THE C-FLAME QUENCHING BY CONVECTIVE BOUNDARY MIXING IN SUPER-AGB STARS AND THE FORMATION OF HYBRID C/O/Ne WHITE DWARFS AND SN PROGENITORS

The evolution of a single star is determined by its initial chemical composition and mass. In this paper, we consider models of super asymptotic giant branch (SAGB) stars of compositions close to solar. By definition, the initial masses of SAGB stars have to be sufficiently high (M > M_up) to burn C in their hydrogen and helium exhausted cores, while being too low (M < M_mas) to ignite neon afterward (Garcia-Berro & Iben 1994). The main products of C burning are oxygen and neon; therefore, the SAGB stars build ONe degenerate cores in their interiors. What will eventually happen with the ONe core depends on the ratio of its growth rate and the stellar mass-loss rate (Poelarends et al. 2008). Like in their less massive AGB counterparts, the cores of SAGB stars can increase in mass because they are surrounded by H and He burning shells. The latter intermittently experiences thermal pulses, possibly followed by dredge-up, after having accumulated a certain amount of fresh He from the overlying H shell. One of the major uncertainties in the core growth rate comes from the poorly constrained depth of the He shell that is reached by the base of the surface convection zone immediately after a He thermal pulse, during the third dredge-up. The details of the cumulative core-growth depend on the convective boundary mixing assumptions for the bottom boundary of the convective envelope that may or may not trigger the third-dredge up. If the core mass reaches the Chandrasekhar limit M_Ch ≈ 1.37 M_☉ before the star has lost its H-rich envelope, then electron-capture reactions, starting on Mg and Ne, will trigger its collapse leading to an electron-capture supernova (SN; Miyaji et al. 1980). Otherwise, an ONe white dwarf (WD) will be born (Pumo et al. 2009).

The neutrino energy losses in high-density CO degenerate cores, predominantly via the photo and plasma neutrinos, cause a temperature inversion with the maximum located some distance from the center. Therefore, just as the He-core flash, C burning in SAGB stars starts off-center if the initial mass is not too close to M_mas. Because of the degenerate conditions, the first episode of C ignition takes the form of a thermal flash that slightly reduces C abundance in the C-shell convective zone and eventually dies out. The following phase of C burning occurs near the bottom of the extinct convective zone under less degenerate conditions. This time, the C flame propagates all the way toward the center in the form of deflagration, pulling along the convective zone (Figure 1). However, this is true only for the case when all other possible mixing processes, except convection operating within Schwarzschild boundaries, are neglected (Siess 2006). The situation radically changes when thermohaline mixing is added below the bottom of the C-shell convective zone, where the mean molecular weight increases with the radius as a result of the off-center C burning. Siess (2009) has shown that the C flame stops propagating toward the center in this case because thermohaline mixing deprives the C-flame precursor of fuel. We can reproduce this behavior (Figure 2) when making the same assumption on the efficiency of thermohaline mixing (see below). In that case, by the end of C burning, as much as 2%–5% of unburnt C (in mass fraction) remains inside the ONe core which, according to Siess (2009), may modify its electron-capture induced collapse later on.

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However, the efficiency of thermohaline mixing has since been investigated in more detail. When it starts, thermohaline mixing has a pattern of rising and sinking fluid parcels that resemble fingers. The latter are often called "salt fingers" because it is an unstable salinity distribution that causes their growth in the ocean. In one-dimensional (1D) computations, the efficiency of thermohaline mixing can be parameterized using the salt-finger aspect ratio a = l/d, where l and d are a finger's length and diameter (Denissenkov 2010). In his simulations, Siess (2009) has actually used a value of a ≈ 7 needed to explain the observed evolutionary declines of the surface C abundance and ¹²C/¹³C ratio in low-mass stars on the red giant branch (RGB) above the bump luminosity, as proposed by Charbonnel & Zahn (2007). However, the recent 2D and 3D numerical simulations of thermohaline convection in a low-mass RGB star have independently produced an estimate of a < 1 (Denissenkov 2010; Traxler et al. 2011; Denissenkov & Merryfield 2011). We will show that this estimate is still valid for the CO cores of SAGB stars and that it renders thermohaline mixing too inefficient to prevent the C flame from reaching the center.

