Earth-baseline VLBI restores the observability of a lunar surface station in joint orbit-and-clock determination

Abstract

Lunar positioning, navigation, and timing (PNT) is moving from concept to hardware, ESA's Moonlight/LCNS, NovaMoon reference stations, LunaNet, and Coordinated Lunar Time, all reducing to one estimation core: fix the orbits and clocks of the lunar infrastructure and tie them to an Earth/inertial frame. We ask which measurements make a surface station's absolute position observable, and prove the answer. In a snapshot batch fit, the internal observables (station-to-satellite and inter-satellite ranging plus clock-sync) constrain only relative geometry and leave a six-dimensional rigid-body datum defect: three translations and three rotations of the cluster. The clocks are fully observable, so the defect is purely positional, and closing it needs a tie to the Earth frame. Two such ties exist and are not interchangeable. An indirect tie (Earth-to-satellite ranging through the constellation) reaches the station only when the satellite geometry is rich; a direct tie (an Earth-baseline VLBI delay to the station beacon) fixes it regardless. This gives a conditional design law, not a single number: VLBI restores absolute observability when the constellation cannot supply it, and merely sharpens the bound when it can. For a sparse three-satellite constellation the station lies in the null space of the Fisher information until VLBI is added, reaching a Cramer-Rao bound of 20.1 m; for a rich six-satellite constellation VLBI tightens the bound from 23.2 m to 9.7 m. A single-epoch baseline informs at most two of three axes, so the datum closes at three non-collinear Earth stations. The Gauss-Newton estimator attains the bound (efficiency 1.02), with a 91x median station-error improvement in the sparse regime. The FIM/CRLB engine is validated against NumPy and published closed forms; the lunar application stays modelled, every figure deterministic and reproducible.