Spectral signatures of H-rich material stripped from a non-degenerate companion by a Type Ia supernova

The single-degenerate scenario for Type Ia supernovae (SNe Ia) should yield metal-rich ejecta that enclose some stripped material from the non-degenerate H-rich companion star. We present a large grid of non-local thermodynamic equilibrium steady-state radiative transfer calculations for such hybrid ejecta and provide analytical fits for the Halpha luminosity and equivalent width. Our set of models covers a range of masses for 56Ni and the ejecta, for the stripped material (Mst), and post-explosion epochs from 100 to 300d. The brightness contrast between stripped material and metal-rich ejecta challenges the detection of HI and HeI lines prior to ~100d. Intrinsic and extrinsic optical depth effects also influence the radiation emanating from the stripped material. This inner denser region is marginally thick in the continuum and optically thick in all Balmer lines. The overlying metal-rich ejecta blanket the inner regions, completely below about 5000A, and more sparsely at longer wavelengths. As a consequence, Hbeta should not be observed for all values of Mst through at least 300 days, while Halpha should be observed after ~100d for all Mst>= 0.01Msun. This contrasts with the case of circumstellar (CSM) interaction, not subject to external blanketing, which should produce Halpha and Hbeta lines with a strength dependent primarily on CSM density. We confirm previous analyses that suggest low values of order 0.001Msun for Mst to explain the observations of the two SNe Ia with nebular-phase Halpha detection, in conflict with the much greater stripped mass predicted by hydrodynamical simulations for the single-degenerate scenario. A more likely solution is the double-degenerate scenario, together with CSM interaction, or enclosed material from a tertiary star in a triple system or from a giant planet. [Abridged]


Introduction
Unambiguous signatures of the single-degenerate scenario for Type Ia supernovae (SNe Ia) are much desired but hard to secure. One such signature is the identification of material stripped from the companion star by the SN Ia ejecta. Analytic explorations (Wheeler et al. 1975;Chugai 1986) and numerical simulations (Marietta et al. 2000;Pakmor et al. 2008;Liu et al. 2012;Pan et al. 2012;Liu & Stancliffe 2017) of such a scenario have been performed in recent decades and have yielded similar conclusions. The key signatures are that 0.1−0.5 M of material should systematically be stripped from the non-degenerate companion star, asymmetrically distributed but limited to low velocities below about 1000 km s −1 in the resulting ejecta. The main radiative signature for this scenario is proposed to be the appearance at late times of a narrow and strong Hα line on top of a normal SN Ia spectrum (Chugai 1986;Mattila et al. 2005;Botyánszki et al. 2018). More sophisticated and complete studies of the progenitor evolution do not alter this picture (see, for example, Liu & Stancliffe 2017).
Narrow Hα emission has been detected in a number of SNe Ia. In some events (e.g., SNe 2002ic or 2005gj), the detection is early, around bolometric maximum, and compatible with circumstellar-material (CSM) interaction (Hamuy et al. 2003;Silverman et al. 2013;Kotak et al. 2004). For two noteworthy events, the Hα emission has been detected later, at ∼ 100 d past maximum or more (for example, SN 2018cqj, Prieto et al. 2020;ASASSN-18tb, Kollmeier et al. 2019;Vallely et al. 2019). These events may be explained by either CSM interaction (in particular when the detection occurs earlier, as for ASASSN-18tb as +39 d; Vallely et al. 2019) or by emission from stripped material from a companion. However, in the latter case, radiative transfer simulations, confirmed by the present work, suggest the stripped material mass M st is on the order of 0.001 M . This is much less than predicted by all hydrodynamical simulations of this scenario, which suggest that at least 0.1 M of material should be stripped (see original work by Marietta et al. 2000 and the most recent complete study by Liu et al. 2012). Another tension for the single-degenerate scenario is that such Hα emission is very rarely seen (stringent upper limits on M st are placed instead), even for the most nearby SN Ia events or those with very high quality observations (Leonard 2007;Lundqvist et al. 2013Lundqvist et al. , 2015Shappee et al. 2013Shappee et al. , 2018Maguire et al. 2016;Graham et al. 2017;Sand et al. 2018;Holmbo et al. 2019;Dimitriadis et al. 2019;Sand et al. 2019;Tucker et al. 2019a,b), while theory predicts that it should instead be systematically detected. In the current framework, this is a major problem for the single-degenerate scenario for SNe Ia. One potential way out of the discrepancy may be to modify the donor star through some mechanism so that the amount of stripped, hydrogen-rich material is vastly reduced. Such mechanisms could be hydrogen stripping by an optically thick wind so that it has a helium-rich envelope (Hachisu Article Pan et al. 2010) or a "spin-up spin-down" phase that strips the envelope and also leaves a much more compact star (Justham 2011;Hachisu et al. 2012).
In this work, we focus on the emission properties that stripped material will have on the resulting spectrum. So far, only a few studies have been published on the non-local thermodynamic equilibrium (non-LTE) radiative-transfer modeling for SNe Ia with stripped material from a companion star. Mattila et al. (2005) and Lundqvist et al. (2013Lundqvist et al. ( , 2015 use parametrized ejecta structures together with the radiative-transfer code of Kozma et al. (2005) to set constraints on the Hα emission from the stripped material. Making a number of simplifications for the radiative transfer (Botyánszki & Kasen 2017), and in particular pure optically-thin line emission, Botyánszki et al. (2018) perform similar (i.e., Monte Carlo) simulations but based on physically consistent 3D hydrodynamical simulations of the ejectacompanion interaction. The study of Botyánszki et al. (2018) predicts Hα luminosities that appear a hundred to a thousand times greater than those predicted by Mattila et al. (2005) for the same stripped material mass and composition. These various studies use the Sobolev approximation, and treat approximately, or ignore, line overlap and multiple scattering.
In this paper, we build on these previous studies and conduct non-LTE steady-state radiative transfer simulations for SNe Ia with stripped material for a range of post-explosion times, SN Ia ejecta (total mass and 56 Ni mass), as well as a large range of stripped material masses M st . The simulations are performed with the standard setup for SN Ia calculations with CMFGEN (see, for example, Dessart et al. 2014b;Wilk et al. 2019a), treat all non-LTE processes, and employ a large model atom. The Sobolev approximation is not used, and line overlap and line blanketing are computed explicitly. This allows for an assessment of the blanketing effects of the metal-rich faster-moving ejecta on the emission from the stripped material, as well as all optical depth effects on continuum and line photons. As in Mattila et al. (2005), we adopt parametrized ejecta structures but also perform tests to evaluate the impact of this shortcoming on our results.
In the next section, we present our numerical approach. Section 3 studies in detail the results for a reference case, chosen to be a standard Chandrasekhar-mass SN Ia model (i.e., with 0.51 M of 56 Ni) with 0.18 M of stripped material. Section 4 elaborates on the processes that control the appearance of Hα. Section 5 presents one clue for the identification of stripped material. Section 6 evaluates the limitations of our approach for the adopted structure of the stripped material. We also discuss the predicted signatures from He i lines in our simulations in Sect. 7. Section 8 presents results for the whole grid of simulations performed in this study, encompassing five epochs from 100 to 300 d, Chandrasekhar as well as sub-Chandrasekhar mass ejecta, 56 Ni masses from about 0.1 to 0.9 M , and stripped material masses from 10 −5 up to 0.5 M . Section 9 emphasizes the distinction between line luminosity and line equivalent width and the relevance for the detection of Hα from stripped material in SN Ia spectra. Section 10 presents our conclusions.

