# A Single-Launch Deployment Strategy for Lunar Constellations
## 3 Design of the Deployment Strategy
### Injection to the Low-Energy Trajectory  
In the first phase of the deployment strategy the launch vehicle, in which the satellites to be deployed are stowed, is transferred from its departure trajectory to the equilibrium region. The initial epoch \(t_{0}\) of the operations is fixed here to the end of the translunar injection (TLI) maneuver. The position of the spacecraft at \(t_{0}\) depends on the performance of the launch vehicle and on the launch profile [52]; therefore, it is reasonable to assume that the semimajor axis \(a_{d}\), eccentricity \(e_{d}\) and true anomaly \(\theta_{d}\) at \(t_{0}\) are given. By contrast, the launch date and the value of the other orbital elements at \(t_{0}\) (the RAAN \(\Omega_{d}\), the argument of perigee \(\omega_{d}\) and the inclination \(i_{d}\)) can be selected in the design phase.  
The injection into the equilibrium region is performed by taking advantage of lunar gravity-braking; therefore, the orbital elements \(\Omega_{d}\), \(\omega_{d}\), and \(i_{d}\) are selected such that the departure trajectory reaches the Moon at a minimum distance \(r_{p,a}\), inclination \(i_{a}\) and negative value for the position coordinate \(y_{a}\). In particular, the condition \(y_{a}<0\) is necessary for the spacecraft energy to decrease after the flyby (i.e., \(y_{a}>0\) would result in a gravity  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Phase:** & **Free** & **Constrained** \\ \hline Lunar flyby & \(\Omega_{d}\) ; \(\omega_{d}\) ; \(i_{d}\) & \(a_{d}\) ; \(e_{d}\) ; \(\theta_{d}\) \\ Injection into equilibrium region & \(\mathbf{V}_{p,a}\) & \(\mathbf{r}_{p,a}\) \\ Targeting capture parameters & \(v_{x,\sigma}\) ; \(v_{y,\sigma}\) ; \(v_{z,\sigma}\) & \(x_{\sigma}\) ; \(y_{\sigma}\) ; \(z_{\sigma}\) \\ Corrections & \(\mathbf{V}_{p,c}\) & \(\mathbf{r}_{p,c}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of free and constrained parameters during the different phases of the deployment.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Phase:** & **Control** & **Target** \\ \hline Lunar flyby & \(\Omega_{d}\) ; \(\omega_{d}\) ; \(i_{d}\) & \(r_{p,a}\) ; \(y_{a}\) ; \(i_{a}\) \\ Injection into equilibrium region & \(\Delta V_{x,a}\) ; \(\Delta V_{y,a}\) ; \(\Delta V_{z,a}\) & \(x_{\sigma}\) (\(\Omega_{c}\)) ; \(y_{\sigma}\) ; \(h_{\sigma}\) \\ Targeting capture parameters & \(v_{x,\sigma}\) ; \(v_{y,\sigma}\) ; \(v_{z,\sigma}\) & \(x_{1}\) ; \(i_{c}\) ; \(r_{p,c}\) \\ Corrections & \(\Delta V_{x,c}\) ; \(\Delta V_{y,a}\) ; \(\Delta V_{z,c}\) & \(e_{c}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of control and target parameters during the different phases of the deployment.  
assist) while the minimization of \(r_{p,a}\) and \(i_{a}\) is pursued to maximize the gravity-braking effect [53].  
Once reached, the pericenter \(\mathbf{r}_{p,a}\), a \(\Delta V_{p,a}\) is provided along the direction of the velocity to reduce the energy level to a target value \(h_{er}\) (see Section 3.2), injecting the launcher and its payload into a low-energy trajectory heading towards the equilibrium region. Finally, the satellites are released from the launcher and, from now on, they are maneuvered independently.