The Moon That Was Always Late

This is the story of the first time anyone measured anything at all about light — how fast it goes, whether it takes any time to get anywhere, whether the question even means anything. It was settled in 1676. Not by an experiment; nobody built an apparatus, and nobody set out to do it. It was settled by a man employed to do something else, who noticed that a column of numbers other people had been dismissing as sloppiness for eight years had a shape in it.

The numbers are below. I am going to show them to you before I explain them, because everything you need is in them.

The job

Ole Rømer was a Dane in Paris, thirty-one years old, employed at the Royal Observatory. He was not there to think about light. He was there to watch a moon.

The moon was Io, innermost of the four large satellites of Jupiter, which Galileo had spotted in 1610 with one of the first telescopes ever pointed upward. Sixty years on, those four moons had become the most closely watched objects in the sky, for a reason that had nothing to do with Jupiter and everything to do with ships.

If you want to know how far east or west you are — your longitude — the sky offers no direct clue. The Earth is symmetrical about its axis, and every line of longitude looks like every other. But the Earth turns, and it turns at a known rate. So if you can establish what time it is where you are, and simultaneously what time it is back in Paris, the gap between those two numbers tells you precisely how far around the world you have come. Longitude is not really a problem of geography. It is the problem of knowing what time it is somewhere you are not.

Sailors had been managing without a solution for a very long time, and it is worth knowing how badly. The standard method was dead reckoning: take your heading from the compass, estimate your speed by throwing a knotted rope over the stern and counting knots against a sandglass, and keep a running tally of where those numbers must have put you. It works, in the sense that it is better than nothing. But every error is permanent — an unsuspected current, a helmsman half a point off his heading, a fortnight of cloud — and the errors compound for as long as the voyage lasts. Many captains dealt with this by declining to trust their easting and westing at all. Sail north or south until you reach the latitude of your destination, which you can measure honestly from the sun, then turn and run along that line until you hit something. It is slow, it wastes weeks, and it funnels every ship onto the same few predictable roads, which is excellent news if you are a pirate.

The clever proposals were older than the telescope, and all of them stalled on the same kind of obstacle. In 1514 Johannes Werner suggested measuring the moon against the fixed stars — the moon moves fast enough to serve as a hand sweeping across the sky’s dial — but neither the instruments nor the theory of the moon’s motion were remotely good enough, and would not be for another two and a half centuries. In 1530 Gemma Frisius proposed the obvious thing: carry a clock set to home time. He was entirely right and completely impractical, since no clock then in existence could keep time on land for a week, let alone at sea for a year. And there was a long-running hope invested in magnetism, because a compass needle does not point at true north and the size of its error varies from place to place — so perhaps a chart of that error could tell you where you stood. Edmond Halley would later spend years at sea mapping it. It failed for a reason nobody could have anticipated: the Earth’s magnetic field drifts, and the chart goes stale.

Jupiter’s moons offered a better answer. They orbit their planet on a schedule of extraordinary regularity, and each time round they slide into the planet’s shadow and wink out. That wink happens at a single moment for the whole of the Earth. If you could publish a table saying this particular eclipse will occur at 11:42 Paris time, then anyone anywhere with a telescope and a clear night could watch for the wink, note their own local time, and read off their longitude from the difference.

The tables were the hard part. Building them took years of patient observation, and the Observatory’s director, Giovanni Domenico Cassini, had made his reputation on them. Rømer’s job was to keep feeding the machine: watch Io, note the moment it vanished or returned, compare against the prediction, hand in the numbers.

The clock in the sky

Io goes round Jupiter once every forty-two and a half hours. It is a fast, tight, tidy orbit, and it does not wander. Each circuit, the moon passes behind the planet into the long cone of shadow the planet throws, and for a couple of hours it is simply gone. Then it emerges on the other side.

That is the tick. A few hundred usable ticks a year, each one visible to anyone on the night side of the Earth with a telescope good enough to separate a fleck of light from a bright disc.

Cassini’s tables predicted those ticks. The way you build such a table is to measure the period once, very carefully, and then multiply: if the moon comes out of shadow at a known moment and the period is 42 hours 28 minutes, you can compute the moment of the eclipse three hundred orbits from now with nothing but arithmetic.

