An analytical study of the powered swing-by to perform orbital maneuvers

In the present paper, we present an analytical description of the powered swing-by maneuver in the three-dimensional space. To perform this task, new analytical equations are derived for the three-dimensional unpowered swing-by, including the variation in velocity, angular momentum, energy and inclination of the spacecraft due to the maneuver. This is done based on the patched-conics approximation. From those general three-dimensional equations, it is possible to confirm some well-known analytical results for the planar case. The equations developed here are verified by numerical integrations, using the restricted problem of three bodies, showing an agreement better than 1%. After that, based on those equations, some new analytical results are obtained for the situation when an impulse is applied to the spacecraft during the close approach. Those new equations are based in the assumption that the impulse applied is small compared to the velocity of the spacecraft, and they allow the calculation of the same variations cited above. Those equations can be simplified for specific cases, and it can generate some conclusions about the direction to apply the impulse to attend specific goals. A study is also executed to investigate in which cases the impulse is more efficient when applied at the periapsis of the swing-by hyperbola or at points far way from the swing-by body.