Celestial Mechanics and Dynamical Astronomy

, Volume 112, Issue 2, pp 191–219 | Cite as

Low-thrust propulsion in a coplanar circular restricted four body problem

  • Marta Ceccaroni
  • James Biggs
Original Article


This paper formulates a circular restricted four body problem (CRFBP), where the three primaries are set in the stable Lagrangian equilateral triangle configuration and the fourth body is massless. The analysis of this autonomous coplanar CRFBP is undertaken, which identifies eight natural equilibria; four of which are close to the smaller body, two stable and two unstable, when considering the primaries to be the Sun and two smaller bodies of the Solar System. Following this, the model incorporates ‘near term’ low-thrust propulsion capabilities to generate surfaces of artificial equilibrium points close to the smaller primary, both in and out of the plane containing the celestial bodies. A stability analysis of these points is carried out and a stable subset of them is identified. Throughout the analysis the Sun-Jupiter-asteroid-spacecraft system is used, for conceivable masses of a hypothetical asteroid set at the libration point L 4. It is shown that eight bounded orbits exist, which can be maintained with a constant thrust less than 1.5 × 10−4 N for a 1000 kg spacecraft. This illustrates that, by exploiting low-thrust technologies, it would be possible to maintain an observation point more than 66% closer to the asteroid than that of a stable natural equilibrium point. The analysis then focusses on a major Jupiter Trojan: the (624) Hektor asteroid. The thrust required to enable close asteroid observation is determined in the simplified CRFBP model. Finally, a numerical simulation of the real Sun-Jupiter-(624) Hektor-spacecraft is undertaken, which tests the validity of the stability analysis of the simplified model.


Restricted problems Stability Periodic orbits Sun-Jupiter-asteroid-spacecraft system 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

10569_2011_9391_MOESM1_ESM.tex (7 kb)
ESM 1 (TEX 8 kb)


