What absorbs the early TeV photons of GRB 221009A?

In this section, we discuss the absorption of TeV photons by low-energy keV photons from the external shock itself. These low-energy photons are also produced by the afterglow. Quantum electrodynamics (QED) calculations indicate that $\sigma_{\gamma\gamma}$ is approximately equal to $\sigma_{\rm{T}}$ and reaches its maximum in a head-on collision and when $\hbar\omega_{0}\hbar\omega_{0}^{\prime}=(m_{\mathrm{e}}c^{2})^{2}$ (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 $\nu/[(1+z)\Gamma_{\rm{b}}]$ and $\nu^{\prime}/[(1+z)\Gamma_{\rm{b}}]$, where $\nu$ and $\nu^{\prime}$ are the frequency of the two photons detected. Taking $\Gamma_{\rm{b}}=560$ (LHAASO Collaboration, 2023) for the interaction involving $\sim 1$ TeV photons at the condition of largest cross-section condition, the lower energetic photon has energy $h\nu^{\prime}\sim 0.08$ 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 $\gamma\gamma$ pair production. We can then calculate the absorption of TeV photons by the external shock keV photons according to the $\gamma\gamma$ reaction channel. The $\tau$ can be approximately calculated as:

The shock radius $R$ is given by:

Notice the bulk Lorentz factor $\Gamma_{\rm{b}}$ is evolving with time. However, since we are considering the very early stage, that is, $t\sim 1-4.85\rm{s}$, it is before the deceleration time, which shows as a peak at $t\sim 18\rm{s}$. Therefore, we consider $\Gamma_{\rm{b}}$ 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 $\alpha\sim-1.3$ and $\beta\sim-0.75$ for $f_{\nu}(t)\propto t^{\alpha}\nu^{\beta}$ (Kann et al., 2023), we can calculate the observed flux density at $t=4$ s and at $\sim 0.08$ MeV (see above), which is about $10^{-25}\ \rm{erg\ cm^{-2}\ s^{-1}\ Hz^{-1}}$. 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 $\tau(t)|_{t=4\,s}\sim 10^{-5}(t/4{\rm s})^{-0.3}$. This optical depth is too small and its decay rate is too slow. Moreover, it cannot explain the start of the TeV emission.

A promising and self-consistent scenario could be that the TeV photons are blocked by the cascade-generated secondary electron- positron. We describe the scenario as follows. During the very early time ($\mathbf{t}<\mathbf{2}$ s), the cascade had not fully initiated, allowing TeV photons to escape. Later on ($\mathbf{t}$ $\sim 2-4.85$ 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 $\dot{N}_{\epsilon}^{\rm esc}$ can be written as follows (Böttcher et al., 2013):

where $\tau_{\gamma\gamma}(\epsilon)$ is the optical depth of photons due to $\gamma\gamma$ absorption, $\dot{N}_{\epsilon}^{0}$ represents the injection rate of TeV photons, and $\dot{N}_{\epsilon}^{\rm sec}$ is the secondary photon component mainly arising from synchrotron radiation, which can be expressed as (Böttcher et al., 2013):

where $\gamma_{{\rm{e}}}$ is the Lorentz factor of electrons, $N_{{\rm{e}}}(\gamma_{{\rm{e}}})$ denotes the distribution of electrons, $b\equiv B/B_{c},$ where $B_{c}=4.4\times 10^{13}$ G and $B$ 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 $\dot{N}_{\epsilon}^{\rm sec}$ (Eq. (8)), the generation rate of electron-positron pairs $\dot{N}_{\mathrm{e}}^{\gamma\gamma}$ can be calculated as (see Eq. (10) of Böttcher et al., 2013):

where the $\dot{\gamma}_{\mathrm{e}}$ denotes the cooling of electrons according to synchrotron radiation, and the $\dot{N}_{\mathrm{e}}(\gamma_{\mathrm{e}})^{\mathrm{esc}}=-N_{\mathrm{e}}(\gamma_{\mathrm{e}})/t_{\mathrm{esc}}$. The $t_{\mathrm{esc}}$ is the electron escape time scale, expressed as $\eta_{\mathrm{esc}}R/c$, where $\eta_{\mathrm{esc}}\geq 1$. These equations give the evolution of $N_{\mathrm{e}}(\gamma_{\mathrm{e}}).$

