# Jet Lorentz factor constraint for GRB 221009A based on the optical depth of
the TeV photons

Duan-yuan Gao School of Physics, Huazhong University of Science and
Technology, Wuhan 430074, China  Yuan-Chuan Zou School of Physics, Huazhong
University of Science and Technology, Wuhan 430074, China

###### Abstract

The recent detection of tera-electronvolt (TeV) photons from the record-
breaking gamma-ray burst GRB 221009A during its prompt phase poses challenges
for constraining its Lorentz factor. We re-evaluate the constraints on the jet
Lorentz factor considering a two-zone model, wherein the TeV photons originate
from the external shock region while the lower energy MeV photons come from
the internal prompt emission region. By properly accounting for the evolution
of the MeV photon spectrum and light curve, we calculate the optical depth for
TeV photons and derive a minimum Lorentz factor about 300. It is consistent
with the afterglow modeling for the TeV emission.

Gamma-ray bursts(629) — High energy astrophysics (739) — Relativistic
jets(1390)

##  1 Introduction

GRB 221009A is the brightest gamma-ray burst ever observed in recorded
history. Existing statistical data suggest that such an event occurs, on
average, once every ten thousand years (Burns et al., 2023). The central
engine of GRB 221009A is very close to Earth, with a redshift of 0.151 (de
Ugarte Postigo et al., 2022) and trigger time ($T_{\rm 0}$) of 2022 October 9
at 13:16:59.99 UTC(Lesage et al., 2023). The prompt burst’s isotropic energy
is exceptionally high, on the order of $10^{55}\,\rm{erg}$, concentrated in
the MeV band (An et al., 2023; Lesage et al., 2023; Frederiks et al., 2023).
This indicates that the main burst emitted a significant amount of MeV
photons, allowing for spectral fitting in different time intervals. An et al.
(2023); Lesage et al. (2023) have all performed band power-law spectral
fitting for different time intervals of GRB 221009A. Approximately 226 seconds
after the burst, LHAASO collaboration (2023) detected 64,000 TeV photons,
including a photon with an energy of 7 TeV. Some researchers propose that the
origin of TeV photons may be attributed to certain hadronic processes in the
prompt burst (Wang et al., 2023). Conversely, others believed that TeV photons
may originate from external Inverse Compton and proton synchrotron radiation
in the external shock (Zhang et al., 2023). However, LHAASO collaboration
(2023) argued that these photons originate from synchrotron self-Compton (SSC)
emission in the afterglow. The observation of very high-energy TeV emission
during the prompt phase suggests that the Lorentz factor might have reached an
extremely large value, which may raise certain concerns regarding the self-
consistency of the overall picture.

Lorentz factor is always an important property of GRB jets. As discussed in
Zhang et al. (2021), the bulk Lorentz factor in the prompt phase and in the
deceleration phase are different. Various methods have been proposed to either
directly obtain or constrain the Lorentz factor of the GRB jet at different
phases. In the external shock scenario, the deceleration time of the material,
reflected in the peak of the light curve, can be used to estimate the Lorentz
factor (Mészáros & Rees, 1997; Sari & Piran, 1999a, b; Kobayashi, 2000). By
comparing X-ray and optical emissions and fitting the parameters of the
emitting region, the Lorentz factor can also be estimated (Sari & Piran,
1999a; Zhang et al., 2003). Considering the prompt emission should not be
Compton scattered by the electrons associated with ejected baryons, one could
set a lower limit (Limit C in Lithwick & Sari, 2001), which is generally less
than one and is useful only for strong GRBs. Additionally, the Lorentz factor
can be estimated through the observation of the thermal component (Nakar et
al., 2005; Pe’er et al., 2007; Zou et al., 2015). Another approach involves
considering that the high-energy gamma-ray photons to escape without being
absorbed by lower-energy gamma-ray photons. Two models have been proposed for
this: the single zone model, where high-energy and low-energy photons
originate from the same region (Lithwick & Sari, 2001), and the two-zone
model, which suggests that high-energy and low-energy photons come from
different regions (Zou et al., 2011). Most high-energy emissions have shown
smooth light curves, consistent with the afterglow model (Ghisellini et al.,
2010). For GRB 221009A, the smooth TeV light curve indicates an external shock
origin. LHAASO collaboration (2023) have successfully modeled the entire light
curve and spectrum with a standard afterglow plus SSC model. This suggests
that the TeV emission should not originate from the same internal shock
regions that produce the prompt MeV emission. Therefore, the constraint on the
Lorentz factor should consider the two-zone effect.

For GRB 221009A, Lesage et al. (2023) estimated the minimum values of the
Lorentz factors at different stages. During the precursor radiation phase, the
minimum bulk Lorentz factor was found to be 188. In the main burst phase,
considering that the emission site must be optically thin, the minimum bulk
Lorentz factor for the minimum variability timescale
($t_{\rm{var}}=0.1\rm{s}$) was calculated to be 1040. According to the single
zone model, the minimum Lorentz factor estimated for the highest-energy photon
at 99.3 GeV was 1560, and considering a more relativistic situation could
lower it to 780. Moving on to the afterglow phase, considering the
interstellar medium model, the minimum Lorentz factor was determined to be
260, while in the stellar wind model, it was 282. Based on support from the
light curve, LHAASO collaboration (2023) adopted the uniform interstellar
medium model to estimate the bulk Lorentz factor of the afterglow as 560. Yang
et al. (2023) used data from the Fermi Gamma-ray Burst Monitor (precursor) and
SATech-01/GECAM-C (main emission and flare) to fit the Lorentz factor of GRB
221009A using the SSC model in different time segments. The Lorentz factor was
estimated to be around 250, reaching a maximum of 741.

In this work, we check the constraint by the $\gamma\gamma$ absorption on the
TeV photons. As the light curve of GRB 221009A has smooth temporal profile
which supports the TeV photons come from external shock, we mainly focus on
the so called two-zone model. In section 2, we introduce the single zone model
and the two-zone model considering spectral and variability corrections, as
well as present the simplified equations for the two-zone model. In section 3,
we calculate the optical depth for GRB 221009A for fixed Lorentz factors and
minimum Lorentz factors. Finally, in the section 4, we provide conclusions and
future prospects.

##  2 restriction to Lorentz factor

###  2.1 Single zone model

If TeV photons and MeV photons are produced in the same location, they may
collide to generate electron pairs. This leads to a limit for the minimal
Lorentz factor of single zone model (Lithwick & Sari, 2001; Gupta & Zhang,
2008). The energy required for photon annihilation is much lower than the peak
energy of the spectrum. Therefore, we treat the spectrum as a single power-law
spectrum, using the photon index in the low-energy range. So, the photon
spectrum of MeV photons (i.e., the number density of photons per unit time per
unit area per unit energy) can be represented as $fE^{-\beta}$, where f is the
normalization factor, and $E$ is the photon energy. The Optical depth of
single power-law spectrum was described in Gupta & Zhang (2008) as case 1,
i.e., equations (13) and (14). By replacing $\beta_{2}$ with the low-energy
index $\beta$, changing the optical depth $\tau_{int}$ to $\tau$, and
replacing the maximum energy of electrons $E_{\gamma h}^{o}$ with $E_{\max}$,
we obtain:

$\tau=\frac{A_{1}(E_{\max})}{R^{2}}\left(\frac{\Gamma}{1+z}\right)^{-2\beta+2},$
(1)

where $R$ is the emitting radius from the central engine, which is different
in different model, $z$ is the redshift, $\beta=1.69$ is the photon spectral
index for GRB 221009A (Lesage et al., 2023), $E_{\max}=\rm{7\ TeV}$, and
$A_{1}$ is defined as

$\displaystyle A_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{(11/180)\sigma_{\rm{T}}fd_{z}^{2}R}{(\beta-1)\Gamma^{2}c}\left(\frac{E_{\max}}{m_{\rm{e}}^{2}c^{4}}\right)^{\beta-1},$

where $\sigma_{\rm T}$ is the Thomson cross section, $m_{\rm e}$ is the rest
mass of an electron, $c$ is the speed of light, $d_{z}$ is the luminosity
distance. As the photon with energy $E_{\max}$ has been observed, it indicates
the optical depth $\tau<1$, which derives a minimum Lorentz factor from
equation (1). Since what we have observed is the photon flux per second rather
than the total flux, for convenience, we replace $f_{1}^{o}$ of equation (14)
in Gupta & Zhang (2008) with $fR/(\Gamma^{2}c)$, and
$fd_{z}^{2}=3.32\times{10}^{54}{\rm erg}^{\beta-1}{\rm s}^{-1}$ which can be
derived from the isotropic luminosity and peak energy. To be consistent with
Lithwick & Sari (2001), we change the coefficient $3/[8(\beta_{1}^{2}-1)]$ in
equation (14) of Gupta & Zhang (2008) to $11/[180(\beta-1)]$. We set the
optical depth equal to 1 to obtain the minimum Lorentz factor. In the case of
internal shocks, emitting radius is $R=2\Gamma^{2}ct_{v}/(1+z)$, where
$t_{v}=0.1{\rm s}$ (Lesage et al., 2023) is the variability time scale of the
prompt emission. We finally get $\Gamma_{\min}=2164$. This value is much
larger than that given by Lesage et al. (2023) ($\Gamma_{\min}=1560$), because
the maximum energy of the high energy photon was 99.3 GeV in Fermi-LAT data.
This is far greater than the estimated value by the afterglow model, which is
unreasonable.

