# GRB 211024B: An Ultra-long GRB Powered by Magnetar

Shao-Yu Fu<fnref n="1" />,<fnref n="2" /> ID, Dong Xu<fnref n="1" /> ID, Wei-Hua Lei<fnref n="3" /> ID, Antonio de Ugarte Postigo<fnref n="4" />,<fnref n="5" /> ID, Daniele B. Malesani<fnref n="6" />,<fnref n="7" /> ID,
David Alexander Kann<fnref n="8" />,<fnref n="9" /> ID, Páll Jakobsson<fnref n="10" />, Johan P. U. Fynbo<fnref n="6" />,<fnref n="11" /> ID, Elisabetta Maiorano<fnref n="12" />, Andrea Rossi<fnref n="12" /> ID, Diego Paris<fnref n="13" /> ID,
Xing Liu<fnref n="1" />,<fnref n="2" />, Shuai-Qing Jiang<fnref n="1" />,<fnref n="2" />, Tian-Hua Lu<fnref n="1" />,<fnref n="2" />, Jie An<fnref n="1" />,<fnref n="2" />, Zi-Pei Zhu<fnref n="1" /> ID, Xing Gao<fnref n="14" />, and Jian-Yan Wei<fnref n="1" />,<fnref n="2" />

## Abstract

Ultra-long gamma-ray bursts are characterized by exceptionally long-duration central engine activities, with characteristic timescales exceeding 1000 s. We present ground-based optical afterglow observations of the ultra-long gamma-ray burst GRB 211024B, detected by Swift. Its X-ray light curve exhibits a characteristic “internal plateau” with a shallow decay phase lasting approximately ~15 ks, followed by a steep decline ($\alpha_{drop} \sim -7.5$). Moreover, the early optical emission predicted by the late r-band optical afterglow is significantly higher than the observed value, indicating an external shock with energy injection. To explain these observations, we propose a magnetar central engine model. The magnetar collapses into a black hole due to spin-down or hyperaccretion, leading to the observed steep break in the X-ray light curve. The afterglow model fitting reveals that the afterglow injection luminosity varies with different assumptions of the circumburst medium density, implying different potential energy sources. For the interstellar medium case with a fixed injection end time, the energy may originate from the magnetar's dipole radiation. However, in other scenarios, relativistic jets produced by the magnetar/black hole system could be the primary energy source.

*Unified Astronomy Thesaurus concepts: Gamma-ray bursts (629); Transient sources (1851)*

## 1. Introduction

Gamma-ray bursts (GRBs) are the most energetic explosions in the universe, releasing immense amounts of energy peak in the high-energy gamma-ray band. Traditionally, GRBs have been classified based on their gamma-ray emission duration, with long GRBs (LGRBs) lasting longer than 2 s ($T_{90} > 2$ s) and short GRBs (SGRBs) lasting less than 2 s ($T_{90} < 2$ s) (C. Kouveliotou et al. 1993). The observed association between LGRBs and Type Ic supernovae strongly suggests that at least some LGRBs arise from the collapse of massive stars (S. E. Woosley & J. S. Bloom 2006). Conversely, the detection of a gravitational wave event GW170817 coincident with GRB 170817A confirmed the theoretical prediction that at least some SGRBs originate from the merger of neutron stars (B. P. Abbott et al. 2017; I. Arcavi et al. 2017; D. A. Coulter et al. 2017; P. S. Cowperthwaite et al. 2017; M. M. Kasliwal et al. 2017; M. Soares-Santos et al. 2017).

In the Swift era, the detection efficiency of GRBs has been greatly improved, and we have a clear understanding of the

morphological features of X-ray light curves (B. Zhang et al. 2006). In addition to the power-law decay predicted by the external shock theory, there are also features such as flares and plateaus, which suggest the presence of an active central engine at the burst center. In recent years, a class of GRBs with extremely long durations has been discovered, known as ultra-long GRBs (ULGRBs), such as GRB 060607A (H. Ziaeepour et al. 2008), GRB 091024A (F. J. Virgili et al. 2013), GRB 101225A (C. C. Thöne et al. 2012; A. J. Levan et al. 2014), GRB 111209A (B. Gendre et al. 2013; G. Stratta et al. 2013; D. A. Kann et al. 2018), GRB 121027A (F.-k. Peng et al. 2013; X.-F. Wu et al. 2013; A. J. Levan et al. 2014), GRB 130925A (P. A. Evans et al. 2014), and GRB 220627A (Y.-Y. Huang et al. 2022; S. de Wet et al. 2023). Some of these GRBs have gamma-ray emission exceeding 1000 s, while others exhibit long-term active central engine features in their X-ray light curves. A. J. Levan et al. (2014; L14) studied three ULGRBs and found that their host galaxies are faint, compact, and highly star-forming dwarf galaxies. They also proposed a criterion for identifying ULGRBs: high-energy emission lasting up to ~$10^4$ s; and rapid decay after plateau phase, with a decay index consistent with high-latitude emission ($\alpha_{drop} \le -3$) and rapid variations (flares or dips) in plateau phase. B.-B. Zhang et al. (2014) studied the Swift GRBs

<figure><img src="image_1.png" /></figure>

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<footnote n="1">¹ Key Laboratory of Space Astronomy and Technology, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China; dxu@nao.cas.cn</footnote>

<footnote n="2">² School of Astronomy and Space Science, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China</footnote>

<footnote n="3">³ Department of Astronomy, School of Physics, Huazhong University of Science and Technology, Wuhan, 430074, People's Republic of China; leiwh@hust.edu.cn</footnote>

<footnote n="4">⁴ Université de la Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Artemis, 06304 Nice, France</footnote>

<footnote n="5">⁵ Aix Marseille Univ, CNRS, LAM Marseille, France</footnote>

<footnote n="6">⁶ Niels Bohr Institute, University of Copenhagen, Jagtvej 155, 2200, Copenhagen N, Denmark</footnote>

<footnote n="7">⁷ Department of Astrophysics/IMAPP, Radboud University, PO Box 9010, 6500 GL, The Netherlands</footnote>

<footnote n="8">⁸ Instituto de Astrofísica de Andalucía, Glorieta de la Astronomía s/n, 18008 Granada, Spain</footnote>

<footnote n="9">⁹ Hessian Research Cluster ELEMENTS, Giersch Science Center, Max-von-Laue-Strasse 12, Goethe University Frankfurt, Campus Riedberg, 60438 Frankfurt am Main, Germany</footnote>

<footnote n="10">¹⁰ Centre for Astrophysics and Cosmology, Science Institute, University of Iceland, Dunhagi 5, 107 Reykjavík, Iceland</footnote>

<footnote n="11">¹¹ The Cosmic Dawn Centre (DAWN), Denmark</footnote>

<footnote n="12">¹² INAF—Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, I-40129 Bologna, Italy</footnote>

<footnote n="13">¹³ INAF—Osservatorio Astronomico di Roma, via Frascati 33, I-00040 Monte Porzio Catone, Italy</footnote>

<footnote n="14">¹⁴ Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, People's Republic of China</footnote>

<footnote>Received 2024 March 11; revised 2024 October 11; accepted 2024 October 16; published 2024 December 12</footnote>
sample and estimated the burst duration $t_{\text{burst}}$ of the sample. They found that $t_{\text{burst}}$ is significantly higher than $T_{90}$, and even 11.5% of GRBs have $t_{\text{burst}} > 10^4$ s, indicating that the active time of the GRB central engine is longer than it appears.

Due to the longer duration of ULGRBs, blue supergiant stars (BSGs) have been proposed as possible progenitor stars of ULGRBs because their radiuses are much larger than those of the progenitor Wolf–Rayet (WR) stars of LGRBs (P. Mészáros & M. J. Rees 2001; D. Nakauchi et al. 2013). R. Perna et al. (2018) performed an end-to-end simulation and found that a BSG explosion can produce an accretion disk with an accretion time-scale of ~$10^4$ s. The fragmentation of a massive star envelope (A. King et al. 2005) or the accretion disk (R. Perna et al. 2006) can produce the flares that we see in X-ray light curves. Some have proposed that active newborn magnetars can produce such long-term radiation and will exhibit a plateau seen in the X-ray light curve of some GRBs (E.-W. Liang et al. 2007; E. Troja et al. 2007; H.-J. Lü & B. Zhang 2014; A. K. Ror et al. 2024).

