Convection during the Late Stages of Simmering in Type Ia Supernovae
aa r X i v : . [ a s t r o - ph ] J a n A CCEPTED FOR PUBLICATION IN T HE A STROPHYSICAL J OURNAL
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CONVECTION DURING THE LATE STAGES OF SIMMERING IN TYPE IA SUPERNOVAE A NTHONY
L. P
IRO AND P HILIP C HANG Astronomy Department and Theoretical Astrophysics Center, University of California, Berkeley, CA 94720;[email protected], [email protected]
Accepted for publication in The Astrophysical Journal
ABSTRACTFollowing unstable ignition of carbon, but prior to explosion, a white dwarf (WD) in a Type Ia supernova(SN Ia) undergoes a simmering phase. During this time, a central convective region grows and encompasses ∼ M ⊙ of the WD over a timescale of ∼ yrs, which sets the thermal and turbulent profile for the subsequentexplosion. We study this time-dependent convection and summarize some of the key features that differ fromthe traditional, steady-state case. We show that the long conductive timescale above the convective zone andthe extraction of energy to heat the WD core leads to a decrease of the convective luminosity and characteristicvelocities near the convective zone’s top boundary. In addition, differences in the composition between theconvective core and the conductive exterior will significantly alter the location of this boundary. In this respect,we find the biggest effect due to complete Ne sedimentation prior to carbon ignition. These effects adddiversity to the possible WD models, which may alter the properties of the SN Ia explosion.
Subject headings: convection — supernovae: general — white dwarfs INTRODUCTIONThe use of Type Ia supernovae (SNe Ia) as cosmic dis-tance indicators has focused attention to the study of whitedwarf (WD) explosions. Of particular importance is deter-mining the parameters that dictate the observed diversity ofSNe Ia. Recent modeling demonstrates that the variationalong the width-luminosity relation (Phillips et al. 1999) maybe explained by large variations in the abundance of stableiron group elements (Kasen & Woosley 2007; Woosley et al.2007) with the dominant cause for diversity likely resid-ing in the explosion mechanism (Mazzali et al. 2007). An-other important variable is the metallicity of the WD core(Timmes et al. 2003).It is critical to explore the initial conditions that may add tothe diversity. Simulations of turbulent thermonuclear flamesin WDs have demonstrated that the composition and energy ofejecta depend sensitively on the competition between flamepropagation, instabilities driven by turbulence, and expan-sion of the WD (Hillebrandt & Niemeyer 2000, and refer-ences therein). These simulations should therefore dependon the thermal and turbulent state of the WD set by the pre-explosive convective simmering phase (Nomoto et al. 1984;Woosley & Weaver 1986).During the early stages of simmering many studies havefocused on the convective Urca process (Paczy´nski 1972;Bruenn 1973; Couch & Arnett 1975; Iben 1978a,b, 1982;Barkat & Wheeler 1990; Mochkovitch 1996; Stein et al.1999; Bisnovatyi-Kogan 2001; Lesaffre et al. 2005;Stein & Wheeler 2006). This occurs when nuclei re-peatedly electron capture and beta decay as they are carriedby convection back and forth across the electron capturethreshold density (the “Urca shell”). Although the energyloss from this process is most likely not great enough tocause global cooling, it can still have a significant effecton the convective motions (Lesaffre et al. 2005). Once thecentral temperature has grown above ≈ (5 - × K Miller Institute for Basic Research, University of California, Berkeley,CA 94720. (which corresponds to ∼ s before the burning wavebegins), there is no longer time for electron captures on Na(Piro & Bildsten 2007), and the convective Urca process willcease. During the last ∼ s any compositional gradientsare mixed homogeneously by subsequent convection.An additional place where simmering is important isfor understanding the conditions within the WD imme-diately prior to the explosion (García-Senz & Woosley1995; Höflich & Stein 2002; Woosley et al. 2004;Wunsch & Woosley 2004). The properties of the tem-perature fluctuations present in the convection set the sizeand distribution of the ignition points, which are crucial fordetermining the success of the subsequent burning wave (seeRöpke et al. 2006, and references therein). The interaction ofconvection with rotation sets the morphology of convectivemotions (Kuhlen et al. 2006) as well as the overall rotationprofile of the WD (Piro 2008).The last way simmering has gained attention is in itsability to enhance the neutron abundance in the WD core(Piro & Bildsten 2007; Chamulak et al. 2007). This hap-pens primarily via the reaction chain C( p , γ ) N( e - , ν e ) C,where the protons are leftover from C burning. Depend-ing on the amount of carbon that is consumed before burningbecomes dynamical, as well as the density at which it takesplace, this neutronization enhancement could very well belarge enough to mask any trend expected with metallicity inenvironments that have roughly sub-solar metallicity.In this present work we focus on the general properties ofthe simmering convection, with the aim of identifying char-acteristics that may introduce diversity to the SN Ia progeni-tors. We begin in §2 by presenting the main features of ourmodels. We illustrate how time-dependent convection in thesimmering phase differs from the familiar case of steady-stateconvection. In this new picture, the convective flux decreasesoutside the central heating zone due to both the heating ofnew material as the convective region grows and the inabilityto transfer significant energy to the conductive exterior. In §3we explore the location of the top of the convective zone. Wepoint out that degeneracy effects enhance the response of the PIRO & CHANGboundary location to changes in composition. We concludewith a summary of our results and a discussion of future workin §4. LUMINOSITY AND CHARACTERISTIC VELOCITIESFOR EXPANDING CONVECTIONWe begin by summarizing the main features of our simmer-ing models. (For further details, the interested reader shouldrefer to Woosley et al. 2004; Lesaffre et al. 2006; Piro & Bild-sten 2007; Piro 2008.) Unstable ignition of C occurs whenthe heating from carbon fusion beats neutrino cooling. Thecentral temperature then rises and a convective zone growsoutward, eventually encompassing ∼ M ⊙ of the WD after ∼ yrs. As the central temperature, T c , increases, carbonburning becomes more vigorous and the heating timescale, t h ≡ ( d ln T c / dt ) - , gets shorter. This timescale in general de-pends on the size of the region responding to the rising centraltemperature at the core.Simmering ends and a burning wave commences once t h . t conv , where t conv is eddy overturn timescale. At these latetimes, individual eddies may experience significant heatingduring their transit (García-Senz & Woosley 1995), so thatthe temperature profile is no longer an adiabat and the en-tire convective core does not respond to the increasing centraltemperature. (In contrast, we show below that during the ma-jority of the time t h depends on the heat capacity of the entireconvective mass.) This makes it difficult to exactly calculatethe precise moment when simmering ends. For this reasonLesaffre et al. (2006) explore t conv = α t h , where α . t conv ≈ t h ≈ c p T c /ǫ , where c p is the specific heat capac-ity at constant pressure, ǫ is the heating from carbon burn-ing, and all these quantities are evaluated at the WD center.This estimates that simmering should end when t h ≈ T c ≈ . × K and ρ c ≈ . × g cm - , which is roughly in agreement withthe results presented by Woosley et al. (2004) using the Ke-pler stellar evolution code (Weaver et al. 1978).We follow the simmering phase by calculating a series ofhydrostatic WD models, each with a different central tem-perature, but at a fixed mass (see Piro & Bildsten 2007; Piro2008). For simplicity we ignore the convective Urca processsince our focus is on the last ∼ s. The timescale for ther-mal conduction across the WD is t th ≡ K c / R ∼ yrs, where K c is the conductivity and R is the radius, which is muchlonger the timescale over which heating is occurring. There-fore the convection efficiently mixes entropy and the convec-tive region nearly follows an adiabat out from the WD center.Outside the convective zone, we assume the WD is isothermalwith a temperature T i .2.1. Convective Luminosity
To understand how time-dependent convection is differentthan from that normally found in steady-state convection, wefocus on the time-dependent entropy equation c p ∂ T ∂ t = ǫ - ∂ L c ∂ M r , (1)where L c is the convective luminosity. This equation omits thework required to expand the WD as the heating takes place,which is a significant amount of energy and thus requires some discussion. For each convective model we compared thetotal change in WD binding energy to the total change in in-ternal energy of the electrons (which are primarily degenerateand relativistic), including the ion-electron Coulomb interac-tion energy (according to Chabrier & Potekhin 1998). Thesetwo quantities are equal to the numerical accuracy of our inte-grations, which demonstrates that all of the work required toexpand the WD comes from changes in the internal energy ofthe electrons. Thus, the entropy created from nuclear burningall goes into convective motions or the internal thermal en-ergy, and we are justified in omitting the binding energy andelectron internal energy terms from equation (1).The temperature profile in the convective zone follows anadiabat with a power law index n ≡ ( ∂ ln T /∂ ln P ) ad . Thetime derivative of the temperature at a given pressure can beexpressed as ∂ T /∂ t = ( P / P c ) n (cid:2) ∂ T c /∂ t + T c ln( P / P c ) ∂ n /∂ t (cid:3) ≈ ( P / P c ) n ∂ T c /∂ t , (2)where P c is the central pressure, and for simplicity we areassuming that it does not change appreciably in time. Fromthis we see that the timescale for the temperature change atany pressure is set by the central temperature change ∂ ln T ∂ t = ∂ ln T c ∂ t ≡ t h . (3)Therefore there is a well-defined, global heating timescale, t h . To account for the changing central pressure in a more rig-orous calculation, we must take partial derivatives at constantmass coordinate, M r . This can be performed by inverting theempirically found M c ( T i , T c ) relation presented in Piro (2008;or see eq. [11] below), where T i is the nearly isothermal tem-perature of the non-convective, conductive region. The resultis T ( T c , M r ) = 0 . T c (cid:20) - (cid:16) µ e (cid:17) M r . M ⊙ (cid:21) , (4)where µ e is the mean molecular weight per electron. This canbe used to find ( d ln T / dt ) M r = d ln T c / dt , which confirms ourconclusion that t h is the same at any depth within the convec-tive zone.Multiplying equation (1) by dM r = 4 π r ρ dr and integrat-ing, Z M r c p ∂ T ∂ t dM r = Z M r ǫ dM r - L c ( M r ) + L c (0) . (5)We pull t h outside of the left-hand integral to find Z M r c p T ∂ ln T ∂ t dM r = 1 t h Z M r c p T dM r = E th ( M r ) t h , (6)where E th ( M r ) is the integrated thermal energy up to a masscoordinate M r . We set L c (0) = 0 and define the nuclear lumi-nosity as L nuc = R ǫ dM r , so that equation (5) becomes E th ( M r ) t h = L nuc ( M r ) - L c ( M r ) (7)Equation (7) expresses that the nuclear luminosity must ei-ther go into thermal heating or convective motions, and it is Note that this timescale is different than the local timescale, c p T c / ǫ , usedabove for estimating the end of simmering. This is because, with the excep-tion of late times, the convective zone is well-coupled. ONVECTION DURING SIMMERING 3 F IG . 1.