A Monte Carlo approach to magnetar-powered transients: II. Broad-lined type Ic supernovae not associated with GRBs

TL;DR

This study systematically analyzes a large sample of broad-lined type Ic supernovae not linked to gamma-ray bursts, demonstrating that a magnetar plus 56Ni model effectively explains their light curves and velocities, with implications for explosion energy and magnetic fields.

Contribution

It provides the first comprehensive analysis of SNe Ic-BL without GRBs using a magnetar+56Ni model, highlighting its effectiveness over alternative models.

Findings

Magnetar+56Ni model fits light curves and velocities well.

Reduced 56Ni mass needed compared to previous models.

Some SNe are better explained by pure magnetar or two-component models.

Abstract

Broad-lined type Ic supernovae (SNe Ic-BL) are a subclass of rare core collapse SNe whose energy source is debated in the literature. Recently a series of investigations on SNe Ic-BL with the magnetar (plus 56Ni) model were carried out. Evidence for magnetar formation was found for the well-observed SNe Ic-BL 1998bw and 2002ap. In this paper we systematically study a large sample of SNe Ic-BL not associated with gamma-ray bursts. We use photospheric velocity data determined in a homogeneous way. We find that the magnetar+56Ni model provides a good description of the light curves and velocity evolution of our sample of SNe Ic-BL, although some SNe (not all) can also be described by the pure-magnetar model or by the two-component pure-56Ni model (3 out of 12 are unlikely explained by two-component model). In the magnetar+56Ni model, the amount of 56Ni required to explain their luminosity is significantly reduced, and the derived initial explosion energy is, in general, in accordance with neutrino heating. Some correlations between different physical parameters are evaluated and their implications regarding magnetic field amplification and the total energy reservoir are discussed.

Tables

Table 1. Table 1 The SNe Ic-BL sample

SN	References		Extinctionaafootnotemark:	Colourbbfootnotemark:	Luminosity distance
	Light curve	Velocity	corrected	used	in referenced paperccfootnotemark: $\left(\mathrm{Mpc}\right)$	methodddfootnotemark:	adopted eefootnotemark: $\left(\mathrm{Mpc}\right)$
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	$428.1$
PTF10vgvfffootnotemark:	C12	M16	G	$-$	$60.3$	C	$63.5$

Table 2. Table 2 Best-fitting parameters of our SNe Ic-BL sample

SN	$M_{\mathrm{ej}}$	$M_{\mathrm{Ni}}$	$B_p$	$P_0$	$v_{\mathrm{sc0}}$	${\kappa }_{\gamma ,\mathrm{Ni}}$	${\kappa }_{\gamma ,\mathrm{mag}}$	$T_{\mathrm{start}}$	$T_{\mathrm{Neb}}$	Constraints on $T_{\mathrm{Neb}}$aafootnotemark:	$E_{K0}$
	$\left(M_{\odot }\right)$	$\left(M_{\odot }\right)$	$\left({10}^{15}G\right)$	$\left(\mathrm{ms}\right)$	$\left(\mathrm{km}{s}^{-1}\right)$	$\left({\mathrm{cm}}^2{g}^{-1}\right)$	$\left({\mathrm{cm}}^2{g}^{-1}\right)$	$\left(\mathrm{days}\right)$	$\left(\mathrm{days}\right)$	$\left(\mathrm{days}\right)$	${10}^{51}\mathrm{erg}$
1997ef	${3.3}_{-0.19}^{+0.21}$	$0.058\pm 0.001$	${11.4}_{-0.5}^{+0.4}$	${5.2}_{-0.3}^{+1.3}$	${1900}_{-1000}^{+2600}$	${0.12}_{-0.02}^{+0.03}$	$-$	$-{16.1}_{-0.5}^{+0.4}$	$153$	$(61.3,119.8)$	$0.07$
2002ap	$1.7\pm 0.2$	$0.049\pm 0.001$	$19.5\pm 2$	$12\pm 4$	${10500}_{-1000}^{+700}$	$0.20\pm 0.02$	$8.8\pm 6$	$1.9\pm 0.1$	$71$	$(44,81)$	$1.13$
2003jd	$2.8\pm 0.3$	$0.05\pm 0.02$	$2.5\pm 0.3$	${18}_{-1.6}^{+1.2}$	$12490\pm 500$	${0.3}_{-0.2}^{+0.3}$	${1.75}_{-1.1}^{+1.5}$	$-14.9\pm 1$	$92$	$(62.5,81.2)$	$2.64$
2007bg	$1.5\pm 0.2$	$0.03\pm 0.004$	${10.5}_{-3}^{+5}$	${20}_{-13}^{+11}$	${11900}_{-2100}^{+1200}$	${0.4}_{-0.3}^{+1.2}$	$-$	$-10\pm 2$	$77$	$(34.7,67.7)$	$1.3$
2007ru	$5.6\pm 0.5$	$0.078\pm 0.006$	${6.2}_{-0.2}^{+0.3}$	$1.6\pm 0.1$	$\sim 0$	${0.19}_{-0.03}^{+0.05}$	$-$	${0.8}_{-0.6}^{+0.9}$	$86$	$(70,200)$	$\sim 0$
2009bb	$2.2\pm 0.07$	$0.025\pm 0.01$	$2.9\pm 0.3$	$28.6\pm 0.6$	${19300}_{-700}^{+600}$	${0.4}_{-0.35}^{+2.8}$	${2.3}_{-1.6}^{+1.8}$	$-12\pm 0.4$	$48$	$(55,295)$	$4.90$
2010ah	${2.58}_{-1.6}^{+2.6}$	${0.14}_{-0.03}^{+0.02}$	${17.5}_{-7.9}^{+20}$	${4.2}_{-2.7}^{+35}$	${14100}_{-9700}^{+10000}$	$\gtrsim 0.06$	$-$	$-0.7\pm 1$	$69$	$>14.5$	$3.1$
2010ay	${6.7}_{-1}^{+1.8}$	$-$	$0.8\pm 0.1$	$11\pm 1$	${24900}_{-2100}^{+300}$	$-$	$-$	${2.0}_{-0.4}^{+0.3}$	$55$	$>44.1$	$33.4$
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	${1.9}_{-0.4}^{+0.6}$	$0.28\pm 0.06$	${9.9}_{-6.6}^{+12}$	${26}_{-19}^{+64}$	${21307}_{-2500}^{+2400}$	${0.3}_{-0.1}^{+1.5}$	$-$	$-{10.5}_{-1.9}^{+2.7}$	$39$	$(38.8,230.4)$	$5.1$
PTF10vgv	${0.7}_{-0.07}^{+0.08}$	$0.059\pm 0.003$	$1.7\pm 0.2$	${28.8}_{-0.7}^{+0.5}$	${10000}_{-600}^{+700}$	$0.19\pm 0.03$	$0.013\pm 0.001$	$6.9\pm 0.04$	$43$	$(46.2,82.6)$	$0.42$

Table 3. Table 3 Comparison of explosion times derived in this work and previous papers.

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

Table 4. Table 4 Best-fitting parameters for the selected SNe using the 1D 56 Ni model.

SN	$M_{\mathrm{ej}}$	$M_{\mathrm{Ni}}$	$v_{\mathrm{sc0}}$	$T_{\mathrm{start}}$	${\kappa }_{\gamma ,\mathrm{Ni}}$	${\chi }^2/\mu$
	$\left(M_{\odot }\right)$	$\left(M_{\odot }\right)$	$\left(\mathrm{km}{s}^{-1}\right)$	$\left(\mathrm{days}\right)$	$\left({\mathrm{cm}}^2{g}^{-1}\right)$	pure-56Ni model	magnetar+56Ni model
2007bg	$1.3\pm 0.5$	${0.10}_{-0.02}^{+0.01}$	${12000}_{-1200}^{+1400}$	$-{17.5}_{-2.8}^{+4.9}$	${0.05}_{-0.02}^{+0.05}$	6.0	0.22
2007ru	$2.6\pm 0.2$	$0.34\pm 0.01$	$16300\pm 700$	$-{4.9}_{-0.07}^{+0.16}$	${0.04}_{-0.004}^{+0.005}$	4.7	0.7
2009bb	$2.5\pm 0.1$	$0.24\pm 0.007$	$19600\pm 700$	$-{15.8}_{-0.5}^{+0.4}$	${0.03}_{-0.002}^{+0.003}$	0.67	0.15
2010ah	${1.0}_{-0.5}^{+0.7}$	$0.17\pm 0.02$	$20600\pm 7000$	$-{2.5}_{-0.6}^{+0.9}$	$\gtrsim 0.5$	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	$1.6\pm 0.2$	$0.11\pm 0.004$	$14900\pm 800$	$-{1.4}_{-0.9}^{+0.8}$	$-$	0.12	0.14
PTF10qts	${1.9}_{-0.3}^{+0.4}$	${0.34}_{-0.02}^{+0.03}$	$23000\pm 2000$	$-{12.1}_{-1.1}^{+0.9}$	${0.35}_{-0.1}^{+1.2}$	1.4	2.0
PTF10vgv	${0.24}_{-0.02}^{+0.03}$	$0.25\pm 0.002$	${4700}_{-400}^{+500}$	$7.1\pm 0.04$	$0.014\pm 0.001$	4.2	0.7
PTF10vgv-27	$0.82\pm 0.04$	$0.26\pm 0.002$	$14100\pm 500$	$6.9\pm 0.03$	$0.027$	1.2	-

Table 5. Table 5 Best-fitting parameters of the SNe in Figures 2 and 3 using the pure-magnetar model.

