Giant X-ray and optical Bump in GRBs: evidence for fall-back accretion model
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Giant X-ray and optical Bump in GRBs: evidence for fall-back accretion model
Litao Zhao, He Gao, WeiHua Lei, Lin Lan, and Liangduan Liu Department of Astronomy , Beijing Normal University, Beijing, China; Department of Astronomy, School of Physics, Huazhong University of Science and Technology, Wuhan, China
ABSTRACTThe successful operation of dedicated detectors has brought us valuable information for understand-ing the central engine and the progenitor of gamma-ray bursts (GRBs). For instance, the giant X-rayand optical bumps found in some long-duration GRBs (e.g. GRBs 121027A and 111209A) imply thatsome extended central engine activities, such as the late X-ray flares, are likely due to the fall-back ofprogenitor envelope materials. Here we systemically search for long GRBs that consist of a giant X-rayor optical bump from the Swift GRB sample, and eventually we find 19 new possible candidates. Thefall-back accretion model could well interpret the X-ray and optical bump for all candidates within areasonable parameter space. Six candidates showing simultaneous bump signatures in both X-ray andoptical observations, which could be well fitted at the same time when scaling down the X-ray flux intooptical by one order of magnitude, are consistent with the standard F ν ∝ ν / synchrotron spectrum.The typical fall-back radius is distributed around 10 − cm, which is consistent with the typicalradius of a Wolf-Rayet star. The peak fall-back accretion rate is in the range of ∼ − − − M (cid:12) s − at time ∼ − s, which is relatively easy to fulfill as long as the progenitor’s metallicity is not toohigh. Combined with the sample we found, future studies of the mass supply rate for the progenitorswith different mass, metallicity, and angular momentum distribution would help us to better constrainthe progenitor properties of long GRBs. Keywords:
Gamma-ray bursts (629) INTRODUCTIONIncreasing evidence suggests that the long gamma-ray bursts (GRBs) are associated with the death of massive stars(Woosley 1993; Paczy´nski 1998; MacFadyen & Woosley 1999; Woosley & Bloom 2006). At the end of a massive star’slife, electron-trapping and photon decomposition will trigger core collapse and form a hyperaccreting black hole (BH)or a rapidly spinning magnetar, which can launch a relativistic jet. The internal dissipation of the relativistic jet fuelsthe prompt emission, and the external shocks (especially the forward shock) due to jet-medium interaction contributesmultiwavelength afterglow emission (see Zhang (2018) for a review).In general, the end of the prompt emission phase means the cease of the central engine. However, the observationsof Neil Gehrels Swift suggest that many GRBs have an extended central engine activity time, manifested throughflares (Burrows et al. 2005a; Zhang et al. 2006; Margutti et al. 2011) and extended shallow plateaus (Troja et al. 2007;Liang et al. 2007; Zhao et al. 2019; Tang et al. 2019) in the X-ray light curves following the MeV emission. It has longbeen proposed that some of these interesting signatures could help us to determine the central engines for particularGRBs (Dai & Lu 1998; Rees, & M´esz´aros 1998; Zhang & M´esz´aros 2001; Zhang et al. 2006; Nousek et al. 2006).For instance, systematic analysis for the
Swift
GRB X-ray afterglow shows that bursts with X-ray plateau featureslikely have rapidly spinning magnetars as their central engines (Liang et al. 2007; Zhao et al. 2019; Tang et al. 2019),especially when the X-ray plateau followed by a very steep decay. The steep decay is difficult to be interpret withinthe framework of a BH central engine, but is consistent within a magnetar engine picture, where the abrupt decay isinterpreted as the collapse of a supramassive magnetar into a BH after the magnetar spins down (Troja et al. 2007;
Corresponding author: He [email protected] a r X i v : . [ a s t r o - ph . H E ] D ec 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) found one candidate, GRB 070110, that showed a small X-raybump following its internal plateau, and Zhao et al. (2020) found another three candidates in the Swift sample, i.e.,GRBs 070802, 090111, and 120213A, whose X-ray afterglow light curves contain two plateaus, with the first one beingan internal plateau. These particular cases provide further support to the magnetar central engine model.For GRBs without shallow decay features, their most promising central engine should be the hyperaccreting BHsystem. In this scenario, the late X-ray flares could be interpreted with four different approaches: 1) part of themassive star envelope mass falling back onto the BH and reactivating the central engine (Kumar et al. 2008a,b); 2)late time features need not necessarily be related to late central engine activity, since they might be due to the lateinternal collisions or refreshed external collisions from early ejected shells (Rees, & M´esz´aros 1998; Sari & M´esz´aros2000; Gao & M´esz´aros 2015); 3) late flares can arise from the interaction of a long-lived reverse shock (RS) with astratified ejecta produced by a gradual and nonmonotonic shutdown of the central engine right after the initial ejectionphase (Uhm & Beloborodov 2007; Genet et al. 2007; Hasco¨et et al. 2017); 4) an outflow of modest Lorentz factor islaunched more or less simultaneously with the highly relativistic jet that produced the prompt gamma-ray emission,so that flares are produced when the slow moving outflow reaches its photosphere (Beniamini & Kumar 2016).For the fallback accretion model, if the fallback accretion rate and fallback duration are large enough, the giant X-rayand optical bump with rapid rising and t − / decaying feature are expected, which can hardly be interpreted with thelatter three candidate models. Up to now, such giant X-ray bumps have been discovered in two GRBs, 121027A and111209A, and both data could be well interpreted under the fall-back accretion model framework (Wu et al. 2013; Yuet al. 2015; Gao et al. 2016b).Thanks to the successful operation of dedicated satellites and ground-based detectors, many GRBs were detectedwith good quality X-ray and optical afterglow observations. In this work, we systematically search for long GRBswith giant X-ray or optical bump from the GRB sample. The data reduction method and the sample selection resultsare presented in section 2. In section 3, we described the fall-back accretion model and apply this model to the giantX-ray and optical bump observed in our selected sample. The conclusion and implications of our results are discussedin Section 4. Throughout the paper, the convention Q = 10 n Q n is adopted in c.g.s. units. DATA REDUCTION AND SAMPLE SELECTIONFor the purpose of this work, we systemically search for long GRBs consisting of a giant X-ray or optical bump fromthe GRB sample (detected between 1997 January and 2019 October). The XRT light-curve data were downloadedfrom the Swift/XRT team website (Belokurov et al. 2009), and processed through HEASOFT (v6.12) software. Theoptical afterglow data were searched from published papers. Compared with typical flares, the duration of the giantbump should be relatively longer. Yi et al. (2016) have analyzed all significant X-ray flares from the GRBs observedby Swift from 2005 April to 2015 March, and obtained an empirical relationship:log T dur = ( − . ± .
11) + (1 . ± . × log ( T peak ) . (1)where T peak is the peak time of the flare, and T dur = T end − T start is the flare duration, with T start and T end being thestart time and end time of flares. For each flare, T peak and T dur could be easily obtained by fitting the light curve witha smooth broken power-law function (representing the flare) superposing on a simple power-law function (representingthe underlying continuum component), see details in Yi et al. (2016). Here we define the flares of 2 σ deviate from the T peak − T dur empirical relationship (Eq. 1) to the longer duration trend as a giant bump.Besides GRB 121027A and GRB 111209A, we find another 19 GRB candidates containing a giant X-ray or opticalbump signature that could meet our selection criteria. We collect the names and optical data references of thesecandidates in Table 1, together with their basic observational properties, such as the prompt emission duration T ,the detected gamma-ray fluences S γ , the power-law photon index of prompt emission spectrum Γ γ , the beginning time t start and ending time t end of the giant X-ray or optical bump, and the redshift z . For the whole sample, the T , S γ ,Γ γ and z are distributed in the ranges of 4 − . × − − . × − erg cm − , 0 . − .
7, and 0 . − . − . × s,and the duration of the bump is in an extended range of 538 − . × s. Here we divide the new candidates intotwo samples: the gold sample that contains simultaneous bump signatures in both X-ray and optical afterglow data (6/19), and the silver sample that shows giant X-ray but without giant optical bump or vice versa (13/19). It is worthnoting that GRB 130831A contains detections from both X-ray and optical band, but only optical bands has enoughdata when the bump signature emerges, and it is thus assigned to the silver sample. Due to the lack of simultaneous Table 1.
Features of the sampleGRB log ( t start ) a log ( t end ) a Instruments T Γ γ S γ Reference b z Reference c Reference d Groups s s 10 − erg cm − .
11 BeppoSAX 20 3.7 ± ± .
01 HETEC2 10 3 . +0 . − . . ± . .
17 Swift 65 +40 − . ± .
17 21 . ± . .