— The convective luminosity, L c and characteristic velocities, V c ,as a function of mass coordinate, M r . The upper (lower) lines in each panelare for a central temperature of T c = 8 × K (6 × K). Solid linesare the estimates for time-dependent convection using eq. (9). The dashedlines are calculations that assume L c ( M r ) = L nuc ( M r ), which are plotted forcomparison. valid at any M r . It contains two unknowns, t h and L c ( M r ). Weset the luminosity at the surface of the convective zone to bezero, L c ( M c ) = 0. This boundary is required since t th is longin the non-convective regions, which prevents significant heattransfer. We can then solve for t h , t h = E th ( M c ) / L nuc ( M c ) , (8)which matches the definition of t h that Weinberg et al. (2006)use in the context of type I X-ray bursts on neutron stars. Wesubstitute t h back into equation (7) to get the convective lumi-nosity L c ( M r ) = L nuc ( M r ) - E th ( M r ) / t h = L nuc ( M r ) (cid:20) - E th ( M r ) E th ( M c ) L nuc ( M c ) L nuc ( M r ) (cid:21) . (9)For steady-state convection, L c ( M r ) = L nuc ( M r ). The ratio E th ( M r ) / E th ( M c ) is the modification due to the growing na-ture of the convection and the L nuc ( M c ) / L nuc ( M r ) term is fromlong thermal time for the conductive exterior, which forces L c ( M c ) = 0.In the upper panel of Figure 1 we plot as solid lines theconvective luminosity found using equation (9). These mod-els all have a composition of 0.5 C, 0.48 O, and 0.02 Ne by mass fraction, with a mass of 1 . M ⊙ and an initialisothermal temperature T i = 10 K. We solve for ρ using thePaczy´nski (1983) fit for the equation of state, and include theCoulomb energy of Chabrier & Potekhin (1998). We presentcentral temperatures of T c = 6 × and 8 × K, which cor-responds to t h = 14 hrs and 170 s, respectively. The t h associ- ated with the latter case is an overestimate since, as mentionedabove, at these late times during simmering only inner por-tion of the core responds to the rising central temperature (ineffect decreasing E th in eq. [8]). The energy generation ratefor C burning is taken from Caughlan & Fowler (1988) withstrong screening included from Salpeter & van Horn (1969).Also plotted in Figure 1 is the convective luminosity for L c ( M r ) = L nuc ( M r ), i.e., for steady-state convection (dashedlines). Near the center, E th ( M r ) is small and grows lessquickly than L nuc ( M r ), so L c ( M r ) is initially ≈ L nuc ( M r ) (seeeq. [9]). At larger M r , L c decreases due to the effects we havehighlighted. 2.2. Convective Velocities
If we take a characteristic eddy scale l c , the thermal con-duction timescale across an eddy is ∼ ( l c / R ) yrs ∼ ( H / R ) yrs ∼ yrs, where H is the pressure scaleheight. Since this timescale is long, the convection is efficient(Hansen & Kawaler 1994). Using estimates from mixing-length theory, the characteristic convective velocity, V conv , isrelated to F conv via V c = (cid:18) Q gl c c p T F c ρ (cid:19) / ∼ (cid:18) F c ρ (cid:19) / , (10)where Q = - ( ∂ ln ρ/∂ ln T ) P and g = GM r / r is the local grav-itational acceleration, and we have set l c ≈ H .In the bottom panel of Figure 1 we have plotted V c , set-ting the mixing-length to the scale height, l c = H ( solid lines ).The shape of these velocity profiles are similar to thoseLesaffre et al. (2005) present in the context of studying theconvective Urca process, which is active at much earlier timesduring the simmering. The convective velocities are differentby as much as ∼
50% from the naïve estimate of L c = L nuc ( dashed lines ) near the top of the convective zone. THE CONVECTIVE BOUNDARY LOCATIONBuoyantly rising eddies ascend until their density matchestheir surroundings. The boundary between the convective andisothermal zones is therefore set by a neutral buoyancy condi-tion. In practice this means that both the pressure and densitymust be continuous. If both the convective and isothermalregions have the same composition, the boundary is simplyset by when the adiabatic temperature of the convective zonereaches the isothermal temperature, T i (i.e., where the entropymatches, Höflich & Stein 2002). If the composition is differ-ent, a buoyantly rising eddy will be prevented from passingvery far into the isothermal region and truncates the size ofthe convective zone. This creates an abrupt change in temper-ature at the boundary of size ∆ T . In actuality this change willbe smoothed by overshoot and mixing, so that the entropy re-mains continuous (see, for example, Kuhlen et al. 2006). Thethermal and compositional structure at this boundary will becomplicated, especially at the last moments of the simmering,when relatively few convective overturns take place for anygiven location of the convective boundary. Since the convec-tion is always sub-sonic, we expect the overshoot to be mod-est and the simplification of a sharp compositional boundaryto be adequate to estimate the size of the convective zone.Piro (2008) presented an empirically derived relationshipfor the convective boundary, M c , as a function of the ratio of PIRO & CHANG TABLE 1C
