Equilibrium points and associated periodic orbits in the gravity of binary asteroid systems: (66391) 1999 KW4 as an example

  • Yu Shi
  • Yue Wang
  • Shijie Xu
Original Article


The motion of a massless particle in the gravity of a binary asteroid system, referred as the restricted full three-body problem (RF3BP), is fundamental, not only for the evolution of the binary system, but also for the design of relevant space missions. In this paper, equilibrium points and associated periodic orbit families in the gravity of a binary system are investigated, with the binary (66391) 1999 KW4 as an example. The polyhedron shape model is used to describe irregular shapes and corresponding gravity fields of the primary and secondary of (66391) 1999 KW4, which is more accurate than the ellipsoid shape model in previous studies and provides a high-fidelity representation of the gravitational environment. Both of the synchronous and non-synchronous states of the binary system are considered. For the synchronous binary system, the equilibrium points and their stability are determined, and periodic orbit families emanating from each equilibrium point are generated by using the shooting (multiple shooting) method and the homotopy method, where the homotopy function connects the circular restricted three-body problem and RF3BP. In the non-synchronous binary system, trajectories of equivalent equilibrium points are calculated, and the associated periodic orbits are obtained by using the homotopy method, where the homotopy function connects the synchronous and non-synchronous systems. Although only the binary (66391) 1999 KW4 is considered, our methods will also be well applicable to other binary systems with polyhedron shape data. Our results on equilibrium points and associated periodic orbits provide general insights into the dynamical environment and orbital behaviors in proximity of small binary asteroids and enable the trajectory design and mission operations in future binary system explorations.


Binary asteroids Restricted full three-body problem Equilibrium points Periodic orbit families Homotopy method 



This work has been supported by the National Natural Science Foundation of China under Grants 11432001 and 11602009, the Young Elite Scientist Sponsorship Program by China Association for Science and Technology, and the Fundamental Research Funds for the Central Universities.

Compliance with ethical standards

Conflict of interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the paper submitted.


