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. For stars with shellular rotation, i.e. stars with a constant angular velocity Omega on horizontal surfaces (cf. Zahn \cite{Za92}), we show that the equation of stellar surface has no significant departures with respect to the Roche model; high radiation pressure does not modify this property. Also, we note that contrarily to some current expressions, the correct Eddington factors Gamma in a rotating star explicitely depend on rotation. As a consequence, the maximum possible stellar luminosity is reduced by rotation. 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_sun at various T_eff are given. Mass loss from rotating stars is anisotropic. Polar ejection is favoured by the higher T_eff at the polar caps (g_eff-effect), while the ejection of an equatorial ring is favoured by the opacity effect (kappa -effect), if the opacity grows fastly for decreasing T_eff.