# GRB afterglows with energy injections in AGN accretion disks

Bao-Quan Huang Department of Astronomy, Xiamen University, Xiamen, Fujian
361005, China  Tong Liu Department of Astronomy, Xiamen University, Xiamen,
Fujian 361005, China  Xiao-Yan Li  Department of Astronomy, Xiamen University,
Xiamen, Fujian 361005, China  Yun-Feng Wei Department of Astronomy, Xiamen
University, Xiamen, Fujian 361005, China

###### Abstract

Active galactic nucleus (AGN) disks are widely considered potential hosts for
various high-energy transients, including gamma-ray bursts (GRBs). The
reactivation of GRB central engines can provide additional energy to shocks
formed during the interaction of the initially ejected GRB jets with the
circumburst material, commonly referred to as energy injections. In this
paper, we study GRBs occurring in AGN disks within the context of energy
injections. We adopt the standard external forward shock (EFS) model and
consider both short- and long-duration GRB scenarios. Light curves for two
types of radiation, namely the radiation from the heated disk material (RHDM)
and GRB afterglows, are computed. We find that the energy injection
facilitates the EFS to break out from the photosphere of the low-density AGN
disk at relativistic velocity. Moreover, the energy injection almost does not
affect the RHDM but significantly enhances the peak flux of the GRB
afterglows.

Active galactic nuclei (16); Galaxy accretion disks (562); Gamma-ray bursts
(629); Relativistic jets (1390); Shocks (2086)

##  1 Introduction

The accretion disks in active galactic nuclei (AGNs) have been extensively
investigated since the discovery of AGNs in the last century (see reviews by
Pringle, 1981; Kato et al., 2008; Abramowicz & Fragile, 2013). It is commonly
accepted that AGN disks harbor numerous stars and compact objects. These stars
and compact objects can either form within the disks (e.g., Goodman, 2003;
Artale et al., 2019; Dittmann & Miller, 2020; Cantiello et al., 2021; Fan &
Wu, 2023) or be captured from outside (e.g., Artymowicz et al., 1993; Fabj et
al., 2020; MacLeod & Lin, 2020; Nasim et al., 2023; Generozov & Perets, 2023;
Wang et al., 2024), and some of them probably exist in binary systems (e.g.,
Baruteau et al., 2011; Pfuhl et al., 2014; Bartos et al., 2017). The
collapses, explosions, or tidal disruptions of the stars, as well as the
collisions/mergers between or among the compact objects, are generally
followed by various transient events across electromagnetic wavelengths, such
as gamma-ray bursts (GRB, e.g., Perna et al., 2021; Zhu et al., 2021; Yuan et
al., 2022; Lazzati et al., 2022; Qi et al., 2022a; Wang et al., 2022; Tagawa
et al., 2023; Ray et al., 2023; Lazzati et al., 2023; Kathirgamaraju et al.,
2023).

GRBs stand as the most brilliantly luminous explosions in the Universe. They
are broadly categorized into two groups: short- and long-duration GRBs (SGRBs
and LGRBs), where SGRBs are generally considered to be associated with the
mergers of compact objects, whereas LGRBs are linked to the collapses of
massive stars (see reviews by Woosley & Bloom, 2006; Nakar, 2007; Zhang,
2018). Additionally, they are widely believed to be generated by
ultrarelativistic jets originating from their central engines, which could be
either a stellar-mass black hole (BH) surrounded by a hyperaccretion disk (for
reviews, see Liu et al., 2017) or a massive millisecond magnetar (e.g., Duncan
& Thompson, 1992; Usov, 1992; Dai & Lu, 1998b; Zhang & Mészáros, 2001).

In general, after the prompt $\gamma$-ray emission, enduring multiband
afterglows arise from synchrotron radiation, which stems from electrons
accelerated by external forward shocks (EFSs) formed as the jets interplay
with the circumburst material (e.g., Zhang et al., 2006). Observationally, a
substantial portion of X-ray afterglows exhibit single or double plateaus
(e.g., Li et al., 2018; Hou et al., 2021; Yi et al., 2022) and single or
multiple flares (e.g., Mu et al., 2016; Liu & Mao, 2019) in light curves.
Regarding these phenomena, primary explanations involve the reactivation of
GRB central engines. This reactivation can directly contribute to the flares
(e.g., Burrows et al., 2005; Dai et al., 2006), or provides energy injections
into EFSs to indirectly form the plateau phase (e.g., Zhang & Mészáros, 2001;
Li et al., 2016; Hou et al., 2018; Lin et al., 2018; Huang & Liu, 2021).
According to the statistical analyses, we consider that most of X-ray plateaus
and flares might have the same physical origin but appear with different
features (e.g., Yi et al., 2022).

Recently, significant attention has been directed towards GRBs occurring in
AGN disks. Unlike typical environments, GRBs within AGN disks can experience
diffuse emergence due to the high densities of disk material (Perna et al.,
2021). Additionally, the diffuse afterglows are fairly weak and peak at very
late stages (Wang et al., 2022). To comprehend the role of reactivated GRB
central engines within AGN disks, we investigate the effects of energy
injections resulting from their reactivation on the dynamical evolution of
EFSs and the characteristics of relevant radiations in this work.

The paper is structured as follows. In Section 2, we present the AGN disk
model, the EFS-dynamics model, and relevant radiation mechanism. The main
results are outlined in Section 3, and the conclusions and discussion are made
in Section 4.

Figure 1: Illustration of a GRB jet propagating in an AGN disk.

##  2 Method

We investigate the physical processes of EFSs with energy injections
propagating in AGN disks. We assume EFSs are generated during the initially
ejected GRB jets interact with the circumburst medium. Here, GRB jets are
assumed to be launched close to the equatorial plane of the disks and to move
vertically upward, as depicted in Figure 1. During the propagation of EFSs in
the disks, on the one hand, the reactivation of GRB central engines pushes out
new jets; these new jets catch up with EFSs, continuously providing them with
energy injections. On the other hand, the radiation emitted by EFSs is
significantly scattered and absorbed by the high-density material of the
disks. For simplicity, we assume that the material eventually absorbs all
radiation from EFSs in regions where the optical depth exceeds one, leading to
heating and subsequent thermal emission, hereafter referred to as radiation
from the heated disk material (RHDM). Conversely, we neglect the material’s
effects on photon propagation in regions where the optical depth is less than
one. Thereby, once EFSs penetrate through the photosphere of the disks, GRB
afterglows become transparent. Note that here the EFSs involve the initially
ejected GRB jets.

