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Compton Scattering of Self-Absorbed Synchrotron Emission

Synchrotron self-Compton (SSC) scattering is an important emission mechanism in many astronomical sources, such as gamma-ray bursts (GRBs) and active galactic nuclei (AGNs). We give a complete presentation of the analytical approximations for the Compton scattering of synchrotron emission with both weak and strong synchrotron self-absorption. All possible orders of the characteristic synchrotron spectral breaks ($ν_{\rm a}$, $ν_{\rm m}$, and $ν_{\rm c}$) are studied. In the weak self-absorption regime, i.e., $ν_{\rm a} < ν_c$, the electron energy distribution is not modified by the self-absorption process. The shape of the SSC component broadly resembles that of synchrotron, but with the following features: The SSC flux increases linearly with frequency up to the SSC break frequency corresponding to the self-absorption frequency $ν_{\rm a}$; and the presence of a logarithmic term in the high-frequency range of the SSC spectra makes it harder than the power-law approximation. In the strong absorption regime, i.e. $ν_{\rm a} > ν_{\rm c}$, heating of low energy electrons due to synchrotron absorption leads to pile-up of electrons, and form a thermal component besides the broken power-law component. This leads to two-component (thermal + non-thermal) spectra for both the synchrotron and SSC spectral components. For $ν_{\rm c} < ν_{\rm a} < ν_{\rm m}$, the spectrum is thermal (non-thermal) -dominated if $ν_a > \sqrt{ν_m ν_c}$ ($ν_a < \sqrt{ν_m ν_c}$). Similar to the weak-absorption regime, the SSC spectral component is broader than the simple broken power law approximation. We derive the critical condition for strong absorption (electron pile-up), and discuss a case of GRB reverse shock emission in a wind medium, which invokes $ν_{\rm a} > {\rm max} (ν_{\rm m}, ν_{\rm c})$.