The Sun Was Eclipsed; It Was Total

What the slider actually does

An eclipse has two halves, and they are kept on two different clocks. Where the Moon and the Sun are is a problem in gravitation, and gravitation runs on TT, a uniform time. Which part of the Earth is turned towards them is a problem in rotation, and the Earth is not a good clock. The difference between the two, ΔT = TT − UT1, is the amount by which the planet is running late.

You cannot look it up for 709 BC. Nobody was measuring. The only way to know how far the Earth had turned on a given morning three thousand years ago is to find something that wrote the answer down without meaning to, and a total eclipse is exactly that: the umbra is a spot roughly two hundred kilometres wide, dragged across the planet in a few hours, and anyone standing inside it knows. A chronicle saying the Sun went out, at a place we can put on a map, pins the Earth's rotational position at that instant to the width of that spot.

So the slider is not a fudge factor. It is the measurement. Drag it until the band covers the city, and read off how late the Earth was.

Two of the answers are worth having in advance. In 136 BC the Babylonians watched Venus, Mercury, Jupiter and Mars come out in the middle of the morning, and their tablets are, in the judgement of the people who read them, the finest description of a total eclipse to survive from antiquity. Compute that eclipse without the clock correction and the umbra is over Morocco, 4,592 km west of the city. The Chinese record from 709 BC is worse: the shadow is in the eastern Mediterranean, 7,841 km away, close to a quarter of the way round the planet.

Notice what the record has to say for this to work. At ΔT = 0 both places still see an eclipse, because the penumbra is thousands of kilometres wide: Qufu gets a respectable partial with a quarter of the Sun bitten out of it. A chronicle that said only "the Sun was eclipsed" would measure nothing. It is the second clause, it was total, that narrows the Earth's rotational position to the width of a shadow.

And the slide is exact, not an approximation

The Earth is a solid of revolution about its spin axis, so changing ΔT rotates the whole computed geometry about that axis and does nothing else. Latitudes do not move. Path widths do not change. Only longitude shifts, by 360 × 1.00273781 / 86400 = 0.0041781° for every second of ΔT, which is the sidereal rotation rate. That is why the band on the map can be computed once and then simply slid: the page is not cheating with a small-angle approximation, it is exploiting a symmetry. The verifier checks it, at ΔT = 3 hours, to better than a nanodegree of latitude.

Each orange bar is a band of ΔT in which the recorded place lies inside the umbra, recomputed here from scratch. The grey curve is what the Earth's rotation would have done if tidal friction were the only thing acting on it. Every single record sits below it. The planet has been slowing, but it has not been slowing as fast as the Moon alone can explain, and the gap has to be paid for by something else.

Stephenson, Morrison and Hohenkerk attribute most of the difference to the solid Earth still rebounding from the weight of the last ice age, which makes the planet less oblate and so speeds it up, with a further correction for the coupling between the core and the mantle. In their words, this work sets firm boundaries for future work on post-glacial rebound and core-mantle coupling. A clay tablet from Babylon is, by this route, a constraint on the viscosity of the mantle.

The three records the parabola misses

The one it gets wrong, on purpose

The records