Physical Parameters of the Afterglows of Grb 980703, 990123, 990510, and 991216 Determined from Modeling of Multi-frequency Data

We model the radio, optical, and X-ray emission for the afterglows of GRB 980703, 990123, 990510, and 991216, within the framework of relativistic jets, to determine the physical parameters of these afterglows. For the last three we find jet energies around 10 ergs and initial opening angles less than 3 deg. The external medium density obtained from fits to the data ranges from 10 cm to less than 1 cm. Our results show that, for these four afterglows, the uncertainty in the remnant isotropic equivalent energy and in the external medium density is about one order of magnitude. If the energy per solid angle in the ejecta is uniform (i.e. has no angular dependence), then the radiative efficiency during the GRB phase must have been at least 0.3% for GRB 980703, 2% for GRB 990510, 4% for GRB 991216, and 8% for GRB 990123. Subject headings: gamma-rays: bursts ISM: jets and outflows methods: numerical radiation mechanisms: non-thermal shock waves