# An Explanation of GRB Fermi-LAT Flares and High-Energy Photons in Stratified
Afterglows

Nissim Fraija ,1 Boris Betancourt Kamenetskaia ,2,3, Antonio Galván-Gámez ,1,
Peter Veres ,4,5, Rosa L. Becerra ,6, Simone Dichiara 7, Maria G. Dainotti
,8,9,10, Francisco Lizcano1, Edilberto Aguilar-Ruiz1
1Instituto de Astronomía, Universidad Nacional Autónoma de México,
Circuito Exterior, C.U., A. Postal 70-264, 04510 México City, México
2Technical University of Munich, TUM School of Natural Sciences, Physics
Department, James-Franck-Str 1, 85748 Garching, Germany
3Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring
6, 80805 Munich, Germany
4Department of Space Science, University of Alabama in Huntsville, Huntsville,
AL 35899, USA
5Center for Space Plasma and Aeronomic Research, University of Alabama in
Huntsville, Huntsville, AL 35899, USA
6Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Apartado Postal 70-264, 04510 México, CDMX, México
7Department of Astronomy and Astrophysics, The Pennsylvania State University,
525 Davey Lab, University Park, PA 16802, USA
8National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan
9Space Science Institute, Boulder, CO, USA
10The Graduate University for Advanced Studies, SOKENDAI, Shonankokusaimura,
Hayama, Miura District, Kanagawa 240-0193, Japan
E-mail<EMAIL_ADDRESS>0000-0002-0173-6453 0000-0002-2516-5739
0000-0001-5193-3693 0000-0002-2149-9846$ 0000-0002-0216-3415
0000-0001-6849-1270 0000-0003-4442-8546

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The second Fermi/LAT gamma-ray burst (GRB) catalog (2FLGC) spanning the first
decade of operations by the LAT collaboration was recently released. The
closure relations of the synchrotron forward shock (FS) model are not able to
reproduce a sizeable portion of the afterglow-phase light curves in this
collection, indicating that there may be a large contribution from some other
mechanism. Recently, synchrotron self-Compton (SSC) light curves from the
reverse shock (RS) regions were derived in the thick- and thin-shell regime
for a constant-density medium, and it was demonstrated that analytical light
curves could explain the GeV flare observed in several bursts from 2FLGC,
including GRB 160509A. Here, we generalise the SSC RS scenario from the
constant density to a stratified medium, and show that this contribution helps
to describe the early light curves exhibited in some Fermi/LAT-detected
bursts. As a particular case, we model a sample of eight bursts that exhibited
a short-lasting emission with the synchrotron and SSC model from FS and RS
regions, evolving in a stellar-wind environment, constraining the
microphysical parameters, the circumburst density, the bulk Lorentz factor,
and the fraction of shock-accelerated electrons. We demonstrate that the
highest-energy photons can only be described by the SSC from the forward-shock
region.

###### keywords:

gamma-rays bursts: individual: [GRB 080916C, 090323, 090902B, 090926A,
110731A, 130427A, 160625B, 180720B] – radiation mechanism: non-thermal –
acceleration of particles

††pubyear: 2023††pagerange: An Explanation of GRB Fermi-LAT Flares and High-
Energy Photons in Stratified Afterglows–An Explanation of GRB Fermi-LAT Flares
and High-Energy Photons in Stratified Afterglows

##  1 Introduction

Gamma-ray bursts (GRBs) are among the most powerful short-duration explosions
in the universe. These sources are often discovered as transient gamma-ray
flashes that last from milliseconds to hours and emit $\sim 10^{51}-10^{54}$
erg (for a review, see Kumar & Zhang, 2015). Most GRBs are usually detected in
the keV-MeV energy range, and their classification is determined by the
duration of the first event called "prompt emission" ($T_{90}$).111$T_{90}$
represents the time during which a GRB emits between $5\%$ and $95\%$ of the
total observed counts from its prompt emission. Typically, GRBs are modelled
by the empirical Band function (Band et al., 1993). After the first event, a
second event, known as afterglow, is observed from radio wavelengths up to TeV
gamma-rays and is explained using the fireball scenario (e.g., see Cavallo &
Rees, 1978). When a relativistic jet comes into contact with the circumburst
medium, the basic model predicts the formation of two shock waves: a forward
shock (FS; Kobayashi et al., 1999; Granot & Sari, 2002; Sari et al., 1998;
Becerra et al., 2019a) and a reverse shock (RS; Kobayashi, 2000; Kobayashi &
Zhang, 2003; Fraija et al., 2016; Becerra et al., 2019b). Both shocks are
predicted to produce synchrotron and synchrotron-self Compton (SSC) photons
due to the acceleration and cool down of the non-thermal electron
distribution. Synchrotron photons are scattered to energies larger than tens
of GeV by the SSC process (Zhang & Mészáros, 2001; Fraija et al., 2019b), even
if the highest photon energy produced by the synchrotron process is $\sim
5-10~{}{\rm~{}GeV}$ (Piran & Nakar, 2010; Abdo et al., 2009b; Barniol Duran &
Kumar, 2011).

According to multi-wavelength data modelling during the afterglow phase,
multiple GRBs are compatible with a circumburst medium with a density profile
$\propto r^{-k}$ with a PL index in the range $0\leq k<3$. For example, Yi et
al. (2013) studied the evolution of the FS and RS emission in a circumstellar
medium represented by $\propto r^{-k}$. They tested their afterglow model on
19 bursts and found that the density profile index ranged from $0.4\leq{\rm
k}\leq 1.4$. Liang et al. (2013), who examined a larger sample of 146 bursts,
reported a range consistent with Yi et al.’s result. Since black hole
accretion of a stellar envelope produces the plateau phase in X-ray light
curves, Kumar et al. (2008) constructed a density profile index with $k>2$.
Using a synchrotron afterglow model in the adiabatic domain without energy
injection, Kouveliotou et al. (2013) evaluated the multiwavelength data of GRB
130427A, one of the most intense bursts with redshift $z<1$. The authors
concluded that these observations were consistent with a density profile index
that ranged between $\rm k=0$ (constant-density medium) and $\rm k=2$ (wind-
like medium). Izzo et al. (2020) modelled the X-ray observations of the nearby
SN 2020bvc. According to the authors, these afterglow observations were
consistent with synchrotron emission evolving in a circumburst medium with a
density profile index of ${\rm k}=1.5$. A complete analysis of Fermi and Swift
observations by Ajello et al. (2018); Cenko et al. (2011); Ackermann et al.
(2013b) found that there is a preference for LAT-detected bursts to evolve in
a wind-like circumstellar medium. This inference was due to the observed
behaviour of the cooling frequency in the synchrotron spectrum in several
GRBs. It was noted that it evolved in time to larger energies, a behaviour
that contrasted with the predicted evolution to lower energies, as expected
from a constant-density medium. As such, the afterglow was best fitted with a
low-density wind-like model (Ajello et al., 2018). The bursts in question were
GRB 090323, GRB 090328, GRB 090902B, GRB 090926A (Cenko et al., 2011), and GRB
110731A (Ackermann et al., 2013b).

Ajello et al. (2019) presented the Second Fermi/LAT GRB Catalogue (2FLGC),
covering the first decade of operations (from 2008 to 2018, August 4). The
data set contains 169 bursts with high-energy emissions greater than $100$
MeV, including 29 bursts higher than $10$ GeV. Additionally, this data set
includes a sample of 86 bursts with temporarily extended emission described
with a power law (PL) function and a broken power law (BPL) function. A subset
of these events (21 GRBs) showed a break in the LAT light curve between 63 and
1250 s. Although long-lasting emission is commonly interpreted as synchrotron
radiation from external FSs (Kumar & Barniol Duran, 2009, 2010), not all LAT
light curves fulfil the relation between the temporal and spectral indices’
evolutions or the closure relations predicted if the FS dominates the emission
(Sari et al., 1998). Furthermore, data suggests that the entire GeV light
curve for multiple bursts cannot be satisfactorily described by FS synchrotron
radiation alone (Maxham et al., 2011).

Fraija et al. (2020) derived the SSC light curves from the RS region for a
constant-density medium. They demonstrated that analytical SSC light curves
from the RS region could explain the GeV flares observed in several bursts
from 2FLGC. The late-time steepening of LAT afterglow observations was caused
by the synchrotron cooling break passing across the LAT energy band when the
FS evolves in a uniform environment. Here, we generalise the SSC light curves
from the RS for a uniform-density medium to a stratified environment with a
profile density $\propto r^{\rm-k}$ with ${\rm k}$ lying in the range of
$0\leq{\rm k}<3$. The paper is structured as follows: in Section §2, we derive
the SSC light curves evolving in a stratified medium. Section §3 presents the
analysis of Fermi/LAT data, the derivation of the Fermi/LAT light curves and
the modelling of the multiwavelength observations of a sample of eight GRBs
with our current model for ${\rm k=2}$. In addition, we present the results
and a discussion. Section §4 provides a summary of our work and some
concluding remarks. We use unprimed (primed) quantities in the observer
(comoving) frame. The subindexes “f" and “r" refer to the quantities derived
in the FS and RS regions, respectively.

##  2 SSC light curves from the reverse shock evolving in a stratified
environment

When the relativistic jet encounters the stratified medium, a RS is formed,
and it propagates back into the outflow (Mészáros & Rees, 1997; Sari & Piran,
1999; Kobayashi & Sari, 2000). We consider a density profile of the stratified
medium $n(r)=A_{\rm k_{s}}r^{-k}$ with $0\leq k<3$, and use the value of
$A_{\rm k_{s}}=\frac{\dot{M}_{\rm W}}{4\pi m_{p}\,v_{\rm W}}A_{\rm
k}=3.0\times 10^{35}\,{\rm cm^{-1}}A_{\rm k}$ for ${\rm k=2}$ to report the
quantities of the theoretical model, unless otherwise indicated. The terms
$v_{\rm W}$ and $\dot{M}_{\rm W}$ correspond to the wind velocity and the
mass-loss rate, respectively, $m_{p}$ is the proton mass, and $A_{\rm k}$ is
the density parameter. The evolution of the bulk Lorentz factor is estimated
as $\Gamma=\left(\frac{\ell}{\Delta}\right)^{\frac{3-k}{2(4-k)}}$, where
$\ell=\left(\frac{(3-k)E}{4\pi m_{p}c^{2}\,A_{\rm
k_{s}}}\right)^{\frac{1}{3-k}}$ and $\Delta=2c(1+z)^{-1}\,t_{\rm x}$ are the
Sedov length and the observed width of the shell, respectively.222The term
$t_{\rm x}$ is the shock crossing time, $z$ is the redshift, and $c$ is the
speed of light. The isotropic gamma-ray energy, $E_{\rm\gamma,iso}$,
determines the isotropic equivalent kinetic energy, $E$, through the kinetic
efficiency $\eta=E_{\rm\gamma,iso}/(E+E_{\rm\gamma,iso})$. The dynamics of the
RS is categorized into two distinct scenarios: the thick-shell regime for
$\Gamma_{c}\lesssim\Gamma$ or $t_{\rm x}\lesssim T_{90}$, and in the thin-
shell regime for $\Gamma<\Gamma_{c}$ or $T_{90}<t_{\rm x}$, where $\Gamma_{c}$
is the critical Lorentz factor, which is calculated considering the evolution
of the bulk Lorentz factor evaluated at the duration of the burst
$\left(\Gamma_{c}\equiv\Gamma(t_{\rm x}=T_{90})\right)$. In this case, the
critical Lorentz factor becomes

$\displaystyle\Gamma_{c}\equiv\left(\frac{3-k}{(4c)^{5-k}\pi
m_{p}}\right)^{\frac{1}{2(4-k)}}(1+z)^{\frac{3-k}{2(4-k)}}E^{\frac{1}{2(4-k)}}A_{\rm
k_{s}}^{-\frac{1}{2(4-k)}}T_{90}^{-\frac{3-k}{2(4-k)}}\,.$ (1)

###  2.1 Dynamics of the thick-shell scenario

The RS in the thick-shell scenario is ultra-relativistic and can decelerate
the shell dramatically. Before the RS crosses the shell, the scale of the
hydrodynamic quantities as a function of the observed time is
$\gamma_{3}\propto t^{-\frac{2-k}{2(4-k)}}$, $n_{3}\propto
t^{-\frac{k+6}{2(4-k)}}$, $p_{3}\propto t^{-\frac{k+2}{4-k}}$ and $N_{\rm
e}\propto t$. Electrons are accelerated and described by a PL distribution
($\propto\gamma_{e}^{-p}\,d\gamma_{e}$ with $p>2$), and the magnetic field is
magnified in the RS zone. The terms $\gamma_{e}$ and $p$ are the Lorentz
factor of the electrons and the spectral index, respectively. We consider that
a fraction $\varepsilon_{\rm e_{r}}$ and $\varepsilon_{\rm B_{r}}$ of the
total energy goes to the electrons and the magnetic field, respectively.
Therefore, the magnetic field, the minimum and cooling electron Lorentz
factors evolve as $B^{\prime}_{\rm r}\propto t^{-\frac{k+2}{2(4-k)}}$,
$\gamma_{\rm m,r}\propto t^{\frac{2-k}{2(4-k)}}$ and $\gamma_{\rm c,r}\propto
t^{\frac{3k-2}{2(4-k)}}$, respectively. Moreover, the synchrotron spectral
breaks and the maximum synchrotron flux in terms of observed time are
$\nu^{\rm syn}_{\rm m,r}\propto t^{-\frac{k}{4-k}}$ and $\nu^{\rm syn}_{\rm
c,r}\propto t^{\frac{3k-4}{4-k}}$ and $F^{\rm syn}_{\rm max,r}\propto
t^{\frac{2-k}{4-k}}$, respectively.

Once the RS crosses the shell, the scale of the hydrodynamic variables becomes
$\gamma_{3}\propto t^{\frac{2k-7}{4(4-k)}}$, $n_{3}\propto
t^{\frac{2k-13}{4(4-k)}}$, $p_{3}\propto t^{\frac{2k-13}{3(4-k)}}$, and
$N_{\rm e}\propto t^{0}$. In this case, the magnetic field, the minimum and
cooling electron Lorentz factors, the synchrotron spectral breaks and the
maximum synchrotron flux in terms of the observed time are $B^{\prime}\propto
t^{\frac{2k-13}{6(4-k)}}$, $\gamma_{\rm m,r}\propto t^{\frac{2k-13}{12(4-k)}}$
and $\gamma_{\rm c,r}\propto t^{\frac{25-2k}{12(4-k)}}$, $\nu^{\rm syn}_{\rm
m,r}\propto t^{-\frac{73-14k}{12(4-k)}}$ and $\nu^{\rm syn}_{\rm c,r}\propto
t^{\frac{1+2k}{4(4-k)}}$ and $F^{\rm syn}_{\rm max,r}\propto
t^{-\frac{47-10k}{12(4-k)}}$, respectively. We derive the expected light
curves before and after the shock crossing time.

####  2.1.1 SSC light curves for $t<t_{\rm x}$

At the deceleration radius $r=4c(1+z)^{-1}\gamma_{3}^{2}\,t$, the same
electron population can up-scatter synchrotron photons up to higher energies
as $h\nu^{\rm ssc}_{\rm i,r}\sim\gamma^{2}_{\rm i,r}h\nu^{\rm syn}_{\rm i,r}$
with ${\rm i=m}$ and ${\rm c}$, reaching a maximum flux of $F^{\rm ssc}_{\rm
max,r}\sim\,\frac{4\sigma_{T}nr}{3g(p)}\,F^{\rm syn}_{\rm max,r}$ with
$g(p)=\frac{p-1}{p-2}$.333The term $\sigma_{T}$ is the Thompson cross-section
and $h$ is the Planck constant. Therefore, the SSC spectral breaks and the
maximum SSC flux are

$\displaystyle h\nu^{\rm ssc}_{\rm m,r}$ $\displaystyle\simeq$ $\displaystyle
0.6\,{\rm~{}GeV}\,\left(\frac{1+z}{2}\right)^{\frac{3(k-2)}{4-k}}g^{4}(2.2)\zeta_{\rm
e,-1}^{-4}\varepsilon^{4}_{\rm e_{r},-1}\varepsilon^{\frac{1}{2}}_{\rm
B_{r},-2}\,\Gamma^{4}_{2.5}A^{\frac{3}{4-k}}_{\rm
k,-1}\Delta^{\frac{k+2}{2(4-k)}}_{11.8}\,E^{-\frac{k+2}{2(4-k)}}_{53}t^{\frac{2(1-k)}{4-k}}_{1}\,,$
(2) $\displaystyle h\nu^{\rm ssc}_{\rm c,r}$ $\displaystyle\simeq$
$\displaystyle 0.7\,{\rm
eV}\,\left(\frac{1+z}{2}\right)^{\frac{2-5k}{4-k}}\left(\frac{1+Y_{\rm
r}}{2}\right)^{-4}\varepsilon^{-\frac{7}{2}}_{\rm
B_{r},-2}A^{-\frac{9}{4-k}}_{\rm
k,-1}\Delta^{\frac{10-7k}{2(4-k)}}_{11.8}\,E^{\frac{7k-10}{2(4-k)}}_{53}\,t^{\frac{6(k-1)}{4-k}}_{1},$
(3) $\displaystyle F^{\rm ssc}_{\rm max,r}$ $\displaystyle\simeq$
$\displaystyle 6.8\,{\rm
mJy}\,\left(\frac{1+z}{2}\right)^{\frac{2k}{4-k}}\,g^{-1}(2.2)\,\zeta_{\rm
e,-1}\varepsilon^{\frac{1}{2}}_{\rm B_{r},-2}\,A_{\rm
k,-1}^{\frac{4}{4-k}}\,d^{-2}_{\rm
z,28.3}\,\Gamma^{-1}_{2.5}\,\Delta^{\frac{12-5k}{2(k-4)}}_{11.8}\,E^{\frac{12-5k}{2(4-k)}}_{53}t^{\frac{4-3k}{4-k}}_{1}\,,$
(4)

where $\zeta_{e}$ is the fraction of electrons accelerated during the shock
(Fan & Piran, 2006), $Y_{\rm r}$ is the Compton parameter and $d_{\rm
z}=(1+z)\frac{c}{H_{0}}\int^{z}_{0}\,\frac{d\tilde{z}}{\sqrt{\Omega_{\rm
M}(1+\tilde{z})^{3}+\Omega_{\Lambda}}}$ (Weinberg, 1972) corresponds to the
luminosity distance, with the constant values of
$\Omega_{m}=1-\Omega_{\Lambda}=0.315$ and $H_{0}=67.4~{}{\rm km}\,{\rm
s}^{-1}~{}{\rm Mpc}^{-1}$ (Planck Collaboration et al., 2020). We report the
proportionality constants of the spectral breaks and maximum fluxes
considering the values of $p=2.2$ and $t_{\rm x}=15\,{\rm s}$, unless
otherwise stated. At a particular observed energy ($h\nu$), the SSC light
curves for the fast- and the slow-cooling regime evolve as

