  We use the planar circular restricted three-body problem in order to
numerically investigate the orbital dynamics of orbits of a spacecraft, or a
comet, or an asteroid in the Saturn-Titan system in a scattering region around
the Titan. The orbits can escape through the necks around the Lagrangian points
$L_1$ and $L_2$ or collide with the surface of the Titan. We explore all the
four possible Hill's regions depending on the value of the Jacobi constant. We
conduct a thorough numerical analysis on the phase space mixing by classifying
initial conditions of orbits and distinguishing between three types of motion:
(i) bounded, (ii) escaping and (iii) collisional. In particular, we locate the
different basins and we relate them with the corresponding spatial
distributions of the escape and crash times. Our results reveal the high
complexity of this planetary system. Furthermore, the numerical analysis shows
a strong dependence of the properties of the considered basins with the total
orbital energy, with a remarkable presence of fractal basin boundaries along
all the regimes. We hope our contribution to be useful in both space mission
design and study of planetary systems.
