Comparative Direct Analysis of Type Ia Supernova Spectra. II. Maximum Light

In the first paper of this series (Branch et al. 2005, hereafter Paper I), the parameterized supernova synthetic‐spectrum code SYNOW was used to fit and interpret a time series of spectra of the well‐observed, spectroscopically normal Type Ia SN 1994D. In this paper, we concentrate on near–maximum‐light spectra of 24 Type Ia SNe (SNe Ia). Maximum‐light spectra are of particular interest because they are the ones that are most likely to be obtained, especially for high‐redshift SNe Ia. In § 2 the spectra are quantified and arranged to illustrate relationships between them. In § 3 to § 6, four groups of spectra are discussed, and selected comparisons with SYNOW synthetic spectra are shown. Results are discussed in § 7.

As in Paper I, we restrict our attention to optical spectra, from the Ca ii H and K feature in the blue (∼3700 Å) to the Ca ii infrared triplet (Ca ii IR3) in the red (∼9000 Å). The 24 spectra selected for study were obtained within 3 days of B‐band maximum light (which occurs about 20 days after explosion; Riess et al. 1999), a time interval short enough that spectroscopic evolution is not strong, yet long enough to allow some events of special interest to be included. All spectra have been corrected for the redshifts of the parent galaxies. The original spectra have a range of slopes, owing to intrinsic differences, differences in the amount of reddening by interstellar dust, and observational error. In this paper, because we are interested only in the spectral features, and not in the underlying continuum slopes, the spectra are tilted by multiplying the flux by λ^α, where α has been chosen to make the flux peaks near 4600 and 6300 Å equally high. The tilting makes it much easier to compare the spectral features. The events of the sample are listed in Table 1, together with parameters discussed below.

Various ways of arranging the spectra according to measurements of selected features were explored. The arrangement shown in Figure 1 appears to be useful. The W(5750) parameter is the equivalent width of the absorption near 5750 Å, usually attributed to Si ii λ5972, and W(6100) is that of the stronger absorption near 6100 Å, produced by Si ii λ6355. (Because of the presence of emission features, these are equivalent‐width–like quantities rather than true equivalent widths.) The wavelength limits of integration are chosen by eye² in the same way as they are for measuring the R(Si ii) parameter, which is the ratio of the fractional depths of these two absorptions (Nugent et al. 1995), and the flux used as the divisor of the integral is the mean of the fluxes at the wavelength limits. The ratio of W(5750) to W(6100) is correlated with, but not equal to, R(Si ii). For our present purpose of arranging the spectra, the measurement errors are not important, even though the fractional error of W(5750) is large when W(5750) is small.

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Plotting W(5750) against W(6100) illustrates the relationships according to only two spectral features. To quantify the relationships over a broader wavelength range, the spectra were interpolated onto a standard wavelength grid with a 2 Å separation, and for each pair of spectra, a χ²‐like figure of merit that was formed from the logarithmic flux differences between 3700 and 6500 Å (or as much of this interval as possible) was minimized by scaling the flux and varying the tilt (Appendix). The results identify the nearest neighbor of each event. In Figure 1, a single‐headed arrow points to a nearest neighbor (e.g., SN 2002bf is the nearest neighbor of SN 1984A) and a double‐headed arrow connects events that are mutual nearest neighbors.

Sections 3 to 6 address four groups of spectra, denoted by different symbols in Figure 1. The group assignments were made on the basis of Figure 1 and on the appearance (depth, width, and shape) of the 6100 Å absorption feature. It should be mentioned here that although we consider four groups, we conclude in § 7 that for the most part, the spectra appear to have a continuous distribution of properties, in which case, exactly how the group boundaries are defined is not an important issue.

Figure 2 shows spectra of seven SNe Ia that are tightly clustered in Figure 1 and have similar 6100 Å absorptions, not only in equivalent width but also in shape. Most other features, even weak ones, are also similar. The members of this highly homogeneous group are referred to here as "core‐normal" SNe Ia. The only conspicuous difference among them is the exceptionally strong high‐velocity Ca ii IR3 absorption near 8000 Å in SN 2001el. Some differences in the Ca ii H and K absorptions at the extreme short‐wavelength ends of the spectra also are noticeable.