$\displaystyle F^{\rm ssc}_{\rm\nu,r}(t<t_{\rm
x})\propto\begin{cases}t^{-\frac{5k-6}{4-k}}\,\nu^{\frac{1}{3}},\hskip
42.67912pt\nu<\nu^{\rm ssc}_{\rm c,r},\cr
t^{\frac{1}{4-k}}\,\nu^{-\frac{1}{2}},\hskip 45.52458pt\nu^{\rm ssc}_{\rm
c,r}<\nu<\nu^{\rm ssc}_{\rm m,r},\cr
t^{\frac{(1-k)p+k}{4-k}}\,\nu^{-\frac{p}{2}},\,\,\,\,\,\hskip
19.91684pt\nu^{\rm ssc}_{\rm m,r}<\nu<\nu^{\rm ssc}_{\rm
KN,m,r}\,,\cr\end{cases}$ (5)

and

$\displaystyle F^{\rm ssc}_{\rm\nu,r}(t<t_{\rm
x})\propto\begin{cases}t^{-\frac{7k-10}{3(4-k)}}\,\nu^{\frac{1}{3}},\hskip
42.67912pt\nu<\nu^{\rm ssc}_{\rm m,r},\cr
t^{\frac{p(1-k)+3-2k}{4-k}}\,\nu^{-\frac{p-1}{2}},\hskip 14.22636pt\nu^{\rm
ssc}_{\rm m,r}<\nu<\nu^{\rm ssc}_{\rm c,r},\cr
t^{\frac{(1-k)p+k}{4-k}}\,\nu^{-\frac{p}{2}},\,\,\,\,\,\hskip
25.6073pt\nu^{\rm ssc}_{\rm c,r}<\nu<\nu^{\rm ssc}_{\rm
KN,c,r}\,,\cr\end{cases}$ (6)

respectively. Furthermore, the Klein-Nishina (KN) break must be considered
since the SSC flux is strongly attenuated if the photon energy lies above this
break. For $\nu^{\rm ssc}_{\rm c,r}<\nu^{\rm ssc}_{\rm m,r}$ and $\nu^{\rm
ssc}_{\rm m,r}<\nu^{\rm ssc}_{\rm c,r}$, the KN breaks ($h\nu^{\rm ssc}_{\rm
KN,m,r}\simeq\frac{\gamma_{3}\gamma_{m,r}}{(1+z)}\,m_{e}c^{2}$ and $h\nu^{\rm
ssc}_{\rm KN,c,r}\simeq\frac{\gamma_{3}\gamma_{c,r}}{(1+z)}\,m_{e}c^{2}$)
become

$\displaystyle h\nu^{\rm ssc}_{\rm KN,m,r}$ $\displaystyle\simeq$
$\displaystyle 3.7\times
10\,{\rm~{}GeV}\,\left(\frac{1+z}{2}\right)^{-1}g(2.2)\zeta_{e,-1}^{-1}\,\varepsilon_{\rm
e_{r},-1}\,\Gamma_{2.5}\,,$ (7) $\displaystyle h\nu^{\rm ssc}_{\rm KN,c,r}$
$\displaystyle\simeq$ $\displaystyle
0.2\,{\rm~{}GeV}\left(\frac{1+z}{2}\right)^{-\frac{k+2}{4-k}}\left(\frac{1+Y_{\rm
r}}{2}\right)^{-1}\varepsilon^{-1}_{\rm B_{r},-2}A^{-\frac{3}{4-k}}_{\rm
k,-1}\,\Delta^{\frac{1-k}{4-k}}_{11.8}\,E^{\frac{k-1}{4-k}}_{53}\,t^{\frac{2(k-1)}{4-k}}_{1}\,,$
(8)

with $m_{e}$ the electron mass. Following Nakar et al. (2009) with the
definition of a new spectral break ($h\nu^{\rm ssc}_{\rm
KN,0,r}\simeq\frac{\gamma_{3}\gamma_{0,r}}{(1+z)}\,m_{e}c^{2}$)444See eqs. 20
and 49 of Nakar et al. (2009) for the definition of $\gamma_{\rm 0,r}$ in the
fast ($\gamma_{\rm m,r}>\gamma_{\rm c,r}$) and slow ($\gamma_{\rm
c,r}>\gamma_{\rm m,r}$) cooling regimes, respectively. the additional PLs in
the SSC light curves due to KN effects are

$\displaystyle F^{\rm ssc}_{\rm\nu,r}(t<t_{\rm
x})\propto\begin{cases}t^{\frac{p-k(p-1)}{4-k}}\,\nu^{-(p-1)},\hskip
50.07669pt\nu^{\rm ssc}_{\rm KN,m,r}<\nu<\nu^{\rm ssc}_{\rm 0,r},\cr
t^{\frac{p-k(p-1)}{4-k}}\,\nu^{-(p-\frac{1}{2})},\hskip 48.08522pt\nu^{\rm
ssc}_{\rm 0,r}<\nu<\nu^{\rm ssc}_{\rm KN,c,r},\cr
t^{-\frac{5-3p-k(8-3p)}{3(4-k)}}\,\nu^{-(p+\frac{1}{3})},\,\,\,\,\,\hskip
25.6073pt\nu^{\rm ssc}_{\rm KN,c,r}<\nu\,,\cr\end{cases}$ (9)

and

$\displaystyle F^{\rm ssc}_{\rm\nu,r}(t<t_{\rm
x})\propto\begin{cases}t^{\frac{2-k}{4-k}}\,\nu^{-(p-1)},\hskip
42.67912pt\nu^{\rm ssc}_{\rm KN,c,r}<\nu<\nu^{\rm ssc}_{\rm 0,r},\cr
t^{\frac{1}{4-k}}\,\nu^{-\frac{p+1}{2}},\hskip 48.08522pt\nu^{\rm ssc}_{\rm
0,r}<\nu<\nu^{\rm ssc}_{\rm m,r},\cr
t^{\frac{1}{4-k}}\,\nu^{-(p+\frac{1}{3})},\,\,\,\,\,\hskip 34.14322pt\nu^{\rm
ssc}_{\rm m,r}<\nu\,,\cr\end{cases}$ (10)

for the fast (weak) - and slow-cooling regimes, respectively.

####  2.1.2 SSC light curves for $t>t_{\rm x}$

The SSC spectral breaks and the maximum SSC flux are given by

$\displaystyle h\nu^{\rm ssc}_{\rm m,r}$ $\displaystyle\simeq$ $\displaystyle
2.7\times 10^{-3}\,{\rm
eV}\,\left(\frac{1+z}{2}\right)^{\frac{17-2k}{4(4-k)}}\,g^{4}(2.2)\,\zeta_{\rm
e,-1}^{-4}\,\,\varepsilon^{4}_{\rm e_{r},-1}\varepsilon^{\frac{1}{2}}_{\rm
B_{r},-2}A^{-\frac{3}{k-4}}_{\rm
k,-1}\,\Gamma^{4}_{2.5}\Delta_{11.8}^{\frac{3(4k-15)}{4(k-4)}}\,E^{\frac{k+2}{2(k-4)}}_{53}t^{\frac{3(11-2k)}{4(k-4)}}_{2},$
(11) $\displaystyle h\nu^{\rm ssc}_{\rm cut,r}$ $\displaystyle\simeq$
$\displaystyle 1.3\,{\rm
eV}\,\left(\frac{1+z}{2}\right)^{\frac{17-2k}{4(4-k)}}\,\varepsilon^{-\frac{7}{2}}_{\rm
B_{r},-2}A^{\frac{9}{k-4}}_{\rm k,-1}\,\left(\frac{1+Y_{\rm
r}}{2}\right)^{-4}\,\Delta^{\frac{29+4k}{4(4-k)}}_{11.8}\,E^{\frac{10-7k}{2(k-4)}}_{53}\,t^{\frac{3(11-2k)}{4(k-4)}}_{2},$
(12) $\displaystyle F^{\rm ssc}_{\rm max,r}$ $\displaystyle\simeq$
$\displaystyle 0.3\,{\rm
mJy}\,\left(\frac{1+z}{2}\right)^{\frac{89-16k}{12(4-k)}}\,g^{-1}(2.2)\,\zeta_{e}\,\varepsilon^{\frac{1}{2}}_{\rm
B_{r},-2}\,\Gamma^{-1}_{2.5}\,A^{\frac{4}{4-k}}_{\rm
k,-1}\,\Delta_{11.8}^{\frac{17-10k}{12(4-k)}}\,d^{-2}_{\rm
z,28.3}\,E^{\frac{12-5k}{2(4-k)}}_{53}\,t^{\frac{41-4k}{12(k-4)}}_{2},\,\,\,\,\,\,\,\,$
(13)

where the cutoff frequency $\nu^{\rm ssc}_{\rm cut,r}=\nu^{\rm ssc}_{\rm
c,r}(t_{\rm x})\,\left(\frac{t}{t_{\rm x}}\right)^{\frac{33-6k}{4(k-4)}}$ is
estimated by the requirement that no electrons are shocked anymore and the
fluid expands adiabatically. For $\nu^{\rm ssc}_{\rm c,r}<\nu^{\rm ssc}_{\rm
m,r}$ and $\nu^{\rm ssc}_{\rm m,r}<\nu^{\rm ssc}_{\rm c,r}$, the spectral
breaks in the KN regime are

$\displaystyle h\nu^{\rm ssc}_{\rm KN,m,r}$ $\displaystyle\simeq$
$\displaystyle
8.9\,{\rm~{}GeV}\,\left(\frac{1+z}{2}\right)^{-\frac{7-2k}{6(4-k)}}g(2.2)\,\zeta_{e,-1}^{-1}\,\varepsilon_{\rm
e_{r},-1}\,\Gamma_{2.5}\,\Delta^{\frac{17-4k}{6(4-k)}}_{11.8}\,t^{-\frac{17-4k}{6(4-k)}}_{2},$
(14) $\displaystyle h\nu^{\rm ssc}_{\rm KN,c,r}$ $\displaystyle\simeq$
$\displaystyle
0.8\,{\rm~{}GeV}\left(\frac{1+z}{2}\right)^{\frac{13-2k}{3(k-4)}}\left(\frac{1+Y_{\rm
r}}{2}\right)^{-1}\varepsilon^{-1}_{\rm B_{r},-2}A^{\frac{3}{k-4}}_{\rm
k,-1}\,E^{\frac{k-1}{4-k}}_{53}\,\Delta^{\frac{2(k-2)}{3(4-k)}}_{11.8}\,t^{\frac{k+1}{3(4-k)}}_{2}.\,\,$
(15)

The SSC light curves for the slow-cooling regime evolve as

$\displaystyle F^{\rm ssc}_{\rm\nu,r}(t>t_{\rm
x})\propto\begin{cases}t^{\frac{k+4}{6(k-4)}}_{1}\,\nu^{\frac{1}{3}},\hskip
63.44952pt\nu<\nu^{\rm ssc}_{\rm m,r},\cr
t^{\frac{17-10k-99p+18kp}{24(4-k)}}\,\nu^{-\frac{p-1}{2}},\hskip
11.38092pt\nu^{\rm ssc}_{\rm m,r}<\nu<\nu^{\rm ssc}_{\rm cut,r},\cr
0,\,\,\,\,\hskip 93.89418pt\nu^{\rm ssc}_{\rm cut,r}<\nu\,.\cr\end{cases}$

The SSC emission generated in the RS region could decay faster because of the
angular time delay effect (Panaitescu & Kumar, 2000; Kobayashi & Zhang, 2003).
During this phase, the evolution of the SSC flux due to high-latitude
afterglow emission is described by $F^{\rm ssc}_{\rm\nu,r}(t>t_{\rm x})\propto
t^{-(\beta+2)}$, where $\beta$ is the spectral index with
$\beta=1/2,\,(p-1)/2$ or $p/2$ for the fast and slow cooling regimes,
respectively.

###  2.2 Dynamics of the thin-shell scenario

In the thin-shell regime, the RS becomes mildly relativistic and hence cannot
decelerate the shell. In this case, the shock crossing time is longer than the
duration of the burst ($T_{90}<t_{\rm x}$) with $t_{\rm
x}=\left(\frac{(3-k)E\,(1+z)^{3-k}}{2^{4-k}c^{5-k}\pi\,m_{p}\,A_{k}}\right)^{\frac{1}{3-k}}\,\Gamma^{-\frac{8-2k}{3-k}}$.
Before the RS crosses the shell, the scaling of the hydrodynamic quantities as
a function of the observed time is $\gamma_{3}\propto t^{0}$, $n_{3}\propto
t^{-3}$, $p_{3}\propto t^{-k}$, and $N_{\rm e}\propto t^{\frac{3-k}{2}}$. In
this regime, the magnetic field, the minimum, and the cooling electron Lorentz
factors evolve as $B^{\prime}_{\rm r}\propto t^{-\frac{k}{2}}$, $\gamma_{\rm
m,r}\propto t^{-(k-3)}$ and $\gamma_{\rm c,r}\propto t^{k-1}$, respectively.
Moreover, the synchrotron spectral breaks and the maximum synchrotron flux in
terms of the observed time are $\nu^{\rm syn}_{\rm m,r}\propto
t^{\frac{12-5k}{2}}$ and $\nu^{\rm syn}_{\rm c,r}\propto t^{\frac{3k-4}{2}}$
and $F^{\rm syn}_{\rm max,r}\propto t^{\frac{3-2k}{2}}$, respectively.

Once the RS crosses the shell, for ${\rm k=2}$, the scaling of the
hydrodynamic variables becomes $\gamma_{3}\propto t^{-\frac{1}{3}}$,
$n_{3}\propto t^{-\frac{8}{7}}$, $p_{3}\propto t^{-\frac{32}{21}}$ and $N_{\rm
e}\propto t^{0}$. In this case, the magnetic field, the minimum and cooling
electron Lorentz factors, the synchrotron spectral breaks and the maximum
synchrotron flux as function of the observed time are $B^{\prime}\propto
t^{-\frac{16}{21}}$, $\gamma_{\rm m,r}\propto t^{-\frac{8}{21}}$ and
$\gamma_{\rm c,r}\propto t^{\frac{6}{7}}$, $\nu^{\rm syn}_{\rm m,r}\propto
t^{-\frac{13}{7}}$ and $\nu^{\rm syn}_{\rm c,r}\propto t^{\frac{13}{21}}$ and
$F^{\rm syn}_{\rm max,r}\propto t^{-\frac{23}{21}}$, respectively. We derive
the expected light curves before and after the shock crossing time.

####  2.2.1 SSC light curves for $t<t_{\rm x}$

For this time interval, the SSC spectral breaks and the maximum SSC flux are

$\displaystyle h\nu^{\rm ssc}_{\rm m,r}$ $\displaystyle\simeq$ $\displaystyle
0.2\,{\rm
eV}\,\left(\frac{1+z}{2}\right)^{\frac{9k-26}{2}}g^{4}(2.2)\zeta_{\rm
e,-1}^{-4}\varepsilon^{4}_{\rm e_{r},-1}\varepsilon^{\frac{1}{2}}_{\rm
B_{r},-2}\,A^{\frac{9}{2}}_{\rm
k,-1}\,\Gamma^{34-9k}_{1.5}\,E^{-4}_{52}\,t^{\frac{24-9k}{2}}_{2.5},$ (16)
$\displaystyle h\nu^{\rm ssc}_{\rm c,r}$ $\displaystyle\simeq$ $\displaystyle
2.7\times 10^{-6}\,{\rm
eV}\,\left(\frac{1+z}{2}\right)^{\frac{6-7k}{2}}\,\varepsilon^{-\frac{7}{2}}_{\rm
B_{r},-2}A^{-\frac{7}{2}}_{\rm k,-1}\left(\frac{1+Y_{\rm
r}}{2}\right)^{-4}\,\Gamma^{7k-10}_{1.5}\,t^{\frac{7k-8}{2}}_{2.5},$ (17)
$\displaystyle F^{\rm ssc}_{\rm max,r}$ $\displaystyle\simeq$ $\displaystyle
0.2\,{\rm
mJy}\,\left(\frac{1+z}{2}\right)^{\frac{4k-3}{2}}\,g^{-1}(2.2)\,\zeta_{\rm
e,-1}\,\varepsilon^{\frac{1}{2}}_{\rm
B_{r},-2}\,\Gamma^{7-4k}_{1.5}\,A^{2}_{\rm k,-1}\,d^{-2}_{\rm
z,28.3}\,E^{\frac{1}{2}}_{52}\,t^{\frac{5-4k}{2}}_{2.5}.$ (18)

In analogy to the description of the SSC light curves for the thick-shell
regime, the SSC light curves before the shock-crossing time for the fast- and
the slow-cooling regimes at a particular observed energy are

$\displaystyle F^{\rm ssc}_{\rm\nu,r}(t<t_{\rm
x})\propto\begin{cases}t^{\frac{23-19k}{6}}\,\nu^{\frac{1}{3}},\hskip
68.28644pt\nu<\nu^{\rm ssc}_{\rm c,r},\cr
t^{\frac{2-k}{4}}\,\nu^{-\frac{1}{2}},\hskip 68.28644pt\nu^{\rm ssc}_{\rm
c,r}<\nu<\nu^{\rm ssc}_{\rm m,r},\cr
t^{\frac{3p(8-3k)+2(4k-11)}{4}}\,\nu^{-\frac{p}{2}},\,\,\,\,\,\hskip
18.49411pt\nu^{\rm ssc}_{\rm m,r}<\nu<\nu^{\rm ssc}_{\rm
KN,m,r}\,,\cr\end{cases}$ (19)

and

$\displaystyle F^{\rm ssc}_{\rm\nu,r}(t<t_{\rm
x})\propto\begin{cases}t^{-\frac{k+3}{2}}\,\nu^{\frac{1}{3}},\hskip
68.28644pt\nu<\nu^{\rm ssc}_{\rm c,r},\cr
t^{\frac{3p(8-3k)+k-14}{4}}\,\nu^{-\frac{p-1}{2}},\hskip 25.6073pt\nu^{\rm
ssc}_{\rm m,r}<\nu<\nu^{\rm ssc}_{\rm c,r},\cr
t^{\frac{3p(8-3k)+2(4k-11)}{4}}\,\nu^{-\frac{p}{2}},\,\,\,\,\,\hskip
17.07182pt\nu^{\rm ssc}_{\rm c,r}<\nu<\nu^{\rm ssc}_{\rm
KN,c,r}\,,\cr\end{cases}$ (20)

respectively. For $\nu^{\rm ssc}_{\rm c,r}<\nu^{\rm ssc}_{\rm m,r}$ and
$\nu^{\rm ssc}_{\rm m,r}<\nu^{\rm ssc}_{\rm c,r}$, the spectral breaks in the
KN regime are

$\displaystyle h\nu^{\rm ssc}_{\rm KN,m,r}$ $\displaystyle\simeq$
$\displaystyle 6.1\times
10^{-2}\,{\rm~{}GeV}\,\left(\frac{1+z}{2}\right)^{k-4}g(2.2)\zeta_{e,-1}^{-1}\,\varepsilon_{\rm
e_{r},-1}\,A_{\rm k,-1}\,\Gamma^{9-2k}_{1.5}\,E_{52}^{-1}\,t^{3-k}_{2.5}\,,$
(21) $\displaystyle h\nu^{\rm ssc}_{\rm KN,c,r}$ $\displaystyle\simeq$
$\displaystyle 3.9\times
10^{-3}\,{\rm~{}GeV}\left(\frac{1+z}{2}\right)^{-k}\left(\frac{1+Y_{r}}{2}\right)^{-1}\varepsilon^{-1}_{\rm
B_{r},-2}\,A^{-1}_{\rm k,-1}\,\Gamma^{2(k-1)}_{1.5}\,t^{k-1}_{2.5}\,.$ (22)

In analogy to the description of the SSC light curves for the thick-shell
regime, the additional PLs in the SSC light curves due to KN effects are
(Nakar et al., 2009)

$\displaystyle F^{\rm ssc}_{\rm\nu,r}(t<t_{\rm
x})\propto\begin{cases}t^{-\frac{2(17-15p)-k(12-11p)}{4}}\,\nu^{-(p-1)},\hskip
42.67912pt\nu^{\rm ssc}_{\rm KN,m,r}<\nu<\nu^{\rm ssc}_{\rm 0,r},\cr
t^{-\frac{2(14-15p)-k(10-11p)}{4}}\,\nu^{-(p-\frac{1}{2})},\hskip
39.83368pt\nu^{\rm ssc}_{\rm 0,r}<\nu<\nu^{\rm ssc}_{\rm KN,c,r},\cr
t^{-\frac{2(47-45p)-k(40-33p)}{12}}\,\nu^{-(p+\frac{1}{3})},\,\,\,\,\,\hskip
34.71239pt\nu^{\rm ssc}_{\rm KN,c,r}<\nu\,,\cr\end{cases}$ (23)

and

$\displaystyle F^{\rm ssc}_{\rm\nu,r}(t<t_{\rm
x})\propto\begin{cases}t^{-\frac{2(9-11p)-k(4-7p)}{4}}\,\nu^{-(p-1)},\hskip
42.67912pt\nu^{\rm ssc}_{\rm KN,c,r}<\nu<\nu^{\rm ssc}_{\rm 0,r},\cr
t^{-\frac{8(1-2p)-k(2-5p)}{4}}\,\nu^{-\frac{p+1}{2}},\hskip 51.21504pt\nu^{\rm
ssc}_{\rm 0,r}<\nu<\nu^{\rm ssc}_{\rm m,r},\cr
t^{-\frac{6(5-11p)-k(8-21p)}{12}}\,\nu^{-(p+\frac{1}{3})},\,\,\,\,\,\hskip
32.72049pt\nu^{\rm ssc}_{\rm m,r}<\nu\,,\cr\end{cases}$ (24)

for the fast (weak) - and slow-cooling regimes, respectively.

####  2.2.2 SSC light curves for $t>t_{\rm x}$ with $k=2$

Following the previous process, the spectral breaks and the maximum flux of
SSC emission are given by

$\displaystyle h\nu^{\rm ssc}_{\rm m,r}$ $\displaystyle\simeq$ $\displaystyle
0.7\,{\rm
eV}\left(\frac{1+z}{2}\right)^{\frac{34}{21}}\,g^{4}(2.2)\,\zeta_{\rm
e,-1}^{-4}\,\varepsilon^{4}_{\rm e_{r},-1}\varepsilon^{\frac{1}{2}}_{\rm
B_{r},-2}\,A^{-\frac{47}{42}}_{\rm
k,-1}\,\Gamma^{-\frac{136}{21}}_{1.5}\,E^{\frac{34}{21}}_{52}\,t^{-\frac{55}{21}}_{3.5}\,,$
(25) $\displaystyle h\nu^{\rm ssc}_{\rm cut,r}$ $\displaystyle\simeq$
$\displaystyle 1.0\times 10^{-4}\,{\rm
eV}\left(\frac{1+z}{2}\right)^{\frac{34}{21}}\,\left(\frac{1+Y_{\rm
r}}{2}\right)^{-4}\varepsilon^{-\frac{7}{2}}_{\rm
B_{r},-2}\,A^{-\frac{383}{42}}_{\rm
k,-1}\Gamma^{-\frac{388}{21}}_{1.5}\,E^{\frac{118}{21}}_{52}\,t^{-\frac{55}{21}}_{3.5}\,,$
(26) $\displaystyle F^{\rm ssc}_{\rm max,r}$ $\displaystyle\simeq$
$\displaystyle 4.9\times 10^{-3}\,{\rm
mJy}\,\left(\frac{1+z}{2}\right)^{\frac{17}{7}}\,g^{-1}(2.2)\,\zeta_{\rm
e,-1}\,\varepsilon^{\frac{1}{2}}_{\rm B_{r},-2}A^{\frac{29}{14}}_{\rm
k,-1}\,\Gamma^{-\frac{5}{7}}_{1.5}\,d^{-2}_{\rm
z,28.3}E^{\frac{3}{7}}_{52}\,t^{-\frac{10}{7}}_{3.5}\,,$ (28)

where the cutoff frequency $\nu^{\rm ssc}_{\rm cut,r}=\nu^{\rm ssc}_{\rm
c,r}(t_{\rm x})\,\left(\frac{t}{t_{\rm x}}\right)^{-\frac{55}{21}}$ is
estimated by the requirement that no electrons are shocked anymore and the
fluid expands adiabatically. For $\nu^{\rm ssc}_{\rm c,r}<\nu^{\rm ssc}_{\rm
m,r}$ and $\nu^{\rm ssc}_{\rm m,r}<\nu^{\rm ssc}_{\rm c,r}$, the spectral
breaks in the KN regime are given by

$\displaystyle h\nu^{\rm ssc}_{\rm KN,m,r}$ $\displaystyle\simeq$
$\displaystyle
0.1\,{\rm~{}GeV}\left(\frac{1+z}{2}\right)^{-\frac{2}{7}}g(2.2)\zeta_{e,-1}^{-1}\varepsilon_{\rm
e_{r},-1}A^{-\frac{5}{7}}_{\rm
k,-1}\Gamma^{-\frac{13}{7}}_{1.5}\,E^{\frac{5}{7}}_{52}\,t^{-\frac{5}{7}}_{3.5},$
(29) $\displaystyle h\nu^{\rm ssc}_{\rm KN,c,r}$ $\displaystyle\simeq$
$\displaystyle 2.4\times
10^{-2}\,{\rm~{}GeV}\left(\frac{1+z}{2}\right)^{-\frac{32}{21}}\left(\frac{1+Y_{\rm
r}}{2}\right)^{-1}\varepsilon^{-1}_{\rm B_{r},-2}\,A^{-\frac{31}{21}}_{\rm
k,-1}\,\Gamma^{\frac{2}{21}}_{1.5}\,E^{\frac{10}{21}}_{52}\,t^{\frac{11}{21}}_{3.5}.\,\,\,$
(30)

The SSC light curves for the slow-cooling regime are

$\displaystyle F^{\rm ssc}_{\rm\nu,r}(t>t_{\rm
x})\propto\begin{cases}t^{-\frac{5}{9}}\,\nu^{\frac{1}{3}},\hskip
48.36958pt\nu<\nu^{\rm ssc}_{\rm c,r},\cr
t^{-\frac{5(11p+1)}{42}}\,\nu^{-\frac{p-1}{2}},\hskip 8.5359pt\nu^{\rm
ssc}_{\rm m,r}<\nu<\nu^{\rm ssc}_{\rm cut,r},\cr 0,\,\,\,\,\,\hskip
64.01869pt\nu^{\rm ssc}_{\rm cut,r}<\nu\,.\cr\end{cases}$

The SSC emission from the RS region could decay faster due to the angular time
delay effect. During the emission at high latitudes, the SSC flux evolves in
the same manner as the flux mentioned above in the case of the thick-shell
scenario.

###  2.3 Synchrotron light curves from forward shocks

The dynamics of the forward shocks for a relativistic outflow expanding into a
constant-density (${\rm k=0}$) and stellar-wind environments (${\rm k=2}$) are
treated in Sari et al. (1998); Chevalier & Li (2000); Panaitescu & Kumar
(2000). For a density profile $\propto r^{\rm-k}$, the evolution of the bulk
Lorentz factor is $\Gamma\propto t^{-\frac{3-k}{2(4-k)}}$, and the electron
Lorentz factors for the minimum and the cooling energy are $\gamma_{\rm
m,f}\propto t^{-\frac{3-k}{2(4-k)}}$ and $\gamma_{\rm c,f}\propto
t^{\frac{k+1}{2(4-k)}}$, respectively. Using the evolution of synchrotron
energy breaks ($\nu^{\rm syn}_{\rm m,f}\propto t^{-\frac{3}{2}}$ and $\nu^{\rm
syn}_{\rm c,f}\propto t^{\frac{4-3k}{2(k-4)}}$) and the maximum flux ($F^{\rm
syn}_{\rm max,f}\propto t^{\frac{k}{2(k-4)}}$), the observed flux in the fast
cooling regime is proportional to $F^{\rm syn}_{\rm\nu,f}\propto
t^{-\frac{1}{4}}\,\nu^{-\frac{1}{2}}$ for $\nu<\nu^{\rm syn}_{\rm m,f}$ and
$\propto t^{-\frac{3p-2}{4}}\,\nu^{-\frac{p}{2}}$ for $\nu^{\rm syn}_{\rm
m,f}<\nu$. In the slow cooling regime, the observed flux is proportional to
$F^{\rm syn}_{\rm\nu,f}\propto
t^{\frac{12-5k-3p(4-k)}{4(4-k)}}\,\nu^{-\frac{p-1}{2}}$ for $\nu<\nu^{\rm
syn}_{\rm c,f}$ and $\propto t^{-\frac{3p-2}{4}}\,\nu^{-\frac{p}{2}}$ for
$\nu^{\rm syn}_{\rm c,f}<\nu$.

###  2.4 The Compton Y-parameter

When KN effects are strong, the Compton parameter changes. In this case, this
parameter in the slow-cooling regime can be estimated with

$\displaystyle Y_{j}(Y_{j}+1)=Y_{0,j}\,\begin{cases}1\,\hskip 85.35826pt{\rm
for}\hskip 5.69046pt{\nu^{\rm syn}_{\rm c,j}<\nu^{\rm syn}_{\rm
KN,c,j}}\cr\left(\frac{\nu^{\rm syn}_{\rm KN,c,j}}{\nu^{\rm
syn}_{c,j}}\right)^{-\frac{p-3}{2}}\,\hskip 8.5359pt{\rm for}\hskip
5.69046pt{\nu^{\rm syn}_{\rm m,j}<\nu^{\rm syn}_{\rm KN,c,j}<\nu^{\rm
syn}_{\rm c,j}}\,\,\,\,\,\,\cr\left(\frac{\nu^{\rm syn}_{\rm m,j}}{\nu^{\rm
syn}_{\rm c,j}}\right)^{-\frac{p-3}{2}}\left(\frac{\nu^{\rm syn}_{\rm
KN,c,j}}{\nu^{\rm syn}_{\rm m,j}}\right)^{\frac{4}{3}}\,{\rm for}\,{\nu^{\rm
syn}_{\rm KN,c,j}<\nu^{\rm syn}_{\rm m,j}}\,,\end{cases}$ (31)

where
$Y_{0,j}=\frac{\varepsilon_{e,j}}{\varepsilon_{B,j}}\left(\frac{\gamma_{\rm
m,j}}{\gamma_{\rm c,j}}\right)^{p-2}$ with ${\rm j=r}$ or ${\rm f}$. For the
RS region, $h\nu^{\rm syn}_{\rm
KN,c,r}\simeq\frac{\gamma_{3}}{(1+z)}\,\frac{m_{e}c^{2}}{\gamma_{c,r}}$ for
$\nu^{\rm syn}_{\rm c,r}>\nu^{\rm syn}_{\rm m,r}$ (see Nakar et al., 2009;
Wang et al., 2010). It is worth mentioning that in the fast cooling regime the
synchrotron break in the KN regime becomes $h\nu^{\rm syn}_{\rm
KN,m,r}\simeq\frac{\gamma_{3}}{(1+z)}\,\frac{m_{e}c^{2}}{\gamma_{m,r}}$ for
$\nu^{\rm syn}_{\rm c,r}<\nu^{\rm syn}_{\rm m,r}$. For the FS region,
$h\nu^{\rm syn}_{\rm
KN,c,f}\simeq\frac{\Gamma}{(1+z)}\,\frac{m_{e}c^{2}}{\gamma_{c,f}}$ for
$\nu^{\rm syn}_{\rm c,f}>\nu^{\rm syn}_{\rm m,f}$ and $h\nu^{\rm syn}_{\rm
KN,m,f}\simeq\frac{\gamma_{3}}{(1+z)}\,\frac{m_{e}c^{2}}{\gamma_{m,f}}$ for
$\nu^{\rm syn}_{\rm c,f}<\nu^{\rm syn}_{\rm m,f}$. To describe the LAT data
above 100 MeV, we must determine the Lorentz factor of electrons that may
produce high-energy photons via the synchrotron process, include a new
spectrum break, and recalculate the Compton value. The new spectral break
becomes $h\nu^{\rm syn}_{\rm KN,c,j}(\gamma_{*})$, and the recalculated
Compton parameter, $Y(\gamma_{*})$, becomes
$Y(\gamma_{*})=Y(\gamma_{c,j})\left(\frac{\nu_{*}}{\nu_{c,j}}\right)^{\frac{p-3}{4}}\left(\frac{\nu^{\rm
syn}_{\rm KN,c,j}(\gamma_{\rm c,j})}{\nu^{\rm
syn}_{c,j}}\right)^{-\frac{p-3}{2}}$ for $\nu^{\rm syn}_{\rm m,j}<h\nu^{\rm
syn}_{\rm KN,c,j}(\gamma_{*})=100\,{\rm~{}MeV}<\nu^{\rm syn}_{\rm
c,j}<\nu^{\rm syn}_{\rm KN,c,j}(\gamma_{\rm c,j})$ (for details, see Wang et
al., 2010).

###  2.5 The maximum photon energy

####  2.5.1 Reverse-shock region

The maximum energy radiated by the SSC process can be estimated by equalising
the acceleration ($\propto\gamma_{e}B^{\prime-1}_{\rm r}$) and the synchrotron
($\propto\gamma_{e}^{-1}B^{\prime-2}_{\rm r}$) timescales. The maximum Lorentz
factor of the electron distribution is $\gamma_{\rm
max,r}=\left(3q_{e}/\xi\sigma_{T}B^{\prime}_{\rm r}\right)^{\frac{1}{2}}$ with
$\xi$ the Bohm parameter555In the Bohm limit, this parameter becomes $\xi\sim
1$. and $q_{e}$ the elementary charge. Therefore, the maximum energy radiated
by the synchrotron model is $h\nu^{\rm syn}_{\rm
max,r}=3q^{2}_{e}\gamma_{3}/2\pi\sigma_{T}m_{e}c(1+z)$. In this case, the
maximum energy generated by the SSC mechanism for $t<t_{x}$ in the thick and
thin regime is

$\displaystyle h\nu^{\rm ssc}_{\rm max,r}$ $\displaystyle\simeq$
$\displaystyle
1.3\,{\rm~{}GeV}\,\left(\frac{1+z}{2}\right)^{-\frac{4}{4-k}}\varepsilon^{-\frac{1}{2}}_{\rm
B_{r},-2}A^{-\frac{2}{4-k}}_{\rm
k,-1}\,\Delta^{-\frac{k}{2(4-k)}}_{11.8}\,E^{\frac{k}{2(4-k)}}_{53}\,t^{\frac{k}{4-k}}_{1}\,,$
(32)

and

$\displaystyle h\nu^{\rm ssc}_{\rm max,r}\simeq
0.4\,{\rm~{}GeV}\,\left(\frac{1+z}{2}\right)^{-\frac{k+2}{2}}\varepsilon^{-\frac{1}{2}}_{\rm
B_{r},-2}A^{-\frac{1}{2}}_{\rm
k,-1}\,\Gamma^{k}_{1.5}\,t^{\frac{k}{2}}_{2.5},$ (33)

respectively. For $t>t_{\rm x}$, the maximum energies are determined by
$\nu^{\rm ssc}_{\rm cut,r}$.

####  2.5.2 Forward-shock region

The maximum Lorentz factor of the electron distribution is $\gamma_{\rm
max,f}=\left(3q_{e}/\xi\sigma_{T}B^{\prime}_{\rm f}\right)^{\frac{1}{2}}$ and
the maximum energy radiated by the synchrotron yields

$\displaystyle h\nu^{\rm syn}_{\rm max,f}$ $\displaystyle\approx$
$\displaystyle
0.2\,{\rm~{}GeV}\left(\frac{1+z}{2}\right)^{\frac{k-5}{2(4-k)}}\,A_{\rm
k,-1}^{-\frac{1}{2(4-k)}}E_{53}^{\frac{1}{2(4-k)}}t_{2}^{-\frac{3-k}{2(4-k)}}\,.$
(34)

###  2.6 The short-lasting bright peak and the long-lasting emission in the
Fermi/LAT band

We display in Figure 1 the expected SSC and synchrotron light curves from the
RS and FS model when the outflow decelerates in a stellar-wind density (${\rm
k=2}$). They are presented in the fast and slow cooling regime, and when the
RS lies in the thick (left column) and thin (right column) shell case.
Possible transitions between the fast- and slow-cooling regimes and from
constant-density to stellar-wind medium are not considered. The effects in the
self-absorption regime are not considered because they are unimportant at the
Fermi/LAT energy range (e.g., see Panaitescu et al., 2014). The SSC RS flux in
blue lines illustrates that a peak is expected at $t=t_{\rm x}$ when it
evolves in $\nu^{\rm ssc}_{\rm c,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm m,r}$ for
the thick-shell case and $\nu^{\rm ssc}_{\rm m,r}<\nu_{\rm LAT}<\nu^{\rm
ssc}_{\rm c,r}$ for the thin-shell case. Otherwise, the flux decreases
monotonically, exhibiting a break at $t=t_{\rm x}$. The temporal break is
associated with the transition time (black line). Under the cooling conditions
$\nu^{\rm ssc}_{\rm m,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm cut,r}$ and $t_{\rm
x}<t$, the light curves in the thick and thin regime could exhibit a temporal
break due to the passage of the cutoff break through the Fermi/LAT band
($\nu^{\rm ssc}_{\rm cut,r}<\nu_{\rm LAT}$).

