Application of the Magnetar Engine to an Intermediate-Luminosity Gamma-Ray Burst Associated with the Supernova GRB 201015A/SN 201015A

Abstract

We present optical photometry for the afterglow of GRB 201015A, which can be classified as a medium-luminosity gamma-ray burst ($L_{\gamma ,\mathrm{iso}}\approx 2.55\times {10}^{49}\mathrm{erg}{\mathrm{s}}^{-1}$ ) and the associated underlying supernova SN 201015A. A millisecond magnetar engine has been widely suggested to exist in gamma-ray burst (GRB) phenomena. In this paper, we study the effects of the magnetar engine on GRB 201015A/SN 201015A by light curve analysis. We use a smooth broken power-law plus magnetar spin-down model to fit the X-ray and optical light curves of GRB 201015A/SN 201015A. The best-fitting results reveal that the magnetar initial spin period and surface magnetic field at the pole are constrained to be $P_0=16.{80}_{-0.47}^{+0.24}\mathrm{ms}$ and $B_{\mathrm{p}}=0.{80}_{-0.32}^{+0.34}\times {10}^{15}\mathrm{G}$, respectively, and the SN ejected a total mass of $M_{\mathrm{ej}}=2.{55}_{-0.37}^{+1.12}{M}_{\odot }$ and an ejecta velocity of $v_{\mathrm{ej}}$ = ${\mathrm{30,000}}_{-2500}^{+4800}\mathrm{km}{\mathrm{s}}^{-1}$, inferring a kinetic energy of $E_{\mathrm{SN},\mathrm{K}}\approx 1.37\times {10}^{52}\mathrm{erg}$. From our analysis, we find that the central engine of GRB 201015A/SN 201015A may well be a magnetar, and the emission from a magnetar central engine can be solely responsible for powering SN 201015A.

1. Introduction

The connection between long-duration gamma-ray bursts (LGRBs) and broad-line Type Ic supernovae (SNe Ic-BL) has been established—the “GRB–SN connection” (e.g., [1,2,3]). However, one of the unresolved mysteries of LGRB phenomena is the nature of their central engines, which could be a stellar-mass black hole (BH; e.g., [4,5,6,7,8]) or a rapidly rotating neutron star with an exceptionally large magnetic field (a magnetar; e.g., [9,10,11,12,13,14,15,16]). Based on the isotropic equivalent luminosity in the gamma-ray band, $L_{\gamma ,\mathrm{iso}}$, some authors have classified GRBs as low-luminosity (LL) GRBs with $L_{\gamma ,\mathrm{iso}}\lesssim 48.5\mathrm{erg}{\mathrm{s}}^{-1}$, intermediate-luminosity (IL) GRBs with ${10}^{48.5}\mathrm{erg}{\mathrm{s}}^{-1}\lesssim L_{\gamma ,\mathrm{iso}}\lesssim {10}^{49.5}\mathrm{erg}{\mathrm{s}}^{-1}$, and high-luminosity (HL) GRBs with $L_{\gamma ,\mathrm{iso}}\gtrsim {10}^{49.5}\mathrm{erg}{\mathrm{s}}^{-1}$ [3,17,18,19]. Some of the nearby GRBs also belong to classes of low/intermediate-luminosity GRBs and ultra-long-duration GRBs, which are outliers that have revealed crucial observational evidence that is used to distinguish potential powering mechanisms from progenitors [20,21].

On 15 October 2020, at 22:50:13 (UTC dates are used herein), the Burst Alert Telescope (BAT; [22]) onboard the Neil Gehrels Swift Observatory (Swift; [23]) triggered an extraordinarily interesting LGRB, GRB 201015A [24]. After they obtained the redshift of GRB 201015A from the 10.4 m Gran Telescopio Canarias telescope (GTC, Canary Island, Spain; [25]), refs. [26,27] encouraged follow-up optical observations to search for the associated supernova. As GRB 201015A optical afterglow observations continued, photometric and spectroscopic evidence for an SN appeared [28,29]. In addition, refs. [30,31] identified this SN as a Type Ic-BL, named SN 201015A.

GRB 201015A has a redshift of $z=0.426$ [26,32], a ${\mathrm{T}}_{90}=9.78\pm 3.47$ s in the [15–350] keV energy range [33], and an isotropic energy of $E_{\gamma ,\mathrm{iso}}=1.{75}_{-0.53}^{+0.60}\times {10}^{50}\mathrm{erg}$ [30]. We therefore calculated the isotropic luminosity as $L_{\gamma ,\mathrm{iso}}=E_{\gamma ,\mathrm{iso}}(1+z)/T_{90}\approx 2.55\times {10}^{49}\mathrm{erg}{\mathrm{s}}^{-1}$. Therefore, GRB 201015A can be classified as ILGRB. The X-ray light curve of GRB 201015A shows a late flattening from $(2.61\pm 1.27)\times {10}^4\mathrm{s}$ to $(1.{67}_{-0.65}^{+1.14})\times {10}^6\mathrm{s}$ after the BAT trigger [30]. Since the discovery of the shallow-decay and plateau afterglows and, in particular, afterglow flares, it has been widely suggested that a magnetar should play a crucial role in causing these afterglow features (e.g., [11,13,34,35,36,37]). To discuss the central engine of GRB 201015A/SN 201015A in this work, we therefore analyze the afterglow of GRB 201015A by fitting the X-ray and optical light curve with a smooth broken power-law plus magnetar spin-down model.