Does this leave us where we were before the work of Siess (2009)? Our answer is no, because another mixing process relevant to the convective C-shell burning in the cores of SAGB stars has yet to be accounted for. Indeed, it is known that at the boundaries of shell-flash convection, such as those in classical novae or in AGB stars, shear motion induced by convective flows and internal gravity waves lead to mixing beyond the Schwarzschild convective boundaries, in which the Kelvin–Helmholtz instability plays an important role (Glasner et al. 1997; Herwig et al. 2006; Casanova et al. 2011). The most relevant case, due to its similarity, is such convective boundary mixing (hereafter CBM) at the bottom of the He-shell flash (or pulse-driven) convection zone in AGB stars (e.g., Herwig et al. 1999; Miller Bertolami et al. 2006; Weiss & Ferguson 2009). The consequences include larger ¹²C and ¹⁶O abundances in the intershell, in agreement with observations of H-deficient post-AGB stars (Werner & Herwig 2006). We have included CBM in our simulations of C burning in the cores of SAGB stars using a prescription that is supported by the recent multi-dimensional hydrodynamic simulations of He-shell flash convection (Herwig et al. 2006, 2007). We have found that CBM, like thermohaline convection with a = 7 and a = 10, forces the C flame to stop propagating toward the center. The main difference between the two mixing processes is that CBM is present only in the vicinity of convective boundaries, and therefore C is left unburnt in the entire core below a narrow region adjacent to the bottom of the convective zone. This leads to a new evolutionary path in which the SAGB stars, at least those with the initial masses not too close to M_mas, build in their interiors degenerate cores composed of a relatively large CO core surrounded by a thick ONe zone with a thin CO layer on the top. Such hybrid C–O–Ne degenerate cores will behave differently when the C-rich material in their central parts, surrounded by thick ONe zones, eventually ignites, or when they become WDs and begin cooling down.

In Section 2, we analyze the physics of C-flame propagation toward the center in the absence of extra mixing processes and identify its most important driving mechanism. In Section 3, we explain why this mechanism does not work when CBM is taken into account. In Section 4, we present the results of our hydrodynamic simulations of thermohaline convection in a region below the C-flame convection zone and apply them to simulate the C-flame propagation in an SAGB star. The effect of heat transport in the CBM algorithm is investigated in Section 5, while Section 6 contains discussion and conclusions.

We have computed the evolution of a star with the initial mass 9.5 M_☉ and metallicity Z = 0.02 from the pre-main sequence to C ignition in its degenerate CO core using the state-of-the-art stellar evolution code of MESA revision 4631 (Paxton et al. 2011, 2013). Thus, our first SAGB model has the same initial parameters as the corresponding model of Siess (2006) with which we compare our results. We have used the MESA equation of state (EOS). It adopts the 2005 update of the OPAL EOS tables (Rogers & Nayfonov 2002) supplemented for lower temperatures and densities by the SCVH EOS that accounts for partial dissociation and ionization caused by pressure and temperature (Saumon et al. 1995). Additionally, the HELM (Timmes & Swesty 2000) and PC (Potekhin & Chabrier 2010) EOSs cover the regions where the first two EOSs are not applicable. There are smooth transitions between the four EOS tables. We have used the OPAL opacities (Iglesias & Rogers 1993, 1996) supplemented by the low temperature opacities of Ferguson et al. (2005), and by the electron conduction opacities of Cassisi et al. (2007). The nuclear network consists of 31 isotopes from H to ²⁸Si coupled by 60 reactions that account for H burning in pp chains, CNO-, NeNa-, and MgAl-cycles, as well as He, C, and Ne burning. By default, MESA uses reaction rates from Caughlan & Fowler (1988) and Angulo et al. (1999), with preference given to the second source (NACRE). It includes updates to the NACRE rates for ¹⁴N(p,γ)¹⁵O (Imbriani et al. 2005), the triple-α reaction (Fynbo et al. 2005), ¹⁴N(α, γ)¹⁸F (Görres et al. 2000), and ¹²C(α, γ)¹⁶O (Kunz et al. 2002).

