Bivariate Infinite Series Solution of Kepler’s Equations

Daniele Tommasini

doi:10.3390/math9070785

Abstract

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

Highlights

It is convenient to define approximate solutions Sn obtained by truncating the infinite series of Equation (2) keeping only the terms with k + q ≤ n, so that n n−k
I described an analytical procedure for the exact computation of all the higher-order partial derivatives of the elliptic and hyperbolic eccentric anomalies with respect to both the eccentricity e and the mean anomaly M
Such derivatives depend implicitly on the solution of Kepler Equations (KEs), they can be computed explicitly by choosing a couple of base values ec and Ec for the eccentricity and the eccentric anomaly, so that the corresponding value Mc of the mean anomaly can be obtained without solving KE

Summary

Introduction

Citation: Tommasini, D. BivariateInfinite Series Solution of Kepler’sEquations. Mathematics 2021, 9, 785.https://doi.org/10.3390/math9070785In the Newtonian approximation, the time dependence of the relative position of two distant or spherically symmetric bodies that move in each other’s gravitational field can be written with explicit analytical formulas involving a finite number of terms only when the eccentricity, e, is equal to 0 or 1, corresponding to circular and parabolic orbits, respectively [1]. For 0 < e < 1 and for e > 1, such evolution can be obtained by solving forE one of the following two Kepler Equations (KEs) (see e.g., Chapter 4 of Ref. [1]), Academic Editor: Richard Linares M = f (e, E) =E − e sin E, for e < 1, e sinh E − E, for e > 1 (1)Received: 19 February 2021

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