# Characterization of Low-Energy Quasiperiodic Orbits in the Elliptic Restricted 4-Body Problem with Orbital Resonance

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Elliptic Restricted 4-Body Problem

#### 2.1. Dynamical Model

- The center of mass of the system is indicated as O;
- The relative motion of ${m}_{2}$ with respect to ${m}_{1}$ describes a Keplerian orbit onto the plane $\mathsf{\Pi}$ with semimajor axis a and eccentricity e;
- The relative motion of ${m}_{3}$ with respect to ${m}_{1}$ describes a Keplerian orbit onto the plane ${\mathsf{\Pi}}_{p}$ with semimajor axis ${a}_{p}$ and eccentricity ${e}_{p}$;
- The orbital plane ${\mathsf{\Pi}}_{p}$ is tilted of an angle ${\epsilon}_{p}$ with respect to $\mathsf{\Pi}$;

#### 2.2. Hamiltonian Formalism

## 3. Classification of Low-Energy Trajectories in the Elliptic Restricted 4-Body Problem

#### 3.1. Persistence of the Topological Properties

#### 3.2. Normal Forms for the Elliptic Restricted 4-Body Problem

#### 3.3. Topological Characterization of Low-Energy Trajectories

- Lissajous quasiperiodic orbits, characterized by ${x}_{1}={y}_{1}=0$, which evolve inside the equilibrium region;
- Transit trajectories, corresponding to the hyperbolic segments ${x}_{1}{y}_{1}<0$, which cross the equilibrium region twice, once towards ${m}_{1}$ and once towards ${m}_{2}$, in a finite interval of time;
- Bouncing trajectories, corresponding to the hyperbolic segments ${x}_{1}{y}_{1}>0$, which never cross the equilibrium region;
- Long-term ballistic captures, characterized by either ${x}_{1}\to 0$ or ${y}_{1}\to 0$, which cross the equilibrium region twice, over an indefinitely long interval of time.

