# External Inverse-Compton Emission from Low-Luminosity Gamma-Ray Bursts:
Application to GRB 190829A

B. Theodore Zhang11affiliation: Department of Physics, Pennsylvania State
University, University Park, Pennsylvania 16802, USA 22affiliation: Department
of Astronomy & Astrophysics, Pennsylvania State University, University Park,
Pennsylvania 16802, USA 33affiliation: Center for Multimessenger Astrophysics,
Institute for Gravitation and the Cosmos, Pennsylvania State University,
University Park, Pennsylvania 16802, USA , Kohta Murase11affiliation:
Department of Physics, Pennsylvania State University, University Park,
Pennsylvania 16802, USA 22affiliation: Department of Astronomy & Astrophysics,
Pennsylvania State University, University Park, Pennsylvania 16802, USA
33affiliation: Center for Multimessenger Astrophysics, Institute for
Gravitation and the Cosmos, Pennsylvania State University, University Park,
Pennsylvania 16802, USA 44affiliation: Center for Gravitational Physics,
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Kyoto
606-8502, Japan , Péter Veres55affiliation: Center for Space Plasma and
Aeronomic Research (CSPAR), University of Alabama in Huntsville, Huntsville,
AL 35899, USA , Péter Mészáros11affiliation: Department of Physics,
Pennsylvania State University, University Park, Pennsylvania 16802, USA
22affiliation: Department of Astronomy & Astrophysics, Pennsylvania State
University, University Park, Pennsylvania 16802, USA 33affiliation: Center for
Multimessenger Astrophysics, Institute for Gravitation and the Cosmos,
Pennsylvania State University, University Park, Pennsylvania 16802, USA

###### Abstract

The detection of TeV gamma-ray bursts (GRBs) brought new opportunities for
studying the physics of particle acceleration at relativistic shocks. The
H.E.S.S. telescopes recently observed very-high-energy (VHE) emission from a
nearby low-luminosity GRB, GRB 190829A. Follow-up observations with, e.g.,
Swift-XRT, revealed unusual flare activities at $\sim 10^{3}\rm~{}s$, which
can be caused by a long-lasting central engine. We show that the VHE emission
during the H.E.S.S. observation time is naturally produced in the external
inverse-Compton (EIC) scenario, where seed photons supplied by the flares or
other late-time dissipation are upscattered to VHE energies by the non-thermal
electrons accelerated at the external forward shock. Our calculations show
that the EIC flare nearly coincides with the late-prompt flare, but extends
$\sim 3-4$ times longer than the duration of the late-prompt flare. The
preferred kinetic energy and initial Lorentz factor used in our model are
$\sim 10^{52}\rm~{}erg$ and $\sim 20$, respectively. Understanding the
mechanisms of the VHE emission from low-luminosity GRBs will help us constrain
the properties of the outflow and the central engine activities, as well as
the particle acceleration mechanism.

non-thermal, supernovae, radiative processes, gamma-ray bursts

##  1 Introduction

Very high energy (VHE) $\gamma$-rays ($\gamma$-rays with energy greater than
$\sim 100\rm~{}GeV$) have been observed by ground-based Cherenkov telescopes
(e.g., Hinton & Hofmann, 2009; Inoue et al., 2013). The sources of VHE
$\gamma$-rays represent extreme astrophysical accelerators in the universe. In
the multimessenger era, the detection of VHE $\gamma$-rays along with
multiwavelength electromagnetic radiation, neutrinos, cosmic rays, as well as
gravitational waves can help to unveil the mechanisms of high-energy
astrophysical processes (e.g., Murase & Bartos, 2019; Mészáros et al., 2019).

The recent detection of two TeV gamma-ray bursts (GRBs), GRB 190114C (Acciari
et al., 2019a, b) and GRB 180720B (Abdalla et al., 2019), has opened a new
window for studying GRBs in the VHE band, providing us with new opportunities
to investigate the acceleration and radiation processes (see Mészáros, 2006;
Kumar & Zhang, 2014, for reviews). VHE $\gamma$-rays originating from GRBs can
be naturally explained with the inverse-Compton (IC) process by relativistic
electrons (Mészáros & Rees, 1994; Dermer et al., 2000; Sari & Esin, 2001;
Zhang & Mészáros, 2001). The origins of VHE $\gamma$-rays from GRB 190114C and
GRB 180720B have been widely discussed in the framework of the synchrotron
self-Compton (SSC) scenario (Derishev & Piran, 2019; Abdalla et al., 2019;
Fraija et al., 2019; Wang et al., 2019; Acciari et al., 2019b; Zhang et al.,
2019; Asano et al., 2020), where the same population of electrons that emit
synchrotron photons can also upscatter these photons to much higher energies
by a factor of $\sim\gamma_{e}^{2}$, where $\gamma_{e}$ is the electron
Lorentz factor (Blumenthal & Gould, 1970; Rybicki & Lightman, 1986).

Recently, the High Energy Stereoscopic System (H.E.S.S.) reported the
detection of VHE $\gamma$-rays from GRB 190829A with a confidence level of
$>5\sigma$ (de Naurois, 2019). The measured redshift of GRB 190829A is
$z=0.0785\pm 0.005$ (Valeev et al., 2019), which is the nearest GRB detected
in the VHE band. GRB 190829A has been discussed as a possible low-luminosity
GRB (LL GRB), which has an isotropic-equivalent luminosity of $L_{\rm iso}\sim
10^{49}\rm~{}erg~{}s^{-1}$ (Chand et al., 2020). The late-time optical
observation revealed an associated broad-line type-Ic supernovae, SN 2019oyw
(Hu et al., 2020). Even though the physical origin of LL GRBs is still
controversial, there is a consensus that LL GRBs are related to the deaths of
massive stars and have properties between classical GRBs and normal supernovae
(Kumar & Zhang, 2014). The observed X-ray light curve indicates an unusual
long-lasting flare during the time interval $\sim 10^{2}-10^{4}~{}\rm s$,
which may originate from a late-time central-engine activity (Chand et al.,
2020).

It has been proposed that VHE $\gamma$-rays can also be produced via the
external inverse-Compton (EIC) process in the framework of GRBs, where the
seed photons (produced in a region different from the acceleration zone) can
be X-ray flares (Wang et al., 2006; He et al., 2012), late-time long-lasting
emission (Murase et al., 2011, 2018; Veres & Mészáros, 2012), prompt emission
(Murase et al., 2010), shock breakout emission (Wang & Mészáros, 2006),
hypernova envelope emission (He et al., 2009), and cocoon emission (Toma et
al., 2009; Kimura et al., 2019). If the long-lasting X-ray flare photons in
GRB 190829A catch up with non-thermal electrons accelerated at the external
forward shock, then these lower-energy photons can be upscattered to the VHE
band. In the case of anisotropic scattering with target photons impinging from
behind, the EIC flux of VHE $\gamma$-rays is reduced by a factor of two or
three compared to the simplest estimate by the product of the scattering
optical depth and seed photon flux (Brunetti, 2000; Fan et al., 2008; Murase
et al., 2011). However, if the target photons have a larger energy density
compared to synchrotron photons as in the case of late-time flares, then VHE
$\gamma$-rays from the EIC emission can be dominant.

In this work, we study the origin of VHE $\gamma$-rays in the EIC scenario,
with an application to GRB 190829A. The paper is organized as follows. In Sec.
2, we analytically discuss the origin of high-energy emission considering both
the SSC and EIC scenarios in the framework of GRB 190829A. In Sec. 3, we show
the numerical results and make a comparison to observations. We discuss the
implications of this work and give a summary in Sec. 5. Throughout the paper,
we use cgs units, and adopt notations such as $Q_{x}\equiv Q/10^{x}$. We use E
represents observed photon energy, $\varepsilon=E(1+z)$ is the photon energy
measured in the cosmic rest frame, and
$\varepsilon^{\prime}=\varepsilon/\Gamma$ is the photon energy measured in the
source comoving frame which have Lorentz factor $\Gamma$.

