Modeling the Light Curves of the Luminous Type Ic Supernova 2007D

As a subclass of core-collapse supernovae (CCSNe), Type Ic SNe (SNe Ic) have long been believed to be the results of explosions of massive stars that had lost all of their hydrogen and all (or almost all) of their helium envelopes, thereby showing no hydrogen and helium absorption lines (see Filippenko 1997; Matheson et al. 2001; Gal-Yam 2017 for reviews). The light curves (LCs), spectra, and physical parameters of SNe Ic are rather heterogeneous. According to their peak luminosities they can be classified into three subclasses: ordinary SNe Ic, luminous SNe Ic, and superluminous SNe Ic (SLSNe Ic; Quimby et al. 2011; Gal-Yam 2012, 2018).¹⁰

Based on their spectra around peak brightness, SNe Ic can be divided into normal SNe Ic and "broad-lined SNe Ic" (SNe Ic-BL; Woosley & Bloom 2006). And, according to their kinetic energy (E_K), they can be split into normal SNe Ic (E_K ≲ 2 × 10⁵¹ erg) and "hypernovae" (E_K ≳ 2 × 10⁵¹ erg; Iwamoto et al. 1998). A minority of SNe Ic-BL are associated with gamma-ray bursts (GRBs) or X-ray flashes (XRFs) and were called "GRB-SNe" (see Woosley & Bloom 2006; Hjorth & Bloom 2012; Cano et al. 2017b and references therein).

Study of the energy sources of SNe Ic-BL and SLSNe-I/Ic is a very important part of time-domain astronomy. The LCs of normal SNe Ic can be explained by the ⁵⁶Ni cascade decay model (⁵⁶Ni model for short; Colgate & McKee 1969; Colgate et al. 1980; Arnett 1982), while the energy sources of luminous SNe and SLSNe are still being debated: they cannot be explained by the ⁵⁶Ni model (e.g., Quimby et al. 2011; Gal-Yam 2012; Inserra et al.2013), so instead researchers often invoke the magnetar model (Maeda et al. 2007; Kasen & Bildsten 2010; Woosley 2010; Chatzopoulos et al. 2012, 2013; Dessart et al. 2012b; Inserra et al. 2013; Chen et al. 2015; Wang et al. 2015a, 2016b; Dai et al. 2016), involving nascent highly magnetized neutron stars (magnetic strength B_p ≈ 10¹³–10¹⁵ G),¹¹ and the circumstellar interaction model (Chevalier 1982; Chevalier & Fransson 1994; Chugai & Danziger 1994; Chatzopoulos et al. 2012, 2013; Ginzburg & Balberg 2012; Liu et al. 2018), in which ejecta kinetic energy is converted to radiation.

In this paper, we study the very nearby Type Ic SN 2007D. The luminosity distance D_L derived from the Tully–Fisher relation and the redshift z of the host galaxy of SN 2007D (UGC 2653) are ${106}_{-8.5}^{+2}\,\mathrm{Mpc}$ (from NED)¹² and 0.023146 ± 0.000017 (recession velocity 6939 ± 5 km s⁻¹; Wegner et al. 1993), respectively. The photospheric velocity (v_ph) of SN 2007D inferred from the Fe iiλ5169 absorption line about 8 days before V-band maximum brightness is ∼13,350 ± 4000 km s⁻¹ (Modjaz et al. 2014, 2016), smaller than the canonical value of SNe Ic-BL (∼22,200 ± 9400 km s⁻¹; Modjaz et al. 2016) and the average values of SLSNe-I (∼15,000 ±2600 km s⁻¹; Liu et al. 2017b) 10 days after peak brightness.