Numerical approach
We investigated the radiative signatures of stripped material in the innermost layers of SN Ia ejecta using a parametrized approach. For the SN Ia ejecta, rich in iron-group elements and H deficient, we adopted the 1D ejecta structure and composition of the delayed-detonation models DDC0, DDC15, and DDC25 from Blondin et al. (2013). To test the influence of the ejecta mass, we also included the sub-Chandrasekhar mass model SCH3p5 from Blondin et al. (2017). In the same model order, the 56 Ni mass prior to any decay is 0.86, 0.51, 0.12, and 0.30 M , and the ejecta mass is 1.41, 1.41, 1.41, and 0.98 M ( Table 1). The simulations of Mattila et al. (2005) and Botyánszki et al. (2018), to which we will compare our results, are based on the pure-deflagration W7 model of Nomoto et al. (1984). This model is very similar to our model DDC15 below an ejecta velocity of about 10000 km s −1 (see discussion in Blondin et al. 2015). Since this is the region that dominates the emission at nebular times, the relevant metal-rich part of our DDC15 model is analogous to that used in those two studies. For the stripped material, we used a similar approach to Mattila et al. (2005) and prescribed a mass (in the range from ∼ 10 −5 up to ∼ 0.5 M ) and defined the maximum velocity that limits its extension in space (typically 1000 km s −1 ). We assumed homologous expansion for both regions so there is a direct correspondence between velocity V and radius R for a given SN age t through R = Vt. The case of a very low mass of stripped material (i.e., 10 −5 M ) was used to gauge the impact of the stripped material in other models. Because these models have the same setup (grid, model atoms etc), flux subtraction can be used to reveal subtle offsets associated with the emission from the stripped material. This helps, for example, to reveal a global but weak flux offset, or the putative presence of a weak Hα line, in particular when it first appears.
To setup a model with stripped material, we took the SN Ia ejecta, cut out the inner regions below a chosen velocity limit (typically 1000 km s −1 ) and replaced what used to be metal-rich material with some H-rich material with a metal composition at the solar value. We also reset the density in this inner region so that it varies as 1/V 2 and yields the desired mass for the stripped material. The ejecta structure and composition of the SN Ia ejecta are left untouched above that velocity limit. To avoid sharp variations at this new interface, we applied a gaussian smoothing to the density and composition. A representative ejecta structure resulting from this procedure is shown in Fig. 1 for the SN Ia model DDC15 with 0.18 M of stripped material. Other models varying in SN Ia ejecta composition (DDC0, DDC25), mass (SCH3p5), or in the mass of stripped material, have a similar structure but shifted up and down from the profiles shown in Fig. 1. This setup is technically unphysical but operationally viable (see discussion in Sect. 6 where a comparison for two different stripped-material configurations is presented). The unphysical aspect arises since a real mass-stripping will yield material offset from the center of the explosion, asymmetrically distributed, and yielding line profiles whose width and centroid will be directionally dependent. This study remains worthwhile since the detectability at late times will not be affected by such considerations.
For the radiative transfer, we used the same technique as described in our SN Ia calculations (see for example Dessart et al. 2014b). However, the model atom was extended to include H and Co iiiv, and Ni iiiv. The model atom used for all these atoms and ions is as described by Dessart & Hillier (2011) and in the appendix A of Dessart et al. (2014b). The large model atom was used for Co ii and Co iii. This implies that our calculations treat 1.66 million bound-bound transitions. We also included the same set of decay routes as in Dessart et al. (2014b) and described in Appendix B. Thus, we included the two-step decay chain of 56 Ni, as well as other two-step and one-step decay chains (see Dessart et al. 2014b for details and decay constants used). Unlike in Dessart et al. (2014b), wherein a Monte Carlo transport solver was used for the γ rays, we here employ a simple pureabsorption model (with κ γ = 0.06 Y e cm 2 g −1 , where Y e is the electron fraction) for the computation of the γ-ray energy deposition (Swartz et al. 1995;Wilk et al. 2019b).
Since we focus on late times, the very high velocity ejecta are optically thin. We thus reduced the maximum velocity of our grid to 15000 km s −1 . In contrast, the stripped material at low velocity is still very dense at 100 − 300 d for the larger values of M st . This material has a continuum optical depth near unity, while Balmer lines remain optically thick, even at 300 d. The configurations explored here are thus distinct, from the point of view of the radiative transfer, from those of standard SNe Ia at a few hundred days. We will discuss these aspects in detail in the Results section below.
Unlike all previous simulations of SNe Ia with stripped material, which employ a Monte Carlo technique and use the Sobolev approximation, we treated the finite intrinsic width of lines and allowed for line overlap (Hillier & Miller 1998). We also introduced a turbulent velocity of 10 km s −1 . This value corresponds approximately to the thermal velocity of H atoms at 6000 K, but overestimates that of metals by a factor of a few. However, this is probably more physical for the radiative transfer (especially for H) than the zero intrinsic line width assumed in the Sobolev approximation. Botyánszki et al. (2018) argued that some of the differences between their results and those of Mattila et al. (2005) stemmed from the higher physical consistency of their approach. There is more than a decade between these two studies, which differ in many ways other than the adopted hydrodynamical structure (one problem is that there is little detail on the radiative transfer and its limitations in either of these studies). Botyánszki et al. (2018) do not compare their results to an equivalent 1D setup. The asymmetric distribution of the stripped material causes a velocity shift of the Hα line but it is not clear that it affects the total line flux, especially since they assume optically thin line formation. Furthermore, the hydrodynamical simulations of Botyánszki et al. (2018) predict 0.1 to 0.5 M of material stripped from the companion, as in previous studies. For lower masses of stripped material, their simulations are scaled-down versions and thus no longer physically consistent. It is not clear in this context that such adopted configurations are superior to the adopted ejecta of Mattila et al. (2005).
Although our work is based on parametrized 1D ejecta, we investigated the effect of a different distribution for the stripped material, by shifting it to larger velocities (outer bound at 2000 rather than 1000 km s −1 ; see Sect. 6). A relevant point to consider is what level of accuracy is needed to contribute to the current debate. Inferred masses of stripped material for ASASSN-18tb and SN2018 cqj suggest very low values of about 0.001 M (Kollmeier et al. 2019;Prieto et al. 2020). Such low values are incompatible with the 0.1 − 0.5 M stripped material masses expected from the standard single-degenerate scenario for SNe Ia, and this holds whether one uses the model results of Mattila et al. (2005) or those of Botyánszki et al. (2018).