SN	$M_{\mathrm{ej}}$	$B_p$	$P_0$	$v_{\mathrm{sc0}}$	${\kappa }_{\gamma ,\mathrm{mag}}$	$T_{\mathrm{start}}$	${\chi }^2/\mu$
	$\left(M_{\odot }\right)$	$\left({10}^{15}G\right)$	$\left(\mathrm{ms}\right)$	$\left(\mathrm{km}{s}^{-1}\right)$	$\left({\mathrm{cm}}^2{g}^{-1}\right)$	$\left(\mathrm{days}\right)$	pure-magnetar	magnetar+56Ni
1997ef	$1.8\pm 0.08$	$1.5\pm 0.02$	$37.4\pm 0.2$	${7674}_{-150}^{+149}$	$\gtrsim 3$	$-{25.5}_{-0.3}^{+0.4}$	$2.25$	$1.51$
2002ap	$0.8\pm 0.1$	$2.3\pm 0.04$	$48\pm 0.4$	${11257}_{-639}^{+648}$	$1.5\pm 0.2$	$-{1.9}_{-0.3}^{+0.2}$	$1.3$	$0.3$
2003jd	$2.6\pm 0.2$	$1.8\pm 0.05$	$20\pm 0.5$	${13134}_{-461}^{+476}$	${1.9}_{-0.9}^{+3.2}$	$-{15.9}_{-0.6}^{+0.8}$	$0.25$	$0.22$
2007bg	$1.8\pm 0.3$	$3.1\pm 0.1$	$38\pm 1$	${12803}_{-992}^{+957}$	$\gtrsim 5$	$-17\pm 1$	$1.96$	$0.22$
2007ru	$2.7\pm 0.13$	$1.9\pm 0.01$	$22\pm 0.15$	${14037}_{-538}^{+543}$	$\gtrsim 5$	$-4.7\pm 0.1$	$3.3$	$3.7$
2009bb	$2.2\pm 0.1$	$2.4\pm 0.06$	$28\pm 0.4$	${19540}_{-757}^{+616}$	${4.9}_{-2.8}^{+3.3}$	$-12.8\pm 0.3$	$0.20$	$0.15$
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	${2.7}_{-0.7}^{+2.0}$	$1.1\pm 0.2$	${20}_{-2}^{+1}$	${20478}_{-1910}^{+2421}$	$\gtrsim 1$	$-13.5\pm 2$	$1.7$	$2.0$
PTF10vgv	${0.29}_{-0.007}^{+0.008}$	${2.4}_{-0.014}^{+0.016}$	$24\pm 0.2$	$\sim 0$	${4.2}_{-2.7}^{+2.5}$	${6.9}_{-0.03}^{+0.04}$	$1.2$	$0.7$

Equations

$$M=m-5\log \left(D_L/10\mathrm{pc}\right)+2.5\log \left(1+z\right),$$

Full text

1. Introduction

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), i.e. 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 is 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 circumstellar medium (Yan et al., 2015; Wang et al., 2016d; Chen et al., 2017) at late times.

For SNe Ic-BL, shortcomings of one-dimensional (1D) 56Ni model (e.g., Iwamoto et al., 2000; Nakamura et al., 2001) stimulated the suggestion for 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, c, 2017), which are 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 in between ordinary SNe Ibc and energetic GRBs and are similar to the sub-energetic GRBs, e.g. GRB 100316D (Margutti et al., 2013) and GRB 140606B (Cano et al., 2015). This may indicate that similar engines were operating in sub-energetic GRBs and SNe 2009bb and 2012ap.

Based on the above findings, here we test the hypothesis that all of 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 a 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 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.

2. SN sample and data analysis

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 passbands UV (ultraviolet), BVRI (optical) and IR (UVOIR) 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. UV 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 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, decide to not include the contribution of $UV$ and/or NIR bands (Tomita et al., 2006; Sahu et al., 2009; Young et al., 2010; Pignata et al., 2011).

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 are used to calculate the bolometric correction. In Table 1 we list the color we used to calculate 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 choice can minimize the errors that may be introduced by interpolation and/or extrapolation. Sometimes data are available only in a single passband for some time duration, e.g. the data of SN 2007bg before 7.2 days given in Table 3 of Young et al. (2010), while these data are crucial to constrain 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 $E(B-V)_{\mathrm{total}}=0.45\mathord{\thinspace\rm mag}$, while Liu et al. (2015) adopted a value of $E(B-V)_{\mathrm{total}}=0.87\mathord{\thinspace\rm mag}$.

