From the pioneer work of Iben (1975), one realized that inside TP-AGB stars,
a very rich nucleosynthesis occurs. During the recurrent third dredge-up
events, the synthesized nuclides are mixed inside the convective envelope
and then ejected into the interstellar medium (ISM) through the strong mass
loss. In conclusion, TP-AGB stars play a crucial role for the chemical
evolution of galaxies, especially for some elements like 
,
, 
, 
, nuclides heavier than
, ...It is the reason why we will give, in Sect. 8.3, the
yields resulting from our intermediate-mass star computations.
Let us analyze the various nucleosynthesis processes occurring inside intermediate-mass TP-AGB stars. In fact, three very different sites have to be distinguished.
As indicated in Table 2 (click here), most of the nuclear energy release on the
AGB is due to the HBS, mainly through the CNO bi-cycle. More precisely, the
six most important nuclear reactions that energetically supply the star are
(
), 
(
), 
(
), 
(
), 
(
)
and 
(
), all occurring inside the
HBS. These numbers are quite similar (within a few %) for all our
modeled stars. Whatever the AGB star total mass, the HBS slightly depletes
and 
and significantly depletes 
,
and 
, while 
is produced. The case
of 
is more difficult: while its abundance is very slightly
increased in the HBS of 
stars (whatever Z), it is somewhat
decreased in the HBS of 
stars.
The rather high temperature inside the HBS also allows the activation of the
NeNa and MgAl chains, even if they are much less influent for the global
energetics. The most interesting consequences of the NeNa chain are the
production of 
and 
in 
stars and
their slight destruction in more massive AGB stars. The MgAl chain is very
sensitive to the AGB total mass. In < 
objects, 
is
destroyed to the benefit of 
(roughly half in 
and half in 
that quite instantaneously decays in
); a small quantity of 
is also destroyed in
. In more massive AGB stars, the HBS temperature is high enough
to substantially produce 
through the
reaction. This in turn leads to a much
important 
production. However, as the HBS mean temperature
always increases with time, 
begins to be partially destroyed
by 
reactions late along the TP-AGB. We refer to
Forestini et al. (1991) for the first detailed study of 
production inside AGB stars, by radiative proton burning (i.e. in the HBS).
In Guelin et al. (1995), more details can be found, including the
possibility to synthesize 
by proton burning at the bottom of
the convective envelope (i.e. by HBB). It was however a parametric study.
First predictions concerning the enrichment of the AGB star surfaces, on
grounds of full evolutionary models are presented in Sects. 7.6 and 8.1.
The comparatively small amount of nuclear energy supplied by the HeBS comes
from the 
(
) and 
(
) reactions. Consequently, the HeBS mainly produces
and, to a lower extent, 
, while during the previous
central He-burning stage, the 
production was preponderant. At
the beginning of a thermal pulse, it is the 
reaction that becomes
the dominant energy source (see also Sect. 7.1.3 for the thermal pulse
energetics).
During the long inter-pulse phase that follows a 3DUP episode, the
coming out of the HBS can be burnt in the inter-shell region,
close to the HeBS top, mainly through the 
reaction. This reaction is highly energetic. The first published work
reporting the existence of this inter-shell 
burning is
Straniero et al. (1995). The produced neutrons are then captured by light
species up to 
mainly, but part on them are also used to
synthesize s elements. By decreasing order of amplitude, the nuclear
energy liberated in the inter-shell region during the inter-pulse phase
comes from these 
reactions up to 
(
), 
(
), the so-called s-process
reactions (
), 
(
) and
(
). Note the important nuclear energy
released by the neutron captures. This energy production creates a third
burning shell between the HeBS and HBS, even if its nuclear activity is
considerably lower than that of both the HeBS and HBS. It continues to exist
until 
has been destroyed there. This happens, in general,
almost when the following thermal pulse begins. As a consequence, when the
convective tongue of a thermal pulse engulfs the inter-shell matter, the
amount of 
that is mixed is reduced compared to what was
previously thought, i.e. 
(
) [H - He],
where [H - He] designates, as before, the inter-shell region. We show, in
Fig. 3 (click here), the nuclear energy production rate (
)
profile in the case of our 
AGB star. This clearly demonstrates the
three-burning shell structure. Such a structure is also found in the more
massive AGB stars. Let us emphasize that this third burning shell could at
least partially explain the production of the s nuclides (see Sect. 7.5)
in a radiative zone. This has been suggested by Straniero et al. (1995),
even if their amount of 
in the inter-shell was artificially
increased in order to study the s-process nucleosynthesis.
Figure 3:
Profile of the nuclear energy production rate per unit mass,
, in the region of the burning shells, in the
case of the 
(Z = 0.02) star, during the inter-pulse phase
between the 18th and the 19th thermal runaways. The dotted curve
corresponds to a time just before the 19th thermal pulse, while the solid
one is plotted roughly at the middle of the inter-pulse phase (see the text
for a discussion)
If the temperature 
at the base of the deep convective envelope is
high enough (i.e. 
