Line-driven Radiative Winds in B-Supergiants: Bridging the Gap between Fast and Slow m-CAK Solutions
The modified Castor, Abbott, and Klein (m-CAK) theory predicts different wind regimes based on the line force parameter for changes in ionization ($δ$) and the rotation parameter ($Ω$). Stationary hydrodynamic studies have reported ''forbidden regions'' or gaps in this parameter space where no steady-state solution exists, suggesting physical instabilities. We investigate the stability of wind solutions within these gaps for B-supergiants to determine if they correspond to physical instabilities or numerical artifacts. We perform 1D time-dependent hydrodynamic simulations, systematically exploring the full $(Ω, δ)$ space for three B-supergiant models ($T_{\rm eff}=15-25$ kK), adopting a fixed density boundary condition. Our simulations reveal stable stationary solutions continuously across the entire parameter space, effectively filling the reported gaps. The transition from fast to slow regimes is smooth but structurally complex. Within the gap, the velocity profile develops a distinct ''kink'' or extended plateau in the supersonic flow, allowing the wind to reach a stable state. The mass-loss rate ($\dot{M}$) varies smoothly without artificial jumps. We find that the $\dot{M}$ gradient depends on the radiative driving strength ($k$): while $\dot{M}$ increases with $δ$ for standard driving ($k \approx 0.32$), it decreases for the weak-driving regime ($k = 0.1$), consistent with stationary predictions. Moreover, in this regime, the final solution depends on the initial flow acceleration, confirming multiple hydrodynamic solutions. We conclude the m-CAK solution space is continuous; reported forbidden regions are artifacts of stationary methods. Time-dependent simulations effectively bridge the regimes, suggesting these transitions correspond to metastable states.