Pulsating reverse detonation models of Type Ia supernovae. I: Detonation ignition

1 MB·

by  Eduardo Bravo

Pulsating reverse detonation models of Type Ia supernovae. I:
Detonation ignition
Eduardo Bravo1,2, Domingo Garc´ıa-Senz1,2
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ABSTRACT
a
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ObservationalevidencespointtoacommonexplosionmechanismofTypeIasupernovaebased

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on a delayed detonation of a white dwarf. Although several scenarios have been proposed and
2
explored by means of one, two, and three-dimensional simulations, the key point still is the

] understandingoftheconditionsunderwhichastabledetonationcanforminadestabilizedwhite
R dwarf. One of the possibilities that have been invoked is that an inefficient deflagration leads to
S the pulsationof a Chandrasekhar-masswhite dwarf, followedby formationofan accretionshock
h. around a carbon-oxygenrich core. The accretion shock confines the core and transforms kinetic
p energyfromthecollapsinghalointothermalenergyofthecore,untilaninwardmovingdetonation
- is formed. This chain of events has been termed Pulsating Reverse Detonation (PRD). In this
o
workweexploretherobustnessofthedetonationignitionfordifferentPRDmodelscharacterized
r
t by the amount of mass burned during the deflagration phase, M . The evolution of the white
s defl
a dwarf up to the formation of the accretion shock has been followed with a three-dimensional
[
hydrodynamicalcodewithnuclearreactionsturnedoff. Wefoundthatdetonationconditionsare

1 achievedforawiderangeofMdefl. However,ifthenuclearenergyreleasedduringthedeflagration
v phase is close to the white dwarf binding energy ( 0.46 1051 erg Mdefl 0.30 M⊙) the
8 accretion shock cannot heat and confine efficiently∼the cor×e and deton⇒ation co∼nditions are not
0
robustly achieved.
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3 Subjectheadings: Supernovae: general–hydrodynamics–nuclearreactions,nucleosynthesis,abundances
.
1
0 1. Introduction mination as required, for example, to determine
9 the equation of state of the dark energy com-
0 Type Ia supernovae (SNIa) are the most en-
ponent of our Universe, it is necessary to un-
: ergetic transient phenomena in the Universe dis-
v derstand the physics of SNIa explosions. From
i playing most of its energy in the optical band the theoretical point of view, the accepted model
X
of the electromagnetic spectrum, where they ri-
of SNIa consists of a carbon-oxygen white dwarf
r val in brightness with their host galaxies during
a (WD) nearthe Chandrasekharmass thataccretes
several weeks. The importance of SNIa in as-
matter from a companion in a close binary sys-
trophysics and cosmology is highlighted by their
tem. This model accounts for the SNIa sam-
use as standard (or, better, calibrable) candles to
ple homogeneity, the lack of prominent hydrogen
measurecosmicdistancesandrelatedcosmological
lines in their spectra, and its detection in ellip-
parameters(Perlmutter et al.1998;Schmidt et al.
tical galaxies. Massive WDs are extremely un-
1998; Riess et al. 2001). However, in order to
stable bodies in which a modest release of en-
achieve a high precision in the distance deter-
ergy can produce a huge expansion (i.e. their
ratio of binding to gravitational energy is only
1Dept. F´ısica i Enginyeria Nuclear, Univ. Polit`ecnica 15%,whileinanormalstaritis 50%). There
de Catalunya, Diagonal 647, 08028 Barcelona, Spain; ed- ∼ ∼
are two main ingredients of the standard model
[email protected], [email protected]
2Institut d’Estudis Espacials de Catalunya, Barcelona, that are still poorly known: the precise configura-
Spain tion of the stellar binary system and its evolution
1
priortothermalrunawayoftheWD(Langer et al. detonation might take place after a long-range
2000; Nomoto et al. 2003; Piersanti et al. 2003; pulsation of the white dwarf, in what is known
Han & Podsiadlowski 2004; Badenes et al. 2007; as the Pulsating Delayed Detonation (PDD)
Hachisu et al.2008),andtheexplosionmechanism scenario. Khokhlov (1991) carefully computed
(Hillebrandt & Niemeyer 2000). Fortunately, as one-dimensional delayed detonation models that
long as a carbon-oxygen WD reaches the Chan- were successfully compared with observations of
drashekar mass, its previous evolution is not crit- SNIa explosions (Ho¨flich et al. 1995; Howell et al.
ical for the explosion because the WD structure 2001; Quimby et al. 2007; Gerardy et al. 2007;
is determined by the state of a degenerate gas of Fesen et al.2007)andtheirremnants(Badenes et al.
fermions that can be described with a single pa- 2006, 2008; Rest et al. 2008). In these one-
rameter, the mass of the star. This fact leaves dimensional models the detonation initiation had
the explosionmechanismas the mostrelevantun- to be postulated, as there was not identified any
known concerning SNIa. sound physical mechanism by which such a tran-
In spite of continued theoretical efforts dedi- sition could happen in an unconfined white dwarf
cated to understand the mechanism behind SNIa, (Niemeyer 1999).
