A Polarization Sequence for Type Ia Supernovae?

We collected the polarizational and spectral data of the Si ii 635.5 nm line for various SNe Ia from published literature. All of the polarization data and most of the spectral data are collected from the literature, while some spectra are taken from the data observed by the LiJiang 2.4 m telescope at Yunnan Observatories. The details of the data are summarized in Table 1. The sample (see Table 1) includes all subclasses of SNe Ia, even a peculiar object, SN 2005hk (Branch et al. 2009).

Generally, the peak level of the polarization of the Si ii 635.5 nm line in the polarization spectrum is taken as the polarization degree of the line, and the peak value usually appears at the absorption minimum of the line in the flux spectrum. The degree of polarization across the silicon line changes with time after the explosion, and the relation between the polarization degree and the time may be fitted by a second order polynomial (Wang et al. 2007). It is found that the polarization degrees of some SNe Ia at −5 days after maximum light are correlated to their maximum light (Wang et al. 2007). In this paper, for the collected polarization data, we normalized the polarization degree to the value at −5 days after maximum light by subtracting $0.041{(t+5)+0.013(t+5)}^{2}$ from the observed degree of polarization, where t is the day after the optical maximum light of a SN Ia (Wang et al. 2007).

In the polarization sample, the polarization observations for four SNe Ia were carried out after their maximum light, and their polarization degrees at −5 days after their maximum light were estimated based on their velocity evolution of the Si ii 635.5 nm absorption feature (Maund et al. 2010a), i.e.,

$$\begin{eqnarray}&&{P}_{\mathrm{Si}{\rm{II}}}=0.267+0.006\times {\dot{v}}_{\mathrm{Si}{\rm{II}}},\end{eqnarray} \tag{ 1 }$$

where ${\dot{v}}_{\mathrm{Si}{\rm{II}}}$ is the expansion velocity gradients of photosphere inferred from the Si ii 635.5 nm absorption feature (see Table 1).

We then looked for the spectrum closest to −5 days in the literature and defined a relative equivalent width (REW), which is defined as the ratio of the pEW to the relative depth (a) of an absorption feature. For the meaning of the REW, please see the next subsection. The measurements of the pEW and a for the absorption feature of Si ii 635.5 nm are exactly the same as the method used in Silverman et al. (2012). The REW values of the four SNe Ia with polarization measurements that were carried out after the maximum light are derived from the spectra closest to the time of −5 days after their maximum light.

In this paper, REW is defined as the ratio of the pseudo-equivalent width (pEW) to the relative depth (a) of an absorption feature (Silverman et al. 2012). Based on the definition of the REW, it can always be written into a form of

$$\begin{eqnarray}&&\mathrm{REW}={c}_{1}({\nu }_{2}-{\nu }_{1})+{c}_{2}({v}_{2}-{v}_{1})\end{eqnarray} \tag{ 2 }$$

in first order approximation (Rybicki & Hummer 1978), where c₁ and c₂ are coefficients that are related to the species of an element and temperature. ν₁ and ν₂ are the corresponding frequencies at the blue and red wings of an absorption feature, and v₁ and v₂ are the expansion velocities of the inner and outer boundary of a given element in the supernova ejecta. Generally, v₁ is equal to the expansion velocity of the photosphere. We show a simple proof as follows. For a line profile from an expanding atmosphere in a co-moving reference system, the line profile can be expressed by

$$\begin{eqnarray}&&F(x)=\displaystyle \frac{1}{2}{\int }_{0}^{{\rm{R}}}{\int }_{0}^{1}I({z}_{\max },p,x)\mu d\mu {dp},\end{eqnarray} \tag{ 3 }$$

where I is the specific intensity at the point (z_max, p) in the direction μ (Huang 1998). z and p are coordinates in the cylindrical coordinate, and z_max is the value at the surface of the atmosphere. x is a dimensionless frequency in the co-moving reference system and can be expressed as the linear combination of frequency ν and the material velocity field v, i.e., $x={b}_{0}+{b}_{\nu }\nu +{b}_{{\rm{v}}}v$, where b₀ and b_ν are correlated with the thermal velocity of a species and the central frequency of a line, and b_v is related to the thermal velocity of the species and μ. For simplicity, we normalized the continuum flux to be 1, and then the REW is defined as

$$\begin{eqnarray}&&\mathrm{REW}=\displaystyle \frac{{\int }_{{x}_{1}}^{{x}_{2}}[1-F(x)]{dx}}{1-F({x}_{0})}=f(x){| }_{{x}_{1}}^{{x}_{2}},\end{eqnarray} \tag{ 4 }$$

