×

zbMATH — the first resource for mathematics

Equilibria in the gravitational field of a triangular body. (English) Zbl 1448.70034
Summary: 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.
MSC:
70F15 Celestial mechanics
70F07 Three-body problems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Herrera-Succarat, E.: The Full Problem of Two and Three Bodies: Application to Asteroids and Binaries. University of Surrey, Department of Mathematics, Surrey (2012)
[2] 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)
[3] 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)
[4] Scheeres, D.J.: Orbital Motion in Strongly Perturbed Environments: Applications to Asteroid, Comet and Planetary Satellite Orbiters. Springer, Berlin (2012)
[5] 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)
[6] 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)
[7] 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)
[8] 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) · Zbl 1371.70054
[9] Nikonov, V.I.: On relative equilibria of mutually gravitating massive point and triangular rigid body. Proc. Int. Astron. Union 9, 170-171 (2014)
[10] 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) · Zbl 1371.70060
[11] 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) · Zbl 1371.70053
[12] 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)
[13] 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) · Zbl 1255.85008
[14] Robe, H.A.G.: A new kind of 3-body problem. Celest. Mech. Dyn. Astron. 16(3), 343-351 (1977) · Zbl 0374.70007
[15] Plastino, A.R., Plastino, A.: Robe’s restricted three-body problem revisited. Celest. Mech. Dyn. Astron. 61(2), 197-206 (1995) · Zbl 1375.70033
[16] 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) · Zbl 1006.70011
[17] Valeriano, L.R.: Parametric stability in Robe’s problem. Regul. Chaotic Dyn. 21(1), 126-135 (2016) · Zbl 1368.70014
[18] 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)
[19] 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) · Zbl 1393.70035
[20] 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)
[21] 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)
[22] Celli, M.: The central configurations of four masses x, -x, y, -y. J. Differ. Equ. 235(2), 668-682 (2007) · Zbl 1157.70008
[23] Piña, E.: Newtonian few-body problem central configurations with gravitational charges of both signs. arXiv:1212.3219, 11 Dec (2012)
[24] Shatskiy, A.A., Novikov, I.D., Kardashev, N.S.: The Kepler problem and collisions of negative masses. Phys. Usp. 54(4), 381-385 (2011)
[25] Poincaré, H.: Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Acta Math. 7, 259-380 (1885) · JFM 17.0864.02
[26] Chetaev, N.G.: Stability of Motion. Nauka, Moscow (1955) · Zbl 0066.33605
[27] Karapetyan, A.V.: The Stability of Steady Motions. Editorial URSS, Moscow (1998)
[28] Zhuravlev, S.G.: Stability of the libration points of a rotating triaxial ellipsoid. Celest. Mech. 6(3), 255-267 (1972) · Zbl 0254.70007
[29] 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) · Zbl 0288.70007
[30] Kosenko, I.I.: On libration points near a gravitating and rotating triaxial ellipsoid. J. Appl. Math. Mech. 45(1), 18-23 (1981) · Zbl 0486.70013
[31] 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) · Zbl 0591.70026
[32] 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)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.