Magnetar Engines in Fast Blue Optical Transients and Their Connections with SLSNe, SNe Ic-BL, and lGRBs

The criteria for our sample selection are as follows: (1) reported rise time above half-maximum t_rise ≲ 10 days, (2) spectroscopic redshift measurement from the host galaxy spectral features, (3) published lightcurves observed in at least two filters, and (4) at least some data available close to the peak.

Following the above criteria, we collect 40 FBOTs from the literature: 7 events from the PS1-MDS (Drout et al. 2014; Inserra 2019); 25 events reported in the Dark Energy Survey Supernova Program (DES-SN; Pursiainen et al. 2018); 3 events collected from the Supernova Legacy Survey (SNLS; Arcavi et al. 2016); 2 events discovered by the Hyper Suprime-Cam SSP (HSCSSP) Transient Survey (Tampo et al. 2020); and 3 well-studied samples, i.e., PTF 10iam, AT 2018cow, and Koala, discovered by the Palomar Transient Factory (Arcavi et al. 2016), the Asteroid Terrestrial-impact Last Alert System (ATLAS) Survey (Prentice et al. 2018; Perley et al. 2019), and the Zwicky Transient Facility One-day Cadence (ZTF-1DC) Survey (Ho et al. 2020), respectively.

Our FBOT sample contains 22 robust cases whose rises were recorded in survey projects. For these events, sufficient data on the rise and decline phases of the lightcurve pose a strict constraint on the model parameters, especially on the ejecta mass. The sample also includes 18 events without any detection during the rise phase of the lightcurve. Due to the lack of observational epochs before the peak, the model parameters are related to the rise time we set.

As usual, the spin-down luminosity of a magnetar can be generally expressed according to the luminosity of the magnetic dipole radiation as

$$\begin{eqnarray}&&{L}_{\mathrm{sd}}(t)={L}_{\mathrm{sd},{\rm{i}}}{\left(1+\displaystyle \frac{t}{{t}_{\mathrm{sd}}}\right)}^{-2},\end{eqnarray} \tag{ 1 }$$

where ${L}_{\mathrm{sd},{\rm{i}}}={10}^{47}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{P}_{{\rm{i}},-3}^{-4}{B}_{{\rm{p}},14}^{2}$ is the initial value of the luminosity, ${t}_{\mathrm{sd}}\simeq 2\times {10}^{5}\,{\rm{s}}\,{P}_{{\rm{i}},-3}^{2}{B}_{{\rm{p}},14}^{-2}$ is the spin-down timescale, and P_i and B_p are the initial spin period and polar magnetic strength of the magnetar, respectively. The total rotational energy of the magnetar can be written as ${E}_{\mathrm{rot}}={L}_{\mathrm{sd},{\rm{i}}}{t}_{\mathrm{sd}}=2\times {10}^{52}\,\mathrm{erg}\,{P}_{{\rm{i}},-3}^{-2}$. Here the conventional notation Q_x = Q/10^x is adopted in cgs units.

We adopt the common analytic solution derived by Arnett (1982) to calculate the bolometric luminosity of an FBOT powered by a magnetar:

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{rad}}(t) & = & {e}^{-{\left(t/{t}_{\mathrm{diff}}\right)}^{2}}(1-{e}^{-{{At}}^{-2}})\\ & & \times \,{\displaystyle \int }_{0}^{t}2{L}_{\mathrm{sd}}(t^{\prime} )\displaystyle \frac{t^{\prime} }{{t}_{\mathrm{diff}}}{e}^{{\left(t^{\prime} /{t}_{\mathrm{diff}}\right)}^{2}}\displaystyle \frac{{dt}^{\prime} }{{t}_{\mathrm{diff}}},\end{array}\end{eqnarray} \tag{ 2 }$$

where t_diff is the photon diffusion timescale of the FBOT ejecta and A is the leakage parameter. For an ejecta of a mass M_ej and velocity v_ej, the diffusion timescale is given by ${t}_{\mathrm{diff}}={(3\kappa {M}_{\mathrm{ej}}/4\pi {v}_{\mathrm{ej}}c)}^{1/2}$, where κ is the optical opacity. Here the dynamical evolution of the ejecta is ignored. The kinetic energy of the ejecta is assumed to be directly determined by the rotational energy of the magnetar so that the ejecta velocity can be estimated as ${v}_{\mathrm{ej}}\simeq \sqrt{2{E}_{\mathrm{rot}}/{M}_{\mathrm{ej}}}$. This assumption is viable as long as t_sd ≲ t_diff and the initial value of the kinetic energy is not much higher than 10⁵⁰ erg. ⁶ By considering that the energy injected into the ejecta could be in the form of high-energy photons, we write the leakage parameter of the ejecta as $A=3{\kappa }_{\gamma }{M}_{\mathrm{ej}}/4\pi {v}_{\mathrm{ej}}^{2}$, where κ_γ is the opacity for gamma rays. The frequency dependence of κ_γ is ignored for simplicity.

Finally, in order to calculate the monochromatic luminosity of the FBOT emission, we define the photosphere temperature as

$$\begin{eqnarray}&&{T}_{\mathrm{ph}}(t)=\max \left[{\left(\displaystyle \frac{{L}_{\mathrm{rad}}}{4\pi {\sigma }_{\mathrm{SB}}{v}_{\mathrm{ph}}^{2}{t}^{2}}\right)}^{1/4},{T}_{\mathrm{floor}}\right]\end{eqnarray} \tag{ 3 }$$

with the Stefan–Boltzmann constant σ_SB, floor temperature T_floor, and photospheric velocity v_ph ≃ v_ej (which is a standard approximation in the literature).

