In this last section, we present our predictions concerning (i) the
evolution of all the surface isotopic ratios along the TP-AGB phase (by
including our evolutionary models and extrapolations up to the convective
envelope removal) and (ii) the total amount of material ejected at
different ages for all the species up to 
. We refer to the
previous sections for detailed explanations. Thus, in the following, we will
just comment these predictions.
Figs. 24 (click here) to 37 (click here) display, for each computed AGB star, the evolution of the isotopic ratios as a function of the total remaining mass that decreases due to mass loss.
Let us stress that we stop our calculations at the last thermal pulse that
occurs before the convective envelope mass is reduced below 
.
This stage indeed roughly corresponds to the PN ejection that we do not
maintain to model here. This explains the quite large remaining total stellar
masses in Figs. 24 (click here) to 37 (click here), especially for those
computed with the highest mass loss rates.
The first comment concerns the formation of carbon stars. 
AGB
stars with Z = 0.02 and 3 and 
AGB stars with Z = 0.005
become carbon stars during their TP-AGB phase. The surface
/
becomes greater than unity more rapidly for
(i) decreasing total mass and/or (ii) decreasing Z. This
metallicity dependence is confirmed by the increase of C stars compared
to M stars from the galactic bulge to LMC and SMC (Blanco et al. 1978).
Our 
(
) AGB stars with Z = 0.02 (0.005) can also become
C stars with our lowest mass loss rates. In such cases indeed, in spite
of very efficient HBB transforming 
in 
and
, many 3DUP events can still occur with very reduced envelope
masses, so that the relatively short inter-pulse phases are not long enough
to substantially destroy 
. We will see however, together with
other isotopic ratios, that these very low mass loss rates seem actually
extreme. Let us emphasize that the only super-lithium-rich star (see
Sect. 7.2) we found to be also a carbon star is the 
with Z =
0.005 one (with our medium mass loss rate). However, as already
mentioned in Sect. 6.2, observations indicate that Z = 0.02 carbon
stars are formed easier (i.e. at lower luminosities) than predicted by
all the models. Let us also stress the very high
/
isotopic ratios that can be expected at the
surface of 
AGB stars, whatever Z. On the other hand,
more massive AGB stars experience HBB that progressively leads the
/
isotopic ratio close to its CN cycle
equilibrium value (formation of J stars). The case of 
stars
with Z = 0.005 very clearly shows the competition between the 3DUP episodes
(increasing 
/
) and HBB (decreasing it). For lower
mass loss rates, the 3DUP events are more numerous for the same total mass
decrease, leading to a more rapid 
/
increase.
However, the corresponding longer TP-AGB phase allows the base of the
convective envelope to reach higher temperatures that finally destroy
through the CN cycle. This efficient transformation of
to 
also reduces the 
/
ratio, i.e. the possibility to form carbon stars. It happens if the
temperatures at the base of the convective envelopes become higher than
K. Such stars have corresponding surface luminosities
> 
(
) for Z = 0.02 (0.005), i.e. 
(-6.45). We consequently predict that C stars cannot be formed at
higher luminosities than this threshold, due to strong HBB. This is in
agreement with Boothroyd et al. (1993) predictions. This also corresponds
to the observed threshold for AGB stars of the LMC (Cohen et al. 1981).
Finally, note the surface pollution in 
that is more important,
compared to that of 
, for decreasing total masses. For 
AGB stars, the surface 
/
ratio is quite independent
of the mass loss rate. This is due to the inter-pulse duration that is
significantly greater than the 
lifetime, so that the
decay is the dominant factor for the 
/
ratio evolution at the surface of such stars. The quite complicated behavior
of the 
/
isotopic ratio at the surface of our most
massive models is again due to the competition between the 3DUP that adds
into the convective envelope and HBB that can partially destroy
it. By extrapolation, it is reasonable to expect even lower
/
ratios in low-mass AGB stars. As mentioned in
Forestini et al. (1996), 
could become detectable for <
evolved AGB stars.
Due to HBB, those stars showing very low surface 
/
ratios are also expected to have very high 
/
ones.
It has to be stressed indeed that in our most massive AGB stars, the CN
cycle operating inside the convective envelope produces large amounts of
.
The 
/
and 
/
surface isotopic
ratios also evolve as expected from our discussions in Sect. 7. Note that
quite low 
/
and very high 
/
ratios, which are clear signatures of the operation of the ON cycle at the
base of the convective envelope (HBB), appear rather simultaneously with the
CN cycle signatures in our most massive objects. Also remark the progressive
/
decrease in our 
(
) AGB star with
Z = 0.02 (0.005). For these objects, the temperature at the base of the
convective envelope finally becomes high enough for 
to
partially burn.
