# Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit
## 4 Phase 1: Two Impulses DRO-Earth Transfer Orbit Design
### Optimal Result of Two-Impulse DRO-Earth Transfer  
In this subsection, the optimal two-impulse DRO-Earth transfer orbits when the spacecraft departs from different DROs are to be discussed. Note that there are numerous DRO orbits. The most commonly discussed types of cisluunar DRO include 3:2 resonance (i.e., the initial position in the Earth-Moon rotating frame is \(x_{0}\approx 0.73\) DU), 2:1 resonance with \(x_{0}\approx 0.81\) DU, and 3:1 resonance with \(x_{0}\approx 0.86\) DU. A comparison of optimal transfer orbits starting from different DROs is shown in Table 9.  
In Table 9, \(\Delta\widetilde{\mathbf{V}}=||\Delta\mathbf{V}_{0}||+||\Delta\mathbf{V}_{1}||\), and \(\Delta\widetilde{\mathbf{I}}=||\Delta t_{0}||+||\Delta t_{1}||\). As Table 9 shows, when the spacecraft departs from the 3:2 resonant DRO, \(\Delta\widetilde{\mathbf{V}}\) is the smallest, being 15.6% less than the \(\Delta\widetilde{\mathbf{V}}\) of the 2:1 resonant DRO and 11.3% less than the \(\Delta\widetilde{\mathbf{V}}\) of the 3:1 resonant DRO. Therefore, the subsequent design is based on the case that the spacecraft departs from the 3:2 resonant DRO. The orbital maneuver parameters of \(\Gamma_{0}(\mathbf{X}_{0})\) and \(\Gamma_{1}(\mathbf{X}_{1})\) when the spacecraft departs from the 3:2 resonant DRO are shown in Table 10. The orbits of the spacecraft in different coordinates are shown in Figure 9.  
From Table 10, it can be seen that the spacecraft departs from the 3:2 resonant DRO at 2025/10/31/23:58:43, and the phase of departure is 236.70\({}^{\circ}\). The impulse of departure \(||\Delta V_{0}||\) is 285 m/s. After 27.46 days, the spacecraft reaches the perilune. Near the perilune, the impulse of the PLF \(||\Delta V_{1}||\) is 185 m/s. Then, after 13.89 days, the spacecraft flies by the first selected piece of target debris, 25733. The relative distance of the flyby is 67.75 km, and the relative velocity is 3.99 km/s. The total velocity increment is 0.47 km/s, and the total transfer time of the DRO-Earth transfer and flyby is 41.35 days.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Depature Time** & \(\varphi_{0}\)\({}^{*}\) & 11\(\Delta\mathbf{V}_{0}\) 11/m s\({}^{-1}\) & \(\Delta t_{0}\)days & 11\(\Delta\mathbf{V}_{1}\) 11/m s\({}^{-1}\) & \(\Delta t_{1}\)days & **Flyby Debris** & 11\(\delta\)r1 1/Km & 11\(\delta\)r1 1/Km - s\({}^{-1}\) \\ \hline
2025/10/31/23:58:43 & 236.70 & 285.0 & 27.46 & 185.0 & 13.89 & 25733 & 69.75 & 3.99 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Parameters of \(\Gamma_{0}(\mathbf{X}_{0})\) and \(\Gamma_{1}(\mathbf{X}_{1})\), and the results of the flyby.  
Figure 9: Optimal DRO–Earth transfer orbit. (**a**) The center of mass rotating frame. (**b**) The J2000 ECI frame. The purple \({}^{\star}\) represents the Earth and the purple + represents the Moon.  
In Phase 1, the spacecraft finishes a DRO-Earth transfer and flies by one piece of SSO debris. On a computer with an AMD 3900X 64-core processor and 128 GB RAM, it takes about 3.5 h and 3.3 days to complete the traversal and pruning calculations of the single-impulse DRO-Earth transfer and the two-impulse DRO-Earth transfer, respectively. Next, the orbits needed for the spacecraft to fly by all the SSO debris in Phase 2 are to be designed.