realistic simulations are still unable to provide a Ignition of a detonation in a white dwarf
satisfactorydescriptionofthedetailsofthesether- can happen in two basic ways, either through
monuclear explosions. Nowadays, there is a con- a deflagration-to-detonation transition (DDT) or
sensus that the initial phases of the explosion in- because of a sudden energy release in a confined
volve a subsonic thermonuclear flame (deflagra- fluid volume (hereafter: confined detonation ig-
tion), whose propagation competes with the ex- nition or CDI). The essential feature of detona-
pansion of the WD. After a while the corrugation tion initiation is the formationof a non-uniformly
of the flame front induced by hydrodynamic in- preheated region with a level of fluctuations of
stabilities culminates in an acceleration of the ef- temperature, density, and chemical composition
fective combustion rate. However, the huge dif- such that a sufficiently large mass burns be-
ference of lengthscales between the white dwarf fore a sonic wave can cross it (Khokhlov 1991;
( 108cm)andtheflamewidth(.1cm)prevents Niemeyer & Woosley1997). Thethermalgradient
∼
that large-scale numerical simulations resolve the needed is (Blinnikov & Khokhlov 1987; Woosley
deflagration front, making it necessary to imple- 1990; Khokhlov 1991):
ment a subgrid model of the subsonic flame. The
precise value of the effective combustion rate is ΘT
T < (1)
currentlyunderdebate,asitdependsondetailsof ∇ Av τ
sound i
the flame model. Recent three-dimensional calcu- whereAisanumericalcoefficient, τ =T/T˙ isthe
i
lations by different groups have shown that pure
induction timescale at temperature T, and Θ
deflagration models always give final kinetic en- ∼
0.04 0.05 is the Frank-Kameneetskiifactor:
ergies that fall short of 1051 ergs, while leaving −
too much unburntcarbonandoxygenclose to the ∂lnT
center (Reinecke et al. 2002; Gamezo et al. 2003; Θ= (2)
−∂lnτ
i
Garc´ıa-Senz & Bravo 2005; Ro¨pke et al. 2007a).
Such fluctuations could be produced by a vari-
Because both signatures are at odds with obser-
ety of mechanisms: adiabatic pre-compression in
vational constraints, Gamezo et al. (2003, 2004,
frontofadeflagrationwave,shockheating,mixing
2005) concluded that the only way to reconcile
ofhotasheswithfreshfuel(Khokhlov1991),accu-
three-dimensionalsimulationswithobservationsis
mulation of pressure waves due to a topologically
to assume that a detonation ignites after a few
complex geometrical structure of the flame front
tenths of a solar mass have been incinerated sub-
(Woosley & Weaver 1994), or transition to the
sonically (delayed detonation).
distributed burning regime (Niemeyer & Woosley
The idea that a delayed detonation is some-
1997). Among the proposedmechanisms of DDT,
how involved in thermonuclear explosions lead-
turbulence pre-conditioninghas receivedthe most
ing to SNIa has been around for many years.
attention. Khokhlov et al. (1997) determined cri-
Ivanova et al. (1974) proposed that a delayed
teria for a DDT in unconfined conditions, such
2
as those realized during the expansion of a white PRD scenario, as a function of the mass burned
dwarf following a deflagration phase. For a DDT during the deflagration phase prior to pulsation,
to be feasible, the turbulent velocity has to ex- M . The evolutionof the white dwarf up to the
defl
ceed the laminar flame velocity by a factor . 8 formationoftheaccretionshockhasbeenfollowed
at a lengthscale comparable to the detonation withathree-dimensionalhydrodynamiccodewith
wave thickness. This criteria was fulfilled for nuclear reactions turned off in order to determine
flame densities in the range 5 106 g/cm3 < themostextremeconditionsachievedbythewhite
×
ρ < 2 5 107 g/cm3 for reasonable assump- dwarf during the re-collapsing phase of the pulsa-
− ×
tions. At densities in excess of 108 g/cm3 a DDT tion. In the next section we summarize the main
is quite unlikely (Khokhlov et al. 1997, but see features of the proposed scenario, and compare
Zingale & Dursiwho pointed to bubbles fragmen- PRD models with one-dimensional PDD models.
tation as a way to increase the flame surface and After that, we analyze if detonation conditions
facilitate a DDT at ρ 2 108 g/cm3). areachievedinourPRDmodels. We then discuss
∼ ×
Small-scale simulations are needed to ascertain the implications that the presence of a small cap
if the necessary conditions for a DDT are ac- of helium on the white dwarf at the moment of
tually achieved during a white dwarf explosion thermal runaway might have for CDI. Finally, we
driven by a deflagration wave. Up to the present, presenttheconclusionsofthiswork. Acompanion
such studies seem to disfavor a DDT in view of paper (Bravo et. al., 2009; hereafter paper II) is
the robustness of subsonic flames against inter- devoted to present the detailed evolution of PRD
action with vortical flow (Ro¨pke et al. 2004) for models after detonation ignition and their final
the maximum expected velocities at the integral outcomes.
scale, 100 km/s (Lisewski et al. 2000). More-
over, N∼iemeyer (1999) estimated that the size of 2. Briefing of the pulsating reverse deto-
the fluctuations that canbe expected fromturbu- nation model
lenceatthe criticaldensitiesis 3ordersofmag-
∼ In the PRD scenario the explosion proceeds in
nitudesmallerthanrequiredforaDDTtohappen.
three steps (Bravo & Garc´ıa-Senz 2006): 1) an
Three-dimensional numerical simulations of
initial pre-conditioning phase of the WD whose
white dwarfs undergoing a slow deflagration have
result is an expanded structure with a positive
provided new ways to achieve a CDI. Plewa et al.
gradient of the mean chemical weight, 2) forma-
(2004) followed the evolution of a white dwarf as-
tion of an accretion shock that confines the fuel
suming ignition in a single slightly off-centered
volume and, 3) launch of a reverse, converging,
bubble, and found that a CDI might result from
detonation wave. We have probed the PRD con-
the convergent flow at the antipodes of the ini-
cept by performing three-dimensional simulations
tial ignition point. In their model, the det-
of the explosion. The initial model consists of a
onation was possible because of the confining
1.38 M⊙ WD made of carbon and oxygen. The
gravitational field, hence it was termed Gravi-
hydrodynamic evolution starts with the ignition
tationally Confined Detonation (GCD, see also
of a few sparks and burns . 0.3 M⊙, mostly to
Ro¨pke et al. 2007b; Plewa 2007; Townsley et al.
