Can Fallback Accretion on the Magnetar Model Power the X-Ray Flares Simultaneously Observed with Gamma Rays of Gamma-Ray Bursts?

The prompt emission, X-ray plateau, and X-ray flares of gamma-ray bursts (GRBs) are thought to be from internal dissipation, and the magnetar as the central engine with propeller fallback accretion is proposed to interpret the observed phenomena of GRBs. In this paper, by systematically searching for X-ray emission observed by Swift/X-ray Telescope, we find that seven robust GRBs include both X-ray flares and plateau emissions with measured redshift. More interestingly, the X-ray flares/bumps for those seven GRBs are simultaneously observed in the gamma-ray band. By adopting the propeller fallback accretion model to fit the observed data, it is found that the free parameters of two GRBs (140512A and 180329B) can be constrained very well, while in the other five cases, more or less, they are not all sufficiently constrained. On the other hand, this requires the conversion efficiency of the propeller to be two or three times higher than that of the spindown dipole radiation of the magnetar. If this is the case, it is contradictory to the expectation from the propeller model: namely, a dirtier ejecta should be less efficient in producing gamma-ray emissions. Our results hint that at least the magnetar central engine with propeller fallback accretion model cannot interpret very well both the GRB X-ray flares simultaneously observed in the gamma-ray band and the X-ray flares of GRBs with a high Lorentz factor.