K), i.e. if the base of the convective
envelope reaches the HBS, proton burning occurs inside it. This HBB occurs
in models with initial masses 
for Z = 0.02 and 
for Z = 0.005. These lower mass limits and this Z dependence
are common features of all the AGB models (see e.g. Boothroyd et al. 1993).
As mixing up to the cool surface is very efficient (the mean turn-over time
of the convective envelope is roughly 0.5 yr), the effective
nuclear reaction rates are lower than in radiative H-Burning, leading to
specific nucleosynthesis signatures.
For 
K already, 
can be produced from
following a scenario that cannot operate in radiative H-burning
(see Sect. 7.2 for more details). This Li synthesis has been qualitatively
suggested by Cameron & Fowler (1971) and quantitatively first demonstrated
on grounds of evolutionary models by Sackmann & Boothroyd (1992).
For 
K, the CN cycle begins to operate partially, i.e.
is converted to 
that accumulates. As first
investigated by Boothroyd et al. (1993), this conversion can efficiently
reduce the 
/
ratio and eventually, if HBB is strong
enough, this can prevent the formation of C stars (see also Sects. 8.1 and
8.2).
If 
> 
K typically, the CN cycle almost operates
at equilibrium through the whole envelope, leading to a
/
isotopic ratio lower than 10 (depending on
). In our models, this happens for the 
(
) AGB
star with Z = 0.02 (0.005) and also, later along the TP-AGB, for our
star with Z = 0.02 (see Sect. 8.1). At very high 
,
other reactions begin to occur; in particular, in extreme cases (i.e. the
end of the TP-AGB phase of our 
models with Z = 0.02),
can eventually be directly produced inside the convective
envelope through the 
reaction on
supplied by 
(see Sect. 8.1).
The thermal runaway, inside the HeBS, leading to form a convective tongue is
principally due to the 
reaction excitement. When this convective
tongue finally penetrates the inter-shell region, the HBS ashes are
ingested, i.e. not only 
but also 
, 
and
mainly. As these elements are partially burnt in the bottom
of the inter-shell region during the inter-pulse phase (globally, we find
that half of the 
is so destroyed), they mainly survive in the
upper part of the inter-shell. They are consequently engulfed by the
convective tongue when it reaches its maximum extension. At that time, the
reaction is already dominant for the energetics. We consequently
find that the energetics associated with the 
burning through
the 
reaction as well as the neutron captures
that follow is rather negligible (less than 1 %). This is also found by
Straniero et al. (1995). This conclusion could however be modified if more
was synthesized in the inter-shell region, as required for the
s-process (see Sect. 7.5). The important energetic effects reported by
Bazan & Lattanzio (1993) concerning the 
burning inside thermal
pulses was probably due to (i) the fact they did not give account of
partial inter-shell 
burning and (ii) the very high
amount artificially ingested into their thermal pulses in order
to study the consequent s-process.
Part of produced neutrons participate to 
reactions, mainly on
and, to a lower extent, on 
. The resulting
protons, at these high temperatures characterizing He-burning during a
thermal pulse (
K at the base of the convective tongue), are
rapidly captured by various elements (see Sects. 7.3 to 7.5 for more
details). Let us stress that the simultaneous presence, inside a mixed
He-burning region (the convective tongue), of neutrons and protons leads to
a very specific (and unique) nucleosynthesis.
We will analyze in more details the nucleosynthesis processes associated with the thermal pulses in Sects. 7.3 to 7.5.
Sackmann & Boothroyd (1992) were the first ones to quantitatively
demonstrate how the Cameron & Fowler (1971) scenario leads to 
synthesis in intermediate-mass AGB stars. This major result was obtained by
using a time-dependent convective diffusion algorithm suited for hot-bottom
convective envelopes. Indeed, at the base of such envelopes, some nuclear
reactions involved in 
production occur faster than mean
turn-over time associated with the convective motions (see Fig.
4 (click here)).
Figure 4:
Nuclear reaction time scales for the reactions involved in the
Cameron-Fowler scenario, inside the convective envelope of the 
(Z = 0.02) star during the 14th inter-pulse. At that time, its base
temperature is 
K. 
(
) and
(
) correspond to the proton capture on 
and 
, respectively. 
(
) and
(
) relate to the 
destruction
by 
and 
capture, respectively. Finally,
(
) is the electron capture rate on 
.
The mean turn-over time of the convective motions is also plotted
(
)
We also use a time-dependent treatment to study the 
synthesis
[see Eqs. (4) in Sect. 2.2.3]. We show in Fig. 5 (click here) the typical
resulting 
and 
profiles inside the convective
envelope of our 
AGB models with Z = 0.02. When HBB occurs (see
Sect. 7.1.2), 
first begins to burn through the
and 
reactions, the second one being significantly slower than the first one. As
in deep stellar interiors, the 
destruction by electron capture
and proton capture are much longer than the time scale associated with the
convective mixing, the produced 
is efficiently up-heaved
towards cooler regions where it progressively decays in 
. Let us
emphasize that without fast convective mixing, 
could not
accumulate at the surface. Indeed, at the base of the convective envelope,
(i) 
would be partially destroyed by electron and proton
captures and (ii) the 
resulting from 
decay
would be quite instantaneously destroyed by proton capture too.