In this paper, we present optical observations of the ULGRB GRB 211024B detected by Swift. Its X-ray light curve shows a typical “internal plateau” feature. Notably, the early-time optical flux is observed to be lower than the predictions based on the extrapolation from late-time afterglow. We propose that these observational features arise from continued activity of the newborn magnetar. In Section 2, we provide a description of the observation and data reduction. In Section 3, we introduce our model and fitting procedure for GRB 211024B. The discussion and conclusion are presented in Section 4 and Section 5, respectively. Throughout this paper, we adopt a conventional cosmological model with the following parameter values: $H_0 = 69.6 \text{ km s}^{-1} \text{ Mpc}^{-1}$, $\Omega_M = 0.286$, and $\Omega_\Lambda = 0.714$ (C. L. Bennett et al. 2014).

# 2. Observation and Data Reduction

## 2.1. Space Observation

GRB 211024B was discovered by the Swift Burst Alert Telescope (BAT) on 2021 October 24 at 22:20:36 UT. The prompt emission shows a single-peak structure with a duration of about 70 s (J. D. Gropp et al. 2021). Owing to the conflict of preplanned observation, X-ray Telescope (XRT) and Ultraviolet/Optical Telescope (UVOT) were unable to slew to the target at the first time. The late-time light curve of the prompt emission observed by BAT shows that the duration of GRB 211024B was longer than ~600 s.

XRT began observation of GRB 211024B at ~5.6 ks after the BAT trigger and gave enhanced XRT position R.A. (J2000) = 10:18:51.04, decl.(J2000) = 24:34:05.7, with an uncertainty of 1′′9 (P. A. Evans et al. 2021). The XRT light curve shows a plateau-like structure until 13 ks followed by a steep decay with index ~5. The PC mode spectrum can be fitted by absorbed power law with photon spectral index $\Gamma = 2.08_{-0.10}^{+0.11}$ and absorption column is $9.7_{-2.5}^{+2.7} \times 10^{20} \text{ cm}^{-2}$ (P. A. Evans et al. 2021).

Optical emission from GRB 211024B was detected by UVOT at ~5.6 ks after the BAT trigger with U-band magnitude 18.8 ± 0.2. We downloaded Swift/UVOT data from the UK Swift Science Data Centre<fnref n="15" /> and analyzed it with the *uvotproduct* pipeline of the HEAsoft software version 6.31.

## 2.2. Ground-base Photometry Observation

The initial observation automatically started at 22:23:13 UT on 2021 October 24, i.e., 157 s after the BAT trigger, using the NEXT-0.6 m optical telescope located at Nanshan, Xinjiang, China. We obtained a series of frames in the Sloan r-filter and covered prompt emission phase for at least the initial 600 s. However, there is no obvious correlation between the optical and the prompt emission light curve. Subsequent photometry follow-up observations were carried out using the 2.56 m Nordic Optical Telescope (NOT; Roque de los Muchachos observatory, La Palma, Spain) equipped with Alhambra Faint Object Spectrograph and Camera (ALFOSC) from 2023 October 26 to December 29. In addition, we observed the burst in the Sloan r and z bands with the Large Binocular Telescope (LBT) on 2021 November 7 (about 13 days after the burst), and performed *griz* multicolor photometry of its host galaxy on 2022 April 25 (about 182 days after the burst).

All optical data are reduced by standard procedures with the Image Reduction and Analysis Facility (IRAF) v2.16 (D. Tody 1986). For each frame of NEXT, bias and dark subtraction, flat-fielding, and cosmic ray removal were implemented. At each epoch of the NOT observation, bias subtraction, flat-fielding, and image combination were performed. Aperture photometry was conducted on each frame with an aperture radius of 2.5 times full width at half-maximum. Flux calibration was calibrated by referencing nearby Pan-STARRS1 field stars (K. C. Chambers et al. 2016).

In addition to our own observations, we collated observation results from the Gamma-ray Coordinates Network (GCN, M. Jelinek et al. 2021; D. A. Kann et al. 2021). All photometry results are reported in Table 1, while the corresponding light curves are shown in Figure 1.

## 2.3. Spectroscopic Observation

We obtained a spectrum of the GRB 211024B host galaxy on 2022 March 27, 153 days after the GRB, using the X-shooter spectrograph mounted on Unit Telescope 3 of the European Southern Observatory’s Very Large Telescope. The observation consisted of four exposures of 1200 s each in the UVB and VIS arms, and eight exposures of 600 s each in the NIR arm, covering a wavelength range of 3000–21000 Å. A portion of the K-band was intentionally blocked by the JH-slit to improve the efficiency of the J and H bands.

The host galaxy continuum is not visible in the spectrum, and only possible weak emission line features are detected at 7875, 10271, and 10582 Å. We tentatively identify the emission line feature at 10582 Å as [O III] 5008 Å, yielding a redshift of $z = 1.113$. This is supported by the weak emission lines of [O II] 3737 Å at 7875 Å and $H_\beta$ at 10271 Å (top panel of Figure 2). The expected $H_\alpha$ at 13868 Å, unfortunately, falls within a telluric absorption region, resulting in nondetection. If the emission line feature at 10582 Å is $H_\alpha$, it corresponds to a redshift of $z = 0.612$, and we would expect to see [O III] 5008 Å at 8073 Å and possibly $H_\beta$ at 7836 Å. In this case, no emission line features are found in the corresponding two-dimensional spectrum (bottom panel of Figure 2). Moreover, a redshift of $z \sim 0.6$ would be a bit low for such a faint galaxy. Finally, if 10582 Å is [O II], we would clearly resolve the [O II] doublet in the NIR arm of the spectrum. Therefore, we conclude that the redshift of the host galaxy is $z = 1.113 \pm 0.001$.

