Periods, Eccentricities and Axes around *L*4,5 in the ER3BP under Radiating and Oblate Primaries

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

In the framework of the elliptic restricted three-body problem, using a semi-analytic approach, we investigate the effects of oblateness, radiation and eccentricity of both primaries on the periodic orbits around the triangular Lagrangian points of oblate and luminous binary systems. The frequencies of the long and short orbits of the periodic motion are affected by the oblateness and radiation of both primaries, so are their eccentricities, semi-major and semi-minor axes.

In the framework of the elliptic restricted three-body problem, using a semi-analytic approach, we investigate the effects of oblateness, radiation and eccentricity of both primaries on the periodic orbits around the triangular Lagrangian points of oblate and luminous binary systems. The frequencies of the long and short orbits of the periodic motion are affected by the oblateness and radiation of both primaries, so are their eccentricities, semi-major and semi-minor axes.

Cite this paper

Umar, A. and Singh, J. (2014) Periods, Eccentricities and Axes around*L*4,5 in the ER3BP under Radiating and Oblate Primaries. *International Journal of Astronomy and Astrophysics*, **4**, 668-682. doi: 10.4236/ijaa.2014.44061.

Umar, A. and Singh, J. (2014) Periods, Eccentricities and Axes around

References

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

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

[3] Poincare (1892) Les Methodes Nouevelles de la Mechanique ce’leste Guthier villars, Paris, Chap. V, 250. (Published in English in Three Volumes).

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

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

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

[7] Radzievskii, V.V. (1950) The Restricted Problem of Three Bodies Taking Account of Light Pressure. Astronomical Journal, 27, 249.

[8] Chernikov, Yu.A. (1970) The Photo Gravitational Restricted Three Body Problem. Soviet Astronomy—AJ, 14, 176-181.

[9] Kunitsyn, A.L. and Perezhogin, A.A. (1978) On the Stability of Triangular Libration Points of the Photo Gravitational Circular Restricted Three-Body Problem. Celestial Mechanics and Dynamical Astronomy, 18, 395-408.

[10] Schuerman, D.W. (1980) The Restricted Three Body Problem including Radiation Pressure. The Astrophysical Journal, 238, 337-342.

http://dx.doi.org/10.1086/157989

[11] Luk’yanov, L.G. (1984) Lagrangian Solutions in the Photogravitational Restricted Circular Three-Body Problem. Astronomicheskii Zhurnal, 789, 94; Soviet Astronomy, 28, 329-333.

[12] Luk’yanov, L.G. (1988) On the Family of Libration Points in the Restricted Three-Body Problem. Astronomicheskii Zhurnal, 65, 422, 432; Soviet Astronomy, 32, 6.

[13] Simmons, J.F.L., McDonald, A.J.C. and Brown, J.C. (1985) The Restricted 3-Body Problem with Radiation Pressure. Celestial Mechanics, 35, 145-187.

http://dx.doi.org/10.1007/BF01227667

[14] Kunitsyn, A.L. and Tureshbaev, A.T. (1985) On the Collinear Libration Points of the Photogravitational Restricted Three-Body Problem. Celestial Mechanics, 35, 105, 112.

[15] Kunitsyn, A.L. (2000) The Stability of Triangular Libration Points in the Photogravitational Three-Body Problem. Journal of Applied Mathematics and Mechanics, 64, 757-763.

http://dx.doi.org/10.1016/S0021-8928(00)00105-2

[16] 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

[17] AbdulRaheem, A. and Singh, J. (2006) Combined Effects of Perturbations, Radiation and Oblateness on the Stability of Equilibrium Points in the Restricted Three-Body Problem. Astronomical Journal, 131, 1880-1885.

[18] AbdulRaheem, A. and Singh, J. (2008) Combined Effects of Perturbations, Radiation and Oblateness on the Periodic Orbits in the Restricted Three-Body Problem. Astrophysics and Space Science, 317, 9-13.

http://dx.doi.org/10.1007/s10509-008-9841-4

[19] Singh, J. and Ishwar, B. (1999) Stability of Collinear Equilibrium Points in the Generalized Photogravitational Elliptic Restricted Three-Body Problem. Bulletin of the Astronomical Society of India, 27, 415.

