  We present the results of detailed dynamical simulations of the effect of the
migration of the four giant planets on both the transport of pre-formed Neptune
Trojans, and the capture of new Trojans from a trans-Neptunian disk. We find
that scenarios involving the slow migration of Neptune over a large distance
(50Myr to migrate from 18.1AU to its current location) provide the best match
to the properties of the known Trojans. Scenarios with faster migration (5Myr),
and those in which Neptune migrates from 23.1AU to its current location, fail
to adequately reproduce the current day Trojan population. Scenarios which
avoid disruptive perturbation events between Uranus and Neptune fail to yield
any significant excitation of pre-formed Trojans (transported with efficiencies
between 30 and 98% whilst maintaining the dynamically cold nature of these
objects). Conversely, scenarios with periods of strong Uranus-Neptune
perturbation lead to the almost complete loss of such pre-formed objects. In
these cases, a small fraction (~0.15%) of these escaped objects are later
recaptured as Trojans prior to the end of migration, with a wide range of
eccentricities (<0.35) and inclinations (<40 deg). In all scenarios (including
those with such disruptive interaction between Uranus and Neptune) the capture
of objects from the trans-Neptunian disk (through which Neptune migrates) is
achieved with efficiencies between ~0.1 and ~1%. The captured Trojans display a
wide range of inclinations (<40 deg for slow migration, and <20 deg for rapid
migration) and eccentricities (<0.35), and we conclude that, given the vast
amount of material which undoubtedly formed beyond the orbit of Neptune, such
captured objects may be sufficient to explain the entire Neptune Trojan
population. (Shortened version)
