Astronomy & Celestial Mechanics
the sky, computed
Concept

Newton's method — where it appears

Root-finding by repeatedly replacing a function with its tangent, which doubles the number of correct digits at each step once it is close. It needs a derivative and a starting guess, and where a problem has several roots it returns whichever one the guess fell nearest without saying so.

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

The three anomalies at E = 1.15 rad. The auxiliary circle construction. The eccentric anomaly E is measured at the centre, the true anomaly ν at the focus, and the mean anomaly M is time expressed as an angle. Here E = 1.150, ν = 1.827 and M = 0.602 radians, related by M = E − e sin E.

The position that has no formula, and is computed anyway

Kepler's equation relates where a body is to when it is there. It cannot be solved in elementary functions, Kepler said so, and nobody has managed it since — which has stopped nothing.

orbits · The ellipse 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

Named alongside it

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

EccentricityKepler's equationAnomalyBarkers equationConditioningConic sectionsEccentric anomalyFocusHyperbolic orbitMean anomalyOrbital elementsOrbital energy

All concepts