As the stars are ascending the AGB, the features of the thermal pulses
evolve, and some of the main characterizing quantities of these events reach
``asymptotic values". The most important feature is the thermal pulse
intensity that can be defined in terms of the HeBS luminosity 
reached at the maximum of the He-shell flash. 
is quite low
during the first thermal pulses, but it rapidly increases with core mass (or
flash number). The increasing rate is greater for lower metallicities or
higher stellar masses. After typically 10 (5) thermal pulses in our Z =
0.02 (0.005) models, the thermal pulse intensity is very high and only
slightly increases from pulse to pulse, monotonously with the core mass.
This second part of the TP-AGB phase is usually called the ``asymptotic
regime'' or ``full amplitude'' of the thermal pulses.
For each of the modeled stars, we have computed at least four full amplitude thermal pulses. Let us briefly discuss the main evolutionary properties of these thermal runaways.
The thermal runaways are stronger inside stars of lower Z, which facilitates dredge-up events (see Sect. 6.2). The intensities of our He-shell flashes are comparable to other published values for stars of about the same core masses (see e.g. Boothroyd & Sackmann 1988b and Table 4).
While the stars evolve along the TP-AGB, the HeBS and HBS advance in mass, and the mean inter-shell mass slightly decreases with time. From one pulse to another, the top of the convective tongue at its maximum extension gets closer from the HBS, even if contact between both regions has never been found due to the fact that the HBS is still active at that time. Moreover, for all the thermal runaways we have computed, the overlap in mass between two successive thermal pulses is ranged between 0.6 and 0.4 for AGB stars of increasing total mass. This overlap is slightly larger for our lower Z models.
The maximum (minimum) temperature 
(
) and density
(
) reached at the base (top) of a thermal
pulse both increase (decrease) from pulse to pulse. For a given He-shell
flash, 
increases with stellar mass and metallicity, while
decreases with stellar mass but increases with metallicity.
In the asymptotic regime, the growth rate of 
also tends to
an asymptotic value. By comparison with other works for the 
(Z =
0.02) star, we found that our 
are typically 4 % higher than
those of Boothroyd & Sackmann (1988b) and (Straniero et al. (1996)]str96. This
could be mainly due to (i) different prescriptions to compute
, (ii) different envelope masses due to
different mass loss rates and (iii) different MLT treatments (see
Sect. 2.1.4). The temperature and density differences between the base
and top of each convective tongue are rather large. This point is important
for nucleosynthesis purposes (see Sect. 7).
The duration of the convective tongue associated with each thermal pulse
(
) and, to a lower extent, that of the inter-pulse phase
(
), are crucial quantities for the nucleosynthesis too. The
stronger a He-flash, the shorter it is and the quickest it deactivates.
Both quantities are thus smaller for higher core masses and lower
metallicities. Moreover, 
tends to decrease as the stars evolve
along the TP-AGB. 
also decreases after the first He-shell
flashes, but it then reaches an almost constant value during the asymptotic
regime, only very slightly decreasing with the increasing core mass.
Along the TP-AGB phase, the bottom of the convective envelope can penetrate deep inside regions that have been nuclearly processed just after a thermal pulse, if it is strong enough (in order to give rise to a large enough entropy increase in the inter-shell region). Contrarily to the first and second dredges-up, this so-called third dredge-up, when it occurs, successively mixes up to the surface (i) material that has experienced H-burning in the thin HBS but also (ii) part of the region where the thermal pulse nucleosynthesis operated just before. Both regions give rise to very different - and sometimes opposite - chemical pollutions of the convective envelope (see Sects. 7.1 and 7.6).
In our Z = 0.02 models, the HBS region is already dredged-up from the
14th, 5th and 3rd thermal pulse for the 3, 4 and 
stars,
respectively, while for the Z = 0.005 models, such a mixing already occurs
from the 2nd and 1st thermal pulse for the 3 and 
stars,
respectively. The 
(
) star with Z = 0.02 (0.005)
continuously mixes H-burning products up to the surface as it experiences
H-burning at the bottom of its convective envelope (the so-called
``Hot-Bottom Burning'' or HBB; see Sect. 7.1). This kind of dredge-up
already modifies the surface composition for some species (like
; see Sects. 7.4 and 7.6).
The complete third dredge-up (one usually simply calls the third
dredge-up, 3DUP hereafter), i.e. reaching the upper part of the previous
thermal pulse region, is more difficult to obtain numerically. Following
models (Wood 1981; Iben 1983; Lattanzio 1987;
Boothroyd & Sackmann 1988c),
the dredge-up of thermal pulse material is easier for stars of lower
metallicity Z. This is supported by observations (see e.g.
