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Astronomy & Astrophysicsmanuscript no. noel c© ESO 2017
April 18, 2017

A new reduced network to simulate detonations in superbursts
from mixed H/He accretors

C. Noël1, S. Goriely1, Y. Busegnies1, and M.V. Papalexandris2

1 Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles, Campus plaine CP 226, Boulevard du Triomphe, 1050
Bruxelles, Belgium
e-mail:cnoel@ulb.ac.be

2 Département de Mécanique, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium

Preprint online version: April 18, 2017

ABSTRACT

Aims. We construct a new reduced nuclear reaction network able to reproduce the energy production due to the photo-disintegration
of heavy elements such as Ru, which are believed to occur during superbursts in mixed H/He accreting systems. We use this network
to simulate a detonation propagation, inside a mixture of C/Ru.
Methods. As our reference, we use a full nuclear reaction network, including 14758 reactions on 1381 nuclides.Until the reduced and
full networks converge to a good level of accuracy in the energy production rate, we iterate between the hydrodynamical simulation,
with a given reduced network, and the readjustment of a new reduced network, on the basis of previously derived hydrodynamical
profiles.
Results. We obtain the thermodynamic state of the material after the passage of the detonation, and the final products of the combus-
tion. Interestingly, we find that all reaction lengths can beresolved in the same simulation. This will enable C/Ru detonations to be
more easily studied in future multi-dimensional simulations, than pure carbon ones. We underline the dependence of thecombustion
products on the initial mass fraction of Ru. In some cases, a large fraction of heavy nuclei, such as Mo, remains after the passage of
the detonation front. In other cases, the ashes are principally composed of iron group elements.

Key words. Hydrodynamics – Nuclear reactions, nucleosynthesis, abundances – Shock waves – Stars: neutron – X-rays: bursts

1. Introduction

Superbursts have been discovered by long-term monitoring of
the X-ray sky, using instruments such as RXTE and BeppoSAX.
Compared to normal type I X-ray bursts, superbursts are 1000
times more energetics, with integrated burst-energies of approx-
imately 1042 ergs; they last 1000 times longer, from hours to half
a day; and they reoccur every one to a few years. They are rare:
only 13 events have been identified from 8 sources (for reviews
see Kuulkers 2004, Cumming 2005, and references therein).
They exhibit similarities with normal type I X-ray bursts, such
as a rapid rise in the light curve, a quasi-exponential decay, and
a hardening of the spectrum during the rise followed by a soften-
ing during the decay. These attributes are well representedby a
blackbody model, which has an effective temperature that grows
during the rise, and decreases during the decay phase (Kuulkers
2004). This suggests that superbursts, like normal type I bursts,
are thermonuclear in origin (Cornelisse et al. 2000). The current
view is that superbursts are due to thermally unstable ignition
of 12C at densities of about 108 - 109 g cm−3 (Cumming 2001,
Strohmayer & Brown 2002, Cumming 2005).

Most superbursts have been observed in systems that accrete
a mixture of H and He (Kuulkers 2004). In these H/He accretors,
the12C mass fraction that remains after the combustion of H and
He, via the rp-process (Wallace & Woosley 1981), is unknown
(Schatz et al. 2001, 2003b, Woosley et al. 2004). Cumming &
Bildsten (2001) showed that a12C mass fraction of 0.05− 0.1

Send offprint requests to: C. Noël

was sufficient for an unstable ignition; however, fits to the ob-
served lightcurves suggest that the actual abundance is much
larger (Cumming et al. 2006).

Schatz et al. (2003) showed that, at the high temperatures
reached in superbursts (T > 109K), the photodisintegration of
the heavy rp-process ashes influences the energetics of the det-
onation. More details about this interesting mechanism arepro-
vided below.

To calculate energy production, a simplified nuclear reaction
network is, in general, used in astrophysical hydrodynamics sim-
ulations (e.g. Noël et al. 2007). In this study, we extend such a
network to include a contribution by rp-process ashes. However,
full nuclear reaction networks are intractable in hydrodynamical
simulations, where some 107 calls to the nuclear reaction net-
work, are required. This would be too time-consuming, in par-
ticular in multi-dimensional simulations that will be considered
in a future work. For this reason, we constructed a reduced nu-
clear reaction network. This network and the detonation profiles
are described in Sect. 2. Our results are discussed in Sect. 3.