Study of the "standard" M ch SN Ia model with stripped material
The left panel of Fig. 2 shows the results from our simulations for the DDC15 model, which is representative of a standard SN Ia with its 56 Ni mass of 0.51 M (other models are discussed in subsequent sections) with M st = 0.18 M . The main signature associated with the stripped material is the narrow Hα line. This is seen as a very weak feature at 100 d that progressively becomes the strongest line in the optical at 300 d, exceeding the strength of Fe iii 4658 Å, which is produced by the outer, faster moving metal-rich ejecta. We will not make a comparison to this Fe iii line in the rest of the paper because its strength varies with 56 Ni mass, the ionization level, clumping, and other factors, and it is thus not a robust physical probe. The Hα line width never changes because the H-rich material is by design confined to the innermost regions, below about 1000 km s −1 . This is done to reflect the results from multi-D hydrodynamical simulations (Marietta et al. 2000). The luminosity decrease occurs at all wavelengths but more slowly for the Hα line, which eventually becomes the strongest optical line.
The right panel of Fig. 2 shows the normalized flux for model DDC15 at 200 d but now for a range of stripped material masses covering from 0.0018 up to 0.58 M . Qualitatively, the evolution with increasing M st is similar to that in time for a fixed value of M st : the Hα line strengthens relative to the rest of the spectrum. However, the changes are now roughly at constant luminosity so only the Hα line changes with varying M st (this is the reason for plotting the normalized flux). The luminosity is only roughly constant because at a given time a greater decay power is absorbed for increasing stripped material mass (see Sect. 8.2).
The physical interpretation for these results has been discussed in the past (Mattila et al. 2005;Botyánszki et al. 2018), although results for multiple epochs have not been shown be-  while the bottom panel shows the contributions (ordinate in logarithmic space) for the full ejecta and those from the stripped material (blue) or the metal-rich ejecta (red). As is evident from these plots, the predominant emerging feature due to stripped material of the companion star at optical wavelengths is Hα.
fore. The power source is 56 Co decay. Of the total decay power absorbed, positrons (whose energy is assumed to be deposited locally) contribute 15% at 100 d and this contribution grows roughly linearly in time to 53% at 300 d. However, this positron contribution exclusively benefits the metal-rich ejecta, from which the γ-rays escape increasingly with time. For the stripped We show the total emergent luminosity L λ as well as that associated with the continuum and bound-bound transition of selected ions (H i, He i, O i, and Ca ii). The calculations for the latter are done by assuming that only the selected ion contributes (together with all continuum processes) and thus ignore the impact of other ions on that selected ion (such as metal-line blanketing on the H i lines arising from the innermost regions associated with the stripped material). The other three models show a similar behavior to that shown here for model DDC15. Hence, while the stripped, low-velocity material is expected to radiate quite strongly at many wavelengths, the primary observable feature at optical wavelengths is due to Hα. material, the situation is very different. Being free of 56 Ni (and 56 Co), it can only be powered by the non-local deposition of γray energy, which is not negligible because of the high density of this H-rich material (this efficiency is also boosted because the opacity to γ-rays is greater for H-rich material owing to its larger electron fraction, which is about 0.8 rather than 0.5 for symmetric nuclear matter). So, as time passes from 100 to 300 d, the metal-rich ejecta receive a decreasing fraction of the total decay power absorbed, leading to a relative strengthening of the emission from the stripped material and in particular Hα (we find that this line radiates typically 10% of the decay power absorbed by the stripped material; see Sect. 8.2). For an increased stripped material mass (see example in the right panel of Fig. 2), more decay power is absorbed and the Hα line flux increases relative to the emission from the metal-rich ejecta.
At 200 d for model DDC15, Hα is seen unambiguously for M st greater than about 0.01 M , and is essentially impossible to detect for M st below 0.001 M . An important property of this stripped material is that if Hα can be detected around 100 d, it should be detected for months thereafter since its relative strength to metal lines like Fe iii 4658 Å grows in time. A disappearance at late times is not expected in the context of stripped material from a non-degenerate companion. It could be caused by an insufficient decay power absorbed by the stripped material, as may occur if M st were low or very low, but this should compromise the Hα detectability at early times as well. Figure 3 shows how the different ejecta regions (indicated here in velocity space) contribute to the emergent luminosity (or flux) at a given wavelength. This illustration is exact in the sense that the sum of the flux contributions at a given wavelength give the total emergent flux at that wavelength. The stripped material contributes through the strong Hα line, as well as some weak and narrow metal lines. The continuum flux is so weak that even weak lines supersede it, with the exception of the spectral region around 6700 − 7000 Å (see also Chugai 1986). The metalline emission from the ejecta appears as broad and strong lines (mostly from Fe ii, Fe iii, and Ca ii). For Ca ii 7300 Å and some Fe ii lines, there is both a broad component from the metal-rich ejecta and a narrow component from the H-rich stripped material (in which all metal mass fractions are at the solar value). Figure 4 gives a complementary perspective to that of Fig. 3. It shows the flux that a given ion would produce in the absence of other ions. This is more artificial than the information presented in Fig. 3 because it ignores any cross-talk between ions, line overlap, and related optical depth effects. By identifying the differences between the full spectrum and the intrinsic contribution of each ion, however, one can assess the importance of optical depth effects (suggested early on by Chugai 1986 and later by Leonard 2007). These optical depth effects are unambiguously present since the individual ion spectra are rich in many narrow lines, yet none but Hα is obviously seen in the total optical flux. The continuum level is 10 to 100 times weaker than the total flux in the emergent spectrum, with an offset that decreases towards longer optical wavelengths. Some lines suffer from overlap with strong line emission from the metal-rich fast moving ejecta. This strong brightness contrast compromises the identification of the contribution from the stripped material. Other lines suffer from strong attenuation by the metal rich regions, like Hβ. In that case, the metal-rich ejecta act as a "curtain" that blocks the incoming radiation from the inner regions, as suggested by Leonard (2007). In terms of detectability in the emergent spectrum, the lines of interest sit in regions of lower opacity, and include Hα, He i 1.083 µm, H i 1.093 µm, and O i 1.129 µm and thus appear as strong narrow features. The resulting emission in the total emergent spectrum is roughly the sum of the individual contributions at the corresponding wavelength. The appearance of narrow lines is in part fortuitous because it arises from the non-uniform distribution of line opacity with wavelength. Overall, the region below 5000 Å remains thick until late times because of metal-line blanketing, even though the continuum optical depth of these regions is negligible. Consequently, lines like Hβ are attenuated persistently.
Such optical depth effects -and the associated absence of line features in the resulting spectrum -have not been treated or discussed by prior studies since strong, narrow features are seen in the predicted synthetic spectrum (see, for example, Botyán-szki et al. 2018). This may lead to artificially restrictive limits on the stripped material based on the non-existence of these lines in the observed spectrum (e.g., Tucker et al. 2019a).
We note that there are weak bumps on the red side of the strongest lines in the individual ion spectrum calculations (see, for example, the blue curve for H i lines in Fig. 4). The electron scattering optical depth is below, but close to, unity for the metalrich ejecta, and this is enough to cause one scattering with a free electron at a large velocity of many thousands of kilometers per second. The associated redshift yields an excess flux between about 10000 and 20000 km s −1 on the red side of the lines. This is compatible with the ejecta kinematics. The effect cannot be caused by other lines since the Hβ and Hα profiles can be overlaid nearly exactly (after a flux scaling). These bumps weaken and move to lower velocities (toward line center) with increasing time, being nearly absent at 300 d. All these properties are compatible with an electron-scattering origin.