## 4. Resonant Terms and Quasiperiodic Solutions

#### 4.1. Identification of Resonant Terms

#### 4.2. Determination of Stationary Points

## 5. Numerical Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Terms of the Power Series Expansion

## Appendix B. Canonical Transformations

#### Appendix B.1. Second Order

#### Appendix B.2. Third Order

## References

- Rathsman, P.; Kugelberg, P.; Bodin, P.; Racca, D.G.; Foing, B.; Stagnaro, L. SMART-1: Development and Lessons Learnt. Acta Astronaut.
**2005**, 56, 455–468. [Google Scholar] [CrossRef] - Lo, M.W.; Williams, B.G.; Bollman, W.E.; Han, D.S.; Hahn, Y.S.; Bell, J.L.; Hirst, E.; Corwin, R.; Hong, R.; Howell, K.; et al. Genesis mission design. J. Astronaut. Sci.
**2011**, 49, 169–184. [Google Scholar] [CrossRef] - Folta, D.C.; Woodard, M.; Howell, K.; Patterson, C.; Schlei, W. Applications of multi-body dynamical environments the ARTEMIS transfer trajectory design. Acta Astronaut.
**2012**, 73, 237–249. [Google Scholar] [CrossRef][Green Version] - Roncoli, R.B.; Fujii, K.K. Mission design overview for the gravity recovery and interior laboratory GRAIL mission. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Toronto, ON, Canada, 2–5 August 2010. [Google Scholar]
- Farquhar, R.W. The Flight of ISEE-3/ICE Origins, Mission History, and a Legacy. J. Astronaut. Sci.
**2001**, 49, 23–73. [Google Scholar] [CrossRef] - Dunham, D.W.; Jen, S.J.; Roberts, C.E.; Seacord, A.W., II; Sharer, P.J.; Folta, D.C.; Muhonen, D.P. Transfer trajectory design for the SOHO libration-point mission. In Proceedings of the 43rd International Astronautical Congress, Washington, DC, USA, 28 August–5 September 1992. [Google Scholar]
- Sharer, P.; Harrington, T. Trajectory Optimization for the ACE Halo Orbit Mission. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, San Diego, CA, USA, 29–31 July 1996. AIAA paper 96-3601-CP. [Google Scholar]
- Franz, H.; Sharer, P.; Ogilvie, K.; Desch, M. WIND Nominal Mission Performance and Extended Mission Design. J. Astronaut. Sci.
**2001**, 49, 145–167. [Google Scholar] [CrossRef] - Uesugi, K.; Matuso, H.; Kawaguchi, J.; Hayashi, T. Japanese first double Lunar swingby mission HITEN. Acta Astronaut.
**1991**, 25, 347–355. [Google Scholar] [CrossRef] - Belbruno, E.; Miller, J. Sun-perturbed Earth-to-Moon transits with balistic capture. J. Guid. Control Dyn.
**1993**, 16, 770–775. [Google Scholar] [CrossRef] - Szebehely, V. Theory of Orbit the Restricted Problem of Three Bodies; Academic Press: London, UK, 1967. [Google Scholar]
- Goḿez, G.; Masdemont, J. Some zero cost transfers between libration points orbits. In Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, Clearwater, FL, USA, 23–26 January 2000. Paper AAS 00-177. [Google Scholar]
- Breakwell, J.V.; Brown, J.V. The halo family of 3-dimensional periodic orbits in the Earth–Moon restricted 3-body problem. Celest. Mech. Dyn. Astron.
**1979**, 20, 389–404. [Google Scholar] [CrossRef] - McCarthy, B.P.; Howell, K.C. Leveraging quasi-periodic orbits for trajectory design in cislunar space. Astrodynamics
**2021**, 5, 139–165. [Google Scholar] [CrossRef] - Farquhar, R.W.; Kamel, A.A. Quasi-periodic orbits about the translunar libration point. Celest. Mech. Dyn. Astron.
**1973**, 7, 458–473. [Google Scholar] [CrossRef] - Moeckel, R. A variational proof of existence of transit orbits in the restricted three-body problem. Dyn. Syst. Int. J.
**2005**, 20, 45–58. [Google Scholar] [CrossRef][Green Version] - Anderson, R.L.; Easton, R.W.; Lo, M.W. Isolating blocks as computational tools in the circular restricted three-body problem. Physica D
**2017**, 343, 38–50. [Google Scholar] [CrossRef] - Giancotti, M.; Pontani, M.; Teofilatto, P. Lunar capture trajectories and homoclinic connections through isomorphic mapping. Celest. Mech. Dyn. Astron.
**2012**, 114, 55–76. [Google Scholar] [CrossRef] - Conley, C.C. Low energy transit orbits in the restricted three-body problem. J. Appl. Math.
**1968**, 16, 732–746. [Google Scholar] [CrossRef] - Conley, C.C. On the Ultimate Behavior of Orbits with Respect to an Unstable Critical Point l. Oscillating, Asymptotic, and Capture Orbits. J. Differ. Equ.
**1969**, 5, 136–158. [Google Scholar] [CrossRef][Green Version] - Koon, W.S.; Lo, M.W.; Marsden, J.E.; Ross, S.D. Low Energy transit to the Moon. Celest. Mech. Dyn. Astron.
**2001**, 81, 63–73. [Google Scholar] [CrossRef] - Giancotti, M.; Pontani, M.; Teofilatto, P. Cylindrical isomorphic mapping applied to invariant manifold dynamics for Earth-Moon Missions. Celest. Mech. Dyn. Astron.
**2014**, 120, 249–268. [Google Scholar] [CrossRef] - Anderson, R.L.; Lo, M.W. Spatial approaches to moons from resonance relative to invariant manifolds. Acta Astronaut.
**2014**, 105, 335–372. [Google Scholar] [CrossRef] - Khaja Fayaz, H.; Khaja Faisal, H.; Carletta, S.; Teofilatto, P. Deployment of a microsatellite constellation around the Moon using chaotic multi body dynamics. In Proceedings of the 71st International Astronautical Congress, Dubai, United Arab Emirates, 25–29 October 2021. [Google Scholar]
- Carletta, S.; Pontani, M.; Teofilatto, P. Earth-Mars microsatellite missions using ballistic capture and low-thrust propulsion. In Proceedings of the 71st International Astronautical Congress, Dubai, United Arab Emirates, 25–29 October 2021. [Google Scholar]
- Szebehely, V.; Giacaglia, G.E.O. On the elliptic restricted problem of three bodies. Astronaut. J.
**1964**, 69, 230–235. [Google Scholar] [CrossRef] - Michalodimitrakis, M. The circular restricted four-body problem. Astrophys. Space Sci.
**1981**, 75, 289–305. [Google Scholar] [CrossRef] - Carletta, S.; Pontani, M.; Teofilatto, P. Station-keeping about sun-mars three-dimensional quasi-periodic collinear libration point trajectories. Adv. Astronaut. Sci.
**2020**, 173, 299–311. [Google Scholar] - Carletta, S.; Pontani, M.; Teofilatto, P. Long-term capture orbits for low-energy space missions. Celest. Mech. Dyn. Astron.
**2018**, 130, 46. [Google Scholar] [CrossRef][Green Version] - Conley, C.; Easton, R. Isolated invariant sets and isolating blocks. Trans. Am. Math. Soc.
**1971**, 158, 35–61. [Google Scholar] [CrossRef] - Graziani, F.; Sparvieri, N.; Carletta, S. A low-cost Earth-Moon-Mars Mission Using a Microsatellite Platform. In Proceedings of the 71st International Astronautical Congress, Online Event, 12–14 October 2020. [Google Scholar]
- Carletta, S. Design of fuel-saving lunar captures using finite thrust and gravity-braking. Acta Astronaut.
**2021**, 181, 190–200. [Google Scholar] [CrossRef] - Carletta, S.; Pontani, M.; Teofilatto, P. Dynamics of three-dimensional capture orbits from libration region analysis. Acta Astronaut.
**2019**, 165, 331–343. [Google Scholar] [CrossRef] - Arnold, V.I. Small denominators and problems of stability of motion in classical and celestial mechanics. In Collected Works; Vladimir I. Arnold—Collected Works; Givental, A.B., Khesin, B.A., Marsden, J.E., Varchenko, A.N., Vassiliev, V.A., Viro, O.Y., Zakalyukin, V.M., Eds.; Springer: Berlin/Heidelberg, Germany, 2009; Volume 1. [Google Scholar]
- Blanc, M.; Alibert, Y.; André, N.; Atreya, S.; Beebe, R.; Benz, W.; Bolton, S.J.; Coradini, A.; Coustenis, A.; Dehant, V.; et al. LAPLACE: A mission to Europa and the Jupiter System for ESA’s Cosmic Vision Programme. Exp. Astron.
**2009**, 23, 849–892. [Google Scholar] [CrossRef][Green Version] - Phillips, C.B.; Pappalardo, R.T. Europa Clipper Mission Concept: Exploring Jupiter’s Ocean Moon. EOS
**2014**, 95, 165–167. [Google Scholar] [CrossRef] - Carletta, S.; Pontani, M.; Teofilatto, P. Design of low-energy capture trajectories in the elliptic restricted four-body problem. In Proceedings of the 70th International Astronautical Congress, Washington, DC, USA, 21–25 October 2019. [Google Scholar]
- Liu, C.; Gong, S. Hill stability of the satellite in the elliptic restricted four-body problem. Astrophys. Space Sci.
**2018**, 363, 162. [Google Scholar] [CrossRef] - Llibre, J.; Piñol, C. On the elliptic restricted three-body problem. Celest. Mech. Dyn. Astron.
**1990**, 48, 319–345. [Google Scholar] [CrossRef] - Meyer, K.R.; Hall, G.R.; Offin, D. Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. In Applied Mathematical Sciences; Springer Science+Business Media: New York, NY, USA, 2009. [Google Scholar]
- Siegel, C.L.; Moser, J.K. Lectures on Celestial Mechanics Reprint of the 1971 Edition; Springer: Berlin/Heidelberg, Germany, 1995. [Google Scholar]
- Koon, W.S.; Lo, M.W.; Marsden, J.E.; Ross, S.D. Dynamical Systems, the Three-Body Problem and Space Mission Design; Marsden Books, 2011; Available online: https://www.researchgate.net/publication/328913173_Dynamical_Systems_the_Three-Body_Problem_and_Space_Mission_Design (accessed on 20 February 2022).
- Moser, J. On the generalization of a theorem of A. Liapounoff. Commun. Pure Appl. Math.
**1958**, 11, 257–271. [Google Scholar] [CrossRef] - Carletta, S.; Pontani, M.; Teofilatto, P. Dynamics of capture orbits from libration region analysis. In Proceedings of the 69th International Astronautical Congress, Bremen, Germany, 1–5 October 2018. [Google Scholar]