2. Nuclear reaction network extension

In a mixed H/He accreting system, the detonation that leads to
the superburst, is expected to propagate into the ashes of the rp-
process, which occurred in the upper atmosphere. For simplic-
ity, as in Cumming & Bildsten (2001), we represent the ashes

http://arxiv.org/abs/0803.1085v1


2 C. Noël et al.: A new reduced network to simulate detonations in superbursts from mixed H/He accretors

using a fiducial heavy nucleus,96Ru1. We simulate the propaga-
tion of a detonation, in a mixture of12C and96Ru, with mass
fractionsX96Ru = 1 − X12C, using the hydrodynamical algo-
rithm described in Noël et al. (2007). This algorithm solves the
adiabatic Euler’s equations for compressible, non-viscous gas-
dynamics with source terms. It is a finite-volume method in the
spirit of the original MUSCL scheme of van Leer (1979). The
algorithm is of second-order in the smooth part of the flow, and
avoids dimensional splitting. Parallelization, which is required
to solve computationally, the problem in hand, is based on the
mpi library, as described in Deledicque & Papalexandris (2006).
The equation of state accounts for partially degenerate andpar-
tially relativistic electrons and positrons. The ions are treated as
a Maxwell-Boltzmann gas, and the radiation, considered to be
at local thermodynamic equilibrium with the matter, follows the
Planck law. Coulomb corrections to the equation of state arenot
included. In our simulations, these corrections would change the
thermodynamic quantities of our ideal plasma by less than 2−3%
(Fryxell et al. 2000), but will be included in future work.

It remains untractable at the present time to perform the hy-
drodynamical simulation with an extented network, so that are-
duced network needs to be constructed. For this purpose, we first
extend the reduced network of thirteen species from12C to 56Ni
used in Noël et al. (2007), with theα-chain between64Ni and
96Ru. We adopt the nuclear data from the Brussels nuclear reac-
tion rate library (BRUSLIB), based on published compilations
of experimental reaction rates, and on the determination ofreac-
tion rates using the statistical Hauser-Feshbach model (Aikawa
et al. 2005, Arnould & Goriely 2006). The photodisintegration
rates (γ, α) are calculated using the reciprocity theorem (Eq. (4)
of Arnould & Goriely 2006). The resulting network is referred
to as net0.

Table 1. Initial conditions on the left- (column 2) and right-hand
(column 3) sides of the discontinuity.

Left Right
ρ (g cm−3) 3.01× 108 108

T (K) 4.46× 109 108

v (cm s−1) 8.07× 108 0

Using net0, we simulate a detonation, propagating in a mixture
X12C = 0.2, X96Ru = 0.8. The ignition conditions are the same as
used for pure12C in Noël et al. (2007) (see Table 1). The length
of the domain is 1000 cm, the initial discontinuity is placedat
x = 100 cm, where x is the distance from the left boundary of
the domain, and the resolution is 1 cm (1000 numerical cells).
We adopt a mixtureX56Ni = 0.1 andX64Ni = 0.9, at the left
of the discontinuity. The nuclear energy generation, tempera-
ture, density and pressure profiles, at timet = 7 × 10−7 s are
presented in Fig. 1, in terms of the distance to the shock. The
detonation velocity isD = 1.19× 109 cm s−1. Using the thermo-
dynamic profiles presented in Fig. 1, we perform a full network
calculation, including 14758 reactions on 1381 nuclides lying
between the proton and neutron drip lines and with charge num-
bers Z≤50. Rates are taken from experiments, whenever avail-
able, and otherwise from the BRUSLIB library. The derived nu-

1 Choosing only96Ru implies naı̈vely that the electron captures in
the high-density environment have not occured yet and that only 96Ru
is produced during the combustion phases preceding the superburst.
Future detailled hydrodynamical simulation in X-ray burstmay provide
us with better initial conditions for the superburst.