  1. Alvarez-Ramirez, M., Vidal, C.: Dynamical aspects of an equilateral restricted four-body problem. Math. Probl. Eng. 2009, 1–23 (2009)Google Scholar
  2. Ambrosetti A., Prodi G.: A Primer of Nonlinear Analysis, pp. 153–159. Cambridge University Press, Cambridge (1993)Google Scholar
  3. Andreu, M.A.: New results on computation of translunar Halo orbits of the real Earth-Moon system. In: Proceedings of the Conference Libration point orbits and applications, Aiguablava, pp. 225–237 (2002)Google Scholar
  4. Arnold V.A., Kozolov V.V, Neishtadt A.I.: Mathematical Aspects of Classical and Celestial Mechanics. (Dynamical systems. III). Springer, Berlin (2006)Google Scholar
  5. Artin M.: Algebra, pp. 543–547. Prentice Hall, New Jersey (1991)Google Scholar
  6. Baig S., McInnes C.R.: Artificial three body equilibria for hybrid low-thrust propulsion. J. Guid. Contr. Dyn. 31(6), 1644–1655 (2008)CrossRefGoogle Scholar
  7. Baig S., McInnes C.R.: Artificial halo orbits for low-thrust propulsion spacecraft. Celest. Mech. Dyn. Astron. 104(4), 321–335 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  8. Baltagiannis A.N., Papadakis K.E.: Equilibrium points and their stability in the restricted four body problem. Int. J. Bifurcat. Chaos. 21(8), 2179–2193 (2011)CrossRefGoogle Scholar
  9. Bombardelli C., Pelaez J.: On the stability of artificial equilibrium points in the circular restricted three-body problem. Celest. Mech. Dyn. Astron. 109(1), 13–26 (2011)MathSciNetADSCrossRefGoogle Scholar
  10. Ceccaroni, M., Biggs, J.: Extension of low-thrust propulsion to the Autonomous Coplanar Circular Restricted Four Body Problem with application to future Trojan Asteroid missions. In: 61st International Astronautical Congress, IAC-10-1.1.3, Prague (2010)Google Scholar
  11. Cronin J., Richards P.B., Russell L.H.: Some periodic solutions of a four-body problem. Icarus 3, 423 (1964)MathSciNetADSCrossRefGoogle Scholar
  12. Dvorak, R., Schwarz, R., Lhotka, C.: On the dynamics of Trojan planets in extrasolar planetary systems. In: Proceedings of the IAU (International Astronomical Union) symposium, N. 249 (2008)Google Scholar
  13. Erdi B., Forgács-Dajka E., Nagy I., Rajnai R.: A parametric study of stability and resonances around L4 in the elliptic restricted three-body problem. Celest. Mech. Dyn. Astron. 104, 145–158 (2009)ADSMATHCrossRefGoogle Scholar
  14. Fearn, D.G., Crookham, C.: The development of ion propulsion in the UK: a historical prespective. In: Proceedings of the 29th IEPC, Princeton (2005)Google Scholar
  15. Gascheau M.: Examen d’une classe d’equations differentielles et application a un cas particulier du probleme des trois corps. C. R. Acad. Sci. 16, 343 (1843)Google Scholar
  16. Gomez, G., Jorba, A., Masdemont, J., Simó, C.: Dynamics and mission design near libration points vol. 4: advanced methods for triangular points. In: World Scientific Monograph Series in Mathematics, vol. 5, pp. 249–253 (2001)Google Scholar
  17. Hadjidemetriou, J.D.: Instabilities in planetary-type orbits: applications to celestial mechanics. Instabilities in Dynamical Systems, Proceedings of the Advanced Study Institute, Cortina d’Ampezzo, pp. 135–163 (1978)Google Scholar
  18. Hadjidemetriou J.D.: On periodic orbits and resonance in extrasolar planetary systems. Celest. Mech. Dyn. Astron. 102, 69–82 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  19. Hadjidemetriou J.D., Psychoyos D., Voyatzis G.: The 1/1 resonance in extrasolar planetary systems. Celest. Mech. Dyn. Astron. 104, 23–38 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  20. Hou X.Y., Liu L.: On quasi-periodic motions around the triangular libration points of the real EarthMoon system. Celest. Mech. Dyn. Astron. 108(3), 301–313 (2010)MathSciNetADSMATHCrossRefGoogle Scholar
  21. Kloppenborg B., Stencel R., Monnier J.D., Schaefer G., Zhao M., Baron F. et al.: Infrared images of the transiting disk in the \({\epsilon}\) Aurigae system. Nature Lett. 464, 870–872 (2010)ADSCrossRefGoogle Scholar
  22. Koon W.S., Lo M., Marsden J.E., Ross S.: Dynamical Systems, The Three-Body Problem and Space Mission Design, pp. 123–130. Marsden Books, London (2008)Google Scholar
  23. Marchal C.: Long term evolution of quasi-circular Trojan orbits. Celest. Mech. Dyn. Astron. 104, 53–67 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  24. Marzari, F., Scholl, H., Murray, C., Lagerkvist, C.: Origin and evolution of Trojan asteroids, vol. 1, pp. 725–738. Asteroids III University of Arizona Press (2002)Google Scholar
  25. Marzari F.: Puzzling Neptune Trojans. Science 313(5786), 451–452 (2006)CrossRefGoogle Scholar
  26. McInnes C.R., McDonald A.J., John F.L., MacDonald E.W.: Solar sail parking in restricted three-body systems. J. Guid. Contr. Dyn. 17(2), 399–406 (1994)ADSMATHCrossRefGoogle Scholar
