Type Ia Supernovae with Bi-Modal Explosions Are Common -- Possible Smoking Gun for Direct Collisions of White-Dwarfs
aa r X i v : . [ a s t r o - ph . H E ] S e p MNRAS , 1–7 (2015) Preprint 2 September 2015 Compiled using MNRAS L A TEX style file v3.0
Type Ia Supernovae with Bi-Modal Explosions AreCommon – Possible Smoking Gun for Direct Collisions ofWhite Dwarfs
Subo Dong, ⋆ Boaz Katz, , Doron Kushnir and Jose L. Prieto , Kavli Institute for Astronomy and Astrophysics, Peking University, Yi He Yuan Road 5, Hai Dian District, Beijing 100871, China Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel Institute for Advanced Study, Einstein Drive, Princeton, NJ, 08540, USA N´ucleo de Astronom´ıa de la Facultad de Ingenier´ıa, Universidad Diego Portales, Av. Ej´ercito 441, Santiago, Chile Millennium Institute of Astrophysics, Santiago, Chile
Accepted September 1, 2015. Received July 11, 2015; in original form July 11, 2015
ABSTRACT
We discover clear doubly-peaked line profiles in 3 out of ∼ type Ia supernovae (SNeIa) with high-quality nebular-phase spectra. The profiles are consistently present inthree well-separated Co/Fe emission features. The two peaks are respectively blue-shifted and red-shifted relative to the host galaxies and are separated by ∼ / s .The doubly-peaked profiles directly reflect a bi-modal velocity distribution of the ra-dioactive Ni in the ejecta that powers the emission of these SNe. Due to their randomorientations, only a fraction of SNe with intrinsically bi-modal velocity distributionswill appear as doubly-peaked spectra. Therefore SNe with intrinsic bi-modality arelikely common, especially among the SNe in the low-luminosity part on the Philipsrelation ( D m ( B ) & . ∼ of all SNe Ia). Such bi-modality is naturally expectedfrom direct collisions of white dwarfs (WDs) due to the detonation of both WDs andis demonstrated in a 3D . M ⊙ - . M ⊙ WD collision simulation. In the future, witha large sample of nebular spectra and a comprehensive set of numerical simulations,the collision model can be unambiguously tested as the primary channel for type IaSNe, and the distribution of nebular line profiles will either be a smoking gun or ruleit out.
Key words: supernovae: general
Type Ia supernovae (SNe Ia) are well-known cosmologi-cal “standard candles” thanks to a tight empirical corre-lation (the Phillips relation established by Phillips 1993)between intrinsic peak luminosities and post-peak bright-ness decline-rates ( D m ( B ) ). SNe Ia are powered by the de-cay of Ni produced from the explosion of Carbon-OxygenWhite Dwarfs (WDs), but the explosion mechanism is un-known. The two popular scenarios, single-degenerate (WDaccretion exceeding the Chandrasekhar limit) and double-degenerate mergers (merger of two close WDs that spiralin due to gravitational radiation), have many theoreticaland observational challenges (Hillebrandt & Niemeyer 2000;Maoz, Mannucci, & Nelemans 2014). For both scenarios, aserious challenge is that a successful ignition of an explosivedetonation has never been convincingly demonstrated. ⋆ E-mail: [email protected]
Direct, head-on collisions of WDs would undoubtedlylead to successful explosions due to the strong shocksformed during the high-velocity impacts (Kushnir et al.2013; Rosswog et al. 2009; Raskin et al. 2009, 2010;Hawley, Athanassiadou, & Timmes 2012), but they hadlong been thought to only occur in dense stellar environ-ments (globular clusters) and responsible for a negligiblefraction of SNe Ia. It was recently shown by Katz & Dong(2012) that the rate of direct collisions in common fieldtriple systems may be as high as the SNe Ia rate. Previ-ously, Thompson (2011) argued that the secular Lidov-Kozaimechanism (Lidov 1962; Kozai 1962) in triples might playan important role in WD-WD mergers via gravitational ra-diation to produce SNe Ia and speculated that collisionsmay sometimes occur. The high collision probability dueto the non-secular corrections to the Lidov-Kozai mecha-nism obtained by Katz & Dong (2012) raised the possibilitythat the majority of SNe Ia result from collisions. Support-ing evidence was provided in Kushnir et al. (2013) in which c (cid:13) Subo Dong et al. high resolution numerical simulations of WD collisions re-produced several robust observational features of SNe Ia,especially establishing that the full range of Ni necessaryfor all SNe Ia across the Phillips relation, from the faint-endof SN 1991bg-like events to luminous 1991T-like events, canbe produced by collisions of typical WDs.Here we report that nebular spectra of some SNe Iashow double-peaked line profiles that suggest bimodal dis-tributions of radioactive Ni in the ejecta. We find that3 out of a sample of 20 SNe show this structure, imply-ing a much larger fraction when viewing angle effects aretaken into account. Using simulations we show that collisionmodels of SNe Ia can produce structures similar to those ob-served due to the existence of two separate detonation seeds,providing new direct evidence for collisions as a significantSNe Ia channel. The bi-modal distributions can serve as atouchstone in testing SN Ia explosion scenarios in general.
