# 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  
This subsection intends to investigate the feasibility of using a single-impulse strategy to perform the SSO object flyby mission. The DRO-Earth transfer orbit with a single impulse is denoted as \(\left\{\mathrm{G}_{0}(\mathrm{\Gamma}_{0})+\mathrm{G}_{1}(0)\right\}\). The spacecraft applies \(\mathrm{\Gamma}_{0}(\mathbf{X}_{0})\) in DRO and flies to the perilune. Then, the spacecraft takes the thrustless lunar flyby, i.e., \(\Delta\mathbf{V}_{1}=0\), and flies towards the Earth. The requirements of the single-impulse DRO-Earth transfer orbit are as follows:  
\[ \begin{array}{l}\Delta\tau\leq\zeta_{1},\ \Delta t_{0}\leq\zeta_{1}\\ \Big{|}\big{|}\Delta\mathbf{V}_{0}\Big{|}\Big{|}\leq\zeta_{2},\ i_{1}\geq\widetilde{ i}_{1}\\ \widetilde{h}_{1}\leq h_{p-moon}\leq h_{SOI}\\ \widetilde{h}_{2}\leq h_{p-earth}\leq\widetilde{h}_{3}\end{array}  \tag{10}\]  
where \(\zeta_{1}\), \(\zeta_{2}\), \(\widetilde{h}_{1}\), and \(\widetilde{h}_{2}\) are identical to those of Table 3, \(h_{SOI}\) is the height of the lunar SOI, and the radius of lunar SOI is set to 61,525 km. \(\widetilde{h}_{3}=2000\) km. \(i_{1}\) is the orbital inclination of the spacecraft when it arrives at the perigee and \(\widetilde{i}_{1}=80^{\circ}\). In order to investigate the feasibility of the single-impulse approach, this section studies whether the requirements in Equation (10) can be satisfied under different parameters determining \(\mathrm{\Gamma}_{0}(\mathbf{X}_{0})\). The parameters determining \(\mathrm{\Gamma}_{0}(\mathbf{X}_{0})\) in the CRTBP system are as follows:  
\[\mathbf{X}_{0}=[\Delta\tau,||\Delta\mathbf{V}_{0}||,\alpha_{0},\beta_{0}] \tag{11}\]  
As shown in Figure 5b, \(\alpha_{0}\) is the angle between \(\Delta\mathbf{V}_{0}\) and the projection of \(\Delta\mathbf{V}_{0}\) in the \(\hat{\mathbf{x}}-\hat{\mathbf{y}}\) plane, and \(\beta_{0}\) is the angle between the projection of \(\Delta\mathbf{V}_{0}\) in the \(\hat{\mathbf{x}}-\hat{\mathbf{y}}\) plane and \(\hat{\mathbf{x}}\). The equation of \(\Delta\mathbf{V}_{0}\) with respect to \((\alpha_{0},\beta_{0})\) is shown in Equation (12). \[\Delta\mathbf{V}_{0}=||\Delta\mathbf{V}_{0}||(\cos\alpha_{0}\cos\beta_{0},\cos\alpha_ {0}\sin\beta_{0},\sin\alpha_{0})^{T} \tag{12}\]  
The calculation procedure includes large-scale ergodicity according to Table 4 and verification of the satisfaction of Equation (10) for each \(\mathrm{\Gamma}_{0}(\mathbf{X}_{0})\), followed by pruning. In Table 4, this paper chooses the stepsizes as \(\delta_{1}=1\) day, \(\delta_{2}=0.005\) km\(/\)s, and \(\delta_{3}=10^{\circ}\). Let us traverse the parameters according to Table 4 and calculate the values of the following variables in \(\left\{\mathrm{G}_{0}(\mathrm{\Gamma}_{0})+\mathrm{G}_{1}(0)\right\}\) for each set of parameters: \(\Delta t_{0}\), the perilune height \(h_{p-moon}\), Jacobi constant \(J_{t}\) for \(\mathrm{G}_{1}\) after one thrustless lunar gravity assist, and the perigee height \(h_{p-earth}\). From Equation (10), we can see that the maximum flight time of the spacecraft from the perilune to the perigee is 31 days. Since there may be more than one perigee, \(\mathrm{G}_{1}\) is the orbit of the spacecraft from the perilune to the first perigee that conforms to the pruning condition of \(h_{p-earth}\). The pruning conditions are shown in Equation (13). \[ \begin{split} J_{t}&\leq C_{1}\\ \widetilde{h}_{1}&\leq h_{p-moon}\leq h_{SOI}\\ \widetilde{h}_{2}&\leq h_{p-earth}\leq 36,000\ \text{km} \end{split}  \tag{13}\]  
where \(C_{1}=3.024150064\) is the Jacobi constant of point \(L_{1}\) in CRTBP. Equation (13) includes three inequalities. The first one means that the cislunar space is connected. The second one means that the spacecraft could use a lunar gravity assist without colliding with the Moon. The third inequality means that the spacecraft could reach the Earth without entering the Earth's atmosphere. Note that, in this subsection, the second maneuver \(\Delta V_{1}\) is assumed to be zero.