Adopting a Markov Chain Monte Carlo method, we use the magnetar engine model described in Section 2.2 with the emcee package (Foreman-Mackey et al. 2013) to fit the multiband lightcurves of the collected FBOTs. For the Milky Way extinction, we take values from the dust maps of Schlafly & Finkbeiner (2011) and fix R_V = 3.1. Because the extinction of the host galaxy is unknown, we set A_V as a free parameter, with a uniform distribution prior between 0 and 0.5 mag. There are eight free parameters: ejecta mass M_ej, initial spin period P_i, magnetic field strength B_p, opacity κ, opacity to high-energy photons κ_γ , floor temperature T_floor, host extinction A_V , and time of explosion relative to the first observed data point t_shift. Ho et al. (2021) recently reported 22 FBOTs with spectroscopic observations, most of which were classified as Type Ib/Ibn/IIb SNe or hybrid IIn/Ibn SNe. Furthermore, a fraction of FBOTs were found to be Type Ic SNe (e.g., Drout et al. 2013; De et al. 2018). Although a major fraction of FBOTs lack spectroscopic classifications, we assume that these collected FBOTs could plausibly contain a large amount of helium, carbon, or oxygen. Thus, the prior of κ for FBOTs is preferably set in the range of 0.05–0.2 cm² g⁻¹, which is suitable for scattering in ionized helium, carbon, or oxygen. For those events without any detection before the peak, we note that the upper limit of the prior for t_shift is defined as the time between the pre-explosion nondetection and the first detection. The priors of these fitting parameters are listed in Table 1.

For each lightcurve fitting, we run the code in parallel using 12 nodes with at least 10,000 iterations where the first 100 iterations are used to burn in the ensemble. We list the fitting results of the derived model parameters in Table 2, while the detailed fittings to the multiband lightcurves for each event are shown in the Appendix. Generally, the FBOT lightcurves can be well fitted by the magnetar engine model with high quality.

Table 2. The Lightcurve Properties and the Fitting Results of the Derived Parameters for the Collected FBOTs

