2 Continuum and line emission in thermal plasma

To compute the continuum emission from a thermal optically thin plasma we adopt the treatment and basic equations of Rybicki & Lightman (1979). To compute the luminosities of K$_\alpha$ lines from a thermal plasma, instead, we adopt the widely used code MEKAL (Mewe et al. 1985; Liedahl et al. 1995) as implemented in XSPEC (Arnaud 1996). We concentrate in particular the elements Mg, Si, S, Ar, Ca and Fe, for which emission lines have been detected in the afterglow of GRBs. In Fig. 1 we show the line production efficiency for an optically thin thermal plasma as a function of the plasma temperature and for several values of metallicity. We plot $\eta _{\rm {line}}$, i.e. the ratio of the K$_\alpha$ emission line for the six elements above, irrespective of their ionization state, over the total luminosity of the plasma. Gray shading highlights regions in which the equivalent width (EW) of the lines is less than 100 eV, a robust lower limit to any emission feature detected in the afterglow so far.

Figure 1: Efficiency of K$_\alpha$ line emission for several elements as a function of the temperature in an optically thin plasma. The parameter $\eta _{\rm {line}}$ is defined as the line luminosity over the bolometric luminosity of the thermal plasma. Each panel shows 5 different lines. From top to bottom these refer to a metallicity 10, 3, 1, 0.3 and 0.1 solar. The gray shading highlights regions in the parameter space where an equivalent width lower than 100 eV is predicted.

When using the MEKAL code to evaluate line emission luminosities, one has to remember that the code does not include any radiation transfer, since it assumes that the medium is optically thin to radiation. The actual optical depth of a cloud of plasma at temperature T depends on the temperature and on the frequency of the radiation considered. When X-ray continuum radiation is considered, the optically thin approximation can be used up to column densities $N_{\rm H}\la\sigma_T^{-1}\sim1.5\times10^{24}$ cm^-2. If, however, line emission is concerned, it must be taken into account that intermediate-high Z elements retain some electrons which may cause the plasma to be optically thick due to photoionization. In Fig. 2 we show the optical depth of a solar metallicity plasma with $N_{\rm H}=1.5\times10^{24}$ cm^-2 as a function of frequency for a range of temperatures between 10⁶ and 10⁸ K. The opacity of a cold gas is also shown for comparison. In the region of the considered emission lines, the plasma can be optically thick up to temperatures of several keV. This will limit the maximum line luminosity and EW: increasing the column density of the plasma will have no effect on the line luminosity since line photons will be able to escape freely only out of the optically thin surface layer of the medium. Instead of properly introducing radiation transfer in the optically thin MEKAL code, which is a formidable task, in the following we will assume that, after the plasma becomes optically thick to radiation at the line frequency, the line luminosity does not increase if the column density of the gas is increased (see Lazzati et al. 2002b for discussions on a similar assumptions in reflection models).