# Lunar Satellite Constellations in Frozen Low Orbits
## 5 Candidate Constellations Performance and Stability  
In the framework of this study, several dozens of LLO constellations have been examined based on the the databases of frozen orbits for \(h_{\mathrm{ref}}=261\) km and \(h_{\mathrm{ref}}=522\) km. The osculating elements of the frozen orbits designed are given in Tables 2 and 3. The initial epoch is midnight, 1 January 2022. Among the number of different designs, we will further focus only on five configurations that seem to be best suited for the goals of global or local lunar surface coverage of a multiplicity high enough to ensure communication and navigation services. To visualize the parameters of the selected constellations, the five star marks are displayed on the CoDe nomogram (see Figure 10), with the \(y\)-value for each of the marks being equal to the total number of satellites (in 100 s) and the labels next to the marks designating the numbers of orbital planes and satellites per plane. Out of the five candidate constellations, three correspond to \(h_{\mathrm{ref}}=261\) km, whereas the other two are designed for \(h_{\mathrm{ref}}=522\) km. The major performance metrics of the candidate constellations are summarized in Table 4. Note that the minumum distance between the constellation satellites is also tracked in addition to earlier mentioned metrics since it is a key safety parameter. Based on the obtained results, one can conclude that three constellations (\(84^{\circ}\): \(54/3/1\), \(84^{\circ}\): \(70/5/1\), \(84^{\circ}\): \(36/3/1\)) are capable of providing navigation for the lunar polar zones, including the targeted craters and the poles themselves. Navigation quality is very good almost all the time. It is especially true for the \(84^{\circ}\): \(70/5/1\) constellation. Apart from high-precision polar navigation, it also provides the global communication service. The same property is shared by the \(84^{\circ}\): \(36/3/1\) constellation. Close performance can also be attained with significantly smaller numbers of satellites and orbital planes, but at the cost of four times higher FSPL due to the doubled altitude. In general, however, the larger the orbital altitude, the greater the minimum intersatellite distance and the safer the mission. So, performance and safety dictate the altitude to be as high as the satellite power budget allows. Figure 9: Quasi-uniform surface grid of 1200 points with 8 sites included by default; among them are the North and South lunar poles and the Boguslawsky (green) and Manzinus (purple) craters. The one-year evolution of the eccentricity vector and the semimajor axis for each orbital plane of the three constellations is shown in Figures 11-13. To avoid overlapping, we displayed the curves for just one satellite in each orbital plane; the other satellites in the same plane have a very similar evolution. The figures confirm that all the constellation orbits are indeed frozen and remain near-circular. Note that a slight variation in the semimajor axis from plane to plane serves as the main instrument for meeting the frozenness condition. If one seeks guaranteed navigation globally, over the entire lunar surface, the constellation size has to be significantly larger. For a three-plane design of the 522 km constellation, the number of satellites in each plane should be increased almost threefold: the \(84^{\circ}\): \(99/3/1\) constellation is the first attaining global continuous coverage. On the other part of the range, among the downsized constellations, the \(84^{\circ}\): \(18/2/1\) constellation at a 261 km altitude seems to be the most promising if a continuous up/downlink for the lunar base in the polar region is required. There are no other lunar constellations orbiting so low that can perform more efficiently.