The expected synchrotron FS fluxes in red lines are shown for a stellar-wind
medium. Since the spectral breaks evolve as $\nu^{\rm syn}_{\rm m,f}\propto
t^{-\frac{3}{2}}$ and $\nu^{\rm syn}_{\rm c,f}\propto t^{\frac{1}{2}}$, the
temporal breaks displayed in the synchrotron FS fluxes correspond to the
transitions from $\nu_{\rm LAT}<\nu^{\rm syn}_{\rm m,f}$ to $\nu^{\rm
syn}_{\rm m,f}<\nu_{\rm LAT}$ for the fast cooling regime, and from $\nu^{\rm
syn}_{\rm c,f}<\nu_{\rm LAT}$ to $\nu_{\rm LAT}<\nu^{\rm syn}_{\rm c,f}$ for
the slow cooling regime.

We argue that a LAT light curve with similar features to that shown in Figure
1 can be described as a superposition of SSC RS and synchrotron FS emissions
evolving in a stellar wind environment. It is worth noting that, depending on
the value of the shock crossing time, the SSC emission would appear during the
prompt emission (thick shell) or later (thin shell).

##  3 Application: Second Fermi/LAT Catalog

###  3.1 Our sample of GRBs

####  3.1.1 GRB 080916C

GRB 080916C triggered the Gamma Burst Monitor (GBM) instrument on board the
Fermi satellite at 00:12:45.613542 UT on September 16, 2008 (Goldstein & van
der Horst, 2008). The duration of prompt emission measured by the GBM
instrument was $T_{90}=62.98\,{\rm s}$ correspondent to a fluence of $(4.0\pm
0.6)\times 10^{-5}\,{\rm erg\,cm^{-2}}$ and an isotropic energy of
$E_{\rm\gamma,iso}=(1.7\pm 0.1)\times 10^{54}\,{\rm erg}$ (Ajello et al.,
2019). This burst was detected at a redshift $z=4.35\pm 0.15$ (Clemens et al.,
2008) at R.A.=$08^{\rm h}07^{\rm m}12^{\rm s}$ and
Dec=-$61^{\circ}18^{\prime}00"$ with a $2.8^{\circ}$ uncertainty at 68
percent. In the energy range of 10 keV to 10 GeV, the apparent isotropic
energy release was measured as $8.8\times 10^{54}\,{\rm erg}$ (Abdo et al.,
2009a). The Swift/XRT instrument began observing GRB 080916C at 17:08 UT, 17
hours after trigger time (Perri et al., 2008). This instrument monitored GRB
080916C in the Photon Counting (PC) mode from $6.1\times 10^{4}$ to $\sim
10^{6}\,{\rm s}$. The best-fit intrinsic absorption column density is
$1.49\times 10^{21}\,{\rm cm^{-2}}$. GRB 080916C was detected by the Gamma-Ray
Burst Optical/Near-Infrared Detector in optical bands (GROND; Clemens et al.,
2008).

####  3.1.2 GRB 090323

At 00:02:42.63 UT on March 23, 2009, GRB 090323 triggered the Fermi/GBM
instrument (Ohno et al., 2009). The duration of the prompt emission measured
by the GBM instrument was $T_{90}=133.89\,{\rm s}$, corresponding to a fluence
of $(0.26\pm 0.07)\times 10^{-5}\,{\rm erg\,cm^{-2}}$ and an isotropic energy
of $E_{\rm\gamma,iso}=(3.0\pm 2.0)\times 10^{53}\,{\rm erg}$ (Ajello et al.,
2019). The Fermi/LAT instrument also detected this burst (Kennea et al., 2009;
Perri & Stratta, 2009). The isotropic energy in the LAT energy range was
$\simeq 4.1\times 10^{54}$ erg (Ackermann et al., 2013a). The Swift/XRT
instrument began observing GRB 090323 at 19:27 UT, 19.4 hours after the
trigger. This instrument monitored GRB 090323 in the PC mode with a spectrum
exposure of 6 ks. The best-fit intrinsic absorption column density is
$2.20\times 10^{22}\,{\rm cm^{-2}}$ with a redshift $z=3.57$, which was
measured by Chornock et al. (2009). McBreen et al. (2010) described the
burst’s optical and infrared observations in great detail.

####  3.1.3 GRB 090902B

GRB 090902B was detected by the Fermi/GBM instrument on 2009 September 2 at
11:05:08.31 UT with coordinates R.A.= 17h38m26s and Dec: 26∘30’. The duration
of the prompt emission in the energy band 50 - 300 keV was $T_{90}=19.33\,{\rm
s}$, corresponding to a fluence $(7.0\pm 1.0)\times 10^{-5}\,{\rm
erg\,cm^{-2}}$, and an isotropic energy of $E_{\rm\gamma,iso}=(3.7\pm
0.3)\times 10^{53}\,{\rm erg}$ (Ajello et al., 2019). The burst was within the
Fermi/LAT field of view initially at an angle of $51^{\circ}$ from the sight
of vision. The X-ray afterglow was detected within the LAT error circle by the
Swift/XRT (Kennea & Stratta, 2009), and Ultraviolet/Optical Telescope (UVOT)
(Swenson & Siegel, 2009), and later by several other ground-based telescopes.
The XRT instrument monitored GRB 090902B in the PC mode with a spectrum
exposure of 13.5 ks. The best-fit intrinsic absorption column density is
$2.3^{+0.8}_{-0.6}\times 10^{22}\,{\rm cm^{-2}}$ with a redshift $z=1.822$,
which was determined by the Gemini-North telescope (Cucchiara et al., 2009).

####  3.1.4 GRB 090926A

GRB 090926A was discovered by the Fermi/GBM at 04:20:26.99 UT on 26 September
2009, with coordinates R.A.=354.5∘ and Dec=-64.2∘ (Ackermann et al., 2011).
The duration of the prompt emission, fluence and isotropic energy measured by
GBM instrument were $T_{90}=13.76\,{\rm s}$, $(2.9\pm 0.3)\times 10^{-5}\,{\rm
erg\,cm^{-2}}$, and $E_{\rm\gamma,iso}=(5.3\pm 0.4)\times 10^{53}\,{\rm erg}$,
respectively (Ajello et al., 2019). GRB 090926A presented a characteristic
high-energy PL component that is different from the known Band function,
according to a joint LAT and GBM data analysis (Ackermann et al., 2011).
Swift/XRT, INTEGRAL SPI-ACS (Bissaldi, 2009), Suzaku/WAM (Noda et al., 2009),
CORONAS-Photon (Chakrabarti et al., 2009), the Konus-wind experiment
(Golenetskii et al., 2009) and the Ultra-violet Optical Telescope (UVOT)
instrument on board Neil Gehrels Swift Observatory (Malesani et al., 2009),
all identified this burst separately. The Swift/XRT instrument began observing
GRB 090926A, 13.9 hours after the trigger. This instrument monitored GRB
090926A in the PC mode since $4.7\times 10^{4}$ to $1.8\times 10^{6}\,{\rm
s}$. The best-fit intrinsic absorption column density is
$3.0^{+4.0}_{-3.0}\times 10^{21}\,{\rm cm^{-2}}$ with a redshift $z=2.1062$,
which was estimated using X-shooter installed on the Very Large Telescope
(Malesani et al., 2009).

####  3.1.5 GRB 110731A

GRB 110731A triggered the Fermi/GBM instrument at 11:09:29.94 UT on 2011 July
31 (Ackermann et al., 2013b). Approximately 30 s after the GBM trigger, the
Burst Alert Telescope (BAT) on board Neil Gehrels Swift Observatory was also
set off, and it located GRB 110731A at R.A.=$18^{\rm h}42^{\rm m}05^{\rm s}$
and Dec=-$28^{\circ}32^{\prime}44"$ with an uncertainty of 3 arcmin. The
duration of the prompt emission measured by GBM instrument was
$T_{90}=7.49\,{\rm s}$, corresponding to a fluence and an isotropic energy of
$(0.930\pm 0.07)\times 10^{-5}\,{\rm erg\,cm^{-2}}$ and
$E_{\rm\gamma,iso}=(1.2\pm 0.3)\times 10^{53}\,{\rm erg}$, respectively
(Ajello et al., 2019). The isotropic energy measured using a PL and a Band
function was $(7.6\pm 0.2)\times 10^{53}\,{\rm erg}$ (Ackermann et al.,
2013b). The Swift/XRT instrument started detecting GRB 110731A at 11:10:36.9
UT, 66.4 s after the BAT trigger (Oates et al., 2011). This instrument
monitored GRB 110731A in the windowed-timing (WT) mode with a spectrum
exposure of 573 s and the PC mode with a spectrum exposure of 7.5 ks. The
best-fit intrinsic absorption column density is $4.4^{+3.1}_{-3.0}\times
10^{21}\,{\rm cm^{-2}}$. The Swift UVOT instrument began observing the burst’s
position 75 s after the BAT trigger (Oates et al., 2011). Following the
Swift/BAT trigger, several efforts were undertaken to observe the burst’s
position, such as those by the Faulkes Telescopes North and South (Bersier,
2011), the Nordic Optical Telescope equipped with ALFOSC (Malesani et al.,
2011), Konus-Wind (Golenetskii et al., 2011), the EVLA (Zauderer et al.,
2011), the Suzaku Wide-band All-sky Monitor (WAM) (Hanabata et al., 2011) and
the SAO RAS and Terskol observatories (Moskvitin et al., 2011). Tanvir et al.
(2011) obtained spectroscopic observations with the GMOS-N instrument on
Gemini-North, which led to an estimation of a redshift of $z=2.83$.

####  3.1.6 GRB 130427A

GRB 130427A triggered the Fermi/GBM instrument at 07:47:06.42 UTC on April 27,
2013 (von Kienlin, 2013). Approximately 50 s after the initial trigger, it
also triggered the Swift/BAT instrument. Following the initial triggers, there
was a substantial follow-up campaign by extraterrestrial observatories and
ground-based experiments (from the ultra-violet, optical and X-ray telescopes
(Maselli et al., 2014); SPI-ACS/INTEGRAL (Pozanenko et al., 2013); AGILE
(Verrecchia et al., 2013); Konus-Wind (Golenetskii et al., 2013); NuSTAR
(Kouveliotou et al., 2013); RHESSI (Smith et al., 2013); MAXI/GSC (Kawamuro et
al., 2013); VLT/X-shooter (Flores et al., 2013)). The duration of the prompt
emission measured by GBM instrument was $T_{90}=138.24\,{\rm s}$ (Ajello et
al., 2019) with an isotropic energy of $E_{\rm\gamma,iso}=(1.7\pm 0.2)\times
10^{52}\,{\rm erg}$ (Ajello et al., 2019). The total fluence and the isotropic
energy measured in the 10 keV - 100 GeV range was $4.9\times 10^{-3}\,{\rm
erg\,cm^{-2}}$ and $1.4\times 10^{54}\,{\rm erg}$, respectively (Ackermann et
al., 2014). The Swift/XRT instrument started observing GRB 130427A at
07:50:17.7 UT, 140.2 s after the BAT trigger. This instrument monitored GRB
130427A in the WT mode with a spectrum exposure of 1.9 ks and the PC mode with
a spectrum exposure of 4.1 ks. The best-fitting absorption column (intrinsic)
is $2.7^{+0.8}_{-0.8}\times 10^{20}\,{\rm cm^{-2}}$ and
$1.1^{+0.4}_{-0.4}\times 10^{21}\,{\rm cm^{-2}}$ for WT and PC modes,
respectively. The Combined Array for Research in Millimeter-wave Astronomy
(CARMA) localized this burst to R.A. = $173.1367^{\circ}$, Dec. =
$27.6989^{\circ}$ (J2000) with an uncertainty of 0.4 arc sec (Perley, 2013).
Optical spectroscopy from Gemini-North conducted by Levan et al. (2013) found
the redshift of the GRB to be $z=0.34$.

####  3.1.7 GRB 160625B

Fermi/GBM triggered and located GRB 160625B at 22:40:16.28 UT, 2016 June 25
(Burns, 2016). Immediately, Fermi/LAT triggered this burst at 22:43:24.82 UT
(Dirirsa et al., 2016). The duration of the prompt emission measured by the
GBM instrument was $T_{90}=453.38\,{\rm s}$, corresponding to a fluence and an
isotropic energy of $(2.5\pm 0.3)\times 10^{-5}\,{\rm erg\,cm^{-2}}$ and
$E_{\rm\gamma,iso}=(1.5\pm 0.1)\times 10^{53}\,{\rm erg}$, respectively
(Ajello et al., 2019). The Swift/XRT instrument monitored GRB 160625B in the
PC mode from $9.8\times 10^{4}$ to $4.1\times 10^{6}\,{\rm s}$. The best-fit
absorption column (intrinsic) is $1.6\pm 0.6\times 10^{21}\,{\rm cm^{-2}}$,
with a redshift $z=1.406$ (Xu et al., 2016). Several optical observations were
performed with background telescopes (Zhang et al., 2016; Troja et al., 2017).

####  3.1.8 GRB 180720B

GRB 180720B was identified and immediately followed by GBM and LAT
instruments, the two instruments onboard the Fermi satellite (Roberts &
Meegan, 2018; Bissaldi & Racusin, 2018), and BAT, XRT, and UVOT instruments
onboard the Swift satellite (Palmer et al., 2018; Barthelmy et al., 2018). The
BAT instrument triggered this burst on July 20, 2018, at 14:21:44 UT and
located it with coordinates: ${\rm R.A.=00^{\rm h}02^{\rm m}07^{\rm s}}\,$ and
${\rm Dec}=-02^{\rm d}56^{\prime}00^{\prime\prime}\,(J2000)$ with an
uncertainty of 3 arcmin. The duration of the prompt emission measured by the
GBM instrument was $T_{90}=48.90\,{\rm s}$ correspondent to a fluence
$(0.19\pm 0.05)\times 10^{-5}\,{\rm erg\,cm^{-2}}$, and an isotropic energy of
$E_{\rm\gamma,iso}=(0.39\pm 0.09)\times 10^{52}\,{\rm erg}$ (Ajello et al.,
2019). The Swift/XRT instrument started observing GRB 180720B at 14:23:11.0
UT, 86.5 s after the BAT trigger. This instrument monitored GRB 180720B in the
WT mode with a spectrum exposure of 2.2 ks and the PC mode with a spectrum
exposure of 3.9 ks. The best-fitting absorption column (intrinsic) is
$3.72^{+0.11}_{-0.11}\times 10^{21}\,{\rm cm^{-2}}$ and
$3.4^{+0.6}_{-0.5}\times 10^{21}\,{\rm cm^{-2}}$ for WT and PC modes,
respectively. GRB 180720B started to be monitored in the optical and near-
infrared (NIR) bands on July 20, 2018, at 14:22:57 UT, 73 s after the trigger
time (Sasada et al., 2018). Vreeswijk et al. (2018) detected the optical
emission of GRB 180720B using the VLT/X-shooter spectrograph, associating a
redshift of $z=0.654$.

###  3.2 Data analysis

The data files used for the Fermi/LAT analysis were obtained from the online
data website.666https://fermi.gsfc.nasa.gov/cgi-bin/ssc/LAT/LATDataQuery.cgi
Fermi/LAT data was analyzed in the 0.1-100 GeV energy range and within the
time-resolved likelihood analysis for each burst ($\rm{t_{LAT,0}}$,
$\rm{t_{LAT,1}}$, see Table 1 from Ajello et al. (2019) for the definition)
with the Fermi Science
tools777https://fermi.gsfc.nasa.gov/ssc/data/analysis/software/ in their
conda-based installation.ScienceTools
2.2.0888https://github.com/Fermi/LAT/Fermitools-conda/wiki For this analysis
we adopt the responses reported for each burst by Ajello et al. (2019),
following the unbinned likelihood analysis presented by the Fermi/LAT
team.999https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/likelihood_tutorial.html
Using the gtselect tool, we select, with an event class 32 nor 8 depending on
the response
chosen101010https://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/
Cicerone_Data/LAT_DP.html, a region of interest (ROI) around the position of
the burst within a radius of 15∘. We apply a cut to the zenith angle above
100∘. Then, we select the appropriate time intervals (GTIs) using the gtmktime
tool on the selected data before considering the ROI cut. To define the model
needed to describe the source, the diffuse components and point sources of
4FGL-DR3 reported by Abdollahi et al. (2022) embedded in ROI
make4FGLxml111111https://fermi.gsfc.nasa.gov/ssc/data/analysis/user/ were
used. We define a point source at the position of this burst, assuming a PL
spectrum, and we define a diffuse galactic component using GALPROP gll_iem_v07
as well as the extragalactic background
iso_P8R3_SOURCE_V3_v1121212https://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html.
For each burst, the spectral index is fixed with the value reported by Ajello
et al. (2019) in their Table 4, letting free the normalisation and the
normalisation of the diffuse component. We use gtdiffrsp to take into account
all of these components. Following the likelihood procedure, we produce a
lifetime cube with the tool gtltcube, using a step $\delta\theta=0.025$, a bin
size of 0.5 and a maximum zenith angle of 100∘. The exposure map was created
using gtexpmap, considering a region of 30∘ around the GRB position and
defining 100 spatial bins in longitude/latitude and 50 energy bins, and we
perform the likelihood analysis with
pyLikelihood131313https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/python_tutorial.html.
Finally, we obtain the photons with a probability greater than 90$\%$ to be
associated to each burst with the gtsrcprob tool.

The upper panels in each plot of Figure 2 show the Fermi/LAT energy flux
(blue) and photon flux (red) light curves, and the lower panels exhibit all
photons with energies $\geq 100$ MeV associated with GRB 080916C, GRB 090323,
GRB 090902B, GRB 090926A, GRB 110731A, GRB 130427A, GRB 160625B and GRB
180720B. The filled circles in black correspond to the individual photons and
their energies with probability $>90\%$ of being associated with the
respective burst and the open circles in grey indicate the LAT gamma transient
class photons.