If we adopt the Internal Collision-Induced Magnetic Reconnection and
Turbulence (ICMART) model, the radius can reach to a larger value of
approximately $R=2\Gamma^{2}ct_{v}/(1+z)$(Zhang & Yan, 2011), where $t_{v}\sim
20{\rm s}$ represents the characteristic timescale of the long pulse. In this
case, the minimum Lorentz factor can be reduced to 808.

###  2.2 Two-zone model

From the TeV light curve given by LHAASO collaboration (2023), it can be well
modeled by the external shock. Therefore, we turn to consider the two-zone
model (Zou et al., 2011). We assume that TeV photons come from the external
shock, while MeV photons originate from the prompt internal shocks, which is
consistent with LHAASO collaboration (2023).

In the two-zone model, we assume that MeV and TeV photons come from distinct
regions. The MeV photons come from an emitting region with a radius
$R_{\rm{M}}$, a Lorentz factor of $\Gamma_{\rm{M}}$, and an angular width of
$\theta_{\rm{M,J}}$. Similarly, the TeV photons come from a different emitting
region with a radius $R_{\rm{T}}$, a Lorentz factor of $\Gamma_{\rm{T}}$, and
an angular width of $\theta_{\rm{T,J}}$. Notice the notation of T (meaning
TeV) was used as G (meaning GeV) in Zou et al. (2011). To get a more accurate
constraint, we extend the T90 in Zou et al. (2011) to a real time varying MeV
light curve.

Figure 1: A schematic diagram illustrates the geometry of the difference in
retarded time between MeV photons and TeV photons. When TeV photons collide
with MeV photons at point A. the MeV photons will not reach Earth as the
direction is not pointing to the Earth, but we can determine their number
density from the photons arriving at Earth with the same retarded time. The
MeV photons move radially at $R$, while the TeV photons move along the x-axis
at $x=R\cos\theta$ when they are colliding.

Figure 1 shows a sketch for the geometry of TeV and MeV photons collision and
propagating. A more detailed illustration can be seen in Zou et al. (2011). We
set the x-axis toward Earth, and the zero point of x is at the central engine.

To describe the distribution of MeV photons, we utilize the photon field,
assuming that the MeV photon comes from a radius $R_{\rm M}$, which is much
smaller than R. In other words, we consider the MeV photons as originating
from the center approximately. Hence, the photon number density is decay with
$R^{2}$ , in the other word, the photon number density times radius square
only related to the radius $R$ and time $t$, denoted as $R^{2}n_{\rm M}(R,t)$.
Since photons always move at the speed of light, we can express $R^{2}n_{\rm
M}$ as a function of the retarded time $t_{\rm R,M}=t-R/c$. The retarded time
does not change with the movement of photons. Taking the retarded time
obtained from the trigger time and the position of Earth as the zero point,
the retarded time is equivalent to the time on the light curve. Similarly, for
TeV photons that come to Earth, retarded time is $t_{\rm R,T}=t-x/c$.

Considering a TeV photon at $(R,\theta)$ that moves toward Earth and collides
with a MeV photon, the time $t$ of the TeV photon and the MeV photon is the
same. So the difference in their retarded time is:

$\delta t_{\rm{R}}=t_{\rm R,T}-t_{\rm
R,M}=(1+z)\left(\frac{R}{c}-\frac{x}{c}\right)=(1+z)\frac{R}{c}(1-\cos\theta)\approx\frac{1}{2}(1+z)\theta^{2}\frac{R}{c}.$
(3)

Due to the TeV photons direction being parallel to the line connecting the
central engine and the Earth, the collision angle between TeV photons and MeV
photons is $\theta$, and the distance of the TeV photons to the x-axis remains
constant. This means:

$R\sin\theta=R_{\rm{T}}\sin\theta_{\rm{T}}={\rm const.}\approx R\theta.$ (4)

Combining equation (3), we can deduce that $\delta
t_{\rm{R}}\propto\frac{1}{R}$. Therefore, at an infinite distance, $\delta
t_{\rm{R}}=0$, indicating that the MeV photons and TeV photons that collide at
Earth will reach Earth simultaneously. This natural consequence confirms the
consistency in the timing of their arrival at Earth.

