Stumpff function explained

In celestial mechanics, the Stumpff functions

were developed by Karl Stumpff for analyzing trajectories and orbits using the universal variable formulation.They are defined by the alternating series: for

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 to open orbits (parabolas and hyperbolas, 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.)

Relations to circular and hyperbolic trigononometric functions

By comparing the Taylor series expansion of the trigonometric functions sin and cos with

and a relationship can be found. For

Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find for

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.

Recursion

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

for

or when

for

Using this recursion, the two further Stumpf functions needed for the universal variable formulation are, for

and for

Relations to other functions

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