*Astron. Astrophys. 360, 952-968 (2000)*
## 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
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