On J2 short-term orbit predictions in terms of KS elements


  • Sharon Sara Saji Department of Aerospace Engineering, Karunya University, Coimbatore - 641114, Tamil Nadu, India
  • Harishkumar Sellamuthu Department of Aerospace Engineering, Karunya University, Coimbatore - 641114, Tamil Nadu, India
  • Ram Krishan Sharma Department of Aerospace Engineering, Karunya University, Coimbatore - 641114, Tamil Nadu, India






Analytical Integration, KS Element Equations, Earth's Oblateness, Short-Term Orbit Prediction, Radial Distance.


Sharma’s singularity-free analytical theory for the short-term orbital motion of satellites in terms of KS elements in closed form in eccentricity with Earth’s zonal harmonic term J2, is improved by using King-Hele’s expression for the radial distance ‘r’ which includes the effect of J2, and is suitable for low eccentricity orbits. Numerical experimentation with four test cases with perigee altitude of 200 km and eccentricity varying from 0.01 to 0.3 for different inclinations is carried out. It is found that the orbital elements computed with the analytical expressions in a single step during half a revolution match very well with numerically integrated values and show significant improvement over the earlier theory. The solution can be effectively used for computation of mean elements for near-Earth orbits, where the short-term orbit perturbations due to J2 play most important role. The theory will be very useful in computing the state vectors during the coast phase of rocket trajectories and flight algorithms for on-board implementation.


[1] Aksnes, K. (1970) A second-order artificial satellite theory based on an intermediate orbit. Astron. J. 75, 1066-1076. https://doi.org/10.1086/111061.

[2] Andrus, J.F. (1977) First-order effects of the earth's oblateness upon coasting bodies, Celest. Mech. 15, 217-224. https://doi.org/10.1007/BF01228463.

[3] Brouwer, D. (1959) Solution of the problem of artificial satellite theory without drag. Astron. J. 64, 378-396. https://doi.org/10.1086/107958.

[4] Celleti, A., Negrini, P. (1995) Non-integrability of the problem of motion around an oblate planet. Celest. Mech. 61, 253-260. https://doi.org/10.1007/BF00051896.

[5] Deprit, A. (1981) The elimination of the parallax in satellite theory, Celest. Mech. 24, 111-153. https://doi.org/10.1007/BF01229192.

[6] Engels, R. C. and Junkins, J. L. (1981) The gravity-perturbed Lambert problem – A KS variation of parameters approach. Celest. Mech. 24 3-21. https://doi.org/10.1007/BF01228790.

[7] Garfinkel, B. (1959) The orbits of a satellite of an oblate planet. Astron. J. 64, 353. https://doi.org/10.1086/107956.

[8] Gurfil, P. and Lara, M. (2014) Satellite onboard orbit propagation using Deprit’s radial intermediary. Celest. Mech. Dyn. Astron. 120, 217-232. https://doi.org/10.1007/s10569-014-9576-1.

[9] Kozai, Y. (1962) Second order solution of artificial satellite theory without air drag. Astron. J. 67, 446-461. https://doi.org/10.1086/108753.

[10] King-Hele, D. G. (1958) The effect of the Earth’s oblateness on the orbit of a near satellite, Proc. R. Soc. London a 247, 49-70. https://doi.org/10.1098/rspa.1958.0169.

[11] Kustaanheimo, P. and Stiefel, E. (1965) Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew Math. 218, 204-219.

[12] Lyddane, R.H. (1963) Small eccentricities or inclinations in the Brouwer theory of the artificial satellite, Astron. J. 68, 555-558. https://doi.org/10.1086/109179.

[13] Sharma, R. K. (1989) Analytical approach using KS elements to short-term orbit predictions including J2, Celest. Mech. Dyn. Astron. 46, 321-333. https://doi.org/10.1007/BF00051486.

[14] Sharma, R. K. (1997) Analytical integration of K-S element equations with J2 for short-term orbit predictions, Planet. Space Sci. 45, 1481-1486. https://doi.org/10.1016/S0032-0633(97)00093-7.

[15] Stiefel, E.L. and Scheifele, G. (1971) Linear and Regular Celestial Mechanics, Springer, Berlin. https://doi.org/10.1007/978-3-642-65027-7.

View Full Article: