A Search for New Candidate Super-Chandrasekhar-Mass
Type Ia Supernovae
in the Nearby Supernova Factory Dataset
We present optical photometry and spectroscopy of five type Ia supernovae discovered by the Nearby Supernova Factory selected to be spectroscopic analogues of the candidate super-Chandrasekhar-mass events SN 2003fg and SN 2007if. Their spectra are characterized by hot, highly ionized photospheres near maximum light, for which SN 1991T supplies the best phase coverage among available close spectral templates. Like SN 2007if, these supernovae are overluminous () and the velocity of the Si ii 6355 absorption minimum is consistent with being constant in time from phases as early as a week before, and up to two weeks after, -band maximum light. We interpret the velocity plateaus as evidence for a reverse-shock shell in the ejecta formed by interaction at early times with a compact envelope of surrounding material, as might be expected for SNe resulting from the mergers of two white dwarfs. We use the bolometric light curves and line velocity evolution of these SNe to estimate important parameters of the progenitor systems, including mass, total progenitor mass, and masses of shells and surrounding carbon/oxygen envelopes. We find that the reconstructed total progenitor mass distribution of the events (including SN 2007if) is bounded from below by the Chandrasekhar mass, with SN 2007if being the most massive. We discuss the relationship of these events to the emerging class of super-Chandrasekhar-mass SNe Ia, estimate the relative rates, compare the mass distribution to that expected for double-degenerate SN Ia progenitors from population synthesis, and consider implications for future cosmological Hubble diagrams.
Subject headings:white dwarfs; supernovae: general; supernovae: individual (SN 2003fg, SN 2007if, SN 2009dc, SNF 20080723-012)
Type Ia supernovae (SNe Ia) have become indispensable as luminosity distance indicators for exploring the accelerated expansion of the universe (Riess et al., 1998; Perlmutter et al., 1999). Their utility is due mainly to their very high luminosities, in combination with a set of relations between intrinsic luminosity, color and light curve width (Riess et al., 1998; Tripp, 1998; Phillips et al., 1999; Goldhaber et al., 2001) which reduces their dispersion around the Hubble diagram to mag. Much recent attention has been given to improving the precision of distance measurements by searching for further standardization relations, with some methods using near-maximum-light spectra to deliver core Hubble residual dispersions as low as 0.12 mag (Bailey et al., 2009; Wang et al., 2009; Folatelli et al., 2009; Foley & Kasen, 2011).
Despite ongoing research, however, many uncertainties remain regarding the physical nature of SN Ia progenitor systems, although observational subclasses can be formed (e.g., Branch et al., 1993; Benetti et al., 2005). Detailed observations of the light curve of the nearby type Ia SN 2011fe shortly after explosion have shown that it must have had a compact progenitor (PTF11kly Nugent et al., 2011; Bloom et al., 2012), in line with expectations that SNe Ia result from thermonuclear explosions of white dwarfs in binary systems. The mass and evolutionary state of the progenitor’s companion star, the events that trigger the explosion, and the circumstellar and host galaxy environment of most normal SNe Ia remain unknown. Next-generation SN Ia cosmology experiments make stringent demands, and any cosmological or astrophysical phenomenon which could bias the measured luminosities of SNe Ia at the level of a few percent has become important to investigate (Kim et al., 2004). If two or more progenitor channels exist corresponding to different peak luminosities or luminosity standardization relations, evolution with redshift of the relative rates of observed SNe Ia from these channels could mimic the effects of a time-varying dark energy equation of state (Linder, 2006).
The two main competing SN Ia progenitor scenarios are the single-degenerate scenario (Whelan & Iben, 1973), in which a carbon/oxygen white dwarf slowly accretes mass from a non-degenerate companion until exploding near the Chandrasekhar mass, and the double-degenerate scenario (Iben & Tutukov, 1984), in which two white dwarfs collide or merge. There are also sub-Chandrasekhar models (Woosley & Weaver, 1994; van Kerkwijk, Chang, & Justham, 2010), in which the explosion is triggered by the detonation of a helium layer on the surface of a sub-Chandrasekhar-mass white dwarf (Sim et al., 2010). Although historically the single-degnerate scenario has been favored, the double-degenerate scenario has recently gained ground both theoretically and observationally. Gilfanov & Bogdan (2010) used X-ray observations of nearby elliptical galaxies to set a stringent limit on the number of accreting white dwarf systems in those galaxies (see also Di Stefano, 2010a), and hence on the single-degenerate contribution to the SN Ia rate, although their interpretation of the measurements has been questioned (Di Stefano, 2010b; Hachisu et al., 2010). Based on searches for an ex-companion star near the site of explosion, Li et al. (2011) ruled out a red giant companion for single-degenerate models of SN 2011fe, and Schaefer & Pagnotta (2012) argue strongly that the supernova remnant SNR 0509-67.5, in the Large Magellanic Cloud, must have had a double-degenerate progenitor; however, single-degenerate scenarios have been put forth in which long delays between formation of the primary white dwarf and the SN Ia explosion could allow the companion to evolve and become fainter, evading attempts to detect them directly (Di Stefano & Kilic, 2012). While a merging white dwarf system could also undergo accretion-induced collapse to a neutron star (Nomoto & Kondo, 1991; Saio & Nomoto, 1998) rather than exploding as a SN Ia, theoretical investigations of SNe Ia from mergers have also progressed. Some merger simulations produce too much unburnt material to reproduce spectra of normal SNe Ia (Pfannes et al., 2010a); other simulations suggest that if the merger process is violent enough to ignite the white dwarf promptly, mergers may produce subluminous SNe Ia (Pakmor et al., 2011) or even normal SNe Ia (Pakmor et al., 2012).
Interest has also been aroused by the discovery of “super-Chandra” SNe Ia which far exceed the norm in luminosity: SN 2003fg (Howell et al., 2006), SN 2006gz (Hicken et al., 2007), SN 2007if (Scalzo et al., 2010; Yuan et al., 2010), and SN 2009dc (Yamanaka et al., 2009; Tanaka et al., 2010; Taubenberger et al., 2011; Silverman et al., 2011). If the light curves of these SNe are powered by the radioactive decay of , as expected for normal SNe Ia, the mass necessary to produce the observed luminosity implies a system mass significantly in excess of the Chandrasekhar mass. The high luminosity of SN 2006gz reported in Hicken et al. (2007) hinges upon an uncertain reddening correction, and late-phase photometry and spectroscopy suggest a smaller mass than that inferred from the dereddened peak luminosity (Maeda et al., 2009); SN 2003fg, SN 2007if, and SN 2009dc are much more luminous than normal SNe Ia even before dereddening. None of these SNe lie on the existing luminosity standardization relations. If clear photometric and spectroscopic signatures of the progenitor system or explosion mechanism can be discovered for these rare SNe Ia, they may help bring to light similar, but weaker, signatures in less-extreme SNe Ia.
In addition to its high luminosity (), SN 2007if showed a red () color at maximum light, C ii absorption in spectra taken near maximum light, and a low ( km s) and very slowly-evolving Si ii absorption velocity. Scalzo et al. (2010) used this information to model SN 2007if as a “tamped detonation” resulting from the explosion of a super-Chandrasekhar-mass white dwarf inside a dense, but compact, carbon/oxygen envelope, as expected in some double-degenerate merger scenarios (Khokhlov et al., 1993; Höflich & Khohklov, 1996). Using the bolometric light curve between 60 and 120 days past explosion to determine the optical depth for gamma-ray trapping in the ejecta (see e.g. Jeffery, 1999; Stritzinger et al., 2006), Scalzo et al. (2010) estimated the total SN 2007if system mass to be , with of this mass bound up in the carbon/oxygen envelope formed in the merger process. Such a high mass is near the theoretical upper mass limit for two carbon/oxygen white dwarfs in a binary system. However, numerical calculations have since confirmed that the very large mass of needed to explain the luminosity of SN 2007if can plausibly be produced in a collision of two high-mass white dwarfs (Raskin et al., 2010), or in the prompt detonation of a single rapidly-rotating white dwarf (Pfannes et al., 2010b). Recent work has suggested that a white dwarf with a non-degenerate companion could be spun up by accretion to high mass, and remain rotationally supported for some time only to explode later (Justham, 2011; Hachisu et al., 2011).
In contrast, the comparably-bright SN 2009dc had a normal color () near maximum light and showed rapid Si ii velocity evolution at km s day (Silverman et al., 2011), difficult to explain by a tamped detonation. Tanaka et al. (2010) group SN 2009dc with the 2003fg-like SNe Ia based on its high luminosity and broad light curve, but they compare it spectroscopically with SN 2006gz, given its strong Si ii and C ii absorption at early phases. Silverman et al. (2011) also noted some spectroscopic differences in the post-maximum spectra of SN 2007if and SN 2009dc. Moreover, the SN was fainter at one year after explosion than expected for the estimated mass (Taubenberger et al., 2011; Silverman et al., 2011), as Maeda et al. (2009) noted for SN 2006gz. Taubenberger et al. (2011) calculated a total system mass of for SN 2009dc by estimating the diffusion time from the width of the bolometric light curve (Arnett, 1982); they explored a number of different thermonuclear and core-collapse explosion scenarios, and found none of them to be completely satisfactory in explaining all the observations.
After the discovery of SN2007if, we remained vigilant for SNe with similar characteristics; when possible, additional follow-up was obtained when such SNe were recognized. Given the high ionization state and predominance of Fe ii and Fe iii absorption in SN 2007if’s spectra up until maximum light, characteristics shared with SN 1991T, we used a 1991T-like spectroscopic classification to select for super-Chandrasekhar-mass SN candidates, triggering follow-up even for more distant, fainter examples. The Nearby Supernova Factory (SNfactory) obtained, as part of its spectroscopic follow-up program on a large sample of nearby SNe Ia, observations of five such SNe Ia besides SN 2007if, the presentation of which is the subject of this paper. Later examination of the full data set of SNfactory spectroscopic time series showed that the SNe in our sample each also show a plateau in the time evolution of the velocity of the Si ii absorption minimum, lasting from the earliest phase the velocity was measurable until 10–15 days after maximum light, as in SN 2007if. The absorption minimum velocities of other intermediate-mass elements in these SNe also show plateau behavior. These events have a different appearance from SN 2006gz and SN 2009dc, which do not show velocity plateaus and show somewhat different behavior in their early spectra and late-time light curves. SN 2003fg was spectroscopically observed only once, making it impossible to determine how it evolved spectroscopically.
Our supernova discoveries, our sample selection, and the provenance of our data are described in §2; the light curves and spectra are presented in section §3. In §4 we model our SNe as tamped detonations, using the formalism of Scalzo et al. (2010) to estimate an envelope mass and total system mass for each SN. We discuss the broader implications of our results in §5, including implications for progenitor systems, explosion mechanisms, and cosmology, and we summarize and conclude in §6.
|SN Name||RA||DEC||Disc. UT Date||Disc. PhaseaaIn rest-frame days relative to -band maximum light, as determined from a SALT2 fit to the -corrected rest-frame light curve.||bbFrom template fit to host galaxy spectrum (Childress et al., 2011, M. Childress et al. 2012, in preparation).||Host TypeccMorpological type from visual inspection.||ddFrom Schlegel et al. (1998).|
|SNF 20070528-003||16:47:31.46||+21:28:33.4||2007 May 05||0.1171||dIrr||0.045|
|SNF 20070803-005||22:26:24.03||+21:14:56.6||2007 Aug 03.4||0.0315||Sbc||0.047|
|SN 2007ifeeDiscovered independently by the Texas Supernova Search (Yuan et al., 2007, 2010) and by SNfactory as SNF 20070825-001 (Scalzo et al., 2010).||01:10:51.37||+15:27:40.1||2007 Aug 25.4||0.0742||dIrr||0.082|
|SNF 20070912-000||00:04:36.76||+18:09:14.4||2007 Sep 12.4||0.1231||Sbc||0.029|
|SNF 20080522-000||13:36:47.59||+05:08:30.4||2008 May 22||0.0453||Sb||0.026|
|SNF 20080723-012||16:16:03.26||+03:03:17.4||2008 July 23.4||0.0745||dIrr||0.062|
This section details the discovery, selection and follow-up data for our sample of candidate super-Chandrasekhar-mass SNe Ia.
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 and 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 . 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.
2.2. Selection criteria
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 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 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 lightcurve 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., 2011). We reiterate that characteristics such as luminosity excess, velocity evolution, lightcurve shape, etc. were not used in our selection — it is purely based on optical spectrophotometry.
