Foretellings of Ragnar\"ok: World-engulfing Asymptotic Giants and the Inheritance of White Dwarfs

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.