  We examine the population of simply periodic orbits in the Hill problem with
radiation pressure included, in order to understand the distribution of
gravitationally bound dust in orbit around a planet. We study a wide range of
radiation pressure strengths, which requires the inclusion of additional terms
beyond those discussed in previous analyses of this problem. In particular, our
solutions reveal two distinct populations of stable wide, retrograde, orbits,
as opposed to the single family that exists in the purely gravitational
problem.
  We use the result of these calculations to study the observational shape of
dust populations bound to extrasolar planets, that might be observable in
scattered or reradiated light. In particular, we find that such dusty clouds
should be elongated along the star--planet axis, and that the elongation of the
bound population increases with $\beta$, a measure of the strength of the
radiation pressure.
  As an application of this model, we consider the properties of the Fomalhaut
system. The unusual orbital properties of the object Fomalhaut~b can be
explained if the observed light was scattered by dust that was released from an
object in a quasi-satellite orbit about a planet located in, or near, the
observed debris ring. Within the context of the model of Hayakawa \& Hansen
(2023), we find that the dust cloud around such a planet is still approximately
an order of magnitude fainter than the limits set by current JWST data.
