Celestial Mechanics and Dynamical Astronomy

, Volume 128, Issue 4, pp 383–407 | Cite as

On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited

  • Alexandre Pousse
  • Philippe Robutel
  • Alain Vienne
Original Article


In the framework of the planar and circular restricted three-body problem, we consider an asteroid that orbits the Sun in quasi-satellite motion with a planet. A quasi-satellite trajectory is a heliocentric orbit in co-orbital resonance with the planet, characterized by a nonzero eccentricity and a resonant angle that librates around zero. Likewise, in the rotating frame with the planet, it describes the same trajectory as the one of a retrograde satellite even though the planet acts as a perturbator. In the last few years, the discoveries of asteroids in this type of motion made the term “quasi-satellite” more and more present in the literature. However, some authors rather use the term “retrograde satellite” when referring to this kind of motion in the studies of the restricted problem in the rotating frame. In this paper, we intend to clarify the terminology to use, in order to bridge the gap between the perturbative co-orbital point of view and the more general approach in the rotating frame. Through a numerical exploration of the co-orbital phase space, we describe the quasi-satellite domain and highlight that it is not reachable by low eccentricities by averaging process. We will show that the quasi-satellite domain is effectively included in the domain of the retrograde satellites and neatly defined in terms of frequencies. Eventually, we highlight a remarkable high eccentric quasi-satellite orbit corresponding to a frozen ellipse in the heliocentric frame. We extend this result to the eccentric case (planet on an eccentric motion) and show that two families of frozen ellipses originate from this remarkable orbit.


Restricted three-body problem Co-orbital motion Quasi-satellite Averaged Hamiltonian 



Rotating frame with the planet


Averaged problem


Reduced averaged problem


Retrograde satellite








“Satellized” retrograde satellite


Binary quasi-satellite


Heliocentric quasi-satellite

List of symbols

\(L_1\), \(L_2\), \(L_3\)

Circular Eulerian aligned configurations

\(L_4\), \(L_5\)

Circular Lagranian equilateral configurations

\({{{\mathscr {L}}}_{4}^{l}}\), \({{{\mathscr {L}}}_{5}^{l}}\)

In the RF, long period families that originate from \(L_4\) and \(L_5\).

\({{{\mathscr {L}}}_{3}}\), \({{{\mathscr {L}}}_{4}^{s}}\), \({{{\mathscr {L}}}_{5}^{s}}\)

In the RF, short period families that originate from \(L_3\), \(L_4\) and \(L_5\).

Family f

In the RF, one-parameter family of simple-periodic symmetrical retrograde satellite orbits that extends from an infinitesimal neighbourhood of the planet to the collision with the Sun. For \({\varepsilon }<0.0477\), it is stable but contains two particular orbits where the frequencies \(\nu \) and \(1-g\) are in 1 : 3 resonance. These two orbits decomposed the neighbourhood of the family f in three domains: sRS, QS\(_{b}~\)and QS\(_{h}~\).

1, \(\nu \), g

Frequencies, respectively, associated with the fast variations (the mean longitudes \(\lambda \) and \(\lambda '\)), the semi-fast component of the dynamics (oscillation of the resonant angle \(\theta \)) and the secular evolution of a trajectory (precession of the periaster argument \(\omega \)).

\({{{\mathcal {N}}}_{L_4}^{u}}\), \({{{\mathcal {N}}}_{L_5}^{u}}\)

In the RAP, the AP and the RF, families of \(2\pi /\nu \)-periodic orbits parametrized by \(|u|\le 0\) and that originate from \(L_4\) and \(L_5\). Moreover, they correspond to \({{{\mathscr {L}}}_{4}^{l}}\) and \({{{\mathscr {L}}}_{5}^{l}}\) in the RF.

\({{{\mathcal {G}}}_{L_3}^{e_0}}\), \({{{\mathcal {G}}}_{L_4}^{e_0}}\), \({{{\mathcal {G}}}_{L_5}^{e_0}}\)

In the RAP, families of fixed points parametrized by \(e_0\) and that originate from \(L_3\), \(L_4\) and \(L_5\). In the AP and the RF, these fixed points correspond to periodic orbits of frequency, respectively, g and \(1-g\). Moreover, they correspond to \({{{\mathscr {L}}}_{3}}\), \({{{\mathscr {L}}}_{4}^{s}}\) and \({{{\mathscr {L}}}_{5}^{s}}\) in the RF.

