Fireball Loading and the Blast-Wave Model of Gamma-Ray Bursts

A simple function for the spectral power P(ɛ,t)≡νL(ν) 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 ɛ=hν/m_ec² is the observed dimensionless photonenergy, 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 Γ₀ and an equipartition term q, which is assumed to be constant in time and independent of Γ₀. Expressions are given for the peak spectral power P_p(t)=P(ɛ_p,t) at the photon energy ɛ=ɛ_p(t) of the spectral power peak. A general rule is that the total fireball particle kinetic energy E₀~Π₀t_d, where t_d~Γ^-8/3₀ is the deceleration timescale, and Π₀≡P(ɛ_p,t_d)~Γ^8/3₀ is the maximum measured bolometric power output in radiation during which it is carried primarily by photons with energy \Escr₀=ɛ_p(t_d)~qΓ⁴₀. This rule governs the general behavior of fireballs with different baryon loading. Clean fireballs with small baryon loading (Γ₀>>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 (Γ₀<<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<~Γ₀<~1000) follows from this rule.