Universal variable formulation: Difference between revisions

Revision as of 16:45, 15 May 2009

Revision as of 17:04, 15 May 2009

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:<math>\frac{ds}{dt} = \frac{1}{r}</math>

where <math>r = r(t)</math> is the time-dependent distance to the center of attraction. The fundamental equation <math>\frac{d^2\mathbf{r}}{dt^2} + \mu \frac{\mathbf{r}}{r^3} = \mathbf{0}</math> is [[regularization|regularized]] by applying this change of variables to yield<ref name=Danby/>:

:<math>\frac{d^2\mathbf{r}}{ds^2} + \alpha\ \mathbf{r} = -\mathbf{P}</math>

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>

ItThe turnsfamily outof solutions to this differential equation<ref name=Danby>eq 6.9.22</ref> thatare eachwritten onesymbolically ofas the functions <math>1,\ s\ sc_1c_1(\alpha s^2), s^2c_22\ c_2(\alpha s^2)</math>, where <math>\ c_k(x)</math> are the [[Stumpff function]]s, isgeneralizations aof solutionsine toand thecosine third-degree differential equationfunctions. 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 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]]:

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\frac{dg}{dt} & = 1 - \left(\frac{\mu}{r}\right)s^2c_2(\alpha s^2)

\end{align}</math>

The values of the f and g functions determine the position of the body at the time <math>t</math>.:

:<math>\mathbf{r} = \mathbf{r}_0\ f(s) + \mathbf{v}_0\ g(s)</math>

where <math>\mathbf{r}</math> is the position at time <math>t</math>, and <math>\mathbf{r}_0</math> and <math>\mathbf{v}_0</math> are the position and velocity, respectively, at arbitrary initial time <math>t_0</math>.

==References==