Eight-shaped Lissajous orbits in the Earth-Moon system
MathematicS In Action, Volume 4 (2011) no. 1, pp. 1-23.

Euler and Lagrange proved the existence of five equilibrium points in the circular restricted three-body problem. These equilibrium points are known as the Lagrange points (Euler points or libration points) L 1 ,...,L 5 . The existence of families of periodic and quasi-periodic orbits around these points is well known (see [20, 21, 22, 23, 37]). Among them, halo orbits are 3-dimensional periodic orbits diffeomorphic to circles. They are the first kind of the so-called Lissajous orbits. To be selfcontained, we first provide a survey on the circular restricted three-body problem, recall the concepts of Lagrange point and of periodic or quasi-periodic orbits, and recall the mathematical tools in order to show their existence. We then focus more precisely on Lissajous orbits of the second kind, which are almost vertical and have the shape of an eight – we call them eight-shaped Lissajous orbits. Their existence is also well known, and in the Earth-Moon system, we first show how to compute numerically a family of such orbits, based on Linsdtedt Poincaré’s method combined with a continuation method on the excursion parameter. Our original contribution is in the investigation of their specific stability properties. In particular, using local Lyapunov exponents we produce numerical evidences that their invariant manifolds share nice global stability properties, which make them of interest in space mission design. More precisely, we show numerically that invariant manifolds of eight-shaped Lissajous orbits keep in large time a structure of eight-shaped tubes. This property is compared with halo orbits, the invariant manifolds of which do not share such global stability properties. Finally, we show that the invariant manifolds of eight-shaped Lissajous orbits (viewed in the Earth-Moon system) can be used to visit almost all the surface of the Moon.

Published online:
DOI: 10.5802/msia.5
Classification: 70F07,  37N05
Keywords: Lagrange points, Lissajous orbits, stability, mission design
@article{MSIA_2011__4_1_1_0,
     author = {Gr\'egory Archambeau and Philippe Augros and Emmanuel Tr\'elat},
     title = {Eight-shaped {Lissajous} orbits in the {Earth-Moon} system},
     journal = {MathematicS In Action},
     pages = {1--23},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {4},
     number = {1},
     year = {2011},
     doi = {10.5802/msia.5},
     mrnumber = {2777006},
     zbl = {1238.70023},
     language = {en},
     url = {https://msia.centre-mersenne.org/articles/10.5802/msia.5/}
}
TY  - JOUR
TI  - Eight-shaped Lissajous orbits in the Earth-Moon system
JO  - MathematicS In Action
PY  - 2011
DA  - 2011///
SP  - 1
EP  - 23
VL  - 4
IS  - 1
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://msia.centre-mersenne.org/articles/10.5802/msia.5/
UR  - https://www.ams.org/mathscinet-getitem?mr=2777006
UR  - https://zbmath.org/?q=an%3A1238.70023
UR  - https://doi.org/10.5802/msia.5
DO  - 10.5802/msia.5
LA  - en
ID  - MSIA_2011__4_1_1_0
ER  - 
%0 Journal Article
%T Eight-shaped Lissajous orbits in the Earth-Moon system
%J MathematicS In Action
%D 2011
%P 1-23
%V 4
%N 1
%I Société de Mathématiques Appliquées et Industrielles
%U https://doi.org/10.5802/msia.5
%R 10.5802/msia.5
%G en
%F MSIA_2011__4_1_1_0
Grégory Archambeau; Philippe Augros; Emmanuel Trélat. Eight-shaped Lissajous orbits in the Earth-Moon system. MathematicS In Action, Volume 4 (2011) no. 1, pp. 1-23. doi : 10.5802/msia.5. https://msia.centre-mersenne.org/articles/10.5802/msia.5/

[1] Abarbanel, H.D.I., Brown, R., Kennel, M.B.: Variation of Lyapunov exponents on a strange attractor. J. Nonlin. Sci. 1, 175–199 (1991) | Article | MR: 1118984 | Zbl: 0797.58053

[2] Anderson, R.L., Lo, M.W., Born, G.H.: Application of local Lyapunov exponents to maneuver design and navigation in the three-body problem. Paper presented at the AAS/AIAA conference, Big Sky, Montana, AAS 03-569, 3–7 August 2003, http://hdl.handle.net/2014/39457

