A&A 448, 717-729 (2006)
DOI: 10.1051/0004-6361:20053043
L. Siess
Institut d'Astronomie et d'Astrophysique, Université Libre de Bruxelles, CP 226, 1050 Brussels, Belgium
Received 11 March 2005 / Accepted 5 November 2005
Abstract
We present new computations of the evolution of solar metallicity
stars in the mass range 9 - 12 .
This first paper of a series
focuses on the propagation of the carbon burning flame front and provides
a detailed analysis of the structural evolution up to the formation of
the neon-oxygen core. Our calculations which do not include overshooting
indicate that off-center carbon ignition is restricted to a small mass
range between 9.0 and 11.3
.
The chemical imprints of the first and
second dredge-ups on the surface composition are analyzed and compared to
"standard'' less massive stars. It results that, aside from being more
luminous and slightly bluer in the HR diagram, massive AGB stars are
almost indistinguishable from their lower mass counterparts, as far as
the chemical composition is concerned. During the second dredge-up, we
note however that the envelope penetrates deeper into the He burning
shell than lower mass stars. Our simulations indicate that above
11.0
,
the depth of the second dredge up is considerably
reduced, marking the transition between low and massive stars. We also
investigate the effects of the nuclear uncertainties associated with
+
reactions and show that it has a little impact
on the core composition. Finally we describe the nucleosynthesis and
chemical structure of the newly formed neon-oxygen core.
Key words: stars: evolution - nuclear reactions, nucleosynthesis, abundances - stars: AGB and post-AGB
The evolution of stars in the mass range 9 - 12
is
characterized by the off-center ignition of carbon in conditions of partial
degeneracy, followed by the development of a flame that will transform the
CO core into a degenerate neon-oxygen (NeO) mixture. However, the fate
of these objects is still subject to large uncertainties mainly associated
with our poor knowledge of the mass loss rate which is critical in this
transition mass range. Massive intermediate mass stars can either enter the
super-AGB (SAGB) phase where recurrent thermal instabilities develop in the
helium burning shell as in standard thermally pulsing AGB stars and end as
NeO white dwarfs or, if the NeO core mass exceeds
1.37
(Nomoto 1987), electron captures can induce core collapse and a supernovae
explosion ensues (e.g., Miyaji et al. 1980; Nomoto 1987; Gutierrez et al.
1996).
In the late 60's and early 70's, Rakavy et al. (1967), Beaudet & Salpeter
(1969) and Boozer et al. (1973) considered the evolution of CO stars in the
mass range 1.03-1.44
and showed that stars with this typical core
mass should ignite carbon off-center at relatively low density (
107 g cm-3). Then Barkat et al. (1974) computed the first consistent
evolution of a 8
star from the main sequence up to the formation a
OSi core and pointed out the possibility that these stars could be the
progenitors of pulsars. Then, a large wealth of efforts was dedicated to
pursue the advanced and final stages of the evolution of the NeO core in
the regime of electron-captures and oxygen burning (e.g. Miyaji et al. 1980;
Hillebrandt et al. 1984; Ray et al. 1984; Burrows & Lattimer 1985), and the
field is still active (e.g. Isern et al. 1991; Gutierrez et al. 1996; Wanajo
et al. 2003). In the 80's, Nomoto and collaborators (Hillebrandt et al. 1984;
Nomoto 1984, 1987; Miyaji & Nomoto 1987; Hashimoto et al. 1993)
explored in more detail the mass range 9 - 12
,
investigating the
evolution of 2.4
and 2.6
"helium balls'' up to the first
stages of core cooling after central carbon exhaustion and the evolution of
a 2.2
"helium ball'' from central helium burning to the shock
bounce following core collapse. These works demonstrated the fundamental
role of electron captures prior to the supernova explosion and gave a very
good estimate of the critical core mass above which collapse must
occur. More recently, Garcia-Berro and collaborators (Garcia-Berro & Iben
1994; Ritossa et al. 1996; Garcia-Berro et al. 1997; Iben et al. 1997; Ritossa
et al. 1999) followed the evolution of a 9 and 10
model (
)
from main sequence up to the first thermal pulses of SAGB phase, and the
evolution of a 10.5
and 11
star (
)
up to the
activation of electron captures. These studies, based on consistent
evolutionary models give a detailed analysis of the structural
evolution with a first description of the deflagration during the
off-center carbon burning phase. In parallel, binary evolution provided
additional paths to the formation of NeO white dwarf (e.g. Iben & Tutukov
1985; Dominguez et al. 1993; Gil-Pons & Garcia-Berro 2002).
