Canonical Treatments for the Two Bodies Problem with Varying Mass Taking into Consideration the Periastron Effect

Affiliation(s)

Astronomy Department, Faculty of Science, King Abdu Aziz University, Jeddah, Saudi Arabia.

Astronomy Department, Faculty of Science, King Abdu Aziz University, Jeddah, Saudi Arabia.

ABSTRACT

In this work, the Hamiltonian of the two body problem with varying mass was developed in the extended phase space taking into consideration the periastron effects. The short period solution was obtained through constructing a second order canonical transformation using “Hori’s” method developed by “Kamel”. The elements of the transformation as well as the inverse transformation were obtained too. The final solution of the problem was derived using “Delva-Hanslmeier” method.

Cite this paper

El-Saftawy, M. and Algethami, A. (2014) Canonical Treatments for the Two Bodies Problem with Varying Mass Taking into Consideration the Periastron Effect.*International Journal of Astronomy and Astrophysics*, **4**, 70-79. doi: 10.4236/ijaa.2014.41008.

El-Saftawy, M. and Algethami, A. (2014) Canonical Treatments for the Two Bodies Problem with Varying Mass Taking into Consideration the Periastron Effect.

References

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[2] Prieto, C. (1995) Publicaciones del Departamento de Matematica Aplicada, Universidade de Santiago de Compostela-Spain.

[3] Hadjidemetriou, J. (1963) Two-Body Problem with Variable Mass: A New Approach. Icarus, 2, 440-451. http://dx.doi.org/10.1016/0019-1035(63)90072-1

[4] Hadjidemetriou, J. (1966) Analytic Solutions of the Two-Body Problem with Variable Mass. Icarus, 5, 34-46. http://dx.doi.org/10.1016/0019-1035(66)90006-6

[5] Jeans, J.H. (1924) Cosmogonic Problems Associated with a Secular Decrease of Mass. MNRAS, 85, 2-11.

[6] Jeans, J.H. (1925) The Effect of Varying Mass on a Binary System. MNRAS, 85, 912-925.

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[8] Prieto, C. and Docobo, J.A. (1997) On the Two-Body Problem with Slowly Decreasing Mass. Celestial Mechanics and Dynamical Astronomy, 68, 53-62. http://dx.doi.org/10.1023/A:1008235630740

[9] Docobo, J.A., Blanco, J. and Abelleira, P. (1999) Monografias de la Academiade Ciencias de Zaragoza, 14, II Jornadas de Mecanica Celeste, Academia de Ciencias de Zaragoza, Zaragoza, 33. (in Spanish)

[10] Andrade, M. and Docobo, J.A. (2000) The Influence of Decreasing Mass on the Orbits of Wide Binaries: An Approach to the Problem. Proceedings of the 4th Scientific Meeting of the Spanish Astronomical Society (SEA), Santiago de Compostela, 11-14 September 2000, 273.

[11] Andrade, M. and Docobo, J.A. (2002) The Influence of Mass Loss on Orbital Elements of Binary Systems by Periastron Effect. International Conference on Classical Nova Explosions, AIP Conference Proceedings, Santiago de Compostela, 82-85.

[12] Andrade, M. and Docobo, J.A. (2003) Orbital Dynamics Analysis of Binary Systems in Mass-Loss Scenarios. Conference to Honor John Dyson, Pátzcuaro, Michoacán, México, Santiago de Compostela, 223-225.

[13] Rahoma, W.A., et al. (2009) Analytical Treatment of the Two-Body Problem with Slowly Varying Mass. Journal of Astrophysics and Astronomy, 30, 187-205. http://dx.doi.org/10.1007/s12036-009-0012-y

[14] Delva, M. (1984) Integration of the Elliptic Restricted Three-Body Problem with Lie Series. Celestial Mechanics, 34, 145-154. http://dx.doi.org/10.1007/BF01235797

[15] Hanslmeier, A. (1984) Application of Lie-Series to Regularized Problems in Celestial Mechanics. Celestial Mechanics, 34, 135-143. http://dx.doi.org/10.1007/BF01235796

[16] Rahoma, W.A., et al. (2011) Two-Body Problem with Varying Mass in Case of Isotropic Mass Loss. Advances in Theoretical and Applied Mechanics, 4, 69-80.

[17] Deprit, A. (1983) The Secular Accelerations in Gylden’s Problem. Celestial Mechanics, 31, 1-22. http://dx.doi.org/10.1007/BF01272557

[18] Hori, G. (1966) Theory of General Perturbation with Unspecified Canonical Variable. Astronomical Society of Japan, 18, 287.

