We have assembled a large-area spectroscopic survey of giant stars in the Sagittarius (Sgr) dwarf galaxy core. Using medium resolution (R ~15,000), multifiber spectroscopy we have measured velocities of these stars, which extend up to 12 degrees from the galaxys center (3.7 core radii or 0.4 times the King limiting radius). From these high quality spectra we identify 1310 Sgr members out of 2296 stars surveyed distributed across 24 different fields across the Sgr core. Additional slit spectra were obtained of stars bridging from the Sgr core to its trailing tail. Our systematic, large area sample shows no evidence for significant rotation, a result at odds with the ~20 km/s rotation required as an explanation for the bifurcation seen in the Sgr tidal stream; the observed small (<= 4 km/s) velocity trend along primarily the major axis is consistent with models of the projected motion of an extended body on the sky with no need for intrinsic rotation. The Sgr core is found to have a flat velocity dispersion (except for a kinematically colder center point) across its surveyed extent and into its tidal tails, a property that matches the velocity dispersion profiles measured for other Milky Way dwarf spheroidal (dSph) galaxies. We comment on the possible significance of this observed kinematical similarity for the dynamical state of the other classical Milky Way dSphs in light of the fact that Sgr is clearly a strongly tidally disrupted system.
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