= Solution
In a <rotating reference frame> at angular frequency $\Omega$, the relevant <energy> difference is approximately $E_{\rm kin}-\Omega L_z$. The vortex-free state has zero <circulation> and zero corresponding <angular momentum>. For $a_0/R_r\ll1$, the preceding shell estimates reduce to
$$
E_{\rm kin}\simeq\frac{\pi H_0\hbar^2n_0}{m}\left[\log\frac{R_r}{a_0}-\frac12\right],\qquad
L_z\simeq\frac{\pi H_0\hbar n_0R_r^2}{2}.
$$
Therefore the <thermodynamic vortex-nucleation frequency> for the first positive <quantum vortex> is
$$
\boxed{\Omega_c\simeq\frac{2\hbar}{mR_r^2}\left[\log\frac{R_r}{a_0}-\frac12\right],\qquad
\Omega>\Omega_c\ \Rightarrow\ E_{\rm kin}-\Omega L_z<0.}
$$
The common factors $H_0,n_0$ cancel explicitly, although the trap radius and core scale still depend on <number density>. In the strict leading-logarithm limit, $\Omega_c\sim2\hbar\log(R_r/a_0)/(mR_r^2)$. The finite constant $-1/2$ is the one obtained from the stated Thomas–Fermi shell <integral>; a resolved vortex core and readjustment of the <number density> at fixed particle number can change the nonlogarithmic constant. This is an energetic threshold, not a dynamical guarantee of nucleation: a surface barrier can delay entry. Higher positive <winding numbers> have flow <energy> proportional to $q^2$ and <angular momentum> proportional to $q$, so their analogous threshold is larger in this approximation.
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