The Expected Duration of Gamma-Ray Bursts in the Impulsive Hydrodynamic Models

Depending upon the various models and assumptions, the existing literature on gamma-ray bursts (GRBs) mentions that the gross theoretical value of the duration of the burst in the hydrodynamical models is τ ~ r_γ/2η²c, where r_γ is the radius at which the blast wave associated with the fireball (FB) becomes radiative and sufficiently strong. Here η ≡ E/Mc², c is the speed of light, E is the initial laboratory frame energy of the FB, and M is the baryonic mass of the same (Rees & Mészáros). However, within the same basic framework, some authors (e.g., Katz and Piran) have given τ ~ r_γ/2ηc. We intend to remove this confusion by considering this problem at a deeper level than has been considered so far. Our analysis shows that none of the previously quoted expressions are exactly correct, and in case the FB is produced impulsively and the radiative processes responsible for the generation of the GRB are sufficiently fast, its expected duration would be τ ~ ar_γ/2η²c, where a ~ O(10). We further discuss the probable change, if any, of this expression, in case the FB propagates in an anisotropic fashion. We also discuss some associated points in the context of the Mészáros & Rees scenario.