Universal variable formulation: Difference between revisions

Revision as of 23:39, 15 June 2009

Revision as of 13:31, 12 August 2009

Line 16:

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>, called [[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^2\ c_2(\alpha s^2) + \mu \ s^3\ 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]] orfor 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}

f(s) & = 1 - \left(\frac \mu {r_0}\right) s^2 c_2(\alpha s^2), \\