Signatures of A Companion Star in Type Ia Supernovae
aa r X i v : . [ a s t r o - ph . S R ] A ug Accepted for publications in ApJ on 15 August 2014.
Preprint typeset using L A TEX style emulateapj v. 05/04/06
SIGNATURES OF A COMPANION STAR IN TYPE IA SUPERNOVAE
Keiichi Maeda , Masamichi Kutsuna , Toshikazu Shigeyama Accepted for publications in ApJ on 15 August 2014.
ABSTRACTWhile type Ia Supernovae (SNe Ia) have been used as precise cosmological distance indicators,their progenitor systems remain unresolved. One of the key questions is if there is a non-degeneratecompanion star at the time of a thermonuclear explosion of a white dwarf (WD). In this paper, weinvestigate if an interaction between the SN ejecta and the companion star may result in observablefootprints around the maximum brightness and thereafter, by performing multi-dimensional radiationtransfer simulations based on hydrodynamic simulations of the interaction. We find that such systemsresult in variations in various observational characteristics due to different viewing directions, whilethe predicted behaviors (redder and fainter for the companion direction) are opposite to what weresuggested by the previous study. The variations are generally modest and within observed scatters.However, the model predicts trends between some observables different from observationally derived,thus a large sample of SNe Ia with small calibration errors may be used to constrain the existenceof such a companion star. The variations in different colors in optical band passes can be mimickedby external extinctions, thus such an effect could be a source of a scatter in the peak luminosity andderived distance. After the peak, hydrogen-rich materials expelled from the companion will manifestthemselves in hydrogen lines. H α is however extremely difficult to identify. Alternatively, we find thatP β in post-maximum near-infrared spectra can potentially provide powerful diagnostics. Subject headings: supernovae: general – radiative transfer – cosmology: distance scale INTRODUCTION
Type Ia supernovae (SNe Ia) are mature standardizedcandles, and have been playing a key role in the obser-vational cosmology (Riess et al. 1998; Permutter et al.1999). The SN Ia cosmology relies on an empiricallyderived relation between the peak luminosity and lightcurve decline rate (e.g., ∆ m defined as a magnitudedecrease from the peak to 15 days after), the so-calledPhillips relation (Phillips et al. 1999). This is also com-plemented by further relations between the luminosity(or decline rate) and the intrinsic colors (most frequently B − V ), as essential in calibrating the external extinction(e.g., Folatelli et al. 2010, and references therein).However, the progenitor and explosion of SNe Ia arenot yet fully understood. There has been a long debateabout the progenitor system(s). The proposed systemsare largely divided into two categories. One is calledthe single degenerate (SD) scenario, where a white dwarf(WD) accretes materials from its binary non-degeneratecompanion star, either a red giant (RG) or a main se-quence (MS), to increase its mass to the (nearly) Chan-drasekhar limit and ignites carbon near the center (e.g.,Whelan & Iben 1973; Nomoto 1982; Hachisu et al. 1999).The companion stars could be even of different types(e.g., Wang et al. 2009; Wheeler 2012; Liu et al. 2013),but in this paper we mainly focus on RG and MS cases.The other one involves a merger of two WDs, and is Department of Astronomy, Kyoto University,Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan;[email protected] . Kavli Institute for the Physics and Mathematics of theUniverse (WPI), Todai Institutes for Advanced Study, Universityof Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan Research Center for the Early Universe, School of Science,University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,Japan called the double degenerate (DD) scenario (e.g., Iben& Tutukov 1984; Webbink 1984). The merger of WDsmay result in a prompt explosion (Pakmor et al. 2010), orelse it may resemble to the final stage of the SD scenariowith a larger accretion rate: a WD with the (nearly)Chandrasekhar mass may evolve hydrostatically towardthe central ignition (Yoon et al. 2007), or such a systemmay experience an accretion induced collapse rather thanSNe Ia (Saio & Nomoto 1985). Different scenarios willlead to different explosion mechanisms, and thus under-standing the origin of the diversity and relations in SNIa luminosity and other observational properties relieson understanding the progenitor and evolution scenarios.Different scenarios may well predict a different evolutionof SN properties as a function of the redshift, thus thisis also a critical question in the SN Ia cosmology.New development on this issue has been achieved in thelast few years from observational viewpoints. So far thestrongest constraints have been placed by direct searchesfor surviving non-degenerate companion stars. A super-nova remnant (SNR) 0509-67.5 in the Large MagellanicCloud (LMC) was found to have no point sources downto M V ∼ . α emission in the late phase(Mattila et al. 2005), while there has been no sign of H α in nearby SNe Ia so far (Mattila et al. 2005; Leonard2007; Shappee et al. 2013; Lundqvist et al. 2013).On the other hand, there are also observational indi-cations for the SD scenario for at least a part of SNe Ia.There is a strong candidate for a surviving G-type dwarfin Tycho’s SNR (Ruiz-Lapuente et al. 2004; Bedin et al.2014) (but see also Ihara et al. 2007; Kerzendorf et al.2009). The discovery of strongly interacting SNe Ia (SNeIa exploding within dense CSM) favors the SD scenariofor these SNe, specifically systems in which the compan-ion is still in the non-degenerate phase at the time of theexplosion (Hamuy et al. 2003; Aldering et al. 2006; Dildayet al. 2012). In particular, Dilday et al. (2012) discoveredevidence for the traces of nova explosions preceding theSN, which had been predicted by the SD scenario. Fromthis we expect that there are also cases where a WDwith a non-degenerate companion explodes but withoutshowing strong CSM interaction signals (Hachisu et al.2012). In sum, the issue is still controversy, and furtherstudy is required. Especially, the arguments based on SNproperties can be model dependent (e.g., see Kutsuna& Shigeyama 2013; Kutsuna 2013, for uncertainties ofthe collision-induced emission in the pre-maximum stage), and thus different ideas based on different physicalprocesses and different observational strategies are quiteuseful. In this respect, we investigate the issue of howthe maximum and post-maximum phases are affected bythe existence of a companion star – while this is the mosteasily accessible observations, indeed the model predic-tions so far are mostly restricted to the early rising phase(a few days after the explosion) or the late-time neb-ular phase (about an year after the explosion). Therehas been virtually only one study on this issue by Kasenet al. (2004). They predicted that the maximum spec-trum is generally not sensitive to a viewing angle, whilethe SN looks blue and peculiar (i.e., 1991T-like) whenviewed from the direction of the hole created by theejecta-companion interaction. There were however somelimitations in their study: (1) It has not been clarifiedif the overall (or angle-averaged) properties are affectedas compared to the non-interaction case, (2) the pre-diction was made only for the maximum spectra, thusthere were no specific features predicted for the post-maximum spectra, spectral evolution, or the multi-bandlight curves, (3) finally, the model was based on a toymodel, which might be missing some ingredients impor-tant in hydrodynamics.In this paper, we explore the maximum and post-maximum properties of SNe in the optical through near-infrared (NIR) wavelength ranges, as a result of inter-action between ejecta and a non-degenerate companionstar. In §
2, we summarize our hydrodynamic models andmethods for radiation transfer simulations. In §
3, we dis-cuss overall properties in spectra and multi-band lightcurves. In §
4, we discuss details on individual spectralfeatures, colors, and their mutual relations predicted by
TABLE 1Models
Model M ( M ⊙ ) a R (10 cm) b A (10 cm) c MS 1 0.01 0.03RGa 1 0.7 2RGb 1 1 3 a The mass of the companion star. b The radius of the companion star. c The binary separation. the simulations. In §
5, we investigate a possibility to de-tect hydrogen lines as diagnostics of the non-degeneratecompanion star. The paper is closed in § METHOD AND MODELS
Hydrodynamic Models
Our input models for the radiation transfer simulationsare taken from Kutsuna & Shigeyama (2013) (see alsoKutsuna 2013). In this section, we summarize main fea-tures of the models, and we refer Kutsuna & Shigeyama(2013) for further details of the models.These models are results of radiation hydrodynamicsimulations of the collision between the expanding SNejecta and a non-degenerate companion star. Thus, theinitial configuration is specified by a few binary param-eters, namely the type of the companion star and theseparation between the WD and the companion. Wetake three models from their simulations, named ModelsMS, RGa, and RGb. Basic features of these models aresummarized in Table 1. Model RGa represents a close bi-nary system with a RG companion. The RG has 0 . M ⊙ of the He core and 0 . M ⊙ of the convective H-rich en-velope. The separation is set to be 2 × cm. ModelRGb is the same with RGa, except for the separationbeing 3 × cm. Model MS represents a close binarysystem with a MS companion. The companion MS massis 1 M ⊙ and the separation is 3 × cm. The compo-sition in the H envelope is set as follows: 75% in H and25% in He. We ignore metal content in the hydrogenenvelope (see § ∼ . M ⊙ in all the three models.We note here that while the hydrodynamic behaviors aregenerally consistent with previous studies (e.g., Mariettaet al. 2000), the companion star in Model MS suffers froma large amount of hydrogen stripping. This is likely dueto an insufficient computational resolution (Pakmor etal. 2008), thus the amount of hydrogen in Model MSshould be regarded as being overestimated (Kutsuna &Shigeyama 2013; Kutsuna 2013). We note however thathow much hydrogen is stripped away in the WD-MS sys-tem is still under debate (Liu et al. 2012). Fig. 1.—
The ejecta structure of Model RGa. The density is scaled to the value at 10 days since the explosion (when the ejecta arealready in a homologous expansion). The companion was initially on the − y direction (i.e., toward the bottom). The viewing angle θ isdefined to be θ = 0 in the companion direction ( − y in this figure), while θ = π in the opposite direction (+ y ). The W7 model (Nomoto et al. 1984) was used for theSN ejecta model. Accidentally, in Kutsuna & Shigeyama(2013) (see also Kutsuna 2013) the innermost stableNi was counted as radioactive Ni in preparing the in-put model, thus this model has ∼ . M ⊙ of Ni, whichis larger than in the original W7 model. After map-ping onto our numerical grids for the radiation transfer,the amount of Ni is 0 . M ⊙ in our ejecta model while0 . M ⊙ in the original W7 model. In any case, sincethe exact explosion mechanism is not yet clarified and M ( Ni) is within the observationally derived range ofBranch-normal SNe Ia (Branch et al. 2006), we take this‘modified’ W7 model as our reference model. Note thatwe are mainly aiming at investigating differences betweenthe SN Ia with and without a non-degenerate compan-ion, variations from different viewing directions – thusdetails of the ‘reference’ model are not important.In Kutsuna & Shigeyama (2013), the collision be-tween the SN ejecta and the companion star was sim-ulated by two-dimensional radiation hydrodynamic sim-ulations, with a few simplifications in the radiation trans-fer scheme (e.g., gray transfer, ignoring the bound-boundtransition, a simplified γ -ray deposition scheme, flux-limited diffusion approximation). After the collision, theexpanding ejecta (affected by the impact to the compan-ion) reach the homologous expansion quickly, and thedensity structure in the homologous phase is little af-fected by radiation transfer effect. Thus we adopt thedensity and composition structures at 35 days after theexplosion as our reference, and set the ejecta structureaccording to the homologous expansion to the initial timefor the detailed radiation transfer simulations (typically10 days after the explosion). While the impact dissipatesthe kinetic energy (Kasen 2010; Kutsuna & Shigeyama2013), the resultant thermal energy is lost in a few daysmostly due to the adiabatic loss. As such, in a few dayssince the explosion, the thermal condition is determinedby the radioactive input. Thus, we neglect the thermalenergy content due to the impact in our radiation simu-lations.In addition to these ‘companion-interaction’ models,we also perform the radiation transfer simulations for theoriginal W7 model and our own 1D reference model with-out interaction. The reference 1D model is constructedas follows - we extract the radial information from ModelMS in the direction opposite to the companion, and thisradial structure is mapped into all the directions in 3Dspace. This model represents the SN ejecta model with-out the interaction. This model is used for a fair compar-ison to the interaction models, better than the originalW7 model. Reasons for this are (1) the SN ejecta modelsused for the interaction simulations have the distribution and mass of Ni slightly different from the original W7model (see above), and (2) this is computed through thesame hydrodynamic code with the interaction models,thus a possible numerical diffusion is taken into accountin this reference model in the same manner as in theinteraction models.