where x₀ is the value at linecore, and $f(x)=\tfrac{\int [1-F(x)]{dx}}{1-F({x}_{0})}$. Via a Taylor series expansion at x₀, we may get

$$\begin{eqnarray}\mathrm{REW} & \approx & [f({x}_{0})+f^{\prime} ({x}_{0})(x-{x}_{0}){]}_{{x}_{1}}^{{x}_{2}}\\ & = & f^{\prime} ({x}_{0})[{b}_{\nu }({\nu }_{2}-{\nu }_{1})+{b}_{{\rm{v}}}({v}_{2}-{v}_{1})],\end{eqnarray} \tag{ 5 }$$

which shares the same form as Equation (2). Therefore, although the definition of REW is simple, it relates to two physical quantities of supernova ejecta, and we expect to see a linear relation between the REW and a kind of velocity difference that can reflect the distribution of an element in supernova ejecta and be derived from an absorption feature.

In Figure 1, we show the correlation between the REW and the full width at half-maximum (FWHM) intensity of the Si ii 635.5 nm absorption feature in velocity space, V_FWHM, for 155 SNe Ia around maximum light. We can see that there is a very good linear relation between the REW and V_FWHM, as expected from the definition of the REW. Since an absorption feature may reflect the properties of an element outside of the photosphere in the supernova ejecta of a SN Ia, REW may present the distribution of silicon outside of the photosphere in the velocity space of the supernova ejecta.

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From Figure 1, we can see that the value of the REW consecutively varies. To get the intrinsic distribution of the REW, we collected the spectra of SNe Ia from the literature; the sample is mainly from Silverman et al. (2012) and Wang et al. (2013). Since the spectrum at −5 day after the maximum light is rare, we collected spectra around maximum light. There was no bias in the collection of the sample. Although it is very likely that the value of REW may evolve with time, the distribution of REW around maximum light may reflect the distribution at −5 day to a great extent, at least in the sense of the distribution's tendency and shape. In Figure 2, we show the distribution of the REW around maximum light. The distribution may be well fitted by a Gaussian with an average value of 159.7 Å and σ = 45.56 Å, and there is no significant signal to show a bimodal distribution. If REW reflects the nature of the explosion mechanism of SNe Ia, the one Gaussian distribution implies that all SNe Ia share the same explosion mechanism.

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In Figure 3, we show the correlation between the polarization and REW of the Si ii 635.5 nm line for 28 SNe Ia. It seems that the SNe Ia are divided into two groups based on their position in the ${P}_{\mathrm{Si}}\mbox{--}\mathrm{REW}$ plane, i.e., one is a high-P_Si high-REW group including only the broad-line (BL) SNe Ia, the other is a low-P_Si low-REW group including core-normal (CN), shallow-silicon (SS), and cool (CL) SNe Ia (Branch et al. 2009). This appearance is mainly derived from the gap around REW ∼ 200 Å. The high-P_Si high-REW group has a value of (P_Si, REW) = (0.973% ± 0.296%, 241.6 ± 16.5 Å), and the low-P_Si low-REW group has a value of (P_Si, REW) = (0.496% ± 0.199%, 160.2 ± 19.6 Å), i.e., on average, the polarization degree of the high-P_Si high-REW group is higher than that of the low-P_Si low-REW group by about 0.5%, and the REW value by about 80 Å. Interestingly, the low-P_Si low-REW group has an almost equal average value of the REW to that of the REW distribution in Figure 2, and the REW value of the high-P_Si high-REW group is clearly beyond the 1σ level of the REW distribution.

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The two-group appearance is mainly derived from the existence of the gap of the REW in Figure 3. It is still not clear why there is the gap in Figure 3. One possible reason is that the dependence of the REW on a parameter in a successful model is bifurcated. However, the distribution of REW is consecutive (see Figure 1), and there is no signal of the bifurcated distribution of the REW. Indeed, the REW distribution may well be fitted by one Gaussian (see Figure 2). The small size of the sample may also contribute to the gap. To test the probability that the gap is derived from a statistical fluctuation, we performed a simple Monte-Carlo simulation. By assuming a Gaussian distribution as shown in Figure 2, we found that the appearance of the gap in the REW can be attributed to a statistical fluctuation of 19.5%, if the observational error of the REW is not considered, and a fluctuation of 72.9% if the observational error is considered. Therefore, considering the Gaussian distribution of the REW, the gap of the REW in Figure 3 is very likely derived from the small size of the polarization sample. Another possible reason is an artificial selection effect, i.e., more attention to polarization observations is paid to special SNe Ia than to CN SNe Ia, which could result in some CN SNe Ia not being located in the gap. For example, SN 2016coj, which is a CN SN Ia (Zheng et al. 2016), has a REW value of ∼190 Å, higher than any other SS, CN, or CL SNe Ia in the sample here.

