Optical Observations of the Young Type Ic Supernova SN 2014L in M99

SN 2014L is well-observed in the standard Johnson UBV and Kron–Cousins RI bands, with a typical FWHM 16 for LJT images, and 25 for the TNT images. High-quality template images were taken with the LJT+YFOSC and TNT at 2–3 years after the explosion. We perform background subtraction to remove the time-invariant background structures and better reveal the time-variant SN signals. The photometry of this SN is transformed into the standard Johnson–Cousin photometry system (as listed in Table 1 and presented in Figure 2) through some local standard stars, as marked in the left panel of Figure 1. The UBVRI magnitudes of these reference stars (as listed in Table 2) are calibrated via observing a series of Landolt (1992) photometric standard stars on three photometric nights.

Figure 3 shows the spectral sequence of SN 2014L over a period of about four months starting from 2014 January 27. The observation journal of these spectra is listed in Table 3, including 13 spectra from the LJT (+YFOSC) and 1 spectrum from the Xing-Long 2.16 m telescope (XLT) with the BFOSC (Beijing Faint Object Spectrograph and Camera). All of these spectra are calibrated in both wavelength and flux, and they are corrected for telluric absorption and redshift. The narrow emission lines (e.g., Hα, Hβ) presented in the spectra are due to the contamination in the host galaxy, which becomes dominating at late phase. Variations of these narrow emissions are due to the slit position and spectral resolution. Two spectra from the ANU WiFeS SuperNovA Program (AWSNAP; Childress et al. 2016) are also included in this figure to fill the observation gaps.

Zoom In Zoom Out Reset image size

Figure 4 displays the UBVRI light curves of SN 2014L compared to those of some typical SNe to better understand the photometric properties of SN 2014L. Three SNe Ic, SN 1994I (Richmo et al. 1996), SN 2007gr (Valenti et al. 2008; Chen et al. 2014), and SN 2004aw (Taubenberger et al. 2006); the board-line SN Ic, SN 1998bw (Clocchiatti et al. 2011); SN Ib, SN 2008D (Modjaz et al. 2009; Tanaka et al. 2009; Bianco et al. 2014); faint and calcium-rich events SN 2005E (Perets et al. 2010) and SN 2012hn (Valenti et al. 2014) are plotted in Figure 4.

Zoom In Zoom Out Reset image size

The light curves of SN 2014L are overall similar to those of SN 2007gr, though the former appears slightly narrower than the latter. It is surprising that SN 2014L, SN 2004aw, SN 2007gr, and SN 1998bw show similar evolution in the U band, with a small scatter of <0.2 mag. In the B band, SN 2014L, SN 1994I and SN 2012hn also show similar light-curve evolution at t < +20 days. After that, these SNe exhibit different decay rates. Larger scatters also exist in the VRI bands at similar phases.

These SNe can be divided into three clusters, depending on the decline rate from 0 < t < +30 days. For example, the slow declining group including SN 2004aw, SN BL-Ic 1998bw, SN Ib 2008D; the fast declining group including SN Ic 1994I and SN Ca-rich 2005E; and the intermediate declining group including SN Ic 2014L, SN Ic 2007gr, and SN Ca-rich 2012hn. In each cluster, however, apparent differences can be found in spectral comparisons at both photospheric and nebular phase of Section 4.

The relatively similar peak properties are often in contrast with the considerable spread in the properties of the late-time tail of the light curve (e.g., Wheeler et al. 2015, hereafter is WJC 15). Clear differences existing among the tails of these light curves might be related to the different optical opacities and γ-ray leakage rates of different SNe Ib/c.

The early spectra of SN 2014L show red continuum and significant absorption of narrow Na i D from the host galaxy. These features suggest substantial line-of-sight reddening towards the SN. The equivalent width (EW) of Na i D absorption in the SN spectra can be used to estimate the reddening according to some empirical correlations between reddening and EW of Na i D, e.g., E(B − V) = 0.16 EW_Na − 0.01 (Turatto et al. 2003), and E(B − V) = 0.25 EW_Na (Barbon et al. 1990). However, these empirical correlations usually exhibit large scatter for the extinction measurement (e.g., Poznanski et al. 2011; Phillips et al. 2013). In SN 2014L, EW(Na i D) is measured as 2.7 ± 0.1 Å, which corresponds to a color excess of E(B − V)_host = 0.42 ± 0.04 or 0.68 ± 0.05 following the above relations.

