The strong dependence of Coulomb energies on nuclear radii makes it possible to extract the latter from calculations of the former. The resulting estimates of neutron skins indicate that two mechanisms are involved. The first one-isovector monopole polarizability-amounts to noting that when a particle is added to a system it drives the radii of neutrons and protons in different directions, tending to equalize the radii of both fluids independently of the neutron excess. This mechanism is well understood and the Duflo-Zuker (small) neutron skin values derived 14 years ago are consistent with recent measures and estimates. The alternative mechanism involves halo orbits whose huge sizes tend to make the neutron skins larger and have a subtle influence on the radial behavior of sd and pf shell nuclei. In particular, they account for the sudden rise in the isotope shifts of nuclei beyond N=28 and the near constancy of radii in the A=40-56 region. This mechanism, detected here for the first time, is not well understood and may well go beyond the Efimov physics usually associated with halo orbits.