[20] Singh, J. and AbdulKarim, A. (2014) Instability of Triangular Libration Points in the Perturbed Photogravitational R3BP with Poynting-Robertson (P-R) Drag. Astrophysics and Space Science, 351, 473-482.

http://dx.doi.org/10.1007/s10509-014-1862-6

[21] Shankaran, S.J.P. and Ishwar, B. (2011) Out-of-Plane Equilibrium Points and Stability in the Generalized Photogravitational Restricted Three-Body Problem. Astrophysics & Space Science, 332, 115.

[22] 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

[23] Singh, J. and Umar, A. (2012) Motion in the Photogravitational Elliptic Restricted Three-Body Problem under an Oblate Primary. The Astronomical Journal, 143, 109-131.

http://dx.doi.org/10.1088/0004-6256/143/5/109

[24] 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

[25] 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

[26] 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

[27] Das, M.K., Narang, P., Mahajan, S. and Yuasa, M. (2009) Effect of Radiation on the Stability of a Retrograde Particle Orbit in Different Stellar Systems. Planetary and Space Science, 57, 836-845.

http://dx.doi.org/10.1016/j.pss.2009.02.007

[28] Sharma, R.K. (1987) The Linear Stability of Libration Points of the Photogravitational Restricted Three-Body Problem When the Smaller Primary Is an Oblate Spheroid. Astrophysics and Space Science, 135, 271-281.

http://dx.doi.org/10.1007/BF00641562

[29] Ishwar, B. and Kushvah, B.S. (2006) Linear Stability of Triangular Equilibrium Points in the Generalized Photogravitational Restricted Three Body Problem with Poynting_Robertson Drag. Journal of Dynamical Systems and Geometric Theories, 4, 79-86.

http://dx.doi.org/10.1080/1726037X.2006.10698504

[30] Tsirogiannis, G.A., Douskos, C.N. and Perdios, E.A. (2006) Computation of the Liapunov Orbits in the Photogravitational RTBP with Oblateness. Astrophysics and Space Science, 305, 389-398.

http://dx.doi.org/10.1007/s10509-006-9171-3

[31] Vishnu Namboori, N.I., Sudheer Reedy, D. and Sharma, R.K. (2008) Effect of Oblateness and Radiation Pressure on Angular Frequencies at Collinear Points. Astrophysics and Space Science, 318, 161-168.

http://dx.doi.org/10.1007/s10509-008-9934-0

[32] Mital, A., Ahmad, I. and Bhatnagar, K.B. (2009) Periodic Orbits in the Photogravitational Restricted Problem with the Smaller Primary an Oblate Body. Astrophysics and Space Science, 323, 65-73.

http://dx.doi.org/10.1007/s10509-009-0038-2

[33] Sahoo, S.K. and Ishwar, B. (2000) Stability of Collinear Equilibrium Points in the Generalized Photogravitational Elliptic Restricted Three-Body Problem. Bulletin of the Astronomical Society of India, 28, 579-586.

[34] Abouelmagd, E.I. (2012) Existence and Stability of Triangular Points in the Restricted Three-Body Problem with Numerical Applications. Astrophysics and Space Science, 342, 45-53.

[35] Singh, J. and Haruna, S. (2014) Periodic Orbits around Triangular Points in the Restricted Problem of Three Oblate Bodies. American Journal of Astronomy and Astrophysics, 2, 22-26.

[36] Sarris, E. (1989) Families of Symmetric-Periodic Orbits in the Elliptic Three-Dimensional Restricted Three-Body Problem. Astrophysics and Space Science, 162, 107-122.

http://dx.doi.org/10.1007/BF00653348

[37] 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

[38] 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

[39] 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

[40] 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

[41] Kumar, S. and Ishwar, B. (2009) Solutions of Generalized Photogravitational Elliptic Restricted Three-Body Problem. AIP Conference Proceedings, 1146, 456-460.

http://dx.doi.org/10.1063/1.3183564

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

http://dx.doi.org/10.4314/ijest.v3i2.68143

[43] 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

[44] Charlier, C.I. (1899) Die Mechanik des Himmels. Walter de Gryter and Co., Berlin and Leipzig.

[45] Plummer, H.C. (1901) On Periodic Orbits in the Neighborhood of Centres of Liberation. Monthly Notices of the Royal Astronomical Society, 62, 6.

[46] Riabov, U.A. (1952) Preliminary Orbits Trojan Asteroids. Soviet Astronomy, 29, 5.

[47] Elipe, A. and Lara, M. (1997) Periodic Orbits in the Restricted Three-Body Problem with Radiation Pressure. Celestial Mechanics and Dynamical Astronomy, 68, 1-11.

http://dx.doi.org/10.1023/A:1008233828923

[48] Hadjidemetriou, J.D. (1984) Periodic Orbits. Celestial Mechanics, 34, 379-393.

http://dx.doi.org/10.1007/BF01235816

[49] Khanna, M. and Bhatnagar, K.B. (1999) Existence and Stability of Libration Points in the Restricted Three Body Problem When the Smaller Primary Is a Triaxial Rigid Body and the Bigger One an Oblate Spheroid. Indian Journal of Pure and Applied Mathematics, 30, 721-723.

[50] Perdios, E.A. (2003) Critical Symmetric Periodic Orbits in the Photogravitational Restricted Three-Body Problem. Astrophysics and Space Science, 286, 501-513.

http://dx.doi.org/10.1023/A:1026328832021

[51] Kushvah, B.S., Kishor, R. and Dolas U. (2012) Existence of Equilibrium Points and Their Linear Stability in the Generalized Photogravitational Chermykh-Life Problem with Power-Law Profile. Astrophysics and Space Science, 337, 115-125.

http://dx.doi.org/10.1007/s10509-011-0857-9

[52] Beevi, A. and Sharma, R.K. (2012) Oblateness Effect of Saturn on Periodic Orbits in Saturn-Titan Restricted Three-Body Problem. Astrophysics and Space Science, 340, 245-261.

http://dx.doi.org/10.1007/s10509-012-1052-3

[53] 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

[54] Abouelmagd, E.I. and El-Shaboury, S.M. (2012) Periodic Orbits under Combined Effects of Oblateness and Radiation in the Restricted Problem of Three Bodies. Astrophysics and Space Science, 341, 331-341.

[55] McCuskey, S.W. (1963) Introduction to Celestial Mechanics. Addison-Wesley, New York.

[56] Ishwar, B. and Elipe, A. (2001) Secular Solutions at Triangular Equilibrium Point in the Generalized Photogravitational Restricted Three Body Problem. Astrophysics and Space Science, 277, 437-446.

http://dx.doi.org/10.1023/A:1012528929233

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

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

[3] Poincare (1892) Les Methodes Nouevelles de la Mechanique ce’leste Guthier villars, Paris, Chap. V, 250. (Published in English in Three Volumes).

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

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

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

[7] Radzievskii, V.V. (1950) The Restricted Problem of Three Bodies Taking Account of Light Pressure. Astronomical Journal, 27, 249.

[8] Chernikov, Yu.A. (1970) The Photo Gravitational Restricted Three Body Problem. Soviet Astronomy—AJ, 14, 176-181.

[9] Kunitsyn, A.L. and Perezhogin, A.A. (1978) On the Stability of Triangular Libration Points of the Photo Gravitational Circular Restricted Three-Body Problem. Celestial Mechanics and Dynamical Astronomy, 18, 395-408.

[10] Schuerman, D.W. (1980) The Restricted Three Body Problem including Radiation Pressure. The Astrophysical Journal, 238, 337-342.

http://dx.doi.org/10.1086/157989

[11] Luk’yanov, L.G. (1984) Lagrangian Solutions in the Photogravitational Restricted Circular Three-Body Problem. Astronomicheskii Zhurnal, 789, 94; Soviet Astronomy, 28, 329-333.

[12] Luk’yanov, L.G. (1988) On the Family of Libration Points in the Restricted Three-Body Problem. Astronomicheskii Zhurnal, 65, 422, 432; Soviet Astronomy, 32, 6.

[13] Simmons, J.F.L., McDonald, A.J.C. and Brown, J.C. (1985) The Restricted 3-Body Problem with Radiation Pressure. Celestial Mechanics, 35, 145-187.

http://dx.doi.org/10.1007/BF01227667

[14] Kunitsyn, A.L. and Tureshbaev, A.T. (1985) On the Collinear Libration Points of the Photogravitational Restricted Three-Body Problem. Celestial Mechanics, 35, 105, 112.

[15] Kunitsyn, A.L. (2000) The Stability of Triangular Libration Points in the Photogravitational Three-Body Problem. Journal of Applied Mathematics and Mechanics, 64, 757-763.

http://dx.doi.org/10.1016/S0021-8928(00)00105-2

[16] 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

[17] AbdulRaheem, A. and Singh, J. (2006) Combined Effects of Perturbations, Radiation and Oblateness on the Stability of Equilibrium Points in the Restricted Three-Body Problem. Astronomical Journal, 131, 1880-1885.

[18] AbdulRaheem, A. and Singh, J. (2008) Combined Effects of Perturbations, Radiation and Oblateness on the Periodic Orbits in the Restricted Three-Body Problem. Astrophysics and Space Science, 317, 9-13.

http://dx.doi.org/10.1007/s10509-008-9841-4

[19] Singh, J. and Ishwar, B. (1999) Stability of Collinear Equilibrium Points in the Generalized Photogravitational Elliptic Restricted Three-Body Problem. Bulletin of the Astronomical Society of India, 27, 415.

[20] Singh, J. and AbdulKarim, A. (2014) Instability of Triangular Libration Points in the Perturbed Photogravitational R3BP with Poynting-Robertson (P-R) Drag. Astrophysics and Space Science, 351, 473-482.

http://dx.doi.org/10.1007/s10509-014-1862-6

[21] Shankaran, S.J.P. and Ishwar, B. (2011) Out-of-Plane Equilibrium Points and Stability in the Generalized Photogravitational Restricted Three-Body Problem. Astrophysics & Space Science, 332, 115.

[22] 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

[23] Singh, J. and Umar, A. (2012) Motion in the Photogravitational Elliptic Restricted Three-Body Problem under an Oblate Primary. The Astronomical Journal, 143, 109-131.

http://dx.doi.org/10.1088/0004-6256/143/5/109

[24] 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

[25] 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

[26] 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

[27] Das, M.K., Narang, P., Mahajan, S. and Yuasa, M. (2009) Effect of Radiation on the Stability of a Retrograde Particle Orbit in Different Stellar Systems. Planetary and Space Science, 57, 836-845.

http://dx.doi.org/10.1016/j.pss.2009.02.007

[28] Sharma, R.K. (1987) The Linear Stability of Libration Points of the Photogravitational Restricted Three-Body Problem When the Smaller Primary Is an Oblate Spheroid. Astrophysics and Space Science, 135, 271-281.

http://dx.doi.org/10.1007/BF00641562

[29] Ishwar, B. and Kushvah, B.S. (2006) Linear Stability of Triangular Equilibrium Points in the Generalized Photogravitational Restricted Three Body Problem with Poynting_Robertson Drag. Journal of Dynamical Systems and Geometric Theories, 4, 79-86.

http://dx.doi.org/10.1080/1726037X.2006.10698504

[30] Tsirogiannis, G.A., Douskos, C.N. and Perdios, E.A. (2006) Computation of the Liapunov Orbits in the Photogravitational RTBP with Oblateness. Astrophysics and Space Science, 305, 389-398.

http://dx.doi.org/10.1007/s10509-006-9171-3

[31] Vishnu Namboori, N.I., Sudheer Reedy, D. and Sharma, R.K. (2008) Effect of Oblateness and Radiation Pressure on Angular Frequencies at Collinear Points. Astrophysics and Space Science, 318, 161-168.

http://dx.doi.org/10.1007/s10509-008-9934-0

[32] Mital, A., Ahmad, I. and Bhatnagar, K.B. (2009) Periodic Orbits in the Photogravitational Restricted Problem with the Smaller Primary an Oblate Body. Astrophysics and Space Science, 323, 65-73.

http://dx.doi.org/10.1007/s10509-009-0038-2

[33] Sahoo, S.K. and Ishwar, B. (2000) Stability of Collinear Equilibrium Points in the Generalized Photogravitational Elliptic Restricted Three-Body Problem. Bulletin of the Astronomical Society of India, 28, 579-586.