Blanco et al.
1978). In our models, we just found a 3DUP from the 9th thermal pulse of
the 
with Z = 0.005 star. However, based on the evolution of
the post-thermal pulse entropy profile from pulse to pulse, we can
predict that such a dredge-up will also occur, within the next two
thermal pulses, for our 5 and 
stars with Z = 0.02 and for the
star with Z = 0.005 (see Table 4 (click here) for the number of
computed thermal pulses for each object). Some additional thermal pulses
are still needed for lower mass objects. Several years ago, the 3DUP was
found to occur too late along the TP-AGB phase for AGB stars of decreasing
mass and/or increasing metallicity (see e.g. Iben 1976,
Fujimoto &
Sugimoto 1979). This was a common problem (see Lattanzio 1989 for a
review). Recent works showed that with the new OPAL opacities, the
3DUP can now be obtained even for low-mass Pop I objects (see e.g.
Frost
& Lattanzio 1996 or Straniero et al. 1995).
One of the clearest signature of the 3DUP is the significant 
pollution of the convective envelope, brought into the inter-shell region by
the convective tongue of a thermal pulse (see Sect. 8.1). As the thermal
pulse intensity (and consequently the probability of 3DUP occurrence)
increases with time, as do the core mass and the total luminosity, the
models tend predict the formation of carbon (C) stars (for which the surface
/
becomes greater than unity thanks to the
repetitive 3DUP events) at too high luminosities, i.e. to late during the
TP-AGB phase, especially for solar metallicity stars. Solutions to increase
the depth of the convective envelope during a dredge-up, e.g. by treating
in some way semi-convection (see e.g. Lattanzio 1986) or by invoking
undershooting (see Alongi et al. 1991) have been suggested. They indeed help
the 3DUP to occur but do not yet solve completely the problem.
Fundamentally, this could be due to our bad knowledge of convection inside
stars. Very recently, Straniero et al. (1996) suggested that by considerably
increasing the number of mass shells and especially the number of time steps
to model thermal pulses, the 3DUP occurs earlier during the
TP-AGB phase. This consists in a real numerical progress in that field. The
fact we find 3DUP latter than them along the AGB phase can be due to our
lower number of time steps (typically by a factor of 3).
It was first discovered by Paczynski (1975) that during the asymptotic
TP-AGB phase, there is a linear relation between the surface luminosity L
and the core mass 
(see Table 2 (click here)) which is only
slightly dependent on the total mass of the AGB star. Blöcker &
Schönberner (1991) however found that this kind of relation does not exist
for the most massive AGB stars for which the bottom of the convective
envelope goes down to the HBS so that part of the H-burning occurs inside
the convective envelope (HBB).
From our models for the last four thermal pulses of the 3 and 
with
Z = 0.02 and the 
with Z = 0.005, we have derived a core
mass-luminosity relation in the form
where 
, 
, 
, 
, 
and 
. It
is in good agreement with other 
relations, for 
and 
(see e.g. Lattanzio 1986 or
Boothroyd & Sackmann 1988a). The more massive stars experience HBB and do
not fit, consequently, to the same relation. In Table 4 (click here), we
however give a linear relation of the form 
for the 
(
) AGB star with Z = 0.02
(0.005) that do not experience a very strong HBB. Let us stress that
during the TP-AGB phase, L suffers very important variations during and
after each thermal pulse. Frequently, the core mass-luminosity relation
is derived by taking solely account of the maximum surface luminosity
preceding each thermal pulse. However, the probability to observe an AGB
star just at that time is very weak (
; see Table 4 (click here)). We thus preferred to derive our relations with
a more realistic surface luminosity that is averaged over each
pulse-inter-pulse cycle.
On the other hand, it is well known that the rate of mass advance of a
burning shell (here 
for the HBS) is related to the rate of nuclear
energy production by
with 
being the luminosity produced by the HBS,
the mean hydrogen mass fraction over this burning shell
and 
the amount of
energy produced by the conversion of 1 g of 
to 
through the CNO bi-cycle. Along the TP-AGB, most of the surface
luminosity L is produced by the HBS. As indicated in Table 2 (click here),
the HeBS contributes much less than the HBS to the nuclear luminosity
(except during a thermal runaway). Gravitation also constitutes a
relatively small energy source, especially for the less massive AGB
stars. In conclusion, we can write 
with 
depending on M. By eliminating 
in Eq. (6) with the core
mass-luminosity relation, one finds that the core mass 
increases
exponentially with time during the asymptotic TP-AGB phase. The time
scale for the core mass growth is 
(
) yr for a
(
) AGB star, that is, interestingly, comparable to the
total duration of the TP-AGB phase. Table 5 (click here) demonstrates that
Eq. (6) predicts rates of HBS advance in mass that are very similar to
those coming from our numerical models, except for the 
(
) object with Z = 0.02 (0.005), probably due to the strong HBB.