13 Swift 4 ± ± ± .
51 Swift 7 ± . ± .
34 8 . ± . .
55 Swift 43.6 ± ± ± .
14 Swift 19 ± ± ± .
73 Swift 6 ± ± ± .
85 Swift 260 ±
40 1.25 ± ± .
41 Swift 270 ±
45 1 . ± .
18 21 ± .
25 Swift 8.8 ± ± ± .
51 Swift 7 ± . ± .
25 3 . ± . .
12 Swift 439 ±
33 1 . ± .
21 21 ± .
41 Swift 13 ± ± ± .
92 Swift 1400 1.48 ± ±
10 P11a 0.677 V11 ... Silver120118B 2.61 4 .
92 Swift 23.26 ± ± ± . ± ± ± .
71 Swift 32 . ± . . ± .
05 65 ± .
22 Swift 23.4 ± ± ± .
85 Swift 35.53 ± ± ± .
93 Swift 58.2 ± ± ± at and t are beginning time and end time of the giant X-ray or optical bump, respectively. b The references of GRB prompt phase observations for our sample. c The references of GRB redshift for our sample. d The references of optical afterglow data for our sample.
Note —References: (A02)Amati et al. (2002);(A08)Amati et al. (2008);(B05)Barbier et al. (2005);(B07)Barbier et al. (2007);(B08a)Barthelmy et al. (2008); (B08b)Berger et al. (2008); (B12)Barthelmy et al. (2012); (B13)Barthelmy et al. (2013);(B16)Barthelmy et al. (2016); (C08a)Cucchiara et al. (2008); (C08b)Cummings et al. (2008); (C09)Cummings et al. (2009);(C10a)Chornock et al. (2010); (C10b)Cucchiara & Fox (2010); (C13)Cucchiara & Perley (2013); (C14)Chornock et al.(2014); (C16)Cano et al. (2016); (C19) Chen et al. (2019); (D10)D’Avanzo et al. (2010); (D18)de Ugarte Postigo et al.(2018); (D19)Dichiara et al. (2019); (F09a)Fugazza et al. (2009); (F09b)Fynbo et al. (2009);(J06a)Jakobsson et al. (2006a);(J06b)Jakobsson et al. (2006b); (K07)Krimm et al. (2007);(K10)Kann et al. (2010); (K12)Kr¨uhler et al. (2012); (K16a)Kuin &Kocevski (2016); (K16b)Kuroda et al. (2016); (K19a)Klose et al. (2019); (K19b)Kumar et al. (2019); (L13)Liang et al. (2013);(L15)Laskar et al. (2015);(L19)Lien et al. (2019); (M08)Miller et al. (2008); (M13)Malesani et al. (2013); (M15)Melandriet al. (2015); (P06a)Palmer et al. (2006);(P06b)Prochaska et al. (2006); (P07)Prochaska et al. (2007); (P11a)Palmer et al.(2011);(P11b)Piranomonte et al. (2011); (Q13)Qin & Chen (2013); (R08)Rumyantsev et al. (2008); (S04a)Sakamoto et al.(2004); (S04b)Soderberg et al. (2004); (S05)Soderberg et al. (2005);(S06a)Sakamoto et al. (2006);(S06b)Sato et al. (2006);(S10)Sakamoto et al. (2010); (S12)Sakamoto et al. (2012);(S13)S´anchez-Ram´ırez et al. (2013); (S14)Stamatikos et al. (2014);(T05)Tueller et al. (2005); (T12)Tanvir et al. (2012); (U10)Ukwatta et al. (2010a); (U11)Ukwatta et al. (2011); (V11)Vreeswijket al. (2011); (V19)Valeev et al. (2019); (X10)Xu & Huang (2010);(Y16)Yanagisawa et al. (2016). bump signatures, we put GRB 121027A and GRB 111209A in the silver sample. Figure 1 shows the Amati relation( E iso ,γ, i − E p,i )(Amati et al. 2002) for GRBs in our sample and other GRBs that have redshift measurements and weredetected up to 2019 October (Minaev & Pozanenko 2020), where E iso ,γ, i is isotropic-equivalent radiation energies ofthe GRB prompt phase in the 1 − keV band and E p,i is the peak energy of the νF ν spectrum ( E p ) in the burstrest frame, i.e., E p,i = E p (1 + z ). It is interesting that bursts in our sample tend to have larger E iso ,γ, i for given E p,i values, which is consistent with the interpretation that these bursts may have more active fall-back accretion rates. log (E p, i (keV)) l o g ( E i s o ,, i ( e r g )) Other GRBsGold sampleSilver sample
Figure 1.
