On the Computation of the Hansen Coefficients

The Hansen coefficients are one of the most important tools in the analytical or semi-analytical methods of celestial mechanics. The problem of efficient computation was investigated in detail in many papers, but this subject is still open and to date there are not standard algorithms to generate the Hansen coefficients with accuracy and good computation time. Although the analytical expressions are known through their accuracy, the formulation by recurrent relations was preferred due to their stability and applicability in the case when a large number of the Hansen coefficients are needed. In this paper we turn our attention to the polynomial approximations and we present a method to generate the polynomials associated to the Hansen coefficients and their derivatives in the particular case used in expression of disturbing function due to central-body. Using the recurrent formulas corresponding to this particular case, we define difference equations and, by their iteration, we generate the polynomials. By computational tests, we compare our polynomial approximations for the Hansen coefficients and their derivatives with the recurrent solutions in order to prove the non-regression of the accuracy and the gain in computation time.