###  2.1 AGN accretion disk model

We adopt the accretion disk model proposed by Sirko & Goodman (2003). This
disk model considers the effect of self-gravity and provides an accurate
description for the radial structure from inner regions out to $\sim 2\times
10^{5}{R_{\rm g}}$, where $R_{\rm g}=2GM_{\rm BH}/c^{2}$ is the Schwarzschild
radius with $G$ being the gravitational constant, $M_{\rm BH}$ being the mass
of BHs, and $c$ being the speed of light. Additionally, within this disk
model, the supermassive BH (SMBH) spin is not involved, and the outer part of
the accretion disks belongs to the non-ionizing region due to significantly
low temperatures. The SMBH masses are set to $M_{\rm BH}=10^{6}$, $10^{7}$,
and $10^{8}\,M_{\odot}$, the SMBH accretion efficiency $l_{\rm Edd}=0.5$
($l_{\rm Edd}=0.1\dot{M}c^{2}/L_{\rm Edd}$, where $\dot{M}$ and $L_{\rm Edd}$
are the accretion rate and the Eddington luminosity, respectively), and the
viscosity parameter $\alpha=0.01$.

For the vertical structure, we utilize a Gaussian function to depict the
density distribution, which is written as follows

$\rho_{\rm disk}(R_{r},R_{z})=\rho_{\rm
0}(R_{r})\exp\left[{-\frac{R_{z}^{2}}{2H(R_{r})^{2}}}\right],$ (1)

where $R_{r}$ represents the radial radius, $R_{z}$ stands for the vertical
distance from the equatorial plane, $\rho_{\rm 0}(R_{r})$ denotes the density
at the equatorial plane, and $H(R_{r})$ signifies the half-thickness of
accretion disks. Further, the photospheric radius of accretion disks, $R_{\rm
ph,disk}$, defined as the location where the optical depth is one from the
surface of accretion disks inwards, can be calculated by

$\displaystyle\tau(R_{r})=\int_{R_{\rm ph,disk}}^{H}\kappa_{0}(R_{r})\rho_{\rm
disk}(R_{r},R_{z})\,dR_{z}=1,$ (2)

where $\kappa_{0}(R_{r})$ is the opacity of accretion disks.

###  2.2 Dynamical evolution of EFSs with energy injections

Figure 2: The initial radii of EFSs in SGRBs ($R_{\rm ES,s}$) and LGRBs
($R_{\rm ES,l}$), the photospheric radius of AGN disks ($R_{\rm ph,disk}$),
and the disk disk half-thickness ($H$) evolving with radial locations for
different SMBH masses, including $10^{6}\,M_{\odot}$ (top panel),
$10^{7}\,M_{\odot}$ (middle panel), and $10^{8}\,M_{\odot}$ (bottom panel).

Assuming the initially ejected GRB jets to be thick shells, the initial
position of EFSs can be written as (Sari & Piran, 1995)

$R_{\rm ES}=R_{\rm Sedov}^{3/4}\Delta^{1/4},$ (3)

where $R_{\rm Sedov}$ represents the Sedov radius defined by the integral

$\int_{R_{0}}^{R_{\rm Sedov}}4\pi R^{2}\rho c^{2}\,dR=E_{\rm iso},$ (4)

with ${R_{0}}$ and $E_{\rm iso}$ being the initial position and isotropic
energy of the initially ejected GRB jets, and $\Delta=cT$ is the lab-frame jet
width with $T$ being the duration of GRBs. $\rho$ is the density of the
circumburst medium with $\rho=\rho_{\rm disk}$ inside the disks and
$\rho=n_{\rm ISM}m_{\rm p}$ outside the disks, where $n_{\rm ISM}$ is the
number density of the interstellar medium (ISM) and $m_{\rm p}$ is the proton
mass. In this work, we consider both SGRB and LGRB situations, and denote the
corresponding initial radii of EFSs as $R_{\rm ES,s}$ and $R_{\rm ES,l}$. $R$
represents the distance from the central engines. Here the jet direction along
the $z$ axis is considered. For SGRBs, we set ${R_{0}}=10^{7}\,{\rm cm}$,
$E_{\rm iso}=10^{51}\,{\rm erg}$, and $T=1\,{\rm s}$. Meanwhile, for LGRBs, we
set ${R_{0}}=10^{10}\,{\rm cm}$, $E_{\rm iso}=10^{53}\,{\rm erg}$, and
$T=10\,{\rm s}$.

We use an approximate dynamical evolution model to discuss the evolution of
EFSs, which contains differential equations listed as follows (Huang et al.,
1999),

$\frac{d\Gamma}{dt}=\frac{-(\Gamma^{2}-1)\frac{dm}{dt}+\frac{1}{c^{2}}\frac{dE_{\rm
in}}{dt}}{M_{\rm j}+\epsilon m+2(1-\epsilon)\Gamma m},$ (5)
$\frac{dm}{dt}=2\pi(1-\cos\theta_{\rm jet})R^{2}\rho\frac{dR}{dt},$ (6)
$\frac{dR}{dt}=\frac{c\beta}{(1-\beta\cos\theta_{\rm v})(1+z)}.$ (7)

