A heat flow that depends on a number nobody can compute

A tide does two things to a body: it deforms it, and it heats it. The deformation is elastic and reversible and carries no energy away. The heating comes from the fact that the deformation lags — the material is not perfectly elastic, so the bulge is raised slightly late, and the lag is where the energy goes.

Every consequence of that lag is proportional to the same combination of two numbers. The Love number $k_2$ says how large the deformation is; the quality factor $Q$ says what fraction of the stored energy is lost per cycle. Every measurable rate — the Moon’s recession, the Earth’s slowing, Io’s heat, the circularisation of a close binary — carries $k_2/Q$ and nothing else.

The combination is so pervasive that it is worth listing where it appears before taking it apart. It sets the rate at which a moon recedes or spirals in; the rate at which a planet’s day lengthens; the period below which a binary is circular; the rate at which a hot Jupiter’s orbit shrinks; the amplitude of a satellite’s forced libration; and the power dissipated inside a body being flexed. Six observables, in five different subfields, measured by five different techniques, and all of them proportional to the same two numbers in the same combination.

Why the two cannot be separated by watching an orbit #

The tidal torque on a satellite comes from the misalignment of the primary’s bulge with the line to the satellite. The bulge’s size is set by $k_2$ and the misalignment by the phase lag, which for a linear body is $1/Q$. The torque is a product of the two, and so is everything that follows from it.

Write it out for the simplest case, a satellite on a circular orbit outside synchronous:

$$\frac{da}{dt} = 3\frac{k_2}{Q}\frac{m}{M}\left(\frac{R}{a}\right)^5 na,$$

with $m$ and $M$ the two masses, $R$ the primary’s radius, $a$ the separation and $n$ the mean motion. Everything on the right except $k_2/Q$ is geometry or a mass, and all of those are measured to several digits.

So a measured recession rate is a measurement of $k_2/Q$ — a single number — and it is a very good one. The Moon’s recession is known to a fraction of a per cent from laser ranging. That precision buys nothing about the interior, because $k_2/Q$ for the Earth is dominated by a shallow-sea process that has almost nothing to do with the planet’s bulk properties.

The situation is worse than a simple two-for-one degeneracy suggests, because the two quantities have completely different physical characters. $k_2$ is a static property of the body’s elastic structure — the same kind of quantity as its moment of inertia, computable from a density and rigidity profile. $Q$ is a property of a dissipative process, it depends on frequency, and it varies by four orders of magnitude between materials that have almost identical elastic properties. Merging them into one measured ratio merges a quantity that is nearly known with one that is nearly unknown.

There is a further sense in which the ratio is a poor summary, and it is arithmetical. $k_2$ ranges over about one and a half orders of magnitude across the solar system — from 0.02 for a small rigid moon to 0.6 for a giant planet — while $Q$ ranges over five, from about twelve for the present Earth to $10^6$ for a cold rigid body. So a measured $k_2/Q$ is overwhelmingly a measurement of $Q$, and the temptation is to treat $k_2$ as approximately known and divide it out. That is exactly what is usually done, and it is defensible for a body whose interior is understood and indefensible for one whose interior is what is being investigated — which is to say for every case where the answer is interesting.

The measurement that supplies the second line #

The way out is to measure the deformation itself, which requires seeing the body’s gravity field change as the tide passes.

A spacecraft in orbit around a body feels the body’s gravity, and if the body is being deformed by a tide then the field changes through the tidal cycle by a fractional amount proportional to $k_2$. Tracking the spacecraft’s Doppler shift through many orbits, and fitting for a periodic variation at the tidal frequency, gives $k_2$ directly with no assumption about dissipation.

That has now been done for several bodies. Juno measured Jupiter’s $k_2$ as 0.59; the Cassini mission measured Titan’s as about 0.6, which is far too large for a fully solid body and is the strongest evidence that Titan has a subsurface ocean; the MESSENGER and BepiColombo tracking of Mercury and the GRAIL mission at the Moon did the same for those.

It is worth noticing what makes this measurement hard, because it explains why the list is short. The tidal variation in a body’s gravity field is a part in $10^{5}$ or smaller of the static field, and it has to be separated from everything else that varies on comparable timescales — the spacecraft’s own non-gravitational accelerations, the station’s clock, the propagation media. The signal is periodic at a known frequency, which is what saves it: fitting for a sinusoid at the tidal period rejects everything that is not at that period. The technique is the same one that finds a planet in a stellar velocity curve, applied to a spacecraft rather than a star.