The 
surface enhancements we predict are displayed in a special
form, corresponding to the observation presentations adopted by Jorissen et
al. (1992). Compared to their Fig. 8, two conclusions can be stated.
excess
and the degree of 
surface pollution, that is perfectly
understandable from the identified nucleosynthetic processes and the
mixing episodes (see Sect. 7). Furthermore, our models clearly indicate a
trend: the less massive the AGB star, the more efficient the
surface enhancement. In fact, our most massive models even
show a surface 
depletion due to its partial destruction by
HBB.
enriched
stars in the [
/
] versus
/
diagram. Based on the trend just mentioned, we
predict that the AGB stars showing very large 
surface
enhancements have initial total masses < 
. We plan to verify
this statement, i.e. that low-mass AGB stars are the most efficient
galactic producers (see Sect. 9).
As expected, the 
/
and 
/
surface isotopic ratios are much less sensitive to the 3DUP episodes and
HBB. More precisely, only our most massive models with the lowest mass loss
rates show large 
depletions due to a very strong HBB. Again,
these are however quite extreme situations.
Last but not least, we predict substantial 
surface
enhancements due to the repetitive dredges-up of the HBS and inter-shell
regions. The 
/
isotopic ratio tends to decrease
with increasing total mass due to (i) the greater dilution as mixing
occurs in a more massive convective envelope and (ii) the thinner HBS
and inter-shell regions (in mass). Nevertheless,
/
could well be increasing again if
was produced inside the convective envelope itself by strong
HBB. Note however that such a very high 
production level is
actually expected to be accompanied by very low Mg isotopic ratios, as
is produced by the MgAl chain in the extreme situations
mentioned above. Let us finally mention that Wasserburg et al. (1994)
presented computations in which the evolution of the 
surface
abundance has been followed self-consistently, by including the effect of
neutron irradiation due to an enhanced amount of 
in the
inter-shell region (to engender a s-process).
Our results compare very well with those obtained by Boothroyd et al.
(1994, 1995) concerning the O isotopic ratios in intermediate-mass stars.
They also mention the problem to reproduce the 
/
ratio, especially in low-mass carbon AGB stars. We cannot go further in this
comparison without having yet computed the TP-AGB phase of low-mass stars.
Boothroyd et al. (1995) already suggested that to conciliate all the data
with observations of low-mass stars, one probably has to take into account
of slow-particle transports inside radiative zones, what they called
``cool-bottom precessing''. That is what we plan to do (see Sect. 9).
Figure 24:
Surface isotopic ratios as a function of the remaining total stellar
mass, that decreases due to the mass loss along the TP-AGB. The solid
line refers to the standard mass loss rate (see Sect. 6.3.2 for the
corresponding 
parameter values in the Reimers relation). Dotted
lines correspond to mass loss rates increased or decreased by a factor
of two. The extrapolated thermal pulses (and 3DUP episodes) are
indicated by the filled circles, and are of course more numerous for
lower mass loss rates. Here are indicated the 
/
,
/
, 
/
,
/
, 
/
and
/
ratios, in the case of the 
(Z =
0.02) star. All ratios are in mass fractions, except for the
/
ratio (given in number abundances)
Figure 25:
/
, 
/
,
/
ratios as well as
[
/
] as a function of 
/
,
in the case of the 
(Z = 0.02) star. All ratios are in mass
fractions and [A/B] means 
Figure 26:
Same as Fig. 24 (click here), in the case of the 
(Z = 0.02) star
Figure 27:
Same as Fig. 25 (click here), in the case of the 
(Z = 0.02) star
Figure 28:
Same as Fig. 24 (click here), in the case of the 
(Z = 0.02) star
Figure 29:
Same as Fig. 25 (click here), in the case of the 
(Z = 0.02) star
Figure 30:
Same as Fig. 24 (click here), in the case of the 
(Z = 0.02) star
Figure 31:
Same as Fig. 25 (click here), in the case of the 
(Z = 0.02) star
Figure 32:
Same as Fig. 24 (click here), in the case of the 
(Z = 0.005) star
Figure 33:
Same as Fig. 25 (click here), in the case of the 
(Z = 0.005) star
Figure 34:
Same as Fig. 24 (click here), in the case of the 
(Z = 0.005) star
Figure 35:
Same as Fig. 25 (click here), in the case of the 
(Z = 0.005) star
Figure 36:
Same as Fig. 24 (click here), in the case of the 
(Z = 0.005) star
Figure 37:
Same as Fig. 25 (click here), in the case of the 
(Z = 0.005) star
Evolutionary models of TP-AGB stars can be strongly constrained by the confrontation between predicted and observed surface isotopic ratios, especially if different ratios are observed for the same stars. For most of the observed AGB stars, only the C and O isotopic ratios have been determined (see e.g. Harris & Lambert 1984; Harris et al. 1985; Lambert et al. 1986; Harris et al. 1987; Harris & Lambert 1987; Kahane et al. 1992). Globally, we found agreements. General features come out.
