The Collinear Libration Points in the Elliptic R3BP with a Triaxial Primary and an Oblate Secondary

Affiliation(s)

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

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

ABSTRACT

This paper examines the motion of a dust grain around a triaxial primary and an oblate companion orbiting each other in elliptic orbits about their common barycenter in the neighborhood of collinear libration points. The positions and stability of these points are found to be affected by the triaxiality and oblateness of the primaries, and by the semi-major axis and eccentricity of their orbits. The stability behavior of the collinear points however remains unchanged; they are unstable in the Lyapunov sense.

Cite this paper

Singh, J. and Umar, A. (2014) The Collinear Libration Points in the Elliptic R3BP with a Triaxial Primary and an Oblate Secondary.*International Journal of Astronomy and Astrophysics*, **4**, 391-398. doi: 10.4236/ijaa.2014.42034.

Singh, J. and Umar, A. (2014) The Collinear Libration Points in the Elliptic R3BP with a Triaxial Primary and an Oblate Secondary.

References

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http://dx.doi.org/10.1196/annals.1370.025

[2] Belbruno, E., Topputo, F. and Gidea, M. (2008) Resonance Transitions Associated to Weak Capture in the Restricted Three-Body Problem. Advances in Space Research, 42, 1330-1351.

http://dx.doi.org/10.1016/j.asr.2008.01.018

[3] Romagnoli, D. and Circi, C. (2009) Earth-Moon Weak Stability Boundaries in the Restricted Three and Four Body Problem. Celestial Mechanics and Dynamical Astronomy, 103, 79-103.

http://dx.doi.org/10.1007/s10569-008-9169-y

[4] Bazso, A. (2012) 43rd Lunar and Planetary Science Conference.

[5] Lagrange (1772) Collected Works Paris. VI, 229.

[6] Laplace and Delaunay (1867) Memoire sur la theorie de la Lune, Mem. De l’des Science, 28 and 29.

[7] Poincare (1892) Les Methodes Nouevelles de la Mechanique ce’leste Guthier villars, Paris, Chap. V, 250.

[8] Birkhoff (1927) Dynamical System. American Mathematical Society, New York.

[9] Szebehely, V.G. (1967) Theory of Orbits. Academic Press, New York.

[10] Danby, J.M.A. (1988) Fundamentals of Celestial Mechanics. 2nd Edition, Willmann-Bell, Inc., Virginia.

[11] Kumar, V. and Choudry, R.K. (1990) Nonlinear Stability of the Triangular Libration Points for the Photo Gravitational Elliptic Restricted Problem of Three Bodies. Celestial Mechanics and Dynamical Astronomy, 48, 299-317. http://dx.doi.org/10.1007/BF00049387

[12] Markellos, V.V., Perdios, E. and Labropoulou, P. (1992) Linear Stability of the Triangular Equilibrium Points in the Photogravitational Elliptic Restricted Problem I. Astrophysics and Space Science, 194, 207-213. http://dx.doi.org/10.1007/BF00643991

[13] Sahoo, S.K. and Ishwar, B. (2000) Stability of Collinear Equilibrium Points in the Generalized Photogravitational Elliptic Restricted Three-Body Problem. Bulletin of Astronic Society of India, 28, 579.

[14] Kunitsyn, A.L. (2001) The Stability of Collinear Libration Points in the Photogravitational Three-Body Problem. Journal of Applied Mathematics and Mechanics, 65, 703-706.

http://dx.doi.org/10.1016/S0021-8928(01)00075-2

[15] Zimovshchikov, A.S. and Tkhai, V.N. (2004) Instability of Libration Points and Resonance Phenomena in the Photogravitational Elliptic Restricted Three-Body Problem. Solar System Research, 38, 155-164. http://dx.doi.org/10.1023/B:SOLS.0000022826.31475.a7

[16] Szenkovits, F. and Mako, Z. (2005) Publications of the Astronomy Department of Eotvos Lorand University, Budapest, 15, 221.

[17] Ammar, M.K. (2008) The Effect of Solar Radiation Pressure on the Lagrangian Points in the Elliptic Restricted Three-Body Problem. Astrophysics and Space Science, 313, 393-408.

http://dx.doi.org/10.1007/s10509-007-9709-z

[18] Kumar, S. and Ishwar, B. (2009) Solutions of Generalized Photogravitational Elliptic Restricted Three-Body Problem, AIP Conference Proceedings, 1146, 456. http://dx.doi.org/10.1063/1.3183564

[19] Kumar, S. and Ishwar, B. (2011) Locations of Collinear Equilibrium Points in the Generalized Elliptic Restricted Three-Body Problem. International Journal of Engineering, Science and Technology, 3, 157-162.

