The Second Plateau in X-ray Afterglow Providing Additional Evidence for Rapidly Spinning Magnetars as the GRB Central Engine
Litao Zhao, Liangduan Liu, He Gao, Lin Lan, WeiHua Lei, Wei Xie
DDraft version May 5, 2020
Typeset using L A TEX default style in AASTeX62
The Second Plateau in X-ray Afterglow Providing Additional Evidence for Rapidly Spinning Magnetars as the GRBCentral EngineSubmitted to ApJ 2020 March 10; Accepted 2020 May 1
Litao Zhao, Liangduan Liu, He Gao, Lin Lan, WeiHua Lei, and Wei Xie Department of Astronomy , Beijing Normal University, Beijing, China; School of Physics, Huazhong University of Science and Technology, Wuhan, China Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Guizhou Normal University, Guiyang, 550001, China.
ABSTRACTEvidence for the central engine of gamma-ray bursts (GRBs) has been collected in the Neil Gehrels
Swift data. For instance, some GRBs show an internal X-ray plateau followed by very steep decay,which is difficult to be interpreted within the framework of a black hole (BH) central engine, but areconsistent within a rapidly spinning magnetar engine picture. The very steep decay at the end ofthe plateau suggests a sudden cessation of the central engine, which is explained as the collapse of asupra-massive magnetar into a black hole when it spins down. Here we propose that some additionalevidence, such as a second X-ray plateau feature would show up, if the fall-back accretion could activatethe newborn BH and sufficient energy could be transferred from the newborn BH to the GRB blastwave. With a systematic data analysis for all long GRBs, we find three candidates in
Swift sample, i.e.,GRBs 070802, 090111, and 120213A, whose X-ray afterglow light curves contain two plateaus, withthe first one being an internal plateau. We find that in a fairly loose and reasonable parameter space,the second X-ray plateau data for all 3 GRBs could be well interpreted with our proposed model.Future observations are likely to discover more similar events, which could offer more information ofthe properties of the magnetar as well as the newborn BH.
Keywords: accretion, accretion disks black hole physics gamma-ray burst INTRODUCTIONGamma-ray bursts (GRBs) have been extensively explored since their discovery more than 50 years ago, but thenature of the GRBs central engine remains a mystery. In the literature, two main kinds of central engine have beenwell discussed: hyper-accreting black holes (BH) or rapidly spinning magnetars (Zhang 2018, for a review). It has longbeen proposed that some interesting signatures showing in some GRB’s X-ray afterglow could help us to determinetheir central engines (Dai & Lu 1998; Rees, & M´esz´aros 1998; Zhang & M´esz´aros 2001; Zhang et al. 2006; Nousek etal. 2006). For instance, systematic analysis for the Swift GRB X-ray afterglow shows that bursts with X-ray plateaufeatures likely have rapidly spinning magnetars as their central engines (Liang et al. 2007; Zhao et al. 2019; Tang et al.2019). In particular, when the X-ray plateau is followed by a steep decay with temporal decay index α (cid:38)
3, hereaftercalled “internal X-ray plateaus”, the sharp decay at the end of the plateau is difficult to be interpreted within theframework of a BH central engine, but are consistent within a magnetar engine picture, where the abrupt decay isunderstood as the collapse of a supra-massive magnetar into a BH after the magnetar spins down (Troja et al. 2007;Lyons et al. 2010; Rowlinson et al. 2010, 2013; L¨u & Zhang 2014; L¨u et al. 2015; Gao et al. 2016a; De Pasquale et al.2016; Zhang et al. 2016).Recently, Chen et al. (2017) proposed that if the sudden drop after the internal plateau indeed indicates the collapseof a supra-massive NS into a BH, signatures from this newborn BH should be expected. For instance, for long GRBs if
Corresponding author: He [email protected] author: Liangduan [email protected] a r X i v : . [ a s t r o - ph . H E ] M a y a fraction of the envelope material fall back and activate the accretion onto the newborn BH (Kumar et al. 2008a,b; Wuet al. 2013; Gao et al. 2016b), the hyper-accreting BH system can launch a relativistic jet via Blandford-Znajek (BZ)mechanism (Blandford & Znajek 1977). Supposing a fraction ξ of the jet energy would undergo internal dissipation,some detectable signals, such as a X-ray bump following the internal plateau is expected (Chen et al. 2017). Searchingthrough Swift -XRT data archive, the authors found a particular case, GRB 070110, which showed a small X-raybump following its internal plateau, and successfully interpreted its multi-band data with their model. But it is worthnoting that for GRB 070110, Troja et al. (2007) found the optical data indicate a decaying light curve feature sittingunderneath the X-ray plateau without showing the rapid drop for the “internal plateau”, which means unlike the X-rayband, the optical emission trend could be hardly disturbed by the late central engine activity.Note that most internal plateaus are not found to be followed by the X-ray bump, it may