There is a small fraction of LGRBs in which the prompt emission shows two or more sub-bursts of emission in the light curves, with quiescent times of up to hundreds of seconds, detected by both Fermi/Gamma-ray Burst Monitor (GBM) and Swift/Burst Alert Telescope (BAT; Koshut et al. 1995;Lazzati 2005;Burlon et al. 2008;Bernardini et al. 2013;Hu et al. 2014;Lan et al. 2018).Lan et al. (2018) investigated the spectral and temporal properties of two sub-bursts of emission in the prompt emission, and they did not find any statistically significant correlation in the duration and peak energy (E p ) of the two sub-bursts of emission.This suggests that those two or more sub-bursts of emission of GRB are likely to be from the same origin.
Thanks to Swift, a multiwavelength GRB mission (Gehrels et al. 2004) has led to great progress in understanding the nature of this phenomenon (for a review, see Zhang 2007).Especially, the prompt slewing capability of the onboard X-ray Telescope (XRT; Burrows et al. 2004) has revealed the discovery of a canonical X-ray light curve following the prompt emission, which shows successively four power-law decay segments (i.e., an initial steep decay segment, a shallow decay segment, a normal decay segment, and a post-jet-break phase), with superimposed erratic flares (Nousek et al. 2006;O'Brien et al. 2006;Zhang et al. 2006).Moreover, a small fraction of GRBs show an internal plateau3 in the X-ray light curve (Liang et al. 2007;Troja et al. 2007;Lyons et al. 2010;Rowlinson et al. 2010Rowlinson et al. , 2013;;Lü & Zhang 2014).
From the theoretical point of view, both the internal plateau and the X-ray flares are inconsistent with any external shock model, but must be attributed to the internal dissipation of a central engine wind.There are two forms of continuous energy injection into the external forward shock.One involves invoking a long-lasting central engine, such as a spinning-down millisecond magnetar (Dai & Lu 1998;Zhang & Mészáros 2001).The other one involves a stratification of the ejecta Lorentz factor in an impulsively ejected fireball (Rees & Mészáros 1998;Sari & Mészáros 2000;Uhm et al. 2012).The widely discussed model for dealing with the energy injection process involves assuming a millisecond magnetar as the central engine (Zhang & Mészáros 2001;Toma et al. 2007;Troja et al. 2007;Lü & Zhang 2014;Rea et al. 2015;Stratta et al. 2018;Fraija et al. 2021).Within the scenario of a magnetar central engine, the energy injection to explain the plateau (or shallow decay) phase is from the dipole radiation of the magnetar spindown.This has previously been studied for LGRBs (e.g., Lyons et al. 2010;Dall'Osso et al. 2011;Lü & Zhang 2014), short GRBs (e.g., Rowlinson et al. 2013;Lü et al. 2015Lü et al. , 2017)), as well as the extended emission of short GRBs (e.g., Gompertz et al. 2013).These works assumed a constant rate of spindown and therefore a constant level of dipole luminosity.
Another component, a flare/bump in the X-ray emission, is also an important clue to understanding the physical process in the framework of the magnetar central engine.This has been discovered in a good fraction of GRBs (Barthelmy et al. 2005;Chincarini et al. 2007;Falcone et al. 2007;Cusumano et al. 2010;Margutti et al. 2010).In most cases, flares/bumps are superposed on either the steep decay phase or the shallow decay phase (Nousek et al. 2006;Zhang et al. 2006).Peng et al. (2014) found that a good fraction of GRB X-ray flares, which were observed simultaneously by both BAT and XRT on board the Swift mission, obeyed the same power-law spectral fit.By studying the properties of both temporal and spectral behaviors, this suggests that the X-ray flares are produced by late central engine activities, and may share the same physical origin as the prompt emission of GRBs (Burrows et al. 2005;Yi et al. 2016).If this is the case, a small fraction of the GRB X-ray flares at least can be used to constrain the Lorentz factors, which range from a few tens to hundreds (Jin et al. 2010;Yi et al. 2015).On the other hand, several models have also been invoked to interpret the X-ray flares, such as the tail of prompt emission with a high-latitude curvature effect (Liang et al. 2006), delayed magnetic dissipation activity as the ejecta decelerates (Giannios 2006), and anisotropic emission in the blast wave comoving frame (Beloborodov et al. 2011).Moreover, the magnetar central engine with the delayed onset of a propeller regime, which accelerates local material via magnetocentrifugal slinging, has also been proposed to explain the X-ray flares in GRBs (Gibson et al. 2017(Gibson et al. , 2018)).This model has also been invoked to account for supernova explosions (Piro & Ott 2011) and X-ray plateaus for short GRBs (Gompertz et al. 2014).However, the main issue of the propeller mechanism for explaining the prompt emission of GRBs is baryon loading.The propeller mechanism expels a lot of unaccreted material from the system, and the outflow must be very dirty and nonrelativistic if the material is expelled in the funnel region.This means that no matter how large the power is, it cannot power a GRB itself.
One interesting question is whether the propeller mechanism with fallback accretion on the millisecond magnetar model can indeed interpret an X-ray light curve of a GRB that is composed of both plateau emission and an X-ray flare/bump.In this paper, by systematically searching for X-ray emission observed by Swift/XRT, we find X-ray light curves of seven GRBs composed of plateau emission and X-ray flares with measured redshift.The basic propeller with fallback accretion model of the magnetar central engine is shown in Section 2. In Section 3, we present the criteria for the sample selection, and the fitting results for those seven GRBs with the propeller model, by adopting the method of Markov Chain Monte Carlo (MCMC; Mackay 2003), are reported in Section 4. Finally, we summarize our discussion and conclusion in Section 5. Throughout the paper, a concordance cosmology with the parameters H 0 = 71 km s −1 Mpc −1 , Ω M = 0.30, and Ω Λ = 0.70 is adopted.

