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.
Similar content being viewed by others
References
Celletti, A., Pucacco, G., Stella, D.: Lissajous and Halo orbits in the restricted three-body problem. J. Nonlinear Sci. 25, 343–370 (2015)
Cushman, R.: Geometry of the bifurcations of the normalized reduced Hénon-Heiles family. Proc. R. Soc. Lond. Ser. A 382, 361–371 (1982)
Deprit, A.: Canonical transformations depending on a small parameter. Celest. Mech. 1(1), 12–30 (1969)
Deprit, A.: The Lissajous transformation. I—basics. Celest. Mecha. Dyn. Astron. 51, 201–225 (1991)
Deprit, A., Elipe, A.: The Lissajous transformation. II—normalization. Celest. Mech. Dyn. Astron. 51, 227–250 (1991)
Deprit, A., Rom, A.: The main problem of artificial satellite theory for small and moderate eccentricities. Celest. Mech. 2(2), 166–206 (1970)
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)
Farquhar, R.W., Kamel, A.A.: Quasi-periodic orbits about the translunar libration point. Celest. Mech. 7, 458–473 (1973)
Ferraz-Mello, S.: Canonical Perturbation Theories—Degenerate Systems and Resonance. Vol. 345 of Astrophysics and Space Science Library. Springer, New York (2007)
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)
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)
Giorgilli, A., Galgani, L.: Formal integrals for an autonomous Hamiltonian system near an equilibrium point. Celest. Mech. 17, 267–280 (1978)
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
Hénon, M.: Numerical exploration of the restricted problem, V. Hill’s case: periodic orbits and their stability. Astron. Astrophys. 1, 223–238 (1969)
Hénon, M.: Numerical exploration of the restricted problem. VI. Hill’s case: non-periodic orbits. Astron. Astrophys. 9, 24–36 (1970)
Hénon, M.: Vertical stability of periodic orbits in the restricted problem. II. Hill’s case. Astron. Astrophys. 30, 317 (1974)
Hénon, M.: New families of periodic orbits in Hill’s problem of three bodies. Celest. Mech. Dyn. Astron. 85, 223–246 (2003)
Hénon, M., Petit, J.-M.: Series expansion for encounter-type solutions of Hill’s problem. Celest. Mech. 38, 67–100 (1986)
Henrard, J.: Periodic orbits emanating from a resonant equilibrium. Celest. Mech. 1, 437–466 (1970)
Hill, G.W.: Researches in the Lunar theory. Am. J. Math. 1, 5–26 (1878)
Hopf, H.: Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104, 637–665 (1931)
Hori, G.: Theory of general perturbation with unspecified canonical variables. Pub. Astron. Soc. Jpn. 18(4), 287–296 (1966)
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)
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)
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)
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
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)
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
Lara, M., San-Juan, J.: Dynamic behavior of an orbiter around Europa. J. Guid. Control Dyn. 28(2), 291–297 (2005)
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)
Marchesiello, A., Pucacco, G.: Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance. Int. J. Bifurc. Chaos 26, 1630011–1562 (2016)
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
Michalodimitrakis, M.: Hill’s problem—families of three-dimensional periodic orbits. I. Astrophys. Sp. Sci. 68, 253–268 (1980)
Miller, B.R.: The Lissajous transformation. III—parametric bifurcations. Celest. Mech. Dyn. Astron. 51, 251–270 (1991)
Petit, J.-M., Hénon, M.: Satellite encounters. Icarus 66, 536–555 (1986)
Richardson, D.L.: Analytic construction of periodic orbits about the collinear points. Celest. Mech. 22, 241–253 (1980)
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)
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)
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)
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)
Villac, B.F., Scheeres, D.J.: Escaping trajectories in the Hill three-body problem and applications. J. Guid. Control Dyn. 26, 224–232 (2003)
Zagouras, C., Markellos, V.V.: Three-dimensional periodic solutions around equilibrium points in Hill’s problem. Celest. Mech. 35, 257–267 (1985)
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10569-017-9778-4