Advances in Astrophysics

Download PDF (400.7 KB) PP. 36 - 46 Pub. Date: May 1, 2016

DOI: 10.22606/adap.2016.11003

• Jaiyeola Sefinat. B.^*
  Department of Physical Sciences, College of Natural Sciences, Al-Hikmah University, P.M.B 1601,Ilorin, Kwara State, Nigeria
• Abdulrazaq Abdulraheem
  Department of Statistics and Mathematical Sciences,Kwara State University, Malete, Kwara State, Nigeria.
• Titiloye, Emmanuel O.
  Department of Mathematics, University,of Ilorin, PMB 1515, Ilorin, Kwara State, Nigeria.

This research work investigates the effect of Poynting-Robertson (P-R) drag on the stability of the triangular libration points of the Restricted Three Body Problem (RTBP) when the primaries are considered to be oblate spheroid as well as sources of radiation. We established the equations of motion, which incorporated the perturbing forces, for the system. Using linear approximations, we obtained the triangular libration points in terms of the perturbing parameters and found that they were affected by them. Due to the nature of the characteristics equation corresponding to the variational equations of motion of the system obtained, we employed Murray’s criteria and concluded that the triangular libration points remained unstable.

Stability, libration points, oblateness, P-R drag, RTBP.

[1] E.I. Abouelmag, “Existence and Stability of Triangular Point in the Restricted Three-Body Problem with Numerical applications,” Astrophysics and space science, vol. 342, pp. 48-53, 2012.

[2] S.B. Akere-Jaiyeola, J. Singh, A, Abdulraheem and J. Braimah, “On the Effects of Perturbation, Radiation and Triaxiality on the Stability of Libration Points in the Restricted Three-body Problem,” IOSR journal of mathematics, vol. 11, no. 1, pp. 69-79, 2015.

[3] A. Abdulraheem, and J. Singh,” Combined Effects of Perturbations Radiations and Oblateness on the Stability of Equilibrium Points in the Restricted Three-body Problem,” Astronomical Journal, vol. 131, pp. 1880-1885, 2006.

[4] J. A. Aredondo, J. Gui and C. Stoica, “On the Restricted Three-body Problem with Oblate Primaries,” Astrophysics and Space Sci, vol. 341, pp. 315 -322, 2012.

[5] K. B. Bhatnagar and P. P. Hallan, “Effects and Perturbations in the Coriolis and Centrifugal Forces on the Stability of Libration Points in the Restricted Problem,” Cellestial Mechanics, vol. 18, pp. 105-112. 1979).

[6] K.B. Bhatnagar and M.L. Khanna, “Existence and Stability of Libration Points in the Restricted Three-Body Problem when the small Primary is triaxle Rigid Body and the Bigger one is an Oblate Sheroid,” Indian Journal of Pure and Applied Mathematics, vol. 30, no. 7, pp. 721-733, 1999.

[7] Y.A. Chernikov.” The Photogravitational Restricted Three-body Problem,” Soviety Astronomy Astrophysics, vol. 14, no. 1, pp. 176 -181, 1970.

[8] M.K. Das, P. Narang, S. Mahajan and M. Yuasa, “On Out of Plane Equilibrium Points in Photogravitational Restricted Three-body Problem,” Journal of Astronomical. Astrophysics, vol. 30, pp. 177-185. 2009.

[9] B. Ishwar and B.S. Kushvah,” Linear Stability of Triangular Equilibrium Points in the generalized Photogravitational Restricted Three-body problem with Poynting-Robertson drag,” Journal of Dynamical system, vol. 4, no.1, pp. 79-86, 2006.

[10] B. Ishwar and J. Singh,” Stability of Triangular Points in generalized Photogravitational Restricted Three-body problem,” Bulletin of Astronomical Society of India, vol.27, pp. 415 -424.1999.

[11] M. Jain, S.B. Chakraborty and Abdullah,” Restricted Three-body Problem with Robertson Drag Effect,” International Journal of Applied Mathematics and Mechanics, vol. 10, no. 3, pp. 32-44,2013.

[12] S. N. Khasan,” Libration Solutions to the Photogravitaional Three-body Problem,” Cosmic Research, vol. 34, no. 2, pp.146 -151, 1996.

[13] V. Kumar and R. K. Choudhry, “On the Stability of the Triangular Libration Points for the Photogravitaional Circular Rescrited Problem of three bodies when both of the attracting bodies are well radiating,” Cellestial Mechanics, vol. 40, pp. 155-170, 1987.

[14] A. L. Kunitsyn and E. N. Polyathara, “The Restricted Photogravitational Three-body problem,” Astronomical Astrophysics Trans, vol. 6, pp. 283-293, 1995.

[15] A. L. Kunitsyn and A. A. Perezhogin, “On the Stability of Triangular Libration Points of the Photogravitational Restricted Circular Three-body problem,” Celestial Mechanics, vol. 18, pp. 395-408, 1978.

[16] A. L. Kunitsyn, “Stability of Triangular Libration Point in the Photogravitational Three-body Problem,” Journal of Applied Mathematics and Mechanics, vol. 65, no. 5, pp. 757-760. 2000.