production (see Sect. 7.2) are significantly lower
than those allowing efficient 
destruction by the CN cycle,
we predict that luminous C stars enriched in 
can exist,
especially for lower Z objects (that have lower 
content).
There are indeed a few % of the observed C stars that are also
super-lithium-rich and there are more numerous in the LMC compared to our
galaxy (see e.g. Smith & Lambert 1990b for the LMC and Feast 1974 for the
Galaxy).
/
< 15. Most of
the J stars show 
/
ratios between 3 et 4, i.e.
very close to the CN cycle equilibrium value.
/
isotopic ratios
while only upper limits have been found for the 
/
ratio. Most of the other AGB stars present 
/
ratios ranged between 300 and 4000 and 
/
ratios
ranged between 500 and 5000. Only stars with rather high
/
ratios are difficult to explain with our
intermediate-mass AGB models (see also Boothroyd et al. 1995). Also note
that the corresponding O isotopic ratios in the ISM are
/
and
/
. Again, there is a problem
with 
. Let us however stress that important uncertainties
remain concerning the nuclear cross sections of proton captures on
. On the other hand, low-mass AGB stars are present among
those observations.
Presently, there is only one star for which all the isotopic ratios have
been determined. It is Cw Leo. This C star is one of the nearest evolved
AGB star ever observed. Its distance is estimated to 
pc
(see e.g. Claussen et al. 1987;
Keady et al. 1988;
Griffin 1990). This
corresponds to a surface luminosity 
and a
present mass loss rate of roughly 
. Such a high wind makes it quite optically thick. A very
extended circumstellar envelope has been detected to surround Cw Leo,
IRC+10216. IRC+10216 shows evidences for a bi-polar structure (see e.g.
Guélin et al. 1993), attesting that Cw Leo is very evolved and close to
the end of its TP-AGB phase. Consequently, its present convective envelope
mass is probably very reduced (
typically).
The circumstellar envelope of Cw Leo allowed very accurate determinations of
the isotopic ratios (see Kahane et al. 1988 and 1992;
Guélin et al. 1995;
Forestini et al. 1996): 
/
,
/
> 1400, 
/
> 5300,
/
, 
/
, 
/
,
/
and
/
(to be confirmed). Furthermore, the
Si isotopic ratios appear to have interstellar medium values, i.e.
unaltered by the star. As derived in Guélin et al. (1995) and
Forestini et
al. (1996), almost all these isotopic ratios seem to be consistent (within
the error bars), following our predictions, with an AGB star that had a main
sequence total mass between 4 and 
(assuming Z = 0.02; see Figs.
26 (click here) and 29 (click here)). However, 
/
is
again not well explained. As various surface isotopic ratios (especially
/
and 
/
) are significantly
changing between 4 and 
AGB models, we are presently computing the
TP-AGB phase of a 
star with Z = 0.02 in order to improve this
last prediction.
Finally, another very interesting way to constraint stellar evolution models
of AGB stars is to compare the predicted surface isotopic ratios with those
determined in primitive meteorites (mainly through grains included inside
carbon rich chondrites). Most of the 
/
,
/
and 
/
isotopic
anomalies measured in grains (see e.g. Ott 1991; Zinner et al. 1991a) are
consistent with those coming from the surface of evolved TP-AGB stars of
intermediate-mass. This is of course in favor of the presence, when the
solar system has been formed, of one or a few AGB stars in its surroundings.
Such grains are indeed well known to be formed in the cool atmosphere of
evolved AGB stars (i.e. when mass loss rates become rather high). Concerning
the measured O isotopic ratios, the situation is not so clear as many grains
have ratios significantly out of the corresponding predicted ranges. Some of
these grains at least can have been formed in low-mass AGB stars. Again, as
mentioned in Sect. 8.1, Boothroyd et al. (1995) recently suggested that in
such stars, cool-bottom processing can help to reduce these discrepancies.
