Explosive Nucleosynthesis in Near-Chandrasekhar-mass White Dwarf Models for Type Ia Supernovae: Dependence on Model Parameters

We construct the initial C + O WD models at the central carbon ignition with the masses from M = 1.30 to 1.39 M_⊙ (and thus the central densities from 5 × 10⁸ g cm⁻³ to 5 × 10⁹ g cm⁻³) for various metallicities and the carbon fraction (see Section 4 for details). The internal temperature is assumed to be 1 × 10⁸ K.

To carry out the 2D simulations, we set the model parameters as follows. For each CO WD, we choose a given central density ρ_c, metallicity Z, and CO mass ratio C/O with an isothermal profile. Then we follow the hydrodynamics simulation without further alternation. In the initial model, we solve the hydrostatic equilibrium of the WD assuming a constant composition and a constant C/O. To model metallicity in the simplified network, we treat ²²Ne as the proxy of metallicity.

The above initial model for the simulation of the explosion is a simplified approximation of the realistic evolutionary model of an accreting WD from its formation through the initiation of a deflagration. To clarify the simplification of our initial model, let us compare with the evolutionary models calculated by Nomoto et al. (1976, 1984).

In Nomoto et al. (1976, 1984), the initial mass of a C + O WD is 1.0 M_⊙, with uniform mass fractions of C, O, and ²²Ne as X(C) = 0.475, X(O) = 0.50, and X(²²Ne) = 0.025. Here ²²Ne is converted from the initial CNO elements during H and He burning so that X(²²Ne) is treated as the proxy of initial metallicity. In the present initial models, we also adopt a uniform abundance distribution with X(O) = 0.50 and X(C) = 0.50–X(²²Ne), where different X(²²Ne) is the proxy of different metallicity. X(22Ne) = 0.025 is regarded as the solar metallicity, although the latest solar abundances correspond to X(²²Ne) = 0.0134 (Asplund et al. 2009).

In Nomoto et al. (1976, 1984), the WD is cooled down to the central temperature of T_c = 10⁸ and 10⁷ K for the two cases, respectively, with almost isothermal distribution. Afterward, mass accretion onto the WD starts with different accretion rates, which give the rate of compressional heating of the WD interior.

In the SD scenario, the WD mass (and thus the central density) increases by mass accretion from the companion star (e.g., Nomoto et al. 1994). The internal temperature depends on the competition between the compressional heating and radiative cooling (e.g., Nomoto 1982a, 1982b). For a higher accretion rate, the central temperature increases faster and carbon is ignited at the center at a lower central density (and thus a smaller WD mass).

For heating, heat inflow from the H/He shell burning is also important (Nomoto et al. 1984). Since the timescale of heat conduction in the WD interior is shorter than the accretion timescale, the WD interior is close to isothermal with a temperature of ∼10⁸ K (Nomoto et al. 1984).

In this way, the adopted WD masses at the carbon ignition correspond to different mass accretion rates from the companion stars (Nomoto et al. 1984) and/or the delay time in uniformly rotating WDs (Benvenuto et al. 2013).

We should note that the highest accretion rate for the central carbon ignition is limited to ∼7 × 10⁻⁷ M_⊙ yr⁻¹ by the rate of steady hydrogen burning above which a strong WD wind blows (Nomoto et al. 2007; Kato et al. 2014), or to ∼3 × 10⁻⁶ M_⊙ yr⁻¹ by the off-center carbon ignition for the accretion of He from an He star companion (Nomoto & Iben 1985). For these limitations, the lowest WD mass at the carbon ignition is ∼1.35 M_⊙ (Nomoto et al. 1984).

After C ignition in the center due to strong screening effects, a simmering phase starts with a developing convective core, which was calculated using the time-dependent mixing length theory (Unno 1967; Nomoto et al. 1976, 1984; for recent works on simmering phase, see, e.g., Piro & Chang 2008; Jackson et al. 2010; Krueger et al. 2012; Ohlmann et al. 2014; Martinez-Rodringuez et al. 2016). In these calculations, so-called convective Urca processes were not included. The extent of the convective core might be limited by convective Urca processes, although it is quite uncertain (e.g., Arnett 1996; Lesaffre et al. 2006), In view of the large uncertainty of convective Urca processes, the exact distributions of the temperature and abundances in the WD should be regarded as highly uncertain, and further study of the simmering phase is necessary (see, e.g., Piro & Chang 2008, for an analytic analysis). We study the effects of the initial C/O ratio as the origin of model diversity.