Figure 3 shows profiles of various stellar structure variables at and near the bottom of the C-flame convection zone in the Schwarzschild-only convective boundary model 4900 from Figure 1. Siess (2006) has demonstrated that for the C flame to continue propagating all the way toward the center, the maximum of the nuclear energy generation rate ε_nuc should precede the maximum on the log T curve. The relative positions of the vertical solid and dashed lines in the upper- and middle-left panels in Figure 3 confirm that this is true for our simulation. In this case, the energy of C burning released at the maximum of ε_nuc pre-heats plasma in front of the maximum of log T, thus facilitating its advancement. Siess (2006) has also noted that the location of the bottom of the C-flame convection zone is tightly bound with the location of the maximum of log T. Indeed, the former is determined by the Schwarzschild criterion, ∇_rad = ∇_ad, where

$$\begin{eqnarray} &&\nabla _\mathrm{rad} = \frac{3\kappa }{16\pi G ac}\frac{P}{T^4}\frac{L_r}{M_r}, \end{eqnarray} \tag{ 1 }$$

and ∇_ad = (∂ln T/∂ln P)_S are the radiative and adiabatic temperature gradients (logarithmic and with respect to pressure), κ is the opacity, and other quantities have their usual meanings. Given that ∇_rad = 0 at the location of the maximum of log T because L_r∝dT/dr = 0 there, the Schwarzschild criterion is satisfied at some small distance above it, where

$$\begin{eqnarray} &&L_r = L_\mathrm{Sch} = \frac{16\pi G ac}{3\kappa }\frac{T^4}{P}M_r\nabla _\mathrm{ad}. \end{eqnarray} \tag{ 2 }$$

The increase of the luminosity from its zero value at the point M_r(T_max), where ∇_rad = 0, to its critical value L_Sch at the convective boundary at M_Sch, where ∇_rad = ∇_ad, is provided by the generation of C-burning nuclear energy in this mass interval,

$$\begin{eqnarray} &&L_\mathrm{Sch} = \int _{M_r(T_\mathrm{max})}^{M_\mathrm{Sch}}\varepsilon _\mathrm{nuc}dM_r. \end{eqnarray} \tag{ 3 }$$

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For the maximum of ε_nuc to precede that of log T, the derivative (dlog ε_nuc/dlog T) must be negative immediately below M_r(T_max), where log T is decreasing (dlog T < 0), while ε_nuc should be increasing (dlog ε_nuc > 0) with depth (the upper- and middle-left panels in Figure 3). For C burning at log T = 8.8–8.9, its energy generation rate can be approximated as ε_nuc∝ρX²(¹²C)Tⁿ, where n ≈ 40. Therefore,

$$\begin{eqnarray} &&\frac{d\log \varepsilon _\mathrm{nuc}}{d\log T} = \frac{d\log \rho }{d\log T} + 2 \frac{d\log X(^{12}\mathrm{C})}{d\log T} + n. \end{eqnarray} \tag{ 4 }$$

From the last equation, it is seen that in order to get (dlog ε_nuc/dlog T) < 0, the other two derivatives have to be negative with relatively large absolute magnitudes to compensate the positive term n ≈ 40. A numerical evaluation of the derivatives on the right-hand side of this equation for the profiles presented in Figure 3 gives the following estimates: (dlog ρ/dlog T) ≈ −0.7 and (dlog X(¹²C)/dlog T) ≈ −20. Therefore, we conclude that it is the steep rise of the ¹²C mass fraction in the direction toward the center immediately below the point M_r(T_max) (the lower-right panel in Figure 3) that secures and maintains the required relative positions of the maxima of ε_nuc and log T in the model without extra mixing processes. In such a case, the C flame propagates all the way down to the center and, as a result, an ONe degenerate core is formed.

Timmes et al. (1994) have pointed out that a physically consistent simulation of the C flame requires a very fine spatial resolution with mass zones thinner than ∼1 km in the burning region. To comply with this requirement, Siess (2006) used as many as ∼50 grid points to describe the precursor flame between the bottom of the C-shell convection zone and the minimum in the luminosity profile below it. In our simulations of the C-flame propagation, the resolution in the region of the C-flame precursor is even better (Figure 4). Here, we have more than 100 mass zones separated by distances less than 1 km. Moreover, our calculated C-flame speed V_cond ≈ 1.1 × 10⁻³ cm s⁻¹ is in a very good agreement with the value of V_cond ∼ (1–4) × 10⁻³ cm s⁻¹ interpolated from Tables 1 and 2 of Timmes et al. (1994) using the bounding values of T₉ = 0.76 and ρ = 1.4 × 10⁶ g cm⁻³ from the model shown in Figure 4. Note that in our model, X(¹²C) increases from ∼0.15 to ∼0.35 in the region of the C-flame precursor, and Timmes et al. (1994) assumed the initial abundances X(¹²C) = 0.2 and X(¹²C) = 0.3 in their Tables 1 and 2.

