Analyzing the Largest Spectroscopic Data Set of Hydrogen-poor Super-luminous Supernovae

Table 1 lists the dates of maximum light, which were provided by the published papers in different filters for the different SLSNe Ic. Most of the rest-frame filters are in u, B, g, r, or R bands. Since there are no established relationships for the peak epochs in different filters of SLSNe Ic, we could not convert these peak epochs to the same filter. Since there is no common filter in which light curves of all SLSNe Ic in our sample are available in the literature, we could not compute peak epochs in the same filter. Based on the relationship between peak epoch in U, B, V, and ${\rm{R}}/{\rm{r}}^{\prime}$ bands for stripped-envelope core-collapse SNe from Bianco et al. (2014), we roughly estimate that compared to peak epoch in V-band (${\mathrm{JD}}_{\mathrm{vmax}}$), the peak epochs used in our analyses will be off by −3 days (peak happened before the ${\mathrm{JD}}_{\mathrm{vmax}}$) and 2 days at the most (peak happened after the ${\mathrm{JD}}_{\mathrm{vmax}}$). This should not impact our comparisons of average spectra (see Section 3.2 and sections that used average spectra of SLSNe Ic), since we use a bin size of five days for each average spectrum. Nor should this have much influence on our analysis of the bulk velocity evolution (see Section 4.1), since we calculate the moving weighted average velocities smoothed with a Gaussian kernel with a standard deviation of five days when studying the velocity evolution of the different SN populations.

The different SN samples in this work are at vastly different redshifts with means of ${z}_{\mathrm{Ic},\mathrm{mean}}=0.02$, ${z}_{\mathrm{Ic} \mbox{-} \mathrm{bl},\mathrm{mean}}=0.09$, and ${z}_{\mathrm{SL},\mathrm{mean}}=0.47$ and medians of ${z}_{\mathrm{Ic},\mathrm{median}}=0.01$, ${z}_{\mathrm{Ic} \mbox{-} \mathrm{bl},\mathrm{median}}\,=0.03$, and ${z}_{\mathrm{SL},\mathrm{median}}=0.33$, thus a valid question is whether the different redshift coverage has any impact on our results. At the broadest level, this should not influence our results too much, since all SN spectra are de-redshifted into the rest frame, and all spectral comparisons are made for the same spectral ranges (e.g., the expected wavelength range for the Fe ii λ5169  line) that are observed in almost all SN spectra in our sample. However, five SLSNe Ic in our sample that have $z\gt 0.9$ do not cover the Fe ii λ5169  region when their spectra are de-redshifted into the rest frame, since the Fe ii λ5169  line will be at infrared wavelengths in the observed frame. Moreover, the higher redshifts of the SLSNe Ic sample does somewhat impact our comparison of average spectra (see Section 3.2), since those of SLSNe Ic have a larger UV and blue coverage in the rest frame (∼3000–7000 Å) than those of SNe Ic (∼4000–8000 Å). Nevertheless, for the important blend of O ii at ∼4300 Å seen in spectra of SLSN I, we have spectra of SNe Ic and SNe Ic-bl at the same rest wavelengths (see Figure 3).

The next question is whether evolutionary effects for their progenitors is expected to be significant within such a redshift range. While the metallicity content of the universe does not change significantly between the redshift ranges for the SN Ic and the SN Ic-bl samples (both samples are within the extent of the star-forming galaxies of SDSS; Tremonti et al. 2004), there is a slight trend toward lower metallicity for galaxies of the same mass at the median redshift of z = 0.3 for SLSNe Ic (e.g., Zahid et al. 2011; Maier et al. 2015), compared to local star-forming galaxies. Indeed, SLSNe Ic host galaxies seem to be of low metallicity (Chen et al. 2013, 2017; Lunnan et al. 2013, 2014; Perley et al. 2016; Schulze et al. 2016), even lower than those of SNe-GRBs (Leloudas et al. 2015), which in turn seem to be lower than those of SNe Ic and SNe Ic-bl without GRBs (Modjaz et al. 2008, 2011; Kelly & Kirshner 2012; Modjaz 2012; Sanders et al. 2012). However, low metallicity should not affect the velocities of the Fe ii λ5169  line per se. Even the Fe ii λ5169  line strengths should be unchanged for our sample of SLSNe Ic, since line strength changes only become important when the metallicity is by factors of 10–100 lower (Sauer et al. 2006), which is a much larger factor than observed based on host galaxy data.

Absorption velocities can provide clues about the dynamics of the explosion. In particular, the absorption velocity of Fe ii λ5169  has been suggested to trace photospheric velocity by Branch et al. (2002).⁴ In turn, the measurement of the photospheric velocity is needed to estimate explosion parameters, such as ejecta masses, from light curves and spectra (Drout et al. 2011; Cano 2013; Nicholl et al. 2015b; Taddia et al. 2015; Lyman et al. 2016). Moreover, the ejecta velocity, as traced by Fe ii λ5169,  can be used to check the prediction of the one-dimensional magnetar model since the model predicts a flat evolution of velocity over time (Kasen & Bildsten 2010; Mazzali et al. 2016).