be because that the latetime fallback processes are intrinsically weak or the energy fraction for the internal dissipation ( ξ ) is relatively small.For the latter situation, a food fraction 1 − ξ of the BZ jet energy would continuously inject into the GRB blast wave,which may generate a second plateau in X-ray afterglow, if the injected energy is comparable or even larger than theblast wave kinetic energy. We thus propose that GRBs with two X-ray plateaus (the first one is internal plateau) mayprovide further support to the magnetar central engine model.In this work, we first study how the energy injection from the fallback accretion onto the newborn BH would alterthe GRB afterglow emission and analyze the influence of model parameters on the theoretical lightcurves. In section3, we systematically search for GRBs with two X-ray plateaus from the Swift-XRT sample. Considering that longGRBs, which are likely related to the core-collapse of massive stars (Woosley 1993; MacFadyen & Woosley 1999), aremore easier to have enough envelop material to provide late time fall-back accretion than short GRBs, that have beenproposed to originate from the merger of two neutron stars (Eichler et al. 1989; Narayan et al. 1992) or the merger of aneutron star and a black hole (Paczynski 1991), here we focus on long GRB samples . We find three candidates, GRB070802, GRB 090111 and GRB 120213A, whose second plateaus could be well fitted by the model. The conclusion andimplications of our results are discussed in Section 4. Throughout the paper, the convention Q = 10 n Q n is adopted inc.g.s. units. MODEL DESCRIPTIONRapidly spinning magnetars have long been proposed as one of the candidate of GRB central engines. In thisscenario, a collimated jet could be launched by invoking (1) hyper-accretion onto the NS (Zhang & Dai 2009, 2010;Bernardini et al. 2013); (2) magnetic bubbles from a differentially millisecond proto-NS (Dai et al. 2006); (3) or froma protomagnetar wind (Metzger et al. 2011). The internal dissipation of the jet could power the prompt gamma rayemission of a GRB and the interaction between the jet and the ambient medium could produce a strong external shockthat gives rise to bright broadband afterglow emission (Gao et al. 2013, for a review). After launching the jet, themagnetar would also eject a near-isotropic Poynting-flux-dominated outflow, the internal dissipation of which couldpower a bright X-ray emission, whose temporal profile would follow the spin-down profile of the magnetar, i.e., t atearly stage and decay as t − after the magnetar spinning down (Zhang & M´esz´aros 2001). If this emission componentis brighter than the external shock afterglow emission, an X-ray plateau would show up. Sometimes the central enginemagnetar might be a supra-massive NS, which would collapse to a BH when a good fraction of its rotational energy islost and the centrifugal support can no longer support gravity. In this case, the X-ray plateau emission would suddenlystop and follows by a sharp decay at the end of the plateau, due to the abrupt cessation of the magnetars centralengine. This explains the internal X-ray plateau discovered in both long and short GRBs (Troja et al. 2007; Lyonset al. 2010; Rowlinson et al. 2010, 2013; L¨u & Zhang 2014; L¨u et al. 2015). After the collapse of the magnetar, ifa fraction of the envelope material (especially for long GRBs) fall back and activate the accretion onto the newbornBH, the rotational energy of the BH could be extracted via BZ mechanism, and a good fraction of the energy wouldeventually inject into the afterglow blast wave. If the injected energy is comparable or even larger than the blastwave kinetic energy, the broadband afterglow lightcurve could be significantly altered, for instance, a second plateauin the X-ray afterglow may emerge. Figure 1 presents a physical picture for several emission components at differenttemporal stages. Here we focus on study how the energy injection from the fallback accretion onto the newborn BHwould alter the GRB afterglow emission and analyze the influence of model parameters on the theoretical lightcurves. Some previous works found some short GRBs with two X-ray plateaus and explained the data with their own model. For instance,Hou et al. (2018) proposed that two plateaus showing in the X-ray light curve of GRB 170714A could be interpreted as the solidificationand collapsing process of a hyper-massive quark star. Zhang et al. (2018) proposed that a second plateau showing in GRB 160821B couldbe interpreted by fall-back accretion process from a magnetically-arrested disk
Figure 1.
Model illustration: the “internal plateau” is powered by the spin-down power from a supra-massive magnetar andthe steep decay marks the collapse of the magnetar into a BH when it spins down. In addition, the second plateau is caused byenergy transfer from new born BH through BZ mechanism to GRB blast wave.
The Fallback Accretion onto the Newborn BH
Assuming the fallback accretion could trigger the energy extraction from the newborn BH via BZ mechanism, inthis case, the BZ power from a BH with mass M • and angular momentum J • could be estimated as (Lee et al. 2000;Li 2000; Wang et al. 2002; Lei et al. 2005, 2013, 2017; McKinney 2005; Lei & Zhang 2011; Chen et al. 2017; Liu et al.2017; Lloyd-Ronning et al. 2018) L BZ = 1 . × a m • B • , F ( a ) erg s − , (1)where m • = M • /M (cid:12) is the dimensionless BH mass and a = J • c/ ( GM • ) is the dimensionless spin parameter of theBH. Here, F ( a ) = [(1 + q ) /q ][( q + 1 /q ) arctan q − q = a/ (1 + √ − a ), and B • , is the magnetic-field strengththreading the BH horizon in units of 10 G. The evolution of the BZ jet power depends on m • , a , and B • .The evolution of the BH spin and mass governed the competition between spin up by accretion and spin down bythe BZ mechanism. The evolution equations of the BH mass M • and the BH spin a are given by Wang et al. (2002) dM • c dt = ˙ M c E ms − L BZ , (2)and dadt = ( ˙ M L ms − T BZ ) cGM • − a ( ˙ M c E ms − L BZ ) M • c , (3)where ˙ M is the BH accretion rate, and T BZ is BZ magnetic torque (Li 2000; Lei & Zhang 2011; Lei et al. 2017) T BZ = 3 . × a q − m • B • , F ( a )g cm s − , (4)where E ms and L ms are the specific energy and the specific angular momentum at innermost radius r ms of the disk(Novikov & Thorne 1973): E ms = (4 (cid:112) R ms − a ) / ( √ R ms ) , (5) L ms = ( GM • /c )(2(3 (cid:112) R ms − a )) / ( √ (cid:112) R ms ) . (6)Here, R ms = r ms /r g and r g = GM • /c . The innermost stable radius of disk is (Bardeen et al. 1972) r ms = r g (cid:104) Z − [(3 − Z )(3 + Z + 2 Z )] / (cid:105) , (7)where Z l ≡ − a ) / (cid:2) (1 + a ) / + (1 − a ) / (cid:3) and Z ≡ (3 a + Z ) / for 0 (cid:54) a (cid:54) B • , B • π ≈ ˙ M c πr • . (8)where r • is the radius of the BH horizon.With this assumption, the BZ jet power could be written as a function mass accretion rate and BH spin, i.e. (Wuet al. 2013; Chen et al. 2017) L BZ = 9 . × a ˙ mF ( a )(1 + √ − a ) erg s − , (9)where ˙ m = ˙ M / ( M (cid:12) s − ) is the dimensionless BH accretion rate. The accretion rate of BH can be estimated byadopting a simple model described in Kumar et al. (2008a)˙ M (cid:39) M d τ vis , (10)where the viscous timescale τ vis ∼ /α Ω K , here α is the standard dimensionless viscosity parameter, Ω K is the Keplerangular velocity of accretion disk.The mass of the disk M d evolves with time, it increases as a result of fall-back from the envelope and decrease as aresult of accretion. Thus, one has (Kumar et al. 2008a; Lei et al. 2017)˙ M d = ˙ M fb − ˙ M . (11)Combining Eqs (10) and (11), one can obtain the accretion rate onto the BH (Kumar et al. 2008a; Lei et al. 2017),˙ M = 1 τ vis e − t/τ vis (cid:90) tt e t (cid:48) /τ vis ˙ M fb dt (cid:48) . (12)The evolution of the fall-back accretion rate is described with a broken-power-law as (Chevalier 1989; MacFadyenet al. 2001; Zhang et al. 2008; Dai & Liu 2012)˙ M fb = ˙ M p (cid:34) (cid:18) t − t t p − t (cid:19) − / + 12 (cid:18) t − t t p − t (cid:19) / (cid:35) − , (13)where t is the start time of the fall-back accretion in the local frame, t p is peak time of fallback and and ˙ M p is thepeak fall-back rate.For the rapid accretion case, τ vis (cid:28) t , the BH accretion rate would follow the fall-back rate, i.e., ˙ M = ˙ M fb . For alarge value of the viscosity timescale τ vis , the BH accretion rate would be flat until t > τ vis , and then starts to declinewith time, see Figure 7 in Lei et al. (2017).2.2. Energy injection into the GRB afterglow blast wave
The energy flow from the BZ process would continuously inject into the external shock, and cause a significant raiseto the Lorentz factor of the blast wave Γ, which may produce the second plateau following the steep decay. Huanget al. (2000) proposed a generic dynamical model to describe the dynamical evolution of GRB outflow, which hasbeen widely applied for modeling the afterglow light curve. Based on their model and taking the energy injection intoaccount, the evolution equation of the outflow’s bulk Lorentz factor can be written as (Liu & Chen 2014) d Γ dM sw = − M ej + 2Γ M sw (cid:20) Γ − − L BZ c dtdM sw (cid:21) , (14) (Time(s)) log )) s ( F l u x ( e r g s c m l og
20 19 18 17 16 15 14 13 12 11
No energy injection=0.1 a =0.5 a =0.9 a (Time(s)) log )) s ( F l u x ( e r g s c m l og
20 19 18 17 16 15 14 13 12
No energy injection s =10 vis τ s =10 vis τ s =10 vis τ (Time(s)) log )) s ( F l u x ( e r g s c m l og
22 20 18 16 14 12
No energy injections × =1 p t s × =1 p t s × =1 p t (Time(s)) log )) s ( F l u x ( e r g s c m l og
20 18 16 14 12
No energy injection=10 η =30 η =100 η (Time(s)) log )) s ( F l u x ( e r g s c m l og
19 18 17 16 15 14 13 12 11 n=0.1 cm n=1 cm n=10 cm (Time(s)) log )) s ( F l u x ( e r g s c m l og
17 16 15 14 13 12 11 =10 B ∈ =10 B ∈ =10 B ∈ (Time(s)) log )) s ( F l u x ( e r g s c m l og
19 18 17 16 15 14 13 12 =0.01 e ∈ =0.05 e ∈ =0.1 e ∈ (Time(s)) log )) s ( F l u x ( e r g s c m l og
20 18 16 14 12 p=2.2p=2.5p=3 (Time(s)) log )) s ( F l u x ( e r g s c m l og
18 16 14 12 10 8 =100 Γ =200 Γ =300 Γ Figure 2.