Appearance of Hα and optical depth effects
In the simulation DDC15 with M st = 0.18 M presented above, the Hα line grows from being very weak relative to the rest of the spectrum at 100 d to becoming the strongest line (in terms of peak flux rather than line flux or equivalent width; see Sect. 9) in the optical at 300 d. This is shown more clearly in Fig. 5 where we isolate the Hα region. When inspecting the luminosity (top row), Hα is a small bump at 100 d. Its luminosity is very large but so is that of the metal-rich ejecta so the contrast between Hα and the overlapping Fe ii lines is small. To better reveal the presence of Hα, we plot in the bottom row the difference in luminosity between a given model and its counterpart in which M st = 10 −5 M . The Hα line is then unambiguously seen, even for M st = 0.0018 M .
At early times, the small luminosity contrast between metalrich regions and the stripped material is one limitation to identifying Hα. At such times, the γ-ray escape from the metal-rich regions is moderate so these regions outshine the inner H-rich ejecta. The contribution is more extended so both the regions rich in iron-group and intermediate-mass elements contribute, which leads to a more widespread presence of lines in the optical. With time, the metal-rich region traps less decay power, and the emission is biased inward in favor of the iron-rich regions, so holes appear in the continuum-free spectrum.
The other limitation is optical depth. There are contributions, of different magnitudes, from electron scattering and lines, and from the metal-rich SN Ia ejecta as well as from the H-rich stripped material. Figure 6 shows the evolution of the electronscattering and Rosseland mean optical depth at 1000, 2000, and 3000 km s −1 in the SN Ia ejecta of model DDC15 from 30 to 100 d after explosion. 1 Because of the significant impact of lines, the latter is well above the former, although one needs to bear in mind that the Rosseland mean is not an ideal representation of opacity in optically thin regions. Nonetheless, one sees that the conditions are only moderately optically thin even at 100 d and that they are optically thick at 50 d. Because of expansion, electron scattering redistributes the line flux but does not destroy photons, so this may not quench line emission from the inner  stripped material. Metal-line blanketing is in part absorptive but the non-uniform distribution of lines implies that even for optically thick conditions according to the Rosseland mean, some spectral regions may remain opacity holes. This is reflected by the strong variation of the photospheric velocity with wavelength and time (Fig. 7). One should bear in mind that the photosphere is not an opaque wall below which no photons escape. Instead, it represents the layer of "median" emission, in other words about 50% of the emerging photons come from below that layer and the rest from above. So, "opaque" regions (i.e., those located at an optical depth greater than unity) do contribute to the emergent radiation. Hence, Fig. 7 is merely illustrative of the non-uniform trapping efficiency through the optical range. Figure 6 does not show the opacity associated with the stripped material itself. For model DDC15 with M st = 0.18 M , the Rosseland-mean (electron-scattering) optical depth at the base of the ejecta (considering the whole column of material above the innermost ejecta layer at 200 km s −1 ) drops from 3.4 to 0.22 (from 2.54 to 0.17) between 100 and 300 d. This is a greater reduction than expected for homologous expansion (wherein τ ∝ 1/t 2 ) because of a simultaneous reduction in ionization. This dependence on ionization level usually applies when electron scattering dominates, but in some cases opacity can rise with decreasing ionization if this shift leads to a greater supply of optically thick lines.
An additional ingredient is the optical depth of the line itself (its intrinsic optical depth). The stripped material is quite abundant (representing up to a third of the SN Ia ejecta mass, for example, if we consider M st of 0.5 M ) and is also located at the lowest velocities, hence the highest densities. For model DDC15 with M st = 0.18 M , we find that Hα has an optical depth of 100 or more in its formation region over the period 100 − 300 d. The Hα optical depth becomes small in regions where the H mass fraction is small or negligible (thus contributing no emission). Hence, Hα is optically thick at all times (even if M st is reduced by a factor of 100). The same holds but to a smaller degree for Hβ. This implies that optically thin line formation for Balmer lines is not a valid assumption. This may be one origin for the differences in the results of Mattila et al. (2005) and those of Botyánszki et al. (2018).
To conclude, our simulations of stripped material in SN Ia ejecta suggest that the Hα line (and emission from the stripped material) suffers from a brightness contrast with the metal-rich ejecta as well as optical depth effects that are both intrinsic and extrinsic. As a consequence, the Hα line, which is the strongest signature from the stripped material, is essentially impossible to detect prior to 50 d (metal-rich ejecta are optically thick and too luminous), hard to detect at 100 d (metal-rich ejecta are not as optically thick but still very luminous) but increasingly strong as time progresses (metal-rich ejecta become increasingly transparent and faint). The easiest line to detect is Hα in part because it lies in a region largely devoid of strong or optically thick metal lines.
Optical depth effects also cause a blueshift of line profiles, which is most obviously seen through the location of peak flux in the line. This effect is well understood (Dessart & Hillier 2005) and routinely observed in a variety of SNe (Blondin et al. 2006;Anderson et al. 2014;Zhang et al. 2012). Here, the effect is present over the period 100 − 200 d after explosion (see Fig. 5 and the next section) for cases in which M st is greater than about 0.1 M . Hence, line shifts may not result exclusively from an asymmetric ejecta. This complicates the identification of multidimensional effects ).