3.1.1. Velocity from the Template Fitting Method

Since the Fe ii features in SLSNe Ic are highly blended, as in SNe Ic-bl, we measured Fe ii λ5169  velocities via the template fitting method (Appendix A of Modjaz et al. 2016), originally developed for spectra of SNe Ic-bl. First, we identify the spectral region of Fe ii λ5169  in SLSNe Ic in our sample based on identifications and SYNOW fits given for the same SLSNe Ic in the literature, as well as spectral modeling for some SLSNe Ic in Dessart et al. (2012b). Then we measure Fe ii λ5169  velocities in SLSNe Ic via the template fitting method described in Appendix A of Modjaz et al. (2016). This method is a data-driven method and the template is the average spectrum of SNe Ic. In our case, we would like to compare the velocities of the non-blended Fe ii λ5169  in SNe Ic with the velocities of the blended Fe ii λ5169  in SNe Ic-bl and SLSN I. We obtain the Fe ii λ5169  velocities in SLSNe Ic by matching a broadened and blueshifted SN Ic template to an SLSN Ic spectrum at similar phases, which is performed in a Markov-chain-Monte-Carlo (MCMC) framework. As discussed in Appendix A of Modjaz et al. (2016), the broadness and blueshift of the line are quantified by the convolution velocity of the Gaussian kernel with which the SN Ic template is convolved and the amount of additional blueshift, respectively. The corresponding error bars are based on the marginalized distribution of parameter values from MCMC samplers. An example of our velocity measurements for SLSNe Ic is shown in Table 2 for guidance.

Table 2.  Measured Absorption Velocities

Phase with respect to maximum light	Fe ii λ5169
(days)	(km s−1)
LSQ12dlf
8	$-{18500}_{-1200}^{+1300}$
15	$-{16900}_{-2600}^{+2600}$
20	$-{13100}_{-1300}^{+1300}$
33	$-{10700}_{-1400}^{+1400}$
43	$-{5000}_{-800}^{+800}$

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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We could not measure Fe ii λ5169  velocities in all 178 spectra of 32 SLSNe Ic in our sample. We only focus on spectra taken during the photospheric phase, which includes 155 spectra, using ${t}_{\max }=70$ days as the threshold. Some SLSNe Ic have multiple spectra at the same phase. In these cases, we included the spectrum that has the largest wavelength coverage or the highest signal-to-noise ratio. As a result, 122 out of 155 spectra are at unique phases. Out of the 122 spectra of 32 SLSNe Ic, we could measure Fe ii λ5169  velocities in 72 spectra of 21 SLSNe Ic, since 10 spectra are too noisy to identify the Fe ii λ5169  feature, 16 spectra do not cover the Fe ii λ5169  region, and 23 spectra have no significant Fe ii λ5169  absorption. We could not use the template fitting method to measure the Fe ii λ5169  velocity in SN 2007bi at phase 48 days either, since the width of its Fe ii λ5169  profile is narrower than that in the SNe Ic average spectrum, though the shape of the former is similar to the latter. If we use the identification method, which is used to measure the Fe ii λ5169  velocity in SNe Ic (Liu et al. 2016), the velocity we obtain for SN 2007bi at phase 48 is at the high end of all SLSN Ic velocities at similar phases. However, for consistency, we did not include the Fe ii λ5169  velocity for SN 2007bi from identification method in our analyses.

We note that the line feature at the expected position of Fe ii λ5169, which is around 5000 Å, can be contaminated by Fe iii lines at early phases if the temperature is sufficiently high. For example, through detailed comparison with identifications and SYNOW fits given for literature SLSNe Ic, Nicholl et al. (2013, 2016) identified Fe iii lines around 5000 Å shortly after peak in PTF12dam, PS1-11ap, SN 2010gx, and SN 2015bn. While, in Table 2, we report all Fe ii λ5169  velocities in SLSNe Ic, assuming that the feature is not contaminated, we are cautious with those measured at ${t}_{\max }\lt 10$ days. Thus, we do not report weighted average Fe ii λ5169  velocities in SLSNe Ic for comparison with other SN types for ${t}_{\max }\lt 10$ days. The fact that for some SLSNe Ic, those velocities measured at ${t}_{\max }\lt 10$ days appear to be increasing, instead of decreasing, indicates that there is certainly some contamination at early times.

