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

Considering a point of polar coordinates (r,νr,ν) 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 Zsn,m, which are the coefficients of the expansion of (r/a)nexpimν(r/a)nexpimν 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 0⩽|m|⩽n0⩽|m|⩽n 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 Zsn,m whereas the other algorithm involves the generalized Laplace coefficients bp,rk (Laskar, 2005). Numerical behavior of the algorithms is investigated for low and high eccentricities. Both algorithms provide a relative accuracy better than 10-1410-14 for n⩽30n⩽30. Also, they are at least 10 time faster than an algorithm based on the FFT method (Klioner et al., 1997).

► We provide two recursive algorithms to compute a Z  -table for 0≤|m|≤n0≤|m|≤n.
► The two algorithms are free of singularities or small divisors.
► Recurrence relations for the Algorithm 1 are presented.
► The two algorithms preserve accuracy and are stable for any eccentricity 0≤e<10≤e<1.
► The numerical schemes are much faster than the Fast Fourier Transform.