This paper is organized as follows: Section 2 presents the observations and data reduction, while Section 3 shows the analysis and results. Our conclusions and implications are presented in Section 4. Throughout this paper, we adopt a concordance cosmology with the parameters $H_0=69.3\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, ${\Omega }_{\mathrm{M}}=0.286$, and ${\Omega }_{\Lambda }=0.714$. The mass and radius of the Sun are denoted by $M_{\odot }$ and $R_{\odot }$, respectively.

2. Observations and Data Reduction

The Swift X-Ray Telescope (XRT; [38]) began observing the GRB at 23:43:47, 3214 s after the BAT trigger, and the revised source position was RA(J2000) = 23:37:16.46, Dec(J2000) = +53:24:57.7 with an uncertainty of 1.7 arcsec (radius, 90 percent containment) [39,40]. The Ultraviolet and Optical Telescope (UVOT; [41]) began observing at 3217 s [42].

The afterglow of GRB 201015A was observed by many ground-based telescopes. The Las Cumbres Observatory Global Telescope Network (LCOGT; [43]) observed the afterglow of GRB 201015A, R-band and I-band images were obtained with the 1 m Sinistro instrument at the McDonald Observatory, Texas, USA. Photometric time series of GRB 201015A at four epochs about +0.111 ($5\times 300\mathrm{s}$), +17.360 ($8\times 300\mathrm{s}$), +19.234 ($8\times 300\mathrm{s}$), and +20.366 ($5\times 300\mathrm{s}$) days after the BAT trigger were observed. Data reduction was carried out following standard routines in the IRAF1 software package. In order to increase the signal-to-noise ratio (S/N) of the detections, individual images from each epochs were then stacked by the imcombine code. Aperture photometry was performed with the apphot code of the daophot package. Several nearby stars were chosen from the Pan-STARRS12 catalog for calibration; their magnitudes were transformed into the Landolt [46] magnitudes using the empirical prescription presented in Equation (6) of [47]. The photometry from LCOGT follow-up observations is reported in

Table 1. Photometry of GRB 201015A ^a.

${\mathit{t}}_{\mathbf{mid}}$ (s) b	${\mathit{t}}_{\mathbf{mid}}$ (Days) b	Mag (Vega) c	$1\mathit{\sigma }$	Filter	Telescope
9607.12	0.111	22.03	0.28	R	LCOGT
1,499,929.58	17.360	>20.93	–	R	LCOGT
1,661,795.90	19.234	22.12	0.09	I	LCOGT
1,759,613.62	20.366	23.04	0.15	I	LCOGT

^a includes contributions from the GRB afterglow, host galaxy, and associated SN 201015A; ^b $t_{\mathrm{mid}}$ is the midpoint of each observation after the BAT trigger; ^c: The data have not been corrected for extinction in the Milky Way Galaxy or the GRB host galaxy.

. The photometry results were corrected for galactic extinction with $E(B-V)=0.229$ mag [48] for analysis. Additionally, the authors of ref. [49] reported that they derive a change in the local extinction $A_{\mathrm{V}}^{\mathrm{local}}$ from ∼$0.8\mathrm{mag}$ to 0.3 mag in ∼2500 s. Later, ref. [31] reported a GRB host extinction of $A_{\mathrm{R}}=0.375\mathrm{mag}$ ($A_{\mathrm{V}}\sim 0.554\mathrm{mag}$), which is consistent with the result of [49]. Therefore, we adopt $A_{\mathrm{R}}=0.375\mathrm{mag}$ to perform GRB host galaxy extinction in this work.

We collected additional photometry data for our analysis from [31,49,50]. XRT data were downloaded from the UK Swift Science Data Center at the University of Leicester [51]3, and we also collected the X-ray data of Chandra observations, unabsorbed 0.3–10 keV fluxes of $(1.26\pm 0.05)\times {10}^{-13}\mathrm{erg}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$ at 8.4 days and $(1.10\pm 0.04)\times {10}^{-13}\mathrm{erg}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$ at 13.6 days, from [52]. Figure 1 shows the multiband light curves of the afterglow in both the optical and X-ray bands.

3. Analysis and Results

3.1. Modeling the Afterglow Light Curves

For the afterglow, one component of our model is a smooth broken power-law (BPL) function (e.g., [53,54,55])

$$F=F_0{\left[{\left({\displaystyle \frac{t}{t_{\mathrm{b}}}}\right)}^{{\alpha }_1\omega }+{\left({\displaystyle \frac{t}{t_{\mathrm{b}}}}\right)}^{{\alpha }_2\omega }\right]}^{-1/\omega },$$

(1)

where ${\alpha }_1$ and ${\alpha }_2$ are the temporal slopes, $t_{\mathrm{b}}$ is the break time, and $\omega$ measures the sharpness of the break (in this paper, we fix $\omega =3$).

