# Integrated Design of Moon-to-Earth Transfer Trajectory Considering Re-Entry Constraints
## 3 Opportunity Determination with Re-Entry Constraints
### Geometry-Based Re-Entry State Determination  
The state of the probe at the re-entry point can be obtained by the longitude and latitude of the landing field, the altitude of the re-entry point, the re-entry velocity, the voyage, and the inclination of the re-entry trajectory.  
As shown in Figure 1, the latitude of the landing field \(F\) (marked with a five-pointed star) is chosen as \(\varphi_{P}\), and the inclination of the re-entry trajectory is set as \(i_{r}\). The argument of the landing field can be determined according to the sine law of the spherical triangle as  
\[U_{F}=\arcsin\frac{\sin\varphi_{F}}{\sin i_{r}}\text{ (Rail-up), }U_{F}=\pi-\arcsin \frac{\sin\varphi_{F}}{\sin i_{r}}\text{ (Rail-down)} \tag{3}\]  
According to the mission constraints, the re-entry trajectory voyage is set as \(S_{r}\), and the argument of the trajectory is  
\[U_{r}=\frac{S_{r}}{R_{e}} \tag{4}\]  
where \(R_{s}\) represents the radius of the Earth. Then, the argument of the re-entry point can be obtained as  
\[U_{P}=U_{F}-U_{r} \tag{5}\]  
The longitude of the landing field is chosen as \(\lambda_{F}\), and the longitude difference from the re-entry point to the ascending node of the trajectory can be obtained according to Napier's Law of the spherical triangle as  
\[\Delta\lambda_{P}=\arctan\frac{\cos i_{r}}{\tan U_{P}} \tag{6}\]  
Therefore, the flight azimuth of the re-entry point is  
\[Q=\arcsin\frac{\sin\Delta\lambda_{P}}{\sin U_{P}} \tag{7}\]  
Based on the landing field argument \(U_{F}\), the longitude difference between the landing field and the ascending node of the re-entry trajectory is calculated as  
\[\Delta\lambda_{F}=\arctan\frac{\cos i_{r}}{\tan U_{F}} \tag{8}\]  
The longitude of the re-entry point is determined as  
\[\lambda_{P}=\lambda_{F}-\Delta\lambda_{F}+\Delta\lambda_{P} \tag{9}\]  
and the latitude of the re-entry point is  
\[\varphi_{P}=\arcsin(\sin U_{P}\sin i_{r}) \tag{10}\]  
The altitude of the re-entry point is chosen as \(H\) according to the mission constraint, and the geocentric distance of the re-entry point is  
\[R=H+R_{e} \tag{11}\]  
Therefore, the position vector of the re-entry point in ECEF is determined as  
\[\mathbf{r}_{e}=[R\cos\varphi_{P}\cos\lambda_{P},R\cos\varphi_{P}\sin\lambda_{P},R \sin\varphi_{P}] \tag{12}\]\(\gamma\) and \(v_{re}\) are denoted as the re-entry angle and the preset magnitude of the re-entry velocity, respectively; the velocity vector of the probe at the re-entry point in the re-entry coordinate frame is  
\[\mathbf{v}_{re}=[v_{re}\cos\gamma,v_{re}\sin\gamma,0] \tag{13}\]  
Then, the velocity vector of the probe at the re-entry point in ECEF is  
\[\mathbf{v}_{e}=\left(\mathbf{M}_{EF}\cdot\mathbf{v}_{re}^{\mathsf{T}}\right)^{\mathsf{T}} \tag{14}\]  
where the superscript \"\(\mathsf{T}\)\" is the transpose symbol.  
Thus, the state of the probe at the re-entry point in ECEF is  
\[\mathbf{X}_{e}=[\mathbf{r}_{e},\mathbf{v}_{e}] \tag{15}\]  
The procedures for geometry-based re-entry state determination are shown as module M1 in Figure 2.