**Figure 2.**Motion of the primaries in rotating-pulsating coordinates. Only one orbit of ${m}_{3}$ is shown.

**Figure 6.**Representation of the vertical oscillator associated to a stationary point in the rotating frame.

**Figure 7.**Representation of the vertical oscillator associated to a stationary point in the inertial frame.

Variable | Symbol | Value |
---|---|---|

Mass of Jupiter | ${m}_{1}$ | 1.899 × 10${}^{27}$ kg |

Mass of Europa | ${m}_{2}$ | 4.799 × 10${}^{24}$ kg |

Mass of Io | ${m}_{3}$ | 8.932 × 10${}^{22}$ kg |

Jupiter–Europa semimajor axis | a | 6.711 × 10${}^{5}$ km |

Jupiter–Io semimajor axis | ${a}_{p}$ | 4.218 × 10${}^{5}$ km |

Jupiter–Europa eccentricity | e | 0.0094 |

Jupiter–Io eccentricity | ${e}_{p}$ | 0.0041 |

Inclination between the orbital planes | $\epsilon $ | 0.430 deg |

Coordinate of the libration point ${L}_{1}$ | ${L}_{1}$ | 6.081 × 10${}^{5}$ km |

Resonant Hamiltonian terms | ${\widehat{H}}_{3}^{res}$ | 10${}^{-15}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Carletta, S.; Pontani, M.; Teofilatto, P.
Characterization of Low-Energy Quasiperiodic Orbits in the Elliptic Restricted 4-Body Problem with Orbital Resonance. *Aerospace* **2022**, *9*, 175.
https://doi.org/10.3390/aerospace9040175

**AMA Style**

Carletta S, Pontani M, Teofilatto P.
Characterization of Low-Energy Quasiperiodic Orbits in the Elliptic Restricted 4-Body Problem with Orbital Resonance. *Aerospace*. 2022; 9(4):175.
https://doi.org/10.3390/aerospace9040175

**Chicago/Turabian Style**

Carletta, Stefano, Mauro Pontani, and Paolo Teofilatto.
2022. "Characterization of Low-Energy Quasiperiodic Orbits in the Elliptic Restricted 4-Body Problem with Orbital Resonance" *Aerospace* 9, no. 4: 175.
https://doi.org/10.3390/aerospace9040175