Populations of rotating stars II. Rapid rotators and their link to Be-type stars

Context. Even though it is broadly accepted that single Be stars are rapidly rotating stars surrounded by a flat rotating circumstellar disk, there is still a debate about how fast these stars rotate and also about the mechanisms involved in the angular-momentum and mass input in the disk. Aims. We study the properties of stars that rotate near their critical-rotation rate and investigate the properties of the disks formed by equatorial mass ejections. Methods. We used the most recent Geneva stellar evolutionary tracks for rapidly rotating stars that reach the critical limit and used a simple model for the disk structure. Results. We obtain that for a 9 M-circle dot star at solar metallicity, the minimum average velocity during the main-sequence (MS) phase to reach the critical velocity is around 330 km s(-1), whereas it would be 390 km s(-1) at the metallicity of the Small Magellanic Cloud (SMC). Red giants or supergiants originating from very rapid rotators rotate six times faster and show N/C ratios three times higher than those originating from slowly rotating stars. This difference becomes stronger at lower metallicity. It might therefore be very interesting to study the red giants in clusters that show a large number of Be stars on the MS band. On the basis of our single-star models, we show that the observed Be-star fraction with cluster age is compatible with the existence of a temperature-dependent lower limit in the velocity rate required for a star to become a Be star. The mass, extension, and diffusion time of the disks produced when the star is losing mass at the critical velocity, obtained from simple parametrized expressions, are estimated to be between 9.4 x 10(-12) and 1.4 x 10(-7) M-circle dot (3 x 10(-6) to 4.7 x 10(-2) times the mass of the Earth), 2000 and 6500 R-circle dot, and 10 and 30 yr. These values are not too far from those estimated for disks around Be-type stars. At a given metallicity, the mass and the extension of the disk increase with the initial mass and with age on the MS phase. Denser disks are expected in low-metallicity regions.