# Closure Relations of Synchrotron Self-Compton in Afterglow stratified medium
and Fermi-LAT Detected Gamma-Ray Bursts

Nissim Fraija,1 Maria G. Dainotti,2,3,4 B. Betancourt Kamenetskaia,5,6 D.
Levine7 and A. Galvan-Gamez1
1Instituto de Astronomía, Universidad Nacional Autónoma de México, Circuito
Exterior, C.U., A. Postal 70-264, 04510 México City, México
2National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan
3Space Science Institute, Boulder, CO, USA
4The Graduate University for Advanced Studies, SOKENDAI, Shonankokusaimura,
Hayama, Miura District, Kanagawa 240-0193, Japan
5Technical University of Munich, TUM School of Natural Sciences, Physics
Department, James-Franck-Str 1, 85748 Garching, Germany
6Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring
6, 80805 Munich, Germany
7Department of Astronomy, University of Maryland, College Park, MD 20742, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The Second Gamma-ray Burst Catalog (2FLGC) was announced by the Fermi Large
Area Telescope (Fermi-LAT) Collaboration. It includes 29 bursts with photon
energy higher than 10 GeV. Gamma-ray burst (GRB) afterglow observations have
been adequately explained by the classic synchrotron forward-shock model,
however, photon energies greater than 10 GeV from these transient events are
challenging, if not impossible, to characterize using this afterglow model.
Recently, the closure relations (CRs) of the synchrotron self-Compton (SSC)
forward-shock model evolving in a stellar wind and homogeneous medium was
presented to analyze the evolution of the spectral and temporal indexes of
those bursts reported in 2FLGC. In this work, we provide the CRs of the same
afterglow model, but evolving in an intermediate density profile ($\propto{\rm
r^{-k}}$) with ${\rm 0\leq k\leq 2.5}$, taking into account the
adiabatic/radiative regime and with/without energy injection for any value of
the electron spectral index. The results show that the current model accounts
for a considerable subset of GRBs that cannot be interpreted in either
stellar-wind or homogeneous afterglow SSC model. The analysis indicates that
the best-stratified scenario is most consistent with ${\rm k=0.5}$ for no-
energy injection and ${\rm k=2.5}$ for energy injection.

###### keywords:

Gamma-rays bursts: individual — Physical data and processes: acceleration of
particles — Physical data and processes: radiation mechanism: nonthermal —
ISM: general - magnetic fields

††pubyear: 2015††pagerange: Closure Relations of Synchrotron Self-Compton in
Afterglow stratified medium and Fermi-LAT Detected Gamma-Ray Bursts–3

##  1 Introduction

The most intense transients in the Universe are gamma-ray bursts (GRBs). These
phenomena, which may last anywhere from milliseconds to a few of hours
(Kouveliotou et al., 1993; Piran, 1999; Kumar & Zhang, 2015) and have a
spectrum given by the empirical Band function (Band et al., 1993), are
frequently seen in the keV-MeV energy range. After the initial burst of gamma
rays dies down, we observe a much longer-lasting phenomenon we call
“afterglow" in wavelengths ranging from radio to GeV. This phenomenon occurs
when a relativistic outflow ejected from the central engine interacts with the
surrounding environment and transfers a significant portion of its energy to
the surrounding circumburst medium (Mészáros & Rees, 1997; Piran, 1999). The
synchrotron radiation, which ranges from radio to gamma rays (Sari et al.,
1998; Kumar & Barniol Duran, 2009, 2010; Ackermann & et al., 2013; Fraija et
al., 2016), and the synchrotron self-Compton (SSC) process, which up-
scattering synchrotron photon at very-high energies (VHE $\geq$ 10 GeV; Fraija
et al., 2019b; Fraija et al., 2019a; Fraija et al., 2021; Fraija et al., 2017;
Zhang, 2019), are the primary mechanisms by which the shock-accelerated
electrons inside it are cooled down. The temporal ($\alpha$) and spectral
($\beta$) indexes of these radiation processes evolving in different
environments (e.g., homogeneous or stellar wind) predict the “closure
relations" (CRs) $F_{\nu}\propto t^{-\alpha}\nu^{-\beta}$, which are unique in
each scenario and therefore used to describe the afterglow observations.
Regarding the analysis of the high-energy emission we can date back the first
emission even to the previous century. The first claim of a VHE photon was
associated with GRB 940217, detected by the Energetic Gamma-Ray Experiment
Telescope (EGRET; Hurley et al., 1994). In that same year, GRB 941017 also
presented a late-time, spectrally-rising GeV component (González et al.,
2003). The Second Gamma-ray Burst Catalog (2FLGC) was provided by Ajello et
al. (2019), and it includes data from the first 10 years of program (from 2008
to 2018 August 4). The collection includes a sample of GRBs with temporarily-
extended emission described with a power-law (PL) and a broken power-law (BPL)
function, and photon energies above a few GeV, as well as 169 GRBs with high-
energy emission higher than $\geq 100$ MeV. Those bursts described with a BPL
function showed a temporal break in the light curve around several hundreds of
seconds. The light curves of the bursts that might be represented with a BPL
function showed a temporal break in time somewhere in the vicinity of many
hundreds of seconds. The evolution of the temporal and spectral indexes of the
synchrotron afterglow model is usually invoked to interpreted the temporarily-
extended emission (Sari et al., 1998; Kumar & Barniol Duran, 2009, 2010). The
high-energy emission is most likely to be dominated by the forward shock, an
idea that is motivated not just by the temporarily-extended LAT emission, but
also by the delayed onset (e.g., Abdo et al., 2009) and the inconsistency of
the GeV flux with a simple extrapolation from the sub-MeV energy range as
observed in a significant fraction of bursts (e.g. Ando et al., 2008;
Beniamini et al., 2011). While it is expected that if the forward shock
dominates the high-energy emission, CR would be present, not all LAT light
curves satisfy this condition.

Tak et al. (2019) systematically analyzed the CRs in a sample of 59 chosen
LAT-detected bursts, taking into account temporal ($\alpha_{\rm LAT}$) and
spectral ($\beta_{\rm LAT}$) indices. While the typical synchrotron forward-
shock emission accomplishes its goals effectively of explaining the spectral
and temporal indices in most instances, they reported that a sizeable minority
of bursts defy this explanation. Tak et al. (2019) reported that a large
fraction of bursts fulfills the CRs when the synchrotron scenario: i) lies in
the slow-cooling regime ($\nu^{\rm syn}_{\rm m}<\nu_{\rm LAT}<\nu^{\rm
syn}_{\rm c}$), ii) evolves in the homogeneous medium, and iii) requires that
magnetic microphysical parameter has an atypically small value of
$\varepsilon_{B}<10^{-7}$. The spectral breaks $\nu^{\rm syn}_{\rm c}$ and
$\nu^{\rm syn}_{\rm m}$ corresponds to the cooling and characteristic ones,
respectively. On the other hand, Ghisellini et al. (2010) analyzed the high-
energy emission of 11 bursts (up until October of 2009) identified by the
Fermi LAT. They concluded that the relativistic forward shock was in the fully
radiative regime for a hard spectral index $p\sim 2$ because the LAT
observations evolved as $\sim t^{-\frac{10}{7}}$ rather than $t^{-1}$ as
predicted by the adiabatic synchrotron forward-shock model. They reported that
three of them displayed this phase, being a slow-cooling regime in homogeneous
interstellar medium (ISM) the most favorable scenario. Furthermore, Dainotti
et al. (2021b) looked into the cases with a broken PL segment with a shallower
slope than the usual decay, the so-called “plateau" phase, in the X-ray and
optical light curves and found three cases with known redshift among a sample
of GRBs deteted by Fermi-LAT from August 2008 until May 2016. One of the
possibilities for interpreting this phase is provided by the the energy
injection scenario (Barthelmy et al., 2005; King et al., 2005; Dai et al.,
2006; Perna et al., 2006; Proga & Zhang, 2006; Burrows et al., 2005;
Chincarini et al., 2007; Dall’Osso et al., 2017; Becerra et al., 2019). They
found that the slow-cooling regime evolving in ISM was the most favourable
scenario. Finally, Fraija et al. (2022) derived the CRs of SSC forward-shock
model evolving in the stellar wind and ISM. The authors showed that a
significant fraction of GRBs could not be interpreted in the synchrotron
scenario, and also that ISM is preferred without energy injection and the
stellar wind with energy injection.

