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In [[orbital mechanics]], the '''universal variable formulation''' is a method used to solve the [[two-body problem|two-body]] [[Kepler problem]]. It is a generalized form of [[Kepler's Equation]], extending it to apply not only to [[elliptic orbits]], but also [[parabolic orbit|parabolic]] and [[hyperbolic orbit]]s common for spacecraft departing from a planetary orbit. It is also applicable to ejection of small bodies in [[Solar System]] from the vicinity of massive planets, during which processes the approximating two-body orbits can have widely varying [[orbital eccentricity|eccentricities]], almost always {{nobr| {{math| [[Orbital eccentricity|''e'']] ≥ 1 }} .}}
==Introduction==
A common problem in orbital mechanics is the following: Given a body in an [[orbit]] and a fixed original time <math>\ t_\mathsf{o}\ ,</math> find the position of the body at some later time <math>\ t ~.</math> For [[elliptical orbit]]s with a reasonably small [[orbital eccentricity|eccentricity]], solving [[Kepler's Equation]] by methods like [[Newton's method]] gives excellent results. However, as the orbit approaches an escape trajectory, it becomes more and more eccentric, [[limit of a sequence|convergence]] of numerical iteration may become unusably sluggish, or fail to converge at all for {{nobr| {{math| [[Orbital eccentricity|''e'']] ≥ 1 }} .}}<ref name=StiefelScheifele>
{{cite book
|first1 = Eduard L. |last1 = Stiefel |author1-link=Eduard Stiefel
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