# Interplanetary patched-conic approximation with an intermediary swing-by maneuver with the moon

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## Abstract

The present work quantifies the fuel consumption of a space vehicle in bi-impulsive interplanetary trajectories with an intermediary swing-by maneuver with the Moon. In this way, an interplanetary patched-conic approximation with a lunar swing-by maneuver is formulated with an important characteristic: the swing-by maneuver is designed before the determination of the trajectory by specifying its geometry. The transfer problem is then solved by a multi-point boundary value problem (MPBVP) with two constraints. The intermediary constraint is related to the geometry of the swing-by maneuver with the Moon, and the terminal constraint is related to the altitude of the arrival at the low orbit around the target planet. The proposed algorithm is built in such way that the MPBVP is split into two-point boundary value problems (TPBVPs): the first one is solved to ensure the satisfying of the intermediary constraint, and the second TPBVP is solved next to satisfy the final constraint. Both TPBVPs are solved by means of Newton–Raphson algorithm. The proposed algorithm is then utilized to determine the Earth–Mars and Earth–Venus trajectories with several geometric configurations. The geometric configuration with the smallest fuel consumption is obtained for both missions and compared to an interplanetary patched-conic approximation without swing-by maneuver with Moon. The results show advantages in performing swing-by maneuver with the Moon for interplanetary missions by saving fuel consumption without much increase of the time of flight.

## Keywords

Earth–Mars trajectory Earth–Venus trajectory Lunar swing-by maneuver Interplanetary patched-conic approximation## Notes

### Acknowledgements

This research is supported by Grant 2012/25308-5, São Paulo Research Foundation (FAPESP), and by CNPq under contract 304913/2013-8.

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