  Instabilities in planetary systems can result in the ejection of planets from
their host system, resulting in free-floating planets (FFPs). If this occurs in
a star cluster, the FFP may remain bound to the star cluster for some time and
interact with the other cluster members until it is ejected. Here, we use
$N$-body simulations to characterise close star-planet and planet-planet
encounters and the dynamical fate of the FFP population in star clusters
containing $500-2000$ single or binary star members. We find that FFPs ejected
from their planetary system at low velocities typically leave the star cluster
40% earlier than their host stars, and experience tens of close ($<1000$ AU)
encounters with other stars and planets before they escape. The fraction of
FFPs that experiences a close encounter depends on both the stellar density and
the initial velocity distribution of the FFPs. Approximately half of the close
encounters occur within the first 30 Myr, and only 10% occur after 100 Myr. The
periastron velocity distribution for all encounters is well-described by a
modified Maxwell-Bolzmann distribution, and the periastron distance
distribution is linear over almost the entire range of distances considered,
and flattens off for very close encounters due to strong gravitational
focusing. Close encounters with FFPs can perturb existing planetary systems and
their debris structures, and they can result in re-capture of FFPs. In
addition, these FFP populations may be observed in young star clusters in
imaging surveys; a comparison between observations and dynamical predictions
may provide clues to the early phases of stellar and planetary dynamics in star
clusters.
