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Solution to the main problem of the artificial satellite by reverse normalization

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Abstract

The nonlinearities of the dynamics of Earth artificial satellites are encapsulated by two formal integrals that are customarily computed by perturbation methods. Standard procedures begin with a Hamiltonian simplification that removes non-essential short-period terms from the Geopotential, and follow with the removal of both short- and long-period terms by means of two different canonical transformations that can be carried out in either order. We depart from the tradition and proceed by standard normalization to show that the Hamiltonian simplification part is dispensable. Decoupling first the motion of the orbital plane from the in-plane motion reveals as a feasible strategy to reach higher orders of the perturbation solution, which, besides, permits an efficient evaluation of the long series that comprise the analytical solution.

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Notes

  1. The advantages of decoupling the motion of the instantaneous orbital plane from the in-plane motion are well known and are commonly pursued in the search for efficient numerical integration methods, q.v. [43] and references therein.

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Acknowledgements

Support by the Spanish State Research Agency and the European Regional Development Fund under Projects ESP2016 -76585-R and ESP2017-87271-P (MINECO/ AEI/ERDF, EU) is recognized.

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  1. Scientific Computing Group–GRUCACI, University of La Rioja, Madre de Dios 53, 26006, Logroño, La Rioja, Spain

    Martin Lara

  2. Space Dynamics Group, Polytechnic University of Madrid–UPM, Plaza Cardenal Cisneros 3, 28040, Madrid, Spain

    Martin Lara

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Appendix: Tables of inclination polynomials

Appendix: Tables of inclination polynomials

Inclination polynomials needed in the evaluation of the analytical perturbation solution up to the third order of \(J_2\) are summarized in this appendix.

Table 2 Inclination polynomials \(\gamma _{2,j,k}\) in Eq. (21)
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Table 3 Nonzero inclination polynomials \(\varGamma _{2,j,k,l}\) in Eqs. (22)–(23)
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Table 4 Inclination polynomials \(\gamma _{3,j,k}\) in Eq. (24)
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Table 5 Inclination polynomials \(\varLambda _{2,j,k}\) in Eq. (33)
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Table 6 Nonzero inclination polynomials \(\varGamma _{3,j,k,l}\) in Eq. (25)
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Table 7 Inclination polynomials \(\lambda _{3,j}\) in Eq. (34)
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Table 8 Nonzero coefficients \(\varPhi _{3,j,k}\) in Eq. (35). \(\varPhi _{3,1,0}=\frac{15}{2}\varPhi _{3,0,3}\), \(\varPhi _{3,1,2}=-\frac{3}{2}\varPhi _{3,0,3}\), \(\varPhi _{3,2,0}=3\varPhi _{3,0,3}\), and \(\varPhi _{3,3,0}=\frac{1}{2}\varPhi _{3,0,3}\)
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Table 9 Nonzero inclination polynomials \(\varLambda _{3,j,k}\) in Eq. (35)
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Table 10 Inclination polynomials \(\varPsi _{i,j}\) in Eq. (45)
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Table 11 Inclination polynomials \(\omega _{i,j}\) in Eq. (46)
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Table 12 Inclination polynomials \(\varOmega _{i,j}\) in Eq. (47)
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Lara, M. Solution to the main problem of the artificial satellite by reverse normalization. Nonlinear Dyn 101, 1501–1524 (2020). https://doi.org/10.1007/s11071-020-05857-3

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