We report here the first study of proper motions of fast filaments in the young, oxygen-rich supernova remnant G292.0+1.8, carried out using a series of [O III] 5007 A emission-line images taken over a period of more than 21 years. Images taken at seven epochs from 1986 to 2008, all from telescopes at the Cerro Tololo Inter-American Observatory, show oxygen-emitting filaments, presumably ejecta fragments, throughout most of the remnant. We have measured the proper motions for 67 discrete filaments through two-dimensional correlations between images from different epochs. While the motions are small, mostly 20 to 100 milli-arcsec, they are nevertheless measurable through a robust technique of averaging measurements from many epoch pairs. The data are qualitatively consistent with a free-expansion model, and clearly show systematic motions outward from a point near the center of the radio/X-ray shell. Global fits using this model indicate an expansion center at R.A.(2000.0) = 11:24:34.4, Dec.(2000.0) = -59:15:51, and a kinematic age of 2990+-60 years. The young pulsar PSR J1124-5916 is located 46 arcsec southeast of the expansion center. Assuming that it was launched by the supernova, we expect the pulsar to be moving southeastward at 16 milli-arcsec, or a transverse velocity of 440 km/s. We find the fastest ejecta along an axis oriented roughly N-S in the plane of the sky, suggesting that a bipolar explosion produced G292.0+1.8, as appears to have been the case for Cas A.
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