  The case of the classical Hill problem is numerically investigated by
performing a thorough and systematic classification of the initial conditions
of the orbits. More precisely, the initial conditions of the orbits are
classified into four categories: (i) non-escaping regular orbits; (ii) trapped
chaotic orbits; (iii) escaping orbits; and (iv) collision orbits. In order to
obtain a more general and complete view of the orbital structure of the
dynamical system our exploration takes place in both planar (2D) and the
spatial (3D) version of the Hill problem. For the 2D system we numerically
integrate large sets of initial conditions in several types of planes, while
for the system with three degrees of freedom, three-dimensional distributions
of initial conditions of orbits are examined. For distinguishing between
ordered and chaotic bounded motion the Smaller ALingment Index (SALI) method is
used. We managed to locate the several bounded basins, as well as the basins of
escape and collision and also to relate them with the corresponding escape and
collision time of the orbits. Our numerical calculations indicate that the
overall orbital dynamics of the Hamiltonian system is a complicated but highly
interested problem. We hope our contribution to be useful for a further
understanding of the orbital properties of the classical Hill problem.
