A SEARCH FOR NEW CANDIDATE SUPER-CHANDRASEKHAR-MASS TYPE Ia SUPERNOVAE IN THE NEARBY SUPERNOVA FACTORY DATA SET

The supernovae are among the 400 SNe Ia discovered in the SNfactory SN Ia search, carried out between 2005 and 2008 with the QUEST-II camera (Baltay et al. 2007) mounted on the Samuel Oschin 1.2 m Schmidt telescope at Palomar Observatory ("Palomar/QUEST"). QUEST-II observations were taken in a broad RG-610 filter with appreciable transmission from 6100–10000 Å, covering the Johnson R and I bandpasses. Table 1 lists the details of the SN discoveries, including SN 2007if.

Upon discovery, candidate SNe were spectroscopically screened using the SuperNova Integral Field Spectrograph (SNIFS; Aldering et al. 2002; Lantz et al. 2004) on the University of Hawaii (UH) 2.2 m on Mauna Kea. Our normal criteria for continuing spectrophotometric follow-up of SNe Ia with SNIFS were that the spectroscopic phase be at or before maximum light, as estimated using a template-matching code similar, e.g., to SUPERFIT (Howell et al. 2005), and that the redshift be in the range 0.03 < z < 0.08. Aware of the potential for discovering nearby counterparts to the candidate super-Chandrasekhar-mass SN 2003fg (Howell et al. 2006), we allowed exceptions to our nominal redshift limit for continued follow-up if the spectrum of a newly screened candidate appeared unusual or especially early.

Our spectroscopic selection was informed by the existing spectra of SN 2003fg and SN 2007if. In particular, the pre-maximum and near-maximum spectra of SN 2007if showed weak Si ii λ6355 and Ca ii H+K absorption and strong absorption from Fe ii and Fe iii, with noted similarity to SN 1991T (Scalzo et al. 2010). We therefore prioritized spectroscopic follow-up for SNe Ia visually similar to, or more extreme (i.e., having weaker intermediate-mass elements (IME) and stronger Fe-peak absorption) than, SN 1991T itself. Here, we have excluded the less extreme 1999aa-likes, which can be separated based on the strength of Ca ii H+K (Silverman et al. 2012). As a cross-check on our initial selection conducted using the initial classification spectra, we have run SNID v5.0 (Blondin & Tonry 2007) on the entire SNfactory sample, using version 1.0 of the templates supplemented by the Scalzo et al. (2010) SNfactory spectra of SN 2007if. We searched for pre-maximum spectra for which the best subtype was "Ia-91T" or "Ia-pec" with the top match being a 1991T-like SN Ia or SN 2007if itself; this yielded the same set of SNe as the visual selection.

Our sample could thus be described as 1991T-like based on their spectroscopic properties. However, this categorization is often used to imply light-curve characteristics—luminosity excess or slow decline rate—that were not part of our selection criteria. Moreover, since our interest is ultimately in the masses of these systems, we will refer to these as "candidate super-Chandra SNe Ia" throughout this paper.

The sample of six objects (including SN 2007if) presented here constitutes all such spectroscopically selected candidate super-Chandra SNe Ia in the SNfactory sample. This was established as part of the classification cross-check described above. An additional 141 SNe Ia discovered by the SNfactory were also followed spectrophotometrically and constitute a homogenous comparison sample. Like SN 2003fg, but unlike SN 2006gz and SN 2009dc, our sample of super-Chandra candidates and our reference sample were discovered in a wide-area search and therefore sample the full range of host galaxy environments. This may prove important in understanding the formation of super-Chandra SNe Ia, e.g., if metallicity plays an important role (Taubenberger et al. 2011; Khan et al. 2011; Hachisu et al. 2012). We reiterate that characteristics such as luminosity excess, velocity evolution, light-curve shape, etc., were not used in our selection—it is purely based on optical spectrophotometry.

Most of the BVRI photometry in this work was synthesized from SNIFS flux-calibrated rest-frame spectra, using the bandpasses of Bessell (1990), and corrected for Galactic dust extinction using E(B − V) from Schlegel et al. (1998) and the extinction law of Cardelli et al. (1988) with R_V = 3.1. Follow-up BVRI photometry for SNF 20070803-005, using the ANDICAM imager on the CTIO 1.3 m, was obtained through the Small and Moderate Aperture Research Telescope System (SMARTS) Consortium. Redshifts were obtained from host galaxy spectra, which were either extracted from the SNIFS datacubes, or taken separately using the Kast Double Spectrograph (Miller & Stone 1993) on the Shane 3 m telescope at Lick Observatory, the Low Resolution Imaging Spectrograph (Oke et al. 1995) at Keck-I on Mauna Kea, or the Goodman High-Throughput Spectrograph at SOAR on Cerro Pachon (see Childress et al. 2011, and M. Childress et al. 2012, in preparation). The redshifts (from spectroscopic template fitting) and morphological types (from visual inspection) are listed in Table 1.

2.3.1. SNIFS Spectrophotometry

2.3.2. SMARTS Photometry

The rest-frame, Milky Way dereddened Bessell BVRI light curves of the SNfactory super-Chandra candidates are given in Table 2 and shown in Figure 1. The color evolution is shown in Figure 2. The light curves of the two SNe at the high end of the SNfactory redshift range, SNF 20070528-003 (z = 0.117) and SNF 20070912-000 (z = 0.123), have less extensive coverage than the other SNe; the S/N is lower, and a significant fraction of the rest-frame I-band transmission lies outside of the observer-frame wavelength range of the SNIFS spectrograph. For a small number of observations, rest-frame B-band or V-band measurements are unavailable due to instrument problems with the SNIFS blue channel. The detailed analysis of these light curves is presented in Sections 3.1 and 3.2.

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Figure 3 presents the subset of spectra that were taken near maximum light for the six SNfactory SNe Ia, including SN 2007if. Before maximum light, the spectra show weak Si ii, S ii, and Ca ii absorption plus strong Fe ii and Fe iii absorption, in accordance with our selection criteria. SN 1991T and the prototype candidate super-Chandra SN Ia 2003fg are shown for comparison. After maximum light, some features noted in SN 2003fg and SN 2007if appear, including the iron-peak blend near 4500 Å, the sharp notch near 4130 Å identified as Cr ii in Scalzo et al. (2010; or tentatively as C ii λ4237 in Howell et al. 2006), and the blended lines near 3300 Å identified as Cr ii and Co ii in Scalzo et al. (2010). (The 4130 Å feature is not clearly present in the z = 0.123 candidate SNF 20070912-000, although this may be due to the lower S/N of the spectrum.) Taubenberger et al. (2011) note that a similar feature in SN 2009dc strengthens with time rather than fading, as one would expect for Cr ii rather than C ii.

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SN 2007if also shows a weak C ii λ6580 line in the post-maximum spectra, which Scalzo et al. (2010) interpreted as a signature of unburned material from the explosion. While this line is not detected unambiguously in the other SNe, the unusually shallow slope of the red wing of Si ii λ6355, e.g., in SNF 20070528-003 and SNF 20080723-012, may be a sign that C ii is present in these SNe (Thomas et al. 2011a). The spectral properties of the full data set are analyzed in Section 3.3.

As discussed in Scalzo et al. (2010), SN 2007if shows no distinct second maximum. While Figure 1 does show second maxima for our other SNe, its prominence is suppressed relative to normal SNe Ia. In SNF 20070803-005 and SNF 20080723-012, the peak-to-trough difference in a quintic polynomial fit to the data from day +7 to day +42 is only 0.11 mag versus 0.23 mag for SNF 20080522-000 and 0.35 mag for the SALT2 model with x1 = 1, c = 0. Kasen (2006) noted three physical effects which could reduce the contrast of the I-band second maximum: low ⁵⁶Ni mass, efficient mixing of ⁵⁶Ni into the outer layers of ejecta, and greater absorption in the Ca ii NIR triplet line source function. Since our SNe are all overluminous with broad light curves, Arnett's rule gives a high nickel mass, as we find in the next section. The I-band first maximum is roughly concurrent with the B-band maximum for our SNe, rather than being significantly delayed (see Figure 14 of Kasen 2006), so it seems unlikely that emission in the Ca ii NIR triplet is contributing significantly. The most likely interpretation, especially given the prominence of Fe-peak lines in early spectra of our SNe, is that ⁵⁶Ni is well-mixed into the outer layers.

Following practice from Scalzo et al. (2010), we use the updated version (v2.2) of the SALT2 light-curve fitter (Guy et al. 2010) to interpolate the magnitudes and colors of each SN around maximum light, and to establish a date of B-band maximum with respect to which we can measure light-curve phase. While we use SALT2 here as a convenient functional form for describing the shape of the light curves near maximum light, and to extract the usual parameters describing the light-curve shape, we do not expect the SALT2 model, trained on normal SNe Ia, to give robust predictions for these peculiar SNe Ia outside the phase and wavelength coverage for each SN. To minimize the impact of details of the SALT2 spectral model on the outcome, we use SALT2 in the rest frame, include SALT2 light-curve model errors in the fitting, and we fit BVR bands only; I band is excluded from the fit. The quantities derived from the SALT2 light-curve fits are shown in Table 3. A cross-check in which cubic polynomials were fitted to each band produces peak magnitudes and dates of maximum in each band consistent with the SALT2 answers, within the errors, for all of the new SNe; we adopt the SALT2 values as our fiducial values for direct comparison with other work.

