Universal variable formulation: Difference between revisions

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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 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 [[solarSolar systemSystem]] 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| [[Orbital 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 [[orbital eccentricity|eccentricity]], solving [[Kepler's Equation]] by methods like [[Newton'ts method]] gives excellent results. However, as the orbit approaches an escape trajectory, it becomes more and more eccentric, [[limit of a sequence|convergence]] of numerical iteration may become unusably sluggish, or fail to converge at all for {{nobr|&thinsp;{{math| [[Orbital eccentricity|''e'']] ≥ 1 }} .}}<ref name=StiefelScheifele>

{{cite book

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>{{cite |author=Danby, J. M. A.|title=Fundamentals of Celestial Mechanics|publisher=Willman-Bell|date=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.

|first1 = Eduard L. |last1 = Stiefel |author1-link=Eduard Stiefel

|first2 = Gerhard |last2 = Scheifele

|year = 1971

|title = Linear and Regular Celestial Mechanics: Perturbed two-body motion, numerical methods, canonical theory

|publisher = Springer-Verlag

}}

</ref><ref name=Danby>

{{cite book

|last = Danby |first = J.M.A.

|year = 1988

|title = Fundamentals of Celestial Mechanics |edition = 2nd

|publisher = Willmann-Bell

|isbn = 0943396204

}}

</ref>

Note that the conventional form of Kepler's equation cannot be applied to [[Parabolic orbit|parabolic]] and [[hyperbolic orbit]]s without special adaptions, to accommodate [[imaginary number]]s, since its ordinary form is specifically tailored to sines and cosines; escape trajectories instead use &thinsp;{{math|sinh}}&thinsp; and &thinsp;{{math|cosh}}&thinsp; ([[hyperbolic function]]s).

==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> \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 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>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]]:

:<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>

(In all of the following formulas, carefully note the distinction between [[scalar (physics)|scalars]] <math>\ r\ ,</math> in ''italics'', and [[vector (physics)|vectors]] <math>\ \mathbf r\ ,</math> in upright '''bold'''.)

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>.

We can [[regularization (mathematics)|regularize]] the fundamental equation

: <math>\ \frac{\ \operatorname d^2 \mathbf{r}\ }{\ \operatorname d t^2\ } + \mu \frac{\ \mathbf{r}\ }{~ r^3\ } = \mathbf{0}\ ,\quad</math>

: where <math>~~ \mu \equiv G \left( m_1 + m_2 \right) ~~</math> is the system gravitational scaling constant,

by applying the change of variable from time <math>\ t\ </math> to <math>\ s\ </math> which yields<ref name=Danby/>

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

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> \alpha =\equiv \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]], however, the unknown constant vector is somewhat inconvenient. Taking the derivative again, we geteliminate the constant vector <math>\ \mathbf P\ ,</math> at the price of getting a third-degree differential equation:

: <math>\ \frac{\ \operatorname d^3 \mathbf r}\ }{ds~\operatorname d s^3\ } + \alpha\frac{\ \operatorname d \mathbf r}\ }{ds\ \operatorname d s\ } = \mathbf{0}\ </math>

The family of solutions to this differential equation<ref name=Danby>eq 6.9.22</ref> are for convenience written symbolically asin terms of the three functions <math>1,\ 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]]. ApplyingThe thischange-of-variable resultsequation in<refmath>\ \tfrac{ \operatorname d t }{\ \operatorname d s\ } name="Danby">Equation 6.9.26r\ </refmath>: gives the scalar integral equation

::<math>\ \int_{\tilde{t}=t_\mathsf{o}}^{t} \operatorname{d}\tilde{t} = \int_{\tilde{r}=r_\mathsf{o},\ \tilde{s}=0}^{r,\ s} ~\tilde{r}(\ \tilde{s}\ ) ~ \operatorname{d}\tilde{s} ~.</math>

After extensive algebra and back-substitutions, its solution results in<ref name=Danby/>{{rp|at=Eq. 6.9.26}}

:<math>\ 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 equation]].

whichThere is theno closed analytic solution, but this universal variable formulationform of Kepler's Equation. This equation can now be solved numerically for <math>\ s\ ,</math> using a [[root-finding algorithm]] such as [[Newton's method]] or [[Laguerre's method]] for a given time <math>\ t\ ~.</math> toThe yieldvalue of <math>\ s\ </math>, which in turnso-obtained is then used in turn to compute the [[<math>\ f\ </math> and <math>\ g\ </math> functions]] and the <math>\ \dot f\ </math> and <math>\ \dot g\ </math> functions needed to find the current position and velocity:

: <math>\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]

\ \dot{f}(s) \equiv \frac{\ \operatorname d f\ }{\ \operatorname d t\ } &= -\left(\frac{\ \mu\ }{\ r_\mathsf{o} r\ }\right) s\ c_1\!\!\left(\ \alpha s^2\ \right)\ , \\[1.5ex]

\ \dot{g}(s) \equiv \frac{\ \operatorname d g\ }{\ \operatorname d t\ } &= 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{vr}(t) = \mathbf{r}_0_\ \dotmathsf{fo}\ f(s) + \mathbf{v}_0_\ \dotmathsf{go}\ 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{rv}(t) = \mathbf{r}_0_\mathsf{o}\ \dot{f}(s) + \mathbf{v}_0_\mathsf{o}\ \dot{g}(s)\ </math>

: where <math>\ \mathbf{r}(t)\ </math> and <math>\ \mathbf{v}(t)\ </math> are respectively the position and velocity vectors at time <math>\ t\ ,</math> and <math>\ \mathbf{r}_\mathsf{o}\ </math> and

where:: <math>\mathbf{r}</math> is the position at time <math>t</math>, and <math>\mathbf{rv}_0</math> and <math>_\mathbfmathsf{vo}_0\ </math> are the position and velocity, respectively, at arbitrary initial time <math>t_0\ t_\mathsf{o} ~.</math>.

==References==

{{reflist|25em}}

<references/>

{{astronomy-stub}}

[[Category:Orbits]]

[[Category:Equations of astronomy]]