In addition to the BPL function, here we also introduce a few models that are helpful for understanding the physics of the GRB and SN.

3.1.1. Spindown of Magnetar

We assume a magnetar with mass $M_{*}$, radius R, initial spin period $P_0$, surface magnetic field at the pole $B_p$, and misalignment angle between the spin axis and the magnetic dipole axis $\pi /2$. The maximum energy is the total rotational energy of a millisecond magnetar, which is defined as

$$E_{\mathrm{rot}}={\displaystyle \frac{1}{2}}I{\Omega }_0^2\approx 2\times {10}^{52}{M}_{1.4}{R}_6^2{P}_{0,-3}^{-2}\mathrm{erg},$$

(2)

where $I=0.35M_{*}{R}^2$ is the moment of inertia, ${\Omega }_0=2\pi /P_0$ is the initial angular frequency of the neutron star, and $M_{1.4}=M_{*}/1.4M_{\odot }$. Hereafter, the convention $Q={10}^x{Q}_x$ is adopted in cgs units for all other parameters except for mass.

The spindown luminosity of the GRB jet evolving with time can be written as [56]

$$L\left(t\right)=L_0{\left(1+{\displaystyle \frac{t}{\tau }}\right)}^{{\displaystyle \frac{1+n}{1-n}}},$$

(3)

where n is the braking index (in this paper, we fix $n=3$ during fitting),

$$L_0=1.0\times {10}^{49}\left(B_{\mathrm{p},15}^2{P}_{0,-3}^{-4}{R}_6^6\right)\mathrm{erg}{\mathrm{s}}^{-1}$$

(4)

is the initial luminosity, and

$$\tau =2.05\times {10}^3\left(I_{45}{B}_{\mathrm{p},15}^{-2}{P}_{0,-3}^2{R}_6^{-6}\right)\mathrm{s}$$

(5)

is the characteristic spindown timescale [13,36,57]. In this paper, we use standard values for a neutron star [58] with mass $M_{*}\approx 1.4M_{\odot }$ and $R_6\approx 1$.

The spindown luminosity generally includes the luminosity of multiband radiation, with

$$\begin{array}{ccc}\hfill L_{\nu }& =& {\eta }_{\nu }{L}_{\nu ,\mathrm{iso}}{f}_b,\hfill \end{array}$$

(6)

where $L_{\nu ,\mathrm{iso}}$ and ${\eta }_{\nu }$ are the isotropic luminosity and the radiation efficiencies of the magnetar spindown in the X-ray, g, r, i, and z bands, respectively. The beaming correction factor is $f_b=1-cos{\theta }_j$, where ${\theta }_j$ is the opening half-angle of the jet [59].

3.1.2. Magnetar-Powered Supernovae

For SNe, the basic form of the magnetar engine model has been described numerous times in the literature (e.g., [60,61,62,63,64,65]). The spindown power of a magnetar by magnetic dipole radiation can be written as

$$P_{\mathrm{mag}}\left(t\right)=L_0{\displaystyle \frac{1}{{(1+t/\tau )}^2}}.$$

(7)

The output luminosity can therefore be written as [66]

$$\begin{array}{cc}\hfill L_{\mathrm{SN}}\left(t\right)=e^{-{(t/t_{\mathrm{diff}})}^2}{\int }_0^{t}2P_{\mathrm{mag}}\left(t^{\prime }\right){\displaystyle \frac{t^{\prime }}{t_{\mathrm{diff}}}}& \\ \hfill e^{{(t^{\prime }/t_{\mathrm{diff}})}^2}(1-e^{-A{t}^{-2}}){\displaystyle \frac{d{t}^{\prime }}{t_{\mathrm{diff}}}},\end{array}$$

(8)

where

$$t_{\mathrm{diff}}={\left({\displaystyle \frac{2\kappa M_{\mathrm{ej}}}{\beta c{v}_{\mathrm{ej}}}}\right)}^{1/2}$$

(9)

is the diffusion time,

$$A={\displaystyle \frac{3{\kappa }_{\gamma }{M}_{\mathrm{ej}}}{4\pi v_{\mathrm{ej}}^2}}$$

(10)

is the leakage parameter [64], and the parameter $\beta$ has a typical value of 13.8 [66]. Here, $M_{\mathrm{ej}}$, $v_{\mathrm{ej}}$, $\kappa$, ${\kappa }_{\gamma }$, and c are, respectively, the ejecta mass, the expansion velocity of the ejecta, Thomson electron scattering opacity, effective gamma-ray opacity, and the speed of light in a vacuum. We assume the velocity at the photosphere $v_{\mathrm{phot}}\approx v_{\mathrm{ej}}$. For a uniform density profile, the kinetic energy is given by $E_{\mathrm{SN}},\mathrm{K}=(3/10)M_{\mathrm{ej}}{v}_{\mathrm{phot}}^2$.