At the same time, there seems to be a trend that sees the polarization degree increase with the REW, and a linear relation fitting the correlation, i.e.,

$$\begin{eqnarray}&&{P}_{\mathrm{Si}{\rm{II}}}=-0.2951+4.988\times {10}^{-3}\times \mathrm{REW},\end{eqnarray} \tag{ 6 }$$

where the correlation coefficient is 0.6641 and the observational error is taken as the weight for the fitting. The vertical deviation of the 1σ statistical error for the linear fit is 0.24%. The observational error bars for most SNe Ia, except for SN 2004dt and SN 2001V, are consistent with the 1σ level. At present, the sample size is small and the observational error is still large, which effects the confidence level of the correlation between the polarization and the REW, as shown by the correlation coefficient. However, by combining it with the Gaussian distribution of the REW a correlation based on the present data is still very possible, e.g., the average value of the polarization degree of the high-P_Si high-REW group is significantly higher than that of the low-P_Si low-REW group, although the error bar is partly overlapped. Moreover, even when the three most deviate SNe Ia, SN 2001V, 2004dt, and 2004ef, are removed, we still cannot use a horizontal line to fit the data points, while a tighter (σ = 0.15%) and smaller slope (3.250 × 10⁻³) line may fit the data points well.

In Figure 3, the polarization observation for four SNe Ia were carried our after the maximum light (purple points in Figure 3). Their positions in the ${P}_{\mathrm{Si}}\mbox{--}\mathrm{REW}$ plane still follow the polarization sequence, except that the polarization of SN 2009dc seems to be slightly beyond the 1σ range (but still consistent with the sequence within observational error). SN 2002bf belongs to the BL subclass, and SN 2003hv and 2004S to the CN subclass.

In Figure 3, the SNe Ia seem to be divided into a high-P_Si high-REW group and a low-P_Si low-REW group, where the high-P_Si high-REW group are only included the BL SNe Ia. Maund et al. (2010a) also found that among normal SNe Ia, the high-velocity-gradient (HVG) SNe Ia show a higher polarization than the low-velocity-gradient (LVG) SNe Ia. Generally, the HVG SNe Ia classified by Benetti et al. (2005) overlap with the BL SNe Ia classified by Branch et al. (2009), except for some peculiar SNe Ia. For example, a HVG SN Ia in Benetti's classification, SN 2001V, belongs to SS SNe Ia in Branch's classification, and SN 2001V has a very low polarization. Here, the high-P_Si high-REW group consists exclusively of BL SNe Ia. At the early phase, BL SNe Ia often exhibit high- and low-velocity components in their absorption profiles, and the high-velocity component has a high polarization (Maund et al. 2013). However, the high-velocity feature (HVF) is a ubiquitous property of SNe Ia (Mazzali et al. 2005a, 2005b). For example, SN 2012fr, which belongs to the SS subclass, shows clear high-velocity components of the spectral features, but its polarization at five days before maximum light is not as high as those BL SNe Ia. Why do the HVF in SS, CL, and CN SNe Ia not show such high polarization as is shown in BL SNe Ia?

If the appearance of the two groups in the ${P}_{\mathrm{Si}}\mbox{--}\mathrm{REW}$ plane is accurate for SNe Ia, the results imply that SNe Ia are derived from at least two explosion models (Hillebrandt & Niemeyer 2000) or two progenitor models (Wang et al. 2013), e.g., the violent merger model (Pakmor et al. 2010) produces the high-P_Si high-REW SNe Ia, while the Chandrasekhar mass model (Khokhlov 1991) and the double-detonation model (Livne 1990) produce the low-P_Si low-REW SNe Ia. However, numerical simulations show that the polarization degree predicted from the violent model is much higher than that of the high-P_Si high-REW SNe Ia, and the Chandrasekhar-mass model may well reproduce the polarization distribution of all SNe Ia (Bulla et al. 2016a, 2016b). As discussed in Section 3, the two-group appearance is mainly derived from the existence of the REW gap in Figure 3, but the gap is very likely attributed to the small sample size.

