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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

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2018

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Vol. 22, nr 4

1175--1195

EN

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.

2

A second order secular J-S planetary theory. Part I, Lemma

Kamel O. M., Soliman A. S.

Mechanics and Mechanical Engineering

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2018

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Vol. 22, nr 4

871--874

EN

A concise lemma is given for the construction of a semi-analytic Hamiltonian second order secular J-S planetary theory using the Jacobi-Radau system of origins and in terms of the non-singular variables of H. Poincaré. We truncate our expansions at the desired power in the eccentricities and the sines of the inclinations.

3

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

Kamel O. M., Soliman A. S.

Mechanics and Mechanical Engineering

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2016

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Vol. 20, nr 4

371--375

EN

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.

4

A Semi-Analytic First Order Jupiter-Saturn Planetary Theory. Part II

Kamel O. M., Soliman A. S.

Mechanics and Mechanical Engineering

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2016

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Vol. 20, nr 3

233--247

EN

In this part we present the algebraic calculus computations related to the Hamiltonian equations of motion for the Jupiter – Saturn subsystem. Also we give a comment on the methods of the solution for the system of linear and nonlinear differential equations describing the motion of this subsystem.

5

A First Order Jupiter Saturn Planetary Theory. Numerical Results

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

Mechanics and Mechanical Engineering

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2016

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Vol. 20, nr 1

59--73

EN

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.

6

A Semi - Analytic First Order Jupiter - Saturn Planetary Theory. Part II

Kamel O. M., Soliman A. S.

Mechanics and Mechanical Engineering

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2015

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Vol. 19, nr 2

103--118

EN

In this part we present the algebraic calculus computations related to the Hamiltonian equations of motion for the Jupiter – Saturn subsystem. Also we give a comment on the methods of the solution for the system of linear and nonlinear differential equations describing the motion of this subsystem.

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A Semi - Analytic First Order Jupiter - Saturn Planetary Theory Part I : Outline

Kamel O. M.

Mechanics and Mechanical Engineering

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2015

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Vol. 19, nr 1

17--22

EN

Herein we lay the broad lines for the construction of a first order w.r.t planetary masses Jupiter-Saturn theory - giving the orbital elements of the two planets at any epoch. This is implemented by the evaluation of the R. H. S. of the original first order Hamiltonian equations of motion. The first order Hamiltonian is composed of the first order secular terms and the first order periodic terms. We restrict the periodic terms to be the commensurate ones for the J-S (Jupiter-Saturn) subsystem. We give throughout the text an idea to the extension of the theory to the case of the four major outer planets J-S-U-N.