The Amati relation ( E iso ,γ, i − E p,i ) in our sample, compared with other GRBs that have redshift measurementsand are detected up to 2019 October. We apply the regression model to GRBs in our sample and other GRBs, yieldedlog ( E iso ,γ, i ) = (50 . ± . . ± .
28) log ( E p,i ) and log ( E iso ,γ, i ) = (48 . ± . . ± .
13) log ( E p,i ), respectively.The violet line mark the best fitting result for our sample and the light violet shadowed region shows the intrinsic scatter tothe population 3 σ . The grey line marks the best fitting result for other GRBs and the light gray shadowed region shows theintrinsic scatter to the population 3 σ .3. FALL-BACK ACCRETION MODEL APPLICATION3.1.
Model Description
In this paper, we intend to use the fall-back accretion model that has been described in Wu et al. (2013) to interpretgiant X-ray and optical bumps in our selected sample. The physical picture is as follows: for our selected GRBs, theirprogenitor stars have a core-envelope structure, as is common in stellar models. At the end of the star’s life, the bulkof the mass in the core part collapses into a rapidly spinning BH, and the rest mass forms a surrounding accretion disk.A relativistic jet is launched by the hyperaccreting BH system and it successfully penetrates the envelope to power theGRB prompt γ -ray and broadband afterglow emission. During the penetration, parts of the jet energy are transferredinto the envelope, which might help the supernova to explode. The bounding shock responsible for the associatedsupernova would transfer kinetic energy to the envelope materials, so that most envelope materials would be ejectedbut with a small portion falling back onto the BH (Kumar et al. 2008a,b). The fall-back of the envelope materials mayform a new accretion disk, powering a new relativistic jet through the Blandford-Znajek (BZ) mechanism (Blandford& Znajek 1977; Lee et al. 2000; Li 2000; Lei et al. 2005, 2013) or neutrino-annihilations mechanism(Popham et al.1999; Narayan et al. 2001; Di Matteo et al. 2002; Janiuk et al. 2004; Gu et al. 2006; Chen & Beloborodov 2007; Liuet al. 2007, 2015; Lei et al. 2009, 2017; Xie et al. 2016). In general cases, a BZ jet is more powerful than a neutrino-annihilation jet, which is more likely accounts for the central engine activities (Kawanaka et al. 2013; Lei et al. 2017;Xie et al. 2017; Lloyd-Ronning et al. 2018). Especially during the fall-back accretion stage, the typical accretion rateis far below the igniting accretion rate , the neutrino-annihilation power cannot explain the late-time X-ray activitiesin GRBs. Finally, a part of the jet energy would undergo internal dissipation and generate the observed giant X-rayand optical bump.According to some analytical and numerical calculations, the evolution of the fall-back accretion rate can be describedby a broken power-law function of time (Chevalier 1989; MacFadyen et al. 2001; Zhang et al. 2008; Dai & Liu 2012)˙ M acc = ˙ M p (cid:34) (cid:18) t − t t p − t (cid:19) − s/ + 12 (cid:18) t − t t p − t (cid:19) − s/ (cid:35) − /s , (2)where t is the beginning time of the fall-back accretion in the cosmologically local frame, t p and ˙ M p are the peaktime and peak rate of fall-back accretion, and s describes sharpness of the peak.The BZ power from a kerr BH with a 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 . × erg s − a • (cid:18) M • M (cid:12) (cid:19) B • , F ( a • ) , (3)where a • = J • c/GM • is dimensionless BH spin parameter, B • , is strength of the magnetic field near the BH horizonin units of 10 G, and F ( a • ) = (cid:20) q q (cid:21) (cid:20)(cid:18) q + 1 q (cid:19) arctan q − (cid:21) , (4)with q = a • / (1 + (cid:112) − a • ).Accretion disk is essential for maintaining a strong magnetic field. If there is no accretion disk magnetic pressure,the magnetic field near the BH horizon will disappear quickly. We estimate the value of B • by balancing the magneticpressure on the BH horizon and ram pressure of the accretion flow at the inner edge of the accretion disk (Moderskiet al. (1997)) B • π = P ram ∼ ρc ∼ ˙ M acc c πr h , (5)where r h = (1 + (cid:112) (1 − a • )) r