FBOT	Mpeak	trise	tdecl	${\mathrm{log}}_{10}{M}_{\mathrm{ej},\odot }$	${\mathrm{log}}_{10}{P}_{{\rm{i}},-3}$	${\mathrm{log}}_{10}{B}_{{\rm{p}},14}$	${\mathrm{log}}_{10}{\kappa }_{\gamma }$	κ	Tfloor,3	AV
AT 2018cow	$-{22.42}_{-0.04}^{+0.03}$	${0.80}_{-0.00}^{+0.04}$	${1.68}_{-0.04}^{+0.04}$	$-{1.78}_{-0.01}^{+0.02}$	${0.82}_{-0.01}^{+0.00}$	${0.84}_{-0.01}^{+0.01}$	${1.83}_{-0.14}^{+0.12}$	${0.19}_{-0.01}^{+0.01}$	${24.58}_{-0.16}^{+0.27}$	${0.50}_{-0.00}^{+0.00}$
DES 13C3bcok	$-{19.53}_{-0.13}^{+0.14}$	${3.01}_{-0.15}^{+0.30}$	${4.96}_{-0.30}^{+0.45}$	$-{0.64}_{-0.14}^{+0.13}$	${1.06}_{-0.07}^{+0.06}$	${1.19}_{-0.08}^{+0.07}$	$-{0.7}_{-1.2}^{+1.8}$	${0.06}_{-0.01}^{+0.02}$	${10.5}_{-1.1}^{+2.0}$	${0.03}_{-0.02}^{+0.04}$
DES 13C3uig*	$-{19.45}_{-0.37}^{+0.33}$	${1.65}_{-0.45}^{+0.30}$	${3.31}_{-0.75}^{+0.60}$	$-{1.46}_{-0.26}^{+0.25}$	${1.27}_{-0.07}^{+0.05}$	${1.19}_{-0.06}^{+0.06}$	${1.07}_{-0.74}^{+0.64}$	${0.10}_{-0.04}^{+0.06}$	${7.9}_{-1.3}^{+1.2}$	${0.15}_{-0.11}^{+0.17}$
DES 13X1hav*	$-{19.62}_{-0.36}^{+0.41}$	${2.0}_{-0.7}^{+5.2}$	${9.5}_{-3.5}^{+5.3}$	$-{1.69}_{-0.33}^{+0.38}$	${1.04}_{-0.17}^{+0.07}$	${0.56}_{-0.50}^{+0.20}$	${1.22}_{-0.78}^{+0.54}$	${0.13}_{-0.05}^{+0.05}$	${12.2}_{-1.4}^{+1.7}$	${0.29}_{-0.18}^{+0.15}$
DES 13X3gms	$-{19.90}_{-0.26}^{+0.23}$	${4.36}_{-0.45}^{+0.60}$	${7.21}_{-0.75}^{+0.90}$	$-{0.49}_{-0.16}^{+0.16}$	${0.95}_{-0.06}^{+0.05}$	${0.87}_{-0.03}^{+0.03}$	${0.5}_{-1.1}^{+1.0}$	${0.08}_{-0.02}^{+0.04}$	${6.1}_{-2.1}^{+1.9}$	${0.14}_{-0.09}^{+0.11}$
DES 13X3npb	$-{19.08}_{-0.29}^{+0.26}$	${4.21}_{-0.45}^{+0.45}$	${6.91}_{-0.75}^{+0.90}$	$-{0.63}_{-0.20}^{+0.24}$	${1.05}_{-0.11}^{+0.09}$	${1.17}_{-0.12}^{+0.07}$	${0.1}_{-1.2}^{+1.3}$	${0.12}_{-0.05}^{+0.05}$	${6.5}_{-2.4}^{+3.1}$	${0.23}_{-0.15}^{+0.17}$
DES 13X3nyg*	$-{21.11}_{-0.34}^{+0.35}$	${1.65}_{-0.30}^{+0.45}$	${3.01}_{-0.45}^{+0.60}$	$-{1.14}_{-0.22}^{+0.25}$	${0.93}_{-0.09}^{+0.06}$	${0.95}_{-0.04}^{+0.04}$	${0.82}_{-0.86}^{+0.78}$	${0.10}_{-0.03}^{+0.06}$	${10.7}_{-4.0}^{+1.5}$	${0.19}_{-0.13}^{+0.14}$
DES 14C3tvw	$-{19.65}_{-0.30}^{+0.27}$	${3.46}_{-0.75}^{+0.90}$	${6.0}_{-1.1}^{+1.5}$	$-{0.77}_{-0.24}^{+0.23}$	${1.07}_{-0.06}^{+0.05}$	${0.95}_{-0.06}^{+0.05}$	${0.5}_{-1.0}^{+1.0}$	${0.08}_{-0.02}^{+0.05}$	${8.7}_{-3.8}^{+2.4}$	${0.10}_{-0.07}^{+0.13}$
DES 14S2anq*	$-{15.27}_{-0.06}^{+0.04}$	${5.71}_{-0.15}^{+0.15}$	${13.37}_{-0.30}^{+0.45}$	$-{0.16}_{-0.11}^{+0.20}$	${0.94}_{-0.10}^{+0.07}$	${2.03}_{-0.02}^{+0.02}$	${0.4}_{-1.2}^{+1.1}$	${0.15}_{-0.05}^{+0.04}$	${5.63}_{-0.24}^{+0.29}$	${0.03}_{-0.02}^{+0.05}$
DES 14S2plb	$-{15.94}_{-0.54}^{+0.83}$	${3.76}_{-0.60}^{+0.60}$	${7.8}_{-2.1}^{+3.0}$	$-{0.72}_{-0.42}^{+0.68}$	${1.39}_{-0.74}^{+0.24}$	${1.95}_{-0.13}^{+0.28}$	${0.2}_{-1.2}^{+1.3}$	${0.13}_{-0.05}^{+0.05}$	${5.7}_{-1.3}^{+1.5}$	${0.35}_{-0.22}^{+0.12}$
DES 14X1bnh*	$-{20.70}_{-0.49}^{+0.44}$	${2.7}_{-0.8}^{+1.1}$	${6.8}_{-2.1}^{+5.4}$	$-{0.99}_{-0.30}^{+0.31}$	${0.85}_{-0.11}^{+0.06}$	${0.61}_{-0.32}^{+0.15}$	${0.85}_{-0.96}^{+0.81}$	${0.11}_{-0.04}^{+0.06}$	${8.6}_{-3.7}^{+5.5}$	${0.25}_{-0.17}^{+0.16}$
DES 15C3lpq	$-{20.08}_{-0.14}^{+0.15}$	${2.85}_{-0.30}^{+0.45}$	${4.96}_{-0.60}^{+0.60}$	$-{0.72}_{-0.13}^{+0.12}$	${1.02}_{-0.03}^{+0.03}$	${0.97}_{-0.02}^{+0.02}$	${0.66}_{-0.83}^{+0.89}$	${0.06}_{-0.01}^{+0.01}$	${7.66}_{-0.76}^{+0.80}$	${0.03}_{-0.02}^{+0.04}$
DES 15C3mgq*	$-{18.20}_{-0.36}^{+0.29}$	${1.35}_{-0.45}^{+0.75}$	${2.55}_{-0.60}^{+0.90}$	$-{1.74}_{-0.35}^{+0.35}$	${1.53}_{-0.10}^{+0.11}$	${1.61}_{-0.02}^{+0.02}$	${1.07}_{-0.75}^{+0.63}$	${0.14}_{-0.05}^{+0.04}$	${5.23}_{-0.70}^{+0.94}$	${0.39}_{-0.17}^{+0.08}$
DES 15C3nat*	$-{19.79}_{-0.39}^{+0.35}$	${2.25}_{-0.60}^{+0.60}$	${3.9}_{-0.9}^{+1.1}$	$-{0.90}_{-0.32}^{+0.34}$	${1.01}_{-0.21}^{+0.14}$	${1.29}_{-0.16}^{+0.16}$	${0.3}_{-1.4}^{+1.2}$	${0.09}_{-0.03}^{+0.06}$	${6.1}_{-2.2}^{+2.6}$	${0.12}_{-0.09}^{+0.15}$
DES 15C3opk	$-{20.79}_{-0.35}^{+0.25}$	${3.01}_{-0.45}^{+0.60}$	${5.1}_{-0.6}^{+1.2}$	$-{0.80}_{-0.17}^{+0.20}$	${0.85}_{-0.07}^{+0.06}$	${0.79}_{-0.09}^{+0.07}$	$-{0.4}_{-0.5}^{+1.4}$	${0.12}_{-0.05}^{+0.05}$	${17.5}_{-1.3}^{+1.3}$	${0.30}_{-0.11}^{+0.10}$
DES 15C3opp*	$-{18.36}_{-0.39}^{+0.33}$	${2.10}_{-0.60}^{+0.75}$	${3.8}_{-1.1}^{+1.1}$	$-{1.27}_{-0.30}^{+0.30}$	${1.41}_{-0.09}^{+0.07}$	${1.49}_{-0.08}^{+0.08}$	${0.4}_{-1.1}^{+1.1}$	${0.08}_{-0.02}^{+0.05}$	${7.1}_{-2.7}^{+2.0}$	${0.09}_{-0.06}^{+0.13}$
DES 15E2nqh*	$-{20.29}_{-0.30}^{+0.28}$	${2.70}_{-0.75}^{+0.60}$	${4.66}_{-0.90}^{+0.90}$	$-{0.88}_{-0.26}^{+0.23}$	${1.00}_{-0.07}^{+0.06}$	${0.95}_{-0.05}^{+0.04}$	${0.68}_{-0.96}^{+0.91}$	${0.08}_{-0.02}^{+0.05}$	${8.3}_{-2.4}^{+1.2}$	${0.09}_{-0.07}^{+0.12}$
DES 15S1fli*	$-{20.06}_{-0.39}^{+0.29}$	${2.25}_{-0.60}^{+0.60}$	${5.9}_{-1.5}^{+2.4}$	$-{1.23}_{-0.27}^{+0.24}$	${1.04}_{-0.07}^{+0.04}$	${0.78}_{-0.15}^{+0.09}$	${1.23}_{-0.68}^{+0.53}$	${0.12}_{-0.05}^{+0.05}$	${11.0}_{-2.7}^{+2.3}$	${0.17}_{-0.12}^{+0.18}$
DES 15S1fll*	$-{18.86}_{-0.45}^{+0.37}$	${2.9}_{-0.9}^{+1.1}$	${4.8}_{-1.4}^{+1.8}$	$-{1.01}_{-0.28}^{+0.32}$	${1.24}_{-0.10}^{+0.08}$	${1.29}_{-0.07}^{+0.05}$	${0.5}_{-1.0}^{+1.1}$	${0.10}_{-0.03}^{+0.05}$	${7.1}_{-2.5}^{+2.7}$	${0.09}_{-0.06}^{+0.10}$
DES 15X3mxf	$-{20.61}_{-0.18}^{+0.22}$	${2.25}_{-0.15}^{+0.30}$	${4.06}_{-0.45}^{+0.45}$	$-{0.53}_{-0.11}^{+0.09}$	${0.74}_{-0.04}^{+0.05}$	${1.24}_{-0.02}^{+0.02}$	${0.69}_{-0.93}^{+0.90}$	${0.06}_{-0.01}^{+0.03}$	${11.29}_{-0.66}^{+0.81}$	${0.03}_{-0.02}^{+0.05}$