SN 2007D was heavily extinguished by its highly inclined (∼70°; Drout et al. 2011) host galaxy UGC 2653 (E(B − V)_host = 0.91 ± 0.13 mag; Drout et al. 2011) and the Milky Way ($E{(B-V)}_{\mathrm{Gal}}=0.335\,\mathrm{mag}$; Schlegel et al. 1998). By performing the extinction correction, Drout et al. (2011) found that the R-band and V-band peak absolute magnitudes (${M}_{R,\mathrm{peak}}$ and ${M}_{V,\mathrm{peak}}$) of SN 2007D are ∼−20.65 ± 0.55 mag and <−20.54 mag, respectively, significantly brighter than all other SNe Ibc.¹³ While Gal-Yam (2012) suggested that the SLSN threshold can be set at −21 mag, Quimby et al. (2018) and De Cia et al. (2018) re-examined the threshold of SLSNe and suggested it is $\sim -20.5\,\mathrm{mag}$, as adopted by Quimby (2014). According to the latter threshold, SN 2007D is a SLSN. However, the extinction values of the host galaxy of SN 2007D and the Milky Way are rather uncertain. For example, using the values of Schlafly & Finkbeiner (2011) for the foreground extinction¹⁴ (which are roughly 20%–30% lower than those of Schlegel et al. 1998) and the K-corrected V-band LC of SN 2007D, we find a peak absolute magnitude ${M}_{V,\mathrm{peak}}$ of only ∼−20.06 mag,¹⁵ ∼0.48 mag dimmer than the value inferred by Drout et al. (2011; <−20.54 mag). In this case, SN 2007D is a luminous SN whose peak luminosity is between that of ordinary SNe and SLSNe (see, e.g., Arcavi et al. 2016). We call these two different LCs "Case A" and "Case B" throughout this paper.

The energy source of SN 2007D has not yet been definitively determined. By assuming that the luminosity evolution of SN 2007D was powered by ⁵⁶Ni decay and supposing that the ejecta velocity is ∼2 × 10⁹ cm s⁻¹, Drout et al. (2011) inferred that the mass of ⁵⁶Ni synthesized in the explosion and the value of (M_ej/M_⊙)^3/4(E_K/10⁵¹ erg)^−1/4 are ∼1.5 ± 0.5 M_⊙ and $\sim {1.5}_{-0.5}^{+0.8}\,{M}_{\odot }$, respectively (see Table 6 of Drout et al. 2011). Supposing v_sc ≈ 2 × 10⁹ cm s⁻¹ for the scale velocity of the ejecta and solving the equation ${({M}_{\mathrm{ej}}/{M}_{\odot })}^{3/4}{({E}_{{\rm{K}}}/{10}^{51}\mathrm{erg})}^{-1/4}={1.5}_{-0.5}^{+0.8}$, however, we find that the mass of the ejecta ${M}_{\mathrm{ej}}={3.5}_{-1.95}^{+4.7}\,{M}_{\odot }$. Then the ratio of the ⁵⁶Ni mass to the ejecta mass (M_Ni/M_ej) is $\sim {0.43}_{-0.31}^{+0.86}$, significantly larger than the upper limit (∼0.20) determined by numerical simulations (Umeda & Nomoto 2008), suggesting that the photometric evolution of SN 2007D cannot be explained by the ⁵⁶Ni model. Therefore, the question of the energy source of SN 2007D deserves detailed study. In fact, Gal-Yam (2012) had discussed SN 2007D and SN 2010ay as "transitional" events between SLSNe-I and SNe Ic and suggested that a "central engine" may power their large observed peak luminosities. However, no quantitative research on this idea has been performed to date.

In this paper, we investigate in detail the energy-source mechanisms powering the luminosity evolution of SN 2007D. In Section 2, we employ the ⁵⁶Ni model, the magnetar model, as well as the magnetar+⁵⁶Ni model to fit the R-band LC and the V − R color evolution of SN 2007D. Discussion and conclusions are presented in Sections 3 and 4, respectively.

