Electron Spectrum for the Prompt Emission of Gamma-ray Bursts in the Synchrotron Radiation Scenario
Kuan Liu, Da-Bin Lin, Jing Li, Yu-Fei Li, Rui-Jing Lu, En-Wei Liang
aa r X i v : . [ a s t r o - ph . H E ] F e b Draft version February 22, 2021
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Electron Spectrum for the Prompt Emission of Gamma-ray Bursts in the Synchrotron RadiationScenario
Kuan Liu, Da-Bin Lin, Jing Li, Yu-Fei Li, Rui-Jing Lu, and En-Wei Liang GuangXi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning 530004,China
Submitted to ApJABSTRACTGrowing evidences indicate that the synchrotron radiation mechanism may be responsible for theprompt emission of gamma-ray bursts (GRBs). In the synchrotron radiation scenario, the electronenergy spectrum of the prompt emission is diverse in theoretical works and has not been estimatedfrom observations in a general way (i.e., without specifying a certain physical model for the electronspectrum). In this paper, we creatively propose a method to directly estimate the electron spectrumfor the prompt emission, without specifying a certain physical model for the electron spectrum in thesynchrotron radiation scenario . In this method, an empirical function (i.e., a four-order B´eziercurve jointed with a linear function at high-energy) is applied to describe the electron spectrum in log-log coordinate. It is found that our empirical function can well mimic the electron spectra obtained inmany numerical calculations or simulations. Then, our method can figure out the electron spectrumfor the prompt emission without specifying a model. By employing our method on observations,taking GRB 180720B and GRB 160509A as examples, it is found that the obtained electron spectraare generally different from that in the standard fast-cooling scenario and even a broken power law.Moreover, the morphology of electron spectra in its low-energy regime varies with time in a burst andeven in a pulse. Our proposed method provides a valuable way to confront the synchrotron radiationmechanism with observations.
Keywords: gamma-ray burst: general — magnetic reconnection — radiation mechanisms: non-thermal— magnetic fields INTRODUCTIONThe radiation mechanism for the prompt emission of gamma-ray bursts (GRBs) remains unclear after decades ofobservations. The radiation spectra of the prompt emission are usually characterized by an exponential-jointed bro-ken power-law function, i.e., Band function (Band et al. 1993). The typical value of parameters in Band functionby fitting the observations are α ∼ − β ∼ − .
2, and E p ∼ α , β , and E p are the low-energy pho-ton spectral index, high-energy photon spectral index, and the peak photon energy, respectively (Preece et al. 2000;Nava et al. 2011; Kaneko et al. 2006; Goldstein et al. 2012). Owing to the lack of physical origin for Band function,one derives the physical implications by inferring what mechanism the fit parameters can be produced by. Syn-chrotron radiation is a very natural candidate to explain the non-thermal feature of Band function (Meszaros et al.1994; Tavani 1996a; Daigne & Mochkovitch 1998; Ghirlanda et al. 2002). However, the most straightforward syn-chrotron model suffers from “fast-cooling problem”, i.e., the typical observed spectrum should have a low-energyphoton spectral index − /
2, which strongly conflicts with observations (Sari et al. 1998; Ghisellini et al. 2000). Manyattempts have been made to alleviate the fast-cooling problem, e.g., adopting a decaying magnetic field in emis-sion region (Pe’er & Zhang 2006; Uhm & Zhang 2014; Zhao et al. 2014), introducing a slow heating mechanism by
Corresponding author: Da-Bin [email protected] magnetic turbulence (Asano & Terasawa 2009), involving the inverse Compton scattering effect at the Klein-Nishinaregime (Derishev et al. 2001; Nakar et al. 2009), considering a marginally fast cooling regime (Daigne et al. 2011;Beniamini et al. 2018; Florou et al. 2021), or invoking a fast-increasing electron energy injection rate (Liu et al. 2020).In addition, it is shown that the synchrotron model could not account for about one third of bursts with α > − / In this paper, we propose a method in thispaper to directly estimate the electron spectrum for the prompt emission, without specifying a certainphysical model or presumptive morphology for the electron spectrum.