2.3. Follow-up Observations
Most of the 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 from Schlegel et al. (1998) and the extinction law of Cardelli et al. (1988) with . Follow-up 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, 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
Observations of all six SNe were obtained with SNIFS, built and operated by the SNfactory. SNIFS is a fully integrated instrument optimized for automated observation of point sources on a structured background over the full optical window at moderate spectral resolution. It consists of a high-throughput wide-band pure-lenslet integral field spectrograph (IFS, “à la TIGER”; Bacon et al., 1995, 2000, 2001), a multifilter photometric channel to image the field surrounding the IFS for atmospheric transmission monitoring simultaneous with spectroscopy, and an acquisition/guiding channel. The IFS possesses a fully filled spectroscopic field of view (FOV) subdivided into a grid of spatial elements (spaxels), a dual-channel spectrograph covering 3200–5200 Å and 5100–10000 Å simultaneously, and an internal calibration unit (continuum and arc lamps). SNIFS is continuously mounted on the south bent Cassegrain port of the UH 2.2 m telescope (Mauna Kea) and is operated remotely. The SNIFS spectrophotometric data reduction pipeline has been described in previous papers (Bacon et al., 2001; Aldering et al., 2006; Scalzo et al., 2010). We subtract the host galaxy light using the methodology described in Bongard et al. (2011), which uses SNIFS IFU exposures of the host taken after each SN has faded away.
2.3.2 SMARTS Photometry
The SMARTS imaging of SNF 20070803-005 and SN 2007if with ANDICAM consisted of multiple 240 s exposures in each of the filters at each epoch. The images were processed using an automated pipeline based on IRAF (Tody, 1993); this pipeline was used in Scalzo et al. (2010) and is described in detail there. We briefly summarize the methods below.
All ANDICAM images were bias-subtracted, overscan-subtracted, and flat-fielded by the SMARTS Consortium, also using IRAF (ccdproc). In each band, four to six final reference images of each SN field were taken at least a year after explosion, and combined to form a co-add. This co-add was then registered, normalized, and convolved to match the observing conditions of each SN image, before subtraction to remove the host galaxy light.
An absolute calibration (zeropoint, extinction and color terms) was established on photometric nights from observations of Landolt (1992) standards, fitting a zeropoint and extinction coefficient for each night separately as well as a color term constant across all nights. The calibration was transferred to the field stars for each photometric night separately using the zeropoint and extinction but ignoring the color terms, producing magnitudes on a “natural” ANDICAM system which agrees with the Landolt system for stars with . These calibrated magnitudes were then averaged over photometric nights to produce final calibrated ANDICAM magnitudes for the field stars.
Each SN’s ANDICAM-system light curve was then measured by comparison to the field stars. To fix the SN’s location, we found the mean position over observations in the same filter, weighting by the signal-to-noise ratio (S/N) of each detection. We then measured the final flux in each image in a circular aperture centered at this mean location. The observer-frame ANDICAM magnitudes were corrected for Galactic extinction using from Schlegel et al. (1998) and the extinction law of Cardelli et al. (1988) with . The SN magnitudes were -corrected (Nugent et al., 2002) to rest-frame Bessell bandpasses (Bessell, 1990) using the SNIFS spectrophotometric time series and a set of ANDICAM system throughput curves, the central wavelengths of which were shifted to match ANDICAM observations of spectrophotometric standard stars to published synthetic photometry (Stritzinger et al., 2005).
The rest-frame, Milky Way de-reddened Bessell 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 () and SNF 20070912-000 (), have less extensive coverage than the other SNe; the S/N is lower, and a significant fraction of the rest-frame -band transmission lies outside of the observer-frame wavelength range of the SNIFS spectrograph. For a small number of observations, rest-frame -band or -band measurements are unavailable due to instrument problems with the SNIFS blue channel. The detailed analysis of these lightcurves is presented in §3.1 and §3.2.
|MJDaaObserver frame .||PhasebbIn rest-frame days relative to -band maximum light.||Instrument|
|MJDaaObserver frame .||PhasebbIn rest-frame days relative to -band maximum light.||Instrument|
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 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 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.
SN 2007if also shows a weak C ii 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 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., 2011). The spectral properties of the full dataset are analyzed in §3.3.
The analysis (this section), modeling (§4) and interpretation (§5) for our sample parallels that made for SN 2007if in Scalzo et al. (2010), with improvements described below. We include a re-analysis of our observations of SN 2007if in this paper using the improved techniques, for a more direct comparison with the other SNe presented in this paper.