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In Branch et al. (2003) and in Paper I, SYNOW fits to the spectra of two core‐normal SNe Ia, SN 1998aq and SN 1994D, were presented and discussed. Descriptions of the SYNOW code and its use can be found in these two papers and references therein. For the present work, we adopted precepts for fitting spectra of core‐normal SNe Ia that differ mildly from those of previous papers. For photospheric‐velocity (PV) ions, the line optical depth e‐folding velocity v_e is 1000 km s⁻¹. For most detached high‐velocity (HV) ions, v_e = 2000 km s⁻¹. (HV Ca ii in SN 2001el is an exception.) The excitation temperature T_exc is 10,000 K. The ions used are PV O i, Mg ii, Si ii, Si iii, S ii, Ca ii, Fe iii, Co ii, and Ni ii, and HV Ca ii and Fe ii. In Paper I, Na i also was used, but the identification (at maximum light) was not considered to be definite. In this paper, Na i is not used for core‐normal SNe Ia, on the grounds that if Na i were present in core‐normal SNe Ia, then it should be much stronger in cooler SN 1991bg–like events, which is not the case. Lines of C ii are also not used, because if present at maximum light, they are weak and difficult to confirm (although they appear to be present at maximum light in at least SN 1998aq [Branch et al. 2003] and will be used in D. Branch et al. [2006, in preparation, hereafter Paper III] on pre–maximum‐light spectra). In Paper I, HV Fe ii was not used in near–maximum‐light spectra. In this paper, HV Fe ii is used³ because we now have stronger evidence for its presence, as discussed below. The SYNOW fitting parameters for the core‐normal SNe Ia are listed in Table 2.

In Figure 3, the spectrum of SN 1994D is compared with two synthetic spectra. As usual, the blackbody continuum of SYNOW makes the synthetic spectrum too high from about 6500 to 7800 Å. (The continuum of SNe Ia cannot be modeled as a single blackbody curve, but an attempt to model the exact continuum shape at the SYNOW level of analysis would be an unrewarding fitting exercise.) One synthetic spectrum includes HV Fe ii, detached at 16,000 km s⁻¹, and the other does not. The main effect of HV Fe ii is produced by two lines of multiplet 42, λ5018 and λ4924. (The third line of the multiplet, λ5169, is in this case overwhelmed by Si ii absorption.) In SN 1994D and most other events of the sample, an absorption near 4430 Å, which we believe to be produced by Si iii λ4550, is not well fitted because the Si iii synthetic absorption is too blue. The fit could be improved by lowering v_phot and slightly detaching all ions other than Si iii, but this would not help us to understand the discrepancy, so we have refrained from doing it. Because the synthetic spectra are usually too high in the red, and because of the presence in some spectra of the telluric absorption near 7600 Å, it is difficult to know how to fit the absorption produced by O i λ7773 (and Mg ii λ7890), so little significance should be attached to the fitting parameters used for O i.

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In Figure 4, the spectrum of SN 2001el is compared with a synthetic spectrum in which HV Ca ii has a high optical depth (τ = 12, compared to τ = 2–5 for other core‐normal SNe Ia), a high value of v_e (12,000 km s⁻¹), and a maximum velocity of 30,000 km s⁻¹. The high value of v_e makes the absorption flat and extends it to the maximum velocity. This fit is not unique, but it demonstrates that HV Ca ii can account for the feature. Polarization observations indicate that asymmetry, which is not taken into account in SYNOW, affects HV Ca ii line formation in SN 2001el (Wang et al. 2003; Kasen et al. 2003). The best fit for SN 2001el is obtained with an HV Fe ii optical depth that is the highest among the core‐normal SNe Ia. Since SN 2001el has strong HV Ca ii absorption, this is regarded as support for the HV Fe ii identification. The normality of all features not affected by HV Ca ii and HV Fe ii indicates that the PV spectrum of SN 2001el is very much like those of the other core‐normal SNe Ia.