3.1.2. Velocity Comparisons with Literature Measurements

Mean spectra and the standard deviation contours can characterize the spectral properties of each SN subtype as a population. They can also be used to check whether spectra of a newly discovered transient are within the statistical diversity of the mean spectra of some SN types at certain phases. In order to compare the average spectra of various SN subtypes, as we will show in Section 4.2, we have used continuum-removed average spectra of SN Ic and of SN Ic-bl from Liu et al. (2016) and Modjaz et al. (2016), and constructed average spectra of SLSNe Ic in the same way. In order to compare two relatively special SLSNe Ic in our sample to average spectra of various SN subtypes, as we will show in Section 6, we have also constructed average spectra where the continuum is included for SN Ic-bl, SN-GRB, and SLSN Ic at needed phases. We have done our best to correct for Milky Way extinction in SLSN Ic spectra using $E(B-V)$ values from the literature (Schlegel et al. 1998; Schlafly & Finkbeiner 2011). We did not correct for host extinction, since SLSNe Ic are generally in metal-poor galaxies, and thus the host extinction is small in most cases (Lunnan et al. 2014; Leloudas et al. 2015). All of the average spectra of SLSNe Ic that we have constructed are available on our group GitHub repository.⁵

Although the temperature in the photosphere of SLSNe Ic affects the presence and strength of spectral lines, we did not group SLSNe Ic by temperature when we construct their average spectra, since at the date of maximum light, SLSNe Ic generally have a blackbody temperature (T_BB) between 10,000 and 16,000 K, except for iPTF13ehe (∼7000 K; Yan et al. 2015), PS1-10awh (∼20,000 K; Chomiuk et al. 2011), and ASASSN-15lh (∼22,000 K; Dong et al. 2016).

We compared the Fe ii λ5169  velocities in SLSNe Ic, SNe Ic-bl, and SNe Ic, since Fe ii λ5169  is a common line in these SN subtypes and has been suggested to trace the photospheric velocity (see the footnote in Section 3.1). As discussed in Section 3.1.1, we measured the Fe ii λ5169  velocities via template fitting as was done in Modjaz et al. (2016) to measure the velocities of Fe ii λ5169  in spectra of SNe Ic-bl. The absorption velocity evolution of the Fe ii λ5169  line for all individual SLSNe Ic in our sample is shown in the left panel of Figure 1, while the weighted average absorption velocities of Fe ii λ5169  for SLSNe Ic, SNe Ic-bl, and SNe Ic are shown in the right panel. We calculated the average velocities as a Gaussian smoothed weighted average, where the weights correspond to the errors in the velocity measurements, and the Gaussian kernel has a standard deviation of five days. This provides a smoother velocity curve for each SN subtype than the method used in Liu et al. (2016; their Appendix D), where the smoothing was achieved with a boxcar kernel (i.e., a rolling mean). In addition, in Liu et al. (2016) and Modjaz et al. (2016), the measurements for each SN were first binned at five-day granularity, while here each velocity measurement is used. Since Fe iii lines may contaminate the Fe ii λ5169  line in SLSNe Ic at early phases, we disregarded measurements at ${t}_{\max }\lt 10$ days when computing the weighted average velocity values for SLSNe Ic.

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There is a lot of diversity in Fe ii λ5169  absorption velocities for SLSNe Ic. For example, at ${t}_{\max }\simeq 10$ days, PTF11rks has velocities around 20,000 km s⁻¹, while LSQ14mo has velocities around 10,000 km s⁻¹. However, we found no systematic difference in velocities for SLSNe Ic that had fast-declining light curves compared to those that had slow-declining light curves.⁶ The error bars on the weighted average velocities, which are calculated as the weighted standard deviations of the data and thus represent the diversity of the SNe velocities, are larger for SLSNe Ic and SNe Ic-bl than for SNe Ic, especially at early times. However, we note that error bars in all these SN subtypes are about 20%–30% of the absolute values. Moreover, the scatter in the weighted average measurement may be affected by the inaccurate measures of the date of maximum, since we have to use different filters, in which the date of maximum was measured, for different SLSNe Ic (see Table 1 and Section 2.1). Thus, the intrinsic diversity in Fe ii λ5169  velocities of SLSNe Ic, SNe Ic-bl, and SNe Ic could be smaller than that indicated by the error bars shown in the right panel of Figure 1. Modjaz et al. (2016) showed that, although there is some overlap between Fe ii λ5169  absorption velocities in SNe Ic-bl and SNe Ic, the weighted average velocities in SNe Ic-bl are systematically higher than those in SNe Ic at almost all phases. Here, we have found similar patterns in Fe ii λ5169  absorption velocities between SLSNe Ic and SNe Ic. Although the Fe ii λ5169  velocities of individual SLSNe Ic are not clearly separated from velocities of individual SNe Ic, as a population, SLSNe Ic have Fe ii λ5169 velocities that are higher on average than those in SNe Ic at almost all phases and are similar to those in SNe Ic-bl. In particular, as shown in Table 3, the weighted average Fe ii λ5169 absorption velocities of SLSNe Ic and SNe Ic-bl are about two times that of SNe Ic at ${t}_{\max }=10$ days. While we can measure the absorption velocity of this line in SLSNe Ic only starting at ${t}_{\max }=10$ days, which at that phase has an average value of −15,000 (±2600) km s⁻¹, we note that this implies that the absorption velocity of SLSNe Ic at the date of maximum light, which is usually the one used for ejecta mass estimates, will have to be even higher. Moreover, Jerkstrand et al. (2016) found that SLSNe Ic and SNe Ic-bl have similar nebular-phase spectra in terms of velocities. These similarities in observations indicate that SLSNe Ic and SNe Ic-bl may have similar explosion engines, which is consistent with a multi-dimensional magnetar model in Suzuki & Maeda (2017), where they claimed that SLSNe Ic and SNe Ic-bl could be produced by a similar engine with different energy injection rates. If SLSNe Ic and SNe Ic-bl also have similar ejecta masses (which may not be the case, e.g., Nicholl et al. 2015b), then they may have similar kinetic energies, given they have broadly similar photospheric velocities as traced by Fe ii λ5169  line.

