Illustrated Physics
physics, drawn
Astrophysics

The moon kept molten by its neighbours

Io is the most volcanic body known, with hundreds of active vents and lava hotter than any erupting on the Earth, on a moon smaller than the Earth's own and far too small to have kept any heat from its formation. The heat is made continuously, by tides: Jupiter stretches Io, and because Io's orbit is slightly eccentric the stretch changes every day and a half, flexing the rock and warming it. Tides should have rounded the orbit off in a hundred thousand years and switched the heating off. They cannot, because two other moons keep tugging the orbit back into its oval, and the heat Io radiates is paid for by Jupiter's spin.
15 min read 5 figures The arrow of timeOnly some values fit

Assumes: The distance that forgets the moon · The term free fall cannot remove

The distance that forgets the moon finds the distance inside which a planet’s tide tears a moon apart, and ends by pointing inward from it: at “the tidal heating that a body inside but not through the limit experiences, which is what keeps Io molten”. Io orbits Jupiter at a little under six Jupiter radii, well outside the Roche limit, so it holds together. But it is close enough, and its circumstances are peculiar enough, that the tide does something to it other than stretch: it heats it, continuously, by about a hundred million million watts.

In March 1979, three days before Voyager 1 flew past Jupiter, Stanton Peale, Patrick Cassen and Ray Reynolds published a paper predicting that Io’s interior should be largely molten and that “widespread and recurring surface volcanism” might be seen. Days later the Voyager images showed a volcanic plume three hundred kilometres high over the limb of the moon, and then eight more. It remains one of the most striking successful predictions in planetary science, and it was made from the orbit alone. The drawings follow the argument.

A tide in solid rock

A tide in solid rock, once an orbit. The height of Io's tidal bulge above its average, in metres, through one 1.77-day orbit, for a body that yields to Jupiter's pull like a fluid: the static bulge, 7.8 km, varies as the inverse cube of Io's distance, which swings by 0.41 per cent either way on its eccentric orbit. The bulge rises and falls by 192 m between the far point of the orbit and the near one; a real Io, stiffer than a fluid, flexes by roughly half as much, the figure of about a hundred metres usually quoted. A body on a circular orbit, turning to keep one face to its planet, would carry a bulge of fixed height pointing the same way and flex not at all; the eccentricity is what makes the rock move, the bulge's height changing once an orbit and its direction rocking to and fro once an orbit too.
Fig. 1 The height of Io’s tidal bulge above its average through one 1.77-day orbit, for a body that yields like a fluid: the static bulge, 7.8 km, varies as the inverse cube of Io’s distance, which swings by 0.41 per cent either way. The bulge rises and falls by 192 m; a real Io, stiffer than a fluid, flexes by roughly half as much, the figure of about a hundred metres usually quoted.

The term free fall cannot remove is the tide: the difference between a planet’s pull on a moon’s near side and on its far side, which stretches the moon along the line to the planet. For Io the stretch raises a bulge several kilometres high, pointing at Jupiter. On a circular orbit, with Io turning once per orbit so as always to present the same face — which it does — the bulge would be frozen into the rock: same height, same place, for ever. A steady stretch does no work, and a moon on a perfectly circular orbit is tidally distorted but not tidally heated.

Io’s orbit is not quite circular. Its eccentricity of 0.0041 carries it about 1,700 kilometres closer to Jupiter at its nearest than at its farthest, and the tide, which goes as the inverse cube of distance, changes by more than two per cent over each orbit. The bulge swells and subsides once an orbit, and because Io spins at a steady rate while its orbital speed varies, the bulge also rocks to and fro in direction, once an orbit, by a fraction of a degree. The solid surface of Io rises and falls by tens of metres twice a week. The drawing shows the height’s swing for a body that yields like a fluid; Io’s rock yields less, and the real swing is somewhat smaller, but it is not small.

The heat an eccentricity buys

The heat an eccentricity buys. The power tides dissipate inside Io against the eccentricity of its orbit, from (21/2)(k₂/Q) G M² R⁵ n e²/a⁶ with k₂/Q = 0.015: it grows as the square of the eccentricity. At Io's forced eccentricity, 0.0041, it is 9.3 × 10¹³ W, against about 1.0 × 10¹⁴ W measured from Io's infrared glow — the same to within the uncertainty of both — and against 4.7 × 10¹³ W for the whole of the Earth's internal heat, from a body forty-three times Io's volume. Halve the eccentricity and the heating falls fourfold; take it away and Io would freeze.
Fig. 2 The power tides dissipate inside Io against orbital eccentricity, from 212(k2/Q) GM2R5ne2/a6\tfrac{21}{2}(k_2/Q)\,GM^2R^5ne^2/a^6 with k2/Q=0.015k_2/Q = 0.015: it grows as the square of the eccentricity. At Io’s forced eccentricity, 0.0041, it is 9.3 × 10¹³ W, against about 1.0 × 10¹⁴ W measured from Io’s infrared glow, and 4.7 × 10¹³ W for the whole of the Earth’s internal heat.

Flexing rock is not perfectly elastic. Part of the work done on it each cycle is lost to internal friction and appears as heat, and the fraction lost is described by a quality factor QQ, the same measure of damping that describes a driven oscillator’s lag behind its drive: a body with QQ of a hundred loses about one part in a hundred of its stored energy each cycle. The tidal heating formula combines that with how much the body deforms, measured by its Love number k2k_2, the strength of the tide, which goes as the planet’s mass over the cube of the distance, the rate of flexing, and the square of the eccentricity, which sets how much the tide varies.

For Io, with the ratio k2/Qk_2/Q inferred from the slow evolution of its orbit, the formula gives 9.3×10139.3\times10^{13} watts. Io’s measured heat output, from spacecraft and telescopes watching its infrared glow over decades, is about 101410^{14} watts. The agreement is as good as the uncertainties allow, and it settles where Io’s heat comes from: tides supply all of it. That output is twice the whole Earth’s, from a body forty-three times smaller in volume. The Earth’s heat comes mostly from the radioactive decay of uranium, thorium and potassium in its rocks, plus heat left over from its formation; Io is too small to have kept its formation heat for four billion years — a small body cools faster than a large one, having more surface for each unit of volume and less far for its heat to travel — and too small for its radioactivity to matter. It is heated from outside, by its orbit.

The square of the eccentricity is what makes the arrangement so sensitive. Halve Io’s eccentricity and its heating falls fourfold; make its orbit circular and the heating stops, and within a few hundred million years Io would be as cold and dead as the Earth’s Moon.

Tidal heating falls steeply with distance

Tidal heating falls as the seven-and-a-halfth power of distance. The tidal heating Io would suffer with its present size, stiffness and eccentricity if it orbited Jupiter at other distances, measured in Jupiter radii: the tide's strength falls as the cube of distance, the rate of flexing as the orbital frequency, and the heating, which goes as their squares and the frequency, as distance to the power −15/2. Moved out to Europa's distance it would be heated 33 times less; to Callisto's, 7.4 × 10⁴ times less. Io is heated so fiercely because it is both the innermost of the large moons and locked into a resonance that keeps its orbit eccentric; Europa, further out and heated far more gently, keeps an ocean liquid under its ice rather than melting its rock.
Fig. 3 The heating Io would suffer with its size, stiffness and eccentricity if it orbited Jupiter at other distances, falling as distance to the power −15/2. At Europa’s distance it would be heated 33 times less; at Callisto’s, 74,000 times less. Io is heated so fiercely because it is the innermost large moon and its orbit is kept eccentric.

The heating falls off extremely steeply with distance. The tide’s strength goes as the inverse cube of distance and the heating as its square, so the inverse sixth power; the rate of flexing, which is the orbital frequency, falls as the inverse three-halves power by Kepler’s third law. Together that is distance to the minus seven and a half. Move Io out to Europa’s orbit, a factor of 1.6 further from Jupiter, and it would be heated thirty-three times less; to Callisto’s, tens of thousands of times less.