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Most of the spectral features of the core‐normal SNe Ia have established line identifications. Ions whose presence is considered to be definite are PV Si ii, S ii, Ca ii, O i, Mg ii, and Fe iii, and HV Ca ii. Co ii and Si iii are probable, while Ni ii is possible but difficult to establish. We return to the issue of HV Fe ii in § 6 below. The issue of C ii will be addressed in Paper III, on premaximum spectra. A few of the weak features of the core‐normal SNe Ia are not reproduced by the SYNOW spectra. Possible identifications of the weak absorptions near 5500 and 5150 Å (seen most clearly, and labeled, in Fig. 3) are Si iii λ5740 and Co ii λ5311, respectively. The weak absorptions near 4150 and 4550 Å (seen clearly in Fig. 2 and labeled with question marks in Fig. 3) remain unidentified (but see § 7 on the 4550 Å absorption).

Figure 5 shows the spectrum of the core‐normal SN 1994D (for comparison) and that of seven SNe Ia that are spectroscopically normal in the traditional sense that their strongest spectral features are the usual ones, rather than the conspicuous Fe iii features of SN 1991T or the blue Ti ii trough of SN 1991bg. However, these seven "broad‐line" SNe Ia have 6100 Å absorptions that are broader and deeper than those of the core‐normal SNe Ia, which puts them to the right of the core‐normal SNe Ia in Figure 1. The spectra of SN 1992A and SN 1981B are not very different from core‐normal, but their lines are broader; e.g., notice the lack of resolution in the absorption complex from 4600 to 5100 Å. Therefore, we regard SN 1992A and SN 1981B as small steps from the core in the direction of SN 1984A. The spectra in Figure 5 do not appear to comprise a simple one‐dimensional sequence. In SN 2002bf, the absorption near 7500 Å is broad and shallow. SN 2001ay has a weak Ca ii IR3 absorption and steep pieces of spectra near 4000 and 4200 Å. (Extreme photometric peculiarities of SN 2001ay are discussed by Howell & Nugent [2004].)

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The fitting parameters for the broad‐line SNe Ia are given in Table 3. The ions are the same as for the core‐normal SNe Ia. Reasonable fits were obtained with v_phot = 12,000 km s⁻¹, a typical value for core‐normal SNe Ia, and higher values of v_e: 2000 km s⁻¹ for both PV and HV ions, or higher as indicated in Table 3. For consistency with the modeling of the core‐normal SNe Ia, PV and HV components were used for Ca ii, but these broad Ca ii features also can be fitted as single, high‐v_e components, so the Ca ii fitting parameters are far from uniquely determined.

In Figure 6, the spectrum of the moderately broad‐line SN 2002bo is compared with two synthetic spectra. HV Fe ii, with a high optical depth of 1.5, affects the synthetic spectrum in several places. In Figure 7, SN 1984A is compared with synthetic spectra in which Si ii has v_e = 5000 km s⁻¹ and a maximum velocity of 25,000 km s⁻¹. The quality of these two fits is about the same as those of the core‐normal SNe Ia.

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Line identifications in the broad‐line SNe are the same as in the core‐normal SNe Ia. Given the nonuniqueness of SYNOW fits (e.g., to an extent, lowering v_e can be compensated by raising the optical depth), most of the differences between the optical depths used for the broad‐line and the core‐normal SNe Ia are not necessarily significant.

Figure 8 shows spectra of the core‐normal SN 1990N and five "cool" SNe Ia, four of which have a conspicuous absorption trough from about 4000 to 4400 Å, due in part to PV Ti ii (Filippenko et al. 1992a; Mazzali et al. 1997; Garnavich et al. 2004). SN 1989B, on the other hand, is spectroscopically normal in the traditional sense and is not far from core‐normal, but its 6100 Å absorption is deeper than in the core‐normal SNe Ia. Its W(6100) value is similar to that of SN 1981B, which is one of the broad‐line events of § 4, but the features of SN 1989B are better resolved than those of SN 1981B. Note, for example, the well‐resolved region between 4600 and 5100 Å in SN 1989B. The 6100 Å absorption of SN 1989B is almost identical to that of SN 1986G. The best‐fitting value of v_phot for SN 1989B is lower than that of the core‐normal SNe Ia and is the same as that of the other members of the cool group (Table 4). For these reasons, we suspect that SN 1989B is a small step from the core in the direction of SN 1991bg–like spectra, and it is included with the cool SNe Ia, even though it does not have the blue absorption trough.