Fe-groupelements, during the first second, releas-
2007; Jordan et al. 2008). Bravo & Garc´ıa-Senz ing<5 1050 ergsofnuclearenergy. Thereleased
(2005) described another CDI scenario in which a ×
energy resides for the most part in the outermost
detonation was triggered by inertial confinement
layers leading to WD pulsation.
due to an accretion shock born around a carbon-
The details of the first phase, that spans
oxygenwhitedwarfcoreafterpulsationcausedby
the first second after thermal runaway, have
burning of 0.1 0.3 M⊙: the PulsatingReverse
∼ − been known for some time (Plewa et al. 2004;
Detonation (PRD)1. In this work, we explore the
Garc´ıa-Senz & Bravo 2005; Livne et al. 2005) so
robustnessofdetonationignitionconditionsinthe
we will only briefly sketch here the evolution dur-
ingthisphase. Eventhoughthepreciseconfigura-
1A conceptually similar model was proposed
tion at thermal runaway is difficult to determine
by Dunina-Barkovskayaetal. (2001). See also
Dunina-Barkovskaya&Imshennik(2003) (Ho¨flich et al.2003),currentworkssuggestamul-
3
tipoint ignition in which the first sparks are lo- the output obtained at the end of the deflagra-
cated slightly off-center (Garcia-Senz & Woosley tion phase computed as in Garc´ıa-Senz & Bravo
1995; Woosley et al. 2004; Wunsch & Woosley (2005). Our working hypotheses are that dur-
2004). If the number of sparks is too small or ing the deflagration phase a mass in the range
the burning rate is too slow the nuclear energy Mdefl . 0.30 M⊙ is burnt and that the final out-
released is not enough to unbind the star and come of the explosion is determined by the value
the explosion fails (Ro¨pke et al. 2007b). This is of M . We disregard any effect due to differ-
defl
known as the ”bubble catastrophe” (Livne et al. ent possible deflagrationhistories that might lead
2005). The bubbles float to the surface before to a given M with different chemical composi-
defl
the combustion wave can propagate substantially tions (e.g. different proportions of Fe-group and
(Plewa et al. 2004; Garc´ıa-Senz & Bravo 2005) intermediate-mass nuclei) and chemical profiles.
and the star remains energetically bound. This Hereafter, the present models will be designated
behavior produces a composition inversion, i.e. withtheacronymDFnn,where’nn’standsforthe
the internal volume is plenty of fuel, cold carbon hundredths of solar mass burned during the de-
and oxygen, while the ashes of the initial com- flagration phase, for instance model DF11 means
bustion, mostly hot iron and nickel, are scattered that Mdefl =0.11 M⊙.
around. InFig.1thereisshowntheevolutionofmodels
The second phase of the explosion starts when DF11, DF18 and DF29 (characterized by M =
defl
the deflagration quenches due to expansion and 0.11 M⊙, Mdefl =0.18 M⊙, and Mdefl =0.29 M⊙,
ends when an accretion shock is formed (several respectively)throughthedeflagrationandthepul-
seconds after initial thermal runaway) by the im- sating phases. These models started from differ-
pact of the in-falling material onto the carbon- ent number of igniting bubbles and used differ-
oxygen core. An important feature is that the entflamevelocitieswith the resultthata rangeof
decompression and further expansion of the cold Mdefl = 0.11 0.26 M⊙ was obtained. The pro-
−
core in his way to regain hydrostatic equilibrium cess of generating the initial model was the same
imparts mechanical work to the iron-rich atmo- as described in Garc´ıa-Senz & Bravo (2005), the
sphere. The core-atmosphereenergy transfer con- location of the center of mass of the bubbles and
tributes to the strengthofthe shock formedwhen their incinerated mass are given in Table 1. Mod-
the atmosphere re-collapses. The evolution from elsDF11andDF18 usedonlythe firstsixbubbles
the time of maximum expansion up to the mo- in the Table, while model DF29 started from the
ment of formation of the accretion shock can be seven bubbles listed. The flame velocities were:
comparedtosimulationsoftheself-gravitatingcol- v =100,150,and200kms−1 formodelsDF11,
defl
lapse of a cold gas sphere, a well-known hydrody- DF18, and DF29 respectively.