We estimate the host reddening of the SNe in two separate ways. First, we fit the B − V color behavior of each SN to the Lira relation (Phillips et al. 1999; Folatelli et al. 2010) for those SNe for which we have appropriate light-curve phase coverage. The Lira relation is believed explicitly not to hold for SN 2007if and the candidate super-Chandra SN Ia 2009dc (Yamanaka et al. 2009; Taubenberger et al. 2011), but its value may nevertheless be useful in studying the relative intrinsic color of these SNe. Additionally, we search for Na i D absorption at the redshift of the host galaxy for each SN. We perform a χ² fit to the Na i D line profile, modeled as two separate Gaussian lines with full width at half-maximum equal to the SNIFS instrumental resolution of 6 Å, to all SNIFS spectra of each SN, as for SN 2007if in Scalzo et al. (2010). In the fit, the equivalent width EW(Na i D) of the Na i D line is constrained to be non-negative. We convert these to estimates of E(B − V)_host using both the shallow-slope (0.16 Å⁻¹) and steep-slope (0.51 Å⁻¹) relations from Turatto et al. (2002, hereafter "TBC"). While the precision of these relations has been called into question when used on their own (e.g., Poznanski et al. 2011), we believe that examining such estimates together with the Lira relation and fitted colors from the light curve can provide helpful constraints on the importance of host reddening. The best-fit Lira excesses, values of EW(Na i D), and derived constraints on the host galaxy reddening are listed in Table 4.

We detect weak Na i D absorption in SNF 20070803-005 and SNF 20080522-000. Neither of these SNe appear to have very red colors according to the SALT2 fits, and we believe it to be unlikely that either are heavily extinguished, so for purposes of extinction corrections to occur later in our analysis, we use reddening estimates from the shallow-slope TBC relation, together with a CCM dust law with R_V = 3.1 (Cardelli et al. 1988). When applied to Milky Way Na i D absorption in our spectra, the shallow-slope TBC relation produces E(B − V) estimates consistent with Schlegel et al. (1998). SNF 20070912-000 shows a marginal (<2σ) detection, though the spectra are noisy and the limits are not strong. We detect no Na i D absorption in the other SNe.

Based on the very strong limit on Na i D absorption from the host galaxy, Scalzo et al. (2010) inferred that the large Lira excess of SN 2007if was not due to host galaxy extinction. Those SNe observed at sufficiently late phases show measured Lira excesses consistent with zero, additional evidence that host galaxy dust extinction is minimal for these SNe if the Lira relation holds.

Allowing for varying amounts of extinction associated with the Lira excess or Na i D absorption, each of the new SNe has maximum-light (B − V) colors consistent with zero. While most of our SNe have well-sampled light curves around maximum light and hence have well-measured maximum-light colors, SN 2007if has a gap between −9 days and +5 days with respect to B-band maximum (phases fixed by the SALT2 fit). The light-curve fit of Scalzo et al. (2010), using SALT2 v2.0 and a spectrophotometric reduction using a previous version of the SNIFS pipeline, suggests a red color B − V = 0.16 ± 0.06. The more recent reduction fit with SALT2 v2.2 gives B − V = 0.12  ± 0.06. However, the directly measured B − V color of SN 2007if at −9 days (−0.07 ± 0.04) and at +5 days (0.15 ± 0.02) is consistent within the errors with the mean values at those epochs for our other five SNe. The systematic error on the maximum-light color of SN 2007if, at least 0.04 mag, may therefore be too large for it to be considered significantly redder at maximum than its counterparts.

As input to our further analysis to calculate ⁵⁶Ni masses and total ejected masses for our sample of SNe, we calculate quasi-bolometric UVOIR light curves from the photometry. To derive bolometric fluxes from SNIFS spectrophotometry, we first deredden the spectra to account for Milky Way dust reddening (Schlegel et al. 1998), then deredden by an additional factor corresponding to a possible value of the host galaxy reddening, creating a suite of spectra covering the range 0.00 < E(B − V)_host < 0.40 in 0.01 mag increments. We then integrate the dereddened, deredshifted, flux-calibrated spectra over all rest-frame wavelengths from 3200–9000 Å. For each quasi-simultaneous set of ANDICAM BVRI observations and each possible value of E(B − V)_host, we multiply the dereddened, deredshifted SNIFS spectrum nearest in time by a cubic polynomial, fitted so that the synthetic photometry from the resulting spectrum matches, in a least-squares sense, the ANDICAM imaging photometry in each band. We then integrate this "warped" spectrum to produce the bolometric flux from ANDICAM. This procedure creates a set of bolometric light curves with different host galaxy reddening values which we can use in our later analysis (see Section 4).

SNF 20070528-003 and SNF 20070912-000 are at a higher redshift than our other SNe, such that SNIFS covers only the rest-frame wavelength range 3000–8500 Å, so we integrate their spectra in this range instead. We expect minimal systematic error from the mismatch, since the phase coverage for these two SNe is such that only the bolometric flux near maximum light, when the SNe are still relatively blue, is useful for the modeling described in Section 4.

To account for the reprocessing of optical flux into the near-infrared (NIR) by iron-peak elements over the evolution of the light curve, we must apply an NIR correction to the integrated SNIFS fluxes. No NIR data were taken for any of the SNe, except SN 2007if; in general, they were too faint to observe effectively with the NIR channel of ANDICAM. Since these are peculiar SNe, any correction for the NIR flux necessarily involves an extrapolation. Since the I-band second maximum has low contrast for all the SNe in our sample, the JHK behavior should be similar for these SNe to the extent that I and JHK are related (e.g., as in Kasen 2006). With these caveats, we therefore use the NIR corrections for SN 2007if derived in Scalzo et al. (2010) for all of our SNe, with the time axis stretched according to the stretch factor derived from the SALT2 x₁ (Guy et al. 2007) to account for the different timescales for the development of line blanketing in these SNe. Our modeling requires knowledge of the bolometric flux only near maximum light (to constrain the ⁵⁶Ni mass) and more than 40 days after maximum light (to constrain the total ejected mass). The NIR correction is at a minimum near maximum light (∼5%), and at a maximum near phase +40 days (∼25%), so it should not evolve quickly at these times and our results should not be strongly affected. We assign a systematic error of ±5% of the total bolometric flux (or about ±30% of the NIR flux itself near +40 days) to this correction.

To estimate ⁵⁶Ni masses, we also need a measurement of the bolometric rise time t_{rise, bol}. We establish the time of bolometric maximum light t_{max, bol} by fitting a cubic polynomial to the bolometric fluxes in the phase range $-10 \,\mathrm{d{\rm ays}} < t < +20 \,\mathrm{d{\rm ays}}$. We then calculate the final bolometric rise time t_{rise, bol} via

$$\begin{equation} t_\mathrm{rise,bol} = t_{\mathrm{rise,}B} - t_{\mathrm{max,}B} + t_{\mathrm{max,bol}}. \end{equation} \tag{ 1 }$$

We find t_{max, bol} to occur about one day earlier than t_max,B for the SNe in our sample. We have a strong constraint on the time of explosion only for SN 2007if (Scalzo et al. 2010), for which this procedure results in a bolometric rise time of 23 days. We have utilized the discovery data from our search along with our spectrophotometry to constrain the rise times of the new SNe presented here. For these, we find t_{rise, B} = 21.2 ± 1.9 days, and so use t_{rise, bol} = 20 ± 2 days in our models. Our value is very similar to the value of t_rise,B = 21 ± 2 days given by the sample of 1991T-like SNe Ia in Ganeshalingam et al. (2011). This approach leads to more conservative uncertainties than the single-stretch correction of Conley et al. (2006); Figure 6 of Ganeshalingam et al. (2011) suggests that the relation between rise time and decline rate Δm₁₅ (or stretch s) may break down at the high-stretch end.

We calculate the ⁵⁶Ni mass, M_Ni, for the six SNe by relating the maximum-light bolometric luminosity L_bol to the luminosity from radioactive decay L_rad (Arnett 1982):

$$\begin{eqnarray} \alpha ^{-1} L_\mathrm{bol} \ & = & \ L_\mathrm{rad} \nonumber \\ & = & \ N_{^{56}\mathrm{Ni}}\lambda _{^{56}\mathrm{Ni}}Q_{{^{56}\mathrm{Ni}},\gamma } \, e^{-\lambda _{^{56}\mathrm{Ni}}t} \nonumber \\ && + N_{^{56}\mathrm{Ni}}\lambda _{^{56}\mathrm{Ni}}\frac{\lambda _{^{56}\mathrm{Co}}}{\lambda _{^{56}\mathrm{Ni}}-\lambda _{^{56}\mathrm{Co}}} (Q_{{^{56}\mathrm{Co}},e^+} + Q_{{^{56}\mathrm{Co}},\gamma }) \nonumber \\ & &\times (e^{-\lambda _{^{56}\mathrm{Co}}t} - e^{-\lambda _{^{56}\mathrm{Ni}}t}), \end{eqnarray} \tag{ 2 }$$

where t (=t_{rise, bol} in this case) is the time since explosion, $N_{^{56}\mathrm{Ni}}= M_{^{56}\mathrm{Ni}}/(56 \,\mathrm{{\rm amu}})$ is the number of ⁵⁶Ni atoms produced in the explosion, $\lambda _{^{56}\mathrm{Ni}}$ and $\lambda _{^{56}\mathrm{Co}}$ are the decay constants for ⁵⁶Ni and ⁵⁶Co (e-folding lifetimes 8.8 days and 111.1 days) respectively, and $Q_{{^{56}\mathrm{Ni}},\gamma }$, $Q_{{^{56}\mathrm{Co}},\gamma }$, and $Q_{{^{56}\mathrm{Co}},e^+}$ are the energies released in the different stages of the decay chain (Nadyozhin 1994). The dimensionless number α is a correction factor accounting for the diffusion time delay of gamma-ray energy through the ejecta, typically ranging between 0.8 and 1.6 for reasonable explosion models (see, e.g., Table 2 of Höflich & Khohklov 1996, where it is called Q). A nominal value of α = 1.2 is often used in the literature (e.g., Nugent et al. 1995; Branch & Khokhlov 1995; Howell et al. 2006, 2009). The tamped detonation models of Khokhlov et al. (1993) and Höflich & Khohklov (1996), on which we will base our modeling later in the paper, have slightly higher values closer to α = 1.3. We therefore adopt a fiducial value of α = 1.3 for our simple estimate here.