[34] Abouelmagd, E.I. (2012) Existence and Stability of Triangular Points in the Restricted Three-Body Problem with Numerical Applications. Astrophysics and Space Science, 342, 45-53.

[35] Singh, J. and Haruna, S. (2014) Periodic Orbits around Triangular Points in the Restricted Problem of Three Oblate Bodies. American Journal of Astronomy and Astrophysics, 2, 22-26.

[36] Sarris, E. (1989) Families of Symmetric-Periodic Orbits in the Elliptic Three-Dimensional Restricted Three-Body Problem. Astrophysics and Space Science, 162, 107-122.

http://dx.doi.org/10.1007/BF00653348

[37] 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

[38] 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

[39] 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

[40] 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

[41] Kumar, S. and Ishwar, B. (2009) Solutions of Generalized Photogravitational Elliptic Restricted Three-Body Problem. AIP Conference Proceedings, 1146, 456-460.

http://dx.doi.org/10.1063/1.3183564

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

http://dx.doi.org/10.4314/ijest.v3i2.68143

[43] 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

[44] Charlier, C.I. (1899) Die Mechanik des Himmels. Walter de Gryter and Co., Berlin and Leipzig.

[45] Plummer, H.C. (1901) On Periodic Orbits in the Neighborhood of Centres of Liberation. Monthly Notices of the Royal Astronomical Society, 62, 6.

[46] Riabov, U.A. (1952) Preliminary Orbits Trojan Asteroids. Soviet Astronomy, 29, 5.

[47] Elipe, A. and Lara, M. (1997) Periodic Orbits in the Restricted Three-Body Problem with Radiation Pressure. Celestial Mechanics and Dynamical Astronomy, 68, 1-11.

http://dx.doi.org/10.1023/A:1008233828923

[48] Hadjidemetriou, J.D. (1984) Periodic Orbits. Celestial Mechanics, 34, 379-393.

http://dx.doi.org/10.1007/BF01235816

[49] Khanna, M. and Bhatnagar, K.B. (1999) Existence and Stability of Libration Points in the Restricted Three Body Problem When the Smaller Primary Is a Triaxial Rigid Body and the Bigger One an Oblate Spheroid. Indian Journal of Pure and Applied Mathematics, 30, 721-723.

[50] Perdios, E.A. (2003) Critical Symmetric Periodic Orbits in the Photogravitational Restricted Three-Body Problem. Astrophysics and Space Science, 286, 501-513.

http://dx.doi.org/10.1023/A:1026328832021

[51] Kushvah, B.S., Kishor, R. and Dolas U. (2012) Existence of Equilibrium Points and Their Linear Stability in the Generalized Photogravitational Chermykh-Life Problem with Power-Law Profile. Astrophysics and Space Science, 337, 115-125.

http://dx.doi.org/10.1007/s10509-011-0857-9

[52] Beevi, A. and Sharma, R.K. (2012) Oblateness Effect of Saturn on Periodic Orbits in Saturn-Titan Restricted Three-Body Problem. Astrophysics and Space Science, 340, 245-261.

http://dx.doi.org/10.1007/s10509-012-1052-3

[53] 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

[54] Abouelmagd, E.I. and El-Shaboury, S.M. (2012) Periodic Orbits under Combined Effects of Oblateness and Radiation in the Restricted Problem of Three Bodies. Astrophysics and Space Science, 341, 331-341.

[55] McCuskey, S.W. (1963) Introduction to Celestial Mechanics. Addison-Wesley, New York.

[56] Ishwar, B. and Elipe, A. (2001) Secular Solutions at Triangular Equilibrium Point in the Generalized Photogravitational Restricted Three Body Problem. Astrophysics and Space Science, 277, 437-446.

http://dx.doi.org/10.1023/A:1012528929233