Analytic orbit theory with any arbitrary spherical harmonic as the dominant perturbation

  • Bharat MahajanEmail author
  • Kyle T. Alfriend
Original Article
Part of the following topical collections:
  1. 50 years of Celestial Mechanics and Dynamical Astronomy


In the gravitational potential of Earth, the oblateness term is the dominant perturbation, with its coefficient at least three orders of magnitude greater than that of any other zonal or tesseral spherical harmonic. Therefore, analytic orbit theories (or satellite theories) are developed using the Keplerian Hamiltonian as the unperturbed solution, oblateness term as the first-order and the remaining spherical harmonics as the second-order perturbations. These orbit theories are generally constructed by applying multiple near-identity canonical transformations to the perturbed Hamiltonian in conjunction with averaging to obtain a secular Hamiltonian, from which the short-period and long-period terms are removed. If the oblateness term is the only first-order perturbation, then the long-period terms appear only in the second- or higher-order terms of the single-averaged Hamiltonian, from which the short-period terms are removed. These second-order long-period terms are separated from the Hamiltonian by the first-order generating function using a second canonical transformation. This results in a secular Hamiltonian dependent only on the momenta. However, in the case of other gravitational bodies with more deformed shapes compared to Earth such as moons and asteroids, the oblateness coefficient may have the same order of magnitude as some of the higher spherical harmonic coefficients. If these higher harmonics are treated as the first-order perturbation along with the oblateness term, then the long-period terms appear in the first-order single-averaged Hamiltonian. These first-order long-period terms cannot be separated using the generating function in the conventional way. This problem occurs because the zeroth-order Hamiltonian, i.e., the Keplerian part, is degenerate in the angular momentum. In this paper, a new approach to the long-period transformation is proposed to resolve this issue and obtain a fully analytic orbit theory when for the perturbing gravitational body, any arbitrary zonal or tesseral harmonic is the dominant perturbation. The proposed theory is closed form in the eccentricity as well. It is applied to predict the motion of artificial satellites for the two test cases: a lunar orbiter and a satellite of 433 Eros asteroid. The prediction accuracy is validated against the numerical propagation using a force model with \(6\times 6\) gravity field.


Orbit theory Artificial satellite theory Zonal harmonics Tesseral harmonics Canonical perturbation methods Lunar orbiter 433 Eros 



The authors would like to thank Dr. Simone D’Amico and his graduate students at Stanford University for the motivation for studying this problem.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

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

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