The resulting ⁵⁶Ni mass estimates are shown in Table 5. The new SNe have M_Ni in the range 0.7–0.8 M_☉, at the high end of what might be expected for Chandrasekhar-mass explosions; the well-known W7 deflagration model (Nomoto et al. 1984) produced 0.6 M_☉ of ⁵⁶Ni, while some delayed detonation models can produce up to 0.8 M_☉ (e.g., the N21 model of Höflich & Khohklov 1996).

The time evolution of the position of the Si ii λ6355 absorption minimum, showing the recession of the photosphere through the ejecta, is shown in Figure 4. The measurements of the absorption minimum were made as follows: bins in each spectrum immediately to the right and left of the line feature were used to fit a linear pseudocontinuum, F_{λ, cont} = a + bλ. This fitted pseudocontinuum was divided into each bin of the spectrum in the region of the line. The resulting spectrum was smoothed with a third-order Savitzsky–Golay filter and the minimum was recorded as the bin with the lowest signal. The error bars on the procedure were determined through a bootstrap Monte Carlo: in the first stage, values of a and b representing possible pseudocontinua F_{λ, cont} were sampled using the covariance matrix of the pseudocontinuum fit; for each candidate pseudocontinuum, fluctuations typical of the measured errors on each spectral bin were added to the smoothed spectrum, and the results were smoothed and the minimum measured again. We have verified that the Savitzsky–Golay filter preserves the line minimum, so that smoothing a spectrum twice introduces negligible systematic error. The final velocity values and their errors were measured as the mean and standard deviation of the distribution of absorption minimum velocities thus generated. While this method has slightly less statistical power than a fit to the entire line profile as in Scalzo et al. (2010), we believe it is more robust to possible bias from line profiles with unusual shapes, and provides more realistic error bars for the line minimum.

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We found that for lines with equivalent widths less than 15 Å, the absorption minima had unreasonably large uncertainties and/or showed large systematic deviations from the trend described by stronger measurements. When measuring these absorption minima, we are probably simply measuring uncertainty in the pseudocontinuum. Blondin et al. (2011) saw similar effects when measuring velocities of very weak absorption minima, to the extent that the Si ii λ6355 velocity would even be seen to increase with time. We therefore reject measurements of such weak absorption features.

Our candidate super-Chandra SNe Ia share a slow evolution of the Si ii velocity, consistent within the errors with being constant in time for each SN from the earliest phases for which measurements are available. Table 6 shows the fitted constant velocities and the chi-square per degree of freedom, χ²_ν, for a fit to a constant. For comparison with earlier work, the velocity gradient $\dot{v} = -dv/dt$, calculated as the slope of the best-fit linear trend of the measurements before day +14, is also listed, along with the formal error from the fit. All of our SNe would be classified as Benetti LVG (Benetti et al. 2005) based on their velocity gradients. While the slope of the straight-line fit to the absorption velocities for SNF 20070803-005 seems to differ from zero at the 2σ level (formal errors), the reduced chi-square for this fit is extremely small (χ²_ν = 0.006) and we conclude that the evolution cannot be reliably distinguished from a constant (χ²_ν = 1.00). Similarly, the best-fit line to the absorption velocities for SNF 20080723-012 has a positive slope at 1.5σ ($\dot{v} = 72 \pm 47\, \mathrm{km\ s}^{-1}$ day⁻¹), but once again, a constant is a good fit to the data (χ²_ν = 0.72).

Table 6. Velocity Gradient Characteristics

SN Name	vpla	$\dot{v}$b	tpl, 0c	Δtpld	χ2νe
SNF 20070528-003	9371 ± 171	−6 ± 29	−4	16	0.85
SNF 20070803-005	9695 ± 81	46 ± 23	−1	14	1.00
SN 2007if	8963 ± 248	31 ± 29	−9	21	0.76
SNF 20070912-000	9201 ± 403	...	...	...	...
SNF 20080522-000	10936 ± 107	5 ± 10	−4	16	0.41
SNF 20080723-012	10391 ± 291	−72 ± 47	−2	10	0.72

Notes. ^aBest-fit constant velocity, in km s⁻¹, with error. ^bSlope of the best-fit line to absorption line velocities from the first reliable measurement until the break associated with Fe ii line blending, in km s⁻¹ day⁻¹. ^cPhase of first available measurement, in days with respect to B-band maximum light, which we interpret to be the start of the plateau. ^dMinimum duration of the visible plateau phase in days, until Si ii becomes blended with Fe ii. ^eChi-square per degree of freedom for fit to a constant.

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The slow Si ii absorption velocity evolution usually lasts until about two weeks after maximum light, after which a break in the behavior occurs and a pronounced decline begins, at a rate of ∼500 km s⁻¹ day⁻¹. At this point, developing Fe ii lines have probably blended with Si ii and made the velocity measurements unreliable (see, e.g., Phillips et al. 1992). The transition occurs concurrently with the onset of the second maximum in the I-band light curve, which may be attributed to light reprocessed from bluer wavelengths by the recombination of Fe iii to Fe ii as the ejecta expand and cool (Kasen 2006). Figure 5 shows the evolution of the Si ii λ6355 line profile for SNF 20070803-005 near the transition. The line profile after the break shows a portion of the Si ii line near the plateau velocity, suggesting that the material responsible for the plateau has thinned but is not yet transparent. We mark the phases of the last v(Si ii) measurement visually consistent with the plateau velocity, and the first measurement inconsistent with it, by vertical dashed lines in Figure 1 to show their correspondence with the I-band second maximum.

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In four of the six SNe, the plateau velocity is low (∼9000 km s⁻¹), inconsistent with the normal range of behavior of the LVG subclass of Benetti et al. (2005). The remaining two SNe, SNF 20080522-000 and SNF 20080723-012, have higher plateau velocities (∼11,000 km s⁻¹), falling roughly into the range of LVG behavior, although their velocity gradients are still flatter than any remarked upon in that work. Due to the lower S/N of our spectra of SNF 20070912-000 and the weakness of the Si ii λ6355 absorption feature, only two measurements of the line velocity before the break, each around 9000 km s⁻¹ with large errors, were extracted; we can at most say that the observed velocity gradient in this SN is consistent with that of the other SNe in our sample.

Figure 4 also shows the line minimum velocities of Ca ii H+K and S ii λλ5456, 5640. Although, as expected, Ca ii H+K stays optically thick longer than Si ii λ6355, its velocities at early times are consistent with the observed plateau behavior shown by Si ii. The S ii lines are weak, but when their absorption minima can be reliably measured, they do not show dramatically higher or lower velocities than the other lines. This supports the interpretation that all of these lines are formed in the same thin, dense layer of ejecta.

The ejecta density structure of SNe Ia is frequently modeled as an exponential ρ(v)∝exp (− v/v_e), where the ejecta are in homologous expansion at velocity v = (r − r_initial)/t since the explosion at time t = 0, and v_e is a characteristic velocity scale. Many hydrodynamic models of SN Ia explosions, including the well-known W7 model (Nomoto et al. 1984), have density profiles which are very close to exponential.

Jeffery (1999) made semi-analytic calculations of the time evolution of the gamma-ray energy deposition in an exponential model SN Ia, with reference to its bolometric light curve. By about 60 days after explosion, virtually all of the ⁵⁶Ni has decayed and the dominant energy source is the decay of ⁵⁶Co (which in turn was produced by ⁵⁶Ni decay at earlier times). The optical depth to Compton scattering of ⁵⁶Co gamma rays behaves as τ_γ = (t/t₀)⁻² with τ_γ = 1 at some fiducial time t₀. The value of t₀ can be extracted by fitting the bolometric light curve for t > 60 days to a modified version of Equation (2), in which α = 1 (rather than its maximum-light value from Arnett's rule) and the term corresponding to ⁵⁶Co gamma rays is multiplied by a factor 1 − exp (− τ_γ). The total mass of the ejecta can then be expressed as

$$\begin{equation} M_\mathrm{WD}= \frac{8\pi }{\kappa _\gamma q} (v_e t_0)^2. \end{equation} \tag{ 3 }$$

Here, κ_γ is the Compton scattering opacity for ⁵⁶Co gamma rays, and q is a form factor describing the distribution of the ⁵⁶Co in the ejecta, which follows the original distribution of ⁵⁶Ni in the explosion. The value of κ_γ is expected to lie in the range 0.025–0.033 cm⁻² g (Swartz et al. 1995), with the low end (0.025) corresponding to the optically thin regime. The value of q can be readily calculated given an assumed distribution of ⁵⁶Ni (see Section 4.4 below).

Stritzinger et al. (2006) used this method to measure progenitor masses for a sample of well-observed SNe Ia with UBVRI light-curve coverage. They constructed "quasi-bolometric" light curves according to the procedure of Contardo et al. (2000), by converting the observed UBVRI magnitudes to monochromatic fluxes at the central wavelengths of their respective filters, then summing them, using corrections for lost flux between filters derived from spectroscopy of SN 1992A. They then applied the semi-analytic approach of Jeffery (1999) to fit for the gamma-ray escape fraction, and hence the ejected mass of the SN Ia progenitor.

Our own work improves on the previous use of this method in two important ways. First, Stritzinger et al. (2006) made no attempt to correct for the NIR contribution to the bolometric flux, simply asserting that it is small during the epochs of interest. In Scalzo et al. (2010), we found that for SN 2007if the NIR contribution was indeed small (∼5%) near maximum light, but was greater than 25% at 40 days after maximum light, and our estimate for its value at 100 days after maximum light is still around 10%. Therefore, at least for SNe Ia like the ones we study here, the method of Stritzinger et al. (2006) underestimates the fraction of trapped ⁵⁶Co gamma rays for a given initial ⁵⁶Ni mass, and hence the ejected mass, as a result of neglecting NIR flux.