That steepness is why the four large moons of Jupiter are so different. Europa, the next one out, is also tidally flexed and also has a forced eccentricity, and its heating, though far weaker than Io’s, is enough to keep a global ocean of liquid water beneath its ice shell; the ocean’s existence is inferred from the magnetic field it induces as Jupiter’s field sweeps past, and it makes Europa one of the prime places to look for life beyond the Earth. Ganymede, further out still, is heated feebly and has a thicker icy crust, and Callisto, outside the resonance and with no forced eccentricity, is barely heated at all and has hardly differentiated into core and mantle. The moons are a graded sequence of how much tide they receive, from one that is too hot to three that grow progressively colder.

The eccentricity the tides cannot remove

The eccentricity the tides remove, and the one they cannot. Io's orbital eccentricity against time, in thousands of years, starting from 0.012, as the tides it raises dissipate energy: any eccentricity Io has of its own decays on a timescale (4/63)(Q/k₂)(m/M)(a/R)⁵/n = 102 thousand years. On its own, with no other moons, Io's orbit would be round within half a million years and its heating would stop. In the Laplace resonance with Europa and Ganymede, whose orbital periods are two and four times Io's, the regular tugs at the same point of every orbit force an eccentricity of 0.0041, and only the free part decays; the forced part is renewed as fast as it is spent, and the 9.3 × 10¹³ W it pays for is drawn from Jupiter's rotation, through the tides Io raises on Jupiter, and shared among the three moons' orbits.
Fig. 4 Io’s eccentricity against time, starting from 0.012, as its tides dissipate energy: any eccentricity of its own decays on a timescale of 102 thousand years. Alone, Io’s orbit would be round within half a million years. In the Laplace resonance with Europa and Ganymede only the free part decays; the forced part, 0.0041, is renewed as fast as it is spent.

Tidal heating takes its energy from the orbit, and it takes it in a particular way: it removes eccentricity. A body on an eccentric orbit that dissipates energy at each close approach loses more energy there than elsewhere, and the orbit relaxes towards a circle. For Io the timescale is short — about a hundred thousand years, by the drawing’s formula — and it is only a moment compared with the age of the Solar System. If Io were Jupiter’s only moon its orbit would have been circular for billions of years and its heating long since over.

Io is not alone. Its orbital period is 1.77 days; Europa’s is almost exactly twice that and Ganymede’s almost exactly four times, a chain of resonances discovered by Pierre-Simon Laplace, who proved that the arrangement is stable. Because the periods are commensurate, Europa passes Io at the same point of Io’s orbit time after time, and the small tug it gives at each passage adds up rather than averaging away. The accumulated tugs force Io’s orbit into an oval of eccentricity 0.0041, and the forcing is renewed as fast as the tides remove it. The drawing shows the result: any eccentricity Io has beyond the forced value decays, but the forced value itself stays. It is the same kind of protection a resonance gives the Trojans at a hilltop: the configuration of the orbits, not any property of the body, fixes a quantity that dissipation would otherwise erase.

Where the energy comes from

The heat has to be paid for, and the resonance is only the channel, not the source. The source is Jupiter’s rotation. Io raises tides on Jupiter as Jupiter raises them on Io, and because Jupiter spins faster than Io orbits, the tidal bulges on Jupiter are carried slightly ahead of the line to Io. Their gravity pulls Io forward along its orbit, adding energy and angular momentum and pushing it outward, while Jupiter’s spin slows imperceptibly. The resonance shares that outward push among the three moons, keeping their periods locked, and the energy flows into Io’s orbit and is dissipated in Io’s interior as heat.

It is the same process — dissipation draining an orbit, as radiation does in the orbit that has to shrink, but here with an outside supply — that is carrying the Earth’s Moon away from the Earth by about four centimetres a year and slowing the Earth’s day by a couple of thousandths of a second per century: tidal friction transferring spin from a planet to a moon’s orbit. The difference is that the Moon’s orbit is not locked in a resonance, so its heating — never large — faded long ago. Measurements of Io’s position, compared over more than a century of observations, show its orbit evolving under this balance, and the direction of the evolution is itself a measure of whether Io’s heating is greater or less than the energy being supplied by Jupiter’s tides at present — a question on which careful analyses have disagreed.

Heat through each square metre of ground

Heat through each square metre of ground. The heat flowing out through the surface, per square metre, on a logarithmic scale: the Moon (Apollo probes), 0.020 W; the Earth, average, 0.087 W; Io, from the tidal formula, 2.2 W; Io, from its infrared glow, 2.4 W. Io's surface loses heat at about thirty times the Earth's average rate, most of it through a few hundred volcanic hot spots, some hotter than any lava erupting on the Earth today; the Moon, with no tides worth mentioning, is nearly cold. The tidal formula's figure is Io's total dissipation spread evenly over its surface.
Fig. 5 The heat flowing out through each square metre of surface: the Moon, about 0.020 W from the Apollo probes; the Earth’s average, 0.087 W; Io, 2.2 W from the tidal formula spread evenly and 2.4 W from its infrared glow. Io loses heat at about thirty times the Earth’s rate, most of it through a few hundred volcanic hot spots.

Spread over Io’s surface, the heat is about two and a half watts through every square metre — thirty times the Earth’s average and a hundred times the Moon’s. The Earth loses most of its heat through the slow creation of new sea floor at mid-ocean ridges; Io, with no plate tectonics, loses it almost entirely by volcanism, through hundreds of active hot spots, some of them lava lakes the size of a small country and some erupting lava hotter than any the Earth produces today. Io’s surface is being repaved so fast that no impact crater has been found on it: any crater is buried by fresh lava within a few million years.

This way of losing heat has a name, heat-pipe volcanism: magma rises through the crust in narrow conduits, erupts, cools at the surface and is buried by later flows, so that the crust is continually carried downward and the heat carried up. The Earth is thought to have lost heat this way in its earliest history, when its own heat flow was far larger, and Io is the one body where the process can be watched working.

Tides elsewhere, and a clock made of circles

Io is the extreme case of a process that runs wherever bodies orbit closely. Saturn’s small moon Enceladus, five hundred kilometres across and only just above the size at which an icy body becomes round, vents jets of water vapour and ice grains from four long fractures near its south pole, and Cassini flew through them: a subsurface ocean, kept liquid by tidal flexing, with an eccentricity forced by a two-to-one resonance with the larger moon Dione. The inner planets of the TRAPPIST-1 system, a chain of seven Earth-sized worlds in a resonant chain round a small red star, are expected to be tidally heated by the same mechanism, some of them enough to matter for whether they could keep liquid water.

Stars do it too. Two stars orbiting each other closely — each a self-gravitating ball of gas with no rigidity at all — raise tides on each other, and the tides circularise their orbit exactly as they would Io’s, but with no third star to force the eccentricity back. The result is a clock. In a cluster of stars of known age, binaries with orbital periods shorter than some cutoff have circular orbits and those with longer periods do not, and the cutoff grows with the cluster’s age, because the circularisation time rises steeply with the separation of the pair. Measuring the period at which binary orbits stop being circular is one of the ways the efficiency of tidal dissipation in stars is calibrated, and it has shown that stars dissipate tides more efficiently than the simplest theories allow.

The Earth’s own tidal dissipation is mostly in its oceans, not its rock: about three and a half million million watts, dissipated largely by tidal currents dragging over the shallow floors of continental shelves and seas, and by internal waves breaking in the deep ocean. It is a tiny fraction of the Earth’s heat budget and irrelevant to its geology, but it is what lengthens the day and pushes the Moon away. The same physics that melts Io is, on the Earth, a slow transfer of the planet’s spin into its satellite’s orbit — and the rate depends on the shapes of the ocean basins, which have changed with continental drift, so the Moon’s recession has not been steady over the Earth’s history.

Why the stiffness hides a number

The drawings use one number for how Io responds, k2/Qk_2/Q, and it hides most of the difficult physics. The rate at which a body dissipates tidal energy depends on how its interior deforms at the forcing period of a day and a half. Rock is elastic on short timescales and flows on long ones, and between the two it is a solid that remembers: its response depends on the ratio of the forcing period to its own relaxation time, and dissipation is greatest when the two are comparable. A partly molten layer, whose relaxation time falls steeply as it melts, can dissipate enormously more than solid rock, and whether Io’s heat is released deep in its mantle or in a shallow layer just under the crust changes where on the surface the volcanoes should cluster.