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The fitting parameters for the cool SNe Ia are listed in Table 4. SN 1989B can be fitted with parameters that differ only mildly from those of the core‐normal SNe Ia. For the other SNe Ia in Figure 8, the PV excitation temperature is lowered from 10,000 to 7000 K, in view of the development of Ti ii lines, which require low temperature. In addition to Ti ii, five more ions, Na i, Mg i, Ca i, Sc ii, and Cr ii, are introduced. Garnavich et al. (2004) used Mg i and Ca i in their study of SN 1999by, and Sc ii is also plausible at low temperature. (Sc ii lines appear in SNe II when they become sufficiently cool.) Si iii and Fe iii require high temperatures, so they are not used for the cool SNe Ia.

The spectrum of SN 1986G cannot be well fitted with the precepts that were adopted for the core‐normal SNe Ia; when the 6100 Å absorption is fitted, the 5750 Å absorption, which is strong in the observed spectrum, hardly appears in the synthetic spectrum. One plausible solution is to impose a maximum velocity on Si ii. In Figure 9, the spectrum of SN 1986G is compared with a synthetic spectrum in which Si ii has a maximum velocity of 15,000 km s⁻¹ and a high optical depth of 100. Owing to the maximum velocity, the 6100 Å absorption saturates, and owing to the high optical depth, the 5750 Å absorption appears.

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It has been suggested (Garnavich et al. 2004) that Ti ii may contribute to the 5750 Å absorption in SN 1991bg–like spectra, but we find that when the Ti ii optical depth is chosen to fit the blue trough, Ti ii makes no significant contribution to the 5750 Å absorption. (If the Ti ii optical depth is raised, or if an improbably high value of T_exc is used, an absorption due to Ti ii lines does appear near, but not near enough, to the 5750 Å absorption.) On the basis of detailed spectrum calculations with the PHOENIX code (Baron et al. 2003 and references therein), S. Bongard et al. (2006, in preparation) also conclude that Ti ii makes no contribution to the 5750 Å absorption. Again, a plausible solution is to impose a maximum velocity on Si ii. This is shown in Figure 10 for SN 1991bg; in the synthetic spectrum the Si ii maximum velocity is 14,000 km s⁻¹, and the optical depth is 200.

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In view of the large number of ions that may be present, line identifications in the cool SNe Ia (other than SN 1989B) are a challenge. Si ii, Ca ii, Mg ii, Ca ii, and Ti ii can be regarded as definite. (Contrary to previous opinion, S ii cannot, because Sc ii has lines that could be wrongly attributed to S ii.) As shown in Figures 9 and 10, Cr ii, Sc ii, Mg i, Ca i, Na i, Co ii, and Ni ii can be combined to obtain reasonable fits; all of these ions are plausible, but none are definite.

Figure 11 shows spectra of the core‐normal SN 1994ae and five "shallow‐silicon" SNe Ia that have smaller values of W(5750) and W(6100), thus occupying the lower left corner of Figure 1. Despite their shallow Si ii absorptions, SN 1999ee and SN 1999aw are spectroscopically normal in the traditional sense, and SN 1999ee is not far from core‐normal. SN 1991T and SN 2002cx are spectroscopically peculiar, because Fe iii produces the strongest features (Filippenko et al. 1992b; Ruiz‐Lapuente et al. 1992; Jeffery et al. 1992; Mazzali et al. 1995; Fisher et al. 1999; Li et al. 2003; Branch et al. 2004b).

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In Figure 12, the spectrum of SN 1999ee is compared with that of the core‐normal SN 2001el. SN 1999ee has weaker Si ii and S ii features, but its HV Ca ii is almost as strong as that of SN 2001el (Mazzali et al. 2005a), and the features attributed to HV Fe ii are about as strong as in SN 2001el. This is additional support for the HV Fe ii identification.

