  Cold outer debris belts orbit a significant fraction of stars, many of which
are planet-hosts. Radiative forces from the star lead to dust particles leaving
the outer belts and spiralling inwards under Poynting-Robertson drag. We
present an empirical model fitted to N-body simulations that allows the fate of
these dust particles when they encounter a planet to be rapidly calculated.
High mass planets eject most particles, whilst dust passes low mass planets
relatively unperturbed. Close-in, high mass planets (hot Jupiters) are best at
accreting dust. The model predicts the accretion rate of dust onto planets
interior to debris belts, with mass accretions rates of up to hundreds of
kilograms per second predicted for hot Jupiters interior to outer debris belts,
when collisional evolution is also taken into account. The model can be used to
infer the presence and likely masses of as yet undetected planets in systems
with outer belts. The non-detection of warm dust with the Large Binocular
Telescope Interferometer (LBTI) around Vega could be explained by the presence
of a single Saturn mass planet, or a chain of lower mass planets. Similarly,
the detection of warm dust in such systems implies the absence of planets above
a quantifiable level, which can be lower than similar limits from direct
imaging. The level of dust detected with LBTI around beta Leo can be used to
rule out the presence of planets more massive than a few Saturn masses outside
of ~5au.