Second, our fitting procedure includes covariances between different inputs to the prediction for the bolometric light curve, constrained by a set of Bayesian priors motivated by explosion physics. Specifically, covariances between q, v_e, and α may influence the interpretation of the fitted value of t₀. Stritzinger et al. (2006) simply fixed the ⁵⁶Ni mass from Arnett's rule, and then fitted for t₀. Similarly, they assume q = 1/3, v_e = 3000 km s⁻¹, and α = 1 for all of their SNe, with no covariance between any of these parameters. Because models with more ⁵⁶Ni need less gamma-ray trapping to produce the same bolometric luminosity at a given time, there is a large fitting covariance between the ⁵⁶Ni mass and t₀, mentioned in Scalzo et al. (2010). The value of α is model dependent, and not a fundamental physical quantity, but as noted in Section 3.2 above, α = 1.2 (±0.1) is also a common choice when no other prior is available from explosion models. Since α affects the nickel mass, a smaller assumed value of α results in a larger ⁵⁶Ni mass but a smaller ejected mass, as interpreted from a given bolometric light curve. In a self-consistent choice of parameters, v_e and q will each depend in part on the mass of ⁵⁶Ni and therefore on α.

The value of v_e is difficult to measure directly, since observed velocities of absorption line minima may depend on temperature as much as density. Since v_e appears squared in Equation (3), its contribution to the error budget on M_WD is potentially quite large if treated as an independent input. However, its value can be constrained within a range of ±300 km s⁻¹ by requiring energy conservation. Following the practice in the literature (Howell et al. 2006; Maeda & Iwamoto 2009), we calculate the kinetic energy E_K as the difference between the energy E_N released in nuclear burning and the gravitational binding energy E_G, and then set v_e = (E_K/6M_WD)^1/2.

Calculating the energy budget of an SN Ia requires us to assume a composition. Our model considers four components to the ejecta.

• 1.  
  ⁵⁶Ni, which contributes to the luminosity, E_N, and E_G.
• 2.  
  Stable Fe-peak elements ("Fe"), which contribute to E_N and E_G.
• 3.  
  Intermediate-mass elements such as Mg, Si, and S ("Si"), which contribute to E_N and E_G.
• 4.  
  Unburned carbon and oxygen ("C/O"), which contribute only to E_G.

The input parameters, which we vary using a Metropolis–Hastings Monte Carlo Markov chain, are the white dwarf mass M_WD, the central density ρ_c (needed in the calculation of E_G), the parameter α from Arnett's rule, the bolometric rise time t_{rise, bol}, and the fractions f_Ni, f_Fe, and f_Si of ⁵⁶Ni, stable Fe, and intermediate-mass elements within M_WD. We fix the fraction of unburned carbon and oxygen f_CO = 1 − f_Fe − f_Ni − f_Si. We use the prescription of Maeda & Iwamoto (2009) to determine E_N:

$$\begin{equation} E_N = \left[1.74 f_\mathrm{Fe}+ 1.56 f_\mathrm{Ni}+ 1.24 f_\mathrm{Si}\right] \left(\frac{M_\mathrm{WD}}{M_\odot } \right) \times 10^{51} \ \mathrm{erg}. \end{equation} \tag{ 4 }$$

We use the binding energy formulae of Yoon & Langer (2005) for E_G = E_G(M_WD, ρ_c), where ρ_c is the white dwarf central density. These ingredients determine v_e. We apply Gaussian priors α = 1.3 ± 0.1 (as for SN 2007if; Scalzo et al. 2010) and t_{rise, bol} = 20 ± 2 days, and on E(B − V)_host according to the "shallow TBC" values in Table 4. We adopt κ_γ = 0.025 cm⁻² g after Jeffery (1999) and Stritzinger et al. (2006).

One limitation with our approach is the use of E_G from Yoon & Langer (2005), derived for supermassive, differentially rotating white dwarfs. This formula remains an easily accessible estimate in the literature for the binding energy of a white dwarf over a wide range of masses, used by several other authors (Howell et al. 2006; Jeffery et al. 2006; Maeda & Iwamoto 2009). The models of Yoon & Langer (2005) have been criticized on the grounds that they may not exist in nature (Piro 2008), nor explode to produce SNe Ia if they do exist (Saio & Nomoto 2004; Pfannes et al. 2010b). However, it seems reasonable to assume that such models could represent a snapshot in time of a rapidly rotating configuration, such as that encountered in a white dwarf merger, which then detonates promptly rather than continuing to exist as a stable object. The merger simulations of Pakmor et al. (2011, 2012), though they produce comparatively little ⁵⁶Ni, show that prompt detonations in violent mergers can occur. Pfannes et al. (2010a) simulated prompt detonations of rapidly rotating white dwarfs with masses up to 2.1 M_☉, and found that the amount of ⁵⁶Ni produced could be as high as 1.8 M_☉, similar to SN 2007if.

In summary, our procedure directly extracts from a fit to the bolometric light curve only the quantities M_Ni and t₀ (Equation (2)), and marginalizes, in effect, over α, t_{rise, bol}, v_e, and q. The front end of our modeling technique varies the mass, composition, and structure of the SN progenitor as physical quantities which we wish to constrain. We then convert these inputs into physically motivated priors on the values of v_e and q, using Equations (2)–(4), and finally calculate the ejected mass M_WD.

The above considerations all apply to conventional exponential-equivalent models of expanding SN Ia ejecta. To explain the velocity plateaus of the SNe in our sample, however, our model has a disturbed density structure where the high-velocity ejecta (included in the mass M_WD which undergoes nuclear burning) are compressed into a dense shell of mass M_sh, traveling at velocity v_sh. In tamped detonation models, such as the explosion models DET2ENV2, DET2ENV4, and DET2ENV6 (Khokhlov et al. 1993; Höflich & Khohklov 1996, hereafter "DET2ENVN"), such a shell is formed at the reverse shock of the interaction of the ejecta of an otherwise normal SN Ia with a compact (∼10¹⁰ cm) envelope of material with mass M_env (external to, and not included in, M_WD). The suffix N in DET2ENVN refers to the envelope mass, so, for example, model DET2ENV2 has M_env = 0.2 M_☉. "Pulsating delayed detonation" models, such as the PDD535 model of Höflich & Khohklov (1996), have similar shells created by non-homologous pulsations of the white dwarf progenitor prior to the final explosion, and hence do not require an external shell of material. However, these models tend to produce fainter events, with much shorter rise times and redder colors, than we observe for our sample, and so we do not consider them here.

In a tamped detonation, the material which will form the shell imparts its momentum to the envelope, which in the DET2ENVN models acquires an average velocity of about 1.5v_sh. The interaction ends within about the first minute after explosion, and the shell then expands homologously with the other ejecta thereafter. We observe v_sh directly as the plateau velocity, allowing us to constrain M_sh and M_env, and, indirectly, the kinetic energy scale v_e of the ejecta. For a given value of v_e and a measured value of v_sh, and neglecting the binding energy of the envelope, conservation of momentum gives (for more detail see Scalzo et al. 2010)

$$\begin{equation} M_\mathrm{env}= \frac{2}{3} \left[ \frac{3 v_e}{v_\mathrm{sh}} Q\left(4,\frac{v_\mathrm{sh}}{v_e}\right) - Q\left(3,\frac{v_\mathrm{sh}}{v_e}\right) \right] M_\mathrm{WD}, \end{equation} \tag{ 5 }$$

where Q(a, x) = γ(a, x)/Γ(a) is the incomplete gamma function. We calculate f_env = M_env/M_WD and f_sh = M_sh/M_WD by solving Equation (5) numerically. In a double-degenerate merger scenario, the total system mass M_tot = M_WD + M_env is then equal to the initial mass of the two white dwarfs undergoing the merger.

For these calculations, we use only the velocity of the v(Si ii) plateau measured from our spectroscopic time series. We do not model the duration of the plateau or the behavior of v(Si ii) after the plateau phase ends. While the detailed evolution of v(Si ii) undoubtedly contains useful information, reproducing it would require detailed calculations of synthetic spectra which are beyond the scope of this paper. However, as long as we have enough measurements of v(Si ii) to show that a given SN exhibits plateau behavior, we can reliably measure v_sh without knowing the opacity of the material in the shell. We will compare our observations to previous numerical models and observations of SNe Ia with interacting shells in Section 5.1.

In Scalzo et al. (2010), the stable iron fraction f_Fe was allowed to vary freely. In this situation, f_Fe and f_Si are nearly degenerate, since the contribution per unit mass of Fe to the nuclear energy released is only about 40% higher than that of Si. However, since Fe is produced by neutronization in the densest parts of the ejecta during the explosion, a high value of f_Fe can greatly reduce q because the formation of a large Fe core displaces ⁵⁶Ni to a higher average velocity and a lower optical depth. It therefore becomes important to constrain Fe production in any model in which we attempt to calculate q.

The DET2ENVN explosion models, on which our models are loosely based, were intended to describe the detonation of a low-density white dwarf merger remnant of mass 1.2 M_☉ inside envelopes of varying mass. The central density ρ_c in these models is 4 × 10⁷ g cm⁻³, substantially lower than typical central densities of ∼10⁹ g cm⁻³ of deflagrations and delayed detonation models in the literature (Nomoto et al. 1984; Höflich & Khohklov 1996; Krueger et al. 2010). These models have no stable Fe cores immediately after explosion, with $X_{^{56}\mathrm{Ni}}= 0.9$ throughout the region where ⁵⁶Ni and Fe are produced.