<footnote n="15">¹⁵ https://www.swift.ac.uk/swift_portal/</footnote>
<table><caption>**Table 1**
Optical Photometry Observation of GRB 211024B</caption><thead><tr><th>T - T<sub>0</sub><br>(days)</th><th>Band</th><th>Mag (AB)</th><th>Err</th><th>Ins.</th><th>References</th></tr></thead><tbody><tr><td>0.0020</td><td>r</td><td>18.40</td><td>0.13</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0028</td><td>r</td><td>18.33</td><td>0.12</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0035</td><td>r</td><td>17.89</td><td>0.09</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0044</td><td>r</td><td>17.72</td><td>0.06</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0053</td><td>r</td><td>17.73</td><td>0.06</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0063</td><td>r</td><td>17.77</td><td>0.06</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0073</td><td>r</td><td>17.91</td><td>0.07</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0084</td><td>r</td><td>18.08</td><td>0.06</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0097</td><td>r</td><td>18.36</td><td>0.08</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0110</td><td>r</td><td>18.32</td><td>0.08</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0123</td><td>r</td><td>18.59</td><td>0.10</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0137</td><td>r</td><td>18.50</td><td>0.09</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0150</td><td>r</td><td>18.72</td><td>0.11</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0163</td><td>r</td><td>18.71</td><td>0.11</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0176</td><td>r</td><td>18.83</td><td>0.12</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0189</td><td>r</td><td>18.98</td><td>0.14</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0203</td><td>r</td><td>19.21</td><td>0.17</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0216</td><td>r</td><td>18.94</td><td>0.13</td><td>NEXT</td><td>this work</td></tr><tr><td>0.0229</td><td>r</td><td>18.97</td><td>0.14</td><td>NEXT</td><td>this work</td></tr><tr><td>1.0308</td><td>r</td><td>20.97</td><td>0.23</td><td>NEXT</td><td>this work</td></tr><tr><td>1.2880</td><td>r</td><td>21.23</td><td>0.05</td><td>NOT/ALFOSC</td><td>this work</td></tr><tr><td>2.3060</td><td>r</td><td>21.69</td><td>0.06</td><td>NOT/ALFOSC</td><td>this work</td></tr><tr><td>4.3070</td><td>r</td><td>22.17</td><td>0.07</td><td>NOT/ALFOSC</td><td>this work</td></tr><tr><td>13.279</td><td>r</td><td>23.17</td><td>0.19</td><td>NOT/ALFOSC</td><td>this work</td></tr><tr><td>29.323</td><td>r</td><td>>23.6</td><td>...</td><td>NOT/ALFOSC</td><td>this work</td></tr><tr><td>65.312</td><td>r</td><td>25.08</td><td>0.29</td><td>NOT/ALFOSC</td><td>this work</td></tr><tr><td>16.291</td><td>i</td><td>23.01</td><td>0.21</td><td>NOT/ALFOSC</td><td>this work</td></tr><tr><td>16.272</td><td>g</td><td>23.87</td><td>0.2</td><td>NOT/ALFOSC</td><td>this work</td></tr><tr><td>13.561</td><td>r</td><td>22.97</td><td>0.03</td><td>LBT</td><td>this work</td></tr><tr><td>13.561</td><td>z</td><td>22.86</td><td>0.07</td><td>LBT</td><td>this work</td></tr><tr><td>182.39</td><td>g</td><td>25.74</td><td>0.31</td><td>LBT</td><td>this work</td></tr><tr><td>182.39</td><td>r</td><td>25.51</td><td>0.39</td><td>LBT</td><td>this work</td></tr><tr><td>182.39</td><td>i</td><td>>25.0</td><td>...</td><td>LBT</td><td>this work</td></tr><tr><td>182.39</td><td>z</td><td>>24.5</td><td>...</td><td>LBT</td><td>this work</td></tr><tr><td>0.2485</td><td>i</td><td>20.27</td><td>0.05</td><td>CAHA/CAFOS</td><td>D. A. Kann et al. (2021)</td></tr><tr><td>0.1520</td><td>i</td><td>19.94</td><td>0.11</td><td>D50</td><td>M. Jelinek et al. (2021)</td></tr><tr><td>0.0757</td><td>B</td><td>>20.01</td><td>...</td><td>Swift/UVOT</td><td>this work</td></tr><tr><td>0.2073</td><td>B</td><td>20.03</td><td>0.24</td><td>Swift/UVOT</td><td>this work</td></tr><tr><td>0.0662</td><td>U</td><td>19.99</td><td>0.24</td><td>Swift/UVOT</td><td>this work</td></tr><tr><td>0.0805</td><td>U</td><td>20.25</td><td>0.31</td><td>Swift/UVOT</td><td>this work</td></tr><tr><td>0.9751</td><td>U</td><td>21.42</td><td>0.23</td><td>Swift/UVOT</td><td>this work</td></tr><tr><td>22.568</td><td>U</td><td>>22.78</td><td>...</td><td>Swift/UVOT</td><td>this work</td></tr><tr><td>0.0710</td><td>UVW2</td><td>>21.34</td><td>...</td><td>Swift/UVOT</td><td>this work</td></tr><tr><td>0.0846</td><td>UVW2</td><td>>20.75</td><td>...</td><td>Swift/UVOT</td><td>this work</td></tr><tr><td>0.1382</td><td>UVW2</td><td>21.47</td><td>0.19</td><td>Swift/UVOT</td><td>this work</td></tr><tr><td>0.2174</td><td>UVW2</td><td>21.96</td><td>0.29</td><td>Swift/UVOT</td><td>this work</td></tr><tr><td>0.2836</td><td>UVM2</td><td>21.26</td><td>0.32</td><td>Swift/UVOT</td><td>this work</td></tr><tr><td>2.3981</td><td>UVM2</td><td>>23.17</td><td>...</td><td>Swift/UVOT</td><td>this work</td></tr></tbody></table>

Note. The photometric data presented in this table have not been corrected for either Galactic or host galaxy extinction.

### 3. Modeling the Afterglow of GRB 211024B

#### 3.1. Temporal Analysis

The X-ray light curve can be divided into two phases: a plateau-like phase before 15 ks, followed by a steep decay, and a regular decay phase with a jet-break at 10 days. Among the ground-based optical observations, the *r*-band data are the most plentiful. It can also be divided into two phases: a bump-like phase before 3 ks and a regular afterglow decay phase after one day. Therefore, we use the smoothly broken power law (SBPL)

to fit the light curves of each phase in these two bands. The definition of the SBPL is as follows<fnref n="16" />:

$$f(t) = A \left(\frac{t}{t_b}\right)^{-\alpha_1} \left\{ \frac{1}{2} \left[ 1 + \left(\frac{t}{t_b}\right)^{1/s} \right] \right\}^{(\alpha_1 - \alpha_2)s} \quad (1)$$

<footnote n="16">¹⁶ https://docs.astropy.org/en/stable/api/astropy.modeling.powerlaws.SMOOTHLYBROKENPOWERLAW1D.html#astropy.modeling.powerlaws.SMOOTHLYBROKENPOWERLAW1D</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Multiband prompt emission and afterglow light curve of GRB 211024B in logarithmic timescale. The inset shows the Swift/BAT light curve in linear timescale. Inverted triangles represent $3\sigma$ upper limits. Dashed lines depict the best-fit smoothly broken power-law models to the light curves. Note that the data have been subtracted for the host galaxy flux contribution and no Galactic extinction correction has been made.</figcaption></figure>

where A is model amplitude at the break time, $t_b$ is the break time, s is smoothness parameter, and $\alpha_1$ and $\alpha_2$ are power-law index before and after break time, respectively. It should be noted that we do not include the LBT observation of the host galaxy at 182 days in the r-band fitting. The fitting results are presented in Figure 1 and Table 2.

As shown in Figure 1, the r-band light curve in Phase 1 shows an initial bump with rising power-law index $\alpha_{r,1} \sim 1.1$, peaking time 417 s, and decaying power-law index $\alpha_{r,2} \sim -0.96$. We also find a shallow decay following this bump. In Phase 2, the r-band light curve exhibits a typical power-law decay with an index of $\alpha_{r,3} \sim -0.75$, followed by a break at $\sim 1.6 \times 10^6$ s and a subsequent decay with a power-law index of $\alpha_{r,4} \sim -2.2$. On the other hand, the X-ray light curve in Phase 1 shows a shallow decay with a power-law index of $\alpha_{X,1} \sim -0.6$ initially, followed by a break at $\sim 15$ ks and a steep decay with a slope of $\alpha_{X,2} \sim -7.5$. The X-ray light curve in Phase 2 is generally consistent with the optical r-band, starting with a typical power-law decay with an index of $\alpha_{X,3} \sim -0.9$ and then decaying with an index of $\alpha_{X,4} \sim -1.82$ after a break at $1.4 \times 10^6$ s.

It is worth noting that the NOT observation at 65 days after the burst may be contaminated by the host galaxy flux. We estimate the host galaxy flux using the LBT observation at

182 days. After subtracting the host galaxy flux contribution, although this leads to a large uncertainty in the afterglow flux, we can still see that the break times as well as the decay index of the r-band and X-ray light curves are consistent. This is likely due to a jet-break effect, which can be used to constrain the half-opening angle of the jet.

## 3.2. Theoretical Interpretation

As shown above, the X-ray light curve of GRB 211024B exhibits a “plateau” followed by a sharp decay, then a normal decay and finally a break. The optical light curve shows an initial bump followed by a shallow decay, and then a normal decay followed by a break. In this section, we will show that the initial optical bump and the X-ray/optical normal decay phase are consistent with the standard afterglow model predictions. However, the interpretation for the X-ray plateau and optical shallow decay demand late-time central engine activity.

### 3.2.1. Internal Plateau in X-Ray and Central Engine Activity

The near-flat plateau ($\alpha_1 \sim 0.6$) followed by a sharp decay ($\alpha_2 \sim 7.5$) in early X-ray afterglow ($t > 2 \times 10^4$ s) cannot be accommodated in any external shock model. Such an internal plateau has to be attributed to the internal dissipation of a
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Cut-outs of VLT/X-shooter two-dimensional spectrum of the host galaxy. The emission line at 10582 Å is [O III] (top right-hand panel), we can derive redshift of z = 1.113, which is supported by the weak signals of [O II] seen at 7875 Å (top left-hand panel) and Hβ at 10271 Å (top middle panel). However, if we assume that the emission line at 10582 Å is Hα (bottom right-hand panel), implying a corresponding redshift of z = 0.612, we should expect to see [O III] at 8073 Å (bottom middle panel) and possibly Hβ at 7836 Å (bottom left-hand panel), but no evidence of emission lines could be found in the expected region of the two-dimensional spectrum.</figcaption></figure>

central engine (H.-J. Lü et al. 2015; W. Chen et al. 2017), most likely a supramassive millisecond magnetar. The abrupt drop can be interpreted as the collapse of the supramassive magnetar into a black hole (E. Troja et al. 2007; H.-J. Lü et al. 2015; W. Chen et al. 2017). In this phase, the jet internal dissipation power $L_{dis}$ and the X-ray isotropic luminosity $L_{X,iso}$ are related as

$$L_{X,iso} = L_{dis}/f_b, \qquad (2)$$

where $f_b = 1 - \cos \theta_j$ is the beaming factor of the jet.