Figure 5:
and 
profiles inside the same convective
envelope as in Fig. 4 (click here)
Let us now compare our predictions with Sackmann & Boothroyd (1992) results
and with observations. A direct comparison with their Figs. 2a and 2b
reveals nearly identical features. In Fig. 6 (click here), we present the
evolution of the surface 
abundance during the TP-AGB phase of
all our modeled stars. In our 
models, the temperature at the base
of the convective envelope is never high enough for 
to be
produced efficiently, and the surface 
abundance remains almost
constant at a very low value resulting from the first dredge-up. We also
see that our 
star with Z = 0.005 begins to produce some
when it enters the asymptotic regime, but the surface abundance
of this light element always stays at least one order of magnitude below
the interstellar medium (or cosmic) value 
(
)
. Note the 
destruction inside the 
star with Z = 0.02 attesting that its
convective bottom temperature is high enough to burn 
through
but not enough to produce 
by
burning. Finally, our 5 and 
models present real high
surface enhancements, compared to the cosmic value. The
maximum abundances we obtain range between 
(
)
to 5.1, depending on the stellar mass and metallicity. Our
super-lithium-rich stars are predicted to appear in a luminosity range
between 
and 
, i.e. for 
between
-6.25 and -6.65. These values are in good agreement with those found by
Sackmann & Boothroyd (1992).
As the evolution proceeds, the surface 
abundances finally
decrease again very rapidly. This indicates that 
has been
almost completely burned in the envelope [remember that the
is dominant], so that the 
production can no more be supported and it is then destroyed by proton
capture. This fact clearly puts an upper limit in luminosity to observe
super-lithium-rich AGB stars. This last feature is also present in Sackmann
& Boothroyd (1992), even if it appears somewhat later during the TP-AGB
phase of their most massive objects. This could be due to different mass
loss rate prescriptions.
How do these predictions compare with observational data (see Fig.
6 (click here))? In the Magellanic Clouds, the large majority of lithium-rich
giants are observed within a narrow luminosity range, 
to 
(Smith & Lambert 1989, 1990b;
Plez et al. 1993; Smith
et al. 1995), in very good agreement with the predictions. The observed
abundance range is also well reproduced, even if no star has been observed
with 
(
) higher than 4.5 in the Clouds. Moreover,
the mass estimate from pulsation theory for some lithium-rich giants of
both Clouds is consistent with the predicted mass range for surface
enhancement (Smith et al. 1995). Comparisons with 
in galactic giants are more delicate because of the larger observational
uncertainties on luminosity and mass. Galactic super-lithium-rich AGB stars
can exhibit 
(
) values as high as 5.4 ((Abia et al.
1991)]abi91, but these carbon stars have 
laying between -5 and -6.2,
i.e. rather lower than the predicted luminosities. This could be due to
a bad luminosity determination, as it seems difficult to synthesize
inside the convective envelope of AGB stars with such
relatively low luminosities. However, most of the galactic
super-lithium-rich AGB stars have lower masses than the ones we model here,
and they will be addressed in a forthcoming paper.
Figure 6:
Resulting surface 
abundances, in term of
(
), for the seven computed TP-AGB phases, as a
function of the stellar surface luminosity. The solid (dotted) lines
refer to the Z = 0.02 (0.005) models. Observations are shown for
bright AGB stars of the SMC and LMC (Smith et al. 1995) and of the
Galaxy (Abia et al. 1991)
We now review all the species (from C) by indicating in which nuclear
reactions they are implicated inside the convective tongue of the successive
thermal pulses, as a function of the initial mass M and metallicity Z of
the computed TP-AGB models. We refer to Figs. 7 (click here) to 9 (click here)
for the discussion. The mean mass fractions 
are calculated
by taking the average of the mass fraction profile of each nuclide a over
the maximum region covered by the convective tongue of each thermal pulse,
just after its disappearance, i.e.
We also refer to Figs. 10 (click here) to 16 (click here) that present the mean mass fractions of key nuclides (calculated as for the convective tongue) in each burning region (i.e. the HBS and inter-shell regions) compared to their convective envelope content, respectively for the seven computed AGB stars. Similarly, Figs. 17 (click here) to 23 (click here) present the abundance variations of the same nuclides inside typical convective thermal pulses.
Figure 7:
Mass fraction of each key nuclide resulting from the thermal pulse
nucleosynthesis, as a function of the pulse number, for the seven
computed TP-AGB phases, up to 
Figure 8:
Same as Fig. 7 (click here), up to 
Figure 9:
Same as Fig. 7 (click here), up to 
During the growth of the convective tongue, the mean 
mass
fraction first decreases due to the 
mixing from the inter-shell
region. Then, at the thermal pulse maximum, 
is produced by the
reaction and also enhanced due to the downwards penetration of
the convective tongue into the He-burnt shell. Although this penetration is
commonly found, its amplitude seems somewhat more important in our case,
bringing more 
into the thermal pulses. Note this increased
amount of 
makes the 3DUP episodes more efficient to convert an
AGB star into a C star. During the first thermal pulses, the amount of
that is produced increases due to the higher temperatures
reached at the base of the convective tongue. This bottom temperature
increase from pulse to pulse is a common feature of all the stars and it
traduces that the thermal pulse intensity increases. Fig. 7 (click here)
shows that the last computed thermal pulses of the 
object with Z =
0.02 are even hot enough to partially destroy 
by the
reaction.