Radiation Transfer
We have performed radiation transfer simulations forthe input models described in § n r , n θ ) =(50 , r and θ represent radial and polar anglecoordinates. This spatial resolution is enough to resolvethe major features in the ejecta structure arising fromthe interaction (Fig. 1), and also sufficient to resolve thespectral features arising from the photon Doppler shift(i.e., the radial spatial resolution corresponds to the pho-ton Doppler shift of ∼
300 km s − finer than typical spec-tral resolution in observations). We have used a multi-dimensional/frequency/epoch radiation transfer code de-veloped by ourselves, HEIMDALL (Handling EmissionIn Multi-Dimension for spectrAL and Light curve calcu-lations) . The full details of the code are presented inAppendix A, and in this section we will provide a sum-mary of the simulation method and description specificto simulations in this paper.The code largely adopts prescriptions presented byLucy (2005), Kasen et al. (2006), and Kromer & Sim(2009). The code solves radiation transfer for densityand composition (taking into account radioactive decays)structures as a function of time given as an input model[ ρ ( ~r, t ) , X i ( ~r, t )]. The radiation field is solved with theMonte-Carlo (MC) method, where the radiation field isdiscretized into photon packets and the interactions be-tween the radiation and matter are treated as individualmicroscopic events in the comoving frame (Lucy 2005,and references therein). To solve the radiation transfer,a mixed-frame approach is adopted, where the transfor-mation from the comoving to the rest frames, and viceversa, automatically takes into account Doppler shift ofradiation with respect to the matter. This is essentialin the SN radiation transfer, since the large velocity gra-dient results in the wavelength shift of photons in thecomoving frame even without interaction, and this wave-length shift can be much larger than the typical sepa-rations of the bound-bound transitions in the frequencyspace. Temperature at each position and time [ T ( ~r, t )]is iteratively solved with radiation field (both in optical-NIR and γ -rays) within a time step, under the assump-tion of radiative equilibrium. Ionization and level pop-ulations are computed under the assumption of LocalThermodynamic Equilibrium (LTE). The new tempera-ture is then used for an initial guess of the temperaturein the next time step, and the radiation field at the endof a given time step is used as the initial radiation fieldin the beginning of the next time step.At the beginning of the simulation, MC packets repre-senting γ − rays are created. They are assigned with thefrequency, energy, spatial position, and emission epoch,following the radioactive decay chain of Ni → Co → Fe. The transfer of γ -rays are then followed with pre-scriptions given by Maeda (2006). The interactions in-clude Compton scattering, photoelectric absorption, andpair creation. As a result, the energy deposition rate by γ -rays is obtained as a function of position and time. To-gether with the positron energy input, which is assumedto take place in situ , the heating/creation rate of ‘optical’(or thermal) photons is obtained.Using the energy deposition rate obtained throughthe γ -ray transfer, thermal photon packets are created.These are then followed by the MC simulation as de-scribed above. For the opacity to thermal (UV throughNIR) photons, we adopt a standard set of the opaci-ties largely used in the radiation transfer simulations inthe expanding SN ejecta – electron scattering, free-free,bound-free, and bound-bound transitions. In this paper,we adopt the expansion opacity prescription and two-level approximation for the discrete transitions. Thisintroduces one parameter in dealing with the discretetransitions called thermalization parameter ǫ . We adopt ǫ = 0 . ∼ in total) to converge thermal condi-tions in ejecta. Then adopting this thermal structure,we have performed the ‘final’ MC transfer with a largernumber of photon packets ( ∼ ) to obtain a sufficientlyhigh Signal-to-Noise ratio in the resulting angle and time-dependent spectra. This large number of photon pack-ets allows us to extract smooth spectra with high S/N -namely, on average the number of photons in each time-wavelength-angle bin is ∼ > LIGHT CURVES AND SPECTRA
Hereafter, we present results of our simulations. Wefrequently comment on the viewing angle to an observer.In the following sections, we denote the viewing angle by θ . The viewing angle θ is defined to be a polar angle asmeasured from the direction toward the companions star(or the hole). Namely, θ = 0 for an observer viewing theSN from the companion direction while θ = π for one inthe opposite direction.Figures 2–4 show synthetic multi-band light curves ofModels RGa, RGb, MS, respectively. Those for our ref-erence non-interaction model are also shown for compar- ison. Despite the large asymmetry in the ejecta struc-ture (Fig. 1), the light curves of the interaction modelsare found to be very similar to those without interac-tion. Difference between different companion types iseven smaller and there are virtually no difference visibledown to the resolution in our simulations. We note thatthe deviation of the non-interaction model light curve inthe J -band from the interaction models is presumably anumerical artifact, as well as a spiky feature in the lightcurves of Model RGa around 30 days. We note that thisspiky feature (apparently non-converged temperature atthis epoch) would not affect the later evolution, since wesolve the temperature convergence at every time step.A close inspection shows that around the peak the non-interaction model is almost identical to the interactionmodel viewed at θ ∼ π (opposite to a companion or ahole). In the later phase, the non-interaction model isfainter than the angle-averaged mean light curves fromthe interaction model. This is consistent with the expec-tation – early on the non-interacting side should look likea totally non-interaction model as one would not see theother side. Later on as the photon diffuses out one willeventually see effects of the interaction, and the bolomet-ric luminosity should eventually follow the γ -ray deposi-tion rate as the ejecta become transparent to optical pho-tons. At this moment, the bolometric luminosity is ap-proximated by a simple gamma-ray deposition rate (see,e.g., Maeda et al. 2003), which is given as L ∝ M /E K (where L is the bolometric luminosity, M ej and E K arethe ejecta mass and the kinetic energy, respectively).Here, while E K should be unaffected by the interactionas the total energy is conserved, the ejecta mass is largerfor the interaction case than the non-interaction case be-cause of the addition of the envelope mass stripped fromthe companion. Namely, there are a larger amount ofmaterials in the interaction case to absorb more γ -raysthan the original ejecta without interaction. Naively, onewould expect that the γ -ray deposition efficiency wouldbe larger in the interaction model ( M ej ∼ . M ⊙ ) thanthe non-interacting model ( ∼ . M ⊙ ) by ∼
60% if thedeposition rate is simply scaled by the ejecta mass. Thiswould lead to the late-time bolometric luminosity of theinteraction model larger than the original by ∼ . ∼ . − . γ -rays than in a sphericalmodel with the same ejecta mass.The variation arising from different viewing directionsis at the level of 0.1 mag, and an SN looks generallyfainter for directions closer to the companion/hole ( θ ∼
0) in all the bands. As we have found little difference fordifferent models, in the rest of the paper we mainly focuson Model RGa.Figures 5 and 6 show the synthetic spectral time se-quence in the optical wavelength and in the NIR wave-length, respectively. As expected from the light curves,the interaction with the companion is found not to createany dramatic effects in the spectra as well. Namely, ex-istence of a close binary non-degenerate companion doesnot leave detectable features in overall spectra around
Fig. 2.—
Simulated multi-band light curves for Model RGa. The color coordinates indicate the light curves from different viewingdirections (red for θ = 0 and blue for θ = π ). The reference model curve is shown by the black-solid curve. The variation due to differentviewing directions is modest, at 0.1 magnitude level in all the bands. Also, it is fainter if viewed from the companion direction ( θ = 0). Fig. 3.—
Simulated multi-band light curves for Model RGb.
Fig. 4.—
Simulated multi-band light curves for Model MS.
