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Initial condition maps of subsets of the circular restricted three-body problem phase space

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Abstract

The six-dimensional phase space of the circular restricted three-body problem offers opportunities to exploit dynamics commonly found in the Solar System for unique spacecraft trajectories. Poincaré first introduced the idea of reducing the phase space by observing a surface of section, or a Poincaré map. In this paper, we show an alternate map parameterization that classifies trajectories according to their qualitative end state and reflects their status onto an initial condition grid. While this method does not allow a visualization of the entire phase space at once, it does allow insight into reduced regimes of the phase space.

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References

  • Anderson, R.L., Lo, M.W.: Virtual exploration by computing global families of trajectories with supercomputers. In: Proceedings of the 15th AAS/AIAA space flight mechanics meeting, number AAS 05-220 in 1, Copper Mountain, Colorado (2005)

  • Anderson, R.L., Parker, J.S.: Survey of ballistic transfers to the lunar surface. J. Guid. Control Dyn. 35(4), 1256–1267 (2012)

    Article  ADS  Google Scholar 

  • Anderson, R.L., Parker, J.S.: Comparison of low-energy lunar transfer trajectories to invariant manifolds. Celest. Mech. Dyn. Astron. 115(3), 311–331 (2013)

    Article  ADS  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. 20, 389–404 (1979)

    Article  MATH  ADS  Google Scholar 

  • Davis, D.C., Howell, K.C.: Trajectory evolution in the multi-body problem with applications in the Saturnian system. Acta Astronaut. 69(11–12), 1038–1049 (2011)

    Article  ADS  Google Scholar 

  • Davis, D.C., Howell, K.C.: Characterization of trajectories near the smaller primary in restricted problem for applications. J. Guid. Control Dyn. 35(1), 116–128 (2012)

    Article  ADS  Google Scholar 

  • Fehlberg, E.: Classical fifth-, sixth-, seventh-, and eighth-order runge-kutta formulas with stepsize control. NASA Technical Report TR R-287, National Aeronautics and Space Administration, George C. Marshall Space Flight Center, Marshall, Alabama (1968)

  • Haapala, A.F., Howell, K.C.: Trajectory design strategies applied to temporary comet capture including Poincaré maps and invariant manifolds. Celest. Mech. Dyn. Astron. 116, 299–323 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  • Haapala, A.F., Howell, K.C.: Representations of higher-dimensional Poincaré maps with applications to spacecraft trajectory design. Acta Astronaut. 96, 23–41 (2014)

    Article  ADS  Google Scholar 

  • Howell, K.C., Breakwell, J.V.: Almost rectilinear halo orbits. Celest. Mech. 32(1), 29–52 (1984)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Howell, K.C., Pernicka, H.J.: Numerical determination of lissajous trajectories in the restricted three-body problem. Celest. Mech. 41, 107–124 (1988)

    Article  ADS  Google Scholar 

  • Jorba, À., Masdemont, J.: Dynamics in the center manifold of the collinear points of the restricted three body problem. Phys. D: Nonlinear Phenom. 132(1–2), 189–213 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Kolemen, E., Kasdin, N.J., Gurfil, P.: Multiple Poincaré sections method for finding the quasiperiodic orbits of the restricted three body problem. Celest. Mech. Dyn. Astron. 112, 47–74 (2012)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Miele, A.: Theorem of image trajectories in the Earth–Moon space. Acta Astronaut. 6(5), 225–232 (1960)

    Google Scholar 

  • Miele, A.: Revisit of the theorem of image trajectories in the Earth–Moon space. J. Optim. Theory Appl. 147(3), 483–490 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Mondelo, J.-M., Ollé, M., de Sousa-Silva, P.A., Terra, M.O.: Families of heteroclinic connections between quasi-periodic libration point trajectories. In: 65th international astronautical congress 2014, Toronto, Canada (2014)

  • Parker, J.S.: Low-Energy Ballistic Lunar Transfers. Ph.D. thesis, University of Colorado (2007)

  • Parker, J.S., Anderson, R.L.: Low-Energy Lunar Trajectory Design. Deep-Space Communications and Navigation Series. Jet Propulsion Laboratory, California Institute of Technology, Wiley (2014)

  • Paskowitz, M.E., Scheeres, D.J.: Robust capture and transfer trajectories for planetary satellite orbiters. J. Guid. Control Dyn. 29(2), 342–353 (2006)

    Article  ADS  Google Scholar 

  • Richardson, D.L.: Analytic construction of perioidic orbits about the collinear points. Celest. Mech. 22, 241–253 (1980)

    Article  MATH  ADS  Google Scholar 

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

    Google Scholar 

  • Utku, A.: Low Cost Capture Using Multi-Body Dynamics. Ph.D. thesis, University of Surrey (2013)

  • Villac, B.F., Scheeres, D.J.: Escaping trajectories in the Hill three-body problem and applications. J. Guid. Control Dyn. 26(2), 224–232 (2003)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Surrey Space Centre, University of Surrey, Guildford, Surrey, GU2 7XH, UK

    A. Utku, L. Hagen & P. Palmer

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  1. A. Utku
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  2. L. Hagen
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  3. P. Palmer
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Correspondence to L. Hagen.

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Utku, A., Hagen, L. & Palmer, P. Initial condition maps of subsets of the circular restricted three-body problem phase space. Celest Mech Dyn Astr 123, 387–410 (2015). https://doi.org/10.1007/s10569-015-9641-4

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  • DOI: https://doi.org/10.1007/s10569-015-9641-4

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