Most of the engulfed 
is destroyed very efficiently through
that produces neutrons. The trend of the
remaining mean 
mass fraction to increase from pulse to pulse is
due to the fact that stronger thermal pulses develop convective tongues
reaching regions always closer to the HBS, i.e. engulfing more material from
the inter-shell zone. As mentioned in Sect. 6.1, the bottom of the HBS
cannot be reached however.
is only produced by the capture of part of the neutrons on the
engulfed 
, through 
. Note that this
reaction contributes to the production of protons. For initial masses above
(
) with Z = 0.02 (0.005) however, the convective tongue
finally becomes hot enough (base temperature 
K) to
significantly destroy the produced 
by the
reaction. The relatively low 
production inside the 
with Z = 0.02 object is due to the
relatively lower amount of 
that is ingested (partially due to
the third 
burning shell; see Sect. 7.1.1), leading to a smaller
quantity of neutrons. However, the last thermal pulses of this star ingest
increasing amounts of 
and 
from the inter-shell,
leading to an increasing production of 
. Due to its short
lifetime (
yr), 
partially decays during the
inter-pulse phases of our less massive AGB models. This depletion is however
reduced by the 
production inside the radiative 
burning shell. For the most massive AGB stars that have inter-pulse
durations shorter that 
, 
mostly behaves like a stable
nuclide, even if strongly destroyed by 
captures. Consequently, as
explained with more details in Forestini et al. (1996), intermediate-mass
AGB stars are globally not important producers of 
.
Apart from the first two thermal pulses (whatever the initial mass or
metallicity), 
is almost completely destroyed inside thermal
pulses through 
. This reaction, as first
noted by Iben (1976), also significantly contributes to the energetics of
a thermal pulse. Only a small part of 
is destroyed through
the 
reaction. Indeed, the inter-shell mean mass
fraction of 
is ranged between 
for the 
star with Z = 0.02 and 
for the 
star with Z =
0.005, while the production level of 
is 
(
), in mass fraction, for the Z = 0.02 (0.005) models
(except for the 
with Z = 0.02 star; see above).
As are 
and 
, 
is engulfed in a thermal
pulse from the inter-shell region. The only significant reaction that
produces this element inside the convective tongue itself is
(see below for the 
production). On
the other hand, 
is mainly destroyed by the
reaction. Its rate is lower than that of
so that 
is efficiently
destroyed in all the stars only in the asymptotic regime, i.e. when thermal
pulses become very intense. Remember that it arrives earlier for lower Z
objects. Note that the 
stars (whatever Z) maintain a relatively
higher 
abundance, especially during the full amplitude thermal
pulses. Indeed, while these thermal pulses efficiently destroy 
whatever the initial mass of the star, its initial abundance inside the
convective tongue is higher for 
objects. This is due to the fact
that the engulfed inter-shell region of 
stars, that contains
, is significantly thicker (in mass) compared to that of more
massive ones.
being partially destroyed in the HBS, the mixing of the
inter-shell region reduces its mean mass fraction. On the other hand, at the
maximum extent of the thermal pulses, it is enhanced, like 
, due
to the downwards penetration of the convective tongue. From pulse to pulse,
it also becomes significantly produced by 
.
This is especially visible for Z = 0.005 stars, the initial 
content of which is lower. This primary 
production by the
thermal pulses finally stabilizes around 
(in mass fraction).
The decrease observed during the last thermal pulses of the 
AGB
star with Z = 0.02 traduces the more important role of mixing of
inter-shell material (already emphasized for 
and 
).
The 
previously contained in the inter-shell region is also
mixed in the convective tongue. It is not produced inside thermal
pulses. On the contrary, it can be destroyed mainly through
and 
, the
first reaction being in general at least ten times faster than the second
one. This weakly contributes to produce neutrons inside the thermal pulses.
This 
destruction already operates from the first thermal
pulses, whatever the initial M and Z. In later thermal pulses, the
mass fraction stabilizes around 4 to 
, that
represents a balance between the amount of 
brought from the
inter-shell region and destroyed by the thermal pulse. Again, our 
models with Z = 0.02 indicate a still dominant role of the inter-shell
mixing. It is less true for the Z = 0.005 models because at lower Z,
thermal pulses are somewhat hotter, that favors 
depletion.
Contrarily to the case of 
, the inter-shell mass fraction of
is very low (
typically). Its presence inside
thermal pulse consequently results from its production by the
reaction. Note that the rapid
decay to 
also significantly contributes to
the thermal pulse energetics. This element is destroyed by three reactions
that are, by decreasing order of nuclear time scales,
, 
and
. This last reaction can even become faster
than 
when significant amounts of protons are
produced. This especially arrives for the late thermal pulses of the
star with Z = 0.02. The convective tongues being hotter for more
massive AGB stars and/or later along the TP-AGB phase of a given star, the
destruction, which is very sensitive to temperature, becomes
more and more efficient from pulse to pulse (see Fig. 8 (click here)). This
point was already noted by Boothroyd & Sackmann (1988b). The thermal pulse
equilibrium mean mass fraction between production and destruction is ranged
between 
and 
depending on M and Z.
