A&A 435, 231-237 (2005)
DOI: 10.1051/0004-6361:20042254
J. Gutiérrez1 - R. Canal2,3,4 - E. García-Berro1,3
1 - Departament de Física Aplicada, Escola Politécnica
Superior de Castelldefels, Universitat Politècnica de
Catalunya, Avda. del Canal Olímpic s/n, 08860
Castelldefels, Spain
2 -
Departament d'Astronomia i Meteorologia, Universitat
de Barcelona, Facultat de Física, Martí i
Franquès 1, 08028 Barcelona, Spain
3 -
Institute for Space Studies of Catalonia, c/Gran Capità
2-4, Edif. Nexus 104, 08034 Barcelona, Spain
4 -
Special Research Center in Astrophysics, Particle Physics
and Cosmology, University of Barcelona, Martí i
Franquès 1, 08028 Barcelona, Spain
Received 26 October 2004 / Accepted 23 January 2005
Abstract
The final stages of the evolution of electron-degenerate
ONe cores, resulting from carbon burning in "heavy weight''
intermediate-mass stars (
)
and
growing in mass, either from carbon burning in a shell or from
accretion of matter in a close binary system, are examined in the
light of their detailed chemical composition. In particular, we have
modelled the evolution taking into account the abundances of the
following minor nuclear species, which result from the previous
evolutionary history: 12C, 23Na, 24Mg, and
25Mg. Both 23Na and 25Mg give rise to Urca processes,
which are found to be unimportant for the final outcome of the
evolution. 24Mg was formerly considered a major component of ONe
cores (hence called ONeMg cores), but updated evolutionary
calculations in this mass range have severely reduced its abundance.
Nevertheless, we have parameterized it and we have found that the
minimum amount of 24Mg required to produce NeO burning at
moderate densities is
23%, a value exceedingly high in the
light of recent evolutionary models. Finally, we have determined that
models with relatively small abundances of unburnt carbon
(
)
could be a channel to explosion at low to
moderate density (
g cm-3). This is clearly
below the current estimate for the explosion/collapse threshold and
would have interesting consequences.
Key words: nuclear reactions, nucleosynthesis, abundances - stars: evolution - stars: interiors - white dwarfs - supernovae: general
The final evolution of electron-degenerate cores made of oxygen and
neon (hereafter ONe) has received little attention in the last few
years despite its potential importance for supernova theory. One of
the reasons for this lack of accurate models is that there are very
few realistic evolutionary calculations leading to the formation of
such cores. This, in turn, is due to the fact that for stars with
masses ranging from 8 to
,
carbon-burning is ignited
off-center and proceeds through a series of mild flashes in partially
degenerate conditions, a situation especially difficult to simulate,
which requires very short time-steps and careful zoning, leading to a
heavy computational demand.
Up to now, no detailed multidimensional calculations of the evolution
of high-end intermediate mass stars in this phase exist.
Nevertheless, there are several possible effects, such as
rotation-induced mixing and overshooting, and, more importantly, the
possibilty of off-center ignitions (García-Senz & Woosley
1995; Woosley et al. 2004; Wunsch & Woosley 2004) that
should be modeled and discussed using multi-dimensional hydrodynamics
- see, however, Höflich & Stein (2002). Some multidimensional
stellar models already exist, but they are in an early stage of
development. Spherically-symmetric models include a degree of detail
that current two- and three-dimensional simulations are still far
from reaching. A discussion of this kind of effect is beyond the scope
of this paper. However, although multidimensional evolutionary
calculations of stars in this mass range do not exist yet, Hirschi et al. (2004) have studied the evolution of rotating stars with masses
- slightly larger than those considered here
- and have found that most of the differences between the
presupernova structures obtained from rotating and non-rotating
stellar models have their origin in the effects of rotation during the
core hydrogen and helium burning phases. Additionally, they
explicitly mention that the advanced stellar evolutionary stages
appear too short in time to allow the rotational instabilities to have
a significant impact during the late stages. Hence, given the lack of
accurate multidimensional calculations for this phase of the evolution
of heavy-weight intermediate mass stars, one should conclude, in
principle, that spherically-symmetric evolutionary calculations can
still provide a reasonable approximation to the real behavior - see
the recent review of Maeder & Meynet (2000) for a thorough discussion
of this topic.
These objects were long ago proposed as clear candidates for the
so-called accretion-induced collapse - a process first described
by Canal & Schatzman (1976). Currently, it is widely accepted that
ONe electron-degenerate objects do collapse to form a neutron star
when the central density is driven beyond a certain threshold - see,
for instance, Gutiérrez et al. (1966, 1997), or Canal &
Gutiérrez (1997), and references therein. In particular, in the
pioneering work of Miyaji et al. (1980) it was shown that all stars
in the mass interval
would
develop electron-degenerate ONeMg cores during carbon shell burning.
These cores would undergo a phase of electron capture on 24Mg and
24Na first, and later on 20Ne and 20F, to finally
ignite Ne and O explosively at central densities higher than
g cm-3. At these very high central densities,
fast electron captures occuring on the nuclear statistical equilibrium
(NSE) material would rapidly drive the Chandrasekhar mass below the
actual mass of the degenerate core and, consequently, gravitational
collapse would ensue. Later, Woosley et al. (1980) and
Nomoto (1984) reduced the above mass range to
.
Stars more massive than
would
ignite Ne in a series of non-explosive flashes and ultimately proceed
through the O-burning and Si-burning stages in a way similar to more
massive stars. Subsequently, Timmes & Woosley (1992, 1994) found
that the density threshold for an ONe core to collapse to neutron star
dimensions was around
g cm-3. On the
contrary, if NSE was reached at densities below this critical density,
the degenerate object would be completely disrupted. Finally,
Gutiérrez et al. (1996) analyzed the role of Coulomb corrections,
both in the equation of state and in the electron-capture threshold
energies, and found that explosive NeO ignition takes place at
densities high enough to ensure gravitational collapse to nuclear
matter densities. The previously described scenario should apply not
only to the cores of Super-Asymptotic Giant Branch (SAGB) stars
(García-Berro & Iben 1994) but also to ONeMg white dwarfs
accreting material from a companion in a close binary system (Miyaji
et al. 1980; Nomoto 1984, 1987).
Most of the studies done so far for this mass range had considered 24Mg as a major constituent of the core. However, the detailed studies of Ritossa et al. (1996), García-Berro et al. (1997), Iben et al. (1997) and Ritossa et al. (1999) have clearly shown that the mass fraction of 24Mg is much smaller than previously thought. Following the trends shown by these modern evolutionary calculations for the stars in this mass range, in this paper we investigate the effects of the chemical composition of the degenerate core on the possible outcome of these objects. In particular, we analyze the impact of the abundance of 24Mg, the presence of 23Na and 25Mg, and the presence of unburnt 12C, which appears to be a specific characteristic of current evolutionary models. The paper is organized as follows. In Sect. 2 we describe the most relevant input physics and the initial model. Section 3 is devoted to the effects of an increasing 24Mg abundance, while in Sect. 4 we analyze the role played by the previously unburnt 12C. The role of the minor chemical species (23Na and 25Mg) is analyzed in Sect. 5. In Sect. 6 we discuss our results and we draw our conclusions.
For the sake of conciseness, and given that the numerical code and most of the physical inputs employed in the calculations discussed in this paper have already been described in previous papers - see Gutiérrez et al. (1996) and references therein - we will not present them here in detail. Instead, in this section we only summarize the most relevant physical ingredients which are relevant for the discussion of the evolutionary sequences analyzed below. These are the electron capture rates, Coulomb corrections and the prescription adopted for convective transport.
Electron capture rates have been taken from Oda et al. (1994), which provides the most up-to-date, and physically consistent determinations. This point seems already fixed, as the rates for sd-shell nuclei have remained almost unchanged since the work of Takahara et al. (1989). However, the data of Oda et al. (1994) are much richer for 24Mg, 20Ne and other sd-shell nuclei relevant for this work, such as 23Na and 25Mg.
Coulomb corrections to the electron capture threshold have been
included as in Couch & Loumos (1974). The net effect on the Coulomb
interaction energy is to lower it, hence reducing the Fermi energy and
increasing the capture density threshold. The exact expression
to compute the energy shift is
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(1) |
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(2) |
Another very important issue is what convective criterion must be
adopted. In the paper of Miyaji et al. (1980), convection was assumed
to set in according to the Schwarzschild criterion: a convective core
started to develop at the onset of the electron captures on
24Mg. Convective heat transport contributed to keep the
temperature below that of explosive Ne ignition (
K) along core contraction until a central density
g cm-3 was reached. That was questioned by
Mochkovitch (1984), who argued that the steep gradient of the electron
mole number,
,
produced by the same electron captures that
create the superadiabatic temperature gradient, should stabilize the
fluid against convective motion and a semiconvective region would
form. Miyaji & Nomoto (1987) tested this situation by assuming
semiconvective mixing to be negligible. They found that explosive NeO
burning was ignited at a central density
g cm-3: heating by the electron captures and cooling by
thermal neutrinos was then purely local. Thus, convective criteria
are essential in ascertaining the final outcome of the evolution of
compact ONe objects. In this paper we adopt a rather pragmatic
approach. That is, we first consider the classical Schwarzschild
criterion
.
However, since this stability
criterion assumes complete chemical homogeneity and this is not the
case when electron-captures start, we also use the Ledoux criterion:
,
where:
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(3) |
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(4) |
Our initial model, used in all the calculations, is a
electron-degenerate object, whose main chemical constituents
are 16O and 20Ne, with the distribution obtained by Ritossa
et al. (1996). Its central density and temperature are, respectively,
g cm-3 and
K. The core is discretized in up to 1000 mass
shells by an adaptive algorithm, which ensures good resolution both in
chemical composition and in entropy throughout the structure. The ONe
object is slowly contracting due to spherical accretion at a rate of
yr -1. Its mass is consistent with the
mass of ONe white dwarfs, and is close to the Chandrasekhar mass,
making this object prone to experience sudden instability leading to
either disruption or collapse.
Since the pioneering work of Miyaji et al. (1980), the astrophysical
nuclear reaction rates have undergone significant changes. In
particular, the amount of 24Mg present in typical ONe cores has
been reduced by almost an order of magnitude (Ritossa et al.
1996). Currently, it seems that the most important rates are already
reasonably well known but, nonetheless, it would be interesting to
ascertain the minimum abundance of 24Mg required to cause an
early explosion of the degenerate core. For this, we have
parameterized the abundance of 24Mg while keeping constant the
ratio O/Ne. While this is not truly consistent, it serves well the
point relevant here: how much 24Mg is the minimum to engage a
explosion at about
g cm-3, which would possibly
lead to the complete disruption of the degenerate object. With the
currently accepted abundances of 24Mg (
3% by mass), the
maximum temperature attained during the electron captures is only
K (Gutiérrez et al. 1996), very far below the
minimum needed to initiate the explosive oxygen burning and to drive
the matter to NSE. The goal of this section is to determine the
minimum abundance of 24Mg required to produce a central
temperature of about
K.
In Fig. 1 several evolutionary sequences with increasing abundances (3%, 10%, 15% and 25%) of 24Mg are shown. We have restricted our study to the Ledoux case because the preclusion of convection implies a stronger heating of the central region, due to the relative inefficiency of the radiative-conductive transport of energy. Evolutionary calculations performed with the Schwarzschild criterion (Fig. 2) confirm this point, even taking into account the mixing induced by the convective transport. As it can be seen in Fig. 1, the higher the abundance of 24Mg, the higher the central temperature achieved during the phase of electron capture on this isotope.
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Figure 1:
Evolutionary sequences of an ONe electron-degenerate object
for increasing abundances of 24Mg. The abundances of
24Mg are, respectively, 3%, 10%, 15% and 25% by
mass. The model with
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Table 1: Ignition density and maximum central temperature reached during the captures on 24Mg and 24Na, for several initial abundances of 24Mg. The cases whose central densities are marked with an asterisk correspond to off-centre explosions. In all the calculations, the Ledoux convective criterion has been adopted.
Table 1 shows the net effect of increasing the abundance of 24Mg,
both for the maximum central temperature and for the central density
at ignition. The first trend obviously appearing in Table 1 is that
failed explosions lead to higher collapse densities - see Fig. 1.
This is due to the fact that electron captures proceed in conditions
of constant
,
and hence higher amounts
of 24Mg imply larger reductions of
.
Consequently,
the collapse density increases from
g cm-3,
when there is no 24Mg in the degenerate core, to about
g cm-3, when the amount of 24Mg is 10%. The
minimum abundance of 24Mg for which NSE develops during the
electron capture phase is about 15%, and the explosion is clearly
off-center. The reason is simple: during the phase of electron
capture, part of the energy generated is released on the mass shells
above that in which the electron capture reactions are proceeding;
thus, the temperature of these shells is constantly increasing. There
is also a second effect: a reduction in
increases the rate
of contraction and, thus, the gravitational enery release, leading to
a still higher temperature of the material. This can be readily seen
in Fig. 1, in which the slope of the evolutionary track in the
plane becomes steeper with increasing
abundances of 24Mg. Eventually, the energy generation in one of
these preheated shells is able to drive the temperature to the point
at which oxygen burning can be self-sustained, and, in fact,
accelerated to produce the transition to NSE. For even larger amounts
of 24Mg the mass shell in which the off-center explosion ensues
is located closer to the center. The explanation is, again, rather
straightforward, as the preheating for increasing abundances of
24Mg is more extreme and the change in temperature caused by the
electron captures on 24Mg is also correspondingly higher.
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Figure 2: Evolution of an ONe object with the Schwarzschild criterion (solid line). The dotted line corresponds to a case in which convective mixing is inhibited. In the case with mixing, the effects of a convective Urca shell are evident. The peak corresponds to the point where the nuclides of mass 24 able to capture electrons (namely 24Mg and 24Na) are exhausted throughout the convective core. |
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A central explosion occurs only when the amount of 24Mg
approaches 25%, which is not credible given the current
nuclear reaction rates relevant for the production of 24Mg during
the carbon burning phase. The same can be said about the minimum
abundance needed to produce off-center explosions: it is extremely
unlikely - and many things would need to change significantly, in
particular the rates of nuclear reactions during carbon burning -
that 24Mg could by itself induce the heating to NSE in a electron
degenerate core at low density.
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Figure 3:
Evolution on the log(
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Recent models (Domínguez et al. 1993; Ritossa et al. 1996;
Gil-Pons & García-Berro 2001, 2002) have convincingly shown
that carbon burning can be incomplete in degenerate cores. The amount
of remaining carbon depends somewhat on the detail of the calculations
but is typically of the order of 1%. Nevertheless,
Domínguez et al. (1993) found a central region of
with a high carbon content, of the order of 25%, that may
eventually lead to an early explosion, and hence to a disruption of
the electron-degenerate core. More specifically, Domínguez et al. (1993) followed the evolution of a
star in a
binary system. The extreme dependence of neutrino cooling on density
was able to cool down the center of the star to temperatures low
enough to preclude the onset of central carbon burning. Although it
has been claimed that this result could be to due the roughness of the
numerical procedure adopted in Domínguez et al. (1993) - in
particular to the poor spatial resolution of their mesh - or to the
very high mass-loss rate adopted, this is a point that undoubtely
deserves more attention, since carbon burning in degenerate conditions
is prone to develop into the explosive regime, even when this element
is found in only relatively small amounts. Woosley (1986) has
estimated that, in the interval T8=2.5-7.5, for each 0.12% of
carbon burned the temperature increases by 108 K.
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Figure 4:
Same as Fig. 3, but using the Schwarzschild convective
criterion. Only the model with
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In this case, given that there is a difference on the atomic mass of
12C and its daughter nuclei (namely, 16O, 20Ne and 24Mg), the Ledoux criterion must include an additional term to
take it into account. The expression employed throughout this work is
then
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(5) |
Table 2: Models with variable 12C abundance. In the table, "L'' stands for the Ledoux criterion, while "S'' stands for the Schwarzschild criterion; "C'' denotes collapse and "E'' disruption. For carbon abundances over 2%, all realistic models imply the complete disruption of the star. The case with a 4% abundance of carbon, and with the convection artificially inhibited (marked by an asterisk) corresponds to an off-center explosion.
In order to determine the minimum amount of remaining carbon needed to completely disrupt the core at low densities, we have parameterized its abundance from 1% to 6%, a broad enough range. We have adopted both the Ledoux and the Schwarzschild criteria. Also, to investigate the worst possible case, we have artificially inhibited the convective motion and energy transport. In this case carbon burning proceeds in a strictly local mode, making the onset of NSE more difficult. This could be similar to what would happen if the thermonuclear timescale were much shorter than the convective turnover and evolutionary timescales.
The first set of calculations is shown in Fig. 3, for which the Ledoux
criterion was used. As it can be seen, the only model for which an
early explosion is avoided is that with carbon abundance
.
In Fig. 4 the evolution in the
plane is shown for the set of models in which the
Swarzschild criterion was adopted. Again, only the models with
do not lead to an early explosion. Finally, in
Fig. 5 the results obtained when convection was artificially inhibited
are shown. In this case, those models in which the carbon abundance is
2% give an explosion at moderate densities. For completeness,
our results are summarized in Table 2, where the maximum temperature
attained during carbon burning (fourth column), the ignition density
(fith column) and the outcome of the evolution (sixth column) are
shown as a function of the carbon abundance and the convective
criterion. Table 2 shows that in the cases where convection is taken
into account, the minimum abundance of 12C should be as high as
1.5%, only a factor of 2 higher than the abundance presently found in
most calculations. Even when the convective processes are
artificially inhibited, the amount of 12C required to achieve NSE
at low densities increases to 4% - and the explosion occurs
slightly off-center. It is clear that the presence of small amounts
of unburned 12C could be a possible channel for the complete
disruption of ONe degenerate objects at very low density unless
non-physically consistent situations are artificially forced. It is
also not surprising that a model adopting the 12C distribution
obtained by Domínguez et al. (1993) results in an explosion at
low density. Nevertheless, Rayleigh-Taylor instabilities would
rapidly mix the 12C-rich matter with the rest of the core. In
this case, the abundance of 12C is already within the range
studied in the present work (about a 4.5%).
Given the low density at which NSE proceeds, it is likely that some of the reactions will be rapidly quenched, thus producing an incomplete burning and a corresponding lower fraction of Fe-peak elements. While this is not compatible with classical type Ia SNe nucleosynthesis, it could be an alternative explanation for subluminous type Ia SNe, like SN1991bg (Ruiz-Lapuente et al. 1993). Clearly, this outcome deserves further consideration, which is beyond the scope of this paper.
Recent evolutionary calculations including detailed nuclear reaction
networks have found that noticeable amounts of 23Na and 25Mg
appear after the carbon burning phase - see, for example, Gil-Pons
& García-Berro (2001). These nuclides have low
electron-capture energy thresholds (Q=4.376 MeV and Q=3.833 MeV,
respectively) and so they will start captures even before 24Mg,
which has a higher energy threshold. The Urca processes for these
nuclides are 23Na(e,Na(e,
Ne and
25Mg(e,
Na(e,
Mg, occurring at densities of
g cm-3 for 23Na and
g cm-3 for 25Mg. The rates for these Urca pairs have
also been taken from Oda et al. (1994). Note that throughout the
calculations presented in this section Coulomb corrections were
applied to the rates, as was done previously. The chemical
composition of the ONe core was
,
,
,
,
and
as
derived from Ritossa et al. (1996). Also note that for simplicity we
have excluded the small abundance of unburnt 12C and renormalized
the resulting abundances.
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Figure 5: Same as Figs. 3 and 4, but with convection artificially inhibited. In this case, even a 2% abundance of 12C fails to trigger a moderate density explosion. |
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As it can be seen in Fig. 6, the main effect of the Urca pairs
consist of small decreases in
and in
.
The
later effect is the only lasting one. These Urca pairs are never in
equilibrium, as the core is slowly contracting and, hence, increasing
the Fermi energy of the electrons. Some 109 s after
formation of the Urca pairs, the rates for electron-capture on the
involved nuclides are much higher than the
-decay
rates. Eventually, all the 23Na becomes 23Ne, and the
25Mg becomes 25Na, with a total reduction on
of
0.00286, which translates into a slightly higher ignition density for
NeO burning.
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Figure 6:
Evolution in the
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When using the Schwarzschild criterion, convection appears during the
electron captures on the nuclides of mass 24 (24Mg and
24Na). Due to the limited extent of the convective region, there
is no convective Urca for 23Ne nor for 25Na, since the upper
layer of the convective region is well above the density for
significant rates of
decay from these nuclides. Thus, the
only durable effect of 23Na and 25Mg on the evolution of our
objects is a small reduction in
(from the original value
of 0.4981 to 0.4952). Hence, due to this slight reduction in
,
the ignition density for NeO burning is
g cm-3,
clearly above the collapse threshold given by Timmes &
Woosley (1992, 1994).
The final fate of pure ONe electron degenerate objects seems at
present well established: they will collapse to a neutron star after
electron captures on 20Ne at a density of about
g cm-3. The main goal of this work has been to elucidate
whether the minor species present in ONe cores - namely, 12C,
23Na, 24Mg and 25Mg - can open a channel to produce
an explosion at moderate densities (about 109 g cm-3) which
would lead to the complete disruption of the star.
We have found that the amount of 24Mg required to produce a low
density explosion is about one order of magnitude above the results of
current, state-of-the-art evolutionary calculations. Hence, it is
very unlikely that electron captures proceeding on 24Mg (and its
daughter nucleus 24Na) can drive the temperature of the star to
the point of igniting the NeO burning at
g cm-3. Even in the (improbable) case that this
actually happened, this density is near the current accepted threshold
for gravitational collapse (Timmes & Woosley 1992,
1994). Consequently, if there is not a very significant change in the
reaction rates leading to the production of 24Mg, this nuclide
can be discarded as a possible trigger of a moderate density
explosion.
On the contrary, even small amounts of unburned 12C are enough to
drive a moderate to low density explosion. The reason is twofold.
First, the energy yield is much higher for carbon burning than for
electron captures on the A=24 nuclei. Secondly, there is much more
12C available, since the ignition is very local and can induce
convective motion that leads to mixing that carries fresh 12C
into the burning region. In this way, even seemingly small abundances
of 12C (about a 1.5%) can produce an explosion at
g cm-3, which would lead to the complete
disruption of the star. This potential channel to complete disruption
would give rise to a thermonuclear supernova. The characteristics of
such a kind of thermonuclear supernova are beyond the scope of this
work, but could explain in part why there seems to be a correlation
between spiral arms and Type Ia supernovae (Bartsunov et al. 1994;
McMillan & Ciardullo 1996). Finally, the effects of
the Urca pairs generated by 23Na and 25Mg are minor. Apart
from a small reduction in the temperature, the only lasting effect is
a small decrease in the electron mole number
.
This will
result in a slightly higher density for the electron captures on the
nuclei of A=24 and A=20.
We note that multidimensional simulations of convection indicate a tendency to increase the amount of convective mixing and to induce rotational mixing to an extent that could slightly modify our conclusions. Many efforts are currently being invested in modeling carbon burning under strongly degenerate conditions using multidimensional codes - see, for instance, Reinecke et al. (2003) and Röpke et al. (2004a,b) for recent developments in this field. A time-dependent numerical procedure for mixing (necessary for coupling core contraction and convection) could slightly change the details of our conclusions, but it is unlikely to cause a qualitative change - that is, transforming a collapse into an explosion or vice versa. Until a complete treatment of multidimensional convection and burning is available, our results can be regarded as a fair approximation to the real evolution of this type of star.
Acknowledgements
Part of this work was supported by the MCYT grant AYA04094-C03-01, by the European Union FEDER funds, and by the CIRIT. We would also like to acknowledge the invaluable advice of J. Isern and J. Labay who largely contributed through his support, suggestions and comments to improve the manuscript.