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 . 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 can be reduced considerably. The difference in 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 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 ) 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:
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.
© European Southern Observatory (ESO) 1999
Online publication: November 26, 1998