And it worked, nearly. Each eclipse arrived a minute or two off the predicted time — sometimes early, sometimes late. Everybody knew this. Everybody put it down to the ordinary grit of the work: the period had been measured imperfectly, the telescopes were long and awkward, the clocks drifted, and the moment of an eclipse is not a snap but a fade, so two competent observers could differ by a minute on when exactly it had happened.

A residue of a minute or two is precisely what a careful person expects from a difficult measurement. So the interesting question was never why is there a residue. It was what shape is the residue.

The data

Here it is. Every dot is one eclipse of Io. The vertical position is the only thing that matters: how many minutes late the eclipse arrived, compared with what the table said. Above the line, late. Below the line, early.

(These timings are reconstructed — computed from the orbits and given the kind of random scatter a seventeenth-century observer would have contributed. The real observations were fewer, gappier and messier, which is a point I will come back to, because it matters. But the pattern in them is the pattern in these.)

Take a minute with it before reading on.

The first panel is the raw record: six and a half years of observations, gaps and all. The second is a tool for asking one specific question, and it is the question that cracks the whole thing open, so it is worth understanding what it does.

If a wobble repeats on some rhythm, you can test which rhythm by stacking. Pick a candidate period — say a year — and lay every observation down according to where it falls within that year, ignoring which year it came from. If you have guessed the right rhythm, every repetition lands on top of every other, and the scatter collapses into a single clean curve. If you have guessed wrong, the repetitions land out of step, and you get a shapeless cloud.

There are three obvious candidates, and there are buttons for all three. Try them.

What it isn’t

Work through the suspects. This is the part Rømer actually did, and there is nothing in it you cannot do yourself.

Perhaps the table is simply set wrong. If the period had been measured a little too long or a little too short, every prediction would be off — but off in a steady, accumulating way, drifting further from the truth as the years went on. That is not what the record shows. The error goes up, comes back down, and goes up again. Something oscillates. A wrong constant cannot oscillate.

Perhaps Io really is speeding up and slowing down. This is the natural reading, and it is what most people assumed: the moon’s orbit is more complicated than the simple table allows, and the table is failing to keep up. If so, whatever governs the variation must belong to the Jupiter system, so the rhythm ought to be Io’s own period, or Jupiter’s year, or some combination. Stack on Io’s forty-two and a half hours: a cloud. Stack on Jupiter’s twelve-year circuit: a cloud. Neither rhythm is in the data.

Perhaps it is us — the clocks. This is the sharp objection, the one a genuinely careful skeptic raises, and it deserves respect. A pendulum clock keeps time by the swing of a rod, and a metal rod is longer in summer than in winter. Observatories knew this. A clock that runs slightly slow in July and slightly fast in January would produce exactly this kind of smooth seasonal wobble in the timings, and it would be an error in the observatory, not a fact about the heavens. So stack on our year — 365 days.

A cloud.

Whatever this is, it is not keeping time with our seasons. That single result kills the clocks, kills the observers’ summer habits, kills anything that runs on the calendar.

Perhaps the observers are sloppy. They were, somewhat. But sloppiness scatters. It does not organise itself into a smooth rise and fall lasting many months. Bad data is noisy; this is noisy and shaped, and the shape is the thing that needs accounting for.

So run the slider and hunt. The scatter score tells you how tightly the data has collapsed, and you can chase it downhill. It bottoms out hard, and not near anything you were expecting.

Three hundred and ninety-nine days

That is the number. Not 365. Not 42.5 hours. Not twelve years. Three hundred and ninety-nine days, and at that period the cloud snaps into a curve so clean that what remains is about a minute of scatter — which is roughly the accuracy with which a man in 1673 could judge the moment a speck of light faded out through a wobbling forty-foot telescope. Fold on 399 days and there is nothing left to explain except the limits of the eyepiece.

So: what in the world takes 399 days?

Nothing does. That is the beautiful part, and it is worth sitting with. There is no object in the solar system with a 399-day period. Not the Earth, not Jupiter, not Io, not anything. The number does not belong to a thing at all.