Krueger et al. (2010) investigated the effects of central density on ⁵⁶Ni yields in three-dimensional simulations of detonations of Chandrasekhar-mass white dwarfs, averaging over an ensemble of realizations for each value of ρ_c. They found f_Ni/(f_Ni + f_Fe) = 0.9 for ρ_c = 10⁹ g cm⁻³, decreasing by 0.047 on average for each 10⁹ g cm⁻³ increase of ρ_c thereafter.

Our model already includes the effects of the central density ρ_c on the binding energy E_G of the white dwarf, through the fitting formula of Yoon & Langer (2005). Although M_WD may be super-Chandrasekhar in our models, the overall extent of nuclear burning should depend on the density, not the mass, and so we may consider extrapolating those results here. Since the link between f_Fe and f_Ni is statistical rather than deterministic, we do not attempt to calculate a definite fraction for each model. Instead, we calculate the ratio of ⁵⁶Ni to total iron-peak elements, η = f_Ni/(f_Ni + f_Fe), as well as the relation from Krueger et al. (2010):

$$\begin{equation} \eta _\mathrm{Kr10} = \min \left\lbrace 0.9, 0.959 - 0.047 \left(\frac{\rho _c}{10^9\ \mathrm{g\ cm}^{-3}} \right) \right\rbrace \end{equation} \tag{ 6 }$$

enforcing a Gaussian prior η = η_Kr10 ± 0.1. For most models calculated, this results in only a small fraction of stable Fe, as appropriate for low-density explosions.

The gamma-ray transport form factor q is the dimensionless ⁵⁶Ni-weighted gamma-ray optical depth through the ejecta. Its value ranges between zero and one, with large values corresponding to high central concentrations of ⁵⁶Ni; the gamma-ray optical depth is proportional to q. For perfectly mixed, exponentially distributed ejecta where the fraction of ⁵⁶Ni is constant throughout, q = 1/3.

In Scalzo et al. (2010), the calculation of the total mass for SN 2007if assumes q = 1/3. We chose this value because the lack of a distinct second maximum in the I-band light curve suggested that a large amount of ⁵⁶Ni had to be mixed to higher velocities (see Section 3.1 above). This is not necessarily true for our other SNe. The use of q = 1/3 also assumes that the reverse-shock shell has a negligible effect on gamma-ray trapping, which may also not be true for very massive shells. Fortunately, q is easy to calculate numerically (Jeffery 1999):

$$\begin{equation} q = \frac{\int _0^\infty dz \int _0^\infty dz_s \int _{-1}^1 d\mu \ z^2 \tilde{\rho }(z) X_{^{56}\mathrm{Ni}}(z) \tilde{\rho }(z^{\prime })}{\int _0^\infty dz \ z^2 \tilde{\rho }(z) X_{^{56}\mathrm{Ni}}(z)}, \end{equation} \tag{ 7 }$$

where z and z' are dimensionless velocity coordinates in units of v_e, $z_s = \sqrt{z^{\prime 2}+z^2+2zz^{\prime }\mu }$ is the beam path length, $\tilde{\rho }(z)$ is a dimensionless density profile normalized to unit mass, and $X_{^{56}\mathrm{Ni}}(z)$ is the (velocity-dependent) ⁵⁶Ni fraction just after explosion. The geometry of the integration is shown in Figure 6.

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To calculate q, we assume a density profile motivated by DET2ENVN which includes an envelope with density profile ρ ∼ z⁻² on the outside, a thin Gaussian shell at z_sh = v_sh/v_e, and an exponential in velocity beneath the shell. The total density is given by

$$\begin{eqnarray} \rho (z) & = & C_\mathrm{bulk} \phi \left(\frac{z-z_\mathrm{sh}}{\sigma _\mathrm{z,sh}}\right) \exp (-z) + C_\mathrm{env} \phi \left(\frac{z_\mathrm{sh}-z}{\sigma _\mathrm{z,sh}}\right) z^{-2} \nonumber \\ && + C_\mathrm{sh} \exp \left[ -\frac{1}{2} \left(\frac{z-z_\mathrm{sh}}{\sigma _\mathrm{z,sh}}\right)^2 \right], \end{eqnarray} \tag{ 8 }$$

where we choose $\phi (z) = \mathrm{erfc}(z/\sqrt{2})/2$ as a smooth function which approaches 1 for z ≪ −1 and 0 for z ≫ +1, and the constants C_bulk, C_env, C_sh are determined so that the mass fractions in each density profile component agree with the input values. Quimby et al. (2007) suggest a nominal width of 500 km s⁻¹ for the reverse-shock shell at velocity v_sh ∼ 10, 000 km s⁻¹. The self-similar shock interaction model of Chevalier et al. (1982) suggests that the reverse-shock velocity width should be about 3% of the velocity at the contact discontinuity for an interaction with an r⁻² envelope and ejecta with a power-law density profile r⁻ⁿ with n ∼ 7 (used to approximate SNe Ia in the context of interaction with a steady wind, e.g., Wood-Vasey et al. 2004). We assume a shell half-width of σ_{z, sh} = 0.015z_sh in line with Chevalier et al. (1982), although the expansion is homologous and no longer self-similar after the interaction. Our results are not sensitive to the exact value of σ_{z, sh}; values in the range (0.01–0.05)z_sh give us the same value of q to within 5% for reverse-shock shells with masses up to 0.5 M_WD, and with much better agreement for less massive shells.

For $X_{^{56}\mathrm{Ni}}(z)$, we use a parameterized composition structure inspired by Kasen (2006):

$$\begin{eqnarray} X_{^{56}\mathrm{Ni}}(z) & = & \phi \left(\frac{m_\mathrm{Fe,core}-m(z)}{a_\mathrm{Fe,core}}\right) \nonumber \\ && \times \phi \left(\frac{m(z)-m_\mathrm{IPE}}{a_\mathrm{IPE}}\right) \times \max (\eta,0.9), \end{eqnarray} \tag{ 9 }$$

where $m(z) = \int _{0}^{z} z^2 \tilde{\rho }(z) \, dz$ is the mass coordinate, and the stable Fe-peak core and the ⁵⁶Ni mixing zone are bounded by mass coordinates m_{Fe, core} and m_IPE with mixing widths a_{Fe, core} and a_IPE, respectively. This allows for some stable Fe-peak elements to be mixed throughout, while maintaining a core in the innermost regions for high central density models (see Figure 7). For our modeling below, we choose a_Fe = 0.025 and a_Ni = 0.35, so that the outward mixing of ⁵⁶Ni corresponds roughly to the "enhanced mixing" case of Kasen (2006), in accordance with the behavior we see in the light curves. As the mass of the shell increases, the results may depend more sensitively on the particular distribution of ⁵⁶Ni in the shell. We proceed with the analysis, but caution that more detailed models of the shock interaction, and/or actual hydrodynamic simulations of the explosion, may be needed to accurately understand gamma-ray transport for cases in which a large amount of ⁵⁶Ni is swept up into the shell.

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In general, the values of q are higher for our SNe (0.45 ± 0.05) than the nominal q = 1/3 value for a completely mixed exponential SN Ia, but there is very little variation with shell mass fraction. The mixing of ⁵⁶Ni into a shell (and potentially above the photosphere) and the displacement of ⁵⁶Ni to higher velocities by a stable Fe core have comparable effects on q, but the former effect is minimized in the "enhanced mixing" model characteristic of our light curves.

The results of our modeling for the SNe in our sample are summarized in Table 7. Figure 8 shows the confidence regions in the M_WD–⁵⁶Ni mass plane for all six SNfactory SNe. While we limit our later statements about relative rates (see Section 5.4) and Hubble residuals (see Section 5.6) to the untargeted SNfactory search, which samples the smooth Hubble flow (z > 0.03) and samples all host galaxy environments in an unbiased manner, we also include our results for two spectroscopically analogous SNe Ia from the literature as useful points of comparison: SN 1991T and SN 2003fg (see the Appendix).

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Our results can be summarized as follows for the SNe in the SNfactory sample.

• 1.  
  We recover the results of Scalzo et al. (2010) for SN 2007if, within the uncertainties. The total system mass has come down slightly, from 2.41 M_☉ to 2.30 M_☉ (probability distribution median), since trapping of ⁵⁶Co gamma rays by the envelope is now included in the mass estimate. The fractions of the total system mass in the shell and in the envelope, set by the plateau velocity, remain the same.
• 2.  
  SNF 20070803-005 and SNF 20080522-000 are consistent with being Chandrasekhar-mass SNe Ia, albeit spectroscopically peculiar ones.
• 3.  
  SNF 20070528-003 and SNF 20070912-000 have data quality sufficient only to obtain lower limits on the total mass, and upper limits on the shell and envelope mass. In particular, the lack of late-time photometry points mean that the lower limits are driven by the ⁵⁶Ni mass from Arnett's rule. The total system mass M_tot, which includes the mass M_env of the envelope which we infer from the plateau velocity, is super-Chandrasekhar-mass at >98% confidence, although the mass M_WD of the central merger remnant is consistent with Chandrasekhar-mass values.
• 4.  
  SNF 20080723-012 appears to be a moderately super-Chandrasekhar-mass object, with ⁵⁶Ni mass, total mass, and other parameters intermediate between SN 2007if and the rest of the population. We place a 98% CL lower limit of 1.41 M_☉ on M_WD, so that the progenitor system is likely super-Chandrasekhar-mass even without including M_env.

We also perform the following cross-checks. First, our results are not strongly sensitive to the assumption of a particular mixing parameter, changing by less than 2% if we instead assume a completely stratified composition with a_Fe = a_Ni = 0. Although changes in the mixing parameters do influence q substantially for models with small amounts of ⁵⁶Ni, they matter much less when the ⁵⁶Ni mass is large. We also confirm that when we discard the velocity information and fix M_env = 0, the resulting median reconstructed masses M_tot are within 2% of the median values of M_WD, and the probability distributions have comparable widths. This is expected, since only the most massive envelopes (as in SN 2007if) make an appreciable contribution to the gamma-ray optical depth as seen from the inner layers of ejecta.