##  2 High-energy afterglow emission from GRBs

We consider a relativistic outflow that has isotropic-equivalent energy
$\mathcal{E}_{k}$ and initial Lorentz factor $\Gamma_{0}$, propagating into an
external medium with constant density $n_{\rm ex}$. In this work, we
numerically calculate afterglow dynamics in detail for both relativistic and
non-relativistic regimes. However, in the following text, we mainly discuss
the results using the self-similar evolution phase of the adiabatic blast wave
(Blandford & McKee, 1976) to explain essential results. We numerically
calculate EIC emission following Murase et al. (2010) and Murase et al.
(2011), and we perform numerical calculations for synchrotron emission as
well.

The radius of the forward shock is estimated to be

$R\simeq 2.2\times 10^{17}~{}\mathcal{E}_{k,52}^{1/4}n_{\rm
ex}^{-1/4}t_{z,4}^{1/4}\rm~{}cm,$ (1)

where $t_{z,4}=t_{4}/(1+z)$ is the redshift-corrected observation time and the
bulk Lorentz factor is

$\Gamma\simeq 14~{}\mathcal{E}_{k,52}^{1/8}n_{\rm ex}^{-1/8}t_{z,4}^{-3/8},$
(2)

where the relation between radius $R$ and observation time $t$ is $R\approx
4\Gamma^{2}ct_{z}$ which takes into account the history of the blast wave
evolution and the contribution from a range of angles within $1/\Gamma$ cone
(Waxman, 1997; Panaitescu & Mészáros, 1998; Sari et al., 1998).

In the external forward shock model, a fraction of thermal electrons can be
injected into the acceleration process and is able to accelerate to higher
energies via the diffusive shock acceleration (DSA) mechanism (e.g., Drury,
1983; Blandford & Eichler, 1987). The lower-energy synchrotron photons and
external photons can then be upscattered to the VHE band by high-energy non-
thermal electrons. The luminosity of the newly accelerated electrons can be
estimated to be $L_{e}\sim\epsilon_{e}4\pi R^{2}c\Gamma^{2}U_{\rm
sh}^{\prime}\sim 2\times
10^{47}\epsilon_{e,-1}\mathcal{E}_{k,52}t_{z,4}^{-1}\rm~{}erg~{}s^{-1}$, where
$\epsilon_{e}$ is the fraction of internal energy that goes into shocked
electrons and $U_{\rm sh}^{\prime}\sim 2\Gamma^{2}n_{\rm ex}m_{p}c^{2}$ is the
comoving internal energy density (Wang et al., 2006; Fan et al., 2008). We
approximate the synchrotron luminosity to be

$L_{\rm syn}\sim\frac{1}{1+Y_{\rm tot}}\zeta_{\gamma}L_{e},$ (3)

where $\zeta_{\gamma}$ is the fraction of electron energy that is radiated,
$Y_{\rm tot}=Y_{\rm SSC}+Y_{\rm EIC}$ is the total Compton parameter, $Y_{\rm
SSC}$ is the SSC Compton parameter defined as the ratio between the SSC
luminosity and the synchrotron luminosity, and $Y_{\rm EIC}$ is the EIC
Compton parameter defined as the ratio between the EIC luminosity and the
synchrotron luminosity. In the fast-cooling regime, the value of
$\zeta_{\gamma}$ equals to 1, while in the slow cooling regime
$\zeta_{\gamma}\sim(\gamma_{e,c}/\gamma_{e,m})^{2-s}$ where $s$ is the
electron spectral index (Sari & Esin, 2001). The minimum electron Lorentz
factor is

$\displaystyle\gamma_{e,m}$ $\displaystyle\approx$
$\displaystyle(\epsilon_{e}/f_{e})g(s)(m_{p}/m_{e})(\Gamma-1)$ (4)
$\displaystyle\sim$ $\displaystyle 400\mathcal{E}_{k,52}^{1/8}n_{\rm
ex}^{-1/8}\epsilon_{e,-1}f_{e}^{-1}t_{z,4}^{-3/8},$

where $\epsilon_{e}$ is the energy fraction of internal energy that goes into
electrons and $f_{e}$ is the number fraction of electrons that are
accelerated, and $g(s)=(s-2)/(s-1)$ for $s>2$ (we adopt $s=2.2$). The maximum
electron Lorentz factor is limited by the cooling process,

$\displaystyle\gamma_{e,M}$ $\displaystyle\approx$ $\displaystyle(6\pi
e/(\sigma_{T}B\eta(1+Y_{\rm tot})))^{1/2}$ (5) $\displaystyle\sim$
$\displaystyle 9\times 10^{8}\eta^{-1/2}(1+Y_{\rm
tot})^{-1/2}\mathcal{E}_{k,52}^{-1/16}n_{\rm
ex}^{-3/16}\epsilon_{B,-5}^{-1/4}t_{z,4}^{3/16},$

where $\eta$ is the acceleration efficiency, which depends on details the
acceleration mechanism (see, e.g., Equation 14 of Asano et al., 2020), and
$\epsilon_{B}$ is the energy fraction of internal energy that is converted
into the magnetic energy. The electron cooling Lorentz factor is,

$\displaystyle\gamma_{e,c}$ $\displaystyle\approx$ $\displaystyle\frac{6\pi
m_{e}c}{(1+Y_{\rm tot})\sigma_{T}\Gamma B^{2}t_{z}}$ (6) $\displaystyle\sim$
$\displaystyle 2\times 10^{7}(1+Y_{\rm
tot})^{-1}\mathcal{E}_{k,52}^{-3/8}n_{\rm
ex}^{-5/8}\epsilon_{B,-5}^{-1}t_{z,4}^{1/8}.$

The non-thermal electrons are in the fast cooling regime for
$\gamma_{e,m}>\gamma_{e,c}$, where nearly all of the injected electrons cool
during the dynamical time. When $\gamma_{e,m}<\gamma_{e,c}$, electrons are in
the slow cooling regime. Note in the following analytical estimates, we mainly
consider the slow cooling regime, while our numerical code can treat both
slow- and fast-cooling cases self-consistently. Note that the fraction of
electron energy that was radiated away in the slow cooling case is estimated
to be $\zeta_{\gamma}\sim 0.1(1+Y_{\rm
tot})^{s-2}\mathcal{E}_{k,52}^{-\frac{2-s}{2}}n_{\rm
ex}^{-\frac{2-s}{2}}\epsilon_{B,-5}^{s-2}\epsilon_{e,-1}^{s-2}f_{e}^{2-s}t_{z,4}^{\frac{2-s}{2}}$.
The characteristic energy of the synchrotron emission is estimated to be
$E_{m}\approx[eB/(2\pi m_{e}c)]\gamma_{m}^{2}\Gamma/(1+z)\simeq 4.2\times
10^{-4}~{}\mathcal{E}_{k,52}^{1/2}\epsilon_{e,-1}^{2}f_{e}^{-2}\epsilon_{B,-5}^{1/2}t_{4}^{-3/2}(1+z)^{1/2}\rm~{}eV$
and the corresponding cooling energy is $E_{c}\approx[eB/(2\pi
m_{e}c)]\gamma_{c}^{2}\Gamma/(1+z)\simeq 10^{6}~{}(1+Y_{\rm
tot})^{-2}\mathcal{E}_{k,52}^{-1/2}n_{\rm
ex}^{-1}\epsilon_{B,-5}^{-3/2}t_{4}^{-1/2}(1+z)^{-1/2}\rm~{}eV$. The observed
X-ray luminosity at the keV band will evolve as $L_{X}\propto t^{(3-3s)/4}$
for $E_{m}<E_{X}<E_{c}$ (Zhang et al., 2006).

