Detonating failed deflagration_model_of_themonuclear_supernovae_explosion_dynamics

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Submitted to the ApJPreprint typeset using LATEX style emulateapj v. 10/09/06

DETONATING FAILED DEFLAGRATION MODEL OF THERMONUCLEAR SUPERNOVAE I. EXPLOSIONDYNAMICS

Tomasz Plewa1,2

(Received; Accepted)Submitted to the ApJ

ABSTRACT

We present a detonating failed deflagration model of Type Ia supernovae. In this model, thethermonuclear explosion of a massive white dwarf follows an off-center deflagration. We conduct asurvey of asymmetric ignition configurations initiated at various distances from the stellar center.In all cases studied, we find that only a small amount of stellar fuel is consumed during deflagrationphase, no explosion is obtained, and the released energy is mostly wasted on expanding the progenitor.Products of the failed deflagration quickly reach the stellar surface, polluting and strongly disturbingit. These disturbances eventually evolve into small and isolated shock-dominated regions which arerich in fuel. We consider these regions as seeds capable of forming self-sustained detonations that,ultimately, result in the thermonuclear supernova explosion.

Preliminary nucleosynthesis results indicate the model supernova ejecta are typically composed ofabout 0.1−0.25 M⊙ of silicon group elements, 0.9−1.2 M⊙ of iron group elements, and are essentiallycarbon-free. The ejecta have a composite morphology, are chemically stratified, and display a modestamount of intrinsic asymmetry. The innermost layers are slightly egg-shaped with the axis ratio≈ 1.2− 1.3 and dominated by the products of silicon burning. This central region is surrounded by ashell of silicon-group elements. The outermost layers of ejecta are highly inhomogeneous and containproducts of incomplete oxygen burning with only small admixture of unburned stellar material. Theexplosion energies are ≈ 1.3 − 1.5 × 1051 erg.Subject headings: hydrodynamics — nuclear reactions, nucleosynthesis, abundances — supernovae:

general

1. INTRODUCTION

Almost half a century ago, Hoyle & Fowler (1960) pro-posed that some supernovae may originate from the de-generate remnants of stellar evolution. These objects areknown as Type Ia supernovae (SN Ia) and are commonlybelieved to be the end points of the evolution of interme-diate mass stars in close binary systems (Whelan & Iben1973).

The two most attractive theories of formation ofType Ia supernovae are the single-degenerate (SD;Whelan & Iben 1973; Nomoto 1982; Starrfield et al.2004; Yoon, Langer, & Scheithauer 2004) and double-degenerate (DD; Iben & Tutukov 1984; Webbink 1984)scenarios. The observational evidence necessary to dis-criminate which formation channel is preferred in natureremains indirect and fragmentary (Branch et al. 1995;Livio & Riess 2003; Mannucci, Della Valle, & Panagia2006, and references therein), in striking contrast to thatof Type II supernovae. Evidence supporting the SDscenario has been collected only recently and in somecases requires more careful analysis. Some examplesinclude observations of accreting massive white dwarfs(Lanz et al. 2005; Suleimanov & Ibragimov 2003), evi-dence of hydrogen-rich material in the vicinity of thesupernova, likely associated with a companion starhamuy+03, and the presence of the fast moving non-degenerate star inside a post-Type Ia supernova rem-

Electronic address: [email protected] Center for Astrophysical Thermonuclear Flashes, University of

Chicago, 5640 South Ellis Avenue, Chicago, IL 606372 Department of Astronomy & Astrophysics, University of

Chicago, 5640 South Ellis Avenue, Chicago, IL 60637

nant (Ruiz-Lapuente et al. 2004), a possible companionsof the supernova progenitor. In the case of the DD for-mation channel, several close binary white dwarf systemswith total mass around Chandrasekhar mass have beenidentified during the last few years (Napiwotzki et al.2003, 2005), essentially confirming double degenerates asprospective progenitors of thermonuclear supernovae.

Here we limit our considerations to a single-degeneratescenario in which the ignition of the degenerate mat-ter takes place in the core of a Chandrasekhar-masswhite dwarf. Only a small difference is expected be-tween the outcomes of the core ignition of a mas-sive white dwarf formed through either the SD or DDchannel. In the latter case, the explosion is perhapsborn in the core of a massive rotating white dwarf(Piersanti et al. 2003), a remnant of the final mergerphase that does not result in an instantaneous explosion(Guerrero, Garcıa-Berro, & Isern 2004).

In defining the initial conditions for multidi-mensional hydrodynamic models, we were guidedin this study by results of the recent analytic(Garcia-Senz & Woosley 1995; Wunsch & Woosley2004; Woosley, Wunsch, & Kuhlen 2004) andpreliminary numerical (Hoflich & Stein 2002;Kuhlen, Woosley, & Glatzmaier 2006) studies ofconditions prevailing in the white dwarf’s core just priorto the thermonuclear runaway. Our limited knowledgeof that evolutionary phase grants us certain freedomin defining starting models. Both the number of theignition points and the timing of the ignition are freeparameters of the current models.

Following the failure of carbon (Arnett 1969;

http://arXiv.org/abs/astro-ph/0611776v1

mailto:[email protected]

2 Plewa

Arnett, Truran, & Woosley 1971; Arnett 1974) andhelium detonation models (Livne & Arnett 1995),explosion modeling has focused on deflagrationmodels and their derivatives. This preference hasbeen firmly established after the apparent suc-cess of parametrized one-dimensional pure defla-gration models (Nomoto, Sugimoto, & Neo 1976;Nomoto, Thielemann, & Yokoi 1984). They, alongwith the later variant known as a delayed-detonationmodel (Khokhlov 1991; Woosley 1990; Arnett & Livne1994b) brought the parametrized models into qualita-tive agreement with observations (Hoflich & Khokhlov1996).

Multi-dimensional studies initially included both2-dimensional models of deflagrations (Muller & Arnett1982, 1986; Livne 1993; Livne & Arnett 1993) anddetonations (Livne & Arnett 1995). During thelast decade, sophisticated 3-dimensional modelshave dominated the scene. Central single-point(Khokhlov 2000; Reinecke, Hillebrandt, & Niemeyer2002; Gamezo et al. 2003) as well as multi-point igni-tion deflagrations (Reinecke, Hillebrandt, & Niemeyer2002; Ropke et al. 2006; Schmidt & Niemeyer 2006;Garcia-Senz & Bravo 2005) seem to produce sublu-minous events with highly mixed ejecta. Althoughmodels with a parametrized deflagration-to-detonationtransition (Gamezo, Khokhlov, & Oran 2004, 2005;Golombek & Niemeyer 2005) appear to address bothdeficiencies, the mechanism behind the transition todetonation demands explanation.

There exists evidence (Hoflich et al. 2002, 2004;Kozma et al. 2005, but see also Blinnikov et al. (2006))that the compositional structure of the ejecta ob-tained in multi-dimensional centrally-ignited deflagra-tions may not be compatible with observations. Atthe same time, spectroscopic and polarization obser-vations of several SNe Ia suggest the existence ofstrongly inhomogeneous outer ejecta layers rich in inter-mediate elements (Chugai, Chevalier, & Lundqvist 2004;Wang et al. 2004; Kasen & Plewa 2005; Leonard et al.2005). These two apparently contradictory requirementscan possibly be reconciled within a class of hybrid mod-els that combine deflagration and detonation within asingle evolutionary sequence. Here we introduce a deto-nating failed deflagration (DFD) model, a generalizationof our early exploratory study (Plewa, Calder, & Lamb2004, hereafter PCL), in which both essential elementsare naturally present. In this model, the inhomogeneitiesin the outer ejecta layers result from the large scale per-turbation of the surface stellar layers induced by an off-center deflagration that fails to unbind the star. In thelong term, that perturbation eventually leads to the for-mation of isolated shock-dominated regions that serveas ignition points for a detonation. The resulting eventis luminous with a composite ejecta structure consist-ing of smoothly distributed detonation products in thecentral regions surrounded by inhomogeneous, turbulent-like outer layers composed of material partially burnedin the deflagration.

Our numerical investigations include certain simplifi-cations with the assumption of axisymmetry being po-tentially the most important one. This, however, willallow us to conduct a small parameter study exploringthe dependence of the explosion parameters on the initial

conditions. In turn, for the first time, we will be able tooffer evidence that the simplifying assumptions regard-ing the geometry may not be the major deficiency of ourmodels.

2. METHODS

We study the hydrodynamic evolution of a massivewhite dwarf using the flash code (Fryxell et al. 2000).flash has been the subject of rather extensive verifi-cation and validation in both subsonic and supersonicregimes (Calder et al. 2002; Weirs et al. 2005). We useda customized version of the code based on the flash2.4release with specialized modules designed to model de-flagrations and detonations. We recorded the history ofindividual fluid elements with tracer particles for the pur-pose of future detailed nucleosynthesis studies.

2.1. Reactive hydrodynamics

We solved the time-dependent reactive Euler equationsof self-gravitating flow in cylindrical geometry assum-ing axial symmetry. The non-reactive set of equationswere extended by an advection-diffusion-reaction (ADR)equation describing the evolution of a deflagration front.The solution to the ADR equation was obtained withthe help of a flame capturing method (Khokhlov 1995).The flash implementation has been the subject of verifi-cation (Vladimirova, Weirs, & Ryzhik 2006) with the re-sults of the application to turbulent flames closely match-ing the original implementation (Zhang et al. 2006).

2.2. Flame model revision

For the present application, several elements of theoriginal ADR scheme were modified. In particular, care-ful analysis of the original three-stage flash burner(Calder et al. 2004, PCL) revealed that it overestimatedthe amount of energy produced by burning the stellarmix to nuclear statistical equilibrium (NSE) at densi-ties typical of the stellar core. As a consequence, thenuclear ashes were both too hot and too rarefied, withbuoyancy effects overestimated by a factor of ≈ 3. Thisled to a much larger acceleration of the burned material,shorter evolutionary time scales (from ignition to bub-ble breakout), a lower consumption of stellar fuel and acorrespondingly lower degree of pollution of the surfacelayers. The current scheme captures the energetics of thedeflagrating material more closely. We also introduced arevised formula for the laminar flame speed that betterapproximates the results of Timmes & Woosley (1992).

2.2.1. Thermonuclear burning

Following Khokhlov (1991, 2000), the evolution of adeflagrating stellar material can be considered as a se-quence of largely independent processes. Carbon burn-ing (stage I) is followed by relaxation toward nuclear sta-tistical quasi-equilibrium (NSQE) that produces silicon-group nuclei (stage II). Eventually, the matter relaxestoward nuclear statistical equilibrium (NSE) producingiron-group nuclei (stage III). In the approximate burner,this is modeled through modifying the composition in allthree stages. Stage I converts carbon into 24Mg while28Si is produced in stage II by the “burning” of 16O and24Mg. We also extended the original scheme by intro-ducing a “light nucleus” that accounts for the presence

Dynamics of Detonating Failed Deflagrations 3

Fig. 1.— Dependence of the energy release on the initial tem-perature obtained with torch47 nuclear network for a 50/50 C/Omixture over a fixed amount of time, tf = τg . The highest

amount of energy is released for Ti = 2 × 109 K and the lowestfor Ti = 1.5 × 109 K. The energy release is essentially insensitiveto the initial temperature for densities & 1 × 107 g cm−3.

of free alpha particles and protons in the very high den-sity and temperature regime (stage III, see below).

