Some possible modifications of Brouwer's theory of the general perturbations in rectangular coordinates

Abstract

The method of general perturbations in rectangular coordinates is the most direct of all methods of expansion of the perturbations into series because it is intimately associated with the computation of ephemerides. In addition, unlike the method of variation of elliptic elements, the method of coordinates does not have the zero eccentricity as a singularity. In Brouwer's theory of the general perturbations in rectangular coordinates the variation of elements in the canonical form is used. However, if the perturbations are being developed into trigonometric series with purely numerical coefficients, the use of canonical elements is not of any advantage. This fact was recognized by Davis, who rewrote Brouwer's formulas in terms of the standard elliptic elements. Davis's formula contains two terms of order −1 in the eccentricity. The presence of these terms causes considerable numerical inconvenience in the study of nearly circular orbits. We suggest here the use of the Eckert‐Brouwer formula for the orbit correction as a foundation of a planetary theory. The application of this formula leads directly to a vectorial expression for perturbations which is free from the disadvantages mentioned above and is also convenient for the numerical computations. The method of iteration is suggested for computing the effects of higher orders. The inclusion of the higher‐order terms is important not only in the planetary case but also in the case of artificial celestial bodies moving in orbits in cislunar space, far away from the earth. Such bodies in their motions resemble planets or comets more than satellites.

Construction of Planetary Theory by Iterative Procedure

“…The denominators in Davis 1 formulas [3] for these coefficients contain the eccentricities. For this reason Musen [4] expressed an opinion that Brouwer's method would lose its effectiveness when small eccentricities are involved. These fictitious peculiarities are eliminated in the present paper by means of trivial transformations and the expressions for the coefficients are given in a simple symmetric form.…”