The SSC luminosity can be estimated as

$\displaystyle L_{\rm SSC}\sim\frac{Y_{\rm SSC}}{1+Y_{\rm tot}}\zeta L_{e}.$
(7)

Note that $\epsilon_{e}\gg\epsilon_{B}$ is one of the conditions for dominant
SSC emission (Sari et al., 1998; Zhang & Mészáros, 2001). In the Thomson
limit, the value of $Y_{\rm SSC}$ remains constant as a function of
$\gamma_{e}$ (Sari & Esin, 2001), and the SSC luminosity will evolve as
$L_{\rm SSC}\propto\zeta_{\gamma}L_{e}\propto t_{z}^{-s/2}$. Thus, we can
expect that the SSC light curve will have a very similar trend as the X-ray
light curve for $s>2$. The characteristic energies of the SSC emission can be
estimated as $E_{m}^{\rm SSC}\approx 2\gamma_{e,m}^{2}E_{m}\simeq 1.4\times
10^{2}~{}\mathcal{E}_{k,52}^{3/4}\epsilon_{e,-1}^{4}f_{e}^{-4}\epsilon_{B,-5}^{1/2}t_{4}^{-9/4}(1+z)^{-5/4}\rm~{}eV$
and $E_{c}^{\rm SSC}\approx 2\gamma_{e,c}^{2}E_{c}\simeq 9.5\times
10^{20}~{}(1+Y_{\rm tot})^{-4}\mathcal{E}_{k,52}^{-5/4}n_{\rm
ex}^{-9/4}\epsilon_{B,-5}^{-7/2}t_{4}^{-1/4}(1+z)^{-3/4}\rm~{}eV$. In the
Thomson limit, the SSC energy spectrum in the slow cooling case is

$F_{E}^{\rm SSC}=F_{E,\rm max}^{\rm SSC}\begin{cases}(\frac{E}{E_{m}^{\rm
SSC}})^{\frac{1}{3}},&E<E_{m}^{\rm SSC}\\\ (\frac{E}{E_{m}^{\rm
SSC}})^{-\frac{(s-1)}{2}},&E_{m}^{\rm SSC}<E<E_{c}^{\rm SSC}\\\
\left(\frac{E_{c}^{\rm SSC}}{E_{m}^{\rm
SSC}}\right)^{\frac{1-s}{2}}\left(\frac{E}{E_{c}^{\rm
SSC}}\right)^{-\frac{s}{2}},&E>E_{c}^{\rm SSC}\end{cases},$ (8)

where $F_{E,\rm max}^{\rm SSC}\sim\tau_{T}F_{E,\rm max}^{\rm syn}$ and
$\tau_{T}\sim(1/3)\sigma_{T}Rn_{\rm ex}f_{e}\sim 5\times
10^{-8}\mathcal{E}_{k,52}^{1/4}n_{\rm ex}^{3/4}f_{e}t_{z,4}^{1/4}$ is the
electron scattering optical depth (Sari & Esin, 2001). The peak synchrotron
flux can be calculated as $F_{E,\rm max}^{\rm
syn}\approx(1+z)N_{e}P_{\varepsilon,\rm max}/4\pi d_{L}^{2}$ where
$P_{\varepsilon,\rm max}\approx
P(\gamma_{e,m})/\varepsilon_{m}=(c\sigma_{T}/6\pi)\gamma_{e,m}^{2}B^{2}\Gamma^{2}/\varepsilon_{m}$
is the synchrotron emission power per electron (Sari et al., 1998) and
$\varepsilon_{m}=E_{m}(1+z)$.

Similarly, the EIC luminosity is

$\displaystyle L_{\rm EIC}\sim\frac{Y_{\rm EIC}}{1+Y_{\rm
tot}}\zeta_{\gamma}L_{e},$ (9)

The EIC light curve depends on the time evolution of external photons. For
prompt or flare photons as targets, the observed EIC light curve usually
appears as an extended bump compared to the SSC light curve (Murase et al.,
2010). On the other hand, for long-lasting photons as target photons, the EIC
light curve is flatter than the SSC light curve which may dominate at later
times (Murase et al., 2011). Assuming that the external photons can be
described as a broken power law with break energy $E_{b}$, then the
characteristic energies of EIC emission are $E_{m}^{\rm EIC}\approx
2\gamma_{e,m}^{2}E_{b}\simeq 3.3\times 10^{7}~{}\mathcal{E}_{k,52}^{1/4}n_{\rm
ex}^{-1/4}\epsilon_{e,-1}^{2}f_{e}^{-2}t_{4}^{-3/4}(1+z)^{3/4}E_{b,2}\rm~{}eV$
and $E_{c}^{\rm EIC}\approx 2\gamma_{e,c}^{2}E_{b}\simeq 8.6\times
10^{16}~{}(1+Y_{\rm tot})^{-1}\mathcal{E}_{k,52}^{-3/4}n_{\rm
ex}^{-5/4}\epsilon_{B,-5}^{-2}t_{4}^{1/4}(1+z)^{-1/4}E_{b,2}\rm~{}eV$. We
model the energy spectrum of the flare as a broken power law without going
into the details of the emission mechanism,

$F^{\rm fl}_{E}=F^{\rm
fl}_{E_{b}}(t)\begin{cases}\left(\frac{E}{E_{b}}\right)^{-\alpha+1},&E<E_{b}\\\
\left(\frac{E}{E_{b}}\right)^{-\beta+1},&E>E_{\rm b}\end{cases},$ (10)

where $E_{b}$ is the break frequency measured in the observer frame, $F^{\rm
fl}_{E_{b}}(t)$ is the peak flux at $E_{b},$ $\alpha$ and $\beta$ are the
spectral indices. The energy spectrum of EIC emission in the Thomson limit is

$F_{E}^{\rm EIC}=F_{E,\rm max}^{\rm EIC}\begin{cases}\left(\frac{E}{E_{m}^{\rm
EIC}}\right)^{1-\alpha},&E<E_{m}^{\rm EIC}\\\ \left(\frac{E}{E_{m}^{\rm
EIC}}\right)^{-\frac{(s-1)}{2}},&E_{m}^{\rm EIC}<E<E_{c}^{\rm EIC}\\\
\left(\frac{E_{c}^{\rm EIC}}{E_{m}^{\rm
EIC}}\right)^{\frac{1-s}{2}}\left(\frac{E}{E_{c}^{\rm
EIC}}\right)^{-\frac{s}{2}},&E>E_{c}^{\rm EIC}\end{cases},$ (11)

where $F_{E,\rm max}^{\rm EIC}\sim\tau_{T}xF_{E_{b}}^{\rm fl}$,
$F_{E_{b}}^{\rm fl}$ is the peak flux of the late-prompt emission, and $x<1$
is a factor due to the anisotropic scattering process (Murase et al., 2011).

The SSC (EIC) Compton parameter is expressed as the ratio of SSC (EIC)
emission power to synchrotron emission power,

$Y_{\rm SSC(EIC)}(\gamma_{e})\approx\frac{P_{\rm SSC(\rm EIC)}}{P_{\rm
syn}}\sim\frac{U_{\rm syn(FL)}^{\prime}[\varepsilon^{\prime}<\varepsilon_{\rm
KN}^{\prime}]}{U_{\rm B}^{\prime}},$ (12)

where $P_{\rm SSC}$ is the SSC emission power, $P_{\rm EIC}$ is the EIC
emission power, $P_{\rm syn}$ is the synchrotron emission power, $U_{\rm
syn}^{\prime}$ is the comoving synchrotron photon energy density, where we
introduce the Klein-Nishina energy $\varepsilon_{\rm KN}^{\prime}\sim
m_{e}c^{2}/\gamma_{e}$ to take into account Klein-Nishina effect, $U_{\rm
FL}^{\prime}$ is the comoving photon density of flares, and
$U_{B}^{\prime}=B^{2}/8\pi$ is the comoving magnetic energy density. If we
neglect EIC cooling, the value of $Y_{\rm SSC}$ can be estimated to be $Y_{\rm
SSC}\sim(\epsilon_{e}/\epsilon_{B})^{1/(4-s)}(\gamma_{e,m}/\gamma_{c,\rm
syn})^{-(s-2)/(2(s-4))}\sim 90$ at $t\sim 10^{4}\rm~{}s$ in the Thomson regime
(Sari & Esin, 2001; Liu et al., 2013).