Here, $\Gamma$, $m$, $\epsilon$, and $\theta_{\rm jet}$ are the Lorentz
factor, swept mass of the external medium, radiation efficiency of EFSs, and
jet-half opening angle, respectively. $M_{\rm j}=E_{\rm
k,iso}(1-\cos\theta_{\rm jet})/2(\Gamma_{0}-1)c^{2}$ is the mass of the
initially ejected GRB jets, where $E_{\rm k,iso}$ is the isotropic kinetic
energy. $\beta=\sqrt{1-1/\Gamma^{2}}$ is the EFS velocity. $t$ is the time
measured in the observer’s frame. $\theta_{\rm v}$ is the viewing angle, which
is the angle between the jet axis and the line of sight, and it is set to
$0^{\circ}$ here, i.e., on-axis viewing. $z$ is the redshift and is set to
0.5. In addition, the functional form of $dE_{\rm in}/dt$ can be simply
described as (Lin et al., 2018)

${\frac{dE_{\rm in}}{dt}}=\left\\{\begin{array}[]{ll}\frac{P_{\rm
iso,in}(1-\cos\theta_{\rm jet})}{2},\quad T_{\rm st,in}<t<T_{\rm st,in}+T_{\rm
st,du},\\\ 0,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad{\rm
others},\end{array}\right.$ (8)

where $P_{\rm iso,in}$, $T_{\rm st,in}$, and $T_{\rm st,du}$ are the isotropic
power, start time, and duration of energy injections, respectively. For
simplicity, in this work we only consider a constant injected power, namely
$P_{\rm iso,in}={\rm constant}$.

Table 1: The parameter values for selected 9 cases. These parameters are the SMBH masses, the radial locations from the centers, the initial radii of EFSs in SGRBs, the initial radii of EFSs in LGRBs, the photospheric radii of AGN disks, and the half-thickness of AGN disks, respectively. $\rm Cases$ |  $M_{\rm BH}\,(M_{\odot})$ |  $R_{r}\,(R_{\rm g})$ |  $R_{\rm ES,s}\,(R_{\rm g})$ |  $R_{\rm ES,l}\,(R_{\rm g})$ |  $R_{\rm ph,disk}\,(R_{\rm g})$ |  $H\,(R_{\rm g})$
---|---|---|---|---|---|---
1 |  $10^{6}$ |  $10^{3}$ |  $2.49\times 10^{0}$ |  $1.563\times 10^{5}$ |  $1.262\times 10^{1}$ |  $1.262\times 10^{1}$
2 |  $10^{6}$ |  $10^{4}$ |  $4.701\times 10^{0}$ |  $2.787\times 10^{1}$ |  $9.703\times 10^{1}$ |  $9.705\times 10^{1}$
3 |  $10^{6}$ |  $10^{5}$ |  $1.708\times 10^{1}$ |  $9.652\times 10^{1}$ |  $1.721\times 10^{3}$ |  $1.726\times 10^{3}$
4 |  $10^{7}$ |  $10^{3}$ |  $3.181\times 10^{-1}$ |  $1.831\times 10^{0}$ |  $8.377\times 10^{0}$ |  $8.379\times 10^{0}$
5 |  $10^{7}$ |  $10^{4}$ |  $9.606\times 10^{-1}$ |  $5.414\times 10^{0}$ |  $1.171\times 10^{2}$ |  $1.171\times 10^{2}$
6 |  $10^{7}$ |  $10^{5}$ |  $5.401\times 10^{0}$ |  $3.038\times 10^{1}$ |  $1.648\times 10^{3}$ |  $3.718\times 10^{3}$
7 |  $10^{8}$ |  $10^{3}$ |  $5.405\times 10^{-2}$ |  $3.043\times 10^{-1}$ |  $7.893\times 10^{0}$ |  $7.895\times 10^{0}$
8 |  $10^{8}$ |  $10^{4}$ |  $3.037\times 10^{-1}$ |  $1.708\times 10^{0}$ |  $2.522\times 10^{2}$ |  $2.524\times 10^{2}$
9 |  $10^{8}$ |  $10^{5}$ |  $1.708\times 10^{0}$ |  $9.605\times 10^{0}$ |  $0.0$ |  $8.010\times 10^{3}$

###  2.3 RHDM

We assume that the disk material begins to be heated as EFSs form. Therefore,
the lower position of the region of the heated disk material is at the initial
radius of EFSs, and the upper position of this region is set to the
photospheric radius of the disks, as shown in Figure 1. To calculate the
radiation from this region, we divide it into a series of shell layers from
bottom to top, which is similar to the treatment of the kilonova or supernova
radiation calculations (e.g., Chen et al., 2022; Qi et al., 2022b; Li et al.,
2023). For each shell layer, the evolution of the internal energy can be
written as

${\frac{dE_{i}}{dt_{\rm
b}}}=\left\\{\begin{array}[]{lll}0,\quad\quad\quad\quad\quad R(t_{\rm
b})<R_{\rm i,1},\\\ L_{\rm inj}-L_{i},\quad R_{\rm i,1}<R(t_{\rm b})<R_{\rm
i,2},\\\ -L_{i},\quad\quad\quad\quad R_{\rm i,2}<R(t_{\rm b}).\\\
\end{array}\right.$ (9)

Here the subscript $i$ represents the $i$th layer, so $E_{i}$ is the internal
energy of the $i$th layer, $L_{i}$ is the observed luminosity contributed by
the $i$th layer, $R_{\rm i,1}$ is the lower radius of the $i$th layer, and
$R_{\rm i,2}$ is the upper radius of the $i$th layer. $t_{\rm b}$ is the time
measured in the lab frame, and the corresponding observer time is equal to
$t_{\rm b}(1+z)$ for the shell layers. Note that the observer time, before
EFSs form, is approximately equal to $R_{\rm ES}(1+z)/2\Gamma_{0}^{2}c$.
$R(t_{\rm b})$ is the radius of EFSs, and $L_{\rm inj}$ is the luminosity of
the radiation escaping from EFSs, which can be expressed as (e.g., Panaitescu
et al., 1998; Piran, 1999)

$L_{\rm inj}=\epsilon(\Gamma-1)c^{2}\frac{dm}{dt_{\rm b}}.$ (10)

The observed luminosity contributed by the $i$th layer can be obtained by the
following formula,

$\displaystyle L_{i}=\frac{E_{i}}{{\rm max}[t_{{\rm d},i},t_{{\rm lc},i}]},$
(11)

where $t_{{\rm lc},i}=R_{i}/c$ represents the light-crossing time, and
$t_{{\rm d},i}$ is the radiation diffusion timescale, which can be written as

$\displaystyle t_{{\rm d},i}=\frac{3\kappa_{0}}{\Omega_{\rm
c}c}\sum_{j=i}^{n}\frac{m_{j}}{R_{j}},$ (12)

where $m_{j}$ and $R_{j}$ are the mass and the middle radius of the $j$th
layer. Here, $m_{j}=\Omega_{\rm c}(R_{j,2}^{3}-R_{j,1}^{3})\rho_{\rm disk}/3$,
and $\Omega_{\rm c}=2\pi(1-\cos\theta_{\rm c})$, with $\theta_{\rm c}$ being
the half opening angle of the heated disk metarial, which is set as
$10^{\circ}$.