  27. McKay, R., Macdonald, M., Bosquillon de Frescheville, F., Vasile, M., McInnes, C.R., Biggs, J.: Non-Keplerian orbits using low thrust, high ISP propulsion systems. In: 60th International Astronautical Congress, IAC-09-1.2.8, Daejeon (2009)Google Scholar
  28. McKay R., Macdonald M., Biggs J., McInnes C.R.: Survey of highly non-Keplerian orbits with low-thrust propulsion. J. Guid. Contr. Dyn. 34(3), 645–666 (2011)CrossRefGoogle Scholar
  29. Melita M.D., Licandro J., Jones D.C., Williams I.P.: Physical properties and orbital stability of the Trojan asteroids. Icarus 195, 686–697 (2008)ADSCrossRefGoogle Scholar
  30. Michalodimitrakis M.: The circular restricted four body problem. Astrophys. Space Sci. 75(2), 289–305 (1981)ADSCrossRefGoogle Scholar
  31. Milani A., Nobili A.M.: On the stability of hierarchical four body systems. Celest. Mech. 31(3), 241–291 (1983)MathSciNetADSMATHCrossRefGoogle Scholar
  32. Morimoto K., Yamakawa M.Y., Uesugi H.: Periodic orbits with low-thrust propulsion in the restricted three-body problem. J. Guid. Contr. Dyn. 29(5), 1131–1139 (2006)CrossRefGoogle Scholar
  33. Morimoto K., Yamakawa M.Y., Uesugi H.: Artificial equilibrium points in the low-thrust restricted three-body problem. J. Guid. Contr. Dyn. 30(5), 1563–1568 (2007)CrossRefGoogle Scholar
  34. Moulton F.R.: On a class of particular solutions of the problem of four bodies. Trans. Am. Math. Soc. 1, 17–29 (1900)MathSciNetMATHCrossRefGoogle Scholar
  35. Moulton F.R.: The straight line solutions of the problem of N bodies. Ann. Math. 12(3), 1–17 (1910)MathSciNetMATHCrossRefGoogle Scholar
  36. Nicolini, D.: LISA pathfinder field emission thruster system development program. In: Proceedings of the 30th International Electric Propulsion Conference, Florence (2007)Google Scholar
  37. Papadakis K.E.: Asymptotic orbits in the restricted four body problem. Planet. Space Sci. 55(10), 1368–1379 (2007)ADSCrossRefGoogle Scholar
  38. Piña E., Lonngi P.: Central configurations for the planar Newtonian four-body problem. Celest. Mech. Dyn. Astron. 108(1), 73–93 (2010)ADSMATHCrossRefGoogle Scholar
  39. Rivkin, A.S., Emery, J., Barucci, A., Bell, J.F., Bottke, W.F., Dotto, E., Gold, R., Lisse, C., Licandro, J., Prockter, L., Hibbits, C., Paul, M., Springmann, A., Yang, B.: The Trojan asteroids: keys to many locks, national academies of science in support of the national academies planetary science decadal survey 2013–2022, SBAG Community White Papers (2009)Google Scholar
  40. Routh E.J.: On Laplace’s three particles, with a supplement on the stability of steady motion. Proc. Lond. Math. Soc. 6, 86–97 (1875)MATHCrossRefGoogle Scholar
  41. Roy, A.E., Walker, I.W., MacDonald, A.J.C.: Studies on the stability of hierarchical dynamical systems. In: Proceedings of the Stability of the Solar System and Its Minor Natural and Artificial Bodies, pp. 151–174. Advanced Study Institute, Cortina d’Ampezzo (1985)Google Scholar
  42. Scheeres D.J.: The restricted Hill four-body problem with applications to the earth moon sun system. Celest. Mech. Dyn. Astron. 70(2), 75–98 (1998)MathSciNetADSMATHCrossRefGoogle Scholar
  43. Schwarz R., Suli A., Dvorak R.: Dynamics of possible Trojan planets in binary systems. Mon. Not. R. Astron. Soc. 398, 2085–2090 (2009)ADSCrossRefGoogle Scholar
  44. Schwarz R., Süli Á., Dvorak R., Pilat-Lohinger E.: Stability of Trojan planets in multi-planetary systems Stability of Trojan planets in different dynamical systems. Celest. Mech. Dyn. Astron. 104(1–2), 69–84 (2009)ADSMATHCrossRefGoogle Scholar
  45. Shoemaker, E.M., Shoemaker, C.S., Wolfe, R.F.: Trojan asteroids: populations, dynamical structure and origin of the L4 and L5 swarms. Asteroids II, Proceedings of the Conference, Tucson (1988)Google Scholar
  46. Simó C.: Relative equilibrium solutions in the four body problem. Celest. Mech. 18(2), 165–184 (1978)ADSMATHCrossRefGoogle Scholar
  47. Steves B.A., Roy A.E., Bell M.: Some special restricted four-body problems—I. Modelling the Caledonian problem. Planet. Space Sci. 46(11–12), 1465–1474 (1998)ADSCrossRefGoogle Scholar
  48. Steves B.A., Roy A.E., Bell M.: Some special solutions of the four body problem—II. From Caledonia to Copenaghen. Planet. Space Sci. 46(11–12), 1475–1486 (1998)ADSGoogle Scholar
  49. Van Hamme W., Wilson R.E.: The restricted four-body problem and epsilon Aurigae. Astroph. J. 306, 33–36 (1986)ADSCrossRefGoogle Scholar
  50. Waters T., McInnes C.R.: Periodic orbits above the ecliptic plane in the solar sail restricted 3-body problem. J. Guid. Contr. Dyn. 30(3), 687–693 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Advanced Space Concepts LaboratoryUniversity of StrathclydeGlasgowUK

Personalised recommendations