1 Introduction: Simple "standard" afterglows

One can understand the dynamics of the afterglows of GRB in a fairly simple manner, independently of any uncertainties about the progenitor systems, using a relativistic generalization of the method used to model supernova remnants. The simplest hypothesis is that the afterglow is due to a relativistic expanding blast wave, which decelerates as time goes on ([Mészáros & Rees 1997a]; earlier simplified discussions were given by [Katz 1994b]; [Paczynski & Rhoads 1993]; [Rees & Mészáros 1992]). The complex time structure of some bursts suggests that the central trigger may continue for up to 100 s. However, at much later times all memory of the initial time structure would be lost: essentially all that matters is how much energy and momentum has been injected; the injection can be regarded as instantaneous in the context of the much longer afterglow. Detailed calculations and predictions from such a model ([Mészáros & Rees 1997a]) preceded the observations of the first afterglow detected, GRB 970228 ([Costa et al. 1997]; [van Paradijs et al. 1997]).

The simplest spherical afterglow model produces a three-segment power law spectrum with two breaks. At low frequencies there is a steeply rising synchrotron self-absorbed spectrum up to a self-absorption break $\nu_{\rm a}$, followed by a +1/3 energy index spectrum up to the synchrotron break $\nu_{\rm m}$corresponding to the minimum energy $\gamma_{\rm m}$ of the power-law accelerated electrons, and then a -(p-1)/2 energy spectrum above this break, for electrons in the adiabatic regime (where $\gamma^{-p}$ is the electron energy distribution above $\gamma_{\rm m}$). A fourth segment and a third break is expected at energies where the electron cooling time becomes short compared to the expansion time, with a spectral slope -p/2 above that. With this third "cooling" break $\nu_{\rm b}$, first calculated in [Mészáros et al. 1998] and more explicitly detailed in [Sari et al. 1998], one has what has come to be called the simple "standard" model of GRB afterglows. This assumes spherical symmetry (also valid for a jet whose opening angle $\theta_{\rm j} \mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$\gt$}}}\Gamma^{-1}$). As the remnant expands the photon spectrum moves to lower frequencies, and the flux in a given band decays as a power law in time, whose index can change as breaks move through it.

  

Figure: Light-curves of GRB 970228, compared to the blast wave model predictions of [Mészáros & Rees 1998a] (from [Wijers et al. 1997])

The standard model assumes an impulsive energy input lasting much less than the observed $\gamma$-ray pulse, characterized by a single energy and bulk Lorentz factor value (delta or top-hat function). Estimates for the time needed for the expansion to become non-relativistic could then be $\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}$ month ([Vietri 1997a]), especially if there is an initial radiative regime $\Gamma\propto r^{-3}$.However, even when electron radiative times are shorter than the expansion time, it is unclear whether such a regime occurs, as it would require strong electron-proton coupling ([Mészáros et al. 1998]). The standard spherical model can be straightforwardly generalized to the case where the energy is assumed to be channeled initially into a solid angle $\Omega_{\rm j}< 4\pi$. In this case ([Rhoads 1997a], 1997b) there is a faster decay of $\Gamma$ after sideways expansion sets in, and a decrease in the brightness is expected after the edges of the jet become visible, when $\Gamma$ drops below $\Omega_{\rm j}^{-1/2}$. A calculation using the usual scaling laws for a single central line of sight leads then to a steepening of the light curve.

The simple standard model has been remarkably successful at explaining the gross features of GRB 970228, GRB 970508, etc. ([Wijers et al. 1997]; [Tavani 1997]; [Waxman 1997]; [Reichart 1997]). Spectra at different wavebands and times have been extrapolated according to the simple standard model time dependence to get spectral snapshots at a fixed time ([Waxman 1997]; [Wijers & Galama 1998]), allowing fits for the different physical parameters of the burst and environment, e.g. the total energy E, the magnetic and electron-proton coupling parameters ${\epsilon}_{\rm B}$ and ${\epsilon}_{\rm e}$ and the external density $n_{\rm o}$. In GRB 971214 ([Ramaprakash et al. 1998]), a similar analysis and the lack of a break in the late light curve of GRB 971214 could be interpreted as indicating that the burst (including its early gamma-ray stage) was isotropic, leading to an (isotropic) energy estimate of 10^53.5 ergs. Such large energy outputs, whether beamed or not, are quite possible in either NS-NS, NS-BH mergers ([Mészáros & Rees 1997b]) or in hypernova/collapsar models ([Paczynski 1998]; [Popham et al. 1998]), using MHD extraction of the spin energy of a disrupted torus and/or a central fast spinning BH. However, it is worth stressing that what these snapshot fits constrain is only the energy per solid angle ([Mészáros et al. 1998b]). The expectation of a break after only some weeks or months (e.g., due to $\Gamma$ dropping either below a few, or below $\Omega_{\rm j}^{-1/2}$) is based upon the simple impulsive (angle-independent delta or top-hat function) energy input approximation. The latter is useful, but departures from it would be quite natural, and certainly not surprising. As discussed below, it would be premature to conclude at present that there are any significant constraints on the anisotropy of the outflow.

  

Figure: Snapshot spectrum of GRB 970508 at t=12 days and standard afterglow model fit (after [Wijers & Galama 1998])