In order to describe the short-lasting peak in the LAT observations, we use
the function

$\displaystyle F_{\rm L,se}(t)=A_{\rm se}\begin{cases}(t-T_{a})^{-\alpha_{\rm
r,bb}}\,\,\hskip 28.45274ptt<t_{\rm r,br},\cr t^{-\alpha_{\rm r,ab}}\,\,\hskip
56.9055ptt_{\rm r,br}<t\,,\cr\end{cases}$ (35)

where $A_{\rm se}$ corresponds to the proportionality constant, $t_{\rm r,br}$
is the temporal break with the temporal indexes before ($\alpha_{\rm r,bb}$)
and after ($\alpha_{\rm r,ab}$) the break. The term $T_{a}$ is the starting
time of the short-term component (Kobayashi & Zhang, 2007; Vestrand et al.,
2006). For the long-lasting component, we use a PL function $F_{\rm
L,ee}(t)=A_{ee}\,t^{-\alpha_{\rm f}}$ or BPL function

$\displaystyle F_{\rm L,ee}(t)=A_{ee}\begin{cases}t^{-\alpha_{\rm
f_{j},bb}}\,\,\hskip 42.67912ptt<t_{\rm f_{j},br},\cr t^{-\alpha_{\rm
f_{j},ab}}\,\,\hskip 42.67912ptt_{\rm f_{j},br}<t\,,\cr\end{cases}$ (36)

where $A_{\rm ee}$ corresponds to the proportionality constant, $t_{\rm
f_{j},br}$ is the temporal break with the temporal indexes before
($\alpha_{\rm f_{j},bb}$) and after ($\alpha_{\rm f_{j},ab}$) the break with
${\rm f=L}$, ${\rm X}$, ${\rm O}$ for LAT, X-ray and optical observations,
respectively. The continuous and dashed lines on the photon energy flux
correspond to the best-fit curves of the short- and long-lasting components,
respectively. We fit each component of the energy flux in the LAT light curve
with a series of broken and simple PLs. To find the best-fit values of the
temporal indexes of PLs, we use the chi-square minimisation $\chi^{2}$ using
the ROOT software package (Brun & Rademakers, 1997). Table 1 lists the best-
fit temporal indices of the short and long-lasting emissions, including the
chi squares ($\chi^{2}$/n.d.f.).

Data sets from the X-ray Telescope (XRT) instrument on board the Neil Gehrels
Swift Observatory were retrieved from the publicly available database on the
official Swift website.141414https://www.swift.ac.uk/burst_analyser/00922968/
The flux density at 10 keV is transformed to 1 keV using the conversion factor
derived in Evans et al. (2010).

###  3.3 Afterglow evolution: LAT and multi-wavelength observations

We report the best fit values for the short and long LAT components using ROOT
Chi-square minimisation $\chi^{2}$ for the start time, the temporal breaks,
and the indexes before and after the break in Table 1. Additionally, with the
chi-square minimisation $\chi^{2}$ of ROOT, we obtain the best-fit values of
the X-ray and optical light curves, which we list in Table 2.

####  3.3.1 GRB 080916C

We note that the short-lasting component occurs during prompt emission
($t_{\rm x}<T_{90}$) because of the best-fit values of the starting time and
the temporal break. Therefore, the RS evolves in the thick-shell regime. The
best-fit value of the starting time indicates that the onset of the afterglow
was very early. When comparing the best-fit values of the temporal indexes
before and after the break of the SSC light curve, we can see that they are
consistent with the cooling conditions $\nu^{\rm ssc}_{\rm c,r}<\nu_{\rm
LAT}<\nu^{\rm ssc}_{\rm m,r}$ for $t<t_{\rm x}$ and $\nu^{\rm ssc}_{\rm
m,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm cut,r}$ for $t_{\rm x}<t$, for $p\approx
2.2$. It should be noted that the decay temporal index is consistent with the
high-latitude emission ($\nu^{\rm ssc}_{\rm cut,r}<\nu_{\rm LAT}$); $\beta+2$.
Other cooling conditions of the SSC light curve cannot explain the best-fit
values of the temporal indexes.

##### High-energy events.

At 3.94 s after the GBM trigger, the first high-energy photon was detected
with a measured energy of 161.4 MeV. The energy range of the photons in this
burst was extensive, with 283 photons over 100 MeV, 17 exceeding 1 GeV, and 2
exceeding 10 GeV. At 40.5 s after the GBM trigger, the highest-energy photon
in the LAT data had a value of 27.43 GeV.

##### Multiwavelength afterglow analysis

Analysis of LAT, X-ray, and optical spectra with PL functions leads to
spectral indexes of $\beta_{\rm LAT}=\Gamma_{\rm LAT}-1=1.20\pm 0.06$ (Ajello
et al., 2019), $\beta_{\rm X}=\Gamma_{\rm X}-1=0.80\pm
0.40$151515https://www.swift.ac.uk/xrt_spectra/00020082/ and $\beta_{\rm
Opt}=0.38\pm 0.20$ (Greiner et al., 2009), respectively. Taking into
consideration the LAT analysis of the long-lasting emission and the best-fit
values of the temporal and spectral indexes of LAT, X-ray and optical
observations, the observed fluxes evolve as $F_{\rm\nu,L}\propto t^{-1.21\pm
0.06}\,\nu^{-1.20\pm 0.06}$, $F_{\rm\nu,X}\propto t^{-1.35\pm
0.06}\,\nu^{-0.80\pm 0.30}$ and $F_{\rm\nu,Opt}\propto t^{-1.41\pm
0.04}\,\nu^{-0.38\pm 0.20}$, respectively. The fact that the temporal
(spectral) indexes for the optical and X-ray observations are larger (lower)
than for the LAT observations suggests that the closure relations of the
synchrotron FS model evolve in a slow-cooling regime through a wind-like
medium ($k=2$) for $p\approx 2.3\pm 0.3$. Note that the spectral and temporal
indices of the optical and X-ray observations are consistent with each other,
so the synchrotron closure relations evolve under the condition $\nu^{\rm
syn}_{\rm m,f}<\nu_{\rm Opt}<\nu_{\rm X}<\nu^{\rm syn}_{\rm c,f}$. Similarly,
LAT observations indicate that the closure relations evolve under the cooling
condition $\nu^{\rm syn}_{\rm c,f}<\nu_{\rm LAT}$.

####  3.3.2 GRB 090323

We see that the short-lasting emission is present during the prompt episode
($t_{\rm x}<T_{90}$), so the RS occurs in the thick-shell regime. According to
the best-fitting value of the starting time, the onset of the afterglow
occurred at the end of the prompt episode. We can infer the evolution of the
LAT frequency in the cooling conditions of the SSC light curve by comparing
the best-fit values of the temporal indices before and after the break. In
this case, $\nu_{\rm LAT}$ evolves as $\nu^{\rm ssc}_{\rm c,r}<\nu_{\rm
LAT}<\nu^{\rm ssc}_{\rm m,r}$ for $t<t_{\rm x}$ and as $\nu^{\rm ssc}_{\rm
m,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm cut,r}$ for $t_{\rm x}<t$, for $p\approx
2.2$. The decay temporal index, $\beta+2$, agrees with the high-latitude
emission ($\nu^{\rm ssc}_{\rm cut,r}<\nu_{\rm LAT}$), which is interesting to
note.

##### High-energy events.

The first high-energy photon, with a measured energy of 363.4 MeV, was
detected 15.05 s after the GBM trigger. In this burst, there were 32 photons
with energy over 100 MeV and 5 with energies above 1 GeV. The highest energy
photon detected in the LAT data was 7.40 GeV, which occurred 145.91 s after
the GBM trigger.

##### Multiwavelength afterglow analysis

We use the best-fit spectral indices presented in 2FLGC ($\beta_{\rm
LAT}=1.30\pm 0.20$; Ajello et al., 2019) and Swift/XRT repository ($\beta_{\rm
X}=0.80^{+0.34}_{-0.21}$)161616https://www.swift.ac.uk/xrt_spectra/00020102/
to evaluate the LAT and X-ray data, respectively. Given the GROND and TLS
observations of the optical/NIR afterglow, McBreen et al. (2010) found a
spectral index of $\beta_{\rm Opt}=0.65\pm 0.13$, considering the dust
extinction similar to Small Magellanic Cloud (SMC) with $A^{\rm
host}_{V}=0.14^{+0.04}_{-0.03}$. Using the LAT analysis of the log-lasting
emission and the best-fit values of the temporal and spectral indexes of LAT,
X-ray, and optical observations, the observed fluxes evolve as
$F_{\rm\nu,L}\propto t^{-1.26\pm 0.40}\,\nu^{-1.30\pm 0.10}$,
$F_{\rm\nu,X}\propto t^{-1.58\pm 0.08}\,\nu^{-0.80^{+0.34}_{-0.21}}$ and
$F_{\rm\nu,Opt}\propto t^{-1.70\pm 0.04}\,\nu^{-0.65\pm 0.13}$, respectively.
The fact that the temporal (spectral) index for the optical and X-ray
observations is larger (lower) than the LAT data suggests that the closure
relations of the synchrotron FS model evolves in a slow cooling regime through
a wind-like medium for $p\approx 2.3\pm 0.3$. It should be noted that the
spectral and temporal indices of X-ray and optical observations are compatible
with each other, and therefore the synchrotron closure relations evolve under
the condition $\nu^{\rm syn}_{\rm m,f}<\nu_{\rm Opt}<\nu_{\rm X}<\nu^{\rm
syn}_{\rm c,f}$. Moreover, LAT observations indicate that closure relations
evolve under the condition $\nu^{\rm syn}_{\rm c,f}<\nu_{\rm LAT}$.

####  3.3.3 GRB 090902B

Comparing the best-fit values of the starting time and the temporal break, one
can see that this component is observed during the prompt episode ($t_{\rm
x}<T_{90}$), then the RS is in the thick-shell regime. The best fit values of
the temporal indices before and after the break of the SSC light curve show
that they are consistent with the cooling conditions $\nu^{\rm ssc}_{\rm
c,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm m,r}$ for $t<t_{\rm x}$ and $\nu^{\rm
ssc}_{\rm m,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm c,r}$ or $\nu^{\rm ssc}_{\rm
cut,r}<\nu_{\rm LAT}$ for $t_{\rm x}<t$, for $p\approx 2.1$. The best-fit
values of the temporal indices cannot be explained by any other cooling
conditions of the SSC light curve.

##### High-energy events.

At 1.86 s after the GBM trigger, the first high-energy photon was detected
with a measured energy of 284.4 MeV. The energy range of the photons in this
burst was extensive, with 469 photons over 100 MeV, 67 exceeding 1 GeV, and 7
exceeding 10 GeV. At 81.7 s after the GBM trigger, the highest-energy photon
in the LAT data had a measured energy of 39.88 GeV.

##### Multiwavelength afterglow analysis

We consider the best-fit spectral index reported in 2FLGC ($\beta_{\rm
LAT}=0.94\pm 0.09$; Ajello et al., 2019). Pandey et al. (2010) analysed the
afterglow X-ray and UV-optical IR observations of GRB 090902B. After the
modelling of the broadband SED, the authors reported spectral indexes of
$\beta_{\rm X}=0.9\pm 0.1$ and $\beta_{\rm Opt}=0.68\pm 0.11$ for X-ray and
UV-optical-IR observations, respectively, considering the SMC-like dust
extinction with $A^{\rm host}_{V}=0.20\pm 0.06$. Given the analysis of the
long-lasting LAT component and the best-fit values of the temporal and
spectral indexes of LAT, X-ray, and optical observations, the observed fluxes
evolve as $F_{\rm\nu,L}\propto t^{-1.35\pm 0.06}\,\nu^{-0.94\pm 0.09}$,
$F_{\rm\nu,X}\propto t^{-1.62\pm 0.15}\,\nu^{-0.80^{+0.34}_{-0.21}}$ and
$F_{\rm\nu,Opt}\propto t^{-1.70\pm 0.04}\,\nu^{-0.65\pm 0.13}$, respectively.
The fact that the temporal (spectral) indexes for the optical and X-ray
observations are larger (lower) than the LAT data suggests that the closure
relations of the synchrotron FS model evolves in a slow cooling regime through
a wind-like medium for $p\approx 2.3\pm 0.3$. It should be noted that the
spectral and temporal indices of X-ray and optical observations are compatible
with each other and, therefore, the synchrotron closure relations evolve under
condition $\nu^{\rm syn}_{\rm m,f}<\nu_{\rm Opt}<\nu_{\rm X}<\nu^{\rm
syn}_{\rm c,f}$, and the LAT observations evolve under condition $\nu^{\rm
syn}_{\rm c,f}<\nu_{\rm LAT}$.

####  3.3.4 GRB 090926A

Given the best-fit values of the onset time and the temporal break, we see
that the short-lasting emission is present during the main episode ($t_{\rm
x}<T_{90}$), so the RS happens in the thick-shell regime. According to the
best-fitting value of the starting time, the onset of the afterglow occurred
at the end of the prompt episode. We can infer the evolution of the LAT
frequency in the cooling conditions of the SSC light curve by comparing the
best-fit values of the temporal indices before and after the break. In this
case, $\nu_{\rm LAT}$ evolves as $\nu^{\rm ssc}_{\rm c,r}<\nu_{\rm
LAT}<\nu^{\rm ssc}_{\rm m,r}$ for $t<t_{\rm x}$ and as $\nu^{\rm ssc}_{\rm
m,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm c,r}$ for $t_{\rm x}<t$, for $p\approx
2.2$. We want to emphasise that the temporal index of decay agrees with the
high-latitude emission that occurs under the cooling condition $\nu^{\rm
ssc}_{\rm cut,r}<\nu_{\rm LAT}$.

##### High-energy events.

A 130.6 MeV photon, the first of several high-energy photons, was detected
2.21 s after the GBM trigger. There was a broad range of photon energies in
this burst, with 339 having energies larger than 100 MeV, 31 having energies
greater than 1 GeV, and 2 having energies greater than 10 GeV. About 24.84 s
from the GBM trigger, the LAT data recorded a photon with a peak energy of
19.46 GeV.

##### Multiwavelength afterglow analysis

We use the best-fit spectral indices presented in 2FLGC ($\beta_{\rm
LAT}=1.14\pm 0.05$; Ajello et al., 2019) and Swift/XRT repository ($\beta_{\rm
X}=0.98^{+0.15}_{-0.14}$)171717https://www.swift.ac.uk/xrt_spectra/00020113/
to evaluate the LAT and X-ray data, respectively. Using the afterglow
observations of GROND and Swift / XRT in three different periods; $\approx
T_{0}$ \+ 84 ks, $\approx T_{0}$ \+ 290 ks and $\approx T_{0}$ \+ 1.3 Ms, Rau
et al. (2010) reported spectral indexes of $\beta_{\rm
Opt}=1.02^{+0.03}_{-0.02}$, $1.05^{+0.04}_{-0.02}$ and $1.04^{+0.08}_{-0.06}$,
respectively. Given the best-fit values of the temporal and spectral indices
of LAT, X-ray, and optical observations, the observed fluxes evolve as
$F_{\rm\nu,L}\propto t^{-1.05\pm 0.22}\,\nu^{-1.14\pm 0.05}$,
$F_{\rm\nu,X}\propto t^{-1.41\pm 0.03}\,\nu^{-0.98^{+0.15}_{-0.14}}$, and
$F_{\rm\nu,Opt}\propto t^{-1.34\pm 0.06}\,\nu^{-1.04^{+0.08}_{-0.06}}$,
respectively. The fact that the temporal (spectral) indexes for the optical
and X-ray observations are larger (lower) than the LAT data suggests that the
closure relations of the synchrotron FS model evolve in a slow cooling regime
through a wind-like medium for $p\approx 2.3\pm 0.3$. It should be noted that
the spectral and temporal indices of X-ray and optical observations are
compatible with each other, and therefore the synchrotron closure relations
evolve under condition $\nu^{\rm syn}_{\rm m,f}<\nu_{\rm Opt}<\nu_{\rm
X}<\nu^{\rm syn}_{\rm c,f}$, and the LAT observations evolve under condition
$\nu^{\rm syn}_{\rm c,f}<\nu_{\rm LAT}$.

####  3.3.5 GRB 110731A

We find that as short-lasting emission is present throughout the main episode,
RS must occur in the thick-shell regime ($t_{\rm x}<T_{90}$), as shown by the
best fits of the onset time and the temporal break. By comparing the best-fit
values of the temporal indices before and after the break, we can deduce in
which cooling conditions of the SSC light curve the LAT frequency evolves. For
$t<t_{\rm x}$, $\nu_{\rm LAT}$ evolves as $\nu^{\rm ssc}_{\rm c,r}<\nu_{\rm
LAT}<\nu^{\rm ssc}_{\rm m,r}$ when the onset time and the temporal break are
not close to each other, and otherwise $\nu_{\rm LAT}$ could evolve $\nu_{\rm
LAT}<\nu^{\rm ssc}_{\rm c,r}$ or $\nu^{\rm ssc}_{\rm c,r}<\nu_{\rm LAT}$. For
$t_{\rm x}<t$, $\nu_{\rm LAT}$ could evolve under the cooling conditions
$\nu^{\rm ssc}_{\rm m,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm c,r}$ for $p\approx
2.1$. The decay temporal index, $\beta+2$, agrees with the high-latitude
emission, which is interesting to note.

##### High-energy events.

The first high-energy photon, with a measured energy of 817.1 MeV, was
observed 3.19 s after the GBM trigger. A large variety of photon energies were
produced during this burst; 40 photons were more than 100 MeV and 4 more than
1 GeV. The highest energy photon recorded by the LAT occurred 1.93 s after the
GBM trigger and had a value of 8.27 GeV.

##### Multiwavelength afterglow analysis

We use the best-fit spectral indices presented in 2FLGC ($\beta_{\rm
LAT}=1.3\pm 0.2$; Ajello et al., 2019) to analyzed the LAT data. Ackermann et
al. (2013b) analysed the X-ray and UV/optical afterglow observations of GRB
110731A. After modelling the broadband SED, the authors reported at 550 s the
spectral indexes of $\beta_{\rm X}=0.95^{+0.07}_{-0.09}$ and $\beta_{\rm
Opt}=0.45^{+0.07}_{-0.09}$ for X-ray and UV/optical observations,
respectively. Given the LAT analysis and the best-fit values of the temporal
and spectral indexes of LAT, late X-ray and optical observations, the closure
relations are $F_{\rm\nu,L}\propto t^{-1.14\pm 0.23}\,\nu^{-1.3\pm 0.2}$,
$F_{\rm\nu,X}\propto t^{-1.30^{+0.07}_{-0.05}}\,\nu^{-0.78^{+0.15}_{-0.13}}$,
and $F_{\rm\nu,Opt}\propto t^{-1.38\pm 0.09}\,\nu^{-0.45^{+0.07}_{-0.09}}$,
respectively. The fact that the temporal (spectral) indexes for the optical
and X-ray observations are larger (lower) than the LAT data suggests that the
closure relations of the synchrotron FS model evolve in a slow cooling regime
through a wind-like medium for $p\approx 2.3\pm 0.3$. It should be noted that
the spectral and temporal indices of X-ray and optical observations are
compatible with each other and, therefore, the synchrotron closure relations
evolve under condition $\nu^{\rm syn}_{\rm m,f}<\nu_{\rm Opt}<\nu_{\rm
X}<\nu^{\rm syn}_{\rm c,f}$, and the LAT observations evolve under condition
$\nu^{\rm syn}_{\rm c,f}<\nu_{\rm LAT}$.

