2 Emission mechanisms

2.1 Recombination

If the plasma can remain cold and dense enough and if the recombination and ionization times are comparable, a strong iron line can be obtained through recombination during the burst with a reasonable amount of iron. A recombination time of almost 10^-2 s is needed to produce a line visible during the afterglow.

The recombination time of an hydrogenic ion of atomic number Z in a thermal plasma is $t_{\rm rec} = (\alpha_{\rm r} \, n)^{-1}$ (Verner & Ferland 1986), where n is the electron density and $\alpha_{\rm r}$ is the temperature dependent recombination coefficient (e.g. Seaton 1959). For a relatively cold and dense iron plasma with T=10⁴ K, $t_{\rm rec} \sim 0.14 \, n_{10}^{-1}$ s while, for T=10⁸ K, $t_{\rm rec} \sim 10^{2} \, n_{10}^{-1}$ s. Hence, while a cold plasma could have the necessary recombination rate to ensure the needed fast photon production, this mechanism would be almost completely damped at higher temperatures with reasonable values for the electron density.

The main problem with this interpretation is that the temperature of the plasma absorbing a sizeable fraction of the burst energy is bound to be large, unless an extreme large density enhances the radiative cooling rates. In fact the Compton equilibrium temperature (assuming typical burst high energy spectra) is of the order of a few times 10⁸ K. Note that the scattering optical depth $\tau_{\rm T}$ is bound to be in the range 0.1-1 since a sizable fraction of the burst energy must be absorbed without an excessive smearing of the emission lines.

We conclude that the case of a strong iron line due to repeated photoionization and recombination events during the burst emission faces the problem of a temperature too large to ensure the required fast recombination rate.

2.2 Thermal emission from the surrounding shell

We assume for simplicity that the shell is homogeneous and compact and that it is heated up to $T=10^8\,T_8$ K by the burst photons. We must require that $\tau_{\rm T} \simeq 0.1 \div 1$,implying a shell radius $R\ge 8\ 10^{15}(M/M_\odot)^{1/2}$.In this conditions the shell would emit a broad band bremsstrahlung continuum with several emission lines overlaid (Raymond & Smith 1977), the most relevant being the 6.7 keV iron blend. Analogous spectra are observed in cluster of galaxies (Sarazin 1988), but the higher iron abundance expected in a supernova shell would enhance line emission.

The equivalent width (EW) of the line in a solar abundance plasma has been carefully computed by Bahcall & Sarazin (1978; see in particular their Fig. 1) and ranges from several tens of eV at high ($5\ 10^8$ K) temperatures to $\sim\! 2$ keV at $2.5 \ 10^7$ K. A very weak line is expected for temperature lower than $5\ 10^6$ K. For temperatures larger than $5\ 10^7$ K the EW dependence on temperature can be reasonably approximated as a power law. Assuming an iron abundance 10 times solar we have:  

$$\hbox{EW}(T) \simeq 3.8 \, T_8^{-1.9}\quad {\rm keV}\quad (T_8 \geq 0.5).$$ (1)

Considering the bremsstrahlung intensity at 6.7 keV, we obtain a line luminosity of:  

$$L_{\rm Fe} \simeq 8 \ 10^{42} \exp\left(-{0.8\over T_8}\righ... ...odot}\right)^2 V_{48}^{-1} T_8^{-2.4} \; \hbox {erg s$^{-1}$}.$$ (2)

Therefore a shell of $M \sim 5 \,M_\odot$, typical for many type II SN (see Raymond 1984; Weiler & Sramek 1988; Woosley 1988; McCray 1993), at a temperature slightly below 10⁸ K can produce a line flux of 10^-13 erg cm^-2 s^-1 for z=1 bursts. The EW with respect to the underlying bremsstrahlung radiation would be a few keV, but any other emission component (e.g. afterglow emission) would decrease the line EW. The line emission process can be stopped after about one day, if the afterglow photons enhance the plasma cooling via inverse Compton, lowering the temperature down to less than 10⁷ K. Line emission can also be quenched by the reheating produced by the incoming fireball.

2.3 Reflection

In Seyfert galaxies we see a fluorescence 6.4 keV iron line produced by a relatively cold (T<10⁶ K) accretion disk, illuminated by a hot corona, which provides the ionizing photons (e.g. Ross & Fabian 1993). In this case we need a scattering optical depth $\tau_{\rm T}\gt 1$ of the fluorescent material and a size large enough to allow the line being emitted even $\sim$ one day after the GRB event (i.e. R > 10¹⁵ cm).

In this model the emission line is produced only during the burst event, but in the observer frame it lasts for R/c. The observed luminosity of the Compton reflection component is equal to the $\sim\!10\%$ of the absorbed energy, divided by the time R/c: $L\sim 3\ 10^{45} \,E_{\rm abs, 51}/R_{15}$erg s^-1. The luminosity in the iron line (see e.g. Matt et al. 1991) is $\sim\! 1\%$ of this, times the iron abundance in solar units. Therefore the reflection component can contribute to the hard X-ray afterglow emission, and the iron line can have a luminosity up to $3\ 10^{44}$ erg s^-1, corresponding to fluxes $\sim\! 10^{-13}$ erg cm^-2 s^-1 for a z=1 burst.