Motivated by significantly non-spherical Ni distributionsexpected from WD collisions, we search for unusual nebularemission features in archival type Ia observations. In thenebular phase, the SN spectrum is emission-line dominated,and the line profiles directly probe the underlying velocitydistribution of the emitting materials along the line of sight(LOS) due to the Doppler effect. By this time, Ni hascompletely decayed into Co and Fe, and the Co/Fe linesretain the velocity distribution of the original Ni.We systematically collect spectra of SNe Iafrom the archival data. The main sources arethe Berkeley Supernova Ia Program (BSNIP)(Silverman, Ganeshalingam, & Filippenko 2013), theCenter for Astrophysics Supernova Program (Blondin et al.2012), Carnegie Supernova Project (Folatelli et al. 2013)and the compilation from various sources by the OnlineSupernova Spectrum Database (SUSPECT). We have alsoincluded the spectra of SN 2011fe taken by Shappee et al.(2013). In the collection, there are 55 SNe with 155 spectracovering the wavelength range of interest (5000 - 7000 ˚A inthe rest frame) and with phase greater than 170 days,which we define in this work as nebular phase. We focuson SNe with the highest Signal-to-Noise-Ratio (SNR)nebular spectra. Note that we exclude from the sampletwo SNe (SN 1986G and SN 2004bv) with a strong Na I Dabsorption feature at ll , ˚A identified at earlierepochs since such a feature hinders clear identification ofdoubly-peak profile at ∼ ˚A. It is also important to becautious about spectra with over-subtraction H- a emissionline at ˚A from the host-galaxy, which may confuse theanalysis of the feature at ∼ ˚A. ∼ suspect/; a public repos-itory to download the spectra: WISeREP (Yaron & Gal-Yam2012). Figure 1.
The nebular spectrum of SN 2007on (top panel, black)has a clear doubly-peaked line profile appearing in three differentCo/Fe emission features. The single-peak nebular spectrum of the“normal” SN 2011fe at a similar epoch is shown for comparison(green). The three double-peak profiles reflect the same bi-modalline-of-sight velocity distribution of the emitting Co/Fe materi-als. This is shown by fitting the spectrum with the convolutionof a template spectrum using the same double-component veloc-ity kernel (inset of middle panel, red). The template is chosento be the narrow-width spectrum of SN 1999by (middle panel,black). The convolved spectrum (bottom panels, red) is com-pared with each of the three features of SN 2007on (bottom pan-els, black) by linear fitting that allows free normalizations andbaseline flux shifts for each feature. The three features are dueto blends of [FeIII] and [FeII] ( ∼ ˚A), [CoIII] ( ∼ ˚A), andblends of [CoIII] and [FeII] ( ∼ ˚A), respectively (Axelrod 1980;Bowers et al. 1997; Turatto et al. 1996). The ∼ ˚A [CoIII]feature is the most reliable among the three, and it is composedof two components ( ˚A and ˚A) narrowly separated by ∼ / s , which is much smaller than the peak-to-peak sep-aration of the double-peak profile ∼ / s . We identify three SNe with clear evidence of doubly-peakedvelocity profiles. Figure 1 shows one example, SN 2007on,which exhibits clear doubly-peaked profiles for three well-spaced Fe and Co emission-line features.A general challenge in interpreting SNe line profiles isthat many spectral features are due to blending from morethan one line. A reasonable concern is that an observeddoubly-peaked line profile could be due to two (or more)adjacent lines within an underlying single-peak velocity dis-tribution. Several lines of evidence show that this is not thecase for SN 2007on, so that the profile requires a bi-modalvelocity distribution:First, the doubly-peaked profiles occur for three widely-separated features emitted by two elements (Co and Fe) withthe same peak-to-peak spacing
D l / l ≈ . . Furthermore,their shapes are all consistent with the same underlying ve- MNRAS , 1–7 (2015) i-modal Explosions of SNe Ia Figure 2.
The nebular spectra of SN 2007on observed by the Claytelescope at 286 days (black) and the Gemini South telescope at353 days (blue) show consistent doubly-peaked line profiles. locity distribution. Figure 1 (bottom panel) shows that thespectra in three regions are well fitted by the convolution ofa template spectrum with the same double-component ve-locity kernel (red line in the sub-panel of the middle panel).We choose the SN 1999by spectrum as the template (blackline in the middle panel) due to its exceptionally narrow linewidth (similar to the famous SN 1991bg with SN 1999byhaving a higher SNR). A remarkable coincidence would berequired in order that three additional lines conspire to pro-duce such consistent profiles.Next, other supernovae at a similar phase (thus a similarratio between decaying Co and stable Fe) have similar lineratios among the three features but do not show the doubly-peaked profile. This is illustrated in the top panel of Figure 1from comparison with SN 2011fe. While the line ratios of thetwo SNe are practically identical, the profiles of SN 2011feare clearly single-peaked. The features from the same SNalso show stable shapes in the nebular phase when multiple-epoch spectra have been taken.Finally, the profile at ∼ − ˚A is overwhelm-ingly dominated by one [CoIII] feature. The line profile isthus “clean” and reliably reflects the underlying velocity dis-tribution without the need of modelling. This is supportedby nebular spectrum modeling (Axelrod 1980; Bowers et al.1997) and re-affirmed by examining the spectra of SNe1991bg and 1999by where the line widths are sufficientlynarrow to allow for clear line identifications (Turatto et al.1996). The Co origin of the line has been firmly establishedfor several SNe by examining the multiple-epoch spectrataken in the intervals 170-400 days, during which the Codecays significantly, and the lines maintain the same shapewhile the relative strengths with respect to Fe lines weakenaccording to the Co nuclear decay rate (Kuchner et al.1994). The other two features at ∼ ˚A and ∼ ˚Aare less reliable due to possible contributions from nearbylines. The frequently studied Fe feature at ∼ ˚A showssignificant blending from neighboring lines and is unsuitable(i.e., not clean enough) to be used for directly deriving the Ni velocity distribution from its profile. We examine thechoice of templates and spectral features in Appendix B.Maeda et al. (2010) reported another nebular-phasespectrum of SN 2007on taken at a different phase ( ) andtelescope (Gemini South) from that in our sample ( atClay). We retrieved archival late-time spectra of SN 2007on
Fe Co Fe Co5000 6000 7000 1991bg1999by 2003gs2003hv 2007on2005am 2004eo2008Q 2011fe2011by 2007sr1990N 1998bu2000cx 1991T1972E 1999aa2006ce
Figure 3.