namical test that can be found, for instance in AscanbeseeninFig.1,mostofthethermonu-
Evrard (1988). The structures seen in the third clearburningtookplaceduringthefirstsecondfor
column of their Fig. 5 qualitatively agree with all models. Two factors determined the quench-
those appearing in Fig. 1 of Bravo & Garc´ıa-Senz ing of the burning (in this context by quenching
(2006) in spite of the different initial conditions. we mean that the nuclear timescale became much
larger than the dynamical timescale): global den-
2.1. Numerical simulations of the pulsat-
sity decrease due to white dwarf expansion, and
ing phase
bubblesmigrationtotheexternallayers. Thesim-
We have simulated the evolution of the white ulations described hereafter belong to the evolu-
dwarf during the second phase of the PRD explo- tion following maximum expansion of the white
dwarf. From that time on the evolution was fol-
sion mechanismwith a Smoothed Particle Hydro-
dynamics (SPH) code whose main features have lowed with the nuclear reactions switched off in
been described elsewhere (Garc´ıa-Senz & Bravo ordertoexposethefueltoextremeconditionsdur-
ing the collapse and check if a stable detonation
2005). The number ofparticles used in the exper-
iments was N = 250000, which provided a good could be achieved. Note that this is nothing but
a convenient approximation to the real situation
spatial resolution, as detailed later. The initial
modelforthese numericalexperiments isgivenby in which some preliminary burning is expected
4
before maximum compression, thereby modifying the range X 0.4 0.8.
fuel
∼ −
the maximum density and temperature achieved
duringpulsation,asshowninpaperII.The three- 2.2. Comparison between PRD and PDD
dimensional evolution of the bubbles during the models
deflagration phase of model DF18 is shown in
Inorderto fully understandthe implicationsof
Fig.2. Thefirstnoticingfeatureisthattheevolu-
thethree-dimensionalcalculationsofthepulsating
tion of the bubbles is similar to each other: all of
phaseofthesupernovaitisconvenienttocompare
themgrowinmassthroughburningandrisetothe
the results of the present PRD simulations with
surface of the white dwarf at the same time that
thoseobtainedwithaone-dimensionalcode(PDD
theirshapebecomestoroidalduetoshearforceson
models). The key difference between the simula-
theirtop(Zingale et al.2005;Ro¨pke et al.2007b).
tions performed in one and three dimensions is
At t = 0.66 s fuel takes up the whole central vol-
the impossibility of the former to change the se-
ume of the white dwarf. At t = 1.05 s the flame
quence of the mass elements. This has two im-
has virtually quenched and the bubbles have ex-
portant consequences. First, in one-dimensional
panded laterally to completely surround the core
calculations the material burned close to the cen-
of the star. Afterwards, the ashes hide the evolu-
ter never catches the fuel located ahead of it.
tion of the inner fuel during the pulsation.
Second, in one-dimensional calculations the un-
The evolution of model DF18 during the first
burned mass always lies above the flame blocking
pulsation, after deflagration quenching, is shown
the expansion of the underlying layers and facil-
in Figs. 3 and 4, and the structure of the white
itating a more efficient burning. In contrast, in
dwarf at the time of accretion shock formation,
three-dimensional models the floatability of ignit-
tacc, is shown in Fig. 5. The inner 1.10 M⊙ ing bubbles allowsthemto reachthe externallay-
∼
pulsate in phase and reach a state of maximum
ers with only a moderate transfer of mechanical
compression at t = 7.16 s, at which time they
worktotheirsurroundings. Thus,inthreedimen-
makeupacompactnearlyhydrostaticcore. Atthe
sional PRD calculations the ashes retain most of
same time, the outermostlayerscontinue expand-
their specific energy (kinetic and thermal), which
ing with velocities largerthan the escape velocity,
enables the formation of a robust accretion shock
while the intermediatelayerscontinue fallingonto
thatultimately triggersadetonation. Incontrast,
the core with velocities as large as 5000 km s−1.
inthePDDmechanism,thedetonationisthought
The successive stretching and shrinking of the C-
to be due to compression of the mixing layer be-
Orichcorecanbebetter seeninFig.4. Although
tween unprocessed and processed material, which
the chemical structure carry the imprint of the
is believed to grow only if there is a long-range
bubbles evolution and lacks spherical symmetry,
pulsation of the white dwarf (Ho¨flich et al. 1995).
but for the central volume, the fuel mass fraction
In Figs. 6 and 7 there are represented the
at radius below 5000 km is high: X &0.4.
∼ fuel angle-averaged specific energy and chemical pro-
The mechanical and thermal structure at t
acc files, respectively, of the three-dimensional model
display a high degree of spherical symmetry, as
DF18 (top panel) and a one-dimensional calcula-
shownbyFig.5. Thepictureoftheradialvelocity
tionofapulsatingdelayeddetonationmodel(bot-
reveals a large gradient above r 3000 km. The
tom panel: model PDDe in Badenes et al. 2003).
∼
opposed gradients of radial velocity and density
In the three-dimensional calculations the energy
leadtoaquitehomogeneousimpactpressure(due
resides for the most part in the outer 0.15 M⊙.
to matter infall onto the core) as high as P =
ram Hence,thepulsatingmotionofthismaterialisde-
ρv2 1023dyncm−2 intherangeofdistancesr
coupledfromtherestofthestructure. Incontrast,
≃ ∼
3000 5000km. This impact pressure takes over
in the one-dimensional calculation the energy is
−
the thermal pressure above r 4000 km, so that
∼ concentrated in the inner 0.2 M⊙, that behaves
thelargestdecelerationandsubsequentheatingof
like a piston driving the pulsation of most of the
the in-falling matter takes place in a ring (in fact,
structure nearly in phase. In the PDD mecha-
a narrowshell inthree dimensions) atr 3000
nism the detonation is initiated atthe edge of the
∼ −
4000 km, as seen in the temperature plot. The
incinerated core during the collapsing phase and
fuel mass fraction in these locations at t lies in
acc propagatesoutwardsthroughthein-fallingmatter
5
(Khokhlov et al. 1993). evolutionofthevelocityprofilesofthePDDmodel
An important requirement of the PDD mecha- is displayed. Except for a few layers at the top of
nism is the necessity to burn a quite precise mass thewhitedwarf,thestarpulsatesinphasewithout
of fuel, 0.3 M⊙, during the initial deflagration development of any shock wave (unless a detona-
phase in∼order to achieve a large amplitude pul- tion is assumed to happen). These calculations
sation (necessary to create a wide enough mixed show that the chemical composition inversion is
layer)yetnotunbindthe white dwarf. Thisis not a crucial ingredient in the evolution of the white
the case in the PRD models, because: dwarf up to the moment of detonation ignition.