Fig. 5.—
Simulated spectra in optical wavelengths for Model RGa. The color coordinates indicate the spectra from different viewingdirections (red for θ = 0 and blue for θ = π ). When viewed from the companion (‘hole’) direction ( θ = 0), the spectra are redder, with thesmaller flux especially in the blue, than viewed from the opposite-side observer ( θ = π ). The difference is at a moderate level so that thespectra would not be classified as peculiar. Fig. 6.—
Simulated spectra in NIR wavelengths for Model RGa. the maximum-light and thereafter (i.e., 10 −
80 days sincethe explosion), and, in other words, this does not conflictwith the observed uniformity of SN Ia spectra in theseepochs.Still, there is a difference. When viewed from the com-panion (‘hole’) direction ( θ = 0), the spectra are red-der, with the smaller flux especially in the blue, thanviewed from the opposite-side observer ( θ = π ). The dif-ference is at a moderate level so that the spectra wouldnot be classified as peculiar and this SN would be clas-sified into the same class irrespective of the viewing di-rections. Rather, this would create diversity, especiallyin the intrinsic color, within the same classification.The temperature and ionization structures are shownin Figures 7 and 8, respectively. Overall, temperatureis lower on the side of the companion, due to a smalleramount of the heating source ( Ni) and also due to asmaller amount of absorbing matter (heavy elements, es-pecially Fe-peak elements) on this side than the others.We note that our results are qualitatively different fromthose found by Kasen et al. (2004), who predicted thatthe SN is bluer and blighter (especially in the shorter wavelengths) when viewed at θ ∼
0. The situation thatan observer at θ ∼ Ni-rich regionwas a main cause of the predicted behavior by Kasen etal. (2004). Kasen et al. (2004) performed a snap-shotspectral synthesis for a maximum-spectrum based on atoy model. In their model, they mimicked the outcomeof the interaction as the SN ejecta with a hole repre-sented by a (nearly) constant opening angle. We notethat their model underestimates the amount of mate-rial in the ‘hole’ – a large amount of the SN ejecta, ei-ther C+O or Si-rich layer, fill up the hole left by theinteraction, and also the H-rich companion materials arenaturally filling the hole as well. Thus the ‘hole’ is notreally a vacuum. The existence of these materials doesnot allow the photosphere to quickly recede to the bot-tom of the hole (i.e., Ni-rich central region). Evenwith only the H-rich envelope from the companion in-truding into this region, electron scattering can becomesignificant (especially for the RG case) to clip the pho-tosphere at a relatively high velocity – with ∼ . M ⊙ of fully ionized hydrogen materials (with 70% in massfraction) confined within a sphere below 1,000 km s − , Fig. 7.—
Temperature distribution of model RGa. Also shown here are the U-band (thick) and R-band (thin) photosphere positions (asdefined by τ = 2 / θ = 0 (red), θ = π/ θ = π (blue). Fig. 8.—
Ionization Structure for Model RGa. Only the region where either hydrogen or the sum of Ni+Co+Fe exceeds 0.1 in massfraction is shown. Also shown are the positions of the U -band (thick) and R -band photospheres (see the caption of Figure 7). the electron scattering optical depth is estimated to be τ ∼ ,
700 ( t/
20 days) − . Thus, this will hide Ni-richregion from the line-of-sight to an observer at θ ∼ γ -rays and optical photons are absorbed efficiently due tothe large density.In our situation based on the hydrodynamic model,the temperature at the photosphere at θ ∼ θ ∼ π , since the companion direction( θ ∼
0) is blocked by the Ni-free materials. This direc-tion lacks Fe-peak elements, accordingly the photosphereat the U -band is at a low velocity. The position of thephotospheres at the R -band is not extremely sensitive tothe viewing direction, supporting the interpretation thatthe main difference in the photosphere in the U -band is caused by the different amount of Fe-peak elements thatare sources of opacity especially in blue bands. The U -band photosphere does not quickly recede in the velocityspace as compared to the R -band, for observers viewingfrom any directions, that reflects the increasing opacityin the blue at the later epochs due to recombination ofFe-peak elements (Fig. 8).Figure 9 shows evolution of the spectral region aroundSi II 6355. Around the peak luminosity (i.e., ∼
15 dayssince the explosion), the absorption minimum (and theemission peak) is at longer wavelength (i.e., lower ve-locity) for observers at smaller θ , as is consistent withthe result by Kasen et al. (2004). We find that the pre-dicted temporal evolution is also different for differentviewing direction (see also Kutsuna & Shigeyama 2013;Kutsuna 2013). Indeed, at ∼
10 days after the explosion(i.e., about a week before the B -band maximum), the Fig. 9.—
Si II 6355 in the simulated spectra. The color coordi-nates indicate the spectra as viewed from different viewing direc-tions (red for θ = 0 and blue for θ = π ). Initially at ∼
10 days theSi II profile is similar for observers at any directions, then observersat smaller θ (toward the companion direction) will observe the SiII at progressively lower velocity than in the opposite direction. line profiles are not sensitively dependent on the viewingdirection. After that, the line velocity decreases morequickly for an observer at θ ∼
0, thus leading to progres-sively lower velocity for this direction. This temporal be-havior is also different from that predicted by Kutsuna& Shigeyama (2013), who predicted the lower velocity ofthe Si II for θ ∼ θ ∼ π already well before themaximum. Figure 10 demonstrates how the line profilesare different for different viewing directions at 16.8 and38.5 days after the explosion. At 38.5 days, it is not easyto identify the Si II in the spectra. We note that theline profile at ∼ , − , θ ∼ θ ∼ π . This wavelength is influenced by the ma-terial moving at the velocity of ∼ , − ,
000 km s − .Figure 10 (right panel) shows that Si II 6355 affects thiswavelength range differently for different viewing direc-tions.Figure 11 shows the Sobolev optical depth of Si II to-gether with the R -band photosphere. Around the peakluminosity, the peak in the opacity distribution is on av-erage at lower line-of-sight velocity for θ = 0, because thehighest velocity materials are missing in this direction.At the epoch of the peak luminosity (at day 16.8), thisparticular optical depth is larger at θ = 0, as the lowertemperature there prefers Si II over Si III. As time goesby, the temperature decrease induces the recombinationof Si II to Si I. The ejecta on the companion side ( θ ∼ θ ∼ π ) still main-tains sufficient amount of Si II to keep its line velocity ashigh as ∼ ,
000 km s − . As a result, at this epoch theflux around 6,200˚A is suppressed for θ ∼ π but not for θ ∼
0. In principle, this small but varying feature in SiII 6355 could be a powerful signature of the presence ofa companion star, and this is further discussed in § SPECTRAL FEATURES AND COLOR In §
3, we showed that the overall light curve and spec-tral behaviors are not much affected by the existence of
Fig. 10.—
The simulated spectra around Si II 6355. The colorcoordinates indicate the prediction for different viewing directions(red for θ = 0 and blue for θ = π ). In this figure, the spectrafor observers at different directions are added with an additionaloffset ( θ = 0 to π , from bottom to top). In he left two panels, thecolor curves are the model curves with hydrogen, while the graycurves are without hydrogen. The color curves shown in the rightpanel is identical to the one in the middle (i.e., model RGa at day38.5 since the explosion) while the gray curves are without silicon.H α is observationally not detectable in both epochs. The viewingangle dependence is small around the B -band maximum, while thevariation arising from Si II 6355 becomes visible in the later epoch. a non-degenerate companion star. Still, we find a smallvariation for different viewing directions, and in this sec-tion we investigate more details on this issue.Figure 12 shows the variations in the colors and the V -band magnitude, and their relations to the B − V colorobtained for Model RGa. The variation in each quantityis at the level of 0.1 mag or even smaller, thus such aneffect is difficult to see in current observations, and itis practically impossible to disentangle this effect fromother possible sources of the scatter. Relations betweendifferent observables, however, have interesting implica-tions. Within the optical range, the relations betweendifferent colors are indeed similar to the effect of exter-nal extinction. This is also true for relations betweenthe color and absolute magnitude. This means that theviewing angle variation can mimic the external extinc-tion, and could be a part of the origins of the scatter inthe luminosity calibration (see, e.g., Maeda et al. 2011).This degeneracy could be solved by adding the NIR in-formation because the optical-NIR colors are found notto follow the extinction vector.The possible effect in the extinction estimate is demon-strated in Figure 13. Here, we show how the intrinsiccolor dispersion predicted for model RGa can be mim-icked by the external extinction. We hypothesized here(while we know it is not the case) that the color vari-ation here is entirely due to a different amount of theextinction, and convolve the external extinction associ-ated with the model B − V color assuming R V = 2 . . § B − V color, and the one between0 Fig. 11.—
Sobolev optical depth of Si II 6355 for Model RGa. Shown here is the R-band photosphere ( τ = 2 /
3) for an observer at θ = 0(red), θ = π/ θ = π (blue). Fig. 12.—
The predicted relations in photometric properties for Model RGa. The color coordinates indicate the prediction for differentviewing directions (red for θ = 0 and blue for θ = π ). The external-extinction vector is shown for R V = 3 . .
7. The variation in eachquantity is at most at the level of 0.1 magnitude. The trend seen in the optical properties is similar to the effect of external extinction,while it is not the case in NIR.