Influence of the model parameters on the X-ray light curve. Except as noted in each subfigure, in the calculation,we take a set fiducial values for the model parameters: E = 10 ergs, Γ = 100, n = 1 cm − , θ = 0 . (cid:15) e = 0 . (cid:15) B = 10 − , p = 2 . a = 0 . τ vis = 10 s, t p = 10 s and η = 30. where M sw is the swept-up mass by shock, M ej is the initial mass of the GRB outflow. The blast wave energycontinuously increase with time, due to the continuously injected energy from the BZ process into the blast wave.The initial kinetic energy of GRB outflow can be estimated as E = Γ M ej c , and the injected energy from the BZprocess can be calculated as E inj = (cid:82) (1 − ξ ) L BZ dt . The total energy in the blast wave could be expressed as E tot = E + E inj . (15)We introduce a parameter η ≡ E inj /E to denote the ratio between the injected energy and the initial energy in blastwave. For η >
1, the injected energy is dominated in the total energy of the blast wave, otherwise the initial kineticenergy of GRB outflow is dominated.In order to obtain the shock dynamical evolution, three additional differential equations are required (Huang et al.2000). The evolution of the radius of shock R , the swept-up mass M sw , and the opening angle of the jet θ are described Since we focus on the case where the BZ jet undergoes weak internal dissipation, here we simply set ξ = 0. by Huang et al. (2000) dRdt = (cid:114) Γ − c Γ(Γ + (cid:112) Γ − , (16) dM sw dR = 2 πR (1 − cos θ ) nm p , (17) dθdt = c s (Γ + √ Γ − R , (18)where n is the number density of the unshocked ISM and c s is the sound speed, c s = ˆ γ (ˆ γ − −
1) 11 + ˆ γ (Γ − c , (19)where the adiabatic index ˆ γ = (4Γ + 1) / (3Γ).For an electron with an energy γ e m e c in the co-moving frame of the shock, the observed frequency from sychrotronemission is (Rybicki, & Lightman 1979) ν ( γ e ) = 34 π Γ γ e q e B (cid:48) m e c , (20)where m e is the electron mass, q e is the electron charge, the bulk Lorentz factor Γ is introduced by transferring theshock co-moving to the observer rest frame and B (cid:48) is the magnetic field strength in the shock co-moving frame. In GRBproblems, one usually assumes the magnetic field density ( B (cid:48) / π ) is a fraction of the internal energy of post-shockedmedium with a shock equipartition parameter (cid:15) B . Therefore, the co-moving magnetic field is in form of (Sari et al.1998) B (cid:48) = (32 πm p (cid:15) B n ) / Γ c. (21)The energy distribution of shock accelerated electron is usually assumed to be a power law, with N ( γ e ) dγ e ∝ γ − pe dγ e .The minimum Lorentz factor of the electron, γ m could be obtained by the conservation laws of particle number andenergy. Assuming a constant fraction (cid:15) e of the post-shock internal energy goes into the electrons, one has (Sari et al.1998) γ m = (cid:15) e (cid:18) p − p − (cid:19) m p m e (Γ − . (22)Another critical electron Lorentz factor is the cooling Lorentz factor γ c . When γ e > γ c , electrons would loss most oftheir energies by synchrotron radiation, otherwise the cooling caused by the radiation can be ignored. If synchrotronradiation is dominated, the cooling Lorentz factor is given by (Sari et al. 1998) γ c = 6 πm e cσ T Γ B (cid:48) t , (23)where t refers to time in the rest frame of observer, σ T is Thomson cross-section.As shown in Eq (20), electrons with different Lorentz factors γ e have different radiation frequencies ν ( γ e ). Twocharacteristic frequencies, ν m = ν ( γ m ) and ν c = ν ( γ c ) would determine the synchrotron spectrum. The evolution ofthese frequencies with time can be derived from shock dynamics.For a given dynamical time t , if γ c > γ m , it means only a small fraction of electrons could be cooled, This is calledslow cooling regime. In this case, for an observational frequency ν , the synchrotron spectrum is described a brokenpower law characterized by ν m and ν c as follows (Sari et al. 1998) F ν = ( ν/ν m ) / F ν, max , ν < ν m ( ν/ν m ) − ( p − / F ν, max , ν m < ν < ν c ( ν c /ν m ) − ( p − / ( ν/ν c ) − p/ F ν, max , ν c < ν , (24)On the other hand, for γ c < γ m , all the accelerated electrons could be cooled in the dynamical timescale t . This isnamed fast cooling regime. The radiation spectrum of the shock is (Sari et al. 1998) F ν = ( ν/ν c ) / F ν, max , ν < ν c ( ν/ν c ) − / F ν, max , ν c < ν < ν m ( ν m /ν c ) − / ( ν/ν m ) − p/ F ν, max , ν m < ν , (25)where F ν, max is the observed peak flux at luminosity distance D L from the source, which can be estimated as (Sari etal. (1998)) F ν, max = N e, tot m e c σ T πq e D L B (cid:48) Γ , (26)where N e, tot = 4 πnR / Swift -XRT energy band (0 . −
10 keV). Note that the dominated radiationmechanism for X-ray afterglow emission is synchrotron radiation (M´esz´aros & Rees 1997; Sari et al. 1998), therefore,inverse Compton mechanism is ignored in our calculation.2.3.
Influence of the Model Parameters on the X-ray Light Curve
In this subsection, we show the numerical results of our model. Here, we explore the influence of the model parameterson the theoretical X-ray light curves. There are several free parameters in our model, which can be divided into twocategories. The first category is related to the BZ process and the newborn BH, including the initial mass of thenewborn BH M • , , the initial BH spin a , the viscosity timescale of disk τ vis , peak time of fallback t p and the peakfallback rate ˙ M p . The second category is associated with the external shock, including initial kinetic energy of GRBoutflow E , the initial bulk Lorentz factor of GRB outflow Γ , the initial opening angle of the jet θ , the equipartitionparameters for the magnetic field and electrons (cid:15) B and (cid:15) e , and the electron distribution index p .For the first category, considering that the newborn BH is produced by the collapse of a supermassive NS, theinitial mass of the BH, M • , should be close to the maximum mass of NS, which should be between 2 to 3 solar mass(Lattimer 2012). Within this range, the exact value of M • , hardly affect the BZ power. Here we adopt M • , = 2 . M (cid:12) as a fiducial value. The influence of a , τ vis and t p are shown in Figure 2. We find that: 1) a larger value of a leads toa more luminous and longer duration plateau; 2) a longer viscosity timescale of disk τ vis results in a longer duration,but lower luminous plateau, which is understandable, since according to Eq (12), a larger value of τ vis corresponds toa slower and weaker BH accretion; 3) a larger value of t p leads to lower luminosity of lightcurves.For the second category, the initial opening angle of the jet hardly affect the final results. Here we adopt θ = 0 . , n , (cid:15) B , (cid:15) e and p are shown in Figure 2. We find that: 1) when n is larger, the blastwave collects more material in a higher density ISM, which leads to more luminous lightcurve; 2) since synchrotronradiation intensity increases with the magnetic energy density and the electrons energy in the external shock, theincrease of (cid:15) B and (cid:15) e would brighten the X-ray flux; 3) the value of Γ mainly affects the early behavior of the lightcurve, but not the late behavior when energy injection happens; 4) the value of p mainly affects the decline slope ofthe light curve.In order to better reflect the injected energy E inj and the initial kinetic energy of the blast wave E , we test theinfluence of parameter η ≡ E inj /E instead of ˙ M p and E . As expected, the larger of η value, the more significant ofthe light curve re-brightening (see Figure 2). SAMPLE SELECTION AND INTERPRETATION3.1.
