Expansion of the Disturbing Function by Factorization

Abstract

We present an expansion of the disturbing function for the thirdbody perturbations in the form

$$ R = \frac{{\mu _3 }} {{a_3 }}\sum\limits_{n = 2}^\infty {\sum\limits_{m = 0}^n {W_{nm} \varepsilon ^n [X_{nm} X_{nm}^{(3)} + Y_{nm} Y_{nm}^3 ],} } $$

where ε is the ratio of semi-major axes (a/a′) and X_nm and Y_nm depend only on the five elements e , i , M , ω , Ω of the satellite while X ⁽³⁾_nm and y ⁽³⁾_nm depend on the corresponding elements of the perturbing body. Each one of the four functions X and Y is represented as a product of two factors as opposed to three factors in our previous publication of this subject (Broucke, 1981). We develop several relations for the construction of the X and Y-series. These series are relatively short and can be computed in advance and stored on magnetic tape, for instance.