0 0.5 1 1.5 2

0

log(Z)

0 0.5 1 1.5 2

0

log(Z)

0 0.5 1 1.5 2

log(Z)

0 0.5 1 1.5 2

log(Z)

Fig. 1. Nuclear energy generation (erg g−1 s−1), temperature (K),
density (g cm−3) and pressure (erg cm−3) profiles for a detona-
tion front in a mixtureX12C = 0.2, X96Ru = 0.8 atT = 108 K and
ρ = 108 g cm−3. Z is the distance to the shock in cm.

Fig. 2. Nuclear mass fractions of some species calculated with
the full network including 14758 reactions on 1381 nuclides
using the profiles of Fig. 1. The time in s is the time elapsed
since the passage of the shock, assuming a constant velocity
D = 1.19 109 cm s−1.

clear mass fractions, for some species are presented in Fig.2. In
Fig. 3 we compare the energy production rates of the reduced
network net0, and of the full network. In this case the reduced
network has reproduced the total energy production as well as
the energy rate, of the full network. This favorable situation may
not however, be possible, for detonations propagating in a mix-
ture composed of a lower amount of12C.



C. Noël et al.: A new reduced network to simulate detonations in superbursts from mixed H/He accretors 3

Fig. 3. Network calculation of the energy production rate (erg
g−1 s−1) and total energy (erg g−1) for a detonation in a mixture
X12C = 0.2, X96Ru = 0.8 using the profiles of fig. 1 for two nu-
clear reaction network. Solid lines: full network including 14758
reactions on 1381 nuclides; dotted lines: reduced network net0.
The time in s is measured since the passage of the shock, assum-
ing a constant velocityD = 1.19 109 cm s−1.

0 0.5 1 1.5 2 2.5

0

log(Z)

0 0.5 1 1.5 2 2.5
0

log(Z)

0 0.5 1 1.5 2 2.5

log(Z)

0 0.5 1 1.5 2 2.5

log(Z)

Fig. 4. Same as Fig. 1 but for a mixtureX12C = 0.1, X96Ru = 0.9.

In a second step, we simulate a detonation propagating in a mix-
tureX12C = 0.1, X96Ru = 0.9. The ignition conditions are similar
to those used above. The nuclear energy generation, tempera-
ture, density, and pressure profiles, at timet = 7 × 10−7 s are
presented in Fig. 4 in terms of the distance to the shock. The
detonation velocity isD = 1.16× 109 cm s−1. Using the thermo-
dynamic profiles of Fig. 4, a full-reaction network calculation is
performed. The derived nuclear mass fractions, for some species,
are presented in Fig. 5.

We compare the energy production rates, of the reduced net-
work net0, and of the full network (Fig. 6). We see that this case

Fig. 5. Same as Fig. 2 but for a mixtureX12C = 0.1, X96Ru = 0.9,
using the profiles of Fig. 4 and assuming a constant velocityD =
1.16 109 cm s−1.

is characterized by an endothermic phase, due to the enhanced
initial abundance of96Ru, and the significant contribution of the
photodesintegration reactions, to the total energy balance, at an
early stage. This phase of the energy generation profile obtained
using the full network, is not reproduced by the reduced network
net0. We therefore construct a more sophisticated reduced net-
work, called net1. Let us describe the methodology used in its
construction. In the reduced networks, only the capture or emis-
sion ofα-particles are taken into account. However, neutrons and
protons can be produced significantly by photo-reactions, and be
recaptured. In particular, the nucleus (Z-2,N-2), where Z is the
charge number, and N the number of neutrons, can be produced
by the photodesintegration of the nucleus (Z,N), followingsix
additional paths, complementing the (γ, α) one; these include
the emission of two neutrons and two protons. The characteristic
timescale, for each of the seven possible paths is given by

λ−1
0 = λ−1

(γ,α),

λ−1
1 = λ−1

(γ,n)(Z,N) + λ−1
(γ,n)(Z,N − 1)+ λ−1

(γ,p)(Z,N − 2)

+λ−1
(γ,p)(Z − 1,N − 2),

λ−1
2 = λ−1

(γ,n)(Z,N) + λ−1
(γ,p)(Z,N − 1)+ λ−1

(γ,n)(Z − 1,N − 1)

+λ−1
(γ,p)(Z − 1,N − 2),

λ−1
3 = λ−1

(γ,n)(Z,N) + λ−1
(γ,p)(Z,N − 1)+ λ−1

(γ,p)(Z − 1,N − 1)

+λ−1
(γ,n)(Z − 2,N − 1),

λ−1
4 = λ−1

(γ,p)(Z,N) + λ−1
(γ,n)(Z − 1,N) + λ−1

(γ,n)(Z − 1,N − 1)

+λ−1
(γ,p)(Z − 1,N − 2),

λ−1
5 = λ−1

(γ,p)(Z,N) + λ−1
(γ,n)(Z − 1,N) + λ−1

(γ,p)(Z − 1,N − 1)

+λ−1
(γ,n)(Z − 2,N − 1),

λ−1
6 = λ−1

(γ,p)(Z,N) + λ−1
(γ,p)(Z,N − 1)+ λ−1

(γ,n)(Z − 2,N)

+λ−1
(γ,n)(Z − 2,N − 1), (1)

so that the total photo-reaction rate of (Z,N), leading to (Z-2,N-
2), can be approximated by the effective rate

λe f f = λ0 + λ1 + λ2 + λ3 + λ4 + λ5 + λ6. (2)



4 C. Noël et al.: A new reduced network to simulate detonations in superbursts from mixed H/He accretors

The reverseα-capture rate can then be calculated using the de-
tailed balance equation (Eq. (4) of Arnould & Goriely 2006).
While λ0, describingα-photoemission, is the only path included
in the net0 approximation, the neutron and/or proton photoe-
mission could potentially dominate in the case of light (Z<40)
species, depending on the nuclei involved. In the neutron-
deficient region (especially along the neutron-shell closures), the
proton-photoemission is dominant, while close to the stability
line the neutron-photoemission usually dominates. Any of these
seven paths, however, can be impeded to a more or less large
extent by the reverse nucleon captures that would modify the
relative significance of a given path. This modification depends
on the thermodynamic and composition conditions, as well as
on the nuclear cross sections. To determine which paths are im-
portant, we use the following method. We first calculate the det-
onation profiles using net0. On the basis of the temperature and
density profiles (for instance those of Fig. 4 forX12C = 0.1,
X96Ru = 0.9), the full reaction network is solved. This calcula-
tion allows to determine the dominant paths and to constructa
reduced network with new effective reaction rates but the same
reduced number of nuclei which is introduced in the hydrody-
namical algorithm. We perform the detonation simulation using
this new reduced network, and the resulting thermodynamical
profiles are used again in a full network calculation. We iterate
between a hydrodynamical simulation with a given reduced net-
work, and the readjustment of a new reduced network, on the
basis of the previously-derived hydrodynamical profiles, until
the reduced and full networks converge to achieve high accu-
racy in the energy production rate. Note that no strict mathe-
matical criteria have been used to estimate the quality of the
newly-developed reaction network. In contrast, we estimate that
the new network must fulfill two major conditions: it must re-
produce the right order of magnitude (roughly within a factor
of two) of the total amount of energy produced, and the possi-
ble appearance of endothermic phases. Indeed any endothermic
phase can play a role of first importance on the detonation prop-
erties. In particular pathological detonations may occur (Fickett
& Davis 1979, Khoklov 1989, Sharpe 1999).

In contrast to the reduced network used in Noël et al. (2007),
which provided an energy source typical of explosive heliumand
carbon burning in the absence of hydrogen (Gamezo et al. 1999),
our reduced network depends on the detonation profiles, and ini-
tial composition. For each new detonation simulation, we con-
struct the adapted reduced network using the iteration procedure
described above.

Convergence is achieved in the case ofX12C = 0.1, X96Ru =
0.9, with the following reduced network, called net1, in which
the equivalent reaction rate, adopted for each photodisintegration
from 92Mo to 64Ni, and from56Ni to 16O is

λnet1 = λ0 + λ1 + λ6, (3)

and for 96Ru, λnet1 = λ0. The inverseα-capture rates are
calculated using Eq. (4) of Arnould & Goriely (2006), and
the five remaining rates for12C(12C,α)20Ne, 12C(16O,α)24Mg,
16O(16O,α)28Si, 4He(2α, γ)12C, and12C(γ, 2α)4He, are similar
to those used in net0.

Figure 6 compares the energy production calculated using
net0, net1, and the full network. Using the profiles obtainedwith
net0 (Fig. 4), we perform a full network calculation, and com-
pute the energy production rate. The same profiles are used for
the reduced network net0, and the energy production rates are
compared. We see that the endothermic part of the profile is not
reproduced by the reduced network net0. A revised reduced net-
work is therefore developed, net1, which simulates more accu-

Fig. 6. Same as Fig. 3 but for a mixtureX12C = 0.1, X96Ru = 0.9,
assuming a constant velocityD = 1.16 109 cm s−1. and for
three nuclear reaction network. Solid lines: full network includ-
ing 14758 reactions on 1381 nuclides; dotted lines: reducednet-
work net0; dashed lines: reduced network net1.

0 0.5 1 1.5 2 2.5

0

log(Z)

0 0.5 1 1.5 2 2.5

0

log(Z)

0 0.5 1 1.5 2 2.5

log(Z)

0 0.5 1 1.5 2 2.5

log(Z)

Fig. 7. Same as Fig. 4, but with net1.

rately, the full network energy production rate. However, new
detonation profiles, using the hydrodynamical algorithm have to
be calculated with this new network. They are shown in Fig. 7,
at timet f = 1.6× 10−6s. The domain length is of 2000 cm, and
the resolution is of 1 cm. The initial discontinuity is positioned
at x= 200 cm. The detonation velocity isD = 1.09×109 cm s−1.

Using these new thermodynamic profiles (Fig. 7), network
calculations of the energy production are made for the full net-
work, and the reduced network net1, which are compared in Fig.
8. One sees that the reduced network reproduces satisfactorily
the energy production given by the full network. We can there-



C. Noël et al.: A new reduced network to simulate detonations in superbursts from mixed H/He accretors 5

Fig. 8. Same as Fig. 6, but with the profiles of Fig. 7 andD =
1.09 109 cm s−1.

Fig. 9. Same as Fig. 5 but with the profiles of Fig. 7 andD =
1.09 109 cm s−1.

fore consider net1 as a suitable reduced network for the simula-
tion of a detonation for a mixtureX12C = 0.1, X96Ru = 0.9, and
the initial conditions shown in Table 1. Adopting the thermody-
namic profiles shown in Fig. 7, a final calculation is performed
using the full network. The derived nuclear mass fractions for
some species, are presented in Fig. 9. A comparaison between
Fig. 5 and Fig. 9, reveals the sensitivity of the final composi-
tion to the choice of nuclear reaction network. For instance, net0
produces much less90Zr and92Mo than net1, at the end of the
reaction zone.

Far more than two reduced networks were tested in the iter-
ation procedure. Figure 10 displays the total energy obtained,
using five different reduced networks, for the thermodynamic
profiles of Fig. 4. It is clear that net1 reproduces most closely
the global energy obtained by the full network simulation.