Table 3.  Weighted Mean Absorption Velocity of the Fe ii λ5169  and Full Width at Half Maximum (FWHM) of the Convolved Gaussian Kernel with the SN Ic Template, at ${t}_{\max }\simeq 10$ Days

SN type	${V}_{\mathrm{absorption}}$	${V}_{\mathrm{conv}\mathrm{FWHM}}$ with Respect to SNe Ic
	(103 km s−1)	(103 km s−1)
SNe Ic	−8.0 ± 1.4	NA
SNe Ic-bl	−18.5 ± 7.4	8.9 ± 2.1
SLSNe Ic	−15.0 ± 2.6	11.3 ± 3.6

Note. The errors are the weighted standard deviations of data that contribute to the weighted average value, which are indicated as the error bars in the figure that show the weighted average values.

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We note that weighted average velocities of Fe ii λ5169  in SLSNe Ic, SNe Ic-bl, and SNe Ic converge to ∼−7000 km s⁻¹ starting at ${t}_{\max }=35$ days. Thus, when we compare absorption velocities of different subtypes, we focus on measurements taken at ${t}_{\max }\lt 35$ days.

We compared spectral features in SLSNe Ic, SNe Ic-bl, and SNe Ic by comparing continuum-removed average spectra at different phases. We divided the continuum out from raw spectra in order to remove the impact of any potential reddening caused by dust and to focus on spectral features. The left panel of Figure 2 compares continuum-removed average spectra of SLSNe Ic with those of SNe Ic, while the right panel compares SLSNe Ic with SNe Ic-bl. Obviously, SNe Ic have stronger and narrower features than SLSNe Ic at many wavelengths. In particular, from 4000 to 6000 Å at ${t}_{\max }=0$, 10, and 20 days, SLSNe Ic mean spectra are up to two standard deviations away from the SNe Ic mean spectra. As listed in Table 3, at ${t}_{\max }=10$ days, the average full width at half maximum (FWHM) of the Gaussian kernel convolved with the SN Ic template is around 11,000 km s⁻¹. In contrast, SLSNe Ic and SNe Ic-bl have similar features in terms of width and strength at most wavelengths, though there are mismatches around 4600 and 6000 Å at ${t}_{\max }=-10$ and 0 days, as well as around 4300 Å at ${t}_{\max }=-10$ as we discuss in detail in Section 4.3. We checked that the broad lines are present in individual SLSN Ic spectra, and thus the broad lines in average spectra are not an artifact of the averaging process. We note that, although there is no clear separation between SLSNe Ic (or SNe Ic-bl) and SNe Ic in terms of width of Fe ii λ5169, SLSNe Ic and SNe Ic-bl have on average systematically broader Fe ii λ5169  lines than SNe Ic (Table 3). The fact that the mean (and standard deviation) spectra of SLSNe Ic and SNe Ic-bl overlap at many wavelengths in spectra at photospheric phases again indicates that SLSNe Ic and SNe Ic-bl may have similar dynamics. This is important because SLSNe Ic are usually compared to SNe Ic (e.g., Pastorello et al. 2010; Inserra et al. 2013). Actually, the average spectrum of SLSNe Ic at ${t}_{\max }=-10$ days is similar to the average spectrum of SNe Ic at ${t}_{\max }=30$ days, thus, we have partly validated the claim in Pastorello et al. (2010) and Inserra et al. (2013), namely that SLSN Ic spectra look similar to SNe Ic with a phase lag of about one month. However, more importantly, at the same phase, we have found that certain spectral characteristics (such as line widths and Fe ii λ5169  velocities) of SLSNe Ic are more similar to those of SNe Ic-bl than those of SNe Ic. We note that a large difference between SLSNe Ic and SNe Ic-bl spectra is that the continuum in SLSNe Ic is much bluer than in both SNe Ic-bl and SNe Ic (not shown in Figure 2).