Near the end of the simmering phase in the models by Arnett (1969) and Nomoto et al. (1976, 1984), it was found that a superadiabatic temperature gradient appears at the central temperature of T_c > 8 × 10⁸ K and increases sharply. The timescale of the temperature rise becomes shorter than the dynamical timescale around T_c > 3 × 10⁹ K. At T_c > 5 × 10⁹ K, nuclear reactions become rapid enough to realize nuclear statistical equilibrium (NSE), and T_c ∼ 10¹⁰ K is reached. The steep temperature jump, i.e., a deflagration front, is formed. Such an evolution through NSE takes place in a timescale shorter than the convective energy transport timescale, and the convective core size and the abundances in the bulk of the convective core are frozen, i.e., nuclear burning products in the center are not well mixed with the outer layers.

In the deflagrated region, NSE is realized so that the details of the abundance change during rapid nuclear reaction are not important. A decrease in Y_e due to electron capture during the simmering phase is also negligibly small compared with electron capture after NSE is realized. Thus, the neglect of the convective region and the composition change during the simmering phase do not much affect the current results.

We also note that, owing to the degenerate electron gas, the mass and radius are less sensitive to the choice of temperature profile. We observe that the masses, radii, and density profiles are still comparable to those presented in the literature (see, e.g., Krueger et al. 2010, for the WD model obtained from the stellar evolutionary model).

In the present study, we extend the WD mass down to the range of 1.30–1.35 M_⊙. Such low WD masses may be called sub-Chandrasekhar masses. The central carbon ignition in such a sub-Chandrasekhar-mass WD would be possible by shock compression of the central region due to the surface He detonation (e.g., Arnett 1996). The important difference from those "double-detonation" models is that, because of the relatively large WD mass and thus the high central density, the central carbon ignition does not necessarily produce strong shock waves to induce the detonation, but rather develops a carbon deflagration due to the large electron-degeneracy pressure compared with the thermal pressure released by nuclear burning (Nomoto et al. 1976). Whether the surface He detonation induces the carbon deflagration or direct detonation will be studied in forthcoming papers.

Here we briefly describe the new input physics used in the code. For the basic data structure of the code and the code test, we refer the readers to Leung et al. (2015b). We also refer the readers to Appendix A for the numerical implementation of the SN Ia physics in our code. Here we only list the parts relevant to SNe Ia. We use the most recent rates we have for describing the microphysics, including the Helmholtz equation of state (Timmes 1999), nuclear reaction rates (Rauscher & Thielemann 2009), the strong screening factor (Kitamura 2000), and electron capture rates (Langanke & Martinez-Pinedo 2001).

In the present study, we assume that the central carbon flash develops into the deflagration as follows. The exact pattern of the initial flame is not well constrained. To trigger the deflagration phase, therefore, we impose a flame by hand in the star. The zone is assumed to be incinerated into NSE. We choose two different morphologies. First, it is a centered flame with some sinusoidal perturbations. This is similar to the c3 flame as used in Reinecke et al. (1999a), which mimics that the flame grows at center and then is perturbed by Rayleigh–Taylor instabilities. Second, it is an off-center "bubble" (in 2D the bubble corresponds to a ring in 3D), similar to b1 in Reinecke et al. (1999a). This pattern mimics the evolution that the convection in the star is rapid enough to bring the hot parcel from the center before it runs away.¹

To determine the DDT, we compare the eddy turnover scale with the flame width, i.e., the Karlovitz number, Ka, which is defined as (Niemeyer et al. 1995)

$$\begin{eqnarray}&&\mathrm{Ka}=\displaystyle \frac{{l}_{\mathrm{flame}}}{{l}_{\mathrm{turb}}}.\end{eqnarray} \tag{ 1 }$$

Here l_flame and l_turb are the representative length scale of the deflagration wave and the turbulent eddy motion (see Appendix A for details). At the end of each time step we scan across the flame surface to see whether the Karlovitz number, Ka, satisfies the DDT condition, for which we pick Ka = 1 to be the required DDT condition. Once this condition is satisfied, we put in the initial C detonation in the form of a 2D bubble (a ring) at that point and allow that detonation to freely evolve. Extra detonation bubbles are added as long as the flame surface is not yet swept by the detonation wave. We follow the evolution until the whole star expands sufficiently that it becomes sparse and cold and all electron capture and major nuclear reactions have stopped.

We emphasize that there still exists theoretical uncertainties in the DDT model, especially related to the robustness of trigger detonation in an unconfined media (see Appendix E for a comparison of how this certainty affects the nucleosynthesis).

