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Astron. Astrophys. 321, 465-476 (1997)

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4. Effects of the rotationally induced mixing

To study the effects of the rotationally induced mixing, we have computed a 40 [FORMULA] model with [FORMULA], including in addition to the hydrostatic effects discussed above, the effects of mixing of the chemical elements. The mixing of the chemical species, modelised through a diffusion equation in our rotating stellar models, intervenes through three hydrodynamical processes: the horizontal and vertical turbulence induced by the shear and the meridional circulation. The diffusion coefficient [FORMULA], which accounts for the effects of the horizontal turbulence induced by the shear and of the meridional circulation, exerts its effect whenever turbulent and circulation motions are sustained. The same is true for the action of [FORMULA], which describes the effects of the vertical turbulence induced by the shear. But its action is in addition submitted to the Richardson criterion which states when the shear is large enough to produce vertical mixing.

Let us recall that for a radiative zone, the Richardson stability criterion becomes (Maeder & Meynet 1996)


where [FORMULA] and [FORMULA], measure the strength of the shear and that of the µ-gradient respectively. In a semiconvective zone, the stability criterion, with respect to shears, is


where [FORMULA]. Current symbols are used (Kippenhahn & Weigert 1990, see also Maeder & Meynet 1996). On Fig. 5, we present the profiles of various physical quantities inside the model with a hydrogen mass fraction at the centre equal to [FORMULA]. Let us first concentrate on the semiconvective zone. We can note the following points:

1) The semiconvective zone encompasses nearly all the region where there is an important µ-gradient just outside the convective core. (see panels a and c). Any efficient mixing in this zone would deeply affect the stellar structure. 2) In the semiconvective zones, the Richardson stability criterion is always fullfilled by a wide margin (see panel d). Indeed [FORMULA] ([FORMULA]) is 2-3 orders of magnitude greater than [FORMULA] ([FORMULA]). Let us note that the term [FORMULA] which, in a semiconvective zone, weakens the inhibiting effect of the µ-gradient, has a negligible effect. Its value is of the order of [FORMULA]. 3) The values of [FORMULA] in this region range between [FORMULA] cm2 s-1, implying a mixing timescale [FORMULA] superior to 109 y ! (R is the radius of the star). Thus, in a semiconvective zone, the vertical turbulence induced by the shear is never strong enough (by a wide margin) to overcome the inhibiting effects of the µ-gradients. At the same time [FORMULA] take too low values to produce a significant mixing. Let us now turn to the radiative zone. One can see that: 1) In zones where [FORMULA], [FORMULA] can reach values as high as 109 -1010 cm2 s-1. For such values the mixing timescale is of the order of [FORMULA] y , i.e. shorter than the main sequence lifetime. But, this occurs in zones where there is almost nothing to mix, the regions being already nearly completely homogeneous. 2) At the base of this radiative zone, the chemical profile steepens and [FORMULA], which can reach values as high as 10-2, stabilizes the medium. 3) Only in the small zone of very strong [FORMULA] -gradient and of µ-gradient nearly zero, which occurs above the main intermediate convective zone (at [FORMULA] between 25 and 26 [FORMULA]) does the shear overcome again the µ-barrier. In this zone, the critical parameter [FORMULA] defined by Maeder & Meynet (1996) is inferior to [FORMULA]. In that case there is no preferred turbulent scale which can be used to define the diffusion coefficient and we choose to set the diffusion coefficient equal to the convective diffusion coefficient as it can be expressed in the frame of the mixing length theory. A posteriori this procedure seems justified. Indeed the zones where this situation occurs are always adjacent to a convective zone and the peak which appear on Fig. 5, panel e) (see also Fig. 6), can be seen as a small extension of an adjacent convective zone. 4) In the radiative regions, the values of [FORMULA] are generally one or two orders of magnitude lower than that of [FORMULA] in agreement with the expectations of Zahn (1992). To resume, in radiative zones, the diffusion coefficients are strong enough to produce an efficient mixing only in regions where there is almost nothing to mix.

