Advances in Astrophysics

Download PDF (546.3 KB) PP. 138 - 160 Pub. Date: November 1, 2016

DOI: 10.22606/adap.2016.13002

• Vivek Kumar Mishra^*
  University Department of Mathematics, B.R.A. Bihar University Muzaffarpur-842001
• Bhola Ishwar
  University Department of Mathematics, B.R.A. Bihar University Muzaffarpur-842001

We examined the non-linear stability of triangular equilibrium points in the photogravitational elliptic restricted three body problem with Poynting-Robertson drag. In this problem, the bigger primary is taken as radiating and smaller primary is assumed to be an oblate spheroid. We performed normalization of Hamiltonian of our problem. Using Whittaker (1965) method we have found that the second order part H2 of the Hamiltonian is transformed into the normal form. We have found normalized Hamiltonian up to fourth order. To find the condition of non-linear stability, we have used KAM theorem. We have found three critical mass ratios. We came to conclusion that triangular equilibrium points are stable in the non-linear sense except at three critical mass ratios at which KAM theorem fails.

Non-linear stability, triangular equilibrium points, photogravitational, ERTBP, oblateness, P-R drag.

[1] Arnold,V.I.(1961), On the stability of positions of equilibrium of Hamiltonian system of ordinary differential equations in the general elliptic case , Sov.Math.Dokl.2,247,1961.

[2] Breakwell,J.V.,Pringle,R.,(1966), Resonances affecting near the earth-moon equilateral libration points in R.L, Celest Mech.& Dyn.Astro,17,55-74.

[3] Bhatnagar,K.B. and Hallan,P.P.,(1983),The effect of perturbations in coriolis and centrifugal forces on the nonlinear stability of equilibrium points in the restricted problem of three bodies,Celest Mech.& Dyn.Astro.,Vol- 30,97-114.

[4] Chernikov,Y.A.(1970), The photogravitational restricted three body problem,Astrn.Z.47,217,1970

[5] Deprit,A.,and Deprit.Bartholome(1967), Stability of the triangular Lagrangian points, AJ,72,173-179.

[6] Elipe,A.,et al.(1985), On the equilibrium solutions in the circular restricted three bodies problem, Celest Mech,37,59-70.

[7] Henrard,J.,(1990), A semi-numerical perturbation method for separable Hamiltonian systems,Celest.Mech. and Dyn. Astro.,49(1),43-67

[8] Ishwar,B.(1997),Non-linear stability in the generalized restricted three body problem, Celest Mech. Dyn. Astron. 65,253-289

[9] Kushvah, B.S.,Ishwar,B.,(2006), Linear stability of triangular equilibrium points in the generalized photogravitational restricted three body problem with Poynting-Robertson drag, Journal of Dynamical system & Geometric Theories 4(1), 79-86

[10] Kushvah et al.(2007),Non-linear stability in the generalized photogravitational restricted three body problem with Poynting –Robertson drag, Astrophys Space Sci,312,279-293

[11] Kumar,S.,Ishwar,B.,(2009), Solutions of generalized photogravitational elliptic restricted three body problem, AIP Proceedings,1146(1),456-461

[12] Moser, J.,1962,On invariant curves of area preserving mappings of annulas, Nachr. Acad. Wiss. Gottingern, Math. Phys. KI,II,1-20.

[13] Murray,C.D.,1994,Dynamical effects of drag in the circular restricted three body problem.1: Location and stability of the Lagrangian equilibrium points,Icarus 112,465-484.

[14] Markellos,V.V.,et al.(1996), Non-linear stability zones around triangular equilibria in the plane circular restricted three-body problem with oblateness,Astrophys.Space Sci,245,157-164.

[15] Mishra,V.K.,Ishwar,B.,(2015),Normalization of Hamiltonian in photogravitational elliptic restricted three body problem with Poynting-Robertson drag, International Journal of Advanced Astronomy.3(1),42-45.

[16] Mishra,V.K., et al.(2016),Stability of triangular equilibrium points in photogravitational elliptic restricted three body problem with Poynting-Robertson drag, International Journal of Advanced Astronomy,4(1),33-38.

[17] Narayan,A.,Kumar,C.R.,(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(2),201-224.

[18] Narayan,A.,et al.,(2015), Characteristics exponents of the triangular solution in the elliptical restricted three body problem under the radiation and oblateness of primaries, IJAA,3(2),107-116.

[19] Poynting, J.H., (1903), Radiation in the solar system : its effect on temperature and its pressure on small bodies, Philos. Trans. R. Soc. Lond., 202: 525–552.

[20] Robertson.H.P.(1937), Dynamical effects of radiation in the solar system, Mon.Not.R.Astron.Soc.97,423-438.

[21] Radzievskii,V.V. (1950), The restricted problem of three bodies taking account of light pressure, Astron.Z,27,250.

[22] SubbaRao,P.V.,et al.(1997), Effect of oblateness on the non-linear stability of in the restricted three body problem, Celest Mech.Dyn.Astron,65,291-312.

[23] Schuerman, D. (1980), The restricted three body problem including radiation pressure, Astrophys. J., 238: 337– 342.

[24] Sahoo,S.K.,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.

[25] Singh,J.,Ishwar,B.,(1999), Stability of triangular points in the generalized photogravitational restricted three body problem,BASI 27,415.

[26] Singh,J.,Umar,A.,2012,On the stability of triangular points in the elliptic R3BP under radiating and oblate primaries,Astrophysics and Space Sci.341,349-358.

[27] Whittaker(1965), A treatise on the analytical dynamics of particles and rigid bodies,Cambridge Cambridge University Press,London,427-430.