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Determining accurate orbits of binary stars with powerful winds ischallenging. The dense outflows increase the effective photospheric radius,precluding direct observation of the Keplerian motion; instead the observablesare broad lines emitted over large radii in the stellar wind. Our analysisreveals strong, systematic discrepancies between the radial velocitiesextracted from different spectral lines: the more extended a line's emissionregion, the greater the departure from the true orbital motion. To overcomethese challenges, we formulate a novel semi-analytical model which encapsulatesboth the star's orbital motion and the propagation of the wind. The modelencodes the integrated velocity field of the out-flowing gas in terms of aconvolution of past motion due to the finite flow speed of the wind. We testthis model on two binary systems. (1), for the extreme case $\eta$ Carinae, inwhich the effects are most prominent, we are able to fit the model to 10 Balmerlines from H-alpha to H-kappa concurrently with a single set of orbitalparameters: 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 thatfor commonly used lines, such as He II and N III/IV/V, we expect deviationsbetween the Keplerian orbit and the predicted radial velocities. Our studyindicates that corrective modelling, such as presented here, is necessary inorder to identify a consistent set of orbital parameters, independent of theemission line used, especially for future high accuracy work.

Uncovering the orbital dynamics of stars hidden inside their powerful winds: application to $\eta$ Carinae and RMC 140