Because $n_{\rm M}$ is zero when $t_{\rm{R}}>T_{90}$, so we can only consider
$R$ smaller than a certain distance (Zou et al., 2011):

$R_{\max}(R_{\rm{T}},\theta_{\rm{T}})=\frac{R_{\rm{T}}}{1-2cT_{\rm{90}}/[(1+z)R_{\rm{T}}\theta_{\rm{T}}^{2}]}.$
(5)

Due to the high brightness of GRB 221009A, the spectrum of MeV photons is very
accurate, even in each time interval. Therefore, we can describe the spectrum
using the Band function, with the number density $n_{\rm{0,MeV}}$, the power
law photon indices $\alpha$ and $\beta$, and peak energy $E_{\rm{P}}$ depend
on time:

$\displaystyle n_{\rm{MeV}}(E_{\rm{M}})\simeq
n_{\rm{0},\rm{MeV}}(R,t_{\rm{R}})\left\\{\begin{array}[]{ll}\left[\frac{E_{\rm{M}}}{E_{\rm{P}}(t_{\rm{R}})}\right]^{-\alpha_{\rm{M}}(t_{\rm{R}})},\phantom{00}&E<E_{\rm{P}}(t_{\rm{R}}),\\\
\left[\frac{E_{\rm{M}}}{E_{\rm{P}}(t_{\rm{R}})}\right]^{-\beta_{\rm{M}}(t_{\rm{R}})},\phantom{00}&E_{\rm{p}}(t_{\rm{R}})<E<E_{{\rm{M}},{\max}},\end{array}\right.$
(8)

where $\alpha_{\rm{M}}(t_{\rm{R)}}$ is the low energy photon index and
$\beta_{\rm{M}}(t_{\rm{R}})$ is the high energy photon index. We aim to use
the spectral indices to obtain a more accurate number density of the MeV
photons. Considering the high-energy power law spectral index $\beta$ is
approximately 2, and the upper energy boundary of HXMT is close to
$E_{\rm{P}}$, we need to consider the upper limit of the integral. The number
density of the MeV photons can be expressed as follows:

$n_{0,\rm{Mev}}=\frac{L_{\rm{M}}(t_{\rm{R}})}{(1+z)^{2}4\pi
R^{2}c\int_{0}^{E_{\max,\rm{o}}}E\left(\frac{E}{E_{\rm{P}}(t_{\rm{R}})}\right)^{-{\alpha}}{\rm
d}E}=\frac{L_{\rm{M}}(t_{\rm{R}})}{(1+z)^{2}4\pi
R^{2}c{E_{\rm{P}}^{*2}(t_{\rm{R}})}},$ (9)

where $E_{\max,\rm{o}}$ is the maximal energy of the observed MeV photon, the
power law photon index $\alpha$ can take either $\alpha_{\rm{M}}$ or
$\beta_{\rm{M}}$ at different $E$, and $E_{p}^{*}(t_{\rm{R}})$ is the
effective peak energy:

$E_{p}^{*}(t_{\rm{R}})=E_{p}(t_{\rm{R}})\left[\frac{1}{2-{\alpha(t_{\rm{R}})}}-\frac{1}{\beta(t_{\rm{R}})-2}\left(1+E_{\rm{P}}^{\beta(t_{\rm{R}})-2}E_{\max}^{2-\beta(t_{\rm{R}})}\right)\right]^{{1}/{2}}.$
(10)

The interaction between a TeV photon and a MeV photon can occur as long as the
total energy in the center-of-mass system is larger than $2m_{\rm{e}}c^{2}$.
Therefore, only photons that are more total energetic than $E_{M,\min}$ can
annihilate a TeV photon with energy $E_{\rm{T}}$:

$E_{M,\min}=\frac{2(m_{\rm{e}}c^{2})^{2}}{(1+z)^{2}E_{\rm
T}(1-\cos\theta)}\simeq\frac{4(m_{\rm{e}}c^{2})^{2}}{(1+z)^{2}\theta^{2}E_{\rm
T}}.$ (11)

For MeV photons that are sufficiently energetic, the cross-section depends on
the energy as follows:

$\sigma=\frac{3}{16}\sigma_{\rm{T}}(1-\hat{\beta}^{2})\left[(2-\hat{\beta}^{2})\ln\frac{1+\hat{\beta}}{1-\hat{\beta}}-2\hat{\beta}(2-\hat{\beta}^{2})\right],$
(12)

where

$\hat{\beta}=\sqrt{1-2\frac{m_{\rm{e}}c^{2}}{(1+z)E_{\rm{M}}}\frac{m_{\rm{e}}c^{2}}{(1+z)E_{\rm{T}}}(1-\cos\theta)^{-1}},$
(13)

$\hat{\gamma}=\sqrt{1-\hat{\beta}^{2}}$ and $\sigma_{\rm{T}}\simeq 6.65\times
10^{-25}\mathrm{cm^{2}}$ is the Thompson cross-section.

