Swing-By in Three Dimensions: Closed Form Solutions

TL;DR

This paper derives analytical equations for three-dimensional gravity assist maneuvers using patched conics, enabling precise calculations of velocity, angular momentum, energy, and inclination changes, including special cases like planar and polar maneuvers.

Contribution

It provides closed-form solutions for 3D swing-by maneuvers, extending previous 2D models and offering insights into maneuver properties and special cases.

Findings

Analytical equations for 3D swing-by calculations

Properties of inclination change limits and symmetry

Special case analysis for polar and planar maneuvers

Abstract

The main goal of the present research is to make a summary of analytical equations that can be found to calculate a swing-by maneuver in the three-dimensional space. Analytical equations based in the patched conics approximation are showed and they allow to calculate the variation in velocity, angular momentum, energy and inclination of the spacecraft that is involved in this maneuver. Based on that, it is possible to obtain expressions for particular cases, like the planar and the polar maneuver. The most important properties of this maneuver can demonstrated using thos equations, like: in the case of planar maneuver the variation in inclination can be only 180, 0, and -180 degrees; the variation in inclination is symmetric with respect to the out of plane angle; a passage by the poles changes only the y-component of the angular momentum, keeping the energy and the inclination of the trajectory unchanged. The results show several maneuvers.