# Lunar Satellite Constellations in Frozen Low Orbits
## 5 Candidate Constellations Performance and Stability
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\mathbf{h_{\text{ref}}}=\mathbf{261}\) km & \(\mathbf{N_{\text{global}}}\) & \(\mathbf{N_{\text{North}}}\) & \(\mathbf{N_{\text{South}}}\) & \(\mathbf{N_{\text{Bogus}}}\) & \(\mathbf{PDOP_{\text{Npole}}}\) & \(\mathbf{PDOP_{\text{Spole}}}\) & \(\mathbf{PDOP_{\text{Bogus}}}\) \\ \hline \(84^{\circ}\): \(18/2/1\) & min 0 & min 1 & min 1 & min 1 & N/A & N/A & N/A \\ \(d_{\text{min}}=1.01\) km & med 0 & med 2 & med 2 & med 2 & & \\ \(84^{\circ}\): \(54/3/1\) & min 0 & min 4 & min 4 & min 4 & 95\% 6.5 & 95\% 6.9 & 95\% 10.9 \\ \(d_{\text{min}}=0.37\) km & med 2 & med 7 & med 6 & med 6 & med 2.0 & med 2.1 & med 3.0 \\ \(84^{\circ}\): \(70/5/1\) & min 1 & min 4 & min 4 & min 4 & 95\% 3.2 & 95\% 3.4 & 95\% 4.8 \\ \(d_{\text{min}}=0.54\) km & med 3 & med 9 & med 8 & med 1.9 & med 2.0 & med 2.4 \\ \hline \(\mathbf{h_{\text{ref}}}=\mathbf{522}\) km & \(\mathbf{N_{\text{global}}}\) & \(\mathbf{N_{\text{North}}}\) & \(\mathbf{N_{\text{South}}}\) & \(\mathbf{N_{\text{Bogus}}}\) & \(\mathbf{PDOP_{\text{Npole}}}\) & \(\mathbf{PDOP_{\text{Spole}}}\) & \(\mathbf{PDOP_{\text{Bogus}}}\) \\ \hline \(84^{\circ}\): \(36/3/1\) & min 1 & min 5 & min 5 & min 5 & 95\% 7.2 & 95\% 7.4 & 95\% 8.2 \\ \(d_{\text{min}}=2.66\) km & med 3 & med 7 & med 6 & med 6 & med 1.8 & med 1.8 & med 2.3 \\ \(84^{\circ}\): \(99/3/1\) & min 4 & min 16 & min 15 & min 16 & 95\% 1.1 & 95\% 1.1 & 95\% 1.5 \\ \(d_{\text{min}}=1.30\) km & med 7 & med 19 & med 18 & med 18 & med 1.0 & med 1.0 & med 1.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Coverage performance for the candidate constellations. Figure 10: Five candidate LLO constellations marked on the CoDe nomogram. We should remark that the conducted analysis is not an exhaustive search for those LLO constellations that can be made frozen for a long time and also possess good performance metrics for lunar communication and navigation. The numerical results play an illustrative role to demonstrate opportunities the proposed techniques and tools provide to a mission designer. Figure 11: One-year evolution of orbital elements for orbits in the 84\({}^{\circ}\) : 54/3/1 constellation (\(h_{\rm ref}=261\) km). Different colors correspond to orbits with different orbital planes. Figure 12: One-year evolution of orbital elements for orbits in the 84\({}^{\circ}\) : 70/5/1 constellation (\(h_{\rm ref}=261\) km). Different colors correspond to orbits with different orbital planes. It was interesting to reveal that, though the frozen LLOs seem to be long-term stable, they are actually unstable, though very mildly so. This instability, induced by the complex effect of lunar gravitational harmonics, can be proved based on the positive sign of the  
Figure 13: One-year evolution of orbital elements for orbits in the \(84^{\circ}:36/3/1\) constellation (\(h_{\rm ref}=522\) km). Different colors correspond to orbits with different orbital planes. maximum Lyapunov characteristic exponent. Specifically, we estimate the leading finite-time Lyapunov exponent (FTLE):  
\[\lambda(t;t_{0})=\frac{\ln\|\boldsymbol{\Phi}(t;t_{0})\|_{2}}{t-t_{0}}\]  
by integrating the variational equations for the state transition matrix \(\boldsymbol{\Phi}(t;t_{0})\) alongside the equations of motion (\(\|\cdot\|_{2}\) is the spectral norm of matrix). When we start increasing the propagation interval, the leading FTLE tends to some limit value. For the set of frozen orbits with \(h_{\text{ref}}=261\) km, this value appears to equal \(3.5\times 10^{-4}\) and almost does not depend on a specific orbit. In the case of \(h_{\text{ref}}=522\) km, the result is similar: \(3\times 10^{-4}\). Converting it to dimensional time units and inverting yields the characteristic instability time (Lyapunov time) of 1.1 to 1.3 months. According to existing studies of spacecraft motion in the unstable dynamical environment, the instability time provides us with a good estimate for the optimal frequency of impulsive station-keeping burns [31] or optimal update time for low-thrust orbit control [32].