A moon heated by not being allowed to relax

In March 1979 a paper appeared arguing that Io ought to be volcanically active, on the grounds that tidal flexing would be dissipating enough energy inside it to melt a substantial fraction of the interior. Three days after publication Voyager 1 photographed a plume two hundred kilometres high.

The prediction is famous partly because it was right and partly because of the timing, but the interesting thing about it is the shape of the argument. It contains no geology at all. It is a statement about an orbit.

Why a locked moon still flexes #

A satellite in synchronous rotation keeps one face towards its primary, so at first sight the tidal bulge it carries is fixed in the body and there is nothing to dissipate.

That is true only for a circular orbit. On an eccentric one two things vary through a period.

The distance varies, so the height of the tide varies: the bulge grows and shrinks by a fraction of order $3e$ each orbit. That is the radial term.

The apparent direction of the primary varies, because the satellite rotates at a constant rate and moves round its orbit at a rate that is not constant. The primary appears to rock back and forth in the satellite’s sky by about $2e$ radians, so the bulge is dragged across the body twice per orbit. That is the librational term.

Both flex the body, and a real body is not perfectly elastic. Some fraction of the strain energy goes into heat each cycle, and the fraction is what the quality factor $Q$ parameterises. The standard result for a synchronous satellite is

$$\dot{E} = \frac{21}{2}\,\frac{k_2}{Q}\,\frac{G M_p^2 R_s^5\, n\, e^2}{a^6},$$

with $k_2$ the satellite’s Love number — how much it deforms for a given tidal potential — and $Q$ the quality factor of its response. Two features of that expression carry the whole argument.

It goes as $R_s^5/a^6$, so it is savage about size and distance: Io at four hundred thousand kilometres from Jupiter radiates thousands of times what Callisto does at nearly two million.

And it goes as $e^2$.

The eccentricity has an expiry date #

The same dissipation that produces the heat also removes the eccentricity. Energy is being taken out of the orbit and the angular momentum is not, and an orbit that loses energy at fixed angular momentum becomes more circular. The rate is

$$\frac{de}{dt} = -\frac{21}{2}\,\frac{k_2}{Q}\,\frac{M_p}{M_s}\left(\frac{R_s}{a}\right)^5 n\,e,$$

an exponential decay with a time constant that can be evaluated for any real satellite.

That is the argument in its sharpest form. A body that is radiating tidal heat is spending an eccentricity it cannot have saved. Either something is forcing it, or the heat is a transient and the body has been caught in the act.

What forces it #

For Io the answer is the Laplace resonance. The orbital periods of Io, Europa and Ganymede stand in the ratio 1 : 2 : 4 to a precision of about one part in $10^5$, and the three are locked so that the critical angle $\lambda_{\text{Io}} - 3\lambda_{\text{Eur}} + 2\lambda_{\text{Gan}}$ librates about 180° rather than circulating. The consequence is that the conjunctions between each pair always happen at the same place in the orbits. A perturbation that would otherwise average away instead acts in the same sense every time, and it pumps the eccentricities up as fast as the tides damp them down. What is observed — 0.0041 for Io, 0.0094 for Europa — is the equilibrium between the two. The energy ultimately comes from Jupiter’s rotation. The tide Io raises on Jupiter is dragged ahead of Io by Jupiter’s rapid spin, and the resulting torque transfers angular momentum outwards — the same mechanism that is pushing the Moon away from the Earth and lengthening the day. Part of that flows into the resonance and comes out as heat inside Io.

A number, to fix the scale #

The expression is worth evaluating once, because the exponents make it wildly unintuitive.

For Io: Jupiter’s mass $1.899\times10^{27}$ kg, Io’s radius 1,822 km, a semi-major axis of 421,800 km, a mean motion of $4.11\times10^{-5}$ s⁻¹, and $e = 0.0041$. With $k_2/Q = 0.015$ the formula returns about $10^{14}$ watts, which spread over Io’s surface is 2.4 watts per square metre.

That number is worth holding against two others. The Earth’s total surface heat flux, from radioactivity and from primordial heat together, is 0.087 W m⁻² — Io is putting out nearly thirty times as much per unit area as a planet a thousand times its mass. And the solar heating at Jupiter’s distance is about 50 W m⁻² at the sub-solar point, averaging to a little over 12; Io’s internal heat is a fifth of what it receives from the Sun, which for any other solid body in the solar system would be a rounding error.

Run the same expression for Ganymede — twice Io’s distance, a third of Io’s eccentricity — and it returns six ten-thousandths of a watt per square metre. The $R^5/a^6\,e^2$ scaling is not gentle: one rung further out in the same resonance chain, and the tidal term stops mattering at all.

The parameter that is not measured #

There is a circularity in all of this that should be said out loud rather than buried.

$k_2/Q$ is not measured for any of these bodies. It is inferred — from the observed heat flux, using the very expression it appears in. Quoting Io’s tidal heating as a prediction and then quoting $k_2/Q = 0.015$ as an input is quoting the same measurement twice.

The way out is to demand an independent determination, and there is one. The tidal transfer of angular momentum changes Io’s orbit at a rate that astrometry can detect: a century and a quarter of eclipse timings and imaging gives a measured orbital acceleration, and that acceleration is a statement about the energy flowing through the system from the other side. Analyses of that record find that Jupiter dissipates considerably more than had been assumed and that Io currently radiates more heat than it is being supplied with, which would mean the system is not in a steady state at all but oscillating on a timescale of tens of millions of years.

That is the honest position: a parameter inferred from one observation, tested against a second of a completely different kind, and the two do not quite agree.

The same circularity is worse for Enceladus, and there it is a known problem rather than a caveat.

Cassini measured Enceladus’s south-polar heat output at something like ten to sixteen gigawatts. For a body 500 kilometres across that is a surface flux twenty or thirty times what its own radioactivity can supply, and solving the dissipation formula for $k_2/Q$ returns a number of order $10^{-2}$ — comparable with Io’s, for a ball of ice. No plausible model of a solid icy interior dissipates that hard. The resolutions on offer all involve the ocean rather than the ice: dissipation in a liquid layer, obliquity tides, or a heat output that is currently above its long-term average.

Two bodies that refuse the argument #

Applying one dissipation parameter to every satellite predicts that several of them should be circular by now, and they are not. That is not a failure of the figure; it is the strongest available evidence about what $k_2/Q$ actually is.

Callisto has an eccentricity of 0.0074 and, at a common $k_2/Q$, a damping time of about $8\times10^8$ years — comfortably shorter than the solar system. It is not in any resonance, and it sits far enough out that the region Jupiter may keep a moon in is the relevant boundary rather than any commensurability. Its eccentricity has therefore survived some six damping times, which requires its dissipation parameter to be at least six times smaller than Io’s. That is exactly what an undifferentiated, cold, largely rigid body ought to give, and Callisto is the one Galilean satellite whose moment of inertia says it never fully separated into a core and a mantle.

Titan is worse: an eccentricity of 0.0288, a nominal damping time of $2.5\times10^8$ years, and no resonance to speak of. Its surviving eccentricity is a standing problem, and the candidate explanations are the usual pair — either $Q$ is far larger than assumed, or something excited the orbit comparatively recently — a resonance crossing during outward migration being the usual suspect, since an inclination can be traded for an eccentricity by mechanisms that leave no trace once they stop.

The Moon settles the question, because there $k_2/Q$ is measured rather than inferred. Lunar laser ranging returns $k_2 = 0.0240$ and $Q \approx 38$, so $k_2/Q = 6.3\times10^{-4}$ — twenty-four times smaller than the value that fits Io — and at that value the damping time is $1.2\times10^{10}$ years, longer than the age of the universe. The Moon’s eccentricity of 0.0549 is a leftover, and it is allowed to be.

So $k_2/Q$ is not a constant of nature but a property of an interior, varying by well over an order of magnitude between bodies of similar size and composition. It is the one number in the whole calculation that carries information about the inside of the body, which is why so much effort goes into measuring it independently, and why a heat flux is the most direct handle anybody has on it.

Where the heat comes out is not where the model puts it #

The expression at the top of this essay gives a total, and a total is compared against a measured total. There is a stronger test available, and the body that supplies it does not pass it cleanly.

Tidal dissipation is not uniform inside a satellite. Where it happens depends on the internal structure, and two limiting cases make very different predictions about where the heat appears at the surface. Dissipation concentrated in a deep, partially molten mantle produces a heat pattern with maxima at the poles. Dissipation concentrated in a shallow, weak layer just under the crust produces maxima at low latitudes near the leading and trailing points.

Io’s volcanoes have been mapped, and their distribution is a measurement of where the heat is coming out. It is concentrated at low latitudes, which favours the second case — and it is also displaced in longitude from where either model puts it, by some thirty to sixty degrees eastward.

The displacement is the awkward part. A rotational offset is not something either dissipation model produces, because both are symmetric about the tidal axes by construction. The proposed explanations are that the heat is transported laterally before it reaches the surface, or that the magma ocean’s own response to the tidal forcing introduces a phase lag of its own, or that the volcano positions are set by the crust’s structure rather than by where the heat arrives.