Previous studies have revealed the complex physics involved in the modeling of massive AGB stars with the propagation of a deflagration front (Timmes et al. 1994; Garcia-Berro et al. 1997) and with the apparition of the URCA process during the pre-supernovae evolution of the most massive NeO cores (e.g. Ritossa et al. 1999). However little is known about their nucleosynthesis and a lot of speculations/uncertainties surround the production of s-elements during thermally pulsing super-AGB phase (Ritossa et al. 1999) and of r-elements (Wheeler et al. 1998; Wanajo et al. 2003) during the supernovae explosion. Their contribution of SAGB stars to the galactic chemical evolution remains also largely unknown.
This paper is the first of a series intended to better understand the
final evolution of massive AGB stars. In the present study, we focus on
the carbon burning phase of solar metallicity stars in the mass range
9 - 12 .
In Sect. 2, we describe the
stellar evolution code and the specific numerical adaptations needed for
these computations. In Sect. 3, we give a brief overview
of the evolution prior to carbon ignition. In Sect. 4 we
explain in details the propagation of a representative carbon burning
flame, give a comparative study of the flame characteristics in the
different stars and describe in Sect. 5 the chemical
composition of the resulting NeO core. In Sect. 6 we discuss the
second dredge-up characteristics and conclude in Sect. 7.
STAREVOL (v2.70) is a lagrangian implicit
stellar evolution code. The independent variables are the mass coordinate Mr and time t. The integration variables are ,
the logarithm
of the temperature,
,
the logarithm of the radius,
a new
variable related to the degeneracy parameter (see below), the lagrangian
velocity u and the luminosity Lr. The equations of the stellar
structure are solved iteratively by a relaxation method. The difference
equations are second order accurate in space and first order in time. The
nucleosynthesis equations are solved after the structure has converged and
the resolution method follows Wagoner (1969).
At low-temperature (T < 8000 K), the atomic and molecular opacities are
given by Alexander & Fergusson (1994). Between
K, we use the OPAL tables (Iglesias & Rogers 1996) and at higher
temperatures radiative opacities were computed using the analytical fits
compiled by Timmes et al. (2000). Conductive opacities are computed from a
modified version of the Iben (1975) fits to the Hubbard & Lampe (1969)
tables for non-relativistic electrons, from Itoh et al. (1983) and Mitake
et al. (1984) for relativistic electrons and from formula of Itoh et al.
(1983) as well as Raikh & Yakovlev (1982) for solid plasmas. Finally,
opacity tables are available for different mixtures which allows us to take
into account CO enrichments.
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Figure 1: STAREVOL nuclear network. |
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The nuclear reaction rates have been updated and new nuclear tables built
by NETGEN, the NETwork GENerator tool available at
http://astropc0.ulb.ac.be. By default we use the NACRE rates (Angulo et al.
1997) for charged induced reactions, Bao et al. (2000) for neutron capture
reactions, Horiguchi et al. (1996) for experimental beta decay rates and
Caughlan & Fowler (1988) otherwise. Our network includes 52 species up to
(see Fig. 1) and all relevant
,
p,
,
n,
reactions plus the
+
and
+
reactions. In particular, the energetics is
correctly accounted for up to the end of neon burning and the
nucleosynthesis of light elements with
is followed
accurately. The neutron abundance is computed using a fake neutron sink
nucleus as described in Jorissen & Arnould (1989). In the regime of
electron captures, relevant for the evolution toward the supernova stage,
we included the following elements (Nomoto 1987; Ritossa et al. 1999):
,
,
,
and
.
We use the analytical formula of Graboske et al. (1973) to
compute the screening factors.
The neutrino energy loss rates are computed according to Itoh et al. (1996) and take into account the effects of plasma, pair, bremsstrahlung, recombination, and photo neutrino emission.
The equation of state (EOS) is based on the principle of Helmholtz
free energy minimization and is described in detail in Siess et al.
(2000). Briefly, we adopt the formalism developed by Eggleton et al.
(1973) and expanded by Pols et al. (1995). In this approach, the new
independent variable ,
which is intimately related to the degeneracy
parameter, is used instead of the density
.
This change of variable
allows a smooth, accurate and analytic derivation of the thermodynamic
quantities in regimes of arbitrary ionization and degeneracy. The
plasma is described by three components photons, ions and electrons,
coupled only through the photoionization and photo-dissociation
processes. Effects of pressure ionization and Coulomb shielding have
been incorporated by means of analytical fits to the non-ideal terms in
the Helmholtz free energy. The analytical expression for the Coulomb
interaction (Pols et al. 1995) allows a smooth transition between the
regime of weak correlation where the Debye-Hückel approximation is
recovered and the regime of strong interactions where it reproduces the
Monte-Carlo simulations of Slattery et al. (1980). In practice, this
expression provides a confident treatment of Coulomb interactions up to
an interaction parameter
.