### 3.2.2. Optical Shallow Decay and Energy Injection to Forward Shock

After the sliding phase, the relativistic blast wave collides with the circumburst medium (CBM), giving rise to long-lived multiband afterglow emission. We consider the forward shock (FS) of a relativistic thin shell with energy $E_{K,iso}$, initial Lorentz factor $\Gamma_0$, and opening angle $\theta_j$ expanding into the CBM with density $n_0$ (M. J. Rees & P. Meszaros 1992; P. Mészáros & M. J. Rees 1997; R. Sari et al. 1998; R. Sari & T. Piran 1999; Y. C. Zou et al. 2005). The dynamics of the shell are described by following equation (Y. F. Huang et al. 2000)

$$\frac{dR}{dt} = \beta c \Gamma (\Gamma + \sqrt{\Gamma^2 - 1}), \qquad (3)$$

$$\frac{dm}{dR} = 2\pi R^2 (1 - \cos \theta_j) n_0 m_p, \qquad (4)$$

$$\frac{d\Gamma}{dm} = - \frac{\Gamma^2 - 1}{M_{ej} + 2\Gamma m}, \qquad (5)$$

where R and t are the radius and time of the event in the burst frame, respectively, m is the swept-up mass, $M_{ej} =$

$E_{K,iso}(1 - \cos \theta_j)/2(I_0 - 1)c^2$ is the ejecta mass, and $m_p$ is the proton mass, and $\beta = \sqrt{\Gamma^2 - 1}/\Gamma$.

The initial optical bump and X-ray/optical normal decay will be interpreted by the FS with the above dynamics (J.-Z. Wang et al. 2014).

As discussed in Section 3.2.1, the internal plateau in X-ray light curve indicates that the central engine restarts during the plateau phase. Besides the energy $L_{dis}$ dissipated into X-ray emission, some part of the jet energy $L_{inj}$ is expected to be injected into the blast wave, thereby increasing the observed afterglow luminosity. Such observational characteristics are reflected exactly in the shallow decay of the r-band light curve: the early flux estimated based on the late afterglow is significantly higher than the observed value after the initial bump. Therefore, we consider the energy injection from GRB central engine in this phase, and Equation (5) should be replaced by (J. J. Geng et al. 2013)

$$\frac{d\Gamma}{dm} = - \frac{\Gamma^2 - 1 - \frac{1-\beta}{\beta c^3} L_{inj} dR/dm}{M_{ej} + 2\Gamma m}, \qquad (6)$$

During the injection time, i.e., $t_{start} < t < t_{end}$, the injected luminosity is $L_{inj} = L_{inj}^0(t/t_{start})^{-q}$, where $L_{inj}^0$ is the initial injection power, q is the decay power-law index, $t_{start}$ and $t_{end}$ are the start and end time for energy injection. By solving these equations with the initial conditions, one can find the evolution of $\Gamma(t)$ and $R(t)$.

Electrons are believed to be accelerated at the shock front to a power-law distribution $N(\gamma_e) \propto \gamma_e^{-p}$ during the dynamical evolution of FS. Assuming a constant fraction $\epsilon_e$ of the shock
<table><caption>**Table 2**
Measurements of the Temporal Index</caption><thead><tr><th>Band</th><th>Phase</th><th>α<sub>1</sub></th><th>α<sub>2</sub></th><th>log t<sub>b</sub></th><th>s</th><th>χ<sup>2</sup><sub>r</sub> (d.o.f)</th></tr></thead><tbody><tr><td>X-ray</td><td>1</td><td>0.60<sup>+0.02</sup><sub>-0.02</sub></td><td>7.50<sup>+0.46</sup><sub>-0.35</sub></td><td>4.18<sup>+0.01</sup><sub>-0.01</sub></td><td>0.06<sup>+0.02</sup><sub>-0.03</sub></td><td>2.15</td></tr><tr><td>X-ray</td><td>2</td><td>0.90<sup>+0.04</sup><sub>-0.04</sub></td><td>1.82<sup>+0.69</sup><sub>-0.60</sub></td><td>6.15<sup>+0.13</sup><sub>-0.15</sub></td><td>0.01*</td><td>1.48</td></tr><tr><td>r</td><td>1</td><td>-1.10<sup>+0.25</sup><sub>-0.37</sub></td><td>0.96<sup>+0.08</sup><sub>-0.07</sub></td><td>2.62<sup>+0.03</sup><sub>-0.04</sub></td><td>0.12<sup>+0.08</sup><sub>-0.06</sub></td><td>0.79</td></tr><tr><td>r</td><td>2</td><td>0.74<sup>+0.06</sup><sub>-0.06</sub></td><td>2.20<sup>+0.55</sup><sub>-0.60</sub></td><td>6.21<sup>+0.22</sup><sub>-0.19</sub></td><td>0.01*</td><td>0.25</td></tr></tbody></table>

Note. Fitting results of X-ray and r-band light curve with the SBPL model. Parameters marked with asterisks indicate that they are fixed during the fitting.

energy $e_2 = 4\Gamma^2 n_1 m_p c^2$ is distributed into electrons, this defines the minimum injected electron Lorentz factor,

$$ \gamma_m = \frac{p-2}{p-1} \epsilon_e (\Gamma - 1) \frac{m_p}{m_e} \quad (7) $$

where $m_e$ is electron mass. We also assume a fraction $\epsilon_B$ of the shock energy resides in the magnetic field generated downstream of the shock. The comoving magnetic field is given by:

$$ B = (32\pi m_p \epsilon_B n_0)^{1/2} c. \quad (8) $$

The synchrotron power and characteristic frequency from electron with Lorentz factor $\gamma_e$ are

$$ P(\gamma_e) \approx \frac{4}{3} \sigma_T c \Gamma^2 \gamma_e^2 \frac{B^2}{8\pi}, \quad \nu(\gamma_e) \approx \Gamma \gamma_e^2 \frac{q_e B}{2\pi m_e c}, \quad (9) $$

where $\sigma_T$ is the Thomson cross-section, $q_e$ is electron charge. The peak spectra power occurs at $\nu(\gamma_e)$

$$ P_{\nu, \text{max}} \approx \frac{P(\gamma_e)}{\nu(\gamma_e)} = \frac{m_e c^2 \sigma_T}{3q_e} \Gamma B. \quad (10) $$

One can define the critical electron Lorentz factor $\gamma_c$ by setting the electron's lifetime equal to the time $t$,

$$ \gamma_c = \frac{6\pi m_e c}{\Gamma \sigma_T B^2 t}. \quad (11) $$

When cooling due to synchrotron radiation becomes significant, the shape of the electron distribution should be adjusted for $\gamma_e > \gamma_c$. Given the radiative cooling and the continuous injection of new accelerated electrons, coupled with the synchronous self-absorption effect, a broken power-law spectrum can be obtained. This spectrum is segmented into several sections based on three characteristic frequencies: $\nu_m$, $\nu_c$, and $\nu_a$ (H. Gao et al. 2013). The $\nu_m$ and $\nu_c$ are defined by $\gamma_m$ and $\gamma_c$, respectively, while $\nu_a$ is characterized by synchronous self-absorption. The peak flux density is $F_{\nu, \text{max}} = N_{e,2}P_{\nu, \text{max}}/4\pi D^2$, where $N_{e,2} = 4\pi R^3 n_0/3$ is the total number of electrons in shocked region of CBM and D is the distance of the source.