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In our 1D simulations of CBM, we use a MESA standard option that allows us to take this mixing into account as a diffusion process. The corresponding diffusion coefficient in radiative layers adjacent to a convective boundary is

$$\begin{eqnarray} &&D_\mathrm{CBM} = D_\mathrm{MLT}(r_0)\exp \left(-\frac{2|r-r_0|}{fH_P}\right)\!, \end{eqnarray} \tag{ 5 }$$

where H_P is the pressure scale height and D_MLT(r₀) is a convective diffusion coefficient calculated using a mixing-length theory (MLT). The MESA mlt module assumes that D_MLT = Λv_conv/3, where Λ = αH_P is the mixing length (we used the value of α = 2) and v_conv is the convective velocity. The radius r₀ is located at the distance fH_P from the Schwarzschild boundary inside the convective zone. The free parameter f should be calibrated through observations, or through hydrodynamic simulations. The exponentially decaying diffusion coefficient (5) has successfully been used to model CBM at the bottom of the He-shell flash convection zone in AGB stars (Herwig et al. 1999) and, more recently, to simulate the interface mixing between a WD and its accreted H-rich envelope during thermonuclear runaway of CO nova (Denissenkov et al. 2013). In the first case, using a value of f ∼ 0.008 produced larger ¹²C and ¹⁶O intershell abundances that were in a better agreement with those observed in the H-deficient post-AGB stars (Werner & Herwig 2006), while in the second case f = 0.004 resulted in the heavy-element enrichment of a nova envelope comparable to those found in multi-dimensional hydrodynamic simulations as well as to the spectroscopic measurements of Z in nova ejecta. For more information on our motivation to use prescription (5) and on the choice of appropriate values of f for specific cases, the interested reader is referred to the work of Denissenkov et al. (2013).

Figure 5 shows the same profiles as Figure 3, but for the case when CBM with f = 0.007 has been included in our computations. The location of the bottom of the C-shell convective zone is still tightly bound to that of the log T maximum, as explained in the previous section, but now CBM penetrates into the convectively stable layers below M_Sch and homogenizes the X(¹²C) distribution, making it almost flat, in these layers. As a result, the steep rise of X(¹²C) is moved away from M_r(T_max), therefore a decrease of log T immediately below this point cannot be compensated by a sufficiently strong increase of the ¹²C mass fraction necessary to place the maximum of ε_nuc at a sufficient distance in front of the maximum of log T. In this situation, C ignites on the inner slope of its profile left from the preceding phase of convective C burning but, instead of advancing toward the center, the C flame is quenched soon after its ignition, when the C abundance is decreasing in the narrowing convective zone. Figure 5 gives a snapshot of such a moment of the C-flame quenching. Although the maximum of ε_nuc is located slightly below M_r(T_max) in this figure, this is simply because of the fact that the plasma has not cooled down here yet after the C flame has passed through it. Therefore, the log T profile is flatter immediately below its maximum in this figure than in Figure 3, and for nearly constant log T and X(¹²C) even a small increment of log ρ with depth increases ε_nuc. The described phase of C ignition followed by the C-flame quenching is repeated many times (Figure 6) until the C abundance behind the C flame is reduced to such a low level that its further ignition becomes impossible. In the end, the C-shell burning with CBM leaves an unburnt CO core (M_CO ≈ 0.2 M_☉), and the final outcome is a hybrid C–O–Ne degenerate core (Figure 6).

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Thermohaline mixing develops in situations when the temperature distribution is convectively stable but the distribution of chemical composition is unstable, provided that heat diffuses faster than the destabilizing chemical component. This can occur in the ocean when warm salty water lies on top of cold fresh water (Stern 1960; Kunze 2003) and in stellar radiative zones when the mean molecular weight μ increases with the radius (e.g., Ulrich 1972; Denissenkov 2010). In the oceanic case, thermohaline mixing usually takes the form of vertically elongated fluid parcels that are called "salt fingers."

