Stumpff function

In celestial mechanics, the Stumpff functions ${\displaystyle \ c_{k}(x)\ ,}$ were developed by Karl Stumpff for analyzing trajectories and orbits using the universal variable formulation.^[1][2][3] They are defined by the alternating series:

${\displaystyle \ c_{k}(x)~\equiv ~{\frac {1}{\ k!\ }}-{\frac {x}{~\left(k+2\right)!\ }}+{\frac {\ x^{2}}{~\left(k+4\right)!\ }}-\cdots ~=~\sum _{n=0}^{\infty }\ {\frac {\ (-1)^{n}\ x^{n}\ }{\ \left(k+2n\right)!\ }}~~}$ for ${\displaystyle ~~k=0,1,2,3,\ \ldots ~~.}$

Like the sine, cosine, and exponential functions, Stumpf functions are well-behaved entire functions : Their series converge absolutely for any finite argument ${\displaystyle \ x~.}$

Stumpf functions are useful for working with surface launch trajectories, and boosts from closed orbits to escape trajectories, since formulas for spacecraft trajectories using them smoothly meld from conventional closed orbits (circles and ellipses, eccentricity e : 0 ≤ e < 1 ) to open orbits (parabolas and hyperbolas, ( e ≥ 1 ), with no singularities and no imaginary numbers arising in the expressions as the launch vehicle gains speed to escape velocity and beyond. (The same advantage occurs in reverse, as a spacecraft decelerates from an arrival trajectory to go into a closed orbit around its destination, or descends to a planet's surface from a stable orbit.)

By comparing the Taylor series expansion of the trigonometric functions sin and cos with ${\displaystyle \ c_{0}(x)\ }$ and ${\displaystyle \ c_{1}(x)\ ,}$ a relationship can be found. For ${\displaystyle ~x>0\ :}$

${\displaystyle {\begin{aligned}c_{0}(x)~&=~~\cos {\sqrt {x\ }}\ ,\\[1ex]c_{1}(x)~&=~{\frac {\ \sin {\sqrt {x\ }}\ }{\sqrt {x\ }}}~.\end{aligned}}}$

Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find for ${\displaystyle ~x<0\ :}$

${\displaystyle {\begin{aligned}c_{0}(x)~&=~~\cosh {\sqrt {-x\ }}\ ,\\[1ex]c_{1}(x)~&=~{\frac {\ \sinh {\sqrt {-x\ }}\ }{\sqrt {-x\ }}}~.\end{aligned}}}$

Circular orbits and elliptical orbits use sine and cosine relations, and hyperbolic orbits use the sinh and cosh relations. Parabolic orbits (marginal escape orbits) formulas are a special in-between case.

For higher-order Stumpff functions needed for both ordinary trajectories and for perturbation theory, one can use the recurrence relation:

${\displaystyle \ x\ c_{k+2}(x)={\frac {1}{k!}}-c_{k}(x)~}$ for ${\displaystyle ~k=0,1,2,\ \ldots \ ~,}$

or when ${\displaystyle \ x\neq 0\ }$

${\displaystyle \ c_{k+2}(x)~=~{\frac {\ 1\ }{x}}\left(\ {\frac {1}{k!}}-c_{k}(x)\ \right)~}$ for ${\displaystyle ~k=0,1,2,\ \ldots ~~.}$

Using this recursion, the two further Stumpf functions needed for the universal variable formulation are, for ${\displaystyle ~x>0\ :}$

${\displaystyle {\begin{aligned}\ c_{2}(x)~&=~~~{\frac {\ 1-\cos {\sqrt {x\ }}\ }{\sqrt {x\ }}}\ ,\\\\\ c_{3}(x)~&=~{\frac {\ {\sqrt {x\ }}-\sin {\sqrt {x\ }}\ }{x}}\ ;\end{aligned}}}$

and for ${\displaystyle ~x<0\ :}$

${\displaystyle {\begin{aligned}\ c_{2}(x)~&=~\quad {\frac {\ 1-\cosh {\sqrt {-x\ }}\ }{\sqrt {-x\ }}}\ ,\\\\\ c_{3}(x)~&=~{\frac {\ {\sqrt {-x\ }}\ -\ \sinh {\sqrt {-x\ }}\ }{-x}}~~.\end{aligned}}}$

The Stumpff functions can be expressed in terms of the Mittag-Leffler function:

${\displaystyle \ c_{k}(x)~=~E_{2,k+1}(-x)~~.}$