  1. Bellerose, J.: The restricted full three body problem: applications to binary asteroid exploration. PhD thesis, University of Michigan (2008)Google Scholar
  2. Bellerose, J., Scheeres, D.J.: Periodic orbits in the vicinity of the equilateral points of the restricted full three-body problem. In: AAS/AIAA Conference, Lake Tahoe, California, AAS-05-295, vol. 711 (2005)Google Scholar
  3. Bellerose, J., Scheeres, D.: General dynamics in the restricted full three body problem. Acta Astronaut. 62(10), 563–576 (2008a)ADSCrossRefzbMATHGoogle Scholar
  4. Bellerose, J., Scheeres, D.J.: Restricted full three-body problem: application to binary system 1999 KW4. J. Guid. Control Dyn. 31(1), 162–171 (2008b)ADSCrossRefGoogle Scholar
  5. Chanut, T., Winter, O., Amarante, A., Araújo, N.: 3d plausible orbital stability close to asteroid (216) Kleopatra. Mon. Not. R. Astron. Soc. 452(2), 1316–1327 (2015)ADSCrossRefGoogle Scholar
  6. Chapman, C., Veverka, J., Thomas, P., Klaasen, K., et al.: Discovery and physical properties of Dactyl, a satellite of asteroid 243 Ida. Nature 374(6525), 783 (1995)ADSCrossRefGoogle Scholar
  7. Chappaz, L., Howell, K.: Trajectory exploration within binary systems comprised of small irregular bodies. In: 23rd AAS/AIAA Space Flight Mechanics Meeting, Kauai, Hawaii (2013)Google Scholar
  8. Chappaz, L., Howell, K.C.: Exploration of bounded motion near binary systems comprised of small irregular bodies. Celest. Mech. Dyn. Astron. 123(2), 123–149 (2015)ADSMathSciNetCrossRefGoogle Scholar
  9. Ćuk, M., Nesvornỳ, D.: Orbital evolution of small binary asteroids. Icarus 207(2), 732–743 (2010)ADSCrossRefGoogle Scholar
  10. Dichmann, D., Doedel, E., Paffenroth, R.: The computation of periodic solutions of the 3-body problem using the numerical continuation software auto. Libration Point Orbits and Applications, pp. 429–488 (2003)Google Scholar
  11. Doedel, E., Keller, H.B., Kernevez, J.P.: Numerical analysis and control of bifurcation problems (i): bifurcation in finite dimensions. Int. J. Bifurc. Chaos 1(03), 493–520 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Doedel, E.J., Romanov, V.A., Paffenroth, R.C., Keller, H.B., Dichmann, D.J., Galán-Vioque, J., et al.: Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem. Int. J. Bifurc. Chaos 17(08), 2625–2677 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Fahnestock, E.G.: The full two-body-problem: simulation, analysis, and application to the dynamics, characteristics, and evolution of binary asteroid systems. PhD thesis, University of Michigan (2009)Google Scholar
  14. Fahnestock, E.G., Scheeres, D.J.: Simulation and analysis of the dynamics of binary near-earth asteroid (66391) 1999 KW4. Icarus 194(2), 410–435 (2008)ADSCrossRefGoogle Scholar
  15. Gómez, G., Mondelo, J.M.: The dynamics around the collinear equilibrium points of the RTBP. Physica D 157(4), 283–321 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. Hénon, M.: Generating Families in the Restricted Three-Body Problem, vol. 52. Springer, Berlin (2003)zbMATHGoogle Scholar
  17. Hou, X., Liu, L.: Bifurcating families around collinear libration points. Celest. Mech. Dyn. Astron. 116(3), 241–263 (2013)ADSMathSciNetCrossRefGoogle Scholar
  18. Howell, K.C.: Three-dimensional, periodic, ‘halo’ orbits. Celest. Mech. 32(1), 53–71 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. Jacobson, S.A., Scheeres, D.J.: Dynamics of rotationally fissioned asteroids: source of observed small asteroid systems. Icarus 214(1), 161–178 (2011a)ADSCrossRefGoogle Scholar
  20. Jacobson, S.A., Scheeres, D.J.: Long-term stable equilibria for synchronous binary asteroids. Astrophys. J. Lett. 736(1), L19 (2011b)ADSCrossRefGoogle Scholar
  21. Jacobson, S.A., Scheeres, D.J., McMahon, J.: Formation of the wide asynchronous binary asteroid population. Astrophys. J. 780(1), 60 (2013)ADSCrossRefGoogle Scholar
  22. Jiang, Y., Baoyin, H.: Periodic orbit families in the gravitational field of irregular-shaped bodies. Astron. J. 152(5), 137 (2016)ADSCrossRefGoogle Scholar
  23. Jiang, Y., Baoyin, H., Li, H.: Periodic motion near the surface of asteroids. Astrophys. Space Sci. 360(2), 63 (2015a)ADSCrossRefGoogle Scholar
