A dynamical definition of the sphere of influence of the Earth

Abstract

The concept of sphere of influence of a planet is useful in both the context of impact monitoring of asteroids with the Earth and of the design of interplanetary trajectories for spacecraft. After reviewing the classical results, we propose a new definition of the sphere which is suitable for the classical patched-conic method: the new definition depends on the position and velocity of the small body for given values C of the Jacobi constant. Here, we compare the orbit of the small body, obtained in the framework of the circular restricted three-body problem, with orbits computed by patching two-body solutions. Our definition is based on an optimisation process, minimising a suitable target function with respect to the assumed radius of the sphere of influence. For different values C we represent the results in the planar case: we show the values of the selected radius as a function of two angles characterising the orbit. In this case, we also produce a database of radii of the sphere of influence for several initial conditions, allowing an interpolation.