3 The evolution of the internal rotation and meridional circulation

There are remarkable differences in the internal distributions of the angular velocity $\Omega(r)$ depending on the stellar metallicity Z. This was already suggested in Paper VII (Maeder & Meynet 2001), when comparing models at Z = 0.004and Z = 0.020. It is extended here with models at Z = 10^-5.

These matters are not academic problems ! Indeed, the distribution of $\Omega(r)$ determines for example the mixing of chemical elements, the size of the convective core and therefore the chemical yields. The results in Sects. 8 and 9 below on the chemical yields are a consequence of the distribution of $\Omega(r)$.

Figure 2 shows the evolution of $\Omega(r)$ during the MS phase of a 15 $M_\odot$ at Z = 10^-5, (this follows the initial convergence of the $\Omega$-profile which is very short, i.e. $\leq$1% of the MS lifetime). We notice that the rotation of the convective core only has a small decrease during the MS phase, much smaller than at higher metallicities. This results from 2 effects. a) The mass loss at Z = 10^-5 is much smaller than at solar composition and thus less angular momentum is removed from the star. b) As we shall see below, the meridional circulation is very slow in the outer regions of the models at very low Zand it transports much less angular momentum outwards than in models at solar composition. In view of these remarks, it is likely that massive stars at lower Z have faster spinning cores.

Figure 4: Evolution of U(r) the radial term of the vertical component of the velocity of meridional circulation for a model of 20 $M_\odot$with Z = 10-5 at various stages during the MS phase. $X_{\rm c}$ is the hydrogen mass fraction at the center. The dashed line shows the values of U(r) inside a 20 $M_\odot$ model at Z = 0.004 when $X_{\rm c}$ = 0.28.

Another significant difference shown by Fig. 2 concerns the gradient of $\Omega$ outside the convective cores. Here, the gradients are steeper and they remain significant up to the stellar surface, while at Z = 0.02 the $\Omega$-distribution becomes very flat in the external layers, as evolution proceeds (Meynet & Maeder 2000). This difference is well illustrated in Fig. 3, where we notice for the 3 and 9 $M_\odot$ models the much steeper $\Omega$-gradients at lower Z, while the models at Z=0.02 show very flat gradients in the outer layers. The reason for the higher $\Omega$-gradients here are the same as for the faster spinning cores. These higher $\Omega$-gradients imply stronger shears and thus more mixing by shear diffusion, which is the main effect for the outward transport of the chemical species. (The differences in $\Omega$ between the 9 and 3 $M_\odot$models result from the fact that we consider stars with the same $v_{\rm ini}$, but different radii).

Figure 4 shows an example at Z = 10^-5 of the evolution of U(r), the vertical component of the velocity of meridional circulation. The size and evolution of U(r) is very different from the case at Z = 0.02. At Z = 0.02, U(r) takes large negative values particularly in the outer layers. This is due to their low density, which makes a large Gratton-Opik term $\frac{-\Omega^2}{2 \pi G \overline{\rho}}$in the expression of U(r), (cf. Maeder & Zahn 1998). At Z = 10^-5, the large negative values of U(r) have disappeared, U(r) is equal to 10^-3 cm s^-1 at the end of the MS phase, while it was 50 times more negative in the corresponding models at Z = 0.02 (Meynet & Maeder 2000). The differences do not concern so much the deep interior, but mainly the outer layers. The physical reason of the above differences is the fact that the star is more compact at lower Z and that the density in the outer layers is not as low as at solar composition.

Figure 4 also shows the curious curve for a model at Z = 0.004. In the interior, U(r) is about the same as in the present models (and this is true for all Z values). The big external dip of U(r), which was present at Z = 0.02 is very much reduced, but still present at Z = 0.004, while at Z = 10^-5 the external dip is fully absent.

The small U(r) in the external layers of the present models is mainly responsible for the presence of an $\Omega$-gradient up to the stellar surface (cf. Fig. 2). Since the mixing of the chemical elements is mainly driven by the shear, the presence of this $\Omega$-gradient in the outer layers enables the large mixing and surface chemical enrichments that are present in the Z = 10^-5 models.