Synchrotron Self-Compton Model of TeV Afterglows in Gamma-Ray Bursts

TL;DR

This work develops a semi-analytic, time-dependent SSC model for GRB afterglows in a radially stratified external medium, including adiabatic cooling, photon escape, and EATS integration with exact Klein–Nishina treatment. The model reproduces broadband synchrotron and SSC spectra with accuracy comparable to kinetic codes but at far lower computational cost, enabling MCMC parameter estimation. Application to GRB 190114C yields a highly energetic, quasi-wind-like external medium with E_{k,iso} ≈ 9.1×10^{54} erg and k ≈ 1.67, implying a non-steady wind or wind-to-ISM transition. The results underscore the critical roles of adiabatic dilution and photon escape in shaping the SSC component and demonstrate that NAS09-type analytic treatments can significantly misestimate the Compton-Y parameter and spectral breaks.

Abstract

The detection of a very-high-energy TeV spectral component in the afterglow emission of gamma-ray bursts (GRBs) has opened a new probe into the energetics of ultra-relativistic blast waves and the nature of the circumburst environment in which they propagate. The afterglow emission is well understood as the synchrotron radiation from the shock-accelerated electrons in the medium swept up by the blast wave. The same distribution of electrons also inverse-Compton upscatters the softer synchrotron photons to produce the synchrotron self-Compton (SSC) TeV emission. Accurate modeling of this component generally requires a computationally expensive numerical treatment, which makes it impractical when fitting to observations using Markov Chain Monte Carlo (MCMC) methods. Simpler analytical formalisms are often limited to broken power-law solutions and some predict an artificially high Compton-Y parameter. Here we present a semi-analytic framework for a spherical blast wave that accounts for adiabatic cooling and expansion, photon escape, and equal-arrival-time-surface integration, in addition to Klein-Nishina effects. Our treatment produces the broadband afterglow spectrum and its temporal evolution at par with results obtained from more sophisticated kinetic calculations. We fit our model to the afterglow observations of the TeV bright GRB\,190114C using MCMC, and find an energetic blast wave with kinetic energy $E_{k, \rm iso} = 9.1^{+7.41}_{-3.13} \times 10^{54} \, \rm erg$ propagating inside a radially stratified external medium with number density $n(r)\propto r^{-k}$ and $k=1.67^{+0.09}_{-0.10}$. A shallower external medium density profile ($k<2$) departs from the canonical approximation of a steady wind ($k=2$) from the progenitor star and may indicate a non-steady wind or a transition to an interstellar medium.

Figures (11)

• Figure 1: Klein-Nishina scattering kernel for different approximations. The solid line is the expression provided in Eq. \ref{['eq_fKN_this_work']} and used in this work. The dashed line is the approximation from Eq. 14 in Nakar+09. The dotted line is the Heaviside approximation with $\widetilde{x}_{\rm KN} = 1$ (see Eq. \ref{['eq_fKN_step_approx']}), which is very crude and employed in most analytical works.
• Figure 2: Electron timescales calculated for the fiducial parameters $E_{k, \rm iso} =4 \times 10^{53} \, \rm erg$, $\Gamma_0 = 200$, $n_0 = 0.3 \, \rm cm^{-3}$, $k=0$, $\epsilon_e=10^{-1}$, $\epsilon_B=10^{-4}$, $p=2.3$, $z=0.43$ ($d_L = 2.45 \, \rm Gpc$), $\gamma_{M} = 1 \times 10^8$. Where $r_L$ is the radius of the shell from which the emission arrives along the LOS at the apparent time $T$ and $r_{\rm dec}$ is the deceleration radius.
• Figure 3: Comparison of slow-cooling synchrotron spectrum from different models with that obtained in this work in the case when $Y_{\rm ssc}(\gamma) \ll 1$, for the following model parameters: $E_{\rm k, iso} = 10^{52} \, \rm erg$, $\Gamma_0 = 200$, $n_0 = 10^{-2} \, \rm cm^{-3}$, $k=0$, $\epsilon_e = 10^{-1}$, $\epsilon_B = 8\times 10^{-2}$, $p=2.4$, $T = 10^5 \, \rm s$, $z=1 \, (6.8 \, \rm Gpc)$. The black curve shows the spectrum from our model after EATS integration, which is compared with the 3D shocked fluid volume and EATS integrated spectrum from Granot-Sari-02 (orange) and the LOS spectrum from Sari+98 (magenta). Dashed and dotted lines are calculated using our model but assuming shorter and longer escape timescale, respectively. This verifies that our chosen escape time (solid black) is optimal.
• Figure 4: Comparison between our model (solid lines) and results obtained using a numerical kinetic code (dashed lines). The parameters used to calculate the spectra are $E_{k, \rm iso} =4 \times 10^{53} \, \rm erg$, $\Gamma_0 = 200$, $n_0 = 0.3 \, \rm cm^{-3}$, $k=0$, $\epsilon_e=10^{-1}$, $\epsilon_B=10^{-4}$, $p=2.3$, $\gamma_{M} = 1 \times 10^8$, $z=0.43$ ($d_L = 2.45 \, \rm Gpc$). The different spectra are shown at different observer-frame times $T$ that corresponds to different $\xi=r_L/r_{\rm dec}$, where $r_L$ is the radius from which emission arrives at the observer at time $T$ along the LOS. Both models include SSC, adiabatic cooling (for particles) and density dilution (particles and photons), and escape of radiation from the emission region.
• Figure 5: Comparison of the resulting spectra obtained from the approach used in this work, considering different effects A) SSC, photon escape and adiabatic expansion (black lines), B) SSC and photon escape (blue lines), and C) SSC with shorter (solid yellow lines) and larger (dashed yellow lines) photon escape timescales. Thin dashed lines represent the synchrotron component and dotted lines represent the IC component. We use the following set of parameters $\Gamma_0 = 1 \times 10^{3}$, $n_0 = 1 \, \rm cm^{-3}$, $k=0$, $\epsilon_e = 10^{-1}$, $p=2.4$ and $z=1$. For slow cooling we set $\epsilon_B = 10^{-4}$ and $E_{k, \rm iso} = 1.08 \times 10^{54} \, \rm erg$ at $T = 2.07 \times 10^4\, \rm sec$. For fast cooling we set $\epsilon_B = 10^{-3}$ and $E_{k, \rm iso} = 4.04 \times 10^{54} \, \rm erg$ at $T = 5.97 \times 10^{2}\, \rm sec$.