[17] A. L. Kunitsyn, “Stability of Colinear Libration Point in the photogravitional Three-Body Problem,”. Journal of Applied Mathematics and Mechanics, vol. 65 no. 4, pp. 703-706, 2001.

[18] B. S. Kushvah and B. Ishwar, “Triangular equilibrium Points in the generalized Photogravitational Restricted three-body Problem with Poynting-Robertson drag,”. Review Bull. Cal. Math. Soc, vol. 12, no. 1 2, pp. 109-114, 2004.

[19] D. C. Liou, H. A. Zook and A.A. Jackson, “Radiation Pressure Poynting Robertson drag and Solar wind drag in the Restricted Three-body” Icarus, vol. 116, pp. 186-201, 1995.

[20] S. W. Mc Cuskey, Introduction to Celestial Mechanics, 1963.

[21] F. Mignard, “Stability of L_4 and L_5 against radiation pressure,” Celestial Mechanics, vol. 34, pp. 275-287, 1984.

[22] C. D. Murray,” Dynamical Effect of Drag in the Circular Restricted Three-body Problem Location and Stability of Lagrangian Equilibrium Points,” Icarus, vol. 112, pp. 465-484, 1994.

[23] J. H. Poynting,” Radiation in the Solar System: Its effect on Temperature and its Pressure on small Bodies,”. MNRAS, vol. 64, A1, 1903.

[24] Radzievskii, V.V, “The Restricted Problem of Three-bodies Taking Account of Light Pressure,” Akad. Nauk. USSR Astronomical Journal, vol. 2, pp. 250-256, 1950.

[25] O, Ragosand F. A. Zafiropoulos, “A Numerical Study of the Influence of the Poynting Robertson Effect on the Equilibrium Points of the Photogravitational Restricted Three-body problem Coplanar case,” Astronomical Astrophysics, vol. 300, pp. 568-578, 1995.

[26] H. P. Robertson, “Dynamical Effects of Radiation in the Solar System,” MNRAS, vol. 97,423.

[27] D. W. Schuerman, “The Restricted Three-body Problem Including Radiation Pressure,” Astrophysics Journal, vol. 238, no. 1, pp. 337-342, 1980.

[28] P. Sengupta and P.Singla,“An Analysis of Stability in the Restricted Three-body Problem,” MEEN 689 project, 2002.

[29] S.M. Shaboury, ‘Equilibrium Solution of the Restricted Problem of 2+2 Axisymmetric Rigid-Bodies,”Celestial Mechanics and Dynamics Astronomy,vol. 50, pp. 199-208, 1991.

[30] R. K.Sharma and P. V. Subbarao, “Stationary Solutions and Their Characteristics Exponents in the Restricted Three-body Problem when the More Massive Primary is an Oblate Spheroid,” Cellestial Mechanics, vol. 13, pp. 137-149, 1976.

[31] R. K. Sharma, Z. A. Taqvi and K. B. Bhatnagar. “Existence and Stability of Libration Points in the Restricted Three-body Problem when the Primaries are Triaxial Rigid bodies,”.Cellestial Mechanics and Dynamical Astrophysics ,vol. 79, pp. 119-133, 2001a.

[32] R. K. Sharma, Z. A. Taqvi and K. B. Bhatnagar, “Existence and Stability of Libration Points in the Restricted Three-body Problem when the Primaries are Triaxial Rigid bodies and Source of Radiations,” Indian Journal of Pure Applied Mathematics,vol. 32, no. 7, pp. 981-994.2001b.

[33] J.F.L. Simmons, A.J.C. Mc Donald,, Brown, J. C. (1985). The restricted 3-body problem with radiation pressure.Celestial Mechanics, 35(2), 145-187.

[34] J. Singh and T. O. Amuda, “Poynting-Roberson Drag and Oblateness Effects on Motion Around the Equilibrium Points in the Photogravitational R3BP, “ Astrophysics and Space Science,vol. 350, pp. 119-126, 2014.

[35] J. Singh, J. Taura, J. Joel, “Stability of Equilibrium Points in a Circular Restricted Three-body Problem with Oblate Bodies Enclosed by a Circular Cluster of Material Points,” Astrophysics and Space Science,vol. 349, pp. 681-697, 2014.

[36] J. Singh, “Combined Effect of Perturbations, Radiation and Oblateness on the Nonlinear Stability of Triangular Point in the restricted Three-Body Problem,” .Astrophysics and Space science,vol. 332,no. 2, pp. 331-339,2011.

[37] V. Szebehely, Theory of Orbits. Academic Press, 1967.

[38] V. Szebehely, “Stability of the point of Equilibrium in the Restricted Problem,” Astronomical Journal, vol. 72, no. 1, pp. 7-9, 1967a.

[39] V. V. Vidyakin, “A Plane Circular Limited Task Pertaining to Three Spheroids,” Astronomical Journal, vol. 51, no. 5, pp. 1087-1094, 1974.