This paper is organized as follows.In Section 2, we describe our proposed empirical function in details. The empirical function is the key point of ourmethod and we focus on the synchrotron radiation scenario . In Section 3, our method is applied to discussBand radiation spectrum and on spectral analysis of observations. In Section 4, we summary our results. PRESCRIPTION OF ELECTRON SPECTRUM AND FITTING METHOD
In this paper, we mainly focus on how to estimate the electron spectrum for the prompt emissionin the synchrotron radiation scenario.
In the synchrotron radiation scenario, the GRBs’ prompt emission isgenerated from a group of relativistic electrons. Therefore, the electron energy spectrum for the prompt emission canbe estimated by fitting the radiation spectrum. For this purpose, we propose an empirical function to picture out thepossible electron spectrum.Intuitively, the electron spectrum shown in Figure 1 can be decomposed into two segments: a high-energy segmentand a low-energy segment jointed at the electron Lorentz factor γ e = γ m . The high-energy segment usually relates tothe electron injection rate Q ( γ e ) ∝ γ − p e and can be described by a power-law function n ( γ e ) ∝ γ − p − e , where ndγ e is thenumber of electrons in [ γ e , γ e + dγ e ]. However, the morphology of the low-energy segment is diversity in theoreticallyand can be very different from a power-law function. Then, we introduce a four-order B´ezier curve to describe thelow-energy electron spectrum in log-log coordinate (i.e., the log γ e − log n plane). Therefore, our empirical functionused to describe the electron spectrum is read aslog n = ( B ( γ e ) , γ e < γ m , log[ y m ( γ e /γ m ) − p − ] , γ e ≥ γ m , (1)where B ( γ e ) is the four-order B´ezier curve and y m is the number density of electrons at γ e = γ m . The four-order B´eziercurve B ( γ e ) is described with a serial of points ( γ e ( t ) , B ( t )), which are calculated with following equation by varying t B´ezier curve is a smooth curve defined by some given control points, which is wildly used in computer graphics and the related fields.In this paper, we adopt a simple four-order, two-dimensional B´ezier curve, which is created by four control points P , P , P , and P intwo-dimensional space. In general, it starts at P going toward P and arrives at P coming from the direction of P . Usually, it wouldnot pass through P and P unless these four points are in a line. However, these two points would determine the behavior of B´ezier curvebetween P and P . from 0 to 1, ( B ( t ) = (1 − t ) log y + 3 t (1 − t ) log y + 3 t (1 − t ) log y + t log y m , log γ e ( t ) = (1 − t ) log γ e , + 3 t (1 − t ) log γ e , + 3 t (1 − t ) log γ e , + t log γ m . (2)Here, P (log γ e , , log y ), P (log γ e , , log y ), P (log γ e , , log y ), and P (log γ m , log y m ) are four control points used tocreate the B´ezier curve. To simply our fittings, we adopt log γ e , = 1, log γ e , = (log γ m , − log γ e , ) / γ e , , andlog γ e , = 2(log γ m , − log γ e , ) / γ e , , where γ m , = 10 is set as the initial value of γ m . Then, the free parametersin our empirical function are y , y , y , γ m , and p . We fit the electron spectra in the left panel of Figure 1 withEquation (2), where the fitting results are shown with solid lines in this panel. One can find that the electron spectrain the left panel of Figure 1 can be well described with our empirical function. Then, our empirical function can beused to figure out the electron spectrum for the prompt emission, without specifying a certain physical model for theelectron spectrum.It should be noted that the electron spectrum, which can be described with our empirical function, should becontinuous. If not, such as the electron spectrum in the figure 3 of Burgess et al. (2011), our empirical function couldnot present a well fit. In addition, freeing the electron spectrum is not equivalent to having an empiricalphoton spectrum in the first place. Firstly, the lowest power-law index of the photon spectrum from ourempirical electron spectrum in the synchrotron radiation scenario