\({{{\mathcal {G}}}_{QS}^{e_0}}\)

In the RAP, family of fixed points parametrized by \(e_0\). In the AP and the RF, these fixed points correspond to periodic orbits of frequency, respectively, g and \(1-g\). Moreover, this family corresponds to a part of the family f that belongs to the QS\(_{h}~\)domain.

\({G_{L_3}}\), \({G_{L_4}}\), \({G_{L_5}}\), \({G_{QS}}\)

In the RAP, fixed points that belong to \({{{\mathcal {G}}}_{L_3}^{e_0}}\), \({{{\mathcal {G}}}_{L_4}^{e_0}}\), \({{{\mathcal {G}}}_{L_5}^{e_0}}\) and \({{{\mathcal {G}}}_{QS}^{e_0}}\) and characterized by \(g=0\). In the AP, sets of fixed points (also denoted as “circles of fixed points”) parametrized by \(\omega (t=0)\). In the RF, sets of \(2\pi \)-periodic orbits parametrized by \(\big (\lambda '-\omega \big )_{t=0}\).

\({G_{L_3,1}^{e'}}\), \({G_{L_3,2}^{e'}}\), \({G_{L_4,1}^{e'}}\), \({G_{L_4,2}^{e'}}\), \({G_{L_5,1}^{e'}}\), \({G_{L_5,2}^{e'}}\), \({G_{QS,1}^{e'}}\), \({G_{QS,2}^{e'}}\)

In the AP with \(e'\ge 0\), families of fixed points that originate from the circles of fixed points \({G_{L_3}}\), \({G_{L_4}}\), \({G_{L_5}}\), \({G_{QS}}\) when \(e'=0\).



This work has been developed during the Ph.D thesis of Alexandre Pousse at the “Astronomie et Systèmes Dynamiques”, IMCCE, Observatoire de Paris.


