Universal variable formulation: Difference between revisions

Revision as of 23:12, 25 May 2024

Revision as of 00:42, 26 August 2024

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==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'''.)

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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>

which is the universal variable formulation of [[Kepler's equation]].

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 <math>\ f\ </math> and <math>\ g\ </math> functions:

which is theThis 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 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]

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\ \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{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 respectivelyvectors at time <math>\ t\ ,</math> and <math>\ \mathbf{r}_\mathsf{o}\ </math> and <math>\ \mathbf{v}_\mathsf{o}\ </math> are the position and velocity, respectively, at arbitrary initial time <math>\ t_\mathsf{o} ~.</math>

==References==