Critical Rotation for the First Vortex in Trapped Condensates and Helium Buckets

The first quantised vortex enters a rotating superfluid only above a critical angular velocity set by the container size and the vortex core size. This paper computes that threshold for a trapped rubidium-87 condensate and for a centimetre-scale helium-4 bucket, compares the two, and examines the shear associated with the single-vortex state against the torsion ceiling. The condensate threshold is 0.22 hertz, comfortably inside trap rotation capabilities, while the helium bucket threshold is 0.46 millihertz, a ratio between the thresholds of 4.90e+02 driven almost entirely by the container-size term. In both systems a single central vortex is a gentle flow state whose gradients are far below those of dense arrays and enormously below the fundamental ceiling; the threshold is a finite-size energetics result, not a bound effect. The calculation is performed from stated geometry, not measured.