[20] Narayan, A. and Ramesh, C. (2011) Effects of Photogravitational and Oblantensss on the Triangular Lagrangian Points in the Elliptical Restricted Three Body Problem. International Journal of Pure and Applied Mathematics, 68, 201.

[21] Narayan, A. and Ramesh, C. (2011) Stability of Triangular Equilibrium Points In Elliptical Restricted Three Body Problem under the Effects of Photogravitational and Oblateness of Primaries. International Journal of Pure and Applied Mathematics, 70, 735.

[22] Singh, J. and Umar, A. (2012) Motion in the Photogravitational Elliptic Restricted Three-Body Problem under an Oblate Primary. The Astronomical Journal, 143, 109. http://dx.doi.org/10.1088/0004-6256/143/5/109

[23] Singh, J. and Umar, A. (2012) On the Stability of Triangular Equilibrium Points in the Elliptic R3BP under Radiating and Oblate Primaries. Astrophysics and Space Science, 341, 349-358.

http://dx.doi.org/10.1007/s10509-012-1109-3

[24] Singh, J. and Umar, A. (2013) On “out of Plane” Equilibrium Points in the Elliptic Restricted Three-Body Problem with Radiating and Oblate Primaries. Astrophysics and Space Science, 344, 13-19. http://dx.doi.org/10.1007/s10509-012-1292-2

[25] Singh, J. and Umar, A. (2013) Collinear Equilibrium Points in the Elliptic R3BP with Oblateness and Radiation. Advances in Space Research, 52, 1489-1496. http://dx.doi.org/10.1016/j.asr.2013.07.027

[26] Singh, J. and Umar, A. (2013) Application of Binary Pulsars to Axisymmetric Bodies in the Elliptic R3BP. Astrophysics and Space Science, 348, 393-402. http://dx.doi.org/10.1007/s10509-013-1585-0

[27] Singh, J. and Umar, A. (2014) On Motion around the Collinear Libration Points in the Elliptic Restricted Three-Body Problem with a Bigger Triaxial Primary. New Astronomy, 29, 36-41.

http://dx.doi.org/10.1016/j.newast.2013.11.003

[28] Arutyunyan, G.G., Sedrakyan, D.M. and Chubaryan, E.V. (1971) Rotating White Dwarfs in the General Relativity Theory. Astrophysics, 7, 274.

[29] Papoyan, V.V., Sedrakyan, D.M. and Chubaryan, E.V. (1971) Newtonian Theory of Rapidly Rotating White Dwarfs. Astrophysics, 7, 55.

[30] Laarakkers, W.G. (1999) Quadrupole Moments of Rotating Neutron Stars. The Astrophysical Journal, 512, 282. http://dx.doi.org/10.1086/306732

[31] Shibata, M. (1998) Effects of the Quadrupole Moment of Rapidly Rotating Neutron Stars on the Motion of the Accretion Disks. Progress of Theoretical Physics, 99, 69-78.

http://dx.doi.org/10.1143/PTP.99.69

[32] Boshkayev, K., Quevedo, H. and Ruffini, R. (2012) Gravitational Field of Compact Objects in General Relativity. Physical Review D, 86, Article ID: 064043.

[33] Heyl, J.S. (2000) Gravitational Radiation from Strongly Magnetized White Dwarfs. Monthly Notices of the Royal Astronomical Society, 317, 310-314.

[34] Sharma, R.K. and Rao, P.V.S. (1976) Stationary Solutions and Their Characteristic Exponents in the Restricted ThreeBody Problem When the More Massive Primary Is an Oblate Spheroid. Celestial Mechanics, 13, 137-149. http://dx.doi.org/10.1007/BF01232721

[35] Elipe, A. and Ferrer, S. (1985) On the Equilibrium Solutions in the Circular Planar Restricted Three Rigid Bodies Problem. Celestial Mechanics, 37, 59-70. http://dx.doi.org/10.1007/BF01230341

[36] Sharma, R.K., Taqvi, Z.A. and Bhatnagar, K.B. (2001) Existence and Stability of Libration Points in the Restricted Three-Body Problem When the Primaries Are Triaxial Rigid Bodies. Celestial Mechanics and Dynamical Astronomy, 79, 119-133. http://dx.doi.org/10.1023/A:1011168605411

[37] Singh, J. and Begha, J.M. (2011) Stability of Equilibrium Points in the Generalized Perturbed Restricted Three-Body Problem. Astrophysics and Space Science, 331, 511-519.

