The Jovilabe

What you are looking at. The big dial is the system seen from above Jupiter’s north pole, and the orbits are drawn to true scale — which is why Jupiter is a bead. Callisto really does ride twenty-six Jupiter radii out. The dark wedge is the planet’s shadow, drawn as the narrowing cone it really is; a moon crossing it is being eclipsed, and you can watch it happen. The dark strip below is the same instant through a telescope, also to true scale, east to the left. The outer ring is the zodiac in heliocentric longitude: the brass pointer is Jupiter, the steel bead the Earth. When the two coincide the Earth stands between the Sun and Jupiter and it is opposition; when they sit opposite each other it is conjunction, Jupiter is on the far side of the Sun, and the light equation runs to its longest. The small round panel magnifies the disc, because a shadow crawling across the cloud tops is the finest thing this instrument predicts and at true scale it would be two pixels wide.

The next seven days

Every beginning and ending of the four phenomena — a moon slipping into the shadow, passing behind the planet, crossing in front of it, or dragging its shadow over the cloud tops. Found by scanning the model at six-minute steps and then bisecting each crossing down to the second.

The limit of a six-minute scan. Anything that begins and ends inside six minutes can be stepped straight over and never appear in this table. That is rare — ingress and egress are what is brief, not the events themselves — but it is a real hole and worth knowing about rather than trusting the list blindly.

The movement

If the four rings are to be carried round by wheels from a single arbor turning once a day, somebody has to choose whole numbers of teeth — and no ratio of whole numbers is 1.769137786. These are the best trains available with pinions of six to twenty leaves and wheels up to a hundred and eighty teeth. They are drawn at one common module, so every mesh in the picture is a mesh that would actually work, and they turn at their true relative rates.

hand	period, days	train	drift/century	a whole turn out in
Io	1.769138	12/11 17/16 29/19	4.09°	8,802 years
Europa	3.551181	14/11 21/11 19/13	5.15°	6,990 years
Callisto	16.689018	16/13 19/13 167/18	0.10°	353,593 years
Jupiter’s own turn (System II)	0.413665	12/7 19/16 19/16 (step-up)	78.46°	459 years
Jupiter’s year round the zodiac	4332.589000	139/6 148/10 139/11	0.00°	never
Earth–Jupiter synodic	398.884050	77/8 121/20 137/20	0.00°	never
Ganymede — not geared but derived	7.154553	bevel differential	0.00°	1° in 5,475,526 years

Why anyone built one

A ship at sea in 1650 could find its latitude with a quadrant and a table. Longitude was the unsolved problem, and it was unsolved for a specific reason: to know how far east or west you are, you need to know what time it is somewhere else at the same moment. There were no clocks that would keep time on a rolling deck.

Galileo saw the answer within two years of first pointing a telescope at Jupiter. The four moons he found there eclipse, one after another, hundreds of times a year, and each eclipse happens at one instant for the entire Earth. Publish the predicted times in Florence; observe the actual time where you stand; the difference is your longitude. He called them the Medicean Stars, and he designed an instrument — the giovilabio — for computing where they would be. Two of them survive in Florence. This is one of their descendants.

It never worked at sea. Finding a moon of Jupiter in a telescope from a pitching deck defeated everyone who tried, Galileo included, and the longitude prize eventually went to Harrison’s chronometers. But on land it worked beautifully, and for a century and a half the map of the world was redrawn by it. France came out noticeably narrower than anyone had thought.

The error that turned out to be physics

The tables were always a little wrong, in a way nobody could shake. Io’s eclipses ran late for part of the year and early for the rest, by up to a quarter of an hour, and the pattern repeated annually. In 1676 Ole Rømer, working at the Paris Observatory, said the obvious and unwelcome thing: the eclipses were not late. The light was.

When the Earth is on the far side of its orbit, the news of an eclipse has an extra three hundred million kilometres to cross, and it arrives about sixteen and a half minutes behind schedule. That is the whole of the discrepancy, and dividing the distance by the delay gives the speed of light — the first time anybody had a number for it. The Light Equation sub-dial is that hand: it reads how long ago the arrangement you are looking at actually happened. It runs from about thirty-three minutes to about fifty-three, and the difference between those two readings is Rømer’s discovery.