It belongs to a relationship. The Earth goes round the sun in 365 days; Jupiter takes about twelve years, so it barely moves by comparison. Which means that we lap it. We come round the inside track, catch it, pass it, and swing away — and by the time we have caught up to it again, 399 days have gone by. That is the only thing in the system that happens on this rhythm: the Earth lapping Jupiter.

The anomaly is keeping time with our position relative to Jupiter, and with nothing else.

Jupiter does not know where we are

Now put the astronomy down for a moment and look at what has just been established.

The claim on the table is that the behaviour of a moon four hundred million miles away depends on where the Earth happens to be standing.

That is an absurd thing to believe. Jupiter takes no interest in us. Io has been ducking into that shadow, on that schedule, since long before there was anyone to watch, and it does not adjust its habits according to which side of the sun we are on this month. If the timing of the eclipse varies in step with our motion, then whatever is varying is not happening at Jupiter.

It is happening between there and here.

And as we lap Jupiter, exactly one thing changes between there and here: the distance. When we pass on the near side, the gap is as small as it gets. When we swing round to the far side of the sun, it is as large as it gets — larger by the full width of our own orbit. Then it closes again. That cycle takes 399 days.

Compare the two curves and the fit is exact. The eclipses run latest when we are furthest. They run earliest when we are nearest. The lateness tracks the distance.

Which leaves one conclusion, and it is the one nobody had been willing to reach for. If the moon’s schedule is fixed, and the eclipse nevertheless looks late when we are further away, then the lateness is not in the event. It is in the news of the event. Light takes time to cross the extra distance.

The boy on the bicycle

Here is the same situation with no astronomy in it at all, because the shape of the argument is more familiar than it looks.

There is a village down the road with a mill, and the mill wheel starts turning at six every morning. It is the most punctual thing in the district. The miller has been starting it at six for thirty years and he is not about to stop.

You live five miles away and you would like to know when the wheel starts. So you pay a boy with a bicycle to wait outside the mill, and the moment the wheel turns he rides out to tell you. He rides at a steady twelve miles an hour, so the news reaches you at twenty-five past six. Every morning, without fail: twenty-five past six.

Now suppose you are not settled. You are working your way along the road, and each week your camp is a little further from the village. The boy still leaves at six. He still rides at twelve miles an hour. But the news now arrives at half past, then at twenty-five to seven, then twenty to, later and later as the weeks go on.

Nothing whatever has happened to the mill. The wheel still starts at six. If you did not know you were moving, you would swear the miller had taken to sleeping in.

And now the part worth reading twice. Suppose you want to work out how fast the boy rides. You do not need to know how far away the village is. You never measured it, and you do not have to. All you need is how much further along the road you have moved, and how much later the news has been getting. Ten extra miles and fifty extra minutes of lateness tells you everything. The distance to the village drops out of the sum entirely.

That is Rømer’s situation, exactly. The mill wheel is Io sliding into shadow at its own unvarying six o’clock, every forty-two and a half hours. The boy on the bicycle is the light. Your progress along the road is the Earth swinging from the near side of its orbit round to the far side. And the total lateness that piles up — the full height of that 399-day wave — is the time light needs to cross the extra distance.

Which is the width of the Earth’s orbit. Not the distance to Jupiter. That cancels, exactly as the village did.

Nothing is wrong with the moon. The delay is in the messenger.

The demonstration

Rømer put this to the Académie des Sciences in the autumn of 1676, and he did something better than argue for it. He used it.

An eclipse of Io was due in early November, and the tables gave a time. Rømer announced in advance that the tables would be wrong — that the eclipse would come about ten minutes later than predicted — and said so publicly, with the number, weeks ahead.

On the night, it came late. By about ten minutes.

That is a far stronger move than fitting an explanation to data you already hold. Anybody can produce a story that accounts for the past; the past sits still and lets you fit things to it. A story that correctly tells you what a moon will do next month, when the standard tables say otherwise, is doing something a merely-fits-the-past story cannot. He had turned a pile of leftovers into a prediction, and the prediction had held.

