# Off-axis MeV and very-high-energy gamma-ray emissions from structured gamma-
ray burst jets

Željka Bošnjak,1 B. Theodore Zhang (张兵),2 Kohta Murase,3,4,5,6,2 and Kunihito
Ioka2
1Faculty of Electrical Engineering and Computing, University of Zagreb, Unska
ul. 3, 10000 Zagreb, Croatia
2Center for Gravitational Physics and Quantum Information, Yukawa Institute
for Theoretical Physics, Kyoto University, Kyoto, Kyoto 606-8502, Japan
3Department of Physics, The Pennsylvania State University, University Park, PA
16802, USA
4Department of Astronomy & Astrophysics, The Pennsylvania State University,
University Park, PA 16802, USA
5Center for Multimessenger Astrophysics, Institute for Gravitation and the
Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
6School of Natural Sciences, Institute for Advanced Study, Princeton, NJ
08540, USA E-mail: zeljka.bosnjak@fer.hr(ZB)E-mail:
<EMAIL_ADDRESS>(BTZ)E-mail<EMAIL_ADDRESS>(KM)E-mail:
<EMAIL_ADDRESS>(KI)

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

###### Abstract

Very-high-energy (VHE) photons around TeV energies from a gamma-ray burst
(GRB) jet will play an essential role in the multi-messenger era, with a fair
fraction of the events being observed off-axis to the jet. We show that
different energy photons (MeV and TeV photons in particular) arrive from
different emission zones for off-axis observers even if the emission radius is
the same. The location of the emission region depends on the jet structure of
the surface brightness, and the structures are generally different at
different energies, mainly due to the attenuation of VHE photons by electron-
positron pair creation. This off-axis zone-shift effect does not justify the
usual one-zone approximation and also produces a time-delay of VHE photons
comparable to the GRB duration, which is crucial for future VHE observations,
such as by the Cherenkov Telescope Array.

###### keywords:

radiation mechanisms: general – relativistic processes – stars: jets –
transients: gamma-ray bursts

††pubyear: 2023††pagerange: Off-axis MeV and very-high-energy gamma-ray
emissions from structured gamma-ray burst jets–Off-axis MeV and very-high-
energy gamma-ray emissions from structured gamma-ray burst jets

##  1 Introduction

The very-high-energy (VHE; $\gtrsim 0.1\rm~{}TeV$) emission has been observed
by Imaging Atmospheric Cherenkov telescopes (IACTs) from several gamma-ray
bursts (GRB 180720B, GRB 190114C, GRB 190829A, GRB 201015A, GRB 201216C and
GRB 221009A; for a recent review see e.g., Noda & Parsons (2022)). Among
these, GRB 190829A and GRB 201015A are classified as low-luminosity (LL) GRBs
based on their prompt emission luminosity. For short gamma-ray bursts (sGRBs),
the MAGIC collaboration has reported a significance of $\sim 3\sigma$
detection of gamma-rays from sGRB 160821B (MAGIC Collaboration et al., 2020).
The detection of GRBs at the VHE band provided new clues for the understanding
of the physics of GRBs, as modeling the emission on a broad energy range from
optical to TeV allows us to constrain the microphysics parameters in the
emission regions (e.g., Derishev & Piran, 2021; Asano et al., 2020). Even
though the origin of VHE gamma-rays from GRBs is consistent with the
synchrotron self-Compton (SSC) afterglow model for current observations (e.g.,
Meszaros & Rees, 1994; Zhang & Meszaros, 2001; Sari & Esin, 2001), the
detection of VHE gamma-rays is expected also during the GRB prompt emission
phase (e.g., Inoue et al., 2013; Vurm & Beloborodov, 2017; Bošnjak et al.,
2009; Banerjee et al., 2022; Gill & Granot, 2022).

