Astronomy & Celestial Mechanics
the sky, computed
Concept

Quality factor — where it appears

A dimensionless measure of how much of the energy stored in a cycle of forcing is dissipated rather than returned. For tides it appears as k₂/Q, the imaginary part of the Love number, and it vanishes at both the elastic and the fluid extremes — so a measured value is met at two viscosities rather than one.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Io's measured heat needs k₂/Q = 0.016, and Enceladus's needs 0.011. Tidal surface heat flux against orbital eccentricity, from Ė = (21/2)(k₂/Q)GM_p²R⁵ne²/a⁶ evaluated at each satellite's own orbit, drawn at a common k₂/Q of 0.015. Every line has slope 2 because the dissipation is quadratic in e and nothing else on this axis varies. The filled marks are each body at its actual eccentricity; the two ringed ones are the bodies with a measured surface heat flux, and they are the only points here that are observations. Solving each of those for k₂/Q gives 0.016 for Io and 0.011 for Enceladus — within a factor of 1.5 of one another, for a warm silicate body and a 500-kilometre ball of ice, which ought to be a coincidence and is instead the sharpest problem in the subject: nothing about Enceladus's ice can plausibly dissipate at 0.011, and the number is what the heat requires all the same. The dashed line is the Earth's measured surface heat flux, 0.087 W/m², which Io exceeds by a factor of 28 — the most volcanically active body in the solar system is the fourth largest moon of the fifth planet, and the reason is entirely in the orbit.

A moon heated by not being allowed to relax

Tidal dissipation goes as the square of an eccentricity that tides themselves destroy, so a moon radiating tidal heat is spending something it cannot have saved. Io's would be gone in a hundred and forty thousand years, and the resonance that keeps putting it back is the reason there are volcanoes.

gravitation · Tidal heating Dissipation against viscosity for Io: a peak at 10^13.5 Pa s, and two solutions at the observed rate. The imaginary part of the Love number — the part that turns tidal work into heat — against the viscosity of the body's interior, for a Maxwell rheology at Io's size, density and forcing period. Both axes are logarithmic and the curve is not monotonic, which is the whole content of the figure. At high viscosity the body is elastic: it stores the energy the tide puts in and gives it back, and dissipates nothing. At low viscosity it is fluid: it deforms all the way and does so in phase with the forcing, and dissipates nothing again. Everything happens in between, at viscosities for which the Maxwell time — viscosity divided by rigidity — is comparable to the orbital period, and the peak here is 0.735 at 10^13.5 pascal seconds. Two consequences follow, and they pull in opposite directions. The peak is an upper limit: a homogeneous body of this size cannot dissipate more than that however its viscosity is chosen, so a measured heat flow above it would refute the model rather than constrain it. And below the peak the observed value is met twice — at 10^11.5 and at 10^15.5 pascal seconds — so a heat flow alone does not say which side of the peak the interior is on. Breaking that degeneracy needs a second observable, and the usual one is the phase of the response rather than its size. The picture treats the body as one homogeneous Maxwell solid, which is certainly wrong for a moon with a molten layer; a partial melt concentrates the dissipation and shifts the peak.

One heat flow, and two viscosities

Io radiates a hundred thousand gigawatts of tidal heat, and that number is supposed to say something about the rock inside it. It does, and not what one would expect — because dissipation vanishes at both extremes of viscosity, the measured heat is produced by two different interiors and cannot choose between them.

gravitation · Tidal heating A wobble that should have stopped seventy years ago. Left, the path of the Earth's rotation pole across its own crust over 13 years, as the sum of two circular motions: the 433-day Chandler wobble at 150 milliarcseconds and the annual wobble at 90. The spiral is a beat, and its period measured off the drawn path is 6.39 years against the 6.39 the two frequencies require. Right, the same path's radius against time. Two numbers in this figure are the argument. The first is the Chandler period itself: a rigid Earth of dynamical ellipticity 0.0032737 would wobble freely at 305 days, and the observed 433 is 42 per cent longer because the Earth deforms under its own wobble and the oceans move with it — the period is a measurement of the planet's elasticity, made by watching a free motion rather than by forcing anything. The second is the damping: at a quality factor of about 100 the wobble should decay in 38 years, and it has been running for as long as anyone has watched. Something is exciting it continuously, and the excitation is fluctuating pressure at the bottom of the ocean and in the atmosphere. What the figure cannot show is the excitation itself, which is not periodic and is only visible statistically.

A wobble that should have stopped

The Earth's rotation pole wanders across its own crust in a circle a few metres wide. A rigid Earth would do it in 305 days; it takes 433, and the difference is a measurement of the planet's elasticity. At the observed damping it should have died out within a human lifetime, and it has not.

sky · Precession One measurement, one line, and every point on it is an interior. The Love number against the quality factor, both logarithmic. An orbital measurement — a moon observed to be receding, a spin observed to be slowing — determines only the ratio of the two, so it picks out a diagonal band rather than a point, and every interior along that band reproduces the observation exactly. A body that deforms twice as easily and dissipates half as efficiently is indistinguishable from one that does the opposite. What breaks it is a measurement of the deformation on its own: a spacecraft tracking the body's gravity field through a tidal cycle measures k₂ directly, which is a vertical line here, and the intersection gives Q = 5.36·10⁴. The uncertainty on that answer is the two fractional errors added in quadrature, 20 per cent, and it is dominated by whichever of the two was worse — which for every body in the solar system is the orbital rate rather than the Love number.

A heat flow that depends on a number nobody can compute

Every tidal rate in astronomy — a moon receding, a spin slowing, an orbit circularising, a satellite melting — is proportional to one combination of two quantities that no orbital measurement can separate. One of them describes how much a body deforms and the other how badly it leaks, and only a spacecraft can tell them apart.

gravitation · Tidal heating What Io dissipates depends on how fast the tide is applied. The dissipative part of the Love number, −Im k₂, against the period of the forcing, for Io at an interior viscosity of 1e+16 Pa s. Two rheologies are drawn: a Maxwell solid, a spring and a dashpot in series, and an Andrade solid, which adds the anelastic creep every real material shows in the laboratory and a Maxwell body does not, with an exponent of 0.3. They agree for tides slower than the Maxwell time of 1.9 days, where the body has time to flow and the transient is irrelevant. They part completely for faster ones: at the shortest period drawn the Andrade body dissipates 20 times what the Maxwell body does. A quality factor quoted without a period is half a number, and which half is missing depends on a rheology measured in a laboratory rather than derived.

A quality factor quoted without a period is half a number

The tidal response of a solid body is not a constant. It is a function of how fast the tide is applied, and two rheologies that agree perfectly about a slow tide disagree by orders of magnitude about a quick one — so the same moon has one Love number at its orbital period and a different one at the period of its own libration.

gravitation · Love numbers

Named alongside it

The objects these essays reach for when they reach for this one.

Love numberTidal dissipationForced eccentricityLaplace resonanceAndrade rheologyHeat flowLibrationMaxwell rheologyRigidityViscosityAnelasticityAngular momentum

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