DES 16C1cbd	$-{19.96}_{-0.32}^{+0.31}$	${2.85}_{-0.60}^{+0.60}$	${5.6}_{-1.1}^{+1.1}$	$-{1.02}_{-0.21}^{+0.20}$	${1.04}_{-0.06}^{+0.06}$	${0.87}_{-0.06}^{+0.06}$	${0.70}_{-0.93}^{+0.89}$	${0.12}_{-0.05}^{+0.05}$	${7.2}_{-2.9}^{+2.4}$	${0.34}_{-0.16}^{+0.11}$
DES 16C3gin	$-{19.81}_{-0.25}^{+0.26}$	${2.70}_{-0.30}^{+0.15}$	${4.36}_{-0.30}^{+0.30}$	$-{0.86}_{-0.15}^{+0.16}$	${1.07}_{-0.06}^{+0.06}$	${1.15}_{-0.01}^{+0.01}$	${0.73}_{-0.86}^{+0.87}$	${0.08}_{-0.02}^{+0.04}$	${5.5}_{-1.7}^{+0.9}$	${0.11}_{-0.07}^{+0.07}$
DES 16E1bir*	$-{22.47}_{-0.29}^{+0.15}$	${3.16}_{-0.45}^{+0.45}$	${5.86}_{-0.75}^{+0.75}$	$-{0.05}_{-0.31}^{+0.26}$	${0.16}_{-0.17}^{+0.21}$	${0.75}_{-0.18}^{+0.11}$	${0.4}_{-1.1}^{+1.1}$	${0.11}_{-0.04}^{+0.05}$	${9.8}_{-4.8}^{+4.7}$	${0.10}_{-0.07}^{+0.14}$
DES 16E2pv*	$-{20.27}_{-0.42}^{+0.34}$	${1.80}_{-0.60}^{+0.63}$	${4.4}_{-1.2}^{+3.0}$	$-{1.36}_{-0.35}^{+0.40}$	${1.02}_{-0.13}^{+0.06}$	${0.88}_{-0.30}^{+0.19}$	${0.7}_{-1.1}^{+0.9}$	${0.11}_{-0.05}^{+0.06}$	${16.7}_{-5.6}^{+2.3}$	${0.18}_{-0.12}^{+0.17}$
DES 16X3cxn*	$-{20.18}_{-0.42}^{+0.34}$	${2.25}_{-0.45}^{+0.60}$	${4.21}_{-0.75}^{+0.90}$	$-{1.09}_{-0.20}^{+0.24}$	${1.05}_{-0.09}^{+0.06}$	${0.96}_{-0.03}^{+0.03}$	${0.69}_{-0.78}^{+0.88}$	${0.10}_{-0.04}^{+0.06}$	${5.4}_{-1.6}^{+2.3}$	${0.24}_{-0.15}^{+0.15}$
DES 16X3ega	$-{19.96}_{-0.03}^{+0.04}$	${3.76}_{-0.15}^{+0.00}$	${6.16}_{-0.15}^{+0.00}$	$-{0.38}_{-0.03}^{+0.03}$	${0.93}_{-0.01}^{+0.01}$	${1.03}_{-0.01}^{+0.01}$	$-{1.20}_{-0.05}^{+0.06}$	${0.05}_{-0.00}^{+0.00}$	${3.45}_{-0.30}^{+0.32}$	${0.16}_{-0.03}^{+0.03}$
HSC 17bhyl*	$-{20.57}_{-0.28}^{+0.21}$	${0.90}_{-0.15}^{+0.30}$	${1.80}_{-0.30}^{+0.45}$	$-{1.89}_{-0.16}^{+0.27}$	${1.27}_{-0.08}^{+0.04}$	${1.30}_{-0.02}^{+0.02}$	${1.57}_{-0.40}^{+0.30}$	${0.14}_{-0.04}^{+0.06}$	${8.13}_{-0.67}^{+0.60}$	${0.08}_{-0.06}^{+0.15}$
HSC 17btum*	$-{18.43}_{-0.27}^{+0.21}$	${4.36}_{-0.30}^{+0.30}$	${8.11}_{-0.75}^{+0.75}$	$-{0.33}_{-0.24}^{+0.22}$	${0.91}_{-0.14}^{+0.15}$	${1.42}_{-0.10}^{+0.08}$	$-{0.6}_{-1.1}^{+1.7}$	${0.10}_{-0.04}^{+0.06}$	${10.6}_{-0.8}^{+1.0}$	${0.10}_{-0.06}^{+0.09}$
Koala	$-{22.85}_{-0.35}^{+0.32}$	${0.90}_{-0.15}^{+0.30}$	${1.65}_{-0.45}^{+0.30}$	$-{1.10}_{-0.25}^{+0.23}$	${0.60}_{-0.08}^{+0.09}$	${1.10}_{-0.07}^{+0.06}$	${0.99}_{-0.73}^{+0.69}$	${0.09}_{-0.03}^{+0.05}$	${15.7}_{-1.6}^{+1.7}$	${0.04}_{-0.03}^{+0.06}$
PS1-10bjp	$-{20.30}_{-0.04}^{+0.04}$	${1.51}_{-0.10}^{+0.00}$	${2.51}_{-0.10}^{+0.00}$	$-{0.88}_{-0.04}^{+0.03}$	${0.95}_{-0.01}^{+0.02}$	${1.47}_{-0.01}^{+0.01}$	${1.03}_{-0.65}^{+0.65}$	${0.05}_{-0.00}^{+0.00}$	${5.44}_{-0.21}^{+0.22}$	${0.48}_{-0.02}^{+0.01}$
PS1-11qr	$-{20.24}_{-0.32}^{+0.26}$	${3.16}_{-0.45}^{+0.60}$	${5.56}_{-0.60}^{+0.75}$	$-{0.81}_{-0.23}^{+0.19}$	${0.81}_{-0.10}^{+0.08}$	${0.31}_{-0.19}^{+0.20}$	$-{1.44}_{-0.38}^{+0.51}$	${0.13}_{-0.05}^{+0.05}$	${13.8}_{-1.9}^{+2.9}$	${0.19}_{-0.11}^{+0.12}$
PS1-12bb*	$-{17.39}_{-0.16}^{+0.19}$	${0.64}_{-0.60}^{+0.24}$	${10.02}_{-0.85}^{+0.85}$	$-{2.71}_{-0.16}^{+0.23}$	${1.61}_{-0.04}^{+0.05}$	${1.11}_{-0.04}^{+0.06}$	${1.61}_{-0.42}^{+0.28}$	${0.13}_{-0.05}^{+0.05}$	${9.56}_{-0.83}^{+0.59}$	${0.35}_{-0.19}^{+0.11}$
PS1-12brf	$-{19.56}_{-0.38}^{+0.33}$	${1.80}_{-0.45}^{+0.45}$	${3.16}_{-0.75}^{+0.60}$	$-{1.27}_{-0.28}^{+0.26}$	${1.21}_{-0.10}^{+0.08}$	${1.31}_{-0.04}^{+0.04}$	${1.11}_{-0.71}^{+0.62}$	${0.11}_{-0.04}^{+0.06}$	${6.4}_{-0.9}^{+1.2}$	${0.11}_{-0.07}^{+0.11}$
PS1-12bv	$-{20.65}_{-0.39}^{+0.32}$	${2.85}_{-0.60}^{+0.60}$	${4.8}_{-0.9}^{+1.2}$	$-{0.76}_{-0.25}^{+0.20}$	${0.88}_{-0.08}^{+0.08}$	${0.91}_{-0.10}^{+0.06}$	${0.3}_{-0.9}^{+1.2}$	${0.09}_{-0.03}^{+0.06}$	${14.3}_{-1.0}^{+1.4}$	${0.18}_{-0.10}^{+0.10}$
PS1-13duy	$-{20.80}_{-0.36}^{+0.42}$	${1.95}_{-0.30}^{+0.30}$	${3.31}_{-0.60}^{+0.90}$	$-{0.81}_{-0.33}^{+0.31}$	${0.74}_{-0.18}^{+0.16}$	${1.24}_{-0.34}^{+0.14}$	$-{0.6}_{-0.9}^{+1.6}$	${0.13}_{-0.05}^{+0.05}$	${6.4}_{-2.3}^{+2.4}$	${0.26}_{-0.13}^{+0.12}$
PS1-13dwm*	$-{18.42}_{-0.56}^{+0.44}$	${1.80}_{-0.60}^{+0.60}$	${3.2}_{-1.1}^{+1.1}$	$-{1.17}_{-0.40}^{+0.32}$	${1.26}_{-0.19}^{+0.15}$	${1.73}_{-0.16}^{+0.14}$	$-{0.1}_{-1.4}^{+1.4}$	${0.11}_{-0.04}^{+0.06}$	${6.7}_{-2.5}^{+2.3}$	${0.17}_{-0.11}^{+0.16}$
PTF 10iam	$-{20.19}_{-0.13}^{+0.14}$	${2.1}_{-0.4}^{+1.7}$	${29.5}_{-2.1}^{+2.6}$	$-{1.54}_{-0.09}^{+0.15}$	${0.83}_{-0.03}^{+0.03}$	${0.10}_{-0.03}^{+0.03}$	${1.83}_{-0.26}^{+0.13}$	${0.13}_{-0.05}^{+0.05}$	${8.95}_{-0.55}^{+0.71}$	${0.40}_{-0.13}^{+0.07}$
SNLS 04D4ec	$-{19.64}_{-0.52}^{+0.45}$	${4.0}_{-2.4}^{+3.4}$	${6.8}_{-0.8}^{+4.4}$	$-{1.92}_{-0.33}^{+0.43}$	${0.48}_{-0.24}^{+0.21}$	$-{0.71}_{-0.48}^{+0.43}$	${1.40}_{-0.44}^{+0.37}$	${0.12}_{-0.05}^{+0.05}$	${8.25}_{-0.13}^{+0.20}$	${0.05}_{-0.04}^{+0.08}$
SNLS 05D2bk	$-{19.48}_{-0.10}^{+0.05}$	${5.02}_{-0.25}^{+0.25}$	${12.54}_{-0.50}^{+0.75}$	$-{0.62}_{-0.21}^{+0.19}$	${0.45}_{-0.09}^{+0.10}$	$-{0.71}_{-0.19}^{+0.20}$	$-{0.65}_{-0.22}^{+0.25}$	${0.11}_{-0.04}^{+0.06}$	${7.35}_{-0.19}^{+0.22}$	${0.07}_{-0.05}^{+0.10}$
SNLS 06D1hc	$-{19.23}_{-0.07}^{+0.03}$	${7.52}_{-0.25}^{+0.25}$	${14.54}_{-0.25}^{+0.50}$	$-{0.02}_{-0.14}^{+0.12}$	${0.44}_{-0.07}^{+0.07}$	$-{0.68}_{-0.13}^{+0.14}$	$-{1.77}_{-0.15}^{+0.18}$	${0.11}_{-0.03}^{+0.04}$	${6.48}_{-0.09}^{+0.11}$	${0.04}_{-0.03}^{+0.06}$
Median	$-{19.9}_{-1.0}^{+1.3}$	${2.5}_{-1.2}^{+1.5}$	${5.0}_{-2.5}^{+3.6}$	$-{0.95}_{-0.64}^{+0.48}$	${0.96}_{-0.28}^{+0.30}$	${1.03}_{-0.50}^{+0.42}$	${0.6}_{-1.0}^{+0.7}$	${0.103}_{-0.033}^{+0.030}$	${7.9}_{-2.7}^{+4.8}$	${0.15}_{-0.11}^{+0.18}$