Fig. 13.—
A demonstration of how the intrinsic color differenceof Model RGa as arising from different viewing directions can bemimicked by the external extinction. The original spectra for var-ious viewing directions are shown in gray, while the ones correctedfor the ‘hypothesized’ extinction are shown in colors. the velocity and the velocity gradients are different thanthe observationally found relation. It has been reportedthat the absorption velocity around the maximum is cor-related with B − V in a way that the higher velocity SNehave redder intrinsic color (Foley & Kasen 2011; Maedaet al. 2011; Blondin et al. 2012). Figure 14 shows thepredicted relation between the Si II velocity and B − V color, overlapped with the observed trend from Blondinet al. (2012) but added an offset of 0.1 mag. Note thatobservationally the zero-point in the intrinsic color is de-pendent on the determination of the reddening and alsoon the sample since the color is also dependent on ∆ m ,and theoretically the absolute scale can be more sensi-tive to numerical details than the ‘relative’ values (seeAppendix B). Here we are interested in the trend anddiversity. Figure 14 shows that the predicted trend isdifferent from the observed trend, while the variation isstill within the 1 σ scatter of the observationally derivedrelation.There is also an observationally found relation betweenthe velocity and ‘velocity gradient’. The velocity gradi-ent is a measure of how quickly the absorption velocitydecreases as a function of time (Benetti et al. 2004, 2005).The velocity gradient is larger (i.e., the velocity decreasesmore quickly) for SNe with larger velocity at maximum(e.g., Silverman et al. 2012). Figure 14 shows the rela-tion predicted for Model RGa as compared to the obser-vationally derived relation from Silverman et al. (2012).Observationally the velocity gradient is defined by a lin-ear or higher order fit to a light curve just after the B -band maximum (Benetti et al. 2004, 2005; Blondin et al.2012). The absolute values of the velocity gradient forspecific objects are dependent on the time interval for thefit as well as the fitting function while overall tendency oflarger gradient for higher-velocity SNe is not sensitive to1 Fig. 14.—
The predicted relations in spectroscopic properties for Model RGa. The color coordinates indicate the prediction for differentviewing directions (red for θ = 0 and blue for θ = π ). The solid lines are for the observationally derived relations with the 1 σ statisticalerrors for the velocity–color (with an additional offset; Blondin et al. 2012, : B12) and for the velocity–velocity gradient (Silverman etal. 2012, : S12). The velocity gradient in the model is defined as the gradient in two epochs, where the first epoch is set at the B -bandmaximum and the second epoch is set either at 4 days or 9 days since the B -band maximum. The relations due to the viewing anglediversity do not follow the observed relations. The expected diversities are within the observed scatters at 1 − σ level. these choices (e.g., Blondin et al. 2012). Here, the veloc-ity gradient in the model is defined as the gradient in twoepochs for the sake of simplicity, where the first epoch isset at the B -band maximum. To check the robustness ofthe result, we vary the second epoch, and it is set eitherat 4 days or 9 days since the B -band maximum in Figure14. While we are forced to adopt the second epoch notvery late (as the model does not reproduce the observedSi II profile around day 25 and thereafter; see AppendixA), we see that our results are not affected much by thedefinition of the velocity gradient (see Figure 14).The velocity predicted for Model RGa is as high asthose of the high velocity (or high velocity gradient) SNe,and the dispersion arising from different viewing anglealone does not explain the observed range of the Si IIvelocity (i.e., 10 , − ,
000 km s − for the low veloc-ity SNe, and reaching to ∼ ,
000 km s − for the highvelocity SNe). Namely, the viewing angle effect arisingfrom the asymmetry introduced by the ejecta-companioninteraction is not a main cause of the diversity in the ve-locity (see, e.g., Maeda et al. 2010, for a possible origin ofthe diversity arising from the asymmetry in the explosionitself). Moreover, the viewing angle effect here predictsthe trend different than the observed one (Fig. 14). Assuch, the observed relation can, in principle, be used as aconstraint on the existence of a non-degenerate compan-ion star. The dispersion predicted for model RGa indeedexceeds the nominal 1 σ error in the observed relation.There are, however, quite a number of outliers in thisrelation (e.g., see Fig. 6 of Silverman et al. 2012), thusthe present sample does not strongly reject the existenceof a RG companion for a majority of SNe Ia. In any case,this relation is potentially a strong diagnostics to limitthe fraction of SNe Ia with a non-degenerate companionstar.In the above arguments, we have dealt with the smalldifferences, due to various viewing directions, at 0.1 mag-nitude level. A question is if our simulations are accurateenough down to this level. In Appendix B, we discuss thisnumerical accuracy issue in details with test calculations,and conclude that the results are not numerical artifacts. HYDROGEN LINES
The hydrogen-rich matter stripped off from the com-panion star, being embedded in the SN ejecta, has beenregarded to be a key to probing (or rejecting) the exis-tence of a non-degenerate companion star at the time ofthe SN Ia explosion. A large fraction of the stripped-offhydrogen is embedded near the center of the SN ejectaat low velocities which are only visible in the late phaseswhen the SN ejecta become fully transparent. As such,searching for the H α emission in late-time spectra hasbeen suggested and performed for a few SNe Ia (Mat-tila et al. 2005; Leonard 2007; Lundqvist et al. 2013;Shappee et al. 2013). A smaller amount of hydrogenare also distributed at high velocities (Marietta et al.2000), which could be in principle probed by more easilyaccessible maximum and post-maximum spectra. How-ever, the latter issue has not been quantitatively inves-tigated in the past. Lentz et al. (2002) investigated ifthe H-rich materials mixed into the high velocity part ofthe SN ejecta ( ∼ > ,
000 km s − ) could be detected inthe pre-maximum spectra, through 1D radiation trans-fer simulations. They concluded that the signatures areexpected to be stronger for the earlier phases, but the sig-nals are generally weak. Recently, Kutsuna & Shigeyama(2013) suggested that one may see H α even just after themaximum light (i.e., ∼ >
10 days after the explosion) forthe interaction with a non-degenerate companion, basedon the hydrodynamic simulation and simplified radiationtransfer. Here, we investigate this issue – based on thesame hydrodynamic simulation models with Kutsuna &Shigeyama (2013) and detailed radiation transfer, we in-vestigate if the existence of hydrogen is visible in themaximum and post-maximum spectra (i.e., up to abouta month since the B -band maximum).Figures 15–17 show the ratio of the synthetic spectrafor the same model(s) but with and without hydrogen in-cluded in the radiation transfer. To create the hydrogen-free reference spectra, we performed the same radiationtransfer simulation based on the temperature structureobtained through the original calculations, but settingthe bound-bound and bound-free hydrogen opacities zeroby hand. In Figures 15-17, the model results for differentdirections are shown by different colors (red for θ = 0,and blue for θ = π ). Since our spectra are extracted2 Fig. 15.—
The ratio of the spectral flux with and without hydrogen lines for Model RGa. The color coordinates indicate the modelsfor different viewing directions (red for θ = 0 and blue for θ = π ). The lines on the bottom show the rest wavelength positions of Balmerseries ( α , β , γ ) and Paschen series ( α , β , γ ). In this plot, the simulated spectra are averaged in three time bins and three wavelength binsto increase the Signal-to-Noise ratio. In later phases, the variations in P α and P β arising from different viewing directions clearly exceedthe MC noise. Fig. 16.—
The ratio of the spectral flux with and without hydrogen lines for Model RGb. Fig. 17.—
The ratio of the spectral flux with and without hydrogen lines for Model MS. Fig. 18.— P β in the simulated spectra. In this figure, the spectrafor an observer at different directions are added with an additionaloffset ( θ = 0 to θ = π , from bottom to top). The color curves arethe model curves with hydrogen, while the gray curves are withouthydrogen. P β is present in these epochs, showing variations in theflux and profile for different viewing directions. from the MC-simulation by binning the emerging photonpackets, we suffer from the MC noise. Especially, whenthe flux is smaller then the noise level becomes larger.For example, this is seen in the larger noise level at thelonger wavelength especially apparent for the maximumspectra, or at the wavelength corresponding to the CaII NIR absorption especially in the later-phase spectra.If there is a ‘real’ feature produced by hydrogen abovethe level of the MC noise, the ratio at the correspondingwavelength should show imbalance with respect to unity(or zero in the logarithmic scale), namely the emissionappears as the increase in the ratio (above unity) whilethe absorption appears as the decrement in the ratio (be-low unity). As such, a P-Cygni profile in the flux spectrashould also appear in the same way in the ‘ratio spectra’shown in Figures 15–17.H α is seen in these figures (above the MC noise). Ir-respective of the epoch, the ratio at the H α is at 10%level (but note that this is probably an overestimate forModel MS). Namely, if the observed Signal-to-Noise ra-tio exceeds this level and if one knows the hydrogen-freereference spectrum a priori , one may detect hydrogenthrough H α . We thus confirmed the suggestion by Kut-suna & Shigeyama (2013), while the latter condition wasnot discussed by them. We investigate this issue later inthis section.We find from these figures that the hydrogen lines inNIR could provide potentially much stronger signals thanH α . Around the B -band maximum brightness, strengthsof P α and P β are below the noise level of our MC sim-ulation in the corresponding wavelength ( ∼
20% level).As time goes by the signal becomes stronger and exceedsthe MC noise. At ∼
25 days after the explosion (i.e.,about a week after the B -band maximum), the P α andP β signals are about 25% level. Later on at ∼
39 and58 days after the explosion ( ∼ B -band maximum), the signals reach 50-60% level. Thisbehavior is seen in all of our models (while it might beoverestimated for the MS model). Note that while theepochs mentioned above are about one month since the B -band maximum, these are around (or just a bit later than) the NIR second-maximum date, i.e., much earlierthan the previously proposed test for H α emission in thelate phase (about an year since the explosion).H α is indeed almost invisible to eyes at this flux level(Figure 10). In this sense, P β is more promising. Fig-ure 18 shows that this feature could be visible even byinspections by eyes. Note that the flux-axis scale in thisfigure is reduced – for example, Figure 6 shows P β moreclearly. Figure 18 shows that the appearance of this fea-ture is also dependent on the viewing angle. At ∼
40 days(around the NIR maximum), the feature is only visiblefor θ = 0. Later on at ∼
60 days the feature is visiblefor all θ , with the larger flux and broader feature to theblue (i.e., larger velocity) for θ = 0.Figures 19 and 20 show the distribution of the Sobolevoptical depths of H α and P β , respectively. Also shownare the underlying photosphere at R and J -bands, re-spectively. Generally, the underlying optical depth issmaller in NIR than in optical, thus the H-rich region ap-pears above the photosphere earlier in NIR. At day 38.5,hydrogen above the photosphere is mostly recombined(Fig. 8), and this explains why P β becomes strongeraround this epoch. At this epoch, the photosphere asseen from an observer with the viewing angle of θ = π does not reach this neutral hydrogen region, thus at day38.5, P β is visible only for an observer at θ = 0. TheSobolev optical depth is higher for H α than P β and thusthe self-absorption is more important in H α . This is onereason why P β is stronger than H α .As we see, in principle one could see hydrogen lines,especially P α and P β , if one knows the hydrogen-free ref-erence spectra a priori . This is observationally a big issuein two respects. (1) One does not know the fraction of SNejecta with hydrogen (through the interaction) – this iswhat we aim to investigate. (2) When a given wavelengthregion is contaminated by other lines, there could be di-versity (by some mechanisms) that is not directly relatedto the existence of hydrogen. For the point (2), even as-suming that the observed SNe Ia are all represented bya series of models presented in this paper, the contam-inating lines could show diversity arising from differentviewing directions – for example, H α is contaminatedby the much stronger Si line, and this line does showthe diversity according to the viewing direction withinour model (Figures 9 & 10). This happens irrespectiveof the existence of hydrogen, even though in this modelboth the asymmetry and hydrogen contamination havethe same origin.If there is a model that would perfectly describe (fit)the SN spectra one could rely on such model spectra.Unfortunately, this is not the case – NLTE effects aresuggested to become strong in post-maximum spectralformation especially in NIR, and so far there are no very‘good’ model spectra for it (e.g., Gall et al. 2012) –while the models generally reproduce main features (in-cluding our synthetic spectra), the flux ratios of differentlines can be strongly dependent on the NLTE treatment.Thus, to confirm the hydrogen lines observationally, onefirst has to define a ‘reference’ (or template) spectrumfrom observed samples, and then one has to see the di-versity of individual SN spectra as compared to the ref-erence spectrum. The reference spectrum can be createdeither from the entire sample or a sub-set of the sample.For example, if one creates the reference spectrum from5 Fig. 19.—
Sobolev optical depth of H α for Model RGa. Also shown here are the R -band photospheres for observers at different directions( τ = 2 /
3; see the caption of Figure 7).
Fig. 20.—
Sobolev optical depth of P β for Model RGa. Also shown here are the J -band photospheres for observers at different directions( τ = 2 /
3; see the caption of Figure 7).