The original flash approximate burning scheme as-sumed the energy release was a simple sum of the nom-inal differences between the binding energy of the ini-tial C/O mix and the burned material, ∆Eb ≈ 7.8 ×1017 erg g−1, independent of density. This is over 70%more than the energy release obtained by Khokhlov(1983, Fig. 2, ∆Eb ≈ 4.5 × 1017 erg g−1) for the den-sities of interest during the initial explosion stages, ρ ≈2 × 109 g cm−3. Our revised scheme uses results ob-tained with the torch47 network (Timmes 2001) in theself-heating mode. The outcome of such calculations de-pends on the initial temperature of the fuel. Calculationsalso need to be conducted for long enough to guaranteea complete burn.

In our first set of calculations, we advanced the net-work for a fixed amount of time, tf = τg, where τg =

446ρ−1/2 s is the hydrodynamic (free-fall) time scale(Fowler & Hoyle 1964; Arnett 1996). Calculations werestarted using several different initial temperatures of themixture, Ti = 1.5 − 2.0 × 109 K. We found a relativelyweak dependence of the energy release on the initial tem-perature (Fig. 1) and adopted Ti = 1.7 × 109 K in sub-sequent calculations.

Once the initial temperature was fixed, we turned ourattention to studying the time-dependence of the energyrelease. We found that for ρ & 1 × 106 g cm−3 the bulkof the energy is released within ∆t = τg (Fig. 2). In thesupernova calculations that follow, we used the energyrelease obtained for Ti = 1.7 × 109 K and tf = τg (thicksolid line in Fig. 2). The energy release was tabulatedas a function of density with resolution 0.1 dex and lin-early interpolated. The flame extinction was modeledby smoothly decreasing the original energy release fordensities < 1 × 106 g cm−3. No energy was released fordensities below 1 × 105 g cm−3.

In contrast to the original flash implementation ofstage III in which 56Ni was the sole product of burn-ing, here we introduced additional “light nuclei” com-

Fig. 2.— Temporal evolution of the energy release in self-heatingtorch47 nuclear network calculations for a 50/50 C/O mix at tem-perature Ti = 1.7 × 109 K. The bulk of the energy is producedwithin a single hydrodynamic time scale, τg . Thick solid lines cor-respond to the energy release adopted in our supernova explosioncalculations.

Fig. 3.— Composition of the “light nuclei” supplementing theNSE composition in stage III of the approximate burning scheme.

posed of alpha particles and protons. This allowed usto better approximate the temperature, and in turn thebuoyancy and dynamics of the Rayleigh-Taylor unstableburning front, especially at early times. As before, weused the torch47 nuclear network in an isochoric self-heating mode of burning to determine the compositionof the light nuclei (Fig. 3). These results appeared insen-sitive to the particular choice of the initial temperatureor the final time provided the network was evolved for atleast τg.

2.2.2. Coupling to Hydrodynamics

In addition to tracing the compositional evolu-tion of the nuclear ashes, the implementation of theKhokhlov’s three-stage burner involves the coupling ofthe energy source term to hydrodynamics. In whatfollows we are primarily concerned with the earlyevolutionary stages when the physical time scalesassociated with nuclear burning are much shorterthan a flame crossing time for computational zones

4 Plewa

(Khokhlov, Muller, & Hoflich 1993; Arnett & Livne1994b; Reinecke, Hillebrandt, & Niemeyer 2002). Whenthe evolution of the burning region is relatively slow andthe expansion of the star is insignificant, as it is duringthe early evolution of centrally ignited deflagrations, astep-wise form of the energy production rate leads tolarge (about two orders of magnitude) rapid fluctuationsin the energy deposition. These localized discrete“explosions” occur in partially-burned material wheredegeneracy is lifted and are a source of pressure wavesand velocity fluctuations of order ≃ 10 km/s. A similarproblem in the context of modeling deflagrations frontswas noted by Khokhlov, Muller, & Hoflich (1993) whoused, however, a completely different procedure tomodel the flame evolution. The origin of the problem,i.e. discrete representation of a thin unresolved burningfront, is common to both studies.

The impact of the numerical artifact just described canbe limited by appropriately scaling the energy depositionin stage II (no such procedure is needed in stage I whichalways operates under strongly degenerate conditionsand no energy is released in stage III). In the simula-tions presented here, the energy generation rate for stageII was kept at 1% of its nominal value throughout sim-ulations. Given the advection time scale ≈ 10−4 s, thisprocedure affected the energy release only at very highdensities for which the problem of spurious numerically-induced flow perturbations was originally identified. Al-ternatively, one can also consider less intrusive proce-dures in which the energy generation rate is limited onlyfor times . 0.5 s. Imposing less strict limitations on theenergy release was found to universally produce velocityperturbations of magnitude comparable to that of largescale flows.

2.2.3. Flame speed calibration

Several factors may cause the numerical flame speedpropagation to differ from the prescribed laminar flamespeed in one dimension. Vladimirova, Weirs, & Ryzhik(2006) have investigated in some detail the influence ofnumerical resolution, evolution with a superimposed con-stant background velocity mimicking hydrodynamic ad-vection (i.e. to verify Galilean invariance), and the im-pact of velocity gradients across the flame front (presum-ably caused by thermal expansion in real applications).

In the supernova context, given the degenerate na-ture of the equation of state, thermal expansion is con-trolled by the energy release which chiefly depends on thefuel density and composition (Timmes & Woosley 1992).Therefore whenever the flame energetics are modified, anappropriate calibration of the numerical parameters con-trolling the numerical flame speed needs to be done forthe numerical flame speed to match the nominal (lami-nar) flame speed.

The calibration procedure is relatively simple, but te-dious because the numerical flame speed depends on den-sity, composition, and numerical resolution. Given ourfocus is on the evolution of progenitors composed of a50/50 carbon/oxygen mix, our flame speed calibrationwas limited to that composition. We performed a num-ber of one-dimensional flame propagation simulations inCartesian geometry. Models were obtained on grids withlengths between 1L and 30L with L = 480 km, and us-ing between 256 and 1024 zones. The typical grid reso-

lution was ≈ 1 km. The results showed only weak de-pendence on the resolution. The flame propagation was,however, somewhat vulnerable to background flow fluc-tuations generated by the flame front motion that freelypropagated and partially reflected off the boundaries. Al-though we used non-reflecting (zero gradient) boundaryconditions, such reflections are understandable given thesubsonic nature of the problem. In our analysis we useddata from carefully selected long evolutionary sequencescomputed on the largest grids and at the highest possibleresolution.

Our calibration procedure was applied towarda slightly modified formula originally proposed byTimmes & Woosley (1992, Eq. 43):

sl,TW = aρb9 ×

(

1.001− e−

“

ρ9−δqσq)

”2)

with a = 35.46538×105, b = 1.110603, δq = 2.6132427×10−2, and σq = 2.9538546×10−2. This improved formulareproduces Timmes & Woosley (1992) results for a C/O50/50 mix to within 15% for densities between 1×107 and1×109 g cm−3. Finally, this has been adjusted to accountfor thermal expansion due to the torch47 energetics:

sl,FLASH = sl,TW × max(0.9, min(1.3, p(log10 ρ))),

where

p(x) = c0 + c1x + c2x2 + c3x

3 + c4x4,

is polynomial density-dependent correction factor basedon our calibration calculations and c0 = 413.6563,c1 = −194.3208, c2 = 34.06912, c3 = −2.633218, andc4 = 7.5665459 × 10−2. The final formula reproducesTimmes & Woosley (1992, Table 3) results to within 5%for densities in the range 1 × 107 and 4 × 109 g cm−3.

2.2.4. Hybrid burning scheme

Our initial investigations into late evolutionary stagesof fizzle off-center deflagrations (PCL) provided strongevidence that the conditions inside the confluence re-gion at the stellar surface are appropriate for creatinga detonation. We observed that both densities and tem-peratures were high enough for the burning time scaleto become shorter than hydrodynamic time scale. Wealso anticipated that the shocked region would remainconfined for an extended period of time sufficient todevelop a self-sustained detonation. We were unable,however, to study that process in detail at that time.The burning module could not reliably discriminate be-tween shocked fuel (a legitimate detonation site) andcompressed partially burned matter. Here we introducea hybrid burning scheme to allow for the simultaneouspresence of deflagrations and distributed nuclear burn-ing in the simulation. Deflagrations are modeled us-ing the ADR flame-capturing scheme (Khokhlov 1995;Vladimirova, Weirs, & Ryzhik 2006) with modificationsas outlined above. The distributed burning is calcu-lated using the flash aprox13t 13-isotope alpha net-work (G. Jordan 2005, private communication) This net-work is an extension of the original flash approx13

network and includes temperature coupling for increasedstability of calculations in the NSE regime Muller (1986).

The first step in our our hybrid burning procedure isto identify shocked zones. This is done using a multi-dimensional shock detection module adopted from the


sppm code (Anderson & Woodward 1995) with pressurejump across the shock, ∆p/p ≥ 0.5. If the tempera-ture in such zones is high enough for nuclear burning(Tnuc,min = 8 × 108 K), a progress variable used bythe ADR flame module is reinitialized and the numer-ical flame speed is set to zero. The flame module issubsequently activated only if the flame speed is greaterthan zero and the material is pure fuel (as indicated by,for example, a small abundance of nuclei not participat-ing in the simplified three-stage burner). Otherwise, thenuclear network is called but only if the following con-ditions are fulfilled simultaneously: (1) both the densityand temperature are in the range appropriate for the net-work calculations, (2) the zone is outside a shock front,(3) the deflagration module was not used, and (4) theADR progress variable is small. The first condition limitsnetwork calculations to high-density (ρ ≥ 1×105 g cm−3)high-temperature (T ≥ 8×108 K) regimes. Condition (2)follows from the conclusions of Fryxell, Muller, & Arnett(1989) who found that the correct speed of detonationwaves can be obtained only in models with burning dis-allowed inside unrelaxed numerical shock profiles. Con-ditions (3) ensures only one physical burning process,either distributed burning following shock heating or de-flagration, operates at a time. Finally, condition (4) pre-vents burning from occurring in regions preheated byan extended diffusive tail of the ADR flame capturingscheme and at the same time allows for burning to con-tinue after the detonation wave sweeps through partiallydeflagrated material. By making the progress variablethreshold for use of nuclear network slowly increasingwith time, we also prevented spurious (distributed) burn-ing in the dense central regions perturbed by ascendingdeflagrating material.

The above scheme captures the essential behavior ofboth deflagrations and wave-induced distributed burn-ing. It allows for the transition to detonation in shock-heated regions away from deflagrating material and en-ables burning in shocked, partially deflagrated material.The above selection rules were developed and improvedbased upon experience accumulated in the course of sev-eral numerical experiments and, as such, offers a prac-tical rather than ideal recipe for modeling deflagrationsand detonations in the same physical setting including apossible deflagration to detonation transition.

Once the detonation wave is launched, a combina-tion of a PPM hydrodynamic solver, a multidimensionalshock detection algorithm, and a aprox13t nuclear burn-ing module was used to advance the evolution in time.The evolution of the detonation wave on a large scale isexpected to be captured correctly given that the thick-ness of the wave is much smaller than the stellar radius(Falle 2000). We did not find it necessary to rescale nu-clear reaction rates in order to obtain the correct propa-gation of the detonation speeds (Arnett & Livne 1994b);excluding the unrelaxed shock profile from burning ap-peared sufficient to yield physical solutions.