We show in Figure 1 the temperature color plot and the deflagration wave fronts. At early phase, the matter density is sufficiently high that most matter is incinerated into NSE (including endothermic photodisintegration of ⁵⁶Ni into ⁴He). In the first 0.8 s, deflagration takes place, where the energy release is slow. The deflagration wave and its subsequent advanced burning release about 10⁵¹ erg s⁻¹.

In the pure turbulent deflagration phase before the DDT, namely, from t = 0 to 1.12 s, deflagration burns about 0.3 M_⊙ of matter. As seen in Figure 2, the deflagration releases nuclear energy slowly, on the order of 10⁵⁰ erg s⁻¹. The nuclear energy production is slow so that the total energy of the WD increases but remains negative.

During the deflagration phase, the star expands considerably. As the flame front reaches the low-density region (∼10⁷ g cm⁻³) beyond t = 0.8 s, the carbon deflagration releases much less energy than what it originally does at the stellar core. The drop of luminosity near t = 1 s suggests that the matter has expanded and cooled down so that the NSE timescale becomes comparable to or even longer than the hydrodynamics timescale.

When the density at the flame front decreases to ≈2.3 × 10⁷ g cm⁻³, the transition to the detonation takes place. We plot in Figure 1 the temperature color plot and the detonation wave fronts. The detonation starts from the tip of the finger shape, around r = 2000 km. The detonation wave is almost unperturbed by the fluid motion, so that the flame structure appears to be almost spherical. The temperature profile shows that most matter is no longer in NSE. Due to the uneven surface of the flame at the moment of DDT, there is unburnt material left behind in the high-density region. At the radius defined by the outermost radius reached by the deflagration wave, there always exists fuel inside. This matter is later burnt into NSE by the detonation wave. This provides an additional source of iron-peak elements. Notice that this feature does not exist in 1D models because the spherical model allows all matter to be burnt inside the same outermost radius reached by the flame. Therefore, the detonation can only burn the low-density matter and produce fewer iron-peak elements.²

The detonation wave quickly burns the remaining material, making the total energy positive. Then the WD expands rapidly and increases its kinetic energy. In contrast to the slow deflagration wave, the detonation is a much more efficient source for producing nuclear energy. It burns the 1.0 M_⊙ matter within the next 0.2 s. The typical luminosity is of the order of 10⁵² erg s⁻¹.

The chemical composition of the ejected matter is presented. To obtain the nucleosynthesis yield, we use the tracer particle scheme to keep track of the thermodynamics history. Then, we calculate nucleosynthesis by using a 495-isotope network, which includes isotopes from ¹H to ⁹¹Tc. Stable neutron-rich isotopes, such as ⁴⁸Ca, ⁵⁰Ti, ⁵⁴Cr, and ⁶⁰Fe, are included so that the nucleosynthesis with electron capture can be consistently calculated for Y_e = 0.45–0.50. Here ${Y}_{{\rm{e}}}={\sum }_{i}{Z}_{{\rm{i}}}/{A}_{{\rm{i}}}$, with Z_i and A_i being the atomic number and mass number, respectively, of the species i. For convenience in this paper, we use Y_e = Z_i/A_i for the individual species i as well. For the numerical details, see Appendix A.

In Figure 4 we plot [X_i/⁵⁶Fe] of stable isotopes, after the decay of short-lived radioactive elements is accounted for. All quantities are given by $[{{\rm{X}}}_{i}{/}^{56}\mathrm{Fe}]={\mathrm{log}}_{10}(({X}_{i}/X{(}^{56}\mathrm{Fe}))/({X}_{i}/X{{(}^{56}\mathrm{Fe}))}_{\odot }))$. It can be seen that in general a considerable number of elements have [X_i/⁵⁶Fe] between −0.3 and 0.3 as marked in the figure. This shows that these elements are consistent with the solar abundances. Notice that many elements in the Sun come from both SNe Ia and SNe II. For the case of underproduction, it is possible that such isotopes may come solely from SNe II, such as the α-chain isotopes. However, for the case of overproduction, it will be a strong constraint for that particular SN Ia model. This is because the typical rate of SNe Ia has the same order of magnitude as SNe II. Any severe overproduction of such an isotope, for instance, 10 times above solar abundance, means that such an explosion model is not a typical one, since that isotope cannot be "diluted" by the underproduction (or null production) of the other type of SN. Representative elements include ²⁸Si, ³²S, ³⁶Ar, ⁴⁰Ca, ^50 −52Cr, ⁵⁵Mn, ^54–57Fe, and ^58–62Ni. However, ⁵⁰Ti and ⁶⁶Zn are underproduced. Furthermore, isotopes with an odd-number atomic number, such as P, Cl, and K, are mostly underproduced, with the exception of ⁵¹V. This is expected, as the system is initially void of hydrogen for proton capture. In this benchmark model, we observed the production of ⁵⁶Ni to be 0.67 M_⊙, 3.34 × 10⁻² M_⊙ neutron-rich species (such as ^48–50Ti, ^53–54Cr, ^57–58Fe, and ^61–64Ni), and 0.385 M_⊙ intermediate-mass elements (IMEs). For the detailed velocity distribution of the products by pure turbulent deflagration, we refer to Section 4.7 for a more detailed discussion.