CRs evolving in ISM ($k=0$) and stellar wind ($k=2$) have been widely used to
test the evolution of LAT light curves. However, studies about the progenitor
and modelling of the multiwavelength observations during the afterglow phase
has suggested that a circumburst medium with an intermediate value of density
profile between ${\rm k}=0$ and ${\rm k}=2$ is consistent in several GRBs
(Dainotti et al., 2023). For instance, Yi et al. (2013) investigated the
evolution of the emission of forward-reverse shocks propagating in a medium
described by the previously mentioned PL distribution. They applied their
model to 19 GRBs and found that the density profile index took values of
$0.4\leq{\rm k}\leq 1.4$, with a typical value of ${\rm k\sim 1}$. This value
was also obtained by Liang et al. (2013), who analyzed a bigger sample of 146
GRBs. A density-profile index with $k>2$ was derived by Kumar et al. (2008)
after looking at the possibility that a black hole’s accretion of a stellar
envelope causes the plateau phase in X-ray light curves. Also in GRBs
presenting radio afterglows a stratified emission can reconcile some of the
observations (Levine et al., 2022). Kouveliotou et al. (2013) analyzed the
multi-wavelength observations of GRB 130427A, one of the most powerful burst
with a redshift $z<1$, and modeled the observations below GeV energy range
with a synchrotron afterglow model in the adiabatic regime without energy
injection for $p>2$. The authors found that an intermediate value of density
profile between ${\rm k}=0$ and ${\rm k}=2$ was consistent with these
observations.

Fraija et al. (2022) presented the CRs for the stellar-wind and ISM in the
forward-shock scenario of a SSC in stratified medium for $k=0$ and $k=2$. They
considered the evolution in both the adiabatic and radiative regimes, with and
without energy injection into the blastwave. Here, we extend this approaches
and take into account the stratified environment with a density profile
$\propto r^{-k}$ with ${\rm k}$ lying in the range of $0\leq k<3$. We apply
the current model to investigate the evolution of the spectral and temporal
indexes of those GRBs reported in 2FLGC. This manuscript will be organized as
follows: in Section §2 we derive the CRs of SSC forward-shock model evolving
in a stratified environment with $0\leq k<3$. In Section §3, we show the
analysis of Fermi-LAT data and the methodology. Section §4 displays the
discussion and finally, in Section §5, we conclude by summing up our findings
and making some general observations.

##  2 SSC Forward-shock model in stratified medium

By decelerating and driving a forward shock into the circumstellar medium, a
relativistic GRB outflow creates the temporally-extended afterglow emission.
During the deceleration phase, the outflow transmits a large portion of its
energy to the external medium. Shocks transmit a significant amount of energy
density to accelerate electrons and increase the magnetic field through the
microphysical parameters $\varepsilon_{e}$ and $\varepsilon_{B}$,
respectively. We take into account a density-profile described by $n(r)=A_{\rm
k}\,r^{\rm-k}$, with $0\leq k<3$ in which accelerated electrons cool down.
Based on the foregoing, we derive the CRs of SSC afterglow in
adiabatic/radiative regime and with/without energy injection. In order to
report the proportionality constant of each quantity, we consider the values
of PL indexes $k=1.0$ for $A_{\rm k}=1.52\times 10^{15}\,{\rm cm^{-2}}$,
$\epsilon=0.5$, $q=0.5$, and $p=1.95$ ($2.15$) for $1<p<2$ ($2\leq p$), unless
otherwise stated.

###  2.1 Closure relations in the adiabatic and radiative scenario

The relativistic forward shock is often assumed to be totally adiabatic in the
conventional GRB afterglow scenario, however it is possible that it is either
partially or completely radiative (Dai & Lu, 1998; Sari et al., 1998; Böttcher
& Dermer, 2000; Ghisellini et al., 2010). Comparing the cooling and dynamical
timescales, the relativistic electrons could lie in the fast or slow-cooling
regime. In the case of the cooling timescale being smaller than the dynamical
timescale, the relativistic electrons are in the fast-cooling regime, and only
when $\varepsilon_{e}$ to be of order unity, the afterglow phase would be in
the radiative regime (Böttcher & Dermer, 2000; Li et al., 2002; Panaitescu,
2019). The deceleration of the GRB fireball by the circumburst medium is
faster when the fireball is radiative than adiabatic. As a result, the shock’s
energetics and synchrotron light curves during the afterglow are altered
(Böttcher & Dermer, 2000; Wu et al., 2005; Moderski et al., 2000).

Once the relativistic jet begins the deceleration phase, the evolution of the
bulk Lorentz factor first in the radiative and then in the adiabatic regime is
$\Gamma=2.8\times
10^{2}\left(\frac{1+z}{1.022}\right)^{\frac{3-k}{[2(4-k)-\epsilon]}}\,A_{k}^{-\frac{1}{[2(4-k)-\epsilon]}}E_{53}^{\frac{1}{[2(4-k)-\epsilon]}}\Gamma_{0,2.8}^{-\frac{\epsilon}{[2(4-k)-\epsilon]}}t_{2}^{-\frac{3-k}{[2(4-k)-\epsilon]}}$
(Wu et al., 2005), where $E$ is the isotropic-equivalent kinetic energy, and
$\Gamma_{0}$ is the initial Lorentz factor. Hydrodynamic evolution of the
forward shock is accounted for in the totally adiabatic ($\epsilon=0$), fully
radiative ($\epsilon=1$), or partly radiative or adiabatic ($0<\epsilon<1$)
regimes, respectively, where $\epsilon$ is the radiative parameter. Hereafter,
we adopt the convention $Q_{\rm x}=Q/10^{\rm x}$ in c.g.s. units. For
electrons with the lowest energy, above which they cool effectively and with
the maximum energy, the Lorentz factors for $p>2$ are

$\displaystyle\gamma_{\rm m}$ $\displaystyle=$ $\displaystyle 8.6\times
10^{3}\,\left(\frac{p-2}{p-1}\right)\left(\frac{1+z}{1.022}\right)^{\frac{3-k}{[2(4-k)-\epsilon]}}\,\varepsilon_{e,-0.3}A_{k}^{-\frac{1}{[2(4-k)-\epsilon]}}\,E_{53}^{\frac{1}{[2(4-k)-\epsilon]}}\,\Gamma^{-\frac{\epsilon}{[2(4-k)-\epsilon]}}_{0,2.8}t_{2}^{-\frac{3-k}{[2(4-k)-\epsilon]}}$
(1) $\displaystyle\gamma_{\rm c}$ $\displaystyle=$ $\displaystyle 1.9\times
10^{4}\,(1+Y)^{-1}\left(\frac{1+z}{1.022}\right)^{-\frac{1+\epsilon+k(1-\epsilon)}{[2(4-k)-\epsilon]}}\,\varepsilon^{-1}_{B,-2}A_{k}^{-\frac{5-\epsilon}{[2(4-k)-\epsilon]}}\,E_{53}^{-\frac{3-2k}{[2(4-k)-\epsilon]}}\,\Gamma^{\frac{(3-2k)\epsilon}{[2(4-k)-\epsilon]}}_{0,2.8}t_{2}^{\frac{1+\epsilon+k(1-\epsilon)}{[2(4-k)-\epsilon]}}\,$
(2) $\displaystyle\gamma_{\rm max}$ $\displaystyle=$ $\displaystyle 5.9\times
10^{7}\,\left(\frac{1+z}{1.022}\right)^{-\frac{6-k\epsilon}{4[2(4-k)-\epsilon]}}\,\varepsilon^{-\frac{1}{4}}_{B,-2}A_{k}^{-\frac{6-\epsilon}{4[2(4-k)-\epsilon]}}\,E_{53}^{-\frac{1-k}{2[2(4-k)-\epsilon]}}\,\Gamma^{\frac{(1-k)\epsilon}{2[2(4-k)-\epsilon]}}_{0,2.8}t_{2}^{\frac{6-k\epsilon}{4[2(4-k)-\epsilon]}}\,,\hskip
170.71652pt$ (3)

respectively, where $Y$ is the Compton parameter and $t$ is the observer time.
The Lorentz factor of the lowest-energy electrons for ${\rm 1<p<2}$ is

$\gamma_{\rm m}=2.1\times
10^{3}\,\left(\frac{2-p}{p-1}\right)^{\frac{1}{p-1}}\left(\frac{1+z}{1.022}\right)^{\frac{6(4-p)-k[4+(2-p)\epsilon]}{4(p-1)[2(4-k)-\epsilon]}}\,\varepsilon^{\frac{1}{p-1}}_{e,-0.3}\varepsilon^{\frac{2-p}{4(p-1)}}_{B,-2}A_{k}^{\frac{2(4-3p)+\epsilon(p-2)}{4(p-1)[2(4-k)-\epsilon]}}E_{53}^{\frac{4-p-k(2-p)}{2(p-1)[2(4-k)-\epsilon]}}\,\Gamma^{-\frac{[4-p-k(2-p)]}{2(p-1)[2(4-k)-\epsilon]}}_{0,2.8}t_{2}^{-\frac{6(4-p)-k[4+(2-p)\epsilon]}{4(p-1)[2(4-k)-\epsilon]}}\,,$
(4)

In the synchrotron afterglow model, the characteristic and cooling spectral
breaks evolve as $\nu^{\rm syn}_{\rm m}\propto
t^{-\frac{6(p+2)-k(2+2p+\epsilon)}{2(p-1)[2(4-k)-\epsilon]}}$
($t^{-\frac{24-k(6+\epsilon)}{2[2(4-k)-\epsilon]}}$) for $1<p<2$ ($2<p$) and
$\nu^{\rm syn}_{\rm c}\propto
t^{\frac{(\epsilon-2)(4-3k)}{2[2(4-k)-\epsilon]}}$, respectively, and the
maximum synchrotron flux as $F^{\rm syn}_{\rm max}\propto
t^{-\frac{k(2-3\epsilon)+6\epsilon}{2[2(4-k)-\epsilon]}}$. When relativistic
electrons are accelerated in stratified environments in the radiative regime,
synchrotron photons have their maximum energy correspond to

$h\nu^{\rm syn}_{\rm max}\approx 5.4\times 10^{2}\,{\rm
MeV}\left(\frac{1+z}{1.022}\right)^{\frac{k+\epsilon-5}{2(4-k)-\epsilon}}\,A_{k}^{-\frac{1}{2(4-k)-\epsilon}}E_{53}^{\frac{1}{2(4-k)-\epsilon}}\Gamma_{0,2.8}^{-\frac{\epsilon}{2(4-k)-\epsilon}}t_{2}^{-\frac{3-k}{2(4-k)-\epsilon}}\,.$
(5)

When the same population of electrons that emits synchrotron photons also up-
scatters them to higher energies, a phenomenon known as SSC emission occurs;
$h\nu^{\rm ssc}_{\rm i}\simeq\gamma^{2}_{\rm i}h\nu^{\rm syn}_{\rm i}$, with
${\rm i=m,c}$ (e.g., see Sari & Esin, 2001). The maximum flux generated by the
SSC mechanism is $F^{\rm ssc}_{\rm max}\sim\sigma_{T}n\,r\,F^{\rm syn}_{\rm
max}$.111$\sigma_{T}$ is the Thompson cross section and $r$ is the forward-
shock radius. Requiring the Lorentz factors for electrons (Eq. 1), and the
relevant quantities (spectral breaks and the maximum flux) for the synchrotron
scenario, the characteristic (for $2<p$) and cooling spectral breaks and the
maximum flux of the SSC scenario are

$\displaystyle h\nu^{\rm ssc}_{\rm m}$ $\displaystyle=$ $\displaystyle
1.7\times 10^{9}\,{\rm
eV}\,\left(\frac{p-2}{p-1}\right)^{4}\left(\frac{1+z}{1.022}\right)^{\frac{2(10+\epsilon)-k(6+\epsilon)}{2[2(4-k)-\epsilon]}}\,\varepsilon^{4}_{e,-0.3}\varepsilon^{\frac{1}{2}}_{B,-2}A_{k}^{-\frac{4+\epsilon}{2[2(4-k)-\epsilon]}}\,E_{53}^{\frac{6-k}{[2(4-k)-\epsilon]}}\,\Gamma^{-\frac{(6-k)\epsilon}{[2(4-k)-\epsilon]}}_{0,2.8}t_{2}^{-\frac{36-k(10+\epsilon)}{2[2(4-k)-\epsilon]}}$
(6) $\displaystyle h\nu^{\rm ssc}_{\rm c}$ $\displaystyle=$ $\displaystyle
4.5\times 10^{10}\,{\rm
eV}\left(1+Y\right)^{-4}\left(\frac{1+z}{1.022}\right)^{-\frac{6(2+\epsilon)+k(6-7\epsilon)}{2[2(4-k)-\epsilon]}}\,\varepsilon^{-\frac{7}{2}}_{B,-2}A_{k}^{-\frac{36-7\epsilon}{2[2(4-k)-\epsilon]}}\,E_{53}^{-\frac{10-7k}{[2(4-k)-\epsilon]}}\,\Gamma^{\frac{(10-7k)\epsilon}{[2(4-k)-\epsilon]}}_{0,2.8}\,t_{2}^{-\frac{4-8\epsilon-k(10-7\epsilon)}{2[2(4-k)-\epsilon]}}$
(7) $\displaystyle F^{\rm ssc}_{\rm max}$ $\displaystyle=$ $\displaystyle
1.0\times 10^{-8}\,{\rm
mJy}\left(\frac{p-2}{p-1}\right)^{-1}\left(\frac{1+z}{1.022}\right)^{\frac{4(7+\epsilon)-k(2+5\epsilon)}{2[2(4-k)-\epsilon]}}\,\varepsilon^{\frac{1}{2}}_{B,-2}D_{\rm
z,26.5}^{-2}\,A_{k}^{\frac{5(4-\epsilon)}{2[2(4-k)-\epsilon]}}\,E_{53}^{\frac{5(2-k)}{[2(4-k)-\epsilon]}}\,\Gamma^{-\frac{5(2-k)\epsilon}{[2(4-k)-\epsilon]}}_{0,2.8}t_{2}^{\frac{4-k(6-5\epsilon)-8\epsilon}{2[2(4-k)-\epsilon]}}\,,$
(8)

respectively, and the characteristic break for ${1<p<2}$ is

$h\nu^{\rm ssc}_{\rm m}=5.7\times 10^{6}\,{\rm
eV}\,\left(\frac{2-p}{p-1}\right)^{\frac{4}{p-1}}\left(\frac{1+z}{1.022}\right)^{\frac{52-2p(8-\epsilon)-2\epsilon-k[10+3\epsilon-p(2+\epsilon)]}{2(p-1)[2(4-k)-\epsilon]}}\,\varepsilon^{\frac{4}{p-1}}_{e,-0.3}\varepsilon^{\frac{3-p}{2(p-1)}}_{B,-2}A_{k}^{\frac{3(4-\epsilon)-p(8-\epsilon)}{2(p-1)[2(4-k)-\epsilon]}}E_{53}^{\frac{6-k(3-p)}{(p-1)[2(4-k)-\epsilon]}}\,\Gamma^{-\frac{[6-k(3-p)]\epsilon}{(p-1)[2(4-k)-\epsilon]}}_{0,2.8}t_{2}^{-\frac{36-kp(2-\epsilon)-3k(2+\epsilon)}{2(p-1)[2(4-k)-\epsilon]}}.$
(9)