Alternatively, the host-galaxy reddening of SNe Ib/c can be estimated by a photometric method. Based on a larger sample of SNe Ib/c, Drout et al. (2011) found that the V − R color of extinction-corrected SNe Ib/c is tightly clustered at 0.26 ± 0.06 mag at t ≈ +10 days after the V-band maximum, and 0.29 ± 0.08 mag at t ≈ +10 days after the R-band maximum. This method yields an estimate of E(B − V)_total = 0.75 ± 0.05 mag for SN 2014L assuming an R_V = 3.1 Milky Way extinction law for the host galaxy. Adopting the Galactic extinction E(B − V)_Gal = 0.04 ± 0.01 (Schlafly & Finkbeiner 2011), the host reddening derived from the V − R color is E(B − V)_host = 0.71 ± 0.05 mag.

Combining the estimations obtained from Na i D absorption and V − R color, an average value of the total reddening, E(B − V) = 0.67 ± 0.11 mag, is adopted in this paper.

The reddening-corrected color curves of SN 2014L are displayed in Figure 5, together with those of the comparison sample shown in Figure 4. The B − V color of SN 2014L is similar to that of SN 1998bw, SN 2008D, and SN 2007gr, but the difference exists in other bands, especially at t > +20 days relative to the V-band maximum. The scatter shown before the peak can be attributed to the difference in temperature and metal abundance of the outer ejecta, while the scatter seen after t ≈ +20 days could be related to the duration of photospheric phase.

Zoom In Zoom Out Reset image size

The large scatter among the colors of these SNe, even in the SNe Ib/c group, might not suggest a uniform color evolution for these SNe. However, as mentioned before, there seems to exist an intrinsic V − R color for regular SNe Ib/c at about ten days after the peak (Drout et al. 2011). Such an inherent color distribution might also exist in B − V and V − I color at a similar phase in Figure 4.

Figure 6 displays the quasi-bolometric light curves of our sample, based on the UBVRI photometry presented in Figure 4. SN 2014L researches the peak quasi-bolometric luminosity of L = (2.06 ± 0.50) × 10⁴² erg s⁻¹ at MJD = 56693.86, which is about one day earlier than the V-band maximum. The fractional contribution of the ultraviolet (UV) and near-infrared (NIR) emission to the bolometric luminosity can be ∼30%–40% at around the peak for SNe Ib/c (e.g., ∼40% in SN 2008D; Modjaz et al. 2009). Considering the missed UV and NIR flux of SN 2014L, the total bolometric flux of this SN could reach ∼3 × 10⁴² erg s⁻¹ around the peak.

Zoom In Zoom Out Reset image size

The early-time spectra (i.e., at a few days after the explosion) contain prominent features to distinguish it among different subclasses of SNe. Figure 7(a) shows the t ∼ −7 day spectra of SN 2014L and some comparison SNe Ib/c including SN 2007gr (Chen et al. 2014), SN 1994I (Filippenko et al. 1995), SN 2004aw (Modjaz et al. 2014), SN 1998bw (Patat et al. 2001), and SN 2008D (Modjaz et al. 2009). Compared to the almost featureless spectrum of SN 2004aw and the board-lined spectrum of SN 1998bw, the spectra of SN 2014L, SN 2007gr, and SN 1994I are characterized by the absorptions of some intermediate-mass elements (IMEs, e.g., Ca, Si, O, and C) and Fe ii. The clear absorption near 6150 Å in the spectra of SN 2014L, SN 2007gr, and SN 1994I can be identified as the feature of Si ii λ6355. The absorption near about 5600 Å is likely attributed to the absorption of Na i D rather than He i λ5876 because of the absence of other helium lines as found in the spectrum of SN Ib 2008D. Overall, SN 2014L shows close resemblances to SN 2007gr and SN 1994I, except that it has more prominent absorption features of Ca ii NIR triplet. The difference in absorptions of Ca ii NIR triplet might indicate difference at the outer layer of ejecta, as discussed in Section 4.5.