We assume that the spectral energy distribution (SED) of our SN is a blackbody. The flux at frequency $\nu$ can be written as

$$f_{\nu }={\displaystyle \frac{2\pi h{\nu }^3}{c^2}}{\displaystyle \frac{1}{e^{h\nu /kT}-1}}\mathrm{erg}{\mathrm{s}}^{-1}{\mathrm{cm}}^{-2}{\mathrm{Hz}}^{-1},$$

(11)

where h is Planck’s constant, k is Boltzmann’s constant, and T is the temperature in Kelvins.

We assume that the photospheric radius expands at a constant velocity, $v_{\mathrm{phot}}$. The temperature and radius are therefore given by [65]

$$T_{\mathrm{phot}}\left(t\right)=\left\{\begin{array}{cc}{\left({\displaystyle \frac{L_{\mathrm{SN}}\left(t\right)}{4\pi \sigma v_{\mathrm{phot}}^2{t}^2}}\right)}^{{\displaystyle \frac{1}{4}}},{\left({\displaystyle \frac{L_{\mathrm{SN}}\left(t\right)}{4\pi \sigma v_{\mathrm{phot}}^2{t}^2}}\right)}^{{\displaystyle \frac{1}{4}}}>T_{\mathrm{f}}& \\ T_{\mathrm{f}},{\left({\displaystyle \frac{L_{\mathrm{SN}}\left(t\right)}{4\pi \sigma v_{\mathrm{phot}}^2{t}^2}}\right)}^{{\displaystyle \frac{1}{4}}}\le T_{\mathrm{f}}\end{array}\right.,$$

(12)

$$R_{\mathrm{phot}}\left(t\right)=\left\{\begin{array}{cc}{v}_{\mathrm{phot}}t,{\left({\displaystyle \frac{L_{\mathrm{SN}}\left(t\right)}{4\pi \sigma v_{\mathrm{phot}}^2{t}^2}}\right)}^{{\displaystyle \frac{1}{4}}}>T_{\mathrm{f}}& \\ {\left({\displaystyle \frac{L_{\mathrm{SN}}\left(t\right)}{4\pi \sigma T_{\mathrm{f}}^4}}\right)}^{{\displaystyle \frac{1}{2}}},{\left({\displaystyle \frac{L_{\mathrm{SN}}\left(t\right)}{4\pi \sigma v_{\mathrm{phot}}^2{t}^2}}\right)}^{{\displaystyle \frac{1}{4}}}\le T_{\mathrm{f}}\end{array}\right.,$$

(13)

where $\sigma$ is the Stefan–Boltzmann constant and $T_{\mathrm{f}}$ is the final plateau temperature, a free parameter that allows us to extend our fits to later times.

3.2. Light-Curve Fitting and Results

In order to detect temporal features of the afterglow light curve, we used 24 parameters to fit multiband light curves. For our analysis, we do not consider the contribution of supernova emission to the X-ray band. We adopt the opening half-angle ${\theta }_j=0.{30}_{-0.12}^{+0.08}\mathrm{rad}$ [31]. We adopt a fiducial value of the optical opacity ($\kappa =0.07{\mathrm{cm}}^2{\mathrm{g}}^{-1}$) (e.g., [67]). The best-fitting parameters are presented in

Table 2. Model parameters.

Parameter	Unit	Best Fit
log $F_{0,\mathrm{X}}$	$\mathrm{erg}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$	$-10.00$ (fit)
log $F_{0,g}$	$\mathrm{erg}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$	$-10.70$ (fit)
log $F_{0,r}$	$\mathrm{erg}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$	$-10.75$ (fit)
log $F_{0,i}$	$\mathrm{erg}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$	$-10.90$ (fit)
log $F_{0,z}$	$\mathrm{erg}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$	$-10.30$ (fit)
${\alpha }_1$		$-1.20$ (fit)
${\alpha }_2$		1.30 (fit)
$t_{\mathrm{b}}$	s	300.00 (fit)
${\theta }_j$	rad	0.30 (fit)
$B_{\mathrm{p}}$	${10}^{15}\mathrm{G}$	$0.{80}_{-0.32}^{+0.34}$
$P_0$	ms	$16.{80}_{-0.47}^{+0.24}$
$\kappa$	${\mathrm{cm}}^2{\mathrm{g}}^{-1}$	0.07 (fit)
log ${\kappa }_{\gamma }$	${\mathrm{cm}}^2{\mathrm{g}}^{-1}$	$-1.{80}_{-0.16}^{+0.59}$
$M_{\mathrm{ej}}$	$M_{\odot }$	$2.{55}_{-0.37}^{+1.12}$
$v_{\mathrm{ej}}$	${10}^9\mathrm{cm}{\mathrm{s}}^{-1}$	$3.{00}_{-0.25}^{+0.48}$
$T_{\mathrm{f}}$	${10}^3\mathrm{K}$	$4.{00}_{-0.32}^{+0.54}$
${\eta }_{\mathrm{X}}$	$\times {10}^{-2}$	$8.{00}_{-0.76}^{+1.12}$
${\eta }_g$	$\times {10}^{-2}$	$0.{80}_{-0.46}^{+0.47}$
${\eta }_r$	$\times {10}^{-2}$	$0.{15}_{-0.13}^{+0.04}$
${\eta }_i$	$\times {10}^{-2}$	$1.{00}_{-0.37}^{+0.39}$
${\eta }_z$	$\times {10}^{-2}$	$1.{80}_{-0.59}^{+0.44}$
$m_g$ a	mag	26.00 (fit)
$m_i$ a	mag	23.20 (fit)
$m_z$ a	mag	22.80 (fit)

^a $m_g$, $m_i$, and $m_z$ are the magnitudes of the host galaxy in the g, i, and z filters, respectively. They are not corrected for Milky Way Galaxy extinction.

, and the results are shown in Figure 1. One can see from the Figure 1 that ${\alpha }_1=-1.20$, ${\alpha }_2=1.30$ and $t_{\mathrm{b}}=300.00\mathrm{s}$ can be constrained by the early data of the g, r, and i bands.

Becuase the data are not corrected for the host galaxy emission and the influence of the host galaxy could not be ignored later in time, we also consider the contribution of the host galaxy in our analysis. We use $r^{\prime }=24.5\mathrm{mag}$ as the contribution for the underlying host galaxy’s light in our analysis, which approximately corresponds to the r band and is measured by [31].

For our best-fitting results, the host galaxy level is roughly constrained to be 26.00, 23.20, and 22.80 mag in the g, i, and z filters, respectively. These have not been corrected for extinction in the Milky Way Galaxy. However, in our analysis, we performed a galactic correction with $E(B-V)=0.229$ mag [48].

The observed X-ray plateau luminosity is $L_{\mathrm{X}}={\eta }_{\mathrm{X}}{L}_{\mathrm{EM}}$ by introducing an efficiency ${\eta }_X$, where ${\eta }_X$ could be defined as [68]

$${\eta }_{\mathrm{X}}\equiv {\displaystyle \frac{{\int }_{0.3\mathrm{keV}}^{10\mathrm{keV}}{L}_{\nu }d\nu }{L_{\mathrm{EM}}}},$$

(14)

where $L_{\mathrm{EM}}$ is the magnetic dipole torque luminosity. Therefor, one can also define

$${\eta }_{\nu }\equiv {\displaystyle \frac{\int L_{\nu }d\nu }{L_{\mathrm{EM}}}}$$

(15)

for other bands. However, the radiation efficiency is quite uncertain, and the bulk saturation Lorentz factor ${\Gamma }_{\mathrm{sat}}<100$ is even smaller, so the X-ray radiation efficiency is possibly closer to 0.1 [68]. The authors of ref. [69] took ${\eta }_{\mathrm{X}}=0.3,0.1,0.01$ for their analysis. Ref. [70] suggested that the efficiency of the conversion of rotational energy from the magnetar into the observed plateau luminosity is ≤$20\%$. We therefore consider our result of ${\eta }_{\mathrm{X}}=8.{00}_{-0.76}^{+1.12}\%$ to be reasonably consistent with theirs. We note that ${\eta }_g$, ${\eta }_r$, ${\eta }_i$, and ${\eta }_z$ are all very small, especially ${\eta }_r$. According to Equation (15), it can be known that this might be because all these bands are narrow bands. In addition, ${\eta }_z>{\eta }_i>{\eta }_r$ might be caused by reddening (high-energy photons are absorbed and transform into lower-energy photons).

The most important physical parameters that we wish to constrain are $P_0=16.{80}_{-0.47}^{+0.24}\mathrm{ms}$ and $B_{\mathrm{p}}=0.{80}_{-0.32}^{+0.34}\times {10}^{15}\mathrm{G}$ of the magnetar. We therefore calculated $\tau \sim 10.20\mathrm{days}$ and $L_0\sim 8.03\times {10}^{43}\mathrm{erg}{\mathrm{s}}^{-1}$.

Table 3. Parameters of GRB-SNe Fitted with a magnetar.

GRB/SN		$\mathit{B}\left({10}^{15}\mathbf{G}\right)$	$\mathit{P}$ (ms)	${\mathit{L}}_0\left(\mathbf{erg}{\mathbf{s}}^{-1}\right)$	$\mathit{\tau }$ (Days)	References
050525A/2005nc	Optical	10.5	18.0	$(1.05\pm 0.05)\times {10}^{46}$	$0.14\pm 0.01$	(1)
050525A/2005nc	X-ray	14.5	8.6	$(3.85\pm 0.18)\times {10}^{47}$	$0.02\pm 0.00$	(1)
091127/2009nz	Optical	5.4	15.2	$(5.31\pm 0.02)\times {10}^{45}$	$0.38\pm 0.00$	(1)
091127/2009nz	X-ray	1.1	2.9	$(1.72\pm 0.12)\times {10}^{47}$	$0.33\pm 0.02$	(1)
111209A/2011kl	Optical	1.1	13.0	$(4.34\pm 0.48)\times {10}^{44}$	$6.51\pm 0.67$	(1)
111209A/2011kl	X-ray	1.3	11.5	$(7.44\pm 1.02)\times {10}^{44}$	$4.82\pm 0.60$	(1)
130831A/2013fu	Optical	12.9	21.3	≈$8.1\times {10}^{45}$	≈$0.13$	(1)
130831A/2013fu	X-ray	7.3	9.2	$(7.62\pm 0.55)\times {10}^{46}$	$0.07\pm 0.00$	(1)
161219B/2016jca	r-band	$2.4\pm 0.3$	$35.2\pm 3.0$	$(3.73\pm 0.26)\times {10}^{43}$	≈$10.25$	(2)
161219B/2016jca	X-ray	$1.0\pm 0.1$	$8.1\pm 0.3$	$(2.25\pm 0.08)\times {10}^{45}$	≈$3.22$	(2)
190829A/2019oyw	X-ray	$60\pm 3$	$1.1\pm 0.1$	$2.93\times {10}^{47}$	≈$0.01$	(3)
201015A/201015A	X-ray/Optical	$0.{80}_{-0.32}^{+0.34}$	$16.{80}_{-0.47}^{+0.24}$	∼$8.03\times {10}^{43}$	∼$10.20$	This work