When the Klein - Nishina (KN) effects are not negligible, the value of the
Compton parameter is not constant and then defined as
$\frac{Y(\gamma_{c})[Y(\gamma_{c})+1]}{Y_{0}}=\left(\frac{\nu^{\rm syn}_{\rm
m}}{\nu^{\rm syn}_{\rm c}}\right)^{-\frac{p-3}{2}}\,\left(\frac{\nu^{\rm
syn}_{\rm KN}(\gamma_{\rm c})}{\nu^{\rm syn}_{\rm
m}}\right)^{\frac{4}{3}}\,\,{\rm for}\,\,{\nu^{\rm syn}_{\rm KN}(\gamma_{\rm
c})<\nu^{\rm syn}_{\rm m}}$, $\left(\frac{\nu^{\rm syn}_{\rm KN}(\gamma_{\rm
c})}{\nu^{\rm syn}_{c}}\right)^{-\frac{p-3}{2}}\,\,{\rm for}\,\,\nu^{\rm
syn}_{\rm m}<\nu^{\rm syn}_{\rm KN}(\gamma_{\rm c})<\nu^{\rm syn}_{\rm c}$,
$1\,\,{\rm for}\,\,\nu^{\rm syn}_{\rm c}<\nu^{\rm syn}_{\rm KN}(\gamma_{\rm
c})$ where
$Y_{0}=\frac{\varepsilon_{e}}{\varepsilon_{B}}\left(\frac{\gamma_{\rm
c}}{\gamma_{\rm m}}\right)^{2-p}$ where $h\nu^{\rm syn}_{\rm
KN}(\gamma_{c})\simeq\frac{\Gamma}{(1+z)}\,\frac{m_{e}c^{2}}{\gamma_{c}}$ for
$\nu^{\rm syn}_{\rm m}<\nu^{\rm syn}_{\rm c}$ (see Nakar et al., 2009; Wang et
al., 2010). To describe the LAT observations above 100 MeV, it is needed to
define the Lorentz factor of those electrons ($\gamma_{*}$) that might emit
high-energy photons via synchrotron process, and a new spectral break
$h\nu^{\rm syn}_{\rm KN}(\gamma_{*})$ would have to be included and the
Compton parameter $Y(\gamma_{*})$ re-computed. For instance, the new Compton
parameter becomes
$Y(\gamma_{*})=Y(\gamma_{c})\left(\frac{\nu_{*}}{\nu_{c}}\right)^{\frac{p-3}{4}}\left(\frac{\nu^{\rm
syn}_{\rm KN}(\gamma_{\rm c})}{\nu^{\rm syn}_{c}}\right)^{-\frac{p-3}{2}}$ for
$\nu^{\rm syn}_{\rm m}<h\nu^{\rm syn}_{\rm KN}(\gamma_{*})=100\,{\rm
MeV}<\nu^{\rm syn}_{\rm c}<\nu^{\rm syn}_{\rm KN}(\gamma_{\rm c})$ (for
details, see Wang et al., 2010).

It is worth noting that the spectral breaks and the maximum flux derived for
$k=0$ and $k=2$ and presented in Fraija et al. (2022) are recovered. Taking
into account the temporal and spectral evolution of spectral breaks together
with the maximum flux of the SSC afterglow model (Eqs. 9), we report the CRs
as shown in Table 1.

####  2.1.1 Evolution of the radiative parameter

The evolution of radiative parameter can be estimated from the relation of
synchrotron and expansion timescale (Huang et al., 2000). Therefore, it
becomes

$\epsilon=\varepsilon_{e}\frac{{t^{\prime}}^{-1}_{\rm
syn}}{{t^{\prime}}^{-1}_{\rm syn}+{t^{\prime}}^{-1}_{\rm ex}}\,,$ (10)

where $t^{\prime}_{\rm syn}=6\pi
m_{e}/(\sigma_{T}{B^{\prime}}^{2}\,(1+Y(\gamma_{\rm c}))\gamma_{m})$ is the
synchrotron cooling time and $t^{\prime}_{\rm exp}=4\,\Gamma t/(1+z)$ is the
comoving frame expansion time (e.g., see Dai & Lu, 1998), with $B^{\prime}$
the comoving magnetic field, $m_{e}$ the electron mass and $\gamma_{m}$ given
in Eqs. 1 and 4 for $2\leq p$ and $1<p<2$, respectively. In order to discuss
the evolution of the radiative parameter, we consider the limiting cases of
synchrotron and expansion timescale. For $t^{\prime}_{\rm syn}\ll
t^{\prime}_{\rm exp}$, the radiative parameter corressponds to
$\epsilon=\varepsilon_{e}(1+t^{\prime}_{\rm syn}/t^{\prime}_{\rm
exp})^{-1}\approx\varepsilon_{e}$ (fully radiative with
$\varepsilon_{e}\approx 1$) and for $t^{\prime}_{\rm exp}\ll t^{\prime}_{\rm
syn}$, the radiative parameter becomes
$\epsilon=\varepsilon_{e}\,t^{\prime}_{\rm exp}/t^{\prime}_{\rm
syn}(1+t^{\prime}_{\rm exp}/t^{\prime}_{\rm syn})^{-1}\approx 0$ (fully
adiabatic). For instance, the radiative parameter becomes
$\epsilon\approx\varepsilon_{e}/2$ when $t^{\prime}_{\rm exp}\approx
t^{\prime}_{\rm syn}$.

###  2.2 Closure relations with energy injection

Refreshed shocks can be generated by the continuous injection of energy by the
progenitor into the afterglow. An expression for the luminosity injected by
the progenitor $L_{\rm inj}(t)\propto L_{0}\,t^{\rm-q}$ where $q$ is the
energy injection index (e.g. Zhang et al., 2006). The equivalent kinetic
energy of the forward shock is given by $E=\int L_{\rm inj}\,dt\propto
L_{0}t^{-q+1}$. The temporal evolution of the bulk Lorentz factor can be
written as $\Gamma=3.3\times
10^{2}\left(\frac{1+z}{1.022}\right)^{\frac{3-k}{2(4-k)}}\,A_{k}^{-\frac{1}{2(4-k)}}E_{53}^{\frac{1}{2(4-k)}}t_{2}^{-\frac{2+q-k}{2(4-k)}}$.

For electrons with the lowest energy, above which they cool effectively and
with the maximum energy, the Lorentz factors for $p>2$ are

$\displaystyle\gamma_{\rm m}$ $\displaystyle=$ $\displaystyle 1.0\times
10^{4}\,\left(\frac{p-2}{p-1}\right)\left(\frac{1+z}{1.022}\right)^{\frac{3-k}{2(4-k)}}\,\varepsilon_{e,-1}A_{k}^{-\frac{1}{2(4-k)}}\,E_{53}^{\frac{1}{2(4-k)}}\,t_{2}^{-\frac{2+q-k}{2(4-k)}}$
(11) $\displaystyle\gamma_{\rm c}$ $\displaystyle=$ $\displaystyle 1.7\times
10^{4}\,(1+Y)^{-1}\left(\frac{1+z}{1.022}\right)^{-\frac{1+k}{2(4-k)}}\,\varepsilon^{-1}_{B,-2}A_{k}^{-\frac{5}{2(4-k)}}\,E_{53}^{-\frac{3-2k}{2(4-k)}}\,t_{2}^{-\frac{2-3q-k(3-2q)}{2(4-k)}}$
(12) $\displaystyle\gamma_{\rm max}$ $\displaystyle=$ $\displaystyle 5.9\times
10^{7}\,\left(\frac{1+z}{1.022}\right)^{-\frac{3}{4(4-k)}}\,\varepsilon^{-\frac{1}{4}}_{B,-2}\,n^{-\frac{3}{4(4-k)}}\,E^{-\frac{1-k}{4(4-k)}}_{53}\,t_{2}^{\frac{2+q+k(1-q)}{4(4-k)}}\,,\hskip
199.16928pt$ (13)

respectively. The Lorentz factor of the lowest-energy electrons for ${\rm
1<p<2}$ is

$\gamma_{\rm m}=2.4\times
10^{3}\,\left(\frac{2-p}{p-1}\right)^{\frac{1}{p-1}}\left(\frac{1+z}{1.022}\right)^{\frac{3(4-p)-2k}{4(p-1)(4-k)}}\,\varepsilon^{\frac{1}{p-1}}_{e,-1}\varepsilon^{\frac{2-p}{4(p-1)}}_{B,-2}A_{k}^{\frac{4-3p}{4(p-1)(4-k)}}E_{53}^{\frac{4-k(2-p)-p}{4(p-1)(4-k)}}\,t_{2}^{-\frac{8+2(2-k)q-p(2+k+q-kq)}{4(p-1)(4-k)}}\,,$
(14)

respectively. In the synchrotron afterglow model, the characteristic and
cooling spectral breaks evolve as $\nu^{\rm syn}_{\rm m}\propto
t^{-\frac{4+(2-k)q+p(2-k+q)}{2(p-1)(4-k)}}$ ($\nu^{\rm syn}_{\rm m}\propto
t^{-\frac{2+q}{2}}$) for $1<p<2$ ($2<p$) and $\nu^{\rm syn}_{\rm c}\propto
t^{\frac{(4-3k)(q-2)}{2(4-k)}}$, respectively, and the maximum synchrotron
flux as $F^{\rm syn}_{\rm max}\propto t^{\frac{8(1-q)-k(4-3q)}{2(4-k)}}$. For
the adiabatic regimes and with energy injection, the maximum energy of
synchrotron photons yields

$h\nu^{\rm syn}_{\rm max}\approx 6.4\times 10^{2}\,{\rm
MeV}\left(\frac{1+z}{1.022}\right)^{\frac{k-5}{2(4-k)}}\,A_{k}^{-\frac{1}{2(4-k)}}E_{53}^{\frac{1}{2(4-k)}}\,t_{2}^{-\frac{2+q-k}{2(4-k)}}\,.$
(15)