It has been shown that both the synchrotron spectrum and the SSC (EIC)
spectrum can be affected in the Klein-Nishina regime (see, e.g., Nakar et al.,
2009; Wang et al., 2010; Murase et al., 2011). The Klein-Nishina effect on the
high-energy IC emission leads to the spectral suppression when the upscattered
photons have energies beyond the critical energy (Blumenthal & Gould, 1970).
In the observer frame, the characteristic break energies are $E_{\rm
KN}^{m}\approx\Gamma\gamma_{e,m}m_{e}c^{2}/(1+z)\simeq 2.9\times
10^{9}~{}\mathcal{E}_{k,52}^{1/4}n_{\rm
ex}^{-1/4}\epsilon_{e,-1}f_{e}^{-1}t_{z,4}^{-3/4}(1+z)^{-1}\rm~{}eV$ and
$E_{\rm KN}^{c}\approx\Gamma\gamma_{e,c}m_{e}c^{2}/(1+z)\simeq 1.5\times
10^{14}~{}(1+Y_{\rm tot})^{-1}\mathcal{E}_{k,52}^{-1/4}n_{\rm
ex}^{-3/4}\epsilon_{B,-5}^{-1}t_{z,4}^{-1/4}(1+z)^{-1}\rm~{}eV$. The Klein-
Nishina break energy can be lower than either $E_{c}^{\rm SSC}$ or $E_{c}^{\rm
EIC}$. The energy spectrum in Equation 8 and Equation 11 can be further
affected by the Klein-Nishina effect, where the spectrum is steepened beyond
the Klein-Nishina break energy (see Eqs. 24 and 25 of Murase et al., 2011).

This work focuses on the case that EIC emission dominates over SSC emission
because the peak flux of the flare emission is much larger than the afterglow
synchrotron emission, i.e., $F_{E_{b}}^{\rm FL}>F_{E,\rm max}^{\rm syn}$.
However, the evolution of the blast wave may still be in the coasting phase
during the stage of the flare emission. In our numerical calculations, we
consider the coasting phase, deceleration phase and non-relativistic phase by
solving a series of partial differential equations. See Appendix A for
details. Note the emission during the non-relativistic evolution phase is
necessary for explaining the late-time radio data (Rhodes et al., 2020). By
extending the method adopted in Murase et al. (2011), we numerically calculate
spectra of synchrotron, SSC, and EIC emission. We take into account the equal-
arrival-time surface (EATS) for not only the EIC component but also the
synchrotron and SSC components. See details in Appendix C.

##  3 Modeling high-energy gamma-ray emission from GRB 190829A

Following Chand et al. (2020), we fit the observed X-ray light curve with two
separate components, the late-time flare emission and external forward shock
emission, respectively. The X-ray light curve drops very quickly during the
flare stage, which is difficult to explain in the standard afterglow model
with typical electron spectral indices of $s\sim 2-2.5$. The late-time X-ray
flare can be fitted with the Norris model (Norris et al., 2005; Chand et al.,
2020),

$F^{\rm fl}_{\varepsilon_{b}}(t)=A\lambda
e^{-\frac{\tau_{1}}{t-t_{i}}-\frac{t-t_{i}}{\tau_{2}}},$ (13)

where $A$ is the pulse amplitude, $t_{i}$ is the pulse start time, $\tau_{1}$
is the pulse r, andise parameter $\tau_{2}$ is the pulse decay parameter,
$\lambda$ is defined as the normalization constant $\lambda={\rm
exp}[2(\tau_{1}/\tau_{2})^{1/2}]$. The best-fit results of the above
parameters are $A=225\rm~{}\mu Jy$, $\tau_{1}=90\rm~{}s$,
$\tau_{2}=3993\rm~{}s$, and $t_{i}=950\rm~{}s$. The corresponding values used
in Equation 10 are $\alpha=1$, $\beta=2.5$ and $E_{b}=100\rm~{}eV$, which is
optimized for brighter EIC fluxes.

In Fig. 1, we show the multi-wavelength light curve from radio, optical, X-ray
to VHE band. The X-ray light curve at 1 keV that has been observed by Swift-
XRT is taken from the public online repository111We convert the observed flux
at 0.3-10 keV to that at 1 keV, assuming a spectral index of $\Gamma_{X}=1.8$.
(Evans et al., 2010). We can see the observed X-ray flux is well explained
with both contributions from the late-prompt flare and forward shock emission.
Assuming the opening angle of the outflow is $\theta_{j}\sim 0.2$ radian, the
jet break occurred at $t\sim 10^{5}\rm~{}s$. Although there is a lateral
expansion, the spectral decline after the jet break is known to be dominated
by the geometrical effect, so we simply multiply a correction factor
$\theta_{j}^{2}\Gamma^{2}$ to the observed flux reduction after the jet break
(Zhang, 2018). The pink triangles are the optical $i$-band data observed by
the Gran Telescopio CANARIAS (GTC) after the correction for both Galactic and
host galaxy extinction (Hu et al., 2020). As done for X-rays, we fit the
optical light curve as a combination of the late-prompt flare and external
forward shock emission. Note that the bump that appeared in the late-time
optical data should be attributed to supernova emission. We also show the
radio data at 1.3 GHz band (yellow squares) and 15.5 GHz band (blue points)
observed by Meer Karoo Array Telescope (MeerKAT) and Arcminute Microkelvin
Imager - Large Array (AMI-LA) one day after the burst, respectively (Rhodes et
al., 2020). In our model, the radio light curve at 1.3 GHz and 15.5 GHz can be
explained in the external forward shock model, even though possible
contributions may also come from the external reverse shock (Rhodes et al.,
2020). Note that we include an additional parameter $f_{e}$ in our model to
consider the case where only a fraction of the electrons are injected into the
acceleration process, which can affect the minimum electron Lorentz factor
$\gamma_{e,m}$ as well as the flux at the radio band (Samuelsson et al.,
2020).

Figure 1: Multi-wavelength light curve from radio, optical, X-ray to TeV
bands. We show both the forward shock (FS) light curve (thick lines) and the
late-prompt (LP) light curve (thin lines). The H.E.S.S. observation time
window is indicated within black dashed vertical lines. The relevant physical
parameters are $\mathcal{E}_{k}=9.8\times 10^{51}\rm~{}erg$, $n_{\rm
ex}=0.09\rm~{}cm^{-3}$, $\epsilon_{e}=0.39$, $f_{e}=0.34$,
$\epsilon_{B}=8.7\times 10^{-5}$, $s=2.1$, $\theta_{j}=0.2$, $\Gamma_{0}=25$,
$\alpha=1$, $\beta=2.5$ and $E_{b}=100\rm~{}eV$

.

We show the SSC and EIC light curves at 500 GeV as dashed and dotted lines,
respectively. The EIC light curve dominates the high-energy emission at $t\sim
10^{3}-10^{4}\rm~{}s$, and gradually declines due to the decrease of the late-
prompt flare emission. The SSC light curve becomes dominant at later times,
$t\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}3\times 10^{4}\rm~{}s$. We define
$\theta_{\rm sc}^{\prime}$ as the scattering angle relative to the photon beam
measured in the external forward shock comoving frame. The EIC emission
diminishes at $\theta_{\rm sc}^{\prime}=0$. See Appendix C for more details.
The EIC emission nearly coincides with the late-prompt flare, but it extends a
factor of $\sim 3-4$ times longer than the duration of the flare (Murase et
al., 2011). Despite the rapid evolution of the late-prompt flare, the EIC flux
decreases slowly due to the effect of the integration over the EATS. Note that
for a given time the angle at which non-thermal electrons produce the dominant
fraction of the EIC flux is $\theta\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}\Gamma^{-1}$ (Murase et al., 2011), which
is smaller than the jet opening angle $\theta_{j}$ before the jet break.

In Fig. 2, we show energy spectra of various components including the flare,
synchrotron, SSC, and EIC at t = $3.5\rm~{}hrs$. The synchrotron emission
peaks around $E_{c}\sim 10^{6}\rm~{}eV$ and the characteristic energy is
$E_{m}\sim 10^{-1}\rm~{}eV$. It is clear that the upscattering of these
higher-energy photons is limited by the Klein-Nishina effect. The maximum EIC
flux is observed at $t\sim 3.5\rm~{}hr$ with a peak flux of $\sim
10^{-10.5}\rm~{}erg~{}cm^{-2}~{}s^{-1}$. At the same time, the SSC flux is
$\sim 5$ times lower than the EIC flux, as shown in Fig. 2. The cutoff of the
VHE emission at $\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}500\rm~{}GeV$ is dominated by the EBL
attenuation during their propagation from the source to Earth. Note that the
effect of the Klein-Nishina break energy is difficult to observe due to the
strong EBL attenuation.

Figure 2: Same as Fig. 1, we show energy spectra of the synchrotron, SSC, EIC,
and late-prompt (LP) flare emission at $t=4.4\rm~{}hr$. The thick green, red
and yellow lines take into account both of the internal $\gamma\gamma$
absorption and EBL attenuation. The H.E.S.S. VHE spectra data on the first
night are taken from Abdalla et al. (2021).