Note. An asterisk (∗) marks an FBOT event that lacked a detection epoch before the peak. The values on the last line represent the median with 1σ deviation of each parameter for the collected FBOTs. The columns from left to right are (1) the FBOT event, (2) the peak absolute magnitude in units of mag, (3) the rise time above half-maximum in units of days, (4) the decline time above half-maximum in units of days, (5) the ejecta mass in units of M_⊙, (6) the initial spin period in units of ms, (7) the polar magnetic field strength in units of 10¹⁴ G, (8) the opacity to high-energy photons in units of cm² g⁻¹, (9) the opacity in units of cm² g⁻¹, (10) the floor temperature in units of 10³ K, and (11) the extinction in units of mag.

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The most important outputs of the lightcurve modeling are the ejecta mass M_ej, initial spin period P_i, and magnetic field strength B_p. We plot M_ej versus P_i and P_i versus B_p in Figures 1 and 2, respectively. The ejecta masses for most of the FBOTs we collect are in the range of ∼0.002–1 M_⊙ with a median value of ∼0.11 M_⊙. For the magnetars, the initial spin periods are centered at $\sim {9.1}_{-4.4}^{+9.3}\,\mathrm{ms}$. The magnetic field strengths, of which the median value is ∼25B_c, have a wide distribution mostly between ∼B_c and ∼200B_c. Here, ${B}_{{\rm{c}}}={m}_{{\rm{e}}}^{2}{c}^{3}/(q\hslash )=4.4\,\times \,{10}^{13}\,{\rm{G}}$ represents the Landau critical magnetic field defined by the electron mass m_e, electron charge q, and reduced Planck constant ℏ.