[3] Arona, L. Masdemont, J.J.: Computation of heteroclinic orbits between normally hyperbolic invariant 3-spheres foliated by 2-dimensional invariant tori in Hill’s problem. Discrete Contin. Dyn. Syst. 2007, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., 64–74 (2007). | Zbl: 1163.70314

[4] Belbruno, E.A., Carrico J.P.: Calculation of weak stability boundary ballistic lunar transfer trajectories. Paper presented at the AAS/AIAA conference, Denver, Colorado, AIAA 2000-4142, August 2000 | Article

[5] Benettin, G., Galgani, L., Giorgilli, A., Strelcyn J.-M.: Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory, and Part 2: Numerical Application. Meccanica 15, 9–30 (1980) | Article | Zbl: 0488.70015

[6] Bonnard, B., Faubourg, L., Trélat, E.: Mécanique céleste et contrôle des véhicules spatiaux. Math. & Appl. 51, Springer Verlag (2006) | Article | Zbl: 1104.70001

[7] Breakwell, J.V., Brown, J.V.: The halo family of 3-dimensional of periodic orbits in the Earth-Moon restricted 3-body problem. Celestial Mechanics 20, 389–404 (1979) | Article | Zbl: 0415.70020

[8] Canalías, E.,Gómez, G., Marcote, M., Masdemont, J.J.: Assessment of mission design including utilization of libration points and weak stability boundaries. ESA publication (electronic). Ariadna id: 03/4103. ESA contract number 18142/04/NL/MV Final report (2004)

[9] Dellnitz, M., Padberg, K. , Post, M., Thiere, B.: Set oriented approximation of invariant manifolds: review of concepts for astrodynamical problems. In: New trends in astrodynamics and applications III, AIP Conference Proceedings 886, 90–99 (2007) | Article | Zbl: 1183.85002

[10] Dieci, L., Russell, R.D., Van Vleck, E.S.: On the computation of Lyapunov exponents for continuous dynamical systems. SIAM J. Numer. Anal. 34(1), 402–423 (1997) | Article | Zbl: 0891.65090

[11] Dunham, D.W., Farquhar, R.W.: Libration Points Missions. In: International Conference on Libration Points and Applications, Girona, Spain, 10-14 June 2002 | Article

[12] Eckhardt, B., Yao, D.M.: Local Lyapunov exponents in chaotic systems. Phys. D 65(1-2), 100–108 (1993) | Article | MR: 1221248 | Zbl: 0781.58024

[13] Euler, L.: De motu rectilineo trium corporum se mutuo attrahentium. Novi commentarii academiae scientarum Petropolitanae 11, 144–151 (1767). In: Oeuvres, Seria Secunda tome XXv Commentationes Astronomicae.

[14] Farquhar, R.W.: Station-keeping in the vicinity of collinear libration points with an application to a Lunar communications problem. Space Flight Mechanics, Science and Technology Series 11, 519–535 (1966)

[15] Farquhar, R.W.: The utilization of Halo orbits in advanced lunar operations. Technical Report X-551-70-449, Goddard Space Flight Center, Maryland (1970)

[16] Farquhar, R.W.: A halo-orbit lunar station. Astronautics & Aeronautics 10(6), 59–63 (1972)

[17] Farquhar, R.W., Dunham, D.W., Guo, Y., McAdams J.V.: Utilization of libration points for human exploration in the Sun-Earth-Moon system and beyond. Acta Astronautica 55(3-9), 687–700 (2004) | Article

[18] Gómez, G., Koon, W.S., Lo, M.W., Marsden, J.E., Masdemont, J., Ross, S.D.: Invariant manifolds, the spatial three-body problem and space mission design. Adv. Astronaut. Sci. 109, 3–22 (2001)

[19] Gómez, G., Koon, W.S., Lo, M.W., Marsden, J.E., Masdemont, J., Ross, S.D.: Connecting orbits and invariant manifolds in the spatial three-body problem. Nonlinearity 17, 1571–1606 (2004) | Article | MR: 2086140 | Zbl: 1115.70007

[20] Gómez, G., Masdemont, J., Simó, C.: Lissajous orbits around halo orbits. Adv. Astronaut. Sci. 95, 117–34 (1997)