####  3.3.6 GRB 130427A

We find that as short-lasting emission is present throughout the main episode,
RS must occur in the thick-shell regime ($t_{\rm x}<T_{90}$), as shown by the
best fits of the onset time and the temporal break. By comparing the best-fit
values of the temporal indices before and after the break, we can deduce in
which cooling conditions of the SSC light curve the LAT frequency evolves. For
$t<t_{\rm x}$, the Fermi/LAT band should evolve under the condition $\nu^{\rm
ssc}_{\rm c,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm m,r}$ when the onset time is
near the temporal break. On the other hand, the Fermi/LAT band could evolve
$\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm c,r}$ or $\nu^{\rm ssc}_{\rm c,r}<\nu_{\rm
LAT}$. For $t_{\rm x}<t$, the Fermi / LAT band could evolve under the cooling
condition $\nu^{\rm ssc}_{\rm m,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm c,r}$ for
$p\approx 2.1$. We emphasise that the temporal index of decay agrees with the
high-latitude emission and also with a contribution from the FS region.

##### High-energy events.

With a measured energy of 189.8 MeV, the first high-energy photon was observed
0.14 s after the GBM trigger. Four hundred sixty-two photons in this burst had
energies greater than 100 MeV, 78 were greater than 1 GeV, and 15 were greater
than 10 GeV, indicating an extensive range of energies. The most energetic
photon seen in the LAT data was clocked in at 94.11 GeV, 243.1 s after the GBM
trigger.

##### Multiwavelength afterglow analysis

According to the LAT and X-ray observations, we consider the best-fit spectral
indexes reported in 2FLGC ($\beta_{\rm LAT}=0.99\pm 0.04$; Ajello et al.,
2019) and Swift repository ($\beta_{\rm X}=0.495\pm 0.016$ for
$T_{0}+1196\,{\rm s}$ and $0.69\pm 0.08$ for $T_{0}+44907\,{\rm
s}$)181818https://www.swift.ac.uk/xrt_spectra/00020113/, respectively. Using
UV/optical/IR afterglow observations collected by several ground telescopes,
Perley et al. (2014) reported a spectral index evolution between $\beta_{\rm
Opt}=0.42^{+0.06}_{-0.06}$ and $0.48\pm 0.04$ for $t<$1 day. Given the best-
fit values of the temporal and spectral indices of LAT, X-ray, and optical
observations, the observed fluxes evolve as $F_{\rm\nu,L}\propto t^{-1.02\pm
0.2}\,\nu^{-0.99\pm 0.04}$, $F_{\rm\nu,X}\propto t^{-1.30\pm
0.05}\,\nu^{-0.495^{+0.015}_{-0.016}}$, and $F_{\rm\nu,Opt}\propto t^{-1.33\pm
0.04}\,\nu^{-0.48\pm 0.04}$, respectively. The fact that the temporal
(spectral) index for optical and X-ray observations is larger (lower) than the
LAT data suggests that the closure relations of the synchrotron FS model
evolve in a slow cooling regime through a wind-like medium for $p\approx
2.3\pm 0.3$. It should be noted that the spectral and temporal indices of
X-ray and optical observations are compatible with each other, and therefore
the synchrotron closure relations evolve under the condition $\nu^{\rm
syn}_{\rm m,f}<\nu_{\rm Opt}<\nu_{\rm X}<\nu^{\rm syn}_{\rm c,f}<\nu_{\rm
LAT}$.

####  3.3.7 GRB 160602B

We see that the short-lived emission is present during the main episode
($t_{\rm x}<T_{90}$), so the RS evolves in the thick-shell regime. According
to the best-fitting value of the starting time, the onset of the afterglow
occurred at the end of the prompt episode. We can infer the evolution of the
LAT frequency in the cooling conditions of the SSC light curve by comparing
the best-fit values of the temporal indices before and after the break. For
$t<t_{\rm x}$, the Fermi/LAT band should evolve under the condition $\nu^{\rm
ssc}_{\rm c,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm m,r}$ when the onset time is
close to the temporal break. On the other hand, the Fermi/LAT band could
evolve under conditions $\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm c,r}$ or $\nu^{\rm
ssc}_{\rm c,r}<\nu_{\rm LAT}$. For $t_{\rm x}<t$, $\nu_{\rm LAT}$ could evolve
under the cooling condition $\nu^{\rm ssc}_{\rm m,r}<\nu_{\rm LAT}<\nu^{\rm
ssc}_{\rm c,r}$ for $p\approx 2.6$. We note that the temporal index of decay
agrees with the high-latitude emission $\nu^{\rm ssc}_{\rm c,r}<\nu_{\rm
LAT}$.

##### High-energy events.

The first high-energy photon was detected 25.6 s after the BAT trigger with an
energy of 160.2 MeV. The energy range of the photons in this burst was very
broad, with 255 photons over 100 MeV, 21 exceeding 1 GeV, and one over 10 GeV.
The highest energy photon was observed 346.2 s after the BAT trigger by the
LAT instrument and had an energy of 15.3 GeV.

##### Multiwavelength afterglow analysis

We use the best-fit temporal and spectral index exhibited in Swift/XRT
repository ($\alpha_{\rm X}=1.27\pm 0.30$ and $\beta_{\rm
X}=0.66^{+0.21}_{-0.14}$)191919https://www.swift.ac.uk/xrt_spectra/00020113/
to evaluate the X-ray data. After modeling the broadband SED, Fraija et al.
(2017a) and Troja et al. (2017) reported spectral indexes of $\beta_{\rm
Opt}=0.71\pm 0.12$ and $\beta_{\rm Opt}=0.50\pm 0.05$, after $\approx$ 3 and 8
hours for optical- NIR observations, respectively, after the trigger time. The
late afterglow (the X-ray and optical fluxes) evolves as $F_{\rm\nu,X}\propto
t^{-1.27\pm 0.30}\,\nu^{-0.66^{+0.21}_{-0.14}}$ and $F_{\rm\nu,Opt}\propto
t^{-0.921\pm 0.163}\,\nu^{-0.71\pm 0.12}$, respectively. Due to the fact that
the temporal and spectral indexes for the late optical and X-ray observations
are compatible with each other, the synchrotron closure relations can evolve
through a wind-like medium for $p\approx 2.3\pm 0.3$ and under the condition
$\nu^{\rm syn}_{\rm m,f}<\nu_{\rm Opt}<\nu_{\rm X}<\nu^{\rm syn}_{\rm c,f}$.

####  3.3.8 GRB 180720B

This component occurs during the prompt emission because of the best-fit
values of the starting time and the temporal break. Therefore, the RS evolves
in the thick-shell regime ($t_{\rm x}<T_{90}$). We can infer the evolution of
the LAT frequency in the cooling conditions of the SSC light curve by
comparing the best-fit values of the temporal indices before and after the
break. For $t<t_{\rm x}$, the Fermi/LAT band should evolve under the condition
$\nu^{\rm ssc}_{\rm c,r}<\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm m,r}$ when the onset
time is close to the temporal break. On the other hand, the Fermi/LAT band
could evolve $\nu_{\rm LAT}<\nu^{\rm ssc}_{\rm c,r}$ or $\nu^{\rm ssc}_{\rm
c,r}<\nu_{\rm LAT}$. For $t_{\rm x}<t$, the Fermi / LAT band could evolve
under the cooling condition $\nu^{\rm ssc}_{\rm m,r}<\nu_{\rm LAT}<\nu^{\rm
ssc}_{\rm c,r}$ for $p\approx 2.6$. The decay temporal index, $\beta+2$,
agrees with the high-latitude emission, which is interesting to note. This
behavior could evolve with high-energy emission from the FS.

##### High-energy events.

The first high-energy photon was detected 12.5 s after the BAT trigger with a
measured energy of 175.2 MeV. The energy range of the photons in this burst
was extensive, with 129 photons over 100 MeV and 8 exceeding 1 GeV. The
highest energy photon was observed 142.4 s after the BAT trigger by the LAT
instrument and had an energy of 4.9 GeV.

##### Multiwavelength afterglow analysis

According to the LAT and X-ray observations, we consider the best-fit spectral
indexes reported in 2FLGC ($\beta_{\rm LAT}=1.23\pm 0.10$; Ajello et al.,
2019) and Swift repository ($\beta_{\rm
X}=0.697^{+0.010}_{-0.010}$)202020https://www.swift.ac.uk/xrt_spectra/00848890/,
respectively. Using optical observations in the R band collected by several
ground telescopes, Fraija et al. (2019c) reported a spectral index evolution
between $\beta_{\rm Opt}=0.68\pm 0.06$ and $0.70\pm 0.05$ corresponding to the
intervals $\sim 2.0\times 10^{2}-2.5\times 10^{3}$ and $\sim 2.5\times
10^{3}-2.6\times 10^{5}$, respectively. Given the best-fit values of the
temporal and spectral indexes of the X-ray and optical observations, while the
X-ray flux varies from $F_{\rm\nu,X}\propto t^{-0.79\pm 0.08}\,\nu^{-0.86\pm
0.03}$ to $F_{\rm\nu,X}\propto t^{-1.26\pm
0.06}\,\nu^{-0.697^{+0.010}_{-0.010}}$, the optical flux evolves
$F_{\rm\nu,Opt}\propto t^{-1.22\pm 0.02}\,\nu^{-0.68\pm 0.06}$ throughout the
interval. Before $t\lesssim 2.5\times 10^{3}\,{\rm s}$, the temporal
(spectral) index for optical observations is larger (lower) than X-ray data,
suggesting that the closure relations of the synchrotron FS model evolve in
slow-cooling regime through a wind-like medium for $p\approx 2.0\pm 0.25$. In
this case, the synchrotron closure relations evolve under condition $\nu^{\rm
syn}_{\rm m,f}<\nu_{\rm Opt}<\nu^{\rm syn}_{\rm c,f}<\nu_{\rm X}$. After
$2.5\times 10^{3}\,{\rm s}\lesssim t$, the spectral and temporal indices of
X-ray and optical observations are compatible with each other, and therefore
the synchrotron closure relations evolve under condition $\nu^{\rm syn}_{\rm
m,f}<\nu_{\rm Opt}<\nu_{\rm X}<\nu^{\rm syn}_{\rm c,f}$. We conclude that the
temporal break at $\simeq 2.5\times 10^{3}\,{\rm s}$ to the passage of the
forward shock cooling break through the Swift/XRT band.

###  3.4 Results and Discussion

All panels in Figure 3 show the LAT, X-ray and optical observations of GRB
080916C, GRB 090323, GRB 090902B, GRB 090926A, GRB 110731A, GRB 130427A, GRB
160625B and GRB 180720B with the best-fit curve of the short- (dotted) and
long- (dotted-dashed) lasting components. Total emission is displayed with a
solid line. We used Markov-Chain Monte Carlo (MCMC) simulations with the eight
parameters used for the complete sample of GRBs to find the best-fit values
that describe the multiwavelength observations with the SSC and synchrotron
from the reverse and forward shock models. To represent all the data in this
case, a total of 15,900 samples and 4,400 tuning steps are used. Figures 4 \-
11 show the best-fit values and the median of the posterior distributions of
the parameters. In Table 3, the best-fit values are shown in red, and the
median of the posterior distributions is presented. Table 4 shows the SSC
spectral breaks from reverse shocks, which are estimated with the best-fit
values reported in Table 3. The SSC spectral breaks from the RS region are
estimated for GRB 080916C, 090323, 090902B, 090926A, 110731A, 130427A,
160625B, and 180720B at $4\,(30)\,{\rm s}$, $100\,(315)\,{\rm s}$,
$3\,(16)\,{\rm s}$, $6\,(20)\,{\rm s}$, $6\,(20)\,{\rm s}$, $6\,(32)\,{\rm
s}$, $100\,(316)\,{\rm s}$ and $63\,(200)\,{\rm s}$ for $t<t_{\rm x}(t_{\rm
x}<t)$, respectively. The synchrotron spectral breaks from the FS region are
estimated to be $10^{5}\,{\rm s}$. Figure 12 exhibits all photons with
energies $>100$ MeV and probabilities $>90$% of being associated with each
burst. Additionally, we show in red lines the maximum photon energies released
by the SSC (dotted) and synchrotron (dashed) from the reverse and forward
afterglow models, respectively, estimated with the best-fit values reported in
Table 3.

####  3.4.1 Microphysical parameters

#####  $\varepsilon_{\rm B}$-parameter

The best-fit values of the magnetic microphysical parameter from the RS lie in
the range of $0.01\lesssim\varepsilon_{\rm B_{r}}\lesssim 0.1$. Taking into
account the values of the microphysical parameters, the magnetisation
parameter is estimated to be in the range $0.01\lesssim\sigma\lesssim 0.1$,
indicating that the ejecta are moderately magnetic and as a result the RS is
expected to be successful. If this was not the case, the acceleration of the
particles in RS would be ineffective and RS would not have occurred (Zhang &
Kobayashi, 2005; Fan et al., 2004). The microphysical parameter of the
magnetic field in the FS and RS regions is different. The ratio of these
parameters lie in the range of $1\lesssim{\mathcal{R}_{B}}\lesssim
20$.212121${\mathcal{R}_{B}}$ is defined as $\varepsilon_{\rm
B_{r}}/\varepsilon_{\rm B_{f}}$. It has been suggested that the high-energy
emission seen in the brightest LAT-detected bursts could be explained by
Poynting flux-dominated models with arbitrary magnetization (e.g., see Uhm &
Zhang, 2014; Zhang & Yan, 2011). The values of the magnetisation parameter
suggest that a substantial amount of Poynting flux was released by the ejecta
during the prompt emission phase, with the internal collision-induced magnetic
reconnection and turbulence (ICMART) event being the most likely mechanism
that accounts for this behaviour (Zhang & Yan, 2011).

#####  $\varepsilon_{\rm e}$-parameter

The best-fit values of the $\varepsilon_{\rm e}$ microphysical parameter in
the forward- and RS model are presented in the 7th and 8th rows of Table 3,
respectively. For the forward shock, we find $2.8\times
10^{-3}\lesssim\varepsilon_{\rm e_{f}}\lesssim 0.12$, while for the RS we have
$0.03\lesssim\varepsilon_{\rm e_{r}}\lesssim 0.13$. As a general rule, we
notice that $\epsilon_{\rm e_{f}}<\epsilon_{\rm e_{r}}$. Moreover, there is a
large variation in the ratio of both quantities between each burst. For
example, in the case of GRB 110731A we find the largest disparity
$\mathcal{R}_{e}\equiv\epsilon_{\rm e_{r}}/\epsilon_{\rm e_{f}}\approx 46$,
while for GRB 160625B $\mathcal{R}_{e}\approx 0.7$, so the variation can occur
over several orders of magnitude. It should be noted that an RS is suppressed
for $\mathcal{R}_{e}\ll 1$ (Fraija et al., 2020).

Although it is usually assumed that the electron microphysical parameter in
both the RS and the FS has a similar value (Zhang & Kobayashi, 2005; Gao et
al., 2015), the possibility of them being different has been explored in
previous work, such as the one by Panaitescu & Kumar (2004), in which the
authors noted that in the case of GRB 990123 for a $k=0$ model $\epsilon_{\rm
e_{r}}<0.1\epsilon_{\rm e_{f}}$, which is not in agreement with our results.
However, given the current uncertainties on the acceleration process that
energises electrons, it proves difficult to determine a relation between this
parameter in both regions.

####  3.4.2 The post jet-break decay phase

After about $T+1$ day (Racusin et al., 2009), during the post jet break decay
phase, the multi-wavelength light curves would evolve as $F^{\rm
syn}_{\nu}\propto t^{-p}$, for $\nu^{\rm syn}_{\rm m,f}<\nu<\nu^{\rm syn}_{\rm
c,f}$ or ${\rm max}\\{\nu^{\rm syn}_{\rm m,f},\nu^{\rm syn}_{\rm c,f}\\}<\nu$
(see e.g. Fraija et al., 2022), which are different from the temporal decay
indexes found for all bursts (Pereyra et al., 2022; Becerra et al., 2019c),
except GRB 160625B (Fraija et al., 2017c). This means that, apart from GRB
160625B, they were likely released from a broad outflow with a large half-
opening angle, as shown by multi-wavelength measurements which show no
indication of late steep decays. Based on the best-fit values of the
circumburst density and the kinetic equivalent energies, we find that the jet
opening angles become $\gtrsim 10^{\circ}$, and for GRB 160625B, the value of
the jet opening angle is $\theta_{j}\approx 2.3^{\circ}$, which is in the
typical interval for $\theta_{j}=2^{\circ}-10^{\circ}$ (Bloom et al., 2001).

####  3.4.3 Efficiency of equivalent kinetic energy

The efficiency provides crucial information on the gamma-ray emitting process.
The best fit values of the equivalent kinetic energies
$10^{53.96^{+1.01}_{-1.05}}\leq E\leq 10^{54.85^{+1.05}_{-1.02}}\,{\rm erg}$
and the isotropic energies in gamma-rays reported by the GBM instrument during
the prompt episode in the range of $(0.39\pm 0.09)\times 10^{52}\leq
E_{\rm\gamma,iso}\leq(1.7\pm 0.1)\times 10^{54}\,{\rm erg}$ (Ajello et al.,
2019) lead to kinetic efficiencies in the range of $0.01\lesssim\eta\lesssim
0.36$, which are typical compared to those values reported in the literature
(Guetta et al., 2001; Zhang et al., 2007; Kumar & Zhang, 2015), and a kinetic
efficiency of $\eta\approx 0.02$ for GRB 180720B (Abdalla et al., 2019; Fraija
et al., 2019c), which is very low. The atypical value of efficiency for GRB
180720B was calculated considering the isotropic energy reported by GBM during
the interval 4.4-53.2 s. If we have considered the isotropic energy reported
by the LAT instrument throughout the interval 11.8-625.0 s, the kinetic
efficiency would have been $\eta\approx 0.002$. Similarly, the second-lowest
efficiency, which corresponds to $\eta=0.01$. If we had considered the
isotropic energy reported by the LAT in the interval 0.1 - 34366.2 s instead
of the GBM instrument, the kinetic efficiency would be $\eta\approx 0.06$.

####  3.4.4 The profile of the circumburst environment

The best-fit values of the wind parameter lie in the range of $10^{-3}\lesssim
A_{\rm k}\lesssim 0.02$, typical for GRBs identified as powerful bursts
(Ackermann et al., 2013b; Perley et al., 2014; Vestrand et al., 2014; Fraija
et al., 2012; Racusin et al., 2008; Fraija et al., 2017c; Becerra et al.,
2017). Recently, Dainotti et al. (2023) considered the GRBs reported in the
2FLGC and analysed the evolution of the relationship between the spectral and
temporal indices using closure relations in a stratified environment. They
found that the afterglow model without energy injection is preferred over that
with energy injection with a clear preference for the cooling condition $\nu>$
max{$\nu_{c},\nu_{m}$}, as derived (see Table 4). Furthermore, the authors
reported that the density profiles $r^{-k}$ with $k=2$ have a higher rate of
occurrence.