The second instrument: the heat coming out #

There is a completely different way to constrain the same dissipation, available for a satellite rather than a primary: measure the heat.

A synchronously rotating satellite on a circular orbit raises a bulge that never moves relative to the body, and dissipates nothing. Give it an eccentricity and the bulge oscillates in amplitude and rocks back and forth once per orbit, and the flexing dissipates. The power goes as the square of the eccentricity, and the coefficient carries the satellite’s own $k_2/Q$.

The measurement is genuinely independent. One side is an orbital rate observed by astrometry over decades; the other is a surface temperature observed by an infrared instrument on a flyby. They share no instrument, no calibration and no model, and they constrain the same dissipation.

They also do not quite agree, and the disagreement is one of the standing problems of the field. Io’s observed heat flow is around $10^{14}$ watts, and the power that the Laplace resonance can supply in a steady state is somewhat less. Either the system is not in a steady state — Io is currently radiating heat stored during a more eccentric epoch — or Jupiter’s $Q$ is lower than the resonance calculation assumes, which the astrometric measurement of Io’s orbital drift now supports.

The two instruments also fail in different places, which is what makes the pair valuable rather than redundant. The orbital rate is exquisite for a body whose motion has been tracked for a long time and useless for one discovered recently. The heat flow is available for a body hot enough to see against its own equilibrium temperature and useless for one where the tidal contribution is a few per cent of the solar input. Io satisfies both conditions by an enormous margin and is the only body in the solar system that does, which is why almost everything quantitative about tidal dissipation in satellites has been learned from one moon.

What was actually measured #

Four measurements, of four different kinds, all constraining the same class of quantity.

The Moon’s recession, 3.83 centimetres a year, from laser ranging to corner reflectors. This is the best-measured tidal rate anywhere and it gives the Earth’s present $k_2/Q$ to better than a per cent. It is also the least representative: the value it gives is an order of magnitude larger than the Earth’s own solid-body dissipation, because most of the loss happens in shallow seas whose geometry is a passing accident.

Io’s orbital drift, from a century of astrometry combined with the Galileo and Juno tracking. Io is moving inward, not outward, which is the signature of a system where the satellite’s own dissipation dominates over the primary’s. Extracting Jupiter’s $Q$ from that measurement gives around $3\times10^4$, an order of magnitude lower — more dissipative — than the classical bound derived from the requirement that the Galilean satellites not have migrated too far in the age of the solar system.

Io’s heat flow, about $10^{14}$ watts, from ground-based and spacecraft infrared mapping. This is a direct measurement of a dissipation rate and it needs no orbital model at all.

Titan’s Love number, about 0.6, from Cassini’s radio tracking through six close flybys. A rigid Titan of the observed density would have $k_2$ near 0.03. The measured value requires a global liquid layer, and it is the reason Titan is believed to have an ocean.

The backwards extrapolation deserves one more sentence, because it is the single most common misuse of a measured tidal rate. Running the Moon’s present recession backwards gives a time to zero separation of about 1.5 billion years, and the Moon is four and a half billion years old. The discrepancy is not a small correction: it is a factor of three, and it says that the Earth’s dissipation has been much weaker for most of history than it is now. The resonance that clears a gap in one place and locks a moon in another is the mechanism proposed for part of the difference, through ocean tides passing in and out of resonance with basin geometry as the continents moved. What is certain is that the present number cannot be used as a constant, and what is not certain is what to use instead.

What a rate cannot tell about a rheology #

Even with $k_2$ and $Q$ separated, there is a further layer of ignorance, and it is the reason the subject is not finished.

$Q$ is not a constant. It depends on the forcing frequency, and the dependence is the signature of the material’s rheology. A simple viscous fluid gives $Q$ inversely proportional to frequency; a Maxwell viscoelastic solid gives a peak at the frequency where the viscous and elastic timescales match; laboratory measurements on real rock at planetary temperatures give something in between and considerably flatter, with a weak power-law dependence.

Which of those is right matters enormously for extrapolation. Every statement about the deep past — how long ago the Moon was formed, whether Io has always been this hot, when a hot Jupiter’s orbit circularised — extrapolates a present-day $Q$ backwards to a different forcing frequency, and the three rheologies give different answers by orders of magnitude.