Measuring this from outside is hard. Juno’s close flybys of Io in 2023 and 2024 measured the tidal change in Io’s gravity field — its k2k_2 — and found it too small for a global ocean of magma beneath the crust, suggesting instead a mostly solid mantle with melt distributed through it. The heating then depends on how the melt is spread, and the distribution of volcanoes on the surface is being compared with models of deep and shallow heating to find out.

Where the heating formula stops

A uniform body. The formula treats Io as a homogeneous sphere with one k2/Qk_2/Q. Io has an iron core, a mantle and a crust, each responding differently, and the total dissipation is a sum over layers whose properties are uncertain.

Small eccentricity. The heating is the leading term in the eccentricity. Io’s eccentricity is small enough that higher terms change it by much less than a per cent.

Steady state. The formula assumes that the heat generated equals the heat lost. Io may not be in balance: coupled calculations of heating and orbital evolution allow it to oscillate between states that are hotter and cooler than average over tens of millions of years, and the present output may not be the long-term mean.

What the drawings cannot show

The drawings show watts and metres; they cannot show Io. What the numbers produce is a world of sulphur-yellow plains and black lava lakes, plumes of sulphur dioxide rising hundreds of kilometres into space and falling back as frost, and a torus of ionised gas stripped from Io and swept round Jupiter by its magnetic field, glowing in ultraviolet light round the planet’s equator. None of it would exist if Europa and Ganymede orbited a few per cent further out, and none of it is visible in a graph of power against eccentricity.

Still open: how the resonance was born

The resonance is what keeps Io hot, and how the three moons came to be locked into it is not settled. One family of explanations has the moons formed out of resonance and driven into it later by tidal evolution: Io, pushed outward fastest by Jupiter’s tides, caught up with the resonance with Europa, and the pair together with Ganymede. Another has the resonance established while the moons were still forming in the disc of gas around the young Jupiter, as they migrated inward and caught each other. The two predict different histories of Io’s heating — a young and recent volcanism in the first case, and ancient in the second — and they are being tested against the ages of the moons’ surfaces and the structure of their interiors.

The habit worth carrying away is to ask what keeps a dissipating system from reaching rest. Tides remove the eccentricity they feed on, so tidal heating should switch itself off — and when it does not, something outside the body is renewing the eccentricity, and the heat is really being drawn from wherever that something gets its energy. For Io it is Europa and Ganymede, and behind them the slow spin-down of Jupiter.

Part 6 of 6

This essay is one argument about Self-gravity. The others:

The distance that does not know how big the moon is. How close a satellite held together by its own gravity can orbit before the tide pulls it apart, in units of the primary's radius, against how much denser the primary is than the satellite. The curve is the distance at which the tidal stretch across the satellite's own body equals the satellite's surface gravity. Setting those two equal cancels the satellite's radius on both sides, so a boulder and a thousand-kilometre moon of the same material break up at exactly the same distance — the limit is a ratio of densities and nothing else, and it goes as the cube root of that ratio, measured here as 0.3333. For ice around a planet of density 687 kg/m³ the rigid limit is 1.15 radii and the limit for a body that can deform under the tide is 2.23, because a satellite pulled into an egg presents a longer body to the tide and gives way sooner. Saturn's rings end at 2.27 radii, just outside that second number, and its innermost round moon orbits at 3.08 — so the boundary between a ring and a moon falls where this calculation puts it. What this calculation leaves out is strength: a body small enough for its material strength to beat its own gravity ignores the limit entirely, which is why Phobos is well inside Mars's and still in one piece, and why the Shoemaker–Levy fragments were held together by nothing at all. The distance that forgets the moon Part 5 — A satellite held together by its own gravity comes apart if it orbits too close, and the distance at which it does contains no reference to its size. Both the tide pulling it apart and the gravity holding it together are proportional to its radius, so the radius cancels twice over and what is left is a ratio of two densities.