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 $\gamma\gamma$ process, energy conservation dictates that the total energy of two $\gamma$ photons must be larger than $2m_{\mathrm{e}}c^{2}$. Since the external shock is moving with a bulk Lorentz factor $\Gamma_{\rm{b}}\sim 560$ (LHAASO Collaboration, 2023), the TeV photons are detected in the observer’s frame. Because of the redshift, a factor of $[(1+z)\Gamma_{\rm{b}}]^{-1}$ 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 $\gamma\gamma$ process must exceed $E_{{\rm{m}}}\simeq 2m_{\mathrm{e}}c^{2}\Gamma_{\rm{b}}(1+z)\sim 660$ MeV. We assume, as an approximation, that all the energy of TeV photons is converted into the rest mass and kinetic energy of electron-positron pairs. In this condition, the number of cascade pairs is the largest. A conservative estimate is that all particles with energy less than $E_{{\rm{m}}}$ cannot continue the cascade reaction. Then, we can approximate that TeV photons will cascade and generate $N$ particles, given by:

where the flux absorbed by photons, $f^{\prime}(t^{\prime})$ can be obtained from observations. By extending the flux of TeV photons from 5 $\sim$ 10 s to 1 $\sim$ 4.85 s and taking the difference (data are from LHAASO Collaboration, 2023), we can estimate the flux ${f^{\prime}(t^{\prime})}$ of absorbed TeV photons in the $\gamma\gamma$ process. Then we can calculate the evolution of the optical depth:

For GRB 221009A, with a luminosity distance of $D_{\rm L}=2.1\times 10^{27}$ cm we find a flux $f^{\prime}(t)\lesssim 10^{-6}\ \rm{erg\ cm^{-2}\ s^{-1}}$. We simplified the integral $\int_{0}^{t}f^{\prime}(t^{\prime})\rm{d}t^{\prime}$ as $f_{\max}^{\prime}(t)t$, where $f_{\max}^{\prime}(t)$ is the maximum of $f^{\prime}(t^{\prime})$. With this simplification, we can estimate the upper limit of $\tau$. With Eq. (1) and Eq. (12), $\tau(t)$ can be written as:

where $Q_{i}=Q\times 10^{i}$, and $E_{{\rm{m}}}$ is in unit of MeV, other quantities are in unit of cgs units. This calculated optical depth at $t=4.85$ 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 $E_{m}=$ 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 $t\sim$ 1.8 s, the absorption begins at the initial time. We discuss this in the next subsections.

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 $\sim\Gamma_{\rm{b}}n_{0}$, where $n_{0}$ is the number density of the circum-burst medium. The optical depth is $\tau=2\sigma_{\rm c}n_{0}c\Gamma_{\rm{b}}t$, which is about $10^{-16}$ for typical values. Moreover, $\tau$ 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.

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 $N_{\rm e}$ is:

where $E_{\rm\gamma,iso}$ is the isotropic equivalent energy of the GRB prompt emission, $m_{{\rm{p}}}$ is the rest mass of the proton, and $\eta$ is the ratio between the kinetic energy and the gamma-ray energy, which is roughly 1.

Taking $E_{\rm\gamma,iso}\sim 1\times 10^{55}$ erg and $\Gamma_{\rm{b}}\approx 560$ (LHAASO Collaboration, 2023), we can calculate the electron number in the internal shock, which is about $2\times 10^{55}$.

The optical depth is $\tau=N_{\rm e}\sigma_{\rm c}/[4\pi(2c\Gamma_{\rm{b}}^{2}t)^{2}]$ ${\approx 3\times 10^{-4}}$ for ${t\sim 10^{-1}}$ s, which is a conservative choice of the internal shock timescale, and the Lorentz factor $\Gamma_{\rm{b}}\approx 560$ is also a conservative choice for the internal shock. Considering the optical depth decreases with time as $t^{-2}$, 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 $l$ of the shocked region increases. However, in the calculation of the optical depth, $l$ is canceled.