Astronomy & Celestial Mechanics
the sky, computed
Concept

True anomaly — where it appears

The angle at the occupied focus from periapsis to the body, which is where the body actually is as seen from the primary. It is what the orbit equation returns a radius for, and it advances fastest at pericentre and slowest at apocentre.

Named by 4 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 Three rotations, applied in order. An orbit of eccentricity 0.45 carried into space by the three orientation angles, one panel per rotation: the argument of periapsis ω = 40°, then the inclination i = 42°, then the longitude of the ascending node Ω = 55°. The last panel applies the same three angles in the reverse order and arrives somewhere else, because rotations about different axes do not commute.

Three rotations that put an orbit in space, and they do not commute

An orbit's orientation takes three angles. Give the same three angles in a different order and the orbit ends up somewhere else — which is why the convention is part of the data.

orbits · Orbital elements Four averages of one distance, and the two of them that are the semi-major axis. The average distance of a body from its primary, against eccentricity and in units of the semi-major axis, computed four ways: averaged over time, over true anomaly, over eccentric anomaly, and as the harmonic mean in time. Every curve is a quadrature over the orbit — 2,048 panels uniform in eccentric anomaly, with Kepler's equation supplying the time weight — and not a closed form. Two of the four are exactly a at every eccentricity, which is why they are drawn as one line: the eccentric-anomaly average, because the mean of cos E over a turn is zero, and the harmonic mean in time, because the time weight cancels 1/r at every node before the sum begins. The other two are not: the time average is a(1 + e²/2), which rises to 1.4050 a at e = 0.9, and the true-anomaly average is a√(1−e²) — the semi-minor axis — which falls to 0.4359 a there. So a is the average distance in two senses out of four, and the ordering b ≤ a ≤ ⟨r⟩ₜ holds at every eccentricity with equality only on the circle. At Earth's e = 0.0167 the four agree to 0.014%, and at Mercury's e = 0.2056 the spread is 2.14%. The distinction is invisible for the planets and unavoidable for a comet, and it is the reason a quoted "mean distance" has to say which mean.

The average depends on what is being averaged

Four ways of averaging one orbit's distance from its primary give four different numbers, and only two of them are the semi-major axis. Which two is not a matter of convention, and the same arithmetic decides how much sunlight a planet receives in a year.

orbits · Orbital averages

Named alongside it

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

AnomalyEccentricityKepler's equationMean anomalyPeriapsisEccentric anomalyOrbital elementsAngular momentumArgument of periapsisAscending nodeAstronomical unit (AU)Bessel functions

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