[19] Kamel, A.A. (1969) Expansion Formulae in Canonical Transformations Depending on a Small Parameter. Celestial Mechanics, 1, 190-199. http://dx.doi.org/10.1007/BF01228838

[20] Ahmed, M.K.M. (1994) On the Normalization of Perturbed Keplerian Systems. The Astronomical Journal, 107, 1900-1903. http://dx.doi.org/10.1086/117001

[1] Polyakhova, E.N. (1994) A Two-Body Variable-Mass Problem in Celestial Mechanics: The Current State. Astronomy Reports, 38, 283-291.

[2] Prieto, C. (1995) Publicaciones del Departamento de Matematica Aplicada, Universidade de Santiago de Compostela-Spain.

[3] Hadjidemetriou, J. (1963) Two-Body Problem with Variable Mass: A New Approach. Icarus, 2, 440-451. http://dx.doi.org/10.1016/0019-1035(63)90072-1

[4] Hadjidemetriou, J. (1966) Analytic Solutions of the Two-Body Problem with Variable Mass. Icarus, 5, 34-46. http://dx.doi.org/10.1016/0019-1035(66)90006-6

[5] Jeans, J.H. (1924) Cosmogonic Problems Associated with a Secular Decrease of Mass. MNRAS, 85, 2-11.

[6] Jeans, J.H. (1925) The Effect of Varying Mass on a Binary System. MNRAS, 85, 912-925.

[7] Prieto C. and Docobo, J.A. (1997) Analytic Solution of the Two-Body Problem with Slowly Decreasing Mass. Astronomy & Astrophysics, 318, 657-661.

[8] Prieto, C. and Docobo, J.A. (1997) On the Two-Body Problem with Slowly Decreasing Mass. Celestial Mechanics and Dynamical Astronomy, 68, 53-62. http://dx.doi.org/10.1023/A:1008235630740

[9] Docobo, J.A., Blanco, J. and Abelleira, P. (1999) Monografias de la Academiade Ciencias de Zaragoza, 14, II Jornadas de Mecanica Celeste, Academia de Ciencias de Zaragoza, Zaragoza, 33. (in Spanish)

[10] Andrade, M. and Docobo, J.A. (2000) The Influence of Decreasing Mass on the Orbits of Wide Binaries: An Approach to the Problem. Proceedings of the 4th Scientific Meeting of the Spanish Astronomical Society (SEA), Santiago de Compostela, 11-14 September 2000, 273.

[11] Andrade, M. and Docobo, J.A. (2002) The Influence of Mass Loss on Orbital Elements of Binary Systems by Periastron Effect. International Conference on Classical Nova Explosions, AIP Conference Proceedings, Santiago de Compostela, 82-85.

[12] Andrade, M. and Docobo, J.A. (2003) Orbital Dynamics Analysis of Binary Systems in Mass-Loss Scenarios. Conference to Honor John Dyson, Pátzcuaro, Michoacán, México, Santiago de Compostela, 223-225.

[13] Rahoma, W.A., et al. (2009) Analytical Treatment of the Two-Body Problem with Slowly Varying Mass. Journal of Astrophysics and Astronomy, 30, 187-205. http://dx.doi.org/10.1007/s12036-009-0012-y

[14] Delva, M. (1984) Integration of the Elliptic Restricted Three-Body Problem with Lie Series. Celestial Mechanics, 34, 145-154. http://dx.doi.org/10.1007/BF01235797

[15] Hanslmeier, A. (1984) Application of Lie-Series to Regularized Problems in Celestial Mechanics. Celestial Mechanics, 34, 135-143. http://dx.doi.org/10.1007/BF01235796

[16] Rahoma, W.A., et al. (2011) Two-Body Problem with Varying Mass in Case of Isotropic Mass Loss. Advances in Theoretical and Applied Mechanics, 4, 69-80.

[17] Deprit, A. (1983) The Secular Accelerations in Gylden’s Problem. Celestial Mechanics, 31, 1-22. http://dx.doi.org/10.1007/BF01272557

[18] Hori, G. (1966) Theory of General Perturbation with Unspecified Canonical Variable. Astronomical Society of Japan, 18, 287.

[19] Kamel, A.A. (1969) Expansion Formulae in Canonical Transformations Depending on a Small Parameter. Celestial Mechanics, 1, 190-199. http://dx.doi.org/10.1007/BF01228838

[20] Ahmed, M.K.M. (1994) On the Normalization of Perturbed Keplerian Systems. The Astronomical Journal, 107, 1900-1903. http://dx.doi.org/10.1086/117001