  The search for planets around White Dwarf stars, and evidence for dynamical
instability around them in the form of atmospheric pollution and circumstellar
discs, raises questions about the nature of planetary systems that can survive
the vicissitudes of the Asymptotic Giant Branch (AGB). We study the competing
effects, on planets at several AU from the star, of strong tidal forces arising
from the star's large convective envelope, and of the planets' orbital
expansion due to stellar mass loss. We, for the first time, study the evolution
of planets while following each thermal pulse on the AGB. For Jovian planets,
tidal forces are strong, and can pull into the envelope planets initially at ~3
AU for a 1M_Sol star and ~5 AU for a 5M_Sol star. Lower-mass planets feel
weaker tidal forces, and Terrestrial planets initially within 1.5-3 AU enter
the stellar envelope. Thus, low-mass planets that begin inside the maximum
stellar radius can survive, as their orbits expand due to mass loss. The
inclusion of a moderate planetary eccentricity slightly strengthens the tidal
forces experienced by Jovian planets. Eccentric Terrestrial planets are more at
risk, since their eccentricity does not decay and their small pericentre takes
them inside the stellar envelope. We also find the closest radii at which
planets will be found around White Dwarfs, assuming that any planet entering
the stellar envelope is destroyed. Planets are in that case unlikely to be
found inside ~1.5 AU of a White Dwarf with a 1M_Sol progenitor and ~10 AU of a
White Dwarf with a 5M_Sol progenitor.
