GRB 201223A: Implication of Fallback Accretion onto the Newborn Black Hole from its Multiband Afterglow

The Swift/BAT data were downloaded from the UK Swift Science Data Centre. The batbinevt was used to generate light curve file and pha file for spectral analysis. The Fermi/GBM payload carries 12 sodium iodide (NaI, 8 keV–1 MeV) and two bismuth germanate (BGO, 200 keV–40 MeV) scintillation detectors (Meegan et al. 2009). Considering the detectors' direction of pointing, we employed two NaI detectors (n7, n8) and one BGO detector (b1) to conduct the spectral analysis. We obtained the Time-Tagged Event (TTE) data covering the time range of this GRB from the Fermi/GBM public data archive. ⁵

The 256 ms time-bin light curves in different energy bands are shown in Figure 3. A joint analysis was performed via threeML (Vianello et al. 2015) for the Swift/BAT data and Fermi/GBM data with Band function model (Band et al. 1993)

$$\begin{eqnarray*}&&N(E)=\left\{\begin{array}{l}A{\left(\displaystyle \frac{E}{100\ \mathrm{keV}}\right)}^{\alpha }\exp \left(-\displaystyle \frac{E}{{E}_{0}}\right),\,E\lt (\alpha -\beta ){E}_{0}\\ A{\left(\displaystyle \frac{(\alpha -\beta ){E}_{0}}{100\ \mathrm{keV}}\right)}^{\alpha -\beta }\exp (\beta -\alpha ){\left(\displaystyle \frac{E}{100\ \mathrm{keV}}\right)}^{\beta },\,E\geqslant (\alpha -\beta ){E}_{0},\end{array}\right.\end{eqnarray*}$$

where A is the normalization of the spectrum, E₀ is the break energy in the spectrum, and α and β are the low-energy and high-energy photon spectral indices, respectively.

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NEXT is an equatorial telescope located at Nanshan, Xinjiang, China. NEXT began its observation on 2017 November. The telescope has a 60 cm aperture and the FOV is $22^{\prime} \times 22^{\prime}$. The size of the CCD is 2048 × 2048 pixels with a pixel size of 15 μm. The pixel scale is 064 pixel⁻¹ (Zhu et al. 2023a). The typical gain is 1.85e⁻/ADU and the usual readout noise is 13e⁻ with 500 kHz readout speed. With images having 30 minute exposures, the typical limiting magnitude can reach 21.5. NEXT is equipped with BV filters in the standard Johnson-Cousins system and griz filters and a white filter in the Sloan system.

NEXT began to obtain the first image of GRB 201223A at 18:01:14 UT, 2.8 minutes after the BAT trigger (Zhu et al. 2020). We obtained 57 images on 2020 December 23rd, all in the r filter. The exposure times were 2 × 40 s, 4 × 60 s, 16 × 90 s, 3 × 200 s, and 30 × 300 s. The observing mid-time and exposure time of each frame are presented in Table 1.

NOT has an aperture of 2.56 m, and is located in La Palma, Canary Islands, Spain. Routine observations started in 1990 with Alhambra Faint Object Spectrograph and Camera (ALFOSC) and NOTCam. The ALFOSC imager can be used for imaging observations, taking low and medium resolution and polarization observations. ALFOSC has an FOV of 64 × 64 in imaging mode and is equipped with a Johnson-Cousins UBVRI filter as well as a Sloan ugriz filter; 1 hr exposure images have typical limiting magnitudes up to 24–25 mag (Djupvik & Andersen 2010).

NOT first obtained 2 × 120 s Sloan r-band frames of GRB 201223A starting at 02:21:52 UT on 2020 December 24th, i.e., 8.4 hr after the BAT trigger (Xu et al. 2020). Unfortunately, the optical afterglow is not detected in the stacked image, down to a limiting magnitude of r ∼ 22.0. About 26.3 days after the BAT trigger, NOT obtained the images again at 00:57:31 UT on 2021 January 19th. Nine raw images were acquired and subsequently combined to enhance the signal-to-noise ratio. The exposure time for each image was 360 s. Through this process, no optical source was detected in our stacked image, down to a limiting magnitude of r ∼ 25.6.