Requiring the Lorentz factors for electrons (Eq. 1), and the relevant
quantities (spectral breaks and the maximum flux) for the synchrotron
scenario, the characteristic (for $2<p$) and cooling spectral breaks and the
maximum flux of the SSC scenario are

$\displaystyle h\nu^{\rm ssc}_{\rm m}$ $\displaystyle=$ $\displaystyle
3.7\times 10^{9}\,{\rm
eV}\left(\frac{p-2}{p-1}\right)^{4}\left(\frac{1+z}{1.022}\right)^{\frac{10-3k}{2(4-k)}}\,\varepsilon^{4}_{e,-1}\varepsilon^{\frac{1}{2}}_{B,-2}A_{k}^{-\frac{1}{4-k}}\,E^{\frac{6-k}{2(4-k)}}_{53}\,t_{2}^{-\frac{6(2+q)-k(4+q)}{2(4-k)}}$
(16) $\displaystyle h\nu^{\rm ssc}_{\rm c}$ $\displaystyle=$ $\displaystyle
2.9\times 10\,{\rm
eV}(1+Y)^{-4}\left(\frac{1+z}{1.022}\right)^{-\frac{3(2+k)}{2(4-k)}}\,\varepsilon^{-\frac{7}{2}}_{B,-2}A_{k}^{-\frac{9}{4-k}}\,E_{53}^{-\frac{10-7k}{2(4-k)}}\,t_{2}^{-\frac{12-10q-k(12-7q)}{2(4-k)}}$
(17) $\displaystyle F^{\rm ssc}_{\rm max}$ $\displaystyle=$ $\displaystyle
2.2\times 10^{-8}\,{\rm
mJy}\left(\frac{p-2}{p-1}\right)^{-1}\left(\frac{1+z}{1.022}\right)^{\frac{14-k}{2(4-k)}}\,\varepsilon^{\frac{1}{2}}_{B,-2}D_{\rm
z,26.5}^{-2}\,A_{k}^{\frac{5}{4-k}}\,E_{53}^{\frac{5(2-k)}{2(4-k)}}\,t_{2}^{\frac{2(6-5q)-k(8-5q)}{2(4-k)}}\,,$
(18)

respectively, and the characteristic break for ${1<p<2}$ is

$h\nu^{\rm ssc}_{\rm m}=1.3\times 10^{7}\,{\rm
eV}\left(\frac{2-p}{p-1}\right)^{\frac{4}{p-1}}\left(\frac{1+z}{1.022}\right)^{\frac{26-k(5-p)-8p}{2(p-1)(4-k)}}\,\varepsilon^{\frac{4}{p-1}}_{e,-1}\varepsilon^{\frac{3-p}{2(p-1)}}_{B,-2}A_{k}^{\frac{3-2p}{(p-1)(4-k)}}E^{\frac{6-k(3-p)}{2(p-1)(4-k)}}_{53}\,t_{2}^{-\frac{6(2+q)-3kq-
kp(2-q)}{2(p-1)(4-k)}}$ (19)

It is important to point out that the spectral breaks and the maximum flux
derived for $k=0$ and $k=2$ and presented in Fraija et al. (2022) are
recovered.

Taking into account the temporal and spectral evolution of spectral breaks
together with the maximum flux of the SSC afterglow model (Eqs. 19), we report
the CRs as shown in Table 2.

##  3 Methods and Analysis

Our sample consists of 86 LAT-detected bursts with temporally extended
emissions considered in Fraija et al. (2022) and reported by the 2FLGC (Ajello
et al., 2019). We analyze the evolution of temporal and spectral indexes using
the same strategy presented in Fraija et al. (2022); Dainotti et al. (2021a,
b); Srinivasaragavan et al. (2020). The errorbars in the plots are in the
1$\sigma$ range, with $\alpha$ and $\beta$ values of the PL segment
$F_{\rm\nu}\propto t^{-\alpha}\nu^{-\beta}$ being dependent; thus, Figures 1
and 2 are displayed with ellipses.

We have derived and listed in Tables 1 and 2 the CRs of SSC afterglow model in
the adiabatic and radiative scenario, and with and without energy injection,
respectively. We consider the CRs as function of the density-profile index
$k$, the radiative parameter $\epsilon$, the energy injection index $q$, and
the electron spectral index for $1<p<2$ and $2\leq p$. We emphasize that
although early analysis of LAT data (using lower energy photons) found CRs
that were consistent with the synchrotron FS model for the cooling condition
$\nu^{\rm syn}_{\rm c}<\nu$ (e.g. Kumar & Barniol Duran, 2010), we only
consider the CRs of the SSC FS model with a sample of 86 LAT-detected bursts
with temporally extended emissions ($\geq 100\,{\rm MeV}$) reported by the
2FLGC (Ajello et al., 2019). Tables 3 and 4 summarize the number and
percentage of bursts following each scenario.

Figure 1 presents the radiative case ($\epsilon=0.5$) in which no energy is
injected. This figure exhibits the CR of SSC afterglow model evolving in a
stratified medium, from top to bottom $k=0.5$, $1$, $1.5$ and $2.5$,
respectively. The fast and slow regimes for each cooling condition are shown
for $1<p<2$ and $p>2$. Table 3 summarizes the number and percentage of bursts
following each cooling condition for $1<p<2$ and $p>2$ and the profile index
$k=0.5$, $1$, $1.5$ and $2.5$. This best-case scenario is most consistent with
${\rm max\\{\nu^{\rm ssc}_{\rm m},\nu^{\rm ssc}_{\rm c}\\}<\nu}$ for $k=0.5$
(22 GRBs, 25.88%), $k=1.0$ (22 GRBs, 25.88%), $k=1.5$ (21 GRBs, 24.71%) and
$k=2.5$ (19 GRBs, 22.35%). There is one burst that satisfies the CRs for each
value of $k$ when the SSC model evolve in the cooling condition ${\rm\nu^{\rm
ssc}_{\rm c}\leq\nu\leq\nu^{\rm ssc}_{\rm m}}$, and few bursts for
${\rm\nu^{\rm ssc}_{\rm m}\leq\nu\leq\nu^{\rm ssc}_{\rm c}}$ for $k=0.5$ (3
GRBs, 3.53%), $1$ (2 GRBs, 2.35%) and $1.5$ (1 GRB, 1.18%).

Figure 2 presents the adiabatic case in which energy ($q=0.5$) is injected.
This Figure exhibits the CR of SSC afterglow model evolving in a stratified
medium, from top to bottom $k=0.5$, $1$, $1.5$ and $2.5$, respectively. The
fast and slow regimes for each cooling condition are shown for $1<p<2$ and
$p>2$. Table 4 summarizes the number and percentage of bursts following each
cooling condition for $1<p<2$ and $p>2$ and the profile index $k=0.5$, $1$,
$1.5$ and $2.5$. The best-case scenario is most consistent with ${\rm
max\\{\nu^{\rm ssc}_{\rm m},\nu^{\rm ssc}_{\rm c}\\}<\nu}$ for $k=0.5$ (32
GRBs, 37.65%), $k=1.0$ (32 GRBs, 37.65%), $k=1.5$ (34 GRBs, 40.0%) and $k=2.5$
(40 GRBs, 47.06%). For the cooling condition ${\rm\nu^{\rm ssc}_{\rm
m}\leq\nu\leq\nu^{\rm ssc}_{\rm c}}$, 28 (25 and 7) GRBs for $k=0.5$ ($1$ and
$1.5$) fulfill the CRs. There is one burst that satisfies the CRs for each
value of $k$ when the SSC model evolve in the cooling condition ${\rm\nu^{\rm
ssc}_{\rm c}\leq\nu\leq\nu^{\rm ssc}_{\rm m}}$.

##  4 Discussion

###  4.1 A particular case: GRB 130427A

The long GRB 130427A has been one of the most energetic bursts reported in the
second LAT burst catalog. Kouveliotou et al. (2013) performed an analysis on
the multi-wavelength data of GRB 130427A and modeled the observations in the
sub-GeV energy range using a synchrotron afterglow model operating in the
adiabatic domain and without energy injection for p values greater than 2.
They showed that an intermediate value of the density profile for constant
($\rm k=0$) and stellar ($\rm k=2$) medium was compatible with these results.
In addition, they hypothesized that the enormous progenitor may have erupted
before the core collapse event, which would have caused a change in the
external density profile.

