Closure Relations of Gamma-Ray Bursts in High Energy Emission

The synchrotron external shock model predicts the evolution of the spectral (β) and temporal (α) indices during the gamma-ray burst (GRB) afterglow for different environmental density profiles, electron spectral indices, electron cooling regimes, and regions of the spectrum. We study the relationship between α and β, the so-called “closure relations” with GRBs detected by Fermi Large Area Telescope (Fermi-LAT) from 2008 August to 2018 August. The spectral and temporal indices for the >100 MeV emission from the Fermi-LAT as determined in the Second Fermi-LAT Gamma-Ray Burst Catalog (2FLGC) are used in this work. We select GRBs whose spectral and temporal indices are well constrained (58 long-duration GRBs and 1 short-duration GRBs) and classify each GRB into the best-matched relation. As a result, we found that a number of GRBs require a very small fraction of the total energy density contained in the magnetic field (ɛ _B ≲ 10^-7). The estimated mean and standard deviation of electron spectral index p are 2.40 and 0.44, respectively. The GRBs satisfying a closure relation of the slow cooling tend to have a softer p value compared to those of the fast cooling. Moreover, the Kolmogorov-Smirnov test of the two p distributions from the fast and slow coolings rejects a hypothesis that the two distributions are drawn from the single reference distribution with a significance of 3.2σ. Lastly, the uniform density medium is preferred over the medium that decreases like the inverse of distance squared for long-duration GRBs.