We have collected all of the published photometry for GRB 990123 and GRB 990510, the first two gamma-ray bursts where breaks were seen in the light curves of their optical afterglows, and determined the shapes of their light curves and the break times. These parameters were used to investigate the physical mechanisms responsible for the breaks and the nature of the ambient medium that the bursts occurred in. The light curve for GRB 990123 is best fit by a broken power law with a break 1.68 +/- 0.19 days after the burst, a slope of alpha1 = -1.12 +/- 0.08 before the break, and a slope of alpha2 = -1.69 +/- 0.06 after the break. This is consistent with a collimated outflow with a fixed opening angle (theta0) of approximately five degrees. In this case the break in the light curve is due to the relativistic fireball slowing to a gamma factor of approximately 1 / theta0. The light curve for GRB 990510 is best fit by a continuous function with an early-time slope of alpha1 = -0.54 +/- 0.14, a late-time slope of alpha2 = -1.98 +/- 0.19, and a slow transition between the two regimes approximately one day after the burst. This is consistent with a collimated outflow with an opening angle of approximately five degrees that is initially radiative, but undergoes a sideways expansion that begins approximately one day after the burst. This sideways expansion is responsible for the slow break in the light curve.
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