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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]], extending themit to apply not only to [[elliptic orbits]], but also [[parabolic orbit|parabolic]] and [[hyperbolic orbit]]s common for spacecraft departing from a planetary orbit. It thus is also applicable to manyejection situationsof insmall thebodies in [[Solar System]] from the vicinity of massive planets, whereduring which processes the approximating two-body orbits ofcan have widely varying [[orbital eccentricity|eccentricities]], almost always {{nobr| {{math| [[eccentricity|''e'']] ≥ 1 are}} present.}}
==Introduction==
A common problem in orbital mechanics is the following: givenGiven a body in an [[orbit]] and a fixed original time ''t''<submath>0\ t_\mathsf{o}\ ,</submath>, find the position of the body at anysome other givenlater time ''<math>\ t'' ~.</math> For [[elliptical orbit]]s with a reasonably small [[Orbitalorbital eccentricity|eccentricity]], solving [[Kepler's Equation]] by methods like [[Newton's method]] gives adequateexcellent results. However, as the orbit approaches an escape trajectory, it becomes more and more eccentric, the numerical iteration may start to [[limit of a sequence|convergeconvergence]] slowlyof numerical iteration may become unusably sluggish, or notfail to converge at all for {{nobr| {{math| [[eccentricity|''e'']] ≥ 1 }} .}}<ref name=StiefelScheifele>{{cite book |author1first1 = Eduard L. |last1 = Stiefel |first2 author2= Gerhard |last2 = Scheifele |year = 1971 |title = Linear and Regular Celestial Mechanics.: Perturbed Twotwo-body Motionmotion Numericalnumerical Methodsmethods Canonicalcanonical Theorytheory | publisher = Springer-Verlag|date=1971 }}</ref><ref name=Danby>{{cite book |authorlast = Danby, |first = J. M. A. |year = 1988 |title = Fundamentals of Celestial Mechanics |edition = 2nd |publisher = Willmann-Bell|date=1988 |isbn = 0943396204|edition=2nd }}</ref> Furthermore, Kepler's equation cannot be directly 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]] ''<math>\ E''\ ,</math> and having a single equation that can be solved regardless of the eccentricity of the orbit. The new variable ''<math>\ s''\ </math> is defined by the following [[differential equation]]:
<math display="block"> \frac{ds \operatorname d s }{dt\ \operatorname d t\ } = \frac{\ 1\ }{ r } </math>
where <math>\ r =\equiv r(t)\ </math> is the time-dependent [[scalar (physics)|scalar]] distance to the center of attraction. The(In fundamentalall equationof the following formulas, carefully note the distinction between [[scalar (physics)|scalars]] <math>\frac{d^2\mathbf{r}}{dt^2} + \mu \frac{\mathbf{r}}{r^3} = \mathbf{0} ,</math> isin ''italics'', and [[Regularizationvector (mathematicsphysics)|regularizedvectors]] by<math>\ applying\mathbf thisr\ change,</math> ofin variablesupright to yield:<ref name=Danby/>'''bold'''.)