SNe with similar peak luminosity (or light curve width),that would effectively reduce a diversity related to thepeak luminosity (i.e., SN Ia spectral features generallycorrelate with the peak luminosity: Nugent et al. 1995).The remaining diversity may come from a combinationof different effects (see, e.g., Maeda et al. 2010, 2011) –a strategy to detect hydrogen would be to see if there isa diversity associated with the wavelengths of the hydro-gen lines.As an experiment, we here follow the same proce-dure as mentioned above to investigate if the hydro-gen lines could be detectable based on Model RGa be-yond other sources of diversities. We compare two cases,with and without hydrogen. In each case, we adopt anangle-averaged spectrum as a reference spectrum (at eachepoch), and then compute a ratio of a spectrum viewedfrom a specific direction and the angle-averaged refer-ence spectrum across the wavelength. This correspondsto the ratio of ‘individual’ SN spectrum and the ‘refer-ence’ spectrum, but using the model spectra. Figure 21shows results of this experiment for Model RGa with hy-drogen (left panels) and without hydrogen (right panels),around the H α at day 38.5. Figure 22 shows the samebut for P β .In general the optical range shows a larger diversitythan NIR, due to the viewing direction difference. Thishighlights more complicated spectrum formation in op-tical (due to many overlapping lines) than in NIR atleast at this epoch. This makes the identification of H α quite difficult – Namely, the ‘diversity’ patterns with andwithout hydrogen are virtually identical, thus practicallyone cannot identify H α . On the other hand, P β shows a clear feature at its characteristics wavelength for thehydrogen-contaminated model spectra – the model pre-dicts that either absorption or emission could be seen atthe wavelength of P β depending on the viewing direction.This feature is not seen in the case of the hydrogen-freemodel spectra. Thus, our model predicts that the diver-sity at the wavelength of P β should arise for model RGa,and for individual SNe it can either appear as emissionor absorption depending on the viewing direction, whencompared with a mean template spectrum.NIR spectra at this epoch are still rare, but the quickdevelopment of NIR detectors and increasing opportuni-ties in NIR observations (Marion et al. 2006, 2009) makeit appealing, potentially powerful, diagnostics. For ex-ample, the magnitude of SN Ia 2003du was about R ∼ R ∼
22 at one yearafter the explosion. It was J ∼
16 at ∼ B -band maximum (Stanishev et al. 2007), and musthave been even brighter in J at about one month af-ter the B -maximum if we apply the Hsiao template lightcurve (Hsiao et al. 2007) (see also Appendix A). Thus,observational requirements for the ‘post-maximum’ P β diagnostics may well be less tight than that for the ‘late-time’ H α diagnostics, in terms of the baseline sensitivityin different band passes. CONCLUSIONS AND DISCUSSION
In this paper, we have investigated possible observa-tional signatures of a non-degenerate companion star inthe progenitor system of SNe Ia. Based on hydrody-namic simulations of the impact between expanding SNejecta and the companion star, we have performed de-tailed radiation transfer simulations. We have focused6
Fig. 21.—
The residual of the synthetic spectra for Model RGa after being divided by the mean spectrum, shown for the optical rangecovering Si II 6355 and H α . The left panels are for the original model, while the right panels are for spectra artificially removing thehydrogen transitions. The panels are divided into four according to the viewing direction ( θ = 0 to θ = π , from bottom to top). Thesynthetic spectra are binned within 3 time bins, but no additional binning is performed in the wavelength and viewing angle directions.The left and right panels are almost identical, showing that it is not observationally feasible to detect H α at this epoch. Alternatively, thesignature of overall ejecta asymmetry could be probed. on the maximum and post-maximum phases (coveringthe first two months) – While the best data set of SNe Iais available for these phases, the issue has been so far in-vestigated mostly for the very early phase and late phasefor which the observations are more challenging.Compared to a previous study by Kasen et al. (2004),our approach is different in the following aspects: (1)we start with the hydrodynamic simulations rather thanassuming a simplified kinetic model, (2) we follow thetemporal evolution, (3) we analyze not only the opti-cal properties but also NIR, with an extended analysisof various observationally testable features. Because ofthe differences, our model predictions indeed differ fromthose by Kasen et al. (2004) even qualitatively.We have found that the overall properties, especiallyphotometric ones, are not much different between thesystems with and without a companion star, even withthe RG companion. Interestingly, we find in our simula-tions that the light curves seen from the companion sideare not bluer and brighter as suggested in the previousstudy (Kasen et al. 2004), but we predict the opposite.The difference is however generally at a 0.1 magnitude level. Therefore, the existence of a non-degenerate com-panion star is not ruled out for individual SNe, by thecurrently available maximum and post-maximum dataof SNe Ia. The model predicts a diversity arising fromdifferent viewing angle (at the level of 0.1 magnitude),showing some correlations between different colors andmagnitudes. In the optical wavelength band, interest-ingly the expected relations are similar to those intro-duced by an external extinction, whose nature is yet tobe clarified. This indicates that this effect, if the progen-itor with a non-degenerate companion explains a goodfraction of observed SNe Ia, can introduce systematic er-rors at the level of 0.1 magnitude in using SNe Ia as stan-dard candles. We have found that the NIR properties donot follow the external-extinction properties, highlight-ing the importance of NIR observations in developing theSN Ia luminosity/distance calibration better than the 0.1magnitude level.The difference between models with and without acompanion is bigger in spectroscopic features than inphotometric features. We predict that the Si II 6355 ve-locity (and other lines) depends on the viewing direction.7 Fig. 22.—
The residual of the synthetic spectra for Model RGa after being divided by the mean spectrum, shown for the NIR rangecovering P β . The left panels are for the original model, while the right panels are for spectra artificially removing the hydrogen transitions.See the caption of Figure 21. The left panel shows the variation for different viewing directions due to P β , while the model withouthydrogen results in virtually no variation (right panel). As such, a possible diversity in the J -band spectra in many SN samples could beused to investigate the presence of hydrogen and a companion. At the maximum brightness, the Si II 6355 velocity issmaller for an observer viewing from the companion side( θ = 0), as is consistent with the result by Kasen et al.(2004). The temporal evolution of the feature shows aneven more interesting behavior. Before the maximum,the Si II feature does not show a strong viewing angledependence. Later on, since just before the maximumdate, the Si II velocity starts decreasing quickly for an ob-server viewing from the companion side. The predictedrelations between the velocity and the optical ( B − V )color, as well as the velocity and the ‘velocity gradient’are found to be different than those inferred from the ob-servations. Thus, the angle variation on the companion-induced asymmetry cannot be a source of these relations.Indeed, we do not try to explain the relations but alterna-tively suggest to use these relations to constrain the exis-tence of a non-degenerate companion star. There shouldbe other mechanisms (e.g., the mass of Ni and otherfactors) that introduce the observed diversity/relation,thus the variation due to the viewing angle based on thepresent model should be regarded to be the ‘minimum’ variation. Therefore the predicted variations should notbe larger than the observational variations. If this is vi-olated, it means that such a model does not account fora bulk of the observed SNe Ia. Comparing the predictedvariations with the observed scatters in the velocity–colorand velocity – velocity gradient relations, we have foundthat the model is marginally consistent with the currentobservations. In the future, observations with better cal-ibrations (especially in photometry) is expected to placea strong constraint on the existence of a non-degeneratecompanion star from this aspect.We have also investigated if there is a chance to probethe non-degenerate companion through the hydrogen fea-tures in maximum and post-maximum spectra. We con-firmed the expectation (while not quantitatively shownbefore) that H α is difficult (practically impossible) to de-tect in these phases. Alternatively, we suggest that P β can potentially be used as a diagnostics around the NIRmaximum phase (or slightly later). We have shown thatdetecting this feature is observationally feasible, and canbe even easier than the search of the H α emission in the8 Fig. 23.—
Observed J -band spectra (bottom) of SNe 1999ee (black) and 2005cf (green) at ∼
30 days since the B -band maximum (left)and ∼
40 days (right). The flux of SN 2005cf is brought to the hypothesized distance of 10 Mpc, assuming the original distance of 28 Mpc.The flux of SN 1999ee is scaled to roughly fit the flux of 2005cf at a similar epoch in the J -band. The P β line in the synthetic spectra isapproximated by a Gaussian profile, extracted from Model RGa (bottom, red for θ = 0 and blue for θ = π ). This is added to the spectrumof SN 1999ee for an observer at θ = 0 (middle) and at θ = π (top). In doing this, two cases are shown (see the labels in the figure) – onewith the original prediction and the one where the synthetic P β flux is multiplied by 0.3 (roughly corresponding to the H-rich envelopemass scaled down to 0 . M ⊙ ). This kind of analysis could be used to constrain the amount of stripped-off hydrogen, thus the existence ofa companion star (see main text for details). later phases.As a demonstration of the observational feasibility, inFigure 23 we show a comparison between NIR spectraof two SNe Ia, 1999ee (Hamuy et al. 2002) and 2005cf(Gall et al. 2012), which have the published NIR dataat similar epochs. The data were obtained through theWeizmann interactive supernova data repository (Yaron& Gal-Yam 2012). The comparison shows that these twoSNe are extremely similar in NIR, and this similarity inNIR provides an ideal situation to investigate particularfeatures (in this case P β ) since defining the ‘continuum’(or template) is relatively straightforward. There is in-deed a hint of the developing difference around P β be-tween the two SNe in the later phase ( ∼
40 days sincethe B -band maximum) while one has to carefully checkthe data reduction process to confirm it is not an arti-fact. This is beyond the scope of this paper, and we willexamine a sample of NIR data in a separate paper (K.Maeda, in prep.).Here as a demonstration we use the spectra of SN1999ee as templates, and investigate the constraint onthe amount of hydrogen contaminated in the ejecta of SN2005cf. Figure 23 also shows the P β at the correspondingepochs, shown in the bottom of both panels, extractedfrom Model RGa for θ = 0 (red) and π (blue). This fluxis then added to the original (observed) spectrum of SN1999ee. This way, we can check if the contamination ofthe H-rich materials in SN 2005cf is consistent with themodel, assuming that there is no H-rich material contam-inated in SN 1999ee. Further, by varying the model flux,we can place a constraint on the amount of H allowed forSN 2005cf.At ∼
30 days since the B -band maximum, Model RGapredicts that P β is visible if viewed from the companionside ( θ ∼
0) while not so from the opposite side ( θ ∼ π ).Adding the P β flux predicted by Model RGa for an ob-server at θ = 0 to the spectrum of SN 1999ee, this exceeds the observed flux of SN 2005cf at that wavelength. Thus,a situation that SN 2005cf had a RG companion in theclose binary system and it was viewed from the compan-ion side is ruled out. If one reduces the predicted P β flux to 30% of the original (corresponding to ∼ . M ⊙ of the mixed hydrogen), this would not conflict with theobservation. Any companion star is not ruled out for anobserver at θ = π . Later on at ∼
40 days since the B -band maximum, a similar but tighter constraint can beobtained: M (H) ∼ < . M ⊙ for an observer at θ = 0 and ∼ < . M ⊙ for θ = π . Note that the SN ejecta contami-nated with 0 . M ⊙ of hydrogen can be a typical featureof the SN ejecta-companion impact (see, e.g., Liu et al.2012), thus the diagnostics we propose here can be quitepowerful to identify/rule out a non-degenerate compan-ion star.Note also that by comparing two SNe, we indeed con-strain difference in the hydrogen content in these twoSNe. Thus, it is necessary to construct a ‘hydrogen-free’template spectrum from a large sample. In doing this,there are several possibilities in the template construc-tion. Dividing the SN sample into subgroups with dif-ferent peak luminosities (or decline rate) is an obviouschoice. One would also be tempted to divide the sam-ple according to the host types or environment proper-ties, then compare the templates for each group as wellas compare individual SNe with the templates. Such astrategy may pick up possible different populations indifferent environments (e.g., Wang et al. 2013) provid-ing a test of how different populations may be related tothe SD and DD scenarios.There are a few limitations in the present study. Wehave adopted the expansion opacity formalism and thetwo-level atom approximation, rather than simulatingthe full details of the fluorescence following the excita-tions. This is a good approximation for ions with compli-cated level structures like Fe-peaks, since the high-rate9of the radiation-matter interactions should establish aquasi-equilibrium which is represented by thermal redis-tribution (see, e.g., § ∼ > time-dependent NLTE effects, which for SNeIIp models could introduce a change in the flux level ofH α by a factor of a few (Dessart & Hiller 2008). Still, thiseffect would not remove all the H α flux predicted in LTEcalculations (Dessart & Hiller 2008), and thus we wouldnot expect that our results will be changed qualitativelydue to the NLTE effects.Related to the NLTE effects, in our formalism the ef-fect of the fluorescence is taken into account by a sin-gle thermalization parameter ( ǫ ), and it has been shownthat the biggest difference from the present prescription( ǫ = 0 .
3) is expected for the pure scattering atmosphere( ǫ = 0) (Baron et al. 1996; Kasen et al. 2006). To checkif this particular choice of ǫ (calibrated for metal lines)would affect the strength of hydrogen features, we haverepeated the same calculations for Model RGa but set-ting ǫ = 0. The result is shown in Appendix C. We con-clude that this does not introduce much difference. Wecaution that the two-level approximation and the ther-malization parameter cannot be exactly calibrated to afull NLTE description, therefore introducing the ther-malization parameter is merely an approximation. Ulti-mately it should be tested by the full NLTE calculationsfor the same models. Still, as mentioned above, the re-sults from the previous studies on SN Ia and IIp models,both relevant to our study, are promising, suggesting thatthis approximation would not introduce significant errorsin the observables of interest in this paper.We note that neglecting non-thermal excitations of hy- drogen by γ -rays might indeed lead to underestimates ofthe hydrogen line fluxes. Also, in our model we omita metal content in the companion envelope, by assum-ing the purely hydrogen and helium in it. This wouldnot change the overall feature, since the main part of theemission is created by the ‘SN ejecta’ and the companionenvelope merely dilutes the emission through Thomsonscattering. An inclusion of the metal would increase theheating of the H-rich region, therefore would keep theionization of hydrogen high for longer time than in ourpresent model. However, as time goes by the γ − ray heating, which is not sensitive to the metal content, be-comes progressively important. As a result, our predic-tion of the appearance of P β in relatively late phaseswould not be dramatically affected by the metal contentin the companion envelope.Besides the hydrogen issue, the ejecta asymmetry isa characteristic feature of a non-degenerate companionsystem, and we predict that this configuration leads toa characteristic ‘diversity pattern’ across the wavelength(Figs. 21 & 22). In Appendix D we show the expecteddiversity patterns at different epochs for model RGa. Inthe same way as we propose for searching P β APPENDIX
A. METHOD OF RADIATION TRANSFER
Our radiation transfer simulation code adopts the Monte Carlo method, where paths of individual photon packetsare computed as a random walk process. This method is broadly adopted in radiation transfer simulations in SN ejecta(Lucy 2005, and references therein). It is suited to treat Doppler shifts of photons due to the velocity gradient in theexpanding/moving medium and resulting enhancement of the bound-bound opacities or ‘expansion opacities’ (Karp etal. 1977; Eastman & Pinto 1993) – due to the successive Doppler shifts in a comoving frame a photon experiences asit flies through the moving medium, it can suffer from the discrete transitions (bound-bound) at frequencies differentfrom the original frequency of a photon at its creation.We largely adopt prescriptions given by Lucy (2005), Kasen et al. (2006), and Kromer & Sim (2009). Our simulationis purely radiation transfer, thus the kinetic and composition structure [ ~v ( ~r, t ) , ρ ( ~r, t ) , X i ( ~r, t )] should be provided asa background (i.e., no feedback process from the radiation to hydrodynamics is taken into account). The radiationtransfer simulation provides iteratively the thermal and ionization conditions and accordingly the distribution ofopacities [ T ( ~r, t ) , n ji ( ~r, t ) , α λ ( ~r, t )] (where n ji is the number density of an ion j at i -th level) so as to be consistentwith the radiation field [ f λ ( ~r, t,~l )] (here ~l is the photon direction vector) under the assumptions of LTE and radiative0equilibrium.The code is applicable for the 1D spherical coordinate, the 2D spherical-polar coordinate, and the 3D spherical-polarand Cartesian coordinates, which can be simply specified in an input parameter file. The 1D and 2D versions assumespherical symmetry and axisymmetry so that the number of photon packets can be reduced to reach to the convergence.We have performed various test calculations to confirm that the imposed asymmetry does not introduce errors in thetransfer simulations (an example is described later).The multi-dimensional/frequency/epoch radiation transfer code,
HEIMDALL (Handling Emission In Multi-Dimension for spectrAL and Light curve calculations) , takes the following steps in simulating the radiation transfer.The main part of the code is written in a general way so that the applicability is not restricted to the radiation transferin the SN ejecta, but below when necessary specific functions for the SN radiation transfer are described. The code iswritten in a hybrid-parallelization mode using openMP and
MPI and its parallelization efficiency has been tested upto 512 cores distributed over 64 cpus.
1: Determining the distribution of initial photon packets:
Over the course of the main MC routine, the properties of a photon packet are described by a set of variables[ ~r ( t ) ,~l ( t ) , λ ( t ) , ε ( t )], i.e., the position, direction vector, wavelength, and the total energy within the packet. Here, wedescribe the photon packet (or a MC quanta) as a group of identical photons (or particles), i.e., ε = n ph hν where n ph is the number of the photons in a packet and ν is the frequency. To compute the change in these variables as afunction of time ( t ) by the main MC routine, we have to determine the initial conditions, i.e., [ ~r ( t ) ,~l ( t ) , λ ( t ) , ε ( t )]where t is the time of the creation of the thermal photon under consideration.Photon packets created by processes other than interactions of already existing thermal photons and matter arespecified at the beginning of simulations (note that photons created by such interactions between already existingthermal photons and matter are treated over the course of the main MC transfer). For simulations performed in thispaper, these are photons created as a result of radioactive decay energy input through the decay chain Ni → Co → Fe. These γ -ray and positron packets are created at the beginning of a simulation based on the distribution of Niand its decay property – the photon packets are assigned with the spectral energy (i.e., branching ratios in the decays)and time of the creation (i.e., decay time) determined with random number generation. If the time at the creationof a γ -ray packet, as determined by the MC random number generation for every packet, is earlier than the startingtime of the whole simulation, the γ -ray packet is assumed to be absorbed by the simulation start time at the positionof the creation in the comoving frame. This deposited energy is converted to optical photons at the starting time ofthe optical photon transfer, taking into account the adiabatic loss of the thermal energy between the deposition timeand the simulation starting time, assuming that the optical photons created here have diffused negligibly to matterin this time interval (Lucy 2005; Kromer & Sim 2009). This treatment is justified by the short mean free paths ofphotons in the early phase, while the approximation becomes less robust if the starting time for the transfer simulationis taken to be later. Our simulations are started at 10 days after the explosion (approximately a week before the B -band maximum), and we have checked the applicability of this approximation in Appendix B, where we find thatour results are not sensitive to this relatively late starting time of the simulations. Transfer of γ -rays are solved takinginto account Compton scattering, pair creation, and photoelectric absorption based on the scheme identical to opticalphotons but without temperature iteration since the cross sections of these interactions are insensitive to the thermalcondition. Positrons are assumed to deposit their energy in situ. The details of the computational method here aregiven by Maeda (2006). During the MC transfer simulation, the energy deposition by γ -rays and positrons, Γ γ ( ~r, t )are tracked as in the same manner with the heating by UV/optical photons (see below).Since the energy deposition by γ -rays and positrons is the only source of the thermal energy (i.e., ultimately theenergy of the thermal radiation) in the present situation, the transfer simulation described above provides the initialcondition for the thermal photon packets to be followed by the main MC routines. With the energy deposition rateΓ γ ( ~r, t ) we thus determine the total energy content of the thermal photon packets emitted at a given spatial bin anda time bin. In the calculations shown in this paper, the energy content of each thermal photon packet ( ε ) is set tobe equal for all the packets at its creation. We first integrate the energy deposition rate in space and time, then thenumber of thermal photons created at a given spatial bin and a given time bin is determined by the relative contributionof Γ γ ( ~r, t ) (as integrated within a spatial bin and time bin) to the total deposited energy. The position and time atits creation within the spatial and time bins are determined with the random number generation (in the comovingframe). The direction vector of the packet is also computed by the random number generation assuming the isotropicemission in the comoving frame. Now we thus have a set of variables to specify the properties of the photon packetsin the comoving frame, except for its wavelength, and these are transformed to the SN-rest (or observer) frame. Thewavelength, λ ( t ), cannot be specified at this step, as it requires the temperature to be known. Thus, the creation ofthe photon packet is coupled with the main MC routine and iteratively solved following the steps described below, tobe self-consistent with the ejecta temperature.