The Magnetar Propeller with Fallback Accretion Model
Previously, the propeller with fallback accretion model of a magnetar central engine has been invoked to explain short GRBs with extended emission (Gompertz et al. 2014;Gibson et al. 2017;Lan et al. 2020), X-ray plateau emission and X-ray flares in LGRBs (Gibson et al. 2018), as well as some strippedenvelope supernovae (Lin et al. 2021).In this section, we will briefly introduce the model of a magnetar propeller with fallback accretion.
The propeller regime is basically defined by the relationship between the Alfvén radius (r m ) and the corotation radius (r c ). Generally speaking, when r c < r m , the materials already within r c accrete to the surface of the magnetar, while the materials within the range of r c and r m are propelled away at r m .If the power of the propeller is not strong enough for the materials, it cannot reach the potential well.Then, the materials will return to the disk without any emission to be detected (called the "propeller regime" by Illarionov & Sunyaev 1975).Based on the paper of Gibson et al. (2017), r m and r c can be defined as follows: where G is the gravitational constant, μ = B p R 3 is the magnetic dipole moment, B p is the surface magnetic field, and M and R are the mass and radius of the magnetar, respectively.t ν = R d /αc s is the viscous timescale, and α and c s are the viscosity prescription and the speed of sound in the disk, respectively.ω is the angular frequency of the magnetar, and M d and R d are the mass and radius of the disk, respectively.In our calculations, we adopt α = 0.1 and c s = 10 7 cm s −1 (Gompertz et al. 2014).
At the Alfvén radius r m , the effect of materials from the accretion is smaller than that of the magnetic field of the magnetar, while at the corotation radius r c , the rotation rate of the matter in the disk is the same as that of the stellar surface, since r m and r c depend on the disk mass and the frequency of the magnetar, respectively.The evolution of the disk mass and frequency with time can be expressed as follows (Gibson et al. 2017): Here, M fb  is the fallback mass changed with time, and M prop  and M acc  are the mass lost via the propeller mechanism and the accretion to the magnetar, respectively.N acc and N dip are the accretion and dipole torques acting on the magnetar, respectively.I = 0.35MR 2 is the magnetar moment of inertia.
Based on the description of Gibson et al. (2017), and M acc  can be defined as follows: where M fb = δM d,0 is the available fallback mass, M d,0 is the initial mass of the disk, and δ is the ratio between the fallback mass and initial disk mass.t fb = òt ν is the fallback timescale, and ò is the ratio between the fallback timescale and viscous timescale.
is the efficiency of the propeller mechanism, and it relates to the "fastness parameter" GM r r r In this work, we adopt n = 1 to do the calculations.
For the dipole torque, we adopt the classical solution as given by Shapiro & Teukolsky (1983) and Piro & Ott (2011): Moreover, N acc has two forms that depend on the relationship between r m and R. It reads as: Then, do the integration of Equations ( 3) and (4), one can estimate the components of propelled luminosity and dipole luminosity: Here, η prop and η dip are the conversion efficiencies of the propeller and dipole energy luminosity, respectively.The total luminosity is therefore defined as follows: where 1/f B is the fraction of the stellar sphere that is emitting, and it is related to the half-opening angle of the jet (θ j ).One has 1999;Sari et al. 1999).By giving some typical parameters of the magnetar propeller with fallback accretion model, such as R = 10 6 cm, M = 1.4 M e , B p = 10 15 G, P 0 = 10 −3 s, M d,0 = 10 −3 M e , R d = 10 7 cm, ò = 0.1, δ = 10 −4 , η prop = 0.8, η dip = 0.1, and 1/f B = 10 1.7 , one can plot the picture of the luminosity as a function of time for both the propeller and dipole radiations (see Figure 1).

Sample Selection and Data Reduction
The XRT data are downloaded from the Swift data archive and the UK Swift Science Data Centre 4 (Evans et al. 2007(Evans et al. , 2009)).Our entire sample includes more than 1718 GRBs observed by Swift/XRT between 2005 January and 2023 June.The magnetar signature typically exhibits a shallow decay phase (or plateau), followed by a normal decay in X-ray emission when it is spinning down, and the delayed onset of a propeller regime that accelerates local material via magnetocentrifugal slinging can produce the X-ray flares.We only focus on the LGRBs with both flare/bump and plateau emissions observed in the X-ray afterglow, and 154 GRBs are too faint to be detected in the X-ray band or do not have enough photons to extract a reasonable X-ray light curve.Then, we select the GRBs whose X-ray emission exhibits the feature where fixed ω = 3 represents the sharpness of the peak and t p , α 1 , and α 2 are the peak time and decay slope of the plateau and normal decay phase, respectively.
In the above scenario, three criteria are adopted for our sample selection: (i) we focus on those LGRBs that show such a transition from shallow decay to normal decay in the X-ray light curves, but require the decay slope of the normal decay segment following the plateau phase to be in the range of −1 to −2 (Lü et al. 2018); (ii) the X-ray flare/bump must be included in the X-ray light curve; and (iii) in order to estimate the intrinsic luminosity of the plateau emission, the redshift needs to be measured.By adopting the criteria for our sample selection, only seven robust cases are identified as satisfying the above three criteria.The X-ray light curves are shown in Figure 3 (also see Table 1 for a summary).
More interestingly, it is found that the X-ray flares/bumps for these seven GRBs were simultaneously observed by Swift/ BAT in the gamma-ray band by searching in the General Coordinates Network (GCN) Circulars Archive case by case.So we downloaded the Swift/BAT data of the seven GRBs In order to search for a lowsignificance signal before and after the duration, the extracted BAT light curves usually cover 300 s before the BAT triggers and after the durations of the GRBs.For more details, please refer to the paper on data analysis with the BBs algorithm (Hu et al. 2014).Phenomenally, we find that the prompt emissions in the gamma-ray band of the seven GRBs are composed of two sub-bursts with a quiescent time.The light curves of the two sub-bursts exhibit different behaviors for our sample, e.g., a softer sub-burst prior to (three out of seven) or following (four out of seven) the stronger sub-burst of prompt emission.
The light curves of both the prompt emission and X-ray flares for these seven GRBs are shown in Figure 2.