References. (1) [71], (2) [19], and (3) [72].

compares our results with other GRB-SNe that fitted with a magnetar. We note that GRB 111209A/SN 2011kl, GRB 161219B/SN 2016jca, and GRB 201015A/SN 201015A have large $\tau$ values. This might be the reason why their peak brightness (see

Table 4. Parameters of SNe Associated with GRBs.

GRB/SN (Name)	${\mathit{t}}_{\mathbf{peak}}$1 (Day)	${\mathit{M}}_{\mathbf{peak}}$1 (Mag)	${\mathit{M}}_{\mathbf{ej}}$ (${\mathit{M}}_{\odot }$)	${\mathit{E}}_{\mathbf{SN},\mathbf{K}}$ (erg)	${\mathit{v}}_{\mathbf{ph}}$ $\mathrm{km}{\mathrm{s}}^{-1}$	References
980425/1998bw	∼17	$-18.86\pm 0.2$	$6.8\pm 0.57$	$(1.3\pm 0.1)\times {10}^{52}$	18,000	(1)–(6)
011121/2001ke	$13\pm 1$	$-18.55\pm 0.55$	$4.44\pm 0.82$	$(1.77\pm 0.88)\times {10}^{52}$	–	(1), (6), (7)
021211/2002lt	∼14	$-18.8\pm 0.4$	$7.16\pm 5.99$	$(2.85\pm 1.3)\times {10}^{52}$	–	(1), (6), (8)
030329/2003dh	$11.5\pm 1.5$	$-18.79\pm 0.23$	$5.06\pm 1.65$	$(1.21\pm 0.39)\times {10}^{52}$	20,000	(1), (2), (6), (9), (10)
031203/2003lw	$21.5\pm 3.5$	$-18.92\pm 0.2$	$8.22\pm 0.76$	$(1.59\pm 0.15)\times {10}^{52}$	18,000	(1), (2), (6), (11)
050525A/2005nc	∼12	$-18.8\pm 0.6$	$4.75\pm 1.08$	$(1.89\pm 0.75)\times {10}^{52}$	–	(1), (6), (12), (13)
060218/2006aj	$10\pm 0.5$	$-18.16\pm 0.1$	$2.58\pm 0.55$	$(6.1\pm 0.14)\times {10}^{51}$	20,000	(1), (2), (6), (14)–(16)
080109/2008d	$19\pm 0.8$	$-16.9\pm 0.2$	$5.3\pm 1$	$(6\pm 3)\times {10}^{51}$	–	(1), (6), (17)–(20)
081007/2008hw	$12\pm 3$	$-18.5\pm 0.5$	$2.3\pm 1$	$(9\pm 5)\times {10}^{51}$	12,600	(1), (6), (21), (22)
091127/2009nz	$15\pm 2$	$-18.65\pm 0.2$	$4.69\pm 0.13$	$(8.1\pm 0.2)\times {10}^{51}$	17,000	(1), (6), (22)–(24)
100316D/2010bh	$8.48\pm 1.06$	$-18.45\pm 0.18$	$2.47\pm 0.23$	$(9.2\pm 0.8)\times {10}^{51}$	35,000	(1), (2), (6), (25)
101219B/2010ma	$10\pm 2$	$-18.5\pm 0.25$	$1.3\pm 0.4$	$(1\pm 0.6)\times {10}^{52}$	–	(1), (6), (22), (26)
111209A/2011kl	$14\pm 0.5$	$-19.8\pm 0.1$	$3\pm 1$	$(5.5\pm 3.5)\times {10}^{51}$	21,000	(1), (6), (27)–(28)
120422A/2012bz	$16.69\pm 1.28$	$-18.56\pm 0.15$	$6.1\pm 0.49$	$(1.53\pm 0.13)\times {10}^{52}$	20,500	(1), (6), (29)–(31)
130215A/2013ez	$6.41\pm 0.34$	$-18.85\pm 0.15$	–	–	6,000	(1),(6),(32)
130427A/2013cq	∼$15.2$	$-18.91\pm 0.2$	$6.27\pm 0.69$	$(6.39\pm 0.7)\times {10}^{52}$	35,000	(1), (6), (33)–(36)
130702A/2013dx	$17.2\pm 0.34$	$-18.4\pm 0.4$	$3\pm 0.1$	$(8.2\pm 0.4)\times {10}^{51}$	21,300	(1), (6), (37)–(39)
130831A/2013fu	$18.53\pm 0.07$	$-18.89\pm 0.05$	$6.71\pm 0.2$	$1.{87}_{-0.62}^{+0.9}\times {10}^{52}$	–	(1), (6), (32)
140606B/iPTF14bfu	$16.32\pm 1.63$	$-19.61\pm 0.27$	$5\pm 2$	$2\pm 1\times {10}^{52}$	19,800	(1), (6), (40)
161219B/2016jca	$10.7\pm 0.3$	$-19.04\pm 0.05$	$5.8\pm 0.3$	$(5.1\pm 0.8)\times {10}^{52}$	–	(1), (6), (41), (42)
171205A/2017iuk	∼11	$-18.4\pm 0.1$	$4.9\pm 0.9$	$(2.4\pm 0.9)\times {10}^{52}$	22,000	(43)
180728A/2018fip	$14.7\pm 2.9$	–	–	–	∼30,000	(44), (45)
190829A/2019oyw	$19.19\pm 0.25$	$-19.04\pm 0.1$	$5.67\pm 0.72$	$(13.55\pm 5.08)\times {10}^{51}$	$20,000\pm 2500$	(46)
201015A/201015A	∼$9.63$	∼$-19.96$	$2.{55}_{-0.37}^{+1.12}$	$1.37\times {10}^{52}$	$30,{000}_{-2500}^{+4800}$	This work
221009A/2022xiw	$11.06$	$-19.21$	3.70	$2.35\times {10}^{52}$	32,600	(47)–(49)
230812B/2023pel	$11.6$	–	3.4	–	17,000	(50), (51)