3.1. Maximum-Light Behavior, Colors and Extinction
|SN Name||MJD()||aaEvaluated from SALT2 rest-frame , with distance modulus at the host galaxy redshift assuming a CDM cosmology with , , Mpc.||bbLight curve decline rate, evaluated directly from the best-fit SALT2 model, accounting for error in the date of -band maximum light.||ccHubble residuals from CDM cosmology, evaluated using equation 2 of Sullivan et al. (2011).|
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, vs. 0.23 mag for SNF 20080522-000 and 0.35 mag for the SALT2 model with , . Kasen (2006) noted three physical effects which could reduce the contrast of the -band second maximum: low mass, efficient mixing of 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 -band first maximum is roughly concurrent with -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 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 -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 bands only; -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 color behavior of each SN to the Lira relation (Phillips et al., 1999; Folatelli et al., 2009) 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 New A absorption at the redshift of the host galaxy for each SN. We perform a fit to the New A 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 (Na i D) of the New A line is constrained to be non-negative. We convert these to estimates of using both the shallow-slope (0.16 Å) and steep-slope (0.51 Å) relations from Turatto, Benetti & Cappellaro (2002) (“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 (Na i D), and derived constraints on the host galaxy reddening are listed in Table 4.
We detect weak New A 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 (Cardelli et al., 1988). When applied to Milky Way New A absorption in our spectra, the shallow-slope TBC relation produces estimates consistent with Schlegel et al. (1998). SNF 20070912-000 shows a marginal () detection, though the spectra are noisy and the limits are not strong. We detect no New A absorption in the other SNe.
|Na i D||Shallow||Steep||Lira|
|SN Name||EWaaMeasured from a simultaneous fit of the New A absorption line profile to all SNIFS spectra of each SN. (Å)||TBCbb derived from New A absorption, using the “shallow” slope (0.16 Å) of Turatto, Benetti & Cappellaro (2002).||TBCcc derived from New A absorption, using the “steep” slope (0.51 Å) of Turatto, Benetti & Cappellaro (2002).||RelationddBest-fit from the Lira relation in the form given in (Phillips et al., 1999). Error bars include a 0.06 mag intrinsic dispersion of normal SNe Ia around the relation, added in quadrature to the statistical errors.|
Based on the very strong limit on New A 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 New A absorption, each of the new SNe have maximum-light 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 days and days with respect to -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 . The more recent reduction fit with SALT2 v2.2 gives . However, the directly measured color of SN 2007if at days () and at days () are each 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.
3.2. Bolometric Light Curve and Synthesis
As input to our further analysis to calculate 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 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 observations and each possible value of , 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 §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 §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 a 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 -band second maximum has low contrast for all the SNe in our sample, the behavior should be similar for these SNe to the extent that and 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 (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 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 (), and at a maximum near phase days (), so it should not evolve quickly at these times and our results should not be strongly affected. We assign a systematic error of of the total bolometric flux (or about of the NIR flux itself near d) to this correction.
To estimate masses, we also need a measurement of the bolometric rise time . We establish the time of bolometric maximum light by fitting a cubic polynomial to the bolometric fluxes in the phase range . We then calculate the final bolometric rise time via
We find to occur about 1 day earlier than 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 days, and so use days in our models. Our value is very similar to the value of 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 (or stretch ) may break down at the high-stretch end.
We calculate the mass, , for the six SNe by relating the maximum-light bolometric luminosity to the luminosity from radioactive decay (Arnett, 1982):
where ( in this case) is the time since explosion, is the number of atoms produced in the explosion, and are the decay constants for and (-folding lifetimes 8.8 days and 111.1 days) respectively, and , and are the energies released in the different stages of the decay chain (Nadyozhin, 1994). The dimensionless number 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 ). A nominal value of 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 . We therefore adopt a fiducial value of for our simple estimate here.
|SN Name||( erg s)||(d)aaCalculated from Equation 1.||bbAssuming fiducial .|
The resulting mass estimates are shown in Table 5. The new SNe have in the range 0.7–0.8 , 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 of , while some delayed detonation models can produce up to 0.8 (e.g., the N21 model of Höflich & Khohklov, 1996).
3.3. Spectral Features and Velocity Evolution
The time evolution of the position of the Si ii 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, . 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 and representing possible pseudocontinua 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.
|SN Name||aaBest-fit constant velocity, in , with error.||bbSlope of the best-fit line to absorption line velocities from the first reliable measurement until the break associated with Fe ii line blending, in day.||ccPhase of first available measurement, in days with respect to -band maximum light, which we interpret to be the start of the plateau.||ddMinimum duration of the visible plateau phase in days, until Si ii becomes blended with Fe ii.||eeChi-square per degree of freedom for fit to a constant.|
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 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, , for a fit to a constant. For comparison with earlier work, the velocity gradient calculated as the slope of the best-fit linear trend of the measurements before day , 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 level (formal errors), the reduced chi-square for this fit is extremely small () and we conclude that the evolution cannot be reliably distinguished from a constant (). Similarly, the best-fit line to the absorption velocities for SNF 20080723-012 has a positive slope at ( day), but once again, a constant is a good fit to the data ().