Note that up to the formation
of the NeO core, this parameter remains small (typically
)
so if the Coulomb interactions play a role, they only become relevant
after the completion of the carbon burning phase when the density is
sufficiently high. The fit to the pressure ionization has been derived
from the MHD EOS (Mihalas et al. 1988) for a mixture X = 0.70 and Z =
0.02. According to Pols et al. (1995), this correction also reproduces
the expected behavior for the pressure ionization of He. However, in the
later stage of evolution to overcome the limitation of this prescription,
we impose the matter to be fully ionized when
K or when
X+Y < 10-2. In any case, we always assume that species other than
hydrogen and helium are fully ionized.
The treatment of convection is based on the standard mixing length theory
and the constant
results from solar fitting
models. The Schwarzschild criterion for convection is used to define the
convective boundaries and an instantaneous mixing of the chemical species
is performed inside each convective zone at each iteration during the
convergence process. Our models are standard in the sense that no extra
mixing processes due to rotation or overshooting are included.
The mass loss rate for these objects is rather unknown since they are intermediate between the AGB and massive stars. In these computations, we arbitrarily choose Vassiliadis & Wood (1993) prescription.
The atmosphere is treated in the grey approximation and integrated up to an
optical depth
.
Finally the initial solar
composition is taken from Grevesse et al. (1996).
As already pointed out by Timmes et al. (1994), an accurate description of the flame propagation requires fine and adapted spatial and temporal discretizations otherwise the characteristics of the phenomenon can be missed. If these conditions are not fulfilled, the flame will not propagate properly and may artificially die out. Our numerical method predicts the location of the flame front at the end of the current timestep so we can guess where the spatial resolution will be needed.
Our algorithm is as follows: first we constrain the time step so that the
flame front moves by no more than a certain fraction of the so-called
precursor flame width
which corresponds to the
distance between the base of the carbon burning convective zone (CBCZ) and
the minimum in the luminosity profile below it (Fig. 2). We
estimate the theoretical flame speed
from the values
computed by Timmes et al. (1994). This quantity is tabulated as a function
of the chemical composition, density and temperature at the base of the
CBCZ. Once
is known, the time step
is chosen so
that
,
where
is the effective flame speed calculated from
the computed models and
a factor of the order of
0.1-0.5. This method limits
to values ranging between
yr.
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Figure 2:
Luminosity profile at the base of the convective carbon burning
zone (hashed area). The precursor flame width
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Spatially, the problem is to account for the thinness of the precursor
flame (at most a few kilometers, compared to the gigantic size of the
overall stellar structure). In this region, the luminosity profile Lrvaries strongly and we impose that between two adjacent grid points the
relative difference
(Li+1-Li)/ Li never exceeds 10%. In addition we
do not allow for shell removal in the precursor flame and add the
constraint that the shell underneath the CBCZ must be split into two whenever
the predicted advance of the flame front
is smaller than
(where
is the width of the shell
underneath the carbon burning convective zone).
This algorithm ensures that the precursor flame is described by 50 grid points. The typical extent of a shell is rather extreme
km and
5000 models are needed to follow the
propagation of the flame to the center.
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Figure 3:
HR diagram of selected models up to the super-AGB phase. By
increasing luminosity and effective temperature, the tracks correspond to
masses equal to 6.0, 9.0, 10, 11, 12 and 15 ![]() ![]() |
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Figure 3 shows the evolution of selected models in the HR
diagram. Compared to an intermediate 6
star, these tracks are
significantly more luminous and bluer which could, in principle, make them
distinguishable from their lower mass counterparts. However, the
probability of observing these stars is reduced due to their shorter
lifetime (Table 1).
Prior to carbon ignition, the evolution of the internal structure is very
similar to that of standard intermediate mass stars: central H and He
burning takes place in a convective core, the mass of which increases with
the initial stellar mass (
). The duration of the burning
phases is also a smoothly decreasing function of
(Table 1).
After central hydrogen exhaustion, all our models experience the first
dredge up (1DUP). As shown in Table 2, the hydrogen mass
fraction decreases to the benefit of helium and the surface abundances
exhibit the chemical signatures of the operation of the incomplete CN and
ON1 cycles:
and
are depleted while
,
and
are increased. The surface enrichments of
and to a lower extent of
concomitant to a
decrease in
witness the marginal activation of the NeNa
cycle
in the H burning shell. Finally, we note that
mass fraction is slightly reduced to the benefit of
as a consequence of the activation of
(
)
,
the most rapid reaction of the MgAl cycle
at low temperature. The number ratios
/
and
/
decrease from 90 to 20 and from 0.475 to 0.319,
respectively.
The chemical signatures of the first dredge-up are the same in the
models as in the standard 6
star. However, for the elements
involved in the NeNa chains, the variations can be as large as a factor of
5. In particular the relative
enrichment and
depletion are respectively a factor of 2 and 5 larger in a
model
as compared to a 6
model. But in the end, these differences are very
small and as far as the chemical composition is concerned, these stars are
almost indistinguishable from their lower mass counterparts.
