  Pollution of white dwarf atmospheres may be caused by asteroids that
originate from the locations of secular and mean-motion resonances in planetary
systems. Asteroids in these locations experience increased eccentricity,
leading to tidal disruption by the white dwarf. We examine how the $\nu_6$
secular resonance shifts outwards into a previously stable region of the
asteroid belt, as the star evolves to a white dwarf. Analytic secular models
require a planet to be engulfed in order to shift the resonance. We show with
numerical simulations that as a planet gets engulfed by the evolving star, the
secular resonance shifts and the rate of tidal disruption events increases with
the engulfed planet's mass and its orbital separation. We also investigate the
behaviour of mean-motion resonances. The width of a mean-motion resonance
increases as the star loses mass and becomes a white dwarf. The $\nu_6$ secular
resonance is more efficient at driving tidal disruptions than mean-motion
resonances with Jupiter. By examining 230 observed exoplanetary systems whose
central star will evolve into a white dwarf, we find that along with an Earth
mass planet at $1\,\rm au$, hot Jupiters at a semi--major axis $a\gtrsim
0.05\,\rm au$ and super--Earths of mass $10\,\rm M_\oplus$ at $a\gtrsim
0.3\,\rm au$ represent planet types whose engulfment shifts resonances enough
to cause pollution of the white dwarfs to a degree in agreement with
observations.

Earth and Planetary Astrophysics;Solar and Stellar Astrophysics