For the stellar case, a linear stability analysis gives the following estimate for a thermohaline diffusion coefficient:

$$\begin{eqnarray} &&D_\mathrm{th} = C_\mathrm{th}\,\frac{\nabla _\mu }{\nabla _\mathrm{rad} - \nabla _\mathrm{ad}}K, \end{eqnarray} \tag{ 6 }$$

where C_th = 2π²a² (Denissenkov 2010). Unfortunately, it is proportional to the square of the aspect ratio, a = l/d, that measures the ratio of the finger's length and diameter. In his work on the C-flame quenching by thermohaline mixing, Siess (2009) used the diffusion coefficient (6) with C_th = 1000 corresponding to a ≈ 7 because that value was known to reproduce the evolutionary decline of the surface C abundance in low-mass red giants above the bump luminosity (Charbonnel & Zahn 2007). However, the 2D and 3D numerical simulations of the ³He-driven thermohaline mixing that Charbonnel & Zahn (2007) proposed to lead to the observed C anomaly all resulted in a < 1 (Denissenkov 2010; Denissenkov & Merryfield 2011; Traxler et al. 2011). Denissenkov (2010) has explained the low a values by the very low viscosity ν in the radiative zones of RGB stars, which facilitates the development of secondary shear instabilities that destroy salt fingers (Radko 2010).

The linear-theory expression (6) is valid only for an ideal gas, which is a good approximation for the H-shell burning in an RGB star. However, the C burning in the cores of SAGB stars occurs under degenerate conditions, in which case the diffusion coefficient (6) has to be multiplied by the ratio (φ/δ), where φ = (∂ln ρ/∂ln μ)_{P, T} and δ = −(∂ln ρ/∂ln T)_{P, μ} (e.g., Charbonnel & Zahn 2007). In the CO cores of SAGB stars, the radiation pressure P_rad contributes only a few percent to the total one, therefore it can be neglected. On the other hand, the pressure of the electron-degenerate gas P_e exceeds that of the nearly ideal gas of ions by a factor of ∼2 immediately below the C flame and by as much as an order of magnitude in the center. When the electron degeneracy increases, the dependence of P_e on T weakens. However, P_e retains a dependence on ρ and μ similar to that for the ideal gas: P_e∝(ρ/μ_e), where (1/μ_e) = (1/μ) − ∑_i(X_i/A_i), the second term on the right-hand side, the inverse atomic mass averaged over the distribution of isotopes in mass fractions, representing less than 16% of the first term in the CO cores. Under the circumstances, we expect that φ will remain close to one, while δ will become very small. Indeed, from the MESA EOS it follows that δ decreases from ∼0.8 to ∼0.03 between the C flame and the center.

Traxler et al. (2011) and Denissenkov & Merryfield (2011) have shown that 2D and 3D numerical simulations of thermohaline convection for the RGB case give almost identical estimates of the diffusion coefficient. Therefore, we present here only the results of 2D numerical simulations of thermohaline convection in the CO core of our 9.5 M_☉ solar-metallicity SAGB model star for parameters extracted from a radiative layer immediately below the C flame. When comparing the simulation parameters for the RGB and CO-core cases, the biggest difference is that between the values of the density ratio R_ρ = (δ/φ)(∇ − ∇_ad)/∇_μ, which are 1700 and 25, respectively. In the CO-core case, the smaller value of R_ρ is caused by the much larger absolute magnitude of the μ-gradient and also by δ ≪ 1. We have used the same code and resolution 1024 × 1024 as in the work of Denissenkov (2010), to which the interested reader is referred for details. The results are presented in Figure 7, where the red curve in the lower panel plots the ratio D_th/K estimated from the numerical simulations, while the dashed black and solid blue lines show the same ratio from Equation (6) calculated for a = 0.35 and a = 1.

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Figure 7 shows that the 2D thermohaline diffusion coefficient in the CO core is very well approximated by the linear-theory one with a = 0.35, much like in the RGB case. In our computations of the C-shell convection, we have used the values of a = 1 and a = 10. The first value is taken a little larger than the one predicted by our numerical simulations to partly compensate for the decrease of δ with depth. The second value is needed, according to our MESA calculations, to reproduce the evolutionary decline of the surface C abundance above the bump luminosity in the metal-poor field and globular-cluster red giants (e.g., Gratton et al. 2000; Shetrone et al. 2010), assuming that the observed pattern is produced by the ³He-driven thermohaline convection. Note that the uncertainty factor C_th = 1000 used by Charbonnel & Zahn (2007) for this purpose is equivalent to a = 7. When we insert a = 10 into Equation (6), we obtain the results similar to those of Siess (2009). In this case, as well as in the case of a = 7, the C flame stops propagating toward the center by the reason explained in that paper. The main difference with the CBM case is that the C flame does not die out soon after being stopped because its life is supported by the fuel from the underlying CO core supplied by thermohaline convection, which is not depicted in Figure 2. As a result, only a few percent of C is left unburnt inside the formed ONe core. On the contrary, in the case with CBM, the C burning leaves a completely untouched CO core below the final position of the quenched C-flame (Figure 6). Finally, when we use the finger aspect ratio a = 1 that is close to a = 0.35 derived from our hydrodynamic simulations,⁵ thermohaline mixing becomes so inefficient that it does not interfere with the C-flame propagation toward the center (Figure 8).

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In deep stellar interiors, the large convective efficiency, Γ ≫ 1, renders the temperature gradient almost adiabatic in convective zones, ∇ ≈ ∇_ad, while ∇ remains close to ∇_rad in radiative zones. However, this is true only if there are no other mixing processes in the radiative zones, or if such extra mixing is present but it cannot compete with radiation and conduction in heat transport. From the MLT (e.g., Weiss et al. 2004), we estimate Γ = γ(D_mix/K), where D_mix is a diffusion coefficient describing heat transport by extra mixing, and K is the thermal diffusivity. For a given value of Γ, the MLT gives ∇ = ζ∇_ad + (1 − ζ)∇_rad, where

$$\begin{eqnarray} &&\zeta = \frac{a_0\Gamma ^2}{1+\Gamma (1+a_0\Gamma)}. \end{eqnarray} \tag{ 7 }$$

In the above equations, γ and a₀ are factors of the order of one that depend on the geometry of fluid elements.

In the case of thermohaline convection, considered in Section 4, our hydrodynamic simulations give Γ ≈ 10⁻³, which results in ∇ ≈ ∇_rad. Therefore, thermohaline convection should not affect the temperature profile below the C-shell convective zone. On the contrary, if prescription (5) is used to also estimate a diffusion coefficient for CBM heat transport, then it will obviously modify ∇ immediately below the C-shell convection zone. This non-radiative CBM will make ∇ almost adiabatic in the vicinity of the convective boundary, where D_CBM ≈ D_MLT ≫ K, but it will restore its radiative value at a larger distance, where the exponential factor reduces D_CBM below the thermal diffusivity.

To see how strongly non-radiative CBM may affect the C-flame propagation, we have computed the evolution of a 7 M_☉ star with Z = 0.01 from the main sequence to the C ignition. To produce a more realistic SAGB model, we have included CBM across the boundaries of the H and He convective cores and across the bottom of the surface convection zone. For these CBM processes, we have used Equation (5) with the value of f = 0.014 that is close to the one constrained by fitting the terminal-age main sequence for a large number of stellar clusters and associations (Herwig 2000). In the case when CBM is included at all evolutionary phases, both the lower and upper limits for the initial masses of SAGB stars shift to lower values. To simulate the C-flame propagation with CBM in the CO core of this SAGB model, we have used the same value of f = 0.007 as in our 9.5 M_☉ model. The CBM heat transport at each convective boundary has been modeled using the above MLT equations, some of which are already incorporated into MESA, while others can easily be added. For the fluid element geometry, we have assumed the following parameters: γ = 2/3 and a₀ = 9/4 (Weiss et al. 2004).

Figure 9 shows the modification of the temperature gradient immediately below the C-shell convection zone produced by the CBM heat transport with the diffusion coefficient shown in the lower panel. The change of the temperature stratification occurs only in the immediate vicinity of the convective boundary because of the short length scale of the exponential decay of D_CBM. Our computations show that this does not prevent the C-flame quenching, and again in this case an unburnt CO core remains in the center (Figure 10).

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We have obtained the following two main results.

• 1.  
  CBM creates conditions in which the C-flame is inhibited from propagating toward the center in the CO cores of SAGB stars.
• 2.  
  Thermohaline mixing driven by the mean molecular weight inversion caused by the off-center C burning does not affect the C-flame inward propagation when one uses a thermohaline diffusion coefficient estimated via hydrodynamic simulations.

6.1. Implications for Supernova and White Dwarf Models

6.2. Uncertainties

We thank Bill Merryfield for letting us use his computer code designed for 2D numerical simulations of salt-fingering convection. F.H. appreciates fruitful conversations with Ken Shen that have prompted us to investigate the effect of heat transport in the CBM region, and that have informed our conclusions on implications for supernova models. F.H. thanks Marten van Kerkwijk for an invitation to Toronto and interesting discussions that have motivated us to complete this project sooner. This research has been supported by the National Science Foundation under grants PHY 11-25915 and AST 11-09174. This project was also supported by JINA (NSF grant PHY 08-22648). F.H. acknowledges funding from NSERC through a Discovery Grant.