4 The evolution of the surface rotation velocities

Figure 5: Evolution of the surface equatorial velocity as a function of time for stars of different initial masses with $v_{\rm ini}=300$ km s-1 and Z =10-5. The track without label corresponds to a 20 $M_\odot$ model.

Figure 6: Same as Fig. 5 for a 2, 3 and 5 $M_\odot$ stellar model.

Figure 7: Evolution of the ratio $\Omega /\Omega _{\rm c}$ of the angular velocity to the break-up angular velocity at the stellar surface for stars of different masses at Z = 10-5.

Figure 8: Same as Fig. 7 for a 2, 3 and 5 $M_\odot$ stellar model.

Figure 9: Evolution of the surface equatorial velocity as a function of time for 60 $M_\odot$ stars with $v_{\rm ini}=300$ km s-1 at different initial metallicities.

It will probably be a certain time until we are able to observe $v \sin i$for stars in galaxies with Z = 10^-5. Nevertheless, these objects have contributed to shape the composition of our universe and they deserve some interest.

Figures 5 and 6 show the evolution of the surface rotational velocities for models with initial masses from 2 to 60 $M_\odot$. Figures 7 and 8 show the corresponding evolution of $\frac{\Omega}{\Omega_{\rm c}}$. We notice the relative constancy of $\frac{\Omega}{\Omega_{\rm c}}$ during the MS evolution for stars with mass between 5 and 20 $M_\odot$. The cases of 40 and 60 $M_\odot$ are noticeable as shown by Figs. 5 and 7. These models reach the break-up velocities near the end of the MS phase. This is completely different from the models at Z = 0.02, where the rotation becomes very small due to the huge losses of mass and angular momentum. Thus, if the initial mass function at low Z extends up to high mass stars, as often supposed, rotation is likely to be a major effect in the course of the evolution of massive stars, since many of them are likely to reach break-up velocities. This would even more be the case for the massive stars which have a blueward evolution as a result of strong internal mixing. Their radii would decrease, thus favouring extreme rotation velocities. We note that for masses between 3 and 20 $M_\odot$, the rotation velocity keeps about constant during the MS phase, before decreasing in the post-MS phases.

Figure 9 clearly illustrates the very different evolution of the rotational velocities of a 60 $M_\odot$ at various metallicities. At low Z like in the models at Z = 10^-5 , the growth of $\frac{\Omega}{\Omega_{\rm c}}$is possible because of the very small mass loss and also it is favoured by the outward transport of angular momentum which is much larger for the more massive stars. As shown by Maeder & Meynet (2001), the values of U(r) are more negative for the larger stellar masses, due to several facts: lower gravity, higher radiation pressure, larger L/M ratio and especially the lower density. Thus, the outward transport is more efficient.

In view of these results, we may somehow precise our suggestion (Maeder & Meynet 2001) that at very low Z a large fraction of the massive stars reach their break-up velocities. This seems true for the highest masses above about 30 $M_\odot$, but not necessarily for the OB stars below this limit. This question is of high importance, because if the massive stars reach their break-up velocity, most of their evolutionary and structural properties will be affected. For example, they could also lose a lot of mass and produce some Wolf-Rayet (WR) stars. They would have a relatively small remaining mass at the time of the supernova explosion, like their counterparts at solar composition.

For the models of 3 $M_\odot$, as illustrated in Figs. 6 and 8, the rotation velocity remains about constant during the MS phase, while $\frac{\Omega}{\Omega_c}$ increases. For 2  $M_\odot$, we notice a net increase. This particular behavior is due to the different shape of the track of the 2 $M_\odot$ model in the HR diagram (Fig. 10), which mimics the tracks of lower masses dominated by the pp chain. This is well explainable, because at Z = 10^-5 the CNO cycle is less important than at solar composition, thus the mass limit where the CNO cycle starts dominating over the pp chain is shifted upward. For this model of 2 $M_\odot$, there is no large increase of the radius during the MS evolution and thus rotation keeps higher.

When one examines the evolution of $v \sin i$ during the MS phase for stars of the same mass but different initial velocities, one usually notes at Z= 0.02 a so-called velocity convergence (Langer 1998). This is due to the fact that the faster rotating stars lose more mass and thus more angular momentum. In the present models, the mass loss rates are in general very small (as long as the stars are not at break-up) and thus there is no velocity convergence, i.e. the stars of different initial velocities finish the MS phase with different velocities as illustrated by Table 1.