VHE $\gamma$-rays from GRB 190829A have been detected by H.E.S.S. with a
confidence level $>5\sigma$ (de Naurois, 2019). The observations started at
$\sim 1.5\times 10^{4}\rm~{}s$ after the GRB trigger, where the integral flux
beyond 200 GeV reached $\sim 4\times 10^{11}\rm~{}erg~{}cm^{-2}~{}s^{-1}$ and
gradually decline following the same trend as X-rays until $t\sim 2\times
10^{5}\rm~{}s$ (Abdalla et al., 2021). Our calculations predict that the high-
energy EIC emission has a flux of $\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}10^{-11}\rm~{}erg~{}cm^{-2}~{}s^{-1}$
consistent with the H.E.S.S. observation in the earlier observation time,
while the late time observation at $t\sim 10^{5}\rm~{}s$ can be explained by
the SSC emission.

##  4 Discussion

VHE $\gamma$-rays from GRB 190829A have been studied based on the SSC
scenario, as in other VHE GRBs (Chand et al., 2020; Fraija et al., 2020;
Rhodes et al., 2020; Hu et al., 2020; Sato et al., 2021). However, the
relative ratio of the EIC component to SSC component depends on both the
energy spectrum of flare emission and synchrotron emission, $F_{E=500\rm
GeV}^{\rm EIC}/F_{E=500\rm GeV}^{\rm SSC}=(F_{E_{b}}^{\rm fl}/F_{E,\rm
max}^{\rm syn})(E_{m}/E_{b})^{-(s-1)/2}\sim Y_{\rm EIC}/Y_{\rm SSC}$ which can
be derived from Equations 8, 11 and 12. In Fig. 3, we show the synchrotron
cooling timescale $t_{\rm syn}$, the SSC cooling timescale $t_{\rm SSC}$, and
the EIC cooling timescale $t_{\rm EIC}$, as a function of $\gamma_{e}$ at
$t=10^{4}\rm~{}s$. We also show the time evolution of the Compton parameters,
$Y_{\rm SSC}(\gamma_{e})$ and $Y_{\rm EIC}(\gamma_{e})$, at various
observation times. We can see that $Y_{\rm SSC}(\gamma_{e})$ is not constant
and it declines with the increase of $\gamma_{e}$ due to the Klein-Nishina
effect. According to Equation 12, the value of $Y_{\rm SSC}(\gamma_{e})$ is
proportional to the comoving synchrotron photon energy density $U_{\rm
syn}^{\prime}[\varepsilon^{\prime}<\varepsilon_{\rm
KN}^{\prime}]\propto{\varepsilon_{\rm KN}^{\prime}}^{1/2}$. Note
$\varepsilon_{\rm KN}^{\prime}$ is proportional to $\gamma_{e}^{-1}$, we can
expect $Y_{\rm SSC}(\gamma_{e})\propto\gamma_{e}^{-1/2}$ consistent with
$Y_{\rm SSC}(\gamma_{e})$ shown in Fig. 3. Unlike $Y_{\rm SSC}(\gamma_{e})$,
the Compton parameter $Y_{\rm EIC}(\gamma_{e})$ remains constant up to
$\gamma_{e}\sim 10^{5}$. The reason is that the energy density of the late-
prompt photons $U_{\rm EIC}^{\prime}[\varepsilon^{\prime}<\varepsilon_{\rm
KN}^{\prime}]$ remains constant, which is dominated by the energy density near
$\varepsilon_{b}\sim 100\rm~{}eV$ as long as $\gamma_{e}\mathrel{\hbox{\hbox
to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}10^{4}$. One visible
feature of $Y_{\rm EIC}(\gamma_{e})$ is the rapid decline following the time
evolution of the X-ray flare.

Figure 3: Upper panel: Dynamical timescale and various electron cooling
timescales for synchrotron, SSC and EIC as a function of electron Lorentz
factor $\gamma_{e}$. Lower panel: The evolution of $Y_{\rm SSC}(\gamma_{e})$
and $Y_{\rm EIC}(\gamma_{e})$ as a function of $\gamma_{e}$ at
$t=10^{3}\rm~{}s$, $t=10^{4}\rm~{}s$ and $t=2.5\times 10^{4}\rm~{}s$.

The physical parameters of LL GRBs are still uncertain due to the limited
number of observations and the values used in Fig. 1 are optimized for fitting
the multi-wavelength light curve. The values of the microphysical parameters
are $\epsilon_{e}\sim 0.4$, $f_{e}\sim 0.3$, and $\epsilon_{B}=9\times
10^{-5}$, assuming uniform microturbulence in the shock vicinity. For GRB
afterglows, the typical value of $\epsilon_{e}$ is $\sim 0.1$, while the value
of $\epsilon_{B}$ varies in a wide range, e.g., $\sim 10^{-5}-10^{-1}$ (Kumar
& Zhang, 2014). Small values of $\epsilon_{B}$ can be consistent with ones
expected around the contact discontinuity, considering the decay of
$\epsilon_{B}$ from a stronger value of $\sim 0.01$ at the shock front
inferred from particle-in-cell simulations (Lemoine, 2013; Vanthieghem et al.,
2020).

In this work, we consider an impulsive relativistic outflow with a kinetic
energy of $\mathcal{E}_{k}=1\times 10^{52}\rm~{}erg$ and an initial Lorentz
factor of $\Gamma_{0}=25$, propagating into an external medium with a constant
density of $n_{\rm ex}=0.14\rm~{}cm^{-3}$. The value of kinetic energy and
$\Gamma_{0}$ is larger than for LL GRBs, e.g. GRB 980425 (Galama et al., 1998;
Kulkarni et al., 1998) and GRB 060218 (Soderberg et al., 2006; Campana et al.,
2006), but smaller than canonical high-luminosity GRBs (Kumar & Zhang, 2014).
The value of $\Gamma_{0}$ should not be much smaller, e.g. $\Gamma<10$,
otherwise the deceleration time is too long. A higher value of $\Gamma_{0}>25$
will enhance the SSC contribution at earlier time $t\mathrel{\hbox{\hbox
to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}10^{4}\rm~{}s$, but
the flux of VHE $\gamma$-rays still dominated by EIC components. We stress
that the advantage of the EIC+SSC model presented in this work is that in the
presence of X-ray flares it gives a better fit to the multi-wavelength light
curve with microphysical parameters that are similar with other TeV GRBs. In
Fig. 4, we show the time evolution of the Lorentz factor and radius of the
blast wave. The deceleration time can be determined when the total mass of the
swept-up matter equals to a fraction $1/\Gamma_{0}$ of the ejecta mass,
$R_{\rm dec}\simeq 2.9\times
10^{17}\mathcal{E}_{k,52}^{1/3}\Gamma_{0,1.4}^{-2/3}n_{\rm
ex,-1}^{-1/3}\rm~{}cm$ and the corresponding deceleration time can be
estimated to be $t_{\rm dec}\approx(1+z)R_{\rm dec}/2\Gamma_{0}c^{2}\simeq
8.4\times 10^{3}(1+z)\mathcal{E}_{k,52}^{1/3}\Gamma_{0,1.4}^{-8/3}n_{\rm
ex,-1}^{-1/3}\rm~{}s$. In addition, we note that the predicted Lorentz factor
of the ejecta is $\Gamma\sim 1.7$ at $\sim 60$ days which is only slightly
smaller than the velocity measured by the Very Long Baseline Interferometry
(VLBI) (Marcello, 2021).

Figure 4: Time evolution of the Lorentz factor and radius of the blast wave as
a function of the observation time since the burst, $t$.

