SUPER-CHANDRASEKHAR-MASS LIGHT CURVE MODELS FOR THE HIGHLY LUMINOUS TYPE Ia SUPERNOVA 2009dc

To construct the homologously expanding models of super-Ch-mass WDs, we apply approximations similar to those adopted by Maeda & Iwamoto (2009). The models are described by the following parameters.

• 1.  
  M_WD: total WD mass.
• 2.  
  M_56Ni (or f_56Ni): mass (or mass fraction) of ⁵⁶Ni. The mass fraction hereafter means the ratio to M_WD, i.e. f_56Ni = M_56Ni/M_WD.
• 3.  
  f_ECE: mass fraction of electron-captured elements (ECEs; mostly ⁵⁴Fe, ⁵⁶Fe, ⁵⁵Co, and ⁵⁸Ni). Stable Fe, Co, and Ni are included, all of which are simply assumed to have the same mass fraction.
• 4.  
  f_IME: mass fraction of intermediate-mass elements (IMEs). Si, S, and Ca are included, whose mass fraction ratio is 0.68:0.29:0.03, similar to that in the Chandrasekhar-mass WD model, W7 (Nomoto et al. 1984; Thielemann et al. 1986).
• 5.  
  f_CO: mass fraction of C+O. C and O are contained equally in mass fraction.

Since the equation

$$\begin{equation} f_{\rm ECE}+f_{\rm 56Ni}+f_{\rm IME}+f_{\rm CO}=1 \end{equation} \tag{ 1 }$$

holds, we eliminate f_IME in the following discussion. In total, we have four independent parameters.

The expansion model of a super-Ch-mass WD is constructed as follows. First, with the above parameters, we calculate the nuclear energy release during the explosion (E_nuc) by

$$\begin{eqnarray} &&\frac{E_{\rm nuc}}{10^{51}\,{\rm erg}}=[0.5f_{\rm ECE}+0.32f_{\rm 56Ni}+1.24(1-f_{\rm CO})]\times \frac{M_{\rm WD}}{M_\odot } \nonumber\\ \end{eqnarray} \tag{ 2 }$$

(e.g., Maeda & Iwamoto 2009). Then, the binding energy of the WD (E_bin) is evaluated by Equations (22) and (32)–(34) described in Yoon & Langer (2005). Here, ρ_c = 3 × 10⁹ g cm⁻³ for all models as in Maeda & Iwamoto (2009), so that E_bin increases almost linearly with M_WD. For M_WD > 2.1 M_☉, we extrapolate the formula of E_bin (Jeffery et al. 2006). These E_nuc and E_bin give the kinetic energy of the exploding WD as

$$\begin{equation} E_{\rm kin}=E_{\rm nuc}-E_{\rm bin}. \end{equation} \tag{ 3 }$$

How much mass fraction of ⁵⁶Ni is synthesized at the deflagration or detonation wave depends mainly on the temperature and thus on the density at the flame front. For W7, for example, ⁵⁶Ni is synthesized at the flame densities of ρ ∼ ρ_c–2 × 10⁷ g cm⁻³. Then the total synthesized ⁵⁶Ni mass depends on the mass contained in this density range, and thus on the presupernova density structure of the WD and the flame speed. The rotating WD is more massive than the non-rotating WD with the same central density (Yoon & Langer 2005). For the same central density, therefore, the density profile is shallower for more massive stars, and thus the mass contained in the density range for ⁵⁶Ni synthesis is larger; i.e., a more massive WD tends to synthesize more ⁵⁶Ni. Also, for the same central density, faster flame produces more ⁵⁶Ni because of less pre-expansion, and the flame speed depends on the WD mass, rotation law, density structure, and possible transition to detonation. To provide constraints on these physical processes, we calculate the LC models with various nuclear yields and E_kin for the same central density models.

Next, the structure of the explosion model of a super-Ch-mass WD is obtained by scaling W7, the canonical Chandrasekhar-mass WD one, in a self-similar way. We scale the density (ρ) and velocity (v) of each radial grid in the model as

$$\begin{equation} \rho \propto M_{\rm WD}^{5/2}E_{\rm kin}^{-3/2} \end{equation} \tag{ 4 }$$

and

$$\begin{equation} v\propto M_{\rm WD}^{-1/2}E_{\rm kin}^{1/2}. \end{equation} \tag{ 5 }$$