20 SNe Ia with high-quality nebular spectra. They spanthe full range of D m ( B ) on the Phillips relation ( . < D m ( B ) < ) (middle panel, black). The spectra are sorted by the D m ( B ) values (shown on the right). Most of the spectra are single-peakedand are well fitted by the convolution of a simple quadratic ve-locity distribution with the template (SN 1999by). The best-fitsingle-peak models are shown in the sub-panels (green) wherethey are scaled to allow comparison with the corresponding spec-tral features (black) due to Fe and Co emissions, respectively. SN1991bg is nearly identical to SN 1999by and is not fitted. Binnedspectrum of SN 2003gs is shown in the right sub-panels to enhanceclarity. Three SNe are identified with doubly-peaked profiles (SN2007on, SN 2005am and SN 2003gs). SN 2008Q shows flat-top lineprofiles. SN 2003hv shows hints of departure from single-peak butis ambiguous to tell whether it has double-peak/flat-top profile.SN 2005am spectrum is taken from two epochs (298d and 381d)since the high SNR spectrum at 298d did not cover the ∼ feature. from the Gemini Science Archive (program GS-2008B-Q-8).The spectra were obtained with Gemini South + GMOS onUT 2008-11-03 with the R150 grating and 1.0” longslit. Wereduced the data using standard routines in the IRAF gem-ini.gmos package, including bias subtraction, flat-fielding,wavelength calibration using a CuAr lamp, 1D spectral ex-traction, and flux calibration. We combined × sec ex-posures to obtain the final spectrum with wavelength cov-erage − ˚A and FWHM resolution of 22 ˚A. Thereduced spectrum is plotted in Figure 2. As can be seen, thedouble-peak features are clearly visible at this epoch, andthey are consistent with those in the 286d spectrum. The sample of 20 SNe with high-SNR nebular spectra areshown in Figure 3 in ascending order of D m ( B ) (if avail-able). To identify doubly-peaked lines, we fit each spectrumusing a simple, single-peak velocity convolution kernel with MNRAS , 1–7 (2015)
Subo Dong et al.
Figure 4.
The three SNe spectra that show double-peak profiles.For each SN, a simple common velocity kernel with two quadraticcomponents is used to fit all three features via convolutions withthe template spectrum of SN1999by (see Fig. 1). The convolvedspectra are in excellent agreement with all the features of SN2007on, SN 2003gs and SN 2005am, suggesting that the underly-ing velocity distribution are bimodal for these SNe. dM / dv los (cid:181) max [ − ( v los / v mod ) , ] . The Fe and Co features ofmany SNe can be well fitted by single-peaks as shown in Fig-ure 3. The use of SN 1999by(/1991bg) as template(s) is justi-fied mainly on an empirical ground, and it could be problem-atic if they result in different explosion physics from otherSNe Ia, as argued in some works (e.g., Mazzali & Hachinger2012). Nevertheless, as discussed in Sec 2.2, we primarilyuse the clean [CoIII] 5900˚A feature for double-peak identi-fication, which does not depend on the convolution method.We identify 3 SNe, SN 2007on, SN 2005am and SN 2003gs,with clearly doubly-peaked spectra that consistently appearin the three features (marked in red). Figure 4 shows thatall three spectra can be well fitted by convolving a two-component velocity profile with the template.SN 2008Q shows a “flat-top” profile that appears inall three Fe/Co features (with low-level of NaI D absorp-tion at ∼ ). SN 2003hv ( D m ( B ) = . ) shows hints of adoubly-peaked or a flat-top profile, but the data do not al-low unambiguous identification. A clear flat-top profile wasreported for SN 2003hv for [Fe II] at 1.644 micron, and itwas regarded as evidence for a “hole” in the Ni distribu-tion (see, e.g., Motohara et al. 2006; H¨oflich et al. 2004). SN2003hv was also reported to have nebular line blue-shiftedby ∼ / s (Maeda et al. 2010). The flat-top profile andthe line shift could be the result of an underlying bi-modelvelocity distribution, but unlike a double-peak profile, flat-top profile does not allow unambiguous identification of bi-modality.Unquantified selection effects exist in this sample, sothe numbers above are not suitable for