Local sound-speed criterion for circular condensate stirring ($D/R>{\omega }_r/\sqrt{{\omega }_r^2+2{\Omega }^2}$)

In a parabolic condensate in the Thomas–Fermi approximation for a condensate, the local sound speed is $c(D)={\omega }_r\sqrt{R^2-D^2}/\sqrt{2}$. Equating it to the obstacle speed $\Omega D$ gives the displayed threshold. This is the stipulated local phonon estimate; a finite strongly depleting obstacle can modify flow and nucleate vortices, so the inequality is not a universal exact onset law for every stirring experiment.

For a two-dimensional Thomas–Fermi condensate at fixed coupling, $N=\pi n_0{R}^2/2\propto {\mu }^2$, so $R\propto N^{1/4}$. Slow exponential atom loss shrinks the cloud while a fixed stirring radius becomes a larger fraction of its size. At $\Omega ={\omega }_r/\sqrt{2}$ and initial $D/R=1/2$, the local-sound-speed threshold is reached at $t_c=2\tau \ln 2$, before the shrinking cloud edge reaches the beam.