Modeling the Multiband Light Curves of the Afterglows of Three Gamma-Ray Bursts and their Associated Supernovae

Some dozen supernovae (SNe) associated with long gamma-ray bursts (GRBs) have been confirmed. Most of the previous studies derive the physical properties of the GRB-SNe by fitting the constructed (pseudo-)bolometric light curves. However, many GRB-SNe only have a few filter data, for which the (pseudo-)bolometric light curves are very difficult to construct. Additionally, constructing (pseudo-)bolometric light curves rely on some assumptions. In this paper, we use the multiband broken power-law plus 56Ni model to fit the multiband light curves of the afterglows and the SNe (SN 2001ke, SN 2013dx, and SN 2016jca) associated with three GRBs (GRB 011121, GRB 130702A, and GRB 161219B). We find our model can account for the multiband light curves of the three GRB-SNe (except for the late-time z-band light curve of two events), indicating that the model is a reliable model. The 56Ni masses we derive are higher than those in the literature. This might be due to the fact that the 56Ni masses in the literature are usually obtained by fitting the pseudo-bolometric light curves whose luminosities are usually (significantly) underestimated. We suggest that the multiband model can not only be used to fit the multiband light curves of GRB-SNe that have many filter observations, but also fit those having sparse data.


Introduction
Gamma-ray bursts (GRBs) are the most powerful explosions in the universe. It is widely believed that GRBs come from the relativistic jet launched by the central engine (Woosley 2011). The interactions between the jets with the surrounding medium would produce X-ray, UV-optical-near-infrared (NIR), and radio afterglows (see Zhang 2018 and references therein). According to the observation of prompt emission duration, GRBs is divided into long-duration bursts (LGRBs) and shortduration bursts (SGRBs) with a dividing line of ∼2 s (Kouveliotou et al. 1993). The observations and analysis for some dozen supernovae (SNe) associated with LGRBs (Hjorth et al. 2003;Matheson et al. 2003;Stanek et al. 2003;Malesani et al. 2004;Deng et al. 2005;Campana et al. 2006;Mirabal et al. 2006;Modjaz et al. 2006;Sollerman et al. 2006;Maeda et al. 2007;Chornock et al. 2010;Starling et al. 2011;Bufano et al. 2012;Melandri et al. 2012Melandri et al. , 2014Melandri et al. , 2019Olivares et al. 2012;Singer et al. 2013;Schulze et al. 2014;D'Elia et al. 2015;Toy et al. 2016;Cano et al. 2017a;Volnova et al. 2017;Ashall et al. 2019;Hu et al. 2021) indicate that most LGRBs are produced by the explosions of massive stars. On the other hand, the confirmation of SSS17a/AT2017gfo, which is a kilonova associated with GW170817 that is a gravitational wave emitted by a merger of a neutron star binary and GRB 170817A that is an SGRB (Abbott et al. 2017;Arcavi et al. 2017;Coulter et al. 2017;Shappee et al. 2017), supports the conjecture that at least a fraction of SGRBs are produced by the mergers of compact binary stars.
On average, one or two GRB-SNe can be found every year. To date, there are about 60 LGRBs that have been confirmed to be associated with SNe. Almost all GRB-SNe are broad-lined Ic (Ic-BL) SNe whose optical spectra are hydrogen deficient and show broad absorption line features. The spectral features indicate that the progenitors of GRB-SNe are highly stripped, and might be Wolf-Rayet stars Sonbas et al. 2008). The broad absorption lines are indicative of huge ejecta velocities 2 × 10 9 cm s −1 . Therefore, a major fraction of GRB-SNe (and the SNe Ic-BL without accompanying GRBs) become so-called hypernovae (HNe) whose kinetic energy is 10 52 erg, which is about 10 times that of normal SNe. The explosion mechanisms of GRB-SNe are still elusive. The most prevailing model adopted to account for the light curves of GRB-SNe is the 56 Ni cascade decay ( 56 Ni → 56 Co → 56 Fe) model (the 56 Ni model, Arnett 1979Arnett , 1980Arnett , 1982Arnett , 1996. Some very luminous GRB-SNe cannot be explained by the 56 Ni model, and alternative or additional energy sources (e.g., the magnetar spinning down, the fallback accretion, etc.) are employed to account for the light curves.
Previous studies focusing on GRB-SNe usually construct the pseudo-bolometric light curves of the SNe and derive the physical properties of GRB-SNe by fitting the constructed pseudo-bolometric light curves. However, it should be noted that the process of constructing the pseudo-bolometric light curves might underestimate the luminosities of the SNe and therefore underestimate the 56 Ni masses.
Recently, the model directly fitting the multiband light curves (Nicholl et al. 2017) has been adopted to fit the light curves of superluminous SNe (Nicholl et al. 2017;Moriya et al. 2018), the tidal disruption events (Mockler et al. 2019), the luminous rapidly evolving optical transients (Wang et al. 2019), and ordinary SNe Ib and Ic (S. Q. Wang et al. 2022, in preparation).
