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

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 Ω-slow solutions (a denser and slower wind) when the effects of gravity darkening and oblateness are considered. This Ω-slow solution appears when the rotational speed is ≳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 Ω-slow solutions. Results. Using our stationary numerical code H<ns0:sc>YDWIND</ns0:sc>, 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 Ω-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.