# A Lunar-Orbiting Satellite Constellation for Wireless Energy Supply
## 5 Orbits  
In this study, three different types of orbits are considered. The first one is optimal to transmit on a receiver located at the same latitude as the Shackleton Crater, with an inclination of \(89.54^{\circ}\). In this case we consider a lunar circular polar orbit with an altitude of 600 km. The satellites are evenly distributed and the true anomaly is computed as follows:\[\
u_{k}=\
u_{1}+\frac{2\pi(k-1)}{N_{satellites}} \tag{5}\]  
where \(k\) indicates the \(k\)-th satellite. This orbit, like all high-altitude lunar orbits, is subject to terrestrial disturbances and so it is astrodynamically inaccurate and would require an oversized propulsive system for station-keeping operations. For the purpose of this study, it is used nevertheless for calculating the limit of transmissible power for a given altitude. In Figure 6, the satellite constellation geometry is shown in this first case. In the following figures, the purple line indicates the orbit while the colored dots indicate the satellites.  
The second scenario considers a low elliptical lunar frozen orbit, found in [48]. The orbital parameters are shown in Table 1. The true anomaly is computed as in Equation (5) and the constellation is shown in Figure 7.  
\begin{table}
\begin{tabular}{c c} \hline \hline \(a\) & 1838 [km] \\ \(e\) & 0.01 \\ \(i\) & 88\({}^{\circ}\) \\ \(\omega\) & 270\({}^{\circ}\) \\ \(RAAN\) & 0\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Scenario 2: orbital parameters.  
Figure 6: Scenario 1: a ten-satellite constellation in lunar circular polar orbit.  
Figure 7: Scenario 2: a ten-satellite constellation in low elliptical lunar frozen orbit.  
In the last scenario, we assume a receiver on the lunar equator. In this case, an Earth-Moon distant retrograde orbit (DRO) is considered, which is neutrally stable with \(A_{x}=2538\) km, as defined in Figure 8.  
The circular restricted three body problem (CR3 BP) is utilized to model the dynamics of the Earth-Moon system, considering the Earth and Moon as the primary masses. This approach allows for the calculation of periodic orbits in close proximity to the Moon. However, many of these orbits, including Lyapunov and halo orbits, are highly unstable and can come dangerously close to the Moon's surface, sometimes within a few hundred meters. Nonetheless, there exist stable periodic orbits that are sufficiently distant from the Moon's surface, known as Earth-Moon DROs.  
Within the CR3 BP framework, periodic orbits can be classified into different types based on their symmetry: axis-symmetric, doubly symmetric, and planar (with most DROs falling into the planar and axis-symmetric category). To determine the initial conditions of a desired Earth-Moon DRO, a differential correction method [57] is employed. This method enables the estimation of a set of initial conditions which when integrated using the equations of motion of the CR3 BP will result in a periodic orbit over a specified time period, with the final state matching the initial conditions.  
Before employing the aforementioned differential correction method, the initial conditions (ICs) for a proposed Earth-Moon DRO are obtained using particle swarm optimization (PSO) [58]. PSO is a heuristic algorithm based on computational swarm intelligence techniques, where a collective group of simple agents, such as schools of fish, bird flocks, or insect colonies, collaboratively solve problems by combining genetic adaptation and social observation to install collective intelligence. The satellite constellation of the third case described above is represented in Figure 9.  
Figure 8: DRO schematization.