http://dx.doi.org/10.1007/s10509-010-0464-1

[38] Singh, J. (2012) Motion around the Out-of-Plane Equilibrium Points of the Perturbed Restricted Three-Body Problem. Astrophysics and Space Science, 342, 303-308. http://dx.doi.org/10.1007/s10509-012-1187-2

[1] Topputo, F., Vasile, M. and Bernelli-Zazzara, F. (2005) Earth-to-Moon Low Energy Transfers Targeting L1 Hyperbolic Transit Orbits. Annals of the New York Academy of Sciences, 1065, 55-76.

http://dx.doi.org/10.1196/annals.1370.025

[2] Belbruno, E., Topputo, F. and Gidea, M. (2008) Resonance Transitions Associated to Weak Capture in the Restricted Three-Body Problem. Advances in Space Research, 42, 1330-1351.

http://dx.doi.org/10.1016/j.asr.2008.01.018

[3] Romagnoli, D. and Circi, C. (2009) Earth-Moon Weak Stability Boundaries in the Restricted Three and Four Body Problem. Celestial Mechanics and Dynamical Astronomy, 103, 79-103.

http://dx.doi.org/10.1007/s10569-008-9169-y

[4] Bazso, A. (2012) 43rd Lunar and Planetary Science Conference.

[5] Lagrange (1772) Collected Works Paris. VI, 229.

[6] Laplace and Delaunay (1867) Memoire sur la theorie de la Lune, Mem. De l’des Science, 28 and 29.

[7] Poincare (1892) Les Methodes Nouevelles de la Mechanique ce’leste Guthier villars, Paris, Chap. V, 250.

[8] Birkhoff (1927) Dynamical System. American Mathematical Society, New York.

[9] Szebehely, V.G. (1967) Theory of Orbits. Academic Press, New York.

[10] Danby, J.M.A. (1988) Fundamentals of Celestial Mechanics. 2nd Edition, Willmann-Bell, Inc., Virginia.

[11] Kumar, V. and Choudry, R.K. (1990) Nonlinear Stability of the Triangular Libration Points for the Photo Gravitational Elliptic Restricted Problem of Three Bodies. Celestial Mechanics and Dynamical Astronomy, 48, 299-317. http://dx.doi.org/10.1007/BF00049387

[12] Markellos, V.V., Perdios, E. and Labropoulou, P. (1992) Linear Stability of the Triangular Equilibrium Points in the Photogravitational Elliptic Restricted Problem I. Astrophysics and Space Science, 194, 207-213. http://dx.doi.org/10.1007/BF00643991

[13] Sahoo, S.K. and Ishwar, B. (2000) Stability of Collinear Equilibrium Points in the Generalized Photogravitational Elliptic Restricted Three-Body Problem. Bulletin of Astronic Society of India, 28, 579.

[14] Kunitsyn, A.L. (2001) The Stability of Collinear Libration Points in the Photogravitational Three-Body Problem. Journal of Applied Mathematics and Mechanics, 65, 703-706.

http://dx.doi.org/10.1016/S0021-8928(01)00075-2

[15] Zimovshchikov, A.S. and Tkhai, V.N. (2004) Instability of Libration Points and Resonance Phenomena in the Photogravitational Elliptic Restricted Three-Body Problem. Solar System Research, 38, 155-164. http://dx.doi.org/10.1023/B:SOLS.0000022826.31475.a7

[16] Szenkovits, F. and Mako, Z. (2005) Publications of the Astronomy Department of Eotvos Lorand University, Budapest, 15, 221.

[17] Ammar, M.K. (2008) The Effect of Solar Radiation Pressure on the Lagrangian Points in the Elliptic Restricted Three-Body Problem. Astrophysics and Space Science, 313, 393-408.

http://dx.doi.org/10.1007/s10509-007-9709-z

[18] Kumar, S. and Ishwar, B. (2009) Solutions of Generalized Photogravitational Elliptic Restricted Three-Body Problem, AIP Conference Proceedings, 1146, 456. http://dx.doi.org/10.1063/1.3183564

[19] Kumar, S. and Ishwar, B. (2011) Locations of Collinear Equilibrium Points in the Generalized Elliptic Restricted Three-Body Problem. International Journal of Engineering, Science and Technology, 3, 157-162.

[20] Narayan, A. and Ramesh, C. (2011) Effects of Photogravitational and Oblantensss on the Triangular Lagrangian Points in the Elliptical Restricted Three Body Problem. International Journal of Pure and Applied Mathematics, 68, 201.

