Radiation-driven stellar winds at the fast-slow transition: new hydrodynamic solutions

Radiation-driven winds of massive stars can be described within the modified CAK theory, which parametrises the radiation force through three key quantities: $\alpha$, $\delta$, and $k$. Different combinations of these parameters, together with rotation, result in three types of stationary solutions, namely fast (or classical), $\delta$-slow, and $\mathrm{\Omega }$-slow solutions. The primary objective of this work is to model radiation-driven winds inside the gap region between the fast and $\delta$-slow regimes, where stationary solutions have proven elusive. In addition, we compute synthetic line profiles of H I, He I, and Si IV to illustrate the morphology of different wind regimes. We employ the time-dependent hydrodynamic code ZEUS-3D, capable of obtaining stationary solutions by progressing through an initial solution. Then we compute the line profiles solving the transfer equation for an expanding atmosphere, assuming spherical symmetry in the comoving frame, under non-local thermodynamic equilibrium (NLTE) conditions. We found new stationary solutions in the gap region, alongside their corresponding line profiles, for a typical B supergiant star model. In this model, the new solutions are stable, and some of them present a kink in the velocity profile at a fixed distance from the star, depending on the $\delta$ value. Perturbations in the wind ionisation may trigger transitions between different hydrodynamic regimes and offer a plausible explanation for structured and variable winds. A systematic investigation of these effects will be the subject of future work. Furthermore, we investigate the resulting line profiles from different hydrodynamic solutions and compare them with those predicted by a velocity profile given by a $\beta$-law using the same global wind parameters.