Existence and Stability of Equilibrium Points in the Robe’s Restricted Three-Body Problem with Variable Masses

Affiliation(s)

Department of Mathematics, Faculty of Science, Ahmadu Bello University, Zaria, Nigeria.

Department of Mathematics, Statistics and Computer Science, College of Science, University of Agriculture, Makurdi, Nigeria.

Department of Mathematics, Faculty of Science, Ahmadu Bello University, Zaria, Nigeria.

Department of Mathematics, Statistics and Computer Science, College of Science, University of Agriculture, Makurdi, Nigeria.

ABSTRACT

The positions and linear stability of the equilibrium points of the Robe’s circular restricted three-body problem, are generalized to include the effect of mass variations of the primaries in accordance with the unified Meshcherskii law, when the motion of the primaries is determined by the Gylden-Meshcherskii problem. The autonomized dynamical system with constant coefficients here is possible, only when the shell is empty or when the densities of the medium and the infinitesimal body are equal. We found that the center of the shell is an equilibrium point. Further, when k﹥1; k being the constant of a particular integral of the Gylden-Meshcherskii problem; a pair of equilibrium point, lying in the -plane with each forming triangles with the center of the shell and the second primary exist. Several of the points exist depending on k; hence every point inside the shell is an equilibrium point. The linear stability of the equilibrium points is examined and it is seen that the point at the center of the shell of the autonomized system is conditionally stable; while that of the non-autonomized system is unstable. The triangular equilibrium points on the -plane of both systems are unstable.

Cite this paper

J. Singh and O. Leke, "Existence and Stability of Equilibrium Points in the Robe’s Restricted Three-Body Problem with Variable Masses,"*International Journal of Astronomy and Astrophysics*, Vol. 3 No. 2, 2013, pp. 113-122. doi: 10.4236/ijaa.2013.32013.

J. Singh and O. Leke, "Existence and Stability of Equilibrium Points in the Robe’s Restricted Three-Body Problem with Variable Masses,"

References

[1] M. Dufour, “Ch.: Comptes Rendus Hebdomadaires de L’,” Accademie de Sciences, Amsterdam, 1886, pp. 840-842.

[2] H. Gylden, “Die Bahnbewegungen in Einem Systeme von zwei Korpern in dem Falle, dass die Massen Ver NderunGen Unterworfen Sind,” Astronomische Nachrichten, Vol. 109, 1884, 1884, pp. 1-6.

[3] I. V. Meshcherskii, “Ueber die Integration der Bewegungsgleichungen im Probleme zweier K?rper von ver nderlicher Masse,” Astronomische Nachrichten, Vol. 159, No. 15, 1902, pp. 229-242. doi:10.1002/asna.19021591502

[4] I. V. Meshcherskii, “Works on the Mechanics of Bodies of Variable Mass,” GITTL, Moscow, 1952, p. 205.

[5] B. E. Gelf’gat, “Modern Problems of Celestial Mechanics and Astrodynamics,” Nauka, Moscow, 1973, p. 7.

[6] A. A. Bekov, “Liberation Points of the Restricted Problem of Three Bodies of Variable Mass,” Soviet Astronomy, Vol. 32, 1988, pp. 106-107.

[7] P. Gurfil and S. Belyanin, “The Gauge-Generalized Gylden-Meshcherskii Problem,” Advances in Space Research, Vol. 42, No. 8, 2008, pp. 1313-1317. doi:10.1016/j.asr.2008.01.019

[8] J. Singh and O. Leke, “Stability of the Photogravitational Restricted Three-Body Problem with Variable Masses,” Astrophysics and Space Science, Vol. 326, 2010, pp. 305-314.