2.3. Some comments on numerical model limitations

One of primary motivations behind development of ahybrid burning scheme was desire to mitigate a risk ofspurious detonation ignitions. The existence of such spu-rious ignitions is a well-documented fact in astrophysi-cal literature (Fryxell, Muller, & Arnett 1989). Here we

only briefly discuss the most typical causes for spuriousignitions and possible ways to prevent them from pollut-ing hydrodynamical models.

Perhaps the most common cause for spurious deto-nation ignitions is a numerically-induced mixing of hotashes and cold fuel. By advecting a material inter-face (contact discontinuity) separating ashes from fuel,Fryxell, Muller, & Arnett (1989) demonstrated that suchmixing may result in artificial preheating of fuel, its ig-nition, and ultimately formation of a combustion wave.Related to this is a problem of species conservation bynonlinear Eulerian advection schemes. Under certainconditions numerical modification of fuel (or partiallyburned material) composition may change burning en-ergetics. Either extinction or enhancement of burningmay follow. To our knowledge, neither numerical speciesdiffusion nor species non-conservation can be completelyeliminated from Eulerian simulations of realistic nonlin-ear systems. Our hybrid burning scheme attempts tolimit possible influence of both effects by constrainingburning to regions occupied by pure fuel.

Transition to detonation can also follow an artifi-cial boosting of an acoustic perturbation. Such per-turbations lead to local variations in temperature andunder degenerate conditions temperature is a sensitivefunction of (relatively small thermal) pressure. Givenstrong dependence of nuclear reactions on temperature,it is conceivable that even small but sustained heat-ing may strengthen acoustic waves and eventually causespurious transition to detonations. Typically, however,small acoustic fluctuations suffer from strong dampingby numerical diffusivity and such spurious transitionsto detonations are likely to occur only if nuclear burn-ing is allowed inside numerical (unrelaxed) shock pro-files. The cure for this problem is to eliminate burningin regions occupied by shocks (Fryxell, Muller, & Arnett1989) and, as we mentioned above, such a filter is em-ployed in our calculations.

Finally, application of nuclear burning source term inour calculations is limited to regions with sufficientlyhigh densities and temperatures. That is, we wish to con-sider a feedback from nuclear burning only if the nucleartimescale is short enough to influence hydrodynamics.This approach not only saves computational time, butmore importantly prevents the nuclear network from be-ing fed with input data representing low-density regionswhere evolution has a highly transient character and isnot correctly captured in our calculations.

To summarize, numerical computations of reactiveflows pose extreme challenges and require very carefultreatment. We attempted to address several known andsome newly emerged problems related to coupling reac-tive source terms to hydrodynamics in great detail. Theimpact of some of these problems could only be limited,but not completely eliminated. For example, in our mod-els no nuclear burning is allowed inside numerical shockprofiles. However, our particular choice of parametersdefining numerical shock profile may not be adequatein all situations affecting evolution of shocks and acous-tics in unwanted manner. Poor numerical resolution onlyadds to the algorithmic inefficiency further limiting pre-dictive abilities of our models.

Clearly, successfully resolving technical problems ofour computations is of high priority and such aspects

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should be remembered when interpreting our results. Atthe same time, however, one should also keep in mindthat our model involves several simplifying assumptions(i.e., we consider a non-rotating, non-convective, non-magnetized chemically homogeneous progenitor), and assuch it is a blend of approximate numerics and uniquechoice of parameters defining physical scenario.

2.4. SN Ia explosion code verification: centraldeflagrations

Given that both the basic hydrodynamic module aswell as the FLASH implementation of the flame captur-ing scheme have been extensively verified and, albeit toa limited extent, also validated in the past (Calder et al.2002; Weirs et al. 2005; Vladimirova, Weirs, & Ryzhik2006), our verification is solely limited to code-to-code comparison in the context of thermonuclear super-nova explosion modeling. Although code-to-code com-parison is widely popular among computational physi-cists, the usefulness of this approach is hotly debated(Trucano, Pilch, & Oberkampf 2003). For one, even per-fect agreement between the simulation results of twocodes does not offer proof of their correctness. More-over, the scope of such an exercise is usually limited bythe specific capabilities of the code, the availability ofthe results, the completeness of documentation, and therelevance of the performed test to the actual problem athand, to name a few. Here we accept the obvious de-ficiencies of the code-to-code comparison approach anduse highly relevant and state-of-the-art calculations as abenchmark. Again, due to the novelty of our ultimate ap-plication, no data are available for comparison, thoughthey will hopefully become available through indepen-dent calculations. Ultimately, the model will be validatedusing accumulated observational evidence. Methodologyand methods required in such assessments are presentedin a companion paper (Kasen & Plewa 2006, see alsoBlinnikov et al. (2006)).

2.4.1. Simulation setup

For our comparison study we selected a family of cen-trally ignited deflagration models obtained by the Garch-ing group (Reinecke, Hillebrandt, & Niemeyer 2002;Ropke & Hillebrandt 2005b, and references therein).These studies are very well-documented and their resultscompare favorably to those obtained by other groups(Gamezo et al. 2003; Gamezo, Khokhlov, & Oran 2004).When comparing results, we considered the overallmorphology of the explosion (flame front structure),the energetics and the approximate ejecta composi-tion of the 2-dimensional explosion calculations re-ported by Reinecke, Hillebrandt, & Niemeyer (2002) andRopke & Hillebrandt (2005a). Our choice of 2-D Garch-ing models is natural given our calculations to follow alsoassume axial symmetry.

In a benchmark study we used flash2.4 customizedfor the thermonuclear supernova explosion problem. Weused a PPM module for real gas inviscid hydrodynamicsand the Helmholtz equation of state required by the de-generate conditions encountered in the white dwarf inte-rior. All calculations were done with Courant factor 0.6.This choice of time step limiter may appear somewhatconservative, but allows for better coupling between dif-ferent physical processes.

Gas self-gravity was accounted for by solving the Pois-son equation through multipole expansion. We foundthat linear momentum is poorly conserved in explosioncalculations when the expansion series is truncated tooearly, especially when the explosion displays significantasymmetries. In test calculations done assuming a spher-ical potential, the bulk of the stellar material displayedmotion of a few hundred km/s after only a few secondsof evolution. The momentum conservation gradually im-proved as additional higher order terms were included inthe expansion. In what follows we used 10 multipole mo-ments and found excellent momentum conservation forall initial conditions considered.

We used a 2-dimensional cylindrical grid (r, z) and as-sumed axial symmetry. This implied imposing a reflect-ing boundary condition at rmin = 0. We applied outflowconditions at the remaining boundaries. In our verifi-cation calculations the computational domain covered arectangular region with rmax = zmax = −zmin = 16, 384km. We used the adaptive capabilities of the flash codeto create several levels of refinement up to a maximumresolution of 8 km. We do not expect the dynamicalevolution of low density gas or at large distances fromthe stellar center to play an important role in explo-sion simulations and therefore adaptive refinement wasused only for radii < 4, 000 km and if the gas density> 1×104 g cm−3. Self-gravity calculations, on the otherhand, require good resolution of dense regions and thegrid refinement was forced to the highest allowed reso-lution whenever the gas density > 3 × 106 g cm−3. Inaddition, the innermost 2, 500 km of the star have al-ways been resolved with the finest zones. In regionswhere adaptivity was allowed, AMR patches were cre-ated dynamically if the local density contrast exceeded0.50 or the total velocity changed by more than 20%.Furthermore, we ensured the flame front was always re-fined to the finest level by forcing refinement wheneverthe fractional change of the flame progress variable ex-ceeded 0.02.

2.4.2. Initial model

The supernova progenitor was an isothermal, T =5 × 107 K, white dwarf composed of equal mass frac-tions of carbon and oxygen. With a central density of2× 109 g cm−3, the progenitor had a radius ≈ 2100 km,total mass ≈ 1.36 M⊙, and total energy ≈ −4.92 × 1050

erg. The progenitor was surrounded by a low den-sity (ρamb = 1 × 10−3 g cm−3) and low temperature(Tamb = 3 × 107 K) medium composed of pure helium.The stellar material and low density ambient mediumwere initially marked with a passively advected tracerthat was set to 1 and 0 in those two regions, respectively.Subsequently, gas gravitational accelerations were multi-plied by the tracer value in the course of the evolution,allowing us to prevent the ambient medium from collaps-ing onto the central object and limiting mass diffusion atthe stellar surface.

The original progenitor model was constructed usinga numerical discretization different from the one usedin the hydrodynamic simulations and assuming a sim-plified equation of state. For that reason, the perfecthydrostatic equilibrium of the original model was de-stroyed as soon as it was interpolated onto the simulationmesh. Even though the mismatch between the two com-


Fig. 4.— Evolution of the root mean square velocity in mod-els without burning obtained with resolution 2 (thick solid), 4(medium solid), 8 (thin solid), 16 (dotted), and 32 km (dashed).The initial models were perturbed by adding random velocitieswith amplitude 200 km s−1 inside the stellar core. Note that thevelocity gently decays in models calculated with a resolution of16 km or better.

putational environments was relatively small, very strongacoustic oscillations quickly developed making such amodel unsuitable for further investigations. Moreover,the oscillations did not decay with time, presumably dueto both low dissipation of the hydrodynamic scheme andthe strong degeneracy of the medium.

We constructed a stable progenitor model using a mod-ified variant of the damping method of Arnett (1994) in a1-D flash simulation in spherical geometry. Our proce-dure combined a very slow diffusion of velocity togetherwith a partial rather than complete (as in the originalmethod) removal of excess momentum after each timestep. We found that the complete removal of momentumprevented the model from reaching equilibrium. Damp-ing process usually lasted several thousands of hydro-dynamic steps. We examined the stability of the relaxedmodel by computing a sequence of hydrodynamic modelswithout burning. Random velocity perturbations withamplitude of 200 km s−1 (Ma ≈ 0.02) were added tothe inner core region of radius 400 km. We observedthe decay of the root mean square velocity with time, asexpected in a stable model, provided the resolution was16 km or better (Fig. 4). We also computed the evolu-tion of an “effective” stellar radius corresponding to thevolume occupied by gas with density > 1 × 104 g cm−3

representing the bulk of the stellar matter. The resultsare shown in Fig. 5. As one can see, the model stellarradius shows significant evolution in computations withresolution 16 km or worse; only a very modest (≈ 0.5%)decrease of radius was observed in the better resolvedmodels, and had no consequence for structure of the stel-lar core (i.e. possible temperature increase).

Unfortunately, close examination of the velocity fieldalso revealed that while the overall stability of modelsimproves with increasing resolution, small velocity per-turbations not only do not decay but are amplified nearthe symmetry axis. This effect was especially strong in2 km resolution model where the velocity near the axisrapidly increased from the initially imposed 200 km s−1

to 800 km s−1 during the first 0.1 s of the evolution

Fig. 5.— Evolution of the stellar radius in models without burn-ing obtained with resolution 2 (thick solid), 4 (medium solid), 8(thin solid), 16 (dotted), and 32 km (dashed). The initial mod-els were perturbed by adding random velocities with amplitude200 km s−1 inside the stellar core. Models computed with resolu-tion no worse than 8 km are stable, displaying only a very smalldegree of radius change. The model using 32 km resolution showsvery strong expansion, making it unsuitable for long-term hydro-dynamic evolution studies.

Fig. 6.— Total velocity profiles at t = 0.1 s near the symmetryaxis (r = 0 cm) in models without burning computed with 2 km(thick line) and 8 km (thin line) resolution. The velocity ampli-tude rapidly and significantly increases in the 2 km model, whileit slowly decreases in the 8 km model.