should be larger than − / . Secondly,the electron spectrum for prompt emission carry the information from the particle accelerating andcooling mechanism. Thus the estimation for electron spectrum could help us to better understand theenergy dissipation process in relativistic jet. For a given electron spectrum, the observed synchrotron radiation flux at a given frequency ν can be calculated as f ν ( ν ) = √ q B Γ2 πd L m e c Z ∞ γ e , F ( ν/ν c ) n ( γ e ) dγ e , (3)where F ( x ) = x R + ∞ x K / ( k ) dk , K / ( k ) is the modified Bessel function of 5/3 order, ν c = 3 q e B γ Γ(1 + z ) / (2 πm e c )is the characteristic frequency of the electron with Lorentz factor γ e in magnetic field B , Γ = 300 is the bulk Lorentzfactor of the jet, d L is the luminosity distance, and q e , m e , and c are the electron charge, electron mass, and lightspeed, respectively.Based on the Equations (1) and (3), we can fit the radiation spectrum of the prompt emission and obtain thecorresponding electron spectrum. This is our proposed spectral-fitting-method used to estimate the electron spectrumfor the prompt emission in the synchrotron radiation scenario. To test our method, we perform a simple testing fittingon a synthetic data. The synthetic data is generated as follows: i ○ We create a synchrotron radiationspectrum based on a bump-shape electron spectrum. As an example, the black-dashed line in themiddle panel of Figure 1, i.e., Equation (1) with log γ e , = 1 , log γ m = 4 , log y = 30 , log y = 43 , log y = 42 , log y m = 40 , and p = − . , is adopted as our electron spectrum. In addition, B = 30 Gs is took. ii ○ Wefold this synchrotron radiation spectrum through the instrumental response of the Fermi Gamma-rayBurst Monitor to create a poisson-distributed synthetic data, where the python source package threeML (Vianello et al. 2015) is used. Then, we perform the spectral fitting based on the synthetic data. The spectralfitting is performed based on the Markov Chain Monte Carlo (MCMC) method to produce posterior predictions for themodel parameters, i.e., log y , log y , log y , log γ m , and p . The python source package emcee (Foreman-Mackey et al.2013) is used for our MCMC sampling, where N walkers × N steps = 10 × is adopted and the initial iterationsare used for burn-in. The priors of log y , log y , log y , log γ m , and p are set as uniform distribution in therange of (-30, 100), (10, 70), (30, 50), (3, 5), and (-5, -3), respectively. The projections of the posteriordistribution in 1D and 2D for the model parameters are presented in the right panel of Figure 1 and theelectron spectra based on the last iterations are also plotted in the middle panel of Figure 1 with redlines. One can find that the obtained values of log y = 41 . +0 . − . , log γ m = 4 . +0 . − . , and p = − . +0 . − . aresimilar to those of our provided electron spectrum. However, the obtained values of log y = 26 . +9 . − . https://github.com/threeML/threeML https://github.com/dfm/emcee/blob/b9d6e3e7b1926009baa5bf422ae738d1b06a848a/docs/index.rst The priors of log y , log y , and log y are set based on the following consideration. With Equation (2), we fit the electron spectra in theleft panel of Figure 1. The fitting results reveal that the values of log y and log y do not deviate from the value of log y significantly.Therefore, we set the priors of log y and log y as (log y − , log y + 10) and (log y − , log y + 30), i.e., (30, 50) and (10, 70),respectively. In addition, the prior of log y may be in a wide range. The reason can be found in the end of Section 2. Then, we set theprior of log y as ( − , and log y = 43 . +2 . − . deviate from those of our provided electron spectrum, especially for the value of log y . It implies that the electron spectrum from our spectral fittings are only robust in the low-energyand high-energy ranges rather than the lowest-energy range. SPECTRAL ANALYSIS3.1.
Comments on Band Function
In this subsection, we investigate the electron spectrum related to Band radiation spectrum in synchrotron radiationscenario. A Band function with typical parameters α = − β = − .
