Exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral harmonics

  • Bharat Mahajan
  • Srinivas R. Vadali
  • Kyle T. Alfriend
Original Article


A novel approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral and sectorial spherical harmonics is presented in this work. It is shown that the exact solution for the Delaunay normalization can be reduced to quadratures by the application of Deprit’s Lie-transform-based perturbation method. Two different series representations of the quadratures, one in powers of the eccentricity and the other in powers of the ratio of the Earth’s angular velocity to the satellite’s mean motion, are derived. The latter series representation produces expressions for the short-period variations that are similar to those obtained from the conventional method of relegation. Alternatively, the quadratures can be evaluated numerically, resulting in more compact expressions for the short-period variations that are valid for an elliptic orbit with an arbitrary value of the eccentricity. Using the proposed methodology for the Delaunay normalization, generalized expressions for the short-period variations of the equinoctial orbital elements, valid for an arbitrary tesseral or sectorial harmonic, are derived. The result is a compact unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes, which is nonsingular for the resonant orbits, and is closed-form in the eccentricity as well. The accuracy of the proposed theory is validated by comparison with numerical orbit propagations.


Artificial satellite theory Tesseral harmonics Canonical perturbation method Delaunay normalization The method of relegation 



This research was partially supported by AFRL Contract FA9453-13-C-0202 with Dr. Alan T. Lovell serving as the Technical Project Monitor. The authors also acknowledge Dr. Martin Lara for private communications and for providing the code for his implementation of the method of relegation.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringTexas A&M UniversityCollege StationUSA

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