The polarization of the silicon line reflects the asymmetry of the silicon distribution in supernova ejecta, while the REW is a measure of the velocity difference of the silicon layer in velocity space. Even for a spherically symmetric structure of supernova ejecta, different REW values are still expected from different supernovae due to their different explosion energy. Thus, the polarization degree and the REW are independent parameters, and a correlation between the two parameters is not necessary. Therefore, if there exists a correlation between P_Si and REW, and the correlation shown in Figure 3 is intrinsic for all SNe Ia, the correlation will put strong constraints on the successful explosion models of SNe Ia, since all kinds of subclasses of SNe Ia seem to obey the same sequence, which can be explained by any successful model. At the present stage of theoretical modeling, deflagration (Nomoto et al. 1984), delayed-detonation (Khokhlov 1991; Livne 1999), gravitationally confined detonation (Plewa et al. 2004), detonating failed deflagration (Plewa 2007), and violent-merger (Pakmor et al. 2010) models can all plausibly be argued to produce clumps in the ejecta, which may contribute to the polarized absorption lines.

As shown in Section 2.2, the REW of an absorption line is proportional to the kind of velocity difference that can reflect the distribution of an element in supernova ejecta and be derived from an absorption feature. As expected, the REW of the Si ii 635.5 nm absorption feature is indeed proportional to the FWHM intensity of the absorption feature in velocity space. Therefore, if the correlation between P_Si and REW is intrinsic for all SNe Ia, we may expect that the greater the asymmetry of the distribution of silicon in the supernova, the wider the velocity interval of the silicon layer. At present, no one simulation clearly shows such a correlation in the literature. However, some simulations of the delayed-detonation model produced interesting results, and may give a reasonable physical explanation, at least in principle. In addition, one fact must be kept in mind: the polarization degree of the continuum of an SN Ia is generally very low, i.e., generally less than 0.2%, which may exclude any progenitor model or explosion model from predicting a highly asymmetric distribution of supernova ejecta (Wang & Wheeler 2008).

Generally, for the delayed-detonation model, a strong detonation is more likely to produce a less turbulent silicon layer, and a later detonation, i.e., the deflagration flame that has propagated sufficiently close to the low-density part of an expanding white dwarf, has to bear the imprint of the complex structure of the prior deflagration. It is then expected that the earlier that the transition from deflagration to detonation occurs, the less asymmetric the silicon. At the same time, the earlier that the deflagration-detonation transition occurs, the narrower the velocity interval of the silicon layer, as shown by some numerical simulations (Gamezo et al. 2005; Blondin et al. 2011). Therefore, it is expected that there exists a correlation between the polarization degree of silicon lines and the velocity interval of silicon distributed in supernova ejecta, as deduced from Figure 3. So, the delayed-detonation model may qualitatively explain the correlation in Figure 3.

Different models use different free parameters to denote the transition from deflagration to detonation. Multi-dimensional numerical simulations generally use a transition time t_tr from deflagration to detonation as a free parameter (Kasen et al. 2009; Gamezo et al. 2005), which translates into a transition density ρ_tr from deflagration to detonation as the free parameter in 1D simulations. The models with different ρ_tr in 1D simulations may provide excellent fits to SN Ia spectra and light curves, and account for the energetics of SNe Ia (Hillebrandt & Niemeyer 2000). If the delayed-detonation model explains the correlation in Figure 3, a correlation between the free parameter denoting the transition from deflagration to detonation and the velocity interval of the silicon layer in supernova ejecta could be expected. To test this idea, we searched for the silicon distribution in velocity space in published literature. The distribution should fulfill the following two criteria: (1) it is the silicon distribution presented in the whole velocity interval, and (2) at least three value of the free parameter was adopted. We found that the 1D simulation in Iwamoto et al. (1999) completely fulfills the above criteria. Here, based on the result of a 1D numerical simulation (Iwamoto et al. 1999), we define the velocity interval of the silicon layer in supernova ejecta as the velocity difference between the two velocity points, where the mass fraction of the silicon is 0.1 (see Figure 25 in Iwamoto et al. 1999). In Figure 4, we show the correlation between the velocity interval of silicon ΔV_0.1 as the function of the transition density ρ_tr. We can see from the figure that the correlation is well fitted by a linear relation. As shown in Section 2.2, REW presents the velocity interval of the silicon layer above the photosphere. At −5 days, the location of the photosphere is still inside the silicon layer, and then the REW is not the exact measure of the velocity interval of the silicon layer in supernova ejecta. However, the REW may still represent the velocity interval to a great extent since a part of silicon layer is on top of the photosphere at −5 days; this is especially true for those high-velocity parts that contribute to a sizable part of the velocity interval of the silicon layer (Iwamoto et al. 1999; Benetti et al. 2005). Therefore, although it is not exactly the same, ΔV_0.1 as defined here may represent V_FWHM to a great extent. Combining the linear relation between the REW and V_FWHM, we expect a linear relation between the REW and the ρ_tr. Considering that the free parameter of the ρ_tr in 1D simulations plays a similar role to the numerical simulation results of the explosion model as the transition time t_tr from deflagration to detonation in multi-dimensional numerical simulations, a linear relation between the t_tr and the REW is expected. Especially, the later the t_tr, the closer the deflagration is to the WD surface, and therefore the following detonation has to bear a more complex density structure. In addition, the later the t_tr, the wider the velocity interval of silicon layer (Gamezo et al. 2005; Blondin et al. 2011). Therefore, a high polarization degree and a large REW are simultaneously expected from a later transition from deflagration to detonation. Moreover, since the delayed-detonation model has close to a spherically symmetric structure for the supernova ejecta, the low polarization degree of the continuum is also not problematic within this model.

