Loaded Fireballs and the Blast Wave Model of Gamma Ray Bursts

A simple function for the spectral power $\nu L(\nu) \equiv P(\epsilon,t)$ is proposed to model, with 9 parameters, the spectral and temporal evolution of the observed nonthermal synchrotron power flux of GRBs in the blast wave model. 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 $\Gamma_0$, and an equipartition term $q$ assumed to be constant in time. Expressions are given for the peak spectral power $P_p(t) = P(\epsilon_p,t)$ at the photon energy $\epsilon = \epsilon_p(t)$ of the spectral power peak. A general rule is that the total fireball particle kinetic energy $E_0 \simeq \Pi_0 t_0$, where $t_0$ is the deceleration time scale and $\Pi_0 = P(\epsilon_p,t_0)$ is the maximum measured bolometric power output in radiation, during which it is carried primarily by photons with energy ${\cal E}_0 = \epsilon_p(t_0)$. For clean fireballs with small baryon loading ($\Gamma_0\gtrsim 300$), GRBs are intense, subsecond, medium-to-high energy $\gamma$-ray events, and are difficult to detect because of inadequate photon counts given the insufficiently large effective areas ($\sim10^3$ cm$^{-2}$) of $> 100$ MeV $\gamma$-ray detectors such as EGRET on {\it CGRO}. Dirty fireballs ($\Gamma_0\lesssim 30$) produce transient emissions which are longer lasting and most luminous at X-ray energies and below, but these events are lost behind the glow of the X-ray and lower-energy background radiations except for rare serindipitous detections by pointed instruments. The existence of short, hard GRBs is explained, though perhaps not as a distinct class.