A Monte Carlo Approach to Magnetar-powered Transients. II. Broad-lined Type Ic Supernovae Not Associated with GRBs

Over the past two decades, the discovery of broad-lined type Ic supernovae (SNe Ic-BL; see Filippenko 1997 for the classification of known SNe) and superluminous SNe (SLSNe) has greatly enlarged the family of known core-collapse SNe (CCSNe). The association between long-duration gamma-ray burst (GRB) 980425 and its spectroscopically associated Ic-BL SN 1998bw (Galama et al. 1998; Patat et al. 2001), that is, the so-called GRB–SN connection (e.g., Woosley & Bloom 2006; Cano et al. 2017), ignited interest in these energetic and rare type of stripped-envelope CCSNe.

To date, the luminosity of most, if not all, GRB-SNe and SNe Ic-BL could be explained by radioactive heating arising from energy deposition from the radioactive decay of nickel and cobalt, which are nucleosynthesized during the explosion, into their daughter products (Cano et al. 2016). However, it appears that the luminosity of many SLSNe cannot be adequately explained in this scenario, and alternative energy sources have been proposed. As a consequence, it is now usually assumed that at least a subclass of SLSNe, type Ic SLSNe, are powered by millisecond magnetars (Kasen & Bildsten 2010; Woosley 2010; Chatzopoulos et al. 2012; Inserra et al. 2013; Nicholl et al. 2014; Metzger et al. 2015; Mösta et al. 2015; Wang et al. 2015a; Dai et al. 2016; Kashiyama et al. 2016), although there is evidence for interaction between ejecta and the circumstellar medium (Yan et al. 2015; Wang et al. 2016d; Chen et al. 2017) at late times.

For SNe Ic-BL, shortcomings of the one-dimensional (1D) ⁵⁶Ni model (e.g., Iwamoto et al. 2000; Nakamura et al. 2001) stimulated the suggestion of a two-component ⁵⁶Ni model (Maeda et al. 2003). In this model, it is assumed that the ejecta are composed of two components: the outer fast-moving component (jet) and the inner slow-moving component (core). The former is responsible for the bright peak of the light curve, while the latter is responsible for the late-time exponential decay. This model is very useful for providing a better description of the ejecta structure and has been very successful in reproducing the luminosity of most SNe Ic-BL.

Recently, the application of the magnetar model to SNe Ic-BL was considered (Cano et al. 2016; Wang et al. 2016b, 2016c, 2017), which is built upon the pioneering works of Ostriker & Gunn (1971), Wheeler et al. (2000), and Thompson et al. (2004). The proposition of the improved magnetar model (Wang et al. 2016c), which takes into account the photospheric recession and acceleration of the ejecta by the spinning-down magnetar, provides an opportunity to examine the magnetar model against SNe Ic-BL in a self-consistent way. It was shown that the spin-down of the magnetar will lose a small fraction of its rotational energy to its light curve (Wang et al. 2016b), while the remaining fraction is transferred into the kinetic energy of the ejecta. Evidence for the formation of stable magnetars following the explosions of SNe Ic-BL was subsequently found by Wang et al. (2017). Such a model can also naturally account for the mysterious origin of the huge kinetic energies of SNe Ic-BL (Wang et al. 2016b).

The discovery of relativistic SNe Ic-BL, 2009bb and 2012ap, through their bright late-time radio emission (Bietenholz et al. 2010; Soderberg et al. 2010; Chakraborti et al. 2011, 2015), places the magnetar model on a more solid ground because such events require central engines to accelerate a tiny fraction of the ejecta to quasi-relativistic velocities (Margutti et al. 2014). Actually there is a continuous distribution of various types of CCSNe on the kinetic energy profile of the ejecta (Soderberg et al. 2006). The relativistic SNe Ic-BL lie between ordinary SNe Ibc and energetic GRBs and are similar to the subenergetic GRBs, such as GRB 100316D (Margutti et al. 2013) and GRB 140606B (Cano et al. 2015). This may indicate that similar engines were operating in subenergetic GRBs and SNe 2009bb and 2012ap.

Based on the above findings, here we test the hypothesis that all SNe Ic-BL are powered by magnetars. Under such hypothesis, we assessed the validity of the derived fitting parameters and consider the statistical characteristics of SNe Ic-BL. Despite the paucity of observed SNe Ic-BL, the accumulation of such events has reached a level where meaningful statistical results can start to be obtained. It is therefore very timely to confront a larger sample (N = 11) of SNe Ic-BL with the magnetar model.

To determine the uncertainties in the fitting parameters, Wang et al. (2017) developed a Markov chain Monte Carlo (MCMC) code on the basis of the magnetar model. This code was applied to SLSNe Ic (Liu et al. 2017) to minimize the total errors arising from fitting the model to the SN light curves, and the evolution of photospheric velocity and temperature, if available. In this paper, we focus on the SNe Ic-BL not associated with GRBs. In what follows, we use the words "SNe Ic-BL" to indicate SNe Ic-BL not associated with GRBs except when specifically mentioned otherwise.

The structure of this paper is as follows. In Section 2 we present the data available in the literature, along with a detailed analysis on the uncertainties of the data. Then in Section 3 we present our fitting results of the known SNe Ic-BL. Section 4 discusses the implications of the results. Particularly, Section 4.2 discusses the estimation of the appearance of nebular features by early light curve modeling; Section 4.3 discusses the correlations between the derived parameters; Section 4.4 discusses the possibility of alternative models to interpret the light curves and velocity evolution of some SNe. A summary is given in Section 5.

Modjaz et al. (2016) listed 12 SNe Ic-BL. However, the light curve of SN 2007D is missing, and we are therefore left with 11 such events, as listed in Table 1. The modeling of SN light curves usually involves the bolometric luminosity. To construct a bolometric light curve, emission in UV (ultraviolet), BVRI (optical), and IR (UVOIR) passbands should be integrated. It is, however, commonplace that only the optical bands are available for the follow-up of an SN from very early times to late times. Ultraviolet emission of an SN Ic-BL is usually strongest only at early stages, and its contribution to the total UVOIR bolometric flux can be more than 20% during the first two weeks (Cano et al. 2011; Lyman et al. 2014), while late-time UV follow-up is frequently missing. IR emission, which is usually strong for the whole evolution stage (and can contribute as much as 50% of the total UVOIR bolometric flux after peak light, e.g., Figure 6 of Tomita et al. 2006, Figure 14 of Valenti et al. 2008, and Figure 7 of Olivares et al. 2015), is only obtained for a few SNe. For this reason, different authors usually resort to different methods to construct the bolometric luminosity. To list some, the observations of SN 2003jd were available only in the BVRI bands, and the contributions from UV and IR bands were added by assuming the same fractional contributions to the bolometric light curve as by SN 2002ap (Valenti et al. 2008). The bolometric light curve of PTF10qts was obtained by increasing the integrated fluxes by 15% to account for the contribution from the unavailable UV and NIR bands (Walker et al. 2014). Some authors, on the other hand, decided to not include the contribution of UV or NIR bands (Tomita et al. 2006; Sahu et al. 2009; Young et al. 2010; Pignata et al. 2011).

