Two algorithms to compute Hansen-like coefficients with respect to the eccentric anomaly Original Research Article

Considering a point of polar coordinates (image) on an elliptic orbit of semi-major axis a, we set up and compare two algorithms based on recurrence relations to compute the Hansen-like coefficients image, which are the coefficients of the expansion of image in Fourier series of the eccentric anomaly. Both Hansen-like coefficients and their derivatives with respect to the eccentricity are considered, with a special focus on the case image arising in the expression of the gravity potential due to a body external to the elliptic orbit. We provide two efficient algorithms to compute a table of coefficients with a simple recursive process. One algorithm uses some recurrence relations linking directly to the image whereas the other algorithm involves the generalized Laplace coefficients image (). Numerical behavior of the algorithms is investigated for low and high eccentricities. Both algorithms provide a relative accuracy better than image for image. Also, they are at least 10 time faster than an algorithm based on the FFT method ().