We perform multiwavelength data fitting for GRB 211024B utilizing the established afterglow model code PyFRS<fnref n="17" /> (see also W.-H. Lei et al. 2016; Z.-P. Zhu et al. 2023; C. Zhou et al. 2024), considering two scenarios for the CBM ($n = n_{18}R^{-k}$): uniform interstellar medium (ISM, $k=0$) and wind-like medium ($k=2$). All data used for fitting have been corrected for Galactic extinction. To explore the parameter space, we utilize the Python-based Markov chain Monte Carlo (MCMC) sampler emcee (D. Foreman-Mackey et al. 2013). For the X-ray band, only data acquired after ~50 ks (Phase 2) were used in the afterglow fitting. If the energy injection of the

afterglow originates from the dipole radiation of a magnetar, the expected end time ($t_{end}$) in the source frame is $t_{end} = t_b/(1+z) \sim 7.1$ ks, where $t_b$ is obtained from fitting of the X-ray light curve in Phase 1. However, if the energy comes from other forms of radiation, the end of injection time becomes uncertain. Therefore, we considered both scenarios in our fitting process, setting the energy injection end time either as a fixed value or as a free parameter.

The MCMC analysis using 120 walkers and 40,000 steps is employed to explore the parameter space, discarding the first 20,000 steps as burn-in. Under this setting, all fits satisfy the condition that the chain length exceeds 80 times the estimated autocorrelation time and the estimate of the autocorrelation time varies by less than 1%, indicating that the fits to have converged sufficiently. We employed the Bayesian Information Criterion (BIC) to assess the goodness of fit. The resulting best-fit parameters are presented in Table 3 and visualized in Figure 3. The posterior probability distribution of the parameters is shown in Figure 4.

Regardless of the scenario, the model fits the observational data well. The BIC values for different scenarios are not significantly different. For the ISM scenario, the best-fit results generally favor wide jet opening angles ($\theta_j > 10^\circ$) and low isotropic kinetic energy ($E_k \sim 10^{51}$ erg). The injection luminosity is around $10^{47}$ erg s$^{-1}$ for both the fixed $t_e$ and free $t_e$ cases, but the former gives a nearly constant luminosity ($q \sim 0$), while the latter shows a gradual decay ($q \sim 0.7$).

The wind-like scenario yields relatively high isotropic kinetic energies ($E_k \sim 10^{53}-10^{54}$ erg) and narrow jet opening angles ($\theta_j \sim 1$), with injection luminosities reaching $10^{50}-10^{52}$ erg s$^{-1}$. For a magnetar with a typical rotational energy of $E \sim 10^{52}$ erg, such high luminosities are difficult to sustain via magnetic dipole radiation, suggesting that the injection energy in this scenario may originate from other sources, such as fallback accretion-powered jets.

# 4. Discussion

## 4.1. Ultra-long Gamma-ray Burst

The classical classification method of GRBs divides them into SGRBs and LGRBs based on the $T_{90}$ criterion of 2 s, corresponding to the compact star merger and collapsar origins, respectively. However, there are some exceptions, such as GRB 060614 (B. Zhang et al. 2007), GRB 160410A (J. F. Agüf Fernández et al. 2023), GRB 211211A (J. C. Rastinejad et al. 2022; J. Yang et al. 2022), and GRB 200826A (A. Rossi et al. 2022). The different energy bands of different detectors can lead to some discrepancies in the $T_{90}$ values of GRBs. Moreover, the peak energy of gamma-rays may quickly drop below the energy band of the detector, resulting in a smaller $T_{90}$, even though the central engine is still active. B.-B. Zhang et al. (2014) studied the Swift GRBs sample and

<footnote n="17">¹⁷ https://github.com/leiwh/PyFRS</footnote>
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 3.** The best-fit afterglow light curve of GRB 211024B with energy injection is shown for two scenarios: (a) a uniform ISM, and (b) a wind-like medium. The solid line represents the case where $t_{end}$ is treated as a free parameter, while the dashed line corresponds to a fixed $t_{end}$.</figcaption></figure>

<table><caption>**Table 3**

Parameters of Afterglow Modeling</caption><thead><tr><th rowspan="2">Parameters</th><th rowspan="2">Unit</th><th rowspan="2">Prior Type</th><th rowspan="2">Prior</th><th colspan="4">Results</th></tr><tr><th>k = 0 (fix t<sub>end</sub>)</th><th>k = 0</th><th>k = 2 (fix t<sub>end</sub>)</th><th>k = 2</th></tr></thead><tbody><tr><td>E<sub>k,iso</sub></td><td>erg</td><td>log-uniform</td><td>[10<sup>51</sup>, 10<sup>56</sup>]</td><td>6.76<sup>+5.26</sup><sub>-2.78</sub> × 10<sup>51</sup></td><td>3.16<sup>+1.85</sup><sub>-1.02</sub> × 10<sup>51</sup></td><td>1.29<sup>+1.28</sup><sub>-0.61</sub> × 10<sup>54</sup></td><td>1.45<sup>+1.18</sup><sub>-0.67</sub> × 10<sup>53</sup></td></tr><tr><td>Γ<sub>0</sub></td><td>l</td><td>log-uniform</td><td>[1, 1000]</td><td>537<sup>+239</sup><sub>-182</sub></td><td>158<sup>+144</sup><sub>-53.8</sub></td><td>219<sup>+69.6</sup><sub>-45.0</sub></td><td>135<sup>+42.9</sup><sub>-30.2</sub></td></tr><tr><td>p</td><td>uniform</td><td>l</td><td>[2.01, 4.0]</td><td>2.09<sup>+0.03</sup><sub>-0.02</sub></td><td>2.21<sup>+0.04</sup><sub>-0.04</sub></td><td>2.43<sup>+0.01</sup><sub>-0.01</sub></td><td>2.43<sup>+0.01</sup><sub>-0.01</sub></td></tr><tr><td>n<sub>18</sub><sup>a</sup></td><td>cm<sup>-3</sup></td><td>log-uniform</td><td>[10<sup>-8</sup>, 10<sup>3</sup>]</td><td>4.90<sup>+41.9</sup><sub>-4.27</sub> × 10<sup>-3</sup></td><td>3.31<sup>+11.8</sup><sub>-2.89</sub></td><td>6.76<sup>+4.20</sup><sub>-3.21</sub> × 10<sup>-2</sup></td><td>6.76<sup>+3.95</sup><sub>-3.13</sub> × 10<sup>-2</sup></td></tr><tr><td>θ<sub>j</sub></td><td>degrees</td><td>uniform</td><td>[0.01, 45]</td><td>11.32<sup>+4.26</sup><sub>-3.07</sub></td><td>22.32<sup>+4.33</sup><sub>-5.10</sub></td><td>1.45<sup>+0.34</sup><sub>-0.32</sub></td><td>0.86<sup>+0.18</sup><sub>-0.18</sub></td></tr><tr><td>θ<sub>obs</sub></td><td>degrees</td><td>uniform</td><td>[0.01, 45]</td><td>5.59<sup>+4.13</sup><sub>-3.51</sub></td><td>10.56<sup>+7.64</sup><sub>-6.67</sub></td><td>2.77<sup>+0.64</sup><sub>-0.61</sub></td><td>2.74<sup>+0.52</sup><sub>-0.57</sub></td></tr><tr><td>ε<sub>e</sub></td><td>l</td><td>log-uniform</td><td>[10<sup>-4</sup>, 1]</td><td>2.19<sup>+1.98</sup><sub>-0.99</sub> × 10<sup>-1</sup></td><td>4.17<sup>+1.72</sup><sub>-1.48</sub> × 10<sup>-1</sup></td><td>2.34<sup>+0.68</sup><sub>-0.65</sub> × 10<sup>-2</sup></td><td>5.13<sup>+2.37</sup><sub>-1.50</sub> × 10<sup>-2</sup></td></tr><tr><td>ε<sub>B</sub></td><td>l</td><td>log-uniform</td><td>[10<sup>-4</sup>, 1]</td><td>6.17<sup>+12.5</sup><sub>-4.35</sub> × 10<sup>-2</sup></td><td>5.50<sup>+14.0</sup><sub>-3.41</sub> × 10<sup>-4</sup></td><td>3.31<sup>+6.02</sup><sub>-1.73</sub> × 10<sup>-4</sup></td><td>2.82<sup>+4.26</sup><sub>-1.34</sub> × 10<sup>-4</sup></td></tr><tr><td>L<sub>inj</sub></td><td>erg s<sup>-1</sup></td><td>log-uniform</td><td>[10<sup>45</sup>, 10<sup>54</sup>]</td><td>1.26<sup>+1.56</sup><sub>-0.76</sub> × 10<sup>47</sup></td><td>7.41<sup>+4.34</sup><sub>-2.40</sub> × 10<sup>47</sup></td><td>3.39<sup>+2.37</sup><sub>-1.35</sub> × 10<sup>52</sup></td><td>1.58<sup>+0.99</sup><sub>-0.61</sub> × 10<sup>50</sup></td></tr><tr><td>t<sub>start</sub></td><td>s (source frame)</td><td>log-uniform</td><td>[10<sup>0</sup>, 10<sup>5</sup>]</td><td>1.15<sup>+0.99</sup><sub>-0.61</sub> × 10<sup>3</sup></td><td>1.62<sup>+0.52</sup><sub>-0.47</sub> × 10<sup>3</sup></td><td>2.51<sup>+0.24</sup><sub>-0.27</sub> × 10<sup>1</sup></td><td>2.24<sup>+6.47</sup><sub>-1.39</sub> × 10<sup>1</sup></td></tr><tr><td>t<sub>end</sub></td><td>s (source frame)</td><td>fixed/log-uniform</td><td>[10<sup>2</sup>, 10<sup>7</sup>]</td><td>7.10 × 10<sup>3</sup></td><td>4.57<sup>+11.6</sup><sub>-2.71</sub> × 10<sup>5</sup></td><td>7.10 × 10<sup>3</sup></td><td>2.04<sup>+0.36</sup><sub>-0.26</sub> × 10<sup>5</sup></td></tr><tr><td>q</td><td>l</td><td>uniform</td><td>[-2.0, 2.0]</td><td>-0.33<sup>+0.35</sup><sub>-0.25</sub></td><td>0.69<sup>+0.06</sup><sub>-0.05</sub></td><td>0.61<sup>+0.04</sup><sub>-0.04</sub></td><td>-0.06<sup>+0.02</sup><sub>-0.02</sub></td></tr><tr><td>BIC</td><td></td><td></td><td></td><td>-1794.92</td><td>-1810.95</td><td>-1762.27</td><td>-1792.66</td></tr></tbody></table>

