Analytical approach to the design of optimal satellite constellations for space-based space situational awareness applications
In recent years, the accumulation of space debris has become an increasingly pressing issue, and adequately monitoring it is a formidable task for designated ground-based sensors. Supplementing the capabilities of these ground-based networks with orbiting sensing platforms would dramatically enhance the ability of such systems to detect, track, identify, and characterize resident space objects -- the primary goals of modern space situational awareness (SSA). Space-based space situational awareness (SBSSA), then, is concerned with achieving the stated SSA goals through coordinated orbiting sensing platforms. To facilitate the design of satellite constellations that promote SSA goals, an optimization approach is selected, which inherently requires a pre-defined mathematical representation of a cost index or measure of merit. Such representations are often analytically available, but when considering optimal constellation design for SBSSA applications, a closed-form expression for the cost index is only available under certain assumptions. The present study focuses on a subset of cases that admit exact representations. In this case, geometrical arguments are employed to establish an analytical formulation for the coverage area provided as well as for the coverage multiplicity. These analytical results are essential in validating numerical approximations that are able to simulate more complex configurations.