In this section, we employ semianalytic models to fit the R-band LC and the V − R color evolution of SN 2007D.¹⁶ To fit these LCs, we neglect the dilution effect (e.g., Dessart et al. 2012a) of the ejecta and assume that the SN radiation is blackbody emission: $F{\left(\nu ,t)=(2\pi h{\nu }^{3}/{c}^{2})({e}^{\tfrac{h\nu }{{k}_{{\rm{b}}}T(t)}}-1\right)}^{-1}({R}^{2}/{D}_{L}^{2})$, where T(t) = (L(t)/4πσ(v_sct)²)^1/4 is the blackbody temperature and L(t) is the bolometric luminosity of an SN. Using the Vega magnitude system ($\mathrm{mag}(\nu ,t)=-2.5\,{\mathrm{log}}_{10}F(\nu ,t)\,-48.598-{zp}({f}_{\nu })$) and Table A2 of Bessell et al. (1998), we can convert the fluxes to magnitudes.¹⁷ Hence, our semianalytic models should simultaneously reproduce the bolometric LC, the temperature evolution, and the multiband LCs of SN 2007D. In adopting a simple blackbody model, we neglect the blue-ultraviolet (UV) suppression that yields a dimmer blue-UV luminosity and a brighter optical luminosity. To get the best-fit parameters and the range, we adopt the Markov Chain Monte Carlo (MCMC) method.

2.1. The ⁵⁶Ni-only Model

We first employ a semianalytic ⁵⁶Ni model to fit the R and V − R LCs. The LCs reproduced by this model are determined by the optical opacity κ, the ejecta mass ${M}_{\mathrm{ej}}$, the initial scale velocity of the ejecta v_sc0, the ⁵⁶Ni mass ${M}_{\mathrm{Ni}}$, the gamma-ray opacity of ⁵⁶Ni decay photons ${\kappa }_{\gamma ,\mathrm{Ni}}$, and the moment of explosion t_expl. We suppose that the initial kinetic energy of the ejecta (${E}_{{\rm{K}}0}=0.3\,{M}_{\mathrm{ej}}{v}_{\mathrm{sc}0}^{2}$) is provided by the neutrino-driven mechanism. Then the upper limit of E_K0 is set to be 2.5 × 10⁵¹ erg as the upper limit of E_K0 provided by the neutrino-driven mechanism is (2.0–2.5) × 10⁵¹ erg (see Janka et al. 2016 and references therein). The upper limit of v_sc0 is adopted to be ∼16,000 km s⁻¹. Without this constraint, MCMC would favor a ${v}_{\mathrm{sc}0}$ value that yields a photospheric velocity significantly larger than the observed one (∼13,350 ± 4000 km s⁻¹) as there is only one velocity point.

The theoretical ⁵⁶Ni-powered R and V − R LCs are shown in Figure 1. The parameters of the ⁵⁶Ni model are listed in Table 1. To match the post-peak R-band LC, the value of ${\kappa }_{\gamma ,\mathrm{Ni}}$ must be ${1.12}_{-0.86}^{+4.01}\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$, larger than the canonical value of 0.027 cm² g⁻¹ (e.g., Cappellaro et al. 1997; Mazzali et al. 2000; Maeda et al. 2003).

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Table 1.  Parameters of the Various Models

	κ	Mej	MNi	Bp	P0	vsc0	${\kappa }_{\gamma ,\mathrm{Ni}}$	${\kappa }_{\gamma ,\mathrm{mag}}$	texpla
	(cm2 g−1)	(M⊙)	(M⊙)	(1014 G)	(ms)	(109 cm s−1)	(cm2 g−1)	(cm2 g−1)	(days)
Case A
56Ni	0.07	${1.39}_{-0.33}^{+0.19}$	${2.66}_{-0.15}^{+0.17}$	⋯	⋯	${1.59}_{-0.02}^{+0.01}$	${1.12}_{-0.86}^{+4.01}$	⋯	$-{9.83}_{-0.26}^{+0.13}$
Magnetar	0.07	${1.23}_{-0.40}^{+0.29}$	0	${3.10}_{-0.35}^{+0.36}$	${7.28}_{-0.21}^{+0.21}$	${1.59}_{-0.02}^{+0.01}$	⋯	${5.01}_{-4.27}^{+35.73}$	$-{9.83}_{-0.28}^{+0.13}$
Magnetar+56Ni	0.07	${1.22}_{-0.39}^{+0.30}$	0.2	${3.04}_{-0.37}^{+0.37}$	${7.43}_{-0.21}^{+0.22}$	${1.59}_{-0.02}^{+0.01}$	0.027	${5.01}_{-4.30}^{+33.89}$	$-{9.84}_{-0.26}^{+0.12}$
Case B
56Ni	0.07	${1.45}_{-0.32}^{+0.17}$	${1.61}_{-0.07}^{+0.08}$	⋯	⋯	${1.58}_{-0.03}^{+0.02}$	${1.55}_{-1.12}^{+3.95}$	⋯	$-{9.67}_{-0.46}^{+0.24}$
Magnetar	0.07	${1.25}_{-0.40}^{+0.29}$	0	${2.81}_{-0.44}^{+0.43}$	${9.00}_{-0.42}^{+0.32}$	${1.58}_{-0.03}^{+0.01}$	⋯	${5.25}_{-4.52}^{+35.49}$	$-{9.80}_{-0.32}^{+0.15}$
Magnetar+56Ni	0.07	${1.24}_{-0.36}^{+0.29}$	0.2	${2.49}_{-0.46}^{+0.49}$	${9.02}_{-0.57}^{+0.44}$	${1.58}_{-0.02}^{+0.01}$	0.027	${5.75}_{-5.05}^{+34.98}$	$-{9.80}_{-0.32}^{+0.15}$