**Note.**

<sup>a</sup> $n_{18}$ is the CBM density at $R = 10^{18}$ cm.

found that the active time of the central engine ($t_{burst}$) is generally longer than $T_{90}$. Therefore, ULGRBs, as a unique category of LGRBs, may have different central engines from LGRBs. It is not appropriate to simply use $T_{90}$ as the sole criterion for classification. The criterion proposed by L14 for ULGRBs is a better solution. In addition to requiring a radiation time (consider γ-ray and X-ray emission both) of more than $10^3$ s, it also imposes two additional constraints: the presence of flares or dips in the X-ray plateau phase, and a steep decay following the plateau phase that is consistent with the prediction of high-latitude emission. Plateaus can be classified into two types: internal plateau and external plateau. Internal plateaus directly originate from the internal dissipation of the central engine. If a steep decay follows the end of the plateau, it characterizes the central engine shutdown timescale. In contrast, external plateaus originate from the

long-lasting energy injection of the central engine into the external shock. Therefore, the decay index following the plateau is predicted by the external shock model. For external plateaus, the end of the plateau may be later than the time of the central engine shutdown, as shown by L.-P. Xin et al. (2016). Therefore, the additional criterion of “plateau followed by a steep decay” directly characterizes the duration of the central engine activity.

Figure 5 presents the X-ray light curves for 1498 GRBs observed by Swift between 2004 December and 2023 May (gray lines). Three ULGRBs reported by L14 are also plotted as colored lines in Figure 5. It is clear that the flux of ULGRBs is significantly higher than that of most “normal” GRBs during the plateau phase (~$10^4$ s), followed by a steep decay after the plateau. The light-curve behavior of GRB 211024B (black line in Figure 5) is very similar to that
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Figure 4. Posterior probability distributions of the afterglow parameters of GRB 211024B for different cases: (a) ISM with fix $t_{end}$; (b) ISM with free $t_{end}$; (c) wind-like medium with fix $t_{end}$; (d) wind-like medium with free $t_{end}$.</figcaption></figure>

of ULGRBs. Therefore, GRB 211024B is a ULGRB according to the definition of L14.

## 4.2. Central Engine Properties

A supramassive magnetar may collapse into a black hole after reaching the maximum gravitational mass. The observational correspondence of this prediction is the internal X-ray plateau that is discovered in some long and short GRBs (H.-J. Lü & B. Zhang 2014; H.-J. Lü et al. 2015; H. Gao et al. 2016). GRB 211024B is likely such a case.

### 4.2.1. Magnetar Collapse Caused by Dipole Radiation Spin-down

A newborn magnetar experiences spin-down, driven by both magnetic dipole radiation and gravitational wave emission (B. Zhang & P. Mészáros 2001). The spin-down rate is given by:

$$ \dot{E} = I\Omega\dot{\Omega} = -\frac{B_p^2 R^6 \Omega^4}{6c^3} - \frac{32I^2\epsilon^2\Omega^6}{5c^5} \quad (12) $$
<figure><img src="image_10.png" /><figcaption>**Figure 5.** X-ray (0.3–10 keV) light curves of Swift GRBs. ULGRBs are represented by colored line, “normal” GRBs by gray lines, and GRB 211024B by black line.</figcaption></figure>

where *I* denotes the moment of inertia, $\Omega = 2\pi/P$ denotes the angular frequency, $\dot{\Omega}$ denotes the time derivative thereof, $B_p$ denotes the surface magnetic field strength at the poles, *R* corresponds to the magnetar’s radius, and $\epsilon$ is the ellipticity of the magnetar.

The spin-down rate can be dominated by either magnetic dipole radiation or gravitational wave radiation. When spin-down dominated by magnetic dipole radiation,

$$t_{\text{md}} = \frac{3c^3I}{B_p^2 R^6 \Omega_0^2} \approx 2.05 \times 10^3 \text{s}(I_{45}B_{p,15}^{-2}P_{0,-3}R_6^{-6}), \quad (13)$$

where $B_{p,15}$ is the magnetic field strength in units of $10^{15}$ G, $P_{0,-3}$ is the initial spin period of newborn magnetar in milliseconds, $R_6$ is magnetar radius in units of $10^6$ cm, $I_{45}$ is the moment of inertia in units of $10^{45}$ g cm$^{-2}$. When spin-down dominated by gravitational wave emission,

$$t_{\text{GW}} = \frac{5c^5}{128GI\epsilon^2\Omega_0^4} \approx 9.1 \times 10^3 \text{s}(I_{45}^{-1}P_{0,-3}^4\epsilon^{-2}). \quad (14)$$

The spin-down luminosity $L_0$ and the spin-down timescale $\tau$ of magnetar can be calculated by (B. Zhang & P. Mészáros 2001),

$$L_0 = 1.0 \times 10^{49} \text{ erg s}^{-1} (B_{p,15}^2 P_{0,-3}^{-4} R_6^6) \quad (15)$$

$$\tau = \min(t_{\text{md}}, t_{\text{GW}}) \quad (16)$$

The relation of spin-down luminosity $L_0$ and the plateau luminosity $L_{X,iso}$ defined as

$$\eta_X L_0 = \eta_X (L_{\text{dis}} + L_{\text{inj}}) = L_{X,\text{iso}} f_b \quad (17)$$

where $\eta_X$ is the radiation efficiency and $L_{\text{inj}}$ is the energy injection luminosity.

As discussed in B. Zhang & P. Mészáros (2001), Y.-Z. Fan et al. (2013), P. D. Lasky & K. Glampedakis (2016), and L. Xie et al. (2022), if a newborn magnetar rotates rapidly ($P_0 \sim 1$ ms) and has a significant ellipticity ($\epsilon > 10^{-3}$), the initial spin-down phase is dominated by gravitational wave emission. Otherwise, the spin-down is dominated by dipole radiation throughout. In a more general scenario, the spin-down mechanism would transition from gravitational wave dominated to magnetic dipole radiation dominated. For simplicity, we assume that the spin-down is dominated by only one of these mechanisms throughout the entire process.