[21] Narayan, A. and Ramesh, C. (2011) Stability of Triangular Equilibrium Points In Elliptical Restricted Three Body Problem under the Effects of Photogravitational and Oblateness of Primaries. International Journal of Pure and Applied Mathematics, 70, 735.

[22] Singh, J. and Umar, A. (2012) Motion in the Photogravitational Elliptic Restricted Three-Body Problem under an Oblate Primary. The Astronomical Journal, 143, 109. http://dx.doi.org/10.1088/0004-6256/143/5/109

[23] Singh, J. and Umar, A. (2012) On the Stability of Triangular Equilibrium Points in the Elliptic R3BP under Radiating and Oblate Primaries. Astrophysics and Space Science, 341, 349-358.

http://dx.doi.org/10.1007/s10509-012-1109-3

[24] Singh, J. and Umar, A. (2013) On “out of Plane” Equilibrium Points in the Elliptic Restricted Three-Body Problem with Radiating and Oblate Primaries. Astrophysics and Space Science, 344, 13-19. http://dx.doi.org/10.1007/s10509-012-1292-2

[25] Singh, J. and Umar, A. (2013) Collinear Equilibrium Points in the Elliptic R3BP with Oblateness and Radiation. Advances in Space Research, 52, 1489-1496. http://dx.doi.org/10.1016/j.asr.2013.07.027

[26] Singh, J. and Umar, A. (2013) Application of Binary Pulsars to Axisymmetric Bodies in the Elliptic R3BP. Astrophysics and Space Science, 348, 393-402. http://dx.doi.org/10.1007/s10509-013-1585-0

[27] Singh, J. and Umar, A. (2014) On Motion around the Collinear Libration Points in the Elliptic Restricted Three-Body Problem with a Bigger Triaxial Primary. New Astronomy, 29, 36-41.

http://dx.doi.org/10.1016/j.newast.2013.11.003

[28] Arutyunyan, G.G., Sedrakyan, D.M. and Chubaryan, E.V. (1971) Rotating White Dwarfs in the General Relativity Theory. Astrophysics, 7, 274.

[29] Papoyan, V.V., Sedrakyan, D.M. and Chubaryan, E.V. (1971) Newtonian Theory of Rapidly Rotating White Dwarfs. Astrophysics, 7, 55.

[30] Laarakkers, W.G. (1999) Quadrupole Moments of Rotating Neutron Stars. The Astrophysical Journal, 512, 282. http://dx.doi.org/10.1086/306732

[31] Shibata, M. (1998) Effects of the Quadrupole Moment of Rapidly Rotating Neutron Stars on the Motion of the Accretion Disks. Progress of Theoretical Physics, 99, 69-78.

http://dx.doi.org/10.1143/PTP.99.69

[32] Boshkayev, K., Quevedo, H. and Ruffini, R. (2012) Gravitational Field of Compact Objects in General Relativity. Physical Review D, 86, Article ID: 064043.

[33] Heyl, J.S. (2000) Gravitational Radiation from Strongly Magnetized White Dwarfs. Monthly Notices of the Royal Astronomical Society, 317, 310-314.

[34] Sharma, R.K. and Rao, P.V.S. (1976) Stationary Solutions and Their Characteristic Exponents in the Restricted ThreeBody Problem When the More Massive Primary Is an Oblate Spheroid. Celestial Mechanics, 13, 137-149. http://dx.doi.org/10.1007/BF01232721

[35] Elipe, A. and Ferrer, S. (1985) On the Equilibrium Solutions in the Circular Planar Restricted Three Rigid Bodies Problem. Celestial Mechanics, 37, 59-70. http://dx.doi.org/10.1007/BF01230341

[36] Sharma, R.K., Taqvi, Z.A. and Bhatnagar, K.B. (2001) Existence and Stability of Libration Points in the Restricted Three-Body Problem When the Primaries Are Triaxial Rigid Bodies. Celestial Mechanics and Dynamical Astronomy, 79, 119-133. http://dx.doi.org/10.1023/A:1011168605411

[37] Singh, J. and Begha, J.M. (2011) Stability of Equilibrium Points in the Generalized Perturbed Restricted Three-Body Problem. Astrophysics and Space Science, 331, 511-519.

http://dx.doi.org/10.1007/s10509-010-0464-1

[38] Singh, J. (2012) Motion around the Out-of-Plane Equilibrium Points of the Perturbed Restricted Three-Body Problem. Astrophysics and Space Science, 342, 303-308. http://dx.doi.org/10.1007/s10509-012-1187-2