Astronomy & Celestial Mechanics
the sky, computed
Concept

Tidal dissipation — where it appears

The conversion of tidal flexing into heat inside a body whose response is not perfectly elastic. For a synchronous satellite the rate goes as e²R⁵/a⁶ times k₂/Q, so it is quadratic in an eccentricity the same process destroys — and a body still radiating must be having it replaced.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

Integrating the measured recession back: the Moon reaches the Earth 1.54 Gyr ago. The Earth–Moon separation and the length of the Earth's day, integrated backwards from the measured present recession rate of 3.83 cm per year. Constant-Q tidal friction makes a^(13/2) linear in time, so the history is a single line in a variable nobody plots, and it is calibrated to the laser-ranging measurement rather than to a modelled k₂/Q — the k₂/Q it implies is 0.0257, or Q = 11.6 for the Earth's k₂ of 0.299, which is a startlingly dissipative Earth. Run back at that rate the separation reaches zero 1.54 Gyr ago and crosses the Roche limit at 2.88 Earth radii only 4 years before it, so the drawing is cut off there rather than extrapolated. The Moon is 4.5 Gyr old, so this is a refutation and not a date: the present rate cannot have been the rate, and a mean Q of 34 — drawn dashed, reaching 4.51 Gyr — is the sort of value the age requires. Tidal rhythmites at 620 Myr put the day at 21.9 h and the Moon at 96.5 per cent of its present distance, and this history reads 20.1 h and 92.4 per cent — too fast and too close, which is the same failure the zero crossing is. Day length follows from total angular momentum, 23.93 h today, 9.84 h at half the present lunar distance and 4.97 h at the Roche limit, and depends on the separation alone: it is the same curve whatever Q is. The rate of lengthening the recession requires is 2.10 ms per century, against a tidal total of about 2.3 including the Sun's tide, which slows the Earth without moving the Moon, and an observed 1.75 from ancient eclipses and occultations — the shortfall being the Earth's moment of inertia falling as the mantle rebounds from the last glaciation.

A day five hours long

The tidal bulge leads, so the Earth's spin is being paid into the Moon's orbit. Run the measured payment backwards and two curves come out of one integration — a timeline that is refuted by the Moon's own age, and a day length that is refuted by nothing.

gravitation · Tides 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 The Love number against central condensation: 3/2 for a uniform body, 2.4e-3 at n = 4. How willingly a body deforms, drawn against how concentrated it is. The horizontal axis is the polytropic index, which is a proxy for the run of density inside — n = 0 is uniform, n = 1.5 is a non-relativistic degenerate gas, n = 3 is a radiative star like the Sun — and the vertical axis is the fluid Love number k₂ on a logarithmic scale. The curve is the Radau equation integrated over each polytrope's own density profile, and its two ends are exact rather than fitted: a uniform incompressible body has k₂ = 3/2 exactly, and a body with all its mass at the centre has k₂ = 0, because a point mass has no quadrupole to offer. Everything real lies between. The fall is steep — four orders of magnitude across the family — which is what makes the number diagnostic: k₂ is not a mild function of structure, it is a sensitive one, and measuring it to ten per cent constrains the interior far better than measuring a mean density to the same precision. The Sun, at n ≈ 3, sits near 2.9e-2. Two conventions collide here and the figure uses one of them: the planetary literature's k₂, for which a uniform body gives 3/2. The stellar literature's apsidal-motion constant is half of this at every point, so a uniform body gives 0.75, and the two are the same quantity. What the picture cannot show is rigidity — every body on it is a fluid, and for anything smaller than a planet that assumption fails badly.

How much a world gives

A body pulled on from one side deforms, and how much it deforms is a single dimensionless number. That number is three halves for a uniform fluid, three hundredths for the Sun, and two thousandths for a moon made of ice — so measuring it is a measurement of what is inside.

gravitation · Love numbers 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 The inflation threshold at 2·10⁵ W m⁻², and the 0.69 R_J above it. Radius against the starlight received, for a population of Jupiter-mass planets. The horizontal line is what a structural model gives for a cold, old, Jupiter-mass ball of hydrogen and helium: 1.06 Jupiter radii, and it hardly depends on mass at all in this range, because degeneracy is beginning to set in and the mass–radius relation is flattening toward its turnover. Planets receiving less than about 2·10⁵ watts per square metre sit on that line, with a median of 1.06, which is the control the rest of the figure depends on: the models are not wrong in general. Above the threshold the radii climb, reaching a median of 1.75 — half again the size a cold planet of the same mass can be — and the onset is sharp enough to be called a threshold rather than a trend. Starlight by itself will not do this. Irradiation is absorbed high in the atmosphere and re-emitted from there; it slows the escape of heat from below, which delays contraction, but it cannot deposit energy beneath the radiative–convective boundary, and it is the interior entropy that sets the radius. So the excess is evidence for a mechanism that carries roughly half a per cent of the incident flux down to pressures of tens of bars — ohmic dissipation of currents driven through a partly ionised atmosphere, breaking gravity waves, and tidally forced turbulence are the candidates, and the threshold is the number each of them has to reproduce. The points are a synthetic population from a seeded generator, not a catalogue; what is real is the threshold, the size of the excess, and the fact that the un-irradiated planets sit exactly where they should.

