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Studying the errors in the estimation of the variation of energy by the “patched-conics” model in the three-dimensional swing-by

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Abstract

The swing-by maneuver is a technique used to change the energy of a spacecraft by using a close approach in a celestial body. This procedure was used many times in real missions. Usually, the first approach to design this type of mission is based on the “patched-conics” model, which splits the maneuver into three “two-body dynamics.” This approach causes an error in the estimation of the energy variations, which depends on the geometry of the maneuver and the system of primaries considered. Therefore, the goal of the present paper is to study the errors caused by this approximation. The comparison of the results are made with the trajectories obtained using the more realistic restricted three-body problem, assumed here to be the “real values” for the maneuver. The results shown here describe the effects of each parameter involved in the swing-by. Some examples using bodies in the solar system are used in this part of the paper. The study is then generalized to cover different mass parameters, and its influence is analyzed to give an idea of the amount of the error expected for a given system of primaries. The results presented here may help in estimating errors in the preliminary mission analysis using the “patched-conics” approach.

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Notes

  1. A fourth-order Runge–Kutta method with monotoring of local truncation error to fifth order was used to ensure accuracy and to adjust stepsize.

  2. The spacecraft never passes too close to the center of the bodies, so regularization is not required to solve this problem.

  3. The dashed red and solid black lines are superposed in almost all figures contained in Fig. 4, due to the symmetry of the problem (in other words: \(max(\Delta E_\mathrm{error})=-min({\Delta E}_\mathrm{error})\)). The exception is for \(\alpha \), which is precisely the variable that creates this symmetry.

  4. The greater errors for lower \(r_p\) and N, for \(\alpha =90\) and \(270^\circ \) , etc., the symmetries.

  5. The percentage is around 15\(\%\) for Jupiter and 6\(\%\) for the Earth.

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Acknowledgements

A special thanks to the reviewers for the ideas that put this paper in a broader context. Also, the author thanks the Grants \(\#\) 406841/2016-0 and 301338/2016-7 from the National Council for Scientific and Technological Development (CNPq), and Grants \(\#\) 2011/08171-3, 2011/13101-4, 2015/24321-6 and 2016/14665-2 from São Paulo Research Foundation (FAPESP).

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  1. Alexander Sukhanov

    Present address: Instituto Nacional de Pesquisas Espaciais, São José dos Campos, Brazil

Authors and Affiliations

  1. Instituto Nacional de Pesquisas Espaciais, São José dos Campos, Brazil

    Rodolfo Batista Negri & Antonio Fernando Bertachini de Almeida Prado

  2. Space Research Institute of Russian Academy of Sciences, Moscow, Russia

    Alexander Sukhanov

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  1. Rodolfo Batista Negri
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  2. Antonio Fernando Bertachini de Almeida Prado
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Correspondence to Rodolfo Batista Negri.

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Negri, R.B., Prado, A.F.B.d.A. & Sukhanov, A. Studying the errors in the estimation of the variation of energy by the “patched-conics” model in the three-dimensional swing-by. Celest Mech Dyn Astr 129, 269–284 (2017). https://doi.org/10.1007/s10569-017-9779-3

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