Radiative Efficiencies of Continuously Powered Blast Waves

We use general arguments to show that a continuously powered radiative blastwave can behave self similarly if the energy injection and radiation mechanismsare self similar. In that case, the power-law indices of the blast waveevolution are set by only one of the two constituent physical mechanisms. Ifthe luminosity of the energy source drops fast enough, the radiation mechanismsset the power-law indices, otherwise, they are set by the behavior of theenergy source itself. We obtain self similar solutions for the Newtonian andthe ultra-relativistic limits. Both limits behave self similarly if we assumethat the central source supplies energy in the form of a hot wind, and that theradiative mechanism is the semi-radiative mechanism of Cohen, Piran & Sari(1998). We calculate the instantaneous radiative efficiencies for both limitsand find that a relativistic blast wave has a higher efficiency than aNewtonian one. The instantaneous radiative efficiency depends strongly on thehydrodynamics and cannot be approximated by an estimate of local microscopicradiative efficiencies, since a fraction of the injected energy is deposited inshocked matter. These solutions can be used to calculate Gamma Ray Burstsafterglows, for cases in which the energy is not supplied instantaneously.Comment: 28 LaTeX pages, including 9 figures and 3 table