One general remark about the frequency dependence before the limits, because it is the reason the three rheologies cannot be told apart by any single measurement. Every observable listed at the top of this essay is a rate at one forcing frequency: the Moon’s recession at twice the Earth’s rotation frequency, Io’s heating at its orbital frequency, a hot Jupiter’s circularisation at its own. A single rate constrains $Q$ at a single frequency, and a model of the frequency dependence is a curve through one point. Distinguishing the models requires two rates on the same body at different frequencies, and the solar system supplies that for exactly one object — the Earth, where the semidiurnal and diurnal tides are both measured, and where the answer is contaminated by the oceans.

Where the picture stops #

Three limits stand out, and the third is the one that makes the whole framework provisional.

The lag is not a constant angle. The “constant $Q$” model used above assumes the phase lag is the same at every frequency, which is convenient and is not what any material does. A “constant time lag” model — in which the bulge is late by a fixed interval rather than a fixed angle — is equally simple, equally arbitrary and gives different answers for eccentric orbits, because an eccentric orbit forces the body at many frequencies at once.

Dissipation may be localised. A body’s $Q$ is treated as a single number describing the whole object, and in reality most of the loss can happen in a thin layer: an ocean, a partial melt zone, an ice shell’s base. Two viscosities in one satellite give one heat flow and completely different distributions of it, and where the heat is deposited decides whether a shell convects, melts or cracks.

And the equilibrium tide is not the only tide. Everything above assumes the body’s response is the static deformation appropriate to the instantaneous forcing. A body with a fluid layer has resonant modes, and if a forcing frequency lands near one the response is enormously amplified — this is the mechanism proposed for Enceladus, whose heat output is far larger than an equilibrium tide in a body of its size can supply. A resonant response is not describable by a $k_2$ and a $Q$ at all.

There is a fourth limit that is easy to overlook and is the reason the Earth is such a poor guide. The dissipation in a body with an ocean depends on the shape of the ocean basins, because the loss happens where a tidal current meets a shallow shelf. The Earth’s present $Q$ of about twelve is a consequence of the current arrangement of continents, and it has varied by a large factor over geological time as the continents moved — the tidal recession rate inferred from rhythmites in ancient sediments is substantially lower than today’s for most of the past. A dissipation parameter that depends on the arrangement of coastlines is not a material property at all, and treating it as one is how the timeline of the Moon’s recession comes out refuted by the Moon’s own age.

Why the degeneracy is the interesting part #

It would be easy to present this as a catalogue of ignorance, and the framing would miss what the degeneracy has actually produced.

The combination $k_2/Q$ is measured superbly well for several bodies. That single number, with no interior model attached, is enough to say that the Moon was much closer in the past, that close binaries circularise below a cut-off period that is an age, and that Io is heated by not being allowed to relax. None of those conclusions requires separating the two factors.

What separating them buys is a different class of statement: not that a body is dissipating, but where and in what. Titan’s $k_2$ says there is an ocean. Mercury’s says the core is liquid. Jupiter’s low $Q$, once separated from its $k_2$, points at dissipation in the dynamical tide rather than in the equilibrium one, which is a statement about the interior’s stratification.

The pattern is one this collection meets repeatedly and it is worth naming. A product of two quantities is often measured far better than either factor, and the product is frequently the thing that governs the observable behaviour. Insisting on the factors is worthwhile only when a question is being asked that the product cannot answer — and the discipline is in knowing which questions those are.

Close on the one measurement that would settle most of this, because it is within reach rather than hypothetical. A spacecraft in orbit around Io, tracked for a few months, would measure Io’s own $k_2$ through the tidal cycle in exactly the way Cassini measured Titan’s. Combined with the heat flow already known and the orbital drift already measured, that would give the satellite’s Love number, its quality factor and its dissipation rate independently — three constraints on two unknowns, which is a test rather than a fit. The reason it has not been done is radiation: Io sits inside Jupiter’s inner magnetosphere, where the trapped-particle environment destroys electronics in weeks. The best-constrained tidal body in the solar system is the hardest one to visit, and that is a fact about engineering rather than about tides.

Where the ladder goes next #

Later rungs on this ladder start with the frequency dependence itself: what a rheology is, how a laboratory measurement on rock at a planetary temperature is turned into a $Q$ at a tidal frequency, and how badly the extrapolation to the deep past can go. The rung after it is the dynamical tide — the resonant response of a fluid layer, which no equilibrium description reaches and which may be where most of the dissipation in the giant planets actually happens.