Near central helium exhaustion, the
(
)
reaction takes over the nuclear energy production in the convective core.
The activation of this reaction decreases the C/O ratio which final value
remains highly dependent on the adopted reaction rate for
(
)
and also on the efficiency (semiconvection, overshooting) and duration of
convective mixing in the late stage of He-core burning when the central He
mass fraction drops below
0.1 (see e.g. Straniero et al. 2003, for a
detailed discussion). The higher temperature found in the core of more
massive stars favors the production oxygen but this effects is partially
compensated by the shorter duration of this phase in these objects. The
final C/O ratio is thus hard to guess but our models indicate that as the
initial mass increases more carbon is burnt and the C/O ratio decreases
(Table 1). We note however that above
11
,
the C/O ratio
drops as carbon ignites at the center.
In stars that develop degenerate CO cores, following central helium
exhaustion, the temperature maximum moves outward as a consequence of
plasma neutrino emission. Eventually, the peak temperature reaches
K and carbon ignites off-center. The large deposition of
nuclear energy favored by the strong dependence of the carbon burning rates
on temperature and the "mild'' response of the structure owing to the
partial degeneracy (
)
leads to an accumulation of heat
at the ignition point. There, the temperature gradient becomes steeper and
convection sets in. Figure 4 shows the evolution of the internal
structure and of relevant quantities of a representative 9.5
model. In all our models in the mass range 9 - 11
,
carbon
burning always proceeds in two steps characterized by a first convective
zone, which we refer to as the carbon flash, followed by the development of
a the flame which propagates all the way to the center. In case of the
11.3
,
carbon ignites so close to the center that only one convective
episode occurs.
Table 1:
Selected features of the evolution of massive intermediate-mass
stars prior to carbon ignition. The quantities shown are: the maximum
extent of the convective core during the central hydrogen (
)
and helium (
)
burning phases, the mass coordinate of the
base of the convective envelope at its deepest extent during the first
dredge-up (
), the mass of the CO core at carbon ignition
(
), the durations of central hydrogen (
)
and helium (
)
burning phases, the time elapsed
between the end of central helium burning and carbon ignition (
)
and the central
and
mass
fractions at the end of He core burning. Masses are in solar unit
(
), times in years (yr).
is defined as the mass
coordinate of maximum nuclear energy production in the He burning shell.
Table 2: Surface abundances mass fractions at the end of the first dredge-up.
To understand why the first convective zone quenches and does not reach the center, a close study of the evolution of the structure in the vicinity of the peak temperature is required.
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Figure 4:
Evolutionary features of a representative 9.5 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Similarly to the occurrence of thermal pulses in AGB stars, the carbon
flash does not develop at the locus of maximum temperature (
)
but
slightly above that location. The reason simply arises from the fact that
convection can only establish if the luminosity Lr is larger than a
critical value defined by
As the temperature increases, carbon is depleted. It results that the
nuclear energy production rate (
)
is not maximum at
the peak temperature but at a mass coordinate located slightly below
where the density is larger (Fig. 5). The deposition of
thermal energy below
heats the material and drags the peak
temperature toward the center.
On the other hand, the distance between
and the base of the carbon
burning convective zone (
)
is primarily determined by the nuclear
energy production in this region. The luminosity at
can be
approximated by
The advance of the convective zone thus depends on the nuclear energy
production on both sides of
:
below this limit it is responsible
for the inward advance of the peak temperature and above it determines the
location of the convective boundary. During the early development of the
convective flash, because of the large increase in
,
both
and
are dragged to the center (Fig. 6).
As degeneracy is lifted, pressure rises and starts pushing the overlying
layers. The core expands and its density decreases. As a result, the
nuclear energy production rate below
,
which indirectly drives the
advance of convective zone, begins to decline and moves away from
(Fig. 7) in regions where both
and the
density are higher. Core expansion induces two important consequences
associated with the decrease in
.
First, it slows
down the advance of
toward the center and second it contributes to
increase (
-
).
Eventually, the distance
traveled by
during a time
interval becomes smaller than the corresponding increase in (
-
)
and the convective zone starts to retreat. Then, the combined effects of a
decreasing temperature at the base of the instability (
in
Fig. 4) and the stopping of fuel injection by growing convection
contribute to accelerate the fall of the nuclear energy production in the
flash. This process is irreversible and leads to the disappearance of
convection
850 yr after it appears. Below the flash, in the
radiative shells left by the receding convective zone, carbon is efficient
depleted and the nuclear energy production die out
(Fig. 7). The peak temperature is now driven by the nuclear
energy production at the base of the convective zone and moves upward. The
flash finally quenches and core contraction rapidly leads to the re-ignition
of carbon at a lower mass coordinate.
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Figure 5:
Profiles of
![]() ![]() ![]() ![]() ![]() |
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Figure 6:
Evolution of the nuclear energy production rate ( bottom),
temperature ( mid) and carbon mass fraction ( top panel) during the
activation of the carbon flash in the 9.5 ![]() |
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Figure 7:
Evolution of the temperature,
![]() ![]() ![]() ![]() |
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Concerning the evolution of the second convective zone, the initial
conditions are substantially different. In the flash, the
abundance is initially rather high (at the end of the helium burning phase
its mass fraction is between
0.25-0.30, see Table 1) and
is kept high as the instability grows in mass. Conversely, when the flame
develops, it grows in regions previously occupied by the flash where carbon
had been efficiently depleted (in the flame at the time of C re-ignition,
). It results that the nuclear energy production
rate (proportional to the square of
)
and subsequent core expansion are
significantly reduced. As a consequence, the peak of
below
is maintained and is able to drive the burning front
to the center. Finally a steady state is achieved where almost all the
energy deposited at the base of the convective flame is instantaneously
carried away by neutrinos (Timmes et al. 1994).
During the flame propagation, the luminosities are constant
(Fig. 4) and the carbon burning region seems to be decoupled
from the rest the star. No indication of the presence of the flame can be
attested from the evolution of ,
or L for
instance. The evolution of the temperature profiles after the onset of the
flame (Fig. 8) clearly shows the establishment of a steady
state where all the quantities seem to be simply translated in mass.
However, as the flame deepens, the density increases and modifies the
opacity profile behind the flame front, resulting in variations of the
outer convective flame boundary. Ahead of the convective flame,
is abundant and
.
In this region, matter is heated and the modifications to
the temperature profile induce the advance of flame. To summarize, contrary
to the first and violent carbon flash, the energy released at the onset of
the convective flame does not induced substantial modifications to the
thermodynamics of the core mainly because
is considerably reduced.
The energy production rate below the convective zone is thus weakly
affected and remains at anytime much larger than what is produced in the
convective flame. As a consequence, the radiative burning of
ahead of the convective flame drives the peak temperature to the center and,
since
varies little, (
-
)
remains
almost constant and convection moves to the center.
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Figure 8:
Profiles evolution during the development and propagation of the
convective flame in the 9.5 ![]() |
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The succession of two convective episodes, a flash and a flame, is
sensitive to the adopted neutrino energy loss rate. As an
illustrative tests, we run a series of computations characterized by a
neutrino energy loss rate
five times larger and
smaller. Although these variations are not realistic, they provide hints
concerning the role of the neutrino losses in this process. If
is lower, less energy escapes the convective instability and this
additional heat is further used to expand the structure. As a result, the
flame quenches rapidly and in our tests we note that 3 flashes occur before
a convective flame forms and propagates to the center. At this point, the
carbon burning proceeds in a very similar way to the helium flash in low
mass stars through the development of successive instabilities. Conversely,
if the neutrino loss rate is increased, the carbon burning zone suffers
less expansion and a unique convective zone appears that propagates as a
convectively bound flame to the center.
The general evolution of our
9 - 11.3
model stars during carbon
burning phase is similar to the evolution of the 9.5
star described
above. In all our models, carbon ignites off center and the first
convective zone appears when the peak temperature reaches
K. The corresponding density and degeneracy are typically
g cm-3 and
.
As Table 3 shows, the carbon flash ignites closer to the center and
is less powerful as
the stellar mass increases. This is a direct consequence of the decreasing core
degeneracy with increasing stellar mass. In the limit of no degeneracy,
carbon ignites quietly at the center as in massive stars. This takes places in our
simulations for M = 11.5 .
The duration of the flash convective zone
(
)
is strongly correlated to the carbon mass fraction in the
instability (
)
and presents a maximum around
10.5
.
This duration depends on the amount of fuel available and on the
rate at which it is depleted which are function of the extent of the
convective zone, of the carbon mass fraction initially present in the core
and of the temperature in the flash. On one hand, the carbon abundance in
the core tends to increase with decreasing mass (Sect. 3)
but on the other hand, it is more effectively depleted during the flash of
lower mass stars where higher temperatures are reached and where the extent
of the convective zone is smaller (
in
Table 3). The duration of the convective zones is thus the result
of the competing effects of burning speed and fuel enrichment. Our models
indicate a maximum around 10.5
.
The flame develops near the
discontinuity left at the deepest
penetration of the flash convective zone. Table 3 indicates that its
duration (
)
is a decreasing function of stellar mass. This is
mainly due to the fact that in more massive stars, the temperature in the
precursor flame is higher and the flame speed, which scales as
(Timmes et al. 1994) is faster. We
also note from Table 3 that the flame, although igniting deeper in
mass (
), is located at relatively large radii from the center
(
), comparable to that of the flash. Typical flame speeds during the
stationary descent of the convective zone increases from
cm s-1 to
cm s-1 as the star
initial mass increases between 9 and 11.0
.
