What absorbs the early TeV photons of GRB 221009A?

The tera-electronvolt (TeV) light curve of gamma-ray burst (GRB) 221009A shows an unprecedentedly rapid rise at the beginning epoch. This phenomenon could be due to the strong absorption of photons and electrons within the emitting region. As the external shock expands outwards and the radius increases, the volume of matter also increases, leading to a gradual decrease in the optical depth for TeV photons. We explore several possibilities for the physical origin of this peculiar behavior. We calculate the optical depth for TeV photons due to annihilation with lower energy photons in the external shock and scattering by electrons produced via cascading of the TeV emission. Even under aggressive assumptions, we find the optical depths for these processes are orders of magnitude too small to explain the observed light curve. Other sources of absorbers, such as electrons in the ejecta or external shock, also do not yield sufficient optical depths. Therefore, the origin of the early peculiar TeV light curve remains uncertain.


INTRODUCTION
Gamma-ray bursts (GRBs) are transient sources caused by highly energetic astrophysical events that emit copious number of highenergy photons, which travel extremely long distances to reach the observer.The emission in GRBs can be divided into two stages: the prompt emission and the afterglow emission, each exhibiting distinct observational characteristics.In recent years, several GRBs have been detected at very high energies (VHE, i.e., > 0.1 TeV), including GRBs 180720B (Abdalla et al. 2019), 190114C (MAGIC Collaboration et al. 2019), 190829A (H.E. S. S. Collaboration et al. 2021), 201015A (Blanch et al. 2020a), and 201216C (Blanch et al. 2020b).
The Large High Altitude Air Shower Observatory (LHAASO), situated in Daocheng, Sichuan Province, China (Cao et al. 2019), also reported the detection of the very early VHE afterglow of GRB 221009A, with more than 64,000 photons above 0.2 TeV observed within the first 3000 seconds (LHAASO Collaboration 2023).Over-★ E-mail: zouyc@hust.edu.cn(Y-CZ) all, as shown in Fig. 3 of LHAASO Collaboration (2023), the TeV light curve of GRB 221009A shows a four-segment shape: a rapid initial rise, a slower rise up to the peak, a slow decay after the peak, and then a steep decay.Each stage can be well-fitted by a power-law function of time ( ∝ , where = − ★ , and ★ is defined as ★ = 0 + 226 s), indicating an external shock origin (LHAASO Collaboration 2023).Although the observed TeV data can be in general explained by the synchrotron self-Compton emission in the external shock, the rapid rise at the very early stage ( ∼ 1−4.85 s) with a temporal slope of ≈ 14.9 is difficult to explain under the standard afterglow scenario, which predicts = 4 in a homogeneous medium or = 1/2 in a wind environment (see Eq. (S13) of LHAASO Collaboration 2023) The rapid rise of TeV flux could be caused by several processes.First, it could be due to the dynamical process of the external shock.At the very early stage, the external shock -driven by the outer ejecta -could be energized by the inner ejecta, leading to the increase of the bulk Lorentz factor, and therefore increasing the TeV flux dramatically (LHAASO Collaboration 2023).Second, the rapid rise behavior could be due to the strong absorption of photons and electrons within the emitting region.In particular, the TeV flux around ∼ 1.8 s is significantly higher than the average flux during ∼ 0−4.85 s, and it can be extrapolated back from the normal slower rise during ∼ 4.85 s to ∼ 18 s (LHAASO Collaboration 2023).This indicates the fast rise phase during ∼ 2 s to ∼ 4.85 s could be due to the severe absorption of the TeV photons within the shock.In this paper, we will explore the possibility of this second scenario for the physical origin of the early stage of the TeV light curve of GRB 221009A.
Initially, in this scenario the external shock is optically thick to TeV photons, but as time progresses, the optical depth ( ) decreases, and the external shock becomes transparent for TeV photons.At the same time, TeV photons could also be blocked by cascade-generated secondary electrons.As the external shock moves outward, the electron density decreases, leading to a rapid increase of the TeV flux at the early stage.We also explore other potential sources of particles that may absorb TeV photons, such as annihilation with keV photons generated by the shock, and scattering by the particles within the ejecta.
This paper is organized as follows: in section 2, we introduce the method of calculating the external shock particles density generated by the cascading process.In section 2, we also describe other sources of electrons that may absorb TeV photons.Then, we use the observational data of GRB 221009A in our calculation of the optical depth.In section 3, we present our discussion and conclusions.

