The Cost of Lunar South-Polar Geometry, and Surface Beacons as the Efficient Fix: A Dilution-of-Precision Analysis
Lunar PNT architectures, NASA's Lunar Augmented Navigation Service (LANS), ESA's Moonlight, and allied concepts, place a small number of satellites in elliptical lunar frozen orbits (ELFO) to serve the south-polar region prioritized for exploration. We report a result that reframes the design trade: for a user at the lunar south pole, the satellite count needed to reach good geometry is roughly double what is currently planned, because the visible satellites cluster into a small solid angle overhead and dilution of precision is limited by their angular spread rather than their number. In a time-averaged simulation, orbit-only ELFO constellations of the planned size (4 to 6 satellites) give a south-polar median geometric DOP (GDOP) of 16 to 21, far worse than the GDOP of about 6 routine for terrestrial GNSS, and the constellation must grow to about 12 satellites before the median GDOP crosses 6. We then show that a small number of surface ranging beacons, a configuration absent from the lunar PNT literature, reaches the same geometric quality far more cheaply by supplying the near-horizon diversity the overhead cluster lacks: three beacons on elevated terrain around a -80 deg latitude user cut the median GDOP from 16.2 to 1.6, a factor of about 10, moving the user from 15% to 100% of the time below GDOP 6, geometry a purely orbital solution reaches only near a 24-satellite fleet. Because there is no atmospheric refraction, surface-to-surface line of sight is bounded by the geometric horizon, so beacon siting on crater rims and elevated terrain is itself a design variable. Surface-beacon augmentation is the lowest-cost, highest-leverage improvement available to lunar south-polar PNT, deployable on assets already planned for the region. The geometry engine is Validated against an independent DOP computation; the constellation and beacon scenario are Modelled.