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The above discussion is a deduction based on some present numerical simulations and an assumption that ΔV_0.1 may represent V_FWHM to a great extent. We also notice that the simulations in the delayed-detonation models are not always consistent with each other. For example, the 3D delayed-detonation model in Seitenzahl et al. (2013) has shown that the models with weaker deflagration have more asymmetric silicon layers, which seems to be exactly opposite to the expectations from Figure 3. Moreover, at −5 days after the maximum light, the Si ii line widths are not exact measures of the velocity intervals of the silicon layer in supernova ejecta. Hence, the above discussion of the delayed-detonation model is just a qualitative analysis. More quantitative simulation is needed to confirm whether or not the delayed-detonation model may reproduce the correlation between the polarization degree and REW as shown in Figure 3.

However, there is still a large scatter of σ = 0.24% for the correlation in Figure 3. For most SNe Ia, their observational error bars are consistent with the 1σ level, except SN 2004dt and SN 2001V. The origin of the scatter is still not clear. One possible origin is the effect of the viewing angle, e.g., the level of the scatter of σ = 0.24% for the correlation is well consistent with that predicted by the delayed-detonation model (Bulla et al. 2016a). It has been verified that the viewing angle may significantly affect the properties of SNe Ia, such as the absolute magnitude, the time of maximum light, the optical spectra, and especially the polarization degree of absorption features (Kasen et al. 2004; Blondin et al. 2011; Bulla et al. 2016b). The effect of the viewing angle on the polarization degree is not a simple monotonic function (Kasen et al. 2004; Bulla et al. 2016b), and the level of the polarization uncertainties due to the viewing angle is heavily model-dependent, especially on the progenitor model. For the single-degenerate model, even if the explosion is spherically symmetric, supernova ejecta may still become asymmetric due to the existence of a companion, i.e., the companion carves out a conical hole in the supernova ejecta (Marietta et al. 2000; Meng et al. 2007; Gray et al. 2016), which leads to an uncertainty of the polarization degree of the silicon line that could be as large as 0.5% (Kasen et al. 2004), which is consistent with the 2σ level of the polarization sequence in Figure 3. As a subclass of the double-degenerate model, the violent-merger model predicts that the difference of the polarization degree of the silicon line from different viewing angels for the same SN Ia may be as large as 1.8, and then the uncertainty of the polarization degree is even higher than the 3σ level obtained here (Bulla et al. 2016b). Considering that the polarization degree of the continuum predicted by the violent merger model is always significantly higher than that from observations (Bulla et al. 2016a, 2016b), the violent-merger model seems not to be a reasonable one to explain the polarization sequence here and the low-continuum polarization of SNe Ia. Similarly, the predicted properties of SNe Ia from the WD–WD collision model is also quite viewing-angle dependent, and the supernova ejecta is also highly asymmetric (Raskin et al. 2009; Rosswog et al. 2009). Therefore, the WD–WD collision model also has no ability to simultaneously explain the polarization sequence and the low continuum polarization of SNe Ia. The polarization degree of SN 2004dt and SN 2001V is beyond 1σ range of the polarization trend, as found in Wang et al. (2007) and Maund et al. (2010a). If the viewing angle is the origin of the polarization uncertainties, then SN 2004dt and SN 2001V would be observed along a very special viewing angle.