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The fitting parameters for the shallow‐silicon SNe Ia are given in Table 5. The value of v_e is 1000 km s⁻¹, except where indicated otherwise in the table. In Figure 13, the spectrum of SN 1999ee is compared to a synthetic spectrum. The ions used are the same as in the core‐normal SNe Ia, but v_e = 4000 km s⁻¹ for HV Fe ii and 6000 km s⁻¹ for O i and HV Ca ii.

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Figure 11 shows that the spectra of SN 1999ee, SN 1999aw, and SN 2000cx have many similarities at wavelengths longer than about 4300 Å. However, compared to SN 1999ee and SN 1999aw, SN 2000cx is strongly depressed from about 3800 to 4200 Å. Until now, SN 2000cx has been thought to be unique among all SNe Ia observed so far, due at least in part to flux blocking by HV Ti ii (Branch et al. 2004a).

In Figure 14, the spectrum of SN 2000cx is compared to a synthetic spectrum in which Si ii is mildly detached at 15,000 km s⁻¹. As in Branch et al. (2004a), HV Ti ii is used to provide blocking in the blue and to account for the sharp flux peak near 4150 Å. Branch et al. were unable to suggest an identification for the distinct absorption feature near 4530 Å, except for HV Hβ, which would be surprising. However, as shown in Figure 14, we now realize that HV Cr ii can account for this feature. If the HV Ti ii identification is correct, then HV Cr ii, at practically the same detachment velocity, is a more plausible identification than Hβ. In the synthetic spectrum of Figure 14, HV V ii is also introduced, but its main effect is in a crowded spectral region, and the identification is not definite.

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SN 1991T may be a continuation of the sequence shown in Figure 11; e.g., it bears some resemblance to SN 1999aw. As further discussed in § 7, even though SN 1991T and SN 2002cx are mutual nearest neighbors spectroscopically, in most other respects they are quite different. It may be that the only thing these two have in common is a high temperature.

The high degree of spectral homogeneity among the core‐normal SNe Ia suggests that they result from a standard, repeatable SN Ia explosion mechanism that does not produce spatially large compositional inhomogeneities near the characteristic photospheric velocity of 12,000 km s⁻¹ (Thomas et al. 2004). The strong HV Ca ii of SN 2001el may involve a spatially large structure near 20,000 km s⁻¹ (Kasen et al. 2003; Kasen & Plewa 2005). On the other hand, the features attributed to HV Fe ii are only mildly enhanced in SN 2001el and rather homogeneous in the others, suggesting that weak HV Fe ii is "natural" in SNe Ia, in the sense that it does not require unusual high‐velocity structures (Höflich et al. 1998; Lentz et al. 2001b). Weak HV Ca ii is also ubiquitous (Mazzali et al. 2005b).

Based on previous work (Branch 1987; Benetti et al. 2004) we expected to fit the broad‐line SNe Ia with high values of v_phot, but better fits were obtained by holding v_phot constant at 12,000 km s⁻¹, a typical value for the core‐normal SNe Ia, and using increased v_e‐values. This may be a consequence of nuclear burning extending to a higher velocity in the broad‐line than in the core‐normal SNe Ia (Lentz et al. 2001a; Benetti et al. 2004; Stehle et al. 2005).

Although the compositional structures of the cool SNe Ia surely differ from those of the core‐normal SNe Ia (e.g., the low luminosity requires a low mass of ⁵⁶Ni), the primary reason for the spectroscopic differences appears to be the lower temperature. The blue trough in cool SNe Ia, usually referred to as the Ti ii trough, is produced not only by Ti ii, but also by other ions that appear at low temperature. Ti ii is not significant for the 5750 Å absorption. The 5750 Å absorption of the cool SNe Ia can be accounted for in terms of a high Si ii optical depth, together with a maximum Si ii velocity. This is a plausible way to simultaneously fit these two absorptions while explaining why R(Si ii) and the ratio of W(5750) to W(6100) go the wrong way with temperature (i.e., increasing with falling temperature), considering that the lower level of λ5972 is the upper level of λ6355. If correct, this maximum velocity interpretation would indicate that in the cool SNe Ia, nuclear burning extends only to unusually low velocities, near 15,000 km s⁻¹.