Table 4:
Some features of our TP-AGB models at the end of the complete stellar
evolution computations. We indicate the total number 
of computed
thermal pulses. The core mass (
) - luminosity (L) relations
indicated for the 
(
) star with
Z = 0.02 (0.005) is marginally valid as these stars already experience
significant hot-bottom burning. No such a relation is given for the
(
) object with Z = 0.02 (0.005). We refer to the text
for the fitting core mass-luminosity relation found for the other
stars. 
is the envelope mass. We also indicate the intensity of
the last computed thermal pulse (in term of the maximum luminosity
produced by the HeBS during this thermal runaway), the
duration of the last inter-pulse (
) and pulse
(
), the time 
elapsed between the maximum
extent of the convective tongue of the last thermal pulse and its
disappearance, the maximum mass extension of the convective tongue
(
) and the mass overlap between the last two thermal pulses
(
). We finally mention the occurrence of the third
dredge-up
Table 5:
Comparison between predictions for the rate of HBS advance in mass from
Eq. (6) and those coming from our computed models. The mean hydrogen
mass fraction 
and HBS luminosity 
are taken during
the inter-pulse phase of the last computed thermal pulse for each star.
is correspondingly calculated over the same inter-pulse
phase
One of our ambitions is to extrapolate our predictions coming from the
complete evolution models not only for the global surface properties but
also for the chemical composition. Eqs. (5) and (6) already allow to
extrapolate rather accurately the core mass and surface luminosity from our
last computed models towards the end of the TP-AGB phase. For the 
(
) star with Z = 0.02 (0.005), independent fits of L and 
as a function of time have been calculated. We also need fits for the
surface radius [in order to follow the mass loss rate increase through Eq.
(1)] as well as the temperature and density at the base of the convective
envelope (to follow the HBB nucleosynthesis). We found, on grounds of the
last four computed thermal pulses for each star, very good fitting relations
for these quantities, which are presented in Table 6 (click here). Note that
they are linear with time during the last part of the TP-AGB phase,
contrarily to 
and L that increase exponentially during this phase.
The main interest of such an extrapolation procedure is that very detailed stellar evolution computations (including a very complete description of the nucleosynthesis) would take prohibitive computation time. However, we had to make two important approximations (compared to full evolution models) that must be mentioned.
, 
and heavier elements to the detriment of
. This increases the mean molecular weight (or Z) and the
mean opacity in the sub-photospheric region, probably leading to an
overestimation of the surface luminosity.
Table 6:
Fitting parameters from the last four computed asymptotic thermal pulses
towards the end of the TP-AGB phase, for the mean total radius and
bottom temperature and density of the convective envelope. For each
quantity A, the relation reads 
, with
time in yr
Such extrapolation procedures have already been realized by Groenewegen &
de Jong (1993 and following papers). These synthetic evolution models are
very powerful to compare observations with computed models concerning the
evolution of the global surface properties of AGB stars. However, they
contain shortcomings that do not allow to predict in detail the evolution of
the surface isotopic composition. Mainly, (i) only the most abundant
elements are followed (
, 
, 
,
and 
) and (ii) the HBB nucleosynthesis is not
followed in detail whereas it becomes a very influent process concerning the
convective envelope composition towards the TP-AGB tip. Recent progresses
have been made by Busso et al. (1995) and (Lambert et al. (1995)]lam95 to follow in
detail the nucleosynthesis as late as possible along the AGB phase.
In Sect. 8, we both present the nucleosynthesis encountered in our full evolution models and the extension of the predictions concerning the convective envelope composition towards the end of the AGB phase corresponding to the seven computed stars. For this extrapolation, we proceed as follows.
. To quantify the adopted depth of each 3DUP, let us
remember the definition of the usual parameter
where 
represents the amount of mass that is
dredged-up (difference between the normal depth of the convective envelope
and its maximum downwards penetration during a dredge-up) and 
the increase in mass of the HBS between two successive thermal pulses. We
adopted a 3DUP depth corresponding to 
for all the stars.
This value has been suggested by Groenewegen & de Jong (1993) because it
reproduces rather well the carbon star luminosity function of the LMC
(that cannot be fitted on grounds of self-consistent evolution models
only; see Sect. 6.2).
