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Abstract
The paper presents the behavior of the motion properties of the variable mass test particle (third body), moving under the influence of the two equal primaries having electromagnetic dipoles. These primaries move on the same circular path around their common center of mass in the same plane. We have determined the equations of motion of the test particle whose mass varies according to Jean's law. Using the system of equations of motion we have evaluated the locations of equilibrium points, their movements and basins of the attracting domain. Finally, we examine the stability of these equilibrium points, all of which are unstable.
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References
- Brouwer, Dirk and Clemence, Gerald M. Methods of celestial mechanics: Elsevier, 2013.
- Szebehely, V. Theory of Orbits: Academic Press, New York, 1967.
- Goudas, C.L. and Petsagourakis, E.G. Motions in the magnetic field of two revolving dipoles. (V.G. Szebehely red.) Stability of the Solar System and Its Minor Natural and Artificial Bodies, by Dr. Reidel Publishing Company, 1985, 349-364.
- Kalvouridis, T.J. Three-Dimensional equilibria and their stability in the magnetic-binary problem. Astrophysics and Space Science, 1989, 159, 91-97.
- Zotos, E.E. Determining the Newton-Raphson basins of attraction in the electromagnetic Copenhagen problem. International Journal of Non-linear Mechanics, 2017, 90, 111-123.
- Benet, L., Trautman, D. and Seligman, T. Chaotic scattering in the restricted three-body problem. I. The Copenhagen problem. Celestial Mechanics and Dynamical Astronomy, 1996, 66, 203-228.
- Perdios, E.A. Asymptotic orbits and terminations of families in the Copenhagen problem. Astrophysics and Space Science, 1996, 240, 141-152.
- Fakis, D. and Kalvouridis, T. The Copenhagen problem with a quasi-homogeneous potential. Astrophysics and Space Science, 2017, 362,102, DOI 10.1007/s10509-017-3077-0.
- Kalvouridis, T.J. On a class of equilibria of a small rigid body in a Copenhagen configuration. Romanian Astronomical Journal, 2008, 18(2), 167-179.
- Kalvouridis, T.J. and Gousidou-Koutita, M.Ch. Basins of attraction in the Copenhagen problem where the primaries are magnetic dipoles. Applied Mechanics, 2012, 3, 541-548.
- Jeans, J.H. Astronomy and Cosmogony: Cambridge University Press, Cambridge. 1928.
- Meshcherskii, I.V. Works on the Mechanics of Bodies of Variable Mass: GITTL, Moscow, 1949.
- Singh, J. and Ishwar, B. Effect of perturbations on the location of equilibrium points in the restricted problem of three bodies with variable mass. Celestial Mechanics, 1984, 32, 297-305.
- Singh, J. and Ishwar, B. Effect of perturbations on the stability of triangular points in the restricted problem of three bodies with variable mass. Celestial Mechanics, 1985, 35, 201-207.
- Lukyanov, L.G. On the restricted circular conservative three-body problem with variable masses. Astronomy Letters, 2009, 35(5), 349-359.
- Zhang, M.J., Zhao, C.Y. and Xiong, Y.Q. On the triangular libration points in photo-gravitational restricted three-body problem with variable mass. Astrophysics and Space Science, 2012, 337, 107-113, doi 10.1007/s10509- 011-0821-8.
- Abouelmagd, E.I. and Ansari, A.A. The motion properties of the infinitesimal body in the framework of bicircular Sun-perturbed Earth-Moon system. New Astronomy, 2019, 73, 101282,
- https://doi.org/10.1016/j.newast.2019.101282.
- Abouelmagd, E.I., Mostafa, A. Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-body problem with variable mass. Astrophysics and Space Science, 2015, 357, 58, doi 10.1007/s10509-015-2294-7.
- Ansari, A.A. Effect of Albedo on the motion of the infinitesimal body in circular restricted three-body problem with variable masses. Italian Journal of Pure and Applied Mathematics, 2017, 38, 581-600.
- Ansari, A.A., Alhussain, Z.A., Sada Nand, P. Circular restricted three-body problem when both the primaries are heterogeneous spheroid of three layers and infinitesimal body varies its mass. Journal of Astrophysics and Astronomy, 2018, 39, 57.
