= Mass-distance degeneracy of an angular stellar orbit
{title2=$\mu/D^3=4\pi^2\alpha^3/P^2$}
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 rescaling
$$
D\mapsto\lambda D,\qquad a\mapsto\lambda a,\qquad \mu\mapsto\lambda^3\mu
$$
leaves 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> is
$$
K=\frac{2\pi a\sin i}{P\sqrt{1-e^2}}.
$$
For $\sin i\ne0$, combining it with the angular fit gives the <orbital parallax> distance
$$
D=\frac{KP\sqrt{1-e^2}}{2\pi\alpha\sin i}.
$$
The 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.
\Image[/stellar-orbit-mass-distance-degeneracy.png]
{title=Identical angular orbits can have different spectral velocity amplitudes}
{description=Original chosen-model family with fixed angular scale, period, eccentricity and orientation. Changing distance changes inferred mass cubically and the line-of-sight velocity amplitude linearly. The coloured velocity curves are predictions, not measured spectra or historical data.}
{source=stellar-orbit-mass-distance-degeneracy.py}
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