[FIGURE] Fig. 5. Profiles of various quantities inside a 40 [FORMULA] stellar model during the H-burning phase: a Profile of the abundance of hydrogen (in mass fraction); b Profile of the angular velocity; c Profiles of [FORMULA] (continuous line), of [FORMULA] (long dashed line) and of [FORMULA] (short dashed line). In the zones where [FORMULA] is zero, [FORMULA] is confounded with [FORMULA] ; d Profiles of [FORMULA], [FORMULA] and [FORMULA] (see text); e profiles of the diffusion coefficients [FORMULA], [FORMULA] and [FORMULA].
[FIGURE] Fig. 6. Profiles at different stages during the main sequence of the diffusion coefficients in a 40 [FORMULA] model computed with an initial [FORMULA]. Hydrostatic effects and rotationally induced mixing of the chemical species have been taken into account. [FORMULA] is the central mass fraction of hydrogen at the different stages considered.

The same situations occur during the whole H-burning phase. Comparing Figs. 6 and 7 which show the profiles of the diffusion coefficients and of the hydrogen in seven models during the H-burning phase, one immediately sees that where µ-gradients are present, i.e. in regions where an efficient mixing would have the strongest impact, the diffusion coefficient [FORMULA] is zero ! As explained above, the peaks of [FORMULA] are small extensions of adjacent convective zones (see Fig. 5 panel e).

[FIGURE] Fig. 7. Same as in Fig. 6 for the profiles of the abundance of hydrogen (in mass fraction).

From these considerations, one can conclude that the rotationally induced mixing will have a very limited impact on the chemical profiles and subsequently on the stellar structure. This is indeed the case. On Fig. 8 the profile of the hydrogen mass fraction, at the end of the MS, inside the stellar model obtained with rotationally induced mixing, is compared with the one in the standard model (without rotation). The action of [FORMULA] in the rotating model gives birth to a series of intermediate convective zones which are nearly totally absent in the standard model. Apart from these differences the general shapes of the two profiles are quite similar. In particular, the sizes of the convective cores are the same and no significant changes of the surface abundances are observed in the rotating model. The main sequence lifetimes of the two models differ by less than 0.4%. The similarity of the two models is a consequence of the very strong inhibiting effect of the µ-gradient. Indeed the condition for mixing (very low µ-gradient) prevents the mixing to have an important impact (in strong µ-gradient regions)!

[FIGURE] Fig. 8. Profiles of the mass fraction of hydrogen at the end of the main sequence in a 40 [FORMULA] model computed with [FORMULA] (long dashed line) and [FORMULA] (continuous line). The hydrostatic effects of rotation and the rotationally induced mixing of the chemical species were considered in the rotating model.

Let us note that this conclusion on the inhibiting effect of the µ-gradient is reinforced by the fact that we did not consider here any transport mechanism of the angular momentum. Indeed, had we done so, the angular velocity gradients would likely have been smoothed and the shear would have been weaker than in the present model. Let us also mention that the choice of another criterion for the convection has no impact on this conclusion. Indeed if the Schwarzschild criterion had been chosen instead of the Ledoux criterion, this would have simply shifted the region of the µ-gradient outwards without removing its inhibiting effect on the turbulent mixing. Thus we are left with the conclusion that even with very high initial angular velocity (here 90% of the critical velocity on the ZAMS), no rotational mixing is expected to occur during the main sequence of a massive star. Although this result confirms previous theoretical investigations which have also shown the strong inhibiting effect of the µ-gradients (Chaboyer et al. 1995), we cannot be fully happy with it. Indeed this result does not account for the observations of Herrero et al. (1992) which show that mixing by rotation is strong enough to operate on timescales shorter than the main sequence lifetime. Thus we consider the above results not as a proof of the inefficiency of rotational mixing but as a sign that some processes are still inadequately described by theory.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998