Approximate Kepler’s Elliptic Orbits with the Relativistic Effects

Author(s)
Leilei Jia

Affiliation(s)

Department of Electrical Engineering, Guilin College of Aerospace Technology, Guilin, China.

Department of Electrical Engineering, Guilin College of Aerospace Technology, Guilin, China.

ABSTRACT

Beginning with a Lagrangian, we derived an approximate relativistic orbit equation which describes relativistic corrections to Keplerian orbits. The critical angular moment to guarantee the existence of periodic orbits is determined. An approximate relativistic Kepler’s elliptic orbit is illustrated by numerical simulation via a second-order perturbation method of averaging.

Cite this paper

L. Jia, "Approximate Kepler’s Elliptic Orbits with the Relativistic Effects,"*International Journal of Astronomy and Astrophysics*, Vol. 3 No. 1, 2013, pp. 29-33. doi: 10.4236/ijaa.2013.31004.

L. Jia, "Approximate Kepler’s Elliptic Orbits with the Relativistic Effects,"

References

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[22] P. J. Torres, A. J. Urena and M. Zamora, “Periodic and Quasi-Periodic Motions of a Relativistic Particle under a Central Force Field,” Bulletin London Mathematical Society, 2012, pp. 1-13. doi: 10.1112/blms/bds076

[23] Q. Liu and D. Qian, “Construction of Modulated Amplitude Waves via Averaging in Collisionally Inhomogeneous Bose-Einstein Condensates,” Journal of Nonlinear Mathematical Physics, Vol. 19, No. 2, 2012, Article ID: 1250017. doi:10.1142/S1402925112500179

[24] Q. Liu, and D. Qian, “Modulated Amplitude Waves with Nonzero Phases in Bose-Einstein Condensates,” Journal of Mathematical Physics, Vol. 52, No. 8, 2011, Article ID: 082702. doi:10.1063/1.3623415

[25] L. Jia, L. Sun and J. Li, “Modulational Instability in Nonlinear Optics,” Journal of Jiangxi Normal University (Natural Science Edition), Vol. 14, No. 3, 2012, pp. 271-275.

[26] J. A. Sanders, F. Verhulst and J. Murdock, “Averaging Methods in Nonlinear Dynamical Systems,” Springer, New York, 2007.

[1] R. Bate, D. Mueller and J. White, “Fundamentals of Astrodynamics,” Dover Publications, New York, 1971.

[2] S. C. Bell, “A numerical Solution of the Relativistic Kepler Problem,” Computers in Physics, Vol. 9, No. 3, 2001, pp. 281-285. doi:10.1063/1.168530

[3] P. Amster, J. Haddad, R. Oterga and A. J. Urena, “Periodic Motions in Forced Problems of Kepler Type,” Nonlinear Differential Equations and Applications, Vol. 18, No. 6, 2011, pp. 649-657. doi:10.1007/s00030-011-0111-8

[4] A. Fonda, R. Toader, and P. J. Torres, “Periodic Motions in a Gravitational Central Field with a Rotating External Force,” Celestial Mechanics and Dynamical Astronomy, Vol. 113, No. 3, 2012, pp. 335-342. doi:10.1007/s10569-012-9428-9

[5] X. GongOu, X. MingJie and Y. YaTian, “Kepler Problem in Hamiltonian Formulation Discussed from Topological Viewpoint,” Chinese Physics Letters, Vol. 22, No. 7, 2005, pp. 1573-1575. doi:10.1088/0256-307X/22/7/004

[6] K. Meyer and G. Hall, “Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,” Springer, New York, 1992.

[7] R. G. Cawley, “Motion of a Charged Light Like Particle in an External Field,” Journal of Mathematical Physics, Vol. 380, No. 14, 1967, pp. 2092-2096.

[8] T. E. Phipps, “Mercury’s Precession according to Special Relativity,” American Journal of physics, Vol. 54, No. 3, 1986, pp. 245-247.

[9] C. Sigismondi, “Astrometry and Relativity,” Nuovo Cimento B Serie, Vol. 120, No. 10, 2005, pp. 1169-1180.

