On the local and global properties of gravitational spheres of influence

Abstract

ABSTRACT We revisit the concept of spheres of gravitational activity, to which we give both a geometrical and a 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) in order to define a first-order ‘boundary’ 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 three-body problem (e.g. Star-Planet-Satellite System). We start with the basic definition of a planet’s sphere of activity as the region of space in which it is feasible to assume the 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. Throughout the paper, we clearly distinguish amongst the sphere of activity, the Chebotarev sphere (a particular case of the sphere of activity), the Laplace sphere, and the Hill sphere. The last two are often wrongfully thought to be one and the same. Furthermore, by taking a closer look at and comparing the ratio of the star’s accelerations (central/perturbing) with that of the planetary accelerations (central/perturbing) as a function of the planeto-centric distance, we have identified different dynamical regimes, which are presented in the semi-analytical analysis.