OMPOSITIONAL S UMMARY FOR N UMERICAL M ODELS
Model Purpose Zone X X X X the isothermal and central temperatures, T i / T c , M c = 1 . M ⊙ (cid:18) µ e (cid:19) (cid:20) - . T i T c (cid:21) . (11)We make the substitutions M c → M c + ∆ M c and T i → T i + ∆ T to solve for the change in the convective mass due to a tem-perature discontinuity, ∆ M c = - . M ⊙ (cid:18) µ e (cid:19) (cid:18) T c / T i (cid:19) ∆ TT i , (12)where we have scaled to a temperature ratio T c / T i = 8, as isappropriate for near the end of the simmering phase.The temperature discontinuity allows a small conductivewave to propagate out from the top boundary of the convectivezone. Since the thermal conduction timescale ( t th ∼ yrs)is long in comparison to the heating timescale ( t h ∼
10 s - yrs), this wave can only travel a distance ∼ ( t h / t th ) / H ≪ H before the growing convective zone overtakes it. For thisreason, we ignore this detail in the following calculations.In the next sections we study the change in the convectiveboundary analytically. We discuss two main ways in whichcompositional discontinuities can be important. These arechanges in the neutron abundance and changes in the compo-sition, which affects the Coulomb corrections to the equationof state. In §3.3 we compare the models summarized in Table1 numerically to confirm these analytic results.3.1. Neutron Abundance Discontinuity
We first study differences in neutron abundance betweenthe convective and isothermal zones. The neutronization istypically expressed as Y e = 1 µ e = X i Z i A i X i , (13)where A i and Z i are the nucleon number and charge of species i with mass fraction X i . The initial metallicity of the SN Iaprogenitor is determined by the isotope Ne, which has twoadditional neutrons (Timmes et al. 2003). A mass fraction X of Ne decreases Y e by an amount ∆ Y e = 2 X / ≈ . × - X / .
02. A large enhancement of Ne could be presentin the convective core if substantial gravitational separationhas occurred (Bildsten & Hall 2001; Deloye & Bildsten 2002;García-Berro et al. 2007). Neutron enhancement in the con-vective zone can also occur from electron captures duringthe simmering, which decreases Y e by an amount ∆ Y e ∼ - - - (Piro & Bildsten 2007; Chamulak et al. 2007).We first consider the simplest equation of state where rel-ativistic degenerate electrons dominate the pressure with cor-rections from the ideal gas of ions (in §3.2 we consider Coulomb corrections), P = K (cid:18) ρµ e (cid:19) / + ρ k B T µ i m p , (14)where K = 1 . × cgs and µ i is the mean molecularweight per ion.We set the temperature to T i + ∆ T and mean molecularweight per electron to µ e + ∆ µ e at the top of the convectionzone, where T i and µ e are the values of these quantities withinthe isothermal zone. We assume that µ i does not change ap-preciably. Note that enhanced neutronization in the convec-tion implies ∆ µ e >
0. We set both P and ρ to be continuousat the convective boundary, and expand to first order, ∆ TT i = 43 µ i m p k B T i K ρ / µ / e ∆ µ e µ e . (15)Recognizing that E F = 4 K ( ρ/µ e ) / m p , this can be writtenmore conveniently as ∆ TT i = 13 µ i µ e E F k B T i ∆ µ e µ e = 13 µ i µ e E F k B T i ∆ Y e Y e , (16)where we have defined ∆ Y e to be positive (i.e., µ e + ∆ µ e = Y e - ∆ Y e ). Setting E F = 1 . /µ e ) / ρ / , where ρ = ρ/ g cm - , this is rewritten as ∆ TT i = 5 . × ρ / T i , (cid:16) µ i . (cid:17) (cid:18) µ e (cid:19) / ∆ Y e Y e , (17)where T i , = T i / K and µ i = 13 . X = X = 0 .