In addition, there are at least two populations contributing to SNe Ia observationally (Wang et al. 2013), which suggests that at least two progenitor scenarios contribute to SNe Ia. Since the sequence in Figure 3 includes all subclasses of SNe Ia, the result here implies that all SNe Ia could share the same explosion mechanism, no matter what their progenitor models are, if the correlation is intrinsic for all SNe Ia.

Among the SNe Ia shown in Figure 3, SN 2005hk was classified as a peculiar SN Ia (Chornock et al. 2006) and was thought to have arisen from the pure deflagration model of a Chandrasekhar-mass CO WD (Kromer et al. 2013). Recently, it was suggested that this peculiar SN Ia arose from the hybrid WDs, where a CO core is surrounded by an oxygen-neon (ONe) mantle (Meng & Podsiadlowski 2014; Wang et al. 2014; Kromer et al. 2015). For a low-carbon abundance in the ONe mantle, the deflagration flame may be switched off in the mantle region and does not translate into detonation. If the switched-off deflagration model is taken as a special case of the delayed-detonation model, the low polarization degree of SN 2005hk is then naturally explained. The following detonation after a deflagration flame may play a great role in the final density structure of supernova ejecta in velocity space, i.e., it would amplify the velocity distribution of a given element, and possibly the asymmetry of supernova ejecta. Due to the absence of the amplification of the following detonation, a low polarization degree and a low REW may be simultaneously expected from a SN Ia arising from the switched-off deflagration model. Whatever, a detailed numerical polarization simulation of this suggestion is needed.

The polarization degree of SN 2009dc is consistent with the 1σ range of the linear relation in Figure 3 within observational error. SN 2009dc is a super-luminous SN Ia and the total mass of the supernova ejecta exceeds the canonical Chandrasekhar mass limit (Howell et al. 2006; Yamanaka et al. 2009a). The supernova is thought to have arisen from the merger of a double-degenerate system, where the final exploding WD is more likely to have a Chandrasekhar mass, surrounded by a large amount of carbon–oxygen circumstellar medium (CSM; Taubenberger et al. 2013). Generally, the supernova ejecta of an SN Ia from the double-degenerate model is highly asymmetric and a high polarization degree is expected, which is inconsistent with the low-continuum polarization degree of SN 2009dc (Tanaka et al. 2010). Two possible reasons may contribute to the low polarization degree of SN 2009dc. One is that its polarization degree at −5 days after maximum light was underestimated, as the ejecta velocity evolution of a super-Chandrasekhar-mass SN Ia is completely different from that of a normal SN Ia, e.g., the ejecta velocity of a super-luminous SNe Ia is generally much lower than that of a normal SN Ia (Howell et al. 2006). Another possible reason is that, if the supernova arises from a Chandrasekhar-mass WD exploding in a dense carbon–oxygen CSM, the WD could experience a short deflagration phase, which would soon become a detonation. Such a short deflagration phase does not produce a highly asymmetric distribution of silicon or a large velocity range. At the same time, early transition from deflagration to detonation results in a large amount of nickel-56 (Blondin et al. 2011). Such a Chandrasekhar-mass explosion is also needed for a low-continuum polarization (Tanaka et al. 2010). If so, the high amount of nickel-56, low polarization degree, and narrow silicon absorption feature may be explained simultaneously, i.e., even if an SN Ia arises from a double-degenerate system with super-Chandrasekhar mass, there are no special circumstances for its explosion mechanism.

The low polarization of SN 2009dc could also arise from the effect of viewing angle, i.e., it was observed from a special viewing angle (Bulla et al. 2016b). However, it seems impossible for the effect of the viewing angle to explain the high luminosity and low polarization degree of SN 2009dc simultaneously, i.e., a high luminosity due to the effect of viewing angle also means a high polarization degree of the silicon line (Hillebrandt et al. 2007; Blondin et al. 2011).

In summary, we found that either SNe Ia are divided into two groups in the P_Si–REW plane, or that all SNe Ia follow a polarization sequence. However, considering the Gaussian distribution of the REW, it is very likely that the two-group result is derived from the small size of the polarization sample here. If the correlation between the polarization degree and the REW is intrinsic for all SNe Ia, it will put strong constraints on the explosion model. At present, no explosion model clearly shows such a correlation, but the delayed-detonation model is an interesting one to qualitatively explain the sequence.