Fitting Results with the Magnetar Propeller with Fallback Accretion Model
One motivation is to test whether the magnetar propeller with fallback accretion model can be invoked to interpret the GRBs that we selected in the sample above.In order to find out the best-fit parameters of the magnetar propeller with fallback accretion model, we adopt the MCMC simulation package (Foreman-Mackey et al. 2013) to fit the X-ray flare/bump and X-ray plateau in the afterglow.There are nine free parameters ( i.e., B p , P 0 , M d,0 , R d , ò, δ, η dip , η prop , and 1/f B ) by invoking the propeller and dipole radiations of magnetar.MCMC is one of the very popular methods in the field of high-energy astronomy, and it is very effective in constraining the free parameters that are in degeneracy between each other.More details can be found in Gibson et al. (2017Gibson et al. ( , 2018)).
Figure 3 shows the MCMC fitting results by using the propeller with fallback accretion model and spindown dipole radiation of the magnetar for the seven GRBs.We find that the free parameters of GRBs 140512A and 180329B can be constrained very well, such as B p and P 0 , which are within a range of several times 10 15 G and tens of milliseconds, respectively.The derived parameters of B p , P 0 , and M d,0 fall into the reasonable range, and they are consistent with the results from Lü & Zhang (2014) and Gibson et al. (2018).However, the free parameters of the other five cases are, more or less, not all constrained well enough, especially the main free parameters of the propeller with fallback accretion model, such as B p , P 0 , and M d,0 .The 2D histograms and parameter constraints of the model fit by invoking the MCMC method for our sample are shown in Figure 4, and the values of the parameter constraints are reported in Table 1.On the other hand, based on the fitting results, we find that the required conversion efficiency of the propeller is two or three times (or even larger) higher than that of the spindown dipole radiation of the magnetar (see Figure 5 and Table 1).Based on the results from Lan et al. (2018) for Fermi/GBM and Peng et al. (2014) for Swift/BAT, this suggests that those two sub-bursts of emission of GRB, together with the observed simultaneous X-ray flare/bump, are likely to share the same physical origin.However, the higher efficiency for the propeller phase in our fits is contradictory to the expectation from the propeller model: namely, a dirtier ejecta should be less efficient in producing gamma-ray emissions.This means at least that the magnetar propeller with fallback accretion model cannot interpret very well the GRB X-ray flares that are simultaneously observed in the gamma-ray band, even though it can be invoked to interpret the X-ray plateaus for short GRBs (Gompertz et al. 2014) and supernova explosions (Piro & Ott 2011).

