  The size distribution and orbital architecture of dust, grains, boulders,
asteroids, and major planets during the giant branch phases of evolution
dictate the preponderance and observability of the eventual debris, which have
been found to surround white dwarfs and pollute their atmospheres with metals.
Here, we utilize the photogravitational planar restricted three-body problem in
one-planet giant branch systems in order to characterize the orbits of grains
as the parent star luminosity and mass undergo drastic changes. We perform a
detailed dynamical analysis of the character of grain orbits (collisional,
escape, or bounded) as a function of location and energy throughout giant
branch evolution. We find that for stars with main-sequence masses of
$2.0M_{\odot}$, giant branch evolution, combined with the presence of a planet,
ubiquitously triggers escape in grains smaller than about 1 mm, while leaving
grains larger than about 5 cm bound to the star. This result is applicable for
systems with either a terrestrial or giant planet, is largely independent of
the location of the planet, and helps establish a radiative size threshold for
escape of small particles in giant branch planetary systems.