(shown with the thin solid line in Fig. 6). No signifi-cant increase in the magnitude of the spurious velocitieswas observed at later times, but the affected region ex-panded from the initial 1 to 3 zones by t = 1 s. Weobserved very similar behavior in the 4 km resolutionmodel, although the velocities near the symmetry axiswere ultimately somewhat smaller (≈ 600 km s−1). Incontrast, the velocities smoothly decay from their initialvalues in the whole perturbed region in the 8 km resolu-tion model (shown with the thick solid line in Fig. 6). Asimilar slow decrease of the velocities near the symmetryaxis was also observed in the 16 km model.

After a stabilized progenitor model was interpolatedonto the simulation mesh, a deflagration front was ini-tialized around a small region at the stellar center. The

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TABLE 1Centrally Ignited Benchmark Deflagration Models

Model nm ∆xc [km] Rc [km] tc [s] Eta [1051 erg]

n7d1r10t15b 7 1 1000 1.5 0.86n7d1r10t15c 7 1 1000 1.5 0.97n9d1r10t15b 9 1 1000 1.5 0.75n9d1r10t15c 9 1 1000 1.5 0.66n11d1r10t15b 11 1 1000 1.5 0.75n11d1r10t15c 11 1 1000 1.5 0.66n11d2r05t10a 11 2 500 1.0 0.27n11d2r05t10b 11 2 500 1.0 0.38n11d2r05t10c 11 2 500 1.0 0.44n11d2r05t10d 11 2 500 1.0 0.38n11d2r05t10e 11 2 500 1.0 0.38n11d2r10t10a 11 2 1000 1.0 0.35n11d2r10t10b 11 2 1000 1.0 0.36n11d2r10t10c 11 2 1000 1.0 0.42n11d2r10t10d 11 2 1000 1.0 0.38n11d2r10t10e 11 2 1000 1.0 0.39n11d2r10t10f 11 2 1000 1.0 0.36n11d2r10t15a 11 2 1000 1.5 0.56n11d2r10t15b 11 2 1000 1.5 0.64n11d2r10t15c 11 2 1000 1.5 0.60n11d2r10t15d 11 2 1000 1.5 0.52n11d2r10t15e 11 2 1000 1.5 0.58n11d2r20t15b 11 2 2000 1.5 0.66n11d2r20t20b 11 2 2000 2.0 0.54n11d2r40t20b 11 2 4000 2.0 0.54n11d2r40t20b3 11 2 4000 2.0 0.55n11d2r40t20b6 11 2 4000 2.0 0.52n11d2r40t15b 11 2 4000 1.5 0.79n11d2r60t15b 11 2 6000 1.5 0.79n13d1r10t15b 13 1 1000 1.5 0.57n13d1r10t15c 13 1 1000 1.5 0.69n15d1r10t15b 15 1 1000 1.5 0.57n15d1r10t15c 15 1 1000 1.5 0.52

flame front was located at

Rf = Rm ×[

1 + am cos(

nm tan−1(z

r

))]

,

where Rm is the unperturbed radius of the burned re-gion (flame radius), am is the flame radius perturbationamplitude, and nn controls the flame radius perturba-tion wavelength. Random velocity perturbation wereadded to the inner core region following the proceduredescribed above. By introducing a small stochastic com-ponent (“cosmic variance”) into the problem, we wereable to examine the robustness of our results. In par-ticular, it is essential to verify that the observables (i.e.the explosion energy) do not show strong dependence onsmall perturbations in the initial conditions, in accordwith the observed intrinsic homogeneity of SN Ia.

2.4.3. Centrally ignited benchmark models

We performed a comprehensive survey of two-dimensional centrally ignited deflagration models. Thedatabase contains 33 models. All models were evolveduntil t = 2.5 s when burning essentially ceased. For eachmodel we varied the flame perturbation wavelength andrandom velocity perturbation pattern (through the seedperturbation). We explored the sensitivity of the resultsto mesh resolution by resolving the central stellar regionof radius Rc t ∆xc for times < tc. Resolution coarseningin this central region was done in equal intervals of time.For times > tc, the resolution was equal to a default valueof 8 km. The initial flame radius and perturbation am-plitude were in all cases fixed at Rm = 100 km and 0.1,respectively. Table 1 presents a complete database of our

benchmark models.3 The model identification tag nameis constructed as a string n--d-r--t--ph where n-- de-notes the perturbation pattern, d- describes the maximalgrid resolution in the core region, r-- denotes a radiusof the core region inside which enhanced resolution wasused, t-- denotes the time up to which enhanced flameresolution was allowed, p distinguishes between differ-ent perturbation patterns, and finally h denotes the de-sired thickness of the flame front (in grid cells) in theADR flame capturing module (see Appendix in Khokhlov1995). The default value of the numerical flame thick-ness was 4; we obtained two models, n11d2r40t30b3 andn11d2r40t30b6, in which the nominal flame thickness wasvaried by −25 and +50%, respectively. The last columnin Table 1 gives the total energy of the model (sum of ki-netic, internal, and potential energies) at the final time,t = 2.5 s.

Evolution in reference model — We use modeln11d2r10t10a to illustrate major characteristics ofa centrally ignited deflagration in our benchmarkconfiguration (Fig. 7). The density in this model at theinitial time is shown in Fig. 7(a). The flame front hasthe shape of the initial regular n = 11 perturbation.At a resolution of 2 km, comparable to that of thehighest resolution Garching group models at early times(Ropke et al. 2006, Fig. 3), the flame region is resolvedinto about 50 zones in radius. This initial configurationundergoes a dramatic evolution during the next second(Fig. 7(b)). After that time only 3 prominent bubblesare clearly identifiable and some parts of the flamebegins forming disconnected regions (e.g. the regionlocated near (r, z) ≈ (650,−550) km). The flame leavesbehind a significant amount of unburned material anda few isolated pockets of fuel can also be identifiedinside rising bubbles. The outermost parts of a highlyconvoluted flame front have reached ≈ 1, 000 km inradius. The star has expanded by ≈ 20% and the typicalexpansion velocities are ≈ 2, 500 km s−1. After anothersecond (t = 2 s; Fig. 7(c)), the expansion of the outerstellar layers becomes slightly nonuniform due to theuneven acceleration caused by individual flame bubbles.The expansion velocity exceeds 10, 000 km s−1 near thestellar surface. At this time one can still identify 3 largebubbles, but these are now more developed and occupya much larger volume fraction. Their morphology doesnot change much at still later times (t = 2.5 s, the finaltime; Fig. 7(d)) as nuclear burning essentially quenchesand the ejecta expansion becomes progressively radial.

Sensitivity to small perturbations — Given the highly non-linear character of the Rayleigh-Taylor unstable defla-grating bubbles it is natural to ask how robust are thepredictions offered by individual models. We studied thisquestion by creating a sequence of models for a given setof primary model parameters (initial flame configurationand numerical resolution). The initial conditions for eachmember of a given sequence differed only by the patternof small amplitude stochastic velocity added to the ini-tially static progenitor model. Our metric for compari-son is to examine the final ejecta morphology and integral

3 The database is available on-line atflash.uchicago.edu/~tomek/Results/DFD/central/.


Fig. 7.— Hydrodynamic evolution of the centrally ignited benchmark supernova model n11d2r10t10a. Panels (a)-(d) show the density inlogarithmic scale together with the outline of the flame front (contour line corresponding to progress variable value of 0.5). Notice the scalechange between panels. (a) The initial conditions at t = 0 s. (b) t = 1.0 s. The flame front is highly convoluted; the star remains sphericalbut expanded by ≈ 20%; the bulk expansion velocity is ≈ 2, 500 km s−1. (c) t = 2.0 s. The flame front is rich in structure with somepockets of unburned material; a large amount of unburned material can be found near the center; individual flame bubbles create smoothlarge scale impressions on the stellar surface; the expansion velocity near the outer stellar edge are just slightly over 10, 000 km s−1. (d)t = 2.5 s. The flame is extinct and the structure of the flame front becomes frozen. Evolution toward homologous expansion begins.

model characteristics (temporal evolution of total energy,burning rate, flame surface area).

As an example we compare select models from a singlesequence. Figure 8 shows the ejecta temperature dis-tribution at the final time in the sequence n11d2r10t15.The isocontour of gas density of 1×104 g cm−3 is shownwith the black line and can be identified with the stellarsurface. Several comments can be made following inspec-tion of Fig. 8.

It is encouraging to notice that the models show noaxis-related bias, a numerical artifact that frequentlypollutes axisymmetric hydrodynamic calculations. Inparticular, the structures developed near the symmetryaxis (r = 0 cm) in models n11d2r10t15a (Fig 8(a)) andn11d2r10t15b (Fig 8(b)) are markedly different. Thereis also no visible difference in the amount and quality ofthe structure developing in the regions above (z > 0 cm)and below (z < 0 cm) the equator. Some models dodevelop structures near the equator (see,e.g., Fig 8(d)),but some others do not (see, e.g., Fig 8(c)). This allowsus to conclude that possible defects due to the geome-try representation do not affect our calculations in anysignificant way.

In all models considered here large scale structures(bubbles) several tens degrees in size dominate in theouter parts of the ejecta. In some models perhaps nomore than 2 (Fig 8(c)) while in some others perhaps asmany as 3 (Fig 8(d)) such large and distinct structuressurvive turbulent burning. These bubbles push aheadunburned material causing relatively mild deformationof the ejecta outer layers as indicated by shape of thedensity isocontour.

As discussed below, the explosion energies also ap-pear sensitive to small perturbations. In the case ofthe n11d2r10t15 models, our limited sample shows thetotal variation in explosion energy ≈ 0.1 foe (1 foe =1 × 1051 erg) or ≈ 10% in the released energy (seeFig. 9(c)). This may indicate that small, naturally oc-curring variations in the internal structure of progenitors(expected to arise from the convective flows developingin their cores prior to runaway) may contribute to theintrinsic diversity of SN Ia. Addressing this interesting

possibility requires more careful study, preferably usingrealistic multi-dimensional progenitor models.

Sensitivity to numerical resolution — From the mod-eler’s point of view, several simulation parameters maypotentially affect the results and therefore need tobe controlled. Given the high degree of complex-ity and highly non-linear character of our applica-tion, it is natural to expect that the model resultswill depend upon the numerical resolution. Conver-gence to the true 3-dimensional solution is not ex-pected to occur in two dimensions due to, for ex-ample, differences between the physics of two- versusthree-dimensional supernova turbulence (Khokhlov 1995;Schmidt, Hillebrandt, & Niemeyer 2006) and Rayleigh-Taylor instability (Kane et al. 2000; Chertkov 2003).Nevertheless, it is still important to examine the sensitiv-ity of our axisymmetric model predictions to numericalresolution.

In adaptive mesh simulations, the computational meshis not necessarily a well-defined entity as the numeri-cal discretization depends on the solution and is usuallyhighly variable both in time and space. The simple pro-cedure of doubling the grid resolution does not have itsusual interpretation. Unlike uniform grid simulations,adaptive mesh refinement computations admit additionalerror into the solution by not resolving smooth or other-wise dynamically insignificant parts of the flow field. Inaddition, some errors, such as flow perturbations arisingat the fine-coarse mesh interfaces, are unique to AMRdiscretization and not easy to characterize (Quirk 1991;Weirs et al. 2005; Pantano et al. 2005).

