Jet Energy and Other Parameters for the Afterglows of Gamma-ray Bursts 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 their physical parameters. The models that yield acceptable fits to the data have jet energies mostly between 10 to 10 erg and initial opening angles between 1 and 4. The external medium density is uncertain by at least one order of magnitude in each case, being around 10 cm for GRB 980703 and 990123, ∼ 10 cm for GRB 990510, and ∼ 3 cm for GRB 991216. If the jets are uniform (i.e. there are no angular gradients of the energy per solid angle) then the 20 keV – 1 MeV radiative efficiency during the GRB phase must have been at least 2-3% for GRB 990510, 20% for GRB 990123, and 30% for GRB 991216. Subject headings: gamma-rays: bursts ISM: jets and outflows methods: numerical radiation mechanisms: non-thermal shock waves