1. The detonation is not launched as a result 3. Confined detonation ignition
of turbulent mixing of fuel and ashes, thus
We now turn to the question of the robustness
avoidingthenecessitytoachievealargeam-
oftheCDIinthePRDscenario. Firstweexamine
plitude of the pulsating motion.
the physical conditions in the accretion shock, as
2. The relative amplitude of the pulsation is obtained in three-dimensional simulations of the
not uniform for all the mass layers of the pulsating phase. By comparing these conditions
white dwarf. The external layers experience with suitable detonation criteria, we then analyze
a large amplitude because they are rich in the most favorable conditions for detonation igni-
hot ashes, freshly synthesized in the float- tion in our three-dimensional simulations.
ing burning bubbles, while the 1 M⊙ core
remains at moderately high density during 3.1. Physical conditionsofshocked matter
the pulsation. Thus, in the PRD scenario a
The locationand physicalconditions at the ac-
large amplitude of the external layers is ob-
cretion shock are very sensitive to the amount of
tained irrespectively of the precise amount
massburned duringthe initialdeflagrationphase.
ofmassburntduringthedeflagrationphase.
A summary of the properties of the accretion
shock for three calculated models is given in Ta-
Finally, we compare the evolution during the
ble 2. The mass at which the accretion shock
pulsation of the structures obtained at the end of
is formed, M , is larger for smaller M be-
the deflagration phase for the three-dimensional acc defl
cause less energy is available to power the ex-
model DF18 and that of a one-dimensionalmodel
pansion of the white dwarf. The relationship be-
constrainedtoburnsubsonicallythesamemassas
tween both masses can be approximated in the
DF18. First, we have mapped the DF18 model
range of M explored by: M 4.18M2
at 4.2 s into an angle-averaged version and its defl acc ≈ − defl−
1.22M +1.50, where all masses are expressed
posterior evolution has been followed with a one- defl
dimensionalhydrocode2withthenuclearreactions in M⊙. Other quantities that can be fitted to an
analytic function of M are: the time of forma-
turnedoff. Asequenceofthevelocityandentropy acc
tion of the accretion shock, t 15.9M +
profiles obtained this way is shown in Fig. 8. In acc acc
≈ −
25.1 s, andthe centraldensity at that time, ρ
this Fig. it can be appreciated clearly the process c8
≈
exp 8.87M2 11.0M +1.50 , where ρ is in
of formation of the accretion shock, whose main units of 108acgcc−m−3. acc c8
distinctive feature is the sudden increase of the
(cid:0) (cid:1)
specificentropy. Second,wehavecomputedaone- Once formed, the accretionshock remains con-
dimensional PDD model with Mdefl = 0.18 M⊙. finedclosetothe hydrostaticcoreduetothe large
By construction,the difference between this PDD ratio of the impact pressure of the in-falling mat-
modelandtheone-dimensionalversionofDF18re- ter with respect to the gas pressure: ρv2/p =
sides in the distribution of fuel and ashes, as well γM2 9 12, where ρv2 is the impact pres-
≈ −
as the specific energy they transport. The con- sure, p is the gas pressure, γ = 5/3 is the adi-
sequences are plain to see in Fig. 9, in which the abatic coefficient, and M is the Mach number
of the flow. The density at the shock is low,
2The one-dimensional hydrocode, based on the finite dif- ρshock (1.4 5.2) 105gcm−3,andthetemper-
≈ − ×
ferences scheme proposed by Colgate&White (1966), is
theonedescribedinBadenes etal.(2003)andBravoetal.
(1993)
6
ature is high, T =T /(1+0.067T ) , (6)
9A 9 9
T = 3 µ v2 (0.5 1) 109 K, (3) Q = 84.165, and T9 = T/109 K. Finally, τi is
shock obtained as:
16kN ≈ − ×
A
c T
whereµisthemeanmolarmass(µ=1.75gmol−1 τ = e , (7)
i
forcompletelyionizedcarbonandoxygenmatter), ρqcAT9YC2exp Q/T91A/3
NA isAvogadro’snumber,andkistheBoltzmann (cid:16) (cid:17)
constant. The accretion shock evolution is there- where q = 4.48 1018 erg mol−1, c is the
c e
after driven by two opposite effects: inertial con- specific heat, A ×= 8.54 1026T5/6T−3/2, and
finement due to mechanical energy deposition by Y =0.5/12molTg9−1 isthe×12Cmol9aArfra9ction(we
C
in-falling matter vs pressure build-up due to nu- assume a 50%-50% by mass carbon-oxygen com-
clear energy release (not included in the present position).
