Jet energy and other parameters for the afterglows of GRB 980703, GRB 990123, GRB 990510, and GRB 991216 determined from modeling of multifrequency data

We model the radio, optical, and X-ray emission for the afterglows of GRB 980703, GRB 990123, GRB 990510, and GRB 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(50) and 10(51) ergs and initial opening angles between 1 degrees and 4 degrees. The external medium density is uncertain by at least 1 order of magnitude in each case, being around 10(-3) cm(-3) for GRB 980703 and GRB 990123, similar to 10(-1) cm(-3) for GRB 990510, and similar to3 cm(-3) 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.