Conclusive evidence for the identification of stripped material
A concern from the observations of Hα in SNe Ia at 100 − 300 d after explosion is whether the emission arises from the presence of stripped material in the innermost ejecta layers (the context explored here) or whether the emission is external to the ejecta and arises from interaction with the H-rich CSM (produced by wind mass loss and mass transfer in the progenitor system). The results described above provide some clues to this problem. The strong metal-line blanketing by the metal-rich ejecta thus offers a clear indication of the presence of stripped material since in that case only Hα should be seen. Hβ photons are destroyed by metal lines at all times prior to about 300 d and should not be seen if the emission arises from stripped material at low velocity. Hence, the detection of Hβ in SNe Ia suggests that the emission arises from regions outside the SN Ia ejecta, free of metal-line blanketing, and thus most likely arising from interaction with CSM. High signal to noise ratio (S /N) observations of the optical and in particular the Hβ region are essential to distinguish both scenarios.

Influence of the adopted stripped material velocity and density
In our simulations, we assumed spherical symmetry and adopted a centrally concentrated distribution for the stripped material. The outer bound of this region is around 1000 km s −1 . Multidimensional hydrodynamical simulations suggest that the stripped material can be offset to larger velocities, as well as offset from the ejecta center (Marietta et al. 2000;Pakmor et al. 2008;Liu et al. 2012;Pan et al. 2012;Botyánszki et al. 2018). To test this effect, we recomputed model DDC15 with M st = 0.18 M but using a distribution of stripped material centered around (rather than bounded by) 1000 km s −1 , extending from 200 km s −1 up to about 2000 km s −1 . The stripped material is now a hollow shell, occupying a larger volume, but still symmetric around the ejecta center. The gaussian smoothing does not conserve the mass exactly in the innermost layers (because of the density profile at the inner boundary) so the stripped material mass is 30% lower in the "offset" model. Figure 9 shows the impact of the stripped material distribution on the ejecta properties. With the offset, the stripped material is shifted to larger velocities, corresponding to lower densities (aggravated by the 30% lower stripped material mass). The electron density shows a broader bump for this H-rich material, with a slightly higher temperature, and a higher mean ionization for H (same ionization for He). The decay power absorbed by the stripped material is comparable in both models.
However, the Hα line in the "offset" model is twice as strong in total line flux and 30% stronger in peak flux (Fig. 10). The Hα line is also broader because of its formation at higher velocities. The change in strength likely arises from a formation at lower optical depth, both in the continuum and in the line. The for-  . It can thus be used to gauge the influence of the metal-rich ejecta on the emission from the stripped material. This influence is moderate or negligible for Hα, but very strong for Hβ. This figure shows that stripped material is not expected to yield any observable Hβ emission for hundreds of days because of metal-line blanketing from the metal-rich (SN Ia) ejecta, whereas Hα emission is prominent. mation over a larger volume probably contributes to desaturate the line since Hα photons are emitted over a broader range of velocities.
The choice of parametrized configurations that we use here, in the spirit of Mattila et al. (2005), can itself lead to significant variations in Hα line strength, so this needs to be borne in mind when estimating the stripped material mass from observations. The factor of two found here, however, is much smaller than the difference between predicted values for M st in the singledegenerate scenario and the inferred values for M st from observations.

He i lines
He i lines are non-thermally excited in our simulations and are predicted to be present but weak in the He i-only spectra that we compute (Fig. 4). In the total emergent spectrum, the weak feature at 5900 Å is due to Na i D, while He i 5875 Å is absent (left panel of Fig. 11). The only unambiguous He signature is He i 10830 Å in the near infrared (right panel of Fig. 11) because this line is usually the strongest under similar SN ejecta conditions (Li et al. 2012). Metal-line blanketing is also weaker in the near infrared and the stripped material is only a few times  9. Impact of the adopted density profile for the stripped material on the gas properties in model DDC15. This model has a M st of about 0.15 M (the value is 30% lower in the model with offset) and at a SN age of 200 d. The simulations are 1D (i.e., assume spherical symmetry), but the simulation "No-offset" uses a more centrally concentrated mass distribution than the simulation "Offset" (see the electron-density profile, which reflects closely the difference in mass density profile). In the inset of the top-middle panel, the normalization of the cumulative decay power absorbed is set to the value obtained for the model with no offset (and limited to the velocity region shown).  This result is not surprising since the properties of the stripped material in a SN Ia ejecta at such late times are analogous to those found in Type II SN ejecta at the same epoch. In general, He i lines are not observed or predicted in optical spectra of type II SNe at nebular times (see, for example, Silverman et al. 2017, Jerkstrand et al. 2012; the models of Dessart et al. (2013) predict the presence of He i 7065 Å and overestimate its strength, but the problem is caused by the adoption of a high turbulent velocity not used here; Dessart & Hillier in prep.).
Our results disagree with the predictions of Botyánszki et al. (2018). This may arise from their neglect of optical depth effects (especially in the optical). We also find that a significant power absorbed by the stripped material emerges in the continuum (or in lines that are later reprocessed by the metal rich ejecta), while they seem to assume that the gas can only cool through (optically thin) line emission.