Data Reduction And Sample Selection
For the purpose of this work, we systemically search sources consisting two X-ray plateaus, where the first oneshould be an “internal plateau” followed by a steep decay. The XRT light curves data were downloaded from the
Swift /XRT team website (Evans et al. 2007, 2009), and processed with HEASOFT v6.12. There were 1291 GRBs (Time(s)) log )) s ( F l u x ( e r g s c m l og ×
10 s × α =0.23 α =5.13 α =0.6 α GRB 070802 (Time(s)) log )) s ( F l u x ( e r g s c m l og
13 12 11 10 9 × × × × α = 0.39 α =3.19 α =0.44 α =1.02 α GRB 090111 (Time(s)) log )) s ( F l u x ( e r g s c m l og
14 13 12 11 10 9 8 7 × × × α =0.02 α =3.01 α =0.59 α GRB 120213A
Figure 3.
Temporal fitting results of XRT 0.3-10 keV light curve for GRB 070802, 090111 and 120213A by using MARStechnique. detected by
Swift /XRT between 2004 February and 2017 July, with 625 GRBs have well-sampled XRT light curves,which including at least 6 data points, excluding the upper limit. XRT light curves for all selected sample are thenfitted with multi-segment broken power law function (in logarithmic scale). Here we adopt the multivariate adaptiveregression spline (MARS) technique (e.g. Friedman (1991)) to fit the light curves. MARS technique can automaticallydetermine both variable selection and functional form, resulting in an explanatory predictive model. Some previousworks have proven that MARS can automatically fit the XRT light curve with multi-segment broken power-law function(results in general consistent with fitting results provided by the XRT GRB online catalog (Evans et al. 2007, 2009)),detect and optimize all breaks, and record all break times and power-law indices for each segment (see Zhang et al.(2014) and Zhao et al. (2019) for details). Here we treat the adjacent segments with index difference smaller than 0.3as one component when calculating the segment time span, in order to avoid the potential over-fitting problem fromMARS technique. With the fitting results provide by MARS, we searched for candidates having two shallow decaycomponents, where segments with decay slope shallower than 0.65 and time span in log scale larger than 0.4 dex aredefined as the shallow decay components . For the purpose of this work, we add one more criteria for the sampleselection, i.e., the decay slop following the first shallow decay component should be (cid:38)
3. Eventually, we find 3 longGRBs (GRB 070802,GRB 090111 and GRB 120213A) meeting all our requirement.GRB 070802 triggered the BAT at 07:07:25 UT on 2007 August 2. T is 16.4 s ± T + 4 . T + 23 . γ = 1 . ± .
27. The fluence in the 15-150 keV band is S γ = 2 . ± . × − ergs / cm (Cummings et al. (2007)).The XRT began taking data 138 s after the trigger (Barthelmy et al. 2007). The fitting result of XRT data providedby MARS is shown in Figure 3. It is worth noting that on this case, the fitting curve given by the on-line XRTcatalogue is a little different from that given by MARS. Since there are few data points in the late stage, the on-linecatalogue take all the data after the first plateau into one segment, which is considered as a normal decay component.According to its own algorithm, however, MARS automatically fits the late data with two segments, and the secondsegment has a decline slope less than 0.65. In this case, we take GRB 070802 as one of our candidates, but withrelatively weak evidence. The X-ray fluence and photon index for the two shallow decay segments are listed in Table3. The UV/Optical Telescope (UVOT; Roming et al. (2005)) start to collect data 141 s after the trigger. No newsource in the UVOT observations at the location of the refined XRT position (Kuin & Immler (2007)). Prochaska etal. (2007) observed the afterglow of GRB 070802 with the ESO VLT + FORS. From the detection of several Fe linesin a 30 minutes spectrum starting on August 2.378 UT, the redshift was measured as z = 2 .
45. Kr¨uhler et al. (2008)presented the optical and near-infrared photometry of the afterglow obtained with the multichannel imager GROND.Unfortunately, the late optical data points are also scarce, and there is no data point around the time when the second Based on the early-year Swift observations, Liang et al. (2007) performed a systematic analysis for the shallow decay component ofGRB X-ray afterglow, and they found that the distribution of shallow decay slope ( α s ) is normal distribution, that is α s = 0 . ± . σ confidence level). Recently, Zhao et al. (2019) revisited the analysis with an updated sample and they foundthat with a larger sample, the distribution of the α s is still normal distribution with α s = 0 . ± .