As for 
, 
is destroyed in the HBS. It is however
produced inside thermal pulses. As first demonstrated by Forestini et al.
(1992), the 
synthesis involves many nuclear reactions, so
that it constitutes a good tracer of the nucleosynthesis conditions
prevailing inside thermal pulses. 
is produced by the
reaction. The rather rapid increase of
the mean 
mass fraction during the first thermal pulses of
each AGB star is due to the increasing importance of the inter-shell
destruction. When convective tongues become hot enough to
completely destroy the engulfed 
, the following thermal pulses
only ingest 
from matter that has not experienced the preceding
thermal pulse (i.e. 
of the convective tongue extension in mass
typically; see Table 4 (click here)). However, as we have explained above,
can be produced by the chain
.
Neutrons are mainly coming from the 
reaction,
and to a lower extent, from 
,
, 
(see below)
and 
. Protons are coming from 
reactions, mostly on 
and 
(see below). However,
(i) only part of the neutrons are available to produce protons as
many 
reactions also occur (see Sect. 7.5) and
(ii) a small (
) part of the protons are also captured by other
nuclides than 
[mainly by 
].
The success of this reaction chain mainly depends on the rate of 
destruction, mainly by 
captures (see above). This destruction
becomes more efficient from pulse to pulse so that only a small fraction
of 
is available for the 
reaction. On the
other hand, when convective tongues become very hot (base temperature 
K), 
begins to be destroyed by
and 
(if significant
amounts of neutrons are available). In conclusion, three facts concur to
considerably decrease the 
production after the first thermal
pulses (whatever M and Z), as shown in Fig. 8 (click here). Indeed, from
pulse to pulse and especially for the most massive AGB stars,
engulfed in the convective tongue of a thermal
pulse decreases (due to the partial destruction of 
in the
inter-shell region during the inter-pulse phase; see Sect. 7.1.1);
ingested from the inter-shell region also
decreases;
captures on 
are more frequent;
destruction becomes significant. This can in particular
lead to a partial destruction of the 
synthesized in the
previous thermal pulses for stars with M > 
.
The mean 
abundance is roughly unchanged in the HBS as well as
by the thermal pulse nucleosynthesis. In number, the contribution of
is rather negligible.
is significantly destroyed in the HBS. Inside the convective
tongues of quite hot thermal pulses (base temperatures above 
K), it is produced by the small proportion of 
that is destroyed
through the 
reaction. There is also a small
contribution by 
. Only a very small
fraction can be destroyed by the 
at the end of very hot thermal pulses (base temperatures above 
K).
is somewhat produced in the HBS. Its abundance is
considerably enhanced inside thermal pulses (by typically a factor of
100) due to its production by 
. However,
its abundance from pulse to pulse becomes constant (or even slightly
decreases inside full amplitudes thermal pulses of the most massive stars
and/or those with lower Z). At their maximum extent, such convective
tongues are effectively hot enough (base temperatures 
K) to activate the 
and
reactions. However, as noted below, only a
small part of the 
nuclides are converted into 
or 
, respectively (typically one hundredth). Nevertheless,
this consists in a non-negligible neutron source for these thermal pulses,
as first noted by Käppeler et al. (1990) and emphasized by
Straniero et al.
(1995).
Concerning 
, one has to distinguish between the 3, 4,
with Z = 0.02 and 
with Z = 0.005 stars on the one
hand, and the 
with Z = 0.02, 4 and 
with Z = 0.005
stars on the other. Inside the former ones, 
is significantly
produced at the bottom of the HBS and very slightly produced by the HeBS
[during the inter-pulse phase, through the 
reaction]. Furthermore, whatever M and Z, 
is almost
unchanged by the thermal pulse nucleosynthesis. As a consequence, its mean
mass fraction inside the convective tongue does not evolve significantly
from pulse to pulse. At the opposite, inside the latter ones, the bottom
of the HBS is hot enough to somewhat destroy 
while it is
produced in its upper part. However, the HeBS substantially produces this
nuclide during the inter-pulse phase. Consequently, when the convective
tongues of such stars penetrate the inter-shell region, they engulf
material that is impoverished in 
; this explains its mean
mass fraction decrease visible on Fig. 9 (click here).
The 
abundance is almost unchanged by the thermal pulse
nucleosynthesis too. Furthermore, the HBS of 
stars is not hot
enough to destroy it significantly. Its mean mass fraction in the convective
tongue region of such stars consequently remains almost constant from pulse
to pulse. The situation is quite different for more massive stars (whatever
Z) as they destroy 
at the bottom of their HBS. Dilution of
inter-shell material by the successive convective tongues thus traduces in a
mean mass fraction decrease with time.
