Restricted three-body problem with stokes drag effect when less massive primary is an ellipsoid

Authors

  • M Javed Idrisi

    School of Physical and Molecular SciencesDepartment of MathematicsAl-Falah UniversityFaridabad (Haryana) - 121004

  • Mamta Jain

    Department of Mathematics, Shri Venkateshwara University, U.P. – 244236

How to Cite

Idrisi, M. J., & Jain, M. (2016). Restricted three-body problem with stokes drag effect when less massive primary is an ellipsoid. International Journal of Advanced Astronomy, 4(1), 61-67. https://doi.org/10.14419/ijaa.v4i1.6140

Received date: April 19, 2016

Accepted date: May 10, 2016

Published date: May 27, 2016

DOI:

https://doi.org/10.14419/ijaa.v4i1.6140

Keywords:

Restricted Three Body Problem, Libration Points, Linear Stability, Stokes Drag.

Abstract

The present paper deals with the effect of Stokes drag force on the existence and stability of collinear and non-collinear libration points in circular restricted three-body problem when less massive primary is an ellipsoid. During the investigation, it is found that there exist five libration points Li (i = 1, 2… 5) out of which three are collinear and two are non-collinear. We observed that the Stokes drag force does not affect the collinear libration points while non-collinear libration points are affected by it and all the libration points either collinear or non-collinear are unstable in Lyapunov sense for the given range of dissipative constant k and mass parameter µ.

References

  1. [1] Danby, J.M.A.: Stability of the triangular points in the elliptic restricted problem of three bodies. The Astronomical Journal, 69 (2), 165 – 172 (1964).http://dx.doi.org/10.1086/109254.

    [2] Szebehely, V.: Theory of orbits, The Restricted Problem of three bodies. Academic Press, New York and London (1967).

    [3] Chernikov, Y.A.: The photogravitational restricted three body problem. Soviet Astronomy-AJ, 14, 176-181 (1970).

    [4] Vidyakin, V.V.: Stability of one particular solution for the motion of three homogeneous spheroids.Soviet Astronomy, 18, 116 (1974).

    [5] Sharma, R.K.: Perturbations of Lagrangian points in the restricted three-body problem. Indian Journal of Pure and Applied Mathematics, 6, 1099-1102 (1975).

    [6] Subbarao, P.V., Sharma, R.K.: A note on the Stability of the triangular points of equilibrium in the restricted three body problem. Astronomy and Astrophysics, 43, 381-383 (1975).

    [7] Sharma, R.K., Subbarao P.V.: Stationary solutions and their characteristic exponents in the restricted three-body problem when the more massive primary is an oblate spheroid. Celestial Mechanics and Dynamical Astronomy, 13, 137-149 (1976). http://dx.doi.org/10.1007/BF01232721.

    [8] Choudhary R.K.: Libration points in the generalized elliptic restricted three body problem. Celestial Mechanics, 16, 411 – 419 (1977). http://dx.doi.org/10.1007/BF01229285.

    [9] Bhatnagar, K.B., Chawla, J.M.: A study of Lagrangian points in the photogravitational restricted three-body problem. Indian J. Pure Appl. Math. 10(11), 1443–1451 (1979).

    [10] Schuerman, D.W.: The restricted three body problem including radiation pressure. The Astrophysical Journal, 238, 337-342 (1980). http://dx.doi.org/10.1086/157989.

    [11] Kunitsyn, A.L., Tureshbaev, A.T.: On the collinear libration points in the photogravitational three-body problem. Celest. Mech. 35(2), 105–112 (1985). http://dx.doi.org/10.1007/BF01227664.

    [12] Cid, R., Ferrer, S., Caballero, J.A.: Asymptotic solutions of the restricted problem near equilateral Lagrangian points. Celestial Mechanics and Dynamical Astronomy, 35, 189-200 (1985). http://dx.doi.org/10.1007/BF01227668.

    [13] Simmons, J.F.L., McDonald, A.J.C., Brown, J.C.: The restricted 3-body problem with radiation pressure. Celestial Mechanics and Dynamical Astronomy, 35, 145-187 (1985). http://dx.doi.org/10.1007/BF01227667.

    [14] Sharma, R.K.: The linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid. Astrophys. Space Sci. 135(2), 271–281 (1987). http://dx.doi.org/10.1007/BF00641562.

    [15] Lukyanov, L.G.: On the family of the libration points in the restricted photogravitational three-body problem. Astron. Zh. 65, 422–432 (1988).

    [16] El-Shaboury, S.M.: Equilibrium solutions of the restricted problem of 2+2 axisymmetric rigid bodies. Celestial Mechanics and Dynamical Astronomy, 50, 199-208 (1991). http://dx.doi.org/10.1007/BF00048764.

    [17] Lang, K. R.: Astrophysical Data: Planets and Stars. Springer-Verlag, New York (1992). http://dx.doi.org/10.1007/978-1-4684-0640-5.

    [18] Xuetang, Z.H.E.N.G., Lizhong, Y.U.: Photogravitationally restricted three-body problem and coplanar libration point. Chinese Phys. Lett., 10 (1), 61 – 64 (1993). http://dx.doi.org/10.1088/0256-307X/10/1/017.

    [19] Bhatnagar, K.B., Gupta, U., Bharadwaj, R.: Effect of perturbed potentials on the non-linear stability of Libration point L4 in the restricted problem. Celestial Mechanics and Dynamical Astronomy, 59, 45-374 (1994). http://dx.doi.org/10.1007/BF00692102.

    [20] Murray, C.D.: Dynamical effects of drag in the circular restricted three body problems: 1.Location and stability of the Lagrangian equilibrium points. Icarus 112, 465-184 (1994). http://dx.doi.org/10.1006/icar.1994.1198.

    [21] Selaru, D., Cucu-Dumitrescu, C.: Infinitesimal orbits around Lagrange points in the elliptic, restricted three-body problem. Celestial Mechanics and Dynamical Astronomy, 61 (4), 333 – 346 (1995). http://dx.doi.org/10.1007/BF00049514.

    [22] Markellos, V.V., Papadakis, K.E., Perdios, E.A.: Non-linear stability zones around triangular equilibria in the plane circular restricted three-body problem with oblateness. Astrophysics and Space Science, 245, 157-164 (1996). http://dx.doi.org/10.1007/BF00637811.

    [23] Subbarao, P.V., Sharma, R.K.: Effect of oblateness on the non-linear stability of L4 in the restricted three-body problem. Celestial Mechanics and Dynamical Astronomy, 65, 291-312 (1997).

    [24] Khanna, M., Bhatnagar, K.B.: Existence and stability of libration points in restricted three body problem when the smaller primary is a triaxial rigid body. Indian Journal of Pure and Applied Mathematics, 29 (10), 1011-1023 (1998).

    [25] Khanna, M., Bhatnagar, K.B.: 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 (7), 721-733 (1999).

    [26] Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999).

    [27] Grün, E., Gustafson, B.A.S., Dermott, S., Fechtig, H.: Interplanetary Dust. Springer, Berlin (2001).http://dx.doi.org/10.1007/978-3-642-56428-4.

    [28] Roberts, G.E.: Linear Stability of the Elliptic Lagrangian Triangle Solutions in the three-body problem. Journal of Differential Equations, 182 (1), 191 – 218 (2002). http://dx.doi.org/10.1006/jdeq.2001.4089.

    [29] Oberti, P.,Vienne, A.: An upgraded theory for Helene, Telesto and Calypso. Astronomy and Astrophysics, 397, 353-359 (2003).http://dx.doi.org/10.1051/0004-6361:20021518.

    [30] Perdiou, A.E., Markellos, V.V., Douskos, C.N.: The Hill problem with oblate secondary: Numerical Exploration. Earth, Moon and Planets, 97, 127-145 (2005). http://dx.doi.org/10.1007/s11038-006-9065-y.

    [31] Sosnytskyi, S.P.: On the Lagrange stability of motion in the three-body problem. Ukrainian Mathematical Journal, 57 (8), 1341-1349 (2005). http://dx.doi.org/10.1007/s11253-005-0266-8.

    [32] Kushvah, B.S., Sharma, J.P., Ishwar, B.: Nonlinear Stability in the Generalized Photogravitational Restricted Three Body Problem with Poynting-Robertson Drag. Astrophysics and Space Science, 312, 279-293 (2007). http://dx.doi.org/10.1007/s10509-007-9688-0.

    [33] Ammar, M.K.: The effect of solar radiation pressure on the Lagrangian points in the elliptic restricted three-body problem. Astrophysics and Space Science, 313 (4), 393 – 408 (2008). http://dx.doi.org/10.1007/s10509-007-9709-z.

    [34] Douskos, C.N.: Collinear equilibrium points of Hill’s problem with radiation and oblateness and their fractal basins of attraction. Astrophysics and Space Science, 326, 263-271 (2010). http://dx.doi.org/10.1007/s10509-009-0213-5.

    [35] Singh, Jagdish, Leke, O.: Stability of the photogravitational restricted three body problem with variable masses. Astrophysics and Space Science, 326, 305-314 (2010). http://dx.doi.org/10.1007/s10509-009-0253-x.

    [36] Celletti, A., Stefanelli, L., Lega, E., Froeschlé, C.: Some Results on the Global Dynamics of the Regularized Restricted Three-Body Problem with Dissipation. Celestial Mechanics and Dynamical Astronomy, 109, 265-284 (2011). http://dx.doi.org/10.1007/s10569-010-9326-y.

    [37] Ershkov, Sergey V.: The Yarkovsky effect in generalized photogravitational 3-body problem. Planetary and Space Science, 70 (1), 221-223 (2012). http://dx.doi.org/10.1016/j.pss.2012.09.002.

    [38] John, A. Arredondo, Jianguang, Guo, Cristina, Stoica, Claudia, Tamayo: On the restricted three body problem with oblate primaries. Astrophysics and Space Science, 341, 315-322 (2012). http://dx.doi.org/10.1007/s10509-012-1085-7.

    [39] Idrisi, M. Javed, Taqvi, Z.A.: Restricted three-body problem when one of the primaries is an ellipsoid. Astrophysics and Space Science, 348, 41-56 (2013). http://dx.doi.org/10.1007/s10509-013-1534-y.

    [40] Idrisi, M. Javed: Existence and stability of the libration points in CR3BP when the smaller primary is an oblate spheroid. Astrophysics and Space Science, 354, 311-325 (2014). http://dx.doi.org/10.1007/s10509-014-2031-7.

    [41] Katour, D.A., Abd El-Salam, F.A., Shaker, M.O.: Relativistic restricted three body problem with oblateness and photo-gravitational corrections to triangular equilibrium points. Astrophysics and Space Science, 351, 143 – 149 (2014). http://dx.doi.org/10.1007/s10509-014-1826-x.

    [42] Idrisi, M. Javed, Amjad, Muhammad: Effect of elliptic angle φ on the existence and stability of libration points in restricted three-body problem in earth-moon system considering earth as an ellipsoid. International Journal of Advanced Astronomy, 3 (2),87-96 (2015). http://dx.doi.org/10.14419/ijaa.v3i2.5313.

    [43] Jain, Mamta, Aggarwal, Rajiv: Restricted three body problem with Stokes Drag effect. International Journal of Astronomy and Astrophysics, 5, 95-105 (2015). http://dx.doi.org/10.4236/ijaa.2015.52013.

    [44] Jain, Mamta, Aggarwal, Rajiv: A study of non-collinear libration points in restricted three body problem with stokes drag effect when smaller primary is an oblate spheroid. Astrophysics and Space Science, 358: 51 (2015). http://dx.doi.org/10.1007/s10509-015-2457-6.

    [45] Singh, J., Amuda, T.O.: Out-of-plane equilibrium points in the photogravitational CR3BP with oblateness and P-R drag. Journal of Astrophysics and Astronomy, 36, 291-305 (2015). http://dx.doi.org/10.1007/s12036-015-9336-y.

    [46] Lhotka, C., Celletti, A.: The effect of Poynting Robertson drag on the triangular Lagrangian points. Astrophysics and Space Science, 250, 249-261 (2015).http://dx.doi.org/10.1016/j.icarus.2014.11.039.

    [47] Mishra, Vivek Kumar, Ishwar, Bhola: Normalisation of Hamiltonian in photogravitational elliptic restricted three body problem with Poynting-Robertson drag. International Journal of Advanced Astroomy, 3, 42-45 (2015). http://dx.doi.org/10.14419/ijaa.v3i1.4534.

    [48] Pal, Ashok Kumar, Kushvah, Badam Singh: Geometry of halo and lissajous orbits in the circular restricted three body problem with drag forces.MNRAS446, 959-972 (2015).

    [49] Idrisi, M. Javed: A general solution to non-collinear equilibria in terms of largest root (κ) of confocal oblate spheroid. International Journal of Advanced Astronomy, 4 (1), 1 – 4 (2016). http://dx.doi.org/10.14419/ijaa.v4i1.5587.

    [50] Idrisi, M. Javed, Shalini, Kumari: Non-collinear libration points in CR3BP when less massive primary is an heterogeneous oblate body with N-layers. International Journal of Advanced Astronomy, 4 (1), 39 – 42 (2016). http://dx.doi.org/10.14419/ijaa.v4i1.5928.

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How to Cite

Idrisi, M. J., & Jain, M. (2016). Restricted three-body problem with stokes drag effect when less massive primary is an ellipsoid. International Journal of Advanced Astronomy, 4(1), 61-67. https://doi.org/10.14419/ijaa.v4i1.6140

Received date: April 19, 2016

Accepted date: May 10, 2016

Published date: May 27, 2016