  Determining accurate orbits of binary stars with powerful winds is
challenging. The dense outflows increase the effective photospheric radius,
precluding direct observation of the Keplerian motion; instead the observables
are broad lines emitted over large radii in the stellar wind. Our analysis
reveals strong, systematic discrepancies between the radial velocities
extracted from different spectral lines: the more extended a line's emission
region, the greater the departure from the true orbital motion. To overcome
these challenges, we formulate a novel semi-analytical model which encapsulates
both the star's orbital motion and the propagation of the wind. The model
encodes the integrated velocity field of the out-flowing gas in terms of a
convolution of past motion due to the finite flow speed of the wind. We test
this model on two binary systems. (1), for the extreme case $\eta$ Carinae, in
which the effects are most prominent, we are able to fit the model to 10 Balmer
lines from H-alpha to H-kappa concurrently with a single set of orbital
parameters: time of periastron $T_{0}=2454848$ (JD), eccentricity $e=0.91$,
semi-amplitude $k=69$ km/s and longitude of periastron $\omega=241^\circ$. (2)
for a more typical case, the Wolf-Rayet star in RMC 140, we demonstrate that
for commonly used lines, such as He II and N III/IV/V, we expect deviations
between the Keplerian orbit and the predicted radial velocities. Our study
indicates that corrective modelling, such as presented here, is necessary in
order to identify a consistent set of orbital parameters, independent of the
emission line used, especially for future high accuracy work.
