A second order Jupiter-Saturn planetary theory

We present a second order secular Jupiter-Saturn planetary theory through Poincaré canonical variables, von Zeipel's method and Jacobi-Radau referential. We neglect in our expansions terms of power higher than the fourth with respect to eccentricities and sines of inclinations. We assume that the disturbing function is composed of secular and critical terms only. We shall deriveF _2si and writeF _2s in terms of Poincaré canonical variables in Part II of this problem.     

Third order uranus-neptune planetary theory

We calculate in this paper the secular and critical terms arising from the principal part of the classical planetary Hamiltonian. This is the first step to establish a third order canonical planetary theory of Uranus-Neptune through the Hori-Lie technique. We truncate our expansions at the second degree of eccentricity-inclination. Our planetary theory is expressed in terms of the canonical variables of H. Poincaré.     

Third-order Jupiter — Saturn planetary theory

The construction of a third order J-S theory is presented. The Hori theory of planetary perturbations is employed. No Critical J-S terms due to the 2:5 commensurabilities and its multiples exist, when we take into account the periodic terms of order 0, 1, 2 with respect to the eccentricity- inclination. In this case the Lie series transformation degenerates and is meaningless. The J-S equations of motion for secular perturbations are solved when we neglect in our treatment, the Poisson terms of degree > 2 in the Poincaré canonical variables H _u , K _u , P _u Q _u (u = 1, 2). The Jacobi-Radau referential is adopted, and the theory is expressed in terms of the canonical variables of H. Poincaré.Now at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, U.S.A.     

The construction of a seventh-order Jupiter-Saturn analytical theory

We extend the construction of the Jupiter-Saturn theory to include all the terms up to the seventh order in the masses. The Hori-Lie transformation technique is employed. The Jacobian coordinates are adopted and the theory is expressed in terms of the canonical non-singular variables of H. Poincaré.     

Elimination of the critical terms in a first-order general planetary theory

We construct a first order canonical general planetary theory, assuming the solar system to be composed of 8 planets excluding Pluto, referring to common fixed plane and applying the Jacobi-Radau set of origins. We eliminated by von Zeipel's method the 2:5 and 1:2 critical terms of Jupiter-Saturn and Uranus-Neptune inequalities. Our variables are those of Poincaré, and we expanded up to power three in the eccentricities and sines of the inclinations.     

A second-order Uranus-Neptune theory by Hori's method

We shall establish a second order - with respect to a small parameter which is of the order of planetary masses - Uranus-Neptune canonical planetary theory. The construction will be through the Hori-Lie perturbation theory. We perform the elliptic expansions by hand, taking into account powers 0, 1, 2 of the eccentricity-inclination. Only the principal part of the planetary Hamiltonian will be taken into consideration. Our theory will be expressed in terms of the canonical variables of Henri Poincaré, referring the planetary coordinates to the Jacobi-Radau system of origin. Only U- N critical terms will be assumed as the periodic terms.     

The construction of a fifth-order Jupiter-Saturn motion: Analytical theory

We construct a fifth-order with respect to masses Jupiter-Saturn secular theory by Hori-Lie canonical technique. The J-S Hamiltonian includes both parts of the perturbing function. The influence of the 2:5 critical terms is taken into consideration. The Jacobi-Radau system of origins is adopted and the theory is expressed in terms of the Poincaré canonical variables.     

The construction of a third-order secular analytical J-S-U-N theory by the Hori-Lie technique

We expand both parts, the principal and indirect, of the Hamiltonian function up to the third order in the masses for the four major planets Jupiter-Saturn-Uranus-Neptune. Accordingly we write down the secular terms ofF ₁,F ₂,F ₃ and the critical terms ofF ₁,F ₂ in terms of the canonical variables of H. Poincaré neglecting powers higher than the second inH, K, P, Q.     

Third-order Uranus-Neptune theory by Hori's method

We construct a U-N secular canonical planetary theory of the third order with respect to planetary masses. The Hori-Lie procedure is adopted to solve the problem. Expansions have been carried out by hand, neglecting powers higher than the second with respect to the eccentricity-inclination. We take into account the principal as well as the indirect part of the planetary disturbing function. The theory is expressed in terms of the Poincaré canonical variables, referring to the Jacobi-Radau set of origins. We assume that the 1:2 U-N critical terms and its multiples are the only periodic terms.     