The raw images obtained from NEXT and NOT were processed by standard processes in the IRAF packages (Tody 1986), including bias and flat correction. The cosmic rays were also removed by the filtering described in Van Dokkum (2001). The measurements of magnitudes were conducted utilizing SourceExtractor (SExtractor, Bertin & Arnouts 1996), employing a circular aperture with a diameter of ten pixels. The magnitude was calibrated with Pan-STARRS1 (PS1, Chambers et al. 2016). All the photometric results are presented in Table 1.

For the Swift/UVOT data (Gropp et al. 2020), the afterglow was detected including white, u, b and v filters. We applied the standard HEAsoft software (version 6.31.1) and utilized the uvotproduct pipeline to reduce UVOT data with a source circular region of 5'' and a background region of 10'' aperture radius.

The duration of GRB 201223A associated with the GBM trigger was T₉₀ = 33 s (50–300 keV) and the event fluence (10–1000 keV) is (2.1 ± 0.3) × 10⁻⁶ erg cm⁻² from T₀ − 17 s to T₀ + 13 s (Wood & Team et al. 2020). We use three different models, namely Band, Blackbody, and Cutoff power-law, to fit the time-averaged spectrum. The best-fit parameters based on the Band model are $\alpha =-{0.62}_{-0.16}^{+0.17}$, $\beta =-{2.93}_{-0.40}^{+0.38}$, and ${E}_{\mathrm{peak}}={137.36}_{-18.25}^{+23.59}$ keV . The spectroscopic redshift was not reported and Xin et al. (2023) found that the redshift should be smaller than 1.85. If we use the redshift of z = 1.85 as an upper limit, then a high isotropic γ-ray energy E_γ,iso < 1.84 × 10⁵² erg and isotropic γ-ray luminosity L_γ,iso < 2.34 × 10⁵¹ erg s⁻¹ are obtained.

We used the cross-correlation function (CCF) (Band 1997; Norris et al. 2000b; Ukwatta et al. 2010) and Monte Carlo simulation (Peterson et al. 1998; Ukwatta et al. 2010) to calculate the spectral lag and uncertainty of the burst (Norris et al. 2000a; Ukwatta et al. 2012). The result of the lag is 282 ± 264 ms between 15–85 and 85–160 keV with BAT data. For the Fermi data, the spectral lag is 225 ± 111 ms between 10–85 and 85–160 keV, which is consistent with the lag of BAT.

We also calculated the minimum variability timescale t_mv = 2.54 s, which represents the rapid variation of prompt emissions in a short period of time (Vianello et al. 2018).

We compared the observed data with the theoretical framework. The r-band optical light curve is best described by a broken power-law (BPL) with α_r,1 = −0.88 ± 1.04, α_r,2 = 1.09 ± 0.01 and break time at t_b = 54.1 ± 28.5 s. The spectral index is β_opt = 1.35 ± 0.13. The single power-law (SPL) is used to fit the X-ray light curve with an index of α_X = 0.94 ± 0.06. The time-averaged spectrum gives the X-ray photon index ${\rm{\Gamma }}={1.91}_{-0.21}^{+0.35}$, and the relation between the spectral index β and photon index Γ is β = Γ −1. Therefore, the X-ray spectral index ${\beta }_{{\rm{X}}}={0.91}_{-0.21}^{+0.35}$ (Evans et al. 2009) which is consistent with the optical to X-ray spectral index β_OX = 0.96 ± 0.03.

We fit the multiband afterglow (optical and early X-ray) of GRB 201223A using PyFRS, ⁶ which can be used to calculate synchrotron light curves and spectra from external shocks (Gao et al. 2013; Wang et al. 2014; Lei et al. 2016; Zhu et al. 2023b; Zhou et al. 2024). In this paper, the top-hat jet-type structure is used to model GRB 201223A. Since the burst redshift cannot be determined, we assume a typical GRB redshift of z = 1.0. We consider eight parameters including the isotropic kinetic energy E_K,iso, the initial Lorentz factor Γ₀, the half-opening angle of the jet θ_j, the viewing angle θ_obs, the number density of the interstellar medium (ISM) n₀, the electron distribution power-law index p, the thermal energy fraction in magnetic field _B, and the thermal energy fraction in electrons _e. The observational angle was not effectively constrained due to the limitations of the available data, so we set θ_obs = 0 for simplicity.