As our earlier investigations have demonstrated, themorphology on small scales of the exploding models ap-pears very sensitive to slight perturbations in the initialconditions, and might be useful only for making qualita-tive statements. A more quantitative comparison of thedifferent models can be made using integral quantities.For example, variations in the final explosion energy areof great interest from the observational point of view,and several possible natural causes for such variationshave been proposed (i.e. differences in the chemical com-

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Fig. 8.— Select centrally ignited benchmark supernova models from sequence n11d2r10t15. Panels (a)-(d) show the temperature inlogarithmic scale at the final time (t = 2.5 s) in models n11d2r10t15a, n11d2r10t15b, n11d2r10t15c, and n11d2r10t15d, respectively. Thedensity isocontour at ρ = 1× 104 g cm−3 is shown as the black solid line. Although the details of ejecta morphology vary strongly betweenmodels, all models produce large bubbles of burned material gently deforming the outer stellar layers. The presence of a symmetry axis inthe simulation does not appear to bias the calculations and no asymmetry between the two hemispheres is observed.

position and/or the rotation of the progenitor).Here we discuss the role of mesh adaption on the evo-

lution of the total energies, flame surface areas and theburning rates. Since as we mentioned before the evolu-tion is rather sensitive to small perturbations, we choseto compare different families of models rather than in-dividual family members. Figure 9 shows the evolu-tion of the integral quantities in three families of then11d2 sequence of benchmark models: n11d2r05t10 (leftpanel), n11d2r10t10 (middle panel), and n11d2r10t15(right panel). The individual members of each familywere obtained using slightly different random velocityperturbations. Compared to the n11d2r10t10 models,in models n11d2r05t10 the innermost region of enhancedresolution (∆x = 2 km) was limited to 500 km in radius(as compared to 1, 000 km). The resolution in this regionwas decreased by a factor of 2 at 0.5 s and by anotherfactor of 2 at t = 1.0 s. In the n11d2r10t15 family ofmodels, this innermost region was derefined in two simi-lar stages with the resolution ultimately decreased to itsnominal value at t = 1.5 s.

Several observations can be made following analysis ofthe the n11d2 sequence.

All n11d2 models produce explosions. The explosionenergies are rather low, ≈ 0.4 − 0.5 foe. In all casesthe nuclear burning is most intense around ≈ 1.2 s afterthe ignition. That period of rapid energy release lastsfor about 0.5 s with the densities dropping rapidly to≈ 1 − 2 × 107 g cm−3 by t = 1.5 s. This is followed bya period of low, almost constant energy release duringwhich in only exceptional cases a slight increase in en-ergy generation was observed. At that time (t ≈ 1.7 s),the burning takes place at densities < 1×107 g cm−3 andchanges its character from active turbulent burning asso-ciated with vigorous creation of flame surface to a muchmilder, distributed mode of burning. By t = 2 s thedensities drop to ≈ 1× 106 g cm−3 and flame quenchingresults in a steady decrease in energy generation.

Models obtained with higher resolution produce moreenergetic explosions. The typical explosion energies varyfrom ≈ 0.45 foe for the least resolved subsequence r05t10to ≈ 0.55 foe for the best resolved subsequence r10t15.Higher resolution also appears to make evolutionary tra-

jectories more similar at early times (the curves run moreclosely in subsequence models r10t10 than in r05t10) butresult in increasing diversity at late times (there are rela-tively large variations in energy generation rates aroundt = 1.5 s in subsequence r10t15).

We found that not only the morphology, but also theintegral quantities are sensitive to small perturbationsin the progenitor. For example, the dispersion of explo-sion energies is about 0.1 foe, even in relatively well-resolved simulations (e.g. subsequence r10t15). Thismay indicate that some of the observed diversity ofsupernovae might be produced by the nonlinear re-sponse of the explosion process to small variations inthe initial conditions. Such variations from one pro-genitor to another are expected to exist in natureespecially given the convective (turbulent) flow con-ditions prevailing in the stellar cores prior to ther-monuclear runaway (see Woosley 1990; Hoflich & Stein2002; Kuhlen, Woosley, & Glatzmaier 2006, and refer-ences therein). The contribution of such a purelystochastic component to the explosion process clearly de-serves more careful study.

2.4.4. Comparison against Garching group models

We found good agreement between the main charac-teristics of our centrally ignited model explosions andthe results of equivalent axisymmetric calculations pre-sented by the Garching group. The overall evolu-tion of the energy generation rate in the n11d2 modelsequence is similar to that in model c3 2d 256 by(Reinecke, Hillebrandt, & Niemeyer 2002, Fig. 3) andto that obtained earlier by (Niemeyer 1995, ; seeReinecke, Hillebrandt, & Niemeyer (1999)). The energygeneration in our model explosions displays a pronouncedmaximum reaching between ≈ 1.1 − 1.2 × 1051 erg s−1

in models n11d2r05t10 and n11d2r10t10. This com-pares very favorably to the result reported by Niemeyer(1995), Reinecke, Hillebrandt, & Niemeyer (2002), andmore recently by Ropke (2005). The latter two studies re-ported peak energy generation rates ≈ 1.2×1051 erg s−1.The rates obtained in models n11d2r10t15 are higherby about 50%. Such higher rates were reported by(Ropke et al. 2006) for some of their centrally ignited


Fig. 9.— Evolution of the total (explosion) energies and burning rates (solid and dashed lines, respectively) in three families of centrallyignited benchmark supernova models from sequence n11d2. (a) models n11d2r05t10; (b) models n11d2r10t10; (c) models n11d2r10t15.Abrupt changes in the energy generation rate visible at t = 1 s (models n11d2r05t10 and n11d2r10t10) and t = 0.75 s (models n11d2r05t15)are caused by mesh derefinement.

models, however these calculations were done in 3-dimensions. The energy generation rate in our modelsis initially smaller and increases at a faster rate than inthe Garching models. The maximum generation rate isachieved at t ≈ 1.25 s, roughly 0.5 s later than in theGarching models. In flash as well as in the Garchingcalculations the burning quenches ≈ 0.4 s after the max-imum.

Our model explosions are on average slightly more en-ergetic than those reported in Garching studies. Thetypical explosion energies are between ≈ 0.3 − 0.7 foein our models while (Reinecke, Hillebrandt, & Niemeyer2002) and (Ropke & Hillebrandt 2005a) reported explo-sions with energies ≈ 0.35 and ≈ 0.45, respectively. Itis conceivable that the higher initial energy generationrates obtained in the Garching calculations may resultin the lower explosion energies of their models since thefaster initial expansion leaves less time for the flame todevelop and burn stellar fuel. Such discrepancies can beexplained by differences in both the adopted flame dy-namics model and the approximations used to describethe nuclear burning and not of serious concern in thecontext of the following results.

3. DETONATING FAILED DEFLAGRATIONS

Limited by the assumption of axial symmetry, we con-sidered off-center bubble ignitions in which the flame ini-tially occupies a small spherically symmetric region(s)positioned at the symmetry axis (r = 0 cm). Two fami-lies of models were constructed, one with a single ignitionpoint and the other with two ignition points. For the lat-ter, we consider only simultaneous ignitions. although inprinciple the multiple ignition process could be extendedin time (see, for example, Schmidt & Niemeyer (2006)).

Following our verification study, off-center explosionmodels were obtained at the maximum resolution of8 km. In models with two ignition points, the flameregions were initialized in different hemispheres. Table 2summarizes the parameters describing the initial flameand mesh configurations of the off-center ignition mod-els. Here zb,1 and zb,2 are the locations of the ignitionspoints along the symmetry axis (r = 0 cm), and Rb

is the radius of flame regions. To keep the flame re-gions well-separated, the mesh resolution in the central

TABLE 2Off-center Ignition Configurations

Model zb,1 [km] zb,2 [km] Rb [km]Y12 12.5 · · · 50Y25 25 · · · 50Y50 50 · · · 50Y100 100 · · · 50Y70YM25 70 -25 35Y100YM25 100 -25 50Y75YM50 75 -50 50

300 km region was increased by a factor of 2 for a shortperiod of time after ignition (0.4 s in models Y70YM25and Y100YM25 and 0.1 s in model Y75YM50). Thisadditional refinement was also necessary to adequatelyresolve the smaller bubbles (Rb = 35 km) used in modelY70YM25.

The computational domain extended up to 131, 072 kmand 524, 288 km in single- and double-ignition point mod-els, respectively. In anticipation of an extended andasymmetric evolution at early times, the region of adap-tive meshing was extended to 6, 000 km in radius. Theinitial conditions did not include random velocity pertur-bations. All other simulation parameters were identicalto those used in the verification study (see § 2.4.1).

3.1. Explosion phase

For no other reason than convenience and limitsin computing power, early multi-dimensional in-vestigations of white dwarf deflagrations assumedperhaps only slightly perturbed but otherwise spheri-cally symmetric ignition conditions (Muller & Arnett1982; Livne 1993; Arnett & Livne 1994a; Khokhlov1995, 2000; Reinecke, Hillebrandt, & Niemeyer 2002;Gamezo et al. 2003). Such a choice is not nec-essarily the most natural one given the whitedwarf core is believed to be convective prior torunaway (Woosley 1990; Garcia-Senz & Woosley1995; Woosley, Wunsch, & Kuhlen 2004), an ex-pectation supported by recent multi-dimensionalhydrodynamic investigations (Hoflich & Stein 2002;Kuhlen, Woosley, & Glatzmaier 2006). This led severalgroups to consider progressively more complex andrealistic (although not necessarily correct!) initialflame configurations (Niemeyer, Hillebrandt, & Woosley1996; Garcia-Senz & Bravo 2005; Ropke et al. 2006;

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Schmidt & Niemeyer 2006).Here we adopt a similar approach, but not being dis-

couraged by the failure of an initial deflagration to pro-duce a supernova, we continue our investigations throughthe following stages of evolution. Our preliminary inves-tigations (PCL) indicated that the energy released in thedeflagration may be used to compress the stellar surfacelayers thereby forming seed points for detonations. Wewere unable, however, to study that process in detailat that time, and only speculated about the possibility.Here, we revisit our original idea of a deflagration to det-onation transition following a slightly off-center ignition.

3.1.1. Failed deflagration phase

In all models ignited off-center, the evolution initiallyproceeds in a way much similar to that described inPCL. Owing to strong buoyancy and the relative slow-ness of laminar burning, the whole burning region isquickly expelled from the core (Garcia-Senz & Woosley1995; Woosley, Wunsch, & Kuhlen 2004) consuming onlya small amount of fuel on its way to the stellar surface.We describe the early evolution of burning regions in theset of single ignition models in terms of mean velocities,the position of centroids, and the effective radii of burnedmatter. The effective radii correspond to spheres of thesame surface area as the burned region.

From their original positions, the initially motionlessbubbles are driven by buoyancy and develop primar-ily vertical velocities, as can be seen by comparing themean vertical (Fig. 10(a)) and radial (Fig. 10(b)) veloc-ities. The initial acceleration is stronger in cases wherethe bubble is placed farther away from the stellar cen-ter (models Y50 and Y100, thick and thin dashed lines inFig. 10) and weaker when the bubble originates deeper inthe core (models Y12 and Y25, thick and thin solid linesin Fig. 10). Although the mean motion of the burnedregion is away from the stellar center, the rich and com-plex flow field includes downflows and outflows develop-ing inside the region that lead to intermittent (apparent)deceleration and acceleration (seen as mild wiggles su-perimposed on the velocity curves). Each such wiggle invelocity is associated with the destruction of the currentgeneration of Rayleigh-Taylor bubbles and the formationof the next. Our data indicates that perhaps two genera-tions of such bubbles are created during the deflagrationphase.