Now, we can estimate the overall optical depth for the interaction of a TeV
photon with the MeV pulse:

$\tau(\theta_{\rm{T}},E_{\rm{T}})=\int_{R_{\rm{T}}}^{R_{\max}(R_{\rm{T}},\theta_{\rm{T}})}{\rm
d}R\int_{E_{M,\min}}^{E_{M,\max}}{\rm d}E_{\rm{M}}\frac{{\rm
d}^{3}n_{\rm{MeV}}(t_{\rm{R}})}{{\rm d}E_{\rm{M}}{\rm
d}\theta_{\rm{M}}}\sigma(1-\cos\theta),$ (14)

where $E_{\max}$ is the upper limit of MeV component.The effective optical
depth for a photon observed with an energy $E_{\rm{T}}$ is averaged over all
angles:

$e^{-\bar{\tau}(E_{\rm{T}})}=\frac{\int_{0}^{\theta_{\rm
T,j}}\mathcal{D}^{-(3+\beta_{\rm{T}})}e^{-\tau(\theta_{\rm{T}},E_{\rm{T}})}\theta_{\rm{T}}{\rm
d}\theta_{\rm{T}}}{\int_{0}^{\theta_{\rm
T,j}}\mathcal{D}^{-(3+\beta_{\rm{T}})}\theta_{\rm{T}}{\rm d}\theta_{\rm{T}}},$
(15)

where $\mathcal{D}=\Gamma_{\rm{T}}({1-\beta_{\rm bulk,T}\cos\theta_{\rm{T}}})$
is the Doppler factor, $\beta_{\rm bulk,T}=(1-1/\Gamma_{\rm{T}}^{2})$ is the
bulk velocity in $c$, and $\beta_{\rm{T}}$ is the photon index of the TeV
emission. Since LHAASO observes a maximum angle of 0.8° corresponding to the
afterglow, which is much larger than $1/\Gamma_{\rm{T}}$, and due to the
beaming effect, we cannot see photons beyond $1/\Gamma_{\rm{T}}$. Therefore,
our integration upper limit is $1/\Gamma_{\rm{T}}$.

By setting the effective optical depth $\bar{\tau}$ to 1, one can solve
equation (15) to get $\Gamma_{\rm T,\min}$.

###  2.3 A simplified model

To improve the feasibility of our model, we make some approximations to derive
an analytic formula. We have found some mistakes in Zou et al. (2011), and we
will correct and address them in the subsequent formula descriptions. We will
later compare these results with the full formula. First, since for a specific
$\theta_{\rm{T}}$, only one power-law index of the spectrum determines the
optical depth, we use a single power-law to describe the spectrum. Second, we
approximate the cross-section as $\sigma_{\rm{T}}/(3\hat{\gamma}^{2})$. Third,
we assume that the angle of MeV photons is larger than the angle of TeV
photons. We define $l\equiv\frac{R}{R_{\rm{T}}}$ and the maximum radius
$l_{\max}\equiv
R_{\max}/R_{\rm{T}}=1/[1-2\Delta_{M}/(R_{\rm{T}}\theta_{\rm{T}}^{2})]$. The
minimal energy of the MeV photon for pair production is (Zou et al., 2011):

$E_{min}\approx\frac{4(m_{\rm{e}}c^{2})^{2}}{(1+z)^{2}E_{\rm{T}}\theta_{\rm{T}}^{2}}l^{2}.$
(16)

The number density of the MeV photons is:

$N(E_{\rm{M}})\approx
n_{0,\rm{Mev}}\left(E_{\rm{M}}\right)=\frac{(1+z)^{2{\alpha}-2}L_{\rm{M}}E_{\rm{T}}^{\alpha}E_{\rm{P}}^{\alpha}\theta_{\rm{T}}^{2{\alpha}}}{4\pi
R_{\rm{T}}^{2}c(2m_{\rm{e}}c^{2})^{2{\alpha}}{E_{\rm{P}}^{*}}^{2}}l^{-(2{\alpha}+2)}y^{-{\alpha}},$
(17)

where $y=E_{\rm{M}}/E_{\rm{M,\min}}=\hat{\gamma}^{2}$. We have corrected
$\hat{\gamma}$ (Zou et al. (2011)) to $\hat{\gamma}^{2}$, and also have
removed the unnecessary $\pi$ in front of $m_{\rm{e}}c^{2}$ in the denominator
of equation (17) (equation (12) in Zou et al., 2011). Considering the
simplified equations above, we have the optical depth for a TeV photon emitted
at $\theta_{\rm T}$:

$\begin{split}\tau(\theta_{\rm{T}})&=\int_{R_{\rm{T}}}^{R_{\max}(R_{\rm{T}},\theta_{\rm{T}})}{\rm
d}R\int_{E_{\rm{{M,\min}}}(\theta,R)}^{E_{\rm{{M,\max}}}}{\rm
d}E_{\rm{M}}\cdot
N(E_{\rm{M}})\times(1-\cos\theta)\frac{\sigma_{\rm{T}}}{3\hat{\gamma}^{2}}\\\
&\approx\frac{(1+z)^{2{\alpha}-4}L_{\rm{M}}\epsilon_{\rm{P}}^{\alpha}\epsilon_{\rm{T}}^{{\alpha}-1}\sigma_{\rm{T}}\theta_{\rm{T}}^{2{\alpha}}}{6\times
4^{\alpha_{\rm{M}}}\pi
R_{\rm{T}}m_{\rm{e}}c^{3}{\epsilon_{\rm{P}}^{*}}^{2}{\alpha}}\left[\frac{1}{2{\alpha}+1}\left(1-l_{\max}^{-(2{\alpha}+1)}\right)-\left(\frac{1}{4}(1+z)^{2}\epsilon_{\max}\epsilon_{\rm{T}}\theta_{\rm{T}}^{2}\right)^{-{\alpha}}\times\left(1-\frac{1}{l_{\max}}\right)\right],\end{split}$
(18)

where $\epsilon_{\rm{P}}\equiv E_{\rm{P}}/m_{\rm{e}}c^{2}$,
${\epsilon}_{\rm{P}}^{*}\equiv{E}_{\rm{P}}^{*}/m_{\rm{e}}c^{2}$,
$\epsilon_{\rm{T}}\equiv E_{\rm{T}}/m_{\rm{e}}c^{2}$ and
$\epsilon_{\max}\equiv E_{\rm M,\max}/m_{\rm{e}}c^{2}$. We have corrected the
power law index of (1+z) in the numerator from $2\alpha-3$ (equation (13) in
Zou et al., 2011) to $2\alpha-4$, which was a typo.

From this equation, we can see that $\tau$ is proportional to
${\theta_{\rm{T}}}^{2{\alpha}}$. By substituting into equation (15), the other
two terms in the numerator of the integrated function are also increasing
functions of $\theta_{\rm{T}}$, so we understand that the effective optical
depth is mainly determined by the maximum value of $\theta_{\rm{T}}$. For GRB
221009A, the outflow angle is larger than $1/\Gamma_{\rm{T}}$, so the maximum
value of $\theta_{\rm{T}}$ is $1/\Gamma_{\rm{T}}$. Since we believe the TeV
photons come from the outflow, $R_{\rm{T}}$ can be expressed as
$R_{\rm{T}}=2\Gamma_{\rm{T}}^{2}ct_{\rm{T}}/(1+z)$. Finally, we obtain the
minimum Lorentz factor:

$\begin{split}\Gamma_{\rm
T,min}&\approx\left[\frac{(1+z)^{2{\alpha}-3}L_{\rm{M}}\epsilon_{\rm{T}}^{{\alpha}-1}\epsilon_{\rm{P}}^{\alpha}\sigma_{\rm{T}}}{12\times
4^{{\alpha}}\pi
m_{\rm{e}}c^{4}t_{\rm{T}}{\epsilon_{\rm{P}}^{*}}^{2}{\alpha}}\right]^{\frac{1}{2{\alpha}+2}}\\\
&\approx 34\left(\frac{30\times 10^{4{\alpha}}}{{\alpha}\times
34^{2{\alpha}+2}}\right)^{\frac{1}{2{\alpha}+2}}\left[\left(\frac{1+z}{2}\right)^{2{\alpha}-3}L_{M,51}\epsilon_{\rm{T}}^{{\alpha}-1}\epsilon_{\rm{P}}^{\alpha}{\epsilon_{\rm{P}}^{*}}^{-2}t_{\rm{T,2}}^{-1}\right]^{\frac{1}{2{\alpha}+2}}.\end{split}$
(19)

##  3 results

Figure 2: The plot shows the optical depth for a photon with $E_{\rm{T}}=7$
TeV as a function of $t_{\rm{T}}$ when the Lorentz factor
$\Gamma_{\rm{T}}=560$. Different lines in different colors and line styles
represent the optical depth as a function of $t_{\rm{T}}$ for different
methods. The black thick line represents the result obtained using the full
formula, i.e., equation (15). This is the most comprehensive and accurate
calculation taking into account all relevant factors. The orange dashed line
is obtained by neglecting changes in luminosity, power-law indices, and peak
energy. In this method, we substitute the variables in equation (9) with their
time-averaged values, while using equation (15) for other parts of the
calculation. The green dashed-dotted line considers only changes in
$t_{\rm{T}}$ and neglects $\delta t_{\rm{R}}$, i.e., $\delta t_{\rm{R}}=0$,
instead of using the full expression from equation (3). Other parts of the
calculation use equation (15). The purple dotted line omits the correction for
the photon number density with the power-law index of the spectrum, i.e.,
$E_{\rm{P}}^{*}=E_{\rm{P}}$, instead of using the full expression from
equation (10). Other parts of the calculation use equation (15). The red thin
solid line is obtained using a simpler method, i.e., equation (18).

Taking all the observational quantities into equation (15) and the related
equations, we can finally calculate the average optical depth for a TeV photon
with certain energy and at a certain observing time. Figure 2 illustrates the
dependence of $\bar{\tau}$ on $t_{\rm{T}}$ for a set of parameters of GRB
221009A: $z=0.151$ (de Ugarte Postigo et al., 2022), $E_{\rm{T}}=E_{\max}=7$
TeV, $\beta_{\rm{T}}=2.3$, $\Gamma_{\rm{T}}=560$ (LHAASO collaboration, 2023),
$E_{\rm\max,o}=10$ MeV, and $L_{\rm{M}}(t_{\rm{R}})$,
$E_{\rm{P}}(t_{\rm{R}})$, $\alpha_{\rm{M}}(t_{\rm{R}})$,
$\beta_{\rm{M}}(t_{\rm{R}})$ obtained from An et al. (2023) and Frederiks et
al. (2023). The data from Frederiks et al. (2023) cover a longer time
interval, while the data from An et al. (2023) have a higher time resolution.
Hence, we use the data from An et al. (2023) to describe the pulse and the
data from Frederiks et al. (2023) for other time intervals. Here,
$t_{\rm{T}}=T-T^{*}$, where $T^{*}=226$ seconds(LHAASO collaboration, 2023).

The parameter $t_{\rm{T}}$ influences $\bar{\tau}$ through $R_{\rm{T}}$ and
the light curve of MeV photons. The three curves (full formula, without
$\delta t$ and $E_{\rm{P}}^{*}=E_{\rm{P}}$) exhibit obvious bends due to the
light curve, while the remaining curves (constant $L_{\rm{M}}$ and simple),
not considering the effect of light curve, are almost straight lines. We see
that $R_{\rm{T}}$ has a larger impact on $\bar{\tau}$ in the case of full
formula (corresponding to black thick line in figure 2) and
$E_{\rm{P}}^{*}=E_{\rm{P}}$ (corresponding to purple dotted line in figure 2),
but the light curve of MeV photons has a larger impact on $\bar{\tau}$ in the
case of without $\delta t$(corresponding to green dashed dotted line in figure
2). From equation (11), we find that $E_{\min}$ is much smaller than
$E_{\rm{P}}$, indicating that low-energy MeV photons contribute more to the
optical depth. In the simple model(corresponding to red thin line in figure
2), we choose the low-energy power-law index as the spectral index. The
difference between the simple models and the full formula is not significant.

The results obtained using constant luminosity (corresponding to orange dashed
line in figure 2) are smaller than the full formula at the beginning and
larger than the full formula towards the end. This is because we used the
average value of the entire pulse, and the number density of MeV photons is
lower than the number density at the peak but higher than the number density
after the pulse.

The results obtained using $E_{\rm{P}}^{*}=E_{\rm{P}}$ are much larger than
the full formula because $E_{\rm{P}}>E_{\rm{P}}^{*}$ at every time, and the
optical depth $\tau\propto{E_{\rm{P}}^{*}}^{-2}$ due to equation (18). The
significant difference between the two curves shows that the correction for
the photon number density with the power-law index of the spectrum is
important.

The difference between the full formula and the method that neglect $\delta t$
correction indicates the importance of this correction. The two curves are
similar in shape. To better illustrate the difference between the two, we need
to make some approximations. The photon density decays with the square of the
radius, so the minimum radius $R_{\rm{T}}$ has the maximum impact on the
optical depth. As mentioned in the previous section, the maximum angle
$\theta_{\rm{T}}=1/\Gamma_{\rm{T}}$ has the greatest impact on the light
depth. Hence, $\delta
t\simeq\frac{1}{2}{\theta_{\rm{T}}}^{2}R_{\rm{T}}/c=t_{\rm{T}}$, in other
words, the complete form is delayed by approximately $t_{\rm{T}}$ time. On the
other hand, the optical depth of the full formula is influenced by the entire
light variation curve, so the curve is smoother. The peak in the two curves is
caused by low $E_{\rm{P}}$, leading to a large number density of MeV photons.

Next, we set $\bar{\tau}=1$ to obtain the minimal Lorentz factor. The results
are shown in figure 2 and a selected time of $t_{\rm T}$ are shown in table 1.
Our minimal Lorentz factor is below 300 for $t_{\rm T}=5{\rm s}$, which is the
earliest time for TeV photons are certainly optically thin seeing from figure
3 of (LHAASO collaboration, 2023). This value is much smaller than the single
zone model, and is consistent with the Lorentz factor estimated by the
afterglow model, where the Lorentz factor of the early TeV emitting region
being around 560 (LHAASO collaboration, 2023). According to equations (18) and
(19), the optical depth $\bar{\tau}$ is more sensitive to changes in
luminosity compared to the minimum Lorentz factor $\Gamma_{T,min}$. Hence, the
difference of optical depth between the full formula, constant $L_{\rm{M}}$,
and the version without $\delta t$ correction is larger than the minimum
Lorentz factor. One can see the minimal Lorentz factor decreases with time.
The reason is that with time increases, the radius of the TeV emitting region
increases, and consequently, the optical depth decreases.

