Angular Size and Emission Timescales of Relativistic Fireballs

The detection of delayed X-ray, optical, and radio emission, ``afterglow,'' associated with γ-ray bursts (GRBs) is consistent with models in which the bursts are produced by relativistic expanding blast waves, driven by expanding fireballs at cosmological distances. In particular, the timescales over which radiation is observed at different wavebands agree with model predictions. It had recently been claimed that the commonly used relation between observation time t and blast wave radius r, t=r/2γ²(r)c , where γ is the fluid Lorentz factor, should be replaced with t=r/16γ²(r)c because of blast wave deceleration. Applying the suggested deceleration modification would make it difficult to reconcile observed timescales with model predictions. It would also imply an apparent source size too large to allow attributing observed radio variability to diffractive scintillation. We present a detailed analysis of the implications of the relativistic hydrodynamics of expanding blast waves to the observed afterglow. We find that modifications due to shock deceleration are small, therefore allowing for both the observed afterglow timescales and for diffractive scintillation. We show that at time t the fireball appears on the sky as a narrow ring of radius h=r/γ(r) and width Δh/h~0.1, where r and t are related by t=r/2γ²(r)c.