Our results can also be affected by the energy spectrum of late-prompt
emission. For example, the peak flux $F_{\varepsilon_{b}}$ used in Equation 10
will decrease for a smaller value of $\beta$, given the constraints at the
X-ray band. In this work, the break energy $\varepsilon_{b}$ of the late-
prompt flare emission is chosen to be constant. In general, the value of
$\varepsilon_{b}$ may decrease with time (Murase et al., 2011). However, the
evolution of $\varepsilon_{b}$ is not essential for our conclusions, since
lower values of $\varepsilon_{b}$ at later times can enhance the scattering
efficiency in the Thomson regime when $\gamma_{e,c}$ becomes larger. Compared
to canonical high-luminosity GRBs, energy spectra of LL GRBs usually have
lower $\varepsilon_{m}$ and higher $\varepsilon_{c}$, with the dominance of
photons with energies around $\varepsilon_{c}$. The Klein-Nishina effect in
the SSC spectrum will be more relevant in such a situation. The IC scattering
between the same population of electrons with external photons can alleviate
the Klein-Nishina suppression if the external photons have the proper energies
that can keep the IC scattering process in the Thomson regime.

GRB 190829A has the lowest radio luminosity in the GRB samples considered in
Rhodes et al. (2020). However, there is no difference in the radio luminosity
among all of the three VHE GRBs and other GRBs. The associated supernova SN
2019oyw appeared to have similar ejecta mass and kinetic energy as SN 1998bw
(Hu et al., 2020) that was associated with a LL GRB. Note that thermal photons
from an associated supernova can also be upscattered by non-thermal high-
energy electrons accelerated by the conventional GRB outflow or trans-
relativistic ejecta (Asano & Mészáros, 2008; Ando & Mészáros, 2008; He et al.,
2009). Hadronic VHE $\gamma$-rays from LL GRBs have also been studied. In
particular, Murase et al. (2008) and Murase & Beacom (2010) proposed heavy-ion
synchrotron emission and VHE $\gamma$-rays from the deexcitaion of the UHECR
nuclei, respectively (see, e.g., Fig. 9 of Murase et al., 2008). On the other
hand, Murase (2012) and Kashiyama et al. (2013) suggested hadronic VHE
$\gamma$-rays originating from $p\gamma$ interactions following prompt and
shock breakout emission, respectively. These mechanisms would compete with
high-energy emission from primary electrons (Ando & Mészáros, 2008). The
detailed modeling of the hadronic processes and the following electromagnetic
cascade is essential (see, e.g., Murase & Beacom, 2010, as an example of
hadronic and leptonic components) to disentangle the hadronic component from
the leptonic component and to obtain constraints both on cosmic-ray
acceleration mechanisms and on radiation processes.

##  5 Summary

In this work, we showed that VHE $\gamma$-rays are naturally produced by the
EIC+SSC mechanism, which may provide an explanation for the observation of VHE
$\gamma$-rays from GRB 190829A by H.E.S.S. . The EIC scenario is seeded by the
observed long-lasting late-prompt flare in the X-ray and optical bands as
target photons. Assuming that the non-thermal electrons are accelerated in the
external forward shock, these can upscatter the late-prompt flare photons, and
our results show that the EIC components can contribute to the H.E.S.S.
observations at $t\sim 1\times 10^{4}\rm~{}s$.

We consider an impulsive relativistic outflow propagating into an external
medium with a constant density, where the physical parameters adopted in this
work are optimized for fitting the multi-wavelength light curve. We showed
that our model can explain the multi-wavelength observations of GRB 190829A ,
given $\mathcal{E}_{k}=10^{52}\rm~{}erg$, $\Gamma_{0}=25$, and $n_{\rm ex}\sim
0.1\rm~{}cm^{-3}$. Our results suggest that GRB 190829A is not a typical LL
GRB but has much higher kinetic energy. The TeV photons observed by H.E.S.S.
can be dominated by EIC emission at earlier times, while the SSC component
becomes dominant at later times. Our results suggest that the EIC emission
nearly coincides with the late-prompt flare, but decreases more slowly than
the evolution of the flare emission, and extends $\sim 3-4$ times longer than
the duration of the flare emission. Future observations of EIC components that
are related to late-prompt flares in GRBs will be helpful for constraining the
properties of the outflow and central engine activities.

The work of K.M. is supported by the Alfred P. Sloan Foundation, NSF Grant No.
AST-1908689, and KAKENHI No. 20H01901 and No. 20H05852. B.T.Z. acknowledges
the IGC fellowship. P.M. acknowledges support from the Eberly Foundation. P.V.
acknowledges support from NASA grants 80NSSC19K0595 and NNM11AA01A.

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We calculate time-dependent spectra of high-energy afterglow emission by
extending the method used in Murase et al. (2011). We numerically calculate
afterglow dynamics to treat not only the relativistic regime (described by the
self-similar solution) but also the non-relativistic regime, and take into
account effects of EATS.

##  Appendix A Afterglow dynamics

We consider an impulsive relativistic outflow with kinetic energy
$\mathcal{E}_{k}$ and initial Lorentz factor $\Gamma_{0}$, propagating into an
external medium of constant density $n_{\rm ex}$. The total energy of the
blast wave which develops is given by (Nava et al., 2013)

$\mathcal{E}_{\rm tot}=\Gamma M_{\rm ej}c^{2}+\Gamma
mc^{2}+\frac{\hat{\gamma}\Gamma^{2}-\hat{\gamma}+1}{\Gamma}\mathcal{E}_{\rm
int}^{\prime},$ (A1)

where $M_{\rm ej}=\mathcal{E}_{k}/\Gamma_{0}c^{2}$ is the mass of the outflow,
$m=(4\pi/3)r^{3}n_{\rm ex}m_{p}$ is the mass of the swept-up external matter,
$\hat{\gamma}=(4+\Gamma^{-1})/3$ is the adiabatic index which is a good
approximation in both relativistic and non-relativistic regimes (Nava et al.,
2013). The comoving internal energy is $\mathcal{E}_{\rm
int}^{\prime}=(\Gamma-1)mc^{2}$. Considering $d\mathcal{E}_{\rm tot}=dmc^{2}$
and neglecting both adiabatic and radiative energy losses (see Nava et al.,
2013; Zhang, 2018, for details), we can derive the following 1D differential
equation,

$\frac{d\Gamma}{dm}=-\frac{\Gamma-5\Gamma^{3}+4\Gamma^{5}}{3M_{\rm
ej}\Gamma^{3}-2m+8\Gamma^{4}m}.$ (A2)

The differential mass of the collected external medium is $dm=4\pi r^{2}n_{\rm
ex}f_{\rm corr}m_{p}dr$. We can derive the well-known relation $\Gamma\propto
r^{-2/3}$ once the blast wave enters into the Blandford-McKee (BM) self-
similar adiabatic evolution regime (Blandford & McKee, 1976). Note that we
multiply a factor of $f_{\rm corr}=9/17$ to the swept-up external matter
density in order to match the normalization of the BM self-similar solution
(Nava et al., 2013). The BM self-similar phase begins after a significant
deceleration occurs, where the kinetic energy of the initial ejecta equals the
sum of the kinetic energy of the swept-up external matter and its internal
energy $\Gamma M_{\rm ej}c^{2}\sim\hat{\gamma}\Gamma^{2}mc^{2}$. In order to
obtain the value of quantities in the observer frame, we adopt the
differential relation, $dr=\beta cdt/(1-\beta)$, which represents the case
that photons propagate with distance $dr$ can be observed in a time interval
$dt$. The differential equation derived in Equation A2 also gives an
appropriate description of the blast wave evolution in both the coasting
($\Gamma=\Gamma_{0},r\propto t$) and non-relativistic deceleration
($\beta\propto t^{-3/5},r\propto t^{2/5}$) regimes (Huang et al., 1999;
Panaitescu & Kumar, 2000; Pe’er, 2012; Nava et al., 2013; Lu et al., 2020).

