3. The stellar evolution code and computations
General aspects of the code. The results presented in this paper have been obtained with the stellar evolution code described by Blöcker Blöcker (1995) with the modifications given in Herwig et al. Herwig et al. (1997). Here, we will add only a few relevant details. The reaction rate has been taken from Caughlan & Fowler Caughlan & Fowler (1988) and multiplied by a factor of 1.7 as recommended by Weaver & Woosley Weaver & Woosley (1993). The reaction rates and have been taken from the compilation of Landré et al. Landre et al. (1990) as described in El Eid El Eid (1994). The model sequence without overshoot has been computed with the latest OPAL opacity tables Iglesias & Rogers 1996. This choice had purely technical reasons but does not affect the comparison.
The adjustment of the numerical resolution (geometry and time) plays an important role in the computation of AGB models Straniero et al. 1997; Frost & Lattanzio 1996; Mowlavi 1999. The resolution is adjusted from model to model according to the changes of the state variables as a function of time and location within the star. For example, the luminosity due to nuclear processing of helium should not change more than between two models. The models have mass grid points which accumulate around the core-envelope interface where the pressure and density drop steeply. The model time steps range from about a dozen years during the quiescent H-burning interpulse phase down to typically a few days during the He-flash and the following phase where dredge-up may occur (Sect. 4.1).
Exponential diffusive overshoot. For the time-dependent treatment of overshoot mixing we follow the prescription of Freytag et al. Freytag et al. (1996). The particle spreading in the overshoot region can be described as a diffusion process and the diffusion coefficient can be fitted by a formula like
where z denotes the geometric distance from the edge of the convective zone and is a characteristic time scale for the considered convection zone. Combining this with the finding of an exponential decay of the velocity field, Freytag et al. Freytag et al. (1996) give the diffusion coefficient in the overshoot region as
where is the velocity scale height of the overshoot convective elements at the convective boundary. can be expressed as a fraction f of the pressure scale height . The quantity is the expression in Freytag et al. (1996: Eq. 9). Here we have approximated by the diffusion coefficient in the convectively unstable region near the convective boundary . Note that is well defined because approaches the convective boundary with a small slope and drops almost discontinuously at . The diffusion coefficient in the convection zone can be derived from the MLT (see Langer et al. 1985). In the radiative zone where the diffusion coefficient has dropped below no element mixing is allowed ().
The free parameter f in Eq. (2) describes the efficiency of the extra diffusive mixing. For larger f the extra partial mixing beyond the convective edge extends further. In their simulations Freytag et al. Freytag et al. (1996) found velocity scale heights of the order of a pressure scale height (). However, under adiabatic conditions the convective flows move so fast that they will not be able to exchange heat efficiently with the surrounding when entering into the stable region. Accordingly a stronger deceleration of the plumes is likely in the stellar interior compared to the situation of the shallow surface convection zones. In order to obtain an order of magnitude for f in the stellar interior it has been scaled to reproduce the observed width of the main sequence. A parameter of was found to reproduce the models of Schaller et al. Schaller et al. (1992) (see end of Sect. 2). Using AGB overshoot models as a starting point for post-AGB models of H-deficient objects has confirmed this assumption a posteriori, at least for the bottom of the He-flash convection zone Herwig et al. 1999a.
The overshoot prescription (Eq. (2)) has been applied irrespective of µ-gradients which may decrease the overshoot efficiency Langer et al. 1983. The analysis of old open clusters even suggests that overshoot may not expand against µ-gradients in low mass stars Aparicio et al. 1991. Theoretically, Canuto Canuto (1998) has recently argued that µ-gradients do decrease the overshoot efficiency, but do not prevent the effect entirely. In our study, this aspect becomes important at the bottom of the envelope convection during the third dredge-up. As explained in Sect. 5.3 the results are not significantly dependent on the amount of overshoot at this convective boundary.
The stellar evolution computations. The findings in this work are based on two model sequences of 3 and 4 with overshoot. For comparison, one 3 TP-AGB sequence has been computed without overshoot. It has been started from a 3 overshoot model taken just before the occurrence of the first TP (Fig. 7, Fig. 9, Fig. 10, Fig. 15). In order to study the dependence of the evolution of the intershell abundance as a function of the parameter f a few more sequences with different f-values have been computed from this starting model (Fig. 12) and followed over three thermal pulses. In addition, the eighth TP cycle of the 3 overshoot case has been recomputed for different f values (Fig. 13, Fig. 14). In Fig. 1 a recomputation of the 14th TP of the 3 sequence initially without overshoot is shown.
A solar-like initial composition, , has been chosen corresponding to Anders & Grevesse Anders & Grevesse (1989). The mixing-length parameter of the MLT is which has been calibrated by reproduction of the solar surface parameters. A Reimers-type mass loss with efficiency has been applied. The Reimers mass loss formula has not been designed for AGB stars, however for the purpose of this study mass loss is not important. The 3 (4) model sequence has a total mass of 2.686 (3.689) at the 12th (11th) TP (see Table 1 and Table 2).
Table 1. Dredge-up properties of the thermal pulses of the sequence with . The table gives the TP number, age (zero point set at first TP which shows dredge-up), peak luminosity of the helium burning shell , amount of dredged-up hydrogen free material , core growth since last TP , dredge-up parameter (see text), mass of hydrogen free core , mass coordinate of top of pulse-driven convective zone , mass coordinate of bottom of pulse-driven convective zone and isotopic and elemental abundance ratios at the surface immediately after the TP.
Table 2. Dredge-up properties of the thermal pulses of the sequence with . For explanation of the displayed quantities see Table 1.
Overshoot has been applied during all evolutionary stages and to all convective boundaries. Overshoot applied to the main sequence core convection has indirect implications for AGB star models as it shifts the core masses at the first thermal pulse upward and decreases the progenitor mass () limit for C/O white dwarfs Alongi et al. 1993. If overshoot is also applied during the second DUP phase this effect is counterbalanced somewhat and (T. Driebe, priv. com., see also Blöcker et al. Blöcker et al. (2000) and Weidemann Weidemann (2000)). This is just above the value obtained from models in which overshoot is only applied during the main sequence phase Bressan et al. 1993.
© European Southern Observatory (ESO) 2000
Online publication: August 23, 2000