Intermediate nuclide of the MgAl chain, 
is modified by the
proton burning inside the HBS. We refer to Sect. 7.1.1 for more details
(dependence on M and Z). However, the evolution of its mean mass
fraction inside the convective tongues of thermal pulses is dominated by its
production through the 
reaction [operating at a
quite comparable rate than 
, following our
present knowledge of the corresponding nuclear reaction rates]. Indeed, even
if this reaction is very slow inside thermal pulses of relatively low mass
AGB stars (especially for their first thermal pulses), 
is much
more abundant than 
(by typically a factor of one hundred) in
that region, that explains the sensible increase of the 
mean
mass fraction.
is slightly (significantly) depleted in the HBS of the Z =
0.02 (0.005) models. On the other hand, it is substantially produced by the
thermal pulse nucleosynthesis (i) by the
reaction (same remark as for 
above) and, to a lower extent, (ii) by the 
reaction (see below). The slope with which its mean mass fraction increases
form pulse to pulse (see Fig. 9 (click here)) is consequently higher for
hotter convective tongues, i.e. steeper for full amplitude thermal pulses of
massive AGB stars.
is one of the most important by-product of the HBS (see
above). With a lifetime 
yr, i.e. much longer than the
inter-pulse duration of intermediate-mass AGB stars, it essentially behaves
like a stable nuclide in the inter-shell region. When engulfed inside the
convective tongue of a thermal pulse, it is mainly destroyed through the
reaction and, at least ten times slower, by
. As a result, 
is the most
important proton source inside thermal pulses, whatever M and Z. Its
destruction is however partial, due to the relatively low neutron abundance
(see Sect. 7.5 below). Note that 
is also the
principal neutron source in the radiative 
burning shell, as
reported by Wasserburg et al. (1994) too.
is significantly produced in the HBS, especially in the most
massive AGB stars. It does not significantly participate to the thermal
pulse nucleosynthesis. As a consequence, it is engulfed by the convective
tongue from the inter-shell region and its corresponding mean mass fraction
slightly increases from pulse to pulse. Note however that with time, most of
the H-burning of the 
AGB star with Z = 0.005 occurs inside its
convective envelope. This considerably reduces the 
production.
Si, P and S are almost not concerned by charged particle nuclear reactions inside the convective tongue of thermal pulses. On the other hand, they are also rather unchanged in the HBS. The only role played by these elements concerns neutron captures that we now briefly discuss.
Many evolved AGB stars show considerable overabundances in elements heavier
than iron (see e.g. Smith & Lambert 1986). The original discovery by
Merrill (1952) of Tc in the spectra of some S stars definitively proved that
these heavy elements are synthesized inside AGB stars. Iben (1975) first
demonstrated how such species can indeed be produced inside the thermal
pulses by slow neutron captures (the so-called s-process), those neutrons
coming from the 
reaction. However, Iben's
computations concerned a very massive AGB star (
), while ever since,
all the observed stars showing s elements enrichment have been identified
as being of lower mass (i.e. < 
and for many of them <
). More recent observations (Aaronson & Mould 1985) and
evolutionary models (Malaney & Boothroyd 1987) then strongly suggested that
the major neutron source had to be the 
reaction.
is engulfed by the convective tongues of thermal pulses from
the inter-shell region where the HBS ashes accumulate. However, the amount
of 
spread out by the HBS is by far (i.e. at least a factor of
10 or more) insufficient to explain the s-process that requires more than
one neutron by seed 
. This so-called ``s-process mystery''
(Sackmann & Boothroyd 1991b) is a common failure of all the recent stellar
evolution models of TP-AGB stars. It is moreover reinforced by the fact that
other lighter nuclides also capture part of the available neutrons (mainly,
by increasing order of atomic mass, 
, 
,
, 
, 
, 
, 
,
, 
, 
, 
, 
,
, ...). One has to note that while the neutron abundance has
to be increased to solve the s-process problem, the present computations
probably slightly overestimate the abundances of the above mentioned
nuclides.
More specifically, in our thermal pulse models, the ratio of the engulfed
mass of 
to the mass of 
inside the convective
tongue at its maximum extent ranges between 0.002 to 0.03. This upper limit
is reached during the asymptotic regime of all our modeled stars and thus
appears to be rather independent of the initial mass or metallicity. Inside
their first thermal pulses however, 
stars already produce neutrons
(ratio roughly equal to 0.02) while at the opposite, the corresponding
thermal pulses of 
stars do not engulf substantial amounts of
. Furthermore, we noted (see Sect. 7.4) that the 
neutron source operates after the maximum of very intense thermal pulses, at
a rate that is growing with the maximum temperature of the convective
tongue, i.e. with initial mass and/or pulse number.
Let us stress that such low quantities of 
ingested by thermal
pulses are partly explained by the partial 
radiative burning,
operating at the bottom of the inter-shell region during the inter-pulse
phase (see Sect. 7.1.1). Consequently, it seems clear that the s-process
occurring inside TP-AGB stars has two distinct origins (in time and space):
;
that is engulfed
by the convective tongue and, to a lower extent (maximum a few percents),
(ii) the 
, close to the end of hot thermal pulses.
Let us emphasize that the 
and accompanying
reactions have non-negligible energetic effects
on the structural evolution of the convective tongues.
