Fireballs Loading and the Blast Wave Model of Gamma Ray Bursts

A simple function for the spectral power $P(ε,t) \equiv νL(ν) $ is proposed to model, with 9 parameters, the spectral and temporal evolution of the observed nonthermal synchrotron power flux from GRBs in the blast wave model. Here $ε= hν/$m$_e$c$^2$ is the observed dimensionless photon energy and $t$ is the observing time. Assumptions and an issue of lack of self-consistency are spelled out. The spectra are found to be most sensitive to the baryon loading, expressed in terms of the initial bulk Lorentz factor $Γ_0$, and an equipartition term $q$ which is assumed to be constant in time and independent of $Γ_0$. Expressions are given for the peak spectral power $P_p(t) = P(ε_p,t)$ at the photon energy $ε= ε_p(t)$ of the spectral power peak. A general rule is that the total fireball particle kinetic energy $E_0 \sim Π_0 t_d$, where $t_d \propto Γ_0^{-8/3}$ is the deceleration time scale and $Π_0 \equiv P(ε_p,t_d) \propto Γ_0^{8/3}$ is the maximum measured bolometric power output in radiation, during which it is carried primarily by photons with energy ${\cal E}_0 = ε_p(t_d) \propto qΓ_0^4$.