  1. Beaugé, C., Roig, F.: A semianalytical model for the motion of the Trojan asteroids: proper elements and families. Icarus 153, 391–415 (2001)ADSCrossRefGoogle Scholar
  2. Benest, D.: Effects of the mass ratio on the existence of retrograde satellites in the circular plane restricted problem. Astron. Astrophys. 32, 39–46 (1974)ADSGoogle Scholar
  3. Benest, D.: Effects of the mass ratio on the existence of retrograde satellites in the circular plane restricted problem. II. Astron. Astrophys. 45, 353–363 (1975)ADSGoogle Scholar
  4. Benest, D.: Effects of the mass ratio on the existence of retrograde satellites in the circular plane restricted problem. III. Astron. Astrophys. 53, 231–236 (1976)ADSGoogle Scholar
  5. Bien, R.: Long-period effects in the motion of Trojan asteroids and of fictitious objects at the 1/1 resonance. Astron. Astrophys. 68, 295–301 (1978)ADSzbMATHGoogle Scholar
  6. Brasser, R., Innanen, K., Connors, M., Veillet, C., Wiegert, P.A., Mikkola, S., Chodas, P.: Transient co-orbital asteroids. Icarus 171, 102–109 (2004)ADSCrossRefGoogle Scholar
  7. Broucke, R.A.: Periodic orbits in the restricted three-body problem with earth-moon masses. JPL Technical report, 32–1168 (1968)Google Scholar
  8. Broucke, R.A.: On relative periodic solutions of the planar general three-body problem. Celest. Mech. 12, 439–462 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. Connors, M., Chodas, P., Mikkola, S., Wiegert, P.A., Veillet, C., Innanen, K.: Discovery of an asteroid and quasi-satellite in an Earth-like horseshoe orbit. Meteorit. Planet. Sci. 37, 1435–1441 (2002)ADSCrossRefGoogle Scholar
  10. Connors, M., Veillet, C., Brasser, R., Wiegert, P.A., Chodas, P., Mikkola, S., Innanen, K.: Discovery of Earth’s quasi-satellite. Meteorit. Planet. Sci. 39, 1251–1255 (2004)ADSCrossRefGoogle Scholar
  11. Couetdic, J., Laskar, J., Correia, A.C.M., Mayor, M., Udry, S.: Dynamical stability analysis of the HD 202206 system and constraints to the planetary orbits. Astron. Astrophys. 519, A10 (2010)ADSCrossRefGoogle Scholar
  12. Danielsson, L., Ip, W.-H.: Capture resonance of the asteroid 1685 Toro by the Earth. Science 176, 906–907 (1972)ADSCrossRefGoogle Scholar
  13. de la Fuente Marcos, C., de la Fuente Marcos, R.: (309239) 2007 RW\(_{10}\): a large temporary quasi-satellite of Neptune. Astron. Astrophys. 545, L9 (2012)Google Scholar
  14. de la Fuente Marcos, C., de la Fuente Marcos, R.: Asteroid 2014 \(\text{ OL }_{339}\): yet another Earth quasi-satellite. Mon. Not. R. Astron. Soc. 445, 2985–2994 (2014)ADSCrossRefGoogle Scholar
  15. Deprit, A., Henrard, J., Palmore, J., Price, J.F.: The Trojan manifold in the system Earth–Moon. Mon. Not. R. Astron. Soc. 137, 311–335 (1967)ADSCrossRefzbMATHGoogle Scholar
  16. Edelman, C.: Construction of periodic orbits and capture problems. Astron. Astrophys. 145, 454–460 (1985)ADSGoogle Scholar
  17. Gallardo, T.: Atlas of the mean motion resonances in the Solar System. Icarus 184, 29–38 (2006)ADSCrossRefGoogle Scholar
  18. Giuppone, C.A., Beaugé, C., Michtchenko, T.A., Ferraz-Mello, S.: Dynamics of two planets in co-orbital motion. Mon. Not. R. Astron. Soc. 407, 390–398 (2010)ADSCrossRefGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. Hadjidemetriou, J.D., Voyatzis, G.: The 1/1 resonance in extrasolar systems. Migration from planetary to satellite orbits. Celest. Mech. Dyn. Astron. 111, 179–199 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. Henon, M.: Exploration numérique du problème restreint. I. Masses égales; orbites périodiques. Annales d’Astrophysique 28, 499 (1965a)ADSzbMATHGoogle Scholar
  22. Henon, M.: Exploration numérique du problème restreint. II. Masses égales, stabilité des orbites périodiques. Annales d’Astrophysique 28, 992 (1965b)ADSzbMATHGoogle Scholar
  23. Henon, M.: Numerical exploration of the restricted problem, V. Astron. Astrophys. 1, 223–238 (1969)ADSzbMATHGoogle Scholar
  24. Henon, M., Guyot, M.: Stability of periodic orbits in the restricted problem. In: Giacaglia, G.E.O. (ed.) Periodic Orbits, Stability and Resonances, Reidel, Dordrecht-Holland, 349–374 (1970)Google Scholar
  25. Jackson, J.: Retrograde satellite orbits. Mon. Not. R. Astron. Soc. 74, 62–82 (1913)ADSCrossRefzbMATHGoogle Scholar
  26. Kinoshita, H., Nakai, H.: Quasi-satellites of Jupiter. Celest. Mech. Dyn. Astron. 98, 181–189 (2007)ADSCrossRefzbMATHGoogle Scholar
  27. Kogan, A.I.: Quasisatellite orbits and their applications. In: Jehn, R. (ed.) Proceedings of the 41st Congress of the International Astronautical Federation, 90–307 (1990)Google Scholar