Note. The uncertainties are 1σ.

^aThe value of t_expl is with respect to the date of the first R-band observation; the lower limit is set to be −10 here.

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For Case A, the inferred ⁵⁶Ni mass is $\sim {2.66}_{-0.15}^{+0.17}\,{M}_{\odot }$. This value is significantly larger than that (∼1.5 M_⊙) derived from a relation linking the R-band peak magnitude M_R,peak and the ⁵⁶Ni mass yield used by Drout et al. (2011). This is because higher peak luminosity and temperature result in a bluer photosphere when the SN peaks and the ratio of the UV flux to the R-band flux is larger than that of the normal SNe Ibc, and more ⁵⁶Ni is needed for powering the SN peak. As shown in Table 1, the derived ejecta mass is ${1.39}_{-0.33}^{+0.19}\,{M}_{\odot }$, smaller than the mass of ⁵⁶Ni. For Case B, the inferred values of the ejecta mass and ⁵⁶Ni mass are ${1.45}_{-0.32}^{+0.17}\,{M}_{\odot }$ and ${1.61}_{-0.07}^{+0.08}\,{M}_{\odot }$, respectively. The ⁵⁶Ni mass is also larger than the ejecta mass.

We note that the value of κ can vary from 0.06 to 0.20 cm² g⁻¹ (see the references listed by Wang et al. 2017c) and was fixed here to be 0.07 cm² g⁻¹. A larger (smaller) value would result in a smaller (larger) value of M_ej (see, e.g., Wang et al. 2015b, 2017c; Nagy & Vinkó 2016). Nevertheless, the inferred ratio of the ⁵⁶Ni mass to the ejecta mass would still be larger than 1.36 (for Case A) or 0.90 (for Case B) even if κ = 0.06 cm² g⁻¹.

These results indicate that the ⁵⁶Ni model cannot explain the multiband LCs of SN 2007D and that there might be other energy sources involved, because the ratio of the ⁵⁶Ni mass to the ejecta mass cannot be larger than ∼0.20 (Umeda & Nomoto 2008).

2.2. The Magnetar Model

Because the modeling disfavors the ⁵⁶Ni-only model, alternative models must be considered. Here we use the magnetar model to fit the R-band LC and the color evolution of SN 2007D. The free parameters of the magnetar model are κ, M_ej, v_sc0, the magnetic strength B_p, the magnetar's initial rotational period P₀, the gamma-ray opacity of magnetar photons ${\kappa }_{\gamma ,\mathrm{mag}}$, and t_expl.

The R and V − R LCs reproduced by the magnetar model are shown in Figure 2, and the corresponding parameters are listed in Table 1. We find that a magnetar with ${P}_{0}\,\approx {7.28}_{-0.21}^{+0.21}\,\mathrm{ms}$ (or ${9.00}_{-0.42}^{+0.32}\,\mathrm{ms}$ for Case B) and ${B}_{p}\,\approx {3.10}_{-0.35}^{+0.36}\times {10}^{14}$ G (or ${2.81}_{-0.44}^{+0.43}\times {10}^{14}$ G for Case B) can power the multiband LCs of SN 2007D.

