Research

The models are computed with the implicit Lagrangian stellarevolution code STAREVOL (v3.00. See Siess et al. 2000; Palacios et al. 2003, 2006; Decressin et al. 2009). In this section, we summarize the main physical ingredients used for the present grid.

• Basic inputs
  The description of the stellar structure rests on the hydrostatic and the continuity equations, and the equations for energy conservation and transport. To solve this system, the following physical ingredients are required:
  • Nuclear reaction rates are needed to follow the chemical changes inside burning sites, and to determine the production of energy by the nuclear reaction, \epsilon_{nuc}, and the energy loss by the neutrino, \epsilon_{\nu}. We follow stellar nucleosynthesis with a network including 185 nuclear reactions involving 54 stableand unstable species from ^{1}H to ^{37}Cl. Numerical tables for the nuclear reaction rates were generated from the NACRE compilation (Arnould et al. 1999; Aikawa et al. 2005) with the NetGen web interface.
  • We mainly use reactions rates from either NACRE or Caughlan & Fowler (1988) when NACRE rates are unavailable. For proton captures on elements higher then Ne, we follow rates from Iliadis et al. (2001) or otherwise Bao et al. (2000). The following reactions rates are computed by:
    – ^{3}He(D, p)^{4}He (Descouvemont et al. 2004);
    – ^{3}He(\alpha\alpha, \gamma)^{12}C (Fynbo et al. 2005);
    – ^{8}B(\beta, \nu)^{24}He; ^{13N(β, ν)13C; ^{22}Na(\beta, \nu)^{22}Ne; ^{26}Alm(\beta, \nu)^{26}Mg; ^{26}Alg(\beta, \nu)^{26}Mg (Horiguchi et al. 1996);
    – ^{14}C(p, \gamma)^{15}N (Wiescher et al. 1990);
    – ^{14}C(p, n)^{14}N (Koehler & O’brien 1989a);
    – ^{14}C(\alpha, n)^{17}O; ^{17}O(n,^{4}He)^{14}C (Schatz et al. 1993);
    – ^{14}C(\alpha, \gamma)^{18}O (Funck & Langanke 1989);
    – ^{14}N(n, p)^{14}C (Koehler & O’brien 1989b);
    – ^{14}N(p, \gamma)^{15}O (Mukhamedzhanov et al. 2003);
    – ^{17}O(n, \gamma)^{18}O (Wagoner 1969);
    – ^{22}Ne(p, \gamma)^{23}Na (Hale et al. 2002);
    – ^{22}Ne(n, \gamma)^{23}Na (Beer et al. 2002);
    – ^{22}Na(n, \gamma)^{23}Na; ^{23}Na(\alpha, p)^{26}Mg; ^{25}Mg(\alpha, p)^{28}Si; ^{26}Mg(\alpha, p)^{29}Si; ^{27}Al(\alpha, p)^{30}Si (Hauser & Feshbach 1952);
    – ^{26}Alm(n, \gamma)^{27}Al; ^{26}Alg(n, \gamma)^{27}Al (Woosley et al. 1978).
  • The screening factors are calculated with the formalism of Mitler (1977) for weak and intermediate screening conditions and Graboske et al. (1973) for strong screening conditions.
  • Opacities are required to compute the radiative gradient \nabla_{rad} and the energy transport by radiative transfer. We generate opacity tables according to Iglesias & Rogers (1996) using the OPAL website3 for T > 8000K that account for C and O enrichments. At lower temperature (T < 8000K), we use the atomic and molecular opacities given by Ferguson et al. (2005).
  • The equation of state relates the temperature, pressure, and density and thus provides different thermodynamic quantities ( \nabla_{ad}, C_p, ...). In STAREVOL, we follow the formalism developed by Eggleton et al. (1973) and extended by Pols et al. (1995), which is based on the principle of Helmholtz free energy minimization (see Dufour 1999; and Siess et al. 2000, for detailed description and numerical implementation): this accounts for the non-ideal effects of Coulombinteractions and pressure ionization.
  • The treatment of convection is needed to compute the temperature gradient inside a convective zone. It is based on a classical mixing-length formalism with \alpha_{MLT}= 1.6, recovered from solar-calibratedmodels that include neither atomic diffusion nor rotation and were computed using Geneva models (see Ekström et al. 2012). We assume instantaneous convective mixing, except when hot-bottom burning occurs on the TP-AGB, which requires a time-dependent convective diffusion algorithm as developed in Forestini & Charbonnel (1997). The boundary between convective and radiative layers is defined with the Schwarzschild criterion. An overshoot parameter dover/Hp is taken into account for the convective core. This parameter is set to 0.05 or 0.10 respectively for stars with masses below or above 2.0 M
  • We use a gray atmosphere where the photosphere is defined as the layer for which the optical depth τ is between 0.005 and 10.We define the effective temperature and radius at the layer where \tau = 2/3.