During such a dredge-up, we assumed that the material coming from the convective tongue of a thermal pulse has abundance profiles corresponding to end of the last computed thermal pulse. Of course, we cannot do better without further computing thermal pulses with detailed nucleosynthesis up to the end of the TP-AGB phase that is, again, not yet realistic. However, this appears to be rather justified for many important chemical elements as they reach almost constant abundances after the few last computed thermal pulses (see Figs. 7 (click here) to 9 (click here) and discussion in Sect. 7).
Taking a constant value for 
is probably not strictly right. As far
as the surface composition is concerned, increasing 
more
efficiently pollutes the convective envelope from pulse to pulse.
There is another quantity that greatly influences the evolution of the
envelope composition: the mass loss rate. Indeed, decreasing the mass loss
rate allows to increase the total number of thermal pulses (and consequently
the total number of 3DUP episodes) before the end of the TP-AGB phase. We
had to increase our 
values in Eq. (1) for each star, in order to
reach the highest mass loss rates (called super-winds) observed among very
evolved AGB stars (see the discussion in Sect. 2.1.5; see also
Zuckerman et
al. 1986; Wannier & Sahai 1986;
Knapp et al. 1989): we adopted 
,
10, 15 and 20 for the 3, 4, 5 and 
AGB models, respectively,
whatever Z. As the exact way following which the mass loss rate increases
with time along the asymptotic TP-AGB phase is still questionable by more
than a factor of two (see Sect. 2.1.5), the resulting uncertainty on the
evolution of the surface composition up to the AGB tip dominates the effect
of 
(see our discussion in Sect. 8.1, where our mass loss rates are
varied by a factor of four). Let us finally mention a very important
observational fact in favor of rather high mass loss rates for
intermediate-mass AGB stars. Oke et al. (1984) and
Bergeron et al. (1991)
have determined the mass distribution of DA and DB white dwarfs, resulting
from the AGB phase. It is rather Gaussian, with a mean mass of 
and a small standard dispersion of 
only. Whatever IMF law is
considered, this clearly rules out intermediate-mass objects ending their
AGB phase with high core masses.
In Table 7 (click here), we present the global mass loss increases we obtained, for the seven TP -AGB stars, from the beginning of the TP-AGB phase up to its end (i.e. including our extrapolation computations).
Table 7:
Increase of the mass loss rates during the whole TP-AGB phase
One of the most important constraint for evolutionary models towards the AGB
tip is the reproduction of the observed correlation between the initial mass
of stars (
) and their final one (
), i.e. their core mass (become a
hot white dwarf) when the convective envelope has been completely removed
after the planetary nebula (PN) ejection. We present in Fig. 2 (click here) the
comparison between observations analyzed by Weidemann & Koester (1983) and
Weidemann (1987) and the final core masses we obtained after our
evolutionary plus extrapolations computations. We note that agreement with
observations is slightly better than for Vassiliadis & Wood (1993),
especially around 
where they only reproduce the upper
limits. On the contrary, the final core mass obtained by Boothroyd &
Sackmann (1988b) is rather low compared to observations. Note we have not
taken into account of the core mass reduction following each 3DUP event.
However, as explained below, such a reduction, even if it increases the
agreement with the observed relation, is rather negligible compared to other
uncertainties (core mass at the first thermal pulse, thermal pulse number
depending on the adopted mass loss rate, ...).
Figure 2:
Relation between the initial mass of stars and the final white dwarf
(WD) mass at the end of the AGB phase. Observational final masses are
derived either from the surface gravity of the WD (filled circle) or
from their radius (open circles); lines are connecting both
determinations for identical objects. Our predictions are the black and
gray rectangles, for the Z = 0.02 and Z = 0.005 modeled stars,
respectively. Their extension is the result of varying the mass loss
rate by a factor of four during the asymptotic TP-AGB phase
We observe that changing the mass loss rate by a factor of four during the
asymptotic TP-AGB phase (see also Sect. 8.1 concerning surface abundance
predictions) does not significantly change the agreement with observations.
We claim that this is due to the fact that while almost 50% of the C-O
core mass is built during the central He-burning phase, roughly 35 to 45%
(in mass) are added during the E-AGB phase. More precisely, during the whole
TP-AGB phase, we found that our core masses increased by 14 (12), 7 (4), 5
(3) and 4% for the 3, 4, 5, and 
stars with Z = 0.02 (0.005),
respectively. The final core mass is consequently almost decided before the
super-wind occurs and this observed relation is not very compulsive for the
empirical mass loss rate to adopt during the TP-AGB phase.