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, 2012ap, redshift-independent methods, e.g. Tully-Fisher measurements, were facilitated to derive the distances, as are available on NASA/IPAC Extragalactic Database (NED). Such linear distances are weightedly averaged, as listed in Table 1. For other SNe that no such linear distances are available, to minimize distance uncertainties, we transform, according to the method described in Cano et al. (2014), the light curves in the literature to a common cosmology, i.e. the latest Plank results: $H_{0}=(67.8\pm 0.9)\mathord{\thinspace\rm km}\mathord{\thinspace\rm s}^{-1}\mathord{\thinspace\rm Mpc}^{-1}$, $\Omega_{m}=0.308\pm 0.012$ (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 $\mu_{\mathrm{PTF10vgv}}=33.9\mathord{\thinspace\rm mag}$, 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 $9.22\mathord{\thinspace\rm Mpc}$, rather than $7.94\mathord{\thinspace\rm Mpc}$ in the original paper. The smallest difference is for SNe 2009bb, for which distances $\sim 40\mathord{\thinspace\rm Mpc}$ 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, e.g. Si II $\lambda$6355, Na I D $\lambda$5891, O I $\lambda$7774, Ca II $\lambda$8579, Fe II $\lambda$5169, usually give different results (Valenti et al., 2008; Modjaz et al., 2016). This difference may be a result of the different depth 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 of measuring velocities for all SNe Ic-BL and SNe Ic in a homogenous way. In this paper we use the velocity data given by Modjaz et al. (2016), when available. Using such a homogenous data 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 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)

[TABLE]

where $D_{L}$ is the luminosity distance and $z$ is the redshift. The last term in 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 $M_{\mathrm{bol}}-M_{R}=-0.496\mathord{\thinspace\rm mag}$ based on the early-time photospheric temperature $T_{\mathrm{phot}}\approx 10^{4}\mathord{\thinspace\rm K}$ 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 $6900\mathord{\thinspace\rm K}$ at peak light. This implies a bolometric correction $M_{\mathrm{bol}}-M_{R}=0.29\mathord{\thinspace\rm mag}$, 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 $20\mathord{\thinspace\rm days}$ before or after peak. For SN 1997ef, only $V$ band data are provided by Iwamoto et al. (2000). According to the effective temperatures ($\sim 6100\mathord{\thinspace\rm K}$) given in Table 3 of Iwamoto et al. (2000), we applied bolometric correction ($M_{\mathrm{bol}}-M_{V}=-0.05\mathord{\thinspace\rm mag}$) 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 root mean squares of the prescription of Lyman et al. (2014) are $\sim 0.06\mathord{\thinspace\rm mag}$, while the measurement errors of the light curve range from $\sim 0.02\mathord{\thinspace\rm mag}$ to $\sim 0.3\mathord{\thinspace\rm mag}$. 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 $0.03-0.06\mathord{\thinspace\rm mag}$, $0.2-0.3\mathord{\thinspace\rm mag}$, and $0.02-0.2\mathord{\thinspace\rm mag}$, 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 measurements 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.111However, given the several sources of uncertainty in the bolometric correction derived from a single filter, we estimate that the minimum error in the latter must be at least 20%, if not larger.

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 contribution from $BVRI$ and $IR$ bands, while the luminosity of SN 2009bb given by the original paper includes contribution only from $BVRI$ bands. Another reason we choose these two SNe is that their luminosity is integrated according to observational data, while the luminosity of some other SNe are calculated in the original papers by assuming some contribution from unavailable bands (frequently the $IR$ band).

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 $\sim 80\mathord{\thinspace\rm days}$, 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 $\sim 80\mathord{\thinspace\rm days}$. The data with $t\gtrsim 80\mathord{\thinspace\rm days}$ 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 stages222The $UV$ flux observed on 2002 February 3 (4 days before $V$-band maximum) by XMM-Neutron contributes only $\sim 4\%$ (Mazzali et al., 2002)., while Figure 1(b) shows that the contribution from the unavailable $UV$ and $IR$ bands cannot be ignored.

3. Fitting result

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 $M_{\mathrm{Ni}}$, grey optical opacity $\kappa$, 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 $B_{p}$, initial rotation period $P_{0}$ 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 grey optical opacity $\kappa$ we take the fiducial value $\kappa=0.1\mathord{\thinspace\rm cm}^{2}\mathord{\thinspace\rm g}^{-1}$, 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 specifically mentioned otherwise.

[FIGURE:]

The magnetar model proposed by Wang et al. (2016c) traces the photospheric recession and therefore the emission from the photosphere and those material outside of 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 nebular phase when the nebula radiates this percentage of emission, we can obtain the time $T_{\mathrm{Neb}}$ (since explosion in rest frame) by fitting the early-time light curve, as listed in Table 2.333This approach is desirable as it is sometimes difficult to identify an eruption as a SN or a tidal disruption event (TDE; Brown et al., 2016; Dong et al., 2016; Leloudas et al., 2016). The indication of the early nebular phase is helpful to confirm the identity of a SN because a TDE does not have a nebular phase. We therefore encourage the modeling of the early-time light curve of a transient to give an estimate of the epoch at which nebular features may appear (if it is a SN) to help constrain the nature of the transient. 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.

To determine the 56Ni mass, it is necessary for the light curve to be observed at least for $\sim 110\mathord{\thinspace\rm days}$. We divide the observed SNe Ic-BL light curves into two classes: those with an observational duration $t\gtrsim 100\mathord{\thinspace\rm days}$ (class I) and with $t\lesssim 100\mathord{\thinspace\rm days}$ (class II). The reason for the choice of $100\mathord{\thinspace\rm days}$ as the dividing boundary is because the lifetime of 56Co is $\sim 110\mathord{\thinspace\rm days}$. If observational duration is longer than $100\mathord{\thinspace\rm days}$, 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 emission from the photosphere and nebula are shown as dashed and dot-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, e.g. 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 dot-dashed lines, respectively. In Figure 4 the light curves were fitted with a pure magnetar model (without 56Ni contribution) because the 56Ni masses of the 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.

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 e.g. Kasen & Bildsten (2010), where the light curves tend to flatten at late time. The decline rate is also faster than 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 to 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, e.g. 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, e.g. SNe 2003jd and 2007bg in Figure 3. We looked into the data and found that the bolometric limit of SN 2003jd is $0.6\mathord{\thinspace\rm days}$ earlier than the explosion date, while the bolometric limit of SN 2007bg is $1.0\mathord{\thinspace\rm days}$ 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 reasonability of the determined parameters such as $M_{\mathrm{ej}}$, $M_{\mathrm{Ni}}$, $B_{p}$, $P_{0}$, $\kappa_{\gamma,\mathrm{mag}}$. We also discussed the possible reasons for a larger value of $\kappa_{\gamma,\mathrm{Ni}}$ than the standard value $\sim 0.027\mathord{\thinspace\rm cm}^{2}\mathord{\thinspace\rm g}^{-1}$.

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, e.g. 2002ap and 2003jd, observed to late stages ($t\gtrsim 300\mathord{\thinspace\rm days}$) 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 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, energy versus dipole magnetic field of the magnetar in Figures 5-8, respectively. In the magnetar model, three forms of energy are considered here, i.e. 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, i.e. 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, e.g. SNe 1997ef, 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.

4. Discussion

4.1. General implications

The high fitting quality both for 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 of getting 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 nebular spectrum than 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 which 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 $M_{\mathrm{ej}}$, $P_{0}$, 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 time scale of the SN (Arnett, 1982; Arnett et al., 2017) because $P_{0}$ 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 $\sim 2.5\times 10^{51}\mathord{\thinspace\rm erg}$, i.e. the theoretical upper limit of explosion energy triggered by neutrino heating (Janka et al., 2016). There is one exception, SN 2010ay, which has explosion energy $\gtrsim 10^{52}\mathord{\thinspace\rm erg}$. 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 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 $0.1M_{\odot}$. The only two values $0.28M_{\odot}$ and $0.14M_{\odot}$ that are above $0.1M_{\odot}$ are for PTF10qts and SN 2010ah, respectively. We note that the observational data of these two SNe are of 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 $0.2M_{\odot}$, i.e. 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 to acquire more clue 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 clue.

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, $0.7M_{\odot}$, in the SNe Ic-BL sample (see Table 2). Its opacity to magnetar photons, $\kappa_{\gamma,\mathrm{mag}}=0.013\mathord{\thinspace\rm cm}^{2}\mathord{\thinspace\rm g}^{-1}$, is also much lower than the values found for the other SNe Ic-BL. Corsi et al. (2012) constrained its progenitor radius to be $R<\left(1-5\right)R_{\odot}$, 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 most excellent case is SN 1998bw, for which the explosion time was constrained to be $-0.009_{-0.36}^{+0.32}\mathord{\thinspace\rm days}$ relative to the GRB trigger time (Wang et al., 2017).444Previously the burst time of GRB 980426 was constrained to coincide with that of SN 1998bw within $\left(+0.7,-2.0\right)\mathord{\thinspace\rm days}$ (e.g., Iwamoto, 1999). 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 non-detection. The explosion time is computed according to the times $T_{\mathrm{start}}$ given in Table 2, after correcting for cosmological time dilation.

In the calculation of the explosion time, we frequently need the time of $V$-band maximum, 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 $\sim 5\mathord{\thinspace\rm days}$ later than that given by Mazzali et al. (2000). Please note that the first bolometric data point in Figure 2 is $4\mathord{\thinspace\rm days}$ later than the discovery date.

4.2. Estimate the appearance of nebular features from early light

curve modeling

In this paper we propose to calculate the time $T_{\mathrm{Neb}}$ when the nebular features begin to emerge in the SN spectra. We searched the spectra from the references listed in Table 1. The lower limit of $T_{\mathrm{Neb}}$ for an SN in question is the latest time at which a spectrum is photospheric, while the upper limit is the earliest time at which the spectrum is nebular. We list these constraints in Table 2.

The spectrum taken on 26 January, 1998 of SN 1997ef is photospheric (Mazzali et al., 2000), while the first nebular spectrum is on $+104\mathord{\thinspace\rm days}$ (in rest frame) post R-band maximum (Young et al., 2010), implying the transition from photospheric to nebular occurred between these two dates. Next, the spectrum obtained $+51\mathord{\thinspace\rm days}$ post B-band maximum of SN 2003jd is photospheric, while the spectrum on $+70\mathord{\thinspace\rm days}$ is nebular (Valenti et al., 2008). For SN 2007bg, the spectrum taken at $+25\mathord{\thinspace\rm days}$ post R-band maximum for SN 2007bg is photospheric, while the spectrum on $+58\mathord{\thinspace\rm days}$ is nebular (Young et al., 2010). For SN 2007ru, the spectrum obtained $70\mathord{\thinspace\rm days}$ after explosion for SN 2007ru is photospheric, while the spectrum on $200\mathord{\thinspace\rm days}$ is nebular (Sahu et al., 2009). Next, the spectrum of SN 2009bb on $+45\mathord{\thinspace\rm days}$ past B-band maximum is photospheric, while the spectrum on $+285\mathord{\thinspace\rm days}$ is nebular (Pignata et al., 2011). For SN 2012ap, the spectrum taken at $+26\mathord{\thinspace\rm days}$ past B-band maximum is photospheric, while the spectrum on $+218\mathord{\thinspace\rm days}$ is nebular (Milisavljevic et al., 2015). For PTF10qts, the spectrum taken $+21\mathord{\thinspace\rm days}$ past R-band maximum is photospheric, while the spectrum on $+230\mathord{\thinspace\rm days}$ is nebular (Walker et al., 2014). Finally, for PTF10vgv the spectrum obtained at $+35\mathord{\thinspace\rm days}$ past R-band maximum is photospheric, while the spectrum on $+72\mathord{\thinspace\rm days}$ is nebular (Corsi et al., 2012).

For SN 2010ah, the spectrum of SN 2010ah on 7, March, 2010 is photospheric (Mazzali et al., 2013), with no data later than this date being published. The same situation applies for SN 2010ay, of which that latest spectrum was obtained on $+24\mathord{\thinspace\rm days}$ past R-band maximum is photospheric (Sanders et al., 2012). For both of these events, the precise timing of the transition from the photospheric phase to the nebular can only be constrained to have occurred after these dates.

We can see from Table 2 that our fitting constraints of $T_{\mathrm{Neb}}$ for SNe 2002ap, 2007ru, 2010ah, 2010ay, 2012ap, and PTF10qts are consistent with observations. For PTF10vgv, $T_{\mathrm{Neb}}$ $\left(43\mathord{\thinspace\rm days}\right)$ is slightly earlier than the lower limit $46.2\mathord{\thinspace\rm days}$. For SN 2009bb the given $T_{\mathrm{Neb}}$ $\left(48\mathord{\thinspace\rm days}\right)$ is $7\mathord{\thinspace\rm days}$ earlier than observation. This may be caused by the helium envelope of this SN because early-time optical spectra showed evidence for the presence of helium in this SN (Pignata et al. 2011; another SN that evidenced with some helium is SN 2012ap, Milisavljevic et al. 2015). The helium envelope will delay the appearance of nebular lines. For SN 2007bg, $T_{\mathrm{Neb}}$ $\left(77\mathord{\thinspace\rm days}\right)$ is $9\mathord{\thinspace\rm days}$ later than the upper limit. This discrepancy for SN 2007bg might be caused by the sparsity of data before peak time, see Figure 3(b).

For 1997ef, $T_{\mathrm{Neb}}$ $\left(153\mathord{\thinspace\rm days}\right)$ is significantly later than the appearance of the first nebular spectrum, i.e. $119.8\mathord{\thinspace\rm days}$. The situation is less significant but also notable for SN 2003jd, for which we have $92\mathord{\thinspace\rm days}$ versus $81.2\mathord{\thinspace\rm days}$. As mentioned in Section 4.1, a plausible reason for this large discrepancy may lie in the failure of our model to fit the early velocity data of these two SNe. We suggest that a fast-moving shell should be introduced for SN 1997ef. Such a shell will contribute a significant fraction of nebular emission and therefore made the appearance of nebular phase earlier.

In summary, we conclude that $T_{\mathrm{Neb}}$ determined in this way is in general a good guide for the emergence of nebular features, although it is not completely accurate. Other factors come into play in determining the emergence of nebular lines aside from the amount of nebular emission.

4.3. Correlations

In the magnetar model, the required 56Ni mass is, unsurprisingly, reduced significantly.555Figure 1 shows that for $t\gtrsim 80\mathord{\thinspace\rm days}$ the luminosity data calculated according to Lyman et al. (2014) are slightly different from that obtained by integrating individual bands. This will affect the derived 56Ni mass. As a result, we call for further study on the luminosity data for $t\gtrsim 80\mathord{\thinspace\rm days}$. It is therefore expected that the 56Ni mass-energy relation will be quite different, as depicted in Figure 5, where we also plot the ordinary type IIb SN 1993J and type Ic SN 1994I. It is clear from this figure that the synthesized 56Ni is consistent with ordinary striped envelope SNe. There is no clear increase of 56Ni mass with increased energy, contrary to earlier findings (Mazzali et al., 2013; Lyman et al., 2016; Toy et al., 2016). In the magnetar+56Ni model, the explosion energies are generally significantly lower than in pure-56Ni models, regardless whether it is the 1D 56Ni model or two-component 56Ni model. The explosion energy is no longer the sole decisive factor for 56Ni synthesis. In this case the synthesis of 56Ni may be determined by other factors, e.g. the radius, and/or density profile of the progenitor star (Smartt, 2009). Such diversity may reflect the mass, binarity, metallicity, mass-loss rate, rotation, and magnetic field of the main sequence star (Smartt, 2009).

For an SN Ic-BL, the bolometric corrections at late times can be $\sim 1\mathord{\thinspace\rm mag}$ larger than that at peak times. This indicates that for SNe 1997ef and PTF10vgv, the light curves at late times should be brighter and flatter than shown in Figures 2(a) and 3(f). As a result, the value of $\kappa_{\gamma,\mathrm{Ni}}$ should be larger. Another impacted parameter is $M_{\mathrm{Ni}}$, although to a less extent. We estimate that $M_{\mathrm{Ni}}$ may be at most a factor of 2 larger. In Figure 5 we show the 56Ni masses of SNe 1997ef and PTF10vgv as points connected by horizontal magenta lines with rightward arrows to indicate the uncertainties introduced by this approximation. As is clear from Figure 5, this does not change our conclusion about the correlation between $M_{\mathrm{Ni}}$ and explosion energy. The 56Ni mass of SN 2010ay cannot be constrained, so the uncertainties introduced by above approximation is irrelevant for SN 2010ay.

Table 2 shows that the explosion energy of SN 2007ru is quite low, but the 56Ni mass $M_{\mathrm{Ni}}$ is not zero. This indicates that the 56Ni of this SN was synthesized by the shock wave generated by the spinning-down magnetar (Nishimura et al., 2015, 2017; Suwa & Tominaga, 2015). This is in agreement with expectations because the rotational energy of the magnetar powering SN 2007ru is the largest in the SNe Ic-BL sample. The magnetic field $B_{p}$ is also strong enough to synthesize the needed 56Ni. In the magnetar model, both the explosion shock and the magnetar-powered shock can synthesize 56Ni. This complicates the 56Ni mass-energy relation of SNe Ic-BL.

From Figure 6 it is clear that the ejecta mass increases with energy. This is similar to earlier findings (Mazzali et al., 2013; Lyman et al., 2016), but the energies are much lower than the values given by pure-56Ni models (Mazzali et al., 2013; Lyman et al., 2016).

We also examined the relation between explosion energy and neutron star rotational energy, given in Figure 7. This figure implies that there is no clear correlation between these two energies. Figure 8 shows magnetic field $B_{p}$ and energy. No clear correlation is seen between these two quantities. Because $B_{p}$ harbours a fraction of the toroidal magnetic field within the neutron star, $B_{p}$ can serve as an indication of the magnetic energy present within the neutron star. If this is true, Figure 8 may imply that the amplification of magnetic field in the neutron star is unrelated with the explosion energy.

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 that 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).

From Figure 9 it can be seen that the 1D 56Ni model can only account for the first $\sim 50\mathord{\thinspace\rm days}$ data (SNe 2010ay, 2012ap).666The 1D 56Ni model can also account for the light curves of SNe 2010ah and PTF10qts with $\sim 100\mathord{\thinspace\rm days}$ data. However, this may be the result of the poor quality of the data. For all well-observed SNe Ic-BL with observational time $\gtrsim 100\mathord{\thinspace\rm days}$ (SNe 2007bg, 2007ru, 2009bb in this figure), two-component model should be invoked.