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 . 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 -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 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 (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 -band second maximum.
In four of the six SNe, the plateau velocity is low ( km s), 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 ( km s), 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 absorption feature, only two measurements of the line velocity before the break, each around 9000 km/s 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 . Although, as expected, Ca ii H+K stays optically thick longer than Si ii , 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.
4. Constraints on Total Mass and Density Structure
The modeling procedure we use here represents a refined version of that used in Scalzo et al. (2010), which we compare and contrast with the similar approach of Stritzinger et al. (2006) in §4.1 for the special case of a simple equivalent exponential density profile. We then describe our extensions to the method, including modeling of density profiles with shells (§4.2), priors on the central density (§4.3), and calculation of the form factor (§4.4). We then present our final modeling results in §4.5.
4.1. SN Ia Ejected Mass Measurements Using the Equivalent Exponential Density Formalism
The ejecta density structure of SNe Ia is frequently modeled as an exponential , where the ejecta are in homologous expansion at velocity since the explosion at time , and 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 semianalytic 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 has decayed and the dominant energy source is the decay of (which in turn was produced by decay at earlier times). The optical depth to Compton scattering of gamma rays behaves as with at some fiducial time . The value of can be extracted by fitting the bolometric light curve for d to a modified version of Equation 2, in which (rather than its maximum-light value from Arnett’s rule) and the term corresponding to gamma rays is multiplied by a factor . The total mass of the ejecta can then be expressed as
Here, is the Compton scattering opacity for gamma rays, and is a form factor describing the distribution of the in the ejecta, which follows the original distribution of in the explosion. The value of is expected to lie in the range 0.025–0.033 cm g (Swartz et al., 1995), with the low end (0.025) corresponding to the optically thin regime. The value of can be readily calculated given an assumed distribution of (see §4.4 below).
Stritzinger et al. (2006) used this method to measure progenitor masses for a sample of well-observed SNe Ia with light curve coverage. They constructed “quasi-bolometric” light curves according to the procedure of Contardo, Leibundgut & Vacca (2000), by converting the observed 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 semianalytic 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 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 () 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 gamma-rays for a given initial 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 , , and may influence the interpretation of the fitted value of . Stritzinger et al. (2006) simply fixed the mass from Arnett’s rule, and then fit for . Similarly, they assume , km s, and for all of their SNe, with no covariance between any of these parameters. Because models with more need less gamma-ray trapping to produce the same bolometric luminosity at a given time, there is a large fitting covariance between the mass and , mentioned in Scalzo et al. (2010). The value of is model-dependent, and not a fundamental physical quantity, but as noted in §3.2 above, () is also a common choice when no other prior is available from explosion models. Since affects the nickel mass, a smaller assumed value of results in a larger mass, but a smaller ejected mass, as interpreted from a given bolometric light curve. In a self-consistent choice of parameters, and will each depend in part on the mass of and therefore on .
The value of is difficult to measure directly, since observed velocities of absorption line minima may depend on temperature as much as density. Since appears squared in Equation 3, its contribution to the error budget on is potentially quite large if treated as an independent input. However, its value can be constrained within a range of km s by requiring energy conservation. Following practice in the literature (Howell et al., 2006; Maeda & Iwamoto, 2009), we calculate the kinetic energy as the difference between the energy released in nuclear burning and the gravitational binding energy , and then set .
Calculating the energy budget of a SN Ia requires us to assume a composition. Our model considers four components to the ejecta:
, which contributes to the luminosity, , and ;
Stable Fe-peak elements (“Fe”), which contribute to and ;
Intermediate-mass elements such as Mg, Si and S (“Si”), which contribute to and ;
Unburned carbon and oxygen (“C/O”), which contribute only to .
