Sphere of influence and gravitational capture radius: a dynamical approach
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For problems in celestial mechanics that involve close encounters, it is necessary to determine the region where the gravitational influence of a body prevails over the influence of other bodies. From this need comes the concept of the sphere of influence. The models most used for the calculation of the radii of these spheres are the Hill sphere and the Laplace sphere. These are determined in terms of constant parameters, resulting in a fixed-size sphere, independent of the conditions of the encounter. In this paper, we present a numerical model for the sphere of influence, whose radius has been defined in terms of the initial relative velocity of the encounter, and of the mass ratio of the system considered. The same idea was applied to the delimitation of the regions where the phenomenon of temporary gravitational capture occurs, for some given initial conditions. With this goal, a numerical study was made through integrations of the restricted three-body problem and by monitoring the energy variation of the two-body problem. This study resulted in a complete mapping of the influence and capture regions, considering systems with amass ratio from 10(-1) to 10(-12), with the empirical functions for the calculation of these limits, called the capture radius and the influence radius.