Analytical theory in terms of J2, J3, J4 with eccentric anomaly for short-term orbit predictions using uniformly regular KS canonical elements


  • Athira Anand P Department of Aerospace EngineeringKarunya UniversityCoimbatore-641114Tamilnadu, India
  • Ram Krishan Sharma Department of Aerospace Engineering, Karunya Institute of Technology and Sciences, Coimbatore - 641114, Tamil Nadu





Hamilton’s Equations of Motion, Uniformly Regular KS Canonical Elements, Earth’s Oblateness, Short-Term Orbit Predictions, Analytical Integration.


A new non-singular analytical theory with respect to the Earth’s zonal harmonic terms J2, J3, J4 has been developed for short-periodic motion, by analytically integrating the uniformly regular KS canonical equations of motion using generalized eccentric anomaly ‘E’ as the independent variable. Only one of the eight equations need to be integrated analytically to generate the state vector, due to the symmetry in the equations of motion, and the computation for the other equations is done by changing the initial conditions. King-Hele’s expression for radial distance ‘r’ with J2 is also considered in generating the solution. The results obtained from the analytical expressions in a single step during half a revolution match quite well with numerically integrated values. Numerical results also indicate that the solution is reasonably accurate for a wide range of orbital elements during half a revolution and is an improvement over Sharon et al. [17] theory, which is generated in terms of KS regular elements. It can be used for studying the short-term relative motion of two or more space objects and in collision avoidance studies of space objects. It can be also useful for onboard computation in the navigation and guidance packages.



Author Biography

Athira Anand P, Department of Aerospace EngineeringKarunya UniversityCoimbatore-641114Tamilnadu, India



[1] Aksnes, K. (1970). A Second-Order Artificial Satellite Theory Based on an Intermediate Orbit, Astron. J, 75, 1066-1076.

[2] Celleti, A. and Negrini, P. (1995). Non-integrability of the problem of motion around an oblate planet,†Celestial Mechanics, 61, 253-260.

[3] Deprit, A. and Rom, A. (1970). The main problem of satellite theory for small eccentricities, Celestial Mechanics, vol. 4, 119-121, 1970.

[4] Deprit, A. (1981). The elimination of parallax in satellite theory, Celestial Mechanics, 24, 111-153.

[5] Engels, R. C. and J. L. Junkins, J. L. (1981). The Gravity-Perturbed Lambert Problem: A KS Variation of Parameters Approach, Celestial Mechanics, 24, 3-21.

[6] Garfinkel, B. (1959). The orbits of a satellite of an oblate planet, Astron. J, 64, 353-366.

[7] Gooding, R. H. (1991). Perturbations, untruncated in eccentricity, for an orbit in an axi-symmetric gravitational field, Journal of Astronautical Science, 39, 65-85.

[8] James Raj, M. X. and Sharma, R. K. (2003). Analytical short-term orbit prediction with J2, J3, J4 in terms of KS uniformly regular canonical elements, Advances in Space Research, 31, 2019-2025.

[9] Jezewski, D. J. (1983). A noncanonical analytic solution to the J2 perturbed two-body problem, Celestial Mechanics, 30, 343-361.

[10] King-Hele, D. G. (1958). The effect of the Earth Oblateness on the Orbit of a Near Satellite. Proceedings of the Royal Society of London, A 247, 49-72.

[11] Kinoshita, H. (1977). Theory of rotation of the rigid Earth, Celestial Mechanics, 15, 277-326.

[12] Kozai, Y. (1959). The Motion of a Close Earth Satellite, Astron. J, 64, 367-377.

[13] Sharma, R. K., and James Raj, X. M. (1988). Long term orbit computations with KS uniformly regular canonical elements with oblateness. Earth, Moon andPlanets, 42, 163-178.

[14] Sharma, R.K. (1989). Analytical approach using KS elements to short-term orbit predictions including J2, Celestial Mechanics and Dynamical Astronomy, 46, 321-333.

[15] Sharma, R. K., (1993). Analytical short-term orbit predictions with J3 and J4 in terms of KS elements. Celestial Mechanics and Dynamical Astronomy, 56, 503-521.

[16] Sharma, R. K., (1997). Analytical integration of K-S element equations with J2 for short-term orbit predictions. Planetary and Space Sciences, 45, 1481-1486.

[17] Sharon Sara Saji, Sellamuthu, H. and Sharma, R. K., (2017). On J2 short-term orbit predictions in terms of KS elements. International Journal of Advanced Astronomy, 5, 7-11.

[18] Smibi, M.. J., Sellamuthu, H., Sharma, R. K. (2017) Analytical integration of uniformly regular KS canonical equations with J2 in terms of eccentric anomaly for short-term orbit predictions. Advances in Astrophysics, 2, 141-150.

[19] Stiefel, E.L., Scheifele, G. (1971) Linear and Regular Celestial Mechanics, Berlin/Heidelberg/New York, Springerâ€Verlag.

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