# A Single-Launch Deployment Strategy for Lunar Constellations
## 3 Design of the Deployment Strategy
### Deployment to the Final Orbit  
The characterization presented in Section 2.2 indicates that, once the satellites have reached the equilibrium region, a small change in the velocity \(\Delta V_{er}\) can be implemented to transfer each one of them to a low-energy capture with desired inclination \(i_{c}\) and periselenium radius \(r_{p,c}\). The components \(\Delta V_{x,er}\), \(\Delta V_{y,er}\) and \(\Delta V_{z,er}\) shall be selected to satisfy the following conditions  
\[\left\{ \begin{array}{l}x_{1}=\varepsilon\\ \frac{h_{2}}{h_{1}}=\frac{1}{\cos^{2}i_{c}}-1\\ \tan^{-1}\left(\frac{Im(x_{2})}{1+Re(x_{2})}\right)=\frac{2r_{p,c}}{r_{SOI}} \\ y_{2}=-j\hat{x}_{2}\\ y_{3}=-j\hat{x}_{3}\end{array} \right. \tag{16}\]  
with \(\varepsilon>0\) arbitrarily small and \(r_{SOI}=a_{M}\left(\frac{m_{2}}{m_{1}}\right)^{\frac{2}{5}}\) is the radius of the sphere of influence of the Moon. Equation (16) represents five constraints in the six transformed Hamiltonian variables \((\mathbf{x},\mathbf{y})\). The first three represent the capture condition and the inclination and periselenium radius at capture, while the last two result from the properties of the canonical transformation \(\mathbf{T}_{N}\) given by Equation (9). For the sake of clarity, Equation (16) can be transformed by making explicit the dependence on the position space variables  
\[\left\{ \begin{array}{l}\frac{\left(1+2\gamma^{2}\right)(x-x_{ L_{1}})+\left(1-\gamma^{2}\right)y+\alpha v_{x}-\alpha\sigma v_{y}}{2[(1- \gamma^{2})\sigma^{2}+\rho^{2}]}=\varepsilon\\ v_{z}=\sqrt{\frac{2h_{1}\tan^{2}i_{c}}{\lambda_{2}}}\\ \frac{t_{5,1}x+t_{5,5}v_{y}}{1+t_{2,2}y+t_{2}\alpha v_{x}}=\tan\left(\frac{2r _{p,c}}{r_{SOI}}\right)\\ h_{1}=2\lambda_{1}(t_{2,2}y+t_{2,4}v_{x})\left(t_{5,1}x+t_{5,5}v_{y}\right)\\ h_{2}=2\lambda_{2}(t_{3,3}z)(t_{6,6}v_{2})=h-h_{1}\end{array} \right. \tag{17}\]  
where \(\gamma^{2}=\left(-\frac{\mu}{|x_{L_{1}}+\mu-1|^{3}}-\frac{1-\mu}{|x_{L_{1}}+ \mu|^{3}}\right)\), \(\sigma=-2\rho/\left(\rho^{2}+\gamma^{2}-1\right)\), \(\tau=\frac{2(\lambda_{2}^{2}-1)}{\lambda_{2}^{2}-3\lambda_{2}^{2}+2}\) and \(t_{5,1}=0.1764\), \(t_{2,2}=1.0387\), \(t_{2,4}=0.1409\), \(t_{5,5}=0.5053\), \(t_{3,3}=1.0651\), \(t_{6,6}=0.4694\) are the \(t_{i,j}\) elements of the \(\mathbf{T}_{N}\) matrix. Once a suitable combination of \(\varepsilon\), \(i_{c}\),\(r_{p,c}\) and \(h\) has been fixed, Equation (17) can be used to select five out of the six position and velocity coordinates. The last degree of freedom allows the selection of the RAAN at capture \(\Omega_{c}\). The value of \(\Omega_{c}\) is strictly related to the time \(t^{\star}\) at which the low-energy capture trajectory crosses the Moon equatorial plane for the first time, which can be delayed or anticipated by changing the time at which the \(\Delta V_{er}\) is provided. Recalling now that all the satellites enter the equilibrium region at the same time, a simple way to select different values of \(\Omega_{c}\) for different satellites is to provide the \(\Delta V_{er}\) at different coordinates \(x\). In any case, because a closed-form solution equivalent to Equation (16) to relate the difference in \(x\) to that in \(\Omega_{c}\) is not available, the problem must be solved iteratively. Nevertheless, this does not represent an actual issue, since numerical solvers can find a solution in a few iterations. As mentioned above, providing \(\Delta V_{\sigma r}\) changes the energy level from the value \(h_{\sigma r}\), before the maneuver, to \(h_{c}\). Therefore, another iterative process can be implemented to minimize the difference between the energy level \(h_{\sigma r}\) and \(h_{c}\). Even though this step is not strictly required, it is useful to minimize the total \(\Delta V\). The low-energy capture trajectory obtained from the process above is inherently chaotic; consequently, the values of its orbital elements change in time affecting the performance of the constellation.