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Stellar evolution with rotation IV: von Zeipel's theorem and anisotropic losses of mass and angular momentum

The von Zeipel theorem is generalised to account for differential rotation in the case of a “shellular” rotation law (cf. Zahn 1992). We write this law in the form Ω = Ω(r), a simplification which does not apply to fast rotation. We find that von Zeipel’s relation contains a small additional term, generally further increasing the radiative flux at the pole and decreasing it at the equator. We also examine the local Eddington factor in rotating stars and notice some significant differences with respect to current expressions. We examine the latitudinal dependence of the mass loss rates Ṁ(θ) in rotating stars and find two main source of wind anisotropies: 1) the “geff” effect which enhances the polar ejection; 2) the “opacity effect” (or “κ–effect”), which favours equatorial ejection. In O–stars the geff effect is expected to largely dominate. In B– and later type stars the opacity effect should favour equatorial ejection and the formation of equatorial rings. We also examine the behaviour of the wind density and notice a strong enhancement at the equator of B– and later type stars. Possible relations with the polar ejections and the skirt of η Carinae and with the inner and outer rings of SN 1987 A are mentioned. If Ṁ(θ) has sharp extrema due to some peaks in the opacity law, non equatorial and symmetrical rings may be produced. We also show that the global mass loss rate of a star at a given location in the HR diagram is rapidly increasing with rotation, which is in good agreement with the numerical models by Friend & Abbott (1986). Anisotropic stellar winds remove selectively the angular momentum. For example, winds passing through polar caps in O–stars remove very little angular momentum, an excess of angular momentum is thus retained and rapidly redistributed by horizontal turbulence. These excesses may lead some Wolf– Rayet stars, those resulting directly from O–stars, to be fast spinning objects, while we predict that the WR–stars which have passed through the red supergiant phase will have lower rotation velocities on the average. We also show how anisotropic ejection can be treated in numerical models by properly modifying the outer boundary conditions for the transport of angular momentum. Finally, in an Appendix the equation of the surface for stars with shellular rotation is discussed.