5. The large prefactor demonstrates that degeneracygreatly enhances the small changes in µ e on the temperaturediscontinuity. This is because the ions provide only a smallcontribution to the pressure, so that a large temperature jumpis needed to offset a small change in density. Using equation(12), we find ∆ M c = - . M ⊙ ρ / T c , / (cid:16) µ i . (cid:17) (cid:18) µ e (cid:19) / ∆ Y e / Y e - , (18)where T c , = T c / K. It is interesting to note that although T i affects the absolute location of M c (see eq. [11]), ∆ M c is infact independent of T i .3.2. Mean Molecular Weight per Ion andCoulomb Correction Discontinuities
We next consider the temperature difference from changesin the composition of the nuclei. This is of relevance be-cause evolution of the progenitor star during the heliumburning stage enhances C versus O with larger radii(Straniero et al. 2003).First we present the effect of an increase of the mean molec-ular weight in the convection zone, ∆ µ i , using the equation ofstate given by equation (14). Setting the density and pressurecontinuous across the convective boundary, ∆ TT i = ∆ µ i µ i . (19)As a concrete example, we consider the composition in Model3 from Table 1, which gives ∆ µ i /µ i ≈ .
06. This implies aONVECTION DURING SIMMERING 5 F IG . 2.— Temperature profiles for the compositional models summarized inTable 1. For each model we consider a central temperature of T c = 6 × K( dashed lines ) and T c = 8 × K ( solid lines ). All models have an isothermaltemperature of T i = 10 K and a mass of 1 . M ⊙ . The convective boundaryis at the temperature break. change in convective mass of ∆ M c = - . × - M ⊙ ρ / T c , / (cid:18) µ e (cid:19) ∆ µ i /µ i . , (20)which is negligible.However, compositional changes also alter the Coulombcontributions to the internal energy. For a multi-componentplasma, the strength of this effect is measured via theCoulomb parameter Γ = h Z i e ak B T = 21 . ρ / T h Z i (cid:18) . µ i (cid:19) / , (21) (Schatz et al. 1999) where a is the mean ion separation de-fined as 4 π a ρ/ (3 µ i m p ) = 1 and h Z i = µ i X i Z i A i X i . (22)We take the fitting function found by Chabrier & Potekhin(1998) in the limit Γ ≫ P = K (cid:18) ρµ e (cid:19) / + ρ k B T µ i m p (cid:20) - A Γ (cid:21) , (23)where A = 0 . Γ , we write Γ = Γ / T . Within the convection, Γ is larger by a fractionalamount ∆ΓΓ = ∆Γ Γ - ∆ TT i , (24)where ∆Γ is strictly from changes in composition (i.e.,changes in h Z i and µ i ). Setting the pressure to be continuousacross the convective boundary we find ∆ TT i = A Γ ∆Γ Γ . (25)Due to the factor of Γ , a small fractional change ∆Γ / Γ ofmerely ∼
10% (as present for Model 3) implies a mass changeof ∆ M c = - . M ⊙ ρ / T c , / h Z i (cid:18) . µ i (cid:19) / (cid:18) µ e (cid:19) ∆Γ / Γ . . (26)Comparing equations (18), (20), and (26) shows that bothchanges in Y e and in Coulomb corrections can make non-negligible corrections to the convective boundary. Which ef-fect is largest depends on the specific progenitor model.3.3. Numerical Models
We now calculate a few models to compare how the con-vective boundary changes due to compositional discontinu-ities. These models use the same microphysics as described in§2.1. Each model is summarized in Table 1 and is motivatedby a plausible progenitor scenario. Model 1 does not have acompositional discontinuity and is included for purposes ofcomparison. Model 2 shows the effects of neutronization byhaving a ∆ Y e ≈ × - , comparable to the level of neutron-ization found by Piro & Bildsten (2007). For simplicity weassume that all the additional neutrons are in C (instead ofheavier elements such as Ne). Model 3 considers a changein the mass fractions of C and O to show the effects ofa compositional gradient. Finally, Model 4 shows the maxi-mum effect of sedimentation by assuming that all of the Nehas sunk into the convective core.In Figure 2 and 3 we compare the four models for centraltemperatures of T c = 6 × K and 8 × K, and a WD massof 1 . M ⊙ . In Figure 2 we set T i = 10 K, while in Figure 3we set T i = 2 × K. While our models are not self-consistentin that we assume a fixed, non-evolving compositional discon-tinuity, they show that the location of the convective bound-ary can vary substantially depending on the progenitor modeland the nuclear reactions that take place during the simmer-ing. Comparing Figures 2 and 3 shows that ∆ M c can be large PIRO & CHANG F IG . 