Skip to main content
Log in

Equilibria in the gravitational field of a triangular body

  • Original Article
  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript


The existence, stability and bifurcation analysis is performed for equilibria of a material point in the gravitational field of three homogeneous penetrable balls fixed in absolute frame. The radii of the balls are assumed finite. In the case when the mass distribution admits a symmetry axis, analytic expressions are written out, allowing one to investigate the properties of equilibrium positions located both on the symmetry axis and outside it. The stability of solutions is studied; domains with different instability degree are described.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others


  • Herrera-Succarat, E.: The Full Problem of Two and Three Bodies: Application to Asteroids and Binaries. University of Surrey, Department of Mathematics, Surrey (2012)

    Google Scholar 

  • Herrera-Sucarrat, E., Palmer, Ph, Roberts, M.: Modeling the gravitational potential of a non-spherical asteroid. J. Guid. Control Dyn. 36(3), 790–798 (2013)

    Article  ADS  Google Scholar 

  • Turconi, A., Palmer, P., Roberts, M.: Efficient modelling of small bodies gravitational potential for autonomous proximity operations. In: Astrodynamics Network AstroNet-II: The Final Conference, Astrophysics and Space Science Proceedings, vol. 44, pp. 257–272 (2016)

    Chapter  Google Scholar 

  • Scheeres, D.J.: Orbital Motion in Strongly Perturbed Environments: Applications to Asteroid, Comet and Planetary Satellite Orbiters. Springer, Berlin (2012)

    Book  Google Scholar 

  • Zeng, X.Y., Jiang, F.H., Li, J.F., Baoyin, H.X.: Study on the connection between the rotating mass dipole and natural elongated bodies. Astrophys. Space Sci. 355, 2187–2200 (2015)

    Google Scholar 

  • Zeng, X., Baoyin, H., Li, J.: Updated rotating mass dipole with oblateness of one primary (i): equilibria in the equator and their stability. Astrophys. Space Sci. 361(14), 1–12 (2016)

    MathSciNet  Google Scholar 

  • Zeng, X., Baoyin, H., Li, J.: Updated rotating mass dipole with oblateness of one primary (ii): out-of-plane equilibria and their stability. Astrophys. Space Sci. 361(15), 1–9 (2016)

    MathSciNet  Google Scholar 

  • Nikonov, V.I.: Relative equilibria in the motion of a triangle and a point under mutual attraction. Mosc.Univ. Mech. Bull. 69(2), 44–50 (2014)

    Article  Google Scholar 

  • Nikonov, V.I.: On relative equilibria of mutually gravitating massive point and triangular rigid body. Proc. Int. Astron. Union 9, 170–171 (2014)

    Article  Google Scholar 

  • Kugushev, E.I., Nikonov, V.I.: An estimate for the number of relative equilibria in the motion of a plane rigid body and a material point under mutual attraction. Mosc. Univ. Mech. Bull. 70(6), 144–148 (2015)

    Article  Google Scholar 

  • Burov, A.A., Nikonov, V.I.: Stability and branching of stationary rotations in a planar problem of motion of mutually gravitating triangle and material point. Russ. J. Nonlinear Dyn. 12(2), 179–196 (2016)

    MathSciNet  MATH  Google Scholar 

  • Gasanov, S.A., Luk’yanov, L.G.: The libration points for the motion of a star inside an elliptical galaxy. Astron. Rep. 46(10), 851–857 (2002)

    Article  ADS  Google Scholar 

  • Avinash, K., Eliasson, B., Shukla, P.K.: Dynamics of self-gravitating dust clouds and the formation of planetesimals. Phys. Lett. A 353(2), 105–108 (2006)

    Article  ADS  Google Scholar 

  • Robe, H.A.G.: A new kind of 3-body problem. Celest. Mech. Dyn. Astron. 16(3), 343–351 (1977)

    Article  Google Scholar 

  • Plastino, A.R., Plastino, A.: Robe’s restricted three-body problem revisited. Celest. Mech. Dyn. Astron. 61(2), 197–206 (1995)