Note that net1 includes photoneutron and photoproton paths
that are found to reproduce globally all possible paths, when in-
cluding neutrons and protons in the network. Both paths (path
1 and path 6 in Eq. 1) are required to simulate the bottle-
necks which are related to the neutron (N=20, 28, 50), or pro-
ton (Z=20, 28) magic numbers, which are crossed by the pho-

Fig. 10. Same as Fig. 6 but for five different networks and only
the total energy produced is presented. The network net2 is iden-
tical to net0 for all species between4He and56Ni, and usesλe f f

for the rates of all reactions between64Ni and 96Ru. The net-
work net3 usesλe f f for the rates of all photo-reactions between
16O and96Ru, and the net0 rates for the other reactions.

tonuclear flow. All nuclei included in the reduced network are
even-even, such that if a first (γ, n) reaction is possible, the re-
sultant (Z,N-1) nucleus will certainly photo-emit a neutron. The
same holds for proton emission. The equivalent rateλnet1 in Eq.
3, simulates the possible inclusion of neutrons and protonsin the
network. The inverse effectiveα-captures rate, that we deduced
from detailed balance expressions, similarly enable to simulate
the possible neutron and proton captures, without having toin-
clude neutrons and protons explicitely in the network. Finally
note that for96Ru, we do not take the (γ,n) and (γ,p) contribu-
tions from Eq. 3, into account to counterbalance the neutronand
proton captures by96Ru, at early time when the matter is almost
entirely made of96Ru.

We emphasize that the reduced network considered for the
detonation of the mixture ofX12C = 0.1, X96Ru = 0.9 cannot
be generalized automatically to cases characterized by differ-
ent initial conditions. The construction of the reduced network
does not only depend on the temperature and density time evolu-
tion, but also on the specific initial composition. Different initial
mass fractions of the existing species, or a different elemental or
isotopic composition, would require the construction of a new
network, with new optimum effective reaction rates. As an ex-
ample, we present a second case study, in whichX12C = 0.05,
X96Ru = 0.95, as considered by Cumming & Bildsten (2001).
The detonation profiles, in this case, are shown in Fig. 11, using
net1. The domain length is 1000 cm, the resolution is 1 cm, the
initial discontinuity is placed at x= 100 cm, and the detonation
velocity is D = 9.88× 108 cm s−1. We see in Fig. 12 that net1
does not reproduce the endothermic phase. A more suited net-
work needs to be defined, for example, by considering a linear
superposition of photorates, as in Eq. 2, that is different for each
nucleus included in the network. Wheter such a scheme should
be developed depends on the outcome of realistic modelling,of
the initial composition of superbursts, in heavy elements.



6 C. Noël et al.: A new reduced network to simulate detonations in superbursts from mixed H/He accretors

0 0.5 1 1.5 2 2.5

0

log(Z)

0 0.5 1 1.5 2 2.5

0

log(Z)

0 0.5 1 1.5 2 2.5

log(Z)

0 0.5 1 1.5 2 2.5

log(Z)

Fig. 11. Same as Fig. 4 but withX12C = 0.05 andX96Ru = 0.95.

Fig. 12. Same as Fig. 6 but with the profiles of Fig. 11,D =
9.88 108 cm s−1 andX12C = 0.05,X96Ru = 0.95.

3. Conclusion

We have shown that a simple extension of anα-chain, might
not be sufficiently accurate to reproduce the energy production
rate of a detonation in a mixture that contains too much heavy
elements. We have developed a new methodology to construct
a reduced network, with adapted effective reaction rates, based
on an iteration procedure between hydrodynamical simulations,
with a chosen reduced network, and a comparison with full net-
work calculations. In particular, we have constructed a newre-
duced network that reproduces satisfactorily, the energy gener-
ation rate of a reference network, including all reactions on all
nuclides, probably involved in the specific case of the propaga-
tion of a detonation atT = 108K, and ρ = 108 g cm−3, in a
mixtureX12C = 0.1, X96Ru = 0.9. Adopting such a reduced net-

work, we have simulated detonation profiles. These profiles are
used in a full network simulation to compute the associated nu-
cleosynthesis. One sees that the presence of96Ru decreases the
total energy of the detonation, due to an endothermic phase at
early stages. For this reason, the propagation velocity of the det-
onation decreases. ForX12C = 0.1, X96Ru = 0.9, essentially only
iron-group elements remain at the end of the reaction zone, with
some addition of N=50 elements, such as90Zr. This fact can in-
fluence the neutron star crust composition and properties, such
as its conductivity, or its neutrino emissivity, as well as the igni-
tion conditions for the superbursts (Weinberg & Bildsten 2007,
Gupta et al. 2007, Haensel & Zdunik 1990).

