Separation of Variables in Rectangular Coordinates
In this chapter we reproduce with slight modifications a technique of Brumberg (1978) to compute the perturbations in rectangular coordinates. This method is similar to the von Zeipel technique of separation of short- and long- period perturbations in elements. The general solution of the homogeneous equations of variations for the two-body problem is presented here only on the basis of a transformation leading to a differential system with constant coefficients in Jordan form. Therefore, the equations of perturbed motion have linear left-hand sides in Jordan form whereas their right-hand sides are holomorphic functions with respect to the unknown variables. A further transformation excludes all short-period terms and leads to a polynomial system with slowly changing coefficients. This system determines the long-period terms. The elements of the intermediate orbit of the two-body problem may be táken either in analytical or numerical form. Moreover, the mean motion always preserves its value in the process of solution. This method extends the technique of the general planetary theory (Brumberg, 1970; Brumberg and Chapront, 1973; see also Chapter 10) to non-planar and non-circular intermediary and may be used for constructing theories of motion for the major planets, asteroids and satellites. For the efficient realization of the method on a computer it is helpful to use the facilities of the PS and Keplerian processors.
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