$t_{\rm T}$ (s) | full formula | constant $L_{M}$ | without $\delta t$ | simple model | $E_{P}^{*}=E_{P}$
---|---|---|---|---|---
5 | 297.92 | 266.92 | 308.42 | 192.88 | 373.35
10 | 162.93 | 222.37 | 154.26 | 158.40 | 201.50
15 | 139.31 | 199.67 | 138.88 | 141.17 | 174.89
20 | 113.06 | 184.90 | 113.00 | 130.09 | 137.51
Table 1: The minimum Lorentz factor as described by different methods in the
two-zone scenario at different time. The code can be found at
https://github.com/qilunuo9/two-zone_model. Figure 3: Minimal Lorentz factor
$\Gamma_{\rm T,min}$ with $E_{\rm{T}}=7\,{\rm TeV}$ as a function of
$t_{\rm{T}}$. Different lines and line style correspond to different method.
The black thick solid line is got by full formula. We neglect change of
luminosity, power law indexes and peak energy when we get orange dashed one.
We neglect $t_{\rm{R}}$, only consider change of $t_{\rm{T}}$ when we get the
green dashed dotted one. The red thin solid one is got by the simple method.
The purple dotted one omits the correction for photon number density with
power law index of spectrum, that is to say, $E_{\rm{P}}^{*}=E_{\rm{P}}$.

##  4 conclusion and discussion

We discussed the constraints on the jet Lorentz factor for the extremely
bright gamma-ray burst GRB 221009A, based on the observation of very high-
energy (TeV) photons during the prompt phase. The detection of TeV photons
during the prompt phase is significant because typically, TeV photons should
collide with lower energy photons and get absorbed. The fact that we observe
TeV photons suggests a very high Lorentz factor that allows the TeV photons to
escape without being significantly absorbed. To estimate the optical depth for
TeV photons, we used a two-zone model, where the TeV photons come from the
external shock region, while lower energy photons come from the internal
prompt emission region. It is crucial to correct for the MeV photon spectrum
evolution and light curve to obtain an accurate estimate of the optical depth.
The most stringent constraint obtained from our calculations is that the
minimum Lorentz factor for the TeV photon emitted at 5 s is about 300. This is
much lower than the estimate from the single-zone model, which highlights the
importance of considering the two-zone model with spectral and light curve
corrections for accurate Lorentz factor estimates. Our two-zone model with
spectral and light curve corrections provides reasonable Lorentz factor
estimates that are consistent with afterglow modeling. Notice that the
constraint is just for the TeV emitting zone.

In this study, we required the gamma-ray burst to be sufficiently bright to
provide an adequate number of photons for accurately fitting the power-law
photon index in each time interval. This ensured that our analysis was based
on robust data. We made certain assumptions to simplify the modeling process.
First, we assumed that the emission radius of MeV photons is much smaller than
that of TeV photons, making them nearly indistinguishable from being emitted
directly from the central engine. Second, we assumed that the emission angle
of MeV photons is larger than that of TeV photons. Although 7 TeV photons are
only observed during a certain time, we made the reasonable assumption that
they remain optically thin throughout the entire period for TeV photons. This
assumption is supported by the fact that $\Gamma_{\rm T,\min}$, which
represents the minimal Lorentz factor, does not sensitively depend on the
energy of the TeV photons, as shown in equation (18).

Dai et al. (2023) proposed that both TeV and MeV photons originate from the
same radius in the ICMART model. They also considered $\gamma-\gamma$
annihilation but reduce the optical depth by assuming that a gamma-ray burst
is composed of multiple sub-pulses. Additionally, they assumed a larger
radiation radius, which results in optical depths less than one and allows for
reasonable Lorentz factors.

In the future, as more gamma-ray bursts with photon power-law indices that can
be fitted in segmented time intervals are observed, we can efficiently
estimate their minimum Lorentz factors using existing codes. Additionally,
with the continuous improvement of instrument precision, the required
brightness threshold for gamma-ray bursts is expected to decrease, leading to
more frequent observations of such bursts.

We thank for the helpful discussions with Aming Chen, Kai Wang, Weihua Lei,
Hao Wang, Jun-Yi Shen, Yuan-Yuan Zuo, Lin Zhou, and Rui-Hang Dong. The
schematic figure was plotted with GeoGebra. The English was polished with
ChatGPT. This work is supported by the National Natural Science Foundation of
China (grant No. U1931203).

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