Astronomy & Celestial Mechanics
the sky, computed
Concept

Eccentric anomaly — where it appears

An angle measured at the centre of an orbit to the projection of the body onto its circumscribed circle, used as an intermediary to elapsed time. It is not an observable and not a direction to anything, and it exists because Kepler's equation is simple in it and in no other variable.

Named by 3 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 One series, two eccentricities, and a limit between them. The Lagrange series for E − M, summed to 1, 2, 3, 6, 22 terms, against the exact solution of Kepler's equation, over half a revolution. Left, at e = 0.5: the partial sums close on the exact curve and the last two are indistinguishable from it. Right, at e = 0.8: they do not, and the 22-term sum is worse than the 1-term one, missing the exact value by 0.24 radians at M = π/2. Nothing about the orbit changes between the two panels — an eccentricity of 0.8 is an ordinary comet — and nothing about the equation changes either. What changes is that a singularity of E as a function of complex e has come inside the circle of radius e, at the Laplace limit 0.6627434, which is the root of e·exp√(1+e²) = 1 + √(1+e²) and has no astronomical meaning whatever. The coefficients are computed from the Bessel expansion in logarithms; the first two are sin M and ½sin 2M exactly, which is what the generator asserts before drawing.

The formula that exists, and is not used

Kepler's equation does have a closed-form solution — an infinite series in the eccentricity, written down by Lagrange. It converges up to e = 0.6627434 and not one part beyond, and that number has nothing to do with astronomy.

orbits · The ellipse 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.

Kepler's equationMean anomalyTrue anomalyAnomalyBessel functionsCollision singularityConditioningEccentricityEnergy driftEquation of the centreFictitious timeFocus

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