Interestingly, it can be seen that forX96Ru > 0.9, the deto-
nation goes through a short endothermic phase (see Figs. 7-12)
that cannot be described by net0 (Fig. 4). The photodesintegra-
tions byα emission are endothermic reactions, and only the re-
combination ofα particles up to Ni compensates for the energy
loss. The energy is therefore released by the captures or recom-
binations of the protons, neutrons andα-particles, liberated by
the photodesintegrations. Furthermore a comparaison between
Fig. 4 and Fig. 7 reveals that the temperature reached duringthe
detonation is lower using net1, than net0, which influences the
nucleosynthesis.

Computationally mixed C/Ru detonations are relatively easy
to handle since the length-scales on which the different elements
burn are comparable. This contrasts with the case of pure car-
bon detonations where a large variety of length-scales is ob-
tained (Noël et al. 2007), and where multiple simulations at dif-
ferent resolutions are required to fully resolve the detonation.
In the C/Ru case, carbon and ruthenium burn on≈ 10 cm, and
the total reaction length is≈ 1000 cm. The length-scales span
only three orders of magnitude compared to six for pure car-
bon detonations. This is an encouraging feature for future multi-
dimensional calculations.

Acknowledgements. The numerical simulations presented herein were per-
formed on the parallel computers of the Intensive ComputingStorage of UCL
and on HYDRA, the Scientific Computer Configuration at the VUB/ULB
Computing Centre. We are grateful to M. Arnould for interesting discussions.

References

Aikawa, M., Arnould, M., Goriely, S. et al. 2005, A&A, 441, 1195
Arnould, M. & Goriely, S. 2006, Nuclear Physics, A, 777, 157
Brown, E., F. & Bildsten, L. 1998, ApJ, 496, 915
Cornelisse, R., Heise, J. & Kuulkers, E. 2000, A&A, 357, L21
Cumming, A. & Bildsten, L. 2001, ApJ, 559, L127
Cumming, A. 2005, Nuclear Physics A, 758, 439
Cumming, A, Macbeth, J., Zand, J.J.M.i. & Page, D. 2006, ApJ,646, 429
Deledicque, V. & Papalexandris, M.V. 2006, Combustion & Flame, 144, 821
Fickett, W. & Davis, W. C. 1979, Detonation, (Univ. California Press, Berkeley)
Fryxell, B.A., Olson, K., Ricker, P. et al. 2000, ApJS,131, 273
Gamezo, V.N. & Wheeler, J.C. & Khokhlov, A.M. & Oran, E.S. 1999, ApJ, 512,

827
Gupta, S. Brown, E. F., Schatz, H. et al. 2007, ApJ, 662, 1188
Haensel, P. & Zdunik, J. L. 1990, A&A, 229, 117
Khokhlov, A. M. 1989, MNRAS, 239, 785
Kuulkers. E. 2004, Nuclear Physics B (Proc.Suppl.), 132, 466
Noël, C. & Busegnies, Y. &. Papalexandris, M. V. et al. 2007,A&A, 470, 653
Schatz, H., Aprahamian, A., Barnard, V. et al. 2001, Phys. Rev. Lett., 86, 3471
Schatz, H., Bildsten, L. & Cumming, A. 2003, ApJ, 583, L87
Schatz, H., Bildsten, L., Cumming, A. & Ouellette, M. 2003b,Nuclear Physics

A, 718, 247
Sharpe, G.J. 1999, MNRAS, 310, 1039
Strohmayer, T. E. & Brown, E. F. 2002, ApJ, 566, 1045
van Leer, B. 1979, J. Comp. Phys., 21, 101
Wallace, R. K. & Woosley, S. E. 1981, ApJS, 45, 389
Weinberg, N. N. & Bildsten, L. 2007, ArXiv e-prints, 0706.3062
Woosley, S.E., Heger, A., Cumming, A. et al. 2004, ApJS, 151,75


	Introduction
	Nuclear reaction network extension
	Conclusion