ner_tags: O O O B-Software O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation O O O O O O O O O O O B-Software O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation O O B-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-CelestialObject O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O B-CelestialObject O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O B-ObservationalTechniques I-ObservationalTechniques O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: computed with the STAREVOL code so far ( Palacios et al. 2003 , 2006 ; Decressin et al. 2009 ; Lagarde et al. 2012 ), while the prescription of Maeder (1997) has been systematically used for the rotating models computed with the Geneva code ( Eggenberger et al. 2012 ; Georgy et al. 2013 ). Meynet et al. (2013) have shown that the choice of the turbulent transport prescriptions may dramatically affect predictions for the structural, chemical, and rotational evolution of massive stars, but they were unable to clearly determine a best combination for fitting the observational chemical constraints. In Sect. 5 we discuss this in a similar way within the framework of the rotational evolution of solar-type stars. Table 3 Parameters and assumptions of the different models computed in this study. Reference. MPZ04 refers to Mathis et al. (2004) , TZ97 to Talon Zahn (1997) ; Matt+15 to Matt et al. (2015) ; Matt+12 to Matt et al. (2012) ; RM 12 to Reiners Mohanty (2012) ; and GB13 to Gallet Bouvier (2013) . SB is for solid-body rotation. 2.4. Transport of chemicals The transport of chemical species in radiative region is computed as a purely diffusive process ( Chaboyer Zahn 1992 ) but accounts for vertical advection and a strong horizontal diffusion. For a chemical species , the concentration follows the equation with the total diffusion coefficient, given bywhere and were defined previously (see Table 2 ). Finally, the term refers to the temporal evolution of the concentration of chemical species due to nuclear burning. 3. Physics of the models For all the models reported in Table 3 , the basic input physics (equation of state, nuclear reaction, opacities) can be found in Lagarde et al. (2012) . The initial abundances and mixing length parameter are calibrated without microscopic diffusion to reproduce a non-rotating Sun with respect to the Asplund et al. (2009) solar mixture with a precision for luminosity and radius at the age of the Sun. We used a mixing length parameter , an initial helium abundance , and an initial metal abundance . These values differ slightly from Lagarde et al. (2012) because we included a non-grey atmosphere treatment based on Krishna Swamy (1966) , which better agrees with observations for Sun-like stars (e.g. Vandenberg et al. 2007 ). In addition to the treatment of transport and loss of angular momentum and mass as described in Sect. 2, we included the modification of the effective gravity by the centrifugal forces and its effect on the stellar structure equations following Endal Sofia (1976) . This effect is slightly visible on the evolutionary track in the Hertzsprung-Russell diagram when the stars rotate fast. Rotation can have a non-negligible impact on the effective temperature for the fastest rotators around the ZAMS. Fig. 2 Evolution of the angular velocity as a function of time for the reference models. Plotted on the left as the rotational periods in days and during the right as the angular velocity in solar