Table 1.  The SNe Ic-BL Sample

SN	References		Extinctiona	Colorb	Luminosity Distance
	Light Curve	Velocity	Corrected	Used	In Referenced Paperc (Mpc)	Methodd	Adoptede(Mpc)
1997ef	I00	M16	None	-	52.3	C	50.6
2002ap	T06	M16	GH	B − I	7.94	L	9.22
2003jd	V08	M16	GH	B − R	78	C	84.3
2007bg	Y10	M16	G	B − R	147	C	157.0
2007ru	S09	M16	G	V − R	67.6	C	69.2
2009bb	P11	M16	GH	V − R	40	L	40.68
2010ah/PTF10bzf	M13, C11	C11	G	B − R	218.8	C	228.5
2010ay	S12	M16	GH	-	297.9	C	311.6
2012ap	M15	M16	GH	B − V	43.05	L	40.37
PTF10qts	W14	M16	G	${g}^{{\prime} }-{i}^{{\prime} }$	415	C	428.1
PTF10vgvf	C12	M16	G	-	60.3	C	63.5

Notes.

^aNone: no extinction was corrected; G: corrected for Galactic extinction; H: corrected for host extinction. ^bThe color used to calculate bolometric magnitude, following Lyman et al. (2014). A hyphen in this column indicates that only one passband is available or no data in individual passbands are provided. ^cThe distance used in the referenced paper, which is calculated according to the given distance modulus. ^dThe method used in this paper to calculate distance. L: linear distance extracted from the NASA/IPAC Extragalactic Database (NED); C: the distance was calculated according to the latest Planck cosmological parameters. ^eThe distance adopted in this paper. ^fCorsi et al. (2012) classified PTF10vgv as SN Ic based on its low Si ii λ6355 absorption velocities, while Modjaz et al. (2016) reclassified it as SN Ic-BL because of its broad-lined optical spectra. Here we follow Modjaz et al. (2016).

References. I00: Iwamoto et al. (2000); T06: Tomita et al. (2006); V08: Valenti et al. (2008); Y10: Young et al. (2010), S09: Sahu et al. (2009); P11: Pignata et al. (2011); M13: Mazzali et al. (2013); C11: Corsi et al. (2011); S12: Sanders et al. (2012); M15: Milisavljevic et al. (2015); W14: Walker et al. (2014); C12: Corsi et al. (2012); M16: Modjaz et al. (2016).

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To reduce the above uncertainty, we decide to use the method developed by Lyman et al. (2014, 2016). In this method, the color defined by two optical bands is used to calculate the bolometric correction. In Table 1, we list the color we used to calculate the bolometric luminosity. In this calculation, we choose the color that has the least rms given in Table 2 of Lyman et al. (2014) and at the same time the longest time coverage in the two passbands defining the chosen color. If these two conditions cannot be met simultaneously, we always choose the passbands that have the longest observational time. Such a choice can minimize the errors that may be introduced by interpolation or extrapolation. Sometimes data are available only in a single passband for some time duration, for example, the data of SN 2007bg before 7.2 days given in Table 3 of Young et al. (2010); while these data are crucial to constraining the fitting parameters, we set their bolometric corrections to the same as that at the closest time.

Another uncertainty in the construction of a bolometric light curve comes from the treatment of extinction. The Galactic extinction is well understood and can be handled properly using the dust maps of Schlegel et al. (1998) and as revised by Schlafly & Finkbeiner (2011). The host extinction, however, can only be estimated for some SNe because of the poor quality of the Na i D lines in the measured spectra (which may be a poor proxy for the host extinction anyways, e.g., Poznanski et al. 2011). We list the extinction treatment in Table 1. Even for the same SN, the determined extinction could be different from different authors. Taking SN 2012ap as an example, Milisavljevic et al. (2015) adopted a total extinction of $E{(B-V)}_{\mathrm{total}}=0.45\,\mathrm{mag}$, while Liu et al. (2015) adopted a value of $E{(B-V)}_{\mathrm{total}}=0.87\,\mathrm{mag}$.

Further uncertainty comes from the different values of the cosmological parameters used in the literature to derive the luminosity distances to the various SNe. For SNe 2002ap, 2009bb, and 2012ap, redshift-independent methods, for example, Tully–Fisher measurements, were facilitated to derive the distances, as are available in the NASA/IPAC Extragalactic Database (NED). Such linear distances are weight-averaged, as listed in Table 1. For other SNe for which no such linear distances are available, to minimize distance uncertainties, we transform, according to the method de-scribed in Cano et al. (2014), the light curves in the literature to a common cosmology, that is, the latest Planck results: ${H}_{0}=(67.8\,\pm 0.9)\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$, ${{\rm{\Omega }}}_{m}=0.308\pm 0.012$ (Ade et al. 2016).

To compare the differences in distances, we also list in Table 1 the distances used in the original papers. Nevertheless, because the redshifts of the SNe studied here are small (see Table 2 in Modjaz et al. 2016), it is found that the errors in the infrared-derived distances introduced by assuming different cosmological parameters are small, typically 3%–5%. Corsi et al. (2012) did not give the distance modulus of PTF10vgv in their derivation of absolute magnitudes. We digitalized their Figure 2 and found ${\mu }_{\mathrm{PTF}10\mathrm{vgv}}=33.9\,\mathrm{mag}$, based on which the light curve transformation was performed. In summary, the largest difference between our adopted distance and that used in the original paper is for SN 2002ap, for which we adopt $9.22\,\mathrm{Mpc}$, rather than $7.94\,\mathrm{Mpc}$ in the original paper. The smallest difference is for SNe 2009bb, for which distances $\sim 40\,\mathrm{Mpc}$ have been adopted in the relevant studies.

The photospheric velocity is another critical quantity that significantly impacts the light curve fitting results. Different velocity indicators in the spectra, for example, Si ii λ6355, Na i D λ5891, O i λ7774, Ca ii λ8579, and Fe ii λ5169, usually give different results (Valenti et al. 2008; Modjaz et al. 2016). This difference may be a result of the different depths of elements in the ejecta, the degree of element mixing, and the amount of deviation from spherical expansion. Recently, Modjaz et al. (2016) developed a way to measure velocities for all SNe Ic-BL and SNe Ic in a homogeneous way. In this paper, we use the velocity data given by Modjaz et al. (2016) when available. Using such a homogeneous set of velocity data reduces the bias in the resulting fitting parameters.

In principle, the above uncertainties all contribute to the errors in bolometric luminosities. In practice, we include errors (all added in quadrature) in bolometric corrections (rms given in Table 2 of Lyman et al. 2014) and in the photometry given in the papers where the observational data were provided.

We calculated the extinction according to Cardelli et al. (1989) by assuming the Milky Way extinction law. Cosmological expansion has been taken into account using the following equation (Hogg et al. 2002; Lunnan et al. 2016):

$$\begin{eqnarray}&&M=m-5\mathrm{log}({D}_{L}/10\,\mathrm{pc})+2.5\mathrm{log}(1+z),\end{eqnarray} \tag{ 1 }$$

where D_L is the luminosity distance and z is the redshift. The last term in the above equation is not a true K correction, but it is a good approximation.

