One heat flow, and two viscosities

Io is heated by not being allowed to relax: a resonance with Europa and Ganymede keeps its orbital eccentricity at 0.0041 when tides would otherwise have circularised it in a hundred and forty thousand years, and the work that goes into flexing it comes out as heat. The measured output is around a hundred terawatts — some two and a half watts per square metre, twenty times the Earth’s — and it makes Io the most volcanically active body in the solar system, heated by a mechanism no other moon of its size has.

That number is a measurement of something. The question is what.

The tidal heating rate can be written exactly, and everything in it except one factor is known: the masses, the radius, the mean motion and the eccentricity are all measured to several figures. What is left is $-\mathrm{Im}\,k_2$, usually written $k_2/Q$, the imaginary part of the tidal Love number. Inverting the measured heat gives $k_2/Q \approx 0.015$.

That is a number about the material Io is made of. Extracting a viscosity from it is where the trouble starts.

Why dissipation has a maximum #

A tide does work on a body only if the body’s response lags the forcing. Two limits make the lag vanish.

If the material is perfectly elastic, it deforms instantly and recovers instantly. The bulge is exactly aligned with the tidal potential, the work done on the way up is recovered on the way down, and no energy is lost.

If the material is perfectly fluid, it deforms to the equilibrium shape with no resistance at all — again in phase, again with no dissipation, though for the opposite reason.

Dissipation lives between the two, and the relevant comparison is between the material’s own relaxation time and the period of the forcing. A Maxwell solid — a spring and a dashpot in series — has a relaxation time $\tau = \eta/\mu$, viscosity over rigidity. When $\omega\tau \gg 1$ the dashpot has no time to move and the body is elastic; when $\omega\tau \ll 1$ the spring is irrelevant and the body is fluid; when $\omega\tau \sim 1$ both matter, and the response lags by an angle of order one.

The peak height depends on how stiff the body is relative to its own weight — the same dimensionless group $19\mu/2\rho g R$ that decides the static Love number and that separates a solid moon from one with an ocean — and for Io that group is about fifty, so the peak sits at $-\mathrm{Im}\,k_2 \approx 0.22$.

It is worth writing out the arithmetic for Io, because the numbers make the position of the peak concrete. Io’s forcing period is its orbital period, 1.77 days, so $\omega = 4.1\times10^{-5}$ per second. Silicate rock at mantle temperatures has a rigidity around sixty gigapascals. Setting $\omega\eta/\mu$ to the value that maximises the response gives a viscosity around $10^{13.5}$ pascal seconds — which is the viscosity of a rock somewhere near its solidus, hot and close to melting but not molten.

That is a suspiciously convenient answer, and it is one of the reasons the subject is difficult. A body that is being heated tidally will tend toward the temperature at which it dissipates most, because dissipation below that temperature warms it toward the peak and dissipation above it cools it back — a thermostat. So Io ought to sit near the peak, and it does not: its observed value is fifteen times below. Either the thermostat is broken, or the rheology is not Maxwell, or the heat is being produced somewhere the homogeneous model does not describe.

Two consequences, pulling opposite ways #

The peak is an upper limit. A homogeneous Maxwell body of Io’s size, density and forcing period cannot dissipate more than the peak value however its viscosity is chosen. That converts into a maximum heat output, and it is a useful bound: a moon observed radiating more than its peak allows is not a homogeneous body, and the excess is evidence of structure — a partially molten layer, say, where the dissipation is concentrated.

For Io the observed value is well below the peak, so the bound is not binding. For Enceladus it very nearly is: the heat coming out of its south polar region is larger than a homogeneous Enceladus can produce at any viscosity, and that is one of the arguments for a global ocean under a thin, flexing shell.

Below the peak, the answer is double. The observed $k_2/Q$ crosses the curve twice, at viscosities that differ by many orders of magnitude. One solution is a stiff, cold interior in the elastic regime; the other is a soft, hot one in the fluid regime. They are entirely different planets, and the heat flow does not distinguish them.

There is a third consequence that is easy to overlook and is arguably the most important. Because the response is a peak, tidal heating is self-limiting in a way that gravitational or radiogenic heating is not. A body warmed past the optimum becomes less dissipative, cools, and returns; a body cooled below it becomes less dissipative too. The equilibrium is therefore not unique — there can be a stable cold state and a stable hot state for the same forcing — and a moon can in principle be in either.

That bistability is a live hypothesis for the outer solar system. Two otherwise identical moons in similar orbits could have taken different thermal histories and settled in different states, one geologically dead and one active, with no difference in their present circumstances to explain it. It is one of the more attractive explanations for why Ganymede and Callisto, of nearly the same size and composition, are so different.

What breaks the degeneracy #

A single number cannot choose between two solutions, so a second observable is needed, and there are three candidates.

The phase of the response. The magnitude of $k_2$ and its phase are different functions of the viscosity, so measuring the static Love number as well as the dissipation picks a branch. That requires gravity science — repeated close flybys measuring the time-variable part of the field — which has been done at Titan and Enceladus and not, so far, at Io.

The pattern of the heat. Where the heat is dissipated depends on where the material is soft. A body dissipating in a deep mantle produces a different surface heat-flux pattern from one dissipating in a shallow asthenosphere, and the two predictions differ in whether the maxima are at the poles or at the sub- and anti-Jovian points. Io’s observed volcano distribution has been compared with both and sits awkwardly between them.

The orbital evolution. Dissipation inside Io transfers angular momentum, and the resulting migration is measurable in the moons’ mean motions over a century of astrometry — the same kind of secular signal in an ephemeris that reveals every other non-gravitational force in this collection. The measured drift says the system is currently moving in the direction that requires Io to dissipate more than it can be radiating in steady state — which means Io is not in equilibrium, and its heat output varies on timescales of tens of millions of years. There is a fourth candidate that is not an observation of Io at all. The same rheology governs how fast Io’s own rotation was damped into synchrony and how fast its obliquity was damped to near zero, and both of those are complete — which sets a lower limit on the dissipation integrated over the past. It is a weak constraint, because the damping timescales are short compared with the age of the system for almost any viscosity, but it is a constraint that requires no present-day measurement.

Maxwell is not the right rheology #

Everything above uses a Maxwell model, which has two parameters and produces the clean curve in the hero figure. Real rocks do not behave that way.

Laboratory measurements on silicates at high temperature show a much broader response: at frequencies above the Maxwell peak the dissipation does not fall as $1/\omega$ but as a weak power, roughly $\omega^{-0.3}$, because a real material has a spectrum of relaxation times rather than one. The standard empirical description is the Andrade model, which adds that transient behaviour with one more parameter.

The difference matters for exactly the question this essay is about. Under Andrade the high-viscosity branch is much less suppressed, so a cold, stiff Io dissipates far more than Maxwell predicts, and the two solutions move closer together. The degeneracy is not removed but it is reshaped, and published viscosities for Io differ between the two treatments by orders of magnitude.

There is a further complication that no simple rheology captures: partial melt. A rock with a few per cent of melt distributed along grain boundaries is dramatically more dissipative than the same rock solid, and Io’s interior is thought to contain a partially molten layer some fifty kilometres thick. Concentrating the dissipation into a thin, very soft layer produces the same total heat as a uniformly moderate body and a completely different depth distribution.

What was actually measured #

The heat flow is the primary observation and it is harder than it sounds.

Io’s thermal emission is measured in the infrared, both from spacecraft and from the ground with adaptive optics, and the total has to be integrated over a body whose emission is dominated by a few hundred hot spots covering a small fraction of the surface. Different treatments of the passive, non-volcanic background — which is warm because the whole surface is heated from below — give totals differing by tens of per cent. The number usually quoted, around a hundred terawatts, has an uncertainty of perhaps twenty-five per cent.

Two things about it are worth noticing. First, it is a global measurement of a quantity that varies enormously across the surface, and the global total is what enters the tidal calculation while the distribution is what would break the degeneracy. Second, it is a snapshot: whether it represents a steady state is exactly what the orbital-evolution argument above puts in doubt.

The masses, radii and orbital elements are known to many figures from spacecraft tracking and from centuries of astrometry, so they contribute nothing to the error. The eccentricity is the one orbital quantity that matters and it is forced rather than free — its value is a property of the resonance, and it is known to about a per cent.

Why the letter Q is a poor name for it #

The quantity inverted from the heat flow is almost always written $k_2/Q$, and the $Q$ in it is borrowed from a different problem in a way that has caused a persistent amount of confusion.

In its original setting, $Q$ describes a resonant oscillator: it is two pi times the energy stored divided by the energy lost per cycle, and it is a property of a system that rings. A tidally forced planet does not ring. It is driven at a frequency set by its orbit, far from any resonance of its own, and there is no stored energy in the sense the definition requires.

So $Q$ here is defined by analogy — as the reciprocal of the phase lag between the forcing and the response, or as an energy ratio computed over one forcing cycle — and different authors have made that analogy differently. The definitions in circulation differ from one another by factors of two, by factors of two pi, and by whether the Love number is included in the quantity or divided out of it.

The consequence is a literature in which the same physical body is assigned $Q$ values differing by an order of magnitude with no disagreement about the physics. Comparing two published numbers requires checking which convention each used, and the papers do not always say.

The deeper problem is that $Q$ is treated as a material constant and is not one. A real rock’s dissipation depends on the forcing frequency, so a body forced at two frequencies has two effective $Q$ values, and a body whose orbit evolves has a $Q$ that changes as the forcing period does. Quoting a single number for a planet is quoting a value at one frequency, and the frequency is usually not stated either.

The quantity that is well defined is the imaginary part of the Love number at a stated frequency, and that is what the calculations actually use. $Q$ is a summary of it that has outlived its usefulness, and the reason it survives is that a dimensionless number between ten and a thousand is easier to remember than a complex response function.

What the figure cannot show #

The hero figure draws one curve for one body, and three of its idealisations are worth naming.

It treats Io as homogeneous. Io is not: it has an iron core of about half its radius, a silicate mantle, and — on the evidence of the induced magnetic field measured during flybys — a partially molten layer beneath the crust. Dissipation in a layered body is concentrated where the material is softest, and the total is not what a homogeneous average would give.

It treats the forcing as a single frequency. The tidal potential on an eccentric synchronous satellite has several components — the radial breathing at the orbital frequency, the libration of the tidal bulge, and terms from the obliquity — and each is felt at a different frequency and sees a different point on the response curve. The curve should really be evaluated at several frequencies and summed.

And it holds the rigidity fixed while sweeping the viscosity, which is not what a real thermal history does. In a rock, viscosity and rigidity both fall as the temperature rises, and the melt fraction changes both together and steeply. The physical trajectory through this diagram as a body warms is not the horizontal line the figure invites, and it can cross the peak in a direction that is not obvious. The dissipation parameter enters every one of these estimates linearly, so it is worth drawing the two conclusions at a value twice the nominal one and seeing which of them survives.

Where the same argument applies #

The structure of this problem — a measured dissipation, a non-monotonic response, and a two-branch answer — recurs wherever tides do work.

Enceladus, where the heat exceeds what a homogeneous body allows and forces a layered interior.

Hot Jupiters, whose circularisation timescales imply a $Q$ for a gas giant and whose inflated radii may be a related symptom, and where the same ambiguity between a nearly-elastic and a nearly-fluid response appears in a completely different material.

The Earth–Moon system, where the Moon’s recession implies a terrestrial $Q$ of about twelve — startlingly dissipative — and the resolution is that the dissipation is not in the solid Earth at all but in shallow seas whose resonant periods happen to sit near the semidiurnal forcing. That is a reminder that a bulk rheology can be entirely the wrong model: sometimes the dissipation is geography.

Neutron stars in binaries, where the deformation before merger depends on the same Love number — measured as a single number in a waveform’s phase — at densities twenty orders of magnitude higher.

And the comparison of heat sources restricted to the three moons where both terms have been measured rather than modelled.

Where the ladder goes #

The first rung of this anchor established that tidal heating is an energy budget: it goes as the square of an eccentricity that tides themselves destroy, so a body radiating tidal heat is spending something that has to be resupplied. This one is about what the number, once measured, can and cannot say about the material.

The next rungs go toward the observations that would break the degeneracy — a Love number for Io from a dedicated orbiter, a heat-flux map good enough to distinguish a deep from a shallow dissipation, and the century-long astrometric record that says whether the system is in equilibrium at all. And there is a rung about the other direction of the same physics: the same imaginary $k_2$ that heats a moon also damps its orbit, so a body’s rheology decides its dynamical history, and the reason some moons are locked in resonance and others are not is at bottom a question about viscosity.