  Irregular satellites (IS) are believed to have been captured during the Solar
system's dynamical history and provide clues for the Solar system's formation
and evolution. IS occupy a large fraction of the Hill sphere of their host
planet and their orbits are highly perturbed by the Sun. We use a novel
formalism developed in paper I to characterise their orbits in terms of an
effective secular Hamiltonian (the Brown Hamiltonian) that accounts for their
large orbital separations. We find that prograde satellites generally follow
the Brown Hamiltonian, while retrograde satellites (which extend further)
deviate more significantly. Nevertheless, the phase portrait is much better
described by the Brown Hamiltonian for all satellites. We construct a
semi-analytic criterion that predicts the librating orbit based on the
effective energy due to the Brown Hamiltonian. We also check our results with
highly accurate N-body integrations of satellite orbits, where initial
conditions are loaded directly from the updated ephemeris from the NASA
Horizons database. Although the retrograde librating orbits occupy more area in
the parameter space, the vast majority of librating IS are prograde. Using our
method we find $13$ librating satellites, $8$ of them previously known to
librate, and the rest shown to librate for the first time. Further observations
of existing and new satellites could shed more light on the dynamical history
of the Solar system and satellite formation and test our results.