[10] N. Jun, “Unification of General Relativity with Quantum Field Theory,” Chinese Physics Letters, Vol. 28, No. 11, 2011, p. 110401. doi:10.1088/0256-307X/28/11/110401

[11] F. JianHui, “Study of the Lie Symmetries of a Relativistic Variable Mass System,” Chinese Physics, Vol. 11, No. 4, 2002, pp. 313-318. doi:10.1088/1009-1963/11/4/301

[12] F. J. Hui and Z. Song-Qing, “Noether’s Theorem of a Rotational Relativistic Variable Mass System,” Chinese Physics, Vol. 11, No. 5, 2002, pp. 445-449. doi:10.1088/1009-1963/11/5/307

[13] O. Coskun, “The Solutions of the Classical Relativistic Two-Body Equation,” Turkish Journal of Physics, Vol. 22, No. 2, 2002, pp. 107-114.

[14] A. Schild, “Electromagnetic Two-Body Problem,” Physical Review, Vol. 131, No. 6, 1963, pp. 2762-2766. doi:10.1103/PhysRev.131.2762

[15] C. M. Andersen and H. C. Baeyer, “Circular Orbits in Classical Relativistic Two-Body Systems,” Annals of Physics, Vol. 60, No. 1, 1970, pp. 67-84. doi:10.1016/0003-4916(70)90482-3

[16] P. Cordero and G. C. Ghirardi, “Dynamics for Classical Relativistic Particles: Circular Orbit Solutions and the Nonrelativistic Limit,” Journal of Mathematical Physics, Vol. 14, No. 7, 1973, pp. 815-822. doi: 10.1063/1.1666401

[17] T. J. Lemmon and A. R. Mondragon, “Alternative Derivation of the Relativistic Contribution to Perihelia Precession,” American Journal of Physics, Vol. 77, No. 10, 2009, pp. 890-893. doi:10.1119/1.3159611

[18] J. M. Potgieter, “An Exact Solution for the Horizontal Deflection of a Falling Object,” American Journal of Physics, Vol. 51, No. 3, 1983, pp. 257-258. doi:10.1119/1.13275

[19] Y. S. Huang and C. L. Lin, “A Systematic Method to Determine the Lagrangian Directly from the Equations of Motion,” American Journal of Physics, Vol. 70, No. 7, 2002, pp. 741-743. doi:10.1119/1.1475331

[20] B. Coleman, “Special Relativity Dynamics without a Priori Momentum Conservation,” European Journal of Physics, Vol. 26, No.4, 2005, pp. 647-650. doi:10.1088/0143-0807/26/4/010

[21] P. Smith and R. C. Smith, “Mechanics,” John Wiley & Sons Ltd., Chichester, 1990.

[22] P. J. Torres, A. J. Urena and M. Zamora, “Periodic and Quasi-Periodic Motions of a Relativistic Particle under a Central Force Field,” Bulletin London Mathematical Society, 2012, pp. 1-13. doi: 10.1112/blms/bds076

[23] Q. Liu and D. Qian, “Construction of Modulated Amplitude Waves via Averaging in Collisionally Inhomogeneous Bose-Einstein Condensates,” Journal of Nonlinear Mathematical Physics, Vol. 19, No. 2, 2012, Article ID: 1250017. doi:10.1142/S1402925112500179

[24] Q. Liu, and D. Qian, “Modulated Amplitude Waves with Nonzero Phases in Bose-Einstein Condensates,” Journal of Mathematical Physics, Vol. 52, No. 8, 2011, Article ID: 082702. doi:10.1063/1.3623415

[25] L. Jia, L. Sun and J. Li, “Modulational Instability in Nonlinear Optics,” Journal of Jiangxi Normal University (Natural Science Edition), Vol. 14, No. 3, 2012, pp. 271-275.

[26] J. A. Sanders, F. Verhulst and J. Murdock, “Averaging Methods in Nonlinear Dynamical Systems,” Springer, New York, 2007.