The evolution of the spin period of magnetars is described by the following equation (S. L. Shapiro & S. A. Teukolsky 1983):

$$P(t) = P_0 \left(1 + \frac{t}{\tau}\right)^{1/2} \quad (18)$$

As the period of a magnetar increases, the centrifugal force cannot resist gravity and collapses into a black hole, resulting a sharp decay in X-ray flux, as observed. The maximum gravitational mass of supramassive magnetars can be expressed as a function of the spin period (P. D. Lasky et al. 2014),

$$M_{\text{max}} = M_{\text{TOV}}(1 + \hat{\alpha}P^{\hat{\beta}}) \quad (19)$$

where $M_{\text{TOV}}$ is the maximum mass of a nonrotating neutron star (NS), $\hat{\alpha}$ and $\hat{\beta}$ depend on the NS Equation of State (EoS).

In this paper, we adopt neutron star EoS GM1 (the maximum mass of a nonrotating NS is $M_{\text{TOV}} = 2.37 M_{\odot}$, the neutron star radius is $R = 12.05$ km, the moment of inertia is $I = 3.33 \times 10^{45}$ g cm$^{-2}$, $\hat{\alpha} = 1.58 \times 10^{-10} \text{s}^{-\hat{\beta}}$ and $\hat{\beta} = -2.84$) in the modeling. Recently, A. Li et al. (2016) suggested that the internal plateau might be produced by a QS. Our model is not sensitive to the EoS, therefore our main results will still hold even if we adopt QS EoSs.

In Section 3, we derived the X-ray plateau flux $F_b \sim 3.0 \times 10^{-10} \text{ erg s}^{-1} \text{ cm}^{-2}$ and constrained GRB afterglow parameters under various assumptions. It is noteworthy that the injection luminosity in the wind-like medium scenario is found to be remarkably high, reaching values of $10^{50}–10^{52} \text{ erg s}^{-1}$. Given that the rotational energy of a typical magnetar is of the order of $10^{52} \text{ erg}$, it is evident that additional energy sources must be invoked to sustain such high luminosity. Fallback accretion-powered jets represent a plausible alternative. Here, we only consider the case of $k=0$ with a fixed $t_e$, where $\tau \sim t_e = t_{\text{col}} = t_b/(1+z) = 7.10 \times 10^3 \text{ s}$. By substituting these afterglow parameters into Equations (15), (16), and (17), we obtain the magnetar’s initial spin period ($P_0$) and magnetic field ($B_p$) for both dipole radiation-dominated and gravitational wave-dominated regimes. For the dipole-dominated case, $P_0 = 8.4 \text{ ms}$ and $B_p = 4.7 \times 10^{15} \text{ G}$. For the gravitational wave-dominated case, $P_0 = 0.79 \text{ ms}$, $B_p = 3.3 \times 10^{14} \text{ G}$. The magnetar’s radiative efficiency ($\eta_X$) is found to be 0.07. It is worth noting that the spin-down timescale typically exceeds the collapse time. Consequently, the derived spin period and magnetic field strength should be interpreted as upper limits.

A. Rowlinson et al. (2014) compiled a sample of Swift X-ray afterglow light curves with known redshifts from 2005 January to 2013 December. They modeled these light curves using a magnetar model, assuming energy release via dipole radiation, deriving magnetar spin periods ranging from 0.2 to 100 ms and magnetic field strengths between $2.0 \times 10^{14} \text{ G}$ and $2.0 \times 10^{17} \text{ G}$ (represented by black dots in Figure 6). Since spin-down luminosity $L_0$ and spin-down timescale are estimated from observational data, it implies that their uncertainties are difficult to estimate. Consequently, the uncertainties of the derived parameters, namely the spin period and magnetic field strength, are also difficult to estimate. The data points in Figure 6 therefore do not include error bars. The magnetar associated with GRB 211024B (marked as stars in Figure 6) exhibits typical values of the whole sample.
<figure><img src="image_11.png" /><figcaption>Figure 6. Magnetic field strengths vs. spin periods of magnetars for Swift GRB samples (black dot, from A. Rowlinson et al. 2014). The magnetar corresponding to GRB 211024B is marked as a red star. It is important to note that all data points in the figure are derived under the assumption of EM-dominated spin-down.</figcaption></figure>

### 4.2.2. Magnetar Collapse Caused by Accretion

Following a supernova explosion, a significant amount of material is ejected into space at extremely high velocities. However, a fraction of this material fails to achieve escape velocity and subsequently falls back onto the central remnant. In such cases, if the central remnant is a magnetar, the evolution of spin and mass deviate from the conventional models presented in Section 4.2.1. Z. G. Dai & R.-Y. Liu (2012) conducted a comprehensive analysis of the influence of hyperaccreting fallback disks on the spin evolution of millisecond magnetars. In this case, the spin evolution of a magnetar is governed by two primary factors: magnetic dipole radiation and hyperaccreting fallback disks. The spin evolution of a magnetar can be given by

$$ \frac{d(I\Omega_s)}{dt} = \tau_{\text{acc}} + \tau_{\text{dip}} \qquad (20) $$

where *I* is the stellar moment of inertia, and $\tau_{\text{acc}}$ and $\tau_{\text{dip}}$ are net torque exerted on the magnetar by the accretion disk and the torque due to magnetic dipole radiation, respectively (the specific form is detailed in Z. G. Dai & R.-Y. Liu 2012). The early spin evolution of a magnetar is primarily influenced by accretion, particularly for low initial spin rates or high accretion rates, resulting in spin-up. In the late stages of evolution, however, magnetic dipole radiation becomes dominant, causing the magnetar to spin-down. Under the dual effects of accretion and spin-down, if the magnetar’s initial mass is sufficiently large or it accretes enough matter, the magnetar’s mass may exceed its maximum gravitational mass, causing it to collapse into a black hole. Before collapse, the dipole radiation of magnetar is

$$ L_{\text{dip}} = 9.6 \times 10^{48} \text{ erg s}^{-1} \sin^2 \chi \left( \frac{B_0 R^3}{10^{33} \text{ G cm}^3} \right)^2 \left( \frac{P}{1 \text{ ms}} \right)^{-4} \qquad (21) $$

where $\chi$ is the inclination angle of the magnetic axis to the spin axis, *R* is the radius of magnetar, $B_0$ is the surface magnetic field strength, and $P = 2\pi/\Omega_s$ is the time-dependence spin

period. The parameterize fallback accretion rate can be described by (A. I. MacFadyen et al. 2001; D. Zhang & Z. G. Dai 2008)

$$ \dot{M} = 10^{-3}[(\eta t^{1/2})^{-1} + (\eta t_p^{13/6} t^{-5/3})^{-1}]^{-1} M_{\odot} s^{-1} \qquad (22) $$

where $\eta$ is a factor that accounts for explosion energy and $t_p$ is the time of peek accretion rate. For more energetic explosion, $\eta$ is smaller and $t_p$ is larger. Therefore, the baryonic mass of the magnetar at time $t$ can be determined as

$$ M_b(t) = M_0 + \int_{t_0}^{t} \dot{M} dt \qquad (23) $$

where $M_0$ is the initial mass and $t_0$ is the start time of fallback accretion. Considering that a portion of the accreted matter is radiated away in the form of neutrinos, the corresponding gravitational mass is calculated to be

$$ M = M_b(t) \left[ 1 + \frac{3}{5} \frac{GM_b(t)}{Rc^2} \right] \qquad (24) $$

Through numerical integration of differential Equation (21), we can determine the evolutionary of the spin and luminosity of magnetar. We adopt an initial mass of $M_0 = 1.4 M_\odot$, an initial radius of $R_0 = 12.05$ km, an initial spin period of $P_0 = 4$ ms, a surface magnetic field of $B_0 = 10^{14}$ G, accretion start time of $t_0 = 100$ s, and $\sin^2 \chi = 0.5$ for the magnetar. Figure 7 presents these evolutionary curves for two specific cases: $\eta = 0.05$, $t_p = 900$ s and $\eta = 0.25$, $t_p = 300$ s. For both scenarios, the magnetar exhibits an initial spin-up phase within the first 1 ks, followed by a subsequent spin-down phase dominated by dipole radiation. Eventually, the magnetar accretes sufficient matter to exceed the maximum gravitational mass, resulting in a collapse into a black hole around 7 ks (source frame time). The precollapse dipole radiation luminosity is estimated to be $\sim 4 \times 10^{47}$ erg s$^{-1}$, consistent with observation results.