The detection of gravitational wave (GW) source GW170817 associated with the
short GRB 170817A (Abbott et al., 2017a, b) opened a new era in the
electromagnetic counterpart search to GRBs (Lamb & Kobayashi, 2017; Ioka &
Nakamura, 2018; Nakar, 2020), as the associated electromagnetic signal
potentially allows the identification of the host galaxy and the redshift
measurement. The properties of sGRB 170817A were rather uncommon: its
$\gamma$-ray luminosity ($\sim$ 1047 erg/s) was four orders of magnitude lower
than typical short GRB, and the prompt emission consisted of two distinct
components showing the unexpected spectral evolution. The first pulse was
fitted with a cutoff power-law model with a hard low-energy spectrum, and it
was followed by the second pulse dominated by a thermal emission (Pozanenko et
al., 2018). The early observations in the optical/near-infrared band were
interpreted as quasi-thermal radiation from a kilonova (Tanaka et al., 2017;
Kasen et al., 2017; Utsumi et al., 2017; Pian et al., 2017). The radio and
X-ray observations exhibited a gradual rise in the emission. Early-time X-ray
and radio observations were consistent with a wide-angle, mildly relativistic
($\Gamma\sim$ 2-3) cocoon emission (Kasliwal et al., 2017). The VLBI
observations of the superluminal motion suggested that the late-time emission
was dominated by the narrowly-collimated ($\theta_{c}\lesssim$ 5∘) jet,
observed from a large viewing angle $\sim$14∘-28∘ (Mooley et al., 2018b;
Ghirlanda et al., 2019). As a viable scenario for the interpretation of the
steadily rising afterglow luminosity in radio to X-ray (Margutti et al., 2018;
Mooley et al., 2018a), the structured jet was discussed (Ioka & Nakamura,
2019; Lazzati et al., 2018; Urrutia et al., 2021; Takahashi & Ioka, 2021).

The angular structure of the GRB jets was proposed early on (e.g., Mészáros et
al., 1998; Zhang et al., 2004; Rossi & Rees, 2003; Kumar & Granot, 2003;
Granot & Kumar, 2003) as it may be arising during the jet launch or during the
interaction of the jet with the dense environment following the merger of a
neutron star binary (Aloy et al., 2005; Murguia-Berthier et al., 2017;
Kathirgamaraju et al., 2018; Preau et al., 2021; Gottlieb et al., 2021); for a
recent review see Salafia & Ghirlanda (2022).

In this work, we study the emission zone of VHE gamma-rays for a structured
jet, similar to GW170817/GRB 170817A when viewed off-axis. We focus on the
model described in Ioka & Nakamura (2019), where the off-axis emission arrives
largely from the off-center jet when the jet luminosity is decreasing sharply
outward as it is required from the observations of GRB 170817A. Matsumoto et
al. (2019) revisited the compactness of the gamma-ray sources given by
Lithwick & Sari (2001) for arbitrary viewing angles, and confirmed that the
relativistic jet core cannot be the origin of the observed emission in GRB
170817A. Future VHE facilities such as CTA will allow the follow-up of the
gravitational events in the VHE band, and GW170817-like objects are promising
sources of off-axis VHE gamma rays (Murase et al., 2018). We apply these
findings in the study of the observed surface brightness of the jet emission
taking into account the opacity of the source to gamma-rays. Recently,
Hendriks et al. (2022) simulated a population of binary neutron stars observed
by GW detectors (LIGO, Virgo, the Einstein Telescope and the Cosmic Explorer)
and made predictions for the detection of sGRBs by Fermi/GBM, Swift/BAT and
GECAM using a top-hat jet model for a GRB. Our study can also be implemented
in such simulations to make predictions for future VHE observations.

This kind of study can be interesting also for LL GRBs (Soderberg et al.,
2006; Liang et al., 2007). For LL GRBs often the relativistic shock breakout
model is discussed (Campana et al., 2006; Nakar & Sari, 2012; Nakar, 2015),
where the energy deposition is done by a narrow jet in the low-mass extended
material. The induced shock is much less relativistic than the jet, and after
the breakout produces the low-luminosity soft gamma-rays which are not
narrowly beamed. The off-axis jet model was proposed for the interpretation of
GRB emission properties in several events (Ioka & Nakamura, 2001; Yamazaki et
al., 2003; Waxman, 2004; Sato et al., 2021). LL GRBs are promising targets for
future VHE facilities due to their predicted high local rate (Wanderman &
Piran, 2010) and consequently, GeV/TeV observations being less affected by the
extragalactic background light (EBL) attenuation (Murase et al., 2008; Rudolph
et al., 2022). The asymmetric collapse of massive stars may also be the source
of GW emission (Shibata et al., 2021), where the interesting candidates are
nearby low-luminosity GRBs (Kobayashi & Mészáros, 2003; Daigne & Mochkovitch,
2007; Nakar, 2015).

