  This paper extends our previous study (Li et al. 2023) of the early evolution
of Jupiter and its two Trojan swarms by introducing the possible perturbations
of a free floating planet (FFP) invading the Solar System. In the framework of
the invasion of a FFP, we aim to provide some new scenarios to explain the
number asymmetry of the L4 and L5 Jupiter Trojans, and some other observed
features. We investigate two different cases: (i) The indirect case, where
Jupiter experiences a scattering encounter with the FFP and jumps outwards at a
speed that is much higher than that considered in(Li et al. 2023), resulting in
a change in the numbers of the L4 (N4) and L5 (N5) Trojans swarms. (ii) The
direct case, in which the FFP traverses the L5 region and affects the stability
of the local Trojans. In the indirect case, the outward migration of Jupiter
can be fast enough to make the L4 islands disappear temporarily, inducing a
resonant amplitude increase of the local Trojans. After the migration is over,
the L4 Trojans come back to the re-appeared and enlarged islands. As for the L5
islands, they always exist but expand even more considerably. Since the L4
swarm suffers less excitation in the resonant amplitude than the L5 swarm, more
L4 Trojans are stable and could survive to the end. In the direct case, the FFP
could deplete a considerable fraction of the L5 Trojans, while the L4 Trojans
at large distances are not affected and all of them could survive. Both the
indirect and direct cases could result in a number ratio of R45=N4/N5~1.6 that
can potentially explain the current observations. The latter has the advantage
of producing the observed resonant amplitude distribution. For achieving these
results, we propose that the FFP should have a mass of at least of a few tens
of Earth masses and its orbital inclination is allowed to be as high as 40
degrees.