During the major part of the descent, the flame speed compares nicely with
the predictions of Timmes et al. (1994). The best agreement is reached in
the 11.0
star and the largest discrepancy is found in the 9.0
star where the flame speed is within a factor
2.5 of the theoretical
value. As already outlined by García-Berro et al. (1997), in their
model Timmes et al. (1994) neglect the gravothermal term in the energy
equation which, in our simulations, represents a non negligible
contribution. This simplification certainly explains the observed
differences.
The comparisons are mostly based on the simulations of Garcia-Berro
and collaborators who provide a detailed description of stellar models in
the same mass range. The results are globally in good agreement. In
particular, off center carbon ignition starts in a 9
in both studies
and ends near the same mass: 11
compared to 11.3
in our
simulations. The morphology of the convective episodes (flashes and flames)
are also very comparable. The evolution of the central temperature and
density are very similar except that our models evolves systematically
slower. Several explanations can be put forward to explain these rather
small differences. First Garcia-Berro et al. did not take into account mass
loss but our tests indicate that suppressing mass loss barely affects our
results since it really becomes efficient later, during the SAGB phase.
Second, and most importantly, are the differences in the input physics. In
particular, they use nuclear reaction rates which date back to Caughlan &
Fowler (1964) and Fowler & Hoyle (1964)! It is thus conceivable that the
main reasons for the discrepancies found in the burning lifetimes find
their origin in the adopted nuclear rates. Up to the end of C burning,
the plasma remains weakly correlated with
.
So if the
degeneracy is treated correctly, the differences in the equation of state
should not be too important at this stage.
In the HR-diagrams, their tracks are also slightly different but this can be easily explained by their use of analytical fits to the opacity in comparison with our use of recent OPAL tables and also because their initial composition is slightly more helium rich than what is given by Grevesse et al. (1996).
However, the value of the transition mass between stars that ignite or
not carbon, usually refered to as
,
can differ substantially from one
simulation to the next. For solar metallicity stars,
ranges between
6 - 9
(for low values: Bertelli et al. 1985; Bressan et al.
1993; Umeda & Nomoto 1999; Bono et al. 2000; and for high values of
:
Becker & Iben 1979, 1980; Maeder & Meynet 1989; Garcia-Berro et al. 1997;
Poelarends & Langer 2005). This critical value depends directly on the
properties of the He and CO cores which in turn are highly dependent on the
mixing prescription (semiconvection, overshooting, rotational mixing...),
on the He content (e.g. Bono et al. 2000) and as we noticed on the numerical
resolution at the He core interface which needs to be relatively high. In
particular, including overshooting can significantly decrease the value of
from
9 to 6
(Bertelli et al. 1985; Bressan et al.
1993). In view of these results and considering the fact that our models do
not include any extra mixing beyond any convective boundaries, our value of
= 9
should be considered as an upper limit.
Table 3:
Selected properties of the convective carbon burning zones as a
function of the initial stellar mass. The superscripts 1 and 2 refer to the
flash and flame, respectively. The quantities shown are the time (t) when
convection sets in, the radius ()
and mass (
)
coordinates where
it develops, the duration
of the convective episode, the maximum
carbon luminosity
,
the mass covered by the instability (
)
and, at the time of peak luminosity, the carbon mass fraction
(
)
in the convective zone.
Carbon burning proceeds mainly through
(
,
)
and
(
,
p)
reactions, the rates of which remain
relatively uncertain, in particular because of the possible presence of
low-energy resonances (Cussons et al. 2003). This effect comes on top of the
large uncertainties already affecting
(
)
which
primarily determines the carbon content inside the white dwarf. Since we
restrict our analysis to the carbon burning phase we only consider the
+
rates. We have run a series of test models
increasing or decreasing both rates by a factor of 2 or 5. The main results
are that for lower carbon burning rates, the onset of the convective
instability is delayed and consequently the mass coordinate of the ignition
point increases. For instance, for a rate twice as large (small), the flash
is triggered 650 yr earlier (latter) than in the reference model and the
mass coordinate
is shifted by -(+)0.06
.
For lower rates, the
peak carbon flash luminosity (
)
is larger since higher temperatures
are reached at the time of ignition. The duration of the first convective
zone is also reduced because the higher flash luminosity produces a larger
expansion and a more efficient quenching of the instability. Concerning
the flame, it also starts at a higher mass coordinate, last longer and is
hotter when the rates are reduced. The effects on the core composition
mainly concern the
profile in the region of the carbon flash
where a stronger depletion is observed at higher rates. However, where the
flame passes through, the chemical changes are modest and, for the most abundant
species, never exceed
.
In conclusion, the effects of
+
rates on the structure and evolution of the core
are weak.
Our analysis of stellar nucleosynthesis is based on the representative
9.5
star and the slight differences present in other mass models do
not affect the general trends depicted here.
