Recursive calculation of hansen coefficients

Hansen coefficients are used in expansions of the elliptic motion. Three methods for calculating the coefficients are studied: Tisserand's method, the Von Zeipel-Andoyer (VZA) method with explicit representation of the polynomials required to compute the Hansen coefficients, and the VZA method with the values of the polynomials calculated recursively. The VZA method with explicit polynomials is by far the most rapid, but the tabulation of the polynomials only extends to 12th order in powers of the eccentricity, and unless one has access to the polynomials in machine-readable form their entry is laborious and error-prone. The recursive calculation of the VZA polynomials, needed to compute the Hansen coefficients, while slower, is faster than the calculation of the Hansen coefficients by Tisserand's method, up to 10th order in the eccentricity and is still relatively efficient for higher orders. The main advantages of the recursive calculation are the simplicity of the program and one's being able to extend the expansions to any order of the eccentricity with ease. Because FORTRAN does not implement recursive procedures, this paper used C for all of the calculations. The most important conclusion is recursion's genuine usefulness in scientific computing.