This paper is organized as follows: In Sec. 2, we describe the details of the
off-axis structured jet model. In Sec. 3, we discuss the origin of VHE gamma-
rays and the optical depth due to the two-photon pair annihilation based on a
structured jet model. Our main results are presented in Sec. 4, where we
arrive at the conclusion that different energy photons arrive from different
emission zones in general. In Sec. 5, we discuss the detectability of the time
delay. In Sec. 6, we study other effects that may affect our results and
discuss the implications of this work. Finally, we give a summary in Sec. 7.

##  2 Structured jet model

Figure 1: Prompt sub-MeV/MeV and VHE emission from a structured jet with
viewing angle $\theta_{v}$. The distance of the emission region from the
center is $R$ when measured in the laboratory frame. We show that different
energy photons generally arrive from different emission zones for an off-axis
observer.

In Fig. 1, we show a schematic picture of the emission from an off-axis
structured jet that we applied: it consists of an energetic and highly
relativistic core, with the energy and Lorentz factor sharply decreasing
outwards. Following Ioka & Nakamura (2019), we consider an off-axis structured
jet with a Gaussian shape,

$E_{\gamma}(\theta)=\epsilon_{\gamma}E_{0}\mathrm{exp}\left(-\frac{\theta^{2}}{2\theta_{c}^{2}}\right),$
(1)

where $E_{\gamma}(\theta)$ is the isotropic-equivalent radiation energy of the
jet at an angle $\theta$ from the jet axis, $E_{0}$ is the isotropic
equivalent energy measured along the jet axis, $\epsilon_{\gamma}$ is the
radiation efficiency, and $\theta_{c}$ is the jet core opening angle (or the
standard deviation of the Gaussian distribution). The angle between the line
of sight and a direction $(\theta,\phi)$ in the jet can be estimated as

$\mathrm{cos}\theta_{\Delta}=\mathrm{sin}\theta~{}\mathrm{cos}\phi~{}\mathrm{sin}\theta_{v}+\mathrm{cos}\theta~{}\mathrm{cos}\theta_{v},$
(2)

where $\theta_{v}$ is the viewing angle and $\phi$ is the azimuth angle with
respect to the jet axis. In the following studies, we adopt fiducial values of
$E_{0}=10^{52.8}\rm~{}erg$, $\theta_{c}=0.059$ and $\theta_{v}=0.38\approx
22^{\circ}$ as inferred from the observations of sGRB 170817A (Troja et al.,
2019). We assume the radiation efficiency for sub-MeV/MeV prompt emissions is
$\epsilon_{\gamma}=0.1$ for simplicity. In reality, the radiation efficiency
should have an angular structure $\epsilon_{\gamma}(\theta)$ which depends on
the details of the radiative processes. In addition, we assume the Lorentz
factor decreases outward with the shape described by the following relation,

$\Gamma(\theta)=\Gamma_{\rm max}\frac{1}{1+(\theta/\theta_{c})^{\lambda}},$
(3)

where $\Gamma_{\rm max}=2000$ and $\lambda=3.8$ (Ioka & Nakamura, 2019). We
assume the energy of the structured jet dissipated at a distance $R$ from the
explosion center in the laboratory frame, where the emission region could be
described by a relativistic shocked shell with comoving width
$\Delta^{\prime}\sim R/\Gamma(\theta)$. The corresponding timescales could be
estimated as $\sim R/\Gamma(\theta)^{2}c$. However, the energy dissipation
radius is difficult to predict without the knowledge of the jet composition
and radial profile. In this work, we consider the energy dissipation that
occurred at the fixed radius $R$, and we will show that different energy
photons generally arrive from different portions even for the same radius.

##  3 VHE prompt emission and optical depth

In recent years, several GRBs have been observed at the VHE band, including
both, high and low luminosity GRBs. All of the present VHE observations are
consistent with the origin at the afterglow phase. The unequivocal VHE
emission during the prompt phase has not been detected yet, though it is
possible that in the early VHE observations by MAGIC telescopes of GRB 190114C
there was a contribution by the late prompt emission (MAGIC Collaboration et
al., 2019). One of the difficulties faced when observing prompt emission is
the duration of the response time following the alert system for IACTs.

In this work, we consider the detection of VHE emission during the prompt
phase. For the synchrotron emission from high-energy electrons, the maximum
photon energy is $\sim 50\Gamma/(1+z)\rm~{}MeV$ due to the limitation of the
synchrotron energy loss process. The maximum photon energy could be somewhat
increased considering the situation where high-energy electrons are
accelerated far from the shock front where the magnetic field strength is
lower but emit efficiently when these electrons travel near the shock front
where the magnetic field strength is larger (Kumar et al., 2012). The inverse-
Compton (IC) process would be more reliable to generate high-energy photons
via upscattering low-energy photons to higher energies. The synchrotron self-
Compton (SSC) process, where the same population of the non-thermal electrons
produces synchrotron emission could also upscatter these photons to higher
energies, and has been successful in explaining the observed VHE emission in
the afterglow phase. However, the SSC process may not be efficient in the
prompt phase due to the Klein-Nishina effect, where the IC cross-section for
scattering decreases significantly. It occurs for the photon energies
comparable to the electron rest mass when measured in the electron rest frame,
$\gamma_{e}\nu_{b}^{\prime}\sim m_{e}c^{2}$, where $\nu_{b}^{\prime}$ is the
target photon energy measured in the comoving frame. Nevertheless, if there
are external photons, i.e., external thermal or non-thermal photons that
overwhelm the prompt Band component in the low-energy range, the external
inverse-Compton (EIC) process will dominate the high-energy photons at the VHE
band. The origin of thermal photons could come from stellar emission or cocoon
emission (e.g., Toma et al., 2009; De Colle et al., 2018; Kimura et al., 2019)
and internal dissipation such as flares and extended emission (e.g., Murase et
al., 2018). The VHE gamma-rays in the prompt phase could also be contributed
by the hadronic processes (Asano & Inoue, 2007; Gupta & Zhang, 2007; Razzaque
et al., 2009; Murase et al., 2012; Rudolph et al., 2023).

The low-energy (LE) sub-MeV/MeV prompt emission could be modeled with a
spectral shape similar to the so-called Band function (e.g.,, Ioka & Nakamura,
2019),

$f(\nu^{\prime},\theta)=\frac{C}{\nu_{0}^{\prime}(\theta)}\left(\frac{\nu^{\prime}}{\nu_{0}^{\prime}(\theta)}\right)^{1+\alpha_{B}}\left[1+\left(\frac{\nu^{\prime}}{\nu_{0}^{\prime}(\theta)}\right)^{2}\right]^{\frac{\beta_{B}-\alpha_{B}}{2}},$
(4)

where the constant $C$ is chosen so that $\int
d\nu^{\prime}f(\nu^{\prime},\theta)=1$. We adopt the following relation for
the observed peak energy $\nu_{0,\rm LE}(\theta,\phi)$ based on the
observations of sGRB 170817A and the requirement to satisfy the Amati
relation,

$\nu_{0,\rm LE}(\theta,\phi)=\delta_{D}\nu_{0,\rm LE}^{\prime}(\theta)\approx
0.15\times
10^{3}\delta_{D}\left[1+\left(\frac{\theta}{\theta_{c}}\right)^{3.4}\right]{\rm~{}eV},$
(5)

where $\delta_{D}=1/(\Gamma(1-\beta{\rm cos}\theta_{\Delta}))$ is the Doppler
factor, and $\nu_{0,\rm LE}^{\prime}(\theta)$ is the peak energy in the
comoving frame.

To model the energy spectrum at the VHE band of the prompt emission, we adopt
a toy model which shares similar spectral properties as the low-energy prompt
emission, with HE spectral peak

$\nu_{0,\rm VHE}(\theta,\phi)=\delta_{D}\nu_{0,\rm HE}^{\prime}(\theta)\approx
0.15\times
10^{9}\delta_{D}\left[1+\left(\frac{\theta}{\theta_{c}}\right)^{3.4}\right]{\rm~{}eV},$
(6)

where $\nu_{0,\rm HE}(\theta,\phi)$ is the observed peak energy of the VHE
prompt emission.

The optical depth for the two-photon pair annihilation process can be
estimated as (e.g., Murase et al., 2016),

$\displaystyle\tau_{\gamma\gamma}(\nu,\theta,\phi)$ $\displaystyle\approx
n^{\prime}(\tilde{\nu}^{\prime})l^{\prime}\sigma_{\gamma\gamma}$
$\displaystyle\approx\frac{1}{4\pi R^{2}\Delta
R^{\prime}\tilde{\nu}^{\prime}}\frac{E_{\gamma}(\theta)}{\Gamma(\theta)}f(\tilde{\nu}^{\prime},\theta)l^{\prime}\sigma_{\gamma\gamma}$
$\displaystyle\simeq
1.6\left(\frac{R}{10^{14}\rm~{}cm}\right)^{-2}\frac{\zeta(\theta)}{\zeta(4.5\theta_{c})},$
(7)

where

$\zeta(\theta)=(E_{\gamma}(\theta)/\Gamma(\theta))(f(\tilde{\nu}^{\prime},\theta)/\tilde{\nu}^{\prime}).$
(8)

Also, $n^{\prime}(\tilde{\nu}^{\prime})$ is the comoving frame target photon
number density at the characteristic energy
$\tilde{\nu}^{\prime}\approx\delta_{D}(m_{e}c^{2})^{2}/\nu$, $R$ is the
distance of the emission region that is measured in the lab frame, $\Delta
R^{\prime}\sim R/\Gamma(\theta)$ is the comoving shell width and the thickness
of the gamma-ray emission region, $E_{\gamma}(\theta)/\Gamma(\theta)$ is the
comoving radiation energy, $l^{\prime}$ is the photon path length before
escaping from the source, and $\sigma_{\gamma\gamma}\sim(7/12)\sigma_{T}$ is
the cross section (e.g., Svensson, 1987). We simply assume
$l^{\prime}\sim\Delta R^{\prime}$ (Matsumoto et al., 2019). In the above
estimates, we assume a broken power-law distribution with $\alpha_{B}=-1$ and
$\beta_{B}=-2.5$. We can see the optical depth depends on the distance of the
emission region with $\tau_{\gamma\gamma}\propto R^{-2}$. The optical depth
also depends on the polar angle, where $\zeta(\theta)$ sharply decreases with
increasing $\theta$ when $\theta\gtrsim 2\theta_{c}$. In the above estimates,
we assume $\theta=4.5\theta_{c}=0.26\approx 15^{\circ}$ and $\nu=1\rm~{}TeV$.
Note that we do not consider here the contribution to optical depth from
scatterings of photons by e± created by the annihilation of photon pairs, as
in case when this contribution was large we would expect the burst to be
optically thick to all photons, independently on their energy (Lithwick &
Sari, 2001).

The mean escape probability, or the attenuation factor, of the VHE emission,
can be derived by solving the radiative transfer equation assuming uniform
slab geometry (Svensson, 1987),

$\xi(\nu,\theta,\phi)=\frac{1-e^{-\tau_{\gamma\gamma}(\nu,\theta,\phi)}}{\tau_{\gamma\gamma}(\nu,\theta,\phi)},$
(9)

which is $\xi\simeq 0.6$ for $\tau_{\gamma\gamma}=1$ and $\xi\simeq 0.1$ for
$\tau_{\gamma\gamma}=10$. In general, the optical depth $\tau_{\gamma\gamma}$
at a certain radius depends on the initial assumptions on Lorentz factor and
energy profile, and on the Doppler factor.

##  4 Different energy photons from different emission zones

Figure 2: The surface brightness distribution at the MeV band (upper panel)
and TeV band (lower panel) of a structured jet for $\Gamma_{\rm max}$ = 2000
and $R=10^{14}\rm~{}cm$. The position of the jet core is indicated as a white
plus symbol and the viewing angle $\theta_{v}$ is marked as a white cross
symbol. We can see that the emission regions with 50% surface brightness are
shifted between the MeV and TeV bands (see also Fig. 4). This is mainly caused
by the high optical depth of TeV gamma-rays ($\tau_{\gamma\gamma}=10$ line
with orange dotted line). The different emission region also leads to
different arrival time in Fig. 5.

Figure 3: Same as Fig. 2, but for $R=10^{15}\rm~{}cm$. The shift of the peak
position of the specific surface brightness between the MeV and TeV bands
becomes much smaller.