[21] Gómez, G., Masdemont, J., Simó, C.: Quasihalo orbits associated with libration points. J. Astronaut. Sci. 46, 135–76 (1998)

[22] Gómez, G., Llibre, J., Martinez, R., Simó, C.: Dynamics and mission design near libration points, Volumes I, II (Fundamentals: The Case of Collinear Points; Advanced Methods for Collinear Points, resp.), World Scientific Monograph Series in Mathematics, Volumes I, II (2001) | Article | Zbl: 0971.70003

[23] Gómez, G., Jorba, A., R., Simó, C., Masdemont, J.J.: Dynamics and mission design near libration points. Volumes III, IV (Advanced methods for collinear points; Advanced methods for triangular points, resp.), World Scientific Monograph Series in Mathematics, Volumes III, IV (2001) | Article | Zbl: 0971.70002

[24] Howell, K.C., Pernicka, H.J.: Numerical determination of Lissajous orbits in the restricted three-body problem. Celestial Mechanics 41, 107–124 (1988) | Article | Zbl: 0648.70004

[25] Jorba, A., Masdemont, J.: Dynamics in the center manifold of the collinear points of the restricted three-body problem. Physica D 132, 189–213 (1999) | Article | MR: 1705705 | Zbl: 0942.70012

[26] Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Dynamical Systems, the three-body problem and space mission design. Springer (2008) | Article | Zbl: 0985.70007

[27] Lagrange, J.-L.: Essai sur le problème des trois corps. Prix de l’académie royale des Sciences de paris, tome IX (1772). In: Oeuvres de Lagrange 6, Gauthier-Villars, Paris, 272–282 (1873).

[28] Lo, M.W., Chung, M.J.: Lunar sample return via the interplanetary highway. Paper presented at the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Monterey, California, 5–8 August 2002, AIAA 2002-4718 | Article

[29] Lyapunov A.M.: The general problem of the stability of motion. Comm. Soc. Math. Kharkow (1892) (in Russian), reprinted in English, Taylor & Francis, London, 1992 | Article | MR: 1154209

[30] Marsden, J.E., Koon, W.S., Lo, M.W., Ross, S.D.: Low energy transfer to the Moon. Celestial Mechanics Dyn. Astron. 81, 27–38 (2001) | Article

[31] 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) | Article | MR: 2114685 | Zbl: 1093.70012

[32] Meyer, K.R., Hall, G.R.: Introduction to Hamiltonian dynamical systems and the N-body problem. Applied Math. Sci. 90, Springer-Verlag, New-York (1992) | Article | Zbl: 0743.70006

[33] Miller J.K., Belbruno E.A.: Sun-perturbated Earth-to-Moon transfers with ballistic capture. J. Guidance, Control Dynam. 16(4), 770–775 (1993) | Article

[34] Moser, J.: On the generalization of a theorem of A. Lyapunov. Commun. Pure Appl. Math. 11, 257–271 (1958) | Article | MR: 96021 | Zbl: 0082.08003

[35] Oseledec, V. I.: A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems. (Russian) Trudy Moskov. Mat. Obsc. 19, 179–210 (1968)

[36] Renk, F., Hechler, M., Messerschmid, E.: Exploration missions in the Sun-Earth-Moon system: a detailed view on selected transfer problems. Acta Astronautica, in Press (2009) | Article

[37] Richardson, D.L.: Analytic construction of periodic orbits about the collinear points. Celestial mechanics 22, 241–253 (1980) | Article | MR: 590496 | Zbl: 0465.34028

[38] Szebehely, V.G.: Theory of orbits: the restricted problem of three bodies. Academic Press, New-York (1967) | Article | Zbl: 1372.70004

[39] Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica 16D, 285–317 (1985) | Article | MR: 805706 | Zbl: 0585.58037

[40] Wolff, R.C.L.: Local Lyapunov exponents: looking closely at chaos. J. Roy. Statist. Soc. Ser. B 54(2), 353–371 (1992) | Article | MR: 1160475

[41] Yazdi, K., Messerschmid, E.: A lunar exploration architecture using lunar libration point one. Aerospace Science and Technology 12, 231–240 (2008) | Article | Zbl: 1273.70046

Cited by Sources: