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3D stable and weakly unstable periodic orbits around the Earth near the retrograde co-orbital resonance with the Moon

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Abstract

Stable or weakly unstable orbits in cislunar space are attractive as potential locations that natural objects including dust particles may be trapped. Identifying such orbits is not straightforward especially in high-dimensional, many-body dynamical systems. The present paper adopts a strategy of limiting the search space around symmetric periodic orbits in the Earth–Moon spatial circular restricted three-body problem. We find a variety of linearly stable or weakly unstable periodic orbits near the \(1:1\) retrograde resonance with the Moon. Characteristics of the periodic orbits are explored and their stabilities under solar gravitational perturbations are assessed to understand representative behaviours of retrograde co-orbital orbits around the Earth.

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The data underlying this article will be shared on reasonable request to the corresponding author.

References

  • Bezrouk, C., Parker, J.S.: Long term evolution of distant retrograde orbits in the Earth–Moon system. Astrophys. Space Sci. 362, 176 (2017)

    Article  ADS  Google Scholar 

  • Boudad, K.K., Howell, K.C., Davis, D.C.: Dynamics of synodic resonant near rectilinear halo orbits in the bicircular four-body problem. Adv. Space Res. 66, 2194–2294 (2020)

    Article  ADS  Google Scholar 

  • Broucke, R.A.: Periodic orbits in the restricted three-body problem with Earth–Moon masses. JPL technical report 32-1168. Pasadena, Jet Propulsion Laboratory, California Institute of Technology (1968)

  • Contopoulos, G., Harsoula, M.: Stickiness effects in chaos. Celest. Mech. Dyn. Astron. 107, 77–92 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  • Dei Tos, D.A., Russell, R.P., Topputo, F.: Survey of Mars ballistic capture trajectories using periodic orbits as generating mechanisms. J. Guid. Control Dyn. 41, 1227–1242 (2018)

    Article  ADS  Google Scholar 

  • Gómez, G., Jorba, À., Masdemont, J., Simó, C.: Dynamics and Mission Design Near Libration Points, Vol IV: Advanced Methods for Triangular Points. World Scientific, Singapore (2001)

    Book  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, Wellington (2011)

    MATH  Google Scholar 

  • Kotoulas, T., Voyatzis, G.: Planar retrograde periodic orbits of the asteroids trapped in two-body mean motion resonances with Jupiter. Planet. Space Sci. 182, 104846 (2020)

    Article  Google Scholar 

  • Kotoulas, T., Voyatzis, G., Morais, M.H.M.: Three-dimensional retrograde periodic orbits of asteroids moving in mean motion resonances with Jupiter. Planet. Space Sci. 210, 105374 (2022)

    Article  Google Scholar 

  • Lara, M., Russell, R., Villac, B.F.: Classification of the distant stability regions at Europa. J. Guid. Control Dyn. 30, 409–418 (2007)

    Article  ADS  Google Scholar 

  • Li, M., Huang, Y., Gong, S.: Centaurs potentially in retrograde co-orbit resonance with Saturn. Astron. Astrophys. 617, A114 (2018)

    Article  ADS  Google Scholar 

  • Li, M., Huang, Y., Gong, S.: Survey of asteroids in retrograde mean motion resonances with planets. Astron. Astrophys. 630, A60 (2019)

    Article  ADS  Google Scholar 

  • Minghu, T., Ke, Z., Meibo, L., Chao, X.: Transfer to long term distant retrograde orbits around the Moon. Acta Astronaut. 98, 50–63 (2014)

    Article  ADS  Google Scholar 

  • Morais, M.H.M., Namouni, F.: Retrograde resonance in the planar three-body problem. Celest. Mech. Dyn. Astron. 117, 405–421 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  • Morais, M.H.M., Namouni, F.: A numerical investigation of coorbital stability and libration in three dimensions. Celest. Mech. Dyn. Astron. 125, 91–106 (2016)

    Article  ADS  Google Scholar 

  • Morais, M.H.M., Namouni, F.: Periodic orbits of the retrograde coorbital problem. Mon. Not. R. Astron. Soc. 490, 3799–3805 (2019)

    Article  ADS  Google Scholar 

  • Morais, M.H.M., Namouni, F., Voyatzis, G., Kotoulas, T.: A study of the \(1/2\) retrograde resonance: periodic orbits and resonant capture. Celest. Mech. Dyn. Astron. 133, 21 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  • Oshima, K.: Retrograde co-orbital orbits in the Earth–Moon system: planar stability region under solar gravitational perturbation. Astrophys. Space Sci. 366, 88 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  • Oshima, K., Yanao, T.: Spatial unstable periodic quasi-satellite orbits and their applications to spacecraft trajectories. Celest. Mech. Dyn. Astron. 131, 23 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  • Russell, R.P.: Global search for planar and three-dimensional periodic orbits near Europa. J. Astronaut. Sci. 54, 199–226 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  • Simó, C., Gómez, G., Jorba, À., Masdemont, J.: The bicircular model near the triangular libration points of the RTBP. In: Roy, A.E., Steves, B.A. (eds.) From Newton to Chaos. Springer, Boston (1995)

    MATH  Google Scholar 

  • Slíz-Balogh, J., Barta, A., Horváth, G.: Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth–Moon Lagrange point L5-I. Three-dimensional celestial mechanical modelling of dust cloud formation. Mon. Not. R. Astron. Soc. 480, 5550–5559 (2018)

    Article  ADS  Google Scholar 

  • Slíz-Balogh, J., Barta, A., Horváth, G.: Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth–Moon Lagrange point L5-II. Imaging polarimetric observation: new evidence for the existence of Kordylewski dust cloud. Mon. Not. R. Astron. Soc. 482, 762–770 (2019)

    Article  ADS  Google Scholar 

  • Szebehely, V.: Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press Inc., New York (1967)

    MATH  Google Scholar 

  • Topputo, F.: On optimal two-impulse Earth–Moon transfers in a four-body model. Celest. Mech. Dyn. Astron. 117, 279–313 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  • Wiegert, P., Connors, M., Veillet, C.: A retrograde co-orbital asteroid of Jupiter. Nature 543, 687–689 (2017)

    Article  ADS  Google Scholar 

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Acknowledgements

This study has been partially supported by JSPS Grants-in-Aid No. 20K14951.

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  1. Hiroshima Institute of Technology, 2-1-1 Miyake, Hiroshima, 731-5193, Japan

    Kenta Oshima

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  1. Kenta Oshima
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Correspondence to Kenta Oshima.

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Oshima, K. 3D stable and weakly unstable periodic orbits around the Earth near the retrograde co-orbital resonance with the Moon. Astrophys Space Sci 367, 42 (2022). https://doi.org/10.1007/s10509-022-04071-4

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