A radius no cold planet is allowed

A Jupiter-mass ball of hydrogen has a maximum size, and it is about 1.06 Jupiter radii however old or young it is. Hundreds of hot Jupiters are half again that, and the excess switches on sharply above a threshold in the starlight they receive — which means something is putting energy in deep.

exoplanets · Planet composition A period below which every orbit is round. Orbital eccentricity against period for binaries in four clusters of 0.125, 0.625, 6, 4 billion years, with the eccentricities drawn from one seeded distribution and then damped by exp(−age/τ), where τ rises as the 5.333 power of the period. Each cluster shows the same thing: below a boundary period nothing survives eccentric, above it the original distribution is untouched, and there is almost nothing in between because the timescale is so steep. The boundary is a clock. It moves as the three-sixteenths power of the age, which the figure checks against the drawn curves, and the calibration puts it at 6.5 days at 125 million years, 8.8 at 625 million and 12.5 at four billion — against measured cut-offs near 7.2, 8.5 and 12.5 days in the Pleiades, the Hyades and M67. The boundaries are also read back off the plotted points rather than trusted, and required to move outward with age. This is the cleanest measurement of tidal dissipation in ordinary stars that exists, and its cleanliness comes from the ages: a cluster's age is read off its main-sequence turn-off and owes nothing whatever to the tide being measured.

A cut-off period that is an age

Plot eccentricity against orbital period for the binary stars of one cluster and the picture has a wall in it. Below a certain period every orbit is circular; above it, the original spread survives untouched. The wall moves outward as the cluster ages, and where it stands is a measurement of how stars dissipate a tide.

orbits · Circularisation 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 A wall measures a ratio, and a ratio is a line. The tidal quality factor against the system's age, with each line the locus of one measured circularisation boundary. For a planet the eccentricity damping time τₑ goes as Q′P^(13/3), so the cut-off period goes as (age/Q′)^(3/13) and only the ratio of the two appears. A measured wall is therefore a straight line of slope exactly 1 in this plane and never a point on it: a 5-day boundary is consistent with Q′ = 2·10⁵ in a one-billion-year-old system and with Q′ = 2·10⁶ in a ten-billion-year-old one, and nothing in the light curve chooses. The stellar version of this measurement escapes because the cluster supplies the age, from a main-sequence turn-off that owes the tide nothing — which is why a cut-off period read off four clusters is a dissipation measurement and the same wall in the hot-Jupiter plane is not. The shaded band is what a field star's age is actually worth: known to a factor of 3, it leaves Q′ known to a factor of 3 and no better, against a quantity whose published values for giant planets span 10⁴ to 10⁹. What would break the degeneracy is a second measurement with a different power of P in it — an orbital decay rate, which goes as Q′⁻¹ with no age in it at all, and which has now been measured for one planet.

A wall measures a ratio, and a ratio is a line

The same circularisation boundary drawn for planets probes the dissipation inside the planet rather than inside the star. But the boundary depends only on age over Q′, so with no cluster to date the system the measurement is a line in a plane and never a point on it.

orbits · Circularisation Two damping times, one crossing, and the slope that separates them. Circularisation timescale against orbital period for the two tidal mechanisms, both logarithmic, normalised to 1.217 Gyr at 10 days. On logarithmic axes a power law is a straight line and the index is its slope, so the figure's content is that the two lines have different slopes and one crossing. The equilibrium tide gives 5.33, the bulge raised on a convective envelope being dragged ahead by a viscosity that is turbulent convection itself; the dynamical tide gives 7, gravity waves launched at the convective boundary carrying angular momentum to wherever they break. The horizontal lines are the ages of populations a boundary can be read in: where each curve crosses one is the wall that population shows — at 0.125 Gyr, 6.5 days against 7.2; at 0.625 Gyr, 8.8 days against 9.1; at 4 Gyr, 12.5 days against 11.9; at 10 Gyr, 14.8 days against 13.5. One cluster measures one number and both theories have a free normalisation, so one cluster cannot choose. The separation across every age available is a factor of 1.11. The discriminator that does not depend on the normalisation is mass: the equilibrium tide needs a convective envelope, and above about 1.3 solar masses there is not one, so the two predict different behaviour on either side of a boundary the theory names in advance. That is a measurement about where the wall stops behaving, not about where it is — and it is the reason the samples had to grow from tens of binaries per cluster to hundreds.

Two damping times, one crossing, and the slope that separates them

The equilibrium tide gives a damping time going as the sixteen-thirds power of the period and the dynamical tide as the seventh. One cluster measures one number and both theories have a free normalisation, so one cluster cannot choose.

orbits · Circularisation

Named alongside it

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

Eccentricity dampingEquilibrium tideLove numberQuality factorForced eccentricityLaplace resonanceLibrationRigidityTidal circularisationTidal quality factorAndrade rheologyAngular momentum

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