Gravity assist in 3D like in Ulysses mission

Abstract

We study the gravity assist in the general case, i.e. when the spacecraft is not in a coplanar motion with respect to the planet's orbit. Our derivation is based on Kepler's planetary motion and Galilean addition of velocities, subjects covered in introductory physics courses. The main purpose of this paper is to illustrate how the gravity assist can be used to deviate a spacecraft outside its original plane of motion. As an example, we use the NASA-ESA's Ulysses mission to "test" our simple model.