A sudden drop in vertical velocity and a rapid increaseof lateral expansion marks the moment of bubble break-out. This phase is not well-defined but occurs roughlyat ≈ 0.7 s in model Y100 and not until t ≈ 1 s in modelY12. The effective radius of the burned regions (definedas a radius of a sphere with surface area equal to theflame surface area) at breakout is very similar betweenthe models, ≈ 400 − 450 km (Fig. 10(c)). However, thecentroid of the burned region is located much farther outin model Y100 than in model Y12. That is understand-able considering there is more time for the RT instabilityto develop structure and for the region to grow laterallywhen the ignition takes place closer to the stellar center.This also has profound consequences for the evolutionof the star. A longer deflagration phase allows for moreburning, causes more matter being lifted from the stel-lar core, and eventually results in stronger stellar expan-sion during the early (t < 2 s) post-breakout evolution

(Fig. 11. In the case of double point ignitions, the earlyevolution proceeds in a very similar way to that of singleignitions with a proportionally increase in the amount ofburned material and stronger stellar expansion.

3.1.2. Detonation phase

One of the motivating factors behind extending ourstudy to double ignition point scenarios was to examinewhether a detonation can be formed when the maximal(and perhaps even boosted by flawed numerics) focusingoffered by the symmetry axis is not present. Althoughwe observed detonations forming in all off-center ignitedmodels, only in one double ignition model, Y100YM25,does the detonation form near the equatorial plane. Inthe remaining two double ignition models, detonationseventually emerge near the symmetry axis. Althoughboth models eventually detonate, we cannot considerthem as examples of successful asymmetric DFDs. Nev-ertheless, both of them provide additional evidence forshock to detonation transition. In what follows we willfirst overview the formation of detonations in single ig-nition models. Then we will discuss the flow dynamicsleading to detonation in model Y100YM25. We will con-clude by presenting the ejecta morphology soon after theshock breakout along stellar surface is complete.

The progenitor structure around the time when a det-onation forms is shown in Fig. 12 for models Y12, Y100,and Y100YM25. In all models the bulk of progenitor hasretained its original spherical characters. We do not findany substantial large-scale deformations, except that thestellar cores in single-ignition models are slightly ellip-soidal in shape with axis ratio 1.2−1.3. Stellar expansionhas decreased the core density to ≈ 9 × 107 g cm−3 inmodel Y12 and ≈ 4 × 108 g cm−3 Y100. This is consis-tent with our expectation that lower central densities areto be found in models that experienced more energeticdeflagrations.

The stellar core is surrounded by an extended stronglyturbulent atmosphere. Comparing three models shownin the upper row in Fig. 12, the atmosphere appears bet-ter developed (more extended and turbulent) in modelsthat release more energy in the deflagration. This at-mosphere formed following the breakout of deflagrationproducts through the stellar surface, at which time theashes accelerated unburned surface layers both radiallyand laterally. This circular wave carried both fuel andproducts of the deflagration along the surface of the star.The following evolution depends on whether there wasone or more ignition points.

If the single ignition case, the surface wave eventuallycompletely engulfs the progenitor and collides with itselfin a region located opposite breakout. A conical shockwave forms in the process that thermalizes the kineticenergy of the incoming flow. This shock can be seen asa vertical structure near r = 0 cm extending down from(r, z) ≈ (8×107,−3.5×108) cm in model Y12 (Fig. 12(a))and (r, z) ≈ (4 × 107,−2.5 × 108) cm in model Y100(Fig. 12(c)).

In the multi-point ignition case, there will presumablybe several breakout points and related surface waves thatwill be colliding with one another. Therefore, it is con-ceivable that several shock-dominated regions might beformed. Some of those shocks might be weaker and oth-ers stronger than the ones found here. The number of


Fig. 10.— Kinematics and growth of the burned region in a set of single ignition off-center deflagrations. (a) mean vertical velocity; (b)mean radial velocity; (c) equivalent radius; (d) vertical centroid position. Data for models Y12, Y25, Y50, and Y100 are shown with thicksolid, thin solid, thick dotted and thin dotted lines, respectively.

possible scenarios and outcomes is certainly much largerthan represented in a limited sample of the initial con-figurations considered in this study. Nevertheless, we ex-pect that our models capture the essential features of theevolution. In the case of model Y100YM25, for example,the material of two waves collides near the equatorialplane forming a jet-like radial inflow and outflow near(r, z) ≈ (2.5 × 108,−5 × 107) cm (Fig. 12(e)). This isessentially the same flow configuration we found in thesingle point ignition models.

In what follows, we first focus our discussion on detailsof the transition to detonation process in three selectedmodels. Then we characterize the evolution of the ex-ploding star during the passage of the detonation wave.

Shock-to-detonation Transition — In all cases consideredin our study, we found explosions following a shock-to-detonation transition, SDT (see, e.g., Bdzil & Kapila1992; Sharpe 2002, and references therein). Although re-gions forming detonations differ greatly in morphology,

the common ingredients of the process include the pres-ence of a strong acoustic wave, dense fuel, and a pro-longed compression of the region. This is illustrated inthe bottom row of Fig. 12 which shows the density distri-bution and major flow structures involved in a transitionto detonation process.

Model Y12 shares the initial conditions with the origi-nal PCL study. In the Y12 model, we found no sign of apossible transition to detonation for t < 3.5 s (Fig. 12(a))in contrast to the PCL study, in which a detonation waveformed at t ≈ 1.9 s This difference in timing is due solelyto the incorrect energetics of approximate deflagrationburner used in the PCL calculations. The overestimateof the energy release and hence buoyancy by a factor of≈ 3 in the PCL model resulted in a much shorter de-flagration phase, a lower overall energy release, a morecompact progenitor and ultimately a significantly ear-lier formation of the detonation. At the same time, theless expanded progenitor allowed for the surface wave to

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Fig. 11.— Evolution of the effective stellar radius (defined asthe radius of a sphere with the same volume occupied by matterwith density > 1 × 104 g cm−3) in the single ignition off-centerdeflagrations. Data for models Y12, Y25, Y50, and Y100 are shownwith thick solid, thin solid, thick dotted and thin dotted lines,respectively.

move at a relatively higher speed (due to a lower orbit)and with higher post-shock temperatures, possibly en-hancing the likelihood of a transition to detonation.

Similar to PCL, we observe the formation of a conicalshock in Y12. However, the conditions inside the shockedregion allow only for residual burning that perturbs theshocked gas (the low density region near symmetry axisat r = 0 cm in the lower section of Fig. 12(a)). Thewave that transits to detonation is an accretion shock(blue isocontour extending horizontally from (r, z) ≈(0,−2.55 × 108) cm in Fig. 12(d)) created by infallingmaterial which is trapped horizontally by the symmetryaxis and the incoming deflagration products, and verti-cally by the bulk of the stellar material and the stagna-tion point formed behind the conical shock. A transitionto detonation takes place at the symmetry axis wherethe ram pressure of the incoming flow and the resultingpost-shock temperature are the highest. (Although theaccretion flow is predominantly along z-direction, thereexists a lateral velocity gradient, dvr/dr ≈ −3, in theflow that focuses the flow toward the symmetry axis.)

In model Y100 (Fig. 12(d)), a detonation wave canbe seen as a nearly vertically propagating shock locatednear (r, z) ≈ (6.5×107,−2.85×108) cm. Unlike in modelY12, here the detonation is not directly associated withthe symmetry axis. The conical shock is visible as anwave originating near (r, z) ≈ (7.5 × 107,−3 × 108) cm,just to the right of the detonation region. The appar-ent closeness of the flame front is coincidental and hasnothing to do with origin of the detonation wave. By thetime of Fig. 12(d) the conical shock is already sweepingthrough deflagration products, however the detonationappears moving through a channel of shocked fuel even-tually connecting to the bulk of stellar material.

Model Y100YM25 displays by far the most complexflow structure in the region where a transition to deto-nation occurs. The configuration of two colliding surface

waves resembles that of a “self-colliding” surface wavefound in models with a single ignition point. Here, how-ever, a symmetry of the problem is broken. Not onlydoes the collision not take place at the symmetry axis,but the ignition points were initially located at differ-ent distances from the core. The timing, energetics, andmorphology of each wave were therefore slightly differ-ent. This difference eventually leads to a shift of thecollision plane ≈ −400 km from the equator (Fig. 12(f)).Furthermore, the collision does not occur “head-on” butrather material from the slower wave (ignition point lo-cated closer to the core) tends to penetrate underneaththat of the faster wave (ignition point located fartheraway from the core). In the end, the whole region showsa slight tendency to roll.

Another difference from the highly symmetric singleignition models is that the broken symmetry offers thepotential for creating more than just one shocked re-gion. This is indeed the case in model Y100YM25.Two shock fronts moving in the radial direction canbe seen inside the collision region: one located near(r, z) ≈ (2.3 × 108,−4 × 107) cm, and another near(r, z) ≈ (2.1 × 108,−4 × 107) cm. Both fronts are cre-ated near the collision plane which might be understoodgiven this is where we expect the thermalization rateof the colliding flows being the greatest. The formershock wave evolves into a self-sustained detonation whilethe latter soon dies off. Once a detonation is formed,the wave travels approximately along a fuel-rich channel(a flame-bounded horizontally extending structure near(r, z) ≈ (2.5 × 108,−3.5 × 107) cm that connects to thebulk of the unburned stellar material.

Some more details and observations can be offered re-garding the SDTs observed in our subset of models. Wefound that transitions to detonation occur in gas withpre-shock densities ≈ 1 − 3× 106 g cm−3 in models Y12and Y100 and > 5×106 g cm−3 in model Y100YM25. Asdemonstrated by (Arnett & Livne 1994b), at these den-sities the typical radius of region that can successfullytransit to detonation is ∼few kilometers. This is smallerthan the numerical resolution in our models. It is con-ceivable that problems with producing SDTs in some ofour double-point ignition models might be attributed toinsufficient resolution. For the same reason, the observedSDTs may require less time to launch detonations aftera strong shock wave forms. However, preconditioningof the fuel for SDT is certainly a temporally extendedprocess requiring both compression of the material andthermalization of the flow. The latter is aided by theconfinement that makes the thermalization process moreefficient. Still, large amounts of energy need to be sup-plied to the region and the typical velocity jumps acrossshock waves are 4, 000− 6, 000 km s−1. This guaranteespost-shock temperatures > 1 × 109 K, high enough fornuclear burning to take control of the flow dynamics.

Evolution through detonation — Transitions to detona-tions occur in different models at different times andlocations although, as we discussed earlier, several nec-essary elements (high density fuel-rich matter, strongwave, extended confinement of the region) are commonlypresent. Figure 13 shows the evolution of the equiva-lent stellar radius (defined as the radius of a sphere withthe same volume as that occupied by matter of density


Fig. 12.— Post-deflagration progenitor structure around the time of transition to detonation. Shown is density in log scale. (left): modelY12, (a) t = 3.575 s, (b) t = 3.6 s; (middle): model Y100, (c) t = 2.0 s, (d) t = 2.125 s; (right): model Y100YM25, (e) t = 1.776 s, (f)t = 1.8 s. Several structures are shown with isocontours in the lower row: T = 1 × 109 K (red), flame front (green), shocks (blue).