POSSIBLE SCENARIOS FOR ABSORBING THE EARLY TEV PHOTONS
A TeV photon can undergo two processes: 1. it can be annihilated by interacting with a low-energy photon (pair production) and 2. it can be scattered by an electron (Compton scattering).For the first "annihilation" process, since the number of TeV photons is significantly lower than low-energy photons, the number deduction of low-energy photons in the annihilation reaction can be ignored.We can constrain the flux of these low-energy photons directly from the observations.After calculating the flux of low-energy photons, we can then calculate the optical depth of TeV photons.For the second "scattering" process, the electrons (and/or positrons) may have diverse origins.The cascade process of TeV photons would generate a large number of electron-positron pairs in the external shock.Additionally, the external shock sweeps up the interstellar medium (ISM) and entrains these particles, causing them to move together with the external shock.This group of electrons may also block TeV photons.Wang et al. (2023) suggested that TeV photons originate from internal shocks, and then, electrons within the internal shock may also absorb TeV photons under this scenario.Also, electrons within the ejecta that come from the central engine may also scatter the TeV photons.In this section, we will describe how to calculate the optical depth of TeV photons in all of these different scenarios.Before delving into the origin of low-energy photons or electrons, we first present general arguments in calculating the optical depth of TeV photons.As mentioned above, TeV photons can interact by several processes such as pair production and Compton scattering.The Compton scattering cross-section, c , can be expressed using the Klein-Nishina equation (Section 2, Chapter 5 in You 1998; Klein & Nishina 1929) where = ℏ / e 2 is the photon energy in units of e 2 , and T is Thomson scattering cross-section.
The process cross-section is described by the following equation in the head-on collision approximation (Section 7, Chapter 5 in You 1998; Gould & Schréder 1967): with where 0 and ′ 0 are the frequency of the two photons in the laboratory frame.
The optical depth of TeV photons can be estimated as where the is the cross-section, either from Eq. (1) or Eq. ( 2), depending on the specific process we will consider below, is the number density of the reaction particles and is the thickness of the region.From Eq. ( 4), we can calculate whether the TeV photon can escape from the region.When ≪ 1, the plasma becomes transparent to TeV photons.In the following, we will consider the optical depth for the TeV photons scattering with various sources of electrons and the annihilation due to pair production.

Absorbed by external shock photons
In this section, we discuss the absorption of TeV photons by lowenergy keV photons from the external shock itself.These low-energy photons are also produced by the afterglow.Quantum electrodynamics (QED) calculations indicate that is approximately equal to T and reaches its maximum in a head-on collision and when ℏ 0 ℏ ′ 0 = ( e 2 ) 2 (refer to Section 7, Chapter 5 in You 1998).In the co-moving frame, the frequencies of the two head-on photons will be reduced to /[(1 + )Γ b ] and ′ /[(1 + )Γ b ], where and ′ are the frequency of the two photons detected.Taking Γ b = 560 (LHAASO Collaboration 2023) for the interaction involving ∼ 1 TeV photons at the condition of largest cross-section condition, the lower energetic photon has energy ℎ ′ ∼ 0.08 MeV, which is falls within the X-ray band.
Based on the discussion above, we can determine the energy band that interacts with TeV photons in the pair production.We can then calculate the absorption of TeV photons by the external shock keV photons according to the reaction channel.The can be approximately calculated as: The shock radius is given by: Notice the bulk Lorentz factor Γ b is evolving with time.However, since we are considering the very early stage, that is, ∼ 1 − 4.85s, it is before the deceleration time, which shows as a peak at ∼ 18s.Therefore, we consider Γ b to be constant in this work for estimates of order of magnitude.The evolution might be considered in the detailed modeling.Kann et al. (2023) presents X-ray afterglow observations of GRB 221009A, which can be seen in their Fig.12. Considering ∼ −1.3 and ∼ −0.75 for ( ) ∝ (Kann et al. 2023), we can calculate the observed flux density at = 4 s and at ∼ 0.08 MeV (see above), which is about 10 −25 erg cm −2 s −1 Hz −1 .We now select an energy range of 0.2 -2000 keV, as this band yields a large cross-section.Even in these favorable conditions, we find an optical depth ( )| =4 ∼ 10 −5 ( /4s) −0.3 .This optical depth is too small and its decay rate is too slow.Moreover, it cannot explain the start of the TeV emission.

