# A Single-Launch Deployment Strategy for Lunar Constellations
## 2 Low-Energy Captures in the Sun-Earth-Moon System
### Characterization of Low-Energy Trajectories Crossing the Equilibrium Region  
It is known that the ultimate behavior of low-energy trajectories characterized by a low energy level \(h\) can be predicted based on their state representation inside the equilibrium region, i.e., the phase space surrounding the libration point \(L_{1}\)[34; 40]. The rigorous definition of the equilibrium region and the representation are developed hereafter using the Hamiltonian formalism and the results are later applied to design trajectories that, after crossing the equilibrium region, reach the Moon with desired semimajor axis, eccentricity, inclination, and RAAN. The Hamilton's equations are given by  
\[ \begin{cases}\dot{\mathbf{q}}=\frac{\partial H}{\partial\mathbf{p}}\\ \dot{\mathbf{p}}=-\frac{\partial H}{\partial\mathbf{q}}\end{cases}  \tag{5}\]  
where \(\mathbf{q}=[x-x_{L_{1}},y,z]^{T}\) and \(\mathbf{p}=[v_{x}-y,v_{y}+x-x_{L_{1}},v_{z}]^{T}\) are the position and conjugate momenta. The Hamiltonian function for the CR3 BP is by definition \(H=-C/2\). Aiming at investigating the dynamics in the neighborhood of \(L_{1}\), the following expression for \(H\) can be conveniently derived from quadratic expansion of Equation (4) about \(L_{1}\)[35]  
\[H(\mathbf{q},\mathbf{p})=\frac{1}{2}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+2p_{1}q_{2} -2p_{2}q_{1})-\frac{K}{2}(2q_{1}^{2}-q_{2}^{2}-q_{3}^{2}) \tag{6}\]  
with \(K=\frac{1-\mu}{|x_{L_{1}}+\mu|^{2}}+\frac{\mu}{|x_{L_{1}}+\mu-1|^{2}}\). The linear system associated with Equations (5) and (6) is characterized by a saddle-center-center type of equilibrium, having one couple of real eigenvalues (\(\pm\rho\)) and two couples of complex conjugate ones (\(\pm j\lambda_{1}\), \(\pm j\lambda_{2}\)). Therefore, according to [PERSON]'s lemma, the Hamiltonian function can be represented as the sum of three local integrals of motion, each dependent on a different pair of state variables and associated with one of the three above-mentioned eigenspaces. This new form of \(H\) can be conveniently derived by applying a canonical transformation \([\mathbf{x},\mathbf{y}]^{T}=\mathbf{T}_{N}[\mathbf{q},\mathbf{p}]^{T}\), originally proposed by [PERSON] and [PERSON] [41], producing  
\[H(\mathbf{x},\mathbf{y})=\rho x_{1}y_{1}+\frac{\lambda_{1}}{2}(x_{2}^{2}+y_{2} ^{2})+\frac{\lambda_{2}}{2}(x_{3}^{2}+y_{3}^{2})=h \tag{7}\]  
with \(x_{1},y_{1}\in\mathbb{R}\), \(x_{2},x_{3},y_{2},y_{3}\in\mathbb{C}\) and \(h>0\) is the arbitrarily small energy level. Equation (7) is the mathematical representation of the equilibrium region, the phase space in the neighborhood of \(L_{1}\). Based on a theorem by [PERSON] [42], [PERSON] proved that if a low-energy trajectory is inside the equilibrium region at a given time \(t_{0}\), then its long-term behavior is fully characterized by its topological location in the equilibrium region and provided the following classification [34]  
* \(x_{1}(t_{0})y_{1}(t_{0})<0\), transit trajectories which evolve alternately around one of the two primaries, crossing the equilibrium region multiple times
* \(x_{1}(t_{0})y_{1}(t_{0})>0\), bouncing trajectories which never cross the equilibrium region, thus evolve only around one of the two primaries
* \(x_{1}(t_{0})\to 0\) or \(y_{1}(t_{0})\to 0\), capture trajectories that cross only once the equilibrium region and then evolve around one of the primaries indefinitely in time
* \(x_{1}(t_{0})=y_{1}(t_{0})=0\), quasi-periodic orbits which never depart from the equilibrium region. Within the scope of this research, trajectories transiting from the Earth to the Moon (i.e., transits or captures) are of interest. In particular, low-energy captures are attractive because they do not require any further maneuvering to keep the satellites in the neighborhood of the Moon, thus ensuring relaxed operation times to perform any eventual corrective maneuver.