# 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
The calculation procedures are composed of two-dimensional ergodicity: traverse \(\Gamma_{0}(\mathbf{X}_{0})\) in the set \(\mathsf{S}_{0}\) and \(\Gamma_{1}(\mathbf{X}_{1})\) according to Table 7. The parameters determining \(\Gamma_{1}(\mathbf{X}_{1})\) in the CRTBP system are as Equation (15) shows:  
\[\mathbf{X}_{1}=[||\Delta\mathbf{V}_{1}||,\alpha_{1},\beta_{1}] \tag{15}\]  
where \(\alpha_{1}\) is the angle between \(\Delta\mathbf{V}_{1}\) and the projection of \(\Delta\mathbf{V}_{1}\) in the \(\hat{\mathbf{x}}-\hat{\mathbf{y}}\) plane, and \(\beta_{1}\) is the angle between the projection of \(\Delta\mathbf{V}_{1}\) in the \(\hat{\mathbf{x}}-\hat{\mathbf{y}}\) plane and \(\hat{\mathbf{x}}\). The equation of \(\Delta\mathbf{V}_{1}\) with respect to \((\alpha_{1},\beta_{1})\) is shown in Equation (16). \[\Delta\mathbf{V}_{1}= \left|\left|\Delta\mathbf{V}_{1}\right|\right|(\cos\alpha_{1}\cos \beta_{1},\cos\alpha_{1}\sin\beta_{1},\sin\alpha_{1})^{T} \tag{16}\]  
Let us traverse the parameters according to Table 7, and calculate the following parameters of \(\{\mathrm{G}_{0}(\Gamma_{0})+\mathrm{G}_{1}(\Gamma_{1})\}\): \(\Delta t_{1}\), the perilune height \(h_{p-moon}\), Jacobi constant \(J_{t}\) for \(\mathrm{G}_{1}(\Gamma_{1})\) after a powered lunar flyby (PLF), and the perigee height \(h_{p-earth}\). For each \(\{\mathrm{G}_{0}(\Gamma_{0})+\mathrm{G}_{1}(\Gamma_{1})\}\), we take these parameters into the pruning conditions shown in Equation (17). \[ \begin{array}{l}J_{t}\leq C_{1}\\ \widetilde{h}_{1}\leq h_{p-moon}\leq h_{SOI}\\ \widetilde{h}_{2}\leq h_{p-earth}\leq\widetilde{h}_{3}\end{array}  \tag{17}\]  
The third condition means that the height of the perigee after the PLF is similar to the orbital height of LEO, and the spacecraft will not enter the Earth's atmosphere. After pruning, there are 334,841 pairs of \(\{\mathrm{G}_{0}(\Gamma_{0})+\mathrm{G}_{1}(\Gamma_{1})\}\). The group composed of all orbits \(\{\mathrm{G}_{0}(\Gamma_{0})+\mathrm{G}_{1}(\Gamma_{1})\}\) is denoted as set \(\mathsf{S}_{1}\). When the parameters of \(\Gamma_{0}(\mathbf{X}_{0})\) are \(\Delta\tau=25\,\mathrm{days}\), \(||\Delta\mathbf{V}_{0}||=0.4\,\mathrm{km}/\mathrm{s}\), and \((\alpha_{0},\beta_{0})=(0,0)\), 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 \(180^{\circ}\). \begin{table}
\begin{tabular}{c c c} \hline
**Parameters** & **Meaning** & **Range of Ergodicity** \\ \hline \(\Gamma_{0}(\mathbf{X}_{0})\) & The first maneuver & \(\Gamma_{0}(\mathbf{X}_{0})\) of each \(\mathrm{G}_{0}(\Gamma_{0})\) in set \(\mathsf{S}_{0}\) \\ \(||\Delta\mathbf{V}_{1}||\) & The modulus of \(\Delta\mathbf{V}_{1}\) & The sweep interval is [0.005, \(\zeta_{3}\)] km/s. The sweep stepsize is \(\delta_{2}\). \\ \((\alpha_{1},\beta_{1})\) & Two direction angles of \(\Delta\mathbf{V}_{1}\) & The sweep interval is \([0,360^{\circ}]\). The sweep stepsize is \(\delta_{3}\). \\ \hline \end{tabular}
\end{table} 
Table 7: Parameters and their ergodic ranges.