At around the maximum, the spectral features of SN 2014L are very similar to those of SN 2007gr. These two SNe Ic show narrower absorptions of Si, O, Na, and Fe than SN 1994I and SN 2004aw. In particular, SN 2014L seems to have unusually strong absorption features for a Ca ii NIR triplet, comparable to that of the Ca-rich events such as SN 2005E (Perets et al. 2010) and SN 2012hn (Valenti et al. 2014). The strong Ca ii NIR triplet in SN 2014L might relate to the abundance enrichment of IMEs at the outer layer. To investigate the relative abundance of Si, Ca, and O measured from the spectra shown in Figure 7(b), we plot the EW ratio of O i λ7774/Ca ii NIR triplet versus Si ii λ6355/Ca ii NIR triplet in Figure 8. One can see that the absorption lines ratio Ca/O of SN 2014L is close to that of SN 1994I, while the Ca/Si ratio of SN 2014L is similar to that of SN 2007gr. Both of these two ratios measured from SN 2014L are smaller than the calcium-rich events like SN 2005E and SN 2012hn, ruling out the calcium-rich subclass. Nevertheless, the synthetic spectra at around maximum by (Iwamoto et al. 1998) suggests that the strength of Ca ii NIR triplet is weaker than that of Si ii and O i.

Zoom In Zoom Out Reset image size

Compared to other subclasses of SNe Ib/c, the BL SNe Ic tend to cluster at the top right region of Figure 8. Note, however, that it is hard to measure the absorptions of Ca ii NIR triplet and O i λ7774 in the spectra of SN 1998bw because these two features are blended before and near the maximum light. Moreover, these absorptions cover the wavelength range of the high-velocity Ca ii NIR triplet and the photospheric components of Ca ii NIR triplet and O i λ7774.

According to the distributions shown in Figures 8 and 6, we can divide our sample of SNe Ib/c into three groups. For example, SN 2005E and SN 2012hn had lower luminosity and are located at the bottom of Figure 8. The SN 1998bw-like objects tend to have a smaller amount of calcium and higher luminosity, while the remaining SNe Ib/c in our sample have a moderate luminosity and lie in the center of Figure 8. Of the selected sample, SN 2004aw can be regarded as the transitional event linking the BL SNe Ic with the ordinary ones. The distribution of SNe Ic in Figures 8 and 6 suggests that the relative strength of Ca ii NIR triplet measured in the near-maximum spectra is inversely related to their peak luminosities. Namely, SNe Ic with a significant amount of calcium tend to have lower luminosities. This result may still suffer from small-number statistics, but it is consistent with the theoretical prediction that the O/Ca ratio increases with the mass of progenitor for core-collapse SNe (Houck & Fransson 1996). Thus, a comparable O/Ca ratio seen in SN 2014L, SN 1994I, and SN 2007gr might imply that they have similar progenitor masses and brightness, while SN 2004aw and SN 1998bw are likely from more massive progenitors. It is possible that the Ca-rich events might have lower-mass progenitor systems given the fact that they tend to occur at positions far from the host galaxies. However, massive stars could go for an outing through tidal stripping during galaxy interactions. And we can not rule out a massive progenitor scenario for the Ca-rich events thoroughly (Kasliwal et al. 2012).

By t ∼ 1 month, the spectra show a clear distinction for different SNe Ic, as shown in Figure 7(c). For example, the t ≈ +26 day spectrum of SN 2014L shows a close resemblance to the t ≈ +35 day of SN 2007gr and the t ≈ +26 day spectrum of SN 1994I, while its t ≈ +36 day spectrum shows less photospheric features compared to the t ≈ +35 day spectrum of SN 2007gr. This implies that SN 2014L has a relatively faster spectral evolution than SN 2007gr. In comparison, the Ca-rich class seems to be the most rapidly evolving spectral feature. At this phase, some nebular features, such as the emission at around 7300 Å (due to the blended lines of O ii and Ca ii) started to emerge. The significant differences in the spectral evolution conform to the scatter seen in the color curves at similar phases. This carbon feature becomes unambiguous in the t ∼ 26 day observed spectrum of SN 1994I due to the diffraction fringing effect of the instrument.

It should be noted that the C i λ9087 absorption is detected in the spectra of SN 2014L and SN 2007gr. The presence of such carbon line is not necessarily in the t ∼ 26 day spectrum of SN 1994I due to the diffraction fringing effect of the instrument. However, detection of this C i line was reported in the t ≈ +10 day spectrum of SN 1994I (Baron et al. 1996). The feature of C i observations indicates that SN 2014L, SN 2007gr, and SN 1994I might arise from carbon-rich progenitors (Baron et al. 1996; Valenti et al. 2008).

