  Of the four giant planets in the Solar system, only Jupiter and Neptune are
currently known to possess swarms of Trojan asteroids - small objects that
experience a 1:1 mean motion resonance with their host planet. In Lykawka et
al. (2009), we performed extensive dynamical simulations, including planetary
migration, to investigate the origin of the Neptunian Trojan population.
Utilising the vast amount of simulation data obtained for that work, together
with fresh results from new simulations, we here investigate the dynamical
capture of Trojans by all four giant planets from a primordial trans-Neptunian
disk. We find the likelihood of a given planetesimal from this region being
captured onto an orbit within Jupiter's Trojan cloud lies between several times
10^-6 and 10^-5. For Saturn, the probability is found to be in the range <10^-6
to 10^-5, whilst for Uranus the probabilities range between 10^-5 and 10^-4.
Finally, Neptune displays the greatest probability of Trojan capture, with
values ranging between 10^-4 and 10^-3. Our results suggest that all four giant
planets are able to capture and retain a significant population of Trojan
objects from the disk by the end of planetary migration. As a result of
encounters with the giant planets prior to Trojan capture, these objects tend
to be captured on orbits that are spread over a wide range of orbital
eccentricities and inclinations. The bulk of captured objects are to some
extent dynamically unstable, and therefore the populations of these objects
tend to decay over the age of the Solar System, providing an important ongoing
source of new objects moving on dynamically unstable orbits among the giant
planets. Given that a huge population of objects would be displaced by
Neptune's outward migration (with a potential cumulative mass a number of times
that of the Earth), we conclude that the surviving remnant of the Trojans
captured during the migration of the outer planets might be sufficient to
explain the currently known Trojan populations in the outer Solar system.