In the direction of this burst, 17 events with photon energies greater than 10
GeV in the rest frame were reported, with the highest at 126.1 GeV and the
lowest at 14.5 GeV. Both events arrived at 243 s and 23.2 s after the trigger
time, respectively. A derivation of the maximum energy of synchrotron
radiation for radiative/injection regime and with/without energy displays
these photons can be hardly explained in the synchrotron afterglow scenario
(see Eqs. 5 and 15). Even while the temporal and spectral indices of GRBs
recorded by Fermi-LAT may be understood in the synchrotron scenario, it cannot
explain photons with energy exceeding the synchrotron limit for any value of
$\epsilon$ and $q$. If the magnetic field decays downstream of the shock on a
sufficiently small scale, the maximum synchrotron frequency may be increased
by a factor of a few over the one-zone limit (Kumar et al., 2012). In
consideration of the above, a new mechanism, such as SSC or hadronic
interactions, is most suggested rather than the synchrotron scenario.
Potentially high-energy photons exceeding the synchrotron limit may be
produced through photo-hadronic interactions (e.g., see Waxman & Bahcall,
1997), but the IceCube collaboration’s reports of no neutrino-GRB coincidences
put a severe limit on how many hadrons may be in play during these collisions
(Abbasi et al., 2012; Aartsen et al., 2016, 2015).

Considering the CRs reported in Tables 1 and 2, we plot the set of parameters
$k$, $\epsilon$ and $q$ that satisfy the temporal and spectral indexes
reported in Ajello et al. (2019) for GRB 130427A. The left-hand panel in
Figure 3 shows the allowed parameter space of $k$ and $\epsilon$ for $p=2.01$
and $2.2$ and the right-hand panel displays the allowed values of $k$ and $q$
for $p=2.2$ and $p=2.4$. These panels show that as $p$ increases $\epsilon$
decreases, and as $p$ increases $q$ increases. Any value of $p>2.4$ ($p<2.2$)
and $\epsilon>0$ ($q>1$) satisfies the CRs. It is worth noting that from this
figure can be inferred that for the adiabatic regime ($\epsilon=0$) and
without energy injection $q=1$, a spectral index of $p\simeq 2.2$ is obtained,
which agree the value reported in Kouveliotou et al. (2013). We want to
highlight that a millisecond magnetar is discarded as progenitor.

It is worth noting that to explain the afterglow of GRB 130427A, Beniamini et
al. (2022) derived a set of new CRs and proposed that the emission results
from an underlying jet with a shallow angular profile in which larger viewing
angles gradually dominate the emission.

###  4.2 Evolution of the magnetic microphysical parameter

We can estimate the microphysical parameter associated with the total energy
given to amplify the magnetic field using Eqs. 6 and 16, and the passage of
the synchrotron cooling break through the Fermi-LAT band ($\nu^{\rm ssc}_{\rm
c}=\nu_{\rm LAT}$). In the radiative and energy injection scenario, this
parameter corresponds to the values of

$\varepsilon_{B}\lesssim 2.5\times
10^{-4}\left(\frac{1+z}{1.022}\right)^{-\frac{6(2+\epsilon)}{7(8-\epsilon)}}\,(1+Y(\gamma_{c}))^{-\frac{8}{7}}A_{k}^{\frac{7\epsilon-36}{7(8-\epsilon)}}E_{53}^{-\frac{20}{7(8-\epsilon)}}\Gamma_{0,2.8}^{\frac{20\epsilon}{7(8-\epsilon)}}t_{3}^{\frac{4(2\epsilon-1)}{7(8-\epsilon)}}\left(\frac{h\nu^{\rm
ssc}_{c}}{100\,{\rm MeV}}\right)^{-\frac{2}{7}}$ (20)

and

$\varepsilon_{B}\lesssim 2.2\times
10^{-4}\left(\frac{1+z}{1.022}\right)^{-\frac{3}{14}}\,(1+Y(\gamma_{c}))^{-\frac{8}{7}}A_{k}^{-\frac{9}{14}}E_{53}^{-\frac{5}{14}}t_{3}^{\frac{5q-6}{14}}\left(\frac{h\nu^{\rm
ssc}_{c}}{100\,{\rm MeV}}\right)^{-\frac{2}{7}}$ (21)

respectively. The values of this parameter are in the range of values
($10^{-5}\leq\varepsilon_{B}\leq 10^{-1}$) reported by several authors after
describing the multiwavelength afterglow observations (Wijers & Galama, 1999;
Panaitescu & Kumar, 2002; Yost et al., 2003; Panaitescu, 2005; Wang & Dai,
2013; Santana et al., 2014; Beniamini et al., 2015).

We can notice that for $Y(\gamma_{c})\ll 1$, Eqs. 20 and 21 are not altered,
and for $Y(\gamma_{c})\gg 1$, $\epsilon_{B}$ increases. For instance, for
$Y(\gamma_{c})\gg 1$ and with the spectral breaks in the hierachy $\nu^{\rm
syn}_{\rm m}<h\nu^{\rm syn}_{\rm KN}(\gamma_{*})<\nu^{\rm syn}_{\rm
c}<\nu^{\rm syn}_{\rm KN}(\gamma_{\rm c})$, the new Compton parameter becomes
$Y(\gamma_{*})\simeq 1$ for typical values of GRB afterglow (see Wang et al.,
2010). In all cases with the chosen parameters, $\varepsilon_{B}$ lies in the
range of values reported in Santana et al. (2014) when the SSC afterglow model
is considered. It is essential to mention that with a different set of
parameters, e.g., earlier times ($t\sim 1\,{\rm s}$), the spectral breaks
change in the hierarchy, and therefore, the KN effect is included. For
instance, Wang et al. (2010) discussed the most general range of parameters:
$10^{-6}<\varepsilon_{B}<10^{-1}$, $10^{-3}<\frac{A_{k}}{\rm cm^{-3}}<10$
(with $k=0$) for $\varepsilon_{e}=0.1$ and $E=10^{54}\,{\rm erg}$, for
$t=1\,{\rm s}$ and $t=10\,{\rm s}$ in a constant-density medium. They showed
regions in their Fig. 1 where the spectral breaks move to other positions in
the hierarchy, and therefore, the KN effect is relevant for $t=1\,{\rm s}$ and
not for $t=10\,{\rm s}$, as discussed.

Tak et al. (2019) methodically examined the CRs of 59 LAT-detected bursts,
using the temporal and spectral indices. The authors showed that these bursts
satisfy the CRs of the slow-cooling regime, but only with the condition of
having an exceptionally small value of $\epsilon_{B}<10^{-7}$. We found that
in the SSC afterglow scenario, with or without energy injection, this
parameter lies in the range of $3.5\times 10^{-5}\leq\epsilon_{B}\leq 0.33$,
similar to reported in other afterglow modelling (e.g., see Santana et al.,
2014).

##  5 Summary

We have extended the work presented in Fraija et al. (2022) and derived the
CRs of the SSC afterglow model evolving in a density-profile described by
$n(r)\propto r^{\rm-k}$ with and without energy injection. Here, we have
considered the values of density-profile index of $k=0.5$, $1$, $1.5$ and
$2.5$, and relativistic electrons decelerated during the adiabatic and
radiative afterglow phase with spectral index of $1<p<2$ and $p>2$. We have
performed an analysis between these CRs and the spectral and temporal indices
of bursts that were reported in 2FLGC. In the radiativo scenario without
energy injection, we have summarized the number and percentage of bursts
following each cooling condition. This best-case scenario is most consistent
with ${\rm max\\{\nu^{\rm ssc}_{\rm m},\nu^{\rm ssc}_{\rm c}\\}<\nu}$ for
$k=0.5$ (22 GRBs, 25.88%), $k=1.0$ (22 GRBs, 25.88%), $k=1.5$ (21 GRBs,
24.71%) and $k=2.5$ (19 GRBs, 22.35%). There is one burst that satisfies the
CRs for each value of $k$ when the SSC model evolve in the cooling condition
${\rm\nu^{\rm ssc}_{\rm c}\leq\nu\leq\nu^{\rm ssc}_{\rm m}}$, and few bursts
for ${\rm\nu^{\rm ssc}_{\rm m}\leq\nu\leq\nu^{\rm ssc}_{\rm c}}$ for $k=0.5$
(3 GRBs, 3.53%), $1$ (2 GRBs, 2.35%) and $1.5$ (1 GRB, 1.18%). In the
adiabatic scenario without energy injection, we have summarized the number and
percentage of bursts following each cooling condition. The best-case scenario
is most consistent with ${\rm max\\{\nu^{\rm ssc}_{\rm m},\nu^{\rm ssc}_{\rm
c}\\}<\nu}$ for $k=0.5$ (32 GRBs, 37.65%), $k=1.0$ (32 GRBs, 37.65%), $k=1.5$
(34 GRBs, 40.0%) and $k=2.5$ (40 GRBs, 47.06%). For the cooling condition
${\rm\nu^{\rm ssc}_{\rm m}\leq\nu\leq\nu^{\rm ssc}_{\rm c}}$, 28 (25 and 7)
GRBs for $k=0.5$ ($1$ and $1.5$) fulfill the CRs. There is one burst that
satisfies the CRs for each value of $k$ when the SSC model evolve in the
cooling condition ${\rm\nu^{\rm ssc}_{\rm c}\leq\nu\leq\nu^{\rm ssc}_{\rm
m}}$.