[9] H. A. G. Robe, “A New Kind of Three Body Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 16, No. 3, 1977, pp. 343-351. doi:10.1007/BF01232659

[10] A. K. Shrivastava and D. N. Garain, “Effect of Perturbation on the Location of Libration Point in the Robe Restricted Problem of Three Bodies,” Celestial Mechanics and Dynamical Astronomy, Vol. 51, No. 1, 1991, pp. 67-73. doi:10.1007/BF02426670

[11] P. P. Hallan and N. Rana, “The Existence and Stability of Equilibrium Points in the Robe’s\Estricted Problem ThreeBody Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 79, No. 2, 2001, pp. 145-155. doi:10.1023/A:1011173320720

[12] P. P. Hallan and K. B. Mangang, “Existence and Linear Stability of Equilibrium Points in the Robe’s Restricted Three Body Problem When the First Primary Is an Oblate Spheroid,” Planetary and Space Science, Vol. 55, No. 4, 2007, pp. 512-516. doi:10.1016/j.pss.2006.10.002

[13] A. Sommerfeld, “Mechanics,” Academic Press, New York, 1952.

[14] M. L. Krasnov, A. I. Kiselyov and G. I. Makarenko, “A Book of Problems in Ordinary Differential Equations,” MIR Publications, Moscow, 1983, p. 255.

[1] M. Dufour, “Ch.: Comptes Rendus Hebdomadaires de L’,” Accademie de Sciences, Amsterdam, 1886, pp. 840-842.

[2] H. Gylden, “Die Bahnbewegungen in Einem Systeme von zwei Korpern in dem Falle, dass die Massen Ver NderunGen Unterworfen Sind,” Astronomische Nachrichten, Vol. 109, 1884, 1884, pp. 1-6.

[3] I. V. Meshcherskii, “Ueber die Integration der Bewegungsgleichungen im Probleme zweier K?rper von ver nderlicher Masse,” Astronomische Nachrichten, Vol. 159, No. 15, 1902, pp. 229-242. doi:10.1002/asna.19021591502

[4] I. V. Meshcherskii, “Works on the Mechanics of Bodies of Variable Mass,” GITTL, Moscow, 1952, p. 205.

[5] B. E. Gelf’gat, “Modern Problems of Celestial Mechanics and Astrodynamics,” Nauka, Moscow, 1973, p. 7.

[6] A. A. Bekov, “Liberation Points of the Restricted Problem of Three Bodies of Variable Mass,” Soviet Astronomy, Vol. 32, 1988, pp. 106-107.

[7] P. Gurfil and S. Belyanin, “The Gauge-Generalized Gylden-Meshcherskii Problem,” Advances in Space Research, Vol. 42, No. 8, 2008, pp. 1313-1317. doi:10.1016/j.asr.2008.01.019

[8] J. Singh and O. Leke, “Stability of the Photogravitational Restricted Three-Body Problem with Variable Masses,” Astrophysics and Space Science, Vol. 326, 2010, pp. 305-314.

[9] H. A. G. Robe, “A New Kind of Three Body Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 16, No. 3, 1977, pp. 343-351. doi:10.1007/BF01232659

[10] A. K. Shrivastava and D. N. Garain, “Effect of Perturbation on the Location of Libration Point in the Robe Restricted Problem of Three Bodies,” Celestial Mechanics and Dynamical Astronomy, Vol. 51, No. 1, 1991, pp. 67-73. doi:10.1007/BF02426670

[11] P. P. Hallan and N. Rana, “The Existence and Stability of Equilibrium Points in the Robe’s\Estricted Problem ThreeBody Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 79, No. 2, 2001, pp. 145-155. doi:10.1023/A:1011173320720

[12] P. P. Hallan and K. B. Mangang, “Existence and Linear Stability of Equilibrium Points in the Robe’s Restricted Three Body Problem When the First Primary Is an Oblate Spheroid,” Planetary and Space Science, Vol. 55, No. 4, 2007, pp. 512-516. doi:10.1016/j.pss.2006.10.002

[13] A. Sommerfeld, “Mechanics,” Academic Press, New York, 1952.

[14] M. L. Krasnov, A. I. Kiselyov and G. I. Makarenko, “A Book of Problems in Ordinary Differential Equations,” MIR Publications, Moscow, 1983, p. 255.