# 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
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Colors of Box** & \begin{tabular}{c} \(\mathbf{l}\,\mathbf{l}\,\mathbf{\Delta\mathbf{V}_{1}\,\mathbf{l}\,\mathbf{l}\,}\) \\ **(km/s)** \\ \end{tabular} & \begin{tabular}{c} \(\mathbf{\alpha_{1}}\)**(\(\mathbf{l}\))** \\ \end{tabular} & \begin{tabular}{c} \(\mathbf{\varepsilon_{1}}\)** \\ \end{tabular} & \begin{tabular}{c} \(\mathbf{\alpha_{1}}\)**(\(\mathbf{l}\))** \\ \end{tabular} & \begin{tabular}{c} \(\mathbf{\alpha_{1}}\)**(\(\mathbf{l}\))** \\ \end{tabular} & \
\begin{tabular}{c} \(\mathbf{\alpha_{1}}\)**(\(\mathbf{l}\))** \\ \end{tabular} &
\begin{tabular}{c} \(\mathbf{M_{1}}\)**(\(\mathbf{l}\))** \\ \end{tabular} \\ \hline red, Figure 8a & 0.2000 & (120, 90) & 118,588 & 0.94 & 13.69 & 114.27 & 235.944 & 0.00 \\ \(\mathrm{G}_{0}(\Gamma_{0})+\mathrm{G}_{1}(\Gamma_{1})\) & is denoted as set \(\mathsf{S}_{1}\). & & & & & & \\ \hline blue, Figure 8a & 0.2000 & (240, 90) & 118,587 & 0.94 & 53.82 & 355.67 & 348.26 & 359.99 \\ \((\alpha_{0},\beta_{0})=(0,0)\) & Figure 8a,b show the distribution of \(\alpha_{1}\) and \(\beta_{1}\) about \(i_{1}\). Each data point represents a \(\mathrm{G}_{1}(\Gamma_{1})\) orbit. \([a_{1},e_{1},i_{1},\Omega_{1},\omega_{1},M_{1}]\) are the orbital elements of the spacecraft when it arrives at the perigee. The orbital elements of the data of the red and blue box in Figure 8a,b are shown in Table 8. Their orbits are shown in Figure 8c-f. From Figure 8a,b, it can be seen that \(\mathrm{G}_{1}(\Gamma_{1})\) is symmetrically distributed about \(\alpha_{1}=90^{\circ}\) and \(\alpha_{1}=270^{\circ}\), and that the period of variation of \(\mathrm{G}_{1}(\Gamma_{1})\) about \(\beta_{1}\) is \(180^{\circ}\). \\ \hline \end{tabular}
\end{table} 
Table 8: Orbit parameters of the data indicated by the red and blue boxes in Figure 8a,b. Figure 8: (**a**) Distribution of \(\alpha_{1}-i_{1}\). (**b**) Distribution of \(\beta_{1}-i_{1}\). (**c**) Orbit of the data indicated by the red box in (**a**), the center of mass rotating frame. (**d**) Orbit of the data indicated by the red box in (**a**), the J2000 ECI frame. (**e**) Orbit of the data indicated by the blue box in (**b**), the center of mass rotating frame. (**f**) Orbit of the data indicated by the blue box in (**b**). (**g**) Distribution of all orbits of set S1, \((||\Delta\mathbf{V}_{0}||+||\Delta\mathbf{V}_{1}||)\) versus \(i_{1}\). (**h**) Distribution of all orbits of set S1, \((||\Delta\mathbf{V}_{0}||+||\Delta\mathbf{V}_{1}||)\) versus \(h_{p-earth}\). The purple * represents the Earth and the purple + represents the Moon. Therefore, when calculating the parameters of \(\Gamma_{0}(\mathbf{X}_{0})\) and \(\Gamma_{1}(\mathbf{X}_{1})\), in order to simplify the computation, the ranges of the ergodicity of \(\alpha_{0}\) and \(\alpha_{1}\) can be set as \([0,90^{\circ}]\cup[180,270^{\circ}]\) and the ranges of the ergodicity of \(\beta_{0}\) and \(\beta_{1}\) can be set as \([0,180^{\circ}]\). Figure 8g,h show the distribution of \(i_{1}\) and \(h_{p-earth}\) about \((||\Delta\mathbf{V}_{0}||+||\Delta\mathbf{V}_{1}||)\). It is known that there are some orbits that satisfy \(i_{1}>80^{\circ}\). Comparing Figure 7g,h and Figure 8g,h shows that \(\Gamma_{1}(\mathbf{X}_{1})\) greatly increases the number of orbits that meet the requirement of DRO-Earth transfer. Within two maneuvers \(\Gamma_{0}(\mathbf{X}_{0})\) and \(\Gamma_{1}(\mathbf{X}_{1})\), the spacecraft is able to transfer from DRO to Earth and fly by the SSO debris.