For SN PTF10vgv, only R-band luminosities were observed (Corsi et al. 2012). To obtain bolometric luminosities, Corsi et al. (2012) assumed a bolometric correction ${M}_{\mathrm{bol}}-{M}_{R}=-0.496\,\mathrm{mag}$ based on the early-time photospheric temperature ${T}_{\mathrm{phot}}\approx {10}^{4}\,{\rm{K}}$ of this SN. We use this bolometric correction to derive the bolometric light curve for SN PTF10vgv. Such a treatment is of course somewhat simplified because the temperature evolves rapidly during the early expansion. Another SN for which only R-band luminosities were observed is SN 2010ay (Sanders et al. 2012). The luminosity and expansion velocity were combined to derive a temperature of $6900\,{\rm{K}}$ at peak light. This implies a bolometric correction of ${M}_{\mathrm{bol}}-{M}_{R}=0.29\,\mathrm{mag}$, according to which the bolometric luminosities are derived here. This treatment should not introduce too much bias because the observation duration of this SN is short, within $20\,\mathrm{days}$ before or after peak. For SN 1997ef, only V-band data are provided by Iwamoto et al. (2000). According to the effective temperatures ($\sim 6100\,{\rm{K}}$) given in Table 3 of Iwamoto et al. (2000), we applied a bolometric correction (${M}_{\mathrm{bol}}-{M}_{V}=-0.05\,\mathrm{mag}$) to SN 1997ef. We will discuss the implications of the approximation in obtaining bolometric light curves for SNe 1997ef, 2010ay, and PTF10vgv in Section 4.3.

The rms of the prescription of Lyman et al. (2014) is $\sim 0.06\,\mathrm{mag}$, while the measurement errors of the light curve range from $\sim 0.02\,\mathrm{mag}$ to $\sim 0.3\,\mathrm{mag}$. Therefore, the uncertainties in the bolometric luminosity constructed by this method are usually dominated by measurement errors in the two individual bands from which bolometric corrections are calculated. The measurement errors of SNe 1997ef, 2010ay, and PTF10vgv are $0.03\mbox{--}0.06\,\mathrm{mag}$, $0.2\mbox{--}0.3\,\mathrm{mag}$, and $0.02\mbox{--}0.2\,\mathrm{mag}$, respectively. As a result, if there had been two bands available for SNe 1997ef, 2010ay, and PTF10vgv, the uncertainties are likely slightly larger than that depicted in Figures 2(a), 4(a), and 3(f) but dominated by measurement errors for those points whose measurement errors are large. Given this fact, for simplicity, we adopt the errors in an individual band as the errors of bolometric luminosity for SNe 1997ef, 2010ay, and PTF10vgv.⁸

In Figure 1 we compare the luminosity data provided by the original papers and that calculated according to Lyman et al. (2014) for two representative SNe. We call the luminosity of these two SNe "representative" because the luminosity of SN 2002ap given by the original paper includes the contributions from BVRI and IR bands, while the luminosity of SN 2009bb given by the original paper includes a contribution only from BVRI bands. Another reason we chose these two SNe is that their luminosity is integrated according to observational data, while the luminosity of some other SNe is calculated in the original papers by assuming some contribution from unavailable bands (frequently the IR band).

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In the comparison in Figure 1, the data given in the original papers are transformed to the distances given in Table 1. From this figure it is evident that the method of Lyman et al. (2014) is accurate for the first $\sim 80\,\mathrm{days}$, since the bolometric correction is calculated according to the luminosity data in this time period. Fortunately, most of the luminosity data in our sample have a time coverage that is not much longer than $\sim 80\,\mathrm{days}$. The data with $t\gtrsim 80\,\mathrm{days}$ are enough to constrain most of the model parameters. Figure 1(a) shows that the contribution from the unavailable UV band is small for SN 2002ap even at very early stages,⁹ while Figure 1(b) shows that the contribution from the unavailable UV and IR bands cannot be ignored.

As explained in Wang et al. (2016c, 2017), the model we have adopted is formulated by eight parameters. Although the model is dubbed a "magnetar model," it also includes a ⁵⁶Ni component. As a consequence, the model includes the usual parameters: the ejecta mass ${M}_{\mathrm{ej}}$, ⁵⁶Ni mass ${M}_{\mathrm{Ni}}$, gray optical opacity κ, initial expansion velocity ${v}_{\mathrm{sc}0}$, and opacity to ⁵⁶Ni decay photons ${\kappa }_{\gamma ,\mathrm{Ni}}$ . In addition, the model includes magnetar parameters, the dipole magnetic field B_p, the initial rotation period P₀, and opacity ${\kappa }_{\gamma ,\mathrm{mag}}$ to account for the leakage (Chen et al. 2015; Wang et al. 2015a) of high-energy photons (Murase et al. 2015; Wang et al. 2016a) from magnetars. Here the subscript "p" in B_p means the dipole field at the pole of the star (Shapiro & Teukolsky 1983). For the gray optical opacity κ, we take the fiducial value $\kappa =0.1\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$, as used in previous investigations (Wang et al. 2016b, 2017). We also include the unknown explosion time ${T}_{\mathrm{start}}$ of the SN in the MCMC code. In what follows, we use the name "magnetar model" to indicate the magnetar+⁵⁶Ni model, except when specifically mentioned otherwise.

The magnetar model proposed by Wang et al. (2016c) traces the photospheric recession and therefore the emission from the photosphere, and those materials outside the photospheric radius (hereafter referred to as nebular component, or nebula for short) can be isolated. We examined the spectra of SNe Ic-BL and try to figure out if the nebular emission is helpful in determining the appearance of nebular features in the spectra. It turns out that when the nebula emitted about 16% of the total emission, nebular features (e.g., forbidden lines) could begin to emerge in the SN spectra. If we assume that an SN begins to transition into a nebular phase when the nebula radiates this percentage of emission, we can obtain the time ${T}_{\mathrm{Neb}}$ (since explosion in the rest frame) by fitting the early-time light curve, as listed in Table 2.¹⁰ In the magnetar model, the early peak of the light curve of an SN is caused by the spin-down of the magnetar. Consequently, the ⁵⁶Ni mass can be ignored for such early-time modeling.

Table 2.  Best-fitting Parameters of Our SNe Ic-BL Sample

SN	${M}_{\mathrm{ej}}$	${M}_{\mathrm{Ni}}$	Bp	P0	${v}_{\mathrm{sc}0}$	${\kappa }_{\gamma ,\mathrm{Ni}}$	${\kappa }_{\gamma ,\mathrm{mag}}$	${T}_{\mathrm{start}}$	${T}_{\mathrm{Neb}}$	Constraints on ${T}_{\mathrm{Neb}}$	EK0
	$({M}_{\odot })$	$({M}_{\odot })$	$({10}^{15}\,{\rm{G}})$	(ms)	(km s−1)	(cm2 g−1)	(cm2 g−1)	(days)	(days)	(days)	${10}^{51}\,\mathrm{erg}$
1997ef	${3.3}_{-0.19}^{+0.21}$	0.058 ± 0.001	${11.4}_{-0.5}^{+0.4}$	${5.2}_{-0.3}^{+1.3}$	${1900}_{-1000}^{+2600}$	${0.12}_{-0.02}^{+0.03}$	-	$-{16.1}_{-0.5}^{+0.4}$	153	$(61.3,119.8)$	0.07
2002ap	1.7 ± 0.2	0.049 ± 0.001	$19.5\pm 2$	12 ± 4	${10500}_{-1000}^{+700}$	0.20 ± 0.02	$8.8\pm 6$	1.9 ± 0.1	71	$(44,81)$	1.13
2003jd	2.8 ± 0.3	0.05 ± 0.02	2.5 ± 0.3	${18}_{-1.6}^{+1.2}$	12490 ± 500	${0.3}_{-0.2}^{+0.3}$	${1.75}_{-1.1}^{+1.5}$	$-14.9\pm 1$	92	$(62.5,81.2)$	2.64
2007bg	1.5 ± 0.2	0.03 ± 0.004	${10.5}_{-3}^{+5}$	${20}_{-13}^{+11}$	${11900}_{-2100}^{+1200}$	${0.4}_{-0.3}^{+1.2}$	-	−10 ± 2	77	$(34.7,67.7)$	1.3
2007ru	5.6 ± 0.5	0.078 ± 0.006	${6.2}_{-0.2}^{+0.3}$	1.6 ± 0.1	∼0	${0.19}_{-0.03}^{+0.05}$	-	${0.8}_{-0.6}^{+0.9}$	86	$(70,200)$	∼0
2009bb	2.2 ± 0.07	0.025 ± 0.01	2.9 ± 0.3	28.6 ± 0.6	${19300}_{-700}^{+600}$	${0.4}_{-0.35}^{+2.8}$	${2.3}_{-1.6}^{+1.8}$	$-12\pm 0.4$	48	$(55,295)$	4.90
2010ah	${2.58}_{-1.6}^{+2.6}$	${0.14}_{-0.03}^{+0.02}$	${17.5}_{-7.9}^{+20}$	${4.2}_{-2.7}^{+35}$	${14100}_{-9700}^{+10000}$	$\gtrsim 0.06$	-	$-0.7\pm 1$	69	$\gt 14.5$	3.1
2010ay	${6.7}_{-1}^{+1.8}$	-	0.8 ± 0.1	11±1	${24900}_{-2100}^{+300}$	-	-	${2.0}_{-0.4}^{+0.3}$	55	$\gt 44.1$	33.4
2012ap	${2.3}_{-0.8}^{+1.7}$	-	${3.1}_{-0.5}^{+1.6}$	${40}_{-11}^{+3}$	${14219}_{-900}^{+1000}$	-	-	$-0.6\pm 1$	77	$(38.5,230.5)$	2.72
PTF10qts	${1.9}_{-0.4}^{+0.6}$	0.28 ± 0.06	${9.9}_{-6.6}^{+12}$	${26}_{-19}^{+64}$	${21307}_{-2500}^{+2400}$	${0.3}_{-0.1}^{+1.5}$	-	$-{10.5}_{-1.9}^{+2.7}$	39	$(38.8,230.4)$	5.1
PTF10vgv	${0.7}_{-0.07}^{+0.08}$	0.059 ± 0.003	1.7 ± 0.2	${28.8}_{-0.7}^{+0.5}$	${10000}_{-600}^{+700}$	0.19 ± 0.03	0.013 ± 0.001	6.9 ± 0.04	43	$(46.2,82.6)$	0.42

Notes. All times in this table are in the rest frame. In these fits, we fixed $\kappa =0.1\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$. A hyphen indicates that this quantity cannot be constrained effectively. The data on the left of the vertical line are fitting parameters, while the data on the right are derived values. The MCMC code does not calculate the errors of these derived values. The references for the spectra are the same as in Table 1.

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To determine the ⁵⁶Ni mass, it is necessary for the light curve to be observed at least for $\sim 110\,\mathrm{days}$. We divide the observed SNe Ic-BL light curves into two classes: those with an observational duration $t\gtrsim 100\,\mathrm{days}$ (class I) and with $t\lesssim 100\,\mathrm{days}$ (class II). We chose $100\,\mathrm{days}$ as the dividing boundary because the lifetime of ⁵⁶Co is $\sim 110\,\mathrm{days}$. If the observational duration is longer than $100\,\mathrm{days}$, the mass of ⁵⁶Ni can be constrained. In this case, we allowed the ⁵⁶Ni mass to be a free parameter. In the opposite case, the ⁵⁶Ni mass cannot be constrained, and the only parameters that can be constrained are the magnetar parameters because it is found that in the magnetar model the early peak of the light curve can be attributed to magnetar spin-down (Wang et al. 2016b, 2017).

We find there are nine SNe that belong to class I, while the remaining two SNe fall in class II. Among the SNe in class I, three SNe, 1997ef, 2002ap, and 2007ru, were studied previously with our magnetar model (Wang et al. 2016b, 2017). SN 2002ap was investigated using an MCMC code (Wang et al. 2017), while SNe 1997ef and 2007ru were studied via manual fitting (Wang et al. 2016b). We included them here to test the sensitivity of fitting parameters to the adoption of different photospheric velocities because the velocities used here (the values given by Modjaz et al. 2016) are different from previous studies (Wang et al. 2016b, 2017), where we used the velocities provided in the original papers. In addition, doing so will give unbiased statistical results.

The newly fitted light curves of the three previously studied SNe 1997ef, 2002ap, and 2007ru are shown in Figure 2, where emissions from the photosphere and nebula are shown as dashed and dotted-dashed lines, respectively. The vertical dotted lines in this figure mark the epochs when nebular emission contributes 16% of the total emission, which we assume to be the time when nebular emission features, for example, forbidden emission lines, begin to appear. The remaining six SNe in class I are shown in Figure 3, where we show the contribution from the magnetar and ⁵⁶Ni as dashed and dotted-dashed lines, respectively. In Figure 4 the light curves were fitted with a pure-magnetar model (without ⁵⁶Ni contribution) because the ⁵⁶Ni masses of these two SNe in class II cannot be determined. The best-fitting values are given in Table 2. Because the contribution of magnetar and ⁵⁶Ni to the total emission for the three SNe depicted in Figure 2 were shown previously (Wang et al. 2016b, 2017), we will not show them in this paper, as we do in Figure 3.

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For SNe 2007bg, 2010ah, and PTF10qts in Figure 3, the magnetar contribution dies away rapidly, which is quite different from the light curves given by, for example, Kasen & Bildsten (2010), where the light curves tend to flatten at late times. The decline rate is also faster than for the light curves where the gamma-ray leakage has been taken into account (Chen et al. 2015; Wang et al. 2015a). This rapid decline of magnetar contribution is due to the rapid spin-down of the magnetar powering the SN Ic-BL. At very late times, the magnetar contribution will eventually flatten, as can be seen from the dashed lines in Figures 1 and 2 of Wang et al. (2017). This rapid spin-down is why the magnetar can convert almost all of its rotational energy into the kinetic energy of ejecta of SNe Ic-BL and why the contribution of ⁵⁶Ni is necessary for SNe Ic-BL in the magnetar model.

The MCMC code can only be run for those SNe for which observational errors are given. For the velocity data given by Modjaz et al. (2016), we adopted the errors given in their paper. Modjaz et al. (2016) did not provide the velocity data for SN 2010ah, for which, following Wang et al. (2017), we set the velocity errors to be half of the measured values to account for the large differences given by different velocity measurements (see, e.g., Valenti et al. 2008).

For some SNe, such as SNe 2007bg and 2007ru, the missing or sparse data coverage before peak luminosity makes the upper limits before discovery indispensable for obtaining reliable results; see Figures 2(c) and 3(b). Sometimes the model light curves almost pass through the bolometric limits; see, for example, SNe 2003jd and 2007bg in Figure 3. We looked into the data and found that the bolometric limit of SN 2003jd is $0.6\,\mathrm{days}$ earlier than the explosion date, while the bolometric limit of SN 2007bg is $1.0\,\mathrm{days}$ earlier than the time when the model light curve reaches the same bolometric luminosity as the upper limit.

The best-fitting parameters listed in Table 2 are generally similar to what we found before for SNe 1997ef, 1998bw, 2002ap, and 2007ru (Wang et al. 2016b, 2017), where we had a detailed discussion on the reasonableness of the determined parameters such as ${M}_{\mathrm{ej}}$, ${M}_{\mathrm{Ni}}$, B_p, P₀, and ${\kappa }_{\gamma ,\mathrm{mag}}$. We also discussed the possible reasons for a larger value of ${\kappa }_{\gamma ,\mathrm{Ni}}$ than the standard value $\sim 0.027\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$.

From Table 2 it is clear that usually the opacity to magnetar high-energy photons ${\kappa }_{\gamma ,\mathrm{mag}}$ can only be determined for SNe, such as 2002ap and 2003jd, observed to late stages ($t\gtrsim 300\,\mathrm{days}$) because only at such late stages (except for the early peak) does the magnetar contribution dominate the ⁵⁶Ni contribution. Table 2 indicates that ${\kappa }_{\gamma ,\mathrm{mag}}$ is also constrained for SNe 2009bb and PTF10vgv, despite their short observation duration. For PTF10vgv, the given value is favored because the ejecta mass is small and high-energy radiation from the magnetar will leak even at early stages. For SN 2009bb, the given value is caused by the significant contribution of the magnetar to the light curve even at late stages.

Given the fitting results in Table 2, some correlations between different parameters can be examined. We show the correlations of energy versus ⁵⁶Ni mass, energy versus ejecta mass, explosion energy versus neutron star rotational energy, and energy versus dipole magnetic field of the magnetar in Figures 5–8, respectively. In the magnetar model, three forms of energy are considered here: the initial explosion energy ${E}_{\exp }$, the neutron star's rotational energy ${E}_{\mathrm{NS}}$, and the sum of these two energies ${E}_{\mathrm{total}}$. In the usual magnetar model that does not take into account the acceleration of the ejecta by the spinning-down magnetar, the kinetic energy of the ejecta is just the initial explosion energy. In our adopted model, the kinetic energy is no longer a constant. Instead, it evolves from its initial value, that is, the initial explosion energy, according to the energy injection of the magnetar. The evolution of kinetic energy can be clearly appreciated by inspecting the rapid rise of the photospheric velocities at early times; see, for example, SNe 1997ef and 2007ru in Figure 2. This is why the reported (initial) velocities of these two SNe are significantly smaller than the maximum values attained in Figure 2.

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4.1. General Implications

The high fitting quality for both luminosities and velocities can be appreciated from Figures 2–4. The only exception may be the velocity fitting result of SN 1997ef and to a lesser extent SN 2003jd. For SN 1997ef, the early-time velocities cannot be fitted because in our one-zone model the velocity increases rapidly at very early times (acceleration phase) and then declines progressively faster because of the photospheric recession. Inspection of the velocity data of SN 1997ef indicates that the velocity evolution is flat during the period 40–65 days, which implies that the earlier time velocity should also be flat in our model. A possible way to get better fitting results may be the introduction of a fast-moving shell in the ejecta, which makes the photosphere begin to recede at early times. After this fast shell becomes transparent, the inner compact component slows down the recession of the photosphere, resulting in the later-time flat evolution of the velocity. This hypothesis is also supported by the earlier appearance of a nebular spectrum than in our prediction (see Section 4.2 for more discussion). The model velocity of SN 2010ah (Figure 3) is also somewhat lower compared with the data. This is caused by the large observational errors that make the MCMC code difficult to differentiate between different fitting parameters.

The effect of adopting different velocity data on the derived parameters can be appreciated by comparing the values of the fitting parameters of SN 1997ef given in Table 1 of Wang et al. (2016b) and Table 2. It turns out that the values of ${M}_{\mathrm{Ni}}$, B_p, ${\kappa }_{\gamma ,\mathrm{Ni}}$, ${\kappa }_{\gamma ,\mathrm{mag}}$, and ${T}_{\mathrm{start}}$ are insensitive to the expansion velocity, while ${M}_{\mathrm{ej}}$, P₀, and ${v}_{\mathrm{sc}0}$ are sensitive to the expansion velocity. This is because the former group of parameters are determined by the light curve slope (${\kappa }_{\gamma ,\mathrm{Ni}}$, ${\kappa }_{\gamma ,\mathrm{mag}}$, and ${T}_{\mathrm{start}}$) or luminosity (${M}_{\mathrm{Ni}}$ and B_p), while the latter group of parameters are determined by the diffusion timescale of the SN (Arnett 1982; Arnett et al. 2017) because P₀ and ${v}_{\mathrm{sc}0}$ affect the expansion velocity.

From Table 2 it is evident that in the magnetar (plus ⁵⁶Ni) model, the initial SN explosion energies are usually smaller than $\sim 2.5\times {10}^{51}\,\mathrm{erg}$, that is, the theoretical upper limit of explosion energy triggered by neutrino heating (Janka et al. 2016). There is one exception, SN 2010ay, which has an explosion energy $\gtrsim {10}^{52}\,\mathrm{erg}$ . We note that the light curve and velocity evolution of SN 2010ay are poorly sampled, and it is possible to attribute a fraction of the energy to a magnetar by tuning up the rotational energy of the magnetar. We conclude that the explosion energy of all well-observed SNe Ic-BL can be explained by neutrino heating.

Table 2 shows that the ⁵⁶Ni masses in this sample of SNe Ic-BL are usually smaller than $0.1\,{M}_{\odot }$. The only two values $0.28\,{M}_{\odot }$ and $0.14\,{M}_{\odot }$ that are above $0.1\,{M}_{\odot }$ are for PTF10qts and SN 2010ah, respectively. We note that the observational data of these two SNe are of the poorest quality, except SNe 2010ay and 2012ap, whose ⁵⁶Ni masses are not determined. The sparse luminosity data and large observational errors of these two SNe indicate that the derived values of ⁵⁶Ni mass should not be taken seriously. This implies that the ⁵⁶Ni masses of SNe Ic-BL have an upper limit of $0.2\,{M}_{\odot }$, that is, the maximal amount of ⁵⁶Ni that can be synthesized by the spin-down of a magnetar (Nishimura et al. 2015; Suwa & Tominaga 2015).

Table 2 also shows that the fitting parameters of the two relativistic SNe, 2009bb and 2012ap, are typical among this SNe Ic-BL sample. It is therefore unlikely we can acquire more clues on the explosion mechanism of relativistic SNe solely from such fitting parameters, if the magnetar model is the right model for such SNe. A thorough comparison between SNe Ic-BL and those associated with GRBs is required to get more clues.

PTF10vgv is peculiar because of its low absorption velocities (typical of ordinary SNe Ic) and broad-lined optical spectra (typical of SNe Ic-BL). It has the lowest ejecta mass, $0.7\,{M}_{\odot }$, in the SNe Ic-BL sample (see Table 2). Its opacity to magnetar photons, ${\kappa }_{\gamma ,\mathrm{mag}}=0.013\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$, is also much lower than the values found for the other SNe Ic-BL. Corsi et al. (2012) constrained its progenitor radius to be $R\lt (1-5)\,{R}_{\odot }$, consistent with a compact Wolf–Rayet star. These peculiarities may indicate that PTF10vgv lies in the gap between SNe Ic and SNe Ic-BL.

The MCMC code can determine the explosion time accurately if the light curve data are of high quality. The best case is SN 1998bw, for which the explosion time was constrained to be $-{0.009}_{-0.36}^{+0.32}\,\mathrm{days}$ relative to the GRB trigger time (Wang et al. 2017).¹¹ In Table 3 we compare the explosion time determined in this work with those given in the literature. Also listed in this table are the discovery date and date of nondetection. The explosion time is computed according to the times ${T}_{\mathrm{start}}$ given in Table 2, after correcting for cosmological time dilation.

Table 3.  Comparison of Explosion Times Derived in This Work and Previous Papers

SN	This Work	Previous Estimate	Discovery Date	Date of Nondetection	References
1997ef	$1997/11/{25}_{-0.5}^{+0.4}$	1997/11/20	1997/11/25	1997/11/16	H97, M00
2002ap	2002/01/27 ± 0.5	2002/01/25.5 ± 0.5	2002/01/29	2002/01/25	M02, T06
2003jd	2003/10/15.7 ± 1	<2003/10/17	2003/10/25	2003/10/16	V08
2007bg	2007/04/05 ± 2	-	2007/04/16.15	2007/04/06	Y10
2007ru	$2007/11/{26}_{-0.6}^{+0.9}$	2007/11/25.5	2007/11/27.9	2007/11/22	S09
2009bb	2009/03/18 ± 0.6	2009/03/19.1 ± 0.6	2009/03/21.11	2009/03/19.2	P11
2010ah	2010/02/20 ± 1	2010/02/17.8−2010/02/23.5	2010/02/23.5	2010/02/19.4	C11
2010ay	2010/02/23.06 ± 1.3	2010/02/21.3 ± 1.3	2010/03/05.45	2010/02/17.45	S12
2012ap	2012/02/04.25 ± 2	2012/02/05 ± 2	2012/02/10.23	-	M15
PTF10qts	$2010/08/{04.6}_{-2}^{+3}$	-	2010/08/05.23	2010/08/02	W14
PTF10vgv	2010/9/13.2 ± 0.04	-	2010/09/14.1	2010/09/12.5	C12

Notes. UT dates are used in this table. A hyphen indicates that the date was not specified in the original paper.

References. H97: Hu et al. (1997); M00: Mazzali et al. (2000); M02: Mazzali et al. (2002); T06: Tomita et al. (2006); V08: Valenti et al. (2008); S09: Sahu et al. (2009); Y10: Young et al. (2010); C11: Corsi et al. (2011); P11: Pignata et al. (2011); C12: Corsi et al. (2012); S12: Sanders et al. (2012); W14: Walker et al. (2014); M15: Milisavljevic et al. (2015)

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In the calculation of the explosion time, we frequently need the time of the V-band maximum, for which we consult Modjaz et al. (2014). Because the explosion time determined in this work is calculated according to the relevant time given in the original paper, the uncertainties of the explosion time are the errors given in the original paper, if available, or the errors of ${T}_{\mathrm{start}}$ given in Table 2, whichever is larger.

It can be seen from Table 3 that the times determined in this work are generally in good agreement with those given in the literature. The only exception is SN 1997ef, for which our determination, which is almost coincident with the discovery date, is $\sim 5\,\mathrm{days}$ later than that given by Mazzali et al. (2000). Please note that the first bolometric data point in Figure 2 is $4\,\mathrm{days}$ later than the discovery date.

4.2. Estimate the Appearance of Nebular Features from Early Light Curve Modeling

4.3. Correlations

4.4. Alternative Models?

In this paper we tested the hypothesis that all SNe Ic-BL are powered by a combination of input from a magnetar central engine and ⁵⁶Ni synthesized during the initial explosion. Next, we ask the question, can a pure-⁵⁶Ni model or a pure-magnetar model give comparable results?

In Figure 9 we show the best-fit 1D ⁵⁶Ni modeling result for the selected SNe, whose fitting parameters are listed in Table 4. It is well known that the 1D ⁵⁶Ni model cannot give a satisfactory description for SNe with long observation durations (Iwamoto et al. 2000; Nakamura et al. 2001; Maeda et al. 2003). As a result, the two-component model was employed to investigate most of the SNe in Figures 2 and 3 (SNe 1997ef, 2002ap, Maeda et al. 2003; SN 2003jd, Valenti et al. 2008; SN 2007bg, Young et al. 2010). In Figure 9, we do not show these well-studied SNe (1997ef, 2002ap, 2003jd).

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Table 4.  Best-fitting Parameters for the Selected SNe Using the 1D ⁵⁶Ni Model

SN	${M}_{\mathrm{ej}}$	${M}_{\mathrm{Ni}}$	${v}_{\mathrm{sc}0}$	${T}_{\mathrm{start}}$	${\kappa }_{\gamma ,\mathrm{Ni}}$	${\chi }^{2}/\mu$
	$({M}_{\odot })$	$({M}_{\odot })$	(km s−1)	(days)	(cm2 g−1)	Pure-56Ni Model	Magnetar+56Ni Model
2007bg	1.3 ± 0.5	${0.10}_{-0.02}^{+0.01}$	${12000}_{-1200}^{+1400}$	$-{17.5}_{-2.8}^{+4.9}$	${0.05}_{-0.02}^{+0.05}$	6.0	0.22
2007ru	2.6 ± 0.2	0.34 ± 0.01	16300 ± 700	$-{4.9}_{-0.07}^{+0.16}$	${0.04}_{-0.004}^{+0.005}$	4.7	0.7
2009bb	2.5 ± 0.1	0.24 ± 0.007	19600 ± 700	$-{15.8}_{-0.5}^{+0.4}$	${0.03}_{-0.002}^{+0.003}$	0.67	0.15
2010ah	${1.0}_{-0.5}^{+0.7}$	0.17 ± 0.02	20600 ± 7000	$-{2.5}_{-0.6}^{+0.9}$	$\gtrsim 0.5$	0.04	0.07
2010ay	${6.4}_{-0.5}^{+0.6}$	${1.2}_{-0.07}^{+0.08}$	${27700}_{-1600}^{+1800}$	${1.7}_{-0.4}^{+0.3}$	-	1.0	1.73
2012ap	1.6 ± 0.2	0.11 ± 0.004	14900 ± 800	$-{1.4}_{-0.9}^{+0.8}$	-	0.12	0.14
PTF10qts	${1.9}_{-0.3}^{+0.4}$	${0.34}_{-0.02}^{+0.03}$	23000 ± 2000	$-{12.1}_{-1.1}^{+0.9}$	${0.35}_{-0.1}^{+1.2}$	1.4	2.0
PTF10vgv	${0.24}_{-0.02}^{+0.03}$	0.25 ± 0.002	${4700}_{-400}^{+500}$	7.1 ± 0.04	0.014 ± 0.001	4.2	0.7
PTF10vgv-27	0.82 ± 0.04	0.26 ± 0.002	14100 ± 500	6.9 ± 0.03	0.027	1.2	-

Notes. In these fits, we fixed $\kappa =0.1\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$. For those SNe where ${\kappa }_{\gamma ,\mathrm{Ni}}$ cannot be constrained (marked as hyphen), we set ${\kappa }_{\gamma ,\mathrm{Ni}}=0.027\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$. For those SNe whose ${M}_{\mathrm{Ni}}$ cannot be constrained, i.e., 2010ay and 2012ap, ${\chi }^{2}/\mu$ is the result of the pure-magnetar model. PTF10vgv-27 has the best-fitting parameters for the first 50 days of luminosity data after fixing ${\kappa }_{\gamma ,\mathrm{Ni}}=0.027\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$.

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From Figure 9 it can be seen that the 1D ⁵⁶Ni model can only account for the first $\sim 50\,\mathrm{days}$ of data (SNe 2010ay, 2012ap).¹³ For all well-observed SNe Ic-BL with observational time $\gtrsim 100\,\mathrm{days}$ (SNe 2007bg, 2007ru, and 2009bb in this figure), the two-component model should be invoked.

We will not examine these SNe within the framework of the two-component model in detail. SN 2010ay in Figure 9 is particularly interesting because its peak luminosity $\sim 3.0\times {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ is comparable to that of some of the SLSNe: PTF10hgi ($3.52\times {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1};$ Inserra et al. 2013), PTF11rks ($4.7\times {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1};$ Inserra et al. 2013), and PS1-14bj ($4.6\times {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1};$ Lunnan et al. 2016); see also Table 1 in Liu et al. (2017). Such SLSNe are usually assumed to be powered by magnetars because of the failure of the ⁵⁶Ni model.

The parameters for SN 2010ay are in tension with a typical CCSN. The ratio of ⁵⁶Ni mass to the ejecta mass is 0.19, close to the upper limit of 0.2 expected for a CCSN (Umeda & Nomoto 2008). The needed ⁵⁶Ni mass $1.2\,{M}_{\odot }$ ¹⁴ is higher than that of all SNe Ib/c but SN Ic-BL 2007D (Drout et al. 2011; Sanders et al. 2012). We therefore conclude that SN 2010ay is unlikely to be explained by a pure-⁵⁶Ni model, including the two-component model.

Another SN that is hard to explain by the ⁵⁶Ni model (including the two-component model) is PTF10vgv because its ejecta mass ${M}_{\mathrm{ej}}=0.24\,{M}_{\odot }$ is smaller than ${M}_{\mathrm{Ni}}=0.25\,{M}_{\odot }$ (see Table 4). Even if we double its expansion velocity to $\sim$ 10,000 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (see Figure 9(h)) so that ${M}_{\mathrm{ej}}$ is doubled, the ratio ${M}_{\mathrm{Ni}}/{M}_{\mathrm{ej}}$ is still larger than 0.5. The situation becomes even worse for the ⁵⁶Ni model if we take into account the uncertainties in the bolometric light curve of PTF10vgv. As discussed in Section 4.3, the light curve of PTF10vgv at late times should be brighter than depicted in Figure 9(h). This indicates that more ⁵⁶Ni is needed. The solid lines in Figure 9(h) are the best-fitting results allowing ${\kappa }_{\gamma ,\mathrm{Ni}}$ to vary. Table 4 shows that ${\kappa }_{\gamma ,\mathrm{Ni}}=0.014\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$, lower than the fiducial value ${\kappa }_{\gamma ,\mathrm{Ni}}=0.027\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$.¹⁵ The dotted-dashed lines in Figure 9(h) are the best-fitting results for the first $50\,\mathrm{days}$ of luminosity data after fixing ${\kappa }_{\gamma ,\mathrm{Ni}}=0.027\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$. In this case, the initial expansion velocity is $\sim$14,000 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$, much higher than the average velocity $\sim 7500\,\mathrm{km}\,{{\rm{s}}}^{-1}$ of this SN. Even with such a high expansion velocity, the ratio ${M}_{\mathrm{Ni}}/{M}_{\mathrm{ej}}=0.32$ is still higher than the theoretical upper limit of 0.2. We therefore conclude that PTF10vgv cannot be explained by the ⁵⁶Ni model. Recently, it was found that iPTF16asu (Whitesides et al. 2017) also cannot be explained by the ⁵⁶Ni model.

For PTF10qts, Walker et al. (2014) obtained ${M}_{\mathrm{Ni}}\,=0.35\pm 0.1\,{M}_{\odot }$, based on a model fit to the nebular spectrum of this SN. Such an estimate of ⁵⁶Ni mass is consistent with the value given in Table 4 in the pure-⁵⁶Ni model. This seems to argue against our hypothesis that all SNe Ic-BL were powered by magnetars. However, on the one hand, as commented by Walker et al. (2014), firm conclusions should not be drawn based on this result because of the low signal-to-noise ratio of the observed spectrum. On the other hand, for any SN Ic-BL with a long observational duration, some amount of ⁵⁶Ni is indeed required, although its amount is significantly lower than in the pure-⁵⁶Ni model.

Now we turn to the pure-magnetar model. For those SNe whose ${M}_{\mathrm{Ni}}$ can be constrained, the small errors associated with ${M}_{\mathrm{Ni}}$, as presented in Table 2, clearly indicate the necessity of including ⁵⁶Ni to give the best-fitting results. The synthesis of ⁵⁶Ni is also expected in a CCSN. Neglecting ${M}_{\mathrm{Ni}}$ will usually result in rather poor fitting quality, as shown in Figure 10 with the best-fitting parameters listed in Table 5. In Figure 10 we do not show the fitting result of SN PTF10qts because the poor data quality always allows for a "good" fitting. In Table 5 we also compare the reduced ${\chi }^{2}$ of the pure-magnetar model and magnetar+⁵⁶Ni model.

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Table 5.  Best-fitting Parameters of the SNe in Figures 2 and 3 Using the Pure-magnetar Model

SN	${M}_{\mathrm{ej}}$	Bp	P0	${v}_{\mathrm{sc}0}$	${\kappa }_{\gamma ,\mathrm{mag}}$	${T}_{\mathrm{start}}$	${\chi }^{2}/\mu$
	$({M}_{\odot })$	$({10}^{15}\,{\rm{G}})$	(ms)	(km s−1)	(cm2 g−1)	$(\,\mathrm{days})$	Pure-magnetar	Magnetar+56Ni
1997ef	1.8 ± 0.08	1.5 ± 0.02	37.4 ± 0.2	${7674}_{-150}^{+149}$	$\gtrsim 3$	$-{25.5}_{-0.3}^{+0.4}$	2.25	1.51
2002ap	0.8 ± 0.1	2.3 ± 0.04	$48\pm 0.4$	${11257}_{-639}^{+648}$	1.5 ± 0.2	$-{1.9}_{-0.3}^{+0.2}$	1.3	0.3
2003jd	2.6 ± 0.2	1.8 ± 0.05	$20\pm 0.5$	${13134}_{-461}^{+476}$	${1.9}_{-0.9}^{+3.2}$	$-{15.9}_{-0.6}^{+0.8}$	0.25	0.22
2007bg	1.8 ± 0.3	3.1 ± 0.1	38 ± 1	${12803}_{-992}^{+957}$	$\gtrsim 5$	−17 ± 1	1.96	0.22
2007ru	2.7 ± 0.13	1.9 ± 0.01	$22\pm 0.15$	${14037}_{-538}^{+543}$	$\gtrsim 5$	−4.7 ± 0.1	3.3	3.7
2009bb	2.2 ± 0.1	2.4 ± 0.06	$28\pm 0.4$	${19540}_{-757}^{+616}$	${4.9}_{-2.8}^{+3.3}$	−12.8 ± 0.3	0.20	0.15
2010ah	${0.76}_{-0.45}^{+1.40}$	${1.4}_{-0.9}^{+0.6}$	${29}_{-8}^{+3}$	${21879}_{-8915}^{+12335}$	$\gtrsim 2.7$	$-{2.4}_{-0.7}^{+1.0}$	0.07	0.07
PTF10qts	${2.7}_{-0.7}^{+2.0}$	1.1 ± 0.2	${20}_{-2}^{+1}$	${20478}_{-1910}^{+2421}$	$\gtrsim 1$	$-13.5\pm 2$	1.7	2.0
PTF10vgv	${0.29}_{-0.007}^{+0.008}$	${2.4}_{-0.014}^{+0.016}$	$24\pm 0.2$	∼0	${4.2}_{-2.7}^{+2.5}$	${6.9}_{-0.03}^{+0.04}$	1.2	0.7

Note. In these fits, we fixed $\kappa =0.1\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$.

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The high fitting quality of the magnetar+⁵⁶Ni model can be most easily appreciated by comparing the light curves of SN 2002ap reproduced by these two models. The situation of SN 2002ap in the pure-magnetar model is similar to SN 1998bw in this same model; see Figure 8 of Moriya et al. (2017). The ejecta masses given by the pure-magnetar model are frequently unreasonable, such as for SNe 2002ap, 2010ah, and PTF10vgv. For SN 1997ef, the magnetar+⁵⁶Ni model is favored not only because of the smaller reduced ${\chi }^{2}$ compared to the pure-magnetar model, but also because of the broad peak of this SN, as found by Iwamoto et al. (2000). Comparing Figures 2(a) and 10(a) indicates that the magnetar+⁵⁶Ni model captures the broad peak of this light curve better than the pure-magnetar model; see Figure 1 in Wang et al. (2016b) for a clearer rendering.¹⁶

For SNe 2003jd, 2007ru, 2010ah, and PTF10vgv, the contribution of ⁵⁶Ni is not necessary to give an acceptable fitting result. However, the velocity fitting result of PTF10vgv (see Figure 10) is not good enough. The low velocities are required for PTF10vgv in the pure-magnetar model because of the slow decline rate of the light curve.

The above results indicate that the magnetar+⁵⁶Ni model is the best model in reproducing the light curves and velocity evolutions of the SNe Ic-BL sample, although some SNe can also be described by the two-component ⁵⁶Ni model (e.g., SN 2002ap), while some others (e.g., SNe 2003jd, 2007ru, and 2010ah) can also be well reproduced by the pure-magnetar model.

We note that different models usually give different explosion times, as can be found by comparing ${T}_{\mathrm{start}}$ presented in Tables 2, 4, and 5. Tables 2 and 5 show that the explosion times determined by the magnetar+⁵⁶Ni model are all later than that determined by the pure-magnetar model, with PTF10vgv the only exception. This can be well understood. To account for the late-time light curves, the spin-down timescales of the magnetars in the pure-magnetar model have to be longer than in the magnetar+⁵⁶Ni model. This will result in a slow rise rate, and therefore the explosion times must be somewhat earlier. The slow rise rates in the pure-magnetar model suffer from some tension with the upper limits of the light curves of SNe 2003jd, 2007bg, and 2007ru, as can be seen in Figure 10.

A comparison of ${T}_{\mathrm{start}}$ in Tables 2 and 4 shows that the explosion times in the magnetar+⁵⁶Ni model are usually later than that in the ⁵⁶Ni model, except for PTF10vgv. This can be understood by comparing the spin-down timescale of the magnetar, ${\tau }_{\mathrm{sd}}$, with the ⁵⁶Ni decay timescale, ${\tau }_{\mathrm{Ni}}=8.8\,\mathrm{days}$. It is found that the spin-down timescales of the magnetars powering these SNe are all shorter than ${\tau }_{\mathrm{Ni}}$, with only one exception, ${\tau }_{\mathrm{sd}}(\mathrm{PTF}10\mathrm{vgv})=15.9\,\mathrm{days}$. If ${\tau }_{\mathrm{sd}}\lt {\tau }_{\mathrm{Ni}}$, the energy of the magnetar is released more rapidly in the magnetar model than in the ⁵⁶Ni model, and the rise time in the magnetar model is shorter than in the ⁵⁶Ni model. This is why the explosion time of PTF10vgv in the magnetar model is earlier than in the ⁵⁶Ni model, because in this case ${\tau }_{\mathrm{sd}}\gt {\tau }_{\mathrm{Ni}}$. In this way, it is not difficult to understand why the explosion time of SN 1997ef is almost coincident with the discovery date in the magnetar+⁵⁶Ni model. From Table 2 it is found that ${\tau }_{\mathrm{sd}}=0.01\,\mathrm{days}$ for this SN. The energy was almost explosively released.

The mechanism for the formation of SNe Ic-BL is still unclear. Recently, there has been evidence that SNe Ic-BL are powered by magnetars (Wang et al. 2017). Indeed, for all of the SNe Ic-BL that were observed to phases $\gtrsim 300\,\mathrm{days}$ when the contribution from ⁵⁶Ni decays significantly, there is evidence for magnetar formation.

Motivated by this evidence, we studied a sample of N = 11 SNe Ic-BL and obtain their light curve fitting parameters. From this study it is evident that the sample of SNe Ic-BL can be reasonably described by the magnetar+⁵⁶Ni model. The magnetar+⁵⁶Ni model naturally reduces the needed ⁵⁶Ni and simultaneously accounts for the origin of the huge kinetic energies observed in SNe Ic-BL, with only one exception, SN 2010ay, whose large explosion energy could be attributed to the large photometric uncertainties. We also examine the possibility for the pure-⁵⁶Ni or pure-magnetar model to explain the light curve and velocity evolution. It is found that SNe 2010ay, PTF10vgv, and iPTF16asu (three out of 12) are unlikely to be explained by the (two-component) ⁵⁶Ni model, while some SNe, 2003jd, 2007ru, and 2010ah (not all in the sample), are compatible with the pure-magnetar model.

Our results indicate that the synthesized ⁵⁶Ni mass does not increase with explosion energy or neutron star rotational energy. The ⁵⁶Ni mass is consistent with ordinary SNe. The relation between magnetic field and explosion energy seems to indicate that the amplification of magnetic field of the neutron star is independent of the explosion energy. To get a more robust statistical result, more high-quality observations are definitely needed.

We thank the anonymous referee for constructive suggestions. We are grateful to Maryam Modjaz and Yuqian Liu for sending us the photospheric velocity data before the publication of their paper. This work is supported by the National Program on Key Research and Development Project of China (Grant No. 2016YFA0400801), the National Basic Research Program of China ("973" Program, Grant No. 2014CB845800), and the National Natural Science Foundation of China (grant Nos. U1331202, 11533033, U1331101, 11673006, 11573014, 11422325, and 11373022). D.X. acknowledges the support of the One-Hundred-Talent Program from the National Astronomical Observatories, Chinese Academy of Sciences), and by the Strategic Priority Research Program "Multi-wavelength Gravitational Wave Universe" of the CAS (No. XDB23040100).