  We revisit the critical penetration of Pearl vortices in narrow
superconducting flat rings cooled in magnetic fields. Scanning superconducting
quantum interference device microscopy measurements showed how magnetic field
penetrates and vortices are trapped in flat rings made of amorphous MoGe thin
films. Counting the number of trapped vortices for each image, we found that
the vortices are completely excluded from the ring annulus when the applied
field $H$ is below a threshold field $H_{\rm{p}}$: Above this field, the
vortices increase linearly with field. The obtained values of $H_{\rm{p}}$
depend on the annulus width $w_{\rm{ring}}$ and follow the relation
$\mu_0H_{\rm{p}} = (1.9 \pm 0.1) \Phi_0/w_{\rm{ring}}^2$ with the
superconducting flux quantum $\Phi_0$. This relationship provides an insight
into the effect of the net-current circulating in the annular region, and also
leads to a precise control to trap or eliminate vortices in flat rings.