statistics. Neverthe-less, the SNe Ia with underlying bi-modal velocity distribu-tions are definitely not rare. It is interesting to note that, allthe SNe with evidences for bi-modal velocity distributionshave relatively large D m ( B ) : ∼ . (SN 2003gs), ∼ . (SN2007on), ∼ . (SN 2005am) and ∼ . (SN 2008Q). The bi-modal velocity distribution must be quite common amongthese fast declining, low-luminosity SNe with D m ( B ) > . ,which comprise ∼ of all SNe Ia in a volume-limited sam-ple (Li et al. 2011). A bi-modal velocity distribution is naturally expected fromdirect WD-WD collisions due to the detonations of bothWDs, which occur for all cases in the high-resolution 2Dsimulations for zero impact-parameter WD-WD collisions(Kushnir et al. 2013). However, bi-modal Ni velocity dis-tributions are rare in these 2D simulations. Figure 5 showsthe results of a 3D simulation of the collision of two . M ⊙ WDs with a non-zero impact parameter of 0.2. The impactparameter is defined as the ratio between r p , the minimalseparation between the WDs along their trajectory thatwould have been obtained if they were point masses, and thesum of the two WD radii. The simulation is performed usingFLASH 4.0 (Dubey et al. 2009; Fryxell et al. 2000) with a13 isotope alpha-chain reaction network (Timmes 1999) anda resolution, comparable to the converged 2D resolu-tions in Kushnir et al. (2013). The upper panel of Figure 5shows the projected velocity distribution of the total ejectamass and the Ni mass in the WD-WD orbital plane. The Ni mass consists of two components separated by severalthousand km/s. The bottom panel shows the LOS Ni dis-tributions for numerous viewing angles. The chance of seeingdoubly-peaked line profiles with similar velocity separationsas the observed ones is significant. It is worth noting that anarrow velocity distribution similar to SN 1999by/1991bg isobserved from directions perpendicular to the line connect-ing the centers of the two Ni components.It is important to note that, even though the Ni bi-modality are commonly expected from WD-WD collisionmodels, not all WD-WD collisions result in bi-modality. Assuggested by the results of 2D simulations, collisions withzero impact-parameter rarely produce bi-modality. Besidesthe impact parameter, the masses of the WDs, and in par-ticular their mass ratio, can conceivably play an importantrole in determining bi-modality. 3D simulations with the fullrange of impact parameters and WD masses are required toderive statistics of the expected velocity distributions (Kush-nir et al., in prep).Our results imply that any proposed scenario to ex-plain the majority of SNe Ia must have a considerable frac-tion of the explosions producing bi-modal velocity distribu-tions of Ni. Direct collisions of WDs is a promising chan-nel to explain the bi-modality, and the Ni distribution canbe definitively calculated with no free parameters from thecollision model. High-quality nebular-phase spectra, empha-sizing SNe with fast post-peak decline D m ( B ) > . , areneeded to quantify the occurrence rate of the doubly-peakedprofiles. Bi-model velocity distributions can sometimes mas-querade as single-peak line profiles, and for some of them,their bi-modal nature may be revealed by detecting the shiftsof line peaks with respect to the rest frame. The continuumpolarization measurements of SNe Ia at early epochs werefound to be . and the inferred maximum departure fromspherical symmetry was suggested to be . (see e.g.,Wang & Wheeler 2008; Patat et al. 2012). Whether the bi-modality of Ni suggested in the nebular-phase spectra andexpected from the collision models are consistent with thepolarization measurements is an open issue, and resolving itwill require radiation transfer calculations. By comparisonof accurate computations and comprehensive observations in
MNRAS , 1–7 (2015) i-modal Explosions of SNe Ia −10 0 10 v LOS[10 km / s] f = 0 o f = 45 o f = 90 o θ = 90 ◦ f = 135 o −10 0 10 v LOS[10 km / s] θ = 60 ◦ −10 0 10 v LOS[10 km / s] θ = 30 ◦ −10 0 10 v LOS[10 km / s] D m =1.92003gs D m =1.82007on D m =1.62005am D m =1.52004eo D m =1.42008Q D m =1.32011fe D m =1.2 Figure 5.