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2.3. The Magnetar plus ⁵⁶Ni Model

It has been proposed that ≲0.2 M_⊙ of ⁵⁶Ni can be synthesized by an energetic SN explosion (Nomoto et al. 2013). We employ the magnetar plus ⁵⁶Ni model whose free parameters are κ, M_ej, v_sc0, B_p, P₀, ${\kappa }_{\gamma ,\mathrm{mag}}$, M_Ni, ${\kappa }_{\gamma ,\mathrm{Ni}}$, and t_expl. It can be expected that the contribution of such a small amount of ⁵⁶Ni is substantially less than that of a magnetar. Therefore, the LCs reproduced by the magnetar and the magnetar plus ≲0.2 M_⊙ of ⁵⁶Ni models cannot be distinguished if we tune the parameters. We add 0.2 M_⊙ of ⁵⁶Ni (see also Metzger et al. 2015; Bersten et al. 2016 for SN 2011kl) and fit the LCs.

The LCs produced by such a magnetar (${P}_{0}\approx {7.43}_{-0.21}^{+0.22}\,\mathrm{ms}$ for Case A, ${9.02}_{-0.57}^{+0.44}\,\mathrm{ms}$ for Case B; ${B}_{p}\approx {3.04}_{-0.37}^{+0.37}\times {10}^{14}\,{\rm{G}}$ for Case A, ${2.49}_{-0.46}^{+0.49}\times {10}^{14}\,{\rm{G}}$ for Case B) plus 0.2 M_⊙ of ⁵⁶Ni as well as the LCs powered by 0.2 M_⊙ of ⁵⁶Ni are plotted in Figure 3, and the corresponding parameters are listed in Table 1. While the photometric evolution of SN 2007D can also be explained by the magnetar plus ⁵⁶Ni model, the contribution of ⁵⁶Ni can be neglected.

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3.1. Bolometric LC and the Temperature Evolution of SN 2007D

In Section 2, we used several models to fit the R and V − R LCs of SN 2007D. To obtain more information, we plot the theoretical bolometric LCs and the temperature evolution; see Figure 4. The derived temperature of SN 2007D in Case A is rather high, >10,000 K when $t-{t}_{\mathrm{peak},\mathrm{bol}}\leqslant 10\,\mathrm{days}$ (${t}_{\mathrm{peak},\mathrm{bol}}$ of SN 2007D is ∼10 days), comparable to that of SLSNe (see, e.g., Figure 5 of Inserra et al. 2013) and significantly higher than that of ordinary SNe Ic at the same epoch (≲7000 K; Liu et al. 2017b). The derived temperature of SN 2007D at the same epoch in Case B is 8000–9000 K, between that of SLSNe-I and ordinary SNe Ic.

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We compare the spectrum of SN 2007D with spectra of three SLSNe-I (LSQ14bdq, SN 2016aj, and SN 2015bn) at the same epoch (see Figure 5), finding that SN 2007D is redder than these SLSNe. This result indicates that the temperature of SN 2007D is lower than the temperature of these three SLSNe-I and that Case B is favored—that is, SN 2007D might be a luminous SN Ic rather than an SLSN-I.

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3.2. Physical Parameters of the Ejecta of SN 2007D and the Magnetar

SN 2007D is a very nearby SN Ic whose luminosity distance and redshift are ${106}_{-8.5}^{+2}\,\mathrm{Mpc}$ and 0.023146 ± 0.000017, respectively. Drout et al. (2011) demonstrated that SN 2007D is a very luminous SN Ic: ${M}_{R,\mathrm{peak}}\approx -20.65\pm 0.55\,\mathrm{mag}$ and ${M}_{V,\mathrm{peak}}\lt -20.54\,\mathrm{mag}$, which are brighter than the SLSN threshold (−20.5 mag) given by Quimby et al. (2018) and De Cia et al. (2018), and inferred that the ⁵⁶Ni powering the luminosity evolution of SN 2007D is 1.5 ± 0.5 M_⊙. Adopting the values of Schlafly & Finkbeiner (2011) for the foreground extinction and the K-corrected V-band LC of SN 2007D, however, we found a peak absolute magnitude ${M}_{V,\mathrm{peak}}$ of only ∼−20.06 mag, ∼0.48 mag dimmer than the LCs of Drout et al. (2011).

