Universal variable formulation: Difference between revisions

Revision as of 16:09, 26 October 2012

Revision as of 16:55, 7 March 2013

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==Introduction==

A common problem in orbital mechanics is the following: given a body in an [[orbit]] and a time ''t₀'', find the position of the body at any other given time ''t''.

For [[elliptical orbit]]s with a reasonably small [[Orbital eccentricity|eccentricity]], solving [[Kepler's Equation]] by methods like [[Newton's method]] gives adequate results. However, as the orbit becomes more and more eccentric, the numerical iteration may start to [[limit of a sequence|converge]] slowly or not at all.<ref name=Danby>{{citecitation |author=Danby, J. M. A.|title=Fundamentals of Celestial Mechanics|publisher=Willman-Bell|dateyear=1988}}</ref> Furthermore, Kepler's equation cannot be applied to [[Parabolic orbit|parabolic]] and [[hyperbolic orbit]]s, since it specifically is tailored to elliptic orbits.

==Derivation==

Although equations similar to Kepler's equation can be derived for parabolic and hyperbolic orbits, it is more convenient to introduce a new independent variable to take the place of the [[eccentric anomaly]] ''E'', and having a single equation that can be solved regardless of the eccentricity of the orbit. The new variable ''s'' is defined by the following [[differential equation]]:

:<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 [[Euclidean vector|vector]] and <math>\alpha</math> is defined by

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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>, 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]] for 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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==References==

<references/>

{{astronomy-stub}}

[[Category:Orbits]]