Third-order Uranus-Neptune theory

In this paper of the third order Uranus-Neptune planetary theory which is the third part of this work for the third order theory, we compute the Poisson brackets in the Lie series which is used to transform canonical variables. We apply Hori-Lie technique in this work and neglect all powers higher than the second in Poincaré variables H, K, P, Q. We restrict this work to the principal part of the disturbing function.     

The elimination of short-periodic terms in a Uranus-Neptune general planetary theory by von Zeipel's method

We eliminate by the method of von Zeipel the short-period terms in a first order-with respect to planetary masses—general planetary Uranus-Neptune theory. We exclude in the expansion terms of eccentricities and sines of inclinations higher than the third power.Our variables are the Poincaré canonical variables. We use the Jacobi-Radau set of origins, and we refer the planes of the osculating ellipses to a common fixed plane, the longitudes to a common origin. The short-periodic terms arising from the indirect and principal parts of the disturbing functions, are eliminated separately. The Fourier series of the principal part of the disturbing function, is reduced to the sum of only the first three terms.     

On the elimination of the critical term in a general first-order Uranus-Neptune theory

A solution of the Uranus-Neptune planetary canonical equations of motion through the Von Zeipel technique is presented. A unique determinging function which depends upon mixed canonical variables, reduces the 12 critical terms of the Hamiltonian to the set of its secular terms. The Poincaré canonical variables are used. We refer to a common fixed plane, and apply the Jacobi-Radau set of origins. In our expansion we neglected terms of power higher than the fourth with respect to the eccentricities and sines of the inclinations.     

First-order secular planetary theory up to fourth power in eccentricities and inclinations

We construct a first-order secular general planetary theory, using the Jacobi-Radau set of origins, referring to common fixed plane and in terms of Poincaré canonical variables. We neglect powers higher than the fourth with respect to the eccentricities and sines of inclinations.     

The construction of a general third-order planetary theory

We review in this part the outline of a third-order general planetary theory established through Von Zeipel's method and in terms of Poincaré's canonical variables We consider our system to consist of the Sun as the primary body, one disturbed planet, and one disturbing planet.     

The elimination of the 1:2 critical terms of a first order theory of Uranus perturbed by Neptune through Von Zeipel method

In this paper we eliminate in a first order U-N theory the 1 : 2 critical terms up to the third degree with respect to eccentricity — inclination in both parts, main and indirect of the U-N planetary Hamiltonian. We operate the Von Zeipel technique. We adopt, in this theory, the Jacobi-Radau coordinates, and the Poincaré canonical variables. We neglect powers higher than the third in the eccentricity-inclination. This paper is related to the two previous articles (Kamel, 1982; 1983).     

An elimination of the critical terms of a first-order Uranus-Neptune theory by Hori's method

We eliminate the 1:2 critical terms — after a previous elimination of the short period terms — in the Hamiltonian of a first order U-N theory. We take into account terms of degree 0, 1, 2, 3, 4 in the eccentricity-inclination. We apply for this elimination the Hori-Lie technique through the Poincaré canonical variables and the Jacobi coordinates. The purely principal first order secular U-N Hamiltonian admits a complete solution. We obtained the U-N equations of motion generated by the principal first order long period U-N Hamiltonian which will be solved later. This part III is closely related to the two previous papers (Kamel, 1982, 1983).     

Expression of the initial Poincaré canonical variables as functions of the new in a ninth order J-S theory

We establish the solution of the ninth order — in masses — canonical J-S equations of motion by Hori-Lie technique — i.e., by expressing the initial Poincaré canonical variables as functions of the new variables through the Hori-Lie canonical transformation. Terms of order higher than 9 in the masses are neglected.     

Expansion of the mutual distance between two planets raised to any negative power

An expression for ^–s in terms of the Poincaré variablesL, , H, K, P. Q has been evaluated. The inclinations of the two planets are referred to a common fixed plane. We neglect in the final formula powers higher than the third of the Poincaré variables.     

Third-order Uranus-Neptune planetary theory

In this part we calculate the secular and critical terms arising from the indirect part of the classical planetary Hamiltonian for Uranus and Neptune. We neglect in our expansions powers higher than the second in the eccentricity-inclination. Our required results, are expressed in terms of Poincaré variables.     

A determination of secular terms in first-order planetary theory

The secular terms of the first-order planetary Hamiltonian is determined, by two methods, in terms of the variables of H. Poincaré, neglecting powers higher than the second in the eccentricity-inclination.