##  Appendix B Non-thermal electron distribution

We calculate afterglow synchrotron and SSC spectra numerically, and we confirm
that the numerical results agree with the analytical results. In general,
electron energy spectra in the downstream of the external forward shock can be
derived by solving the kinetic equation (Blumenthal & Gould, 1970),

$\frac{\partial n_{\gamma_{e}}(t^{\prime})}{\partial
t^{\prime}}+\frac{\partial}{\partial\gamma_{e}}\left(n_{\gamma_{e}}(t^{\prime})\dot{\gamma}_{e}\right)+\frac{n_{\gamma_{e}}(t^{\prime})}{t^{\prime}_{\rm
esc}}=\dot{n}_{\gamma_{e}}^{\rm inj}(t^{\prime}),$ (B1)

where $n_{\gamma_{e}}(t^{\prime})$ is the number density of electrons per
electron Lorentz factor,
$\dot{\gamma}_{e}=d\gamma_{e}/dt^{\prime}=\gamma_{e}{t^{\prime}}_{\rm
cool}^{-1}$ is the electron energy loss rate (divided by $m_{e}c^{2}$),
$t^{\prime}_{\rm esc}$ is the possible escape time, $\dot{n}_{\gamma_{e}}^{\rm
inj}(t^{\prime})=\mathcal{C}\gamma_{e}^{-s}$ is the electron injection rate
where $\mathcal{C}\approx(s-1)\gamma_{e,m}^{s-1}n_{e}/t^{\prime}_{\rm dyn}$
(for $s>2$), $\gamma_{e,m}$ is the electron minimum Lorentz factor, $n_{e}$ is
the non-thermal electron number density in the comoving frame which is
normalized based on $N_{e}=4\pi r^{2}n_{e}t^{\prime}_{\rm
dyn}c=(4\pi/3)r^{3}n_{\rm ex}f_{e}$, and considering an on-axis observer we
use $t^{\prime}_{\rm dyn}\approx\Gamma t/(1+z)$ as the dynamical timescale.
The electron minimum Lorentz factor is given by
$\gamma_{e,m}\approx(\epsilon_{e}/f_{e})[(s-2)/(s-1)](m_{p}/m_{e})(\Gamma-1)$
for $s>2$, where $\epsilon_{e}$ is the energy fraction of thermal energy that
goes into non-thermal electrons and $f_{e}$ is the number fraction of thermal
electrons that are injected into the acceleration process. The total cooling
rate consists of radiative cooling and possible adiabatic losses. The
synchrotron loss rate is ${t^{\prime}}_{\rm
syn}^{-1}=(4/3)\sigma_{T}c(\gamma_{e}/m_{e}c^{2})\beta^{2}U_{\rm B}$, where
$U_{B}=B^{2}/8\pi$ is the magnetic energy density (Rybicki & Lightman, 1986).
The IC loss rate ${t^{\prime}}_{\rm IC}^{-1}$ is implemented as in Equation
(B3) of Murase et al. (2011), which includes both SSC and EIC processes. The
adiabatic loss rate is ${t^{\prime}}_{\rm ad}^{-1}\sim{t^{\prime}}_{\rm
dyn}^{-1}$. In general, the electron distribution is obtained by solving the
above differential equation in a time-dependent manner. We call this method
“numerical, time-dependent”, but we demonstrate the results using
$t^{\prime}_{\rm esc}=t^{\prime}_{\rm dyn}$, by which we approximately take
into account adiabatic energy losses or particle escape after the decay of
magnetic fields in the downstream. The steady-state solution via $\partial
n_{\gamma_{e}}/\partial t^{\prime}=0$ can also be found as in Equation (C.11)
in Dermer & Menon (2009). On the other hand, for results presented in the main
text, we take the iteration method as in Murase et al. (2011). However,
different from the previous work that calculated $\gamma_{e,c}$ with
analytical synchrotron spectra and used broken power-law electron
distributions, we determine the electron distribution using numerical
synchrotron spectra through the following function,

$n_{\gamma_{e}}(t^{\prime})=\frac{1}{{t^{\prime}}_{\rm
dyn}^{-1}+{t^{\prime}}_{\rm cool}^{-1}}\frac{1}{\gamma_{e}}\int
d\gamma_{e}^{\prime}\dot{n}_{\gamma_{e}^{\prime}}(t^{\prime}).$ (B2)

This function is motivated by the steady-state solution for ${t^{\prime}}_{\rm
cool}^{-1}={t^{\prime}}_{\rm syn}^{-1}+{t^{\prime}}_{\rm
IC}^{-1}+{t^{\prime}}_{\rm ad}^{-1}$ in the no escape limit. In the fast
cooling case, the electron distribution is mostly described by the steady-
state distribution for ${t^{\prime}}_{\rm cool}^{-1}={t^{\prime}}_{\rm
syn}^{-1}+{t^{\prime}}_{\rm IC}^{-1}$. In the slow cooling case, it is
essentially the injection distribution with spectral steepening by radiative
cooling. This method was also used in Asano et al. (2020) and Zhang et al.
(2021). In Fig. 5, we compared the electron energy spectrum derived using
iteration method mentioned above and the spectrum via numerically solving
time-dependent kinetic equation Equation B1.

For comparison, we will also show analytical afterglow synchrotron and SSC
spectra, and this method is called “analytical”, where $\gamma_{e,c}$ is
determined by the iteration method to evaluate the Compton $Y$ parameter.

Figure 5:  Comparison of the electron energy spectra derived by the iteration
method used for the main results and by solving the kinetic equation at
$t^{\prime}=100\rm~{}s$. We use $\mathcal{E}_{k}=1\times 10^{52}\rm~{}erg$,
$n_{\rm ex}=1\rm~{}cm^{-3}$, $\epsilon_{e}=0.3$, $f_{e}=1.$, $s=2.5$,
$\Gamma_{0}=50$. Left panel: slow-cooling regime, $\epsilon_{B}=10^{-3}$.
Right panel: fast-cooling regime, $\epsilon_{B}=10^{-1}$.

##  Appendix C Radiative processes

The observed radiation flux from a relativistically moving object can be
derived via integration over the equal-arrival-time surface (EATS) (e.g.,
Granot et al., 1999; Woods & Loeb, 1999)

$F_{E}(t)=\frac{1+z}{d_{L}^{2}}\int_{0}^{2\pi}d\phi\int_{-1}^{1}d\mu\int_{0}^{\infty}drr^{2}\frac{j_{\varepsilon^{\prime}}(\varepsilon^{\prime},\Omega^{\prime},\hat{t},\textbf{r})}{\Gamma^{2}(1-\beta{\rm
cos}\theta)^{2}},$ (C1)

where $z$ is the source redshift, $d_{L}$ is the luminosity distance of the
source,
$j_{\varepsilon^{\prime}}(\varepsilon^{\prime},\Omega^{\prime},\hat{t},\textbf{r})$
is the comoving emissivity at r and time $\hat{t}$, $\mu={\rm cos}\theta$,
$\Gamma$ is the Lorentz factor. The observation time is
$t=(1+z)(\hat{t}-r\mu/c)$, where we assume $t=0$ is the arrival time of photon
emitted at origin at $\hat{t}=0$. We assume instantaneous emission at
$\hat{t}_{i}$,
$j_{\varepsilon^{\prime}}=j_{\varepsilon^{\prime}}\delta(\hat{t}-\hat{t}_{i})\Delta\hat{t}_{i}=j_{\varepsilon^{\prime}}\delta(\hat{t}-\hat{t}_{i})(r_{i}/\beta
c\Gamma)$. Using the relation
$\delta(\hat{t}-\hat{t}_{i})=\delta(\hat{\mu}-\hat{\mu}_{i})c/r_{i}$, Equation
C1 can be simplified as

$F_{E}(t)=\frac{(1+z)2\pi}{d_{L}^{2}}\int_{0}^{\infty}drr^{2}\frac{j_{\varepsilon^{\prime}}(\varepsilon^{\prime},r,\hat{t})}{\Gamma^{3}\beta(1-\beta{\rm
cos}\theta)^{2}}.$ (C2)

The comoving synchrotron emissivity can be calculated using following formula,

$\displaystyle j_{\varepsilon^{\prime}}^{\rm
syn}=\frac{\sqrt{3}}{4\pi}\frac{e^{3}B}{m_{e}c^{2}2\pi\hbar\varepsilon^{\prime}}\int
d\gamma_{e}n_{\gamma_{e}}G(x),$ (C3)

where

$G(x)\approx\frac{1.81e^{-x}}{(x^{-2/3}+(3.62/\pi)^{2})^{1/2}},$ (C4)

$x=\varepsilon^{\prime}/\varepsilon^{\prime}_{c}$ and
$\varepsilon^{\prime}_{c}=(3e\hbar B/2m_{e}c)\gamma_{e}^{2}\beta^{2}$ is the
critical energy (Rybicki & Lightman, 1986; Aharonian et al., 2010).

