Lesson plan
Stumpff function
In celestial mechanics, the Stumpff functions \ c_k(x)\ , were developed by Karl Stumpff for analyzing trajectories and orbits using the universal variable formulation. They are defined by the alternating series: : \ 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 ~~ k = 0, 1, 2, 3,\ \ldots ~~. Like the sine, cosine, and exponential functions, Stumpff functions are well-behaved entire functions : Their series converge absolutely for any finite argument \ x ~. Stumpff 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.)
Article summary
In celestial mechanics, the Stumpff functions \ c_k(x)\ , were developed by Karl Stumpff for analyzing trajectories and orbits using the universal variable formulation. They are defined by the alternating series: : \ 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 ~~ k = 0, 1, 2, 3,\ \ldots ~~. Like the sine, cosine, and exponential functions, Stumpff functions are well-behaved entire functions : Their series converge absolutely for any finite argument \ x ~. Stumpff 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.)
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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