3.— The same as Fig. 2, but with T i = 2 × K. even when T i is increased. This is roughly consistent with theanalytics of §§3.1 and 3.2 that shows ∆ M c is independent of T i . CONCLUSION AND DISCUSSIONWe have highlighted some important properties of the con-vection present during the pre-explosive carbon simmeringphase of SNe Ia. The convective velocities near the top bound-ary are decreased significantly because the convective lumi-nosity is extracted to heat and grow the convective zone andthe long conductive timescale of the non-convective exteriorenforces L c = 0 at the top of the convective zone. The size ofthe convective zone can change depending on compositionalgradients within the progenitor WD.Perhaps the most severe effect depends on whether there is time available for sedimentation of Ne prior to carbon igni-tion. There remains considerable uncertainty in the timescalefor this process to occur. For a 1 . M ⊙ WD, complete sed-imentation requires ∼ B / π ∼ ρ V c impliesfields on the order of B ∼ G. In addition, the motion ofthese convective eddies may stochastically excite waves (both g -modes and p -modes). These waves propagate away fromthe convective zone and transfer energy to shallower regions.Both of these processes may prove important in understand-ing the evolution toward explosive burning, and we plan toinvestigate them in more detail in a future study.Multidimensional simulations are needed to study thephysics of this simmering phase not captured by our simplecalculation. These simulations are very challenging in that theflow must be followed for many turnover times. As a result,implicit Eulerian schemes (see for instance in two dimensionsof Höflich & Stein 2002; Stein & Wheeler 2006), anelasticcodes (Kuhlen et al. 2006) or a low Mach number formu-lation (Almgren et al. 2006a,b) are required. These simula-tions will be helpful in characterizing the flow topology (e.g.,Kuhlen et al. 2006) and the number and spatial distribution ofignition points.The simmering phase helps set the nature of the eventualrunaway and subsequent explosion. Woosley et al. (2004)and Wunsch & Woosley (2004) demonstrate that the sum-mering is paramount for setting the initial ignition pointsfor explosive burning. The large scale structure of the con-vective turbulence may also be important for flame propa-gation. For instance, Höflich & Stein (2002) argue that theinitial velocities of the burning fronts are determined bythe background convective motions and not by the laminarflame speed or Rayleigh-Taylor instabilities. This suggeststhat the initial explosive burning is likely to be off-centered(García-Senz & Woosley 1995; Woosley et al. 2004). Thebackground turbulent state will also likely subsequently af-fect the motion of the bubbles by acting as a viscous drag(Zingale & Dursi 2007).Finally, the material above the convective core is devoidof this turbulence. The burning properties may change inan interesting manner as the flame passes into this relatively"quiet" region. Some have argued that a delayed detonationtransition (DDT) of the burning may be needed to match ob-servations (Plewa et al. 2004; Livne et al. 2005). Althoughthe concept of a DDT has been considered for some time(Khokhlov 1991; Woosley & Weaver 1994), how and if it oc-curs is still uncertain (Niemeyer & Woosley 1997; Niemeyer1999; Woosley 2007). The position of the convective bound-ary may be an important detail. At the end of simmering ittypically lies at a density of ∼ × - g cm - , whichmay become near the density usually invoked for a DDT( ∼ g cm - ) once the WD has expanded from the propa-ONVECTION DURING SIMMERING 7gation of the deflagration wave.We thank Lars Bildsten, Eliot Quataert, Nevin Weinberg, and Stan Woosley for helpful discussions. P. C. is supportedby the Miller Institute for Basic Research.) once the WD has expanded from the propa-ONVECTION DURING SIMMERING 7gation of the deflagration wave.We thank Lars Bildsten, Eliot Quataert, Nevin Weinberg, and Stan Woosley for helpful discussions. P. C. is supportedby the Miller Institute for Basic Research.