# A Single-Launch Deployment Strategy for Lunar Constellations
## 4 Analysis of the Constellation Deployment
### Application of the Deployment Strategy  
Phase 1, consisting of the transfer to the equilibrium region, is common to all the satellites. Steps 1-2 of the algorithm reported in Section 3 are implemented in GMAT after selecting the following target parameters for the lunar flyby \(r_{p,a}=2037.1\pm 50\) km, \(y_{a}<0\) and \(i_{a}=30\pm 2\) deg, while the initial epoch is fixed to 17 October 2023, 00:00. The value of \(r_{p,a}\), corresponding to an altitude of \(300\pm 50\) km from the lunar surface, is selected because it represents an effective trade-off between the gravity-braking effect and the harmonic perturbations of the lunar gravitational field [55]. The analysis provides the orbital elements of the departure trajectory corresponding to the above-mentioned target parameters, both reported in Table 4. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Element** & **Value at Departure**1  & **Value at Arrival2 ** \\ \hline \(a\) & 202,700 km & \(-5694\) km \\ \(e\) & 0.968 & 1.369 \\ \(i\) & 33.72 deg & 28.50 deg \\ \(\Omega\) & 42.97 deg & 355.04 deg \\ \(\omega\) & 62.67 deg & 173.02 deg \\ \(\theta\) & 29.00 deg & 0 deg \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Orbital elements of the departure trajectory. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Element** & **Sat-1** & **Sat-2** & **Sat-3** & **Sat-4** & **Sat-5** \\ \hline \(a_{c}\) & 6541.4 km & 9873.0 km & 7500.0 km & 7494.7 km & 6541.4 km \\ \(e_{c}\) & 0.6 & 0.2 & 0.05 & 0.05 & 0.6 \\ \(i_{c}\) & 56.2 deg & 45 deg & 40 deg & 40 deg & 56.2 deg \\ \(\Omega_{c}\) & 50 deg & 30 deg & 50 deg & 10 deg & 140 deg \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Orbital elements for the satellites of the constellation. At the periselenium, the first maneuver performed by the launch vehicle concludes Phase 1, transferring it to a trajectory heading towards the equilibrium region. Based on previous analyses on capture in the Earth-Moon system [44], for a low-energy capture to incline the range reported in Table 3, an energy value \(h_{er}=2\times 10^{-2}\) is required. To achieve it, a \(\Delta V_{p}=228.4\) m/s is provided in the direction opposite to the orbital velocity. Figure 1 shows the trajectory of Phase 1 represented in the rotating frame \(\mathcal{F}_{r}\) with the origin shifted to \(x_{L_{1}}\). The injection into the equilibrium region is achieved on 1 November 2023, at 12:39, about 15.5 days after the departure epoch. The iterative process reported in Step 3 of the algorithm is implemented to determine, independently and for each satellite, the coordinate \(x_{er}\) at which the capture maneuver shall be performed to achieve the desired \(\Omega_{c}\) at capture. The following parameters are considered:  
1. the coordinate \(x_{er}\) is selected within the interval \([200,1200]\) km, to ensure that the satellites are maneuvered close enough to \(L_{1}\), where the linear model provides an accurate description of the dynamics;
2. after propagation to \(x_{er}\), the velocity variation \(\Delta V_{c}\) required to achieve \(i_{c}\) and \(r_{p,c}\) reported in Table 3 is calculated, assuming a tolerance of, respectively, 2 deg, 10 km and \(\varepsilon=10^{-10}\);
3. the capture trajectory is propagated to the periselenium and the value of the RAAN is verified, the process is repeated until the RAAN converges to \(\Omega_{c}\pm 2\) deg;
4. a final iteration allows refining the values of \(x_{er}\) and \(\Delta V_{c}\) to achieve the target orbital elements within a tolerance of 0.1 deg and 1 km. The process described above was applied to the five satellites obtaining the state variables reported in Table 5.