The specific surface brightness per solid angle per frequency can be expressed
as (Ioka & Nakamura, 2018)

$\frac{dE_{\gamma,\mathrm{iso}}}{d\Omega
d\nu}=\frac{1}{4\pi}\frac{E_{\gamma}(\theta)[f(\nu,\theta,\phi;\nu_{0,\rm
LE})+f_{\rm HE}(\nu,\theta,\phi;\nu_{0,\rm
HE})]\xi(\nu,\theta,\phi)}{\Gamma(\theta)^{4}[1-\beta(\theta)\mathrm{cos}\theta_{\Delta}]^{3}},$
(10)

where $\int d\nu f(\nu,\theta,\phi;\nu_{0,\rm LE})=1$ and $\int d\nu f_{\rm
HE}(\nu,\theta,\phi;\nu_{0,\rm HE})=1/5$, and we assume the total energy
radiated in the VHE energy band takes only $\sim 20\%$ of the total energy
radiated in the low-energy sub-MeV/MeV band (this assumption would slightly
increase the adopted value for radiative efficiency $\epsilon_{\gamma}$). Here
$f_{\rm HE}(\nu,\theta,\phi;\nu_{0,\rm HE})$ has the same form as the low-
energy spectrum given by Eq. 4, with the spectral peak in the comoving frame
as in Eq. 6.

In Figs. 2 and 3, we show the surface brightness distribution, $\nu
dE_{\gamma,\mathrm{iso}}/d\Omega d\nu$, on the jet surface at the MeV and TeV
bands, respectively. The peak position of the specific surface brightness is
indicated by a green star at the MeV band (upper panel) and a red star at the
TeV band (lower panel). Due to the effect of the Doppler boost, the observed
brightness for the off-axis structured jet is dominated by a small patch
centered on the peak position with typical angular size $\Delta\Omega$,

$\frac{dE_{\nu,\mathrm{iso}}}{d\nu}\approx\frac{dE_{\nu,\mathrm{iso}}}{d\Omega
d\nu}\Delta\Omega,$ (11)

where $\Delta\Omega$ could be approximated as the region surrounded by the
solid contours, which represents the position where the surface brightness
decreases by a factor of $50\%$ compared to the peak value. The dotted circle
in the lower panels is the position where the optical depth for TeV photons
equals $\tau_{\gamma\gamma}=10$. While the peak position of the surface
brightness is shifted at the TeV band with respect to MeV band for smaller
radius ($R$ = 1014 cm, see Fig. 2), this shift becomes much smaller at higher
radii, see e.g. Fig. 3 for $R$ = 1015 cm. However, the region where the
surface brightness decreased by a factor of 50$\%$ becomes apparently larger,
indicating the larger time spread of the arrival times for the TeV emission
(this effect adds up to the difference in photon arrival times due to the
different emission radii, see Fig. 5). Note that we did not include the
evolution of different parameters (e.g. Lorentz factor, spectral properties)
with radius in our calculation, while this may be expected with the jet
propagation.

In conclusion, we could expect different energy photons to come from different
emission zones for a structured jet when viewed off-axis. This is because the
surface brightness distribution is different at different frequencies, mainly
due to the different optical depth with more attenuation at the TeV band than
the MeV band (and partly due to the different segment of the observed
spectrum).

An important implication is that a popular one-zone approximation in the
spectral analysis is not justified at all in the off-axis jet case. With the
current facilities it is impossible to resolve the emission region for GRBs.
One of the observable effects of such a phenomenon is the arrival time of
photons from different emission regions, which we will discuss in the
following section.

##  5 Photon arrival time and possible time delay

For a relativistic structured jet, photons emitted at the same lab frame time
$t$ at different locations may arrive at the observer at the same observed
time T (e.g., Zhang, 2018). The observed time $T$ is related to the time in
the laboratory frame $t$ as

$T=t-\frac{R}{c}{\rm cos}\theta_{\Delta},$ (12)

where $R$ is the radius of the emitting shell measured in the laboratory
frame. The laboratory frame time $t$ can be estimated as

$t=\int_{0}^{R}\frac{dr}{\beta(\theta)c}\approx\frac{R}{\beta(\theta)c}.$ (13)

The above approximation in Eq. 13 is valid if there is no acceleration or
deceleration of the relativistic shell during the propagation. The observed
time $T$ is

$\displaystyle T$
$\displaystyle=\frac{R}{c\beta(\theta)}\left(1-\beta(\theta)\mathrm{cos}\theta_{\Delta}\right)$
$\displaystyle\approx\begin{cases}\frac{R}{2c\Gamma(\theta)^{2}}&(\theta_{\Delta}<1/\Gamma(\theta))\\\
\frac{R\theta_{\Delta}^{2}}{2c}&(\theta_{\Delta}>1/\Gamma(\theta))\end{cases},$
(14)

where $\beta(\theta)\approx 1-1/2\Gamma^{2}$ and
$\mathrm{cos}~{}\theta_{\Delta}\approx 1-\theta_{\Delta}^{2}/2$. From Eq. 5,
we can see that the observed time $T$ reaches a minimum value when
$\theta_{\Delta}\sim 1/\Gamma$.