g is radius of BH horizon and r g = GM • /c . Therefore, the BZ power can be rewrittenas L BZ = 9 . × a • F ( a • )(1 + (cid:112) − a • ) ˙ M acc ( M (cid:12) s − ) erg s − . (6)We introduce the X-ray and optical radiation efficiency η ( X,O ) and jet beaming factor f b = 1 − cos θ j ( θ j is the jetopening angle) to connect the observed X-ray and optical luminosity L ( X,O ) and the BZ power L BZ by η ( X,O ) L BZ = f b L ( X,O ) . (7)Note that the BZ process extracts rotational energy and angular momentum from the BH, but the accretion pro-cess brings disk energy and angular momentum into the BH. According to the conservation of energy and angularmomentum, the evolution equation of BH under two processes are given as follows (Wang et al. (2002)): dM • c dt = E in ˙ M acc c − L BZ , (8) Accretion in a fall-back disk can occur via two distinct modes. For high accretion rates, the disks are dense and hot enough in theinner regions to cool via neutrino losses. However, for lower fall-back rates and/or at larger radii, the accretion is radiatively inefficientand has an advection-dominated accretion flow. The accretion rate at the transition that the inner disk from being neutrino dominatedto advection dominated is defined as igniting the accretion rate. As discussed in Lei et al. (2017), the igniting accretion rate would be0 . M (cid:12) /s for a nonspinning BH and 0 . M (cid:12) s − for a fast spinning BH. and dJ • dt = J in ˙ M acc − T BZ . (9) T BZ is the torque applied to BH by BZ process, which could be estimated by (Li 2000; Lei & Zhang 2011; Lei et al.2017) T BZ = 2 L BZ Ω H , (10)where Ω H = c GM • a • (cid:112) − a • ) , (11)is the angular velocity of BH horizon.The evolution of dimensionless BH spin parameter a • could be derived from Equations 8 and 9, da • dt = (cid:16) ˙ M acc J in − T BZ (cid:17) cGM • − a • (cid:16) ˙ M acc c E in − L BZ (cid:17) M • c . (12)Here E in and J in are represent the specific energy and angular momentum at the radius of the innermost inner edgeof accretion disk R in , respectively, which are defined as (Novikov & Thorne 1973): J in = GM • c √ R in − a • ) √ R in , (13)and E in = 4 √ R in − a • √ R in , (14)where R in can be obtained from (Bardeen et al. 1972) R in = r in r g = 3 + Z − [(3 − Z )(3 + Z + 2 Z )] / , (15)where Z ≡ − a • ) / (cid:2) (1 + a • ) / + (1 − a • ) / (cid:3) and Z ≡ (3 a • + Z ) / for 0 ≤ a • ≤ Model Application to Selected GRBs
In the following, we apply the fallback accretion model to fit giant X-ray and optical bumps in our selected sample.Here we adopt the beginning time and ending time of the giant X-ray or optical bump ( t start , t end ) as the beginning timeand ending time of the fall-back accretion. Since the initial BH mass M hardly affects the BZ power, here we adopt M = 3 M (cid:12) as a fixed value for all GRBs in our sample. Considering that a BH may be spun up by accretion or spundown by the BZ mechanism, the BH spin will reach an equilibrium value a eq ∼ .
87 (Lei et al. 2017). For simplicity,we adopt a = 0 . η X = 0 .
01 and f b = 0 . η O = ξη X . We take the dimensionlessfallback accretion peak ˙ m p ( ˙ m p = ˙ M p /M (cid:12) s − ), the peak sharpness of the fallback accretion s , the peak time offallback accretion t p and ξ as our free parameters. We then use the Markov chain Monte Carlo (MCMC) methodto fit the data. In our fitting, we use a Python module emcee (Foreman-Mackey et al. 2013) to get best-fit valuesand uncertainties of free parameters. The allowed range of the four free parameters are set as log ( ˙ m p ) ≡ [ − , s ≡ [0 , t p ≡ [ t start , t end ], and log ( ξ ) ≡ [ − , σ errors level asthe fitting results, which are listed in Table 2. We also calculate the total accretion mass M acc during the fall-backprocess, the BH magnetic field strength B p, at t p and the fall-back radius r p corresponding to t p . We find that theX-ray and optical data could be well fitted simultaneously when ξ is in order of 10 − . It is interesting to note that theoptical and X-ray flux ratio is consistent with the standard F ν ∝ ν / synchrotron spectrum below E p . Therefore, wetake η X = 0 .
01 and η O = 0 .
001 to fit the X-ray or optical bumps in the silver sample.The best-fitted light curves for the silver sample are shown in Figure 4. The corner plots of the free parameterposterior probability distribution for the fitting are shown in Figure 5. MCMC Fitting results for the silver sample arelisted in Table 3. It is worth noting that the optical data of GRB 051016B shows a simple power-law decay segmentduring the X-ray bump epoch. In this case, we can give an upper limit for ξ < . ξ value for GRB 130831A would be roughly 0.02, which is alsolower than the value in the gold sample. If the giant bumps found in this work are indeed from the internal dissipationof fall-back accretion powered jets, the dissipation ratio between X-ray and optical bands might be diverse. For otherbursts in the silver sample, there is no good joint band data to make any interesting constraints on their ξ values.We display the distributions of best fitting values of ˙ m p , t p , M acc , B p, and r p for both the gold and total samplesin Figure 6. We find that for the total sample, ˙ m p , t p , M acc B p, and r p accord the lognormal distribution in theranges of 5 . × − < ˙ m p < . × − , 512 .
86 s < t p < . × s, 2 . × − M (cid:12) < M acc < . M (cid:12) ,1 . × − < B p, < .
23, and 1 . × cm < r p < . × cm, respectively, with log ( ˙ m p ) = − . ± . [ t p ( s )] = 3 . ± .
81, log [ M acc ( M (cid:12) )] = − . ± .
31, log ( B p, ) = − . ± .
81 and log [ r p (cm)] = 11 . ± . z = 1 for those GRBs and also use MCMC method to fit the giant X-ray bumps for thoseGRBs, we find that the distributions for all the parameters barely change by adding those GRBs into the sample.Based on the fitting results, we can draw conclusions as follows: 1) a reasonable parameter space can interpret thegiant X-ray and optical bumps for both the gold and silver GRB samples. 2) the lower limit of total accretion masscould be as low as ∼ − M (cid:12) , which means even with a very small fraction of the progenitor’s envelop falling back, itis still possible to generate the giant X-ray and optical bump feature. 3) when the fall-back mass rate reaches its peak,the corresponding radius r p is around 10 − cm, which is consistent with the typical radius of a Wolf-Rayet star.It is worth checking whether the mass supply rate of progenitor envelope material at t p could meet the accretionrate requirement for our fitting. We calculate the mass supply rate of the progenitor envelope materials as (Suwa &Ioka 2011; Woosley & Heger 2012; Matsumoto et al. 2015; Liu et al. 2018)˙ M fall = dM r dt fb = dM r /drdt fb /dr = 2 M r t fb(r) (cid:18) ρ r ¯ ρ − ρ r (cid:19) , (16)where ρ r is the mass density at radius r , ¯ ρ = 3 M r / πr is the average mass density within r , t fb(r) ∼ ( π r / GM r ) / is freefall timescale at r , and M r = M + (cid:90) rr πr ρdr, (17)is the total mass within r . We assume that the jet is launched when the central accumulated mass reach the initialmass of BH ( M ) and set r = r and t = 0 at this time, i.e. t = t fb ( r ) − t fb ( r ). On the other hand, Liu et al. (2018)gave the relationship between the density ρ and radius r for progenitor with different masses and metallicities. We thuscalculate the evolution of the mass supply rate for various progenitor masses and metallicities. As shown in Figure7, we find that for most cases provided in Liu et al. (2018), the mass supply rate of progenitor envelope material at t p is large enough to meet the accretion rate requirement for our fitting. In other words, the constraints set by theprogenitor’s mass and metallicity are not strong, so that the interpretation presented in this work should be justified.It is also of interest to study the relation between the jet energy from the fall-back accretion ( E BZ ) and isotropic-equivalent radiation energies of GRB prompt phase in the 1 − keV band ( E iso ,γ, i ). As shown in Figure 8, we applythe regression model to our sample and obtain the best fit as log ( E iso ,γ, i ) = (14 . ± .
57) + (0 . ± .
