We briefly repeat that part of the synthetic evolution model that is related to the chemical evolution on the AGB. Full details on the model can be found in GJ. The evolution model is started at the first TP, taking into account the changes in mass and abundances prior to the first TP, and is terminated when the envelope mass has been lost due to mass loss or if the core reaches the Chandrasekhar mass. The latter situation never occurs in the best fitting models for the Galaxy and the Large Magellanic Cloud (see GHJ, GJ).
The first dredge-up occurs when the convective envelope moves inwards
as a star becomes a red giant for the first time.
Description of the mass loss during the RGB can be found in GJ.
The convective motion dredges up material that was previously located near the
hydrogen burning shell. The increase in the helium abundance, 
,
is given by (cf. Sweigart et al. 1990):
Results are linearly interpolated for a given Y while the small dependence of
Y on Z for a given Y is neglected.
The change in hydrogen is opposite to the change in helium:
Changes in 
C, 
N, and 
O are calculated from:
The number ratio 
after the first dredge-up
does not vary much with mass or composition (Sweigart et al. 1989) and
is set to 23.
The second dredge-up is related to the formation of the electron-degenerate CO core after central helium exhaustion in stars more massive than a critical mass (see Becker & Iben 1979, 1980) which depends both on the core mass and main-sequence abundances. The base of the convective envelope moves inward through matter pushed outwards by the He-burning shell. The treatment of the second dredge-up follows that of RV closely. No mass loss is assumed during second dredge-up.
Following GJ (and references therein), the abundances after the second
dredge-up can be obtained from the abundances prior to the second
dredge-up and the abundances of the material that is dredged up using
the relation:
where the coefficients a and b are functions of the total mass as well
as the core
mass before and after second dredge-up and are given by Eq. (30) in GJ.
The abundances prior to the second dredge-up are known and the
abundances of the dredged up material are given by RV and Iben & Truran
(1978):
The hydrogen abundance after second dredge-up was
calculated from X = 1 - Y - Z, with Y and Z the helium and metal
abundance after second dredge-up. This was done to ensure that 
at all times.
As discussed above, the pre-AGB evolution with respect to mass loss and chemical evolution during the first and second dredge-up is either calculated according to the recipes in GJ (see previous subsections), or is taken from the theoretical evolution tracks provided by the Geneva group. In the latter case, we use the stellar surface abundances as well as the stellar mass at the end of these tracks.
The synthetic AGB evolution model starts at the first thermal pulse.
In brief, we account for the dependence of core mass on initial
stellar metallicity and assume that third dredge-up occurs only if the
core mass is larger than a critical value 
. In
GJ we argued that a value of 
is
required to fit the low-luminosity tail of the observed carbon star
luminosity function in the LMC (see below).
The time scale on which thermal pulses occur is a function of core
mass as discovered by Paczynski (1975). In GJ and GHJ, we use the
core-mass-interpulse relation presented in Boothroyd & Sackmann
(1988) where the increase in core mass during the interpulse period
(
) is given by:
A certain fraction of this amount is assumed to be dredged up:
The free dredge-up parameter 
is assumed to be a constant.
In GJ we found that a value of 
= 0.75 is required to fit the
peak of the observed carbon stars LF in the LMC (see below).
In principle, the composition of the dredged-up material is determined
by the detailed chemical evolution of the core. For simplicity, we
assume that the composition of the material dredged-up after a TP is:
He = 0.76, 
C = 0.22, and 
O = 0.02 (cf. Boothroyd
& Sackmann 1988). The carbon is formed through incomplete helium
burning in the triple 
process and the oxygen through the
C(
O reaction.
Newly dredged-up material can be processed at the base of the
convective envelope in the CNO-cycle, a process referred to as hot
bottom burning (HBB) and extensively discussed by RV. To a large
extent, HBB determines the composition of the material in the stellar
envelope of thermal pulsing AGB stars. The process of HBB is able to
slow down or even prevent the formation of carbon stars (e.g. Groenewegen &
de Jong 1994a). Since
C is converted into 
C and 
N, it also gives rise
to the formation of 
C-rich carbon stars (usually referred to as
J-type carbon stars) and 
N-rich objects (e.g. Richer et al.
1979).
RV treated HBB in considerable detail as a function of the mixing
length parameter (e.g. 
= 0, 1.0, 1.5, 2). In GJ (see for
details their Appendix A) it was decided to approximate in a
semi-analytical way the results of RV for their 
= 2 case as
it gave the largest effect of HBB. Since then new results regarding
HBB have been obtained, both theoretically (Boothroyd et al. 1993,
1995) and observationally for AGB stars in the Magellanic
Clouds
(Plez
et al. 1993; Smith et al. 1995). These results suggest that HBB is a
common phenomenon that occurs at a level roughly consistent with that
predicted by RV in case 
= 2. In particular, Boothroyd et
al. (1995) estimate that the initial stellar mass above which HBB
takes place is 
which is similar to the value of
predicted by RV (
). Observations indicate
that virtually all stars brighter than 
mag
undergo envelope burning (Smith et al. 1995). These luminosities are
reached for stars with initial masses slightly below 4 
and larger
(Boothroyd et al. 1993).