We performed a parameter search with 30 walkers over 20,000 iterations, discarding the first 10,000 as burn-in steps. The prior types and ranges of each model parameter are listed in Table 2, and the optical afterglow light curves for GRB 201223A as well as the best-fit model are displayed in Figure 4. The corner plot of the model parameters is shown in Figure 5. The best fit of each parameter is given in Table 2 as: ${E}_{{\rm{K}},\mathrm{iso}}={5.01}_{-1.70}^{+1.91}\times {10}^{54}$ erg, ${{\rm{\Gamma }}}_{0}={426.58}_{-138.18}^{+148.86}$, ${\theta }_{{\rm{j}}}={25.98}_{-10.54}^{+9.67}$ deg, ${n}_{0}={0.30}_{-0.26}^{+3.78}$ cm⁻³, $p={2.32}_{-0.01}^{+0.01}$, ${\epsilon }_{{\rm{e}}}={3.31}_{-0.86}^{+1.59}\times {10}^{-5}$, ${\epsilon }_{{\rm{B}}}={3.47}_{-2.62}^{+4.12}\times {10}^{-1}$.

Table 2. The Input Parameters, Prior Type, Prior Range, and Best-fit Value of Multiband Modeling of GRB 201223A were Generated with PyFRS

Parameter	Prior Type	Prior Range	Best Fit
EK,iso (erg)	log-uniform	[1051, 1056]	${5.01}_{-1.70}^{+1.91}\times {10}^{54}$
Γ0	log-uniform	[10, 2000]	${426.58}_{-138.18}^{+148.86}$
θj (deg)	uniform	[0.01, 40]	${25.98}_{-10.54}^{+9.67}$
n0 (cm−3)	log-uniform	[10−4, 103]	${0.30}_{-0.26}^{+3.78}$
p	uniform	[2.01, 3]	${2.32}_{-0.01}^{+0.01}$
e	log-uniform	[10−6, 1]	${3.31}_{-0.86}^{+1.59}\times {10}^{-5}$
B	log-uniform	[10−6, 1]	${3.47}_{-2.62}^{+4.12}\times {10}^{-1}$

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Due to the lack of spectroscopic observations for this burst, we cannot determine its exact redshift. However, since the target was detected in the Swift u-band, indicating the absence of Lyα absorption in this band, an upper limit of z < 1.85 can be placed on the redshift (Xin et al. 2023).

Amati et al. (2002) discovered a correlation between the isotropic-equivalent energy E_γ,iso and the intrinsic peak energy E_p,i of GRB prompt emission

$$\begin{eqnarray}&&{E}_{{\rm{p}},{\rm{i}}}=k{\left({E}_{\gamma ,\mathrm{iso}}/{10}^{52}\mathrm{erg}\right)}^{m},\end{eqnarray} \tag{ 1 }$$

where k and m are constants, and E_p,i = (1 + z)E_p,obs. By plotting these two characteristic energies on a two-dimensional coordinate plane, two clusters of GRBs can be identified: short GRBs (SGRBs) with lower E_γ,iso but higher E_p,i, and long GRBs (LGRBs) with higher E_γ,iso but lower E_p,i. Since both characteristic energies require a precise redshift, conversely, we can estimate the redshift range of GRBs by the evolution of these two energies with redshift. We construct a E_p,i − E_γ,iso sample containing 207 LGRBs and 33 SGRBs, as depicted in Figure 6.

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GRB 201223A is identified as a typical LGRB based on its duration T₉₀ = 33 s and the spectral lag 225 ms derived from Fermi data. Therefore, according to the 2σ range of LGRBs in Figure 6, we can obtain its redshift lower limit z_low ≈ 0.26. Of course, it cannot be ruled out that it is a peculiar GRB with special parameter values different from those of LGRBs. Thus, we propose a redshift range of 0.26–1.85 for GRB 201223A.

The X-ray light curve shows a shallow decay from 200 to 1000 s. The expected flux from the afterglow model is significantly lower than the observed value (see the red dashed line in Figure 4). Two central engine models were considered to explain this light curve behavior: one is the spin-down of a magnetar, and the other is the fallback accretion onto a newborn BH. We will inspect these two central engine models by comparing with the afterglow data.

4.2.1. Spin-down of a Magnetar

The X-ray plateaus can be produced by the spin power of a millisecond magnetar (Dai & Lu 1998; Liang et al. 2007; Tang et al. 2019; Zhao et al. 2019). The characteristic spin-down luminosity L₀ can be written as

$$\begin{eqnarray}&&{L}_{0}\simeq 1.0\times {10}^{49}\mathrm{erg}\ {{\rm{s}}}^{-1}{B}_{{\rm{p}},15}^{2}{P}_{0,-3}^{-4}{R}_{6}^{6},\end{eqnarray} \tag{ 2 }$$

where B_p,15 is the magnetic field strength in units of 10¹⁵ G, P_0,−3 is the initial spin period in millisecond, and R₆ is the radius of the magnetar in units of 10⁶ cm.

The evolution of the magnetar spin period due to dipole radiation is given by

$$\begin{eqnarray}&&P(t)={P}_{0}{\left(1+\displaystyle \frac{t}{{t}_{\mathrm{md}}}\right)}^{(1/2)},\end{eqnarray} \tag{ 3 }$$

where the spin-down is dominated by the dipole radiation with timescale ${t}_{\mathrm{md}}\simeq 2.0\times {10}^{3}s({I}_{45}{B}_{{\rm{p}},15}^{-2}{P}_{0,-3}^{2}{R}_{6}^{-6})$. We consider only the energy loss due to dipole radiation in this work. As the spin-down of magnetar, it may leave behind a stable NS, or collapses into a BH if it is temporarily supported by rigid rotation. The latter will lead to a sharp decay in X-ray flux as observed. The maximum gravitational mass of supermassive magnetars can be expressed as (Lasky et al. 2014)

$$\begin{eqnarray}&&{M}_{\max }={M}_{\mathrm{TOV}}(1+\hat{\alpha }{P}^{\hat{\beta }}),\end{eqnarray} \tag{ 4 }$$

where M_TOV is the maximum mass for a nonrotating NS. For the NS equation of state (EoS), in accordance with recent studies utilizing data from GRBs, we have chosen to utilize the EoS GM1, which specifies the radius of the magnetar R = 12.05 km, the rotational inertia I = 3.33 × 10⁴⁵ g cm⁻², $\hat{\alpha }=1.58\times {10}^{-10}{s}^{-\hat{\beta }}$ and $\hat{\beta }=-2.48$ (Lü et al. 2015; Gao et al. 2016b).

First, we assume that the re-brightening of the X-ray emission originates from energy injection into the forward shock, and find that the injection luminosity via X-ray afterglow fitting is 1.6 × 10⁵³ erg s⁻¹ at a redshift of z = 1.0. This value, however, severely exceeds the energy expected from a magnetar. Even adopting the lower redshift z = 0.26, the result does not change too much. Another point is that such energy injection will also lead to re-brightening in optical which is absent from observations.

We therefore consider that the re-brightening of the X-ray emission comes from internal dissipation of a magnetar, and derive the luminosity L_X ∼ 3.35 × 10⁴⁵ erg s⁻¹, 5.37 × 10⁴⁶ erg s⁻¹ and 1.73 × 10⁴⁷ erg s⁻¹ for redshift of 0.26, 1.0 and 1.85, respectively.

Substituting these three X-ray luminosities as the spin-down luminosities of the magnetar into the above equation, i.e., L_X ∼ L₀, and assuming spin-down timescale t_md = 3000/(1 + z) s, we obtain the spin periods of the magnetar to be 411 ms, 129 ms, and 86 ms for the redshift of 0.26, 1.0 and 1.85, respectively, and the surface magnetic fields of the magnetar to be 3.98 × 10¹⁷ G, 1.58 × 10¹⁷ G, and 1.25 × 10¹⁷ G for the redshift of 0.26, 1.0 and 1.85, respectively. Rowlinson et al. (2014) compiled a sample of GRB magnetars, with a maximum magnetar period of 83 ms and a surface magnetic field range of ∼3 × 10¹⁴ to ∼2 × 10¹⁷ G in the sample. While the results in the sample may vary due to different computational methods, we derived that the magnetic field and period of the magnetar associated with GRB 201223A are both outliers, lying at the edge of the sample. In our calculations, we assumed a relatively large spin-down timescale; if this value is decreased, the derived results would deviate even further from the sample. Therefore, the spin-down magnetar model for the re-brightening of X-ray emission is disfavored.

