# Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit
## 4 Phase 1: Two Impulses DRO-Earth Transfer Orbit Design
### Single-Impulse DRO-Earth Transfer
After a thrustless lunar gravity assist, the resulting height of the perigee in the ergodicity varies in a large range. Therefore, the upper limit of \(h_{p-earth}\) in the third condition is set to be 36,000 km, which is a rather large value. In addition, because the \(i_{1}\) of all \(\left\{\text{G}_{0}(\Gamma_{0})+\text{G}_{1}(0)\right\}\) are less than \(\widetilde{i}_{1}\), the requirement of \(i_{1}\) in Table 4 is not added to the pruning conditions. After traversing all parameters shown in Table 4, the \(h_{p-moon}\), \(J_{t}\), and \(h_{p-earth}\) of each \(\left\{\text{G}_{0}(\Gamma_{0})+\text{G}_{1}(0)\right\}\) are obtained. The traverse results are pruned according to Equation (13). There are 2872 pairs of \(\left\{\text{G}_{0}(\Gamma_{0})+\text{G}_{1}(0)\right\}\) after pruning. Let \(\text{S}_{0}\) denote the group composed of all orbits \(\left\{\text{G}_{0}(\Gamma_{0})+\text{G}_{1}(0)\right\}\) after being pruned by Equation (12). \(\left[a^{*},e^{*},i^{*},\Omega^{*},\omega^{*},M^{*}\right]\) are the orbital elements of the spacecraft when it exits the Lunar SOI. Figure 6 shows the distribution of \(\alpha_{0}\) and \(\beta_{0}\) about \(i^{*}\) in set \(\text{S}_{0}\). Each data point represents a \(\left\{\text{G}_{0}(\Gamma_{0})+\text{G}_{1}(0)\right\}\) in \(\text{S}_{0}\). In Figure 6a, the orbit of the data indicated by the red box and blue box are shown in Table 5 and Figure 7a-d. It can be seen that \(\left\{\text{G}_{0}(\Gamma_{0})+\text{G}_{1}(0)\right\}\) is symmetrically distributed about \(\alpha_{0}=90^{\circ}\) and \(\alpha_{0}=270^{\circ}\) when \(a^{*}\in\left[0,180\right]^{\circ}\) and \(\alpha_{0}\in\left[180,360\right]^{\circ}\), respectively. In Figure 6b, the parameters of the data indicated by the red box and blue box are shown in Table 5. It can be known that the period of variation of \(\left\{\text{G}_{0}(\Gamma_{0})+\text{G}_{1}(0)\right\}\) about \(\beta_{0}\) is \(180^{\circ}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Meaning** & **Range of Ergodicity** \\ \hline \(\Delta\tau\) & Days between \(\overline{\tau}\) and \(\tau_{0}\) & The sweep interval is [1, \(\zeta_{1}\)] days. The sweep stepsize is \(\delta_{1}\). \\ \(||\Delta V_{0}||\) & The modulus of \(\Delta V_{0}\) & The sweep interval is [0.005, \(\zeta_{2}\)] km/s. The sweep stepsize is \(\delta_{2}\). \\ \((\alpha_{0},\beta_{0})\) & Two direction angles of \(\Delta V_{0}\) & The sweep interval is [0\({}^{\circ}\) 360\({}^{\circ}\)]. The sweep stepsize is \(\delta_{3}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of \(\Gamma_{0}(\boldsymbol{X}_{0})\) and their ergodic ranges. From Figure 6, we can see that lunar gravity can reduce the orbit altitude and raise the orbital inclination of the spacecraft. Figure 7g,h show the distribution of \(i_{1}\) and \(h_{p-earth}\) about \(||\Delta V_{0}||\). Figure 7g shows that orbital inclinations of all \(\{\mathrm{G}_{0}(\Gamma_{0})+\mathrm{G}_{1}(0)\}\) in \(\mathrm{S}_{0}\) are \(i_{1}<80^{\circ}\). Orbits of spacecraft are symmetrically distributed in the plane of \(||\Delta V_{0}||-i_{1}\) about \(i_{1}=24.88^{\circ}\), and \(24.88^{\circ}\) is the orbit inclination of the Earth-Moon plane when the spacecraft departs from DRO. This is because of the fact that \(i_{1}\) is related to the departure impulse. When \(\alpha_{0}=0\), the spacecraft's orbit is in the Earth-Moon plane both before and after the thrustless lunar gravity assist. Figure 7h shows that the minimum height of the perigee is 3559 km, which does not conform to the orbital height requirement of the flyby. It can be seen from Figure 7g,h that it is impossible for the DRO spacecraft to fly by the SSO target by applying a single impulse while satisfying the requirements shown in Equation (10). Thus, it is necessary to investigate strategies with at least two impulses.