Astronomy & Celestial Mechanics
the sky, computed
Series

The ellipse — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. An orbit at eccentricity 0.6. An orbit of eccentricity 0.6. The primary sits at a focus, offset from the centre by 0.6 of the semi-major axis, and the closest and furthest points differ by a factor of 4.00.

    The orbit is an ellipse, and the Sun is not in the middle of it

    Kepler's first law is usually drawn wrong. The interesting content is not the ellipse — it is the focus, and the fact that one of the two is empty.

    part 1 · orbits
  2. 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.

    part 2 · orbits
  3. What an eccentricity does to the shape, and what it does to the offset. The fractional flattening 1 − b/a and the fractional focal offset c/a, against eccentricity. The offset is first order in e and the flattening is second, so at Earth's e = 0.0167 the outline is 0.014% from a circle while the Sun sits 1.67% of the semi-major axis off the centre — a factor of 120.

    An orbit can look exactly like a circle and still not be one

    Earth's orbit departs from a circle by fourteen parts in a hundred thousand. The Sun's offset from its centre is a hundred and twenty times larger, and everything interesting is in the offset.

    part 3 · orbits
  4. 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.

    part 4 · orbits

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