Line-driven Radiative Winds in B-Supergiants: Bridging the Gap between Fast and Slow m-CAK Solutions

Abstract

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.

Figures (7)

• Figure 1: Maps of terminal velocity ($v_\infty$, left panels) and mass-loss rate ($\dot{M}$, right panels) in the $(\Omega, \delta)$ parameter space. From top to bottom: Model T15 (late B-supergiant), T19 (mid B-supergiant), and T25 (early B-supergiant). The color gradient indicates the magnitude of the stable stationary solution found by our time-dependent simulations. The white solid rectangles overlaid on the T15 and T19 maps indicate the "forbidden regions" (gaps) reported in previous stationary studies Venero2016, where no solution was found. Our results demonstrate continuous stable solutions across the entire domain, filling these gaps.
• Figure 2: Quantitative comparison between our time-dependent simulations (blue lines/circles) and the stationary results from Venero2016 (black squares) for model T19 with no rotation ($\Omega=0.0$). Top panel: Terminal velocity vs. ionization parameter $\delta$. Bottom panel: Mass-loss rate vs. $\delta$. The gray hatched area indicates the "gap" reported in the stationary study. Note the inflection in the velocity curve across the gap and the bounded increase in $\dot{M}$ in the slow wind regime, contrasting with the stationary jump (see text for discussion).
• Figure 3: Wind structure for model T19 ($\Omega=0.0$) in the fast regime ($\delta=0.10$, blue solid line), gap region ($\delta=0.26$, green dashed line), and slow regime ($\delta=0.40$, red dotted line). Top panel: Radial velocity profiles. The solution within the gap exhibits a velocity plateau or "kink" extending from $2\,R_\star$ to $10\,R_\star$. Bottom panel: Mass density profiles (log scale). The density profiles are consistent with their corresponding velocity regimes, where lower velocities result in higher density levels as required by mass conservation.
• Figure 4: Global existence maps for the weak driving regime (Model T19 with $k=0.1$). Left panel: Terminal velocity $v_\infty$. Right panel: Mass-loss rate $\dot{M}$. Unlike the standard models, $\dot{M}$ decreases drastically as $\delta$ increases, dropping by orders of magnitude into a "dead wind" state for $\delta > 0.30$. However, stable solutions are found across the entire domain.
• Figure 5: Wind structure for the weak driving regime ($k=0.1, \Omega=0.0$). Top panel: Terminal velocity vs. $\delta$. Note the smooth inflection in the slope inside the grey region (the gap reported by Venero+2024). Bottom panel: Mass-loss rate vs. $\delta$. The simulation captures the steep decline in $\dot{M}$ associated with the structural discontinuity in weak winds.