The best-fit values of the initial bulk Lorentz factor lie in the range
$290\lesssim\Gamma\lesssim 10^{3}$, which is similar to the values found in
other bursts detected by the LAT instrument and those expected in numerical
simulations (Tchekhovskoy et al., 2008). Since our GRB sample displayed the
most energetic photons, the values of the bulk Lorentz factor are predicted to
fall within the same range as the bursts observed by the LAT that are the
brightest (Ackermann et al., 2011; Veres & Mészáros, 2012; Ackermann et al.,
2013b; Abdo et al., 2009b; Ackermann & et al., 2010; Ackermann et al., 2014;
Fraija et al., 2019a, c), as found. Given the best-fit values, the critical
Lorentz factors are consistent with the thick-shell scenario.

####  3.4.5 Fraction of the shock-accelerated electrons

In the standard GRB afterglow model, it is assumed that the fraction of
electrons accelerated by shock is $\zeta_{e}\sim 1$. However, the validity of
this assumption has been questioned by several authors. For example, Bykov &
Meszaros (1996) considered the increase in the electron Lorentz factor taking
a value $\zeta_{e}\sim 10^{-3}$, which would lead to a change in the
synchrotron spectrum. On the other hand, Eichler & Waxman (2005) concluded
that, while observations suggest that $\zeta_{e}\sim 1$, the range
$m_{e}/m_{p}\approx 5\times 10^{-4}\leq\zeta_{e}\leq 1$ cannot be ruled out
because models with different values of $\zeta_{e}$ in the aforementioned
range lead to very similar predictions.

In the last row of Table 3, we present the parameter values of $\zeta_{e}$
that we obtained for our sample of GRBs through our MCMC estimation. We notice
that our lowest value is for GRB 080916C with $\zeta_{e}\approx 0.02$, while
the largest corresponds to GRB 180720B with $\zeta_{e}\approx 0.62$. This
means that all estimates are within the allowed range of Eichler & Waxman
(2005).

####  3.4.6 The highest energy photons

Figure 12 shows that with the exception of GRB 110731A, the SSC and
synchrotron models radiated in the RS and FS regions cannot explain the
highest-energy photons exhibited. Therefore, a different process needs to be
invoked to interpret these photons. The number of photons ($N_{\gamma}$) with
energy $h\nu$ that reach the Fermi-LAT instrument during a specific time
interval ($\Delta t$) can be estimated considering the effective area of LAT
($A$) and the observed flux at the determined time. The number of photons is

$\displaystyle N_{\gamma}\sim 1\,{\rm
ph}\left(\frac{h\nu}{10\,{\rm~{}GeV}}\right)\left(\frac{10\,{\rm s}}{\Delta
t}\right)\left(\frac{F^{\rm
ssc}_{\nu,j}}{10^{-9}\,{\rm\frac{erg}{cm^{2}\,s}}}\right)\left(\frac{10^{4}\,{\rm
cm^{2}}}{A}\right)\,.$ (37)

In order to estimate the number of photons we discuss hadronic and SSC
scenarios. In the hadronic scenarios, high-energy gamma-ray emission has been
interpreted via photo-hadronic interactions; ultrarelativistic protons
accelerated in the jet with internal synchrotron photons (Asano et al., 2009;
Dermer et al., 2000), inelastic proton-neutron collisions (Mészáros & Rees,
2000), and relativistic neutrons with seed photons coming from the outflow
(Dermer & Atoyan, 2004; Alvarez-Muñiz et al., 2004). Even though GRBs are
among the most plausible candidates to accelerate cosmic rays up to ultra-high
energies ($\gtrsim 10^{18}$ eV; Waxman, 1995; Vietri, 1995) and thus,
potential candidates for neutrino detection, the IceCube collaboration
reported no coincidences between neutrinos and GRBs after analyzing years of
data (Abbasi et al., 2022, 2012; Aartsen et al., 2016, 2015). Because of this,
we rule out hadronic models as an explanation for the observed properties of
GRBs and conclude that the number of hadrons is too small for hadronic
interactions to efficiently generate observable gamma-ray signals in GRBs. On
the other hand, the SSC scenario has been successfully applied to interpret
the highest-energy photons (e.g., see Fraija et al., 2019c; Wang et al., 2019;
Zhang, 2019; Fraija et al., 2019b; Fraija et al., 2021; Fraija et al., 2017b).
In this scenario, the same electron population can up-scatter synchrotron
photons up to higher energies as $h\nu^{\rm ssc}_{\rm i,f}\sim\gamma^{2}_{\rm
i,f}h\nu^{\rm syn}_{\rm i,f}$, reaching a maximum flux of $F^{\rm ssc}_{\rm
max,f}\sim\,\frac{4\sigma_{T}nr}{3g(p)}\,F^{\rm syn}_{\rm max,f}$ with
$r\simeq 4c\Gamma^{2}t/(1+z)$. The spectral breaks and the maximum flux for
SSC emission in the forward shock can be expressed as

$\displaystyle h\nu^{\rm ssc}_{\rm m,f}$ $\displaystyle\simeq$ $\displaystyle
4.3\times
10^{2}\,{\rm~{}MeV}\,\left(\frac{1+z}{2}\right)^{\frac{10-3k}{2(4-k)}}\,g^{4}(2.2)\,\zeta_{e}^{-4}\,\varepsilon_{\rm
e_{f},-1}^{4}\,\varepsilon_{\rm
B_{f},-2}^{\frac{1}{2}}\,A^{-\frac{1}{4-k}}_{\rm
k,-1}\,E_{53}^{\frac{6-k}{2(4-k)}}t^{-\frac{18-5k}{2(4-k)}}_{2},$ (38)
$\displaystyle h\nu^{\rm ssc}_{\rm c,f}$ $\displaystyle\simeq$ $\displaystyle
4.6\times 10^{2}{\rm
eV}\left(\frac{1+z}{2}\right)^{-\frac{3(2+k)}{2(4-k)}}\left(\frac{1+Y_{\rm
f}}{2}\right)^{-4}\varepsilon_{\rm
B_{f},-2}^{-\frac{7}{2}}A^{-\frac{9}{4-k}}_{\rm
k,-1}\,E_{53}^{-\frac{10-7k}{2(4-k)}}\,t^{-\frac{2-5k}{2(4-k)}}_{2}\,,$ (39)
$\displaystyle F^{\rm ssc}_{\rm max,f}$ $\displaystyle\simeq$ $\displaystyle
8.1\times 10^{-3}\,{\rm
mJy}\left(\frac{1+z}{2}\right)^{\frac{14-k}{2(4-k)}}\,g^{-1}(2.2)\,\zeta_{e}\,\varepsilon_{\rm
B_{f},-2}^{\frac{1}{2}}\,A^{\frac{5}{4-k}}_{\rm k,-1}\,d^{-2}_{\rm
z,28.3}\,E_{53}^{\frac{5(2-k)}{2(4-k)}}\,t^{\frac{2-3k}{2(4-k)}}_{2}\,.$ (40)

In the fast- and slow-cooling regime the SSC light curve are given by (Sari &
Esin, 2001)

$\displaystyle F^{\rm
ssc}_{\nu,f}\propto\begin{cases}t^{\frac{2-k}{4(4-k)}}\nu^{-\frac{1}{2}},\hskip
79.6678pt\nu^{\rm ssc}_{\rm c,f}<\nu<\nu^{\rm ssc}_{\rm m,f},\hskip
7.11317pt\cr
t^{\frac{2(10-3k)+p(5k-18)}{4(4-k)}}\,\nu^{-\frac{p}{2}},\,\,\,\,\hskip
31.2982pt\nu^{\rm ssc}_{\rm m,f}<\nu\,,\cr\end{cases}$ (41)

and

$\displaystyle F^{\rm
ssc}_{\nu,f}\propto\begin{cases}t^{\frac{11(2-k)+p(5k-18)}{4(4-k)}}\nu^{-\frac{p-1}{2}},\hskip
34.14322pt\nu^{\rm ssc}_{\rm m,f}<\nu<\nu^{\rm ssc}_{\rm c,f},\hskip
7.11317pt\cr
t^{\frac{2(10-3k)+p(5k-18)}{4(4-k)}}\,\nu^{-\frac{p}{2}},\,\,\,\,\hskip
31.2982pt\nu^{\rm ssc}_{\rm c,f}<\nu\,,\cr\end{cases}$ (42)

respectively. For $\nu^{\rm ssc}_{\rm c,f}<\nu^{\rm ssc}_{\rm m,f}$ and
$\nu^{\rm ssc}_{\rm m,f}<\nu^{\rm ssc}_{\rm c,f}$, the spectral breaks due to
the KN effect are

$\displaystyle h\nu^{\rm ssc}_{\rm KN,m,f}$ $\displaystyle\simeq$
$\displaystyle 1.2\times
10^{2}\,{\rm~{}GeV}\left(\frac{1+z}{2}\right)^{-\frac{1}{4-k}}g(2.2)\zeta_{e}^{-1}\varepsilon_{\rm
e_{f},-1}\,A^{-\frac{1}{4-k}}_{\rm
k,-1}\,E^{\frac{1}{4-k}}_{53}\,t^{-\frac{3-k}{4-k}}_{2}\,,$ (43)

$\displaystyle h\nu^{\rm ssc}_{\rm KN,c,f}$ $\displaystyle\simeq$
$\displaystyle
3.8\,{\rm~{}GeV}\left(\frac{1+z}{2}\right)^{-\frac{3}{4-k}}\left(\frac{1+Y_{\rm
f}}{2}\right)^{-1}\varepsilon_{\rm B_{f},-2}^{-1}\,A^{-\frac{3}{4-k}}_{\rm
k,-1}\,E^{-\frac{1-k}{4-k}}_{53}\,t^{-\frac{1-k}{4-k}}_{2}\,.$ (44)

The maximum energy radiated by the SSC process is estimated by $h\nu^{\rm
ssc}_{\rm max,f}\simeq\gamma^{2}_{\rm max,f}(h\nu^{\rm syn}_{\rm max,f})$.

Given the best-fit values listed in Table 3, the number of photons at the
highest energies are: 0.9, 0.2, 1.2, 1.6, 31.2, 0.6 and 1.7 for GRB 080916C,
090323, 090902B, 090926A, 130427A, 160625B and 180720B, respectively.
Therefore, although there is an excess from GRB 130427A due to the spectral
index, the most energetic photons could be explained by the SSC mechanism from
the forward shock.

####  3.4.7 Spectral breaks in the Klein-Nishina regime

The values of the spectral breaks derived from the best fit parameters and
listed in Table 4 show that, with the exception of GRB 090323 and GRB 090926A,
the KN effects are neglected. For GRB 090323, the spectral breaks are in the
order $\nu^{\rm syn}_{\rm c}<h\nu^{\rm syn}_{\rm KN}(\gamma_{*})<\nu^{\rm
syn}_{\rm m}<\nu^{\rm syn}_{\rm KN}(\gamma_{\rm m})$ with $h\nu^{\rm syn}_{\rm
m,r}=1.1\times 10^{3}\,{\rm eV}$, $h\nu^{\rm syn}_{\rm c,r}=8.2\,{\rm eV}$,
$h\nu^{\rm syn}_{KN,m,r}=1.2\times 10^{5}\,{\rm eV}$, $h\nu^{\rm
syn}_{KN,c,r}=6.3\times 10\,{\rm eV}$ and $h\nu^{\rm
syn}_{KN,r}(\gamma_{*})=3.5\times 10^{2}\,{\rm eV}$. In this case, the Compton
parameter was recalculated (for details, see Wang et al., 2010). A similar
procedure was developed for GRB 090926A.

##  4 Summary

We have extended the SSC RS model in a medium of constant density initially
proposed to explain the LAT GeV flare observed in GRB 160509A (Fraija et al.,
2020). In the current model, we have generalised and derived the SSC light
curves from the RS region in the thick and thin shell scenarios for a
stratified environment with a density profile $\propto r^{\rm-k}$ with
$0\leq{\rm k}<3$, including the maximum energy radiated by this process in
both regimes. In order to apply our theoretical model to a sample of eight
bursts (GRB 080916C, 090323, 090902B, 090926A, 110731A, 130427A, 160625B and
180720B) reported in the 2FLGC (Ajello et al., 2019), which exhibited
interesting short-lasting bright peaks, we have obtained the Fermi-LAT light
curves together with the photons with energies $\geq 100$ MeV associated with
each burst. We have used the multiwavelength observations to constrain the
parameters in the model and fitted the LAT light curves of the sample of GRBs
with a joint model which considers synchrotron and SSC emission from both the
FS and the RS through MCMC simulations. We have shown that the emission in the
thick-shell scenario could describe the short-lasting bright peaks exhibited
in our GRB sample. Because the shock crossing time is smaller than the
duration of the prompt emission in the thick-shell regime, the bright peak in
this regime is expected at the beginning of the FS emission. By contrast, if
the shock crossing time is longer than the duration of the prompt emission in
the thin-shell scenario, the bright peak is separated from the prompt emission
and expected during the FS emission.

We show that at the beginning of the LAT observations, the SSC flux from the
RS region is larger than the synchrotron flux from the FS. Therefore, the
closure relations of the synchrotron FS model are not expected when the LAT
light curves are a superposition of the SSC and synchrotron from RS and FS. It
is worth highlighting that, depending on the parameter values, the short-
lasting bright peaks could be hidden by the long-lasting emission.

The best-fit values of the magnetic microphysical parameter for our sample of
bursts were consistent with moderate magnetic ejecta and $\mathcal{R}_{e}\ll
1$, which indicated that the RS was successful. With respect to the electron
microphysical parameter, we noticed that there was a large variation between
the values of the FS and the RS, which suggests that the acceleration
mechanisms are substantially different. We found that in all bursts except GRB
160625B, the radiation was most likely released from a broad outflow with a
large opening angle. We calculated the kinetic efficiency for every burst in
our sample and found that most of the values are consistent with those of
previous studies. However, GRB 180720B presented a value that was far below
the standard. We have calculated the best-fit values of the wind parameter for
our sample of bursts. We have found that all of the results are consistent
with powerful GRBs. This conclusion is further strengthened by the fact that
our sample displayed the most energetic photons, which are also expected from
the most powerful bursts. We tested the assumption that the fraction of
accelerated electrons in the shock is close to unity. We have found that this
is not the case in all the bursts of our sample. However, an accurate
determination of this parameter is difficult, as different values lead to
similar predictions.

Given the best-fit values, we have shown that the first high-energy photons
are consistent with the starting time of the short-lasting emission.
Similarly, we have shown that, unlike GRB 110731A, the highest energy photon
cannot be described neither by the standard synchrotron emision from RS nor
from the FS, and a different process from the ones considered have to be
invoked to interpret these photons. We show that unlike GRB 090323 the
highest-energy photons are consistent with SSC from the FS region. We have
shown that in all but two bursts from our sample, KN effects are neglected.
However, in the case of GRB 090323 and GRB 090926A, they are important, and so
we have recalculated the Compton parameter appropriately in both cases.

## Acknowledgements

NF acknowledges financial support from UNAM-DGAPA-PAPIIT through grant
IN106521. M.G.D. acknowledges funding from the AAS Chretienne Fellowship and
the MINIATURA2 grant. R.L.B. acknowledges support from the CONAHCyT
postdoctoral fellowship and the financial support from UNAM-DGAPA-PAPIIT
through grant IN105921.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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Table 1: The best-fit values for the modeling of the short and long-lasting components of the LAT light curves of our GRB sample using PL functions. Event | LAT | Parameter |  Best-fit value |  $\chi^{2}$/ndf
---|---|---|---|---
GRB 080916C |  | $\alpha_{\rm r_{L},bb}$ |  $-1.05\pm 0.84$ |  $1.22$
| Short-lasting component | $\alpha_{\rm r_{L},ab}$ |  $3.53\pm 0.70$ |
|  |  $T_{\rm a}$ (s) |  $2.40\pm 2.20$ |
|  |  $t_{\rm r,pk}$ (s) |  $5.76\pm 0.84$ |
|  | $\alpha_{L,1}$ |  $0.54\pm 0.27$ |  $1.18$
| Long-lasting component | $\alpha_{L,2}$ |  $1.47\pm 0.10$ |
|  |  $t_{\rm f_{L},br}$ (s) |  $22.08\pm 0.02$ |
|  | $\alpha_{L}$ |  $1.21\pm 0.06$ |  $1.12$
GRB 090323 |  | $\alpha_{\rm r_{L},bb}$ |  $-1.11\pm 0.74$ |  $0.96$
| Short-lasting component | $\alpha_{\rm r_{L},ab}$ |  $4.00\pm 1.90$ |
|  |  $T_{\rm a}$ (s) |  $55.19\pm 50.25$ |
|  |  $t_{\rm r,pk}$ (s) |  $106.60\pm 72.82$ |
|  | $\alpha_{L,1}$ |  $1.84\pm 1.12$ |  $0.86$
| Long-lasting component | $\alpha_{L,2}$ |  $1.00\pm 0.70$ |
|  |  $t_{\rm f_{L},br}$ (s) |  $576.10\pm 798.60$ |
|  | $\alpha_{L}$ |  $1.26\pm 0.40$ |  $0.67$
GRB 090902B |  | $\alpha_{\rm r_{L},bb}$ |  $-2.00\pm 1.86$ |  $0.92$
| Short-lasting component | $\alpha_{\rm r_{L},ab}$ |  $1.77\pm 0.39$ |
|  |  $T_{\rm a}$ (s) |  $4.52\pm 1.22$ |
|  |  $t_{\rm r,pk}$ (s) |  $8.74\pm 0.52$ |
|  | $\alpha_{L,1}$ |  $1.90\pm 0.2$ |  $1.45$
| Long-lasting component | $\alpha_{L,2}$ |  $1.30\pm 0.01$ |
|  |  $t_{\rm f_{L},br}$ (s) |  $100\pm 0.08$ |
|  | $\alpha_{L}$ |  $1.35\pm 0.06$ |  $1.32$
GRB 090926A |  | $\alpha_{\rm r_{L},bb}$ |  $-1.37\pm 0.96$ |  $0.92$
| Short-lasting component | $\alpha_{\rm r_{L},ab}$ |  $4.00\pm 1.82$ |
|  |  $T_{\rm a}$ (s) |  $3.00\pm 1.81$ |
|  |  $t_{\rm r,pk}$ (s) |  $6.35\pm 0.23$ |
|  | $\alpha_{L,1}$ |  $1.65\pm 0.10$ |  $1.43$
| Long-lasting component | $\alpha_{L,2}$ |  $1.08\pm 0.10$ |
|  |  $t_{\rm f_{L},br}$ (s) |  $106.5\pm 0.08$ |
|  | $\alpha_{L}$ |  $1.05\pm 0.22$ |  $1.38$
GRB 110731A |  | $\alpha_{\rm r_{L},bb}$ |  $-1.50\pm 1.05$ |  $1.34$
| Short-lasting component | $\alpha_{\rm r_{L},ab}$ |  $2.67\pm 0.062$ |
|  |  $T_{\rm a}$ (s) |  $2.63\pm 1.75$ |
|  |  $t_{\rm r,pk}$ (s) |  $5.62\pm 0.03$ |
|  | $\alpha_{L,1}$ |  $1.57\pm 0.40$ |  $0.79$
| Long-lasting component | $\alpha_{L,2}$ |  $2.04\pm 1.34$ |
|  |  $t_{\rm L,br}$ (s) |  $71.64\pm 0.54$ |
|  | $\alpha_{L}$ |  $1.14\pm 0.23$ |  $1.76$
GRB 130427A |  | $\alpha_{\rm r_{L},bb}$ |  $-0.71\pm 0.28$ |  $1.07$
| Short-lasting component | $\alpha_{\rm r_{L},ab}$ |  $2.00\pm 0.60$ |
|  |  $T_{\rm a}$ (s) |  $1.01\pm 0.76$ |
|  |  $t_{\rm r,pk}$ (s) |  $19.78\pm 0.01$ |
|  | $\alpha_{L,1}$ |  $0.81\pm 0.05$ |  $0.89$
| Long-lasting component | $\alpha_{L,2}$ |  $1.67\pm 0.52$ |
|  |  $t_{\rm f_{L},br}$ (s) |  $1182.00\pm 1071.00$ |
|  | $\alpha_{L}$ |  $1.27\pm 0.20$ |  $1.08$
GRB 160625B |  | $\alpha_{\rm r_{L},bb}$ |  $-0.94\pm 0.45$ |  $0.87$
| Short-lasting component | $\alpha_{\rm r_{L},ab}$ |  $2.50\pm 1.73$ |
|  |  $T_{\rm a}$ (s) |  $63.20\pm 11.96$ |
|  |  $t_{\rm r,pk}$ (s) |  $130.20\pm 18.96$ |
GRB 180720B |  | $\alpha_{\rm r_{L},bb}$ |  $-0.71\pm 0.28$ |  $1.15$
| Short-lasting component | $\alpha_{\rm r_{L},ab}$ |  $1.92\pm 0.20$ |
|  |  $T_{\rm a}$ (s) |  $31.03\pm 4.16$ |
|  |  $t_{\rm r,pk}$ (s) |  $65.78\pm 0.21$ |
|  | $\alpha_{\rm r_{L},ab,1}$ |  $1.420\pm 0.60$ |  $1.18$
|  | $\alpha_{\rm r_{L},ab,2}$ |  $3.15\pm 0.60$ |
|  |  $t_{\rm r,ab}$ (s) |  $281.84\pm 2.24$ |

