1 Introduction

Generally in stellar evolution, it is considered that "details'' concerning the stellar surface, like the anisotropies of the stellar flux, do not affect the internal stellar evolution. The well known reason, as evidenced in basic textbooks, is that the outer boundary conditions have less and less influence as we consider deeper stellar layers. We shall show here that the anisotropies of the radiation flux at the stellar surface may in some cases significantly affect the evolution of a massive star.

The anisotropies of the radiation flux in a rotating star result from the von Zeipel theorem (von Zeipel 1924; for the case of differential rotation, see also Maeder 1999). This theorem essentially says that the local flux on a rotating star is proportional to the effective local gravity $g_{{\rm eff}}$. Thus, the flux is higher at the pole and weaker at the equator, and so does also the effective temperature $T_{{\rm eff}}$.

Such differences of $T_{{\rm eff}}$ over the stellar surface also influence the local flux of mass loss (see also Pelupessy et al. 2000). Models of radiation driven winds from rotating stars with account of 2-D and non-LTE effects have also been recently developed by Petrenz & Puls (2000). The analytical expressions of the theory of stellar winds on a rotating star have been developed by Maeder & Meynet (2000). The main result is that the mass flux shows large deviations from spherical symmetry, with 2 main factors causing the anisotropies: 1) The higher polar $T_{{\rm eff}}$favours polar mass ejections ( $g_{{\rm eff}}$-effect). 2) The lower equatorial $T_{{\rm eff}}$may lead to larger opacities, thus favouring higher equatorial mass loss or even leading to an equatorial ring ejection ($\kappa$-effect). The relative importance of these two effects depends on rotation and on the average stellar $T_{{\rm eff}}$. In O-type and early B-type stars, the opacity is mainly electron scattering opacity and thus there is little or no significant equatorial enhancement of the opacity, and thus in general the main effect is the $g_{{\rm eff}}$-effect, which leads to bipolar outflows. An equatorial ejection with the formation of an equatorial ring is likely to occur when the equatorial regions become cooler than $T_{{\rm eff}} = 21~500$ K (corresponding to spectral type B1.5). Then there are rapid opacity growths or jumps (called bi-stability limits) where the force multipliers (see Lamers et al. 1995; Kudritzki & Puls 2000) characterising the opacities undergo strong changes.

Most nebulae ejected by stars are asymmetric. This is the case for the nebulae around LBV stars, like $\eta$ Carinae or AG Carinae (cf. Nota & Clampin 1997; cf. also Lamers et al. 2001), which show large bi-polar outflows or "peanut shaped'' nebulae. Asymmetries are also generally present in the nebulae around WR stars, in the shell ejection by Be stars and in the planetary nebulae, etc. A polar ejection removes only a little amount of angular momentum, while an equatorial ejection removes a lot of angular momentum. Thus, the amount of angular momentum remaining in massive stars after a phase of heavy mass loss depends very much on the anisotropies of the mass loss rates. As rotation influences the output of stellar evolution (cf. Meynet & Maeder 2000), we conclude that the anisotropies of the stellar winds may in some cases also influence the course of stellar evolution. The object of this work is to give the basic equations for accounting for the anisotropies of the stellar winds, in a way consistent with the equations expressing the transport of angular momentum by meridional circulation and shears. We want also to make some numerical tests to examine the feasibility of the proposed scheme and to explore some first consequences of the anisotropies of the mass loss. Section 2 gives the basic equations for the general case. Section 3 examines some stationary solutions. Section 4 gives the expression of the external torque in the case of anisotropic wind compared to the case of isotropic wind. In Sect. 5 we make some numerical tests and illustrations.