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The expansion of the inverse of the mutual distance between two planets raised to any real integer using Taylor theorem and elliptic expansions Part I, Theoretical results

Elmabsout B., Kamel O. M., Soliman A. S.

Mechanics and Mechanical Engineering

2018

Vol. 22, nr 4

1175--1195

We introduce in this part the method to obtain the literal expansion of the mutual distance between two planets of the solar system raised to any negative real integer.

On the Expansion of the Direct Part of the Disturbing Planetary Function

Kamel O. M., Soliman A. S.

Mechanics and Mechanical Engineering

2016

Vol. 20, nr 4

371--375

We generalize the expansion of Murray–Dermott for the direct part of the disturbing function using Taylor‘s theorem. We present the values of Δ-s for s = 1, 3, 5, . . . which is essential for high order planetary theories. Murray – Dermott executed the expansion for s = 1 which is necessary for only first order theories.

A First Order Jupiter Saturn Planetary Theory. Numerical Results

Kamel O. M., Soliman A. S., El Mabsout B.

Mechanics and Mechanical Engineering

2016

Vol. 20, nr 1

59--73

We present the numerical analysis solution of the eight ordinary non linear differential equations of a first order secular J – S planetary theory. There is no general solution for these equations. We deal with the Poincare’ variables Hu, Ku, Pu, Qu; u=1,2 only. The solution is approximative, since we confine our treatment to a first order secular theory and truncate the Poisson series expansions at the fourth power in eccentricity – inclination.

Determination of the classical orbital elements of the perturbed and perturbing planets from first principles. Part I: Outline

Kamel O. M.

Mechanics and Mechanical Engineering

2011

Vol. 15, nr 2

193-195

In this outline we present a rather simple method to solve the planetary perturbation problem. We do not avoid the introduction of the expansion of the planetary disturbing function, the formulae of the elliptic expansions and the truncation of the Poisson series at the desired degree. We should remark that all orders of magnitude of the masses of the planets are taken into consideration, which is a very important result of this approach which we encounter in the order by order approach of planetary theory.