A quality factor quoted without a period is half a number

The Love number of a body is defined as the ratio of the quadrupole it acquires to the quadrupole imposed on it, and nothing in that definition mentions time. For a fluid it need not: a fluid has no memory, the response is instantaneous, and one number is the whole story.

A solid has memory. Push on rock or ice and it deforms immediately by an elastic amount, then continues to deform slowly as the material creeps, and how much of each it does depends on how long the push lasts. The Love number is therefore a function of frequency, and it is a complex one — a magnitude, which is how much the body gives, and a phase, which is how far behind the forcing it gives it.

The phase is where all of tidal dissipation lives, and it is the half of the number that is routinely quoted without saying at what frequency it was evaluated.

Two elements in series, and what they cannot do #

The Maxwell model is the simplest viscoelastic material there is: a spring and a dashpot in series, so the strains add and the stress is common to both.

$$\frac{1}{\tilde\mu(\omega)} = \frac{1}{\mu} + \frac{1}{i\omega\eta}$$

Its behaviour has two limits and they are both right. Force it faster than the Maxwell time $\tau = \eta/\mu$ and the dashpot has no time to move: the body is elastic, it stores the energy and returns it, and it dissipates nothing. Force it slower and the spring is irrelevant: the body flows, deforms fully in phase with the forcing, and again dissipates nothing. All the dissipation is at periods near the Maxwell time.

That is the two-viscosity result, and it follows from a model with two elements and no more.

The trouble is that real materials do not behave that way at short periods. A Maxwell body’s dissipation falls as $1/\omega$ once the forcing is faster than $\tau$ — a steep collapse — and laboratory measurements on ice, on olivine, and on essentially every polycrystalline solid show dissipation falling far more slowly, as roughly $\omega^{-0.3}$ over many decades of frequency.

The difference is not a detail. Between an orbital period of a day and a Maxwell time of a year, the two predictions differ by more than two orders of magnitude in the heat produced.

The transient that a spring and a dashpot leave out #

What is missing is anelasticity: a recoverable deformation that takes time. Apply a stress to a polycrystalline solid and it does not jump to an elastic strain and then creep steadily; it jumps, then approaches a larger strain over a spread of timescales, and if the stress is removed the extra strain comes back. It is not viscous flow, because it is recovered; it is not elastic, because it takes time.

The physical origin is grain boundaries. A polycrystal is a mosaic of grains that slide against one another under stress, and the sliding relaxes on a range of timescales set by the distribution of grain sizes and boundary properties. A spread of relaxation times produces a power law in frequency rather than a single corner.

The standard parameterisation is Andrade’s, from a creep experiment on lead wire in 1910:

$$\frac{1}{\tilde\mu} = \frac{1}{\mu} + \frac{1}{i\omega\eta} + \beta\,(i\omega)^{-\alpha}\Gamma(1+\alpha)$$

The third term is the transient. The exponent $\alpha$ falls between 0.2 and 0.4 for the materials that matter, and it is measured in a laboratory rather than derived from anything. At long periods the term is negligible and Andrade reduces to Maxwell exactly; at short ones it dominates, and it keeps the body dissipating where a Maxwell body has gone quiet.

The same body at three frequencies #

A single moon is forced at several periods at once, and the response is different at each. Enumerating them makes the point that a Love number is a spectrum rather than a number.

The orbital period carries the eccentricity tide: a synchronously rotating moon on an eccentric orbit sees the tidal bulge advance and retreat and swing east and west once per orbit. For Io that is 1.77 days, for Europa 3.55, for Titan 16.

The libration period is shorter and it is a different tide. A moon oscillating about synchronous rotation deforms at the libration frequency, which for a shell decoupled by an ocean is set by the shell’s own inertia and can be days or hours.

The forced obliquity tide works at the nodal precession period, which is years to centuries, and it is what makes a moon’s spin axis wobble against its orbit normal. It is enormously slower than the other two.

And for a body whose orbit evolves, the secular tide operates over millions of years, where every material flows and the fluid limit is the right one.

Four forcings spanning ten orders of magnitude in period, and the Love number is a different complex number at each. A model that fits the heat flow at the orbital period and then uses the same $Q$ for the obliquity tide is making a claim about the rheology across all ten.

What is actually measured, and it is never a Q #

No measurement returns a quality factor. What is measured is one of four things, and each is converted to a $Q$ through a different chain.

A heat flow. Io’s thermal emission is measured from its infrared brightness and is about $10^{14}$ watts, and equating that to the tidal dissipation rate gives $k_2/Q$ directly. The conversion assumes the heat is in steady state — that what is generated now is what is escaping now — and Io’s orbital evolution makes that doubtful at the factor-of-two level.

An orbital acceleration. Tidal dissipation in a primary transfers angular momentum to a satellite, so the satellite’s mean motion changes at a rate proportional to the primary’s $k_2/Q$. For the Earth–Moon system this is measured to a fraction of a per cent by laser ranging; for Jupiter’s moons it comes from astrometry over a century and from spacecraft, and it gave the surprising result that Jupiter’s $Q$ is far lower than expected.

A phase lag directly. The solid Earth’s tidal bulge lags the Moon by a measurable angle, recovered from satellite tracking, and this is the only case where the imaginary part of $k_2$ is observed rather than inferred from an energy.

A libration amplitude, which is affected by dissipation as well as by structure.

Each of those is evaluated at one frequency. Quoting the result as “the $Q$ of Io” and using it elsewhere is the step this essay is about.

Why the Earth is the awkward case #

The best-measured body is also the one where none of this applies cleanly, and the reason is worth stating because it is often used as a counterexample.

The Earth’s tidal $Q$ inferred from the Moon’s recession is about 12, which is far lower than any solid-body rheology gives. The dissipation is not happening in the solid Earth at all; it is happening in the oceans, in shallow seas where the tidal flow is turbulent and in the deep ocean where internal waves are generated at topography.

That is a fluid-dynamical dissipation with nothing viscoelastic about it, and it is enormously sensitive to the geometry of the continents. The Earth’s $Q$ was substantially higher in the past, which is the resolution of the awkward arithmetic that running the Moon’s recession backwards at the present rate puts it at the Earth’s surface 1.5 billion years ago rather than 4.5.

A quality factor is not a material property of a body; it is a property of whatever mechanism happens to dissipate the most, and for one of the two bodies in this collection where it is measured best, that mechanism is the shape of the sea floor.

The solid Earth’s own $Q$, measured separately from the phase lag of the body tide and from the damping of free oscillations after large earthquakes, is a few hundred and shows the frequency dependence this essay describes — falling roughly as $\omega^{-0.3}$ between the tidal band and the seismic one, which is the Andrade behaviour measured over five decades of frequency in the one body where it can be.

The one body where the slope has been measured #

There is exactly one object whose tidal dissipation has been measured at more than one frequency, and it is not the Earth. It is the Moon.

Lunar laser ranging measures the Moon’s distance to better than a centimetre, continuously, since 1969. The solid Moon’s tidal deformation affects its orbit and its rotation, and the fit to five decades of ranging returns both the Love number and the dissipation — at the monthly forcing period, where the quality factor comes out near 30.

Separately, the Apollo seismometers recorded moonquakes for eight years, and a body’s free oscillations are a sounding of its interior, and the decay of the reverberations gives a quality factor for the lunar mantle at periods of a second or so. That value is above a thousand.

Two measurements, six orders of magnitude apart in frequency, on one body. Fitting a power law through them gives $Q \propto \omega^{0.19}$ — an exponent squarely inside the laboratory range for the Andrade transient, obtained from astronomy rather than from a laboratory, on an object nobody has a sample of.

That is the closest thing this subject has to a direct measurement of the curve every calculation in it assumes, and it is two points.

It is also not quite a clean measurement of a rheology, and the reason is worth stating. The monthly value is low enough that a purely solid mantle struggles to produce it, and the favoured explanation is dissipation at the boundary between the solid mantle and a partially molten layer above the lunar core — a localised mechanism rather than a bulk material property. So the two points may not lie on one curve at all, and the exponent fitted between them may be a coincidence of two different mechanisms.

Why a distribution of timescales is forced #

There is an argument that the frequency dependence must be roughly a power law, and it does not depend on any particular model of what grain boundaries do.

The seismic quality factor of the Earth’s mantle is very nearly constant across the whole seismic band — from a second to a thousand seconds, three decades of frequency. A single relaxation mechanism cannot do that: one Maxwell element gives a peak, and a peak is not a plateau. What produces a plateau is a superposition of relaxation times spread over the band, weighted so that each decade contributes about equally.

A weighting that is flat in the logarithm of the relaxation time gives a quality factor independent of frequency; a slightly tilted one gives the observed weak power law. So the observation of near-constant $Q$ over decades is itself the evidence that the relaxation spectrum is broad, and the Andrade law is a convenient parameterisation of a broad spectrum rather than a mechanism.

That reframing has a consequence for extrapolation. The plateau has edges — an absorption band, bounded at high frequency where the fastest relaxation runs out and at low frequency where steady flow takes over. Extrapolating a power law beyond the band is not conservative, and every application of a tidal $Q$ to a body with a different forcing period is an assumption about where the edges are.

What the exponent costs #

It is worth being explicit about how much a rheological assumption is worth in the quantities people care about.

Tidal heating in a satellite goes as $k_2/Q$ times a function of the orbit. At Io’s orbital frequency with a Maxwell rheology and a mantle viscosity of $10^{19}$, the predicted heating is orders of magnitude below the observed $10^{14}$ watts. With an Andrade rheology at the same viscosity it is within a factor of a few. The two models differ by more than the entire range of viscosities anybody proposes.

The same choice propagates into orbital histories. The rate at which a satellite’s orbit circularises, the rate at which it migrates, and the time it takes to become tidally locked all carry $k_2/Q$ evaluated at the relevant frequency, and that frequency changes as the orbit evolves. A calculation that holds $Q$ fixed while the period changes by a factor of ten is assuming a rheology flat over a decade in frequency — which is Maxwell’s assumption on the slow side and nobody’s on the fast side.

And it propagates into exoplanets, where the same arithmetic sets the tidal circularisation timescale of a hot Jupiter and the locking time of a planet in the habitable zone of an M dwarf. Those predictions are quoted routinely with a $Q$ borrowed from Jupiter, at a forcing period two orders of magnitude different.

What one frequency actually fixes #

It is worth saying precisely what a single measurement at a single period determines, because the answer is a curve rather than a pair of numbers and the curve has a useful shape.

The dissipation depends on the interior through two quantities — a rigidity and a viscosity — and on the forcing through one. A measurement at one frequency therefore fixes one combination of the two interior quantities, and the combination it fixes depends on which side of the Maxwell corner the measurement sits on.

On the slow side, where the body flows, the response is dominated by the viscosity and the rigidity barely enters: the measurement is essentially a viscometer. On the fast side under a Maxwell rheology, the dissipation goes as the inverse of the viscosity, so the measurement is again nearly a viscometer but with the opposite sign of dependence — which is exactly the two-solution ambiguity the peak produces. Under an Andrade rheology the fast side depends on the transient’s amplitude and exponent as well, and the measurement constrains a three-way product.

So the practical value of a second frequency is not that it doubles the information. It is that the ratio of two measurements depends on the shape of the curve and not on its height, which removes the overall scaling that every uncertain quantity enters through. A ratio between the orbital and libration tides on one moon would bound the exponent directly, with the rigidity, the viscosity and the layering all cancelling to first order.

That is the same reason a line ratio measures a pressure while a line depth does not, and it is worth stating as a habit: when a quantity is uncertain by an overall factor, measure a ratio that the factor cancels out of.

Where the picture stops #

A homogeneous body is not a body. Every curve here uses one rigidity and one viscosity for the whole object, and a real moon has a rheology varying by many orders of magnitude between a cold crust and a warm interior. The dissipation is concentrated wherever the local Maxwell time is nearest the forcing period, which is a thin layer, and its position depends on the temperature profile that the dissipation itself is setting.

The feedback is unmodelled here and it is strong. Dissipation heats, heat lowers the viscosity, lower viscosity moves the Maxwell time, and the response changes. A body can therefore settle into a self-regulated state, or oscillate, and the steady solutions of that system are not the points on any curve in this essay.

And the laboratory is not the interior. The Andrade parameters are measured on samples centimetres across at strain rates far higher than a tide imposes, and extrapolated over ten orders of magnitude in frequency and to grain sizes nobody can reproduce. That extrapolation is the largest assumption in the subject and it is invisible in any published number.

Still open: a rheology that is fitted, not derived #

Every number in this essay’s second half is an empirical creep law. Andrade’s exponent, the amplitude of the transient, the way both vary with temperature and grain size and melt fraction — all are fits to laboratory data, and none follows from a theory of what grain boundaries do.

That leaves tidal heating in the same position as a mixing length or a Coulomb logarithm: a physically motivated framework with a parameter that absorbs what is not understood, calibrated on the few systems where an answer is known and applied where it is not. The difference is that a mixing length is one number and a rheology is a function, so the extrapolation is over a range rather than to a point.

What would change it is a measurement of the phase lag at more than one frequency on the same body. That has been done for the Earth and for nothing else. A mission that measured a moon’s tidal response at both its orbital period and its libration period would, for the first time, measure the slope of the very curve every calculation in the field assumes.