Our simple estimate shows that the ratio of ⁵⁶Ni to the ejecta mass of SN 2007D is unrealistically large ($\sim {0.43}_{-0.31}^{+0.86}$). To verify the validity of the ⁵⁶Ni cascade decay model, we use the ⁵⁶Ni model to fit its R and V − R LCs and find that the required ⁵⁶Ni mass ($\sim {2.66}_{-0.15}^{+0.17}\,{M}_{\odot }$ for Case A or ${1.61}_{-0.07}^{+0.08}\,{M}_{\odot }$ for Case B) is larger than the inferred ejecta mass ($\sim {1.39}_{-0.33}^{+0.19}\,{M}_{\odot }$ for Case A or $\sim {1.45}_{-0.32}^{+0.17}\,{M}_{\odot }$ for Case B) if its multiband LCs were solely powered by ⁵⁶Ni, indicating that the ⁵⁶Ni model cannot account for the LCs of SN 2007D. Alternatively, we employ the magnetar model and find that the LCs can be fitted and the parameters are reasonable if the initial period P₀ and the magnetic strength B_p of the putative magnetar are ${7.28}_{-0.21}^{+0.21}\,\mathrm{ms}$ (or ${9.00}_{-0.42}^{+0.32}\,\mathrm{ms}$ for Case B) and ${3.10}_{-0.35}^{+0.36}\times {10}^{14}$ G (or ${2.81}_{-0.44}^{+0.43}\times {10}^{14}$ G for Case B), respectively.

By comparing the LCs reproduced by the magnetar model and the magnetar plus ⁵⁶Ni model (the mass of ⁵⁶Ni is set to be 0.2 M_⊙), we find that the contribution of ⁵⁶Ni was significantly lower than that of the magnetar and can be neglected; it is very difficult to distinguish between the LCs reproduced by these two models. Nevertheless, a moderate amount of ⁵⁶Ni is needed as the shock launched from the surface of the protomagnetar would heat the silicon shell located at the base of the SN ejecta and ≲0.2 M_⊙ of ⁵⁶Ni would be synthesized. According to these results, we suggest that SN 2007D might be powered by a magnetar or a magnetar plus $\lesssim 0.2\,{M}_{\odot }$ of ⁵⁶Ni.

Adopting the SLSN threshold (−20.5 mag) given by Quimby et al. (2018) and De Cia et al. (2018), and assuming that the peak magnitudes of R and V LCs of SN 2007D are −20.65 ± 0.55 mag and <−20.54 mag (respectively), one can conclude that SN 2007D is an SLSN. If we use the values of Schlafly & Finkbeiner (2011) for the foreground extinction, however, the luminosity of SN 2007D would be ∼0.48 mag dimmer, and thus only a luminous SN rather than an SLSN. The spectrum provides additional evidence to discriminate these two possibilities. We find that the extinction-corrected premaximum spectrum of SN 2007D is redder than that of three comparison SLSNe-I (LSQ14bdq, SN 2016aj, and SN 2015bn) at a similar epoch, indicating that the temperature of SN 2007D is lower than that of these objects. This fact favors the possibility that SN 2007D is a luminous SN rather than an SLSN.

We would like to thank an anonymous referee for constructive suggestions that have allowed us to improve our manuscript. This work is supported by the National Key Research and Development Program of China (grant No. 2017YFA0402600), and the National Natural Science Foundation of China (grant Nos. 11573014 and 11833003). S.Q.W. and L.D.L. are supported by the China Scholarship Program to conduct research at U.C. Berkeley and UNLV, respectively. L.J.W. is supported by the National Program on Key Research and Development Project of China (grant 2016YFA0400801). A.V.F.'s supernova group is grateful for financial assistance from the Christopher R. Redlich Fund, the TABASGO Foundation, and the Miller Institute for Basic Research in Science (U.C. Berkeley). This research has made use of the CfA Supernova Archive that has been funded in part by the National Science Foundation through grant AST 0907903, the Weizmann Interactive Supernova Data Repository (WISeREP), and the Transient Name Server.