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Corresponding to our sample selection criterion of t_rise ≲ 10 days, the upper limit of the ejecta masses of the FBOTs can be set to around 1 M_⊙, which hints that FBOTs could have the following types of origins with the formation of a rapidly rotating magnetar: (I) BNS mergers that produce massive NS remnants (i.e., the mergernova model; Yu et al. 2013); (II) mergers of an NS and a WD (Zenati et al. 2019); (III) AIC of WDs, including both single- and double-degenerate cases ⁷ (Yu et al. 2019a, 2019b); and (IV) SN explosions of ultrastripped stars (i.e., ultrastripped SNe; e.g., Tauris et al. 2015; Hotokezaka et al. 2017; Sawada et al. 2022). For Case I, since the derived masses here are generally higher than the masses that can be produced by BNS mergers (e.g., Radice et al. 2018), the mergernova model could be ruled out for most FBOTs. Nevertheless, the model could account for some special sources, such as PS1-12bb, which have the fastest evolution and relatively low luminosities that are consistent with the prediction of the mergernova model. Furthermore, if a larger opacity is taken into account, which can be caused by lanthanides, then more samples could be classified as mergernova candidates as their ejecta masses become ≲0.01 M_⊙. In any case, the relatively low event rate of BNS mergers definitely makes them only account for a very small faction of the observed FBOTs. Alternatively, in comparison, the relatively wide range of ejecta masses of the FBOTs most favors the ultrastripped SN model in close binaries, although it is unclear how newborn NSs formed via this channel can have an initial spin period as long as P_i ∼ 2–40 ms. We infer that the compact companion in a close binary can increase the angular momentum of the ultrastripped star by the tidal torque, possibly resulting in the rapid rotation of the NS. In addition, for the FBOTs with ejecta masses of ∼0.1 M_⊙, WD-related models still cannot be ruled out, which have some advantages in explaining the multiwavelength features of some FBOTs.