¹ The peak time and peak magnitute in the SN light curve. References. (1) [73], (2) [2], (3) [74], (4) [7], (5) [75], (6) [3], (7) [76], (8) [75], (9) [77], (10) [78], (11) [79], (12) [80], (13) [81], (14) [82], (15) [83], (16) [84], (17) [85], (18) [86], (19) [87], (20) [88], (21) [89], (22) [90], (23) [91], (24) [92], (25) [93], (26) [94], (27) [95], (28) [96], (29) [97], (30) [98], (31) [99], (32) [100], (33) [101], (34) [102], (35) [103], (36) [104], (37) [105], (38) [106], (39) [107], (40) [108], (41) [109], (42) [19], (43) [110], (44) [111], (45) [112], (46) [113], (47) [114], (48) [115], (49) [116], (50) [117], (51) [118].

) is higher. The luminosity of a magnetar-powered SN is directly related to how long the central engine is active, where central engines with longer durations give rise to brighter SNe [71].

Figure 2 shows $\log \tau$ as a function of $\log L$ for magnetars in our sample (Table 3), which includes both optical and X-ray. We use $\log L=a+b\log \tau$ to fit all the data [70,119], where a is a normalization constant and b is the slope. The best-fitting results are $a=51.{12}_{-1.28}^{+1.37}$ and $b=-1.{15}_{-0.29}^{0.27}$. Our results are consistent with the intrinsic slope $-1.{07}_{-0.14}^{+0.09}$ [119]. Ref. [120] proposed that a higher plateau luminosity is associated with a shorter spin-down timescale. In such cases, the collapse time might be closer to the spin-down timescale, meaning that most of the energy is released before the magnetar collapses into a black hole. On the other hand, a lower plateau luminosity corresponds to a longer spin-down timescale, where the collapse time could be much shorter than the spin-down timescale, allowing only a fraction of the total energy to be released before the collapse.

We further constrained the total SN ejecta, the ejecta velocity, and the total SN kinetic energy to be $M_{\mathrm{ej}}=2.{55}_{-0.37}^{+1.12}{M}_{\odot }$, $v_{\mathrm{ej}}={\mathrm{30,000}}_{-2500}^{+4800}\mathrm{km}{\mathrm{s}}^{-1}$, $E_{\mathrm{SN}},\mathrm{K}\approx 1.37\times {10}^{52}\mathrm{erg}$. Our results are consistent with the average value inferred by [3] ($M_{\mathrm{ej}}=5.90\pm 3.80M_{\odot }$, $v_{\mathrm{ej}}=\mathrm{20,200}\pm 8500\mathrm{km}{\mathrm{s}}^{-1}$, $E_{\mathrm{SN}},\mathrm{K}=2.52\pm 1.79\times {10}^{52}\mathrm{erg}$). We estimate the SN bolometric magnitude $M_{\mathrm{bol}},\mathrm{AB}\sim -19.96$ mag at SN peak time $t_{\mathrm{peak}}\sim 9.63\mathrm{days}$. Our results are consistent with the results of $t_{\max }=8.54\pm 1.48\mathrm{days}$ and $M_V=-19.{49}_{-0.47}^{+0.85}$ mag reported by [31]. Table 4 shows the parameters of SNe associated with GRBs, where one can see that the $t_{\mathrm{peak}}$ value of SN 201015A is the smallest in our sample.

The energy partition of GRB/SN events can be denoted as an efficiency of $\eta =E_{\mathrm{GRB}}/(E_{\mathrm{GRB}}+E_{\mathrm{SN}},\mathrm{K})$ (e.g., [73,114]). Ref. [31] reported an isotropic kinetic energy of $E_{\mathrm{k}},\mathrm{iso}=2.0_{-1.35}^{+3.8}\times {10}^{53}\mathrm{erg}$, and the SN kinetic energy is given by $E_{\mathrm{SN}},\mathrm{K}=(3/10)M_{\mathrm{ej}}{v}_{\mathrm{ej}}^2\approx 1.37\times {10}^{52}\mathrm{erg}$. Using this approach, we calculated $\eta \sim 0.39$ for GRB 201015A. However, ref. [73] suggested that $\eta$ is typically less than 0.3, and the center value of $\eta$ is $\approx 0.1$. Figure 3 shows the distribution of $\eta$ for the GRB-SN events. We note that GRB 201015A/SN 201015A is another event with a high $\eta$ value, following GRB 111209A/SN 2011kl ($\eta =(75.87\pm 16.6)\%$).

4. Conclusions and Discussion

We have presented optical observations of the medium-luminosity gamma-ray burst, GRB 201015A. We used a smooth broken power-law plus magnetar model to fit the X-ray and optical light curves. For our best fitting, the magnetar initial spin period and surface magnetic field at the pole are constrained to be $P_0=16.{80}_{-0.47}^{+0.24}\mathrm{ms}$ and $B_{\mathrm{p}}=0.{80}_{-0.32}^{+0.34}\times {10}^{15}\mathrm{G}$, respectively. The best-fitting results reveal that the SN ejected a total mass of $M_{\mathrm{ej}}=2.{55}_{-0.37}^{+1.12}{M}_{\odot }$, an ejecta velocity of $v_{\mathrm{ej}}={\mathrm{30,000}}_{-2500}^{+4800}\mathrm{km}{\mathrm{s}}^{-1}$, and a total kinetic energy of $E_{\mathrm{SN}},\mathrm{K}\approx 1.37\times {10}^{52}\mathrm{erg}$. We estimate the energy partition to be $\eta \sim 0.39$.

Furthermore, previous works have also analyzed the light curves. Ref. [30] discussed that the shallow decay phase of the X-ray light curve can be explained by the process of continuous energy injection from a central engine (e.g., a magnetar), even though it is difficult to explain the origin of this central engine activity. After comparing to the known light curves of GRB–SN cases (GRB 980425/SN 1998bw, GRB 060218A/SN 2006aj, GRB 130702A/SN 2013dx, GRB 161219B/SN 2016jca, and GRB 171205A/SN 2017iuk), ref. [31] suggested that the light curves of the SN 2016jca and the SN 2006aj cases have the most similarity with the light curve of the SN 201015A case. GRB 161219B is an ILGRB, and SN 2016jca is likely powered in part, or perhaps exclusively, by radioactive ⁵⁶Ni decay [19]; but one possibility is that a magnetar plays an important role in the kinetic energy of a supernova and has a mass of ⁵⁶Ni [109].

Ref. [121] suggested that GRB 060218/ SN 2006aj was produced by the core collapse of a star with an initial mass of ∼$20M_{\odot }$, in which case a neutron star, rather than a black hole, is formed. In addition, Zhang et al. [122] discussed the effects of a magnetar engine on GRB 060218/SN 2006aj, demonstrating that the primary SN emission is also partially powered by energy injection from the magnetar.

Therefore, we think that the central engine of GRB 201015A/SN 201015A may be a magnetar. In addition, we found that the emission from a magnetar central engine can be solely responsible for powering SN 201015A, which is similarly seen in SN 2011kl [71].

For SN 2011kl, ref. [123] reported that $M_U=-20.39\pm 0.06$ mag, $M_B=-19.65\pm 0.07$ mag, $M_V=-19.80\pm 0.10$ mag, and $M_{R_C}=-20.23\pm 0.09$ mag. Moreover, ref. [95] reported that $t_{\mathrm{peak}}\sim 14$ days and a bolometric magnitude $M_{\mathrm{bol}}\sim -20$ mag.

We estimate the absolute AB mag of the SN peak to be $M_g\sim -19.26$ mag, $M_r\sim -19.70$ mag, $M_i\sim -19.86$ mag, and $M_z\sim -19.92$ mag for SN 201015A. We note that their peak brightness is similar.

We found that SN 201015A and SN2011kl have some similar features: Their peak brightness is similar. They have an odd $\eta$ (particularly large, see Figure 3) value compared to other events, and they can be completely powered by magnetars. We think that SN 201015A and SN 2011kl may have a similar explosion mechanism.

With future missions such as SVOM and the Einstein Probe searching lower energy gamma rays and X-rays, we will find more bursts like GRB 201015A, which will help us establish a larger dataset for studying the explosion mechanism and the energy source of GRB–SN events.