We then consider the distribution of four element groups (ECEs, ⁵⁶Ni, IMEs, and C+O). We assume that ⁵⁶Ni in a super-Ch-mass WD model is mixed. The inner boundary of the mixing region is set to be v = 5000 km s⁻¹ for all the models because the observed line velocities of Si ii are >5000 km s⁻¹ (Yamanaka et al. 2009). And we set the outer boundary of the mixing region based on the mass coordinate: M_r = 1.13(M_WD/M_Ia) M_☉. Here the reference mass coordinate (M_r = 1.13 M_☉) is the outer boundary of the ⁵⁶Ni distribution in the W7 model. This outer boundary corresponds to that of the ⁵⁶Ni-produced zone in the W7 model. By these definitions, the mixing region is uniquely set when we choose four model parameters. Note that, for our models, the velocity at the outer boundary of the mixing region differs among models even with the same M_WD due to the scaling by Equation (5). This mixing is important to account for the observed velocities of the Si ii line (see Section 3.2).

To demonstrate the scaling and mixing, we plot the structure and abundance distribution of the two super-Ch-mass WD models in Figure 1. The model plotted on the left panels had no IMEs at 5000 km s⁻¹ ≲ v ≲ 11, 000 km s⁻¹ before being mixed; the mixing has extended ⁵⁶Ni outward and IMEs inward. In some models, C+O are also mixed inward as in the model on the right in Figure 1.

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We set the parameter range as follows to compare the models with the observations of SN 2009dc.

• 1.  
  M_WD/M_☉ = 1.8, 2.0, ..., 2.8.
• 2.  
  M_56Ni/M_☉ = 1.2, 1.4, ..., 1.8, 1.9, 2.0.
• 3.  
  f_ECE = 0.1, 0.2, 0.3.
• 4.  
  f_CO = 0.1, 0.2, 0.3, 0.4.

Here, f_IME is obtained by using Equation (1). Any models which have f_IME ⩽ 0 are not constructed (e.g., models with M_WD = 2.0 M_☉, f_ECE = 0.2, and f_CO ⩾ 0.2). We also excluded a model where the inner boundary of the IMEs layer without mixing is located further out than the mixing region (e.g., a model with M_WD = 2.2 M_☉, f_ECE = 0.3, and f_CO = 0.1), because no IMEs extend inward by mixing. The parameters and E_kin of our models are listed in Columns 2–5 of Table 1.

Multi-color LCs for the constructed models are calculated with STELLA code (Blinnikov et al. 1998, 2000, 2006), which solves the one-dimensional equations of radiation hydrodynamics. STELLA was first developed to calculate LCs of Type II-L SNe (Blinnikov & Bartunov 1993). Its applications to SNe Ia are described in Blinnikov & Sorokina (2000), Blinnikov et al. (2006), and Woosley et al. (2007). In the present study, we use homologously expanding ejecta as input (see Section 2). Then the radiation hydrodynamics calculation is performed for each model. Maeda & Iwamoto (2009) used the one-dimensional gray radiation transfer code (Iwamoto 1997; Iwamoto et al. 2000), where they assumed simplified opacity for line scatterings. STELLA, on the other hand, considers 155,000 spectral lines in LTE assumption to calculate the opacity with expansion effect more realistically, as well as free–free/bound–free transitions and Thomson scattering.

We note here that the "bolometric" luminosity reported from the observations is the uvoir luminosity (L_uvoir). Yamanaka et al. (2009) derived L_uvoir by assuming that (actually observed) integrated BVRI luminosity (L_BVRI) is 60% of L_uvoir. This is commonly applied for normal SNe Ia (Wang et al. 2009b). Since it is still unknown whether this assumption is applicable to super-Ch candidates, we directly compare theoretical and observed L_BVRI rather than L_uvoir. This is an advantage of our multi-color calculations. In the calculations, the bolometric, uvoir, and BVRI LCs cover 1–50000 Å, 1650–23000 Å, and 3850–8900 Å, respectively.

To compare the calculated and observed LCs, the observational data must be corrected for extinction. In order to correct the extinction, three values are needed: the B − V excesses for the Milky Way (E(B − V)_MW) and the host galaxy of SN 2009dc (E(B − V)_host), and the ratio of the V-band extinction to the B − V excess (R_V) of the host galaxy (R_V is set to be 3.1 for the Milky Way). Of these values, E(B − V)_host and R_V of the host galaxy are somewhat difficult to estimate.

The observed Na i absorption line in the host galaxy suggests that the reddening caused by it is not negligible. To estimate E(B − V)_host, one may use the observed color. However, the observed B − V of SN 2009dc is significantly different from normal SNe Ia, which may suggest that the Lira–Phillips relation (Lira 1996) should not be applied. The other method to derive E(B − V)_host is using the equivalent width of the Na i D absorption line (EW_NaID; e.g., Turatto et al. 2003). But it is also noted that the observed EW_NaID has a (relatively) large error (Silverman et al. 2011; Taubenberger et al. 2011). Also for R_V, a non-standard, smaller value (<3.1) may be preferred for normal SNe Ia (e.g., Nobili & Goodbar 2008; Wang et al. 2009a; Folatelli et al. 2010; Yasuda & Fukugita 2010). If the host galaxy of SN 2009dc also has R_V < 3.1, its extinction is overestimated.

To cover most of the possible ranges of extinction, we consider two extreme cases, where the extinction by the host galaxy is negligible (E(B − V)_host = 0 mag) and significant (E(B − V)_host = 0.14 mag), respectively. We set E(B − V)_MW = 0.71 mag, and R_V = 3.1 for the host galaxy (same as for the Milky Way). An extinction law by Cardelli et al. (1989, Table 3) is applied.

The left panel of Figure 2 shows the calculated BVRI LCs for the super-Ch-mass WD models with different M_WD. The other parameters are set to be the same (M_56Ni = 1.2 M_☉, f_ECE = 0.1, and f_CO = 0.2). A clear relation is seen between the BVRI LCs and M_WD from this panel: a more massive model shows a broader BVRI LC.

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Such a mass dependence is consistent with the relation between the timescale of the bolometric LC (t_bol) and M_WD,

$$\begin{equation} t_{\rm bol}\propto \bar{\kappa }^{1/2}M_{\rm WD}^{3/4}E_{\rm kin}^{-1/2} \end{equation} \tag{ 6 }$$

(Arnett 1982), where $\bar{\kappa }$ is the opacity averaged in the ejecta (although $\bar{\kappa }$ does differ among models and with time). Note that E_kin anticorrelates with M_WD for the models plotted in the left panel of Figure 2 because f_ECE and f_CO are fixed (cf. Table 1).

We consider a timescale to see quantitatively if the BVRI LCs depend on M_WD and E_kin, analogous to Equation (6). For this purpose, we take the declining timescale (t_+1/2), which is defined as the time for the BVRI LC to halve its luminosity after the peak. In the left panels of Figure 3 are shown the M_WD–E_kin (large panel), M_WD–t_+1/2 (small, top), and E_kin–t_+1/2 (small, right) plots for the models listed in Table 1. The BVRI LCs of the massive and less energetic models tend to be wider (i.e., larger t_+1/2), which is expected from the left panel in Figure 2. For the dependence of t_+1/2 on M_WD and E_kin, we plot t_+1/2 against M^3/4_WDE_kin^−1/2 in the right panel of Figure 3. An almost linear relation is seen between t_+1/2 and M^3/4_WDE_kin^−1/2. We thus confirm a relation similar to Equation (6) for the BVRI LCs.

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The calculated and observed BVRI LC are shown in the right panel of Figure 2, by setting the dates of maximum in the B band of the models and observations at the same day as t_B = 0 day. We plot the observational data provided by Yamanaka et al. (2009), neglecting the extinction by the host galaxy (open squares). The open triangle corresponds to the early R-band detection by Silverman et al. (2011), where we simply assume that the R-band luminosity is equivalent to 20% of L_uvoir, i.e., 1/3 of L_BVRI, assuming L_uvoir = L_BVRI/0.6 (cf. Wang et al. 2009b, Figure 24). The BVRI LCs of the super-Ch-mass WD models in the right panel are as luminous as the observations around the maximum, while some have relatively broader BVRI LCs or shorter rising time than the observations. Especially, the two less massive models (M_WD = 1.8–2.0 M_☉) are not preferred because they are too faint at t_B ∼ −20 days.

In order to find the well-fitted models, we calculate the weighted root-mean-square residual (WRMSR) of the BVRI LC,

$$\begin{equation} {\rm WRMSR}=\sqrt{\frac{1}{N}\sum _{i=1}^{N}\left(\frac{L_{{\it BVRI},i}^{\rm (calc)}-L_{{\it BVRI},i}^{\rm (obs)}}{\delta L_{{\it BVRI},i}^{\rm (obs)}}\right)^2}, \end{equation} \tag{ 7 }$$

for each model. We use the observational data for −10 days < t_B < 40 days taken from Yamanaka et al. (2009), where the total number of observations, N, is 23 for this time range. The early detection by Silverman et al. (2011) is excluded in calculating the WRMSR. Here, L^(obs)_{BVRI, i} and L^(calc)_{BVRI, i} respectively denote L_BVRI of the observation and calculation obtained at the ith epoch. δL^(obs)_{BVRI, i} is the observational error of L^(obs)_{BVRI, i}, which is set to 20% of L^(obs)_{BVRI, i}. Maeda & Iwamoto (2009) used the decline rate of the bolometric LC after maximum for comparison between their models and observations. We use the above WRMSR instead of the decline rate, so that brightness can also be taken into account. The WRMSR is calculated with the observational error and shows which model fits relatively to the observed BVRI LC.

