Solution of Kepler’s equation for rectilinear motion

Abstract

The properties of the solution of the kinematic equation (Kepler’s equation) presented as a series in powers of a function of time are considered for the case of unperturbed, rectilinear elliptical and rectilinear hyperbolic motion. Kepler’s equation for unperturbed, rectilinear elliptical motion has the form E − sinE = z 3/6, where E is the eccentric anomaly determining the position in the orbit, z 3/6 is the mean anomaly, which is proportional to the time measured from an encounter, and its solution can be represented as a series in powers of z. It is established that the coefficients of the series are positive. The asymptotic for the coefficients in the region of convergence of the series is found, which covers the entire orbit. The series continues to converge over the entire boundary of the circle of convergence. The kinematic equation for unperturbed, rectilinear hyperbolic motion has the form sinhH − H = ζ 3/6, where, as before, ζ 3 is proportional to time. The substitution E = iH, z = iζ reduces one equation to the other. The series for the solution in the hyperbolic case differs from the series for the elliptical solution only in its alternating-sign coefficients. However, the region of convergence covers only part of the orbit in the hyperbolic case.