  The presence of planets around solar-type stars suggests that many white
dwarfs should have relic planetary systems. While planets closer than $\sim$
5~AU will most likely not survive the post-main sequence lifetime of its parent
star, any planet with semimajor axis $>$ 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.