We compute again some thermal pulses by treating together nucleosynthesis
and time-dependent convective mixing through Eqs. (4). We observed an
abundance gradient inside the convective tongue for a few nuclides that are
involved in very fast nuclear reactions (neutrons, protons and some unstable
nuclides like 
or 
). Globally however, the final
abundances resulting from the thermal pulse nucleosynthesis were similar to
within 
compared to standard computations. More specifically,
neutrons were systematically found to be much more abundant (by typically
five orders of magnitude) at the base of the convective tongues, meaning
that the ingested 
is first transported down before it is
destroyed by 
. The liberated neutrons are very
rapidly captured, as attested e.g. by a greater proton abundance at the base
of the convective tongue too. This leads us to the conclusion that matter
that is actually irradiated by the neutron flux is only the bottom part of
each convective tongue. The number of free neutrons by seed 
is
so enhanced. Such an approach has already been suggested by
Malaney et al.
(1988). The conclusion, however, remains that in order to reproduce the
observed distribution of s elements in the primitive solar system, higher
neutron fluxes are still needed.
The most natural way to conciliate observations and theoretical models
concerning the production of the s elements should be to increase the
amount of 
in the inter-shell region. This can be done if
protons are transported down, e.g. during the 3DUP. 
could then
be produced by the 
reaction. Such a scenario
would enhance the s-process in both sites where it can occur. In our
opinion, the present failure to build this extra-amount of 
could be related to our bad treatment of convective boundaries, especially
when they rapidly penetrates very inhomogeneous regions, like it is the case
during the 3DUP events. The possible occurrence of such a slow-particle
transport process able to transport protons down into the inter-shell region
and its exact efficiency will be quantitatively investigated in the context
of low-mass AGB stars. These stars have indeed longer inter-pulse and 3DUP
durations that could allow to transport significant amounts of hydrogen
downwards (see Sect. 9).
Straniero et al. (1995) have recently shown that such enhanced amounts of
in the inter-shell region could indeed allow the radiative
s-process to occur. Most interestingly, they found that the resulting
signatures are rather similar to those coming from classical s-process
computations and are even reached faster. This work is very important as
it clearly demonstrates that this so-called ``
pocket''
allows to explain the observed s element synthesis in AGB stars.
In order to summarize our detailed discussion of nucleosynthesis in the previous sections, we now identify which specific nuclear region, inside a TP-AGB star, mainly contributes to the surface abundance change of each nuclide, when the third dredge-up occurs.
Concerning the light elements, let us just recall (see Sect. 7.2) that
can be produced by HBB in the convective envelope of >
(
) AGB stars with Z = 0.02 (0.005).
When a third dredge-up occurs, the convective envelope first penetrates the
inter-shell region that is depleted in 
and then reaches a
deeper region previously enriched in 
by the convective tongue.
As a result, 
globally increases in the convective envelope. The
situation is reversed for 
. After a 3DUP, the envelope is
somewhat impoverished in 
as this nuclide is almost completely
destroyed inside the convective tongue. Consequently, the isotopic ratio
/
significantly increases at each 3DUP, all the more
as the convective envelope mass is reduced. However, in our 
(
) TP-AGB stars with Z = 0.02 (0.005), HBB is strong enough to
partially convert 
in 
through the CN cycle so that
the surface 
/
drastically decreases. For these
objects, the 3DUP and HBB have opposite effects. Last but not least,
, somewhat produced by thermal pulses and the third burning
shell, is up-heaved to the surface during 3DUP events, where it however
substantially decays in the less massive AGB stars.
is produced in the inter-shell region but is strongly destroyed
inside thermal pulses. 
is destroyed in the inter-shell region
while it is somewhat produced by the first few thermal pulses and then
destroyed inside full amplitude thermal pulses, i.e. those that can be
followed by a 3DUP. As a consequence, the surface isotopic ratio
/
is expected to slightly increase from 3DUP to
3DUP. Again, as the CN cycle operates inside the convective envelope of the
most massive AGB stars, 
is produced from 
and
is destroyed, thus leading to very high
/
ratios. If HBB occurs at very high temperatures
(i.e. 
K) a rather long time enough (i.e. for the lowest mass loss
rate), the approach towards the CNO bi-cycle equilibrium leads to an
increase of the 
abundance and consequently, a decreasing
/
ratio. This mainly concerns our 
(
) AGB star with Z = 0.02 (0.005).
, like 
, is depleted in the inter-shell region, but
is produced inside the thermal pulses. During 3DUP events, its surface
abundance very slightly increases. 
is always considerably
depleted in both the inter-shell and thermal pulse regions, so that a 3DUP
decreases its surface abundance. Finally, at the end of full amplitude
thermal pulses, 
has a lower abundance as inside the convective
envelope. As it is also completely destroyed in the inter-shell region, its
abundance drastically decreases from 3DUP to 3DUP. Consequently, both
/
and 
/
surface isotopic
ratios significantly increase during each dredge-up. If HBB is strong enough
however, the operation of the ON cycle increases the amount of 
to the expense of 
and substantially destroys 
. So,
in massive AGB stars, 
/
can decrease while
/
further increases much more than in stars that do
not experience HBB.
is destroyed in the HBS but it is significantly produced by
thermal pulses. As a consequence, 3DUP events must lead to correlated
and 
surface enhancements, since both elements are
produced in the same region.