Upper Panels: Ejecta at late, homologously-expandingphase from a 3D simulation of a . M ⊙ - . M ⊙ WD collision withan impact parameter of 0.2. The projected velocity distributionof the Ni mass and total mass in the orbital plane are shown inthe top right and left panels. Ni mass is concentrated in two wellseparated components, which are related to the two ignition spotsin the two WDs. Bottom Left: the resulting line of sight velocitydistributions that are observed at numerous viewing angles (blue).The viewing direction is described by spherical coordinates with q the polar angle with respect to the z axis (direction of theangular momentum) and f is the azimuthal angle in the x − y plane with respect to the x axis. Bottom Right: the resultingvelocity distribution can be directly compared to the line profilesnear ˚A , which is a “clean” [CoIII] line with minor blend atthe red side. The observations are shown in velocity space, v = c ( l − ) / ˚A for several SNe shown in Fig. 3. the near future, the collision models can be unambiguouslytested as the primary channel for type Ia SNe. The rich anddetailed structure of the nebular line profiles will either bea smoking gun of the collision model or rule it out. ACKNOWLEDGEMENTS
We thank Avishay Gal-Yam, Andy Gould and the reviewerMark Phillips for helpful comments. We are grateful toBen Shappee for providing the spectral files for SN 2011feand Jeffrey Silverman for help with BSNIP. We thank theCarnegie Supernova Project, Berkeley Supernova Ia Pro-gram, CfA Supernova Group for making their data publicand the Online Supernova Spectrum Database (SUSPECT)for collecting a comprehensive set of archival SNe spectra –without these efforts, this work would have not been pos-sible. S.D. is supported by “the Strategic Priority ResearchProgram-The Emergence of Cosmological Structures” of the Chinese Academy of Sciences (Grant No. XDB09000000). D.K. gratefully acknowledges support from Martin A. and He-len Chooljian Founders’ Circle. Support for J.L.P. is in partprovided by FONDECYT through the grant 1151445 and bythe Ministry of Economy, Development, and Tourism’s Mil-lennium Science Initiative through grant IC120009, awardedto The Millennium Institute of Astrophysics, MAS. FLASHwas in part developed by the DOE NNSA-ASC OASCRFlash Center at the University of Chicago. Computationswere partly performed at PICSciE and IAS clusters. Thiswork used the Extreme Science and Engineering DiscoveryEnvironment (XSEDE), which is supported by NFS grantACI-1053575.
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Subo Dong et al. λ ( ˚ A ) f SN 1991bg (203d)SN 1999by (183d)
Figure B1.
Nebular-phase spectra for SN 1991bg (blue) and SN1999by (red) normalized in flux for comparison.
APPENDIX A: THE CONVOLUTION KERNELFOR THE BI-MODAL VELOCITYDISTRIBUTION
In Figure 4, the velocity convolution kernel to fit doubly-peaked profiles is given by dMdv
LOS (cid:181) P + r × P , where P = max − ( v LOS − v shift , ) v , , ! and P = max − ( v LOS − v shift , ) v , , ! . (A1)There are 5 free parameters: two shifts v shift , , , the twowidths v mod , , and the peak ratio of the components, r . Theshifts can be alternatively described as a velocity shift v shift = . ( v shift , + v shift , ) and velocity separation v sep = v shift , − v shift , . The best-fit parameters ( v shift , v mod , , v mod , , v sep , r ) are ( − , , , , . ) , ( , , , , . ) and ( , , , , . ) for SN 2007on, 2003gs and2005am, respectively. APPENDIX B: ON THE CHOICE OF THETEMPLATE AND SPECTRAL FEATURES FORCONVOLUTION
The direct convolution method applied in this work makesuse of multiple nebular-phase Co and Fe spectral features tostudy the underlying velocity profile of Ni in the ejecta.Among all nebular-phase spectra in our collection, the spec-tra of SN 1999by and SN 1991bg have the narrowest linewidths, which make them suitable as the templates for con-volution. The spectra of SN 1999by (183 d) and SN 1991bg(203 d) are plotted in Fig. B1. For the work presented inthe main text, we used the spectrum of SN 1999by as thetemplate due to relatively high SNR compared to that of SN1991bg.The ideal nebular spectral features to use shall havelittle blending from nearby spectral lines and exhibit goodconsistency in line shapes for different SNe. The shapes ofspectral features in the range of ∼ ˚A and ∼ ˚Aare essentially identical between the two SNe. The [CoIII]feature at ˚A is especially clean with its wings fallingclose to zero at both blue and red ends. Moreover, theoretical Figure B2.