That last possibility is the one that undermines the test. A volcano is where magma reaches the surface, which depends on the crust as well as on the supply, so a map of volcanoes may not be a map of heat. The direct measurement — the thermal emission itself, integrated over the surface — is less sharply structured than the volcano map and is consistent with a broader range of models.

The total heat is a measurement and its distribution is a harder measurement of a more informative thing, and the second is where the argument currently stands rather than where it is settled.

Where the heat goes, and what it makes #

The consequences are geological rather than dynamical, and they are the reason the subject is more than an exercise.

Io has no impact craters at all. Its surface is being repaved fast enough that nothing survives, and it loses about a tonne per second of sulphur and sulphur dioxide to a torus around Jupiter. Its heat flux of about 2.4 watts per square metre is some thirty times the Earth’s.

Europa radiates far less, but enough to matter: the same calculation at its orbit gives a few tenths of a watt per square metre, which is comfortably sufficient to keep a saltwater ocean liquid under an ice shell tens of kilometres thick. The evidence for the ocean is magnetic rather than thermal — Jupiter’s rotating field induces a response that requires a conducting layer — but the reason such a layer can persist for billions of years is on this plot.

Enceladus vents water into space through fissures at its south pole, and the plume was found by a magnetometer noticing a bend in Saturn’s field before anybody saw it.

How old the arrangement is #

The resonance was described above as maintaining Io’s eccentricity, which raises a question the essay has so far avoided: how long it has been doing so, and whether it will continue.

Two accounts are on offer and they differ in whether the resonance is primordial.

In the first, the moons formed near their present configuration and the resonance was assembled early by differential tidal migration: Io, closest to Jupiter and most strongly torqued, migrated outwards fastest, caught Europa in a commensurability, and the pair then swept Ganymede up. On that account the resonance is nearly as old as the system and the heating has been running for billions of years — which is consistent with Io having lost essentially all of its water and a substantial fraction of its more volatile material.

In the second, the resonance is a recent arrangement and the system has been in and out of it. That possibility is what the measured orbital acceleration raises: if Io is currently radiating more than it is being supplied with, the system is not at equilibrium, and a system not at equilibrium is oscillating about one on some timescale. Estimates put that timescale at tens to hundreds of millions of years, with the eccentricity and the heat output rising and falling together.

Distinguishing them requires a record, and there is one candidate. Europa’s surface is young — a few tens of millions of years by crater counting — and its features record a history of stress that a fluctuating tidal heating would write differently from a steady one. Reading that record is a matter of mapping fracture patterns and inferring the stress state that produced each, which has been done and is not conclusive.

A resonance is a state rather than an event, and asking how old one is means asking what it has done to the bodies in it — which returns the argument to geology after having left it entirely.

What the model leaves out #

Three simplifications are load-bearing.

$Q$ is treated as a constant, and it is not. A real body’s response depends on the frequency at which it is forced and on its temperature, and both are coupled to the heating: a warmer body dissipates more, which makes it warmer. That feedback can run away or can stabilise, and which it does decides whether a satellite’s thermal state is steady or episodic.

The interior is treated as one body. Almost all the dissipation in Io probably happens in a partially molten layer, and almost all of it in Enceladus probably happens in an ocean rather than in ice. A single $k_2/Q$ is a fiction that summarises a stratified body, and its inferred value is not a material property of anything.

The obliquity is ignored. A satellite with a small tilt has an additional flexing term at the same frequency, and for some bodies it is not small compared with the eccentricity term. One more dissipation parameter shows how the eccentricity’s expiry date moves with it.

Every number in this essay is a ratio of two rates, and the reason the answer is robust is that the same ratio appears in the heat flow, in the damping and in the resonance — three measurements of one quantity, made in three different units.

Where this ladder goes next #

This rung has established the mechanism and the arithmetic: a heating rate quadratic in an eccentricity with a short lifetime, a resonance that maintains it, and a dissipation parameter inferred rather than measured.

The rung above is the coupled thermal–orbital problem, where $Q$ depends on the temperature and the temperature depends on the heating, and where the solutions are limit cycles rather than steady states — a satellite that heats, softens, dissipates faster, circularises, cools, stiffens and starts again.

Beside it lies tidal heating outside the solar system: a close-in exoplanet on a modestly eccentric orbit receives the same treatment, and for some of them the tidal power exceeds the stellar irradiation.

And below it, the habit: a body that is doing something energetic is spending something, and the first question is how long the account lasts. Comparing a rate against a reservoir is what turned a heating calculation into a prediction of volcanoes.