  24. Jiang, Y., Yu, Y., Baoyin, H.: Topological classifications and bifurcations of periodic orbits in the potential field of highly irregular-shaped celestial bodies. Nonlinear Dyn. 81(1–2), 119–140 (2015b)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Dynamical Systems, the Three-Body Problem and Space Mission Design. Marsden Books, Wellington (2008)zbMATHGoogle Scholar
  26. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112. Springer, Berlin (2013)Google Scholar
  27. Liu, X., Baoyin, H., Ma, X.: Periodic orbits in the gravity field of a fixed homogeneous cube. Astrophys. Space Sci. 334(2), 357–364 (2011)ADSCrossRefzbMATHGoogle Scholar
  28. Margot, J.L., Nolan, M., Benner, L., Ostro, S., Jurgens, R., Giorgini, J., et al.: Binary asteroids in the near-earth object population. Science 296(5572), 1445–1448 (2002)ADSCrossRefGoogle Scholar
  29. McMahon, J., Scheeres, D.: Detailed prediction for the BYORP effect on binary near-earth asteroid (66391) 1999 KW4 and implications for the binary population. Icarus 209(2), 494–509 (2010a)ADSCrossRefGoogle Scholar
  30. McMahon, J., Scheeres, D.: Secular orbit variation due to solar radiation effects: a detailed model for BYORP. Celest. Mech. Dyn. Astron. 106(3), 261–300 (2010b)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. Ni, Y., Jiang, Y., Baoyin, H.: Multiple bifurcations in the periodic orbit around Eros. Astrophys. Space Sci. 361(5), 170 (2016)ADSMathSciNetCrossRefGoogle Scholar
  32. Ostro, S.J., Margot, J.L., Benner, L.A., Giorgini, J.D., Scheeres, D.J., Fahnestock, E.G., et al.: Radar imaging of binary near-earth asteroid (66391) 1999 KW4. Science 314(5803), 1276–1280 (2006)ADSCrossRefGoogle Scholar
  33. Peng, H., Xu, S.: Stability of two groups of multi-revolution elliptic halo orbits in the elliptic restricted three-body problem. Celest. Mech. Dyn. Astron. 123(3), 279–303 (2015)ADSMathSciNetCrossRefGoogle Scholar
  34. Pravec, P., Harris, A.W.: Binary asteroid population: 1. Angular momentum content. Icarus 190(1), 250–259 (2007)ADSCrossRefGoogle Scholar
  35. Scheeres, D.J.: Orbital Motion in Strongly Perturbed Environments: Applications to Asteroid, Comet and Planetary Satellite Orbiters. Springer, Berlin (2016)Google Scholar
  36. Scheeres, D., Williams, B., Miller, J.: Evaluation of the dynamic environment of an asteroid: applications to 433 Eros. J. Guid. Control Dyn. 23(3), 466–475 (2000)ADSCrossRefGoogle Scholar
  37. Scheeres, D.J., Fahnestock, E.G., Ostro, S.J., Margot, J.L., Benner, L.A., Broschart, S.B., et al.: Dynamical configuration of binary near-earth asteroid (66391) 1999 KW4. Science 314(5803), 1280–1283 (2006)ADSCrossRefGoogle Scholar
  38. Scheeres, D., Van Wal, S., Olikara, Z., Baresi, N.: The dynamical environment for the exploration of Phobos, ists-2017-d-007. International Symposium on Space Technology and Science. Ehime, Japan, pp. 3–9 (2017)Google Scholar
  39. Szebehely, V.: Theory of Orbit: The Restricted Problem of Three Bodies. Elsevier, Amsterdam (2012)zbMATHGoogle Scholar
  40. Vaquero, M., Howell, K.C.: Design of transfer trajectories between resonant orbits in the Earth–Moon restricted problem. Acta Astronaut. 94(1), 302–317 (2014)ADSCrossRefGoogle Scholar
  41. Werner, R.A., Scheeres, D.J.: Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia. Celest. Mech. Dyn. Astron. 65(3), 313–344 (1996)ADSzbMATHGoogle Scholar
  42. Woo, P., Misra, A.K.: Equilibrium points in the full three-body problem. Acta Astronaut. 99, 158–165 (2014)ADSCrossRefGoogle Scholar
  43. Woo, P., Misra, A.K.: Bounded trajectories of a spacecraft near an equilibrium point of a binary asteroid system. Acta Astronaut. 110, 313–323 (2015)ADSCrossRefGoogle Scholar
  44. Yu, Y., Baoyin, H.: Generating families of 3d periodic orbits about asteroids. Mon. Not. R. Astron. Soc. 427(1), 872–881 (2012)ADSCrossRefGoogle Scholar
  45. Yu, Y., Baoyin, H., Jiang, Y.: Constructing the natural families of periodic orbits near irregular bodies. Mon. Not. R. Astron. Soc. 453(3), 3269–3277 (2015)ADSCrossRefGoogle Scholar
  46. Zamaro, M., Biggs, J.: Natural motion around the Martian moon Phobos: the dynamical substitutes of the libration point orbits in an elliptic three-body problem with gravity harmonics. Celest. Mech. Dyn. Astron. 122(3), 263–302 (2015)ADSMathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of AstronauticsBeihang UniversityBeijingChina

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