Table 2: The best-fit values found from modeling the X-ray and optical light curves using PL functions from our GRB sample. Event | Band | Parameter |  Best-fit value |  $\chi^{2}$/ndf
---|---|---|---|---
GRB 080916C | X-rays | $\alpha_{\rm X}$ |  $1.35\pm 0.06$ |  $1.16$
| Optical | $\alpha_{\rm opt}$ |  $1.41\pm 0.04$ |  $1.21$
GRB 090323 | X-rays | $\alpha_{\rm X}$ |  $1.58\pm 0.08$ |  $1.08$
| Optical | $\alpha_{\rm opt}$ |  $1.70\pm 0.04$ |  $1.15$
GRB 090902B | X-rays | $\alpha_{\rm X}$ |  $1.62\pm 0.15$ |  $0.91$
| Optical | $\alpha_{\rm opt}$ |  $1.70\pm 0.04$ |  $1.23$
GRB 090926A | X-Rays | $\alpha_{\rm X}$ |  $1.41\pm 0.03$ |  $1.47$
| Optical | $\alpha_{\rm opt}$ |  $1.34\pm 0.06$ |  $1.31$
GRB 110731A | X-Rays | $\alpha_{\rm X,1}$ |  $1.15\pm 0.02$ |  $1.61$
|  | $\alpha_{\rm X,2}$ |  $1.30\pm 0.03$ |
|  |  $t_{\rm f_{X},br}$ (s) |  $(3.00\pm 0.98)\times 10^{3}$ |
| Optical | $\alpha_{\rm opt}$ |  $1.38\pm 0.09$ |  $1.54$
GRB 130427A | X-Rays | $\alpha_{\rm X,1}$ |  $1.23\pm 0.04$ |  $2.23$
|  | $\alpha_{\rm X,2}$ |  $1.30\pm 0.05$ |
|  |  $t_{\rm f_{X},br}$ (s) |  $(2.36\pm 0.72)\times 10^{4}$ |
| Optical | $\alpha_{\rm opt,1}$ |  $1.16\pm 0.12$ |  $1.63$
|  | $\alpha_{\rm opt,2}$ |  $0.89\pm 0.01$ |
|  | $\alpha_{\rm opt,3}$ |  $1.33\pm 0.04$ |
|  |  $t_{\rm f_{O},br_{1}}$ (s) |  $(1.78\pm 0.02)\times 10^{3}$ |
|  |  $t_{\rm f_{O},br_{2}}$ (s) |  $(4.50\pm 0.01)\times 10^{4}$ |
GRB 160625B | X-Rays | $\alpha_{\rm X,1}$ |  $1.15\pm 0.13$ |  $1.65$
|  | $\alpha_{\rm X,2}$ |  $1.27\pm 0.30$ |
|  |  $t_{\rm f_{X},br}$ (s) |  $(3.12\pm 0.91)\times 10^{3}$ |
| Optical | $\alpha_{\rm opt}$ |  $0.921\pm 0.163$ |  $1.21$
GRB 180720B | X-Rays | $\alpha_{\rm X,1}$ |  $0.79\pm 0.08$ |  $0.92$
|  | $\alpha_{\rm X,2}$ |  $1.26\pm 0.06$ |
|  |  $t_{\rm f_{X},br}$ (s) |  $(2.36\pm 0.72)\times 10^{3}$ |
| Optical | $\alpha_{\rm opt,1}$ |  $1.16\pm 0.12$ |  $1.62$
|  | $\alpha_{\rm opt,2}$ |  $0.89\pm 0.01$ |
|  | $\alpha_{\rm opt,3}$ |  $1.22\pm 0.02$ |
|  |  $t_{\rm f_{O},br_{1}}$ (s) |  $(1.78\pm 0.02)\times 10^{3}$ |
|  |  $t_{\rm f_{O},br_{2}}$ (s) |  $(4.50\pm 0.01)\times 10^{4}$ |

Table 3: Median values of parameters of our GRB sample with symmetrical quantiles. The SSC and synchrotron models from reverse and forward region, respectively are used to constrain the parameters. GRB | 080916C | 090323 | 090902B | 090926A | 110731A | 130427A | 160625B | 180720B
---|---|---|---|---|---|---|---|---
$\rm{log_{10}}(E/{\rm erg})$ | $54.85_{-1.02}^{+1.05}$ | $53.96_{-1.05}^{+1.01}$ | $54.53_{-1.03}^{+1.01}$ | $53.83_{-0.99}^{+1.03}$ | $53.87_{-1.06}^{+0.95}$ | $54.14_{-1.03}^{+1.00}$ | $54.29_{-1.00}^{+1.03}$ | $54.36_{-0.95}^{+1.00}$
$\rm{log_{10}}(A_{\rm k})\,$ | $-2.83_{-0.96}^{+1.13}$ | $-2.29_{-1.06}^{+1.05}$ | $-2.82_{-1.01}^{+1.01}$ | $-2.21_{-0.99}^{+1.02}$ | $-1.77_{-1.06}^{+0.84}$ | $-1.96_{-1.01}^{+0.92}$ | $-2.19_{-1.15}^{+1.10}$ | $-1.68_{-0.98}^{+1.06}$
$\rm{log_{10}}(\Gamma)$ | $2.80_{-0.98}^{+1.09}$ | $2.92_{-0.96}^{+0.99}$ | $2.94_{-0.99}^{+1.09}$ | $2.92_{-0.85}^{+1.00}$ | $2.54_{-0.96}^{+0.87}$ | $2.52_{-1.04}^{+1.05}$ | $2.46_{-0.91}^{+1.06}$ | $2.46_{-0.99}^{+0.89}$
$p$ | $2.22_{-1.01}^{+0.92}$ | $2.68_{-0.98}^{+1.08}$ | $2.27_{-1.02}^{+1.03}$ | $2.50_{-0.98}^{+1.14}$ | $2.35_{-1.08}^{+1.04}$ | $2.10_{-1.00}^{+0.98}$ | $2.16_{-1.10}^{+0.96}$ | $2.09_{-0.93}^{+1.03}$
$\rm{log_{10}}(\varepsilon_{\rm B_{f}})$ | $-1.54_{-1.09}^{+1.07}$ | $-2.13_{-0.96}^{+1.03}$ | $-2.05_{-1.16}^{+1.05}$ | $-1.86_{-1.08}^{+0.91}$ | $-2.20_{-1.05}^{+1.02}$ | $-2.64_{-1.08}^{+1.09}$ | $-2.36_{-1.02}^{+0.95}$ | $-2.22_{-1.05}^{+1.04}$
$\rm{log_{10}}(\varepsilon_{\rm B_{r}})$ | $-1.32_{-0.88}^{+0.95}$ | $-1.03_{-1.01}^{+0.94}$ | $-0.96_{-0.98}^{+1.04}$ | $-0.98_{-0.99}^{+1.00}$ | $-1.99_{-1.05}^{+1.04}$ | $-1.48_{-1.07}^{+1.00}$ | $-1.06_{-1.06}^{+0.98}$ | $-1.54_{-1.10}^{+1.01}$
$\rm{log_{10}}(\varepsilon_{\rm e_{f}})$ | $-1.84_{-1.04}^{+0.93}$ | $-1.24_{-1.03}^{+1.11}$ | $-1.38_{-1.03}^{+0.94}$ | $-1.15_{-0.99}^{+0.91}$ | $-2.55_{-0.94}^{+0.85}$ | $-1.76_{-0.96}^{+0.95}$ | $-0.92_{-1.02}^{+0.98}$ | $-2.02_{-0.92}^{+0.97}$
$\rm{log_{10}}(\varepsilon_{\rm e_{r}})$ | $-1.25_{-1.01}^{+1.11}$ | $-1.11_{-1.04}^{+1.01}$ | $-1.10_{-0.99}^{+1.13}$ | $-1.04_{-0.92}^{+1.07}$ | $-0.89_{-1.06}^{+0.93}$ | $-1.44_{-0.98}^{+0.90}$ | $-1.11_{-0.91}^{+1.05}$ | $-1.51_{-0.92}^{+0.96}$
$\rm{log_{10}}(\zeta_{e})$ | $-1.81_{-1.09}^{+0.97}$ | $-0.91_{-1.02}^{+1.00}$ | $-0.47_{-0.93}^{+1.00}$ | $-0.90_{-1.11}^{+1.00}$ | $-0.16_{-1.10}^{+1.02}$ | $-0.81_{-0.97}^{+0.94}$ | $-1.20_{-0.98}^{+1.07}$ | $-0.33_{-0.91}^{+1.02}$
Table 4: Derived quantities from the best-fit parameter values of the reverse shock. GRB | 080916C | 090323 | 090902B | 090926A | 110731A | 130427A | 160625B | 180720B
---|---|---|---|---|---|---|---|---
|  |  |  | $(t<t_{\rm x})$ |  |  |  |
$h\nu^{\rm ssc}_{\rm m,r}\,({\rm~{}GeV})$ | $7.7$ | $4.1\times 10^{2}$ | $1.8\times 10^{-1}$ | $5.9\times 10^{3}$ | $1.1\times 10^{-8}$ | $9.1\times 10^{-8}$ | $5.8\times 10^{-4}$ | $2.9\times 10^{-7}$
$h\nu^{\rm ssc}_{\rm c,r}\,({\rm~{}GeV})$ | $9.3\times 10^{-2}$ | $1.2\times 10^{-3}$ | $2.9\times 10^{-5}$ | $2.2\times 10^{-6}$ | $1.1$ | $5.5\times 10^{-6}$ | $1.8\times 10^{-2}$ | $1.1\times 10^{-4}$
$F^{\rm ssc}_{\rm max,r}\,({\rm mJy})$ | $1.6\times 10^{-3}$ | $2.0\times 10^{-4}$ | $1.3\times 10^{-1}$ | $2.3\times 10^{-2}$ | $3.6\times 10^{-1}$ | $2.7\times 10$ | $4.7\times 10^{-3}$ | $3.7$
$h\nu^{\rm ssc}_{\rm KN,m,r}\,({\rm~{}GeV})$ | $4.8\times 10^{2}$ | $7.2\times 10^{2}$ | $9.9\times 10$ | $1.0\times 10^{3}$ | $8.2$ | $3.2$ | $2.8\times 10$ | $3.4$
$h\nu^{\rm ssc}_{\rm KN,c,r}\,({\rm~{}GeV})$ | $2.4\times 10^{2}$ | $1.5\times 10$ | $1.2\times 10$ | $4.5$ | $6.9\times 10^{2}$ | $8.9$ | $6.7\times 10$ | $1.5\times 10$
|  |  |  | $(t_{\rm x}<t)$ |  |  |  |
$h\nu^{\rm ssc}_{\rm m,r}\,({\rm~{}GeV})$ | $7.4\times 10^{-2}$ | $6.0\times 10^{2}$ | $1.1\times 10^{-1}$ | $5.8\times 10^{2}$ | $1.1\times 10^{-9}$ | $8.3\times 10^{-9}$ | $7.3\times 10^{-5}$ | $4.0\times 10^{-8}$
$h\nu^{\rm ssc}_{\rm cut,r}\,({\rm~{}GeV})$ | $1.6$ | $8.1\times 10^{-2}$ | $6.1\times 10^{-3}$ | $8.8\times 10^{-6}$ | $1.2\times 10^{2}$ | $2.0\times 10^{-4}$ | $1.2\times 10^{-1}$ | $1.2\times 10^{-3}$
$F^{\rm ssc}_{\rm max,r}\,({\rm mJy})$ | $1.2\times 10^{-4}$ | $5.3\times 10^{-5}$ | $2.1\times 10^{-2}$ | $5.4\times 10^{-3}$ | $8.3\times 10^{-2}$ | $4.4$ | $4.7\times 10^{-3}$ | $9.7\times 10^{-1}$
$h\nu^{\rm ssc}_{\rm KN,m,r}\,({\rm~{}GeV})$ | $1.4\times 10^{2}$ | $6.3\times 10^{2}$ | $9.5\times 10$ | $6.1\times 10^{2}$ | $1.8$ | $2.3$ | $2.8\times 10$ | $2.3$
$h\nu^{\rm ssc}_{\rm KN,c,r}\,({\rm~{}GeV})$ | $8.8\times 10^{2}$ | $3.8\times 10$ | $7.7\times 10$ | $1.1\times 10$ | $2.6\times 10^{3}$ | $3.8\times 10$ | $6.7\times 10$ | $4.2\times 10$

Figure 1: The expected SSC flux from the RS region (blue lines) and the
synchrotron flux from the FS region (red lines) when the outflow decelerates
in the stellar wind medium. From top to bottom, the panels show the SSC RS
light curves for $\nu^{\rm ssc}_{\rm c,r}<\nu<\nu^{\rm ssc}_{\rm m,r}$,
$\nu^{\rm ssc}_{\rm m,r}<\nu$, $\nu^{\rm ssc}_{\rm m,r}<\nu<\nu^{\rm ssc}_{\rm
c,r}$ and $\nu^{\rm ssc}_{\rm c,r}<\nu$. The expected SSC fluxes are presented
in the thick- (left column) and thin- (right column) shell case. The double-
dotted dashed black line corresponds to the shock crossing time. The breaks
displayed in the solid red lines are the transitions between $\nu^{\rm
syn}_{\rm m,f}<\nu<\nu^{\rm syn}_{\rm c,f}$ and $\nu^{\rm syn}_{\rm c,f}<\nu$.

(a) Sub a) GRB 080916C

(b) Sub b) GRB 090323

(c) Sub c) GRB 090902B

(d) Sub d) GRB 090926A

(e) Sub e) GRB 110731A

(f) Sub f) GRB 130427A

(g) Sub g) GRB 160625B

(h) Sub h) GRB 180720B

Figure 2: The photon flux (blue) and flux energy (red) detected by Fermi/LAT
are shown in top panels. The photon energy during the entire time window are
shown in bottom panels. The filled black dots correspond to a photon with a
probability greater than 90% of belonging to the GRB, while the open gray dots
correspond to a photon with a probability lower than 90% of belonging to the
burst.

(a) Sub a) GRB 080916C

(b) Sub b) GRB 090323

(c) Sub c) GRB 090902B

(d) Sub d) GRB 090926A

(e) Sub e) GRB 110731A

(f) Sub f) GRB 130427A

(g) Sub g) GRB 160625B

(h) Sub h) GRB 180720B

Figure 3: The LAT, X-ray and optical observations of our sample of GRBs with
the best-fit curve of the short- (dotted) and long- (dotted-dashed) lasting
components. The total emission is displayed with a solid line and $T_{a}$
corresponds to the starting time obtained from the best-fit parameters listed
in Table 1. Figure 4: Results of our MCMC parameter estimation are shown in
the corner plot for the synchrotron FS and SSC RS model in GRB 080916C. The
marginalized posterior densities for each parameter are shown by the
histograms on the diagonal, the contours are shown for 1$\sigma$, 2$\sigma$
and 3$\sigma$, and the median values in red lines. Figure 5: Same as Figure 4,
but for GRB 090323. Figure 6: Same as Figure 4, but for GRB 090902B. Figure 7:
Same as Figure 4, but for GRB 090926A Figure 8: Same as Figure 4, but for GRB
110731A. Figure 9: Same as Figure 4, but for GRB 130427A. Figure 10: Same as
Figure 4, but for GRB 160625B. Figure 11: Same as Figure 4, but for GRB
180720B.

(a) Sub a) GRB 080916C

(b) Sub b) GRB 090323

(c) Sub c) GRB 090902B

(d) Sub d) GRB 090926A

(e) Sub e) GRB 110731A

(f) Sub f) GRB 130427A

(g) Sub g) GRB 160625B

(h) Sub h) GRB 180720B

Figure 12: All the photons with energies $>100$ MeV and probabilities $>90$%
of being associated with each burst in our sample. The red lines correspond to
the maximum photon energies released by the SSC (dotted) and synchrotron
(dashed) from the reverse and forward afterglow model.