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The narrow "w" feature around 4300 Å in SLSN Ic spectra before ${t}_{\max }=0$ day is observed and identified as O ii lines (even sometimes seen as a number of distinct features; Quimby et al. 2011; Gal-Yam 2017) in the literature (Quimby et al. 2007, 2011; Gal-Yam 2012; Nicholl et al. 2015b; Mazzali et al. 2016; Gal-Yam 2017). We have checked the spectra of all 32 SLSNe Ic in our sample, and found that all SLSNe Ic have a narrow "w" feature around 4300 Å, except for 12 SLSNe Ic that do not have spectra taken before the date of maximum light, 4 SLSNe Ic that do not have spectra covering the 4300 Å region, and 1 SLSN Ic (SN 2011kl) with a spectrum that is too noisy. We have also checked the spectra of all 23 SNe Ic-bl and 17 SNe Ic in our sample. There are 18 SNe Ic-bl and 13 SNe Ic that have spectra before ${t}_{\max }=0$ days and cover the 4300 Å region. However, none of them show a similar narrow "w" feature as seen in SLSN Ic spectra.⁷

We show spectra of all SLSNe Ic, SNe Ic-bl, and SNe Ic in our sample that were taken around ${t}_{\max }=-10$ days in Figure 3. The shaded area indicates the expected positions of the narrow "w" feature due to O ii lines produced at high temperatures and via non-thermal excitation (Mazzali et al. 2016). Spectra of SLSNe Ic show the narrow "w" feature between ∼4000 Å and ∼4600 Å, while spectra of SNe Ic-bl and SNe Ic do not show such narrow "w" features within a similar wavelength range. Thus, we suggest the use of the narrow "w" feature around 4300 Å, along with other spectral and photometric properties, to identify SLSNe Ic at early times, i.e., before maximum light, as initially proposed by Quimby et al. (2011) based on a smaller sample. We note that not all narrow "w" features in SLSNe Ic before the peak epoch are identified as O ii lines. For example, a low temperature SLSN Ic iPTF13ehe (${T}_{\mathrm{BB}}\sim 7000$ K) shows a narrow "w" feature at ${t}_{\max }=-10$ days, but is identified as being due to Mg ii and Fe ii lines (Yan et al. 2015). Except for iPTF13ehe, the narrow "w" feature in other SLSNe Ic is likely due to blended O ii lines (see Table 2 in Mazzali et al. 2016), which may make it hard to measure the absorption velocity of the feature.

In the interaction model for SLSNe, the SN ejecta interacts with circumstellar material that was ejected before the explosion, thus converting the large kinetic energy of the ejecta into radiation (e.g., Ofek et al. 2014 and references therein). While this model can reproduce SLSNe IIn well (i.e., those with hydrogen emission lines; Smith et al. 2010), this model has a number of challenges for SLSNe Ic: the circumstellar material would have to be completely free of both hydrogen and helium (if helium were present, one would expect to observe helium emission lines,⁸ as seen in SNe Ib-n, e.g., Foley et al. 2007; Pastorello et al. 2008, 2016; Shivvers et al. 2016). However, even for hydrogen-free and helium-free material, one would still expect to see narrow lines of oxygen and other elements (e.g., as observed by Ben-Ami et al. 2014 in SN 2010mb), which are not seen in any SLSNe Ic, though another explanation may be that the abundances or ionization/excitation conditions are not sufficient to give rise to them. In addition, the observed large absorption velocities have not been reproduced by the interaction model, which at the same time has to reproduce the observed broad light curves (Sorokina et al. 2016). Sorokina et al. (2016) speculate that the observed spectra of SLSNe Ic could be explained by interaction with a separate shell that is expanding rapidly with a peculiar velocity structure. That velocity structure would be such that the inner radius of the shell is almost at zero velocity, while the outer radius is at the velocity observed in absorption in the spectra. However, since we find that the absorption velocities of SLSNe Ic (see Section 4.1) are even higher than the previously adopted velocity of 10,000 km s⁻¹ for constraining models (Sorokina et al. 2016), there is even more tension—more importantly, we note the SLSN Ic spectra also have broad spectral features (see Section 4.2), indicating a relatively large mass at a high velocity, not just a thin layer, as would be the case in the shell model. Alternatively, as was suggested for the SN Ia 2009dc by Hachinger et al. (2012), the interaction layer may not be responsible for the absorption lines, but only lead to their dilution and only contribute to the continuum in the blue. In any case, we suggest that future works, which attempt to explore interaction models for SLSNe Ic, employ sophisticated hydrodynamical models and synthesize spectra at higher resolution than done in Sorokina et al. (2016) in order to try to reproduce the average spectra of the full SLSNe Ic population.

One promising power source model for SLSNe Ic is the magnetar model (e.g., Kasen & Bildsten 2010; Woosley 2010; Inserra et al. 2013; Nicholl et al. 2013; Metzger et al. 2015; Chen et al. 2016; Suzuki & Maeda 2017). In the one-dimensional magnetar model, there will be a mass shell formed due to the magnetar bubble. Based on radiation hydrodynamic calculations, the mass shell is supposed to cause an observational imprint via a prolonged period of flat velocity evolution, i.e., a velocity plateau, in the observed spectra (Kasen & Bildsten 2010). Some observational papers claim that they have detected such a velocity plateau in their analyses of SLSN Ic spectra. For example, the velocity plateau is claimed in PS1-10ky and PS1-10awh using lines of C ii λ2330, Si iii λ2540, and Mg ii λ2800 (Chomiuk et al. 2011). Nicholl et al. (2014) reported velocity plateaus in SN 2013dg using Mg ii λ4481 and Mg i] λ4571. Nicholl et al. (2015b) have suggested that the Fe ii λ5169  velocity of the following SLSNe Ic has a flat evolution as a function of time: PTF11rks, PTF12dam, SN 2013dg, and LSQ14mo.