In this paper, we collect published data of the (UV-)optical-NIR counterparts (GRB 011121/SN 2001ke, GRB 130702A/ SN 2013dx, GRB 161219B/SN 2016jca) of three GRBs (GRB 011121, GRB 130702A, GRB 161219B; here, the GRBs represent their afterglows) and use the broken power-law plus 56 Ni model to fit their multiband light curves. In Section 2, we model the multiband light curves of the three GRB-SNe using the 56 Ni model. In Section 3, we compare the parameters and bolometric properties of the SNe to those in the literature. We draw some conclusions in Section 4. The values of of the foreground reddening of the Milky Way (E(B − V ) MW ) are from Schlafly & Finkbeiner (2011).   Table 1. The flux of the host galaxy of GRB 011121/ SN 2001ke is negligible, and the flux of the host galaxies of GRB 130702A/SN 2013dx and GRB 161219B/SN 2016jca have been subtracted (the magnitudes of the two host galaxies are from Volnova et al. 2017 andLaskar et al. 2018, respectively, andlisted in Table 2). Then the flux of the (UV-)optical-NIR counterpart of a GRB-SN (F ν,tot (t)) can be divided into that of the GRB afterglow (F ν,AG (t)) and that of the ) associated with the GRB, i.e., The flux density of an afterglow is proportional to a broken power-law decay function ( Beuermann et al. 1999, in the representation of Zeh et al. 2004) and ν − β (F ν,AG (t) ∝ ν − β ), and can be expressed as . The definitions of α 1 , α 2 , t b , n, and β are presented in Table 3.
Assuming the early-time photosphere radii of the SNe is proportional to the time, and the ejecta cool to constant temperatures (T f ), the temperatures and radii can be given by (Nicholl et al. 2017):   , 9, 10, 11, 12, 13, 14 Notes.   To fit the multiband light curves of the SN components, we suppose that the spectral energy distributions (SEDs) of the SNe can be described by the UV-absorbed blackbody model (Nicholl et al. 2017;Prajs et al. 2017), where λ CF = 3000 Å is the cutoff wavelength (Nicholl et al. 2017;Prajs et al. 2017).
The definitions, the units, and the priors of the parameters of the model are listed in Table 3. The values of A V,host (or E(B − V ) host ) of GRB 130702A/SN 2013dx and GRB 161219B/SN 2016jca have been given by the literature, and can be set to be constants. Hence, the multiband 56 Ni model fitting GRB 130702A/SN 2013dx and GRB 161219B/ SN 2016jca has five free parameters (M ej , v ph , M Ni , κ γ , and T f ). GRB 011121/SN 2001ke is far away from the host galaxy (Bloom et al. 2002); Greiner et al. (2003) suggest that it has no host galaxy extinction, while Küpcü Yoldaş et al. (2007) get an upper limit of E(B − V ) host (0.08 mag). We assume that A V,host of GRB 011121/SN 2001ke is an additional free parameter whose range is 0-0.248 mag. Additionally, we assume that the ratio of M Ni to M ej is 0.2 (Umeda & Nomoto 2008). We adopt the Markov Chain Monte Carlo method by using emcee of Python package (Foreman-Mackey et al. 2013) to fit the data to obtain the best-fitting parameters and 1σ parameter range.
The fits of the three GRB-SNe and the best-fit parameters are presented in Figure 1 and Table 4, respectively. The corresponding corner plots are shown in Figures A1-A3 Table 4. We find that the new fit is better than the first fit since the UV bands are also well matched by the model.
There are two (possible) reasons that might explain the bad quality of the fits for the z-band light curves of the two GRB-SNe. (1). Their late-time z-band light curves show fluctuation features that cannot be fully fitted by the theoretical light curves, which are smooth. , 2.61 ± 0.02 × 10 9 , and 2.17 ± 0.03 × 10 9 cm s −1 . The parameters are roughly consistent with the parameter ranges in the literature.

Discussion
Here, we compare the values of the 56 Ni masses, the ejecta masses, the ejecta velocity, and the kinetic energy of the ejecta of the three GRB-SNe to that in the literature and discuss the reasons causing the discrepancies. Moreover, we discuss the theoretical bolometric light curves of the three GRB-SNe. Notes. a Based on the fits of Greiner et al. (2003), the ranges of α 1 , α 2 , and t b of the afterglow of GRB 011121are set to be    The 56 Ni mass of GRB 130702A/SN 2013dx is rather large, but comparable to the 56 Ni mass of SN 1998bw, which is 0.4-0.7 M e (Iwamoto et al. 1998;Nakamura et al. 2001) or -+ 0.54 0.07 0.08 M e (Lyman et al. 2016). Therefore, we suggest that the 56 Ni mass is reasonable.
Our derived early-time photospheric velocities of SN 2013dx and SN 2016jca are 2.61 ± 0.02 × 10 9 and 2.17 ± 0.03 × 10 9 cm s −1 , respectively. The former is between the two values adopted by Toy et al.