During the thermal pulsing AGB, stars lose most of their mass: typically
0.4 
for a 1 
star and 
4.8 
for a 6 
star (at solar
initial metallicity; see GJ). Clearly, stellar yields for intermediate
mass stars are dominated by the mass loss and chemical evolution during this
phase. After gradual ejection of their outer envelope, most AGB stars leave a
white dwarf remnant usually accompanied by the formation of a planetary nebula
(PN).
We briefly discuss assumptions and uncertainties involved with the
synthetic evolution model described above.
In GJ and GHJ we used observations of AGB stars in the LMC and Galaxy
to constrain the synthetic AGB model. The main model parameters are the
minimum core mass for third dredge-up 
, the third
dredge-up efficiency 
, and the scaling law for mass loss on
the AGB 
.
In GJ the following observational constraints were considered: the
luminosity function of carbon stars, the observed number ratio of
carbon to oxygen-rich AGB stars, the birth rate of AGB stars, the
abundances observed in PNe, the initial-final mass relation, and the
frequency of carbon stars in clusters of a given mass. Values of
= 0.58 
and 
= 0.75 were determined
predominantly by fitting the carbon star LF. A value of 
(assuming a Reimers mass loss law) was derived by fitting
the high-luminosity tail of this LF. This set of model parameters
resulted in a birthrate of AGB stars consistent with other
determinations and predicts that carbon stars can form only from
main-sequence stars more massive than 1.2 
, consistent with
observations of carbon stars in LMC clusters. In Groenewegen & de
Jong (1994a) we showed that the model with the aforementioned
parameters predicts the correct observed abundances in LMC PNe. In
Groenewegen & de Jong (1994b) we considered two alternative mass loss
formula to the Reimers one. We found that the mass loss formula
proposed by Vassiliadis & Wood (1993) could less well explain the
observed abundances while a scaled version of the law proposed by
Blöcker & Schönberner (1993) could equally well fit the
observations.
In GHJ we applied the synthetic evolution model to carbon stars in the
Galactic disk. As there yet exists no reliable carbon star LF, we used
observations of carbon stars in open clusters and in binaries to
determine 
= 0.58 
. Using similar
constraints as for AGB stars in the LMC, values of 
and 
were found in optimal agreement with the
observations. Thus, models for the observed luminosity function of
carbon stars in both the Galactic disk and LMC do favour a high mass
loss coefficient 
. This range in
is consistent with additional constraints such as the
initial-final mass relation for 
white dwarfs in the solar
neighbourhood (see Weidemann & Koester 1983;
Weidemann 1990).
In conclusion, a wide range of
observations of AGB stars both in the LMC and Galactic disk can be
explained by one and the same set of model parameters, i.e. 
= 0.58 
, 
, and 
. In the following, we will refer to this set of parameters as the
standard model.
It should be noted that observations indicate that some stars do not
obey the standard model predictions. In particular, the
C/
C ratio after the first dredge-up is often lower than
predicted in stars of low mass, down to 
C/
C 
10. Rotationally induced mixing (e.g. Sweigart & Mengel 1979;
Charbonnel 1995;
Denissenkov & Weiss 1996) or initial abundances different from
those adopted here (see Sect. 4) may play a role.
Clearly, the assumptions of a fixed critical core mass as well as of a
constant dredge-up efficiency and mass loss parameter for all AGB
stars (independent of their initial mass and AGB phase) are first
order approximations. There is much debate whether or not material is
dredged up at every thermal pulse, and how much. Furthermore, it
seems possible that the dredge-up process is turned off when a star
becomes a carbon star (e.g. Lattanzio 1989). Notwithstanding, this
simple three parameter model can explain essentially all present-day
observations of AGB stars so there appears no need for a more
complicated model (although this does not prove that our model
assumptions are correct). We will investigate the sensitivity of the
stellar yields on the adopted values of M
,
, and 
in Sect. 4.
An additional source of uncertainty is associated with the number of
atoms (and isotopes) taken into account, i.e. H, He, 
C,
C, 
N and 
O. The first and second dredge-up
abundance changes are either taken directly from the model tracks of
the Geneva group, or, in the synthetic model, through parametrisation
of other model calculations. All these works include a much larger
chemical network than considered here. The third dredge-up is
simplified in the sense that only the 
C(
O
is included and that the abundances of 
C, 
O and He in
the convective zone after a TP are taken from Boothroyd & Sackmann
(1988) and are assumed to be constant. In particular, we do not
consider s-process reactions which take place in the convective
inter-shell. The most important effect of this process on the species
we consider is through the 
C(
O
reaction. However, the amount of matter burnt in this reaction is
probably small (Marigo et al. 1996) and apparently depends on the
amount of semi-convection assumed in the models (Busso et al. 1992,
1995).
Other uncertainties concern the detailed inclusion of HBB. In
particular, the temperature structure of the envelope, the fraction of
dredged up material processed in the CNO cycle, and the amount of
envelope matter mixed down and processed at the bottom of the envelope
may vary among AGB stars differing in initial mass, composition, and
age. Nevertheless, although the details on HBB are poorly understood
yet, good agreement is obtained between the standard model predictions
including HBB and observations related to HBB in massive AGB stars.
We will investigate the effect of HBB on the stellar yields by
introducing the parameter 
, which defines the core mass
at which HBB is assumed to operate (according to the recipes outlined
in the Appendix in GJ). The default value used in the standard model
is 
= 0.8 
which de facto is the value used in GJ and
GHJ. Other values of 
are discussed below.