The comoving SSC emissivity is estimated to be

$\displaystyle j_{\varepsilon^{\prime}}^{\rm
SSC}=\frac{3}{16\pi}\sigma_{T}c\int
d\gamma_{e}\frac{1}{\gamma_{e}^{2}}n_{\gamma_{e}}\int d\varepsilon_{\rm
syn}^{\prime}\frac{\varepsilon^{\prime}}{\varepsilon_{\rm
syn}^{\prime}}\frac{dn_{\rm syn}^{\prime}}{d\varepsilon_{\rm
syn}^{\prime}}f(q,w),$ (C5)

where $\sigma_{T}$ is the Thomson cross section, $dn_{\rm
syn}^{\prime}/d\varepsilon_{\rm syn}^{\prime}$ is the comoving synchrotron
photon density, and

$f(q,w)=2q{\rm ln}q+(1+2q)(1-q)+\frac{1}{2}\frac{(wq)^{2}}{1+wq}(1-q),$ (C6)

$q=\varepsilon^{\prime}/(4\gamma_{e}\varepsilon_{\rm
syn}^{\prime})(\gamma_{e}-\varepsilon^{\prime}/m_{e}c^{2})$, and
$w=4\varepsilon_{\rm syn}^{\prime}\gamma_{e}/m_{e}c^{2}$ (Blumenthal & Gould,
1970). Note the Thomson limit corresponds to $w\ll 1$, while the above
expression is valid for all the value of $w$ as long as $\gamma_{e}\gg 1$
(Blumenthal & Gould, 1970).

The comoving EIC emissivity can be calculated as

$\displaystyle j_{\varepsilon^{\prime}}^{\rm EIC}$ $\displaystyle=$
$\displaystyle\frac{3}{8\pi}\sigma_{T}c(1-{\rm cos}~{}\theta_{\rm
sc}^{\prime})\int d\gamma_{e}n_{\gamma_{e}}\int dyA(y)\varepsilon_{\rm
ph}^{\prime}\frac{dn_{\rm ph}^{\prime}}{d\varepsilon_{\rm ph}^{\prime}},$ (C7)

where $\varepsilon^{\prime}$ is the comoving EIC photon energy,
$\varepsilon_{\rm ph}^{\prime}$ is the comoving seed photon energy, $dn_{\rm
ph}^{\prime}/d{\varepsilon_{\rm ph}^{\prime}}$ is the comoving density of seed
photons, $\theta_{\rm sc}^{\prime}$ is the scattering angle measured in the
forward shock comoving frame relative to the direction of the photon beam,
$y\equiv\xi m_{e}c^{2}/(2(1-{\rm cos}~{}\theta_{\rm
sc}^{\prime})\gamma_{e}\varepsilon_{\rm ph}^{\prime}(1-\xi))$,
$\xi\equiv\varepsilon^{\prime}/(\gamma_{e}m_{e}c^{2})$, and
$A(y)\equiv(1-\xi)[1-2y+2y^{2}+\xi^{2}/(2(1-\xi))]$ (Aharonian & Atoyan, 1981;
Toma et al., 2009; Murase et al., 2011). Assuming the direction of the photon
beam at the scattering point following the radial expansion of the jet, the
observed flux can be evaluated at $\theta_{\rm sc}^{\prime}=\theta^{\prime}$
in the jet comoving frame where ${\rm cos}\theta_{\rm sc}^{\prime}={\rm
cos}\theta^{\prime}=(\mu-\beta)/(1-\beta\mu)$ with $\mu={\rm cos}\theta$.

Figure 6: Left pannel: Comparison of energy spectra of synchrotron and SSC
emission at $t=10^{4}\rm~{}s$ calculated in the single-zone model with both
the numerical method (solid line) and the analytical method (dot-dashed line).
The dotted line is calculated by the numerical method with a time-dependent
electron energy spectrum via solving Equation B1. We also use the relation
$r=4\Gamma^{2}ct$ in this panel. Right pannel: Comparison of energy spectra of
synchrotron and SSC emission at $t=10^{4}\rm~{}s$ calculated with EATS (solid
line) and without EATS (dotted-dashed line) using the detailed dynamics
described in Appendix A. The dashed line is calculated by the single-zone
model using the same dynamics as in left figure. The relevant physical
parameters are $\mathcal{E}_{k}=1\times 10^{53}\rm~{}erg$, $n_{\rm
ex}=1\rm~{}cm^{-3}$, $\epsilon_{e}=0.05$, $f_{e}=1$, $\epsilon_{B}=1\times
10^{-2}$, $s=2.2$, $\Gamma_{0}=100$, and $z=0.0785$.

In Figure C6 left, we compare results of different methods for the single-zone
model. The iteration method used in this work agrees with the method of
solving the time-dependent equation. Both of the numerical results agree with
the analytical formula. The analytical synchrtron and SSC spectra are taken
from Murase et al. (2011), where the BM solution with $r=4\Gamma^{2}ct$ is
used for dynamics (Blandford & McKee, 1976; Sari et al., 1998; Waxman, 1997)
and the synchrotron peak flux is evaluated with a correction factor introduced
in Wijers & Galama (1999). Note that the SSC spectrum in the Thomson limit is
shown here, although Murase et al. (2011) also considered spectral suppression
due to the Klein-Nishina effect analytically. In Figure C6 right, we compare
results of the single-zone model with $r=4\Gamma^{2}ct$ to those of the
detailed afterglow dynamics moel with/without EATS. This panel demonstrates
that the detailed model with EATS agrees with the standard single-zone
afterglow model in the relativistic limit, although our detailed method has an
advantage that it can be used both relativistic and non-relativistic regimes
consistently. Note that we our synchrotron energy spectra calculated with EATS
are also consistent with the results of Granot & Sari (2002).

Photons with energy beyond the pair production threshold will undergo internal
absorption by ambient photons inside the source and external absorption by
extragalactic background light (EBL) during their propagation to Earth. The
threshold energy can be estimated from the kinematic condition,
$\varepsilon^{\prime}\varepsilon_{\rm seed}^{\prime}\mathrel{\hbox{\hbox
to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}(m_{e}c^{2})^{2}$,
where $\varepsilon^{\prime}$ is the comoving high-energy photon energy and
$\varepsilon_{\rm seed}^{\prime}$ is the comoving target photon energy. As in
Murase et al. (2011), we use the $\gamma\gamma$ optical depth,

$\displaystyle\tau_{\gamma\gamma}=\frac{\tilde{\Delta}}{2}\int_{-1}^{1}d\mu(1-\mu)\int
d\varepsilon_{\rm seed}^{\prime}\frac{dn}{d\varepsilon_{\rm
seed}^{\prime}}\sigma_{\gamma\gamma}(S),$ (C8)

where

$\sigma_{\gamma\gamma}(S)=\frac{3}{16}\sigma_{T}(1-\beta_{\rm
cm}^{2})[(3-\beta_{\rm cm}^{4}){\rm ln}\left(\frac{1+\beta_{\rm
cm}}{1-\beta_{\rm cm}}\right)-2\beta_{\rm cm}(2-\beta_{\rm cm}^{2})]$ (C9)

is the two-photon annihilation cross section, $\beta_{\rm
cm}=\sqrt{1-S^{-1}}$, and $S=(1/2)\varepsilon_{\rm
seed}^{\prime}\varepsilon^{\prime}(1-\mu)$ is the Mandelstam variable. Here,
$\tilde{\Delta}$ is the comoving width, and the attenuation at a given EATS
includes from contributions with different values of $\tilde{\Delta}$. For
simplicity, we adopt $\tilde{\Delta}\approx ct/(1+z)$ and ignore effects of
electromagnetic cascades. The photon spectrum after the internal
$\gamma\gamma$ absorption can be estimated as
$F_{\varepsilon}^{\gamma\gamma}=F_{\varepsilon}/(1+\tau_{\gamma\gamma})$. The
EBL is mainly composed of infrared and optical photons, and we use the low-IR
model calculated in Kneiske et al. (2004). The observed energy spectrum is

$\displaystyle F_{E}^{\rm ob}=F_{E}^{\gamma\gamma}e^{-\tau_{\gamma\gamma}^{\rm
EBL}(E,z)},$ (C10)

where $\tau_{\gamma\gamma}^{\rm EBL}(E,z)$ is the optical depth at given
observed photon energy $E=\varepsilon/(1+z)$ and redshift $z$.