Generator

The love-response generator

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.

4 essays call love-response. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

Gravitation

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.

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.

Gravitation

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.

The ceiling a measured response has to be read against. The fluid Love number of a layered body — the value it would have if its outer shell offered no resistance whatever — against the radius of its core, for cores 1×, 1.6×, 2.4×, 3.6× the density of the material outside. Every curve starts at 3/2, because a core of no size is a uniform body, and falls as the core grows: central condensation stiffens a fluid body without giving it any strength, since the tidal forcing is strongest out where there is then very little mass to move. A body 0.6 of the way core at 3.6× density has a fluid ceiling of 0.787. A measurement above the relevant curve is impossible for any interior of that layering; a measurement below it says only that something is resisting, and does not say what.

Gravitation

An ocean is detected and its depth is not

A tidal response thirty-six times too large for any solid body proves a moon has a liquid layer under its crust. It does not say how deep the liquid is, how thick the crust above it is, or what the core beneath it is made of — because the same one number is produced by a whole surface of interiors.

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.

Gravitation

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.

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