The Red Spot dial

What it is

A moonphase dial is a disc carrying a moon behind a plate with a hole in it. The plate hides most of the moon and the shape of its edge is the phase. That is the entire machine, and it has been the entire machine since the sixteenth century.

This is the same machine with one substitution. The plate is no longer a pair of stylised clouds — it is Jupiter, seen from inside its own weather. The observer stands at the centre of the Great Red Spot, twenty-two degrees south of the equator, carried round once every nine hours and fifty-six minutes, and the edge of the plate is that observer's horizon. A moon slides behind the plate because it has set.

Why the centre of the dial is a point underfoot

Jupiter's north pole stands at the middle of this dial, and from twenty-two degrees south the north pole is twenty-two degrees below the horizon, due north. It is the centre anyway, because it is the axis. Project the sky stereographically from the pole and two things happen at once: every daily track becomes a circle about that point, and a steady rotation stays steady. Centre the drawing anywhere else and the discs would have to speed up and slow down to keep the moons in the right place. This is the astrolabe's oldest trick and it is borrowed here intact.

So the four moons ride four circles about one arbor line, each turning once per its own synodic day — the beat between Jupiter's rotation and the moon's own orbital motion, which is what sets how often a moon comes round again:

The arbor offset is the part worth staring at. A disc turning about the dial's exact centre does not quite fit the sky; shifting its arbor a little does. The offsets were fitted blind, four independent least squares, and they are not free parameters that happened to help — every one of them comes out at half the tangent of that moon's parallax, to within one per cent. (Half, because that is the scale the stereographic projection happens to have where these tracks lie.) Which is to say: the dial is off-centre because the observer is, and by exactly as much. What is left over, the last few degrees, is second-order parallax, and no rigid circle can carry it: at worst it puts Io eleven minutes early or late in a thirteen-hour day, and Callisto three.

How to read the rings

The order of the rings is not the order of the orbits. It is the order of parallax. Io is nearest, so standing on the planet throws Io furthest out of place, and Io's ring is furthest in. In truth all four tracks lie within five per cent of each other, because all four moons hug Jupiter's equator; the dial magnifies that spread about forty-fold so they can be told apart, and the magnification is linear. Watch Io over an hour or two and it visibly swings in and out along its ring — that is not Io moving. That is you, being carried sideways underneath it.

Angles are not magnified. The top of the dial is the meridian, so a moon at the top is at its highest, due north, about sixty-five degrees up. The right-hand edge of the plate is where things rise and the left-hand edge is where they set. The two edges lean, slightly, and by different amounts at different radii — that lean is this dial's version of the moonphase plate's humps, and it is cut where each track truly crosses the horizon rather than where a straight line would put it.

The moons are drawn at their true angular sizes relative to one another, and they swell and shrink as they cross, being nearer overhead than at the horizon. Io subtends about 34 arcminutes from the cloud tops — a shade larger than our own Moon from the ground. They show phases, because of course they do. And when one enters Jupiter's shadow it goes coppery rather than black: Jupiter's air bends a little red sunlight into its own umbra, in the way ours does during a lunar eclipse. That last is an inference from the physics, not something anyone has photographed.

The colour the window takes when the Sun is up is a drawing convention and nothing more. It is set from the Sun's altitude so that the dial reads at a glance, but nobody has stood on those cloud tops, and what little is known — a thin hydrogen-and-helium sky under an ammonia haze, lit twenty-seven times more faintly than noon on Earth — does not settle the question. Treat the tint as an index finger, not as a photograph.

What was checked, and against what

Three things could be wrong independently, so three were tested separately.

Which way Jupiter is facing. The satellite theory lands in a frame whose x-axis is where Jupiter's equator crosses the ecliptic; the IAU's rotation angle is measured from a different line entirely. The offset between them was fitted to 758 sub-observer longitudes from JPL Horizons spanning 2016 to 2099, and over that span the residual is worst-case 0.006° — six-tenths of a second of Jupiter's rotation.

The sky from the cloud tops. Horizons will put an observer on Jupiter, so it will state the altitude and azimuth of Io from a site at 22.4° south. Against 289 of those places over four days, this page's whole chain — theory, frame, rotation, spheroid, local vertical — agrees to within about a minute and a half of arc, which is a few seconds of time. The residue is not the chain: it is the satellite theory's own error, which is about two thousand kilometres in a moon's place, a twentieth of Jupiter's width, and invisible from Earth. Stand a thousand times closer and the same error is suddenly big enough to see.

Where the Red Spot is. This is the soft one, and it is soft because of Jupiter rather than because of arithmetic. The spot drifts — currently about sixteen degrees a year, westward, through the very longitude system that was defined from its own rotation a century ago. Its longitude here was measured by inverting 918 published transit times for 2026 back through this same rotation model, which recovers it to 0.18° — exactly the scatter you would expect from times printed to the nearest minute, so the inversion adds nothing of its own. But published sources disagree with each other by about ten degrees at any given moment, and the spot is some twelve degrees of longitude wide, so "the centre of the Red Spot" is only good to a few degrees in the first place.

Two allowances

Jupiter has no surface, so "horizon" means the top of the surrounding cloud deck. The Red Spot stands about 8 km above that deck, which depresses its horizon by 0.86°, and Jupiter's air bends light over the edge much as ours does — roughly 0.55°, estimated from the refractivity of hydrogen and helium at the cloud tops and a 25 km scale height, and good to about a factor of two. Together they let the moons rise about four minutes early. Both are named constants with their workings written out; neither is large enough to argue about, and both are included so that leaving them out is not a decision made silently.

What standing there would actually do to you

Everything above is computed. This part is not the instrument's claim, it is simply what is known, and it is worth writing down because the dial quietly asks you to imagine a thing that cannot happen.

You would not burn. You would freeze — the cloud tops sit at about −145 °C. You would not breathe, because the air is nine parts hydrogen to one part helium and no parts oxygen. You would weigh two and a half times what you weigh now. The wind at the Spot’s rim runs at something like 430 km/h, though the middle of it — where this dial stands — is comparatively calm, which is the sort of comfort that stops being comforting when you think about it.

And there is nothing to stand on. Jupiter has no surface. You would sink, and it would get warmer and heavier the whole way down, until somewhere in the dark the pressure finished the argument. So: freeze, suffocate, crush — in that order, with the burning saved for last and far too late to matter.

The sky in this dial is real. Point a telescope at Jupiter and the moons are where it says they are; the arithmetic is checked against JPL Horizons and the errors are printed above. The observer is a fiction. That is the only thing on this page that has not been measured, and it seemed better to say so plainly than to let a handsome dial imply otherwise.