circ}\), and 50\({}^{\circ}\), respectively. It should be noted that the re-entry inclination cannot be smaller than the latitude of the landing field.