Effect of gravity darkening and oblate factor in rapidly rotating massive stars

TL;DR

The paper investigates how gravity darkening and stellar oblateness in rapidly rotating massive stars affect the transition between fast and $\Omega$-slow winds within the 1D equatorial m-CAK framework. Using the Hydwind code, it maps the co-existence region of wind solutions and analyzes the separate and combined impacts of gravity darkening and oblateness on wind topology, mass-loss rates, and terminal velocities, including sensitivity to the line-force parameter $\alpha$. Key findings show that gravity darkening shifts the fast-to-$\Omega$-slow transition to higher rotational speeds and can alter mass-loss trends in the $\Omega$-slow regime, while oblateness alone does not produce a co-existence region; when both effects are included, the transition shifts further and the co-existence is generally suppressed. The work highlights the limitations of 1D modeling for the full global wind morphology and calls for multidimensional simulations that incorporate non-radial line forces and viscous processes to realistically assess equatorial disk formation around Be- and B[e]-type stars.

Abstract

Context. Rapid rotation in massive stars leads to gravity darkening and oblateness, significantly affecting their radiation-driven winds. These effects can alter wind dynamics and play a role in forming slowly equatorial outflowing winds. Aims. This work investigates the transition region where the fast solution (i.e. high terminal velocities) of radiation-driven winds in a massive rotating star, in the frame of the modified-CAK theory, switches to the Omega-slow solutions (a denser and slower wind) when the effects of gravity darkening and oblateness are considered. This Omega-slow solution appears when the rotational speed is higher and equal to 75% of the critical rotation speed. Methods. To explore the transition region for various equatorial models of B-type stars, we focus on the co-existence interval where both solutions simultaneously exist and the transition point where fast solutions switch to Omega-slow solutions. Results. Using our stationary numerical code Hydwind, we first analyse the individual effects of gravity darkening and stellar oblateness caused by high rotational speeds and then examine their combined impact on the wind solutions. Conclusions. We find that for a certain range of rotational speeds, both the fast and Omega-slow solutions can co-exist, and the co-existence range strongly depends on the initial conditions. When only gravity darkening is considered, the co-existing interval shifts towards higher rotational speeds. While in the presence of the oblateness, the co-existing interval also occurs at higher rotational speeds; however, it is less than the gravity darkening effect. We also explored how line-force parameters affect the critical point, the location of the co-existing interval, and where the solution switches.

Figures (6)

• Figure 1: Mass-loss rate and terminal velocity as a function of the rotational speed. The upper panel shows the mass-loss rate, normalised to the non-rotating case. The lower panel shows the terminal velocities. These values are obtained using a stationary hydrodynamic solution with the Hydwind code, accounting for the gravity darkening (GD) effect.
• Figure 2: Line-force parameter effects on wind solutions. The upper panel displays mass-loss rates normalised to the non-rotating case, and the lower panel shows terminal velocities. Solutions were derived from the B0 IV model in the presence of the GD, with systematically varied line-force parameter $\alpha$ at constant $k=0.3$.
• Figure 3: Line-force parameter effects on wind solutions. Same as Fig. \ref{['2']}, but for the B2.5 V model.
• Figure 4: Line-force parameter effects on wind solutions. Same as Fig. \ref{['2']}, but for the B3 I model.
• Figure 5: Stellar oblateness effect on wind parameters. The upper panel shows the mass-loss rates, and the lower panel shows the terminal velocities as a function of the rotation parameter $\Omega$. Calculations use the original B0 IV model parameters.