 Mass-distance degeneracy of an angular stellar orbit

Angular astrometry determines the combination displayed by substituting $a=D\alpha$ into Kepler's third law. Keeping the fitted shape, orientation and orbital period fixed, the rescalingleaves the entire angular time series unchanged. A longer physical orbit around a larger mass can trace precisely the same angular path. More angular samples improve the fitted combination; by themselves they cannot break this scaling symmetry of the model.

The radial-velocity semi-amplitude isFor $\sin i\ne 0$, combining it with the angular fit gives the orbital parallax distanceThe spectral velocity scale supplies the missing physical ruler. For a face-on orbit, $\sin i=0$, this radial signal vanishes and that method cannot supply a distance. In a partial or noisy data set, fitted orientation and systemic motion also correlate with the inferred scale.