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

Abstract: Radiation-driven winds of massive stars can be described within the modified CAK theory, which parametrises the radiation force through three key quantities: $α$, $δ$, and $k$. Different combinations of these parameters, together with rotation, result in three types of stationary solutions, namely fast (or classical), $δ$-slow, and $Ω$-slow solutions. The primary objective of this work is to model radiation-driven winds inside the gap region between the fast and $δ$-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 $δ$ 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 $β$-law using the same global wind parameters.

• The paper identifies new intermediate stationary wind solutions in B supergiants that seamlessly bridge classical fast and δ-slow regimes.
• It employs a time-dependent m-CAK framework with ZEUS-3D to reveal critical velocity profile kinks influenced by stellar rotation and line-force parameters.
• Synthetic spectral diagnostics across UV, optical, and IR confirm that subtle differences in wind structure significantly affect observable line profiles.

Hydrodynamic Structure of Radiation-Driven Winds at the Fast–Slow Transition in B Supergiants

Introduction

The structure and stability of radiation-driven winds in massive stars are critical for understanding both the evolution of individual objects and their feedback on galactic scales. This study systematically investigates the region of parameter space in modified CAK (m-CAK) theory where stationary solutions have hitherto been absent—specifically, the fast–$\delta$-slow transition in B supergiant winds. Leveraging the time-dependent capacities of ZEUS-3D, the work overcomes known numerical deficiencies in steady-state solvers and characterizes new families of solutions. In parallel, detailed non-LTE synthetic spectra are computed for diagnostic lines to demonstrate how the wind regime transitions are manifested observationally.

Time-Dependent m-CAK Framework and ZEUS-3D Approach

The authors approach the hydrodynamic modelling of radiation-driven winds using the full time-dependent equations, as opposed to reliance on stationary critical point analysis. The key parameters in this context are the line-force multipliers ($k$, $\alpha$, $\delta$) which encode the distribution and ionization sensitivity of radiative acceleration. In the m-CAK regime, depending on rotation rate $\Omega$ and $\delta$, classical fast and $\delta$-slow (high $\delta$) solutions exist with markedly different asymptotic velocities and mass-loss characteristics.

Historically, integration of the wind equations using codes such as Hydwind leaves a region of the ($k$, $\alpha$, $k$0) manifold inaccessible where neither fast nor $k$1-slow solutions can be enforced with regular critical point crossing. By utilizing ZEUS-3D, initial conditions are allowed to self-adjust via time evolution, which proves crucial in identifying physically acceptable stationary profiles within this "gap" region.

Figure 1: Temporal evolution of the wind velocity field in ZEUS-3D, demonstrating convergence from different initial guesses to stable stationary flows.

Gap Solutions: Structure and Properties

By exploring B supergiant models (e.g., $k$2 K, $k$3), the study delineates the solution topology as a function of $k$4 and $k$5. The classical regimes are clearly reproduced at extreme $k$6, but ZEUS-3D identifies previously unattainable, stable stationary profiles interpolating between the two. These "gap" solutions are neither abrupt nor pathological; rather, they exhibit a continuous morphing of velocity and density structures.

A distinctive feature of several intermediate solutions is a pronounced "kink" in the radial velocity profile, whose location and steepness are $k$7-dependent and modulated by rotation. These inflection points appear robust under extended temporal evolution, suggesting they are physically admissible and not numerical artifacts.

Figure 2: Hydrodynamic solutions for the T19 model, illustrating the continuous transition in wind structure with increasing $k$8 and rotation.

The corresponding mass-loss rates and terminal velocities show a monotonic progression across the transition, with the intermediate solutions partially bridging the gap between fast and $k$9-slow regimes in both observables.

Figure 3: Terminal velocity and mass-loss rate as functions of $\alpha$0 for varying rotation, highlighting the location of new solutions in the former "gap" region.

Synthetic Line Profiles and Spectroscopic Diagnostics

Synthetic spectra were computed via NLTE radiative transfer solutions in the comoving frame, including detailed models for H, He I, and Si IV. Across UV, optical, and IR transitions, line profile morphologies reflect the underlying wind regime. As the wind transitions from fast to $\alpha$1-slow, the blue absorption component of P Cygni profiles in the UV narrows and emission weakens, while Balmer and He I lines develop enhanced emission characteristics consistent with increased mass loss.

Figure 4: Synthetic profiles for Si IV, He I 5876 Å, H$\alpha$2, and He I+Br$\alpha$3 for representative fast, transition, and $\alpha$4-slow hydrodynamic solutions.

The intermediate "gap" solutions generate line morphologies that interpolate smoothly between the classical cases. This establishes that spectroscopic observations can potentially distinguish subtle underlying changes in wind hydrodynamics—especially when multiwavelength diagnostics are considered.

$\alpha$5-Law Prescriptions Versus Hydrodynamic Solutions

By comparing synthetic spectra computed with both direct hydrodynamic solutions and phenomenological $\alpha$6-law velocity structures (with equivalent global wind parameters), degeneracies are found in the predicted line profiles. Notably, H$\alpha$7 profiles with high $\alpha$8 can mimic those produced by $\alpha$9-slow or intermediate solutions, but the velocity stratification is markedly different. However, lines such as IR He I 4.049 μm are more sensitive to the underlying wind structure, revealing diagnostic leverage in disentangling the physical wind regime from spectroscopic observations.

Figure 5: Comparison of synthetic line profiles from hydrodynamic models and $\delta$0-type laws, demonstrating degeneracies (for H$\delta$1) and distinctions (for He I+Br$\delta$2).

Implications and Future Prospects

A key implication of this work is that the supposed absence of stationary solutions in the gap region is an artifact of prior stationary-state numerical schemes. In reality, the wind properties—terminal velocity, mass-loss rate, velocity profile—vary continuously as a function of $\delta$3 and stellar rotation. The localized kinks in the velocity profile may be observationally linked to wind variability and the development of discrete absorption components, phenomena already observed in OB stars.

The demonstrated sensitivity of line profiles to the exact hydrodynamic regime (and not just to global parameters) justifies the use of time-dependent hydrodynamic simulations in future analyses. It is suggested that time-variable ionization, which can modify effective $\delta$4, may drive regime changes in unstable or clumpy winds. More realistic non-isothermal and multidimensional models are needed to fully exploit upcoming UV/optical/IR spectroscopic facilities for wind diagnostics.

Conclusion

By implementing time-dependent, physically self-consistent hydrodynamics with ZEUS-3D, the longstanding gap between fast and $\delta$5-slow wind solutions in B supergiants is shown to be filled by a continuous succession of stationary flows. These intermediate solutions possess distinctive velocity stratifications, impact all major observational diagnostics, and their discovery resolves previous ambiguities in the interpretation of spectroscopic variability and mass-loss measurements. The methodology and insights provided by this study are broadly applicable to wind models of luminous early-type stars and form the foundation for more comprehensive time-dependent and multidimensional investigations into stellar wind physics.