simulations). Eventually, the nuclear energy re-
Using the above equations with the values
leasedbytheburningofamassM becomeslarge
pri of ρ and T from Table 2 the nuclear
shock shock
enough for pressure build-up to take over the me-
timescale is larger than the hydrodynamical
chanical energy deposition, causing the accretion
timescale, hence a detonation cannot begin just
shock to expand and decouple from the underly-
behind the accretion shock. However, the condi-
ing dense hydrostatic core. If this primer mass is
tion τ < τ can be met during the journey
nuc hydro
large enough to ignite a detonation, then nuclear
of the shocked matter along its path towards the
processingofmostofthecorewillbeensured,oth-
surface of the hydrostatic core. In order to obtain
erwisenuclearburningwillceaseshortlyafterdue
ananalyticalestimateofthedistancefromtheac-
to adiabatic cooling of the core.
cretion shock at which the timescale requirement
The rate of mechanical energy deposition at can be fulfilled we need to make some simplifying
the accretion shock, εmech = 2πr2ρ0v3, stays in assumptions concerning the shocked flow:
the range (0.5 1) 1049 erg s−1 during approx-
− ×
imately ∆t 0.3 s. Assuming that nuclear burn- Steady state during times small compared
ingproceeds∼justto28Si(becauseρ<107gcm−3) • with τhydro.
the specific nuclear binding energy releasedis q
5.8 1017 erg g−1. An estimate of the maximum∼ • Spherical symmetry and radial flow (we ig-
× nore any instability in the flow).
primer mass that can be accumulated before ex-
pansionofthe accretionshock canbe obtainedby Optically thick matter (we assume thermal
equatingthenuclearenergyreleasetothemechan- • equilibrium with radiation).
icalenergydepositedbyin-fallingmatterM q
pri
≈
ε ∆t, which gives M (2.6 5.2) 1030 g. Adiabatic evolution (we ignore the nuclear
mech pri ≈ − × •
energy input and any energy transfer mech-
The radius at which a detonation could ignite
anism,plusopticallythickmatterwhichim-
can be estimated independently from the above
plies that radiative cooling is inefficient).
calculationsinthefollowingway. Anecessarycon-
dition for a detonation to be initiated is that the
Under these assumptions, the structure of the
nucleartimescalemustbelowerthanthehydrody-
shocked flow in between the accretion shock and
namical timescale, τhydro = 446/√ρshock ≈ 0.6− the core surface can be obtained by solving the
1.2 s. The nuclear timescale of carbon burning,
following set of equations:
τ , can be calculated as (Blinnikov & Khokhlov
nuc
1987; Khokhlov 1989): 1 p GM
e= v2+ =constant, (8)
2 (γ 1)ρ − r
τ =Θτ , (4) −
nuc i
ρvr2 =constant, (9)
where Θ is the Frank-Kamenetskii factor,
p ργ, (10)
2/3 −1 ∝
Θ= QT9A , (5) The starting point for the integration of these
3T9 ! equations is just the physical state behind the
7
shock (see Table 2): ρ , and T . The formspheredensity,ρ,andthemassoftheprimer,
shock shock
hydrodynamical and nuclear timescales obtained M . A larger primer mass implies a larger en-
pri
fromthesolutiontotheaboveequationsformodel ergyreservoirandastrongerpistoneffect,making
DF18 are plotted in Fig. 10, where it can be seen it easier to build a detonation wave. The detona-
that carbon burns in less than a hydrodynamical tioninitiationconditionsinpureC-Omatter(50%
time at a radius of 3000 km. The density and eachbymass)obtainedbyArnett & Livne(1994);
∼
temperature at that point are given by the cross Niemeyer & Woosley(1997);Ro¨pke et al.(2007b)
symbol in Fig. 11. are summaryzed in Table 3. Their results can be
As shown in Table 2, the distance from the ac- interpreted either as the minimum primer mass
cretion shock to the detonation ignition point is for which a detonation is initiated in C-O mat-
sensitive to the mass burned subsonically. The ter at a given density and peak temperature, or
Lagrangianmass at which a detonation could be- as the minimum temperature for which a detona-
gin, Mdeto, goes from 1.03 M⊙ for model DF26 tionisobtainedatagivendensityandmassofthe
to 1.29 M⊙ for model DF11, which is 0.06 M⊙ primer. In any case, the qualitative trend is that
to 0.03 M⊙ inwards from the accretion shock, re- the smaller the density the harder is to initiate a
spectively. The value of M quoted in Table 2 detonation while, at fixed density, the higher the
deto
for model DF29, Mdeto = 0.38 M⊙ is far too low peaktemperaturethelesserprimermassisneeded
to be minimally realistic. However,we stress that todetonatethesphere. FromTable3itstemsthat
theanalyticcalculationsoftheformationofadet- at densities in the range 107 108 g cm−3 a tem-
−
onation presented in this section are just approx- perature of 2 109 K is enough to initiate a
∼ ×
imations because of the many simplifications in- detonation. At lower densities the primer mass
troduced, particularly the assumption of spher- necessarytoinitiate adetonationat2 109 Kbe-
×
ical symmetry and steady state. In the three- comes a non-negligible fraction of the star mass,
dimensionalpicturetherearetobeexpectedsome hence achievement of detonation initiation condi-
deviations from the results of these analytic cal- tionsismuchmoredifficult. Atρ=3 106gcm−3
×
culations. and T = 2.3 109 K a detonation was obtained
×
forM =2 1028 g,implyingathermalgradient
pri
3.2. Detonation ignition conditions T =200 K×cm−1, of the same order of the max-
∇
imum thermal gradient estimated by using Eq. 1:
Small-scalesimulationsareneededtodetermine T =5 150 K cm−1.