The standard synchrotron forward-shock model, which is valid up to the
synchrotron limit, is able to provide an adequate explanation for both the
afterglow of a GRB as well as the values of its spectral and temporal indices.
Both of these phenomena can be shown to be adequately described by the model.
Although, we have analyzed GRB 130427A, there have been 29 bursts in 2FLGC
that have reported photons with energies more than 10 GeV, which cannot be
well defined by the synchrotron scenario. We propose that the SSC afterglow
model is more suitable for comprehending this sample of bursts than the
synchrotron model, which predicts that the greatest photon energy released
during the deceleration phase would be roughly $\approx 1\,{\rm GeV}$. Photo-
hadronic interactions are ruled out since neutrino non-coincidences with GRBs
have been recorded by the IceCube collaboration (Abbasi et al., 2012; Aartsen
et al., 2016, 2015). This sample of 29 bursts may be best explained by SSC
process, although the CRs of the synchrotron standard model in some cases
could describe them.

After selecting 59 LAT-detected bursts, Tak et al. (2019) methodically
examined their CRs using temporal and spectral indices. For example, they
found that although the standard synchrotron emission describes the spectrum
and temporal indices in most instances, there is still a large fraction of
bursts that can poorly be described with this model. They showed, however,
that many GRBs satisfy the CRs of the slow-cooling regime, but only when the
magnetic microphysical parameter has an exceptionally small value of
$\epsilon_{B}<10^{-7}$. Here, we show that the SSC afterglow model, with or
without energy injection, produces a value of $3.5\times
10^{-5}\leq\epsilon_{B}\leq 0.33$ (e.g., see Santana et al., 2014). Therefore,
the CRs of SSC afterglow models are needed, together with the research
provided by Tak et al. (2019), to account for the bursts in the 2FLGC that
cannot be explained in the synchrotron scenario (e.g., those with photon
energies above 10 GeV).

## Acknowledgements

We thank Peter Veres and Tanmoy Laskar for useful discussions. NF acknowledges
financial support from UNAM-DGAPA-PAPIIT through grant IN106521.

## Data Availability

The data used by this article was taken from the Second Gamma-ray Burst
Catalog (2FLGC; Ajello et al., 2019).

