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Astron. Astrophys. 341, 181-189 (1999)

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6. Conclusion

We have calculated the evolution of the rotation profile accompanying the angular momentum transport by meridional circulation and turbulent diffusion in the radiative envelopes of MS model stars of 10 and [FORMULA]. In these calculations we have closely followed the theoretical scheme of assumptions and simplifications proposed for the first time by Zahn (1992) and elaborated upon later by Talon & Zahn (1997).

It should be noted that all our numerical results do not account for the gradient of the mean molecular weight. At the same time, as shown by expression (5), in a medium with a non-zero µ-gradient [FORMULA] can be reduced considerably. The difference in [FORMULA] with respect to the case of constant µ considered in the paper may amount to one or two orders of a magnitude. Thus the values of [FORMULA] plotted in Figs. 1d and 2d may be greatly underestimated, especially near the convective core border. However, here we meet an apparent disagreement of the current theory with the observations of the He overabundances in O-B stars showing that in real stars additional mixing succeeds in overcoming the µ-gradient barrier. Unfortunately, this disagreement remains unexplained as yet.

One of our simplifications has been ignoring any mass loss by the stars. We consider this simplification as an unavoidable one at this stage of the analysis because it is still not clear which outer boundary conditions for Eq. (4 ) one should use in the presence of a stellar wind. Available semi-empirical formulae for the mass loss rates on the MS allow one to calculate only the angular momentum loss rate, i.e. actually give an outer boundary condition for the integral of Eq. (4). Bearing in mind that in the absence of a strong magnetic field an extended envelope of a mass losing massive MS star is most likely to possess a differential rotation it remains unclear which part of the angular momentum being lost is transfered to the stellar atmosphere by the meridional circulation and which one by the turbulent diffusion.

The situation becomes even more complicated if one wants to take into account the possibility that massive MS stars (presumably, those with [FORMULA]) spend a considerable part of their MS life as objects embedded into a protostellar cocoon and thus acrete material instead of losing it (Beech & Mitalas 1994; Bernasconi & Maeder 1996).

On the basis of the results of calculations presented in the paper we have come to the following conclusions:

the relaxation time [FORMULA] required for a massive MS star to arrive at the state of stationary rotation is much shorter than the star's MS life-time [FORMULA] for large enough values of [FORMULA] which can be estimated a priori provided the quantities L, R and [FORMULA] are known for an appropriate stellar model; if [FORMULA] the assumption of the star being in the state of stationary rotation at every moment from the beginning of its MS evolution used by Talon et al. (1997) is quite reasonable and justified by our calculations; on the other hand, for sufficiently low values of [FORMULA] the above assumption is no longer correct and we have to follow the evolution of the angular velocity profile by solving Eq. (4 ) simultaneously with the stellar evolution calculations;
qualitatively, the nonstationary solutions do not greatly differ from the stationary one in their ability to mix chemical elements; even for nearly uniform rotation the rate of mixing (in this case mainly sustained by the meridional circulation) is found to be much lower than the classical estimate [FORMULA], the reason for this being the effective horizontal erosion by the turbulent diffusion; no complete mixing of the envelope is possible;
despite of the reduced mixing rate, for a sufficiently large [FORMULA] the turbulent diffusion (which becomes a dominant mixing mechanism for rotation close to the steady-state one) succeeds to build-up diffusion-like abundance profiles in the radiative envelope of a massive MS star before it leaves the MS; the surface abundances begin to decline from the initial ones after some delay time which is required for the diffusion wave to reach the stellar surface; recently, Lyubimkov (1996) has reported a similar delay in the appearance of He overabundances in the atmospheres of OB-stars; if additional mixing penetrates the convective core then the evolution of the surface He abundance may look like that of N shown in Fig. 3b;
internal gravity waves generated by convective eddies near the convective core border can successfully compete with the meridional circulation and turbulent diffusion in redistributing the angular momentum, especially in the inner part of the radiative envelope; the IGWs tend towards establishing a state of uniform rotation, however, this does not help to accelerate the mixing of chemical elements considerably (see point (ii ) above); nonlinear effects associated with the IGWs may also initiate mixing of chemical elements in stellar radiative zones (Press 1981; García López & Spruit 1991); we are going to discuss some of these effects in a forthcoming paper.

We finish these conclusions without carrying out any detailed comparison of theory with observations because, from the one hand, the mixing mechanisms considered in the paper need further elaboration (cf. Maeder & Zahn 1998; Ringot 1998) before they can be incorporated as input physics into stellar evolution codes. On the other hand, observational data on the abundance anomalies in massive MS stars are still not definite enough to constrain a particular mixing mechanism (Lyubimkov 1996). Work in both directions is encouraged.

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© European Southern Observatory (ESO) 1999

Online publication: November 26, 1998