Astrophysics and Space Science

, Volume 129, Issue 1, pp 19–38 | Cite as

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

X. Analysis of the general periodic function\(X_n^{\left( r \right)} \left( {\gamma ,\beta ,u} \right) = \left\{ {\sum\limits_{i - 1}^{n + 1} {\gamma _i \cos ^{i - 1} u + } sin u \sum\limits_{i = 1}^n {\beta _i } \cos ^{i - 1} u} \right\}^r \)
  • Mohamed Adel Sharaf


In this paper of the series, the third step of the author's regularization approach will be started by establishing the expansions of the functionX n (r) (γ, β,u) in terms of the sectorial variablesθ j (i) introduced in Paper IV (Sharaf, 1982) to regularize the highly-oscillating perturbation force of some orbital systems. The literal analytical expressions for the Fourier expansion of the function will be explored in terms ofθ j (i) for anyn positive integer,r any real number whatever the types and the number of sectors forming the divisions situation of the elliptic orbits may be. The basic computational materials of the theory will also be given and for which the method of solution, the recurrence formulae, and the general computational sequence for the coefficients are considered.


Fourier Positive Integer Real Number Orbital System Fourier Expansion 
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  1. Broucke, R.: 1974,Celes. Mech. 10, 469.Google Scholar
  2. Brouwer, D. and Clemence, G.: 1961,Methods of Celestial Mechanics, Academic Press, London.Google Scholar
  3. Sharaf, M. A.: 1981a,Astrophys. Space Sci. 74, 211 (Paper I).Google Scholar
  4. Sharaf, M. A.: 1981b,Astrophys. Space Sci. 78, 359 (Paper II).Google Scholar
  5. Sharaf, M. A.: 1982a,Astrophys. Space Sci. 84, 53 (Paper III).Google Scholar
  6. Sharaf, M. A.: 1982b,Astrophys. Space Sci. 84, 73 (Paper IV).Google Scholar
  7. Sharaf, M. A.: 1983,Astrophys. Space Sci. 93, 377 (Paper V).Google Scholar
  8. Sharaf, M. A.: 1984,Astrophys. Space Sci. 104, 264 (Paper VI).Google Scholar
  9. Sharaf, M. A.: 1985a,Astrophys. Space Sci. 112, 51 (Paper VII).Google Scholar
  10. Sharaf, M. A.: 1985b,Astrophys. Space Sci. 116, 251 (Paper VIII).Google Scholar
  11. Sharaf, M. A.: 1986,Astrophys. Space Sci. 125, 259 (Paper IX).Google Scholar

Copyright information

© D. Reidel Publishing Company 1987

Authors and Affiliations

  • Mohamed Adel Sharaf
    • 1
  1. 1.Dept. of AstronomyUniversity of CairoEgypt

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