# Integrated Design of Moon-to-Earth Transfer Trajectory Considering Re-Entry Constraints
## 3 Opportunity Determination with Re-Entry Constraints
### Daily Optimal Transfer Trajectory Design  
Since the Earth rotation period is one day, which is much smaller than the revolution period of the Moon, there is a daily optimal Moon-to-Earth transfer trajectory that can be obtained by inverse integration from the re-entry point to the vicinity of the moon. Due to the existence of the inclination between the Moon's orbit and Equator, the perilune  
Figure 2: Flowchart for feasible transfer trajectory and opportunity determination. **M1** shows the procedures for re-entry state determination; **M2** shows the steps for optimal Moon-to-Earth trajectory design; **M3** reveals the transfer opportunity screening process. altitudes of the Moon of the optimal trajectories on some dates can reach tens of thousands of kilometers or even larger. An excessive perilune altitude means that too much fuel is consumed for adjustment, which cannot be achieved in actual missions. Therefore, the Moon-to-Earth transfer opportunity can be screened according to the altitude of the perilune every day. The Julian date range of the default re-entry time is \(D_{1}\) 0:00 to \(D_{1}+10\):00, and the selected re-entry point time is \(T_{0}\in[D_{1},D_{1}+1]\). The duration of the Moon-to-Earth transfer is given as \(\Delta T\); then, the time of the perilune is \(T_{m}=T_{0}-\Delta T\), and the position of the Moon at that time obtained from the ephemeris is \(\mathbf{r}_{M}=[x_{M},y_{M},z_{M}]\). The Greenwich time corresponding to the re-entry time is GST; then, the rotation matrix from ECEF to ECI is  
\[M_{FI}=\left[ \begin{array}{ccc}\cos(GST)&-\sin(GST)&0\\ \sin(GST)&\cos(GST)&0\\ 0&0&1\end{array} \right] \tag{16}\]  
Then, the state of the probe at the re-entry point in ECI is  
\[\mathbf{X}_{i}=\left[\left(\mathbf{M}_{FI}\cdot\mathbf{r}_{e}^{\mathrm{T}}\right)^{ \mathrm{T}}\!\!,\left(\mathbf{M}_{FI}\cdot\mathbf{v}_{e}^{\mathrm{T}}\right)^{\mathrm{ T}}\right] \tag{17}\]  
The probe is reversely integrated from the re-entry point to the perilune of the Moon to obtain a feasible transfer trajectory with an actual transfer duration of \(\Delta T\). In order to make the actual transfer duration equal to the set value \(\Delta T\), the magnitude of the re-entry velocity needs to be adjusted. If \(\Delta T<\Delta T\), this means that the re-entry velocity is set too high, and the probe will reach the perilune ahead of time; if \(\Delta T>\Delta T\), this means that the re-entry velocity is set too low, and the probe lags behind the perilune. The magnitude of the re-entry velocity is adjusted iteratively so that the transfer duration is exactly \(\Delta T\). Determining the adjustment amount of the re-entry velocity in each iteration is the key, so a method is proposed to determine the adjustment step according to the deviation between the actual and the expected duration. Assuming that the deviation between the actual transfer duration and the set transfer duration in the iteration is \(\delta t=\Delta T-\Delta T\), the adjustment step magnitude of the re-entry velocity in the next iteration is  
\[\Delta v_{re}=\delta t/100 \tag{18}\]  
where the unit of \(\delta t\) is \"day\". The magnitude of the re-entry velocity is adjusted to  
\[v_{re}=v_{re}+\Delta v_{re} \tag{19}\]  
When \(\delta t<0\), \(\Delta v_{re}<0\), the re-entry velocity is reduced, and when \(\delta t>0,\Delta v_{re}>0\), the re-entry velocity is increased. Based on Equation (19), the re-entry velocity is iterated until the transfer duration accuracy requirement is met as  
\[|\Delta T-\Delta T|<\varepsilon_{T} \tag{20}\]  
where \(\varepsilon_{T}\) is the error limit of the accuracy of the transfer duration, which can be set according to the actual demands. Thus far, the feasible Moon-to-Earth transfer trajectory with the transfer duration fixed as \(\Delta T\) and the re-entry time as \(T_{0}\) has been obtained. The Earth rotates once a day. When \(T_{0}\) changes in one day, the orbital direction of the feasible transfer will rotate once.