As already explained, 
is almost unchanged in the HBS and
thermal pulse nucleosynthesis. 
is destroyed inside the
inter-shell region but produced inside thermal pulses, this production
slightly increasing from pulse to pulse, whatever M or Z. Third
dredges-up consequently slightly decreases the 
/
isotopic ratio. In our 
(
) AGB models with Z = 0.02
(0.005) however, 
is partially destroyed by HBB. On the other
hand, the 
/
ratio substantially decreases from
3DUP to 3DUP due to large 
production in the HBS and especially
inside the thermal pulses.
During 3DUP events, the surface abundance of 
slightly
increases for AGB stars with initial masses up to 
(
) and
Z = 0.02 (0.005), due to its production in the HBS. The hotter HBS of
more massive AGB stars do not produce 
, globally. Our
(
) models for the same respective Z even slightly
destroy 
by HBB.
is somewhat destroyed in the HBS of our most massive AGB
stars, especially those with Z = 0.005, while its abundance does not
change significantly after the thermal pulse nucleosynthesis. 
is depleted in the inter-shell region of our 
models with Z =
0.02 as well as in the Z = 0.005 models of all masses. It is however
produced in the HBS of > 
AGB stars with Z = 0.02.
Nevertheless, its surface abundance evolution following 3DUP events is
mainly conditioned by its large abundance (compared to the envelope one)
in the region where thermal pulses occur, especially inside the most
massive AGB stars. Therefore, the surface 
/
isotopic ratio necessarily decreases in all AGB stars. The same is true
for the 
/
ratio that roughly behaves in a
similar way. Reasons are also rather similar, even if 
is
less modified than 
in the HBS. At the surface of our
(
) with Z = 0.02 (0.005) AGB stars, the abundances of
both 
and 
can be enhanced to the expense of
, if HBB is strong enough, leading to very low
/
and 
/
ratios. As we
will see in Sect. 8.1 however, this only concerns the end of the TP-AGB
phase in extreme situations allowing HBB to occur a long time enough (e.g.
in case of relatively low mass loss rates).
Only a very small amount of 
is present in the convective
envelope of E-AGB stars as a result of the first (and eventually the second)
dredge(s)-up. Along the TP-AGB phase, 
is largely produced in
the HBS and most of it accumulates in the inter-shell region. Inside full
amplitude thermal pulses however, it is somewhat destroyed. For its part,
is slightly produced in the HBS, especially inside the most
massive AGB stars. Consequently, 
/
significantly
increases with time during the TP-AGB phase, due to the repetitive 3DUP
episodes. Of course, dilution by convective mixing being more important for
increasing M, the highest ratios are expected at the surface of the less
massive AGB stars. However, the same extreme cases mentioned in the case of
the Mg isotopic ratios can lead to very high 
/
ratios for the most massive AGB stars.
From Si, the 3DUP does not change significantly the surface isotopic ratios. However, as stressed in Sect. 7.5, when models will be able to reproduce the s element enhancements at the surface of evolved AGB stars, Si and S isotopic ratios could be slightly modified by the high neutron flux.
Figure 10:
Mean mass fractions of the 20 most relevant nuclides inside the convective
envelope, the HBS and the inter-shell region, for the 
(Z = 0.02)
star
Figure 11:
Same as Fig. 10 (click here) for the 
(Z = 0.02) star
Figure 12:
Same as Fig.10 (click here) for the 
(Z = 0.02) star
Figure 13:
Same as Fig. 10 (click here) for the 
(Z = 0.02) star
Figure 14:
Same as Fig. 10 (click here) for the 
(Z = 0.005) star
Figure 15:
Same as Fig. 10 (click here) for the 
(Z = 0.005) star
Figure 16:
Same as Fig. 10 (click here) for the 
(Z = 0.005) star
Figure 17:
Variation of the mean mass fractions of the same 20 nuclides as in Figs.
10 (click here) to 16 (click here), due to the thermal pulse
nucleosynthesis, for the 
(Z = 0.02) star, inside an early
thermal pulse (the 3d) and an asymptotic one (the 18th). Arrows indicate
the mean abundance evolution from the beginning of a thermal pulse to
its end
Figure 18:
Same as Fig. 17 (click here) for the 
(Z = 0.02) star (for the
3d and 13th thermal pulses)
Figure 19:
Same as Fig. 17 (click here) for the 
(Z = 0.02) star (for the
3d and 13th thermal pulses)
Figure 20:
Same as Fig. 17 (click here) for the 
(Z = 0.02) star (for the
4th and 14th thermal pulses)
Figure 21:
Same as Fig. 17 (click here) for the 
(Z = 0.005) star (for the
3d and 8th thermal pulses)
Figure 22:
Same as Fig. 17 (click here) for the 
(Z = 0.005) star (for the
3d and 9th thermal pulses)
Figure 23:
Same as Fig. 17 (click here) for the 
(Z = 0.005) star (for the
3d and 9th thermal pulses)