  • For mass loss, we use Reimers (1975) formula (with \eta_{R} = 0.5) from the ZAMS up to central helium exhaustion On the AGB, we adopt the mass-loss prescription of Vassiliadis & Wood (1993).
• Transport processes in radiative zones
  • Thermohaline instability develops along the red giant branch (RGB) at the bump luminosity in low-mass stars and on the early-AGB in intermediate-mass stars, when the gradient of molecular weight becomes negative ( \Delta\nu=\frac{dln \nu}{dln P}<0 ) in the external wing of the thin hydrogen-burning shell surrounding the degenerate stellar core (Charbonnel & Zahn 2007b,a; Siess 2009; Stancliffe et al. 2009; Charbonnel & Lagarde 2010). This inversion of molecular weight is created by the 3He(3He, 2p)4He reaction (Ulrich 1971; Eggleton et al. 2006, 2008). The present grid of models is computed using the prescription advocated by Charbonnel & Zahn (2007b) and Papers I, II. It is based on Ulrich (1972) with an aspect ratio of instability fingers \alpha=6 , in agreement with laboratory experiments (Krishnamurti 2003). It includes the correction for non-perfect gas (including radiation pressure, degeneracy) in the diffusion coefficient for thermohaline mixing that is given by: where K is the thermal diffusivity; \phi= (\delta ln \rho)/(\delta ln\mu)_{P,T} ; \delta= -(\delta ln \rho)/(\delta ln\nu)_{P,\mu} The value of α in actual stellar conditions was recently questionned by the results of two- and three-dimentional hydrodynamical simulations of thermohaline convection for which α is close to unity (Denissenkov 2010; Denissenkov & Merryfield 2011; Rosenblum et al. 2011; Traxler et al. 2011). However, these simulations are still far from the stellar regime, hence we decided to use in this series the prescription described above since it successfully reproduces the abundance data for evolved stars of various masses and metallicities (see Charbonnel & Lagarde (2010) and Lagarde et al. 2011 for a more detailed discussion).
  • Pre-main sequence evolution along the Hayashi track is computed in a standard way (i.e., without accounting for rotationinduced mixing), and solid-body rotation is assumed on the Zero Age Main Sequence (ZAMS). On the main sequence, the evolution of the internal angular momentum profile is accounted for using the complete formalism developed by Zahn (1992), Maeder & Zahn (1998), and Mathis & Zahn (2004), which takes into account advection by meridional circulation and diffusion by shear turbulence (see Palacios et al. 2003, 2006; Decressin et al. 2009, for a description of the implementation in STAREVOL).We do not take into account the inhibitory effects of μ gradients in the treatement of rotation. We assume solid-body rotation in the convective regions, as we did in Papers I and II. In addition, we assume that the transport of angular momentum is dominated by the large amount of turbulence in these regions that instantaneously flattens out the angular velocity profile as it does for the abundance profiles. This hypothesis leads to a minimum shear-mixing approach in the underlying radiative layers, as discussed in Palacios et al. (2006) and Brun & Palacios (2009). On the other hand, the transport of angular momentum in stellar radiative layers obeys the advection/diffusion equation where ρ, r, and Ω have their usual meaning, Ur is the vertical component of meridional circulation velocity, and νν is the vertical component of the turbulence viscosity. The transport of chemical species resulting from meridional circulation and both vertical and horizontal turbulence is computed as a diffusive process (Chaboyer & Zahn 1992). The vertical transport of a chemical species i of concentration ci is described by a pure diffusion equation where c_i represents the variations in the chemical composition due to nuclear reactions. The total diffusion coefficient Dtot for chemicals can be written as the sum of three coefficients with Dth the thermohaline coefficient (Sect. 2.2.1), Deff the effective diffusion coefficient (∝1/Dh), and Dv the vertical turbulent diffusion coefficient (Talon & Zahn 1997), the last of which is proportional to the horizontal diffusion coefficient Chaboyer & Zahn 1992). The expression of the horizontal diffusion coefficient is taken from Zahn (1992), with its expression that prevents numerical divergence We do consider neither possible interactions between thermohaline and rotation-induced mixing, nor magnetic diffusion. Under the present assumptions, the thermohaline diffusion coefficient is several orders of magnitude higher than both the total diffusion coefficient related to rotation and the magnetic diffusivity in the advanced phases where the thermohaline instability develops (see e.g., Charbonnel & Lagarde 2010; Cantiello & Langer 2010, and Paper II). The complete treatment is applied up to either the RGB tip or the second dredge-up for stars undergoing the helium-flash episode.