> 1 × 104 g cm−3) in our sample of DFD models. Thisinitial steady stellar expansion is due to energy deposi-tion by a failed deflagration. In most cases a detonationoccurs when the progenitor either approaches or beginsto recollapse. This is understandable since at later timesthe energy of surface waves is likely to quickly dissipate,thus decreasing the likelihood of strong hydrodynamicinteractions taking place (e.g. in multi-point ignitions,model Y100YM25). Alternatively, the accretion flowscan develop only once expansion stops and high accre-tion luminosities (required for SDT) can be expectedonly shortly after accretion flows develop. In particular,in models Y70YM25 and Y75YM50 (thick dotted anddotted line in Fig. 13, respectively) no SDT occurredduring the collision of surface waves. Instead, flow per-turbations accumulated in the regions near the symmetryaxis, evolved into jet-like flows and eventually triggereddetonations. With our verification study providing evi-dence that the evolution of perturbation near the sym-metry axis cannot be entirely trusted, we do not considerthese two models as successful DFDs produced by dou-ble point ignitions. (One probably could still considerthem members of single point ignition families, perhapsobtained from different initial conditions, and providedtheir radii prior to SDT were similar to original singleignition models. We do not consider this inelegant pos-

sibility any further.)Figure 14 shows the morphology of the exploding

model supernovae Y12, Y100, and Y100YM25 shortlyafter the central density drops below 1×106 g cm−3 andnuclear burning essentially quenches. Several importantobservations can be made. All model supernova ejectaare stratified. The ejecta are composed of a featurelesscore surrounded by inhomogeneous external layers. Thiscomposite structure of the ejecta is a direct result of thetwo stages of the explosion, each involving a diametri-cally different dominant process. The core is the rem-nant of the original progenitor which has been expandedby energy released during the deflagration phase. thedensity distribution in the core displays slight asymme-try reflecting the character of the initial conditions (theshift in the density maximum to the lower hemispherepresent in single ignition models is absent in double ig-nition model) . The outer layers may show global asym-metry, especially in models with a single ignition point,but are always rich in structures reflecting the turbulentnature of the deflagration and the violent evolution ofsurface waves.

The density stratification of the ejecta is accompaniedby compositional stratification. This is due to the deto-nation wave which synthesized iron peak elements whensweeping through the dense central regions and interme-

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Fig. 13.— Evolution of effective stellar radius (defined as ra-dius of a sphere with the volume occupied by matter with den-sity > 1 × 104 g cm−3) in model Y12 (thick solid), Y25 (mediumsolid), Y50 (solid), Y100 (thin solid), Y70YM25 (thick dotted),Y100YM25 (medium dotted), and Y75YM50 (dotted). Evolutiontoward SDT proceeds through overall continuous expansion fol-lowed by a possible period of recollapse. Shock breakout is followedby a very rapid expansion of stellar material at approximately con-stant velocity.

diate group elements when encountering the less denseouter layers. The detonation is driven by essentially in-stantaneous energy release due to carbon burning fol-lowed by approach to NSQE (silicon-group elements) andfinal relaxation to NSE (iron-group elements). With rel-atively crude resolution, only the relaxation to NSE andapproach to NSQE at the lowest densities can be con-sidered spatially resolved in our simulations (Khokhlov1989, Fig. 9). This problem, however, is not related tothe energy release and so does not influence the overalldynamics of the detonation front. The low resolution ofour models prevents us from considering possible effectsrelated to the curvature of the detonation front. Onceagain, such effects affect the structure of the detonationwave only on scales . 10 km (Sharpe 2001) and are notexpected to influence the large scale dynamics of the det-onation wave.

The outermost ejecta are composed of unburned fuelmixed with the deflagration products, most likely inter-mediate group elements with locally entrained iron-richmaterial. Close to the core, those layers were overrunby the detonation wave that additionally modified theircomposition. Although definitely present in our calcula-tions, we estimate this effect to be small. To calculate thedetailed composition of the ejecta, including the deflagra-tion mix, requires postprocessing nucleosynthesis (see forexample Travaglio, Hillebrandt, & Reinecke 2006) and isbeyond the scope of this presentation.

Detonations are known to be susceptible to trans-verse perturbations and developing cellular structure(Fickett & Davis 2001, Chap. 7). Potential sources forsuch perturbations are abundant in our models and in-clude upstream flow perturbations due to preexistingconvection and turbulence (not considered here), initialdeflagration, or possible numerical oscillations of grid-

aligned shock fronts, i.e. odd-even decoupling (Quirk1994). We found no clear evidence for cellular structurein our calculations. One possibility is that the numeri-cal resolution is insufficient to resolve detonation cells ofsub-km size (Gamezo et al. 1999; Falle 2000). Also, thetime available for cellular structure is very limited as lowdensity material rapidly expands following the passage ofthe detonation wave and the burning quickly quenches.

The evolution of the explosion energy (kinetic + in-ternal + potential) and burned mass from the momentof ignition until the shock breakout is shown in Fig. 15.With plenty of unburned fuel available to the detonation,DFD supernovae are energetic events with typical post-detonation energies ≈ 1.3 − 1.5 foe. The deflagrationphase typically supplies only 0.06 to 0.15 foe of energyin burning < 0.1 M⊙ of stellar fuel. The bulk of the en-ergy is released during about 0.4 s when the detonationwave sweeps through the white dwarf. These findings re-semble the results presented by (Arnett & Livne 1994b,Fig. 3) although the mechanism behind the transition todetonation is different in the two models.

3.2. Homologous expansion: final properties

The direct comparison of hydrodynamic explosionmodels to observations is accomplished through the cal-culation of synthetic model light curves, spectra, andpossibly also spectrum polarization. These radiativetransfer calculations take as an input the model super-nova ejecta with complete specification of the densitydistribution, ejecta chemical composition, and typicallymake the simplifying assumption that velocity linearlyincreases with distance from the ejecta center. Thislast assumption is satisfied to different degrees in vari-ous parts of the ejecta and generally does not hold trueduring the early stages of supernova expansion. The rea-son is that linear expansion requires establishing a finebalance of accelerations between neighboring fluid ele-ments in the ejecta, and this requires time. For example,our post-detonation explosion models contain large re-gions where energy of the flow is dominated by internalenergy. This indicates the potential of fluid elementsperforming some work, possibly adjusting their motionrelative to their neighbors. To allow for that process tooperate and establish homologous expansion, the post-explosion needs to be continued for an extended periodof time. Detailed discussion of this process in the contextof Type Ia supernova modeling was recently presented byRopke (2005).

We obtained a complete set of homologously expandingmodel ejecta using flash and its adaptive mesh capabil-ities. Post-detonation models were carefully transportedto a high-resolution uniform mesh with typical relativeerrors of total mass, total energy, and abundances notexceeding 0.1%, 0.5%, 5%, respectively. Given the sev-eral sources of uncertainties and variations in the originalmodels, this accuracy is sufficient for any practical pur-poses. The interpolated models were then used to definethe initial conditions in the flash calculations.

We used a ratio of kinetic energy to the sum of inter-nal plus gravitational energies to monitor the approachof ejecta to homology. In course of several trial compu-tations, we established that continuing calculations for100 s after explosion guarantees that the energy ratio >100 anywhere in the ejecta except for the innermost half


Fig. 14.— Post-detonation structure of the exploding supernova at the time when the central density drops to 1 × 106 g cm−3 andburning effectively quenches. Panels (a)-(c) show the density in log scale in models Y12 (t = 4.90 s), Y100 (t = 3.375 s), and Y100YM25(t = 2.878 s). Abundance isocontours X(56Ni) = 0.95 and X(28Si) = 0.2 are shown with black and gray lines, respectively.

Fig. 15.— Evolution of the total (explosion) energy through theend of detonation phase in model Y12 (thick solid), Y25 (mediumsolid), Y50 (solid), Y100 (thin solid), Y70YM25 (thick dotted),Y100YM25 (medium dotted), and Y75YM50 (dotted). Only smallamount of energy, ≈ 0.06− 0.15 foe is released during deflagrationphase. Detonation phase lasts . 0.4 s.

of the central nickel-rich core (and parts of essentiallymassless shocked ambient medium). This required usinga computational grid extending to 1.68 × 1012 cm. Cal-culations were performed using an automatic mesh dere-finement scheme that kept the computational cost ap-proximately constant as the ejecta expanded. The effec-tive mesh size varied from 8, 192 initially to 2, 048−4, 096zones per dimension at a final time.

The density and compositional structure of the homol-ogously expanding model ejecta are shown in Fig. 16.Here we show only the innermost part of the grid withexpansion velocities reaching ≈ 50, 000 km s−1 at r =5× 1011 cm. The remaining part of the volume containsa low-density shocked ambient medium and the super-nova shock. The bulk of the ejecta material displays es-sentially the same structure as in early post-detonationmodels discussed in the previous section. Most visi-ble differences can be found in the outermost regions

that here were already swept by the shock. The com-posite character of the bulk of the ejecta is preservedwith a featureless core rich in iron group elements andthe outer strongly mixed layers rich in silicon group el-ements. The compositional dichotomy of the outer lay-ers also reflects the contribution of two processes to theexplosion. The inner well-defined silicon-rich ring alsocontains an admixture of calcium (shown with white iso-contours in Fig. 16). This is due to the detonation wavenucleosynthesis calculation done with aprox13t nuclearnetwork. The outer silicon-rich and rather fragmentedshell is devoid of calcium as this species was not consid-ered in the approximate 3-stage flame burner. Improvingupon the approximate nucleosynthesis is one of the ur-gent future tasks, especially given the importance of theouter ejecta layers in formation of supernova spectrum(Kasen & Plewa 2005). It is also interesting to note thatthe each deflagration region seems responsible for form-ing its own outer silicon-rich ring. This is evident inmodel Y100YM25 (Fig. 16(c)).

Table 3 presents approximate nucleosynthetic yieldsand final explosion energies for the complete set of ho-mologous DFD models. The homologous character of themodels is confirmed by the consistently small fraction ofpotential and internal energies as compared to the totalenergy. In most models, explosion energies are in therange 1.3 − 1.5 foe. These produce between 0.9 to al-most 1.2 solar masses of 56Ni and between 0.1 and 0.24solar masses of intermediate elements. Although modelnickel masses may appear relatively high at first, suchhigh nickel masses might be typical for significant frac-tion of objects (Stritzinger et al. 2006). Furthermore,our estimates of nickel mass are likely the upper lim-its given aprox13t nucleosynthesis does not account forproduction of other iron-group elements, e.g. stable iso-topes like 54Fe. Model Y75YM50 is the least energetic(Et ≈ 1.08 × 1051 erg), produces the least amount ofnickel (≈ 0.51 M⊙) and more than a half solar mass ofintermediate mass elements. As we discussed earlier, webelieve that this model should not be considered as aDFD, as the shock to detonation transition was likelybeing promoted by numerics.

4. DISCUSSION

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Fig. 16.— Homologously expanding model ejecta ≈ 100 s after explosion. Panels (a)-(c) show density in log scale in models Y12, Y100,and Y100YM25. Abundance isocontours X(56Ni) = 0.95, X(28Si) = 0.2, and X(40Ca) = 0.05 are shown with black, gray, and whitelines, respectively. Expansion velocity is ≈ 50, 000 km s−1 at distance 5 × 1011 cm from the ejecta center. Only the innermost part ofcomputational grid is shown.