3, and E p = 400 keV is discussed in this subsectionand shown in Figure 2 with black line. In general, it is believed that such radiation spectrum is originated from thesynchrotron radiation of a broken power-law electron spectrum with p low = 2( α + 1) − p = 2( β + 1) −
1, where p low and p are the low-energy and high-energy power-law indexes, respectively. The synchrotron radiation spectrumof such electron spectrum is shown in Figure 2 with green dashed line. Obviously, the radiation spectrum generatedfrom such kind of broken power-law electron spectrum is very different from Band radiation spectrum, especially forthe part around the transition from low-energy spectral segment to high-energy spectral segment. The transitionis apparently sharp for Band function compared with the synchrotron radiation spectrum. This behavior has alsobeen found in many previous works, e.g., Zhang et al. (2016) and Burgess (2019). This result suggests that the Bandradiation spectrum may not be produced by a broken power-law electron spectrum.In the following, we search for the most suitable electron spectrum for Band radiation spectrum by fitting it withEquations (1) and (3). The obtained electron spectrum and its radiation spectrum are shown in Figure 2 with reddashed line. Although the obtained radiation spectrum is basically consistent with Band radiation spectrum, it is a bitweird for the unexpected sharp peak at γ m and the strange bump at the low-energy regime of electron spectrum. Wepoint out that this kind of electron spectrum may be unnatural to some degree. The reasons are shown as follows. (1)The peak at γ m is mainly related to the exponential-connected break in Band function, whereas the physical origin ofthis break is no clear yet. (2) Although the obtained electron spectrum can produce a Band-like synchrotron radiationspectrum, the position of low-energy bump and γ m -peak in electron spectrum should be fine-tuned, which may hardlyexist in real situation. (3) The shape of this electron spectrum is very different from those in the left panel of Figure 1,except the one shown with green line, which has a similar peak at ∼ γ m . However, one should note that such kind ofelectron spectrum mainly appears without making significant contribution to the observed flux (e.g., Uhm & Zhang2014). Therefore, we would like to believe that the γ m -peak in the electron spectrum corresponding to Band functionmay be an unnatural outcome. Then, the exponential transition in Band function may not well describe the transitionbehavior in the radiation spectrum of the prompt emission if the synchrotron radiation does work.This subsection is dedicated to study the electron spectrum corresponding to Band radiation spectrum in thesynchrotron radiation scenario. We found that the electron spectrum of the Band radiation spectrum may be hardlyreproduced in a physical model, e.g., the models producing the electron spectrum in Figure 1. It suggests that the Bandradiation spectrum may be not intrinsic to the prompt emission of GRBs, especially to the transition segment (betweenlow-energy regime and high-energy regime) in the radiation spectrum. We would like to point out that to understandthe characteristics of the Band radiation spectrum, fitting the synthetic observed data of the synchrotron radiation withthe Band function is necessary. For example, Burgess et al. (2015) simulate synchrotron or synchrotron+blackbodyspectra and fold them through the instrumental response of the Fermi Gamma-ray Burst Monitor. They then performa standard data analysis by fitting the synthetic data with both Band and Band+blackbody models to investigate theability of the Band function to fit a synchrotron spectrum within the observed energy band.3.2. Application on GRBs 180720B and 160905A
In this subsection, we fit the radiation spectrum of GRBs 180720B and 160905A with Equations (1) and (3) toestimate the electron spectrum in the synchrotron radiation scenario. In our spectral analysis, we use the datafrom the
Fermi /GBM. GBM has 12 sodium iodide (NaI) scintillation detectors covering the 8 keV-1 MeV energyband, and two bismuth germanate (BGO) scintillation detectors being sensitive to the 200 keV-40 MeV energy band(Meegan et al. 2009). The brightest NaI and BGO detectors are used in our analyses. The python source package gtBurst is used to extract the light curves and source spectra. Xspec (Arnaud 1996; Atwood et al. 2009) is used to https://github.com/giacomov/gtburst perform spectral analysis , where the “Poisson-Gauss” fit statistic (i.e., pgstat) is adopted. The theoretical electronspectra from numerical calculations or simulations are almost a bump or power-law shape in its low-energy regime (seethe left panel of Figure 1). Then, Equation (1) is restricted to be a bump or power-law shape in our fittings. That isto say, the point P ( P ) should be above or on the line of P P ( P P ). GRB 180720B Analysis
GRB 180720B is a long burst with a redshift z = 0 .
654 and detected by
Fermi and
Swift satellites (Roberts & Meegan 2018, Bissaldi & Racusin 2018, Siegel et al. 2018, Vreeswijk et al. 2018). The obtainedNaI 6 light curve of the prompt emission is shown in the bottom inset of each panel in Figure 3, where the brightest NaI(i.e., NaI 6 and NaI 8) and BGO (i.e., BGO 0) detectors are used in our analyses. As an example, we first select a timeperiod of [7 . , .
19] s after the burst triggered for our analysis, which is marked with blue color in the bottom insetof the left panel in Figure 3. This time period is also used in the spectral analysis of Ravasio et al. (2019), of whichthe results can be used to compare with ours. The spectral fitting result is shown with black line in the upper insetof the left panel. The corresponding electron spectrum is shown with blue solid line in this panel and also reportedin Table 1.
Inspired by the fit result in Section 2, such kind of electron spectrum can be decomposed intothree segments: the lowest-energy segment (marked with pink shadow), the low-energy segment (marked with yellowshadow), and the high-energy segment (marked with cyan shadow).
It should be note that only the low-energysegment and the high-energy segment are robust in our analysis. The reason is presented at the endof this section.
One can find that the low-energy segment at γ e ∼ γ m can be approximated as n e ∝ γ − , which isthe low-energy electron spectrum in the standard fast-cooling pattern and is shown with black dashed line in Figure 3.This result is consistent with what reported in Ravasio et al. (2019). Therefore, our method is applicable to estimatethe electron spectrum for the prompt emission in the synchrotron radiation scenario.For the pulse in [7.14, 9.00] s, we also perform detail spectral analysis on the remaining time periods, e.g., [8.19,8.70] s and [8.70, 9.00] s, which are marked with red and green colors in the inset of middle panel of Figure 3,respectively. The obtained electron spectra for these three time segments are shown in the middle panel of Figure 3.The robust low-energy and high-energy segments in the electron spectra are also marked with yellow and cyan shadow,respectively. From this panel, one can find that the morphology of the electron spectra varies with time in a pulse,especially the morphology of the low-energy segment. In terms of this pulse, the electron spectra in its low-energyregime can be very different from the standard fast-cooling pattern and even a broken power-law function. Besides, wealso perform similar spectral analysis for four pulses in this burst, which are in [7.8, 11.2] s (marked with red color),[15.6, 17.0] s (marked with green color), [29.7, 31.5] s (marked with blue color), and [49.0, 52.4] s (marked with graycolor), respectively. Please see the details in the inset of the right panel of Figure 3. The obtained electron spectraare shown in the right panel of this figure with the same color as that marking on the studied time period. In termsof these pulses, the low-energy electron spectra can be also very different from the standard fast-cooling pattern andeven a broken power-law function, e.g., the pulse marked with green color. GRB 160509A Analysis
It is clear that GRB 180720B consists of multiple emission episodes. In this paragraph,we would like to perform the spectral analysis for a burst with single contiguous and pulse-like structure, takingGRB 160509A as an example. GRB 160509A is a long burst with redshift z = 1 .
17 and detected by
Fermi and
Swift satellites. The obtained NaI 0 light curve of the prompt emission is shown in the inset of Figure 4, where the brightestNaI detector (NaI 0 and NaI 3) and BGO (BGO 0) detectors are used for our analyses. Four different time periods areselected, [10-13.35]s, [13.35-14.65]s, [14.65-20]s, and [20-25]s, which are marked with green, red, blue, and gray colors,respectively. The obtained electron spectrum from spectral fitting for each time period is shown with the same colorin this figure and also reported in Table 1. Same as Figure 3, the robust low-energy and high-energy segments in theelectron spectra are also marked with yellow and cyan shadow, respectively. One can find that the low-energy electronspectra are very different from the standard fast-cooling pattern. The low-energy electron spectra in the time periodsof [10-13.35]s, [13.35-14.65]s, and [14.65-20]s are presented as a narrow bump rather than a power-law function. Theelectron spectrum in the time period of [20-25]s is rather soft compared with other three electron spectra. However,its low-energy segment is presented as a power-law function with index ∼ − . − The initial values of y , y , y , and p are set as follows. Firstly, the prompt emission is fitted with Band function to obtain the optimumvalue of α , β , and E . Then, B can be set by solving ν b ≡ . × q e B γ , Γ(1 + z ) / (2 πm e c ) = E b ≡ E ( α − β ), where E b is the breakphoton energy of Band function. In addition, the electron spectrum is initially set as a broken power law with p low = ( α + 1) × − p = ( β + 1) × − y m is set by equaling f ν ( ν b ) /ν b to the photon flux of the Band function at E b . In our fitting, y , y , y , γ m and p arethe free parameters. Based on the above settings, we perform a tentative spectral fit to roughly estimate parameters in a relatively wideparameter areas. With the obtained optimum fitting results from the tentative fitting, we further perform a fine spectral fitting based ona narrow parameter areas. At