The overall bolometric luminosity of the heated disk metarial finally can be
calculated by accumulating the contributions from each shell layer,

$L_{\rm ph}=\sum_{i=1}^{n}L_{i}.$ (13)

Assuming that this radiation approximately satisfies the blackbody spectrum
and is emitted from the photosphere of the disks, the effective temperature
can be expressed as

$T_{\rm eff}=\left(\frac{L_{\rm ph}}{\Omega_{\rm c}R_{\rm
ph,disk}^{2}\sigma_{\rm SB}}\right)^{1/4},$ (14)

where $\sigma_{\rm SB}$ is the Stephan-Boltzmann constant. In addition, the
flux density can be written as

$F_{\nu_{\rm obs}}(t)=\frac{2\pi
h\nu^{3}}{c^{2}}\frac{(1+z)}{\exp(h\nu/kT_{\rm eff})-1}\frac{\Omega_{\rm
c}R_{\rm ph,disk}^{2}}{4\pi D_{\rm L}^{2}},$ (15)

where $\nu$ represents the local frequency, related to the observer frequency
$\nu_{\rm obs}$ by a factor of $(1+z)$, $k$ denotes the Boltzmann constant,
and $D_{\rm L}$ is the luminosity distance in the standard $\Lambda$CDM
cosmology model ($\Omega_{M}=0.27$, $\Omega_{\Lambda}=0.73$, and
$H_{0}=71~{}\rm km~{}s^{-1}~{}Mpc^{-1}$).

###  2.4 Synchrotron radiation

After EFSs break out of disk photospheres, afterglows become transparent. At
this stage, we calculate the synchrotron radiation emitted by the electrons of
the circumburst medium accelerated by the EFSs.

Based on the above-mentioned EFS dynamics, the evolution of the shock-
accelerated electrons is obtained by the following formula (Fan et al., 2008),

$\frac{\partial}{{\partial
R}}\left(\frac{{dN^{\prime}_{\rm{e}}}}{{d\gamma_{\rm{e}}^{\prime}}}\right)+\frac{\partial}{{\partial\gamma_{\rm{e}}^{\prime}}}\left(\frac{d\gamma_{\rm{e}}^{\prime}}{{dR}}\frac{{dN^{\prime}_{\rm{e}}}}{{d\gamma_{\rm{e}}^{\prime}}}\right)={Q},$
(16)

where $dN^{\prime}_{\rm{e}}/d\gamma_{\rm{e}}^{\prime}$ is the instantaneous
electron energy spectrum, $\gamma^{\prime}_{\rm e}$ is the Lorentz factor of
the shock-accelerated electrons, $d\gamma_{\rm{e}}^{\prime}/dR$ is the cooling
term of electrons with the Lorentz factor $\gamma^{\prime}_{\rm e}$, and $Q$
is the injection term for newly shocked-accelerated electrons. Since the
shock-accelerated electrons are assumed to follow a power-law distribution,
one can express $Q=\bar{K}\gamma_{\rm{e}}^{\prime-p}$, with $\bar{K}\approx
2\pi(1-\cos\theta_{\rm jet})(p-1)R^{2}n\gamma_{\rm e,min}^{\prime p-1}$, for
$\gamma^{\prime}_{\rm e,min}\leq\gamma^{\prime}_{\rm
e}\leq\gamma^{\prime}_{\rm e,max}$, where $p$ ($>2$), $\gamma^{\prime}_{\rm
e,min}$, and $\gamma^{\prime}_{\rm e,max}$ represent the power-law index, the
minimum Lorentz factor, and the maximum Lorentz factor of the shock-
accelerated electrons, respectively. Here, the minimum and maximum Lorentz
factor can be expressed as $\gamma_{\rm e,min}^{\prime}=\epsilon_{\rm
e}(\Gamma-1)(p-2)m_{\rm p}/(p-1)m_{\rm e}+1$ and $\gamma^{\prime}_{\rm
e,max}=\sqrt{{9m_{\rm e}^{2}c^{4}}/{8B^{\prime}e^{3}(1+Y)}}$, respectively,
with $B^{\prime}=\sqrt{32\pi\Gamma(\Gamma-1)\rho\epsilon_{B}c^{2}}$ (Kumar et
al., 2012), where $\epsilon_{\rm e}$ is the fraction of the thermal energy
density that is shared by accelerated elections, $\epsilon_{B}$ is the
fraction of the thermal energy density that is shared by magnetic filed, $e$
is the electron charge, $m_{\rm e}$ is the electron masses, and $Y$ is the
Compton parameter (Fan & Piran, 2006). Additionally, the radiation efficiency
of EFSs can be expressed as $\epsilon=\epsilon_{\rm rad}\epsilon_{\rm e}$,
with $\epsilon_{\rm rad}={\rm min}[1,(\gamma_{\rm e,min}^{\prime}/\gamma_{\rm
c}^{\prime})^{(p-2)}]$, where $\gamma_{\rm c}$ is the efficient cooling
Lorentz factor of electrons (Sari & Esin, 2001). Note that quantities with a
superscript accent sign are defined in the comoving frame of EFSs.

The spectral power of synchrotron radiation at frequency $\nu^{\prime}$ can be
expressed as (Rybicki & Lightman, 1979)

$P^{\prime}_{\rm syn}({\nu^{\prime}})=\int\nolimits_{\gamma^{\prime}_{\rm
e,min}}^{\gamma^{\prime}_{\rm e,max}}P^{\prime}_{\rm
e}({\nu^{\prime}},\gamma^{\prime}_{\rm e})\frac{dN^{\prime}_{\rm
e}}{d\gamma^{\prime}_{\rm e}}d\gamma^{\prime}_{\rm e},$ (17)

where

$P^{\prime}_{\rm e}({\nu^{\prime}},\gamma^{\prime}_{\rm
e})=\frac{\sqrt{3}e^{3}B^{\prime}}{m_{\rm
e}c^{2}}F\bigg{(}\frac{\nu^{\prime}}{\nu^{\prime}_{\rm c}}\bigg{)},$ (18)

represents the power of a single election with the Lorentz factor
$\gamma^{\prime}_{\rm e}$. Here, $\nu_{\rm c}^{\prime}=3eB^{\prime}\gamma_{\rm
e}^{\prime 2}/4\pi m_{\rm e}c$, $F({\nu^{\prime}}/{\nu_{\rm
c}^{\prime}})=(\nu^{\prime}/\nu_{\rm
c}^{\prime})\int\nolimits_{\nu^{\prime}/\nu_{\rm
c}^{\prime}}^{+\infty}K_{5/3}(x)dx$, and $K_{5/3}(x)$ is a modified Bessel
function of order 5/3. Since elections with $\gamma^{\prime}_{\rm e}<3$ are
thought to be difficult to produce synchrotron radiation, the lower
integration limit in Equation (17) is set to 3 if $\gamma^{\prime}_{\rm
e,min}<3$. In addition, the self-absorption effect of synchrotron radiation is
taken into account in this work, and the corresponding optical depth is given
by (Rybicki & Lightman, 1979)

$\tau_{\rm tot}({\nu^{\prime}})=\frac{1}{8\pi\nu^{\prime 2}m_{\rm e}f}\int
P^{\prime}_{\rm e}({\nu^{\prime}},\gamma^{\prime}_{\rm e})\gamma^{\prime
2}_{\rm e}\frac{d}{d\gamma^{\prime}_{\rm e}}\left(\frac{dN^{\prime}_{\rm
e}/d\gamma^{\prime}_{\rm e}}{\gamma^{\prime 2}_{\rm
e}}\right)d\gamma^{\prime}_{\rm e}$ (19)

with $f=2\pi(1-\cos\theta_{\rm jet})R^{2}$. Consequently, the spectral power
of synchrotron radiation considering the self-absorption effect can be written
as

$\displaystyle P^{\prime}_{\rm syn,sa}({\nu^{\prime}})=\int_{0}^{\tau_{\rm
tot}}\frac{P^{\prime}_{\rm syn}({\nu^{\prime}})}{\tau_{\rm
tot}({\nu^{\prime}})}\exp{(-\tau)}d\tau.$ (20)

We consider a top-hat jet and divide it into a number of emitters. Therefore,
the observed flux density is calculated by accumulating the flux density of
the emitters at the same observer time, which can be derived as (e.g., Granot
et al., 1999)

$\displaystyle F_{\nu_{\rm obs}}=\frac{1+z}{4\pi D_{\rm
L}^{2}}{\int}\kern-15.0pt\int\limits_{(\rm EATS)}\frac{P^{\prime}_{\rm
syn,sa}(\nu^{\prime})}{2\pi(1-\cos\theta_{\rm jet})}{D^{3}}d\Omega,$ (21)

where “EATS” refers to the equal-arrival time surface corresponding to the
same observer time, $\nu^{\prime}=(1+z)\nu_{\rm obs}/D$, and $D$ represents
the Doppler factor of the emitters.

##  3 Results

Table 2: The chosen values of EFS parameters for both SGRBs and LGRBs. Parameter |  Unit |  SGRB |  LGRB
---|---|---|---
$E_{\rm k,iso}$ |  ${\rm erg}$ |  $10^{51}$ |  $10^{53}$
$\Gamma_{\rm 0}$ |  |  $2\times 10^{2}$ |  $2\times 10^{2}$
$\epsilon_{\rm e}$ |  |  $10^{-1}$ |  $10^{-1}$
$\epsilon_{\rm B}$ |  |  $10^{-3}$ |  $10^{-3}$
$n_{\rm ISM}$ |  ${\rm cm}^{-3}$ |  $1$ |  $1$
$p$ |  |  $2.5$ |  $2.5$
$\theta_{\rm jet}$ |  degree |  $5$ |  $5$
$T_{\rm st,in}$ |  ${\rm s}$ |  $10^{1}$ |  $10^{1}$
$T_{\rm du,in}$ |  ${\rm s}$ |  $2\times 10^{2}$ |  $2\times 10^{3}$
$P_{\rm iso,in}$ |  ${\rm erg/s}$ |  $10^{49}$ |  $10^{51}$
Figure 3: R and $\Gamma$ of evolving EFSs in 9 cases. In each panel, green
(purple) lines represent the evolution of $R$ in SGRBs (LGRBs), and the red
(yellow) lines represent the evolution of $\Gamma$ in SGRBs (LGRBs), where
solid (dashed) lines denote the results with (without) energy injections. In
addition, the black dot-dash lines show the photospheric radius of AGN disks
($R_{\rm ph,disk}$).