It is worth noting that Equation (21) does not account for gravitational wave radiation. In fact, when the magnetar’s spin period accelerates to 1 ms, gravitational wave radiation may become non-negligible (B. Zhang & P. Mészáros 2001). At this point, spin-up becomes difficult and the period evolution enters a brief plateau phase, leading to a plateau in the radiation luminosity as well. As the spin-down dominates at later times, both gravitational wave radiation and accretion effects become negligible.

Studies of the X-ray light curves of short GRBs have shown (e.g., A. Rowlinson et al. 2013; N. Sarin et al. 2020) that these rapidly rotating magnetars, born from binary mergers, collapse due to spin-down with collapse timescales ranging from $\sim 10^2$ to $\sim 10^3$ s. This statistical result differs from our derived collapse timescale of $t_{\text{col}} \sim 7$ ks, which suggests that the collapse mechanism of the magnetar responsible for GRB 211024B may differ from the standard spin-down scenario. Following a supernova explosion, a fraction of the material that fails to escape the gravitational potential well eventually falls back onto the central object, increasing its mass. In such cases, if the central object is a magnetar, its collapse would be induced by accretion. Furthermore, the torque from an early accretion disk can accelerate the magnetar’s spin initially, followed by spin-down due to dipole radiation, which can extend the magnetar’s collapse timescale.
<figure><img src="image_12.png" /><figcaption>**Figure 7.** (a) Temporal evolution of magnetic dipole luminosity (solid line) and spin period (dashed line); (b) temporal evolution of gravitational mass (solid line) and maximum gravitational mass (dashed line, described by Equation (19)). The blue lines correspond to case of $\eta = 0.05$, $t_p = 900$ s, and the red lines correspond to case of $\eta = 0.25$, $t_p = 300$ s.</figcaption></figure>

<figure><img src="image_13.png" /></figure>

Both of these factors can contribute to the observed long collapse timescales.

### 4.3. The Origin of Early Optical Emission

A subset of GRB early optical afterglow observations exhibit light-curve flux profiles that are characterized by an initial rise followed by a decline. This feature can be attributed to either a forward or reverse shock. If the emission is attributed to a FS, the time corresponding to the peak flux represents the deceleration timescale of the jet, which can be used to constrain the initial Lorentz factor. If the emission is attributed to a reverse shock, the time corresponding to the peak flux represents the crossing timescale of the reverse shock (M. J. Rees & P. Meszaros 1992; P. Mészáros & M. J. Rees 1997; R. Sari et al. 1998; R. Sari & T. Piran 1999; Y. C. Zou et al. 2005). S. Kobayashi & B. Zhang (2003) studied in detail the light-curve features of reverse shocks in the thin-shell and thick-shell cases. Under typical afterglow parameters, for the ISM the typical rise and decay indices of the light curve are 5 or 1/2 and -2, respectively; and for the Wind model the typical rise and decay indices of the light curve are 5/2 or 1/2 and -3, respectively (see Figure 1 in S. Kobayashi & B. Zhang 2003). In Section 3.1, we obtain rise and decay indices of 1.1 and -0.96 for the early optical emission of GRB 211024B, which are inconsistent with theoretical expectations. Of course, it is also possible that its afterglow parameters are rather special. On the other hand, the early optical emission could be the result of a combined contribution from a reverse shock and a FS (Z.-P. Zhu et al. 2023). The presence of an extra component can lead to the fitted decay index not accurately reflecting the emission characteristics of the reverse shock. Due to the lack of earlier and multiband observational data, we cannot further explore these cases.

## 5. Conclusion

We present optical follow-up observations of ULGRB GRB 211024B from ~200 s to ~65 days. The early phase X-ray light curve exhibits a two-part structure characterized by

an initial, slowly declining plateau followed by a very steep decay, which is a typical internal plateau feature. This is often taken as evidence for a magnetar as the central engine. In the optical r-band, the early bump-like light curve flux is lower than the extrapolation from the late-time afterglow. To account for the observed increase in late-time afterglow flux, we propose an energy injection mechanism into the external shock. We incorporated an energy injection term into our afterglow fitting, finding injected luminosities ranging from $10^{47}$ to $10^{52}$ erg s⁻¹ depending on the CBM. For the ISM with a fixed $t_e$ scenario, energy injection from magnetar dipole radiation provides a plausible explanation, with derived magnetar parameters consistent with the general magnetar population. However, for the other three scenarios, the energy injection termination time or luminosity deviates from magnetar model predictions, suggesting alternative energy sources.

If the magnetar is surrounded by a hyperaccretion disk, neutrino annihilation and magnetic activity from the stellar surface could produce relativistic jets along the magnetic poles (D. Zhang & Z. G. Dai 2009, 2010), injecting energy into the external shock. Additionally, fallback accretion onto a black hole formed from magnetar collapse could generate relativistic jets through the Blandford–Znajek mechanism (R. D. Blandford & R. L. Znajek 1977). These processes could explain the observed discrepancies between the energy injection parameters and the magnetar dipole radiation model.

The possibility that the early bump-like feature is caused by a reverse shock cannot be ruled out. In this case, the entire optical light curve can be explained without the need for a magnetar energy injection model. Unfortunately, due to the lack of early and multiband observational data, we are unable to confirm whether this is the case with GRB 211024B. For the current observational data, the most natural explanation is still the magnetar model discussed in this paper.

The case of GRB 211024B, along with previously studied ULGRBs (e.g., GRB 101225A, GRB 111209A), highlights the crucial role of an active central engine in these events. Multiband observations can reveal the signatures of the central engine, enabling us to improve our understanding of ULGRBs and constrain the physical parameters of the engine itself.
## Acknowledgments

The data presented here were obtained in part with ALFOSC, which is provided by the Instituto de Astrofísica de Andalucía (IAA) under a joint agreement with the University of Copenhagen and NOT. This research has made use of the Spanish Virtual Observatory (http://svo.cab.inta-csic.es) supported by the MINECO/FEDER through grant AyA2017-84089.7. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. D.X. acknowledges the science research grants from the China Manned Space Project with No. CMS-CSST-2021-A13 and CMS-CSST-2021-B11. This work is supported by the National Natural Science Foundation of China under grant 12473012. D.B.M. has been supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 725246) and by the European Union (ERC, HEAVYMETAL, 101071865). Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant DNRF140. E.M. and A.R. acknowledge support from PRIN-MIUR 2017 (grant 20179ZF5KS). A. Rossi also acknowledges support from INAF project Premiale Supporto Arizona & Italia.

Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme 106.21T6.015 (PI Tanvir, Vergani, Malesani); and with the LBT (program IT-2021B-018, PI Rossi); and with the Nordic Optical Telescope, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and the University of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofísica de Canarias (program 62-508, PI Malesani, Fynbo). We thank the staff of the VLT, NOT, LBT for their critical help in obtaining the data presented here. The LBT is an international collaboration of the University of Arizona, Italy (INAF: Istituto Nazionale di Astrofisica), Germany (LBTB: LBT Beteiligungsgesellschaft), The Ohio State University, representing also the University of Minnesota, the University of Virginia, and the University of Notre Dame.

**Facilities:** XMO:NEXT, NOT (ALFOSC), VLT:Kueyen (X-Shooter), LBT, Swift (BAT, XRT and UVOT).

**Software:** Astropy (Astropy Collaboration et al. 2013, 2018), Source Extractor (E. Bertin & S. Arnouts 1996), emcee (D. Foreman-Mackey et al. 2013), IRAF (D. Tody 1986).

## ORCID iDs

Shao-Yu Fu https://orcid.org/0009-0002-7730-3985

Dong Xu https://orcid.org/0000-0003-3257-9435

Wei-Hua Lei https://orcid.org/0000-0003-3440-1526

Antonio de Ugarte Postigo https://orcid.org/0000-0001-7717-5085

Daniele B. Malesani https://orcid.org/0000-0002-7517-326X

David Alexander Kann https://orcid.org/0000-0003-2902-3583

Johan P. U. Fynbo https://orcid.org/0000-0002-8149-8298

Andrea Rossi https://orcid.org/0000-0002-8860-6538

Diego Paris https://orcid.org/0000-0002-7409-8114

Zi-Pei Zhu https://orcid.org/0000-0002-9022-1928

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