# A Single-Launch Deployment Strategy for Lunar Constellations
## 2 Low-Energy Captures in the Sun-Earth-Moon System
### Extension to the Sun-Earth-Moon System  
The design of low-energy captures by the Moon, as well as the deployment strategy derived from it, cannot ignore two conditions that are overlooked in the CR3 BP model: neither the eccentricity of the Earth and Moon orbits nor the gravitational field of the Sun are negligible in the real environment [45; 46]. The topological characterization presented in Section 2.2 is here extended to the more accurate dynamic framework of the elliptic restricted four-body problem (ER4 BP) that is used to model the Sun-Earth-Moon system. The dynamic equations of motion are developed in a reference frame \(\mathcal{F}_{r}^{(e)}=[\hat{x},\hat{y},\hat{z}]\) centered in the center of mass of the system, with \(\hat{x}\) pointing from the Sun (\(m_{0}\)) to the center of mass of the Earth-Moon system, \(\hat{z}\) orthogonal to the ecliptic plane and \(\hat{y}\) completing the rectangular frame [47]. Denoting by \(r_{E}\) and \(\theta_{E}\) the distance and the true anomaly of the Earth-Moon center of mass \(O\) with respect to the Sun, the equations can be expressed as follows [48]  
\[ \begin{cases}x^{\prime}=v_{x}\\ y^{\prime}=v_{y}\\ z^{\prime}=v_{z}\\ v_{x}^{\prime}=2v_{y}+\tau\Big{(}\frac{\partial u}{\partial x}+x\Big{)}\\ v_{y}^{\prime}=-2v_{x}+\tau\Big{(}\frac{\partial u}{\partial y}+y\Big{)}\\ v_{z}^{\prime}=\tau\Big{(}\frac{\partial u}{\partial z}+z\Big{)}-z\end{cases}  \tag{13}\]where \(\tau=\left(1+e_{M}\cos\theta_{E}\right)^{-1}\), \(\mu_{i}=\frac{m_{i}}{m_{0}+m_{1}+m_{2}}\), \(m_{0}\) is the mass of the Sun, \(u=\sum\limits_{i=1}^{3}\frac{\mu_{i}}{r_{i}}\), and the prime \({}^{\prime}\) indicates the derivative with respect to \(\theta_{E}\). Equation (13) are expressed in terms of the units of distance \(DU(e)=r_{E}(\theta_{E})\) and time \(TU(e)=\sqrt{\frac{r_{E}^{3}(\theta_{E})}{\mathfrak{C}(m_{0}+m_{1}+m_{2})}}\) where \(G\) is the gravitational constant. Because of its dependence on time, the ER4 BP does not admit equilibrium points. Nevertheless, their instantaneous dynamic substitutes can be computed at any given time considering the corresponding geometric configuration of the primaries [48]. Operating as in Section 2.2, the set of Equation (13) can be linearized about the instantaneous libration point \(L_{1}\) of the Earth-Moon system and expressed using the Hamiltonian formalism using the following Hamiltonian function [49]  
\[ \begin{array}{c}H^{(e)}=\frac{1}{2}\big{(}p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\big{)} +p_{1}q_{2}-p_{2}q_{1}-\frac{K}{2}\big{(}2q_{1}^{2}-q_{2}^{2}-q_{3}^{2}\big{)} +\\ +e_{M}c_{1}\cos\theta_{E}+\frac{\mu_{0}}{r_{0}}c_{2}\end{array}  \tag{14}\]  
where \(r_{0}\) denotes the mean distance between the Sun and the center of mass of the Earth-Moon system and the expressions for the coefficients \(c_{i}\) are reported in Appendix A. It can be observed that the expressions of \(H\) and \(H^{(e)}\) are equivalent, except for the last two terms of Equation (14), which represent the effects of the eccentric motion of the primaries (\(e_{M}\)) and the solar gravitational perturbation (\(\frac{\mu_{0}}{r_{0}}\)). Examining in detail the two coefficients, it is easy to verify that for the Earth-Moon system \(e_{M}<<1\) and in the neighborhood of \(L_{1}\) also \(\frac{\mu_{0}}{r_{0}}<<1\), so they can be regarded as small perturbations acting onto the CR3 BP. [PERSON] and [PERSON] proved that the basic topological properties of the CR3 BP in the neighborhood of \(L_{1}\) are preserved in the presence of small perturbations [50].