We list the WRMSR of the models for the two extinction cases in Columns 7 and 8 of Table 1. Good agreement is found between the calculated and observed BVRI LCs for the whole range of M_WD in this study, if the extinction by the host galaxy is not considered. In the case where the extinction is corrected, most models have larger WRMSR (due to smaller L_BVRI), while some models with large M_WD and M_56Ni are better fitted.

Next, we compare the photospheric velocity (v_ph) of the models and the observed line velocity of Si ii. In the last column of Table 1 are listed the v_ph ranges of our models at −5 days ⩽ t_B ⩽ 25 days. This period covers the whole phases of the spectroscopic observations by Yamanaka et al. (2009). We estimate the position of the photosphere (hence, v_ph) in the model from the optical depth at the R band, where the rest-frame wavelength of the observed Si ii line (6355 Å) is located.

The velocity of the observed Si ii line is reported as <9000 km s⁻¹ at the period (Yamanaka et al. 2009). Since the Si ii absorption line is formed by the Si above the photosphere, the calculated v_ph should be smaller than the observed Si ii line velocity (for further details, see Tanaka et al. 2011, Figure 1). For the models with M_WD = 1.8 M_☉ and 2.0 M_☉, however, the calculated v_ph are >10,000 km s⁻¹, much larger than the observed line velocity of Si ii. Thus, these less massive models are far from consistent with SN 2009dc. On the other hand, many models with M_WD ⩾ 2.2 M_☉ have v_ph < 10, 000 km s⁻¹ during −5 days ⩽ t_B ⩽ 25 days, which is somewhat compatible with observations. These models with small v_ph commonly have such a large C+O mass fraction as f_CO = 0.2–0.4. From Equation (2), E_nuc and thus E_kin are affected mainly by f_CO rather than f_ECE. A model with larger f_CO has smaller E_nuc, thus smaller E_kin and v_ph, being preferred for SN 2009dc.

In order to find the most plausible model for SN 2009dc, we first select models based on BVRI LC and v_ph discussed above. We use criteria of WRMSR ≲ 1 and v_ph < 10, 000 km s⁻¹ during the −5 days ⩽ t_B ⩽ 25 days. By these criteria, models with a label A–O in Column 1 of Table 1 are selected. Among them, models A–M are selected for the case where the host-galaxy extinction is neglected; only models N and O are selected for the significant extinction. In Figure 4, we plot WRMSR and maximum v_ph during the period, where the dotted lines indicate the criteria.

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We further analyze multi-color LCs of these models. Figures 5–9 show the BVRI (left), v_ph (right top), and monochromatic LCs (right bottom) of these models. In these figures, the BVRI, B-, V-, R-, and I-band LCs by Yamanaka et al. (2009) and the early detection and U-band LC by Silverman et al. (2011) are plotted as squares and triangles, respectively. Filled and open symbols show the observed LCs with and without the extinction correction.

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With Figures 5–9, we make more detailed comparisons of models A–O with the observational data of SN 2009dc. The comparisons are summarized as follows.

• 1.  
  The early detection in Silverman et al. (2011) enables us to constrain the rising time of the models. The R-band LCs of models E, H, and M are too bright at t_B ∼ −20 days regardless of whether the host-galaxy extinction is corrected or not. These models can be excluded because their LCs evolve too slowly.
• 2.  
  The remaining models for the case of the neglected host-galaxy extinction (A–D, F, G, and I–L) have M_56Ni as small as 1.2 M_☉ and 1.4 M_☉, which is consistent with the analytical estimates. While model J has the smallest WRMSR, its v_ph is relatively larger than the observations. Models D, G, and L also have larger v_ph around t_B ∼ 0 day. Among the other models (A–C, F, I, and K), the models with M_WD ⩽ 2.4 M_☉ (A–C) are preferred, because their BVRI LCs reproduce the observations well around the peak. Especially, model B has the smallest v_ph of the three models, so we suggest that model B is the most plausible model for SN 2009dc without the host-galaxy extinction.
• 3.  
  If the extinction by the host galaxy is significant, we have only two candidate models with M_WD = 2.8 M_☉ (N and O). They both have similar properties: slightly larger L_BVRI at t_B ∼ −20 days, but a bit smaller L_BVRI around t_B ∼ 0 day. We regard model N as the most plausible model for the extinction-corrected case because the rising time is shorter than model O.