Our most recent point of comparison in the literature for the application of explosion models with interacting shells to observations of SNe Ia is Quimby et al. (2007). They noted the presence of a velocity plateau in their observations of SN 2005hj and compared them to delayed detonation, pulsating delayed detonation and tamped detonation models (Khokhlov et al. 1993; Höflich & Khohklov 1996; Gerardy et al. 2004).

For a shell mass fraction f_sh = 0.07, similar to our less extreme SNe, the predicted B − V color is around 0.05-0.1. Quimby et al. (2007) note that the systematic uncertainty in the absolute value of (B − V)_max may be as large as 0.1 mag for the DET2ENVN models and other shell models. The color they report for SN 2005hj is (B − V)_max = 0.04 ± 0.06, consistent with the mean color in our sample, for a claimed shell mass fraction f_sh = 0.14, comparable to SNF 20070803-005.

According to Quimby et al. (2007) and references therein, one might expect to see cooler photospheres, and hence redder (B − V)_max, with increasing shell mass fraction. Our modeling predicts that SN 2007if, with the lowest plateau velocity and the most massive progenitor, has the most massive shell in both absolute and relative terms. This SN may be somewhat redder than the others near maximum light, but the lack of light-curve coverage near maximum makes it difficult to say exactly how much. Table 7 also shows that SN 2007if has the largest expected fraction of iron-peak elements in its shell, corresponding to the material near the photosphere around maximum light. To the extent that SN 2007if is intrinsically redder than our other SNe, this may be due in part to line blanketing by iron-peak elements, rather than a low-temperature photosphere, which would be inconsistent with the weakness of Si ii λ5800 in all the SNe in our sample.

While we cannot predict from theory how long the plateau phase should last without more sophisticated modeling of our SNe, the durations of our plateaus are also broadly consistent with expectations. For models of Chandrasekhar-mass events with envelope masses of about 0.1 M_☉(Quimby et al. 2007), the plateau phase is expected to last about 10 days. In most of our SNe, it lasts at least 15 days (and at least 10 days for SNF 20070912-000). Quimby et al. (2007) derive a plateau duration of 20 ± 10 days for SN 2005hj.

SNF 20080522-000 shows the longest-lived plateau in our data set, stretching as early as 10 days before B-band maximum and lasting as long as 30 days. A more conservative estimate would be that the plateau phase is confirmed to last from day −3, when Si ii λ6355 becomes strong enough that the error bars on the velocity of the absorption minimum drop below 500 km s⁻¹, to day +15, where Fe ii lines begin to develop near Si ii λ6355 and blending may become a concern. All of our velocity measurements in this 18 day time window are contained in a narrow range just 250 km s⁻¹ wide.

Fryer et al. (2010) ran three-dimensional smoothed particle hydrodynamics simulations of double-degenerate mergers. They found that the central merger remnants are indeed surrounded by an envelope with an approximate radial density profile ρ(r)∝r^−β with 3 < β < 4. They then performed radiation hydrodynamics simulations of the interaction of the SN ejecta with the envelope, calculating synthetic spectra and light curves of the resulting explosions. For sufficiently massive envelopes (more massive than about 0.1 M_☉), energy advected by the shock is released over the evolution of the SN, producing non-negligible luminosity near maximum light and extending emission into UV wavelengths. Based on these findings, Fryer et al. (2010) argued that these "enshrouded" systems would, in all likelihood, look nothing like SNe Ia. Blinnikov & Sorokina (2010) performed analogous calculations using the STELLA radiation hydrodynamics code, finding that, in general, a radial density profile as steep as r⁻⁴ looked similar to a normal SN Ia in optical wavelengths (UBVRI), whereas a profile varying as r⁻³ would be dominated by shock emission.

These findings put significant constraints on the radial extent and density profile of any envelope which might have enshrouded the progenitors of the SNe in our sample. In particular, the blue, mostly featureless spectrum seen in SN 2007if at phase −9 days is characteristic of what Fryer et al. (2010) expect, and the optical emission could be powered in part by advected heat energy from the shock interaction at early times, resulting in a broader light curve. However, the fact that v(Si ii) in our events is seen to assume its plateau value as early as a week before maximum light, and that the colors appear similar to those of normal SNe Ia near maximum light, argue strongly against any ongoing interaction with an extended envelope or wind.

As noted in Scalzo et al. (2010), if some fraction of the maximum-light luminosity is due to shock heating, whether advected or resulting from a fresh interaction, instead of ⁵⁶Ni decay, then our mass estimates would tend to increase. This is because the influence of the shock interaction would diminish substantially by about 50 days after explosion (Blinnikov & Sorokina 2010), and more ⁵⁶Co gamma rays would have to be trapped in order to reproduce the observed light curve. Any model in which shock heating produces a large amount of luminosity more than 50 days after explosion would probably not look like an SN Ia, and could not explain the appearance of the SNe Ia discussed in this paper.

One straightforward way to address the extent of any ongoing consequences of shock interactions in future studies would be to obtain early-phase UV light curves of a candidate super-Chandra SN Ia from a satellite such as Swift. The signature of any strong influence of shock heating would be evident therein.

The large inferred masses of our candidate super-Chandra SNe Ia, taken together with the observed plateaus in the Si ii velocity, are both naturally explained by the tamped detonation model we put forth in this paper, and by the underlying double-degenerate merger scenario it represents. Some recent work, however, points toward the possibility of explaining these events in terms of asymmetric single-degenerate explosions.

Maeda et al. (2009, 2010a, 2010b, 2011) invoke large-angular-scale asymmetries to explain the diversity of velocity gradients observed in normal SN Ia explosions and the low/high velocity gradient (LVG/HVG) dichotomy (Benetti et al. 2005). They suggest that an asymmetric explosion may cause an overdensity in the ejecta on one side of the explosion, causing LVG behavior when viewed from that side, with the other side relatively less dense and exhibiting HVG behavior. See also Foley & Kasen 2011 and Foley et al. 2011 for further observational explorations of this theme. Additionally, Hachisu et al. (2012) suggested that optically thick winds blown from an accreting white dwarf could strip mass from the outer layers of its donor star, regulating the accretion rate and potentially allowing the white dwarf to accrete without exploding until reaching masses as large as 2.7 M_☉. Such a white dwarf would have to rotate differentially, as in the models of Yoon & Langer (2005), with the attending uncertainty in the evolutionary history of such objects. If a differentially rotating white dwarf were to explode asymmetrically, then this could present an explanation for our observations within the single-degenerate scenario. A scenario like this one, if correct, could also explain SN 2007if and the HVG SN 2009dc as being similar objects viewed from different angles. Tanaka et al. (2010) interpreted the low continuum polarization of SN 2009dc as evidence for a nearly spherical explosion, but low polarization could also be observed in an axisymmetric explosion viewed along the symmetry axis, as those authors note.

While we cannot at this time conclusively rule out the possibility that the Si ii velocity plateaus we observe in our SNe result from asymmetry, no asymmetry is needed as yet to explain them, as argued, e.g., by Maeda & Iwamoto (2009) for super-Chandrasekhar-mass explosions. The physical cause of the plateau—overdensity in the ejecta—is the same in symmetric and asymmetric models. If the velocity plateaus are indeed the result of asymmetric explosions, then the shell mass fractions derived in Table 7 would still have meaning in terms of disturbances to the density structure along the line of sight, but the inferred envelopes would not be present, i.e., M_env would be zero and M_tot would equal M_WD for the SNe analyzed in this paper.

We can use the relative rate of supernovae spectroscopically similar to those in our sample (see Section 5.4 below) to make some general statements about how well they can be explained by lopsided asymmetric explosions. If the SNe Ia in our sample belonged to the same population as normal SNe Ia, and the spectroscopic peculiarity and brightness were due entirely to viewing angle effects (see Kasen 2004, for an asymmetric model for which this is true), then a relative rate of ∼2% would imply a range of viewing angles no more than ∼15° from the symmetry axis. For models in which the asymmetry does not translate into a very peculiar spectrum, as seems likely for the Maeda et al. (2010a, 2011) models with high ⁵⁶Ni mass, we should look instead at the diversity of velocity gradients among spectroscopic analogs. Since all of the (spectroscopically similar) SNe Ia in our sample have velocity gradients at the slowly evolving extreme of what the model of Maeda et al. (2010a) claims to produce, it seems plausible that, within the context of this class of models, the density enhancements in the ejecta of our SNe are roughly isotropic.

Asymmetries resulting in overdensities in SN Ia ejecta along the line of sight could also affect gamma-ray trapping for that line of sight only. Most of the trapping happens in low-velocity ejecta (see Section 4.5), so the form factor q should not be strongly affected; we think it unlikely that q, and our ejected mass estimates, could vary by more than 10%. Our ⁵⁶Ni mass estimates from Arnett's rule should also be robust, since according to Maeda et al. (2011), the effects of asymmetry on peak brightness are least pronounced for the most luminous SNe Ia. Based on these considerations and those of the preceding paragraph, the super-Chandrasekhar-mass status of SN 2007if and SNF 20080723-012 seems secure to us.

A different class of asymmetries might arise in bipolar, rather than lopsided, explosions. An axially symmetric merger geometry, featuring a disk of disrupted white dwarf material in the equatorial plane, could in principle result in velocity plateaus and peculiar spectra when viewed edge-on, but a more normal SN Ia appearance when viewed pole-on. This possibility was recently suggested by Livio & Pringle (2011), who also suggested that SNe Ia from merger events might show an increasing diversity in their velocity gradients with an increasing mass of carbon. In this scenario, SN 2009dc could be an event like SN 2007if viewed pole-on, consistent with its low continuum polarization (Tanaka et al. 2010); SN 2007if should then correspond to the edge-on case, and should show substantial polarization. Livio & Pringle (2011) make no predictions for relative rates in different SN Ia subgroups, which may depend on details of the hydrodynamic interaction of the SN ejecta with the disk. Nevertheless, an occluded region of ejecta narrow enough to explain their rare incidence would subtend only a small solid angle at the source, with only a small fraction of the ejecta decelerated. This would lead to broader, more complex line profiles than the ones we observe. It therefore once again seems reasonable that our SNe should be more or less spherically symmetric, with only moderate large-angular-scale asymmetries. Furthermore, any asymmetries compatible with our observations should influence only the inferred envelope masses M_env, and should leave our estimates of M_WD intact at the 10% level, as we noted above.

Future observations, including nebular spectra of candidate super-Chandra SNe Ia for direct comparison with the above-mentioned studies, may allow us to make more specific statements about their asphericity. Spectropolarimetry, as for SN 2009dc (Tanaka et al. 2010), may also be helpful.

Mass modeling shows that all six of our SNfactory-discovered SNe Ia have a high probability of having super-Chandrasekhar-mass progenitors. Because ours was a wide-field search and our spectroscopy screening was conducted impartially, that is without pre-selection based on colors, luminosity, light-curve shape, host galaxy environment, etc., we may estimate the relative fraction of such objects.

A total of 400 spectroscopically confirmed SNe Ia were discovered by the SNfactory. We did not spectroscopically screen candidates found to be fainter on the discovery image than on any pre-discovery detections that may have existed, as such candidates were assumed to be after maximum light. Furthermore, we did not perform spectroscopic screening in the cases where a host galaxy had a previously known redshift beyond z = 0.08, though such cases were rare.

We consider two subsamples—one that is purely flux-limited and another that is further limited to z ⩽ 0.08. For both subsamples, we consider only SNe Ia discovered at or before maximum light, when the characteristic spectral features of SN 2003fg, SN 2007if, and SN 1991T are clearest (Li et al. 2011b). The flux limit is represented by the survey efficiency as a function of magnitude rather than as a fixed magnitude. A subset redshift-limited to z ⩽ 0.08 is also considered as a cross-check, since here details of the survey efficiency become unimportant and the spectroscopic typing is very secure.

Among the super-Chandra candidates, all survive the pre-maximum and flux-limit cuts, while SNF 20070528-003 and SNF 20070912-000 are eliminated by the redshift cut. The sample cuts applied to the overall sample leave 240 SNe Ia in the flux-limited sample and 141 in the redshift-limited sample. Due to their enhanced brightness and generally longer rise times, the super-Chandra candidates are enhanced by factors of 1.30 ± 0.06 and 1.11 ± 0.01 in each of the two subsamples. These enhancements are calculated using the control time in the SNfactory search, accounting for both apparent magnitudes and light-curve widths of these SNe relative to normal SNe Ia. The assigned uncertainties are illustrative of the effect of a generous ±0.25 mag shift in the detection efficiency curve. As expected, this effect is small and has essentially no impact for the volume-limited subset. Including these correction factors, the rates are 1.9^+1.1_{− 0.5}% and 2.6^+1.9_{− 0.7}% for the flux-limited and redshift-limited subsamples, respectively. The difference in these relative rates is consistent with Poisson fluctuations, including accounting for the SNe Ia in common.

Among the super-Chandra candidates, all survive the pre-maximum and flux-limit cuts, while SNF 20070528-003 and SNF 20070912-000 are eliminated by the redshift cut. The sample cuts applied to the overall sample leave 240 SNe Ia in the flux-limited sample and 141 in the redshift-limited sample. Due to their enhanced brightness and generally longer rise times, the super-Chandra candidates are enhanced by factors of 1.30 and 1.11 in each of the two subsamples. Including these correction factors, the rates are 1.9^+1.1_{− 0.5}% and 2.6^+1.9_{− 0.7}% for the flux-limited and redshift-limited subsamples, respectively. The difference in these relative rates is consistent with Poisson fluctuations, including accounting for the SNe Ia in common.

These are the relative rates under the assumption that all of our candidates have super-Chandrasekhar-mass progenitors, as their estimated masses under the tamped detonation model indicate. Alternatively, these can be taken as the rates for SNe Ia classified as like SN 1991T by using SNID. Although our 1.9–2.6% relative rate is somewhat lower than the "volume-limited" rate of 9.4^+5.9_{− 4.7}% (5-day cadence) from Li et al. (2011b), we note that it is consistent with the ∼1% "Ia-91T" rate of Silverman et al. (2012) from the same search. The difference between the two is related to how the events are classified. Li et al. (2011b) use classifications from the IAU Circulars, rather than a homogeneous set of spectra over a given range of wavelengths and light-curve phases subclassified with a single method; in fact, Silverman et al. (2012) imply that some of the Li et al. (2011b) subclassifications may be photometric (based on light-curve width) rather than spectroscopic. Li et al. (2011b) also treat SNe with type "Ia-99aa" as being "Ia-91T" rather than breaking them into separate subclasses, which led to their much higher rate. Using SNID, Silverman et al. (2012) classify only one of the Li et al. (2011b) 1991T-likes (SN 2004bv) as "Ia-91T" and the rest as "Ia-99aa" or normal, leading to an updated "volume-limited" rate of 1.4^+3.0_{− 0.0} for the Li et al. (2011b) sample, consistent with our rate. As a point of comparison from an untargeted search similar to SNfactory's, rather than a search targeting known galaxies, we also consider the SDSS spectroscopic sample (Östman et al. 2011). Seventy-eight of the 141 SNe Ia in the sample of Östman et al. (2011) have spectra with light-curve phases at or before maximum light, proof against the 1991T-like "age bias" (Li et al. 2011b). Using SNID, Östman et al. (2011) type only two of these 78 as "Ia-91T", leading to a relative rate of 3.4^+2.4_{− 1.7}%, again consistent with our own rate. We note that our rate estimates have considerably smaller uncertainty than these other 1991T-like SNe Ia rates. The combination of all three surveys, using our redshift-limited rates, gives a net 1991T-like rate of 2.3^+1.2_{− 0.6}%.

The statistical confidence intervals from our models indicate the possibility that not all candidates are super-Chandrasekhar-mass. The probability density extending below the Chandrasekhar mass represents the equivalent of 0.2 SNe Ia for both subsamples. If we correct for this excess probability, then the rates are lowered by factors of 5% and 3% for the flux-limited and redshift-limited subsamples. A more precise correction would depend on the true parent mass distribution function, but it is evident from the above calculation that such details are well below the Poisson uncertainties.

Finally, we consider the rates under the possibility that the shell model is not appropriate. In this case, there would be no contribution from the shell mass, and the best mass estimate would be that of the central merger remnant, M_WD. In this circumstance, SNF 20070803-005 and SNF 20080522-000 could be considered more likely to have been Chandrasekhar-mass events. The rates resulting from their removal would then become 1.5^+1.1_{− 0.4}% and 1.0^+1.4_{− 0.4}%. While we view this situation as unlikely, for reasons discussed above in Section 5.3, the rates remain dominated by Poisson uncertainties rather than modeling assumptions.

These relative rates are very low. Therefore, if double-degenerate mergers are to contribute significantly to the total SN Ia rate, then most such mergers would need to escape our spectroscopic selection. Moreover, the SNfactory has not found overluminous SNe Ia in addition to those presented here or SN 2005gj (Aldering et al. 2006) that would qualify for our redshift-limited sample.

While our events are overluminous with large ⁵⁶Ni masses, theoretical mergers (e.g., Pakmor et al. 2011, 2012) thus far predict a range of smaller ⁵⁶Ni masses, resulting in fainter explosions with lower photospheric temperatures. As discussed in Section 5.2 above, a delay in the release of radiant energy from a shock interaction could also contribute to heating the photosphere at early times in events like SN 2007if, so that mergers with less massive envelopes would have more normal spectra.

Merger events with less massive envelopes should also have less disturbed density structures, and hence may not display the low Si ii velocity plateau behavior we observe in our sample. The small envelope mass fractions implied for the majority of merger events would then represent a constraint on either the dynamics of the merger in the case of a prompt explosion, or the post-merger evolution of the system in the case of a delayed explosion triggered by accretion onto, or by post-accretion spin-down of, the central merger remnant. Unambiguous detection of a density enhancement signature for less massive shells would be difficult. Construction of their mass distribution function would require a parent sample of SNe Ia all followed spectroscopically such that any merger candidates could be selected by their velocity plateau behavior rather than spectroscopic peculiarity. Moreover, such a study would require spectra at very early phases (−10 days or younger), since a less massive reverse-shock shell would become transparent at earlier phases, and might not subsume most of the intermediate-mass elements in the explosion.

Determining the relative importance of any of these factors, and thus understanding the true relation of candidate super-Chandra SNe Ia to the general SN Ia population, will require further work and new spectroscopic data sets. However, we can say that a range of envelope masses are probably needed to explain the range of plateau velocities we observe in our data.

Figure 9 presents the aggregate PDF of the masses of the SNe in our sample, each weighted inversely by their control time in the SNfactory search, accounting for both apparent magnitude and light-curve width. Its lower edge is roughly consistent with the Chandrasekhar mass, with a tail toward higher masses.

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This distribution is qualitatively similar to expectations for super-Chandrasekhar-mass merging white dwarf system masses from population synthesis models (Ruiter et al. 2009; Fryer et al. 2010). The limited statistics and theoretical uncertainties make it difficult to draw a firm, quantitative correspondence between observations and theory, but we can nevertheless look for gross inconsistencies. Figure 9 therefore also includes three theoretical distributions of the progenitor system mass (M_tot) shown in Fryer et al. (2010), cut off below 1.4 M_☉ and convolved with a two-sided Gaussian (σ₋ = 0.08 M_tot, σ₊ = 0.10 M_tot) to model our reconstruction precision.

