Quaternionic vector space - Revision history

en>David Eppstein: dab Finitely generated module

2013-02-14T19:03:56Z

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			In [[orbital mechanics]], the '''universal variable formulation''' is a method used to solve the [[two-body problem\|two-body]] [[Kepler problem]]. It is a generalized form of [[Kepler's Equation]]s, extending them to apply not only to [[elliptic orbits]], but also [[parabolic orbit\|parabolic]] and [[hyperbolic orbit]]s. It thus is applicable to many situations in the [[solar system]], where orbits of widely varying [[orbital eccentricity\|eccentricities]] are present.

			==Introduction==
			A common problem in orbital mechanics is the following: given a body in an [[orbit]] and a time ''t<sub>0</sub>'', 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>{{citation \|author=Danby, J. M. A.\|title=Fundamentals of Celestial Mechanics\|publisher=Willman-Bell\|year=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.

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			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
			:<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>
			The family of solutions to this differential equation<ref name="Danby"/> 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]]:
			:<math>\begin{align}
			f(s) & = 1 - \left(\frac \mu {r_0}\right) s^2 c_2(\alpha s^2), \\
			g(s) & = t - t_0 - \mu s^3c_3(\alpha s^2), \\
			\frac{df}{dt} & = \dot{f}(s) = -\left(\frac{\mu}{r r_0}\right)s c_1(\alpha s^2), \\
			\frac{dg}{dt} & = \dot{g}(s) = 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>
			In addition the velocity of the body at time <math>t</math> can be found using <math>\dot{f}(s)</math> and <math>\dot{g}(s)</math> as follows:
			:<math>\mathbf{v} = \mathbf{r}_0\ \dot{f}(s) + \mathbf{v}_0\ \dot{g}(s)</math>

			where <math>\mathbf{r}</math> and <math>\mathbf{v}</math> are the position and velocity respectively 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==
			<references/>

			[[Category:Orbits]]

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2011-12-21T18:10:29Z

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