We present projected velocity dispersion profiles for the Draco and Ursa Minor (UMi) dwarf spheroidal galaxies based on 207 and 162 discrete stellar velocities, respectively. Both profiles show a sharp decline in the velocity dispersion outside ~30' (Draco) and ~40' (UMi). New deep photometry of Draco reveals a break in the light profile at ~25'. These data imply the existence of a kinematically cold population in the outer parts of both galaxies. Possible explanations of both the photometric and the kinematic data in terms of both equilibrium and nonequilibrium models are discussed in detail. We conclude that these data challenge the picture of dwarf spheroidal galaxies as simple, isolated stellar systems
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