Plate 01 — Totality

Bessel’s method, plainly

The dumb way to compute an eclipse is to stand at a spot on Earth, work out where the Sun is in your sky and where the Moon is in your sky, and see whether one covers the other. Then do it again a minute later. Then do the whole thing again for the next spot. It is spherical trigonometry, twice, for every observer and every instant, and none of the work you did for one person helps the next one.

Friedrich Bessel’s move in 1824 was to stop looking at the sky and start looking at the shadow.

Draw the line from the center of the Sun through the center of the Moon and keep going — that is the shadow axis. Now build a coordinate frame around that line instead of around the observer: origin at the center of the Earth, one axis pointing straight down the shadow. The plane through the Earth’s center at right angles to it is the fundamental plane.

Everything gets simple in that frame, because the shadow is no longer a cone sweeping through space at an angle. It is a circle sitting in a plane:

• x, y — where the axis pierces the plane. Two numbers.
• L1, L2 — the radii the penumbra and the umbra cut on it. Two more.
• d, mu — which way the axis is pointing. Two more.
• tan f1, tan f2 — how fast the two cones taper. Two constants.

Eight quantities, each published as a short polynomial in time, good for the few hours the eclipse lasts. That is the whole table this page runs on — twenty-two numbers a row, and thirty-four rows carries it to 2050.

And here is why it pays. You, wherever you are reading this also have a position in that same frame — call it (ξ, η, ζ), which is nothing but your latitude and longitude run through the same rotation. Your distance from the shadow axis is m = √((x−ξ)² + (y−η)²). Compare that one number against the shadow radii at your height and you have your whole answer: inside L1 and you are in partial, inside L2 and the Moon covers it completely, and the contact times are the instants m crosses each radius. Plane geometry. A circle sliding across a flat surface and a dot sitting on it.

The part that makes it worth your attention if you do this for a living: it is a change of basis. Same trick as restating everything into one currency before you compare, instead of converting pair by pair at every comparison. The hard, expensive work — where are those two bodies, actually — is done once, by the people who own that problem, and handed to you as a small table of coefficients. What is left for the browser is arithmetic a pocket calculator could do. Two hundred years old, still what NASA publishes, and deterministic: same inputs, same answer, every time.

One wrinkle worth knowing because it bites. mu is given in Terrestrial Dynamical Time, which runs uniformly, and the Earth does not. By the time your wall clock agrees, the ground has turned a little further — 0.00417807 degrees for every second of ΔT. Leave that correction out and every longitude comes back a degree or so wrong, which is small enough to look right and big enough to be wrong. It is the one line in there that looks like a fudge factor and is not.

None of those elements are taken on trust, either. Every row is recomputed from scratch — greatest eclipse, gamma, the sub-point, the magnitude, the sun’s altitude and the central duration — and held against NASA’s Five Millennium Catalog before the site is allowed to build. That check is written in Python rather than in the JavaScript running on this page, on purpose: a second engine in a second language cannot inherit the mistakes of the file it is checking. A bad digit does not ship.

I audit the decoration. Ask me what I do with the accruals. The rule came from here — the lowest-stakes thing on this site