It has been widely suggested that SLSNe and SNe Ic-BL associated/unassociated with lGRBs, or at least a good fraction of them, are also driven by millisecond magnetars (e.g., Kasen & Bildsten 2010; Lü & Zhang 2014; Mazzali et al. 2014; Metzger et al. 2015; Kashiyama et al. 2016; Liu et al. 2017; Nicholl et al. 2017; Yu et al. 2017; Lü et al. 2018). Therefore, it is necessary and interesting to investigate the possible connection and differences between FBOTs and these explosion phenomena, as has been done in previous research (e.g., Milisavljevic et al. 2013; Yu et al. 2017; Margutti et al. 2019; Pian et al. 2020) for connecting normal SNe Ib/c, SLSNe, and lGRBs. For comparison, we display the FBOTs along with SLSNe, GRB-SNe, and normal SNe Ic-BL in the M_ej versus P_i space in Figure 1, and in the P_i versus B_p space in Figure 2. The P_i values for GRB-SNe and normal SNe Ic-BL are collected from Lyman et al. (2016), Lü et al. (2018), and Taddia et al. (2019) assuming that the kinetic energy of the ejecta is derived from the rotational energy of the magnetar.

As shown in Figure 1, the combination of the four different types of transients shows a clear universal correlation between ejecta mass and initial spin period as follows:

$$\begin{eqnarray}&&{P}_{{\rm{i}}}\propto {M}_{\mathrm{ej}}^{-0.41},\end{eqnarray} \tag{ 4 }$$

with a Pearson correlation coefficient ρ = −0.84, which is well consistent with the results discovered by Yu et al. (2017) for the SLSN sample only. This anticorrelation strongly indicates that these explosion phenomena may share a common origin. It can be also found that the clearest criterion defining FBOTs could be their small ejecta masses, which generally correspond to relatively long initial spin periods because of the strong anticorrelation. Therefore, on the one hand, FBOTs very likely originate from stellar collapse, but just having a progenitor much lighter and much more stripped than those of SLSNe and SNe Ic-BL. On the other hand, the M_ej–P_i anticorrelation indicates that more massive progenitors have larger angular momenta. In Section 3.1, we suspect that the FBOT progenitors could be ultrastripped stars in close binary systems, which can be spun up by their compact companions. Following this consideration, the M_ej–P_i anticorrelation could be a natural result of the interaction between the progenitor and the compact companion (see also Blanchard et al. 2020; Fuller & Lu 2022; R.-C. Hu et al. 2022, in preparation). If this hypothesis is true, then it is expected that the progenitors of SLSNe and SNe Ic-BL can also be substantially influenced by a compact companion.

Yu et al. (2017) have found that the primary difference between SLSNe and lGRBs could be the magnetic field strengths of their magnetar engines. Specifically, SLSNe have B_c ≲ B_p ≲ 10B_c, while lGRBs have B_p ≳ 10B_c. Therefore, in Yu et al. (2017), it was suspected that the ultrahigh magnetic fields can play a crucial role in launching a relativistic jet to produce GRB emission. Here, it is however found that the surface magnetic fields of more than half of the FBOTs can be higher than 10B_c, but no GRB has been detected to be associated with the FBOTs. One possibility is that the GRB emission associated with these FBOTs is highly beamed and the emission beam largely deviates from the line of sight. This, however, is disfavored by the difference between the event rate density of FBOTs and lGRBs. Therefore, a more promising explanation is that these FBOT magnetars intrinsically cannot produce GRB emission, even though their magnetic fields satisfy B_p > 10B_c. The probable reason is that the FBOT magnetars rotate too slowly to provide sufficiently high energy for a relativistic jet. Additionally, in view of the small masses of FBOT ejecta, the possible fallback accretion onto the magnetar is potentially weak and thus cannot help to launch a jet.

