3 Saturated Compton cooling model

The saturated Compton cooling (SCC) model interprets the spectral break as the Wien peak caused by multiple Compton upscattering of soft photons. In this case $E_{\rm pk}$corresponds roughly to the average lepton energy <E>. At very high Thomson depths $\tau_{\rm T}(\gg 10)$ the Band index $\alpha$ is >0. But as the Thomson depth decreases $\alpha$ decreases to <0 (Rybicki & Lightman 1979; Liang et al. 1997). However for pure nonthermal power-law lepton distributions $E_{\rm pk}$ is always in the hundreds of keVs in the emitter frame. To bring it down to keVs in the emitter frame of a relativistic shell of bulk Lorentz factor $\Gamma=$ hundreds, we need to invoke a hybrid thermal-nonthermal lepton distribution with most of the particles in the thermal population with comoving kT = keVs and only a few percent of the leptions in the nonthermal power-law tail. For typical magnetic fields and lepton densities discussed below, most of the soft photons are still produced by the nonthermal leptons via the synchrotron mechanism. Since the bulk of the leptons are keV thermal particles, the shock conversion efficiency from bulk kinetic energy to internal energy is likely low since the impulsive energization will be concentrated in the small number of nonthermal leptons. These and other physics issues of the SCC model need further investigation.

Since $<E\gt~=E_{\rm pk}$ in this model the Liang-Kargatis (1996) decay law is a natural consequence of radiative cooling plus energy conservation in the comoving frame, and the LK decay constant $\Phi_{\rm o}$ is simply a measure of the total number of emitting particles modulo $\Omega$ (the solid angle filling factor of the ejecta shell) times the distance squared, independent of the bulk Lorentz factor $\Gamma$. Liang (1997) then showed that the ratio of source distance d to the relativistic shell curvature radius R becomes $d/R=10^{12}(\tau_{\rm T}/\Phi_{\rm o})^{1/2}$. Since both $\tau_{\rm T}$ and $\Phi_{\rm o}$ are directly extractable from spectral fitting, we can deduce all relevant physical parameters of the shell once we know the distance. For example, consider a hypothetical shell at a distance of 1 Gpc, with $\Phi_{\rm o}=10=\tau_{\rm T}$ and a pulse rise time of 1 s. This rise time, if interpreted as the time delay caused by the curvature of the forward visible patch, limits the bulk Lorentz factor to $\Gamma\geq 224$ (Liang 1997). We find that the shell has very large aspect ratio, $(R/H\leq 4.5\ 10^3)$, high density (comving lepton density $\geq 2.2\ 10^{13}~{\rm cm}^{-3})$, comoving magnetic field B=1-100 G, and is matter-dominated $(U/Nm_{\rm e}c^2=10^{-3})$ where U is the total internal energy. The total number of leptons $N=10^{57}~\Omega/(4\pi)$ and the total lepton kinetic energy $\Gamma Nm_{\rm e}c^2\geq 10^{53}~\Omega/(4\pi)~{\rm ergs}$.Due to the high comoving density such shells are likely associated with internal shocks of ejecta material (rather than external shocks).