22. In this work, we adopt 1- σ regionupper boundary of α s as the selection criteria for the first and second plateau feature. X-ray plateau emerges. The late optical-IR data could be basically consistent with a single decay segment, so thatthe optical emission trend might be different with the X-ray band, just like what is found for the case of GRB 070110(Troja et al. 2007). On the other hand, as shown in the next section, the late optical-IR data could also be well fittedby our proposed model simultaneously with the X-ray data.The BAT triggered and located GRB 090111 at 23:58:21 UT on 2009 January 11. T is 24.8 ± T − . T + 25 . γ = 2 . ± .
17. The fluence in the 15-150 keV band is S γ = 6 . ± . × − ergs / cm (Stamatikos et al. (2009)). XRT observations started at 76.6s after the trigger (Hoversten & Sakamoto 2009). Thefitting result of XRT data provided by MARS is shown in Figure 3. The X-ray fluence and photon index for the twoshallow decay segments are listed in Table 3. Note that GRB090111 has data more difficult to interpret mainly due tothe orbital gap around thousands of seconds, where the data could also be interpreted as a flare followed by a decay(this would also be consistent with the variation in hardness ratio seen in the XRT data).The UVOT to collect datastarting 86 s after the trigger. No source was detected by the UVOT at the X-ray afterglow position (Hoversten &Sakamoto (2009)). No prompt ground-based observation was reported, probably due to the vicinity (46 ◦ ) to the Sun(Margutti et al. (2009)).GRB 120213A triggered the BAT at 00:27:19 UT on 2012 February 12. T is 48.9 ±
12 s. The time-averagedspectrum from T − .
31 s to T + 74 .
46 s is best fitted by a SPL function. The power law index of the time-averagedspectrum is Γ γ = 2 . ± .
09. The fluence in the 15-150 keV band is S γ = 1 . ± . × − ergs / cm (Baumgartneret al. (2012)). The XRT began collect data 54 s after the trigger (Oates & Sakamoto 2012). The fitting result ofXRT data provided by MARS is shown in Figure 3. The X-ray fluence and photon index for the two shallow decaysegments are listed in Table 3. For this case, there is a fairly clear evidence for a second shallow decay component,but unfortunately the data do not exist to know when the shallow decay stops and at what slop the even later timeemission decays. The UVOT to collect data starting 58 s after the trigger. No optical afterglow consistent with theXRT position and no prompt ground-based observation was reported (Oates & Sakamoto (2012)).3.2. Model Application to Selected GRBs
In this section, we apply the model described in Section 2 to interpret the X-ray light curve data of GRBs 070802,090111, and 120213A. Since the first X-ray plateau data could be well explained with magnetar spin-down power(Troja et al. 2007; Lyons et al. 2010; Rowlinson et al. 2010, 2013; L¨u & Zhang 2014; L¨u et al. 2015), here we focuson fitting the second X-ray plateau data by consider that the second plateau is produced by energy injection from thenew born BH driving BZ power. In order to minimize the χ of the fitting, a Markov Chain Monte Carlo (MCMC)method is adopted. In our MCMC fitting, emcee code (Foreman-Mackey et al. 2013) is used with a walkers number 160and 10 burn-in iterations in the ensemble. Considering that the total number of observational data points availableare not enough to constrain all the model parameters. In order to reduce the number of the free parameters in ourfitting, we fix several parameters at their typical values. For instance, we set E = 10 ergs, Γ = 100, n = 1 cm − , θ = 0 . (cid:15) e = 0 . (cid:15) B = 10 − , p = 2 .
5, and we only take the initial BH spin a , the viscosity timescale of disk τ vis , thepeak time of the fallback t p , and the ratio between the injected energy and initial kinetic energy η as free parameters.We set the allowed ranges for the four free parameters in our fitting as: a ≡ [0 , τ vis ≡ [ t , t ], log t p ≡ [ t , t ], Table 1.
The X-ray fluence of the plateau and photon index of the plateau and its follow-upsegmentname S X a Γ X, Γ X, S X b Γ X, Γ X, − ergs cm − − ergs cm − GRB 070802 0.25 ± . +0 . − . . +0 . − . ± . +0 . − . ...GRB 090111 0.03 ± . +1 . − . . +0 . − . ± . +0 . − . . +0 . − . GRB 120213A 1.5 ± . +0 . − . . +0 . − . ± . +0 . − . ... a For the first plateau. b For the second plateau. η ≡ [ − , t is the ending timescale of internal plateau, which can also be used as the start time of thefall-back accretion and t is the time of last observational data point, the values of them for each burst are shown inFigure 3.Figure 4 shows our fitting results for 3 selected GRBs, where the upper panel shows the fitting light curves and thelower panel shows the corresponding corner plot of the posterior probability distribution for the fitting. We can seethat, the second plateau for all 3 GRBs could be well fitted with our proposed model. GRB 070802 has a redshiftmeasurement z = 2 .