TABLE 3Homologous DFD modelsa

Model Y12 Y25 Y50 Y100 Y75YM25 Y100YM25 Y75YM50Et 1.357 1.496 1.515 1.516 1.464 1.384 1.075Ei 1.59×10−4 8.38×10−5 7.15×10−5 7.09×10−5 5.34×10−4 2.87×10−5 1.97×10−3

−Ep 2.52×10−3 2.39×10−3 2.38×10−3 2.38×10−3 2.31×10−3 2.30×10−3 2.56×10−3

4He 8.03×10−3 1.13×10−2 1.15×10−2 1.10×10−2 1.03×10−2 8.36×10−3 2.25×10−3

12C 8.73×10−3 5.49×10−3 3.30×10−3 4.56×10−3 1.29×10−2 2.05×10−2 2.52×10−2

16O 0.107 4.65×10−2 4.48×10−2 3.91×10−2 7.54×10−2 9.82×10−2 0.23720Ne 4.41×10−4 3.79×10−4 3.28×10−4 4.78×10−4 1.04×10−3 9.53×10−4 9.73×10−4

24Mg 8.70×10−2 3.40×10−2 3.42×10−2 2.81×10−2 4.51×10−2 6.74×10−2 0.19428Si 0.127 7.28×10−2 6.07×10−2 5.74×10−2 8.00×10−2 0.137 0.20232S 7.03×10−2 3.65×10−2 3.06×10−2 3.18×10−2 4.21×10−2 8.75×10−2 0.12436Ar 1.64×10−2 8.26×10−3 6.91×10−3 7.36×10−3 3.97×10−3 2.07×10−2 2.95×10−2

40Ca 1.82×10−2 8.95×10−3 7.53×10−3 8.09×10−3 1.02×10−2 2.20×10−2 3.24×10−2

44Ti 1.41×10−5 9.35×10−6 3.02×10−5 1.35×10−5 2.71×10−5 2.71×10−5 2.58×10−5

48Cr 2.96×10−4 1.49×10−4 1.42×10−4 1.42×10−4 1.78×10−4 3.43×10−4 4.83×10−4

52Fe 6.50×10−3 3.43×10−3 3.01×10−3 2.91×10−3 3.49×10−3 6.85×10−3 1.03×10−2

56Ni 0.926 1.147 1.173 1.186 1.075 0.895 0.510

We studied the fate of a massive carbon/oxygen whitedwarf following an off-center mild ignition. We foundthat such initial configurations do not produce direct ex-plosions. Only a small amount of stellar fuel is initiallyconsumed and the released energy is used to expand theprogenitor. This is in agreement with several previous in-dependent studies in which the deflagration was either in-trinsically weak (Arnett & Livne 1994b) or was initiatedslightly off-center (Niemeyer, Hillebrandt, & Woosley1996; Livne, Asida, & Hoflich 2005).

We found that the following evolution of the perturbedstellar material leads to the formation of isolated wave-dominated regions inside unburned material. We consid-ered these regions capable of launching a detonation wavethrough a shock-to-detonation transition. We observedthe resulting detonations eventually consuming the bulkof the unburned progenitor. These detonating failed de-flagrations are energetic events with explosion energies≈ 1.3 − 1.5 foe.

The model DFD ejecta appear composite, reflectingthe presence of two different physical processes contribut-ing to the explosion. The central parts of the ejecta arecomposed of a mildly deformed but completely feature-less central core rich in iron peak elements. Strongerdeformations may require different physics, e.g. rotation

(Hoflich 2005). The core is surrounded by a slightly in-homogeneous inner ring rich in silicon group elements, aproduct of the detonation burning at low densities. Fi-nally, the outermost layer is highly turbulent containinga mix of deflagration products and unburned material.Preliminary nucleosynthesis results indicate that DFDmodels typically produce over 0.9 M⊙ of iron group el-ements and 0.3M⊙ intermediate elements. The burn isalmost complete leaving essentially no carbon.

Our conclusions are based on calculations using a re-vised numerical scheme that contains substantial im-provements. We found that the energetics of deflagra-tion stage originally considered in (Calder et al. 2004,PCL) tended to overestimate buoyancy effects by a fac-tor ≈ 3. We used a set of self-heating nuclear networkcalculations and implemented a density-dependent en-ergy release scheme. Additional modifications to theapproximate nucleosynthesis were included to improvethe dynamics of the early phases still further. Withthe revised energetics, we calibrated the numerical flamespeed to match the results of detailed calculations ofTimmes & Woosley (1992).

The revised deflagration code was subsequently ver-ified against an independent set of results of centrallyignited deflagrations obtained with the Garching super-


nova code. Good qualitative agreement was found be-tween the two codes. The database of computer modelsis offered on-line to facilitate future verification (code-to-code comparison) studies.

Furthermore, a numerical procedure to stabilize modelprogenitors has been developed. These stabilized progen-itors not only provided initial conditions for supernovasimulations but also allowed us to examine the fidelity ofhydrodynamic advection in axisymmetric situations. Inparticular we found that, on the one hand, no numeri-cally stable progenitors can be obtained if resolution istoo low and, on the other hand, small perturbations arestrongly amplified near the symmetry axis in highly re-solved models. This analysis allowed us to identify theoptimal resolution for our supernova calculations.

We analyzed the observed pattern of shock-to-detonation transitions (SDTs) in some detail. We iden-tified the presence of sufficiently dense fuel, strongly ki-netic flow, wave formation, and a persistent confinementof the region with additional pressure increase due to nu-clear burning in the shocked gas as necessary conditionsfor a SDT. We found that the phenomenon of SDT isnot exclusively associated with the presence of a symme-try axis. We also found that SDTs can occur in regionscompletely free of possible geometrical boosting effects,i.e. near the equatorial plane.

However, transitions to detonations were not a robustprediction of such models. There are some possible rea-sons for that. For example, by assuming axisymmetry weeliminate an angular direction in which additional per-turbations may develop. Such perturbations will enrichthe flow field creating more seed points for SDT and,at the same time, increase the amplitude of fluctuations.That is, the assumed symmetry is likely limiting the pos-sible wave interactions and presumably denying extremeevents such as a SDT. In addition, for stability reasons,our calculations had to be performed using suboptimalnumerical resolution. This caused a strong numericaldamping and limited sampling of the perturbed regionsharming chances for observing SDTs still further. Onthe other hand, we did not include physics that may, ef-fectively, make the system behavior appear more viscous(e.g. magnetic fields) inhibiting formation of small scalestructures.

Our findings also hinge on the assumption that theSDT process is relatively insensitive to details of evolu-tion on scales unresolved in our simulations. This re-mains to be demonstrated, ideally in the course of dedi-cated highly-resolved model calculations of compression-ally heated fuel-rich degenerate mixtures. It will bea daunting task. If any parallel can be drawn, expe-rience accumulated by modelers of inertially confinedfusion systems might be of great help in such studies(Atzeni & Meyer-ter-Vehn 2004; Drake 2006). Even ifsuch models are successfully constructed, many doubtswill remain regarding the outcome of such calculationsgiven how limited our knowledge about real systems is.For example, as we mentioned before (PCL), although weconsider a pure carbon/oxygen progenitor it is almostcertain that in nature progenitor’s surface layers con-tain admixture of helium (Nomoto 1982; Yoon & Langer2005b). Compositional changes will affect energetics ofnuclear burn adding entirely new dimension to the prob-lem.

Being mindful of numerous simplifying assumptionsand model inaccuracies, the essential findings of thiswork are, therefore, rather modest and can be sum-marized as evidence of strong, localized, and prolongedshock heating in regions rich in fuel. We note that theseare necessary conditions for shock-to-detonation transi-tion to occur. We believe this observation is independentof particular details of our model, especially numerics,making SDT one of prime suspects for triggering deto-nations in SN Ia.

However, even if no seed point forms a detonationthrough SDT, this second, after the initial deflagration,failure to unbind the star in no way automatically im-plies supernova will not occur. Perhaps just the opposite.The extensive large-scale mixing of deflagration productswith unburned outer stellar layers combined with abun-dantly present strong acoustic perturbations appear theconditions are ripe for the Zeldovich gradient mechanism(Khokhlov, Oran, & Wheeler 1997). We may expectthat for rotating progenitors (Yoon & Langer 2005a) per-turbations of surface layers will be even stronger due topresence of additional shear component. If all these op-portunities are missed, the white dwarf might still begiven another chance to produce the supernova. A failedattempt to explode would then be a beginning of a cy-cle that repeats, perhaps several times, as the expandedwhite dwarf eventually cools down, shrinks, and preparesfor another ignition. That is, the explosion process mightbe a lengthy one, a kind of laborious slow death.

As the observations improve, we are also beginningto collect evidence that the degree of diversity of SN Iamight be greater than original anticipated (or desired!).Recent observations of the peculiar supernova SN 2002cxby Jha et al. (2006) argue in favor of low energy ex-plosion and large degree of mixing in this object, twocharacteristics that essentially preclude any involvementof a detonation in the explosion process. Other objectslisted by Jha et al. (2006) may belong to SN 2002cx class.These rare peculiar supernovae might be genuine exam-ples of pure deflagrations. Or perhaps these are objectsthat underwent several failed deflagrations leaving onlysmall amount of material to fuel the detonation. If so,normally bright supernovae might be DFDs that suc-ceeded early, with the occurrence of a detonation (or lackthereof) being the primary element determining the ob-servational properties of a given event.

On the other hand, SNe Ia display some characteristicsthat we find difficult to explain in the framework of puredeflagrations. One example are iron-rich clumps foundin Type Ia SNRs (Warren & Hughes 2004; Warren et al.2005). In the model proposed here, the inner ring ofintermediate elements seems to be a natural site forproducing nickel-rich blobs that may float away fromthe central core (Blondin, Borkowski, & Reynolds 2001;Wang & Chevalier 2001). Those radioactive blobs mayalso be explained in pure deflagration models that nat-urally produce clumpy ejecta. However, dominant andisolated regions rich in iron-group elements like the oneobserved in Tycho SNR (Vancura, Gorenstein, & Hughes1995; Warren et al. 2005) can hardly be produced in apure deflagration in which several such regions are ex-pected to be simultaneously present. In DFDs, such anisolated clump located near the outer edge of the super-nova remnant might be a material burned deep in the

20 Plewa

progenitor core and transported to the stellar surface byone of the deflagrating bubbles.

We cannot address the above question without detailednucleosynthesis calculations. This is one of the possi-ble future directions for research. In addition, the re-laxation of the assumption of axial symmetry, althoughcostly, will be necessary. But even with only the currentapproximate nucleosynthesis and simplified geometry ofthe problem, we are in a position to conduct prelimi-nary validation studies against observations for a subsetof DFD models. This will be the subject of the nextcommunication in this series.

Todd Dupont and Dan Kasen provided me with bothsupport and encouragement for continuing this work. Iwould like to thank Timur Linde for contributing theflame surface integrator, Frank Timmes for providingthe initial white dwarf model, Bronson Messer for help-ing in verifying and developing the approximate defla-

gration network, Cal Jordan for extending the originalnuclear burning network, and the anonymous referee forcomments that led to the improvements of the initialversion of the paper. I enjoyed stimulating and help-ful discussions with Carles Badenes, Peter Hoflich, DanKasen, Alexei Khokhlov, Jens Niemeyer, and Joe Shep-herd. This work is supported in part by the U.S. Depart-ment of Energy under Grant No. B523820 to the Centerfor Astrophysical Thermonuclear Flashes at the Univer-sity of Chicago. It benefited from the INCITE award andlater computing allocations provided by the National En-ergy Research Scientific Computing Center, which is sup-ported by the Office of Science of the U.S. Department ofEnergy under Contract No. DE-AC03-76SF00098. Ad-ditional computations were performed on the Teraportcluster, part of the Teraport project at the Universityof Chicago funded through National Science FoundationGrant No. 0321253.

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Detonating failed deflagration_model_of_themonuclear_supernovae_explosion_dynamics

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