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

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.