As shown in Fig. 4, the peak position of the surface brightness along the
direction of $\phi=0$ is $\theta_{\rm MeV}\simeq 13.8\rm~{}deg$ at the MeV
energy band and $\theta_{\rm TeV}\simeq 15.0\rm~{}deg$ at the TeV energy band,
respectively.

Figure 4: Similar to Fig. 2, we show the surface brightness distribution as a
function of the polar angle $\theta$ when $\phi=0$.

The relative difference in the arrival time of TeV and MeV photons is shown in
Fig. 5 as a function of the emission radius. Our results indicate that the
arrival of the TeV photons is typically delayed compared to MeV photons, and
the value reaches a maximum of approximately $T_{\rm pk,delay}\sim 8\rm~{}s$
when $R=10^{13.75}\rm~{}cm$, see the purple thick solid curve in Fig. 5.

The time delay between the TeV photons and MeV photons is caused by the fact
that the TeV photons and MeV photons have different surface brightness, mainly
due to the two-photon annihilation optical depth, see Fig. 2. The reason is
that the emission zone of MeV photons is typically located at $\theta_{\rm
MeV}\sim\theta_{\rm min}$, where $T$ reaches a minimum at $\theta_{\rm min}$.
However, the emission zone of TeV photons is located at $\theta_{\rm
TeV}>\theta_{\rm min}$, where $T$ is larger. At a smaller radius, the whole
VHE emission region is significantly attenuated due to the larger optical
depth. With the increase of the emission radius, the VHE emission region near
the center is still optical thick, but the outer region becomes transparent.
Under such a situation, only VHE photons from the outer region could escape
which are delayed compared to MeV photons. The time delay between TeV and MeV
photons becomes smaller for larger radius, i.e. $R\gtrsim 10^{15}\rm~{}cm$,
where both the VHE emission region and sub-MeV/MeV emission region are
optically thin.

We also show the time spread of the arrival times ( duration) of the MeV
prompt emission from the interior of the half-maximum surface brightness line
as a function of radius in dashed green line. Similarly, the duration of the
TeV emission is shown in dashed orange line. We can see that it is possible
that the time delay between TeV and MeV photons could be larger than the
typical duration of the MeV prompt emission. The typical time delay between
TeV and MeV photons could be significant for energetic events when
$E_{0}=10^{53.8}\rm~{}erg$ because the VHE emission region move outwards due
to the higher optical depth of TeV gamma-rays, see the brown thin solid curve.

Figure 5: The arrival time delay between TeV and MeV photons and the duration
of GRB prompt emission at the MeV energy band as a function of the emission
region radius. The spectral and jet parameters are the same as in Fig. 2. Note
that the viewing angle is set to $\theta_{v}=0.38$.

With the current and near-future VHE gamma-ray facilities, especially with the
operation of CTA (The CTA Consortium, 2019), one may expect co-incident
detections of nearby GRBs at the VHE band with gravitational waves (e.g.,
Murase et al., 2018; Bartos et al., 2019). The study in this work suggests
that the TeV emission pulse could lag behind the main pulse of the prompt
emission at the MeV band, which could compensate for the large slewing time of
CTA, which is $t_{\rm slew}\sim 20\rm~{}s$ for CTA-LST and $t_{\rm slew}\sim
90\rm~{}s$ for CTA-MST (Banerjee et al., 2022). This effect, therefore,
increases the probability of observing the TeV emission from short gamma-ray
bursts during the prompt phase by CTA. Also, the possibility of the prolonged
duration of the prompt phase may be interesting for interpreting the observed
features of long GRBs linked to the compact object binary mergers, e.g. GRB
211211A (Troja et al., 2022; Yang et al., 2022; Mei et al., 2022; Rastinejad
et al., 2022).

##  6 Discussion and implications

When considering the off-axis structured jets, we found that different energy
photons could arrive from different emission zones mainly due to the effect of
the two-photon pair annihilation process. The main reason is that the optical
depth for VHE photons is much higher in the core region on the jet surface,
which gradually decreases outwards allowing VHE photons to escape. In
addition, we showed that the optical depth for VHE photons is sensitive to the
emission radius, where the corresponding time delay between the typical
arrival time of the TeV and MeV emission decreases with the increase of the
emission radius. Such a phenomenon could be prominent if the optical depth
sharply decreases across the emission zone, such as in the case of the
Gaussian jet adopted in this work, where $\zeta(\theta)$ strongly depends on
$E_{\gamma}(\theta)$. A similar effect is possible for the power-law structure
of the jet energy if the power-law index is steep. Note that the angular
dependence of the optical depth on the Lorentz factor $\Gamma(\theta)$ and
energy spectrum $\nu_{0,\rm HE}^{\prime}(\theta)$ could also significantly
affect the surface brightness distribution for different energy photons. The
delayed arrival of $\gtrsim$ TeV photons is also expected in the EIC model
(e.g., Murase et al., 2010; Kimura et al., 2019; Zhang et al., 2023), without
considering the structured jet.

Another important factor is the viewing angle $\theta_{v}$. If $\theta_{v}$ is
close to or smaller than the jet core, we cannot resolve different emission
regions which are similar to the on-axis case. However, if $\theta_{v}$ is too
large, the received flux would be lower than the detection threshold. The
gravitational-wave data could provide an independent measurement of the
inclination angle between the direction of the line-of-sight and jet axis
(Biscoveanu et al., 2020). At present, we can only say that for an off-axis
structured jet with properties similar to sGRB 170817A, different energy
photons could originate from different emission zones. Banerjee et al. (2022)
estimated the minimum isotropic energy $\sim$ 1047 ergs required for the
detection of an event at z $\sim$ 0.1 (up to which the current GW detectors
are expected to observe BNS mergers), and that would allow for this effect to
be observed by the CTA.

##  7 Summary

The off-axis model for the short gamma-ray burst GRB 170817A predicted that
the most luminous region arises neither from the jet core around the primary
axis, nor at the line of sight at the viewing angle $\theta_{v}$, but from the
off-centre jet (Ioka & Nakamura, 2019). Adopting the same assumptions in this
study, we showed that different energy photons could arrive to the observer
from different emission zones for off-axis structured jets, and that the
typical arrival time of VHE photons could be delayed compared to the typical
arrival time of prompt sub-MeV/MeV photons. We discussed how the change in the
emission radius could affect the VHE emission region and related arrival
times. Our results depend on the angular evolution of the total radiation
energy $E_{\gamma}$ and of the Lorentz factor, and on the energy spectrum
(currently the spectral evolution with radius was not accounted for). The off-
axis structured jet could also be observed with a smaller or larger viewing
angle depending on the energetics and detector threshold. One of the
predictions of our model is the difference between the observed arrival time
of prompt MeV emission and high-energy TeV emission. In general, the
observation of the time arrival difference brings information on the emission
radius.

This model could be applicable to nearby short GRBs, VHE afterglow emission
for energetic bursts, but also to LL GRBs, which are interesting as possible
TeV emission and neutrino/UHECR sources (Murase et al., 2008; Murase & Beacom,
2010; Boncioli et al., 2019; Rudolph et al., 2022). As LL GRBs have presumably
lower ejecta velocities and larger opening angles (Bromberg et al., 2011; Cano
et al., 2017; Rudolph et al., 2022), the different assumptions should be taken
into account for the estimates of the emission zones for MeV/TeV photons.

## Acknowledgements

Ž.B. acknowledges the support by FY2022 JSPS Invitational Fellowship for
Research in Japan (Short-term) S22013. This work is partly supported by
KAKENHI No. 23H05430, 23H04900, 22H00130, 20h01901, 20H01904, 20H00158 (K.I.).
The work of K.M. is supported by the NSF Grant No. AST-1908689, No.
AST-2108466 and No. AST-2108467, and KAKENHI No. 20H01901 and No. 20H05852.

## Data Availability

The data developed for the calculation in this work is available upon request.

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