Afterglows as Diagnostics of Gamma Ray Burst Beaming

If gamma ray bursts are highly collimated, radiating into only a small fraction of the sky, the energy requirements of each event may be reduced by several (up to 4–6) orders of magnitude, and the event rate increased correspondingly. The large Lorentz factors (Γ ∼ 100) inferred from GRB spectra imply relativistic beaming of the gamma rays into an angle ∼ 1/Γ. We are at present ignorant of whether there are ejecta outside this narrow cone. Afterglows allow empirical tests of whether GRBs are well-collimated jets or spherical fireballs. The bulk Lorentz factor decreases and radiation is beamed into an ever increasing solid angle as the burst remnant expands. It follows that if gamma ray bursts are highly collimated, many more optical and radio transients should be observed without associated gamma rays than with them. In addition, a burst whose ejecta are beamed into angle ζm undergoes a qualitative change in evolution when Γζm ∼ 1: Before this, Γ ∝ r, while afterwards, Γ ∝ exp(−r/rΓ). This change results in a potentially observable break in the afterglow light curve. Successful application of either test would eliminate the largest remaining uncertainty in the energy requirements and space density of gamma ray bursters. The ejecta from gamma ray bursts must be highly relativistic to explain the spectral properties of the emergent radiation [1,4]. The gamma rays we observe are therefore only those from material moving within angle 1/Γ of the line of sight, and offer no straightforward way of determining whether the bursts are isotropic emitters or are beamed into a small angle. (Here Γ is the bulk Lorentz factor of expansion.) Afterglow emission at longer wavelengths is expected to arise later in the evolution of the burst than the original gamma rays. It therefore offers at least two ways of testing the burst beaming hypothesis. 1) Kitt Peak National Observatory is part of the National Optical Astronomy Observatories, operated by the Association of Universities for Research in Astronomy. Burst and Afterglow Event Rates First, because Γ is lower at the time of afterglow emission than during the GRB itself, the afterglow cannot be as collimated as the GRB can. This implies that the afterglow event rate should exceed the GRB event rate substantially if bursts are strongly beamed. Allowing for finite detection thresholds, N12 N2 ≤ Ω1 Ω2 ≤ N1 N12 , (1) where N1, N2 are the measured event rates above our detection thresholds at our two frequencies; N12 is the rate of events above threshold at both frequencies; and Ω1, Ω2 are the solid angles into which emission is beamed at the two frequencies. A full derivation of this result and discussion of its application is given in [6]. Rather than reproduce it, I will refer the reader to that paper and will here discuss the second test more fully than was possible in [6]. Dynamical Calculations: Numerical Integrations The second test is based on differences between the dynamical evolution of beamed and isotropic bursts. We explore the effects of beaming on burst evolution using the notation of [5]. Let Γ0 and M0 be the initial Lorentz factor and ejecta mass, and ζm the opening angle into which the ejecta move. The burst energy is E = Γ0M0c ζ m/4, where we assume a unipolar jet geometry. Let r be the radial coordinate in the burster frame; t, tco, and t⊕ the time from the event measured in the burster frame, comoving ejecta frame, and terrestrial observer’s frame; and f the ratio of swept up mass to M0. The key assumptions in our beamed burst model are that (1) the energy and mass per unit solid angle are constant at angles θ < ζm from the jet axis and zero for θ > ζm (see [2] for an alternative model); (2) the energy in the ejecta is approximately conserved; (3) the ambient medium has uniform density; and (4) the cloud of ejecta + swept-up material expands in its comoving frame at the sound speed cs = c/ √ 3 appropriate for relativistic matter. The last of these assumptions implies that the working surface of the expanding remnant has a transverse size ∼ ζmr+ cstco. The evolution of the burst changes when the second term dominates over the first. The full equations describing the burst remnant’s evolution are then