Universal variable formulation: Difference between revisions

Revision as of 17:04, 15 May 2009

Revision as of 17:11, 15 May 2009

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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>

The family of solutions to this differential equation<ref name=Danby>eq 6.9.22</ref> are written symbolically as the functions <math>1,\ s\ c_1(\alpha s^2),\ s^2\ c_2(\alpha s^2),</math>, where the functions <math>\ c_k(x)</math>, arecalled [[Stumpff function]]s, are generalizations of sine and cosine functions. 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_22\ c_2(\alpha s^2) + \mu \ s^3c_33\ c_3(\alpha s^2)</math>

which is the universal variable formulation of Kepler's Equation. This equation can now be solved numerically using a [[root-finding algorithm]] such as [[Newton's method]] or [[Laguerre's method]] or a given time <math>t</math> to yield <math>s</math>, which in turn is used to compute the [[f and g functions]]:

:<math>\begin{align}