Results for the grid of models
We now turn to the presentation of results for the whole grid of simulations. We have performed radiative transfer calculations at 100, 150, 200, 250, and 300 d for models DDC0 ( 56 Ni mass is 0.86 M ), DDC15 ( 56 Ni mass is 0.51 M ), DDC25 ( 56 Ni mass is 0.12 M ), and SCH3p5 ( 56 Ni mass is 0.30 M ) and stripped material masses of 0.0018, 0.018, and 0.18 M . For each SN Ia model and epoch, we also computed the cases of a very low stripped material mass of 10 −5 M to facilitate the assessment of the contribution of the stripped material when its influence is weak. For models DDC0, DDC15, and DDC25, additional simulations were also done to better cover the region around M st = 0.1 M . In total, the grid comprises more than a hundred simulations and encompasses a much wider range of values in 56 Ni mass and M st than studied so far. Our Hα luminosities are lower by a factor of five to ten than those of Botyánszki et al. (2018), which are given at a SN age of 200 d (green solid line in Fig. 12). This offset probably stems in part from the differences in the hydrodynamical structure of the ejecta, since their computed ejecta configurations tend to have the stripped material offset at a higher velocity than adopted in our work (see results and discussion in Sect. 6). It is likely that differences also result from optical depth effects (Sect. 4), or differences in the model atom and metal composition for the stripped material (metals can play an important part in the cooling of the gas even at low abundance). This yields a lower Hα luminosity by up to a factor of two at times prior to 200 d, but makes little difference later on (see Fig. 8 and discussion in Sect. 5).

Hα luminosity versus decay power absorbed
We find that the Hα line radiates about 10% (extending from 5% up to 30% for a few outliers) of the decay power absorbed by the stripped material (Fig. 13) (Fig. 14).
Here, we quote the Hα flux from an H i spectrum calculation, thus without any influence from the metal-rich ejecta. By doing this, we can gauge the cooling power of Hα for the stripped For most models, Hα radiates about 10% (the distribution covers from 5% to 30% due to several outliers, all for a low stripped material mass) of the decay power absorbed by the stripped material (this material is 56 Ni deficient and thus the power stems from non-local γ-ray energy deposition). Fig. 14. Decay power absorbed by the stripped material vs. its mass for our set of models. As before, the color coding distinguishes the SN ages, which cover from 100 to 300 d. The decay power absorbed is greater at earlier times (because the total decay power is greater earlier) or for greater masses of stripped material (because of the greater trapping). However, this does not imply that the Hα line is more easily seen earlier since the SN luminosity is also greater. The line luminosity may be high but the equivalent width, or pseudo-equivalent width, may be small and even null.
material. This implies that 90% of the decay power absorbed by the stripped material is radiated by other means. A fraction of this power goes in lines, and the rest in continuum radiation (see Fig. 4 for one representative case). Alternatively, we could quote the Hα luminosity from the total emergent spectrum. This is more useful to compare to observations, but prevents a proper evaluation of the cooling power of Hα since the Hα emission from the stripped material is reprocessed by the metal-rich ejecta. The fraction L(Hα)/ė abs (H-rich) of about 10% holds over three orders of magnitude in decay power absorbed by the stripped material. This is lower than the value of 30% found by Botyánszki et al. (2018) for most of their models. These effects can produce an Hα luminosity that evolves very differently from that expected for 56 Co decay, even though 56 Co decay is the power source for the emission from the stripped material.

Evolution of the Hα luminosity
We have seen that the Hα luminosity represents about 10% of the decay power absorbed (Fig. 13). This dependence implies a connection to the 56 Co characteristic decay time. However, there are a number of complications that can yield an Hα luminosity that deviates from the evolution of the decay power emitted or absorbed. These complications can be, for example, a different luminosity evolution for the metal-rich ejecta and for the stripped material, combined with the evolution of the blanketing effect from the metal-rich ejecta. The top and middle panels of Fig. 15 show the evolution of the Hα luminosity for M st of 0.0018, 0.018, and 0.18 M , first based on the H i spectrum and then based on a difference between the model counterparts with M st = 10 −5 M . This evolution is smooth and qualitatively similar for all cases, largely irrespective of 56 Ni mass. However, the models with lower M st decline faster, along a nearly constant slope (in the log, so in magnitude). The models with a higher M st show a plateau from 100 to 200 d (or even a rise if the flux difference is used), and then decline, but more slowly. The different behavior seen at early times in both panels arises from optical depth effects. It shows that up to 200 d, a very slow decline in the Hα luminosity can arise even if the power source is 56 Co decay; it does not necessarily imply CSM interaction, as proposed by Vallely et al. (2019).
When normalized to the bolometric luminosity (which is equal to the total decay power absorbed at these late epochs), the Hα luminosity is constant or growing for models with a high M st but decreases with time otherwise. This is in part related to the fact that at low M st , γ-rays are not efficiently trapped by the stripped material, while the SN Ia ejecta are increasingly powered by local positron energy (which the stripped material cannot receive since it is 56 Ni deficient). A corollary is that the stripped material allows the SN to be more luminous since it contributes to trapping γ-rays that would have otherwise escaped. This is not necessarily seen in Hα, but may yield a continuum flux excess throughout the optical. This is visible in the bottom-row panels of Fig. 5 where the models with higher M st are offset to higher luminosities at all wavelengths, not just in strong lines like Hα.
In the upper panel of Fig. 15, we computed the line luminosity from the H i spectrum in order to understand how this luminosity evolves without the corrupting effect of photon reprocessing by the metal-rich ejecta. However, this reprocessing is moderate after about 200 d for Hα. After that time, the results shown in Fig. 15 reflect closely the evolution of the emergent Hα luminosity (see Fig. 8). Figure 16 shows the ratio of Hα and Hβ line luminosities computed from the H i-only spectrum. Apart from cases of very low M st , the ratio is not strongly dependent on M st but it varies strongly with time, increasing from about 2 at 100 d up to 10−20 at 300 d. Such a Balmer decrement is much greater than the value of 2.86 for Case B recombination (Osterbrock & Ferland 2006) but the conditions here are also very different from those in a photoionized nebula.

Hα versus Hβ luminosities
A fundamental difference in SNe Ia with stripped material is that the nebula is powered not by ionizing photons from a hot central star but instead from within the ejecta and by radioactive decay. The associated non-thermal effects influence both the excitation and ionization of H and other elements. The process is therefore quite different from photoionization followed by recombination.
The detectability of Hβ in the total emergent spectrum is another issue. As discussed earlier, the blanketing by the metal-rich ejecta affects the Balmer decrement severely in our simulations.  Fig. 16. Ratio of the Hα to Hβ luminosity for our set of models, adopting only H i lines in the spectrum calculation. Values are scattered, reflecting the influence of intrinsic optical depth effects and the non-thermal process of formation of these lines. This ratio ignores optical depth effects caused by other elements such as iron. Consequently, in the total emergent spectrum (resulting from the contribution of all elements and ions), this ratio is much greater, and potentially infinite.
Consequently, Hβ is absent or very weak in our simulations of the total emergent spectrum.