11) log ( E BZ )(Spearman correlation coefficient r = 0 .
82 and significance level p < − for N = 21). The result infers that thefall-back accretion is correlated with the prompt phase accretion.Finally, we would like to note that besides the internal dissipation, a good fraction of the fall-back jet energy wouldeventually inject into the GRB afterglow blast wave. Depending on the ratio between the injected energy and theinitial kinetic energy in the blast wave, the afterglow light curve following the giant bump could become shallower ifthe late injected energy is larger, or not shallower if the initial kinetic energy is larger (Zhao et al. 2020). Late timejet break effect could make the situation even more complicated. In our sample, for GRBs 050814, 051016B, 081028and 140515A, we find that the segment following the X-ray bump tends to become shallower, but not for other bursts. (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og Figure 2.
Modeling results for the giant X-ray and optical bumps of the gold sample. The red solid lines and pink solid linespresents theoretical light curve produced by our model within the the Swift-XRT energy band (0 . −
10 keV) and optical band,respectively. 4.
DISCUSSION AND CONCLUSIONThe giant X-ray bumps discovered in the afterglow of GRB 121027A and 111209A, have been proposed as the directevidence to support that the late central engine activity of long GRBs is likely due to the fall-back accretion process.In this work, we systemically searched all long GRBs detected between 1997 January and 2019 October, and foundanother 19 candidate GRBs showing giant X-ray or optical bumps in their afterglows. We have applied the fall-backaccretion model to interpret the X-ray and optical bump data for the whole sample. We summarize our results asfollows: • We find that the X-ray and optical bump data for the gold sample and silver sample could be well interpretedby the fall-back accretion model within a fairly flexible and reasonable parameter space. For the six GRBs inthe gold sample showing a simultaneous bump signature in both X-ray and optical observations, the X-ray andoptical data could be well fitted simultaneously with ξ = η O /η X ∼ .
1, which happens to be consistent with thestandard F ν ∝ ν / synchrotron spectrum below E p . Figure 3.
The corner plots of the free parameters posterior probability distribution for the fitting of giant X-ray and opticalbump in the gold sample. • The fall-back accretion rate reaches its peak value ∼ − − − M (cid:12) s − at time ∼ − s, the constraintsset by the progenitor’s mass and metallicity are not strong, which makes the mass supply rate of the progenitorenvelope material at late time fulfill the fall-back accretion rate requirement. The lower limit of total accretionmass could be as low as ∼ − M (cid:12) , which means even with a very small fraction of the progenitor’s envelopfalling back, it is still possible to generate the giant X-ray bump and optical feature. Table 2.
MCMC multiband fitting results for the gold sampleGRBname log ( ˙ m p ) s log ( t p ) log ( ξ ) M acc B P, r p χ s 10 − M (cid:12) − cm060206 − . ± .
017 0 . ± .
01 3 . ± . − . ± .
016 11723.21 ± . ± .
31 10 . ± .
71 310.77060906 − . ± .
03 1 . +0 . − . . ± . − . ± .
02 2020 . ± .
32 2665.05 ± ± − . ± .
04 0 . +0 . − . . ± . − . ± .
03 214.52 ± ± . ± .
45 41.87081029 − . ± .
02 0 . ± .
05 4 . ± . − . ± .
02 6903.02 ± ± . ± .
34 327.28100418A − . ± .
03 0 . ± .
01 4 . ± . − . ± .
02 112.91 ± ± . ± .
71 112.31100901A − . ± .
02 0 . ± .
07 4 . ± . − . +0 . − . ± ± . ± .
23 256.94 (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og R BandGRB 970508 (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og R BandGRB 020903 (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og Figure 4.
