# Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit
## 4 Phase 1: Two Impulses DRO-Earth Transfer Orbit Design
### Two-Impulse DRO-Earth Transfer  
This subsection investigates the two-impulse strategy \(\{\mathrm{G}_{0}(\Gamma_{0})+\mathrm{G}_{1}(\Gamma_{1})\}\). The spacecraft applies \(\Gamma_{0}(\mathbf{X}_{0})\) in DRO and flies to the perilune. Then, the spacecraft takes a PLF \(\Gamma_{1}(\mathbf{X}_{1})\) and flies to the Earth. The requirements of the two-impulse DRO-Earth transfer orbit are as follows:  
\[ \begin{split}&\Delta\tau\leq\zeta_{1},\;\Delta t_{0}\leq\zeta_{1},\;|| \Delta\mathbf{V}_{0}||\leq\zeta_{2}\\ &\Big{|}\Big{|}\Delta\mathbf{V}_{1}\Big{|}\Big{|}\leq\zeta_{3},\; \Delta t_{1}\leq\zeta_{1},\;i_{1}\geq\widetilde{i}_{1}\\ &\widetilde{h}_{1}\leq h_{p-moon}\leq h_{SOI},\;\widetilde{h}_{ 2}\leq h_{p-earth}\leq\widetilde{h}_{3}\end{split}  \tag{14}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Epoch** & \(\mathbf{a}\) **(km)** & \(\mathbf{e}\) & \(\mathbf{i}\) **(\(\mathbf{\circ}\))** & \(\mathbf{\Omega}\) **(\(\mathbf{\circ}\))** & \(\mathbf{\omega}\) **(\(\mathbf{\circ}\))** & \(\mathbf{M}\) **(\(\mathbf{\circ}\))** \\ \hline When spacecraft enters the Moon’s SOI & 373,802 & 0.69 & 20.98 & 7.04 & 3.20 & 296.40 \\ When spacecraft exits the Moon’s SOI & 209,944 & 0.82 & 62.71 & 43.32 & 9.93 & 248.97 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Orbit elements when the spacecraft enters and exits the Moon’s SOI. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Colors of Box** & \(\Delta\tau\) **(day)** & \(\mathbf{l}\) \(\mathbf{\Delta V_{0}}\) \(\mathbf{l}\) **(km/s)** & \(\mathbf{(\alpha_{0},\beta_{0})}\) **(\(\mathbf{\circ}\))** & \(\mathbf{a}^{*}\) **(km)** & \(\mathbf{e}^{*}\) & \(\mathbf{i}^{*}\) **(\(\mathbf{\circ}\))** & \(\mathbf{\Omega}^{*}\) **(\(\mathbf{\circ}\))** & \(\mathbf{\omega}^{*}\) **(\(\mathbf{\circ}\))** & \(\mathbf{M}^{*}\) **(\(\mathbf{\circ}\))** \\ \hline red, Figure 6a & 23 & 0.3650 & (80,0) & 224,818 & 0.78 & 38.97 & 348.60 & 319.72 & 240.08 \\  & 23 & 0.3650 & (100,180) & 224,818 & 0.78 & 38.97 & 348.60 & 319.72 & 240.08 \\ \hline blue, Figure 6a & 24 & 0.3950 & (240,180) & 217,626 & 0.85 & 46.97 & 350.77 & 342.92 & 202.02 \\  & 24 & 0.3950 & (300,0) & 217,626 & 0.85 & 46.97 & 350.77 & 342.92 & 202.02 \\ \hline red, Figure 6b & 11 & 0.1850 & (190,40) & 221,927 & 0.82 & 57.17 & 351.40 & 161.45 & 226.10 \\  & 11 & 0.1850 & (350,220) & 221,927 & 0.82 & 57.17 & 351.40 & 161.45 & 226.10 \\ \hline blue, Figure 6b & 9 & 0.1100 & (190,130) & 197,036 & 0.86 & 52.04 & 356.46 & 162.54 & 218.22 \\  & 9 & 0.1100 & (350.310) & 197,036 & 0.86 & 52.04 & 356.46 & 162.54 & 218.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Orbit parameters of the data indicated by the boxes in Figure 6a,b. Figure 7: (**a**) Data of the red box in Figure 6a, the center of mass rotating frame. (**b**) Data of the red box in Figure 6a, the J2000 ECI frame. (**c**) Data of the blue box in Figure 6a, the center of mass rotating frame. (**d**) Data of the blue box in Figure 6a, the J2000 ECI frame. (**e**) Orbit of the spacecraft when \(i^{*}=62.70^{\circ}\), the center of mass rotating frame. (**f**) Orbit of the spacecraft when \(i^{*}=62.70^{\circ}\), J2000 ECI frame. (**g**) Distribution of \(||\Delta V_{0}||-i_{1}\). (**h**) Distribution of \(||\Delta V_{0}||-h_{p-earth}\). The purple * represents the Earth and the purple + represents the Moon. In order to investigate the feasibility of the two-impulse approach, this section studies whether the requirements in Equation (14) can be satisfied.