Equivalent width of the Hα line
Unlike luminosity, the equivalent width (EQW) of an emission line -or, more commonly here, an upper limit on its possible strength -is a quantity that is directly derivable from an observed SN spectrum. That is, it is independent of the SN brightness, distance, and extinction, all of which are required to establish the luminosity of such a line or to place limits on it. Indeed, for all but two of the more than 100 nebular-phase SNe Ia that have so far had their spectra scrutinized for evidence of stripped material, it is an upper limit on the EQW of a line that has served as the fundamental, measured parameter that is derived from the spectrum itself.
Because our models compute the full, nebular-phase SN Ia spectrum in addition to the expected emergent line luminosities produced by stripped material, they permit estimates of specific line EQWs to be made. This can potentially be used to directly assess -or to place limits on -the amount of stripped material entrained in an SN ejecta without requiring contemporaneous photometry or estimates of its distance and extinction.
With this in mind, we measured the EQW of the Hα linehereafter EQW(Hα) -in all of our Chandrasekhar-mass model spectra using a procedure designed to mimic its derivation from an actual SN spectrum. First, to approximate the "continuum" underlying the emission line, points on the spectrum shortwards and longwards of the Hα feature were chosen by hand and connected with a linear fit. 2 The line's emission profile was then normalized through division by this fitted continuum. We then calculated EQW(Hα) by taking the normalized flux, subtracting one from it, and summing the flux up over the range 6541 − 6585 Å.
This range encompasses nearly all of the line emission (Fig. 5), as it includes ±1000 km s −1 from the nominal line center. We display the results in Fig. 17 for all models containing more than 0.001 M ; for the two models that contained less than this -M st = 1.7 × 10 −4 M and M st = 1.7 × 10 −5 M -the Hα line was very difficult to measure (if apparent at all), and in all measured cases yielded EQW(Hα) < 1 Å. From the figure, three trends are immediately apparent. First, EQW(Hα) increases with M st at all epochs and for all models. Second, EQW(Hα) increases with decreasing 56 Ni mass at all epochs. This arises primarily because of the reduced light contamination from the SN ejecta for lower 56 Ni mass (this contamination corresponds to the "pseudo-continuum" by which we normalize the flux when measuring the EQW). Finally, it increases with time for all models with M st ≥ 0.059 M , consistent with our earlier finding that when sufficient stripped material exists it receives an increasing fraction of the decay power relative to the metal-rich ejecta (Sect. 3). However, EQW(Hα) does exhibit more complicated temporal behavior for the two models with lower M st . While it generally increases up to day ∼ 200, it levels off, or even decreases, beyond that time. As discussed in Sect. 8.3, this is likely related to the decreasing efficiency of γ-ray trapping by the stripped material at low M st . We provide analytical fits to the correlations between EQW(Hα) and M st in Appendix B, along with a suggested prescription for deriving conservative limits on the line's possible strength in observed SN Ia spectra.
The essential results of this exercise can be summarized succinctly. First, if a late-time SN Ia spectrum is obtained with sufficient sensitivity to rule out any Hα emission for which EQW(Hα) ≥ 1 Å, a confident limit can be set on the amount of stripped material of M st < 0.001 M , which is well below the amount expected in the single-degenerate scenario. Second, balancing the changing effects of relative Hα strength, a fading SN and line luminosity, and opacity effects at low M st , suggests that the ideal time period to obtain spectra and seek Hα emission lies between days ∼ 150 and 200.