Absorbed by the cascaded pairs
A promising and self-consistent scenario could be that the TeV photons are blocked by the cascade-generated secondary electronpositron.We describe the scenario as follows.During the very early time (t < 2 s), the cascade had not fully initiated, allowing TeV photons to escape.Later on (t ∼ 2 − 4.85 s), the effect of the cascade appeared, and the cascaded electron-positron pairs further blocked the TeV photons via the Compton scattering process.This is why we observed the dip in the TeV light curve.After that, with the increase of the radius of the external shock, the optical depth drops to less than unity, and the TeV light curve returned to a regular GRB afterglow in the optically thin case.The cascade process can be studied using Monte Carlo simulations (Mücke et al. 2000).Böttcher et al. (2013) developed a semi-analytical method to simplify the calculation of the number of cascade particles, which we follow below.
We roughly consider that only TeV photons contribute to system energy injection.The observable photons esc can be written as follows (Böttcher et al. 2013): where ( ) is the optical depth of photons due to absorption, 0 represents the injection rate of TeV photons, and sec is the secondary photon component mainly arising from synchrotron radiation, which can be expressed as (Böttcher et al. 2013): where e is the Lorentz factor of electrons, e ( e ) denotes the distribution of electrons, ≡ / , where = 4.4 × 10 13 G and is the magnetic field strength, and the normalization is given by (Böttcher et al. 2013): By utilizing Eq. ( 2) and Eq. ( 7) as well as the expression for sec (Eq.( 8)), the generation rate of electron-positron pairs e can be calculated as (see Eq. ( 10) of Böttcher et al. 2013): e ( e e ( e )) = e ( e ) + e ( e ) esc , (10) where the e denotes the cooling of electrons according to synchrotron radiation, and the e ( e ) esc = − e ( e )/ esc .The esc is the electron escape time scale, expressed as esc / , where esc ≥ 1.These equations give the evolution of e ( e ).
Before presenting the detailed semi-analytical calculation, we can make two estimations for the number of cascaded pairs, i.e., a conservative estimation and an aggressive estimation, to get the lower and the upper bounds.In the process, energy conservation dictates that the total energy of two photons must be larger than 2 e 2 .Since the external shock is moving with a bulk Lorentz factor Γ b ∼ 560 (LHAASO Collaboration 2023), the TeV photons are detected in the observer's frame.Because of the redshift, a factor of [(1 + )Γ b ] −1 compared to its energy detected on Earth (see above) should be considered.Therefore, in the observer's frame, the total energy of two photons in the process must exceed m ≃ 2 e 2 Γ b (1 + ) ∼ 660 MeV.We assume, as an approximation, that all the energy of TeV photons is converted into the rest mass and kinetic energy of electronpositron pairs.In this condition, the number of cascade pairs is the largest.A conservative estimate is that all particles with energy less than m cannot continue the cascade reaction.Then, we can approximate that TeV photons will cascade and generate particles, given by: where the flux absorbed by photons, ′ ( ′ ) can be obtained from observations.By extending the flux of TeV photons from 5 ∼ 10 s to 1 ∼ 4.85 s and taking the difference (data are from LHAASO Collaboration 2023), we can estimate the flux ′ ( ′ ) of absorbed TeV photons in the process.Then we can calculate the evolution of the optical depth: For GRB 221009A, with a luminosity distance of L = 2.1 × 10 27 cm we find a flux ′ ( ) 10 −6 erg cm −2 s −1 .We simplified the integral ∫ 0 ′ ( ′ )dt ′ as ′ max ( ) , where ′ max ( ) is the maximum of ′ ( ′ ).With this simplification, we can estimate the upper limit of .With Eq. ( 1) and Eq. ( 12), ( ) can be written as: where = × 10 , and m is in unit of MeV, other quantities are in unit of cgs units.This calculated optical depth at = 4.85 s is significantly lower than 1, indicating that TeV photons are free to leave this area.Therefore, the conservative estimation of the optical depth is not sufficient to block TeV photons.
For an aggressive assumption, we choose = 0.511 MeV, i.e., assuming the TeV photons are transferred into pairs with no energy waste.In this condition, the number density of electron-positron pairs is highest.However, it is impossible for the cascade matter to be so dense, since using Eq. ( 13), we have: We can see that even with the most aggressive estimation, the number density of electrons is not enough to block the TeV photons.Therefore, without knowing the details of the cascade, we can see that the cascaded electron-positron pairs are not dense enough to block the early TeV photons.If we ignore the data at ∼ 1.8 s, the absorption begins at the initial time.We discuss this in the next subsections.

