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On optimal two-impulse Earth–Moon transfers in a four-body model

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Abstract

In this paper two-impulse Earth–Moon transfers are treated in the restricted four-body problem with the Sun, the Earth, and the Moon as primaries. The problem is formulated with mathematical means and solved through direct transcription and multiple shooting strategy. Thousands of solutions are found, which make it possible to frame known cases as special points of a more general picture. Families of solutions are defined and characterized, and their features are discussed. The methodology described in this paper is useful to perform trade-off analyses, where many solutions have to be produced and assessed.

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Notes

  1. http://moon.mit.edu/design.html, retrieved on 13 February 2012.

  2. The patched-conics solutions (Hohmann, bielliptic, and biparabolic) have been calculated by assuming the gravitational parameters of the Earth and Moon equal to \(3.986\times 10^{5}\,{\mathrm{km}}^{3}\mathrm{s}^{-2}\) and \(4.902\times 10^{3}\,{\mathrm{km}}^{3}\mathrm{s}^{-2}\), respectively.

  3. See Appendix 1 for the coordinate transformation behind (9) and (10).

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Acknowledgments

The author is grateful to Pierluigi Di Lizia, Alexander Wittig, and Koen Geurts for having proof-read the paper, to Mauro Massari and Franco Bernelli-Zazzera for having shared their workstations, and to Roberto Armellin for the computation of the Pareto-efficient solutions.

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  1. Department of Aerospace Science and Technology, Politecnico di Milano, Via La Masa, 34, 20156 , Milan, Italy

    Francesco Topputo

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Appendices

Appendix 1

An orbit in the rotating frame, \(\varvec{x}(t) = \{x(t), y(t), \dot{x}(t), \dot{y}(t)\}\), is converted to an orbit expressed in the \(P_1\)-centered (or Earth-centered), inertial frame, \(\varvec{X}_1(t) = \{X_1(t), Y_1(t), \dot{X}_1(t), \dot{Y}_1(t)\}\), through

$$\begin{aligned}&X_1(t) = (x(t)+\mu ) \cos t - y(t) \sin t \nonumber \\&Y_1(t) = (x(t)+\mu ) \sin t + y(t) \cos t \nonumber \\&\dot{X}_1(t) = (\dot{x}(t)-y(t)) \cos t - (\dot{y}(t)+x(t)+\mu ) \sin t \nonumber \\&\dot{Y}_1(t) = (\dot{x}(t)-y(t)) \sin t + (\dot{y}(t)+x(t)+\mu ) \cos t \end{aligned}$$
(34)

where \(t\) is the present, scaled time. The transformation into the \(P_2\)-centered (i.e. Moon-centered), inertial frame is obtained from (34) by replacing ‘\(\mu \)’ with ‘\(\mu -1\)’.

Appendix 2

The figures corresponding to the solutions characterized in Sect. 5 are reported below (see Figs. 24, 25, 26).

Fig. 24
figure 24

Negative solution samples belonging to family \(a\) and \(b\) in the Earth-centered inertial frame. (left) Family \(a^-\); (right) Family \(b^-\)

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Fig. 25
figure 25

Negative solution samples belonging to family \(c\) in the Earth-centered inertial frame. (left) Solutions \(c_1^-, c_2^-, c_4^-\); (right) \(c_3^-, c_5^-\)

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Fig. 26
figure 26

Sample solutions (top-left) \(f_1^+\); (top-right) \(h_3^-\); (center-left) \(m_2^+\); (center-right) \(p_5^-\); (bottom row) \(q_2^+\)

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Topputo, F. On optimal two-impulse Earth–Moon transfers in a four-body model. Celest Mech Dyn Astr 117, 279–313 (2013). https://doi.org/10.1007/s10569-013-9513-8

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