Fundamental orbital separation limit from stable mass transfer (red dashed line) seen in the population of BH+He-star systems formed in MESA binary models. Separations below ∼8−10 R_⊙ are excluded in the SMT channel due to mass transfer instability (dashed region, physical origin in Sect. 4.1). As a result, BBH mergers from SMT evolution have long delay times of ≳1 Gyr, and their BH+He-star progenitors tend to be outside the regime of tidal spin-up (P ≳ 0.8 day). Here, the orbital evolution during SMT assumes enhanced AM loss due to L2 outflows with γ ≈ 0.8γ_BH + 0.2γ_L2. However, models assuming a different γ yield a similar range of BH+He-star separations (Fig. 4). The solid red line indicates a hard lower limit for the separation of BH+He-star systems in the CE channel, set by the size of a naked helium star. The figure is based on a grid of BH+O-star models with M_BH; 1 = 10 M_⊙ and Z = 0.1 Z_⊙ metallicity, with similar results found also for larger BH masses (Sect. 3.4).

Fig. 4.

Narrow window in the parameter space for the SMT channel converging to a well-defined narrow range of final orbits regardless of uncertainties in binary physics. We plot the orbital periods of BBH merger progenitors from before the mass transfer (BH+O-star stage, left) and after the SMT phase has taken place (BH+He-star stage, right). The figure is based on grids of detailed binary models with M_BH; 1 = 10 M_⊙ and varying assumptions on the orbital shrinkage during mass transfer (i.e., γ values, shown in different colors). Changing the assumption on γ significantly shifts the pre-SMT periods of BBH merger progenitors, but has little effect on the post-interaction orbits at the BH+He-star stage (or the properties of BBH mergers). The dashed red line shows the lower limit on the period of a BH+He-star systems, such that the naked helium star does not overflow its Roche lobe. We mark the periods and mass ratios of the known BH+OB-star systems (left panel, including supergiant high mass X-ray binaries) and of BH+Wolf-Rayet X-ray binaries (right). Cyg X-3 cannot be reproduced by SMT in our models, suggesting its origin in CE evolution.