max
which are the suitable conditions for detonation ∇ −
The results of the small-scale simulations al-
ignition in white dwarf matter. Arnett & Livne
lowustolookforregionsinthethree-dimensional
(1994); Niemeyer & Woosley (1997); Ro¨pke et al.
hydrodynamic simulations that meet some gross
(2007b) performed such studies following the hy-
criteria for detonation ignition. A comparison of
drodynamical and nuclear evolution of a uniform
the results of the present three-dimensional simu-
density microsphere. This thermonuclear bomb
lations of white dwarf pulsation with the detona-
waslitusingaprimerconsistinginathermalpro-
tion ignition conditions summarized in Table 3 is
file characterized by a central peak temperature
shown in Fig. 11 where we plot in the ρ-T plane
andaconstantnegativethermalgradientdownto
the evolution of the fuel particles that achieve
108 K at a Lagrangian mass coordinate M , be-
pri
the most favorable conditions for detonation ig-
yond which the temperature was uniform. Once
nition, i.e. those particles that attain the max-
ignited, the sphere was subject to two opposite
imum temperature after being shocked. Shown
effects: it expanded and cooled due to the ex-
in this Fig. are the results for five models for
cess of pressure derived from the sudden inciner-
ation of the hot primer but, at the same time, which Mdefl spans the range from 0.11 M⊙ to
the released nuclear energy helped keeping a high 0.29 M⊙. The two big dots belong to the two
first rows in Table 3, which are used to delimit
value of temperature. Which one of the two ef-
the ρ-T conditions for which a C-O detonation
fects wins is what determines if a detonation is
is the most probable outcome. As can be seen
successfully launched or not. These numerical ex-
in this Fig., such conditions are clearly reached
perimentsarethereforecharacterizedby threepa-
in all the three-dimensional pulsating models ex-
rameters: the central temperature, T , the uni-
c
8
cept for DF29, characterized by a mass burned comparableto the maximumprimer mass derived
subsonically close to the mass needed to unbind in Sect. 3.1 for model DF29: M = 2.6 1030 g
pri
×
the white dwarf (0.30 M⊙). The path followed (see also Table 2).
by the particles belonging to the different mod- For comparison purposes we show in Fig. 11
els is nearly identical, and very close to adiabatic the path followed during the recontraction phase
evolution (dot-dashed line in Fig. 11). The point by the PDD model described in Section 2.2, but
at which model DF18 (green points) crosses the with the nuclear reactions turned off. The dis-
line defined by τnuc =τhydro is in good agreement playedpathis that drawnby afuel particlein the
with the estimate based on Eqs. 8-10 (dark-green neighborhood of the quenched flame. The path
cross). InallcasesbutmodelDF29,theC-Odeto- of the particle penetrates the C-O detonation re-
nationignitionlineiscrossedat 3 106gcm−3. gion at ρ 7 107 g cm−3 and T 1.2 109 K,
The peaks of temperature and∼de×nsity span a andafterw∼ard×sreachesamaximum∼tempe×ratureof
nρmararxow (r4ang8e:) Tm10a6xg≈cm(2−.43,−hi2g.h9)en×ou1g0h9 tKo caonnd- Tofmtahxe=ρ-1T.5p×la1n0e9hKa.sHnoowtebveeern,nthooterotuhgahtltyheisxprelogrioend
≈ − ×
fidently assume that a CDI will be produced in in order to determine the conditions suitable for
these cases. detonation ignition (see Table 3 and references
In Fig. 12 there are shown the thermodynamic therein). Thus, in this region the threshold line
and chemical structure in a slice of model DF18 is justanextrapolationfromlowerdensity points.
in the neighborhood of the hottest shocked fuel It is therefore clear that the conditions achieved
particle before (top row) and after (bottom row) during the re-contracting phase of PDD models
it reaches detonation ignition conditions (T are quite different from those obtained with PRD
2.3 109 K, ρ 3 106 g cm−3). The deto∼- models.
× ∼ ×
nation conditions are first reached in a hot spot
of radius 200 km (mass 1029 g) affect- 3.3. Resolution issues
∼ ∼
ing a region with non-uniform chemical compo-
Our numerical resolution is much coarser than
sition,where the fuelmassfractionis inthe range
the expected detonation width at the densities
X(C+O) 0.4 0.8. A rough estimate of
∼ − of detonation ignition. Thus, we have explored
the thermal gradient across the hot spot gives
the sensitivity of the maximum temperature and
T 109 K/200 km = 50 K cm−1. Thus, al-
∇ ∼ density reached during white dwarf pulsation to
thoughthe situationismuchmorecomplexinour
the resolution of the numerical models. To this
modelsthanthataddressedinthesmall-scalesim-
end we have computed additional simulations of
ulations, we can confidently expect that a stable
the pulsating phase of models DF18 and DF29
detonation will be obtained a few tenths of a sec-
with increased resolution. The starting point of
ond after the formation of the accretion shock.
these additional models was the time of maxi-
Note that the hot spot that appears close to the
mum expansion during the pulsation. The par-
center inthe toprowimage is due to the presence
ticles that fulfilled certain selection criteria were
of a clump of ashes, which avoids a detonation to
splitted into N children particles, and the star
sp
be initiated at its position. Note also that, be-
evolutionwasagainfollowedwith the same three-
causewedidnotallowfornuclearreactionsinthe
dimensional SPH code described before (the term
presentmodelsoncethedeflagrationphaseended,
particle splitting was used for the first time by
the chemical changes between both snapshots are
Kitsionas & Whitworth 2002). In order to save
onlyduetoadvectionofparticlesofdifferentcom-
computertime,splittingwasconfinedtotheneigh-
position.