the end of this section, we present the reason why only the low-energy and high-energy segment in our obtainedelectron spectra are robust. This is owing to that the synchrotron emission of the electrons at the lowest energysegment makes a negligible contribution to the total radiation spectrum. The synchrotron radiation spectrum of anindividual electron is f ν ∝ ν / for ν << ν c . Thus the electron spectrum with power-law index being much larger than − / To differentiatethe lowest-energy segment from the robust low-energy segment, here we propose another simpler butmore general method. Taking the spectral analysis in [7.19, 8.17] s of GRB 180720B as an example,we fix log y at two different values around its first best fit value (for example, log y = 5 and − in here)and perform twice independent fit again. The electron spectra obtained from twice fit are shown astwo blue dash lines around the electron spectrum of the first fit result. The overlap region of thesethree spectra would be recognized as the robust low-energy segment. Conversely, the divergence regionwould be recognized as the lowest-energy segment. CONCLUSIONS AND DISCUSSIONSMore and more evidences indicate that synchrotron radiation is a promising mechanism for the prompt emission ofGRBs. However, the electron spectrum for the prompt emission is diverse in numerical calculations or simulations. Inthis paper, we propose a method to estimate the electron spectrum using an empirical function, which is a four-orderB´ezier curve (low-energy regime) jointed with a linear function (high-energy regime) in log-log coordinate. In thesynchrotron radiation scenario with electron spectrum described by our empirical function, the following two worksare studied in this paper. (1) The electron spectrum corresponding to Band radiation spectrum is investigated. Wefind that the exponential transition of Band radiation spectrum is more abrupt compared with that of the synchrotronradiation spectrum based on a broken power-law electron spectrum. Moreover, such exponential transition required afine-tuned electron spectrum, which is hardly produced in real situation. Then, we suggest that it may be inappropriateto use Band function to estimate the electron spectrum for the prompt emission of GRBs. (2) We perform thespectral analysis on the observations of the prompt emission to estimate the electron spectrum. GRB 180720B andGRB 160509A are studied as examples. By performing spectral analysis for a series of time periods in these two bursts,we find that the morphology of the electron spectrum in its low-energy regime evolves with time in a burst and evenin a pulse. In addition, it can be curved in some time periods, which is very different from the standard fast-coolingpattern (i.e., n ∝ γ − ) and even a power-law function. Our proposed method is used to estimate the electron spectrum for the prompt emission, withoutspecifying a certain physical model for the electron spectrum. In this paper, we focus on the synchrotronradiation scenario. Actually, one could imagine convolving this electron spectrum with other emissionkernels may also get equally well-fitting solutions (pointed out by the referee). It would be veryinteresting to investigate the shape of the electron spectrum with other emission kernels.
ACKNOWLEDGMENTSWe thank the anonymous referee of this work for useful comments and suggestions that improved the pa-per. We also thank for Qi Wang and Zhi-Lin Chen for the useful discussions and suggestions. This work issupported by the National Natural Science Foundation of China (grant Nos. 11773007, 11533003, U1938106,11851304, U1731239), the Guangxi Science Foundation (grant Nos. 2018GXNSFFA281010, 2017AD22006, 2018GXNS-FGA281007, 2018GXNSFDA281033), and the Innovation Team and Outstanding Scholar Program in Guangxi Colleges.We acknowledge the use of public data from the Fermi Science Support Center (FSSC).
Software:
Xspec (Arnaud 1996; Atwood et al. 2009), gtBurst (https://github.com/giacomov/gtburst), SciPy(Jones et al. 2001–), emcee (Foreman-Mackey et al. 2013), threeML (Vianello et al. 2015)
Table 1.
Optimum value of parameters and the corresponding pgstat / d . o . f . in each time period.Burst Time Period (s) log y log y log y log y a log γ m p B a pgstat/d.o.f. [7 . − .
19] 0 36 .
94 40 .
86 37 .
28 4 . − .
94 1473 .
89 343 . / . − .
70] 0 36 .
94 40 .
86 37 .
28 4 . − .
94 2000 .
09 343 . / . − .
00] 0 36 .
94 40 .
86 37 .
28 4 . − .
94 1275 .
28 343 . / . − .
2] 0 38 .
07 37 .
23 36 .
36 4 . − .
18 1863 .
36 428 . / . − .
0] 0 27 .
05 38 .
60 37 .
04 4 . − .
75 1347 .
31 422 . / . − .
5] 22 .
77 38 .
39 37 .
24 36 .
