Mission design through averaging of perturbed Keplerian systems: the paradigm of an Enceladus orbiter. (English) Zbl 1223.70047

Summary: Preliminary mission design for planetary satellite orbiters requires a deep knowledge of the long term dynamics that is typically obtained through averaging techniques. The problem is usually formulated in the Hamiltonian setting as a sum of the principal part, which is given through the Kepler problem, plus a small perturbation that depends on the specific features of the mission. It is usually derived from a scaling procedure of the restricted three body problem, since the two main bodies are the Sun and the planet whereas the satellite is considered as a massless particle. Sometimes, instead of the restricted three body problem, the spatial Hill problem is used. In some cases, the validity of the averaging is limited to prohibitively small regions, thus, depriving the analysis of significance. We find this paradigm at Enceladus, where the validity of a first order averaging based on the Hill problem lies inside the body. However, this fact does not invalidate the technique as perturbation methods are used to reach higher orders in the averaging process. Proceeding this way, we average the Hill problem up to the sixth order obtaining valuable information on the dynamics close to Enceladus. The averaging is performed through Lie transformations and two different transformations are applied. Firstly, the mean motion is normalized whereas the goal of the second transformation is to remove the appearance of the argument of the node. The resulting Hamiltonian defines a system of one degree of freedom whose dynamics is analyzed.


70F15 Celestial mechanics
70M20 Orbital mechanics
Full Text: DOI HAL


[1] Aiello, J.: Numerical Investigation of Mapping Orbits about Jupiter’s Icy Moons. Paper AAS 2005-377, Aug 2005
[2] Broucke R.A.: Stability of periodic orbits in the elliptic restricted three-body problem. AIAA J. 7(6), 1003–1009 (1969) · Zbl 0179.53301 · doi:10.2514/3.5267
[3] Broucke R.A.: Long-term third-body effects via double averaging. J. Guid. Control Dyn. 26(1), 27–32 (2003) · doi:10.2514/2.5041
[4] Campbell J.A., Jefferys W.H.: Equivalence of the perturbation theories of Hori and Deprit. Celest. Mech. 2(4), 467–473 (1970) · Zbl 0205.55005 · doi:10.1007/BF01625278
[5] Casotto, S., Padovani, S., Russell, R.P., Lara, M.: Detecting a Subsurface Ocean from Periodic Orbits at Enceladus, AGU 2008 Fall Meeting, 15–19 Dec 2008, San Francisco, CA
[6] Coffey S., Deprit A.: Third-order solution to the main problem in satellite theory. J. Guid. Control Dyn. 5(4), 366–371 (1982) · Zbl 0508.70010 · doi:10.2514/3.56183
[7] Coffey S., Deprit A., Deprit E.: Frozen orbits for satellites close to an earth-like planet. Celest. Mech. Dyn. Astron. 59(1), 37–72 (1994) · Zbl 0840.70020 · doi:10.1007/BF00691970
[8] Cushman R.: Reduction, Brouwer’s Hamiltonian and the critical inclination. Celest. Mech. 31, 409–429 (1983) Errata: id. 33, 297 (1984) · Zbl 0559.70026
[9] Cushman R.: Reduction, Brouwer’s Hamiltonian and the critical inclination. Errata 33, 297 (1984) · Zbl 0559.70026
[10] Deprit A.: Canonical transformations depending on a small parameter. Celest. Mech. 1(1), 12–30 (1969) · Zbl 0172.26002 · doi:10.1007/BF01230629
[11] Deprit A.: Delaunay normalisations. Celest. Mech. 26(1), 9–21 (1982) · Zbl 0512.70016 · doi:10.1007/BF01233178
[12] Deprit A., Rom A.: The main problem of artificial satellite theory for small and moderate eccentricities. Celest. Mech. 2(2), 166–206 (1970) · Zbl 0199.60101 · doi:10.1007/BF01229494
[13] Folta, D., Quinn, D.: Lunar Frozen Orbits, Paper AIAA 2006-6749, Aug 2006
[14] Gómez G., Marcote M., Mondelo J.M.: The invariant manifold structure of the spatial Hill’s problem. Dyn. Syst. Int. J. 20(1), 115–147 (2005) · Zbl 1062.70026
[15] Hamilton D.P., Krivov A.V.: Dynamics of distant moons of asteroids. Icarus 128(1), 241–249 (1997) · doi:10.1006/icar.1997.5738
[16] Hénon M.: Exploration Numérique du Problème Restreint. II. Masses Égales, Stabilité des Orbites Périodiques. Annales d’Astrophysique 28(2), 992–1007 (1965)
[17] Hénon M.: Numerical exploration of the restricted problem. V. Hill’s case: periodic orbits and their stability. Astron. Astrophys. 1, 223–238 (1969) · Zbl 0177.27703
[18] Hénon M.: Numerical exploration of the restricted problem. VI. Hill’s case: non-periodic orbits. Astron. Astrophys. 9, 24–36 (1970) · Zbl 0233.70007
[19] Hénon M.: Vertical stability of periodic orbits in the restricted problem II. Hill’s case. Astron. Astrophys. 30, 317–321 (1974) · Zbl 0359.70021
[20] Hill G.W.: Researches in the Lunar theory. Am. J. Math. 1, 129–147 (1878) · doi:10.2307/2369304
[21] Hori G.-I.: Theory of general perturbations with unspecified canonical variables. Publ. Astron. Soc. Jpn. 18(4), 287–296 (1966)
[22] Jefferys W.H.: A new class of periodic solutions of the three-dimensional restricted problem. Astron. J. 71(2), 99–102 (1966) · doi:10.1086/109862
[23] Kovalevsky J.: Sur la Théorie du Mouvement dun Satellite à Fortes Inclinaison et Excentricité. In: Kontopoulos, G.I. (eds) The Theory of Orbits in the Solar System and in Stellar Systems, IAU Symp. No. 25, pp. 326–344. Academic Press, London (1966)
[24] Kozai Y.: Secular perturbations of asteroids with high inclination and eccentricity. Astron. J. 67(9), 591–598 (1962) · doi:10.1086/108790
[25] Kozai Y.: Motion of a lunar orbiter. Publ. Astron. Soc. Jpn. 15(3), 301–312 (1963)
[26] Kozai Y.: Stationary and periodic solutions for restricted problem of three bodies in three-dimensional space. Publ. Astron. Soc. Jpn. 21(3), 267–287 (1969) · Zbl 0185.52301
[27] Lam, T., Whiffen, G.J.: Exploration of Distant Retrograde Orbits Around Europa, Paper AAS05-110, Jan 2005
[28] Lara M.: Simplified equations for computing science orbits around planetary satellites. J. Guid. Control Dyn. 31(1), 172–181 (2008) · doi:10.2514/1.31107
[29] Lara M., Palacián J.F.: Hill problem analytical theory to the order four: application to the computation of frozen orbits around planetary satellites. Math. Probl. Eng. 2009, 753653–753671 (2009) · Zbl 1188.70050
[30] Lara, M., Palacián, J.F., Russell, R.P.: Averaging and Mission Design: the Paradigm of an Enceladus Orbiter, Paper AAS09-199, Feb 2009, 20 pages. · Zbl 1223.70047
[31] Lara M., Russell R.P.: On the family g of the restricted three-body problem. Monografías de la Real Academia de Ciencias de Zaragoza 30, 51–66 (2007) · Zbl 1316.70009
[32] Lara M., Russell R.P., Villac V.: Classification of the distant stability regions at Europa. J. Guid. Control Dyn. 30(2), 409–418 (2007) · doi:10.2514/1.22372
[33] Lara M., San-Juan J.F.: Dynamic behavior of an orbiter around Europa. J Guid. Control Dyn. 28(2), 291–297 (2005) · doi:10.2514/1.5686
[34] Lara M., San-Juan J.F., Ferrer S.: Secular motion around triaxial, synchronously orbiting, planetary satellites: application to Europa. Chaos Interdisciplin. J. Nonlinear Sci. 15(4), 1–13 (2005) · Zbl 1144.37366
[35] Lara M., Scheeres D.J.: Stability bounds for three-dimensional motion close to asteroids. J. Astronaut. Sci. 50(4), 389–409 (2002)
[36] Lidov M.L.: The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies. Planet. Space Sci. 9(10), 719–759 (1962). Translated from Iskusstvennye Sputniki Zemli 8, 5–45 (1961) · doi:10.1016/0032-0633(62)90129-0
[37] 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. Kosmicheskie Issledovaniya 12, 155–170 (1974)
[38] Lo, M.W., Williams, B.G., Bollman, W.E., Han, D., Hahn, Y., Bell, J.L., Hirst, E.A., Corwin, R.A., Hong, P.E., Howell, K.C.: Genesis Mission Design, Paper AIAA-1998-4468, Aug 1998
[39] Mersman W.A.: A new algorithm for the Lie transformation. Celest. Mech. 3(1), 81–89 (1970) · Zbl 0222.70021 · doi:10.1007/BF01230434
[40] Orlov A.A.: Second-order short-period solar perturbations in the motion of the satellites of planets. Bull. Inst. Theo. Astron. 43(12), 302–309 (1970)
[41] Palacián J.F.: Dynamics of a satellite orbiting a planet with an inhomogeneous gravitational field. Celest. Mech. Dyn. Astron. 98(4), 219–249 (2007) · Zbl 1136.70316 · doi:10.1007/s10569-007-9078-5
[42] Palacián J.F., Yanguas P., Fernández S., Nicotra M.A.: Searching for periodic orbits of the spatial elliptic restricted three-body problem by double averaging. Physica D 213(1), 15–24 (2006) · Zbl 1089.70004 · doi:10.1016/j.physd.2005.10.009
[43] Paskowitz, M.E., Scheeres, D.J.: Orbit mechanics about planetary satellites including higher order gravity fields. Paper AAS 2005-190, Jan. 2005
[44] Russell R.P.: Global search for planar and three-dimensional periodic orbits near Europa. J. Astronaut. Sci. 54(2), 199–226 (2006) · doi:10.1007/BF03256483
[45] Russell R.P., Brinckerhoff A.T.: Circulating eccentric orbits around planetary moons. J. Guid. Control Dyn. 32(2), 423–435 (2009)
[46] Russell R.P., Lara M.: On the design of an Enceladus science orbit. Acta Astronautica 65(1–2), 27–39 (2009) · doi:10.1016/j.actaastro.2009.01.021
[47] San-Juan J.F., Lara M.: Normalizaciones de orden alto en el problema de Hill. Monografías de la Real Academia de Ciencias de Zaragoza 28, 23–32 (2006)
[48] 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) · doi:10.2514/2.4778
[49] Szebehely V.: Theory of Orbits. The restricted problem of three bodies. Academic Press, New York (1967) · Zbl 0158.43206
[50] Vashkov’yak M.A.: A numerical-analytical method for studying the orbital evolution of distant planetary satellites. Astron. Lett. 31(1), 64–72 (2005) · doi:10.1134/1.1854797
[51] Vashkov’yak M.A., Teslenko N.M.: Refined model for the evolution of distant satellite orbits. Astron. Lett. 35(12), 850–865 (2009) · doi:10.1134/S1063773709120056
[52] Villac, B., Lara, M.: Stability maps, global dynamics and transfers. AAS Paper 05-378, Aug 2005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.