Stellar evolution with rotation. IX. The effects of the production of asymmetric nebulae on the internal evolution

The anisotropies of the mass loss by stellar winds, which lead to asymmetric nebulae, influence the loss of angular momentum. Polar enhanced mass loss is embarking less angular momentum than isotropic mass loss, while equatorial mass loss is removing more angular momentum. Thus, the evolution of a star and of its rotation is also influenced by the anisotropies. We give the basic equations expressing the evolution of the angular momentum for a rotating star experiencing mass loss by anisotropic stellar winds, with account for differential rotation, meridional circulation and shear diffusion. In the general case, the outer layers must be studied with a time dependent boundary conditions. However, for low enough mass loss rates, a stationary situation can be established at the stellar surface. It implies a positive Omega -gradient for polar mass loss and a negative Omega -gradient for a dominant equatorial mass loss. At the opposite, for extremely high mass loss rates (like for LBV and WR stars), the outer layers are removed before the torque associated to the anisotropies has the time to be transmitted inward. We show that for the fastest rotating O-type stars (between 25 and 60 M_sun and for an average rotation velocity during the MS phase v >= 300 km s^-1), the anisotropies of the mass loss may significantly influence the evolution of the stellar velocities and may lead the stars to break-up.