# Lunar Satellite Constellations in Frozen Low Orbits
## 4 Constellation Design and Performance Metrics  
The preliminary step in the design of frozen LLO constellations is to select an orbital configuration that we subsequently attempt to make frozen. In this study, we restrict ourselves to Walker-Mozhaev symmetric constellations in identical circular orbits distributed uniformly around the Moon. Four major parameters are to be defined for such a constellation:  
1. \(N\), the total number of satellites;
2. \(P\), the number of orbital planes;
3. \(h_{\mathrm{ref}}\), the reference altitude of orbits in all the planes;
4. \(F\), the phase shift (in pattern units) between the respective satellites in adjacent planes. Recall that the pattern unit \(PU=360^{\circ}/N\) was introduced by [PERSON] ([25], p. 7) as a convenient unit to characterize the geometric pattern of a symmetric constellation. Without loss of generality, only values between \(0\) and \(P-1\) can be considered for the phase shift because any other value of F describes the same pattern as \(F\) mod \(P\). Moreover, for near-polar constellations, the \(F\) and \(P-F\) patterns are almost identical ([25], p. 20). The zero shift is almost never applied due to its inferior coverage and collision safety properties. Thus, for two-plane and three-plane constellations, only the case of \(F=1\) is worth being considered. For four-plane and five-plane constellations, patterns with \(F=1\) and \(F=2\) are to be studied, but we did not observe any significant difference between the performance characteristics of these options. So, using the standard notation \(i^{\circ}\colon N/P/F\), we will further focus on \(84^{\circ}\colon N/P/1\) constellations. The design of symmetric constellations in identical circular orbits uniformly distributed around the spherical celestial body has been widely studied in a large number of papers since the pioneering works of [PERSON] [25; 26] and [PERSON] [27; 28]. If a satellite orbits the body of radius \(R\) at altitude \(h\), it covers the spot with the central half-angle \(\varphi\) (referred to as the footprint size in the below text) such that:  
\[\cos\beta=\frac{R+h}{R}\sin\frac{\alpha}{2}=\frac{R+h}{R}\cos(\varphi+\beta)\]  
where \(\beta\) is the minimum elevation angle at which an orbiting satellite is still considered visible (Figure 7). Taking the value of \(\beta=5^{\circ}\) typically adopted for atmosphereless bodies such as the Moon, we have:  
\[\frac{h}{R}=\frac{\cos 5^{\circ}}{\cos(\varphi^{\circ}+5^{\circ})}-1\]  
This gives a one-to-one correspondence between the orbital altitude and the footprint size. The other quantities uniquely determined by the footprint size are the lower bound for the number of orbital planes:  
\[P_{\mathrm{min}}=\left\lceil\frac{180^{\circ}}{2\varphi^{\circ}}\right\rceil\]  
and the minimum number of satellites in each plane:  
\[\left(N/P\right)_{\mathrm{min}}=\left\lceil\frac{360^{\circ}}{2\varphi^{\circ }}\right\rceil\]required for one-fold global continuous coverage. As for the total number of satellites in a single-coverage constellation, its static estimate from below:  
\[N_{1f}=2+\frac{60^{\circ}}{60^{\circ}-\arctan\sqrt{3}\cos\varphi}\]  
is derived in the classical paper of [PERSON] [29]. Note that a simpler, but very close estimate can also be used:  
\[N_{1f}\approx P_{\min}\cdot\left(N/P\right)_{\min}=162\cdot\left[\left(\frac{10 ^{\circ}}{\varphi^{\circ}}\right)^{2}\,\right]\]  
The lower bound estimate for four-fold global continuous coverage by Walker-Mozhaev constellations is much harder to get analytically. To the best of our knowledge, it has not been published yet. The simplified estimate:  
\[N_{4f}=4\cdot N_{1f}\]  
derived for street-of-coverage constellations is, therefore, utilized, despite such an estimate being slightly conservative for Walker-Mozhaev constellations.