The fundamental equation
<math display="block">\frac{d^2\mathbf{r}}{ds^2} + \alpha\ \mathbf{r} = -\mathbf{P}</math> ▼: <math>\ \frac{\ \operatorname d^2 \mathbf{r}\ }{\ \operatorname d t^2\ } + \mu \frac{\ \mathbf{r}\ }{~ r^3\ } = \mathbf{0}\ ,\quad</math> where <math>\quad\ \mu \equiv G \left( m_1 + m_2 \right)\ \quad</math> is the system gravitational scaling constant,
where '''P''' is a constant [[Euclidean vector|vector]] and <math>\alpha</math> is defined by ▼is [[Regularization (mathematics)|regularized]] by applying this change of variables that yields:<ref name=Danby/>
<math display="block">\alpha = \frac\mu a</math> ▼▲: <math display="block"> \frac{ \ \operatorname d^2 \mathbf{r} \ }{ ds~ \operatorname d s^2 \ } + \alpha\ \mathbf{r} = - \mathbf{P} \ </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: ▼▲where '''<math>\ \mathbf P '''\ </math> is asome [[to be determined|t.b.d.]] constant [[ Euclidean vector (physics)|vector]] and <math> \ \alpha \ </math> is the orbital energy, defined by
<math display="block">\frac{d^3\mathbf r} {ds^3} + \alpha\frac{d\mathbf r} {ds} = \mathbf{0}</math> ▼▲<math display="block"> \alpha =\equiv \frac {\ \mu \ }{ a } ~.</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"/>{{rp|at=Eq. 6.9.26}} ▼▲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 geteliminate the constant vector <math>\ \mathbf P\ ,</math> at the price of getting a third-degree differential equation:
<math display="block">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> ▼▲<math display="block"> \ \frac{ \ \operatorname d^3 \mathbf r }\ }{ ds~\operatorname d s^3 \ } + \alpha\frac{ \ \operatorname d \mathbf r }\ }{ ds\ \operatorname d s\ } = \mathbf{0} \ </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: ▼▲The family of solutions to this differential equation<ref name=Danby/> are for convenience written symbolically asin terms of the four functions <math> \ 1 \ </math> ([[constant function|constant]]), <math>\ s\ c_1 \!\!\left( \ \alpha s^2 \ \right) \ , \ </math> <math>\ s^2 \ c_2 \!\!\left( \ \alpha s^2 \ \right) \ , </math> and <math>\ s^3 c_3\!\!\left(\ \alpha s^2\ \right)\ ;\ </math> where the functions <math>\ c_k \!(x) \ ,</math> , called ''[[Stumpff function]]s '', which are truncated generalizations of [[sine and cosine #series_defs_anchor|sine functionsand cosine series]]. Applying this results in:<ref name= "Danby "/>{{rp|at=Eq. 6.9.26}}
▲<math display="block"> \ t - t_0t_\mathsf{o} = r_0r_\mathsf{o}\ s\ c_1 \!\!\left( \ \alpha s^2 \ \right) + r_0r_\mathsf{o} \frac{ dr_0~ \operatorname d r_\mathsf{o}\ }{ dt\ \operatorname d t\ }\ s^2 \ c_2 \!\!\left( \ \alpha s^2 \ \right) + \mu \ s^3 \ c_3 \!\!\left( \ \alpha s^2 \ \right) \ </math>
▲which is the universal variable formulation of Kepler's Equationequation. 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 <math>\ f \ </math> and <math>\ g \ </math> functions:
<math display="block">\begin{align}
\ f(s) & = 1 - \left( \frac{\ \mu\ }{r_0~ r_\mathsf{o}\ } \right) s^2 c_2\!\!\left(\ \alpha s^2\ \right)\ , \\[1.5ex]
\ g(s) & = t - t_0t_\mathsf{o} - \mu s^3c_33 c_3\!\!\left(\ \alpha s^2\ \right)\ , \\[1.5ex]
\frac{df}{dt} = \dot{f}(s) &= -\left(\frac{\ \mu\ }{\ r_\mathsf{o} r\ r_0}\right) s\ c_1\!\!\left(\ \alpha s^2\ \right)\ , \\[1.5ex]
\frac{dg}{dt} = \dot{g}(s) &= 1 - \left( \frac{\ \mu\ }{ r } \right)\ s^2 c_2\!\!\left(\ \alpha s^2\ \right)\ \\[-1ex]\, ~.
\end{align} </math>
The values of the <math>\ f\ </math> and <math>\ g\ </math> functions determine the position of the body at the time <math>\ t\ </math>:
<math display="block">\ \mathbf{r} = \mathbf{r}_0_\mathsf{o}\ f(s) + \mathbf{v}_0_\mathsf{o}\ 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 display="block">\ \mathbf{v} = \mathbf{r}_0_\mathsf{o}\ \dot{f}(s) + \mathbf{v}_0_\mathsf{o}\ \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_\mathsf{o}\ </math> and <math>\ \mathbf{v}_0_\mathsf{o}\ </math> are the position and velocity, respectively, at arbitrary initial time <math>t_0\ t_\mathsf{o} ~.</math>.
==References==
{{Reflistreflist|25em}}
[[Category:Orbits]]
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