Figure 2 illustrates the evolutions of the initial EFS radii, disk
photospheric radius, and disk half-thickness concerning the radial radius. The
top, middle, and bottom panels present the results corresponding to SMBH
masses of $M_{\rm BH}=10^{6}$, $10^{7}$, and $10^{8}\,M_{\odot}$,
respectively. In Figure 2, it is evident that EFSs in LGRBs exhibit larger
initial radii compared to those in SGRBs. Additionally, at smaller radial
radii, the initial EFS radii surpass the disk half-thickness, indicating the
formation of EFSs outside the disks. However, with the increase in radial
radius, the disk half-thickness undergoes a significant increase. At a
specific location, the initial EFS radii suddenly become smaller than the disk
half-thickness, implying the inception of EFS formation within the disks from
that radial location onwards. Moreover, irrespective of the SMBH masses, the
photospheric radius of the disks (yellow lines) essentially aligns with the
disk half-thickness (red lines) within the region of $R_{\rm r}<10^{5}R_{\rm
g}$. This alignment suggests considerable opacity of the disks in this area,
with the photospheres almost coinciding with the disk surfaces. Beyond this
region, however, the disks quickly become completely transparent. In this
study, to encompass all scenarios, namely EFSs forming in both the optically
thin and thick regions of the disks as well as outside the disks, while
minimizing computational efforts, we adopt three radial radii: $R_{r}=10^{3}$,
$10^{4}$, and $10^{5}R_{\rm g}$ for each SMBH mass, according to the results
of Figure 2. Consequently, our investigation encompasses 9 cases, as outlined
in Table LABEL:table1. Our focus lies on studying the EFS dynamical evolution
and the characteristics of relevant radiations across these 9 cases, with
particular emphasis on the effects of energy injections on the EFSs.

In general, there are two forms of continuous energy injection into EFSs, one
due to a long-lived central engine (e.g., Dai & Lu, 1998a; Zhang & Mészáros,
2001; Kumar et al., 2008; Huang & Liu, 2021) and the other due to a
stratification of the ejecta Lorentz factor (e.g., Rees & Mészáros, 1998; Sari
& Mészáros, 2000; Granot et al., 2003). Regardless of whether the progenitor
is the merger of compact binaries or the collapse of massive stars, either a
rapidly rotating stellar-mass BH or a massive millisecond magnetar is involved
in the center. In the case of BH engines, continuous energy injections are
attributed to material accretion, and in the case of magnetar engines,
continuous energy injections are supplied by the Poynting flux/electron-
positron wind. The energy of the magnetar wind cannot exceed the maximum
rotational energy of $\sim 2\times 10^{52}\,{\rm erg}$, but this limit does
not exist in the BH accretion scenario. Compared to the normal circumstellar
environment, in AGN disks, more massive progenitor stars prefer to produce BHs
and more massive fallback accretion material should prolong the duration of
energy injections and improve the total energy injected. Thus, continuous
energy injections in the BH accretion scenario are effective in helping EFSs
to break out of the disk photospheres. However, to be conservative, we do not
consider specific scenarios and only base on the results of the observed
sample statistics to set $E_{\rm k,iso}=10^{51}\,{\rm erg}$, $P_{\rm
iso,in}=10^{49}\,{\rm erg\,s^{-1}}$, and $T_{\rm du,in}=2\times 10^{2}\,{\rm
s}$ for SGRBs, and $E_{\rm k,iso}=10^{53}\,{\rm erg}$, $P_{\rm
iso,in}=10^{51}\,{\rm erg\,s^{-1}}$, and $T_{\rm du,in}=2\times 10^{3}\,{\rm
s}$ for LGRBs, additionally $T_{\rm st,in}=10\,{\rm s}$ for both. The detailed
values of EFS parameters are listed in Table LABEL:table2.

Figure 4: Optical light curves of RHDM. The purple (green) solid lines
represent the results of SGRBs (LGRBs) with the energy injections, and the red
(yellow) dashed lines denote the results of SGRBs (LGRBs) without the energy
injections.

###  3.1 Evolutions of EFSs

Figure 3 displays the evolutions of $R$ and $\Gamma$ across observer time in 9
cases, with each panel corresponding to a distinct case. Within each panel,
two sections are present: the upper section delineates the $R$ evolutions in
SGRBs (green lines) and LGRBs (purple lines), while the lower section exhibits
the $\Gamma$ evolutions in SGRBs (red lines) and LGRBs (yellow lines). In both
sections, solid lines depict the results considering the energy injections,
whereas dashed lines represent the results without the energy injections.
Furthermore, the black dot-dashed line in the upper section denotes the
photospheric radius of the disks, i.e., $R_{\rm ph,disk}$.

In Figure 3, the evolution of $R$ reveals that EFSs with the energy injections
penetrate the disk photospheres earlier than those without the injections.
This effect is notably more pronounced in LGRBs compared to SGRBs, indicating
that higher injection powers and longer injection durations expedite EFSs’
breakout from the disk photospheres. In addition, the $\Gamma$ evolution
reveals that in some cases with relatively low disk densities and relatively
small radii of the disk photospheres, the EFSs remain at relativistic
velocities after breaking out of the disk photospheres due to the energy
injections, such as the EFS with the energy injections in LGRBs of case 2
maintaining the Lorentz factor $\Gamma\sim 3$. However, these cases constitute
a minority among all cases. In the majority, despite the energy injections,
EFSs still exit the photospheres at non-relativistic velocities due to the
hindrance posed by the high-density disk material. This suggests that, in most
cases, higher injection powers and longer durations than those considered in
this study would be necessary for EFSs to still maintain relativistic
velocities after leaving the photospheres.

There are two exceptional cases, cases 1 and 9. In case 1, EFSs in LGRBs have
initial radii outside the disk. In our study, outside the disk, the assumed
environment is the ISM. Consequently, LGRBs in case 1 belong to typical LGRBs.
Hence, we can observe that the corresponding $\Gamma$ exhibits significantly
larger values in the initial phase and decays more slowly over time, in
contrast to SGRBs (within the same case) where EFSs form inside the disk and
their velocities decrease rapidly as they propagate in the high-density disk.
In case 9, we can firstly see in Table LABEL:table1 that the photospheric
radius is 0. This indicates that the corresponding disk region is optically
thin and low-density. Thus, in this case, there is no the RHDM, and the
afterglows would appear earlier and brighter compared to those with EFSs
formed in the optically thick regions. In addition, due to a low disk density,
$\Gamma$ reach about 20 for SGRBs and 30 for LGRBs in the initial phase, much
larger than those in cases 7 and 8 with higher disk densities.

