Appendix E: Emission of self absorbed radiations

We argue that the energy of the self absorbed radio waves and ionizing photons in the CB is radiated mainly by thermal bremsstrahlung and line emission from the CB contributing significantly to the observed X-ray afterglow. The absorbed radio power is roughly equal to the integrated emissivity of the CB over all frequencies below $\rm\nu_a$, defined by Eq. (23). For the spectrum of Eq. (4), and the normalization of Eq. (5), the absorbed power is:

$${{\rm d}E_{\rm ab}\over {\rm d}t}\simeq {\eta\, \pi\, R_{\rm ... ...1)}\, \left[{\nu_{\rm a}\over \nu_{\rm b}}\right]^{1/2}\cdot$$ (51)

The CB self absorbed radio energy becomes part of the thermal energy of the CB plasma. It is radiated by the plasma as thermal bremsstrahlung at optical wavelengths (in the CB rest frame). For our reference parameters, $\rm\nu_a\sim$ 100 MHz and $\eta<1$, this absorbed power is smaller than the power absorbed by photoionization.

The recombination energy is radiated at a rate $\approx$ $R_{\rm rec}\, x\, N_{\rm cb} \, I$ in the CB rest frame where $x=\bar n_{e}/\bar n_{\rm b}$ is the fraction of ionized hydrogen in the CB and $I=13.6\,\rm eV$ is the binding energy of hydrogen in its ground state. In the distant observer frame, the observed radiation is boosted and collimated by the highly relativistic motion of the CB and redshifted by the cosmological expansion to:

$${{\rm d}E_{\rm rec}\over {\rm d}t} \simeq {R_{\rm rec}\, x\,... ...\, I\, (1+z)\, [\delta(t)]^4\over 4\, \pi\, D_{\rm L}^2}\cdot$$ (52)

For our reference parameters, hydrogen recombination produces X-ray lines with a total energy flux of $\sim$ $x^2\, \times 10^{-12}\,\rm erg\, s^{-1}\,cm^{-2}\, .$

Due to their large Doppler shift $\delta$, the hydrogen emission lines (and the emission lines from the swept up ISM and supernova shell material) as well as the CB's thermal bremsstrahlung, are shifted to the observer's X-ray band. They contribute significantly to the X-ray afterglow and may provide a simple alternative explanation (Dar & De Rújula 2000) to the commonly assumed Fe-line origin of the X-ray lines observed in the afterglows of GRB 970508: Piro et al. (1998), GRB 970828: Yoshida et al. (1999; 2001), GRB 991216: Piro et al. (2000) and GRB 000214: Antonelli et al. (2000).