Our observations correspond well with the Ruiter et al. (2009) models on the leading edge, particularly the γ = 1.5 model. The differences are more pronounced on the high-mass tail; the agreement may improve with a larger sample of SNe. The other two models agree less well, predicting more high-mass mergers than we see in our sample. In the γ = 1.5 model, the relative rate of events dN/dM_tot is about a factor of three higher at 1.4 M_☉ than at 2.1 M_☉, consistent with our observations. The other two models predict more high-mass mergers than we observe; we see no evidence for a separate formation channel or peak near 2 M_☉ as suggested by Fryer et al. (2010). If such massive mergers do exist, then they must therefore produce less ⁵⁶Ni, and be less luminous at maximum light, than the SNe in our sample.

Table 3 also contains the Hubble residuals of each SN in our sample from a standard ΛCDM cosmology:

$$\begin{equation} \Delta \mu = m_B \,{-}\, 25 \,{-}\, 5\, \log _{10} (d_L/\mathrm{Mpc}) \,{-}\, \alpha (s-1) \,{+}\, \beta c \,{-}\, M_B, \end{equation} \tag{ 10 }$$

where m_B, s, and c are derived from the SALT2 fits, d_L is calculated using Wright (2006), and we use values of α, β, and M_B from the w = −1 fit of Sullivan et al. (2011). (We use M_B = M¹_B, i.e., the absolute magnitude appropriate for SNe in host galaxies with stellar mass less than 10¹⁰ M_☉.)

Figure 10 plots the relation Δμ versus M_WD (instead of M_tot, so that it does not depend on interpreting the velocity plateau in terms of an envelope) for the six SNfactory SNe Ia. For comparison, we also plot SN 1991T and SN 2003fg, modeled in a similar fashion (see the Appendix). The Hubble residuals show a correlation with increasing mass (Pearson r = −0.76, p = 0.03). SN 2007if is 1.3 mag overluminous for its stretch and color, and by 1 mag even if no correction is made for its redder-than-average color. SNF 20080723-012 is about 0.6 mag too luminous for its stretch and color, putting it off the Hubble diagram but to a lesser degree than SN 2007if. At the low-mass end, SNF 20070803-005 and SNF 20080522-000, while slightly overluminous, are not alarmingly discrepant, being within two standard deviations of the ΛCDM Hubble diagram assuming an intrinsic dispersion of 0.15 mag. The higher-redshift (z ∼ 0.12) SNF 20070528-003 and SNF 20070912-000 are too luminous for their stretch and color by about 0.5 mag; the residual decreases to about 0.4 mag if no correction is made for the color.

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If the correlation shown in Figure 10 is real, and the large Hubble residuals of SN 2003fg and SN 2007if are related to their large progenitor masses, then the large Hubble residuals of SNF 20070528-003 and SNF 20070912-000 provide further circumstantial evidence that they too may have super-Chandrasekhar-mass progenitors. One might expect deviations of super-Chandra candidate SNe Ia from the width–luminosity relation for normal SNe Ia, either because the large ejected mass could increase the diffusion time of radiation through the ejecta relative to Chandrasekhar-mass explosions, or because the enhanced mixing of ⁵⁶Ni into the outer layers of ejecta in these events affects the ionization state at the photosphere (see Kasen & Woosley 2007 and references therein).

Previous authors have warned about overluminous SNe Ia that could skew the cosmological parameters. Reindl et al. (2005) found that the SN 1991T subclass gives a mean Hubble residual of −0.4 mag. The sample presented here has a mean Hubble residual of −0.56 mag, dropping to −0.41 mag if SN 2007if were excluded on the grounds that it would be easy to eliminate. However, without a spectroscopic veto at high redshift, or by relying only on photometry at any redshift, it will be difficult to tag and remove SNe such as these from the Hubble diagram. Although SN 2007if is excessively overluminous and would probably be excluded from any Hubble diagram on that basis, somewhat less luminous analogs like SNF 20080723-012 probably could not be safely removed based on their overluminosity alone. It would also be difficult, without spectroscopy, to distinguish intrinsically overluminous SNe Ia from normal SNe Ia magnified by gravitational lensing; since lensing conserves photons in general relativity (Weinberg 1976), all events must be included on the Hubble diagram to avoid bias in the reconstructed cosmological parameters.

Suppose candidate super-Chandra SNe Ia occur at the relative rate we calculate for our redshift-limited subsample in Section 5.4 (2.6^+1.9_{− 0.7}) and have a mean Hubble diagram residual of −0.41 mag, the mean of the Table 3 entries excluding SN 2007if. The mean absolute magnitude of SNe Ia is then 0.01–0.02 mag higher (68% CL) than it would be if these SNe followed the normal relations. This number is already comparable to the 2% error budget suggested in Kim et al. (2004) for Stage IV SN Ia cosmology experiments, and would be realized if the rate evolves by either doubling or going to zero at high redshift.

SN 1991T is the most well-known spectroscopic archetype of our sample, although not the most spectroscopically extreme example. SN 1991T has good wavelength coverage in published spectra taken between maximum light and phase +40 days, allowing us to use the same analysis techniques as for the rest of our sample. In addition, modeling of SN 1991T allows a useful direct comparison to the earlier work of Stritzinger et al. (2006).

SN 1991T's remarkably flat velocity evolution was pointed out by Phillips et al. (1992), and it has the second-lowest velocity gradient $\dot{v}$ in the original sample of Benetti et al. (2005). The DET2ENV2 model fits SN 1991T (Höflich & Khohklov 1996), and indeed SN 1991T was once suspected to have had a super-Chandrasekhar-mass progenitor, mostly because of its high luminosity (Fisher et al. 1999). Later measurements of the distance (Richtler et al. 2001; Gibson & Stetson 2001) to the host galaxy, NGC 4527, have decreased the SN's luminosity considerably, and hence its inferred mass.

We can make a new estimate of the progenitor mass of SN 1991T using our reconstruction method, which considers the gamma-ray trapping in more detail than previous studies. We use the BVRI light curve of Lira (1998) and the spectral time series of Mazzali et al. (1995) obtained through the SUSPECT online supernova spectrum archive. Kanbur et al. (2003) give the Cepheid-based distance modulus to NGC 4527, after correction for metallicity, as 30.71 ± 0.13 (LMC period–luminosity relation) and 30.78 ± 0.13 (Milky Way period–luminosity relation); we adopt the mean of these two measurements (30.74 ± 0.13) for our work. From a joint fit to the spectral time series, we extract EW(Na i D) = 0.84 ± 0.18 Å, giving E(B − V)_host = 0.13 ± 0.03 from the shallow TBC relation, in good agreement with the estimates of Phillips et al. (1992, 1999). Our model represents a good fit to the available data, with χ²_ν = 0.83 (P_fit = 0.60). We recover a ⁵⁶Ni mass of 0.77 ± 0.15 M_☉ and a white dwarf mass 1.50^+0.18_{− 0.13} M_☉. If we suppose that the plateau is caused by interaction with a uniform spherical carbon/oxygen envelope, then this envelope has a mass of about 0.1 M_WD, and the total system mass estimate goes up to 1.65^+0.22_{− 0.16} M_☉. The results are similar to those for the nominal Chandrasekhar-mass SNe in our sample, SNF 20070803-005 and SNF 20080522-000.

Stritzinger et al. (2006) measured a mass of 1.21 ± 0.36 M_☉ for SN 1991T. Our median derived mass for SN 1991T exceeds that of Stritzinger et al. (2006) by over 0.4 M_☉; our 98% CL lower limit on M_WD is 1.28 M_☉. However, we find that we can reproduce Stritzinger et al.'s (2006) numbers for t₀, ⁵⁶Ni mass, and ejected mass if we neglect NIR corrections and ignore covariances in our fitting procedure, particularly that between q and v_e. In particular, by including NIR corrections we find t₀ = 48 days, much larger than the Stritzinger et al. (2006) value of 34 days and suggesting more massive ejecta. Our larger median value for q (0.42) and slightly lower v_e (2850 km s⁻¹) pull in the other direction, resulting in a mass closer to the Chandrasekhar mass.

In summary, SN 1991T shows the observational hallmarks of a tamped detonation and may well have been a double-degenerate merger. Because of uncertainties in the host distance and reddening correction, we are unable to establish with confidence whether SN 1991T was itself super-Chandrasekhar-mass.

SN 2003fg (Howell et al. 2006) has a comparatively limited data set for our purposes, with only one spectrum at maximum light and no photometry at sufficiently late times for us to constrain the gamma-ray transparency. Nevertheless, there are noted similarities to SN 2007if. The single spectrum available is a good match to SN 2007if at a comparable phase, though with stronger Si ii and S ii. The observed i'-band light curve of SN 2003fg shows no significant inflection, like SN 2007if and unlike the less massive SNfactory SNe Ia. The low (∼8000 km s⁻¹) Si ii velocity near maximum light is also like SN 2007if at similar phases, and makes SN 2003fg a good candidate for a tamped detonation, although SN 2009dc showed a similarly low velocity without a discernable plateau (Yamanaka et al. 2009).

We can obtain a lower limit on the mass of SN 2003fg, assuming it can be fit by a tamped detonation. We use the value of the bolometric absolute magnitude at maximum light (M_bol = −19.87) and the velocity of the Si ii λ6355 absorption minimum (∼8000 ± 500 km s⁻¹) derived by Howell et al. (2006). Assuming no host galaxy reddening, we find a ⁵⁶Ni mass of 1.18 ± 0.16 M_☉ and a 98% CL lower limit on the total system mass of 1.77 M_☉ (1.46 M_☉ if the envelope mass is neglected). These are somewhat looser constraints than those of Howell et al. (2006), but are free of the assumption that the photospheric velocity should be related directly to the kinetic energy released in the explosion, which, as we have seen, may not be a good approximation for some density structures.