Figure 5: Afterglow light curves for cases 1 (first column), 2 (second
column), and 9 (third column). The first, second, and third rows show the
X-ray, optical, and radio light curves, respectively. Green and red lines
represent the results of SGRBs and LGRBs, where solid and dashed lines denote
the results with and without the energy injections, respectively.

###  3.2 Light curves

There are two types of radiation discussed in our work. As mentioned in
Section 2, one is the RHDM and the other is the afterglow synchrotron
radiation from EFSs after they break out of the disk photospheres. Here we
show their light curves.

####  3.2.1 RHDM

Figure 4 shows the RHDM optical ($5\times 10^{4}\,{\rm Hz}$) light curves. The
purple and red lines denote the light curves in SGRBs, and the green and
yellow lines represent the light curves in LGRBs. Meanwhile, the solid and
dashed lines exhibit the results with and without the energy injections,
respectively.

In the figure, the light curves with and without the energy injections are
overlapped in all cases, indicating that the effects of the energy injections
caused by the reactivation of GRB central engines are minimal for this type of
radiation. The reason is that the observed luminosity from the shell layers
heated before EFSs received the energy injections dominates the total observed
luminosity, compared to the observed luminosity from the shell layers heated
during EFSs received the energy injections, this is because the Lorentz
factors of EFSs have been decayed to relatively lower values in the energy
injection phase than in the initial formation phase due to the resistance of
the high-density disk material. In addition, the fluxes of the light curves do
not obviously change with the observer time. This is attributed to the fact
that the densities of the disk material are large enough to lead to
significantly long diffusion times and thus to observed luminosities much
smaller than the injection luminosities, i.e., $L_{\rm ph}\ll L_{\rm inj}$,
which means that the energy of the heated disk material cannot be effectively
radiated. Moreover, since the EFSs form outside the disk, there is no the RHDM
in the LGRB scenario of case 1.

It can be found that the fluxes improve significantly overall with the
increasing radial radius by comparing the cases with the same SMBH masses but
different radial radii, such as case 1 ($10^{6}M_{\odot}$, $10^{3}R_{\rm g}$),
case 2 ($10^{6}M_{\odot}$, $10^{4}R_{\rm g}$), and case 3 ($10^{6}M_{\odot}$,
$10^{5}R_{\rm g}$), which indicating RHDM fluxes are directly proportional to
the radial radius of the disks. The reason is that, given the model parameter
values, the larger the radial radius, the larger the photospheric radius of
the disks (before about $R_{\rm r}<10^{5}R_{\rm g}$), which hints that EFSs
break out of the disk photospheres with more time, and thus more energy from
the EFSs is absorbed by the disk material.

####  3.2.2 Afterglows

Figure 5 shows the afterglow light curves at X-ray ($1\,{\rm keV}$), optical
($5\times 10^{14}\,{\rm Hz}$), and radio ($10^{9}\,{\rm Hz}$) bands for cases
1, 2, and 9. The top, middle, and bottom rows represent the X-ray, optical,
and radio light curves, respectively. The green and red lines depict the light
curves for SGRBs and LGRBs, while the solid and dashed lines differentiate the
results with and without the energy injections.

In Figure 5, light curves with the energy injections exhibit notable
distinctions in both profiles and peak fluxes compared to those without,
suggesting evident effects of the energy injections on the afterglow light
curves. This stands in contrast to the RHDM.

In case 1, the light curves in LGRBs peak considerably earlier and exhibit
significantly larger peak fluxes compared to those in SGRBs. This is because,
in this case, EFSs in LGRBs form outside the disk, whereas EFSs in SGRBs form
inside the disk and have become unrelativistic as they break out of the disk
photosphere. In case 2, although both the EFSs in SGRBs and LGRBs form inside
the disk, the EFS in LGRBs, supported by the energy injections, maintains
relativistic velocities after breaking out of the disk photosphere (as seen in
Figure 3). Consequently, an earlier peak time and a larger peak flux are
observed in this scenario compared to that without the energy injections in
LGRBs and to those in SGRBs. In case 9, the disk reaches an optically thin
state, allowing the afterglow radiation to nearly insusceptibly pass out of
the disk. However, significantly strong synchrotron self-absorption occurs in
the radio wavelengths when the radiation is produced by the EFSs so that the
radio fluxes are markedly lower. In addition, also due to the higher
synchrotron self-absorption, the flux gap between the X-ray and optical bands
is larger than that in case 1 with the afterglows produced outside the disk
and than that in case 2 with the afterglows contributed by the EFSs after the
breakout from the disk photospheres. Finally, the light curves in the other
cases, regardless of whether they are in LGRBs or SGRBs, resemble those of
SGRBs in cases 1 and 2, i.e., rather weak afterglows. Therefore, we have not
included the afterglow light curves for these cases here.

##  4 Conclusions and discussion

In this paper, we focus on the effects of energy injections supported by the
reactivation of GRB central engines on the dynamical evolution of EFSs
propagating in AGN disks with a Gaussian density profile in the vertical
direction, as well as on the light curves of two types of radiation, the RHDM
and the afterglows. As results, we find that EFSs with the energy injections
can break out of the disk photospheres faster than those without, and the EFSs
with the energy injections in LGRBs occurring at locations with low densities
and small photospheric radii still retains relativistic velocities after
penetrating the disk photospheres. Furthermore, we find that the light curves
with and without the energy injections are not distinct for the RHDM but are
notable differences in peak fluxes for the afterglows.

Figure 6: Spectral energy distributions of the disk, RHDM, and synchrotron
afterglows for $M_{\rm BH}=10^{6},10^{7}\,M_{\odot}$, $l_{\rm E}=0.5$, and
$\alpha=0.01$, where solid and dashed lines denote the results with $M_{\rm
BH}=10^{6}\,M_{\odot}$ and $10^{7}\,M_{\odot}$, respectively.