Where this goes next

The distance that does not know how big the moon is. How close a satellite held together by its own gravity can orbit before the tide pulls it apart, in units of the primary's radius, against how much denser the primary is than the satellite. The curve is the distance at which the tidal stretch across the satellite's own body equals the satellite's surface gravity. Setting those two equal cancels the satellite's radius on both sides, so a boulder and a thousand-kilometre moon of the same material break up at exactly the same distance — the limit is a ratio of densities and nothing else, and it goes as the cube root of that ratio, measured here as 0.3333. For ice around a planet of density 687 kg/m³ the rigid limit is 1.15 radii and the limit for a body that can deform under the tide is 2.23, because a satellite pulled into an egg presents a longer body to the tide and gives way sooner. Saturn's rings end at 2.27 radii, just outside that second number, and its innermost round moon orbits at 3.08 — so the boundary between a ring and a moon falls where this calculation puts it. What this calculation leaves out is strength: a body small enough for its material strength to beat its own gravity ignores the limit entirely, which is why Phobos is well inside Mars's and still in one piece, and why the Shoemaker–Levy fragments were held together by nothing at all. The distance that forgets the moon A satellite held together by its own gravity comes apart if it orbits too close, and the distance at which it does contains no reference to its size. Both the tide pulling it apart and the gravity holding it together are proportional to its radius, so the radius cancels twice over and what is left is a ratio of two densities. How large a box may be before gravity shows in it. The largest freely falling laboratory in which nothing can be detected, against how long the experiment runs, for an instrument resolving 10⁻⁹ m. Free fall removes the field and leaves the gradient: two masses released a distance apart converge across the fall and separate along it, by (GM/r³)Lt²/2, so the box that stays undetectably flat shrinks as the square of the time. At one second the limits are 1.30 mm at the Earth's surface, 5.07 mm at the Sun's surface, 8.61 nm at a white dwarf, 1.86·10⁻¹⁷ m at a neutron star, 3.88·10⁻¹⁷ m at a stellar black hole, 388 mm at a giant black hole. Two things are worth reading off it. The equivalence principle is local in time as much as in space, on the same footing and with a worse exponent — patience is more expensive than room. And the tidal parameter GM/r³ falls with mass at a horizon, so a large enough black hole is one an experimenter could fall through with a metre-sized laboratory and a very good interferometer and detect nothing at all: the curve for the giant lies above the Earth's, not below it. The term free fall cannot remove Fall freely and gravity disappears. It disappears only to the extent that the falling laboratory is small — what survives is the gradient, which pulls two released masses together across the fall and apart along it. Given an instrument, the size of the box in which nothing is detectable is computable, and that number is the whole content of the word "locally". Tadpoles and a horseshoe, drawn in the turning frame. Orbits of small bodies near the circle of a planet's orbit, integrated in the frame turning with it, with the distance from that circle exaggerated 12 times so the shapes can be seen. With Jupiter's mass ratio, 0.000954, a body started 10° and 25° ahead of L4 at rest in the turning frame circles round L4 rather than falling off the hill: it swings between 51° and 70°, and between 42° and 85° from the planet — a tadpole. With a mass ratio of 10⁻⁴, a body started opposite the planet sweeps round through L4, L3 and L5 and back, never coming within 24° of the planet — a horseshoe, the kind of orbit Saturn's moons Janus and Epimetheus follow relative to each other, exchanging places every four years. In every case the Jacobi constant, the one quantity conserved in the turning frame, is held by the integration to better than a part in a million. The hilltop that holds the Trojans Sixty degrees ahead of Jupiter and sixty behind it, thousands of asteroids share the planet's orbit, held at the corners of the equilateral triangles that Jupiter and the Sun make. Seen from the frame that turns with Jupiter, those corners are the tops of two hills in the effective potential — the worst places to try to keep anything still. They hold asteroids anyway, because in a turning frame a body sliding off a hill is swung sideways by the Coriolis force into an orbit round the top. The trick works only if the planet is light enough: above a mass ratio of about one in twenty-six, the hilltops let go.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

DissipationEccentricityIoOrbital resonanceQuality factorTidal heatingTidesViscoelasticity