A short account appeared in the Journal des Sçavans that December. The figure Rømer gave was around twenty-two minutes: the time light needs to cross the full width of the Earth’s orbit.

What he could not do

He did not say how fast light travels. Not from caution or modesty, and not because he missed the point — because he could not, and neither could anyone else then alive.

To turn twenty-two minutes into a speed you must divide a distance by it, and the distance in question is the width of the Earth’s orbit expressed in ordinary units: miles, or leagues, or whatever you please. That number was not reliably known. The astronomy of the day was superb at ratios — it could tell you that Jupiter sits about five times further from the sun than we do, and be right — and much shakier on how far we are from the sun in units you could pace out. A campaign to pin that down had run only four years earlier, and its answer was around seven percent low.

So the honest thing to report was a time, and a time is what he reported. It was Christiaan Huygens who did the division a few years later and produced an actual velocity, because Huygens was building a theory of light as a travelling wave, and a wave needs a speed to travel at. Rømer supplied the measurement. Somebody else supplied the ambition.

Rømer’s twenty-two minutes was too long, as it happens. The true figure is nearer sixteen and a half. He was out by a third — a real error, worth being plain about. But the error lives entirely in the difficulty of judging the exact instant a fuzzy speck fades out through a long, unsteady telescope, and in the scatter of the older observations he had to lean on. The reasoning was sound. Feed the same argument better numbers and it gives the right answer, which is what you want from an argument.

How thin this was

The data you just played with was cleaner than his. That difference is not a detail; it is most of the reason nobody had done this already.

He had no way to compare a clock in Paris with a clock anywhere else, in real time, at all. He had no photograph of anything. He could not record an eclipse and go back to check it; the observation existed only as a time written down in the moment by a person who might have blinked. For weeks at a stretch each year Jupiter sits too close to the sun in our sky to observe, so his wave had holes torn in it exactly where the turning points were. And depending on which side of the sun the Earth stood, the geometry let him watch Io vanish into the shadow or re-emerge from it, but never both — so he was stitching two different kinds of event into a single curve and hoping they were commensurable.

Nor did he have a scatter score, or a way to stack data on a trial period, or any of the machinery that makes the pattern jump out of the screen at you. He had ink, paper, arithmetic, and years.

The whole result rests on a scatter of eye-and-ear timings, taken through instruments that were long, dim and unsteady, against clocks that had only just become good enough for the job.

That it worked at all is not a small thing. It is the first measurement of anything about light, ever, and it was extracted from the noise in somebody else’s table.

The thing to take away

Every measurement leaves a residue — the part of the data that will not sit on the line. Nearly always the residue is you: your instrument, your clock, your tired eye at three in the morning. Learning to attribute it to yourself is most of what it means to become competent at anything.

But not always. Sometimes the leftover has a shape, and the shape keeps time with something, and the something turns out to be a fact about the world that nobody has noticed yet. The hard part was never being clever enough to explain the residue. It was being suspicious enough to look at it properly in the first place, when every instinct and all your training says it is only grit.

Rømer looked at the grit and found a year in it. The wrong year, which was the clue.

Jupiter is up for most of the year, and it is the brightest thing in the night sky that is neither the moon nor Venus. Ordinary binoculars, braced against a wall, will show the four Galilean moons as a short line of sparks beside it — the same specks Galileo saw, and the same ones Rømer spent years timing.

They are still eclipsing, every forty-two and a half hours. Free planetarium software will give you the next one to the second, and if you point something at Jupiter at the right moment you can watch Io go out.

When you do, remember what you are looking at. Depending on where we are in our orbit, the light landing in your eye left Jupiter somewhere between about thirty-three and fifty-three minutes ago. The eclipse you are watching finished before you sat down. You are not seeing Jupiter. You are seeing a message from Jupiter, and it is late — which is the entire discovery, still happening, every clear night, over your own back garden.

This is the first in a series. Everything above depended on things that had to be invented first: glass clear enough to see through, a lens nobody understood, a clock that could hold a minute, a war-driven obsession with knowing where your ships were, and a number for the size of the solar system that had been measured only four years earlier and was wrong. The rest of the series is about where those came from.