A hollow cone would appear behind the EFS as an EFS driven by an initially
ejected GRB jet sweeps up the disk material, and the hollow cone should
gradually fill with the heated disk material. If the hollow cone is completely
filled before new ejected jets catch up with the EFS, there would be no energy
injections. Thus, here we estimate the time scale of the fully filled hollow
cone to evaluate the rationality of the adopted start time of energy
injections. This time scale can be estimated by $t_{\rm c}\approx R_{\rm
c}/\upsilon_{\rm c}$, where $R_{\rm c}$ is the inner radius of the cross
section of the heated disk material, and $\upsilon_{\rm c}=P_{\rm c}/\rho_{\rm
disk}$ is the lateral expansion velocity (Begelman & Cioffi, 1989; Bromberg et
al., 2011). $P_{\rm c}=E_{\rm c}/2V_{\rm c}$, $E_{\rm c}=\int_{t_{\rm
s}}^{t_{\rm e}}L_{\rm inj}dt_{\rm b}$, and $V_{\rm c}=\pi R_{\rm e}^{2}(R_{\rm
e}-R_{\rm ES})(\tan^{2}\theta_{\rm c}-\tan^{2}\theta_{\rm jet})$ are the
pressure, energy, and volume of the heated disk material, respectively.
$t_{\rm s}$ is the time of EFS formation in the local rest frame, ${t_{\rm
e}}$ is the start time of the energy injections in the local rest frame, and
$R_{\rm e}$ is the location of the EFS at the start time. In this work, the
start time of energy injections is set to $10\,{\rm s}$ in the observer frame,
i.e., $T_{\rm st,in}=10\,{\rm s}$. Using the SGRB with the energy injections
in case 1 for an example, one have ${t_{\rm s}}\approx 25\,{\rm s}$, ${t_{\rm
e}}\approx 67\,{\rm s}$, and $R_{\rm e}=1.8\times 10^{12}\,{\rm cm}$. Further,
we obtain the time scale of the fully filled hollow cone $t_{\rm c}\approx
98\,{\rm s}$, which is larger than the start time of the energy injections
(${t_{\rm e}}\approx 67\,{\rm s}$), indicating that setting $T_{\rm
st,in}=10\,{\rm s}$ is reasonable.

Using the same disk parameters as mentioned in Section 2.1, we estimate the
spectral energy distributions of the disk model (Goodman, 2003), RHDM, and
synchrotron afterglows, and compare the optical ($5\times 10^{14}\,{\rm Hz}$)
luminosity from the RHDM and afterglows and the X-ray ($1\,{\rm keV}$)
luminosity from the afterglows with the corresponding disk luminosity.
Regarding the RHDM, the optical luminosity at smaller radial radii is
overshadowed by the optical luminosity of the disk model. However, at larger
radial radii, the optical luminosity can outshine that of the disk model. For
example, in case 3, the optical luminosity of the RHDM reaches $2.4\times
10^{43}\,{\rm erg\,s^{-1}}$, exceeding the corresponding disk luminosity
values of $8.3\times 10^{42}\,{\rm erg\,s^{-1}}$. Although the optical
luminosity of the RHDM outshines that of the disk model at larger radial
radii, they may be indistinguishable due to two factors: (1) the luminosity
excess is not markedly obvious, and (2) the diffuse timescale is sufficiently
large, causing the optical luminosity of the RHDM not to decay significantly
over relatively short times. Regarding the afterglows, the optical luminosity
is encompassed by the optical luminosity of the disk model in optically thick
regions. However, it can surpass the corresponding disk luminosity in
optically thin regions (e.g., case 9) or when EFSs form outside the disks
(e.g., the LGRBs in case 1). The X-ray luminosity scenario is similar to the
optical luminosity scenario. But, if EFSs still maintain relativistic
velocities after breaking out of the disk photospheres, such as the LGRB with
the energy injections in case 2, the X-ray luminosity can exceed the
corresponding disk luminosity. Here, in contrast to the RHDM, the afterglows
can be easily distinguished from the disk radiation due to their relatively
short durations. Finally, as an example, Figure 6 shows the spectral energy
distributions of the disk model (red), RHDM at $R_{\rm r}=10^{5}\,R_{\rm g}$
(purple), and synchrotron afterglows at $R_{\rm r}=10^{4}\,R_{\rm g}$ (green)
for $M_{\rm BH}=10^{6},10^{7}\,M_{\odot}$, $l_{\rm E}=0.5$, and $\alpha=0.01$.
It can be seen that the spectral energy distributions of the afterglows are
significantly broader than those of the disk model and RHDM. However, for the
same accretion rate, the afterglow luminosities decrease dramatically with the
SMBH masses increasing. They are well below the luminosity of the AGN disk,
which implies that it is more difficult to observe the afterglows for the more
massive SMBHs.

Moreover, different values of parameters of accretion system can significantly
affect the above results. For the given accretion rate and viscosity
parameter, the SMBH mass has inverse correlation with the disk densities and
positive correlation with the disk half-thicknesses and luminosity (e.g., Kato
et al., 2008). This implies that EFSs require more time to penetrate the disk
photosphere for larger SMBH mass. Similarly, the accretion rate is inversely
proportional to disk density and proportional to disk half-thickness and
luminosity. In contrast, the viscosity parameter is inversely proportional to
disk density, half-thickness, and luminosity. Therefore, a low-mass SMBH, the
low accretion rate, as well as the large viscosity parameter, could favor the
appearance of GRB afterglows.

In this work, the assumed durations of energy injections in SGRBs and LGRBs
are referenced to times obtained from GRBs occurring in typical environments.
However, in the AGN disk, the long-duration fallback accretion to the central
BH might be maintained to prolong the timescale of the energy injection.
Besides, if cavities with radii close to the disk half-thickness are involved
(Kimura et al., 2021; Yuan et al., 2022), EFSs will encounter less
deceleration while propagating within these cavities, enabling them to
maintain relativistic velocities as they escape from the disk photospheres,
which suggests the anticipation of stronger afterglows.

We thank Bing Zhang, Li Xue, Da-Bin Lin, and Yan-Qing Qi for helpful
discussions. This work was supported by the National Key R&D Program of China
(Grant No. 2023YFA1607902), the National Natural Science Foundation of China
(Grant Nos. 12173031, 12221003, and 12303049), and the National Fund for
Postdoctoral Researcher Program (Grant No. GZC20231424).

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