A general solution to non-collinear equilibria in terms of largest root (κ) of confocal oblate spheroid

  • Authors

    • M Javed Idrisi School of Physical and Molecular SciencesDepartment of MathematicsAl-Falah UniversityFaridabad (Haryana) - 121004
    2016-01-13
    https://doi.org/10.14419/ijaa.v4i1.5587
  • Celestial Mechanics, Restricted Three-Body Problem, Libration Points, Oblate Spheroid, Confocal Oblate Spheroid.

  • Abstract

    This paper deals with the existence of non-collinear equilibria in restricted three-body problem when less massive primary is an oblate spheroid and the potential of oblate spheroid is in terms of largest root of confocal oblate spheroid. This is found that the non-collinear equilibria are the solution of the equations r1 = n-2/3 and κ = 1 – a2, where r1 is the distance of the infinitesimal mass from more massive primary, n is mean-motion of primaries, a is semi axis of oblate spheroid and κ is the largest root of the equation of confocal oblate spheroid passes through the infinitesimal mass.

  • References

    1. [1] MacMillan, W. D.: The theory of the potential. Dover Publications, New York (1958).

      [2] 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.

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

      [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., Hallan, P. P.: Effect of perturbed potentials on the stability of Libration points restricted problem. Celestial Mechanics and Dynamical Astronomy, 20(2), 95-103 (1979). http://dx.doi.org/10.1007/BF01230231

      [10] 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.

      [11] 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.

      [12] 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.

      [13] 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.

      [14] 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.

      [15] 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).

      [16] 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).

      [17] 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).

      [18] 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.

      [19] 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.

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

      [21] 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.

      [22] John, A. Arredondo, Jianguang, Guo, Cristina, Stoica, Claudia, and 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.

      [23] 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.

      [24] 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.

      [25] 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.

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  • How to Cite

    Idrisi, M. J. (2016). 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. https://doi.org/10.14419/ijaa.v4i1.5587

    Received date: 2015-11-29

    Accepted date: 2015-12-21

    Published date: 2016-01-13