Cislunar navigation is an observability problem: arc-length thresholds, Gramian conditioning, and the Doppler lever

Résumé : Dilution of precision is the standard way to describe navigation geometry and it does not apply in cislunar space: its derivation presumes a snapshot solution in which simultaneous measurements determine the state at one instant. A spacecraft tracked by inter-satellite ranging in the Earth-Moon three-body problem has no such solution: range is a function of position only, so the velocity columns of the measurement Jacobian vanish identically and the state is unobservable however many links are in view. The correct figure of merit is dynamical, because the three-body flow carries velocity information into later range measurements. We demonstrate this on four differential-corrected planar distant retrograde orbits, reporting the observability Gramian rather than a dilution of precision. Instantaneous range-only geometry has rank 2 of 4; the full state becomes observable only after a tracking arc of 2.09 h, the first sampled arc at full rank on a 0.26 h grid, and a square-root information filter recomputing the same quantity by a different route becomes well-posed at exactly that arc. Beyond it, conditioning improves monotonically with arc length, the Gramian condition number falling from 1.2e11 at 3 h to 1.4e5 at 48 h. Adding range rate lifts the instantaneous rank from 2 to 4, so a Doppler-capable link makes a snapshot solution exist where a range-only link cannot. Lifting the known-reference assumption, the joint problem of estimating a station and the satellites' states and clocks together is rank-deficient by exactly one without an external interferometric observable in all twenty realizations tested, and full rank with it; the deficiency persists only to four satellites. For the configurations studied, cislunar navigation requirements are better written as arc-length, conditioning and measurement-type budgets than as dilution-of-precision thresholds, which are undefined here.