Context. Cygnus X-3 is the only known Galactic high-mass X-ray binary with a Wolf-Rayet companion. Recent X-ray polarimetry results with the Imaging X-ray Polarimetry Explorer have revealed it is a concealed ultraluminous X-ray source. It is also the first source for which pronounced orbital variability of X-ray polarization has been detected, notably with only one polarization maximum per orbit.

Aims. Polarization caused by scattering of the source X-rays can only be orbitally variable if the scattering angles change throughout the orbit. Since this requires an asymmetrically distributed medium around the compact object, the observed variability traces the intrabinary structures. The single-peaked profile further imposes constraints on the possible geometry of the surrounding medium. Therefore, the X-ray polarization of Cygnus X-3 offers an opportunity to study the wind structures of high-mass X-ray binaries in detail. We aim to uncover the underlying geometry through analytical modeling of the variable polarization. Knowledge of these structures could be extended to other sources with similar wind-binary interactions.

Methods. We studied the variability caused by single scattering in the intrabinary bow shock, exploring both the optically thin and optically thick limits. We considered two geometries for the reflecting medium: the axisymmetric parabolic bow shock and the parabolic cylinder shock. Finally, we determined which geometry offers the best match to the X-ray polarimetric data.

Results. Qualitatively, we find that the peculiar properties of the data can only be replicated with a cylindrical bow shock with asymmetry across the shock centerline and significant optical depth. This geometry is comparable to shocks formed by the jet-wind or outflow-wind interactions. In addition, the orbital axis is slightly misaligned from the observed orientation of the radio jet in all our model fits. © The Authors 2025.