In most of the shallow‐silicon SNe Ia, the spectral features appear to be produced by the same ions as in the core‐normal SNe Ia, although the optical depths are quite different, especially in SN 1991T and SN 2002cx. Most of the differences from core‐normal SNe Ia may be caused by a higher temperature. Like SN 2001el among the core‐normal SNe Ia, SN 1999ee provides support for the HV Fe ii identification. The cause of the depression of the spectra of SN 2000cx in the blue from about 3800 to 4200 Å appears to be extra HV line blocking. If HV Ti ii is present, then HV Cr ii is a more plausible identification than HV Hβ (Branch et al. 2004a) for the 4530 Å absorption in SN 2000cx. As noted in § 3, a less distinct absorption appears near 4550 Å, even in the core‐normal SNe Ia; HV Cr ii is a possibility.

One of the most interesting issues that can be addressed by studies of this kind is whether SNe Ia have a continuous distribution of properties, or whether they consist of discrete subgroups. On the whole, we have a sense of continuity. SN 1992A and SN 1981B may be small steps from the core in the direction of SN 1984A; SN 1989B may be a step in the direction of SN 1986G and SN 1991bg; SN 1999ee may be a step in the direction of SN 1991T. Most SNe Ia may be variations on the basic core‐normal theme, with the diversity reflecting differences of degree, rather than differences of kind. If this is correct, then how narrowly the boundaries of the core‐normal group are defined is not important.

The cool SNe Ia (apart from SN 1989B) do stand apart from the others in Figure 1, but this in itself does not establish that they differ in kind from the others. As noted by Hatano et al. (2002) and Höflich et al. (2002), there is a temperature threshold below which, owing to abrupt changes in key ionization ratios, line optical depths change abruptly (Hatano et al. 1999). Therefore, the temperature interval required to go from SN 1989B through SN 1986G to SN 1991bg–like events may be narrow. SN 1991bg–like events are also quite subluminous, but especially in the B band, which is strongly blocked by the blue trough. Similarly, there is a temperature threshold above which Fe iii features become conspicuous (Hatano et al. 2002). A larger sample will be required to determine whether there is continuity extending from the core‐normal to the cool SNe Ia.

Evidence has recently been presented showing that in terms of stellar population age, there are two SN Ia groups of roughly equal size, one from a young (∼10⁸ yr) population and another from an older (∼3 Gyr) population (Mannucci et al. 2005, 2006; Scannapieco & Bildsten 2005). However, the SNe Ia of our sample do not manifestly split into two distinct groups of roughly equal size. It may be that there are two different binary star evolution paths to SNe Ia, one requiring a short time and the other a long time, but with the two paths producing similar outcomes (i.e., carbon‐oxygen white dwarfs that explode as they approach the Chandrasekhar mass).

The one event of our sample that does seem to be of a different kind is SN 2002cx, which we have placed in the shallow‐silicon group. Like SN 1991T it had conspicuous Fe iii lines, but unlike SN 1991T it did not have a slow light‐curve decline rate, and it was subluminous (Li et al. 2003). Its late‐time spectra were unlike those of normal SNe Ia, being dominated by very low velocity (∼700 km s⁻¹) permitted lines of Fe ii, and perhaps O i. Several other SN 2002cx–like events have been discovered (Jha et al. 2006). On the basis of the first‐season observations by Li et al., Branch et al. (2004b) suggested that SN 2002cx may have been a pure deflagration, while most other SNe Ia, even those like SN 1991bg (Höflich et al. 2002), are delayed detonations (or gravitationally confined detonations; Plewa et al. 2004; Kasen & Plewa 2005). However, the late‐time spectra of SN 2002cx (Jha et al. 2006) do not resemble the spectra calculated by Kozma et al. (2005) for one particular three‐dimensional deflagration model, so the nature of SN 2002cx remains uncertain. Weaker three‐dimensional deflagration models should be considered for SN 2002cx.

The normality of PV features in events such as the core‐normal SN 2001el and the shallow‐silicon SN 2000cx, both of which have strong HV features, suggests that HV and PV are somewhat independent (with HV Fe ii and weak HV Ca ii being ubiquitous and natural, as discussed above). An interesting issue for further study is how much of the photometric diversity of SNe Ia is caused by HV features (which may not correlate with luminosity). Another issue is whether the strong HV blue blocking in SN 2000cx occurs only in the shallow‐silicon group. Strong HV Ca ii IR3 absorption occurs not only in the core‐normal SN 2001el, but also in the shallow‐silicon SNe Ia.

Our study is complementary to that of Benetti et al. (2005, hereafter Be05), who did not consider the strengths of spectral features. Sixteen events in our sample were included in the Be05 study, which assigned SNe Ia to three groups, on the basis of absolute magnitude M_B, the light‐curve decline rate Δm₁₅ (Phillips 1993), the blueshift of the 6100 Å absorption 10 days after maximum, V₁₀(Si ii) (Branch & van den Bergh 1993), the rate at which the 6100 Å absorption drifts to the red, v˙(Si ii) (Be05), and R(Si ii). With a few borderline exceptions (SN 1989B and SN 1992A), our cool SNe Ia correspond to the Be05 FAINT group, which is to be expected, since both temperature and luminosity are controlled mainly by the ⁵⁶Ni mass. Our broad‐line group corresponds to the Be05 HVG (high temporal velocity gradient) group. This also makes sense, because broad 6100 Å absorption requires high Si ii optical depth over a substantial velocity range, which makes it possible for the absorption minimum to shift appreciably with time. The broad‐line SNe Ia may have thicker silicon layers than the core‐normal SNe Ia. Our core‐normal and shallow‐silicon SNe Ia correspond to the Be05 LVG (low temporal velocity gradient) group. The lower velocity range over which Si ii has a high optical depth permits only a smaller shift in the absorption minimum with time.

Polarization observations (Wang et al. 2001, 2003; Howell et al. 2001; Leonard et al. 2005) show that some spectral features are affected by asymmetric structures. A straightforward way to rule out asymmetry as the sole cause of the differences between groups of SNe Ia would be to demonstrate a correlation with parent galaxy type (Be05) or other indicators of the parent galaxy population (Gallagher et al. 2005). If asymmetry were the sole cause of SN Ia differences, SN Ia subtypes would not correlate with galaxy types. But in fact there are correlations and emerging correlations. SN 1991bg–like events tend to occur in older populations (Howell 2001). In the Be05 sample, the mean host galaxy morphological type (the T‐type) was found to increase from the FAINT to HVG to LVG groups. Similarly, for our sample the mean T‐type increases from cool to core‐normal to broad to shallow SNe Ia, but the statistics are too poor to allow definite conclusions.

The expected rapid increase in the number of well‐observed SNe Ia in the near future (e.g., Wood‐Vasey et al. 2004) will allow the relationships among SNe Ia to be clarified. We expect that our Paper III, on premaximum spectra, will also clarify the view presented here.

In order to quantify difference in spectral features, we use the scaled and tilted spectrum expressed by flux values y˜_ℓi for wavelengths λ_i, parameterized as

where the y_ℓi‐values are the observed spectral flux values of spectrum ℓ, the factor S_ℓ is a scale factor, and the factor (λ_i/λ₀)^α_ℓ produces tilt; λ₀ is just a reference wavelength that need not actually be specified. (This scaling and tilting of spectra is more elaborate than the tilting we describe in the main text [see § 1] for spectra that we present and fit with SYNOW spectra.)

For two spectra labeled by indices 1 and 2, we formulate the χ²‐like function

where

To minimize the χ², we find the partial derivatives

Setting these equations equal to zero and solving, we find the s‐ and α‐values that give the stationary point of χ²:

Since the stationary point is unique and χ² must have an analytic global minimum with respect to s and α, equation (A8) gives the global minimizing values of s and α.