Absorbed by electrons from the external shock
The TeV photons are believed to originate from the external shock, where many electrons may scatter the TeV photons.We check the optical depth for TeV photons in this case.The number density of electrons in the external shock is ∼ Γ b 0 , where 0 is the number density of the circum-burst medium.The optical depth is = 2 c 0 Γ b , which is about 10 −16 for typical values.Moreover, is proportional to the observed time, which is contrary to the expectation.Therefore, these electrons cannot absorb the TeV photons.Wang et al. (2023) studied the potential for the prompt phase to generate the TeV photons emission from a hadronic process.Zhang et al. (2023) considered the possibility that the very high energy photons are from the reverse shock of the external shock.In both scenarios, the ejecta from the central engine may radiate the TeV photons.

Absorbed by electrons from the ejecta
Here, we assume that the TeV photons are generated in the internal shock as Wang et al. ( 2023) described.We consider the condition of whether the internal shock electrons can block the TeV photons.In the internal shock, the number of electrons e is: e = ,iso where ,iso is the isotropic equivalent energy of the GRB prompt emission, p is the rest mass of the proton, and is the ratio between the kinetic energy and the gamma-ray energy, which is roughly 1. Taking ,iso ∼ 1 × 10 55 erg and Γ b ≈ 560 (LHAASO Collaboration 2023), we can calculate the electron number in the internal shock, which is about 2 × 10 55 .
The optical depth is = e c /[4 (2 Γ 2 b ) 2 ] ≈ 3 × 10 −4 for ∼ 10 −1 s, which is a conservative choice of the internal shock timescale, and the Lorentz factor Γ b ≈ 560 is also a conservative choice for the internal shock.Considering the optical depth decreases with time as −2 , it becomes even smaller at a later time.Therefore, in the internal shock scenario, the electrons from the ejecta cannot block the TeV photons.
For the reverse shock scenario, as suggested by Zhang et al. ( 2023), the condition is the same.The difference is that the number of electrons in reverse shock evolves with time, and consequently, the total number of shocked electrons increases, i.e., the thickness of the shocked region increases.However, in the calculation of the optical depth, is canceled.

DISCUSSION AND CONCLUSIONS
We have examined several scenarios to explain the unprecedented rapid rise of the TeV light curve of GRB 221009A, including the absorption and scattering of the TeV photons by the photons or electrons generated within the shock.However, even if all the TeV photons' energy is converted to the rest mass of electrons, it is still not dense enough to block the TeV photons.Additionally, we attempted to use external shock-accelerated electrons to block the TeV photons, but it was not successful either.We also checked using the afterglow itself to explain this phenomenon, but the flux of afterglow in the hundred keV photons is not high enough to absorb TeV photons.In conclusion, the early dip in the TeV light curve still remains to be explained.
There are some other processes that may absorb the TeV emission.For example, the early TeV photons may also collide with the cosmic microwave background and/or the galactic infrared background.However, these sources do not change over time, which is not consistent with the idea that only the early TeV photons cannot escape.
The prompt MeV photons may also absorb the TeV photons.Though this is generally considered unlikely as we have seen a full afterglow-like light curve.However, in the early times, as the TeV radiation radius can be small, it makes the absorption possible.We are not able to estimate the optical depth simply similar to the treatment in section 2.1, as the MeV photons are believed to originate from internal shocks, which is different from the TeV origin.For this possibility, one should consider carefully the geometry configuration of the MeV-TeV collision, as well as the time and spectral evolution of the prompt MeV emission.
It could be that there was no absorption at all if we neglect the very first emission at around 1 s.The early light curve before 5 s can be taken as a fast rise.Such a fast rise might be explained by energy injection to the external shock, which has also been suggested in Khangulyan et al. (2023) very recently.If this is the case, one should consider what kind of energy injection can power such a fast rise.