The convolutions of the bi-modal velocity kernels(Appendix A) of the template SN 1999by (183 d) are extendedto the ∼ feature, and the convolved spectra are shown inred. The results for SN 2003gs (201 d) are shown in the upperpanel (black for the un-binned and green for the binned spec-trum to enhance display) and SN 2007on (286 d) shown in thelower panel (black). The convolved spectra are in good agree-ment with SN 2003gs, which was observed at a similar phase tothe template spectrum of 1999by, while the agreement is not goodfor SN 2007on, which was observed at a much later phase than SN1999by. For fitting each spectrum, the velocity kernel is commonto all spectral features while the normalization and baseline areallowed to vary freely for each feature. For SN 2003gs, we alsotest the convolution of the entire SN 1999by template spectrumwith the bi-modal velocity kernel (rather than fitting individualspectral regions), and the convolved spectrum shows remarkableagreement with that of SN 1999by across the entire spectrum.MNRAS , 1–7 (2015) i-modal Explosions of SNe Ia study of nebular-phase spectrum by Axelrod (1980) showsthat the ˚A [CoIII] feature is devoid of blending fromany other lines. The Fe feature at ∼ ˚A (due to theblend of [FeIII] and [FeII]) and the [CoIII] feature at ∼ ˚A (blended with [FeII]) are less clean, and they are used inthe study only to aid the line profile analysis mainly basedon the ˚A feature.In contrast, the spectra features in the range of ∼ − ˚A show significant difference in shape betweenSN 1999by and SN 1991bg, in particular for the prominentFe feature at ∼ ˚A which is frequently used in manystudies of SNe Ia nebular spectra. The ∼ ˚A featureseems to be significantly affected by complicated blends ofneighbouring lines. This makes this feature unsuitable fordirect template convolution.Nevertheless, it is instructive to examine the ∼ ˚Afeatures for the supernovae with bi-model velocity distribu-tion deduced from the ∼ ˚A features. In our sample,the nebular-phase spectra for SN 2005am do not possess theregion near ∼ ˚A. The spectrum of SN 2003gs is themore promising case to be modelled by convolution as itis taken at 201 d, similar to the phase of the template SN1999by spectrum (183 d). The upper panel of Fig. B2 showsthe convolution of the SN 1999by template spectrum withthe bi-modal velocity convolution kernel given in AppendixA (i.e., extending the convolutions shown in Figure. 4 in themain text to the ∼ ˚A feature). There is a remarkableagreement in the line shape at ∼ ˚A between the con-volved spectra and that of SN 2003gs. The nebular-phasespectrum of SN 2007on is taken at 286 d, much later thanthe template spectrum at ∼ d. In the lower panel of Fig.B2, we compare the spectra between SN 2007on taken at286 d and the convolved SN 1999by spectra (183 d), and inthis case the agreement is not good for the ˚A feature.This is probably due to differences in the complex blendin this region between the two supernovae (SN 1999by and2007on) and/or at the different phases.Despite the complexity of the ˚A feature demon-strated above, the case of SN 2003gs suggests that thereis some regularity within this spectral region that can beexplored. For SN 2003gs, we also convolve the whole SN1999by template spectrum with the bi-modal velocity ker-nel (rather than fitting individual spectral regions), and theconvolved spectrum agrees remarkably well with that of SN1999by across the entire spectrum. Observationally, a largersample of high quality nebular spectra spanning a largerrange of epochs beyond 200 d, in particular for the 1991bg-like events, would help greatly to elucidate the situation em-pirically. Further theoretical works would be needed explainthe difference in line profiles for the ˚A feature betweenSNe (such as SN 1991bg and 1999by). The velocity profilededuced from the ˚A can be helpful in the modeling en-deavours to interpret the more complex nebular lines suchas the ˚A feature. This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS000