  We revisit the concept of sphere of gravitational activity, to which we give
both a geometrical and physical meaning. This study aims to refine this concept
in a much broader context that could, for instance, be applied to exo-planetary
problems (in a Galactic stellar disc-Star-Planets system) to define a first
order "border" of a planetary system. The methods used in this paper rely on
classical Celestial Mechanics and develop the equations of motion in the
framework of the 3-body problem (e.g. Star-Planet-Satellite System). We start
with the basic definition of planet's sphere of activity as the region of space
in which it is feasible to assume a planet as the central body and the Sun as
the perturbing body when computing perturbations of the satellite's motion. We
then investigate the geometrical properties and physical meaning of the ratios
of Solar accelerations (central and perturbing) and planetary accelerations
(central and perturbing), and the boundaries they define. We clearly
distinguish throughout the paper between the sphere of activity, the Chebotarev
sphere (a particular case of the sphere of activity), Laplace sphere, and the
Hill sphere. The last two are often wrongfully thought to be one and the same.
Furthermore, taking a closer look and comparing the ratio of the star's
accelerations (central/perturbing) to that of the planetary acceleration
(central/perturbing) as a function of the planeto-centric distance, we have
identified different dynamical regimes which are presented in the
semi-analytical analysis.
