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We employ a fireball model of the GRB explosion to constrain intrinsic andenvironmental parameters of four events with good broadband afterglow data;GRB970508, GRB980329, GRB980703, and GRB000926. Using standard assumptions ofconstant circumburst density and no evolution of the fraction of the explosionenergy in the post-shock magnetic field, we investigate the uniformity of thederived explosion and shock physics parameters among the events. We finddensities that range from those of the ISM to diffuse clouds, energiescomparable to the total GRB gamma-ray energy, collimations from near-isotropyto 0.04 rad, substantial electron energy fractions of 10-30% with energydistribution indices of 2.1-2.9 and magnetic energy fractions from 0.2-25%. Wealso investigate the level to which the data constrain standard modelassumptions, such as the magnetic field evolution, and the allowed densityprofiles of the medium. Fits generally improve slightly with an increasingmagnetic energy fraction epB. Good fits can be produced with epB accumulatingor decaying with the shock strength over the afterglow as epB~gamma^x;-2<=x<=+1. The data are not very sensitive to increasing density profiles,allowing good fits even with density ~r^10. Some parameter values change by upto an order of magnitude under altered assumptions; parameters of even goodfits cannot be taken at face value. The data are sensitive to decreasingdensities; r^-2 profiles may produce reasonable fits; steeper profiles, evenr^-2.5, will not fit the data.Comment: Submitted to the Astrophysical Journal, 39 pages (incl 8 figures, 6  tables

A Study of the Afterglows of Four Gamma‐Ray Bursts: Constraining the Explosion and Fireball Model