At helium ignition, the second most abundant element in the core is
(1.3%) which will be entirely converted into
after two successive
captures. During the entire core He burning
phase, the temperature remains too low for an efficient production of
neutrons by
(
,
n)
.
The neutron density never
exceeds
cm-3 and at He exhaustion the neutron
exposure remains lower than
0.015 mbar-1. Therefore no
s-process nucleosynthesis is expected to take place at this stage. We note
however a slight production of
,
and
as a result of neutron capture reactions. At central helium
exhaustion, the core is mainly composed of
(63%),
(34%) and
(1.2%) and helium burning proceeds
radiatively in a shell surrounding the inert core.
Eventually, the increasing temperature reaches
K and
carbon ignites. The highest nuclear fluxes come from
(
,
)
and
(
,
p)
followed
by
(
)
.
These reactions will convert the CO core
into a NeO core and will supply protons and
particles for a richer
nucleosynthesis.
Ahead of the flame where the temperature is maximum (
K), the
particles that are not captured by
will
participate to the complete destruction of
by (
,
n) and (
)
reactions. Some
is therefore produced. Locally a neutron
density of
cm-3 can be achieved but the irradiation is
too short for a noticeable production of s-elements. The released neutrons
that are not captured by the iron seeds react with the abundant
and to a lower extent with
,
and
,
leading the production of
,
,
and
.
The protons, mainly produced by
(
,
p)
also participate to the production of
and
by (p,
)
and (
)
reactions on
,
and to the synthesis of
by (
)
reactions on
.
Note also that the chain of reactions
(p,
)
(
)
(
)
(
)
contribute to the recycling of
which mass fraction remains
always larger than 0.008. These elements are then engulfed in the
convective flame where the same nucleosynthesis takes place. However, due
to the lower temperature in the convective flame, the neutron density never
exceeds 106 cm-3 in this region.
After the flame has reached the center, the convective zone disappears and
carbon burning proceeds radiatively. Due to the exhaustion of combustible
and increasing neutrino energy losses, the maximum temperature moves
outward. When it reaches the chemical discontinuity left at the mass
coordinate corresponding to the maximum extend of the flame, a third carbon
flash of milder magnitude (
)
develops followed
by a series of instabilities of decreasing strength until a NeO core of
1.108
is formed.
The central composition of the NeO white dwarf of our 9.5
model is
showed in the lower right panel of Fig. 9. The core is mainly
composed of
(
55-60%) and
(
28-32%). The main trace elements are
(
5.5%),
(
3.3%),
(
1.5%),
(
0.7%),
(
0.9%),
(
0.8%),
(
0.7%) and some unburnt
remains. Apart
from
,
the abundance profiles are very similar among our models
but some features deserve additional comments:
First,
is more abundant than
by a factor of
1.8 throughout the core in all our models. We therefore get an
NeONa core rather than an NeOMg white dwarf as input model for the
potential pre-supernova evolution. So far, the initial models used in the
computations of electron capture induced supernova were NeOMg cores
(e.g. Nomoto 1987, and references therein; Canal et al. 1992; Gutierrez et al.
1996). Since the critical density for electron capture on
is
lower than that on
,
an NeONa core may have a different
evolution through the electron capture stages.
may be ignited
at a lower density than previously expected, leading to a disruption of the
core rather than a collapse. It is therefore important to include the
appropriate abundances of both
and
for
collapse-explosion studies.
Second, as one can see from Fig. 9, some unburnt
is
left over. It presents an off-center peak as high as
in the 9.0
star which decreases with increasing mass to reach
in the 11.3
star. The exact shape of
the remaining
profile depends on the details of the carbon
burning flame propagation, but as a general trend, the
peak
location moves closer to the center and is shallower as the stellar mass
increases. Garcia Berro et al. (1997) pointed out that, if the star is in a
binary system and accretes enough mass to reach the Chandrasekhar limit,
this remaining fuel could alter the supernovae evolution from core collapse
to thermonuclear disruption as a result of C and O ignitions.
The central abundances at the end of the carbon burning phase are in
qualitatively good agreement with those obtained by Ritossa et al. (1996,
1999), Garcia-Berro et al. (1997) and Iben et al. (1997). The main
differences concern
which core abundance is smaller in our
models. This discrepancy is already present at the end of the core helium
burning phase and is probably a consequence of the different
(
)
rate adopted in these studies. The higher flame front
temperature achieved in our simulations also contributes to burn
more efficiently and to produce more
.
At the outer edge of the CO core (where ,
i.e. above
in Fig. 9) our profiles differ from those
of Garía-Berro et al. (1997, see their Fig. 32). In particular, we
find that
is the most abundant species after C and O rather
than
.
This difference can be explained by the fact that the
authors did not take into account neutron capture reactions beyond
from which
is made. Finally, we can not
explain their absence of
which, in our models, naturally
comes from the
(
)
at the base of the He
burning shell nor the absence of
where it is also
synthesized.
