Direct Determination of a Non-Cooperative LEO Satellite’s Orbital State from Ground-Station Signals: A Cramér–Rao Analysis

This paper considers passive determination of the orbital state of a non-cooperative low-Earth-orbit satellite from radio signals received at several ground stations. Reversing the roles of direct geolocation in which a satellite with known ephemeris locates a ground emitter, the satellite’s six-dimensional epoch state is estimated in a single step from the received data. Because the trajectory is a deterministic function of this state, every candidate is a valid orbit. The Cramér–Rao bound is derived; it factors into signal, geometry and orbital-dynamics terms, and a closed-form Keplerian state-transition block is obtained by implicit differentiation of the universal Kepler equation. The delay model includes the light-time correction, which leaves the bound essentially unchanged but removes a bias of more than a hundred meters. The unknown emission epoch is treated as a nuisance; it roughly doubles the three-station bound, whereas four stations recover the known-epoch accuracy. Simulations validate the dynamics implementation and demonstrate post-acquisition efficiency under a Gaussian delay/Doppler surrogate of the known-waveform model. Under an independent-cell model of acquisition failures, an ideal exhaustive search predicts a threshold about 6 dB below that of a two-step baseline with hard per-window peak decisions. The paper also quantifies how accuracy scales with arc length, station count and station layout.