By these results, we suggest that the mass of progenitor WD of SN 2009dc is 2.2–2.4 M_☉ if the extinction by its host galaxy is negligible, and ∼2.8 M_☉ with the extinction if the extinction is significant. The ⁵⁶Ni mass needed for the SN is 1.2–1.4 M_☉ for the former case, and 1.8–1.9 M_☉ for the latter. For the latter case with extinction, the estimated masses are consistent with those suggested by Taubenberger et al. (2011), though they assume that the mean optical opacity of SN 2009dc is similar to (normal) SN 2003du.

For SN 2007if, one of the super-Ch candidates, Scalzo et al. (2010) estimate the total mass of its progenitor to be ∼2.4 M_☉, as massive as we estimate for SN 2009dc. They consider a shell-structured model to explain the low Si ii line velocity and its plateau-like evolution, where the massive envelope decelerates the outer layers of the ejecta. Our calculations also reproduce the lower Si ii line velocity and similar evolution of SN 2009dc (the right top panels of Figures 5–9), and suggest that the low line velocity and flat evolution of Si ii can be explained by scaled super-Ch-mass WD models, as well as the shell-shrouding models.

4.1.1. Multi-color Light Curves

4.1.2. Bolometric, uvoir, and BVRI Light Curves

In Figure 10, the bolometric, uvoir, and BVRI LCs are plotted for models W7, B, and N. The uvoir LCs peak earlier than the BVRI LCs, by ∼2 days for W7 model and by ∼10 days for models B and N. As for the peak luminosity, the uvoir LC of W7 is brighter than the BVRI LCs by ∼0.2 dex, while those of the two super-Ch-mass WD models, by ∼0.3 dex. These shifts are understood by considering the ultraviolet (UV) radiation in the early phase (Blinnikov & Sorokina 2000).

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Most of the BVRI LCs for the super-Ch-mass WD models in this paper seem to be fainter for t_B < 0 day, and slightly brighter after that, than the observations (Figures 5–10). This discrepancy could be solved by changing the ⁵⁶Ni distribution in the model, which powers its radiation. More ⁵⁶Ni outside could shorten the rising time, or make it more luminous before the peak, while less ⁵⁶Ni inside could make it fainter after that. Further detailed modeling is needed since we simply assume the ⁵⁶Ni distribution analogous to W7.

In Figure 10, we add the observation data for t_B > 90 days plotted in Figure 7 of Taubenberger et al. (2011) as filled pentagons, which are also corrected for the host-galaxy extinction by using their values. The calculated bolometric LC fits the observed LC tail quite well. The observed tail lies just on model N, which means that the estimate of M_56Ni for the model is consistent.

We calculate E_bin by just extrapolating the formula, valid only for M_WD ≳ 2.1 M_☉ (Yoon & Langer 2005). Using their Equations (22) and (29)–(31), we can calculate the ratio of the rotational energy (T) to (the magnitude of) gravitational one (W) of a super-Ch-mass WD with our parameter ρ_c. This ratio T/W indicates the stability of the WD. If T/W is small enough, the WD is stable against rotation. If T/W reaches 0.14, the WD suffers the non-axisymmetric instability (e.g., Ostriker & Bodenheimer 1973). The above equations show that a WD with M_WD ∼ 2.4 M_☉ has T/W = 0.14 for our ρ_c. A super-Ch-mass WD with M_WD ⩾ 2.4 M_☉ might not form. However, T/W as a function of M_WD differs with the rotation law (e.g., Hachisu 1986), so that it is possible that even such a massive WD as M_WD ⩾ 2.4 M_☉ has T/W < 0.14 and thus forms. We thus consider the models with M_WD ⩾ 2.4 M_☉.

Our results in this paper suggest that the progenitor WD mass of SN 2009dc exceeds 2.2 M_☉ even for the case where the host-galaxy extinction is negligible. Such a large M_WD might suggest that the explosion of this super-Ch-mass WD is triggered when its mass reaches the critical mass for the above instability of the rotating WDs (Hachisu et al. 2012).

The progenitor WD mass for SN 2009dc in our estimate largely exceeds M_Ch, thus putting severe constraints on the presupernova evolution of the binary system. This also should have important implications for the progenitor scenarios of ordinary SNe Ia, both the single degenerate (SD) and double degenerate (DD) scenarios.

4.3.1. Single Degenerate Scenario

4.3.2. Double Degenerate Scenario