Third Order Hamiltonian for a Binary System with Varying Masses Including Preastron Effect

  • Doaa Saleh Al-Johani King Abdul-Aziz University
  • M. I. El-Saftawy National Research Institute of Astronomy and Geophysics
Keywords: Binary-system, Tow-body problem, Varying mass, Pertubation, Periastron effects

Abstract

This work concerns of the effects of the variation in the masses for two attracting bodies on the orbiter orbital elements. The formulation of the problem was done in different kind of mechanics, Newtonian, Lagrangian, and Hamiltonian. Moreover, constructing the Hamiltonian function of the varying masses of a binary system including, periastron effect, in canonical form in the extended phase space, up to third order of the small parameter ?, to be able to solve using canonical perturbation techniques. Canonical perturbation method based on Lee transformation was developed by Kamel used to remove the short periodic terms from the Hamiltonian to be able to solve the system of equations. The Hamiltonian of the system was transformed to the extended phase space by introducing two variable represents the variation of the masses and their conjugate momenta. Finally, Hamilton's equation of motions was used to drive general formula to calculate the variations in the elements due to the variations in their masses and what so called periastron effects.

Author Biographies

Doaa Saleh Al-Johani, King Abdul-Aziz University

King Abdul-Aziz University, Faculty of Science, Department of Astronomy and Space, Jeddah, K.S.A

M. I. El-Saftawy, National Research Institute of Astronomy and Geophysics

National Research Institute of Astronomy and Geophysics, Helwan, Cairo, Egypt.

References

Gyldén, H. (1884):Astronomische Nachrichten, volume 109, Issue 1, p.1

Rahoma, W. A. et al. (2009): J. Astrophys. and Astr., Vol 30, pp. 187–205.

Delva, M. (1984): Celest. Mech., Vol. 34, pp. 145.

Halslmeier, A. (1984) : Celest. Mech., vol.3, p.107.

El-Saftawy, M. I. and Algethami, A. R. (2014): International Journal of Astronomy and Astrophysics, Vol. 4, pp. 70-79.

Hori, G (1966): "Space Mathematics " Vol. 1 part 3, Amer. Math. Soc.

Kamel, A. A. (1969): Celest. Mech., Vol. 1, No 2, pp. 190-199.

M. I. El-Saftawy and F. A. Abd El-Salam (2017): " Second Order Theory for the Two-Body Problem with Varying Mass Including Periastron Effect."; Journal of Nonlinear Dynamics., Springer Science, DOI 10.1007/s11071-017-3341-4.

Deprit, A. (1983): Celest. Mech., Vol. 31, pp. 1-22G. Eason, B. Noble, and I. N. Sneddon, “On certain integrals of Lipschitz-Hankel type involving products of Bessel functions,” Phil. Trans. Roy. Soc. London, vol. A247, pp. 529–551, April 1955. (references)

Andrade, M. & Docobo, J. A. Estudio de la estabilidad en sistemas estelares triples conp´erdida de masa. Mon. Acad. Cienc. Zaragoza, 25, 13–22, (2004).

Jeans, J. H. (1924): MNRAS, Vol. 64, No. 1, pp.2-16.

Jeans, J. H. (1925): MNRAS, Vol. 85, No. 9, pp. 912 – 925.

Published
2019-05-30
How to Cite
Al-Johani, D., & El-Saftawy, M. (2019). Third Order Hamiltonian for a Binary System with Varying Masses Including Preastron Effect. JOURNAL OF ADVANCES IN MATHEMATICS, 16, 8423-8429. https://doi.org/10.24297/jam.v16i0.8299
Section
Articles