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

A simple function for the spectral power P(ε,t) ≡ ν L(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/mₑc² 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 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 Pₚ(t) = P(εₚ,t) at the photon energy ε = εₚ(t) of the spectral power peak. A general rule is that the total fireball particle kinetic energy E₀ ~ Π₀ td, where td ∝ Γ₀-8/3 is the deceleration time scale and Π₀ ≡ P(εₚ,td) ∝ Γ₀8/3 is the maximum measured bolometric power output in radiation, during which it is carried primarily by photons with energy E₀ = εₚ(td) ∝ qΓ₀⁴.