The presence of planets around solar-type stars suggests that many white dwarfs should have relic planetary systems. While planets closer than ～5 AU will most likely not survive the post-main-sequence lifetime of their parent star, any planet with semimajor axis greater than 5 AU will survive, and its semimajor axis will increase as the central star loses mass. Since the stability of adjacent orbits to mutual planet-planet perturbations depends on the ratio of the planet mass to the central star's mass, some planets in previously stable orbits around a star undergoing mass loss will become unstable. We show that when mass loss is slow, systems of two planets that are marginally stable can become unstable to close encounters, while for three planets the timescale for close approaches decreases significantly with increasing mass ratio. These processes could explain the presence of anomalous IR excesses around white dwarfs that cannot be explained by close companions, such as G29-38, and may also be an important factor in explaining the existence of DAZ white dwarfs. The onset of instability through changing mass ratios will also be a significant effect for planetary embryos gaining mass in protoplanetary disks.
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