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where '''P''' is a constant [[vector]] and <math>\alpha</math> is defined by
:<math>\alpha = \frac\mu a</math>
The equation is the same as the equation for the [[harmonic oscillator]], a well-known equation in both [[physics]] and [[mathematics]]. Taking the derivative again, we get a third-degree differential equation:
:<math>\frac{d^3\mathbf r} {ds^3} + \alpha\frac{d\mathbf r} {ds} = \mathbf{0}</math>
It turns out<ref name=Danby>eq 6.9.22</ref> that each one of the functions <math>1, sc_1(\alpha s^2), s^2c_2(\alpha s^2)</math>, where <math>c_k(x)</math> are the [[Stumpff function]]s, is a solution to the third-degree differential equation. Applying this results in<ref name="Danby">Equation 6.9.26</ref>:
:<math>t - t_0 = r_0 s c_1(\alpha s^2) + r_0 \frac{dr_0}{dt}s^2c_2(\alpha s^2) + \mu s^3c_3(\alpha s^2)</math>
which is the universal variable formulation of Kepler's Equation. This equation can now be solved numerically for a given time <math>t</math> to yield <math>s</math>, which in turn is used to compute the [[f and g functions]]:
:<math>f = 1 - (\frac \mu {r_0}) s^2 c_2(\alpha s^2),</math>
:<math>g = t - t_0 - \mu s^3c_3(\alpha s^2),</math>
:<math>\frac{df}{dt} = -(\frac{\mu}{r r_0})s c_1(\alpha s^2),</math>
:<math>\frac{dg}{dt} = 1 - (\frac{\mu}{r})s^2c_2(\alpha s^2)</math>
The values of the f and g functions determine the position of the body at the time <math>t</math>.
==References==
<references/>
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