  We consider the motion of individual two-dimensional vortices in general
radially symmetric potentials in Bose-Einstein condensates. We find that
although in the special case of the parabolic trap there is a logarithmic
correction in the dependence of the precession frequency $\omega$ on the
chemical potential $\mu$, this is no longer true for a general potential $V(r)
\propto r^p$. Our calculations suggest that for $p>2$, the precession frequency
scales with $\mu$ as $\omega \sim \mu^{-2/p}$. This theoretical prediction is
corroborated by numerical computations, both at the level of spectral
(Bogolyubov-de Gennes) stability analysis by identifying the relevant
precession mode dependence on $\mu$, but also through direct numerical
computations of the vortex evolution in the large $\mu$, so-called
Thomas-Fermi, limit. Additionally, the dependence of the precession frequency
on the radius of an initially displaced from the center vortex is examined and
the corresponding predictions are tested against numerical results.