On the contrary, some other grains perfectly agree with our predictions.
This is for example the case of a 
grain of the Bishunpur LL3.1 chondrite by Huss et al. (1994). The derived
ratios, 
/
, 
/
and 
/
, are in perfect agreement with ratios corresponding to a 3 -
AGB star of nearly solar metallicity. Finally, other SiC
-rich grains raise problems, like that discovered by
Zinner
et al. (1991b), exhibiting a very high 
/
ratio
of 
. It seems hard to explain such a value from our models.
However, this grain also exhibits an anomalously large
/
isotopic ratio that seems to exclude its AGB
star origin.
Tables 8 (click here) to 10 (click here) give the total mass ejected by the wind
(in 
) for various elements (i) at the top of the first RGB,
(ii) at the beginning of the TP-AGB phase and (iii) at the end
of the TP-AGB phase (thus including our extrapolated nucleosynthesis
calculations). As such data mainly concerns the chemical evolution of
galaxies, we also present the net yield (positive or negative) for each
element. The data of Tables 8 (click here) to 10 (click here) correspond to the
medium mass loss rates (see Sect. 8.1), i.e. the solid lines of Figs.
24 (click here) to 37 (click here). Let us briefly comment.
Table 8:
Total mass ejected (in 
) in the wind for some nuclides at the RGB
top, at the end of the E-AGB phase and at the AGB tip, just before the
ejection of the planetary nebula. Also given is the net contribution for
these elements, i.e. the total final mass ejected reduced by the amount
of matter that would have been ejected with the initial composition.
Here are presented elements from He to N
Table 9:
Same as Table 8 for elements from O to Na
Table 10:
Same as Table 8 for elements from Mg to Si
is produced during the pre-main sequence and the main
sequence phases. For the stars with masses lower than 
(
)
with Z = 0.02 (0.005), 
survives during the following phases
of evolution and is injected in the ISM by stellar wind, mainly during
the TP-AGB ultimate phase. These objects thus contribute, while more
modestly than low-mass (< 
) stars to the galactic enrichment
of 
. In more massive stars however, 
is
progressively burnt at the benefits of 
and 
by
HBB. At the end of their TP-AGB phase, 
can even be completely
destroyed, so that the corresponding net yields are negative.
Intermediate-mass stars contribute to the galactic enrichment of
. This pollution mainly occurs at the end of the TP-AGB phase
and is higher for the highest stellar masses and lower metallicity models.
Indeed, 
is brought to the surface during the successive 3DUP
episodes. Moreover, in our most massive stars, HBB further contributes to a
non-negligible production of this element inside the convective envelope.
TP-AGB stars appear to be major contributors to the 
galactic
enrichment. The very nuclearly fragile 
is destroyed at very low
temperatures in stellar interiors. As was discussed in Sect. 7.2, it can
however be produced at the expense of 
when HBB is effective. As
a result, the global yield of 
is positive in our 5 and 
stars with Z = 0.02, whereas it is negative for the other stars. Note that
the 
production in our 
AGB star with Z = 0.005 is not
large enough to compensate its depletion during the previous stellar
evolution stages.
As already mentioned (see Sects. 6.2 and 8.1) 3DUP events significantly
pollute the convective envelope in 
. However, if HBB occurs,
CNO-burning leads to primary production of 
and 
at
the expense of 
(and possibly 
). The net yields of
thus decrease for increasing stellar masses and even become
negative for our 5 and 
(
) stars with Z = 0.02 (0.005).
Owing to that, the net yields of 
and 
rapidly
increase with initial stellar mass and metallicity.
The more massive an AGB star, the less efficient the 
production
(see Sects. 7.3 and 8.1). Due to HBB, our most massive models even present
negative net yields for that element. As explained in Sect. 8.1, low-mass
AGB stars are probably the most efficient 
producers in
galaxies.
As a net result of the TP-AGB phase, intermediate-mass stars somewhat
contribute to the ISM enrichment in 
and, to a still lower
extent, 
, to the expense of 
.
Massive AGB stars (higher than 
) which experience strong HBB are the
most efficient sources of 
among the AGB stars. They even
significantly destroy 
. In addition, 
is also
substantially produced by all our models, except the 
star with Z =
0.005. We also note a non-negligible 
production inside the
same massive AGB stars.
