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]], extending it 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 is also applicable to ejection of small bodies in [[Solar System]] from the vicinity of massive planets, during which processes the approximating two-body orbits can have widely varying [[orbital eccentricity|eccentricities]], almost always {{nobr| {{math| [[Orbital eccentricity|''e'']] ≥ 1 }} .}}

==Introduction==

A common problem in orbital mechanics is the following: Given a body in an [[orbit]] and a fixed original time <math>\ t_\mathsf{o}\ ,</math> find the position of the body at some later time <math>\ t ~.</math> For [[elliptical orbit]]s with a reasonably small [[orbital eccentricity|eccentricity]], solving [[Kepler's Equation]] by methods like [[Newton's 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
|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> Furthermore, Kepler's equation cannot be directly applied to [[Parabolic orbit|parabolic]] and [[hyperbolic orbit]]s, since it specifically is tailored to elliptic orbits.

|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 display="block"> \frac{ \operatorname d s }{\ \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.

(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'''.)

The fundamental equation

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

is [[Regularization (mathematics)|regularized]] by applying this change of variables that yields:<ref name=Danby/>

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

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

where <math>\ \mathbf P\ </math> is some [[to be determined|t.b.d.]] constant [[vector (physics)|vector]] and <math>\ \alpha\ </math> is the orbital energy, defined by

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

<math display="block"> \alpha \equiv \frac{\ \mu\ }{ a } ~.</math>

: <math display="block"> \frac{\ \operatorname d^2 \mathbf{r}\ }{~ \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 eliminate the constant vector <math>\ \mathbf P\ ,</math> at the price of getting a third-degree differential equation:

where <math>\ \mathbf P\ </math> is some [[to be determined|t.b.d.]] constant [[vector (physics)|vector]] and : <math>\ \alpha\ </math> is the orbital energy, defined by

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

:: <math display="block"> \alpha \equiv \frac{\ \mu\ }{ a } ~.</math>

The family of solutions to this differential equation<ref name=Danby/> are for convenience written symbolically in terms of the three functions <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 and cosine series]]. Applying this results in:<ref name=Danby/>{{rp|at=Eq. 6.9.26}}

<math display="block">\ t - t_\mathsf{o} = r_\mathsf{o}\ s\ c_1\!\!\left(\ \alpha s^2\ \right) + r_\mathsf{o} \frac{~ \operatorname d r_\mathsf{o}\ }{\ \operatorname d t\ }\ s^2 c_2\!\!\left(\ \alpha s^2\ \right) + \mu \ s^3 c_3\!\!\left(\ \alpha s^2\ \right)\ </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 eliminate the constant vector <math>\ \mathbf P\ ,</math> at the price of getting a third-degree differential equation:

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

The family of solutions to this differential equation<ref name=Danby/> are for convenience written symbolically in terms of the three functions <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 and cosine series]]. ApplyingThe thischange-of-variable resultsequation in:<ref name=Danby/math>\ \tfrac{ \operatorname d t }{rp|at=Eq.\ \operatorname d s\ 6.9.26}} = r\ </math> 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 display="block">\ t - t_\mathsf{o} = r_\mathsf{o}\ s\ c_1\!\!\left(\ \alpha s^2\ \right) + r_\mathsf{o} \frac{~ \operatorname d r_\mathsf{o}\ }{\ \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]].

ThisThere is no closed analytic solution, but this universal variable form of Kepler's 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> The value of <math>\ s\ </math> so-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 display="block">\begin{align}

\ f(s) & = 1 - \left( \frac{\ \mu\ }{~ r_\mathsf{o}\ } \right) s^2 c_2\!\!\left(\ \alpha s^2\ \right)\ , \\[1.5ex]

\ g(s) & = t - t_\mathsf{o} - \mu\ s^3 c_3\!\!\left(\ \alpha s^2\ \right)\ , \\[1.5ex]

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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}(t) = \mathbf{r}_\mathsf{o}\ f(s) + \mathbf{v}_\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}(t) = \mathbf{r}_\mathsf{o}\ \dot{f}(s) + \mathbf{v}_\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
:: <math>\ \mathbf{v}_\mathsf{o}\ </math> are the position and velocity at arbitrary initial time <math>\ t_\mathsf{o} ~.</math>

==References==