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A Hopf variables view on the libration points dynamics

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Abstract

The dynamics about the libration points of the Hill problem is investigated analytically. In particular, the use of perturbation theory allows to reduce the problem to a one degree of freedom Hamiltonian depending on two dynamical parameters. The invariant manifolds structure of the Hill problem is then disclosed, yet accurate computations are limited to energy values close to that of the libration points.

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References

  • Celletti, A., Pucacco, G., Stella, D.: Lissajous and Halo orbits in the restricted three-body problem. J. Nonlinear Sci. 25, 343–370 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Cushman, R.: Geometry of the bifurcations of the normalized reduced Hénon-Heiles family. Proc. R. Soc. Lond. Ser. A 382, 361–371 (1982)

    Article  ADS  MATH  Google Scholar 

  • Deprit, A.: Canonical transformations depending on a small parameter. Celest. Mech. 1(1), 12–30 (1969)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Deprit, A.: The Lissajous transformation. I—basics. Celest. Mecha. Dyn. Astron. 51, 201–225 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Deprit, A., Elipe, A.: The Lissajous transformation. II—normalization. Celest. Mech. Dyn. Astron. 51, 227–250 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Deprit, A., Rom, A.: The main problem of artificial satellite theory for small and moderate eccentricities. Celest. Mech. 2(2), 166–206 (1970)

    Article  ADS  MATH  Google Scholar 

  • Doedel, E.J., Paffenroth, R.C., Keller, H.B., Dichmann, D.J., Galán-Vioque, J., Vanderbauwhede, A.: Computation of periodic solutions of conservative systems with application to the 3-body problem. Int. J. Bifurc. Chaos 13, 1353–1381 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Farquhar, R.W., Kamel, A.A.: Quasi-periodic orbits about the translunar libration point. Celest. Mech. 7, 458–473 (1973)

    Article  ADS  MATH  Google Scholar 

  • Ferraz-Mello, S.: Canonical Perturbation Theories—Degenerate Systems and Resonance. Vol. 345 of Astrophysics and Space Science Library. Springer, New York (2007)

    MATH  Google Scholar 

  • García Yárnoz, D., Scheeres, D.J., McInnes, C.R.: On the “a” and “g” families of orbits in the Hill problem with solar radiation pressure and their application to asteroid orbiters. Celest. Mech. Dyn. Astron. 121, 365–384 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  • Giorgilli, A., Delshams, A., Fontich, E., Galgani, L., Simó, C.: Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem. J. Differ. Equ. 77, 167–198 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Giorgilli, A., Galgani, L.: Formal integrals for an autonomous Hamiltonian system near an equilibrium point. Celest. Mech. 17, 267–280 (1978)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Gómez, G., Jorba, A., Masdemont, J., Simó, C.: Study Refinement of Semi-analytical Halo Orbit Theory. Technical Report Contract 8625/89/D/MD(SC), European Space Operations Center, Robert-Bosch-Strasse 5, 64293 Darmstadt, Germany (1991)

  • Gómez, G., Marcote, M., Mondelo, J .M.: The invariant manifold structure of the spatial Hill’s problem. Dyn. Syst. 20(1), 115–147 (2005). doi:10.1080/14689360412331313039

    Article  MathSciNet  MATH  Google Scholar 

  • Hénon, M.: Numerical exploration of the restricted problem, V. Hill’s case: periodic orbits and their stability. Astron. Astrophys. 1, 223–238 (1969)

    ADS  MATH  Google Scholar 

  • Hénon, M.: Numerical exploration of the restricted problem. VI. Hill’s case: non-periodic orbits. Astron. Astrophys. 9, 24–36 (1970)

    ADS  MATH  Google Scholar 

  • Hénon, M.: Vertical stability of periodic orbits in the restricted problem. II. Hill’s case. Astron. Astrophys. 30, 317 (1974)

    ADS  MATH  Google Scholar 

  • Hénon, M.: New families of periodic orbits in Hill’s problem of three bodies. Celest. Mech. Dyn. Astron. 85, 223–246 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Hénon, M., Petit, J.-M.: Series expansion for encounter-type solutions of Hill’s problem. Celest. Mech. 38, 67–100 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Henrard, J.: Periodic orbits emanating from a resonant equilibrium. Celest. Mech. 1, 437–466 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Hill, G.W.: Researches in the Lunar theory. Am. J. Math. 1, 5–26 (1878)

    Article  MathSciNet  MATH  Google Scholar 

  • Hopf, H.: Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104, 637–665 (1931)

    Article  MathSciNet  MATH  Google Scholar 

  • Hori, G.: Theory of general perturbation with unspecified canonical variables. Pub. Astron. Soc. Jpn. 18(4), 287–296 (1966)