  28. Kortenkamp, S.J.: An efficient, low-velocity, resonant mechanism for capture of satellites by a protoplanet. Icarus 175, 409–418 (2005)ADSCrossRefGoogle Scholar
  29. Kortenkamp, S.J.: Trapping and dynamical evolution of interplanetary dust particles in Earth’s quasi-satellite resonance. Icarus 226, 1550–1558 (2013)ADSCrossRefGoogle Scholar
  30. Lidov, M.L., Vashkov’yak, M.A.: Theory of perturbations and analysis of the evolution of quasi-satellite orbits in the restricted three-body problem. Kosmicheskie Issledovaniia 31, 75–99 (1993)ADSGoogle Scholar
  31. Lidov, M.L., Vashkov’yak, M.A.: On quasi-satellite orbits for experiments on refinement of the gravitation constant. Astron. Lett. 20, 188–198 (1994a)ADSGoogle Scholar
  32. Lidov, M.L., Vashkov’yak, M.A.: On quasi-satellite orbits in a restricted elliptic three-body problem. Astron. Lett. 20, 676–690 (1994b)ADSGoogle Scholar
  33. Meyer, K.R., Hall, G.R.: Introduction to Hamiltonian dynamical systems and the N-body problem, vol. 90 of AMS. Springer, New York (1992)Google Scholar
  34. Mikkola, S., Innanen, K.: Orbital stability of planetary quasi-satellites. In: Dvorak, R. & Henrard, J. (eds.) The Dynamical Behaviour of our Planetary System. Kluwer Academic Publishers, 345–355 (1997)Google Scholar
  35. Mikkola, S., Brasser, R., Wiegert, P.A., Innanen, K.: Asteroid 2002 VE68, a quasi-satellite of Venus. Mon. Not. R. Astron. Soc. 351, L63–L65 (2004)ADSCrossRefGoogle Scholar
  36. Mikkola, S., Innanen, K., Wiegert, P.A., Connors, M., Brasser, R.: Stability limits for the quasi-satellite orbit. Mon. Not. R. Astron. Soc. 369, 15–24 (2006)ADSCrossRefGoogle Scholar
  37. Moeller, J.P.: Zwei Bahnklassen im probleme restreint. Publikationer og mindre Meddeler fra Kobenhavns Observatorium 99, 1–II (1935)Google Scholar
  38. Morais, M.H.M.: Hamiltonian formulation of the secular theory for Trojan-type motion. Astron. Astrophys. 369, 677–689 (2001)ADSCrossRefzbMATHGoogle Scholar
  39. Namouni, F.: Secular interactions of coorbiting objects. Icarus 137, 293–314 (1999)ADSCrossRefGoogle Scholar
  40. Namouni, F., Christou, A., Murray, C.: New coorbital dynamics in the solar system. Phys. Rev. Let. 83, 2506–2509 (1999)Google Scholar
  41. Nesvorný, D., Thomas, F., Ferraz-Mello, S., Morbidelli, A.: A perturbative treatment of the co-orbital motion. Celest. Mech. Dyn. Astron. 82, 323–361 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. Robutel, P., Gabern, F.: The resonant structure of Jupiter’s Trojan asteroids-I. Long-term stability and diffusion. Mon. Not. R. Astron. Soc. 372, 1463–1482 (2006)ADSCrossRefGoogle Scholar
  43. Robutel, P., Pousse, A.: On the co-orbital motion of two planets in quasi-circular orbits. Celest. Mech. Dyn. Astron. 117, 17–40 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. Robutel, P., Niederman, L., Pousse, A.: Rigorous treatment of the averaging process for co-orbital motions in the planetary problem. Comp. Appl. Math. 35, 675–699 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  45. Sidorenko, V.V., Neishtadt, A.I., Artemyev, A.V., Zelenyi, L.M.: Quasi-satellite orbits in the general context of dynamics in the 1:1 mean motion resonance: perturbative treatment. Celest. Mech. Dyn. Astron. 120, 131–162 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. Strömgren, E.: Connaisance actuelle des orbites dans le problème des trois corps. Bull. Astron. 9, 87–130 (1933)ADSzbMATHGoogle Scholar
  47. Wajer, P.: 2002 \(\text{ AA }_{29}\): Earth’s recurrent quasi-satellite? Icarus 200, 147–153 (2009)ADSCrossRefGoogle Scholar
  48. Wajer, P.: Dynamical evolution of Earth’s quasi-satellites: 2004 \(\text{ GU }_{9}\) and 2006 \(\text{ FV }_{35}\). Icarus 209, 488–493 (2010)ADSCrossRefGoogle Scholar
  49. Wajer, P., Królikowska, M.: Behavior of Jupiter non-Trojan co-orbitals. Acta Astron. 62, 113–131 (2012)ADSGoogle Scholar
  50. Wiegert, P., Innanen, K., Mikkola, S.: The stability of quasi satellites in the outer Solar System. Astron. J. 119, 1978–1984 (2000)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Alexandre Pousse
    • 1
    • 2
  • Philippe Robutel
    • 2
  • Alain Vienne
    • 2
  1. 1.Dipartimento di Matematica Ed Applicazioni “R.Caccioppoli”Università di Napoli “Federico II”NapoliItaly
  2. 2.IMCCE, Observatoire de ParisPSL Research Univ., UPMC Paris 6, Univ. Lille 1, CNRSParisFrance

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