2 "Post-standard" afterglow models

In a realistic situation, one could expect any of several fairly natural departures from the simple standard model to occur. The first one is that departures from a delta top-hat approximation (e.g. having more energy emitted with lower Lorentz factors at later times, still shorter than the gamma-ray pulse duration) would drastically extend the afterglow lifetime in the relativistic regime, by providing a late "energy refreshment" to the blast wave on time scales comparable to the afterglow time scale ([Rees & Mészáros 1998]). The transition to the $\Gamma < \theta_{\rm j}^{-1}$ regime occurring at $\Gamma\sim$ few could then occur as late as six months to more than a year after the outburst, depending on details of the brief energy input.

Another important effect is that the emitting region seen by the observer resembles a ring ([Waxman 1997b]; [Panaitescu & Mészáros 1998b]; [Sari 1998]). A numerical integration over angles ([Panaitescu & Mészáros 1998d]) shows that the sideways expansion effects are not so drastic as inferred from the scaling laws for the material along the central-angle line of sight. This is because even though the flux from the head-on part of the remnant decreases faster, this is more than compensated by the increased emission measure from sweeping up external matter over a larger angle, and by the fact that the extra radiation, arising at larger angles, arrives later and re-fills the steeper light curve. Thus, the sideways expansion (even for a simple impulsive injection) actually mitigates the flux decay, rather than accelerating it. Combined with the possibility of an extended relativistic phase due to nonuniform injection, and the fact that numerical angle integrations show that any steepening would occur over factors $\sim 2-3$ in time, one must conclude that we do not yet have significant evidence for whether the outflow is jet-like or not.

  

Figure: Ring-like equal-arrival time T surfaces of an afterglow, based on []

One expects afterglows to show a significant amount of diversity. This is expected both because of a possible spread in the total energies (or energies per solid angle as seen by a given observer), a possible spread or changes in the injected bulk Lorentz factors, and also from the fact that GRB may be going off in very different environments. The angular dependence of the outflow, and the radial dependence of the density of the external environment can have a marked effect on the time dependence of the observable afterglow quantities ([Mészáros et al. 1998]). So do any changes of the bulk Lorentz factor and energy output during even a brief energy release episode ([Rees & Mészáros 1998]).

  

Figure: Optical light-curve of GRB 970508, fitted with a non-uniform injection model ([Panaitescu et al. 1998])

Strong evidence for departures from the simple standard model is provided by, e.g., sharp rises or humps in the light curves followed by a renewed decay, as in GRB 970508 ([Pedersen et al. 1998]; [Piro et al. 1998]). Detailed time-dependent model fits (Panaitescu et al. 1998) to the X-ray, optical and radio light curves of GRB 970228 and GRB 970508 show that, in order to explain the humps, a non-uniform injection or an anisotropic outflow is required. These fits indicate that the shock physics may be a function of the shock strength (e.g. the electron index p, injection fraction $\zeta$ and/or $\epsilon_{\rm b},~\epsilon_{\rm e}$change in time), and also indicate that dust absorption is needed to simultaneously fit the X-ray and optical fluxes. The effects of beaming (outflow within a limited range of solid angles) can be significant ([]), but are coupled with other effects, and a careful analysis is needed to disentangle them.

Spectral signatures, such as atomic edges and lines, may be expected both from the outflowing ejecta ([Mészáros & Rees 1998a]) and from the external medium ([Perna & Loeb 1998]; [Mészáros & Rees 1998b]; [Bisnovatyi-Kogan & Timokhin 1997]) in the X-ray and optical spectrum of afterglows. These may be used as diagnostics for the outflow Lorentz factor, or as alternative measures of the GRB redshift. An interesting prediction ([Mészáros & Rees 1998b]; see also [Ghisellini et al. 1998]; [Böttcher et al. 1998]) is that the presence of a measurable Fe K-$\alpha$ emission line could be a diagnostic of a hypernova, since in this case one can expect a massive envelope at a radius comparable to a light-day where $\tau_{\rm T} \mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}1$, capable of reprocessing the X-ray continuum by recombination and fluorescence.

The location of the afterglow relative to the host galaxy center can provide clues both for the nature of the progenitor and for the external density encountered by the fireball. A hypernova model would be expected to occur inside a galaxy, in fact inside a high density ($n_{\rm o}\gt 10^3-10^5$). Some bursts are definitely inside the projected image of the host galaxy, and some also show evidence for a dense medium at least in front of the afterglow ([Owen et al. 1998]). On the other hand, for a number of bursts there are strong constraints from the lack of a detectable, even faint, host galaxy ([Schaefer 1998]). In NS-NS mergers one would expect a BH plus debris torus system and roughly the same total energy as in a hypernova model, but the mean distance traveled from birth is of order several Kpc ([Bloom et al. 1998]), leading to a burst presumably in a less dense environment. The fits of [Wijers & Galama 1998] to the observational data on GRB 970508 and GRB 971214 in fact suggest external densities in the range of $n_{\rm o}=0.04-0.4$ cm^-1, which would be more typical of a tenuous interstellar medium (however, [Reichart & Lamb 1998] report a fit for GRB 980329 with $n_{\rm o}\sim 10^4$cm^-3). These could arise within the volume of the galaxy, but on average one would expect as many GRB inside as outside. This is based on an estimate of the mean NS-NS merger time of 10⁸ years; other estimated merger times (e.g. 10⁷ years, [van den Heuvel 1992]) would give a burst much closer to the birth site. BH-NS mergers would also occur in timescales $\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}10^7$ years, and would be expected to give bursts well inside the host galaxy ([Bloom et al. 1998]).