45, and the fitting results at 1 σ confidence level are are a = 0 . +0 . − . , log τ vis = 4 . +0 . − . s, log η = 1 . +0 . − . and log t p = 4 . +0 . − . s. It is interesting to note that the late optical-IR afterglow data ofGRB 070802 could also be well fitted by our proposed model simultaneously with the X-ray data (in Figure 4, weshow the data and fitting result for Ks band as an example). For GRB 090111 and GRB 120213A, due to the lackof redshift measurement, here we adopt z = 1 in our analysis. In this case, for GRB 090111, the model parametersat 1 σ confidence level are a = 0 . +0 . − . , log τ vis = 4 . +0 . − . , log η = 1 . +0 . − . and log t p = 2 . +0 . − . ; for GRB120213A, the results are a = 0 . +0 . − . , log τ vis = 5 . +0 . − . , log η = 1 . +0 . − . and log t p = 3 . +0 . − . .From the fitting results, we find that the constraints on model parameters are relatively loose, mainly due to thelack of enough high quality observation data. Even in this case, some general conclusions could still be made: for all 3GRBs, 1) the fallback accretion model could easily explain the second X-ray plateau data with fairly loose parameterrequirements. Note that in the fitting, we have fixed several model parameters, which means the parameter constraintscould become even looser if these parameters were also released. 2) The constraints on η are relatively tight. η wasconstrained to the order of 10-100, inferring that the BZ power much be 10 or 100 times larger than the initial GRBblast wave kinetic energy, in order to produce the second X-ray plateau feature. 3) Although the allowed parameterspace are wide, a tends to have a large value, the distribution peaks are larger than 0.6, which is expected sincelarger a would easily give larger BZ power. 4) The distributions of the viscosity timescale of disk τ vis are also wide.The distribution peaks of τ vis are relatively large, inferring that the fall-back accretion all falls into the slow accretionregime. 5) The distributions for the peak time of the fallback t p tend to peak around the ending time of internalplateau, which are around 10 − s. Taken this as the start time of the fall-back accretion, the minimum radiusaround which matter starts to fall back could be estimated as r fb ∼ . × ( M • / . M (cid:12) ) / ( t / s) / , which isconsistent with the typical radius of a Wolf-Rayet star.The fitting mass fallback rate ˙ M fb for these three GRBs reaches the peak value around 10 − M (cid:12) s − at the timeabout 10 - 10 s . It is interesting to check whether such a mass fallback rate at that time could be supplied by theprogenitor envelope. Here we estimate the mass supply rate from the envelope with the presupernova structure models(e.g., Suwa & Ioka 2011; Woosley & Heger 2012; Matsumoto et al. 2015; Liu et al. 2018), i.e.,˙ M pro = dM r dt ff = dM r /drdt ff /dr = 2 M r t ff (cid:18) ρ ¯ ρ − ρ (cid:19) , (27)in which ¯ ρ = 3 M r / (4 πr ) is the average density within radius r , M r is the mass coordinate of a shell, t ff = (cid:112) π/ (32 G ¯ ρ ) = (cid:112) π r / (8 GM r ) denotes the free-fall timescale. By taking some representative progenitor densityprofiles with different metallicities and masses from Liu et al. (2018) and the references therein, we reproduce themass supply rate changing with time for those progenitor models. In calculation, we set the time when the centralaccumulated mass reaches M r = M = 2 . M (cid:12) (our fiducial value of the mass of a new born magnetar) as the zerotime reference point, i.e., we take t = t ff ( r ) − t ff ( r ), M r = M + (cid:82) rr πr ρdr , here r is the radial coordinate where theenclosed mass is M . As shown in Figure 5, we find that our fitting resulted mass fallback rates are compatible withthe theoretical mass supply rate of some low metallicity massive progenitor stars such as those ones with ( Z (cid:46) − , M (cid:38) M (cid:12) ), ( Z (cid:46) − , M (cid:38) M (cid:12) ), ( Z (cid:46) − , M (cid:38) M (cid:12) ), and so on. While the solor metallicity stars mightnot be so possible to play as the progenitors for our sample.Based on the fitting results, the magnetic field strength of the new-born BH ( B • ) for these three GRBs reachesthe peak value around 10 − G at the time about 10 - 10 s . According to the dipole spin-down model, one canmake estimation for the surface magnetic field B p and the initial spin period P of the rapidly spinning magnetarwith the first plateau data for all 3 GRBs in our sample (Rowlinson et al. 2013; L¨u et al. 2015). Here we adopt theconstant values of the moment of inertia I = 1 . × g cm and radius R=10 km for a typical neutron star. ForGRB 070802, the plateau luminosity and break time are L b = 1 . × erg s − , t b = 1 × s, respectively, one can1 (Time(s)) log )) s ( F l u x ( e r g s c m l og
18 17 16 15 14 13 12 11 10 s KNo energy injectionEnergy injectionParameters Values of energy injection= 0.8 a s × = 5.13 vis τ =100 η s × = 1.12 p t GRB 070802 (Time(s)) log )) s ( F l u x ( e r g s c m l og
16 15 14 13 12 11 10 9 8 a s × = 1.23 vis τ =12.6 η s × = 6.17 p t GRB 090111 (Time(s)) log )) s ( F l u x ( e r g s c m l og
16 14 12 10 8 a s × = 1.7 vis τ = 47.9 η s × = 6.61 p t GRB 120213A
Figure 4.
Fitting results of the second plateau for GRB 070802, 090111 and 120213A. thus derive P < . × − s and B p < . × G. For GRB 090111, the plateau luminosity and break timeare L b = 2 . × erg s − , t b = 427 s, respectively, one can derive P < . × − s and B p < . × G. ForGRB 120213A, the plateau luminosity and break time are L b = 2 . × erg s − , t b = 4 . × s, respectively, onecan derive P < . × − s and B p < . × G. We find that the magnetic field strength of the new-born BHrequired to power BZ jets is comparable or slightly lower than the magnetic field strength of the magnetar, which isunderstandable if we consider that the magnetic flux should be roughly conserved when the magnetar collapse into theBH, and some magnetic energy might dissipate due to the interaction between the magnetosphere and the fall backflow (Lloyd-Ronning et al. 2019). DISCUSSION AND CONCLUSIONIn the 15 years of
Swift ’s operation, it has brought us a lot of observations of GRB X-ray afterglow, which providesvaluable information for understanding the GRB central engine. One particular example is the discovery of internalX-ray plateau, (a plateau followed by a very steep decay phase), which is commonly taken as the smoking gun evidenceof a rapidly spinning magnetar as the central engine. The very steep decay at the end of the plateau suggests a suddencessation of the central engine, which is explained as the collapse of a supra-massive magnetar into a black hole whenit spins down. If this interpretation is correct, the fall-back accretion from the envelope of progenitor star into thenewborn BH could generate some detectable signatures.Here we propose that the energy extracted from the newborn BH could be continuously injected into the GRB blastwave. In this scenario, We find that with appropriate parameters for the fall back accretion and new born BH, it ispossible to produce a second plateau following the steep decay phase of the internal plateau. With a systematical Here we take the break time t b of the first plateau as the lower limit of characteristic spin-down time, and take the plateau luminosity L b as the characteristic spin-down luminosity (see detailed methods in L¨u et al. 2015). Redshift for GRB 070802 is taken as z = 2.45 andthe redshift for GRB 090111 and GRB 120213A is taken as 1. GRB 070802GRB 090111GRB 120213A o40o80 s20s80u20 v40 v800 2 4 6 - - - - ( Time ( s )) l og ( M • p r o ( M ⊙ s - )) Figure 5.
