Astronomy & Celestial Mechanics
the sky, computed
Concept

Stumpff functions — where it appears

The series that appear when the Kepler problem is written in a universal variable. They reduce to trigonometric or hyperbolic functions according to the sign of the orbital energy, which is what allows a single solver to handle every conic without being told which one it is dealing with.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Four conics through one periapsis, drawn by one solve. Distance from the Sun against time for four orbits sharing a periapsis of 0.5 AU, at e = 0.6, e = 1, e = 1.0001, e = 1.4, over 900 days. Every point on every curve came from the same universal Kepler solve — no branch on the conic class anywhere in it — and each curve was then checked against the classical solution of its own kind at the midpoint: Kepler's equation at e = 0.6 agrees to machine precision; Barker's cubic at e = 1 agrees to machine precision; e sinh H − H at e = 1.0001 agrees to machine precision; e sinh H − H at e = 1.4 agrees to machine precision. The curve to read twice is e = 1.0001: over this arc it lies within 0.02% of the parabola and is indistinguishable from it, and it is the only one of the four whose fate the drawing cannot show. What the classical parameterisation costs there is arithmetic rather than impossibility: at the midpoint of this arc, e sinh H − H throws away 3.3 of its sixteen digits to cancellation, against 0.2 at e = 1.4 — enough to matter to an ephemeris and not enough to stop a plot.

The one solve that does not ask which conic it is

Kepler's equation is for ellipses, Barker's cubic for parabolas, and a hyperbolic sine for the rest — three parameterisations of one motion, each worst exactly where its neighbour takes over. The universal variable removes the question, and the removal is not a convenience.

orbits · Universal variables One squaring, and the singularity is gone. Left, a harmonic oscillator: an ellipse centred on the origin, marked at 24 equal steps of its own phase. Right, the same points after squaring as complex numbers, u ↦ u². The image is an ellipse with the origin at a focus — checked here by the focal property, r₁ + r₂ = 2a to nine figures at four points — with semi-axes (A²+B²)/2 = 2.5000 and AB = 1.5000 and the focus at (A²−B²)/2 = 2.0000, which is ae exactly. Three things follow at once. The angle doubles, so one turn of the oscillator is one whole orbit traversed twice as fast in phase; the equal phase steps on the left arrive as the eccentric anomaly on the right, which is why that anomaly and not the true one is what the equations want; and the collision at r = 0, where the inverse square is infinite, is the point u = 0, where the oscillator has a perfectly ordinary velocity. The singularity was a property of the coordinates.

The singularity that is a change of variable

The Kepler problem blows up at zero separation, and a fixed-step integrator falls apart long before it gets there. Divide time by the radius and the equations become a harmonic oscillator — exactly, for every conic at once.

orbits · Universal variables

Named alongside it

The objects these essays reach for when they reach for this one.

ConditioningBarkers equationCollision singularityConic sectionsEccentric anomalyEccentricityEnergy driftFictitious timeHarmonic oscillatorHyperbolic orbitKepler's equationLevi civita transformation

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