7 Brief comments on the afterglows

The discovery of afterglows has not only has extended observations to longer time scales and other wavebands, making the identification of counterparts possible, but also provided confirmation for much of the earlier work on the fireball shock model of GRB, in which the $\gamma$-ray emission arises at radii of 10¹³-10¹⁵ cm [38, (Rees & Mészáros 1992, 1994]; [20, Mészáros & Rees 1993]; [33, Paczynski & Xu 1994]; [15, Katz 1994]; [45, Sari & Piran 1995)]. In particular, this model led to the prediction of the quantitative nature of the signatures of afterglows, in substantial agreement with subsequent observations [21, (Mészáros & Rees 1997a]; [4, Costa et al. 1997]; [50, Vietri 1997a]; [46, Tavani 1997]; [52, Waxman 1997]; [39, Reichart 1997]; [55, Wijers et al. 1997)].

Astrophysicists understand supernova remnants reasonably well, despite continuing uncertainty about the initiating explosion; likewise, we may hope to understand the afterglows of gamma ray bursts, despite the uncertainties about the "trigger'' that I have already emphasised. The simplest hypothesis is that the afterglow is due to a relativistic expanding blast wave. The complex time-structure of some bursts suggests that the central trigger may continue for up to 100 seconds. 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, its distribution in angle, and the mass fractions in shells with different Lorentz factors.

The simplest spherical afterglow model - where a relativistic blast wave decelerates as it runs into ambient matter, leading to a radiative output with a calculable spectrum, and a characteristic power law decay - has been remarkably successful at explaining the gross features of the GRB 970228, GRB 970508 and other afterglows (e.g. [55, Wijers et al. 1997)]. The gamma-rays we receive come only from material whose motion is directed within one degree of our line of sight. They therefore provide no information about the ejecta in other directions: the outflow could be isotropic, or concentrated in a cone of any angle substantially larger than one degree (provided that the line of sight lay inside the cone). At observer times of more than a week, the blast wave would however be decelerated to a moderate Lorentz factor, irrespective of the initial value. The beaming and aberration effects are thereafter less extreme, so we observe afterglow emission not just from material moving almost directly towards us, but from a wider range of angles.

The afterglow is thus a probe for the geometry of the ejecta - at late stages, if the outflow is beamed, we expect a spherically-symmetric assumption to be inadequate; the deviations from the predictions of such a model would then tell us about the ejection in directions away from our line of sight. It is quite possible, for instance, that there is relativistic outflow with lower $\Gamma$ (heavier loading of baryons) in other directions (e.g. [55, Wijers et al. 1997)]; this slower matter could even carry most of the energy [32, (Paczynski 1998)]. An argument for a broad beaming angle in the energy outflow is that, if the energy were channeled into a solid angle $\Omega_j$ then, [41, (Rhoads 1997)], one expects a faster decay of $\Gamma$ after it drops below $\Omega_j^{-1/2}$. A simple calculation using the usual scaling laws leads then to a steepening of the flux power law in time. The lack of such an observed downturn in the light curve has been interpreted as further supporting the sphericity of the entire fireball. There are several important caveats, however. The first one is that the above argument assumes a simple, impulsive energy input (lasting $\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}$than the observed $\gamma$-ray pulse duration), characterized by a single energy and bulk Lorentz factor value. Estimates for the time needed to reach the non-relativistic regime, or $\Gamma < \Omega_j^{-1/2} \mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}$ few, could then be under a month [51, (Vietri 1997b]; [13, Huang et al. 1998)], especially if an initial radiative regime with $\Gamma\propto r^{-3}$ prevails. (It is however unclear whether, even when electron radiative time scales are shorter then the expansion time, such a regime applies, as it would require strong electron-proton coupling [26, (Mészáros et al. 1998)]. [53, Waxman et al. (1998)] have also argued on observational grounds that the longer lasting $\Gamma \propto r^{-3/2}$(adiabatic regime) is more appropriate.) Furthermore, even the simplest reasonable departures from this ideal model (e.g. a substantial amount of energy and momentum ejected behind the main shell with a lower Lorentz factor) 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 [23, (Mészáros & Rees 1998a)]. Anisotropy in the burst outflow and emission affects the light curve at the time when the inverse of the bulk Lorentz factor equals the opening angle of the outflow. If the critical Lorentz factor is less than 3 or so (i.e. the opening angle exceeds 20$^\circ$) such a transition might be masked by the transition from ultrarelativistic to mildly relativistic flow, so quite generically it would difficult to limit the late-time afterglow opening angle in this way if it exceeds 20$^\circ$. Since some afterglows are unbroken power laws for over 100 days (e.g. GRB970228), if the energy input were indeed just a a simple impulsive shell the opening angle of the late-time afterglow at long wavelengths is probably greater than 1/3, i.e. $\Omega_{\rm opt}\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$\gt$}}} 0.4$. However, even this still means that the energy estimates from the afterglow assuming isotropy could be 30 times too high.

The beaming angle for the gamma ray emission could be far smaller, and is much harder to constrain directly. The ratio of $\Omega_\gamma /\Omega_x$ has been considered by [11, Grindlay (1998)] using data from Ariel V and HEAO-A1/A2 surveys, who did not find evidence for a significant difference between the deduced gamma-ray and X-ray rates, and concluded that higher sensitivity surveys would be needed to provide significant constraints. More promising for the immediate future, the ratio $\Omega_\gamma/\Omega_{\rm opt}$ can also be investigated observationally (see also [41, Rhoads 1997)]. The rate of GRB with peak fluxes above 1 phcm^-2s^-1 as determined by BATSE is about 300/yr, i.e. 0.01/sq deg/yr. According to [54, Wijers et al. (1998)] this flux corresponds to a redshift of 3. If the gamma rays were much more narrowly beamed than the optical afterglow there should be many "homeless'' afterglows, i.e. ones without a GRB preceding them. The transient sky at faint magnitudes is poorly known, but there are two major efforts under way to find supernovae down to about R=23 [10, (Garnavich et al. 1998]; [35, Perlmutter et al. 1998)]. These searches have by now covered a few tens of "square degree years'' of exposure and would be sensitive to afterglows of the brightness levels thus far observed. It therefore appears that the afterglow rate is not more than a few times 0.1/sq deg/yr. Since the magnitude limit of these searches allows detection of optical counterparts of GRB brighter than 1 ph cm^-2 s^-1 it is fair to conclude that the ratio of homeless afterglows to GRB is unlikely to exceed $\sim 20$. It then follows that $\Omega_\gamma\gt.05\ \Omega_{\rm opt}$, which combined with our limit to $\Omega_{\rm opt}$ yields $\Omega_\gamma\gt.02$. The true rate of events that give rise to GRB is therefore at most 600 times the observed GRB rate, and the opening angle of the ultrarelativistic, gamma-ray emitting material is no less than $5^\circ$. Combined with the most energetic bursts, this begins to pose a problem for the neutrino annihilation type of GRB energy source.

Obviously, the above calculation is only sketchy and should be taken as an order of magnitude estimate at present. However, it should improve as more afterglows are detected and the modelling gets more precise.