4.2.2. Fallback Accretion onto the Newborn BH

In the framework of BH central engine model, the X-ray plateau or bump seen in GRB 201223A is explained by the fallback accretion (Wu et al. 2013). An accretion system can generate relativistic jets through neutrino-antineutrino annihilation (Popham et al. 1999; Narayan et al. 2001; Janiuk et al. 2004; Gu et al. 2006; Chen & Beloborodov 2007; Lei et al. 2009; Xie et al. 2016) or the BZ mechanism (Blandford & Znajek 1977; Lee & Kim 2000; Li & Paczyński 2000; Lei et al. 2005b). We assume that the evolution of fallback accretion rate is described with a smooth BPL function (Chevalier 1989; MacFadyen et al. 2001; Zhang et al. 2008; Dai & Liu 2012),

$$\begin{eqnarray}&&\dot{M}={\dot{M}}_{{\rm{p}}}{\left[\displaystyle \frac{1}{2}{\left(\displaystyle \frac{t-{t}_{0}}{{t}_{{\rm{p}}}-{t}_{0}}\right)}^{-\displaystyle \frac{1}{2}}+\displaystyle \frac{1}{2}{\left(\displaystyle \frac{t-{t}_{0}}{{t}_{{\rm{p}}}-{t}_{0}}\right)}^{\displaystyle \frac{5}{3}}\right]}^{-1},\end{eqnarray} \tag{ 5 }$$

where t₀ is the beginning time of the fallback accretion in the local frame and t_p is the peak time of the fallback accretion. The early-time fallback accretion behavior follows t^1/2 and late-time fallback accretion behavior follows t^−5/3.

The BZ power can be rewritten as a function of mass accretion rate (Lei et al. 2013; Wu et al. 2013).

$$\begin{eqnarray}&&{\dot{E}}_{{\rm{B}}}=9.3\times {10}^{53}\displaystyle \frac{{a}_{\bullet }^{2}\dot{m}F({a}_{\bullet })}{{\left(1+\sqrt{1-{a}_{\bullet }^{2}}\right)}^{2}}\mathrm{erg}\ {{\rm{s}}}^{-1},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&F({a}_{\bullet })=[(1+{q}^{2})/{q}^{2}][(q+1/q)\arctan q-1].\end{eqnarray} \tag{ 7 }$$

Here $q={a}_{\bullet }/(1+\sqrt{1-{a}_{\bullet }^{2}})$, ${a}_{\bullet }={J}_{\bullet }c/({{GM}}_{\bullet }^{2})$ is the BH spin parameter, and $\dot{m}=\dot{M}/({M}_{\odot }\,{{\rm{s}}}^{-1})$ is the accretion rate.

Then we connect the observed X-ray luminosity to the BZ power through

$$\begin{eqnarray}&&\eta {\dot{E}}_{{\rm{B}}}={f}_{{\rm{b}}}{L}_{{\rm{X}}},\end{eqnarray} \tag{ 8 }$$

where η is the efficiency of converting BZ power to X-ray radiation and ${f}_{{\rm{b}}}=(1-\cos {\theta }_{{\rm{j}}})/2$ is the beaming factor of the jet.

In the case of GRB 201223A, we assume a BH with a mass of M_• = 3M_⊙, a spin of a_• = 0.9, efficiency η = 0.1 and the calculation starts from t₀ = 100/(1 + z) s. We utilize the optimal parameters obtained from the afterglow fitting, which indicate a jet opening angle of 25° (Table 2), and thus f_b ≃ 0.05. The parameters for the fitting are ${\dot{M}}_{{\rm{p}}}\simeq 2.76\times {10}^{-9}{M}_{\odot }\,{{\rm{s}}}^{-1}$, t_p = 1637/(1 + z) s, and the total fallback accreted mass M_fb ≃ 1.41 × 10⁻⁶ M_⊙ for the burst redshift of z = 1.0. Thus, we can estimate the minimum radius around which matter starts to fall back (r_fb) from the following equation (Wu et al. 2013)

$$\begin{eqnarray}&&{r}_{{\rm{fb}}}\simeq 3.5\times {10}^{10}{\left({M}_{\bullet }/3{M}_{\odot }\right)}^{1/3}{\left({t}_{0}/360\,{\rm{s}}\right)}^{2/3}\,{\rm{cm}}.\end{eqnarray} \tag{ 9 }$$

Then we estimate the value of r_fb ≃ 9.4 × 10⁹ cm.

The best-fit X-ray light curve component due to fallback accretion is shown as a dotted line in Figure 4. The red solid line signifies the total emission by including the contributions from both the BZ jet (dotted line) and the external shock (dashed line). Across the redshift range of 0.26–1.85, the derived accretion rate and fallback radius fall within reasonable ranges. Consequently, the fallback accretion scenario provides a more plausible explanation for the X-ray light curve behavior of GRB 201223A.