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In Figure 5 we plot the velocity distribution of major isotopes for the benchmark model at the end of simulations for the polar slide from 0° to 9° (Slice 1). The chemical composition is again obtained from the post-processing nucleosynthesis. The figure is consistent with the standard framework that the inner part, namely, matter with low velocity, contains mostly ⁵⁶Ni. ⁵⁴Fe and ⁵⁸Ni are located at the innermost part. In matter with v ≈ 7200 km s⁻¹, IMEs such as ²⁸Si and ³²S become prominent. This corresponds to the flame entering the low-density region. For v ≈ 6600 km, there is an abrupt jump of ⁵⁶Ni again, which results from detonation near the flame edge, where part of the matter has a density of ∼10⁸ g cm⁻³. Close to v ≈ 9000 km s⁻¹, IMEs becomes prominent again. At the outermost part, the density is too low for nuclear reaction beyond carbon burning, even for detonation. A trace of ¹⁶O is left behind.

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In Figure 6, we make a plot similar to Figure 5, but for the polar slide of 36°–45° (Slice 5). We choose this slide so as to provide a contrast on the time difference between the quenching of deflagration and the arrival of the detonation wave. As shown in Figure 1, detonation starts from the two opposite "fingers" of the far end of the flame, but not the central "finger." This means that, before the detonation wave reaches the central "finger," the matter around has a certain time to expand before being incinerated. Similar to the previous case, isotopes with Y_e < 0.50 are mostly found in the core, where v < 3000 km s⁻¹. The velocity space up to v ≈ 6000 km s⁻¹ is filled with ⁵⁶Ni. The IME gap in this case is larger than that of slice 1, for which almost no ⁵⁶Ni is detected from v = 5400 to 7200 km s⁻¹. The second peak of ⁵⁶Ni appears near v = 7800 km s⁻¹. Close to v ≈ 9000 km s⁻¹, the IMEs become prominent. Different from Slice 1, ¹⁶O appears in matter with a velocity slightly less than 8000 km s⁻¹ and also beyond for two distinctive reasons. For v < 7800 km s⁻¹, the remaining ¹⁶O comes from the tip of deflagration, while for v between 7800 and 9000 km s⁻¹, ¹⁶O appears owing to the longer expansion time between the end of deflagration and detonation. The amount of unburnt ¹⁶O is comparatively higher than that in Slice 1.

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We further classify the chemical yields of the tracer particles by checking whether they reach their runaway by being swept by the deflagration or detonation wave. Notice that it is possible that the tracer particles, first swept by the deflagration wave, are reheated by the shock collision from the detonation wave. In these cases we still regard the chemical yield to be contributed by the deflagration wave. In Figure 7 we plot their corresponding chemical composition ratio to ⁵⁶Fe scaled with solar abundance, together with the total yield. It can be seen that the deflagration wave, similar to the 1D model, contributes mostly to the formation of iron-peak elements, especially neutron-rich ones. For example, it has a higher $[{{\rm{X}}}_{i}{/}^{56}\mathrm{Fe}]$ fraction for ⁵⁴Cr, ⁵⁵Mn, ⁵⁴Fe, ⁵⁸Ni, and ⁵⁹Co. On the other hand, detonation, which swept mostly the low-density region, produces less massive isotopes. IMEs such as ²⁸Si, ³²S, ³⁶Ar, and ⁴⁰Ca are mostly produced in the detonation wave. Some lighter iron-rich elements, such as ^46,48,49Ti and ⁵⁴Cr, are also produced by detonation. As mentioned before, the unburnt field surrounded by the detonation wave is most of the time swept by the detonation wave, which produces the necessary heating for producing iron-peak elements. As a result, one can observe its contribution to iron-peak elements, including ⁶²Ni and ⁶⁶Zn.

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