Finally, in view of the significant similarity between FBOTs and SLSNe, it would be reasonable to regard them as a unified phenomenon with different progenitor masses. From this view, the separation between FBOTs and SLSNe is empirical but does not imply fundamentally different physics. For example, the FBOTs DES 16E1bir (M_ej ∼ 0.9 M_⊙) and SNLS 06D1hc (M_ej ∼ 1 M_⊙) in our sample could in fact be classified as SLSNe. In any case, by combining the FBOT and SLSN data, we can find a weak correlation between P_i and B_p, as presented in Figure 2. Such a correlation could also exist in the lGRB data, but with a shift in B_p. This indicates that GRB magnetars have magnetic fields statistically higher than those of SLSNe and FBOTs for the same initial spin period P_i.

According to the parameter values constrained from the fittings, we can calculate the peak absolute magnitude (or peak luminosity), the rise and decline timescales above the half-maximum luminosity of the FBOTs, and their 1σ uncertainties, which are also listed in Table 2. These parameters determine the basic shape of the lightcurves of the transients and can be measured directly from observational data, which therefore are very useful for classifying the transients. As presented in the left panel of Figure 3, it seems reasonable to set the boundary between FBOTs and SLSNe at t_rise ∼ 10 days, where the data are relatively sparse. So we adopt it as a sample selection criterion. Strictly speaking, it cannot be ruled out that the ambiguous gap between FBOTs and SLSNe could just be a result of selection effects and the distribution between these two phenomenological types of explosion phenomena could in fact be intrinsically continuous.

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Generally speaking, FBOTs together with SLSNe can be separated from SNe Ic-BL including GRB-SNe by the separation line

$$\begin{eqnarray}&&{M}_{\mathrm{peak}}=\left\{\begin{array}{l}{a}_{1}{\tilde{t}}_{\mathrm{rise}}-{b}_{1}{\mathrm{log}}_{10}{\tilde{t}}_{\mathrm{rise}}-{c}_{1},\,\mathrm{for}\,{\tilde{t}}_{\mathrm{rise}}\lesssim 30,\\ {a}_{2}{\tilde{t}}_{\mathrm{rise}}-{b}_{2}{\mathrm{log}}_{10}{\tilde{t}}_{\mathrm{rise}}-{c}_{2},\,\mathrm{for}\,{\tilde{t}}_{\mathrm{rise}}\gtrsim 30,\end{array}\right.\end{eqnarray} \tag{ 5 }$$

corresponding to a nickel mass M_Ni = 0.3M_ej, where ${\tilde{t}}_{\mathrm{rise}}\,={t}_{\mathrm{rise}}/\mathrm{days}$ and the numerical coefficients read as a₁ = 0.083, b₁ = 5.3, c₁ = 14.94, a₂ = 0.0089, b₂ = 5.2, and c₂ = 12.63. This separation line is plotted using the M_ej−E_K relationship derived from the M_ej–P_i relationship, i.e., Equation (4), by assuming that all the kinetic energy of the ejecta comes from the rotational energy of the magnetar. It is commonly believed that the mass of the ⁵⁶Ni synthesized during core-collapse SNe rarely reaches a few tens of percent of the total mass of the SN ejecta (e.g., Suwa et al. 2015; Saito et al. 2022). Based on radiation transport calculations, Ertl et al. (2020) found that current models employing standard assumptions of explosions and nucleosynthesis predict radioactive decay powered lightcurves that are less luminous than commonly observed SNe Ib and Ic. So, both FBOTs and SLSNe of magnitudes above Equation (5) definitely cannot be primarily powered by radioactive decay of ⁵⁶Ni and an engine power is required. Nevertheless, some outliers exist in our sample, e.g., DES 14S2anq and DES 14S2plb. In comparison, the peak luminosity of SNe Ic-BL is relatively lower, which reduces the energy requirement and, in principle, makes the radioactive power model available. However, considering the continuous transition between the different phenomena, it could still be natural to suggest that the emission of a fraction of SNe Ic-BL including GRB-SNe is also partly powered by the magnetar engine, although the majority of the spin-down energy of the magnetar has been converted to the kinetic energy of the SN ejecta (e.g., Lin et al. 2021; Zhang et al. 2022).

As analyzed in Yu et al. (2015, 2017), the emission fraction of the spin-down energy is primarily determined by the relationship between the timescales of t_sd and t_diff, which can be basically reflected by the ratio of the lightcurve rise time to the decline time. As shown in the right panel of Figure 3, the FBOT and SLSN data can be well fitted by the line of t_decl ≈ 1.8t_rise, which corresponds to t_sd = t_diff. This is a reason why we can regard FBOTs and SLSNe as a unified phenomenon. In comparison, the SN Ic-BL data are obviously in the t_sd < t_diff region, as expected. For GRB-SNe, although they are usually classified as SNe Ic-BL, their distribution in the t_rise−t_decl plane is actually more diffusive than that of normal SNe Ic-BL.