Comparing the fitting mass fallback rate with the mass supply rate of the progenitor stars with different metallicitiesand masses. The fitting mass fallback rate of the three GRBs, i.e., GRB 070802, GRB 090111, and GRB 120213A are denotedby the dashed line, the dash-dotted line and the dotted line separately. The mass supply rate of the progenitor stars are denotedby the solid colored lines, in which the nomenclature is the same as Liu et al. (2018), i.e, the signs s, o, v, and u represent themetallicity values Z = Z (cid:12) , 10 − Z (cid:12) , 10 − Z (cid:12) , and 10 − Z (cid:12) , and the numbers beside the signs denote the progenitor masses inunit of solar mass. search through the Swift -XRT sample, we find three interesting long GRBs, i.e., GRBs 070802, 090111, and 120213A,whose X-ray afterglow light curves contain internal X-ray plateau followed by a second plateau. Here we focus on fittingthe second X-ray plateau for these 3 GRBs with our proposed model. We find that in a fairly loose and reasonableparameter space, the second X-ray plateau data could be well interpreted with our model.It is worth noting that the quality of current observation data is not good enough to completely eliminate thedegeneracy of parameters. In our sample, GRB 090111 has the most observational data points in the period of thesecond plateau. GRBs 070802 and 120213A only have four data points in the period of the second plateau. Evenin this case, some general conclusions could still be reached, for instance, in order to interpret the second plateaus,the initial spin of the new-born BH tends to have a large value (the peak of its posterior probability distribution islarger than 0.6), the later injected energy should be 10 or 100 times larger than the initial GRB blast wave kineticenergy, and the viscosity timescale of disk tends to be large, inferring that the fall-back accretion all falls into theslow accretion regime (Lei et al. 2017). The mass fall-back rate reaches the peak value around 10 − M (cid:12) s − at thetime about 10 - 10 s , which is compatible with the late mass supply rate of some low metallicity massive progenitorstars. Based on the fitting results, one can infer the total accreted masses M acc ∼ . − . M (cid:12) and the fallbackradii r fb ∼ a few × cm, which is consistent with the typical radius of a Wolf-Rayet star. Combining the X-ray dataof the internal plateaus of the three GRBs with the magnetic dipole radiation model of the magnetar, we coarselyestimate the strength of magnetic field of the magnetar before its collapsing to a BH. We find the magnetic field isstrong enough to drive a BZ jet.If our interpretation is correct, the 3 GRBs with two X-ray plateaus provide additional evidence for rapidly spinningmagnetar as the GRB central engine. It is worth noticing that most GRBs with internal plateaus do not show a secondX-ray plateau, which means for most cases the fallback accretion may be relatively weak, so that the injected energy issmaller than initial kinetic energy of GRB blast wave. Future observations are likely to discover more similar events,which could offer more information of the properties of the magnetar as well as the newborn BH.In this paper, we ignore the mass into the outflow from disk, which will reduce the accretion rate onto BH horizonduring late central engine activity. Usually, the distribution of accretion rate with disk radius is simply described witha power-law model due to the poor knowledge of the disk outflow. The effects of such outflow are thus highly reliedon the uncertain power-law index parameter. The existence of disk outflow may also be important to comprehend thebaryon loading into GRB jet (Lei et al. 2013, 2017) and Ni synthesis for associated supernovae (Song & Liu 2019).We hope future general-relativistic magnetohydrodynamic (GRMHD) simulation for a better understanding.3We adopt a simple model to describe the evolution of the fall-back accretion rate. For long GRBs, the envelope ofthe progenitor star is considered as the mass supply of the fall-back accretion (Kumar et al. 2008a). The evolutionof fall-back accretion rate is thus a good tracker of the structure of the progenitor envelope (Liu et al. 2018). Wewill explore the time-dependent fall-back accretion rate and the expected afterglow lightcurves from long GRBs withprogenitors of different masses, angular velocities and metallicities in future, and constrain the characteristics of starsby comparing the second plateau data with our model.We thank Bing Zhang for helpful discussion and the anonymous referee for the helpful comments that have helpedus to improve the presentation of the paper. This work is supported by the National Natural Science Foundation ofChina (NSFC) under Grant No. 11722324,11690024,11633001,11773010 and U1931203, the Strategic Priority ResearchProgram of the Chinese Academy of Sciences, Grant No. XDB23040100 and the Fundamental Research Funds forthe Central Universities. LDL is supported by the National Postdoctoral Program for Innovative Talents (Grant No.BX20190044), China Postdoctoral Science Foundation (Grant No. 2019M660515) and “LiYun” postdoctoral fellow ofBeijing Normal University.
Software:
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