![]() |
Figure 9:
Evolution of the chemical profiles in the interior of a 9.5 ![]() |
Open with DEXTER |
![]() |
Figure 10: Kippenhahn diagrams during the carbon burning phase and second dredge-up. The short-dashed (red), dashed (blue) and dotted (green, if present) lines indicate the mass coordinate where the nuclear energy production by carbon, helium and hydrogen burning is maximum, respectively. The hashed areas represent convective regions and the stellar mass is indicated inside each panel. |
Open with DEXTER |
Table 4: Surface abundances mass fractions at the end of the second dredge-up.
At the end of central helium burning, the gravothermal energy released
by core contraction is imparted into heating that may lead to carbon
ignition and into mechanical work that will trigger the envelope expansion
and the subsequent second dredge-up. In the most massive stars (
), the envelope stalls near the hydrogen-helium discontinuity
while, in lower mass stars (our 9.0, 9.5, 10 and 10.5
models) it
penetrates into the He burning shell during the C burning phase. These
differences mainly result from evolutionary timescale considerations: in massive stars, the
evolution is faster and the envelope cannot cross
this discontinuity before core collapse. In addition, lower mass stars undergo a stronger
contraction needed to raise the lower initial core temperature to the
carbon ignition threshold. So more energy is deposited at the base of the
envelope, the expansion is there stronger and the subsequent deepening enhanced.
As for the bump in globular clusters, the
crossing of this chemical barrier during the 2DUP provokes some structural readjustments
induced by the modification of the opacity in the envelope and, in the HR diagram, the star temporarily settles at the base of the AGB.
Note that the occurrence of secondary carbon flashes can also affect the depth of the
dredge-up. In particular, if the instability develops close enough to the
H/He interface, the nuclear energy produced in the flash can freely escape
from the core and feed the envelope. In the expanding layers above the
instability, the luminosity and concomitantly
increase, thus
favoring the penetration of the envelope in the He burning shell. This
happens e.g. in our 10 and 10.5
models during the late carbon flashes that follow the formation of the NeO
white dwarf (Fig. 10). In this case, the top of the instability is
located very close to the base of the envelope (
0.2
).
By the time the 2DUP is completed, the convective envelope has
engulfed the matter processed by the hydrogen burning shell and the surface
abundances of
,
,
,
and
decrease while
mass fraction rises. The marginal
activation of the NeNa cycle is also apparent in the increase of
and
while
is depleted to the
benefit of
as a consequence of the activation of
(
)
.
In our models with
,
the
envelope penetrates into the He burning shell producing a surface abundance
enhancement of
mainly, but also of
,
,
and
.
The surface abundances after
the second dredge-up phase are summarized in Table 4.
Our models reveal an abrupt change in the depth of the 2DUP around
11.0
(Table 5) indicating the transition between stars
that will develop massive cores subject to hydrodynamical collapse and
those that will form a degenerate white dwarf. For the 11.0 and 11.3
,
at the end of carbon burning, neon ignites off-center. It becomes the
dominant source of energy in the 11.3
while in the 10.8 and
11.0
model electron captures come into play. These interesting
aspects of the final evolution of massive AGB stars will be presented in a
forthcoming paper.
Comparisons of our models with previous works reveal that the critical mass
for degenerate off-center carbon ignition
is very similar to the
previous results of Garcia-Berro et al. (1997, and reference therein) and is
around 9
.
However this values should be considered as an upper limit
since no extra-mixing processes were included in our simulations. We also show that
the impact of
+
rates is marginal. Concerning the
evolution of the surface composition, stars in the mass range 9 -12
do not differ substantially from their lower mass
counterpart. Their most distinguishable characteristic is their higher
luminosity. We also find that the depth of the second dredge-up changes
abruptly around
11.0
,
marking the transition between AGB and
massive stars that develop cores massive enough to collapse. However, this
transition mass depends critically on the adopted mass loss rate which
governs the core growth and on the mixing prescriptions which determines
the NeO core mass. Their still remain a lot to explore and in first place,
the effect of metallicity and of the physical parameters such as mixing and
mass loss. These aspects, as well as the subsequent evolution through the
TP-SAGB phase and electron capture regime, will be presented in forthcoming papers.
Table 5:
At the end of the carbon burning phase, stellar mass (
)
and masses of the NeO (
)
and CO cores (
), defined as the mass coordinate where
and
respectively, as a function of the
initial mass
.
corresponds to the mass
coordinate of the deepest extent of the envelope during the second
dredge-up.
Acknowledgements
The author acknowledges financial support from the French "Programme National de Physique Stellaire (PNPS)'', from Geneva Observatory and from the Laboratoire d'Astrophysique de l'Observatoire de Grenoble (LAOG). L.S. is research associate at the FNRS.