Abstract
This chapter illustrates Newton's method for solving transcendental equations (Kepler's equation) for relating position to time for elliptical, parabolic, and hyperbolic paths. The different forms of Kepler's equation are combined into a single universal Kepler's equation by introducing universal variables. The different forms of Kepler's equation are combined into a single universal Kepler's equation by introducing universal variables. Implementation of this appealing notion is accompanied by the introduction of an unfamiliar class of functions known as Stumpff functions. The universal variable formulation is required for the Lambert and Gauss orbit determination algorithms.
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