Figure 7(d) displays the spectra of these SNe at the beginning of nebular phase when the spectra of core-collapse SNe are dominated by a few emissions of IMEs (e.g., O, Mg, Ca). For example, the latest spectrum of SN 2014L is dominated primarily by the emission lines [O i] λλ6300, 6364, a blend of [Ca ii] λλ7291, 7324 and [O ii] λλ7300, 7330, and the Ca ii NIR triplet. However, some photospheric components (e.g., the absorptions of Na i D, O i λ7774 and Ca ii NIR triplet) are still evident in this phase. Similar features are also found in the spectra of SN 1994I, and SN 2007gr. The BL-Ic SN 1998bw shows similar emissions but contains more photospheric features. It indicates a slower evolution in the broad-lined event.

The spectrum of SN 2012hn at t ≈ +34 days keeps some photospheric features and dominated by the emissions of [Ca ii] λλ7291, 7324 and [O ii] λλ7300, 7330, and the Ca ii NIR triplet. Similar emissions are also found in the spectrum of SN 2005E at t ≈ +53 days with less photospheric components. The difference between Ca-rich event, and SNe Ib/c at the early nebular era is the emission of [O i] λλ6300, 6364 which is very weak in the former. Besides, SN 2005E did not develop clear emission of Mg i] and SN 2012hn develops this feature rather slowly. These imply different progenitor systems or explosion mechanism between SNe Ib/c and Ca-rich events. The spectrum of SN 2012hn at t ≈ +151 days evolved to the later nebular phase is similar to the spectra of SNe Ic at about one year after the explosion. It confirms the fact of fast evolution in Ca-rich events.

The apparent and almost symmetric double-peaked feature is found in the [O i] λλ6300, 6364 emissions of SN 2008D. The trace of double-peaked [O i] might exist in that of SN 2007gr and SN 1998bw. However, it is not clear in the spectra of SN 2014L. These might indicate a different geometric structure of the ejecta.

The ejecta velocities derived by the absorption minima of Ca ii H&K, Na i D, Si ii λ6355, C ii λ6580, O i λ7774, Ca ii NIR triplet, and C i λ9087 of SN 2014L are shown in Figure 9. In the early phase, the Ca ii shows a much higher velocity relative to Na i, Si ii, C ii, and O i, and this discrepancy is similarly seen in type Ia supernovae (i.e., Zhao et al. 2015). This might be due to the fact that Ca ii has relatively lower excitation/ionization energy and it can be formed at more distant regions (higher velocities). At t > +10 days, the velocities of Ca ii, Na i, and O i decline slowly and they all tend to reach a velocity plateau of about 8000–10,000 km s⁻¹. We note that the velocity of Na i D shows a significant increase from 8500 km s⁻¹ to 10000 km s⁻¹ during the period from t ∼ 10 days to t ∼ 20 days. This is seen in some SNe Ic (e.g., Hunter et al. 2009) and might be due to the contributions from other spectral lines (e.g., Na ii or He i λ5876).

Zoom In Zoom Out Reset image size

Figure 10 shows the comparison of the velocities inferred from Si ii λ6355 and Ca ii NIR triplet for different SNe Ic, including SN 2014L, SN 2007gr, SN 1994I, and SN 2004aw. One can see that SN 2014L shows a higher Si ii velocity than that of SN 2007gr. Before the maximum light, the velocity of the Ca II lines in SN 2014L exhibits a remarkably high decline rate ∼ 600 km s⁻¹ day ⁻¹. The more significant decline rate of Ca ii velocity makes a lower velocity plateau of SN 2014L than the comparison.

Zoom In Zoom Out Reset image size

The rise time t_r is essential to infer the mass of synthesized ⁵⁶Ni via Equation (1). It can be measured from the very early light curve.

Three groups reported the detection of SN 2014L in their pre-discovery images after the discovery report of Zhang et al. (2014b). The first detection was obtained by Koichi Itagaki (KI) through a 0.5 m reflector telescope at the Takamizawa station on January 24.85 (Yamaoka 2014) in a clear band image (roughly R = 18.15 ± 0.50 mag). Later on, Pan-STARRS1 reported the detection for this SN, marked as PS1-14kd, with the r = 17.01 mag on UT 2014 January 26.49. Gregory Haider (GH) also detected it on January 26.54 in the clear images (roughly R = 17.00 ± 0.40 mag) using the 12.5 inches reflector telescope at Christmas Valley. Combining these pre-discovery detections with the early R-band light curve obtained by LJT and TNT, the shock breakout date of SN 2014L was derived on 2014 January 22.5 (MJD = 56679.5 ± 1.0) through the t² law approximation of the fireball model, see Figure 11. It suggests that SN 2014L reached its bolometric maximum at about 14.5 days after the shock breakout.