The input parameters, which we vary using a Metropolis-Hastings Monte Carlo Markov chain, are the white dwarf mass , the central density (needed in the calculation of ), the parameter from Arnett’s rule, the bolometric rise time , and the fractions , , and of , stable Fe, and intermediate-mass elements within . We fix the fraction of unburned carbon and oxygen . We use the prescription of Maeda & Iwamoto (2009) to determine :
We use the binding energy formulae of Yoon & Langer (2005) for , where is the white dwarf central density. These ingredients determine . We apply Gaussian priors (as for SN 2007if; Scalzo et al., 2010) and days, and on according to the “shallow TBC” values in Table 4. We adopt cm g after Jeffery (1999) and Stritzinger et al. (2006).
One limitation with our approach is the use of 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., 2010a). 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) and Pakmor et al. (2012), though they produce comparatively little , show that prompt detonations in violent mergers can occur. Pfannes et al. (2010b) simulated prompt detonations of rapidly rotating white dwarfs with masses up to 2.1 , and found that the amount of produced could be as high as 1.8 , similar to SN 2007if.
In summary, our procedure directly extracts from a fit to the bolometric light curve only the quantities and (Equation 2), and marginalizes, in effect, over , , , and . 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 and , using Equations 2, 3, and 4, and finally calculate the ejected mass .
4.2. Including the Effects of a Shell
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 which undergoes nuclear burning) are compressed into a dense shell of mass , traveling at velocity . 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 ( cm) envelope of material with mass (external to, and not included in, ). The suffix in DET2ENVN refers to the envelope mass, so for example model DET2ENV2 has . “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 . The interaction ends within about the first minute after explosion, and the shell then expands homologously with the other ejecta thereafter. We observe directly as the plateau velocity, allowing us to constrain and , and, indirectly, the kinetic energy scale of the ejecta. For a given value of and a measured value of , and neglecting the binding energy of the envelope, conservation of momentum gives (for more detail see Scalzo et al., 2010)
where is the incomplete gamma function. We calculate and by solving Equation 5 numerically. In a double-degenerate merger scenario, the total system mass 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 (Si ii) plateau measured from our spectroscopic time series. We do not model the duration of the plateau or the behavior of (Si ii) after the plateau phase ends. While the detailed evolution of (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 (Si ii) to show that a given SN exhibits plateau behavior, we can reliably measure 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 §5.1.
4.3. Including the Effects of Central Density on Yields
In Scalzo et al. (2010), the stable iron fraction was allowed to vary freely. In this situation, and 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 can greatly reduce because the formation of a large Fe core displaces 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 .
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 inside envelopes of varying mass. The central density in these models is g cm, substantially lower than typical central densities of g cm 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 throughout the region where and Fe are produced.
Krueger et al. (2010) investigated the effects of central density on yields in 3-D simulations of detonations of Chandrasekhar-mass white dwarfs, averaging over an ensemble of realizations for each value of . They found for g cm, decreasing by on average for each g cm increase of thereafter.
Our model already includes the effects of the central density on the binding energy of the white dwarf, through the fitting formula of Yoon & Langer (2005). Although 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 and is statistical rather than deterministic, we do not attempt to calculate a definite fraction for each model. Instead, we calculate the ratio of to total iron-peak elements, , as well as the relation from Krueger et al. (2010):
enforcing a Gaussian prior . For most models calculated, this results in only a small fraction of stable Fe, as appropriate for low-density explosions.
4.4. Calculation of for SN Ia Models with a Shell
The gamma-ray transport form factor is the dimensionless -weighted gamma-ray optical depth through the ejecta. Its value ranges between 0 and 1, with large values corresponding to high central concentrations of ; the gamma-ray optical depth is proportional to . For perfectly mixed, exponentially distributed ejecta where the fraction of is constant throughout, .
In Scalzo et al. (2010), the calculation of the total mass for SN 2007if assumes . We chose this value because the lack of a distinct second maximum in the -band light curve suggested that a large amount of had to be mixed to higher velocities (see §3.1 above). This is not necessarily true for our other SNe. The use of 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, is easy to calculate numerically (Jeffery, 1999):
where and are dimensionless velocity coordinates in units of , is the beam path length, is a dimensionless density profile normalized to unit mass, and is the (velocity-dependent) fraction just after explosion. The geometry of the integration is shown in Figure 6.
|SN Name||aaTotal system mass of the two merged white dwarfs.||bbMass of the central white dwarf merger product which undergoes nuclear burning in the explosion.||cc mass synthesized in the explosion.||ddRatio of envelope mass to central merger product mass.||eeFraction of mass of burnt ejecta which is compressed into the reverse-shock shell.||ffMass fraction of iron-peak elements (|