2: Computing thermal and ionization structures and opacity distributions:
At given time and for given temperature T ( ~r, t ), the ionization and level populations are computed under the LTE assumption, i.e., through the Saha equation and the Boltzmann distribution. Then, the opacities are computed as afunction of wavelength, including bound-bound, bound-free, free-free, and electron scattering. The electron scatteringopacity is computed with the electron number density, n e ( ~r, t ), which is given by the ionization condition: α e ( ~r, t ) = σ T n e ( ~r, t ) , (A1)1where σ T is the Thomson cross section. The free-free absorption opacity is computed as follows: α ff ( ~r, t, ν ) = 4 e m e hc (cid:18) π km e (cid:19) / T ( ~r, t ) − / X j Z j n e n j ν − (cid:20) − exp (cid:18) hνkT (cid:19)(cid:21) g ff , (A2)where Z j is the number of free electrons associated with the ion j , and n j is the number density of the ion j . TheGaunt factor ( g ff ) is set to be unity. For the bound-free transitions, cross sections [ α ff ( ~r, t, λ )] are taken from Verneret al. (1996) and Verner et al. (1996). For given λ , the bound-free cross sections are summed over for different ions(with the ionization states determined by the Saha equation).The bound-bound transitions are treated in the Sobolev approximation, where the ‘line’ optical depth is given asfollows: τ lu ( ~r, t ) = πe m e c f lu λ lu t n l ( ~r, t ) (cid:20) − g l n u ( ~r, t ) g u n l ( ~r, t ) (cid:21) , (A3)where subscripts l and u denote the lower and upper levels of a transition under consideration (here we omit thesuperscript j to specify the ion). f lu and λ lu are the oscillator strength and the wavelength of the transition. g l and g u are statistical weights of the lower and upper level, respectively. For the line list, we adopt a standard set of 5 × bound-bound transitions from Kurucz & Bell (1995).With the Sobolev optical depth, the escape probability of the photon out of the resonance region is β lu ( ~r, t ) = 1 − e − τ lu ( ~r,t ) τ lu ( ~r, t ) . (A4)We treat the bound-bound transitions within the expansion opacity formalism, i.e., combining the transitions into adiscrete frequency grid (Karp et al. 1977; Eastman & Pinto 1993). Here, the total cross section at wavelength λ isgiven as α bb ( ~r, t, λ ) = 1 ct X l,u λ lu ∆ λ (1 − e − τ lu ) , (A5)where the sum runs over the bound-bound transitions whose energy difference is within the wavelength bin underconsideration (∆ λ ). The purely absorptive component is defined within the two-level atom approximation, i.e., S λ = (1 − ǫ lu ) J λ + ǫ lu B λ ( T ) , (A6)where the source function is divided into the scattering component (i.e., treated as a resonance line) and into theabsorptive component (i.e., thermalized after multiple scatterings and fluorescence). Generally, this treatment of thebound-bound transitions is shown to provide a good approximation for the thermal conditions appropriate to SNeIa, since the large opacities and many transitions lead to the thermal redistribution. This has been calibrated withmore detailed transfer where the fluorescence is directly treated (Baron et al. 1996; Kasen et al. 2006). While ǫ lu isdependent on different transitions, the result of the radiation transfer is indeed insensitive to the exact value of ǫ lu aslong as ǫ lu ∼ ǫ ≡ ǫ lu = 0 . α bb , abs ( ~r, t, λ ) = 1 ct X l,u λ lu ∆ λ ǫ lu β lu + ǫ lu (1 − β lu ) (1 − e − τ lu ) . (A7)The total opacity is given as the sum of the different components described above: α ( ~r, t, λ ) = α e ( ~r, t ) + α ff ( ~r, t, λ ) + α bf ( ~r, t, λ ) + α bb ( ~r, t, λ ) . (A8)The purely absorptive component is defined as follows: α abs ( ~r, t, λ ) = α ff ( ~r, t, λ ) + α bf ( ~r, t, λ ) + α bb , abs ( ~r, t, λ ) . (A9)
3: Propagation of photon packets through the MC simulation:
With the background condition including opacity distribution (in space and frequency) now specified, the propagationof photon packets is computed from time t n to t n +1 . When the photon packet already exists at t n from the previoustime step, its path until t n +1 or until it escapes out of the ejecta before t n +1 is followed by the MC simulation foreach photon packet, and this procedure is repeated for all the photon packets. If the photon packet is created at t between t n and t n +1 (due to the γ -ray and positron deposition) its path is followed from t .At each step, photon path lengths for several numerical and physical events are computed, then the event withthe minimal length is adopted as the real event. These events include the following cases. (1) A photon reachesto a boundary between the current spatial grid and one of the neighboring grids. (2) The physical time the packetexperiences reaches to the next time step ( t n +1 ), (3) The photon comoving spectral frequency is redshifted to come2 Fig. A1.—
Synthetic multi-band light curves as compared with SN Ia template light curves. The SN Ia template light curves (stars) areconstructed from the Hsiao spectral template (Hsiao et al. 2007) convolved with standard filter functions. Our synthetic light curves basedon W7 model are shown by red curves (thick solid). For comparison, the synthetic light curves computed by STELLA are shown by bluecurves (thin solid), from U to I band. Our 2D ‘reference’ W7 model is shown by green curves (thick dashed). For the ‘reference’ model, weapply an offset of 0.34 magnitudes for all the bands for fair comparison, since the model has a larger amount of Ni (see the main text).The amount of the offset here reflects the difference in the mass of Ni (0 . M ⊙ in the ‘reference’ model and 0 . M ⊙ in the original W7model). into the next frequency bin, (4) A photon suffers from either scattering or absorption. This procedure is repeated forall the photon packets.The item (3) is specific for the radiation transfer in moving medium like the SN ejecta. Since we assume thehomologous expansion, the Doppler shift is simply computed as ∆ λ = λv/c , and this inversely gives the distancethe packet travels before suffering from the Doppler shift of the amount ∆ λ . The item (4) is evaluated through thestandard MC formula as follows: α ′ ( ~r, t, λ ) ρ ( ~r, t ) δs ′ = − ln z , (A10)where z is the random number (between 0 and 1) and δs is the path length. Here the prime indicates the rest-framequantities.When the packet experiences the interaction, its fate after the interaction is again determined through the randomnumber generation, proportional to cross sections to each event. Specifically, we judge if this is a scattering orabsorption, according to the ratio of α abs ( ~r, t, λ ) and α scat ≡ α ( ~r, t, λ ) − α abs ( ~r, t, λ ).A scattering is treated as an isotropic and coherent scattering in the comoving frame, a good approximation forresonance transitions (and Thomson scattering). A new photon direction in the comoving frame is chosen randomlyfollowing a standard MC procedure and it is transferred to the rest frame. It is elastic in the comoving frame, and thetransfer from the comoving frame to the rest frame automatically takes into account the adiabatic loss in a microscopicsense.An absorption and re-emission is treated within the thermalization approximation, and thus a new wavelength atits re-emission is determined through the local thermal emissivity: j λ ( ~r, t ) = B λ ( T ) α abs ( ~r, t, λ ) . (A11)Here the emission is treated as isotropic in the comoving frame.Note that the propagation is treated in the rest frame, using the cross sections originally computed in the comovingframe but transferred into the rest frame. The physical events (scattering and absorption/re-emission) are treatedin the comoving frame, i.e., first the rest frame quantities are transferred into the comoving frame, computing the3 Fig. A2.—
Synthetic spectra as compared with the Hsiao SN Ia template. The zero-point in the epoch is the explosion time, and the B -maximum date is assumed to be 17 days since the explosion to label the Hsiao template spectra. The W7 model spectra (as computedunder the assumption of spherically symmetry) are shown by red, while the 2D ‘reference’ model spectra are shown by blue. The Hsiaotemplate spectra are shown by black. Synthetic spectra of W7, but computed with the radiation transfer scheme in 2D space under theassumption of axisymmetry is shown by (thick) gray lines, to show that our scheme of the radiation transfer in 2D does not introduce anyartifact. The values of the offset applied to each spectra are the same for all the models (i.e., no additional offset is applied to providethe best match between the models and the templates). For the spectra at late epochs, a blue portion of the spectra is truncated in thepresentation, where the MC noise is large due to the small amount of UV photons in the late phase. outcome in the comoving frame, and then the result is transferred back into the rest frame. The formalisms forthe transformation are given by Castor (1972). This way, the Doppler shift in the moving medium is appropriatelyhandled, and for example it results in the P-Cygni profile for bound-bound transitions.
4: New Temperature determination and iteration for the temperature convergence:
Over the course of the photon propagation, the heating rate between t n to t n +1 at each spatial grid is tracked, usingthe MC estimator: Γ opt ( ~r, t ) = 1∆ tV X k α abs ( ~r, t, λ ) ε k δs k , (A12)where the quantities are given in the comoving frame, and this estimator runs over all the packet (as specified by k )which passes through a given grid between t n and t n +1 (∆ t ≡ t n +1 − t n ). Here, V is the volume of the spatial grid.With the heating rate by γ -rays and positrons obtained in the same manner (Γ γ ), the heating-cooling balance underthe radiative equilibrium provides a constraint on the temperature,Λ( T ) = Γ opt ( ~r, t ) + Γ γ ( ~r, t ) , (A13)4where the cooling rate at each spatial grid is given asΛ( T ) = 4 π Z α λ ( T ) B λ ( T ) dλ , (A14)where the absorptive opacity α λ ( T ) is approximated by the one estimated with the ‘previous’ temperature [i.e., α abs ( ~r, t, λ )]. These equations give new temperature estimate.The steps 2 - 4 are repeated for a given time step (between t n and t n +1 ) until the temperature converges at all themeshes simultaneously. Once this happens, the converged temperature is used for the initial guess for the temperaturein the next time step, and the photon packets’ properties are used as the initial conditions for the propagationcalculations in the next time step (at t n +1 ). Then, the procedures 2 - 4 are repeated in the next time step until thetemperature convergence. This way, we proceed with time, following the radiation field and thermal condition in aself-consistent manner.
