# Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit
## 6 Numerical Simulations and Analysis
\(\mathbf{v}_{0}\) is the initial velocity of P\({}_{1}\), and the velocity increment is as follows:  
\[\Delta\mathbf{v}=\mathbf{v}_{1}-\mathbf{v}_{0} \tag{45}\]  
Decomposing \(\mathbf{v}_{1}\) in the chord direction and in the radial direction yields the following:  
\[\mathbf{v}_{1}=\mathbf{v}_{p1}+\mathbf{v}_{c1} \tag{46}\] \[\mathbf{v}_{\rho 1}=\big{|}\big{|}|\mathbf{v}_{\rho 1}|\big{|}|e_{r1},\mathbf{v}_{ c1}=||\mathbf{v}_{c1}||\mathbf{e}_{c}\] \[\big{|}\big{|}\mathbf{v}_{\rho 1}\big{|}\big{|}=\frac{c\sqrt{p}}{|| \mathbf{r}_{1}||\cdot||\mathbf{v}_{2}||\sin\theta},\big{|}\mathbf{v}_{c1}\big{|}=\sqrt{ \frac{p}{p}\frac{1-\cos\theta}{\sin\theta}}\]  
where  
\[\mathbf{e}_{r1}=\frac{\mathbf{r}_{1}}{||\mathbf{r}_{1}||},\mathbf{e}_{c}=\frac{ \mathbf{r}_{2}-\mathbf{r}_{1}}{||\mathbf{r}_{2}-\mathbf{r}_{1}||} \tag{47}\] \[c=\big{|}|\mathbf{r}_{2}-\mathbf{r}_{1}|\big{|},p=a_{1}(1-\epsilon_{1}^ {2})\]  
The optimal \(\mathbf{v}_{1}\) should satisfy the following:  
\[(\mathbf{v}_{\rho 1}-\mathbf{v}_{c1})\bullet(\mathbf{v}_{p1}+\mathbf{v}_{c1}-\mathbf{v}_{0})=0 \tag{48}\]  
which means that the optimal \(\Delta\mathbf{v}\) should satisfy the following:  
\[(\mathbf{v}_{\rho 1}-\mathbf{v}_{c1})\bot\Delta\mathbf{v} \tag{49}\]  
Equations (45)-(49) present the conditions that need to be satisfied for the optimal single-impulse transfer in the field of the two-body system. At \(\tau_{2}\), the spacecraft completes the 0 th revolution. In the 1 st revolution, \(\tau_{2m}=\tau_{2}+dt_{1}\), \(\tau_{3}=\tau_{2m}+dt_{2}\). The spacecraft's position at \(\tau_{2m}\) on the 1 st revolution orbit is denoted as P\({}_{1}\), and it applies an 8.7 m/s impulse at P\({}_{1}\). Then, the spacecraft's orbit is integrated from \(\tau_{2m}\) to \(\tau_{3}\) in the two-body system. The position at \(\tau_{3}\) is denoted as P\({}_{2}\). Substituting these parameters into Equations (45)-(49) yields that the angle between \(\Delta\mathbf{v}\) and \((\mathbf{v}_{\rho 1}-\mathbf{v}_{c1})\) is 91.24\({}^{\circ}\) or 271.24\({}^{\circ}\). Therefore, in the two-body system, the impulse in the two-body system is near optimal. Figure 16b shows the spacecraft's orbit from \(\tau_{2}\) to \(\tau_{3}\) in the CRTBP system and the two-body system, respectively. The spacecraft applies an impulse at \(\tau_{2m}\) and flies by the target debris 25732 at \(\tau_{3}\). During this period, the Moon is far away from the spacecraft; as a result, the Moon has little influence on the spacecraft. Thus, the impulse calculated by the semi-analytical optimization method is near optimal in the CRTBP system. It is worthwhile to compare the transfer strategy from DRO to the debris orbit in this paper to the other low-energy transfer. For the problem of LEO constellation design, reference [52] proposes a strategy in which all satellites carried by several vehicles transfer  
Figure 16: (**a**) Schematic of optimal single-impulse transfer; (**b**) the 1 st revolution orbit in the CRTBP system and the two-body system. from a single launch to the lunar L\({}_{1}\) halo orbit; each vehicle needs 0.6 km/s impulse for halo orbit insertion. Each satellite can return to the desired LEO with small propulsive burns (minimal \(\Delta V\) in Earth return trajectory and 0.12 km/s to adjust the perigee). Compared with reference [52], the DRO-Earth transfer strategy presented in this paper is more fuel-efficient because the DRO parking orbit is not only long-term stable but also requires less than 0.3 km/s impulse to depart. Based on reference [52], reference [53] proposes a similar transfer strategy that uses the manifold, with six vehicles carrying 36 satellites transferring to L\({}_{1}\) halo orbit. Then, the satellites perform station-keeping and transfer back to LEO in different epochs when the orbital planes of the desired orbits are appropriate. The average total time of flight of the return trajectories is 15.98 days (12.34 days for station-keeping and 3.66 days for transfer).