borsof the shockedfuel particlesthat reachedthe
Themaximumtemperatureanddensityreached
maximumtemperature(Fig.11),andtotheneigh-
in model DF29, T =1.42 109 K and ρ =
max max borsoftheirneighbors. Weperformedsimulations
1.3 106gcm−3,areseemingl×ytoolowforigniting
for N =2,4, 10,and100. In everycasethe origi-
× sp
a stable detonation. Ro¨pke et al. (2007b) did not
nal particle mass wassharedevenly between their
obtainastabledetonationforρ=106 gcm−3 and
children, which resulted in a maximum mass res-
atemperatureofT =3.0 109Kevenforaprimer olution of 1.1 1026 g. The initial position of the
×
mass as high as M =3.0 1030 g. This mass is ×
pri children particles was randomly selected within a
×
9
spherecenteredonthepositionoftheparentparti- from the nuclear processing of accreted hydro-
cleandradiusgivenbyitssmoothinglength. They gen. Ho¨flich & Khokhlov (1996) suggested that
werethereforeassignedthesametemperatureand the presence of a 0.01 M⊙ cap of He would im-
∼
chemicalcompositionastheparentparticle,while prove the match between calculated and observed
theirvelocitywasobtainedbyinterpolationofthe light curves. This amountof He might come from
velocity field at the position of the children. thesurroundingaccretiondiskiftheaccretionrate
The results of our resolution study are shown were not constant, for instance if it dropped at
in Table 4, where there are given the maximum some time below the critical rate for steady burn-
density and temperature reached by children par- ing. Cumming et al.(1994b,a)detectedlinescoin-
ticles aswellas their maximumspatialresolution, cidentwithtwoHeIlinesat2.04µmand1.052µm
∆x . Even though the maximum temperature in the spectra of the spectroscopically normal SN
min
and density for a given M show some scatter 1994D. The absorption feature at 1.05 µm has
defl
withincreasingresolutiontheyallstayatthesame been since detected in the early spectra of many
side of the C-O detonation line. The departure otherSNIa(seeNomoto et al.2003;Pignata et al.
from the conditions obtained with our standard 2008), although its identification is problematic
non-splitted models is quite modest and does not because that line could rather be due to Mg II
affect our conclusions: all the variants of model (Wheeler et al. 1998; Hatano et al. 1999). How-
DF18fulfilledtheconditionsforaC-Odetonation, ever, as pointed out by Mazzali & Lucy (1998),
while all the variants of model DF29 did not. He I lines may be blended with magnesium lines
as well as with lines of other intermediate-mass
In our highest resolution simulations of model
DF18, the increase in resolution does not lead to elements. Hence, up to ∼ 0.01 M⊙ of He might
remain undetected in SNIa, hidden by stronger
a too steep thermal gradient that might compro-
magnesium lines.
misethe achievementofdetonationconditions. In
model DF18sp100, the primer mass (calculated If there was actually such small cap of he-
as the total mass of fuel particles with ρ 3 lium,itwouldhaveseveralchancesofmixingwith
106 gcm−3 and T 200Kcm−1 thatare≥neigh×- the C-O outer layers of the white dwarf. The
bors of the hotte∇st c≤hildren) rises to 1.8 1028 g. first opportunity would arise during the convec-
Thismassissimilartotheminimummass×required tive simmering phase shortly before the hydrody-
to detonate C-O, as given in Table 3 for the same namic event (Kuhlen et al. 2006; Piro & Bildsten
density but for a peak temperature substantially 2008; Chamulak et al. 2008). Even though con-
lowerthanthe maximumtemperatureachievedin vectionduringsimmeringisnotexpectedtoreach
model DF18sp100. As we have not been able to the white dwarf surface, it can excite pulsating
achieveconvergenceofthe maximumtemperature and non-spherical mode instabilities of the star
anddensity,itisnotcleariffurtherincreaseofthe that might allow diffusion of He into the un-
resolution might result in a thermal gradient too derlying C-O layers. A second chance for mix-
steep to launch a detonation. ing He would take place during the deflagra-
tion phase of the explosion itself. Turbulence
4. A helium cap on the white dwarf? and, especially, Rayleigh-Taylorinstability induce
large radial excursions of huge volumes of ashes
One way to favor a detonation ignition in C-O that destabilize the whole white dwarf, and can
matter is to mix it with a small amount of He, drive the mixing of an eventual external He layer
whichbothis morereactiveandreleasesmorenu- with the C-O beneath it. Hence, even though it
clear energy than carbon. To allow such a mix- may seem a speculative scenario, one cannot dis-
ing, some mass of He must be present in the card the possibility that He is present in some
white dwarfat the moment ofthermonuclearrun- small mass fraction within the matter that is
away. Indeed, a very small cap made of He might prone to detonate after a white dwarf pulsation.
rest atop of the white dwarf as a result of pre- Mixing of He within a detonating volume seems
vious accretion from the companion star in the more likely in scenarios in which the transition
binary system. This helium could either have to detonation develops in the outermost layers
been accreted directly from a He star or result of the white dwarf, e.g. those CDI described in
10