Conclusions
We have presented a grid of radiative transfer calculations for SN Ia ejecta that enclose some stripped material from a nondegenerate companion star. Our set of models covers from faint to luminous SNe Ia, to test the influence of 56 Ni, as well as M ch and sub-M ch progenitors to test the influence of the ejecta mass on the results. The ejecta structure is spherically symmetric and parametrized, but uses some constraints from multidimensional hydrodynamics simulations. We extend previous calculations by covering a broader parameter space, by treating the non-LTE radiative-transfer problem in detail. Optical depth effects are treated. The influence of the fast-moving metal-rich ejecta on the radiation emanating from the slower moving stripped material is treated. We cover a range of stripped material masses, from the values of 0.1 − 0.5 M obtained in multidimensional hydrodynamics simulations down to the low values of about 0.001 M that have been inferred from observations. Our simulations suggest that the emission from the stripped material first appears in the emergent spectrum some time between 50 and 100 d after explosion for the largest values of M st . This delay is wavelength dependent and results from a combination of effects. Metal-line blanketing blocks the radiation from the stripped material in the region below about 5000 Å as well as in isolated regions (for example over the near infrared Ca ii triplet). This prevents the emergence of Hβ photons even at 300 d (the associated Balmer decrement is thus infinite). In contrast, Hα emerges much earlier (for example around 100 d for M st ∼ 0.1 M ) because it sits in a spectral region that is relatively free of optically thick metal lines. The same holds for lines like He i 1.083 µm or O i 1.129 µm. The second effect is related to the brightness contrast between the emission from the stripped material and that of the overlying metal-rich ejecta. At earlier times, the metal-rich ejecta outshine the stripped material and makes its detection challenging. The problem is less severe for Hα because it is intrinsically the strongest optical emission line radiated by the stripped material but also because it is located in a spectral region where the metal-rich ejecta have a moderate brightness. Because of these two effects, Hα remains the main observable signature from the stripped material at any ultraviolet, optical, or near-infrared wavelength.
At a given time, the Hα luminosity increases with M st because of the greater decay power absorbed by the stripped material (in all our simulations, about 10% of this power is radiated by Hα). Being powered by radioactive decay, the Hα luminosity generally decreases with time, but this drop is steeper for lower M st because these configurations are less efficient at trapping γrays. Growing γ-ray escape from the metal-rich ejecta favors the strengthening of Hα, which becomes the strongest optical line in our simulations at 300 d for large M st . Prior to 200 d, the Hα luminosity can exhibit different behaviors (plateau, rise, or drop), primarily because of optical depth effects. The EQW(Hα) is found to increase with M st , increase with decreasing 56 Ni mass, and generally increase with time except at the lower values of M st . Optical depth effects significantly impact the radiation from SN Ia ejecta with stripped material. The Balmer lines are intrinsically optically thick even at 300 d, and emission from the stripped material is also attenuated by the metal-rich ejecta. Optical depth effects caused by the metal-rich ejecta are a fundamental feature of SN Ia with stripped material. These effects are in a large part absent from SN Ia ejecta interacting with CSM since the emission is external to the ejecta rather than deeply embedded within it (optical depth effects influence the receding part of the ejecta but leave intact the front part).
There seems to be a number of ways to distinguish SN Ia ejecta with stripped material and those with CSM interaction. With CSM interaction, Hα is observed earlier (it may be observed at any time), Hβ is often detected even at the earliest times (see for example Hamuy et al. 2003 andSilverman et al. 2013), the CSM revealed by high-resolution spectroscopy moves slowly (100 km s −1 ; Kotak et al. 2004), and there may be the presence of symmetric electron-scattering wings on Hα. With 0.1 − 0.5 M of stripped material as predicted by hydrodynamical simulations (see for example Marietta et al. 2000), Balmer lines cannot be seen much earlier than 100 d after explosion, Hα should strengthen with time relative to the rest of the optical spectrum, Hβ and higher transitions in the series should not be detected during the first year, and Hα should be broader but cannot exhibit symmetric electron scattering wings.
Overall, our simulations are in rough agreement with the previous works of Mattila et al. (2005) and Botyánszki et al. (2018). However, our results suggest that optical depth effects produce some important signatures and thus should not be neglected. We also predict weak or absent He i lines for stripped material with a solar composition, with the exception of He i 10830 Å, in tension with the numerous optical He i lines predicted by Botyánszki et al. (2018). This may be caused by the different non-thermal and non-LTE treatment, and the neglect of optical depth effects.
The junction between optical and near infrared ranges might also reveal some interesting properties. Since metal-line blanketing is weak in this region, the stripped material can be seen through H i, He i, or O i lines. It is not clear whether such lines are predicted in the case of CSM interaction or whether they have ever been observed in SNe Ia with Hα detection.
The hydrodynamical models of stripped material from a companion in a SN Ia seem unsuited to explain the observations of SNe Ia with an Hα detection (this discrepancy, however, could be resolved by adopting different, and perhaps more physical, initial conditions for the companion, non-degenerate star at the time of explosion; see for example Justham 2011). Even with the large uncertainties inherent to the radiative transfer calculations, these inferred masses are typically 100 times smaller than predicted by hydrodynamical simulations (Kollmeier et al. 2019;Prieto et al. 2020). Figure 18  An alternative scenario for the production of Hα is interaction with an extended H-rich CSM, so that Hα is detected some time after the SN Ia explosion and ceases after a few months once the CSM has been swept up completely by the ejecta. Variations in CSM density structure (or wind mass rate of the progenitor) could yield a wide range of Hα luminosities, both in value at a given time and in the evolution of this value until late times. Another possibility, tied to the double-degenerate scenario, is that the inferred 0.001 M of H-rich material at low velocity could come from a swept-up giant planet (Soker 2019), or from a non-degenerate star in a triple system (Thompson 2011;Kushnir et al. 2013;Vallely et al. 2019).
The properties of SNe Ia with Hα emission reveal an intriguing dichotomy. Events that exhibit strong Hα are associated with SNe Ia having a high peak luminosity (or high 56 Ni mass), interacting with a dense H-rich CSM, and are located in relatively young stellar populations. In contrast, the two events that exhibit weak Hα emission are associated with SNe Ia with faint peak luminosity (or low 56 Ni mass), have fast-declining light curves, and are located in relatively old stellar populations. There is a dearth of events in between these two extremes. Detectability might play a role here (detecting a weak Hα is easier at low 56 Ni; see Sect. 9), but this is probably not the core reason. A distinct progenitor or explosion scenario may be the cause (see, for example, Kollmeier et al. 2019;Vallely et al. 2019).
A final issue not directly addressed in studies of SNe Ia with Hα detection is whether Chandrasekhar-mass ejecta are suitable to explain the nebular properties of the SN in terms of brightness, color, ionization, line ratios, and other factors. There is indeed growing evidence that the majority of SNe Ia are better explained by the properties of sub-Chandrasekhar-mass ejecta (see, for example, van Kerkwijk et al. 2010;Kromer et al. 2010;Kushnir et al. 2013;Pakmor et al. 2013;Scalzo et al. 2014;Blondin et al. 2017Blondin et al. , 2018Botyánszki et al. 2018;Shen et al. 2018;Flörs et al. 2019;Polin et al. 2019;Wygoda et al. 2019). In Fig. 18, a sub-Chandrasekhar-mass model yields a better match to the observed optical spectrum and brightness than a Chandrasekharmass model with a comparable 56 Ni mass. The single-degenerate scenario, intimately tied to the Chandrasekhar-mass for the exploding white dwarf, seems to struggle both in matching the observed Hα line strength and the optical radiation from the metalrich ejecta.

Appendix A: Analytical fits to the correlation between Hα luminosity and M st
It is useful to perform polynomial fits to our results for the Hα luminosity for the various models, epochs, and stripped material masses. Maintaining the same approach as used in Botyánszki et al. (2018), we fit a second order polynomial to the distribution of Hα luminosity versus M st at each epoch and for each model (which reflects a given 56 Ni mass). The results for the polynomial coefficients are given in Table A.1 and an illustration of the fit for model DDC15 at 200 d (which corresponds to the configuration closest to that of Botyánszki et al. 2018) is shown in Fig. A.1.  DDC15,and DDC25 at 100,150,200,250, and 300 d. The polynomial has the form Y = a 0 + a 1 X+a 2 X 2 , where Y = log 10 (L(Hα) diff / erg s −1 ) and X = log 10 (M st /M ).

detection-limit prescription
We have fit polynomials to our results for EQW(Hα) in a manner identical to those fitted to its luminosity (see Appendix A). Figure B.1 provides an illustration of the fit for our standard explosion model DDC15 at 200 d. The polynomial coefficients for all models are reported in Table B.1.  Table B.1 gives the definitions of a 0 , a 1 , a 2 . Past research tells us that it is nearly always an upper limit, and not a detection, that is established on the strength of the Hα line. The procedure for estimating an upper limit on a line's possible strength -as measured by the EQW -has evolved over the years. The basic methodology was proposed by Leonard & Filippenko (2001) based on the seminal work of Hobbs (1984). The technique was then implemented specifically for the case