. Generalized Fermi-Dirac integrals have
been very accurately computed for a wide range of temperatures and
densities;
pair annihilations
and bremsstrahlung neutrinos. The formalism used to evaluate their
respective number densities and energy loss contributions comes from
Munakata et al. (1985), Itoh et al. (1992) and
Kohyama et al. (1993).
At low temperatures, i.e. below 8000 K, we use the Alexander & Ferguson
(1994) opacity tables, very well suited for cool red giant envelopes and
atmospheres. They include (1) continuous opacities for H, 
,
, 
, He, C, N, O, Na, Mg, Al, Si, Ca and Fe,
(2) line opacities for C, N, O, F, Ne, Na, Mg, Al, Si, P, S, Cl, Ar, Ca,
Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and 59 million molecular lines for HF,
HCl, CH, 
, CN, CO, CS, NH, OH, MgH, AlH, SiO, SiS, SH, TiO, ZrO,
YO, 
, HCN, and 
, (3) electron and
Rayleigh scattering for H, He and 
and (4) absorption and
scattering by dust grains with radii between 0.005 and 
(oxygen-rich silicates, iron condensates, amorphous carbon and
silicon carbides).
At temperatures above 8000 K, we use the OPAL opacity tables computed by Rogers & Iglesias (1992), including bound-bound, bound-free and free-free transitions for 12 elements (from H to Fe), the abundance distribution of the heavy elements being scaled from the solar composition given by Anders & Grevesse (1989) and updated by Grevesse (1991), as for low-temperature opacities.
For both sets of opacity tables, we first interpolate bi-linearly in T
and 
for each table the composition of which is
close to the model composition and then linearly in X, Z (and X(O))
to obtain the right opacity coefficient corresponding to each shell.
Finally, we used the Hubbard & Lampe (1969) program to generate conductive opacity tables corresponding to the same chemical compositions [X, Y, Z and X(O)] and to the same T and R grid points as for the OPAL radiative opacity tables.
The stellar structure is integrated from the center to a very low optical
depth (
) in the atmosphere. However, at low optical depth (in
practice, for 
), the 
temperature profile and derivative
quantities (such as the radiative pressure and gradient) no more correspond
to the Eddington approximation. We consequently constraint the atmospheric
temperature profile to correspond to those coming from realistic atmosphere
models obtained by integrating the radiative transfer equation. This
technique was already used by Paczynski (1969). More specifically, we use
atmosphere models (1) from Plez (1992) for 
< 3900 K, (2)
from Eriksson (1994, private communication, using a physics similar to Bell
et al. 1976) up to 5500 K and (3) computed with the Kurucz atmosphere
program above 5500 K. At each time step, we linearly interpolate a 
profile corresponding to our model 
, 
and Z from these grids
of atmosphere models. Of course, we took care to adopt the same 
definition as for these models (see e.g. Plez 1992).
The structure of the convective regions is computed using the classical
Mixing-Length Theory (MLT), as prescribed by Kippenhahn et al. (1968). Our
models are standard in the sense that (1) the Schwarzschild criterion
is considered to delimit the convective zones and (2) neither
overshooting nor semi-convection have been considered in the present
computations. The ratio 
of the mixing-length free parameter over
the pressure scale height has been put to a value of 1.5, i.e. rather close
to 
with which we fit the solar structure. Such a
value is similar to that found by Schaller et al. (1992)
and Richard et al. (1996), while it is lower than that obtained by Sackmann
& Boothroyd (1991a) and Chieffi et al. (1995). Previously by using the old
(Huebner et al. (1977)]hue77 opacity tables, we fitted the Sun with 
. Such a 
increase with the new opacities is a common
feature among the above-mentioned works. Note that among those works, a
rather wide 
range is required to fit the Sun; this is
probably due to the different prescriptions used to solve the MLT equations.
Finally, various semi-empirical algorithms have been developed to better
estimate the convection boundaries when convection penetrates inside regions
of very different chemical composition (i.e. during dredge-up phases; see
e.g. Lattanzio 1986). We did not used any of them. This can lead to
disadvantage the third dredge-up occurrence along the AGB phase.
From the Main Sequence up to the thermally pulsing AGB phase, we used the empirical Reimers (1975) formula
with 
up to the central He exhaustion.
From the beginning of the AGB phase, we adopted variable 
values,
depending on the total mass of the star. More specifically, we took 
, 3, 3.5 or 4 for our 3, 4, 5 or 
models respectively,
whatever Z. This 
increase with the total mass was found by Bryan
et al. (1990) to give rather good results along the AGB. However, later
during the thermally pulsing phase, Blöcker (1995) recently showed that
the Bryan et al. (1990) formula underestimates the observed mass loss
rate. This leads us to further increase 
(see Sect. 6.3).
Vassiliadis & Wood (1993) presented another empirical formula based on
the relation between the mass loss rate and the pulsational period of AGB
stars. This still increases the agreement with the observed initial-final
mass relation. This last formula also indicates that the only virtue of
the Reimers formula is that its mass dependence 
reproduces rather well the mass loss rate increase during the AGB phase.
However, as mentioned by Vassiliadis & Wood (1993), whatever empirical
formula, the mass loss rate increase used in the models along the
thermally pulsing AGB phase still remain uncertain by more than a factor
of two. As demonstrated in Sect. 8, this important uncertainty on the mass
loss rate formula conditions the AGB phase duration and consequently, the
surface composition evolution through the successive third dredge-up events.
We note that along the thermally pulsing AGB phase, the thermal pulse
features slightly depend on the way one estimates the rate
of gravitational energy release or gain (see e.g.
Kippenhahn & Weigert 1991). So, let us specify that we wrote
where u is the specific (i.e. by unit mass) internal energy.
We follow the abundance evolution, through the whole stars, of 45 nuclides,
namely the neutrons, all the 31 stable nuclides up to 
as well
as 
, 
, 
, 
, 
,
, 
, 
, 
, 
,
, 
and 
.
These nuclides interact through a network containing 172 nuclear (neutron,
proton and 
captures) and decay reactions. Many nuclear reaction
rates are taken from Caughlan & Fowler (1988). Let us just mention the
nuclear reactions for which we selected another rate prescription.
from Nagai et al. (1991),
from Caughlan et al. (1985),
from Raman et al. (1990),
from Beer et al. (1991),
from Wiesher et al. (1990),
from Funck & Langanke (1989),
, 
,
and 
from Fowler
et al. (1967),
from Brehm et al. (1988),
from De Oliveira (1995),
from Allen et al. (1971),
from (Wagoner (1969)]wag69,
and 
from Landré
et al. (1990),
from Thielemann (1991),
and 
from Bao &
Käppeler (1987),
from Kious (1990),
, 
,
, 
,
and 
from Beer
et al. (1991),
and 
from
Hammer (1991),
, 
,
and 
from
Woosley et al. (1978),
and 
from
Champagne et al. (1988),
and 
from Illiadis
(1990),
and 
from Vogelaar (1989).
On the other hand, the nuclear reaction rates for 
,
, 
,
, 
,
, 
,
as well as the 45 nuclear reactions involving
Si, P and S are fitted with a procedure developed by Rayet (1993, private
communication). Let us finally mention recent improvements not yet
incorporated in our network: various 
reactions rates on
light nuclei (Nagai et al. 1995) and 
capture rates on 
by Käppeler et al. (1994).
Finally, the nuclear screening factors are parameterized by using the Graboske et al. (1973) formalism, including weak, intermediate and strong screening cases.
Let us stress that this full network has been used to follow the nucleosynthesis in each shell at each time step.
Neutrons can be produced during the thermally pulsing AGB phase. They can,
in particular, be captured by nuclides heavier than 
, mainly
through 
reactions. As we do not follow the abundance of
such elements, we created an additional nuclide, called
, the mass fraction of which is set to
This sum counts in fact 253 stable nuclides up to 
.
In order to have a good estimation of the free neutron abundance, we follow
a method described by Jorissen & Arnould (1989). It consists in adding a
nuclear reaction 
to which
we attribute a Q-value and a nuclear reaction cross-section that is
averaged (and weighted by the seed nucleus number abundance) over all the
individual Q-values and 
cross-sections from
. This procedure differs for reactions involving elements
lighter or heavier than 
.
If nuclear reactions occur inside a convective zone, most of them proceed in general with a longer time scale than the turn-over time scale associated to the motion of convective cells. In such a case, one usually consider the convective mixing as instantaneous. This allows to treat nucleosynthesis in one shot, by taking mass-weighted averages of the number abundances and nuclear reaction rates over the whole convective region.
However, if some key nuclides nuclearly evolve more rapidly than they are mixed, we have to consider the convection transport together with the nuclear reactions, so that nucleosynthesis equations become diffusion equations of the form
where 
, the ratio of the mass fraction over
the atomic mass of each nuclide i, and 
,
the convective diffusion coefficient, with 
and 
being
the mean velocity and turn-over time of the convective cells, respectively.
It is very CPU time consuming to treat the nucleosynthesis and convective
mixing together through such diffusion equations for each nuclide in each
convective shell for all the time steps. Consequently, as usual,
instantaneous mixing has been assumed by default for our stellar evolution
computations, except in two cases that potentially require a time-dependent
treatment of the convective mixing: (1) the convective tongue
associated with thermal pulses and (2) the base of the convective
envelope when it is hot enough, both along the AGB phase.
We verified that inside thermal pulses, the elements that are the most
affected by a diffusion treatment of the convective mixing (i.e. that do not
have a flat profile inside the convective tongue of such thermal pulses) are
the very unstable nuclides (like 
), neutrons and protons.
However, this do not change very significantly the nucleosynthesis global
results (see however Sect. 7.5).
On the contrary, the mean turn-over time scale of the convective motions is
much longer inside the convective envelope than inside a thermal pulse. If
nuclear burning occurs at the base of the convective envelope, some
important nuclear reactions can, depending on the bottom temperature, occur
faster than the convective mixing so that a diffusion treatment is required.
This is in particular the case for some of the reactions involved in the
synthesis (see Sect. 7.2).
We have built initial models of 3, 4, 5, 6 and 
by solving the
Lane-Emden equation with a polytropic index n = 1.5. The initial radius of
these polytropic models has been chosen large enough in order to get central
temperatures below 
K. This roughly corresponds, in the
Hertzsprung-Russell diagram, to the arrival on the Hayashi track marking the
beginning of the pre-main sequence phase.
For our Z = 0.005 (0.02) models, the hydrogen mass fraction has been set to X = 0.745 (0.687) and that of helium to Y = 0.250 (0.293). The abundances of the heavier elements have been scaled on the Anders & Grevesse (1989) distribution that characterizes the elemental and isotopic composition of the material from which the solar system has been formed.