Stellar evolution with rotation VI: The Eddington and Omega-limits, the rotational mass loss for OB and LBV stars

Several properties of massive stars with large effects of rotation and radiation are studied. We show that there are 2 roots for the equation giving the rotational velocities at break-up: 1) The usual solution, which is shown to apply when the Eddington ratio $\Gamma$ of the star is smaller than formally 0.639. 2) Above this value of $\Gamma$, there is a second root, inferior to the first one, for the break-up velocity. This second solution tends to zero, when $\Gamma$ tends towards 1. This second root results from the interplay of radiation and rotation, and in particular from the reduction by rotation of the effective mass in the local Eddington factor. The analysis made here should hopefully clarify a recent debate between Langer (\cite{La97,La98}) and Glatzel (\cite{Gla98}). The expression for the global mass loss-rates is a function of both $\Omega$ and $\Gamma$, and this may give raise to extreme mass loss-rates ($\Omega \Gamma $-limit). In particular, for O-type stars, LBV stars, supergiants and Wolf-Rayet stars, even slow rotation may dramatically enhance the mass loss rates. Numerical examples in the range of 9 to 120 M$_{\odot}$ at various $T_\mathrm{eff}$ are given.