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Analytic Modeling of Synchrotron-Self-Compton Spectra: Application to GRB 190114C

Observations of TeV emission from early gamma-ray burst (GRB) afterglows revealed the long sought for inverse Compton (IC) upscattering of the lower energy synchrotron. However, it turned out that the long hoped for ability to easily interpret the synchrotron-self-Compton (SSC) spectra didn't materialize. The TeV emission is in the Klein-Nishina (KN) regime and the simple Thomson regime SSC spectrum is modified, complicating the scene. We describe here a methodology to obtain an analytic approximation to an observed spectrum and infer the conditions at the emitting region. The methodology is general and can be used in any such source. As a test case, we apply it to the observations of GRB 190114C. We find that the procedure of fitting the model parameters using the analytic SSC spectrum suffers from some generic problems. However, at the same time, it conveniently gives a useful insight into the conditions that shape the spectrum. Once we introduce a correction to the standard KN approximation, the best fit solution is consistent with the one found in detailed numerical simulations. As in the numerical analysis, we find a family of solutions that provide a good approximation to the data and satisfy roughly $B\propto Γ^{-3}$ between the magnetic field and the bulk Lorentz factor, and we provide a tentative explanation why such a family arises.