5: Extraction of synthetic spectra and light curves:
In the MC packet propagation routine, the paths of every photon packet are followed. When the photon packets escapeout of the SN ejecta (or the numerical domain), the information is recorded. This provides the escaping radiation fluxas a function of the viewing direction, time, and wavelength [ f λ ( ~l, t )] (here ~l is the photon direction vector). From thiswe extract angle-dependent spectra as a function of time. The light curves in multi band passes are then extracted byconvolving the filter functions to the synthetic spectral sequence. In this paper, we use the Johnson and Kron-Cousinssystems for UBVRI and 2MASS system for NIR.Figure A1 shows an example of the synthetic light curves for the W7 model. We find a reasonable agreement betweenour result and a result obtained by an independent simulation code STELLA (Blinnikov et al. 1998, 2006). Whiledifferent codes generally agree to reproduce overall behaviors, details are different depending on specific treatments(see, e.g., Kromer & Sim 2009). Our results are well within these variations, and similar to the result of Kasen et al.(2006). We find a reasonable agreement between the W7 model prediction and the Hsiao template light curves (Hsiaoet al. 2007). The discrepancy is larger in NIR than in optical, but this is a general issue in the radiation transfersimulations for SNe Ia (see, e.g., Kromer & Sim 2009; Gall et al. 2012). We note that ‘our reference’ model (themodified W7) also shows a behavior similar to the original W7, justifying to use this model as our reference model.Figure A2 shows a synthetic spectral sequence for the W7 model (red; computed in 1D under the assumption ofspherical symmetry), the ‘reference’ model (blue), and the Hsiao template spectra (black) (Hsiao et al. 2007). We seea reasonable agreement between the W7 model and the Hsiao templates. There are deviations especially in the laterphases – while the model does predict spectral features at correct wavelengths, the strengths of the features can bedifferent from the observed templates. This is suggested to be caused by
NLTE effects (e.g., Kromer & Sim 2009),and is a generic issue for spectrum synthesis in SNe (see, e.g., Sim et al. 2013, for 3D delayed-detonation models).Our ‘reference’ model shows a larger flux than the W7 and the Hsiao templates due to the large amount of Ni, butotherwise the predicted features are very similar to the W7 model. Therefore, using this model as our reference isjustified. We also show the same W7 models but mapped onto the 2D grids and computed in 2D, under the assumptionof axisymmetry (but no symmetry with respect to the equatorial direction). We see a perfect match between the 1Dand 2D calculations, proving that our 2D radiation transfer scheme does not introduce any errors in the transfersimulation.
B. ROBUSTNESS OF THE PREDICTED MAXIMUM-PHASE BEHAVIORS
In this paper, we deal with the diversity in the magnitudes and colors around the B -band maximum at 0.1 magnitudelevel. In this section, we show that our simulations are accurate to this level to claim the diversity arising from thedifferent viewing directions. Especially, we address the following two points: (1) If the relatively late starting time inour simulations (10 days) affects the claimed behaviors, and (2) if the predicted diversities and correlations are notaffected by possible numerical instabilities.Figure B1 shows the multi-band light curves of the ‘reference’ W7 model from the simulation starting at day 5,as compared to our standard run starting at day 10. The same but for the RGa model is shown in Figure B2. Itis seen that the two calculations with different starting time converge quickly toward the B -band maximum date inboth models. A substantial difference is seen in the U -band light curve around its peak date (before the B -bandpeak), while in the other bands the difference is small. This test also shows that (late-phase) kinks seen in the originalcalculations especially in the J -band (i.e., ∼
50 days for the reference model and ∼
30 days in the RGa model) arenumerical artifacts. The fact that these kinks appear much later than the B -band peak where the treatment of thestarting time should be unimportant indicates that this late-phase stability can be sensitive to small variation in thethermal conditions, but fortunately it seems that this unstable behavior does not appear around the maximum phase(see also below).Figure B3 shows the variations in the colors and the V -band magnitude, and their relations to the B − V colorobtained for Model RGa, by the simulation starting at day 5. This should be compared to Figure 12 where thesimulation is started at day 10. While a small offset in the absolute scale is seen for the B − V and V − I colors andthe V -band magnitude at the level of 0.05 magnitude, the trend as a function of different viewing directions and theamount of the resulting diversity are consistent with the original simulation.5 Fig. B1.—
Synthetic multi-band light curves of the ‘reference’ W7 model with different starting time in the simulations. The originalsimulation (starting at day 10) is shown in gray, while the simulation starting at day 5 is shown in black.
Fig. B2.—
Synthetic multi-band light curves of the RGa model with different starting time in the simulations. Shown here are theangle-averaged mean light curves. The original simulation (starting at day 10) is shown in gray, while the simulation starting at day 5 isshown in black.
Fig. B3.—
The predicted relations in photometric properties for Model RGa as same as Figure 12, but for the simulation starting at day5.
The spectral features are even less sensitive to the starting time of the simulation. Figure B4 compares the spectraaround Si II 6355 in both simulations. It is seen that the results are almost identical, and therefore the relationsinvolving the Si velocity should also be unaffected by the treatment of the starting time.We note that while the absolute magnitude can be affected by the treatment of the starting time (which turns out tobe 0.05 mag level), the behavior in the colors and magnitudes arising from different viewing direction should be muchless sensitive here. The spectral features are expected even less sensitive. These are confirmed by the test simulation,6
Fig. B4.—
The simulated spectra around Si II 6355 (from the RGa model) as same as Figure 10 but for the simulation starting at day5 (color curves). The color coordinates indicate the prediction for different viewing directions (red for θ = 0 and blue for θ = π ). In thisfigure, the spectra for observers at different directions are added with an additional offset ( θ = 0 to π , from bottom to top). The graycurves here are the spectra obtained by the original simualtion starting at day 10. justifying our claims regarding the correlation and diversities arising from different viewing directions.Another issue is if the trend and diversity arising from the different viewing directions around the B -band maximumare affected by numerical instability similar to the one seen in the J -band light curves in the later phase. First,the investigation of the sensitivity on the starting time suggests that such an instability takes place less likely inthe earlier phase than the later phase; the different thermal conditions due to the different starting time may mimicnumerical instability in the thermal condition, but we see that the thermal condition quickly converges before the B -band maximum. This is expected, since the radiation–matter coupling is quite strong in the early phase which shouldsuppress the numerical instability quickly (note that the iteration in the thermal condition is performed in every timestep under the assumption of radiative equilibrium). Next, for such an instability to affect the viewing-angle dependence ,the instability itself should create a strong angle-dependent effect, which should be seen as an angle-dependent suddenrise and fall in the multi-band light curves. Even for the possible instability found in the later phases (in the J -band),this effect does not show strong angle-dependence. Therefore, even if a similar numerical instability would take placein the earlier phase, it is unlikely that such a putative effect should affect the angle-dependent effects we claim in thepresent paper.To support these arguments, we have performed a test calculation based on the ‘reference’ W7 model. Here, on day13.6 (i.e., ∼ B -band maximum), we artificially reduce the temperature in the ejecta only within θ = 165 −
180 degree to the 40% of the ‘real’ converged temperature. As such, if the instability does not fade away, thisshould create artificially–introduced viewing–angle dependence for the expected observables. The setup here is chosen,following a trial and error, so that the inserted artificial effect on the viewing angle variation exceeds the variationseen in the RGa model.Figure B5 shows the variation of the V -band and H -band magnitudes in the RGa model due to different viewingdirections, as compared to the angle-averaged mean magnitude in each band. The evolution for given θ is quitecontinuous, and we do not see any sudden change in the viewing angle dependence which would announce any possiblenumerical instabilities. This should be compared with Figure B6 which shows the same quantities for the referenceW7 model where the numerical instability is artificially inserted in the cone (in the direction of θ = 165 −
180 degree).We see here a sudden ‘broadening’ of the angle dependence especially in the H -band. Such a behavior is not seen inthe RGa model, suggesting that any instability which creates numerically-introduced viewing dependence larger thanthe ‘real’ physical behavior is not present in the simulation for the RGa model.Furthermore, the artificially–introduced variation for different viewing directions quickly fades away, and the remain-ing variation (due to the MC noise) becomes much smaller than the variations seen in the RGa model. This furthersupports that the viewing angle dependence around the B -band maximum is not affected by any instabilities, and theMC noise in our simulations is sufficiently small. In sum, we conclude that our claim on the viewing-angle dependenceon the B -band maximum observables (e.g., Figures 12 and 14) are not due to numerical artifacts. C. TREATMENT OF HYDROGEN LINES
As we have adopted the expansion opacity formalism and the two-level atom approximation, a question is if thepredicted fluxes of the hydrogen features can be significantly affected by these assumptions ( § ǫ = 0).To do this, we have taken the thermal condition computed from the original calculations ( ǫ = 0 .
3) so that we can7
Fig. B5.—
Evolution of the difference between the magnitude for observers at various directions and the angle-averaged magnitude forthe RGa model. Shown here are the V -band and H -band magnitudes. Fig. B6.—
Evolution of the difference between the magnitude for observers at various directions and the angle-averaged magnitude forthe ‘reference’ W7 model but with the instability in the thermal condition artificially introduced at θ = 165 −
180 degree on day 13.6 (i.e., ∼ B -band maximum). Shown here are the V -band and H -band magnitudes. purely pick up the effect of ǫ on the line formation. Then, the model calculations with and without hydrogen (forlatter we artificially set the cross sections of hydrogen to zero) are performed, and the ratio is investigated as we didin §
5. Figure C1 shows the result. We thereby confirmed that this does not affect the detectability of the hydrogenlines. As we expect that the scattering dominated atmosphere is an extreme condition, we believe that the hydrogenline formation we find in this paper would not be much affected by the treatment of hydrogen line transitions.
D. DIVERSITY PATTERNS FROM THE COMPANION-INDUCED ASYMMETRY
Irrespective of the hydrogen content in the expanding SN ejecta, the companion-induced asymmetry can createcharacteristic diversity patterns as a function of the wavelength. This can in principle be tested by observations, bystudying a diversity seen in spectra of individual SNe as compared to a reference/template spectrum. The referencespectrum should be created as a mean of the spectra of SNe in a specific sub-group (or all SNe Ia), and the comparisoncan be performed between an individual and the reference spectra. It will allow to see possible diversity of individualSNe from the average behavior of the group. The diversity pattern predicted from the companion-induced asymmetryis different for different viewing directions, and it evolves with time. A specific example of the expected diversitypattern is shown in Figures 21 and 22 for an epoch of 38.5 days since the explosion. In Figures D1–D3, we providethe model predictions for different epochs.8
Fig. C1.—
The ratio of the spectral flux with and without hydrogen lines in Model RGa. This is the same with Figure 15, except thatthe hydrogen bound-bound transitions are treated as fully resonance lines (i.e., ǫ = 0). Fig. D1.—
The diversity patterns from the companion-induced asymmetric configuration. Shown here is the residual of the syntheticspectra for Model RGa after being divided by the mean spectrum, shown for the optical range (left) and for the NIR (right). The epochis 16.8 days after the explosion. The panels are divided into four according to the viewing direction ( θ = 0 to θ = π , from bottom totop). The synthetic spectra are binned within 3 time bins, but no additional binning is performed in the wavelength and viewing angledirections. Fig. D2.—
The diversity patterns from the companion-induced asymmetric configuration, as is the same with Figure D1. The epoch is25.4 days since the explosion. Fig. D3.—
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