# Lunar Satellite Constellations in Frozen Low Orbits
## 1 Introduction  
The steadily growing worldwide interest in lunar exploration and numerous plans to establish a lunar base encourage the study and development of multi-spacecraft constellations around the Moon for communication and navigation purposes. A low-altitude constellation of a cheap small spacecraft that provides low signal latency and is well suited for both optical and laser short-range intersatellite links would be a very attractive option. However, pursuing this, one should deal with two key problems. The first problem is the efficient constellation deployment. In contrast to the case of Earth-orbiting constellations, the multiple-launch deployment scheme, implying at least one dedicated launch followed by a high-energy transfer for each of the orbital planes, can be prohibitively expensive for large distributed systems around the Moon. Therefore, most of the researchers have been focused on constellations in high lunar orbits [1; 2; 3; 4] and libration point orbits [5; 6], with a low number of spacecraft and cheaper launches. A promising solution that potentially allows one not to abandon the use of low lunar orbits (LLOs) seems to be in leveraging a ballistic lunar transfer (BLT), a special kind of low-energy lunar transfers exploiting the solar gravitational perturbation. In the course of a long BLT, several groups of spacecraft can be successively separated from a single platform by small impulses and then captured into orbits with different right ascension of the ascending node values [7]. The second problem to be solved in order for a small satellite lunar constellation to be feasible is to select long-term stable orbits that uniformly surround the Moon. Its irregular gravity field substantially limits the range of suitable lunar orbits--especially of low altitude--with a reasonable lifetime. Due to the fact that traditional approaches developed for the design of Earth constellations were not aimed at exploiting complex naturaldynamics in the circumlunar space, they require much propellant to reject gravitational perturbations. For example, the operational lifetime of the lunar constellation made of 3U CubeSats with a standard monopropellant thruster does not exceed 100 days [8]. The only viable solution is represented by frozen orbits, a special class of orbits with nearly constant mean values of the eccentricity vector components, \(e_{x}=e\cos\omega\) and \(e_{y}=e\sin\omega\). Here, \(e\) is the eccentricity and \(\omega\) is the argument of periapsis. The Earth's gravity is a dominant source of perturbation in high lunar orbits, whereas in LLOs, the condition of frozenness is fully dictated by the Moon's gravity. Medium lunar orbits experience both effects. The present paper further develops and expands the study of frozen orbit lunar constellations [9]. A two-stage non-gradient optimization procedure is devised to search for frozen LLOs. At the first stage, a coarse global minimum search for eccentricity vector periodic behavior over the one-year time interval is done by the robust Bayesian optimizer augmented with the sequential domain reduction. After that, at the second stage, the solution found is fine-tuned by the Nelder-Mead method. Such an approach is quite flexible; it allowed us to design lunar frozen orbits so that their orbital elements meet user-defined box constraints. A useful nomogram with basic visibility parameters and lower bound global coverage curves, first presented below, is employed for the assessment whether a given constellation design is suitable for communication or navigation needs. If so, a set of frozen orbits as much close to those from the considered design as possible is generated; their coverage and stability characteristics are evaluated. The paper has the following structure. The next chapter introduces the dynamical model used in the study, including a simple approximation rule of how to truncate the lunar gravitational potential in case of propagating low to medium orbits.