- Ansari, A.A., et. al. Effect of charge in the circular restricted three-body problem with variable masses. Journal of Taibah University for Science, 2019, 13(1), 670-677.
References
Brouwer, Dirk and Clemence, Gerald M. Methods of celestial mechanics: Elsevier, 2013.
Szebehely, V. Theory of Orbits: Academic Press, New York, 1967.
Goudas, C.L. and Petsagourakis, E.G. Motions in the magnetic field of two revolving dipoles. (V.G. Szebehely red.) Stability of the Solar System and Its Minor Natural and Artificial Bodies, by Dr. Reidel Publishing Company, 1985, 349-364.
Kalvouridis, T.J. Three-Dimensional equilibria and their stability in the magnetic-binary problem. Astrophysics and Space Science, 1989, 159, 91-97.
Zotos, E.E. Determining the Newton-Raphson basins of attraction in the electromagnetic Copenhagen problem. International Journal of Non-linear Mechanics, 2017, 90, 111-123.
Benet, L., Trautman, D. and Seligman, T. Chaotic scattering in the restricted three-body problem. I. The Copenhagen problem. Celestial Mechanics and Dynamical Astronomy, 1996, 66, 203-228.
Perdios, E.A. Asymptotic orbits and terminations of families in the Copenhagen problem. Astrophysics and Space Science, 1996, 240, 141-152.
Fakis, D. and Kalvouridis, T. The Copenhagen problem with a quasi-homogeneous potential. Astrophysics and Space Science, 2017, 362,102, DOI 10.1007/s10509-017-3077-0.
Kalvouridis, T.J. On a class of equilibria of a small rigid body in a Copenhagen configuration. Romanian Astronomical Journal, 2008, 18(2), 167-179.
Kalvouridis, T.J. and Gousidou-Koutita, M.Ch. Basins of attraction in the Copenhagen problem where the primaries are magnetic dipoles. Applied Mechanics, 2012, 3, 541-548.
Jeans, J.H. Astronomy and Cosmogony: Cambridge University Press, Cambridge. 1928.
Meshcherskii, I.V. Works on the Mechanics of Bodies of Variable Mass: GITTL, Moscow, 1949.
Singh, J. and Ishwar, B. Effect of perturbations on the location of equilibrium points in the restricted problem of three bodies with variable mass. Celestial Mechanics, 1984, 32, 297-305.
Singh, J. and Ishwar, B. Effect of perturbations on the stability of triangular points in the restricted problem of three bodies with variable mass. Celestial Mechanics, 1985, 35, 201-207.
Lukyanov, L.G. On the restricted circular conservative three-body problem with variable masses. Astronomy Letters, 2009, 35(5), 349-359.
Zhang, M.J., Zhao, C.Y. and Xiong, Y.Q. On the triangular libration points in photo-gravitational restricted three-body problem with variable mass. Astrophysics and Space Science, 2012, 337, 107-113, doi 10.1007/s10509- 011-0821-8.
Abouelmagd, E.I. and Ansari, A.A. The motion properties of the infinitesimal body in the framework of bicircular Sun-perturbed Earth-Moon system. New Astronomy, 2019, 73, 101282,
https://doi.org/10.1016/j.newast.2019.101282.
Abouelmagd, E.I., Mostafa, A. Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-body problem with variable mass. Astrophysics and Space Science, 2015, 357, 58, doi 10.1007/s10509-015-2294-7.
Ansari, A.A. Effect of Albedo on the motion of the infinitesimal body in circular restricted three-body problem with variable masses. Italian Journal of Pure and Applied Mathematics, 2017, 38, 581-600.
Ansari, A.A., Alhussain, Z.A., Sada Nand, P. Circular restricted three-body problem when both the primaries are heterogeneous spheroid of three layers and infinitesimal body varies its mass. Journal of Astrophysics and Astronomy, 2018, 39, 57.
Ansari, A.A., et. al. Effect of charge in the circular restricted three-body problem with variable masses. Journal of Taibah University for Science, 2019, 13(1), 670-677.