## 3 Dynamical Results
### The Circular Restricted Three-Body Problem
#### 3.1.2 Orbit Selection  
For the coverage analysis, the concept of coverage time percentage (CTP) has been followed. The CTP of a region refers to the time that the region is covered by a satellite divided by the total time of simulation or a reference time, in our case the orbital period; a region is continuously covered if the CTP equals 100% [30]. As an initial analysis, the south polar region is considered as the area between latitudes 60 and 90. For example, a CTP value of 75% means that a satellite orbits above the south polar region for 75% of its orbital period; thus, the subsatellite point has a latitude component below the threshold value for 75% of the considered period.  
The CTP evaluation has been conducted for the entire halo family and led to the selection of three orbits (Table 2) as the result of a compromise between a suf ciently elongated shape tied to the CTP value, stability for proximity to the NRHO family, and  
Figure 4: Largest eigenvalue modulus as a function of the perilune distance for the L1 ( right ) and L2 (left ) Near-Rectilinear Halo Orbits. Credit: [29].  
Figure 5: Representation of the southern family of halo orbits around the lunar L1 and L2 Lagrange points. The orbits within the interval of stability are marked with a red colour. On the right side, a lateral (x, y) view is highlighted.  
avoiding lowering the orbit perilune overly (e.g., below 1000 km altitude); the rst one is chosen for its excellent performance in coverage, the second one for belonging to the NRHO family, and the latter for its sidereal resonance equal to 3. The integer values of resonance result in a more stable orbital behaviour.  
However, the high- delity model demonstrates how only the rst and third are viable solutions because the NRHO tends to lose the correct phase angle between satellites with the developed mission sequence. Figure 6 highlights this problem; the apolume oscillations cause a variation in the orbital period of each satellite that after some time results in latitude overlaps and the loss of continuous coverage properties. This type of constellation demands corrective manoeuvres to adjust the phasing, which add to those needed to ensure stability, thus resulting in an overall increase in cost.  
As can be seen from the CTP values in Table 2, a constellation of satellites with a precise initial true anomaly is required to ensure 24 h coverage. An example of two possible combinations of satellites that can theoretically ensure a 100% CTP is presented in Figure 7 by dividing the orbit into ten equidistant parts in time; clearly, the chosen orbit around L1 requires at least two satellites while the one around L2 requires at least three.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline Orbit & x\({}_{0}\) & z\({}_{0}\) & y\({}_{0}\) & Period & Sinodic & Sidereal & CTP [\%] & Perilune \\  & & & & & & & & Distance \\ \hline L1 & 0.9246 & 0.2180 & 0.1232 & 7.8781 & 3.75 & 3.48 & 80.31 & 3905.9 \\ \hline L2 & 1.0694 & 0.20105 & 0.1860 & 9.4966 & 3.11 & 2.88 & 42.32 & 13,798.6 \\ \hline L2 & 1.0634 & 0.2003 & 0.1770 & 9.1155 & 3.24 & 3.00 & 47.22 & 12,005.7 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Main orbital features of the three selected orbits from the preliminary design (low delity). Only the two highlighted are the subject of the high- delity analysis.  
Figure 6: Latitude of the subsatellite point as a function of elapsed days for a constellation of 3 equally angular spaced satellites on the chosen L2 NRHO. After about 180 days (red circled), it is possible to observe the total loss of phasing.