Bivariate Infinite Series Solution of Kepler’s Equations

A class of bivariate infinite series solutions the elliptic and hyperbolic Kepler equations is described, adding to handful 1-D that have been found throughout centuries. This result based on an iterative procedure for analytical computation all higher-order partial derivatives eccentric anomaly with respect eccentricity e mean M in a given base point (ec,Mc) (e,M) plane. Explicit examples such are provided, corresponding different choices (ec,Mc), their convergence studied numerically. In particular, polynomials obtained by truncating up fifth degree reach high levels accuracy significantly large regions parameter space (e,M). Besides theoretical interest, these can be used designing 2-D spline numerical algorithms efficiently solving Kepler’s values anomaly.