In our work, we have explored the velocity evolution of Fe ii λ5169  using our own measurements that utilize our novel technique (see Section 3). First, we define flat (or slow) velocity-evolution SNe as SNe whose spectra span more than 10 days and for which the Fe ii λ5169  velocities decrease by less than 2000 km s⁻¹. With this definition, we only identify two slow velocity-evolution SLSNe Ic (PTF10hgi and PS1-11ap) out of 13 SLSNe Ic that satisfy the phase requirement and plot them in Figure 4. In contrast, we also show fast velocity-evolution SLSNe Ic (PTF11rks, LSQ12dlf, and SN 2013dg) whose spectra cover more than 10 days, but whose velocities decrease by more than 8000 km s⁻¹. Changing the requirement for the time range over which the SLSNe Ic should have spectra from 10 days to a more lenient 5 days does not change our conclusions, since the only SLSNe Ic fulfilling the new requirement are the same objects as before. The slow-evolution SLSNe Ic are consistent with a dense shell model, while the fast-evolution SLSNe Ic are consistent with a simple spherical model. A dense shell can be formed in a magnetar model but there are other channels, e.g., via eruptions before SNe explode. We note that even for PTF10hgi and PS1-11ap, it is not clear that their velocity evolution should be used to argue for the one-dimensional magnetar model: PTF10hgi has a flat evolution in velocity, but like ordinary SNe Ic, which show velocity plateaus at similar phases, this could simply be indicating that the flat evolution is due to radiative transfer effects and does not reflect a physical ejecta structure; PS1-11ap has a large error bar for the first data point, making it consistent with a high velocity value, and thus no flat evolution.

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In the following, we compare Fe ii λ5169  velocities measured in this work with those in Nicholl et al. (2015b) for seven SLSNe Ic we both have in common, and most of our measurements are consistent within the error bars. However, there are different systematic trends that lead to qualitatively different conclusions for both works. For four SLSNe Ic (PTF11rks, PTF12dam, SN 2013dg, and LSQ14mo), we do not find a flat velocity evolution for the Fe ii λ5169  line as claimed by Nicholl et al. (2015b). One reason is that we have measurements for the earliest and latest spectra that are not presented by Nicholl et al. (2015b). For PTF11rks, we have a Fe ii λ5169  velocity at ${t}_{\max }\simeq 50$ days, which is much lower than velocities at ${t}_{\max }\lt 20$ days, while Nicholl et al. (2015b) only presented Fe ii λ5169  velocities before ${t}_{\max }=20$ days. A similar reason applies to PTF12dam as well. Another reason is that the way we measure the Fe ii λ5169  velocity is different from that in Nicholl et al. (2015b). We fit the Fe ii λ5169  region using the similar region in broadened and blueshifted SN Ic spectra in an MCMC framework, while Nicholl et al. (2015b) fit the whole Fe ii λ5169  region (which for SLSNe Ic is a blend and includes two additional Fe ii lines, namely Fe ii $\lambda \lambda$ 4924, 5018) with only one Gaussian profile. As discussed in Section 3.1.2, though both methods have advantages and disadvantages, the template fitting method in our work can better capture the Fe ii λ5169  line when it is blended with the other two Fe ii lines as shown in Modjaz et al. (2016) for SNe Ic-bl.

We note that a flat evolution in velocities is observed in SNe other than SLSNe Ic. In particular, the right panel of Figure 1 shows that the average velocity evolution of SNe Ic becomes flat after ${t}_{\max }=20$ days. The velocity plateau of He i lines in SNe Ib (SN 2006el, SN 2009 mg, and SN 2011dh) and SNe IIb (SN 1998dt, SN 1999ex, SN 2005bf, and SN 2007Y) is also observed by Folatelli et al. (2014) and Liu et al. (2016). However, the magnetar model is unlikely to be the reason for the velocity plateaus in these stripped SNe. Thus, we suggest that any claim of the magnetar model based on the flat velocity evolution of a line observed in spectra of SLSNe Ic has to be checked by comparing to the velocity evolution of the same line in normal SNe Ic over the same time period. If a line exhibits a velocity plateau in both normal SNe and in SLSNe Ic over the same time period, then it may simply be due to radiative transfer effects, and thus should not be used as a diagnostic for the central power source.

