Fireball loading and the blast-wave model of gamma-ray bursts

A simple function for the spectral power P(epsilon, t) = nu L(nu) is proposed to model, with nine parameters, the spectral and temporal evolution of the observed nonthermal synchrotron power flux from gamma-ray bursts (GRBs) in the blast-wave model. Here epsilon = h nu/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 baryon loading, expressed in terms of the initial bulk Lorentz factor Gamma(0) and an equipartition term a, which is assumed to be constant in time and independent of Gamma(0). 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 similar to II0 t(d), where t(d) proportional to oc Gamma(0)(-) (8/3) is the deceleration timescale, and II0 = P(epsilon(p), t(d)) proportional to Gamma(0)(8/3) is the maximum measured bolometric power output in radiation during which it is carried primarily by photons with energy (0) = epsilon(p)(t(d)) proportional to q Gamma(0)(4). This rule governs the general behavior of fireballs with different baryon loading. Clean fireballs with small baryon loading (Gamma(0) much greater than 300) are intense, subsecond, medium-to-high-energy gamma-ray events and are difficult to detect because of dead time and sensitivity limitations of previous gamma-ray detectors like EGRET on the Compton Gamma Ray Observatory. Dirty fireballs with large baryon loading (Gamma(0) much less than 300) produce transient emissions that 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 serendipitous detections by pointed instruments. The correlation between hardness and duration of loaded GRB fireballs (100 less than or similar to Gamma(0) less than or similar to 1000) follows from this rule.