Modeling results for the giant X-ray or optical bumps of the silver sample. The red solid lines and pink solid linespresent a theoretical light curve produced by our model within the the Swift-XRT energy band (0 . −
10 keV) and optical band,respectively. The pink dotted lines present the upper limit ξ . • The typical fall-back radius is around 10 − cm, which is consistent with the typical radius of a Wolf-Rayetstar.1 (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og c RI Band Band s KR BandWhite BandGRB 161129A (Time(s)) log )) - ( Lu m i no s i t y ( e r g s s l og Figure 4. -Continued • The jet energy from the fall-back accretion is linearly correlated with the isotropic-equivalent radiation energiesof GRB prompt phase in the 1 − keV band, implying that the fall-back accretion is correlated with theprompt phase accretion.In conclusion, our results provide additional support for core collapse from Wolf-Rayet star as the progenitors oflong GRBs, whose late central engine activity is very likely caused by the fall-back accretion process.There are two possible reasons why most long GRBs do not show a giant X-ray and optical bump: firstly, if theprogenitor of long GRBs produce a high-energy supernova when its core collapses into a BH, the supernova shock mayeject most of envelope materials and leave too little material falling back into the BH; secondly, during the fall-backaccretion progress, most of the jet energy injects into the GRB blast wave, while the energy that undergoes internaldissipation is weak.It is worth noting that in Figure 7, we make a direct comparison between the required peak accretion rate ( ˙ M acc )and the fall-back mass rate ( ˙ M fall ). If considering that a good fraction of fall-back mass could be taken away by theaccretion disk outflow, ˙ M fall > ˙ M acc is required, so that low metallicity progenitor stars would become more favored.On the other hand, when calculating the fall-back mass rate, we did not consider the angular momentum distributionof the progenitor, which is approximately valid for a slowly rotating progenitor. In the future, detailed studies for theevolution model of the mass supply rate for the progenitors with different mass, metallicity and rotation speed wouldhelp us to better constrain the progenitor properties of long GRBs.2 Figure 5.
The corner plots of the free parameters posterior probability distribution for the fitting of the giant X-ray or opticalbump in the silver sample.
We thank the anonymous referee for the helpful comments that have helped us to improve the presentation ofthe paper. This work is supported by the National Natural Science Foundation of China (NSFC) under GrantNo. 11722324,11690024,11633001,11773010 and U1931203, the Strategic Priority Research Program of the ChineseAcademy of Sciences, Grant No. XDB23040100 and the Fundamental Research Funds for the Central Universities.3
Figure 5. -Continued
LDL is supported by the National Postdoctoral Program for Innovative Talents (Grant No. BX20190044), China Post-doctoral Science Foundation (Grant No. 2019M660515) and “LiYun” postdoctoral fellow of Beijing Normal University.
Software:
XSPEC(Arnaud (1996)), HEAsoft(v6.12;Nasa High Energy Astrophysics Science Archive Research Center(Heasarc)),root(v5.34;Brun&Rademakers(1997)),emcee(v3.0rc2;Foreman-Mackeyetal.(2013)),corner(v2.0.1;Foreman-Mackey (2016)) REFERENCES
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Figure 6.
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Figure 7.
The evolution of the mass supply rate of progenitor with different masses and metallicities. Blue square and blackstar symbols represent the best fitting required dimensionless peak accretion rate ˙ m p with respect to the peak accretion time t p . The Z (cid:12) is the metallicities of the sun.
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Figure 8.
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MCMC fitting results for the silver sampleGRBname log ( ˙ m p ) s log ( t p ) M acc B P, r p χ s 10 − M (cid:12) − cm970508 − . ± .
02 1 . +0 . − . . ± .
01 763.32 ± ± . ± .
42 157.54020903 − . +0 . − . . +3 . − . . ± .
02 2.22 ± ± . ± .
01 11.29050814 − . +0 . − . . +0 . − . . ± .
02 4938 .12 ± ± . ± .
16 35.66051016B − . +0 . − . . +0 . − . . +0 . − . . ± .
21 890.27 ± . ± .
67 13.94070103 − . ± .
08 1 . +3 . − . . +0 . − . . ± .
31 4435.12 ± . ± .
31 7.22081028 a -5.73 ± . +0 . − . ± ± ± ± − . ± .
06 1 . +1 . − . . +0 . − . . ± ± . ± .
31 6.13110715A − . ± .
02 0 . +0 . − . . ± .
02 640 . ± ± . ± .
21 72.48111209 A b -3.7 1 . − . ± .
03 0 . ± .
04 3 . ± .
03 2873.32 ± ± . ± .
51 27.36121027 A c -3.21 1 . − . ± .
06 2 . +2 . − . ± ± ± . ± .
23 39.76140515A − . ± .
02 1 . +0 . − . . ± .
02 13357.62 ± ± . ± .
23 51.32161129A − . ± .
06 1 . +0 . − . . +0 . − . . ± .
91 1224.51 ± . ± .
74 18.42190829A − . ± .
07 2 . +4 . − . . ± .
01 11.02 ± ± . ± .
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