10 3 . − .
21 2603 .
91 409 . / . − .
4] 20 .
56 38 .
56 37 .
62 36 .
42 4 . − .
99 1248 .
34 412 . / . − . −
30 5 .
95 41 .
83 37 .
34 3 . − .
64 3099 .
20 534 . / . − . −
20 10 .
21 40 .
23 36 .
65 3 . − .
08 3001 .
39 342 . / . − . −
10 14 .
69 38 .
77 36 .
33 3 . − .
70 1521 .
75 561 . / . − .
0] 23 .
24 39 .
29 37 .
69 36 .
10 3 . − .
09 2521 .
53 400 . / a The value of the quantities are fixed in the fitting. n g e Uhm & Zhang (2014) with t=1.0s and b=1.0 Uhm & Zhang (2014) with t=3.0s and b=1.5
Standard "fast-cooling" electron spectrum Liu et al. (2020)
Guo et al. (2014)
Sironi & Spitkovsky (2009)
Lloyd & Petrosian (2000)
Figure 1.
Testing of our empirical function (left panel) and the spectral-fitting-method (middle and right panel).
Left-panel ,the electron spectra collected from different works and the corresponding best fitting results with our empirical function areshown with dashed and solid lines, respectively. Here, the purple, dark green, green, red, orange, and blue dashed lines arethe electron spectra obtained from the figure 1 of Guo et al. (2014) with ω pe t =700, the figure 10 of Sironi & Spitkovsky (2009)with θ = 30 ◦ , the second panel of the figure 1 in Uhm & Zhang (2014) with t obs = 1 . t obs = 3 . q = 1 . p = 3 .
0, and the figure 1of Liu et al. (2020) with t obs = 1 . Middle-panel , the electron spectra based on the last iterationsfrom MCMC sampling are plotted with red lines, where the blue and black dashed line represent the givenelectron spectrum and the best fitting result for the electron spectrum from MCMC sampling. In addition,the upper inset shows zoomed-in view for the electron spectrum at γ e ∼ − and the bottom inset showsthe best fitting result on the synthetic data. Right-panel , the posterior probability density functions by applying ourspectral-fitting-method on the synthetic data.
Radiation spectrum from broken power-law electron spectrum Radiation spectrum from optimum fit result Band radiation spectrum The radiation spectrum n F n ( e r g / c m ^ / s ) E (keV) Electron Spectrum n g e Figure 2.
Band radiation spectrum (black line) and the related electron spectra (inset panel). Here a Band function with α = − β = − .
3, and E p = 400keV is discussed. The inset plots the broken power-law electron spectrum and the electronspectrum obtained based on Equations (1) and (3). The corresponding synchrotron radiation spectra are shown with green andred dashed lines, respectively. Electron Spectrum n (cid:181) g e - n g e Time since trigger (s)
Counts/s -4-2024 E (keV) -4 -2 Model NaI8 NaI6 BGO1 pho t on s c m - s - k e V - Time-resolved spectrumTime bin: [7.17-8.19]
Radiation Spectrum
Residuals n g e [7.17, 8.19]s [8.19, 8.70]s [8.70, 9.00]s n (cid:181) g e - Time since trigger (s) C oun t s / s n g e [15.6, 17.0]s [29.7, 31.5]s [7.8, 11.2]s [49.0, 52.4]s n (cid:181) g e - C oun t s / s Time since trigger (s)
Figure 3.
Electron spectra from our spectral fittings on GRB 180720B, where the bottom inset in each panel shows the timeperiods (marked with different colors) for spectral fittings and the corresponding electron spectrum is shown with solid linesand with the same color as that marking on the studied time period. The dashed line below and above each solid lines are usedto constrain the low-energy and high-energy segments in our obtained electron spectrum. In addition, the standard fast-coolingelectron spectrum n ∝ γ − is shown with black dashed line in each panel. For convenient, the electron spectra of [8.19, 8.70] s,[8.70, 9.00] s, [7.8, 11.2] s, [15.6, 17.0] s, [29.7, 31.5] s, and [49.0, 52.4] s are shifted by timing 20, 30, 15, 0.1, 200, and 2000factors, respectively. [10-13.35]s [13.35-14.65]s [14.65-20]s [20-25]s n g e n (cid:181) g e - Time since trigger (s) C oun t s / s Figure 4.