This instrument applies the correction rather than suffering from it. Each moon is even shown at its own instant, because light from one in front of Jupiter left it a few seconds later than light from one behind — the same effect as Rømer’s, on a baseline a thousand times shorter.

The resonance that gears itself

Io, Europa and Ganymede are locked together. Io goes round almost exactly twice for each turn of Europa, and Europa almost exactly twice for each turn of Ganymede — but only almost, and the near-misses are not independent. What holds exactly is a relation Laplace published in 1805 between the three mean motions:

n₁ − 3n₂ + 2n₃ = 0

It is true to about a part in a hundred billion. The three moons are held there by their own gravity; nudge one and the others pull it back.

The Resonantia sub-dial has three coloured hands at the three moons’ true longitudes, and two needles for the combination. The black one takes it from the mean longitudes: it sits at half a turn and never stirs — over five years it does not move in the fifth decimal place. The red one takes it from the true longitudes, with every periodic term left in, and that one breathes: measured over the same five years it swings between 176.06° and 183.87°. Run the instrument for a month and watch the two of them. The black needle is the law; the red needle is the three moons actually obeying it, tugging at the leash by about four degrees either way.

And it has a mechanical consequence that is too good to pass up. If the relation holds, then Ganymede’s rate is not an independent quantity — it is (3n₂ − n₁) / 2, and a bevel differential whose carrier turns at the mean of its two inputs computes exactly that. So in this movement Ganymede is not geared at all. It is derived, from Europa geared up three times and Io running backwards. Where the other three rings drift by degrees a century, that one is out by a degree in five million years. The physics does the engineering’s work.

How wrong it is

The satellites come from Lieske’s E5 theory, in the form Meeus sets out; Jupiter’s and the Earth’s own places come from a compact series fitted here against JPL’s DE440 ephemeris. Neither is taken on trust. The whole chain was run against JPL Horizons — the actual, published, best-available positions of Jupiter and its four moons — and the difference measured in Jupiter radii, the unit the dial is drawn in.

window checked	epochs	worst error, RJup	in kilometres
2016-03-01 to 2016-06-01	31	0.0275	1,968 km
2026-06-01 to 2026-09-01	31	0.0280	2,005 km
2040-03-01 to 2040-06-01	31	0.0303	2,167 km
2070-03-01 to 2070-06-01	31	0.0454	3,249 km
2099-03-01 to 2099-06-01	31	0.0556	3,975 km

So: better than 0.028 Jupiter radii near the present, degrading to about 0.056 by the end of the century as the theory drifts away from the epoch it was fitted to. For scale, that worst case is about a twentieth of the planet’s own width — smaller than the beads this page draws the moons with.

Two things went wrong on the way here and are worth writing down. The planetary series is referred to the ecliptic of date, and for a while the code precessed it a second time on the way to the satellites. A third of a degree is invisible on the sky and it is a fifth of a Jupiter radius in Callisto’s projected place; the error looked exactly like a plausible theory limitation until it was measured against Horizons and turned out to track the precession angle to four figures. Second, the first fit of the planetary series was garbage — an eighty-five-year span cannot tell two frequencies apart if they differ by less than about 0.012° a day, and the candidate list was full of unresolvable near-duplicates crowding out the real terms. Both were found by measuring, not by reading the code.

What is drawn true and what is not

True: the orbit radii and Jupiter’s globe on the plan dial, to one common scale. The telescopic strip, likewise. The shadow’s width, and its taper — the Sun is not a point from Jupiter, so the umbra closes by about a fortieth of its width by the time it reaches Callisto. Jupiter’s flattening, one part in fifteen, which is why the disc is visibly an ellipse and why transit timings need it. Every tooth count in the movement, and every mesh.

Enlarged, and said so: the moons themselves, which at true scale would be under a pixel; and the magnified disc panel, at about six times the strip’s scale.

Indicative only: the Great Red Spot. The globe turns at the true System II rate, 9h 55m 40.6s, but where the spot sits in that system is not something any theory predicts — it wanders tens of degrees a year and has to be re-set from observation. So it is placed at a stated longitude and left to run, which is exactly what you would have to do with a real one.

Nothing here is loaded from the network. No fonts, no images, no scripts from elsewhere. Open the file with the wifi off and the instrument works, which is the least an instrument should promise.