Conclusion and Discussion
The prompt emission, X-ray plateau, and X-ray flares of GRBs are thought to be from internal dissipation (Zhang 2011).
The magnetar as the central engine of a GRB has been proposed to interpret the observed X-ray flares and plateau emission (Gompertz et al. 2014;Gibson et al. 2017).By systematically searching for X-ray emission observed by Swift/XRT, we find that seven robust GRBs include both X-ray flares and plateau emissions with redshift measured.More interestingly, it is found that the X-ray flares/bumps for those seven GRBs are simultaneously observed by Swift/BAT in the gamma-ray band.The prompt emission in the gamma-ray band of the seven GRBs is composed of two sub-bursts with a quiescent time.In particular, the second sub-burst emission in the prompt emission is observed simultaneously with the X-ray flare.
Then, by adopting the MCMC method, we invoke the propeller with fallback accretion model and spindown dipole radiation of the central magnetar to fit the X-ray data of the plateau emissions and flares for our sample.It is found that the free parameters of two GRBs (GRBs 140512A and 180329B) can be constrained very well, such that B p and P 0 have ranges of several times 10 15 G and tens of milliseconds, respectively.However, the free parameters of the other five cases, more or less, are not all constrained well enough, especially the main free parameters of the propeller with fallback accretion model, such as B p , P 0 , and M d,0 .On the other hand, this requires the conversion efficiency of the propeller to be two or three times higher than that of the spindown dipole radiation of magnetars.If this is the case, the higher efficiency for the propeller phase in our fits is contradictory to the expectation from the propeller model: namely, a dirtier ejecta should be less efficient in producing gamma-ray emissions.Our results suggest that at least the magnetar propeller with fallback accretion model cannot interpret very well the GRB X-ray flares that are simultaneously observed in gamma-rays.
We believe that the magnetar propeller with fallback accretion model can interpret very well the X-ray plateau emissions of some GRBs.However, this model presents significant challenges and flaws for the baryon-loading issue when it is related to gamma-ray emission and high relativistic outflow.From the observational point of view, the X-ray flares are observed in a good fraction of the GRB afterglow, and the constrained bulk Lorentz factor (or just the upper limit) of the X-ray flare outflows ranges from a few tens to hundreds (Jin et al. 2010;Yi et al. 2015).This means that the dirty nature of the baryon-loading issue still exists if we invoke the magnetar propeller with fallback accretion model to interpret the X-ray flares with a high Lorentz factor.Furthermore, several physical models without a magnetar central engine have been invoked to interpret the observed X-ray flare, plateau emission, as well as the two sub-bursts of gamma-ray emission, such as late central engine reactivity (Burrows et al. 2005;Fan & Wei 2005;Dai et al. 2006), the relativistic jet and cocoon emissions (Ramirez-Ruiz et al. 2002;Lazzati & Begelman 2010;Nakar & Piran 2017), two steps in the collapse of the progenitor star (Lipunova et al. 2009), the collapse of a rapidly rotating stellar core leading to fragmentation (King et al. 2005), fragmentation in the accretion disk (Perna et al. 2006), the magnetic barrier around the accretor (Proga & Zhang 2006), gravitational lensing (Paynter et al. 2021;Yang et al. 2021;Lin et al. 2022), and the jet precession model (Gao et al. 2023).However, more or less, each model above cannot fully interpret all properties of the observations for the two sub-bursts of emission of GRB from the light curve, spectrum, as well as the quiescent time.

Figure 1 .
Figure 1.X-ray light curve generated by the propeller with fallback accretion model (blue dashed-dotted line) and the dipole radiation of spindown (green dashed line) of the magnetar central engine.The black solid line is the sum of the propeller and dipole radiation of spindown.

Figure 2 .
Figure2.BAT light curves (black) of the prompt emission and X-ray flares (blue) for our sample.The solid red line is the background of prompt emission.

Figure 3 .
Figure 3. X-ray light curves of GRBs for our sample (red points), as well as the fits by the magnetar propeller with fallback accretion model (red dashed-dotted line) and the magnetar spindown (blue dashed line).The black solid line is the total luminosity of the propeller and spindown.
of a plateau followed by a normal decay and adopt a smooth broken power-law function to fit(Liang et al. 2007):

Figure 4 .
Figure 4. 2D histograms and parameter constraints of the model fits for our sample.The 1D histograms show the distributions for each parameter.The dashed lines indicate the median and ±2σ uncertainty of the values.

Figure 5 .
Figure 5. Efficiency correlation between the propeller and dipole radiation.The dotted line corresponds to η prop = η dip .

Table 1
The Values of the Derived Parameters from the MCMC for Our Sample