Rings and arcs around evolved stars. II. The Carbon Star AFGL 3068 and
  the Planetary Nebulae NGC 6543, NGC 7009 and NGC 7027
  We present a detailed comparative study of the arcs and fragmented ring-like
features in the haloes of the planetary nebulae (PNe) NGC 6543, NGC 7009, and
NGC 7027 and the spiral pattern around the carbon star AFGL 3068 using
high-quality multi-epoch HST images. This comparison allows us to investigate
the connection and possible evolution between the regular patterns surrounding
AGB stars and the irregular concentric patterns around PNe. The radial proper
motion of these features, ~15 km/s, are found to be consistent with the AGB
wind and their linear sizes and inter-lapse times (500-1900 yr) also agree with
those found around AGB stars, suggesting a common origin. We find evidence
using radiative-hydrodynamic simulations that regular patterns produced at the
end of the AGB phase become highly distorted by their interactions with the
expanding PN and the anisotropic illumination and ionization patterns caused by
shadow instabilities. These processes will disrupt the regular (mostly spiral)
patterns around AGB stars, plausibly becoming the arcs and fragmented rings
observed in the haloes of PNe.