We will not examine these SNe within the framework of two-component model in detail. SN 2010ay in Figure 9 is particularly interesting because its peak luminosity $\sim 3.0\times 10^{43}\mathord{\thinspace\rm erg}\mathord{\thinspace\rm s}^{-1}$ is comparable to some of the SLSNe, PTF10hgi ($3.52\times 10^{43}\mathord{\thinspace\rm erg}\mathord{\thinspace\rm s}^{-1}$; Inserra et al., 2013), PTF11rks ($4.7\times 10^{43}\mathord{\thinspace\rm erg}\mathord{\thinspace\rm s}^{-1}$; Inserra et al., 2013), and PS1-14bj ($4.6\times 10^{43}\mathord{\thinspace\rm erg}\mathord{\thinspace\rm s}^{-1}$; 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 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 limits 0.2 expected for a CCSN (Umeda & Nomoto, 2008). The needed 56Ni mass $1.2M_{\odot}$777Sanders et al. (2012) estimated the 56Ni mass of SN 2010ay to be $0.9\pm 0.1M_{\odot}$. However, if the most luminous data point in the light curve of SN 2010ay is adopted, a 56Ni mass of $1.2M_{\odot}$ was derived (Sanders et al., 2012), consistent with our result. Wang et al. (2015b) found that $M_{\mathrm{Ni}}=2M_{\odot}$ is required to meet the peak luminosity of SN 2010ay for a 56Ni model. This higher value results from adopting a different bolometric correction by Wang et al. (2015b). 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 $M_{\mathrm{ej}}=0.24M_{\odot}$ is smaller than $M_{\mathrm{Ni}}=0.25M_{\odot}$ (see Table 4). Even if we double its expansion velocity to $\sim 10000\mathord{\thinspace\rm km}\mathord{\thinspace\rm s}^{-1}$ (see Figure 9h) 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 result allowing $\kappa_{\gamma,\mathrm{Ni}}$ to vary. Table 4 shows that $\kappa_{\gamma,\mathrm{Ni}}=0.014\mathord{\thinspace\rm cm}^{2}\mathord{\thinspace\rm g}^{-1}$, lower than the fiducial value $\kappa_{\gamma,\mathrm{Ni}}=0.027\mathord{\thinspace\rm cm}^{2}\mathord{\thinspace\rm g}^{-1}$.888As discussed in Section 4.3, the late-time light curve of SN 1997ef should be slightly flatter than shown in Figure 9(h). After taking into account this fact the derived $\kappa_{\gamma,\mathrm{Ni}}$ would be close to the fiducial value. The dot-dashed lines in Figure 9(h) are the best-fitting result to the first $50\mathord{\thinspace\rm days}$ of luminosity data after fixing $\kappa_{\gamma,\mathrm{Ni}}=0.027\mathord{\thinspace\rm cm}^{2}\mathord{\thinspace\rm g}^{-1}$. In this case the initial expansion velocity is $\sim 14000\mathord{\thinspace\rm km}\mathord{\thinspace\rm s}^{-1}$, much higher than the average velocity $\sim 7500\mathord{\thinspace\rm km}\mathord{\thinspace\rm s}^{-1}$ 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 $0.2$. We therefore conclude that PTF10vgv cannot be explained by 56Ni model. Recently, it is found iPTF16asu (Whitesides et al., 2017) cannot also be explained by 56Ni model.

For PTF10qts, Walker et al. (2014) obtained $M_{\mathrm{Ni}}=0.35\pm 0.1M_{\odot}$, 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), any 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.

The high fitting quality of the magnetar+56Ni model can be most easily appreciated by comparing the reproduced light curves of SN 2002ap 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, e.g. 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.999We note that the two-component model cannot capture the broad peak of SN 1997ef, see the inset (circles versus dotted line) of Figure 4 of Maeda et al. (2003). This indicates that the magnetar+56Ni model is the best to account for SN 1997ef.

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 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, 2007ru, as can be seen from Figure 10.

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, $\tau_{\mathrm{Ni}}=8.8\mathord{\thinspace\rm days}$. 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, $\tau_{\mathrm{sd}}\left(\mathrm{PTF10vgv}\right)=15.9\mathord{\thinspace\rm days}$. If $\tau_{\mathrm{sd}}<\tau_{\mathrm{Ni}}$, the energy of the magnetar is released more rapidly in the magnetar model than in the 56Ni model, 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 $\tau_{\mathrm{sd}}>\tau_{\mathrm{Ni}}$. By 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 $\tau_{\mathrm{sd}}=0.01\mathord{\thinspace\rm days}$ for this SN. The energy was almost explosively released.

5. Conclusions

The mechanism for the formation of SNe Ic-BL is still unclear. Recently there is 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 $\gtrsim 300\mathord{\thinspace\rm days}$ 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 (3 out of 12) are unlikely 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), 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).