Theoretical Bolometric Light Curves
We use the derived best-fitting parameters to yield the bolometric light curves of the three GRB-SNe we study, see By comparing our derived peak bolometric luminosities of SN 2001ke, SN 2013dx, and SN 2016jca to their peak (pseudo-)bolometric luminosities in the literature, we find that the former are respectively 2.28, 1.92, and 1.65 (or 2.26) times that the latter. 3 The SN velocities inferred from the spectra evolve (usually decrease) with the time. Toy et al. (2016) find that the spectral velocity of SN 2013dx inferred from the Si II lines at days 9.3, 11.3, 14.2, 31.3, and 33.3 are 2.81, 2.52, 2.13, 1.17, and 1.08 × 10 9 cm s −1 , respectively. D' Elia et al. (2015) find that the velocity of SN 2013dx declines from ∼2.7 × 10 9 cm s −1 at day 8 to ∼3.5 × 10 8 cm s −1 at day 40. Previous studies fitting the (pseudo-)bolometric light curves usually adopt the velocity derived from the spectra obtained around maximum light or earlier epochs.
The discrepancies of the peak luminosities of bolometric light curves we derive and those of the pseudo-bolometric light curves might be due to the fact that the latter omit the flux in UV and/or IR bands. Toy et al. (2016) construct the pseudobolometric light curve of SN 2013dx by integrating the flux in the ¢ ¢ ¢ ¢ g r i z yJ bands, and more flux is neglected. Cano et al. (2017a) use the griz band data to construct the pseudobolometric light curve of SN 2016jca, and the flux might also be underestimated.
Our derived rise time of SN 2001ke and SN 2013dx are respectively 11.8 and 13.7 days, which are respectively smaller than and comparable to the rise time of the two SNe in the literature, which are ∼17.5 days (Cano et al. 2017b) and ∼14 days (Toy et al. 2016). Our derived rise time of SN 2016jca is 10.7 days, which is slightly larger than in the literature, which is ∼10 days (Ashall et al. 2019).

Conclusions
In the past two decades, a few dozen LGRBs have been confirmed to be associated with SNe Ic, most of which are SNe Ic-BL and HNe. While the kinetic energy of most GRB-SNe is 10 times that of normal SNe Ic, their average peak luminosities are not significantly higher than those of SNe Ic. Therefore, the 56 Ni model adopted to account for the light curves of normal SNe Ic have also been used to explain the light curves of GRB-SNe. However, many studies exploring the energy sources of GRB-SNe construct the pseudobolometric light curves and fit them. This method might underestimate the 56 Ni masses needed to power the light curves of SNe.
We collected photometric data of three well-observed GRB-SNe (GRB 011121/SN 2001ke, GRB 130702A/SN 2013dx, GRB 161219B/SN 2016jca) and use the multiband broken power-law plus 56 Ni model to fit the multiband light curves of the total flux, which is the sum of those of the afterglows of the GRBs and the SNe. The multiband model we use fits the observed multiband data, rather than the pseudo-bolometric light curves constructed by making some assumptions. A larger data set could pose more stringent constraints on the physical parameters.
We find that the multiband light curves of GRB 011121/ SN 2001ke can be fitted by the model we use; the multiband light curves of GRB 130702A/SN 2013dx and GRB 161219B/ SN 2016jca can be fitted by the model (except their late-time zband light curves). This indicates that the UV-optical-NIR SEDs of SNe associated with GRBs can be well described by the UV-absorbed blackbody model, and that our model can account for the multiband light curves of the three GRB-SNe.
Our derived 56 Ni masses of SN 2013dx and SN 2016jca are 0.74 ± 0.01 and 0.33 ± 0.00 M e , respectively. The former is about ∼2.0 and ∼3.7 times those of the values derived by Toy et al. (2016) andD'Elia et al. (2015), while the latter is -+ 1.50 0.40 0.86 and~-+ 1.22 0.19 0.28 times those of the values derived by Cano et al. (2017a) and Ashall et al. (2019). This might be due to the fact that the constructed pseudo-bolometric light curves of SN 2013dx and SN 2016jca omit a fraction of the total flux. Therefore, we suggest that the 56 Ni masses of at least a fraction of GRB-SNe have been underestimated, and the multiband 56 Ni model can make it possible to avoid underestimating the luminosities of SNe and therefore the 56 Ni masses.
Our study demonstrates the validity of the multiband afterglow plus 56 Ni model for the the fits of the multiband light curves of GRB-SNe. The model can be regarded as an independent model that do not rely on the (pseudo-)bolometric light curves constructed. Although the GRB-SNe we fit have ample data at many bands, we expect that the model can also be used for the multiband light curves of GRB-SNe observed in only one, two, or three bands at some or all epochs. For the GRB-SNe with sparse data, the multiband model can play a key role in determining their physical properties by fitting their multiband light curves, since constructing the (pseudo-) bolometric light curves is very difficult.