Zoom In Zoom Out Reset image size

However, the fireball model might not be valid in the case of SNe Ic. The physics behind this model is that the increasing internal heating roughly compensates the adiabatic cooling of the ejecta due to Ni-decay. Since the envelope of SNe Ic is shock-heated, the Ni-decay plays less of a role than in it does in the case of SNe Ia (e.g., SNe Ia mag have 5–10 times more nickel than SNe Ic). Thus, the fireball model may not be capable of giving an accurate prediction for the moment of shock breakout. Furthermore, some poor-fit results of the fireball model in the case of very-young SNe Ia would suggest a steeper power-law index (e.g., Zheng et al. 2013, 2014).

We noted that some previous works (e.g., Valenti et al. 2008; Chen et al. 2014; WJC 15) suggested that the rise time of SN 2007gr is about 13–14 days. Given the narrower light curve of SN 2014L, as presented in Figure 4, its rise time might be shorter than that of the SN 2007gr. Moreover, the light-curve modeling in Section 5.3 suggests t_r = 13 days of SN 2014L. Meanwhile, we liberalized the exponent of the power law and found a value of 1.4 with a t_r = 13 days. Considering the light-curve comparison of SN 2007gr, the following light-curve modeling, and the result of t^1.4 fit, we adopted t_r = 13 days as the rise time of SN 2014L in the related calculations. The rise time suggests that the shock breakout time of this SN is on 2014 January 24.8 (MJD = 56682.8 ± 1.0). It indicates that the first detection of SN 2014L taken by KI is only a few hours after the shock breakout. Thus, SN 2014L might be the earliest detected SN Ic to date.

The t_r, L_max, v_ph, and T₀ parameters of SN 2014L, SN 1994I, and SN 2007gr derived from the light curve and spectra are listed in Table 5. A fiducial optical opacity κ = 0.1 cm² g⁻¹, and a fiducial gamma-ray opacity κ_γ = 0.03 cm² g⁻¹ are taken to calculate M_ej and E_ke. In Table 6, as noted by WJC 15, parameters derived from the peak do not match those arising from the tail. That relates to the fiducial value of κ and κ_γ adopted in the calculation. If we set the results derived from the peak equal to that from the tail, a relation about κ and κ_γ can be written as

$$\begin{eqnarray}\kappa & = & \displaystyle \frac{5}{3}(\beta c)C\displaystyle \frac{{\kappa }_{\gamma }}{{v}_{\mathrm{ph}}}{\left(\displaystyle \frac{{t}_{r}}{{T}_{0}}\right)}^{2}\\ & = & 0.01\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}\displaystyle \frac{{\kappa }_{\gamma }/0.03\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}}{{\text{}}{v}_{\mathrm{ph},9}}\left(\displaystyle \frac{{\text{}}{t}_{r,10}}{{\text{}}{T}_{\mathrm{0,100}}}\right).\end{eqnarray} \tag{ 6 }$$

WJC 15 assumed that κ_γ is less dependent on explosion parameters than κ. Thus, κ derived from Equation (6), where κ_γ = 0.03 cm² g⁻¹, is close to 0.02–0.03 cm² g⁻¹. This is much smaller than the assumed κ = 0.1 cm² g⁻¹.

Table 5.  Explosion Parameters of Some Well-observed SNe Ic

SN	tra	Lpeak	vph	T0b	${M}_{{}^{56}\mathrm{Ni}}$ c
	(days)	(1042 erg s−1)	(km s−1)	(days)	(M⊙)
2014L	13.0	2.06	7650	90	0.075
1994I	12.0	2.63	9900	85	0.089
2007gr	14.0	1.72	6700	125	0.066

Notes. Observed parameters of SNe Ic, 1994I, 2014L, and 2007gr. The typical errors for the measurement are 10%–20%.