    Article  ADS  Google Scholar 

  • Hallan, P.P., Rana, N.: The existence and stability of equilibrium points in the robes restricted three-body problem. Celest. Mech. Dyn. Astron. 79(2), 145–155 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  • Valeriano, L.R.: Parametric stability in Robe’s problem. Regul. Chaotic Dyn. 21(1), 126–135 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  • Burov, A.A., Guerman, A.D., Nikonov, V.I.: Collocation of equilibria in gravitational field of triangular body via mass redistribution. Acta Astronaut. 146, 181–184 (2018)

    Article  ADS  Google Scholar 

  • Burov, A.A., Guerman, A.D., Kosenko, I.I., Nikonov, V.I.: On the gravity of dumbbell-like bodies represented by a pair of intersecting balls. Russ. J. Nonlinear Dyn. 13(2), 243–256 (2017)

    MATH  Google Scholar 

  • Celli, M.: Homographic three-body motions with positive and negative masses. In: Symmetry and Perturbation Theory, Proceedings of the SPT 2004 Conference (Cala Gonone, Italy, 30 May 6 June 2004), World Sci., pp. 75–82 (2004)

  • Celli, M.: Sur les mouvements homographiques de N corps associés à des masses de signe quelconque, le cas particulier où la somme des masses est nulle, et une application à la recherche de chorégraphies perverses. PhD thesis. Paris 7 University (2005)

  • Celli, M.: The central configurations of four masses x, -x, y, -y. J. Differ. Equ. 235(2), 668–682 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  • Piña, E.: Newtonian few-body problem central configurations with gravitational charges of both signs. arXiv:1212.3219, 11 Dec (2012)

  • Shatskiy, A.A., Novikov, I.D., Kardashev, N.S.: The Kepler problem and collisions of negative masses. Phys. Usp. 54(4), 381–385 (2011)

    Article  ADS  Google Scholar 

  • Poincaré, H.: Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Acta Math. 7, 259–380 (1885)

    Article  MathSciNet  Google Scholar 

  • Chetaev, N.G.: Stability of Motion. Nauka, Moscow (1955)

    MATH  Google Scholar 

  • Karapetyan, A.V.: The Stability of Steady Motions. Editorial URSS, Moscow (1998)

    Google Scholar 

  • Zhuravlev, S.G.: Stability of the libration points of a rotating triaxial ellipsoid. Celest. Mech. 6(3), 255–267 (1972)

    Article  ADS  Google Scholar 

  • Zhuravlev, S.G.: On the stability of the libration points of a rotating triaxial ellipsoid in the three-dimensional case. Sov. Astron. 51(6), 1330–1334 (1974)

    MathSciNet  MATH  Google Scholar 

  • Kosenko, I.I.: On libration points near a gravitating and rotating triaxial ellipsoid. J. Appl. Math. Mech. 45(1), 18–23 (1981)

    Article  MathSciNet  Google Scholar 

  • Kosenko, I.I.: Non-linear analysis of the stability of the libration points of a triaxial ellipsoid. J. Appl. Math. Mech. 49(1), 17–24 (1985)

    Article  MathSciNet  Google Scholar 

  • Wang, X., Jiang, Y., Gong, S.: Analysis of the potential field and equilibrium points of irregular-shaped minor celestial bodies. Astrophys. Space Sci. 353(1), 105–121 (2014)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Anna D. Guerman.

Ethics declarations

Conflict of interest

The authors certify that they have no conflict of interest to declare.

Additional information

This research is partially supported by RFBR, Grants 16-01-00625 and 18-01-00335, project EMaDeS (Centro-01-0145-FEDER-000017), and the Portuguese Foundation for Science and Technologies via Centre for Mechanical and Aerospace Science and Technologies, C-MAST, POCI-01-0145-FEDER-007718. V. I. Nikonov: On leave from FRC “Computer Science and Control” of the Russian Academy of Science.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burov, A.A., Guerman, A.D. & Nikonov, V.I. Equilibria in the gravitational field of a triangular body. Celest Mech Dyn Astr 130, 58 (2018).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: