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Mirror Trajectories in Space Mission Analysis

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Abstract

The theorem of mirror trajectories was proven almost six decades ago by Miele, and states that for a given path in the restricted problem of three bodies (with primaries in mutual circular orbits) there exists a mirror trajectory (in two dimensions) and three mirror paths (in three dimensions). This theorem regards feasible trajectories and proved extremely useful for investigating the natural spacecraft dynamics in the circular restricted problem of three bodies. Several trajectories of crucial importance for mission analysis and design can be identified by using the theorem of mirror trajectories, i.e. (a) free return paths, (b) periodic orbits, and (c) homoclinic connections. Free return paths have been designed and flown in the Apollo missions, because they allow safe ballistic return toward the Earth in case of failure of the main propulsive system. These trajectories belong to the class of mirror paths with a single orthogonal crossing of the axis that connects the Earth and the Moon in the synodic reference system. Instead, if two orthogonal crossings of the same axis exist, then the resulting path is a periodic orbit. A variety of such orbits can be found, encircling either both primaries, a single celestial body, or a collinear libration point. In all cases, periodic orbits represent repeating trajectories that can be traveled indefinitely, (ideally) without any propellant expenditure. Homoclinic connections are special paths that belong to both the stable and the unstable manifold associated with a periodic orbit. These trajectories depart asymptotically from a periodic orbit and can encircle both primaries (even with close approaches) before converging asymptotically toward the initial periodic orbit. After almost six decades, the theorem of mirror trajectories, which clarified the fundamental symmetry properties related to the motion of a third body (or, more concretely, a space vehicle), still excerpts a considerable influence in space mission analysis and design.

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References

  1. J. V. Breakwell and J. Brown, “The Halo family of three dimensional periodic orbits in the Earth-Moon restricted three body problem”, Celestial Mechanics, Vol. 20, No. 4, pp. 389–404, 1979

    Article  MATH  Google Scholar 

  2. R. A. Broucke, “Periodic Orbits in the Restricted Three-Body Problem with Earth-Moon Masses”, JPL Technical Report 32-1168, Pasadena, CA, 1968

    Google Scholar 

  3. G. Darwin, “Periodic Orbits”, Acta Mathematica, Vol. 21, pp. 99–242, 1897

    Article  MathSciNet  MATH  Google Scholar 

  4. K. Davis, D. Scheeres, R. Anderson and G. Born, “Optimal transfers between unstable periodic orbits using invariant manifolds”, Celestial Mechanics and Dynamical Astronomy, Vol. 109, No. 3, pp. 241–264, 2011

    Article  MathSciNet  MATH  Google Scholar 

  5. R. W. Farquhar, “Lunar Communications with Libration-Point Satellites”, Journal of Spacecraft and Rockets, Vol. 4, No. 10, pp. 1383–1384, 1967

    Article  Google Scholar 

  6. D. W. Farquhar and A. A. Kamel, “Quasi-periodic orbits about the translunar libration point”, Celestial Mechanics, Vol. 7, No. 4, pp. 458–473, 1973

    Article  MATH  Google Scholar 

  7. D. C. Folta, M. Woodard, K. Howell, C. Patterson and W. Schlei, “Applications of multi-body dynamical environments: The ARTEMIS transfer trajectory design”, Acta Astronautica, Vol. 73, pp. 237–249, 2012

    Article  Google Scholar 

  8. M. Giancotti, M. Pontani and P. Teofilatto, “Lunar Capture Trajectories and Homoclinic Connections through Isomorphic Mapping”, Celestial Mechanics and Dynamical Astronomy, Vol. 114, No. 1–2, pp. 55–76, 2012

    Article  MathSciNet  Google Scholar 

  9. M. Giancotti, M. Pontani and P. Teofilatto, “Cylindrical isomorphic mapping applied to invariant manifold dynamics for Earth-Moon missions”, Celestial Mechanics and Dynamical Astronomy, Vol. 120, No. 3, pp. 249–268, 2014

    Article  MathSciNet  Google Scholar 

  10. G. Gomez, J. Masdemont and C. Simo, “Quasihalo Orbits Associated with Libration Points”, Celestial Mechanics and Dynamical Astronomy, Vol. 46, No. 2, pp. 135–176, 1998

    MathSciNet  Google Scholar 

  11. G. Gomez and J. Masdemont, “Some zero cost transfers between libration points orbits”, AAS/AIAA Space Flight Mechanics Meeting, Clearwater, FL, 2000

    Google Scholar 

  12. G. Gomez and M. Marcote, “High-Order Analytical Solutions of Hill’s Equations”, Celestial Mechanics and Dynamical Astronomy, Vol. 94, No. 2, pp. 197–211, 2006

    Article  MathSciNet  MATH  Google Scholar 

  13. V. M. Guibout and D. J. Scheeres, “Periodic orbits from generating functions”, Advances in the Astronautical Sciences, Vol. 116, No. 2, pp. 1029–1048, 2004

    Google Scholar 

  14. M. Hénon, “Vertical stability of periodic orbits in the restricted problem I. Equal Masses”, Astronomy and Astrphysics, Vol. 28, pp. 415–426, 1973

    MATH  Google Scholar 

  15. M. Hénon, “New families of periodic orbits in Hill’s problem of three bodies”, Celestial Mechanics and Dynamical Astronomy, Vol. 85, No. 3, pp. 223–246, 2003

    Article  MathSciNet  MATH  Google Scholar 

  16. G. W. Hill, “Review of Darwin’s periodic orbits”, Astronomical Journal, Vol. 18, p. 120, 1898

    Article  Google Scholar 

  17. K. C. Howell, “Three dimensional periodic Halo orbits”, Celestial Mechanics, Vol. 32, No. 1, pp. 53–72, 1984

    Article  MathSciNet  MATH  Google Scholar 

  18. K. C. Howell and H. J. Pernicka, “Numerical determination of Lissajous trajectories in the restricted three-body problem”, Celestial Mechanics, Vol. 41, No. 1–4, pp. 107–124, 1987

    Article  MATH  Google Scholar 

  19. E. Kolemen, N. J. Kasdin and P. Gurfil, “Multiple Poincaré sections method for finding the quasiperiodic orbits of the restricted three body problem”, Celestial Mechanics and Dynamical Astronomy, Vol. 112, No. 1, pp. 47–74, 2012

    Article  MathSciNet  MATH  Google Scholar 

  20. M. W. Lo, B. G. Williams, W. E. Bollman, D. S. Han, Y. S. Hahn, J. L. Bell., E. Hirst, R. Corwin, R. Hong, K. Howell, B. Barden and R. Wilson “Genesis mission design”, Journal of the Astronautical Sciences, Vol. 49, No. 1, pp. 169–184, 2011

    Google Scholar 

  21. J. E. Marsden, W. S. Koon, M. W. Lo and S. D. Ross, “Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics”, Chaos, Vol. 10, No. 2, pp. 427–469, 2000

    Article  MathSciNet  MATH  Google Scholar 

  22. J. E. Marsden, W. S. Koon, M. W. Lo and S. D. Ross, “Constructing a low energy transfer between Jovian Moons”, Contemporaneous Mathematics, Vol. 292, pp. 129–145, 2000

    MathSciNet  MATH  Google Scholar 

  23. R. Marson, M. Pontani, E. Perozzi and P. Teofilatto, “Using space manifold dynamics to deploy a small satellite constellation around the Moon”, Celestial Mechanics and Dynamical Astronomy, Vol. 106, No. 2, pp. 117–142, 2011

    Article  MathSciNet  MATH  Google Scholar 

  24. C. Martin, B. A. Conway and P. Ibanez, “Optimal Low-Thrust Trajectories to the Interior Earth-Moon Lagrange Point”, Space Manifold Dynamics, Springer, New York, NY, pp. 161–184, 2010

    Chapter  Google Scholar 

  25. J. Masdemont, G. Gomez and C. Simo, “Quasihalo orbits associated with libration points”, Journal of the Astronautical Sciences, Vol. 46, No. 2, pp. 135–176, 1998

    MathSciNet  Google Scholar 

  26. A. Miele, “Theorem of Image Trajectories in Earth-Moon Space”, Astronautica Acta, Vol. 6, No. 5, pp. 225–232, 1960

    MATH  Google Scholar 

  27. A. Miele, “Revisit of the Theorem of Image Trajectories in Earth-Moon Space”, Journal of Optimization Theory and Applications, Vol. 147, No. 3, pp. 483–490, 2010

    Article  MathSciNet  MATH  Google Scholar 

  28. Z. P. Olikara and D. J. Scheeres, “Numerical Method for Computing Quasi-Periodic Orbits and Their Stability in the Restricted Three-Body Problem”, Proceedings of the 1st IAA Conference on Dynamics and Control of Space Systems, Porto, Portugal, 2012

    Google Scholar 

  29. M. A. C. Perryman, “Overview of the Gaia Mission”, Proceedings of the Symposium The Three Dimensional Universe with Gaia, Paris, France, 2004

    Google Scholar 

  30. M. Pontani, C. Martin and B. A. Conway “New numerical methods for determining periodic orbits in the circular restricted three-body problem”, Proceedings of the 61st International Astronautical Congress, Prague, Czech Republic, 2010

    Google Scholar 

  31. M. Pontani and P. Teofilatto, “Low-energy Earth-Moon transfers involving manifolds through isomorphic mapping”, Acta Astronautica, Vol. 91, pp. 96–106, 2013

    Article  Google Scholar 

  32. M. Pontani and A. Miele, “Periodic Image Trajectories in Earth-Moon Space”, Journal of Optimization Theory and Applications, Vol. 157, No. 3, pp. 866–1887, 2013

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Pontani, M. Giancotti and P. Teofilatto, “Manifold dynamics in the Earth-Moon system via isomorphic mapping with application to spacecraft end-of-life strategies”, Acta Astronautica, Vol. 105, pp. 218–229, 2014

    Article  Google Scholar 

  34. M. Pontani and A. Miele, “Theorem of Optimal Image Trajectories in the Restricted Problem of Three Bodies”, Journal of Optimization Theory and Applications, Vol. 168, No. 3, pp. 992–1013, 2016

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Pontani and P. Teofilatto, “Invariant manifold connections via polyhedral representation”, Acta Astronautica, Vol. 137, pp. 512–521, 2017

    Article  Google Scholar 

  36. P. Rathsman, J. Kugelberg, P. Bodin, G. D. Racca, B. Foing and L. Stagnaro, “SMART-1: Development and Lessons Learnt”, Acta Astronautica, Vol. 57, No. 2–8, pp. 455–468, 2005

    Article  Google Scholar 

  37. D. L. Richardson, “Analytic construction of periodic orbits about the collinear points”, Celestial Mechanics, Vol. 22, No. 3, pp. 241–253, 1980

    Article  MathSciNet  MATH  Google Scholar 

  38. R. B. Roncoli and K. K. Fujii, “Mission design overview for the gravity recovery and interior laboratory (GRAIL) mission”, AIAA/AAS Astrodynamics Specialist Conference, Toronto, Canada, 2010

    Book  Google Scholar 

  39. V. Szebehely, Theory of Orbits - the Restricted Problem of Three Bodies, Academic Press, New York, NY, pp. 7–25, 1967

    MATH  Google Scholar 

  40. K. Uesugi, H. Matuso, J. Kawaguchi and T. Hayashi, “Japanese first double Lunar swingby mission ‘HITEN’”, Acta Astronautica, Vol. 25, No. 7, pp. 347–355, 1991

    Article  Google Scholar 

  41. M. Vaquero and K. C. Howell, “Poincaré maps and resonant orbits in the restricted three-body problems”, AAS Astrodynamics Specialist Conference, Girdwood, AK, 2011

    Google Scholar 

Download references

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  1. Dipartimento di Ingegneria Astronautica, Elettrica ed Energetica, “Sapienza” - Università di Roma, Italy

    M. Pontani

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Pontani, M. Mirror Trajectories in Space Mission Analysis. Aerotec. Missili Spaz. 96, 195–203 (2017). https://doi.org/10.1007/BF03404754

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