    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, 189–213 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kasdin, N.J., Gurfil, P., Kolemen, E.: Canonical modelling of relative spacecraft motion via epicyclic orbital elements. Celest. Mech. Dyn. Astron. 92, 337–370 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kummer, M.: On resonant non linearly coupled oscillators with two equal frequencies. Commun. Math. Phys. 48, 53–79 (1976). http://projecteuclid.org/euclid.cmp/1103899811

  • Lara, M.: Simplified equations for computing science orbits around planetary satellites. J. Guid. Control Dyn. 31(1), 172–181 (2008)

    Article  ADS  Google Scholar 

  • Lara, M., Palacián, J., Russell, R.: Mission design through averaging of perturbed Keplerian systems: the paradigm of an Enceladus orbiter. Celest. Mech. Dyn. Astron. 108(1), 1–22 (2010). doi:10.1007/s10569-010-9286-2

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Lara, M., Palacián, J.F., Yanguas, P., Corral, C.: Analytical theory for spacecraft motion about Mercury. Acta Astronaut. 66(7–8), 1022–1038 (2010). http://www.sciencedirect.com/science/article/pii/S0094576509004974

  • Lara, M., Peláez, J.: On the numerical continuation of periodic orbits. An intrinsic, 3-dimensional, differential, predictor-corrector algorithm. Astron. Astrophys. 389, 692–701 (2002)

    Article  ADS  MATH  Google Scholar 

  • Lara, M., Russell, R.P., Villac, B.: Fast estimation of stable regions in real models. Meccanica 42(5), 511–515 (2007). doi:10.1007/s11012-007-9060-z

    Article  MathSciNet  MATH  Google Scholar 

  • Lara, M., San-Juan, J.: Dynamic behavior of an orbiter around Europa. J. Guid. Control Dyn. 28(2), 291–297 (2005)

    Article  ADS  Google Scholar 

  • Lidov, M.L., Yarskaya, M.V.: Integrable cases in the problem of the evolution of a satellite orbit under the joint effect of an outside body and of the noncentrality of the planetary field. Cosm. Res. 12, 139–152 (1974)

    ADS  Google Scholar 

  • Marchesiello, A., Pucacco, G.: Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance. Int. J. Bifurc. Chaos 26, 1630011–1562 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Masdemont, J.J.: High-order expansions of invariant manifolds of libration point orbits with applications to mission design. Dyn. Syst. 20(1), 59–113 (2005). doi:10.1080/14689360412331304291

    Article  MathSciNet  MATH  Google Scholar 

  • Michalodimitrakis, M.: Hill’s problem—families of three-dimensional periodic orbits. I. Astrophys. Sp. Sci. 68, 253–268 (1980)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Miller, B.R.: The Lissajous transformation. III—parametric bifurcations. Celest. Mech. Dyn. Astron. 51, 251–270 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Petit, J.-M., Hénon, M.: Satellite encounters. Icarus 66, 536–555 (1986)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Russell, R.P., Lara, M.: On the design of an Enceladus science orbit. Acta Astronaut. 65(1–2), 27–39 (2009). http://www.sciencedirect.com/science/article/pii/S0094576509000587

  • San-Juan, J.F., Lara, M., Ferrer, S.: Phase space structure around oblate planetary satellites. J. Guid. Control Dyn. 29, 113–120 (2006)

    Article  ADS  Google Scholar 

  • Scheeres, D.J., Guman, M.D., Villac, B.F.: Stability analysis of planetary satellite orbiters: application to the Europa orbiter. J. Guid. Control Dyn. 24(4), 778–787 (2001)

    Article  ADS  Google Scholar 

  • Simó, C., Stuchi, T.J.: Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem. Phys. D Nonlinear Phenom. 140, 1–32 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Szebehely, V.: Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press Inc., New York (1967). http://www.sciencedirect.com/science/book/9780123957320

  • Vashkov’yak, M.A.: On the special particular solutions of a double-averaged Hill’s problem with allowance for flattening of the central planet. Astron. Lett. 22, 207–216 (1996)

    ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  • Zagouras, C., Markellos, V.V.: Three-dimensional periodic solutions around equilibrium points in Hill’s problem. Celest. Mech. 35, 257–267 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author acknowledges partial support by the Spanish State Research Agency and the European Regional Development Fund under Projects ESP2013-41634-P, ESP2014-57071-R and ESP2016-76585-R (AEI/ERDF, EU).

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  1. GRUCACI, University of La Rioja, and Space Dynamics Group – UPM, Logroño, Spain

    Martin Lara

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Correspondence to Martin Lara.

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Lara, M. A Hopf variables view on the libration points dynamics. Celest Mech Dyn Astr 129, 285–306 (2017). https://doi.org/10.1007/s10569-017-9778-4

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