^aRise time from the shock breakout to the peak of quasi-bolometric curves in Figure 6. ^bDerived from the light curves at t > +50 days. ^cDerived from Equation (1) and the quasi-bolometric curves in Figure 6.

Download table as:  ASCIITypeset image

The bolometric light curve of SN 2014L, SN 1994I, and SN 2007gr have been modeled by applying the LC2.2 code¹³ to check the explosion parameters estimated at above. This code is the most recent version of the LC2 code presented by Nagy & Vinkó (2016). LC2.2 computes the bolometric light curve of a homologous expanding supernova using the radiative diffusion approximation as introduced by Arnett & Fu (1989). The list of model parameters includes the initial radius (R₀), the ejecta mass (M_ej), the initial ⁵⁶Ni mass (M_Ni), the total kinetic and thermal energy of the ejecta (E_ke and E_th, respectively), the type and the exponent of the density profile (power-law or exponential), and the average value of the Thompson-scattering opacity (κ) in the envelope.

Recombination of hydrogen and/or helium, which is built-in the code, is not relevant in the case of the Type Ic SNe. Thus, it has been turned off, similar to the additional energy input from magnetar spin-down. LC2.2 assumes fixed gamma-ray and positron opacities, set to their nominal values of κ_γ = 0.027 and κ₊ = 7 cm² g⁻¹, respectively. Thus, they are not adjustable parameters, unlike in the previous version described by Nagy & Vinkó (2016). Note also that while LC2.2, in principle, can model a two-component ejecta having different masses, radii, density profiles, and opacities, we applied only a single component fit to the light curve of these SNe Ic.

The initial radius is not a sensitive parameter in this code. Thus, a constant R₀ = 4 × 10¹¹ cm is adopted for the following calculations. Table 7 lists the input parameters of SN 2014L, SN 1994I, and SN 2007gr. M_Ni is derived from Equation (1) based on the optical flux because the UBVRI bolometric light curves are modeled here. M_ej and E_ke in Model A and B are derived from the peak (i.e., Equations (2) and (3)) and tail (i.e., Equations (4) and (5)) of the light curves, respectively. A fiducial optical opacity κ = 0.1 cm² g⁻¹ is adopted in Model A. E_th is a free parameter to match the peak between observation and calculation.

Figure 12 displays the modeling results based on the input parameters in Table 7. The U- and B-band flux are decreased much more quickly than that in the VRI bands. Thus, the full UBVRI band photometry only covers the first 70 days of SN 2014L. On the other hand, it is reasonable to assume that the R-band light curve is a good proxy of the bolometric light curve within a constant scaling factor (WJC 15). For comparison, we over-plotted the scaled R-band light curve by assuming the R-band luminosity scales with the bolometric luminosity.

Zoom In Zoom Out Reset image size

These two models yield a broader light curve than the observation. We noted that the width of light curve produced by LC2.2 code is inversely proportional to the scale of κ and the ratio of E_ke and E_th. Thus, we liberalized the parameters κ, E_ke, and E_th, to find a better fit for the observation. In this case, M_ej follows κ based on the relation of Equation (2).

The best fitting recorded and plotted as Model C, matches the observations well at τ ≲ 40–70 days. For example, Model C gives a good fit for SN 2014L and SN 2007gr at about τ ≲ 40–50 days, and the latter curves of the model are fainter than the data. This might be due to the leakage of gamma-rays. The model C of SN 1994I gives a good fit during all of the observational epochs. It is uncertain after τ > 70 days because of the lack of observation. However, the tendency of the observed light curve is to follow the prediction of model C. In brief, model C yields a good fit at τ ≲ 40–70 days, and model A gives a good fit for the tail at τ > 90 days. The mismatch between these models is probably due to the oversimplifying assumption of constant opacity. As mentioned before, the rise time derived from model C (i.e., t_r = 13 days) is adopted in the above calculation because this model gives proper fitting during the rising phase.

It is remarkable that the ratio of E_ke/E_th in model C is much bigger than that of model A and B. Besides, SN 1994I shows the narrowest light curve in this figure, and it has the highest E_ke/E_th ratio. It seems that the narrower light curve could be the consequence of the faster speed of ejecta. However, this is vigorously challenged by BL SNe Ic, which shows the highest velocity and the broadest light curve. On the other hand, the speeds derived from model C are much higher than those observed, which indicates an overestimation of E_ke for these SNe.