We conclude that as a whole population, the velocity evolution as traced by Fe ii λ5169  of SLSNe Ic is not consistent with that predicted by the one-dimensional magnetar model (Kasen & Bildsten 2010), while individual SLSNe Ic might be. The obvious next step of performing magnetar calculations in two dimensions has recently been tackled. Chen et al. (2016) and Suzuki & Maeda (2017) found that the dense shell, which is commonly seen in one-dimensional simulations and accounts for the predicted velocity plateau, is destroyed by multi-dimensional effects, including turbulence. Thus, while we await spectral synthesis calculations based on two-dimensional models for their predicted velocity evolution, it is reasonable to expect that two-dimensional models will not show such a velocity plateau and thus, will be more in line with our observational results. Moreover, Inserra et al. (2013) showed that a semi-analytical diffusion model in the magnetar scenario can reproduce light curves of SLSNe Ic well, but the predicted photospheric velocity evolution is not flat.

In our analysis, we have used Fe ii λ5169  to trace the velocity structure of the SN Ic family. While using this line to perform inter-comparisons between different SN subtypes (Section 4.1) is valid and indeed required, since different lines have systematically different velocity structures (Modjaz et al. 2016), we now note some of the caveats when using that line to constrain theoretical models. Theoretical models assume a "photospheric velocity" and the question is which observed line traces it best. On the one hand, Branch et al. (2002) suggest that Fe ii λ5169  traces the photospheric velocity well based on their SYNOW models for stripped-envelope core-collapse SNe (stripped SNe). On the other hand, Dessart et al. (2015) conduct a careful exploration using progenitor, explosion, and non-LTE radiative transfer models for stripped SNe and find that while the photosphere, and therefore its associated velocity, is poorly defined, the O i λ7774 velocity is a good indicator of the representative ejecta velocity for SNe Ib and Ic at the date of maximum light. We showed in Liu et al. (2016) that while the absolute values of the Fe ii λ5169  and O i λ7774 velocities are somewhat different for SNe Ib and Ic, the systematic offset between the two subtype velocities was shown by both lines, i.e., the relative velocity values are correct. In this work, we did not use O i λ7774 as an indicator for photospheric velocity since most of our SLSN Ic spectra do not cover its wavelength range.

Thus, our conclusions about the one-dimensional magnetar model in the preceding section may not stand if another line were to be used, since different lines have different velocity behaviors (e.g., Si ii λ6355 versus Fe ii λ5169, see Figure 4 in Modjaz et al. 2016). However, as we note in Section 5.2, the same line needs to be used for both SNe Ic and SLSNe Ic to ensure that any plateau behavior seen in the velocities of SLSNe Ic is not due to radiative transfer effects, which would be reflected in the same plateau behavior for ordinary SNe Ic.

From a theoretical point of view, a few assumptions have to be adopted for the comparison of theory and observations. Specifically for the one-dimensional magnetar model, we used the following prediction based on Equation (7) and Figure 2 in Kasen & Bildsten (2010): if the equation predicts a shell velocity of 10,000 km s⁻¹, then based on the figure, the velocity plateau will start 10 days after the explosion. Of course if the magnetar energy were to be lower by a factor of 12–13, the phase range of the velocity plateaus would be at 100 days after the explosion, a regime not probed by the optical spectra in our sample.

Determining the geometry of SLSNe Ic explosions is crucial for differentiating between the different powering mechanisms. For example, an explosion with a central engine may be an intrinsically asymmetric explosion and would leave imprints on polarimetric observations. As discussed in Section 2, due to the lack of observations, it is hard to conclude whether all SLSNe Ic are non-spherical explosions. Indeed, our observations disfavor the spherically symmetric magnetar model in one dimension (see Section 5.2), though this does not necessarily mean that SLSNe Ic have to harbor large-scale asphericities. If SLSNe Ic are powered by highly aspherical engines, then one may speculate that the discovered SLSNe Ic appear luminous because they are seen on-axis, and thus the SLSNe Ic in our sample would be biased toward high luminosity explosions. Even if this conjecture entails that the population properties of SLSNe Ic may be biased, the velocity measurements of the individual SNe would still be valid, as well as our discussions in Sections 5.1 and 5.2, except that we would have to compare the velocities of SLSNe Ic to those of SNe Ic-bl with observed GRBs only, not the whole population of SNe Ic-bl, a modification that would not change our conclusions.

Dong et al. (2016) claimed ASASSN-15lh (AKA SN 2015L) to be the most luminous SN ever discovered. They found that the spectra of ASASSN-15lh are blue and featureless except for a deep and broad trough near 4100 Å. They also found that the trough in ASASSN-15lh resembles the O ii feature in PTF10cwr (AKA SN 2010gx). However, whether ASASSN-15lh is an SN or not is still debated. On the one hand, Leloudas et al. (2016) found that their observations are more consistent with a tidal disruption event (TDE), based on temperature evolution, transient location in the host galaxy, and other evidence. They also found that if the trough in ASASSN-15lh were due to O ii lines, there would need to be an additional strong feature around 4400 Å, which is not observed in the spectra of ASASSN-15lh. Moreover, Margutti et al. (2016) favored a TDE origin for ASASSN-15lh based on an analysis of X-rays from the host galaxy, though more monitoring of the X-rays is needed to truly distinguish between the two origins. On the other hand, spectra of ASASSN-15lh lack both hydrogen and helium emission lines, something that has not been observed in TDE spectra so far (e.g., Arcavi et al. 2014). The host galaxy does not resemble the typical hosts of SLSNe (Godoy-Rivera et al. 2017) and is too massive for a TDE (Dong et al. 2016). Moreover, the rebrightening in UV is also unexpected for both SLSNe and TDEs (Brown et al. 2016; Godoy-Rivera et al. 2017).

In our work, we partly address the question of whether ASASSN-15lh is an SN or not by comparing its spectra to average spectra (with continua) of SLSNe Ic and SNe Ic-bl. In Figure 5, we compare the spectrum of ASASSN-15lh at ${t}_{\max }=15$ days to average spectra and their corresponding standard deviations of SLSNe Ic at ${t}_{\max }=15$, and −10 days, as well as to the average spectrum of SNe Ic-bl at ${t}_{\max }=15$ days. The left panel shows that there is no similar feature in average spectra of SLSNe Ic and SNe Ic-bl as the trough in ASASSN-15lh at ${t}_{\max }=15$. Moreover, the continuum of ASASSN-15lh does not resemble those in average spectra of SLSNe Ic at short wavelengths and SNe Ic-bl at optical wavelengths. The right panel shows the same ASASSN-15lh spectrum at ${t}_{\max }=15$ days and the SLSN Ic average spectrum and its corresponding standard deviations at ${t}_{\max }=-10$ days. While the trough in ASASSN-15lh is still not consistent with features in the average spectrum within 1 standard deviation, the continua in both spectra are similar. We have several more spectra of ASASSN-15lh at later phases until ${t}_{\max }=39$ days. However, they all resemble the spectrum at ${t}_{\max }=15$ days in continuum and spectral features. Thus, even if ASASSN-15lh were an SLSN I, it would be a very slowly evolving SLSN Ic with peculiar features.

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SN 2011kl is another unusual SLSN Ic since it constitutes the first detection of an SN explosion associated with an ultra-long-duration GRB, namely ULGRB 111209A (Greiner et al. 2015; Kann et al. 2016). SN 2011kl is more luminous than SNe Ic-bl associated with long-duration GRBs (SN-GRBs; for a recent review, see, e.g., Modjaz et al. 2011; Hjorth & Bloom 2012; Cano et al. 2017), but not as luminous as most, if not all, SLSNe Ic (Greiner et al. 2015; Kann et al. 2016). In Figure 6, we compare the spectrum of SN 2011kl to the average and standard deviation spectra of SN-GRBs and SLSNe Ic at similar phases. Note that we could not measure the Fe ii λ5169  velocity in SN 2011kl due to the noisy spectra. Thus, we did not compare the Fe ii λ5169  velocities. We find that the continuum of SN 2011kl's spectrum does not resemble those of the population of SN-GRBs, but is fully consistent with those in the population of SLNe I; however, we find the spectrum of SN 2011kl to be too noisy to robustly evaluate whether the widths of the line features are as broad as in other SLSNe Ic and SN-GRBs. While Greiner et al. (2015) came to a similar conclusion based on fewer objects in their comparison set (only three SLSNe Ic and one SN-GRB), we have conducted this spectroscopic comparison with the full populations of SLSNe Ic and SN-GRBs. Thus, SN 2011kl could be a true SLSN I.

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SN 2007bi has been studied extensively and several models have been proposed for its origin: pair-instability explosion (Gal-Yam et al. 2009), core-collapse explosion (Moriya et al. 2010), and magnetar model (Dessart et al. 2012b). We note that the first spectrum taken for SN 2007bi is at ${t}_{\max }=48$ days. Thus, the velocity diagnostics, as discussed in Section 5, could not be applied in this case. Some works claimed that SN 2007bi spectra are similar to other SLSN Ic spectra (Nicholl et al. 2013, 2016), while some concluded that SN 2007bi has redder spectra than other SLSNe Ic (Mazzali et al. 2016). Here, we try to address the question of whether SN 2007bi is a typical SLSN Ic from the spectral perspective by comparing its spectrum at ${t}_{\max }=48$ days to the average spectrum of SLSNe Ic at ${t}_{\max }=48\pm 2$ days in Figure 7. It is obvious that the spectrum of SN 2007bi is within one standard deviation of the SLSN Ic mean spectrum at most wavelengths, except for wavelength ranges 4500–5100 Å and 7200–7600 Å, which are identified as due to Fe ii lines and [Ca ii] lines (Gal-Yam et al. 2009), respectively. In particular, the Fe ii region in the spectrum of SN 2007bi forms a distinct "W" feature, which is very different from the broad "U" feature in the average spectrum of SLSNe Ic. Thus, future works on SN 2007bi should be aware of the narrow and strong Fe ii feature in absorption at ∼4800 Å, as well as the extremely strong [Ca ii] feature in emission at ∼7400 Å.

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