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Table 1: CRs of the SSC forward-shock scenario evolving in a stratified environment and adiabatic/radiative regime. CRs are shown for fast- and slow-cooling regime without energy injection |  | $\beta$ | $\alpha(p)$ | $\alpha(p)$ | $\alpha(\beta)$ | $\alpha(\beta)$
---|---|---|---|---|---|---
|  |  | $(1<p<2)$ | $(p>2)$ | $(1<p<2)$ | $(p>2)$
|  |  | Slow cooling |  |  |
1 | $\nu^{\rm ssc}_{m}<\nu<\nu^{\rm ssc}_{c}$ | $\frac{p-1}{2}$ | $\scriptsize{\frac{4(7+4\epsilon)+k[6-p(2-\epsilon)-13\epsilon]}{4[2(4-k)-\epsilon]}}$ | $-\frac{4(11-9p-4\epsilon)-k[(22-9\epsilon-p(10+\epsilon)]}{4[2(4-k)-\epsilon]}$ | $\frac{2(7+4\epsilon)+k[2-6\epsilon-\beta(2-\epsilon)]}{2[2(4-k)-\epsilon]}$ | $-\frac{4(1-2\epsilon-9\beta)-k[6-5\epsilon-\beta(10+\epsilon)]}{2[2(4-k)-\epsilon]}$
2 | $\nu^{\rm ssc}_{c}<\nu$ | $\frac{p}{2}$ | $\frac{8(4+\epsilon)-k[4+p(2-\epsilon)+6\epsilon]}{4[2(4-k)-\epsilon]}$ | $-\frac{40-36p-2k(6-\epsilon)-8\epsilon+kp(10+\epsilon)}{4[2(4-k)-\epsilon]}$ | $\frac{4(4+\epsilon)-k[2+\beta(2-\epsilon)+3\epsilon]}{2[2(4-k)-\epsilon]}$ | $-\frac{20-36\beta-k(6-\epsilon)-4\epsilon+\beta k(10+\epsilon)}{2[2(4-k)-\epsilon]}$
|  |  | Fast cooling |  |  |
3 | $\nu^{\rm ssc}_{c}<\nu<\nu^{\rm ssc}_{m}$ | $\frac{1}{2}$ | $-\frac{4-k(2-3\epsilon)-8\epsilon}{4[2(4-k)-\epsilon]}$ | $-\frac{4-k(2-3\epsilon)-8\epsilon}{4[2(4-k)-\epsilon]}$ | $-\frac{[4-k(2-3\epsilon)-8\epsilon]\beta}{2[2(4-k)-\epsilon]}$ | $-\frac{[4-k(2-3\epsilon)-8\epsilon]\beta}{2[2(4-k)-\epsilon]}$
4 | $\nu^{\rm ssc}_{m}<\nu$ | $\frac{p}{2}$ | $\frac{8(4+\epsilon)-k[4+p(2-\epsilon)+6\epsilon]}{4[2(4-k)-\epsilon]}$ | $-\frac{40-36p-2k(6-\epsilon)-8\epsilon+kp(10+\epsilon)}{4[2(4-k)-\epsilon]}$ | $\frac{4(4+\epsilon)-k[2+\beta(2-\epsilon)+3\epsilon]}{2[2(4-k)-\epsilon]}$ | $-\frac{20-36\beta-k(6-\epsilon)-4\epsilon+\beta k(10+\epsilon)}{2[2(4-k)-\epsilon]}$
Table 2: CRs of the SSC forward-shock scenario with energy injection evolving in a stratified environment. CRs are shown for fast- and slow-cooling regime. |  | $\beta$ | $\alpha(p)$ | $\alpha(p)$ | $\alpha(\beta)$ | $\alpha(\beta)$
---|---|---|---|---|---|---
|  |  | $(1<p<2)$ | $(p>2)$ | $(1<p<2)$ | $(p>2)$
|  |  | Slow cooling |  |  |
1 | $\nu^{\rm ssc}_{m}<\nu<\nu^{\rm ssc}_{c}$ | $\frac{p-1}{2}$ | $-\frac{12-k[16-p(2-q)-13q]-26q}{4(4-k)}$ | $-\frac{36-14q-6p(2+q)-k[20-9q-p(4+q)]}{4(4-k)}$ | $-\frac{6-k[7-6q-\beta(2-q)]-13q}{2(4-k)}$ | $-\frac{12-10q-6\beta(2+q)-k[8-5q-\beta(4+q)]}{2(4-k)}$
2 | $\nu^{\rm ssc}_{c}<\nu$ | $\frac{p}{2}$ | $\frac{16q+k[4-p(2-q)-6q]}{4(4-k)}$ | $-\frac{4(6-q)-6p(2+q)-k[2(4-q)-p(4+q)]}{4(4-k)}$ | $\frac{8q+k[2-\beta(2-q)-3q]}{2(4-k)}$ | $-\frac{2(6-q)-6\beta(2+q)-k[4-q-\beta(4+q)]}{2(4-k)}$
|  |  | Fast cooling |  |  |
3 | $\nu^{\rm ssc}_{c}<\nu<\nu^{\rm ssc}_{m}$ | $\frac{1}{2}$ | $-\frac{12-10q-4k+3kq}{4(4-k)}$ | $-\frac{12-10q-k(4-3q)}{4(4-k)}$ | $-\frac{[12-10q-4k+3kq]\beta}{2(4-k)}$ | $-\frac{[12-10q-4k+3kq]\beta}{2(4-k)}$
4 | $\nu^{\rm ssc}_{m}<\nu$ | $\frac{p}{2}$ | $\frac{16q+k[4-p(2-q)-6q]}{4(4-k)}$ | $-\frac{4(6-q)-6p(2+q)-k[2(4-q)-p(4+q)]}{4(4-k)}$ | $\frac{8q+k[2-\beta(2-q)-3q]}{2(4-k)}$ | $-\frac{2(6-q)-6\beta(2+q)-k[4-q-\beta(4+q)]}{2(4-k)}$
Table 3: Number and percentage of bursts satisfying each CR, summarized for the SSC forward-shock model in partially radiative regime ($\epsilon=0.5$) without energy injection. $\nu$ Range | $k$ | CR: 1 $<$ p $<$ 2 | CR: 2 $<$ p | GRBs Satisfying Relation | Proportion Satisfying Relation
---|---|---|---|---|---
${\rm\nu_{m}^{\rm ssc}<\nu<\nu_{c}^{\rm ssc}}$ | $0.5$ | $\frac{10(3+\epsilon)-\beta(2-\epsilon)}{4(7-\epsilon)}$ | $-\frac{2-13\epsilon-\beta(62-\epsilon)}{4(7-\epsilon)}$ | 3 | 3.53%
| 1.0 | $\frac{2(8+\epsilon)-\beta(2-\epsilon)}{2(6-\epsilon)}$ | $\frac{2+3\epsilon+\beta(26-\epsilon)}{2(6-\epsilon)}$ | 2 | 2.35%
| 1.5 | $\frac{2(17-\epsilon)-3\beta(2-\epsilon)}{4(5-\epsilon)}$ | $\frac{10+3\epsilon+3\beta(14-\epsilon)}{4(5-\epsilon)}$ | 1 | 1.18%
| 2.5 | $\frac{2(19-7\epsilon)-5\beta(2-\epsilon)}{4(3-\epsilon)}$ | $\frac{22-9\epsilon+\beta(22-5\epsilon)}{4(3-\epsilon)}$ | 0 | 0.00%
${\rm\nu_{c}^{\rm ssc}<\nu<\nu_{m}^{\rm ssc}}$ | 0.5 | $-\frac{(6-5\epsilon)\beta}{2(7-\epsilon)}$ | $-\frac{(6-5\epsilon)\beta}{2(7-\epsilon)}$ | 1 | 1.18%
| 1.0 | $-\frac{(2-5\epsilon)\beta}{2(6-\epsilon)}$ | $-\frac{(2-5\epsilon)\beta}{2(6-\epsilon)}$ | 1 | 1.18%
| 1.5 | $-\frac{(2+\epsilon)\beta}{2(5-\epsilon)}$ | $-\frac{(2+\epsilon)\beta}{2(5-\epsilon)}$ | 1 | 1.18%
| 2.5 | $-\frac{(7\epsilon-2)\beta}{2(3-\epsilon)}$ | $-\frac{(7\epsilon-2)\beta}{2(3-\epsilon)}$ | 1 | 1.18%
${\rm max\\{\nu_{m}^{\rm ssc},\nu_{c}^{\rm ssc}\\}<\nu}$ | 0.5 | $\frac{8(4+\epsilon)-[2+\beta(2-\epsilon)+3\epsilon]}{4(7-\epsilon)}$ | $-\frac{20-36\beta-\frac{1}{2}(6-\epsilon)-4\epsilon+\frac{\beta}{2}(10+\epsilon)}{2(7-\epsilon)}$ | 22 | 25.88%
| 1.0 | $\frac{4(4+\epsilon)-[2+\beta(2-\epsilon)+3\epsilon]}{2(6-\epsilon)}$ | $-\frac{20-36\beta-(6-\epsilon)-4\epsilon+\beta(10+\epsilon)}{2(6-\epsilon)}$ | 22 | 25.88%
| 1.5 | $\frac{8(4+\epsilon)-3[2+\beta(2-\epsilon)+3\epsilon]}{4(5-\epsilon)}$ | $-\frac{20-36\beta-\frac{3}{2}(6-\epsilon)-4\epsilon+\frac{3\beta}{2}(10+\epsilon)}{2(5-\epsilon)}$ | 21 | 24.71%
| 2.5 | $\frac{8(4+\epsilon)-5[2+\beta(2-\epsilon)+3\epsilon]}{4(3-\epsilon)}$ | $-\frac{20-36\beta-\frac{5}{2}(6-\epsilon)-4\epsilon+\frac{5\beta}{2}(10+\epsilon)}{2(3-\epsilon)}$ | 19 | 22.35%
Table 4: Number and percentage of bursts satisfying each CR, summarized for the SSC forward-shock model in adiabatic regime with energy injection ($q=0.5$). $\nu$ Range | $k$ | CR: 1 $<$ p $<$ 2 | CR: 2 $<$ p | GRBs Satisfying Relation | Proportion Satisfying Relation
---|---|---|---|---|---
${\rm\nu_{m}^{\rm ssc}<\nu<\nu_{c}^{\rm ssc}}$ | $0.5$ | $\frac{7q-5-\beta(2-q)}{7}$ | $-\frac{12-10q-6\beta(2+q)-\frac{1}{2}[8-5q-\beta(4+q)]}{7}$ | 28 | 32.94%
| 1.0 | $\frac{7q+1-\beta(2-q)}{6}$ | $-\frac{12-10q-6\beta(2+q)-[8-5q-\beta(4+q)]}{6}$ | 25 | 29.41%
| 1.5 | $\frac{9-5q-3\beta(2-q)}{5}$ | $-\frac{12-10q-6\beta(2+q)-\frac{3}{2}[8-5q-\beta(4+q)]}{5}$ | 7 | 8.24%
| 2.5 | $\frac{13-17q-5\beta(2-q)}{3}$ | $-\frac{12-10q-6\beta(2+q)-\frac{5}{2}[8-5q-\beta(4+q)]}{3}$ | 0 | 0.00%
${\rm\nu_{c}^{\rm ssc}<\nu<\nu_{m}^{\rm ssc}}$ | 0.5 | $\frac{(17q-20)\beta}{14}$ | $\frac{(17q-20)\beta}{14}$ | 1 | 1.18%
| 1.0 | $\frac{(7q-8)\beta}{6}$ | $\frac{(7q-8)\beta}{6}$ | 1 | 1.18%
| 1.5 | $\frac{(11q-12)\beta}{10}$ | $\frac{(11q-12)\beta}{10}$ | 1 | 1.18%
| 2.5 | $\frac{(5q-4)\beta}{6}$ | $\frac{(5q-4)\beta}{6}$ | 1 | 1.18%
${\rm max\\{\nu_{m}^{\rm ssc},\nu_{c}^{\rm ssc}\\}<\nu}$ | 0.5 | $\frac{13q+2-\beta(2-q)}{14}$ | $-\frac{2(6-q)-6\beta(2+q)-\frac{1}{2}[4-q-\beta(4+q)]}{7}$ | 32 | 37.65%
| 1.0 | $\frac{5q+2-\beta(2-q)}{6}$ | $-\frac{2(6-q)-6\beta(2+q)-[4-q-\beta(4+q)]}{6}$ | 32 | 37.65%
| 1.5 | $\frac{7q+6-3\beta(2-q)}{10}$ | $-\frac{2(6-q)-6\beta(2+q)-\frac{3}{2}[4-q-\beta(4+q)]}{5}$ | 34 | 40.00%
| 2.5 | $\frac{q+10-5\beta(2-q)}{6}$ | $-\frac{2(6-q)-6\beta(2+q)-\frac{5}{2}[4-q-\beta(4+q)]}{3}$ | 40 | 47.06%
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Figure 1: The panels display the closure relations of SSC model in fast and
slow cooling regimes for $\epsilon=0.5$ without energy injection with the
spectral and temporal indexes from 2FLGC. Panels from top to bottom are
$k=0.5$, $1$, $1.5$ and $2.5$, respectively. The purple ellipses are shown
when the CR are satisfied within 1 sigma error bars.
---
Figure 2: The same as Fig. 1, but with energy injection ($q=0.5$) and in the
adiabatic regime.
---
Figure 3: The allowed parameter spaces that satisfy the temporal and spectral
evolution of the LAT light curve of GRB 130427A. The left-hand panel shows the
allowed values of $k$ and $\epsilon$ for $p=2.01$ and $2.2$ and the right-hand
panel displays the allowed values of $k$ and $q$ for $p=2.2$ and $2.4$.